INTEGRATED SUPPLY CHAIN: MULTI PRODUCTS LOCATION ROUTING
PROBLEM INTEGRATED WITH INVENTORY UNDER STOCHASTIC
DEMAND
A Dissertation by
Seyed Reza Sajjadi
M.S., Azad University (Tehran South Campus), Tehran, Iran 2000
B.S., Azad University (Tehran South Campus), Tehran, Iran 1995
Submitted to the Department of Industrial and Manufacturing Engineering
and the faculty of the Graduate School of
Wichita State University
in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
December 2008
Copyright 2008 by Seyed Reza Sajjadi
All Rights Reserved
INTEGRATED SUPPLY CHAIN: MULTI PRODUCTS LOCATION ROUTING
PROBLEM INTEGRATED WITH INVENTORY UNDER STOCHASTIC
DEMAND
The following faculty have examined the final copy of this dissertation for form and
content, and recommend that it be accepted in partial fulfilment of the requirement for
the degree of Doctor of Philosophy with a major in Industrial Engineering.
S.Hossein Cheraghi, Committee Chair
Janet Twomey, Committee Member
Gamal Weheba, Committee Member
Behnam Bahr, Committee Member
Mehmet Bayram Yildirim, Committee Member
Abu Masud, Committee Member
Accepted for the College of Engineering
Zulma Toro-Ramos, Dean
Accepted for the Graduate School
J. David McDonald, Associate
for Research and Dean of the
Graduate School
iii
DEDICATED TO
My wife, Hedieh, who has been patient during fulfilling this dissertation.
iv
ACKNOWLEDGMENTS
I would like to thank my advisor, Dr.Seyed Hossein Cheraghi, for his
invaluable time, encouragement, guidance, and patient in the past several years.
Without his support, this research might never have been finished. I would also like to
thank the other committee members, Dr. Abu Masud, Dr. Gamal Weheba, Dr. Bayram
Yildirim, Dr. Janet Twomey, and Dr. Behnam Bahr, for their helpful comments and
suggestions on all stages of this study.
I also want to thank my parents Mr.Ali Sajjadi and Mrs.Simin Jazayeri for
their support and encouragement. I owe my life to them.
I would also like to appreciate Dr.Mohammad Jazayeri who helped me to
travel to the United State.
v
When two logistical decisions of the supply chain namely location (a strategic
decision) and routing (a tactical decision) are combined, the location- routing problem
(LRP) is formed. LRP deals with simultaneously locating one or more supply facility
among a set (or sets) of potential facilities and assigning customers to the selected
facility (or facilities). To further improve the operation of the supply chain process, it
is presented in this dissertation to integrate inventory control, as another tactical
decision, with the LRP. The presented model considers the multi-product network
under the fixed- interval inventory policy where stochastic demands represent the
customers' requirements. Moreover, the third party logistics allows excess space for
selected warehouses if needed. A two phase heuristic simulated annealing is presented
as solution methodology. Test problems are designed and solved by the developed
algorithm to evaluate the presented solution approach. Results show that the
integrated decision leads to saving on the network cost. Furthermore, two case studies
are discussed to show how the presented model can be used in practice.
vi
TABLE OF CONTENTS
Chapter
Page
1
INTRODUCTION .............................................................................................. 1
2
BACKGROUND AND LITERATURE REVIEW .............................................. 4
2.1
Background............................................................................................................................. 4
2.2
Literature Review ................................................................................................................. 11
2.2.1 Static LRP with Exact Solution ................................................................................. 11
2.2.2 Static LRP with Heuristic Solution ............................................................................ 14
2.2.3 Dynamic and Stochastic LRP .................................................................................... 19
2.2.4 LRP and Inventory..................................................................................................... 24
2.2.5 Other Types of LRP ................................................................................................... 28
3
MATHEMATICAL MODELING AND SOLUTION METHODOLOGY ........ 31
3.1.
Problem Definition ............................................................................................................... 31
3.1.1. Presented LRP ........................................................................................................... 31
3.1.2. Inventory Strategy ..................................................................................................... 34
3.2.
Mathematical Model ............................................................................................................. 36
3.2.1. Inventory Model ........................................................................................................ 36
3.2.2. LRP Model Integrated With Inventory ...................................................................... 42
3.2.3. Sample Problem ......................................................................................................... 50
3.2.4. Mathematical Model Verification.............................................................................. 56
3.3.
Solution Methodology .......................................................................................................... 59
3.3.1. Initial Solution ........................................................................................................... 59
3.3.2. Improvement .............................................................................................................. 62
4
IMPLEMENTATION AND ANALYSIS ......................................................... 84
4.1.
Implementation ..................................................................................................................... 84
vii
TABLE OF CONTENTS (continued)
Chapter
Page
4.1.1. Coding…………………………………………………………………………….....84
4.1.2. Program Verification ................................................................................................. 87
4.2.
Results and Analysis ............................................................................................................. 93
4.2.1. Comparing Network Cost Under Integrated and Individual Decision ....................... 93
4.2.2. Tests……………. ...................................................................................................... 97
4.2.3. Case study................................................................................................................ 110
5
CONCLUSIONS .............................................................................................124
5.1.
Conclusions ........................................................................................................................ 124
5.2.
Contributions ...................................................................................................................... 125
5.3.
Future Research .................................................................................................................. 125
REFERENCES……………………………………………………………………...128
APPENDICES……………...………………………………………………………137
APPENDIX A
APPENDIX C1
APPENDIX C2
APPENDIX D
APPENDIX E
APPENDIX F
viii
LIST OF TABLES
Table
Page
3-1: Variables of the Mathematical Model…………………………………………43
3-2: Capacity of Plant s for Product p (Ksp)………………………………………..48
3-3: Distance Matrix………………………………………………………………..49
3-4: Annual Demand of Customer i for Product P (Dip)……………………………49
3-5: Demand During Order Interval Time …………………………………………49
3-6: Variance of Demand During Order Interval Time…………………………….49
3-7: Optimal Solution of the Example……………………………………………..53
3-8: Distance Matrix of Figure 3-7………………………………………………...64
3-9: Saving Matrix…………………………………………………………………66
3-10: Initial Levels of Presented SA Parameters…………………………………..74
3-11: Four Factor Design of Experiment Considering Two Responses; time and %
improvement (small)………………………………………………………...75
3-12: Four Factor Design of Experiment Considering Two Responses; time and %
improvement (large)…………………………………………………………75
3-13: Analysis of Variance against Percentage of Cost Improvement (small)…….76
3-14: Analysis of Variance against Running Time (small)………………………..77
3-15: Analysis of Variance against Running Time without Interactions (small)….77
3-16: Analysis of Variance against Percentage of Cost Improvement (large)…….77
3-17: Analysis of Variance against Running Time (large)………………………...77
3-18: Setting the Equilibrium Value ()……………………………………………79
4-1: Model Parameters for a LRP Integrated with Inventory Problem (Problem 4)……….85
4-2: Initial versus Final Value of Fixed Interval Service Time (Problem 4)………85
4-3: Model Parameters for a LRP Integrated with Inventory Problem (Problem 5)………86
ix
LIST OF TABLES (continued)
Table
Page
4-4: Initial versus Final Value of Fixed Interval Service Time (Problem 5)………..87
4-4b: Comparing Developed Algorithm Solution to Optimal Solution…………….88
4-4c: Annual and Order Interval Demand…………………………………………..88
4-4d: Depot Capacity………………………………………………………………..88
4-5: Test Problem Level……………………………………………………………..88
4-6: Test Problems Data……………………………………………………………89
4-7: Integrated and Individual Inventory Cost Comparison……………….………90
4-8b: Test Sample Result for 250 Customers, 30 Products Network……………….91
4-8c: Mathematical Network Parameters and Their Values/Range for Test
Problems………………………………………………………………………..94
4-9: Major Network Parameters and Their Levels for Test Problems………………95
4-10: Depot-product Allocation for Case Study I…………………………………..108
4-11: Vehicle-depot Allocation for Case Study I………………………………..…109
4-12: Customer Priority on the Vehicles for Case Study I…………………………110
4-13: Fixed Service Interval for Case Study I……………………………….……..110
4-14: Comparison between Required and Available Space for Case Study I…..…..111
4-15: Vehicles Inventory Levels for Case Study I………………………………….112
4-16: Depots Inventory Levels for Case Study I……………………………………113
4-17: Modified depot Capacity for Redesigning Problem (Case Study II)…..……..117
4-18: Comparison of the Current and Modified (future) Network Configuration…117
4-19: Comparison of the Current and Modified Network Cost……………………118
x
LIST OF FIGURES
Figure
Page
2-1: Interdependence among location, allocation, and routing (Perl [10] and Min et al
[6])……………………………………………………………………………..…..5
2-2: Location Routing Problem (LRP) with Five Potential Facilities
and 6 Customers………………………………………………………………….7
3-1: The pictorial of presented LRP (3/R/T)…………………………………………31
3-2: The Solution Procedure to Form and Solve the Integrated LRP and Inventory
Model……………………………………………………………………………47
3-3: Network of the Example………………………………………………………...53
3-4: Initial Solution Algorithm………………………………………………………58
3-5: Improvement phase algorithm…………………………………………………..60
3-6: Depot Improvement Phase……………………………………………………...62
3-7: Current Route Sequence………………………………………………………..65
3-8: Candidate Route sequence……………………………………………………..66
3-9: Conditions on Applying Clark-Wright Algorithm……………………………..67
3-10: General simulated annealing procedure………………………………………71
4-1: Rejection Area for t Test……………………………………………………….92
4-2: Comparing the Average Cost Improvement against Number of Customer and
Type of Vehicles……………………………………………………….……….97
4-3: Comparing the Average Cost Improvement against Customer and Vehicle Levels
at Product Level When the Number of Products is set to 10…………….……..99
4-4: Comparing the Average of Cost Improvement against Customer and Vehicle
Levels When the Number of Product is set to 20………………………………99
4-5: Comparing the Average of Cost Improvement against Customer and Vehicle
Levels When the Number of Product is set to 40………………………………100
4-6: Effect of changes in Vehicle Capacity and Number of Customers on Total
Network Cost…………………………………………………………………..102
xi
LIST OF FIGURES (continued)
Figure
Page
4-7: Ratio of High to low Vehicle Network Cost against Different Levels of Products
& Customers……………………………………………………………………103
4-8: Average Run time (second) against Product and Customer Levels…………..104
4-9: Average Run Time (second) against Product, Vehicle level, and Customer…105
4-10: Relation between Run Time and Problem Size……………………………...105
xii
CHAPTER 1
INTRODUCTION
1.1.
Research Focus and Objectives
As defined by Chopra and Meindl [1] “a supply chain consists of all parties
involved, directly or indirectly, in fulfilling a customer request. The supply chain
includes not only the manufacturer and suppliers, but also transporters, warehouses,
retailers, and even customers themselves”. Blanchard [2] reports that in the United
States companies spend more than $1 trillion every year on their supply chain
networks. Improvement in supply chain can be made through its two main activities:
production and logistics. While the former is associated with in plant activities such as
production planning, material handling, and shop floor tasks, the later considers the
procurement of raw materials from suppliers to manufacturing plants and then
transportation of finished goods from manufacturing plants to retailers. This study
considers the logistics part of the supply chain.
Although the improvement of the supply chain process is possible by
optimizing its drivers such as facility, inventory, and transportation individually [1],
the work of Burns et al [78], Daskin et al [71], Shu et al [73], and Miranda & Garrido
[74] and [75] indicate that integrating of the supply chain drivers will lead to
minimizing the network cost. Locating plants and distribution facilities and
determining the routings as two logistical decisions in the design and operation of the
supply chain can be integrated to improve the supply network. Location and routing
when combined form the location- routing problem (LRP). LRP models allow
decision makers make three decisions simultaneously: determine the locations of
facilities, allocate customers to those facilities, and determine routings from facilities
1
to customers. While LRP considers the integrity, it relaxes the direct customer
deliveries allowing multiple visits.
Although many LRP models exist in the literature, integrating other
components of logistic decisions such as inventory has not been studied well.
Therefore, it is desirable to enhance the improvement on the supply network by
integrating the inventory decisions with the LRP.
It is also noticeable that most of works in the literature on LRP consider a
situation with a single product. However, in practice, multi product condition
describes the actual circumstances better. Another practical issue which has not been
integrated with LRP is the decision that companies make on the third party logistics to
overcome their space shortage constraint.
This work is a study of location-routing problem integrated with an inventory
model where decisions on location, routing, and inventory are made at the same time.
A three layer logistic network including plants at the first level, distribution centers
(DC) at the second level, and customers at the third level is considered. The goal of
this study is to select one or more DCs among a set of potential DCs, assigning
customers to selected DCs, determining inventory levels at DCs, and identifying the
routing order intervals to minimize total cost of the system. In order to capture reality,
it is assumed that customers demand their multi- products requirements under
uncertainty while third party logistics will cover any storage capacity shortages.
Moreover, homogenous vehicle fleets transfer goods from distribution centers to
customers, which means that all vehicles have the same capacity.
This dissertation has been organized into five chapters. The first chapter,
discussed previously, provides a brief overview of the problem at hand and the
research objective. The second chapter presents a background on the topic and
2
common definition of the classic LRP. It also provides a review of the literature from
very early LRP models to the most recent ones. Chapter Three is divided into two
sections. The first section describes the definition and characteristics of the problem,
proposes an appropriate inventory strategy, and presents a mathematical model for the
defined problem. In the second section, a heuristic simulated annealing (SA) as
solution methodology is presented. In Chapter Four the implementation of the
presented SA is discussed and analysed. Finally, Chapter Five briefly reviews and
concludes the previous chapters while it indicates the future research directions.
3
CHAPTER 2
BACKGROUND AND LITERATURE REVIEW
2.1
Background
As it implies, location routing problem (LRP), is involved with both classical
facility location and vehicle routing problems. The general facility location problem is
defined as: given a set of facility locations and a set of customers who are served from
the facilities, then which facilities should be used? , and which customers should be
served from which facility so as to minimize total cost of serving all the
customers?[3]. On the other hand, as it is stated in the VRP website [4] “The Vehicle
Routing Problem (VRP) is a generic name given to a whole class of problems in
which a set of routes for a fleet of vehicles based at one or several depots must be
determined for a number of geographically dispersed cities or customers. The
objective of the VRP is to deliver a set of customers with known demands on
minimum-cost vehicle routes originating and terminating at a depot”. The simplest
form of the LRP can be represented by traveling salesman location problem (TSLP)
where the salesman's home is to locate among a subset of n customers who are to be
served by him to minimize the expected tour length [5]. In fact, LRP is a generalized
form of both pure location and routing problems. If it is assumed that the locations of
facilities are pre-determined, the LRP is changed to the pure vehicle routing problem.
On the other hand, if it is assumed that customers are directly connected to facilities,
the LRP is changed to pure facility location problem.
There is no specific definition for LRP in the literature. Min et al [6] define the
combined location-routing problem as follows: "in general terms, the combined
4
location-routing model solves the joint problem of determining the optimal number,
capacity and location of facilities (domiciles) serving more than one
customer/supplier, and finding the optimal set of vehicle schedules and routes. Its
major aim is to capitalize on distribution efficiency resulting from a series of
coordinated, non-fragmented movements and transfer of goods". Nagy and Salhi [7]
imply that LRP is an approach rather than a definition such as the Weber problem1.
They follow Burns [8] definition "location planning with tour planning aspects taken
into account". Tour planning means the existence of multiple stops on routs which
occurs when customer demands are less than a full truckload (LTT) allowing the
vehicle to meet more than one customer demand at a time. Siravasta [9]defines the
location problem as "given a set of potential locations, select as facilities those which
will satisfy the given constraints, while meeting the required objective." He also
defines the routing as "finding the sequence of pick-up (delivery) points which may
be visited by a delivery vehicle starting and ending at some depot".
Perl [10] indicates the inter-dependency among location, allocation, and
routing as shown in Figure 2-1.
1
Weber problem is defined as locating one point among a set of sample points so as the summation of
distances from located point to sample points is minimized.
5
Allocation
Location
Routing
Figure 2-1: Interdependence among location, allocation, and routing (Perl [10] and
Min et al [6])
Some of the LRP applications include postal delivery, news paper delivery,
blood distribution, food distribution, school bus pick- up and delivery, waste disposal
transportation, mobile health care, and wherever the distribution centers (DCs) or
retailers are close to customers or in situation that the customer demands are not too
much (LTT). Nagi and Salhi [7] list the applications of LRP.
Figure 2-2 illustrates an example of a two layer LRP system. As shown, there
are five potential facility locations (a- e) and six demand points (customers)
represented by squares (1-6). The objectives in this example are to find the numbers
and location of facilities, assign customers to selected (open) facilities, and identify
the routings from open facilities to customers to minimize the total costs. For
example, one feasible solution can be as follows:
1) Two facilities are selected (b and e). The selected facilities are called "opened
facilities" and their fixed establishing cost is called "fixed facility opening
(establishing) cost"
2) While customers 1 to 3 are assigned to facility b, customers 4, 5, and 6 are
covered by facility e.
6
3) One truck is assigned to customers 1 and 2, one truck is assigned to customer
3, and one truck is assigned to customers 4, 5 and 6.
Note that the truck is full either in outgoing trip (delivery) or ingoing trip (pick
up) which is different from the pick up-delivery problem in which services are given
to the customers in the form of pick-up and delivery at the same time (post pick-up
and delivery).
Facilities
a
Demand
1
b
2
c
3
d
4
e
5
6
Points
Figure 2-2: Location Routing Problem (LRP) with Five Potential Facilities and 6
Customers
While Perl & Daskin [11] introduced the warehouse LRP in a three tiered problem,
Laporte [12] generalized LRP formulation. He presented a survey on LRP
representing five versions of the problem. Each version is different from others in
either formulation or assumptions. He used a λ/M1/…/Mλ-1 symbol to represent an
LRP and its routing structure. In this definition, λ is the number of tiers; Mt describes
the mode of transportation which can be either R or T; Mt =R means return and Mt=T
means trip. R indicates direct visit while T allows multiple visits starting from a
facility and returning to the same facility. For example, a 3/R/T system consists of
three layers (λ=3)(facilities at the first layer, distribution centers (DCs) at the second
layer, and customers at the third layer). Products are transported from facility at the
first level directly (round trip) to distribution centers (R) and then from DC to
customer where multiple visiting is allowed (T). We consider a 3/R/T system in this
7
proposal. Min et al [6] represented more precise characteristics of combined LRP.
They classified LRP with regard to its problem structure and solution method. The
most recent survey is the work of Nagy and Salhi [7] in which they proposed another
classification for the LRP. Some of the LRP structural characteristics are as follows:
•
Solution space: Although in most LRP models it is assumed that locations of
facilities are selected from a set of potential facilities (also called discrete space), they
can be selected from a planar space (Euclidean distance) where selected facilities
belong to a plane rather than discrete nodes. Moreover, the location of a facility can
be placed on a network such as a railroad system.
•
Static / dynamic/ stochastic demand: If demand is fixed over time, it is known
as static. If it is not fixed over time but variations can be predicted precisely, the
demand is known as dynamic. If demand follows a probabilistic behavior, it is known
as stochastic.
•
Homogenous/ heterogeneous vehicle: This characteristic indicates that
whether the capacities of all vehicles in the LRP model are the same. If so, the
vehicles are called homogenous. Otherwise, vehicles are heterogeneous (also known
as non-homogenous).
•
Capacitated/ un-capacitated: It is sometimes assumed that the capacity of a
vehicles and/or facilities is limited. Such a case is known as capacitated vehicles or
facilities. If both vehicles and facilities are capacitated, it is called capacitated LRP.
•
Time windows (distance limit): Some cases are constrained by time or
distance. For example, in food industry requested demand should be delivered in a
limited time or vehicles' drivers may charge the company with a high price if the
8
length of traveling is more than an agreed distance. These limitations are also known
as side constraints.
•
Single/multiple objective: The regular goal of LRP models is to minimize
facilities establishing plus operational cost and traveling cost. However, in some
applications such as disposal industry optimizing multiple conflicting objectives is the
goal of the LRP.
•
Single/ multiple products: This indicates if customers request a single product
or multiple products.
•
Integrity: This implies the degree of integrity which may exist in the LRP.
LRP can be integrated with inventory, packaging, etc. It is noticeable that although
adding integrity to the LRP will amplify the complexity, it leads to saving more costs
in the supply chain.
LRP models can be classified against their solution approaches as exact
methods and heuristic methods. While exact methods guarantee the optimal solution,
heuristic solutions lead to good- enough solutions which are not necessarily optimal.
In computational complexity theory, LRPs belong to nonstatic polynomial time hard
(NP- hard) in which as the size of the problem increases, the running time and
solution space will increase exponentially. As a result, due to the computational
complexity associated with LRP, exact algorithms can not handle over mid-size
problems effectively. Instead, heuristic approaches are most common methods to
solve the problem with a reasonable computational effort.
Although in the literature both terms "integrated supply chain" and LRP
basically combine location and routing in their models, they represent different
concepts. Term "integrated supply chain" implies the combination of location and
9
routing under the direct delivery circumstances rather than multiple- drop (multiplevisit) condition. In fact, the former implies the situation in which the customer
demand is at least one full truckload. In contrast, LRP considers a condition in which
customers demand less than one truckload of product allowing the truck to visit more
than one customer in each trip.
10
2.2
Literature Review
Balakrishnan [13], Laporte [12], Min et al [6], and Nagy & Salhi [7] reviewed
LRP models. In another study, as a part of his PhD dissertation, Burks [14] relaxed
some of LRP assumptions previously presented by Perl and Daskin [11] to introduce a
generalized formulation for LRP. To the best of our knowledge except for the works
of Bowerman [15], Burks[14], Gunnarsson et al [16], and Yi and Ozdamar [17], other
works on LRP consider a single product situation unless it is mentioned. Burks [14]
generalized LRP the formulation proposed by Perl and Daskin [11] to show
mathematical formulation for multi products LRP.
The reviewed literature has been classified in the following categories:
1. Static LRP with exact solution
2. Ststic LRP with heuristic solution
3. Dynamic and stochastic LRP (exact and heuristic solutions)
4. LRP and Inventory
5. Other types of LRP
Appendix A tabulates the LRP literature based on the above classification. It is
notable that neither all reviewed papers in the text are indicated in the table (such as
PhD dissertations or books) nor all the represented papers in the table are described in
the text.
2.2.1 Static LRP with Exact Solution
When demand is fixed or determined over time, the LRP is called static. If the
solution algorithm of the LRP leads to an optimal result, it is known as exact solution
(ex. Simplex method). Static LRP has been well studied. Laporte and Nobert [18]
presented locating of a single facility among a set of potential facilities with the
11
objective of minimizing the cost of the facility and routing at the same time. They
assumed that customer demands are fixed and determined over time. The study solves
n times (n is number of customers) m traveling salesman problem (TSP)
simultaneously. A global integer programming was proposed as the modeling
approach. Laporte et al [19] proposed an LRP in which customers were weighted and
homogenous vehicle capacity was limited. Moreover, they assumed that the set of
potential facilities was disjoint from the set of customers. However, multiple passages
through the same site were allowed. Representing the problem by a two index integer
linear programming variables, optimal solution for up to 20 customers in a reasonable
number of iterations were found. Laporte et al [20] presented LRP as a graph
representation, then transformed the graph into an equivalent constraint assignment
and solved the problem considering up to 80 customers using the branch and bound
algorithm.
Cox [21] formulated the hub-and-spoke problem using LRP configuration.
His problem which relates to the aviation industry consists of departure nodes,
transition points, and destinations. He compared the efficiency of two different
transportation systems: customer multiple visiting system and direct delivery system.
In the case of multiple visiting, an aircraft meets more than one destination while in
the direct delivery there is only a single destination for the aircraft.
Some applications of LRP are presented in the design of the waste disposal
logistics system. Studies assumed that the demands for the waste disposals are static
Zografos and Samara [22] proposed a three objective capacitated vehicle-facility LRP
for a disposal and treatment network. They used goal programming to solve their
problem. They assumed that the number of disposal facilities is a predefined
parameter in the problem. In another study, List and Mirchandani [23] proposed an
12
un-capacitated vehicle and facilities LRP considering multi objective function for
hazardous materials on a network. They solved the problem by using route generation
method. Revelle et al [24] combined methods of shortest path and weighted multi
objective programming to minimize the risk of accident and to locate costs of a
nuclear disposal system. Giannikos [25] presented a two layer- two index LRP
formulation for locating capacitated disposal centers and routing hazardous wastes
through an underlying transportation network. He applied a weighted goal
programming (WGP) solution presented for the proposed multi objective function.
Alumur and Kara [26] formulated a hazardous waste management problem as an
LRP. Considering total cost and transportation risk as a multi objective problem, their
three level mixed integers LRP is subjected to disposal facilities capacity. Moreover, a
minimum amount of requirement constraint is considered in the model. This
constraint ensures that a facility (treatment center) will be open if the amount of waste
is greater than a predefined value.
Singh and Shah [27] improved a tendupatta leaf collection system by defining
an LRP problem subjected to time windows limitation. They solved and compared
three problems: facility location problem (FLP), vehicle routing problem (VRP) and
integrating two former problems as LRP. They showed that when facility location and
vehicle routing problems are combined, the results will be improved as compared to
individual problems.
Presenting two new formulations for the capacitated facility- capacitated
location vehicle routing problem (CLRP), Belenguer et al [28] presented two branch
and cut algorithms to achieve exact solution for small instances and good upper bound
for up to 134 customers and 10 depots. They presented new form of capacity
limitation constraint and vehicle capacity constraint which made the solution
13
approach easier. Berger et al [29] presented a set partitioning formulation of an uncapacitated LRP with distance constraint. They applied the alternate column
generation technique to dramatically decrease the number of constraints, then, used a
branch and bound algorithm to solve the LRP.
2.2.2 Static LRP with Heuristic Solution
Due to the complexity of LRP problems, heuristic algorithms have been
extensively used as solution approach. Nagi and Salhi [7] divided the heuristic
solutions to LRP into three categories: iterative, clustering, and hierarchical based
algorithm. Iterative algorithms improve the solution by repeating the algorithm.
Clustering algorithms start by classifying the customer set into clusters and then
assigning clusters to potential depots (or vehicle routes). Then, the algorithm solves
location (routing) and routing (location) sequentially. Hierarchical based algorithm is
consisting of two sub algorithms: main algorithm and subroutine. The main algorithm
solves location problem and subroutine which receives feedback from main algorithm
at each step to solve the routing part. Besides of these three heuristics solutions, LRP
models have been solved using another method called sequential. The method
considers location and routing problem as two individual problems and solves two
problems separately leading to sub-optimal solution. For this reason sequential
solution are not considered as LRP solution method although it is used some times in
the literature.
Jakobsen and Madsen [30] formulated the newspaper distribution problem as a
three layers vehicle capacitated LRP subject to time windows limitation. They
suggested three heuristics for the problem (one iterative and two sequential) and
compared the results to a benchmark problem. Srivastava and Benton [31] studied the
14
impact of external environmental factors such as cost structure of vehicle, customer
spatial distribution, and number of depots on the performance of selected heuristic
solution method. They showed that the clustering methods will result in better
improvement over the sequential algorithms. Sirivastava [9] presented two iterative
algorithms and one clustering method to solve static LRP. He compared these three
algorithms with a sequential algorithm and concluded that the proposed methods
produced better solution for LRP.
Chein [32] presented a heuristic sequential algorithm for the un-capacitated
facility LRP subject to distance limitation. He approximated multi routing costs by
introducing two estimators for practical-size problems. Bowerman et al [15] presented
an urban school bus routing problem (USBRP) as an LRP where they considered the
school at the first level, depots (stations) and students at the second and third levels
respectively. Using the cluster-first, route- second decomposition method, they
minimized their proposed multi objective function using a heuristic solution. To solve
their problem, they used the districting algorithm to group students in clusters. The
interesting point of the proposed heuristics is that in contrast to most LRP objective
functions which consider only single objective, it is applicable to a multi objective
condition too. Moreover, the model considers multi products condition in its
mathematical formulation.
Nagi and Salhi [33] presented a nested heuristic for the capacitated vehicle
subject to distance limitation LRP. In their hierarchical approach the location is the
master problem while the routing is the sub problem. In other words, the routing stage
is embedded within the location phase. The routing length estimation (RLE) is used to
estimate the routing costs. The fact behind using RLE is that computing the sum of
radial distances is much easier to perform than determining the total routing costs.
15
Computational results indicate that the proposed method produced better results than
the sequential algorithm.
Tuzun and Burk [34] proposed a two phase taboo search (TA) algorithm for
the capacitated LRP that iterates between location and routing phase in order to search
for better solutions. They showed that the proposed algorithm is able to solve the LRP
with 20 facilities and 300 customers efficiently. To prove the effectiveness of their
algorithm, they compared it to ASVI algorithm proposed by Sirvastava [9].
Redesigning a bill delivery logistic system, Lin et al [35] formulated the
system as the LRP. In contrast to classical LRP which considers a single route to a
vehicle, their problem allows assigning multiple routs to a vehicle as long as the total
trip time does not exceed the trip time limit. To solve the problem, they proposed four
heuristic algorithms (two simulated annealing and two traveling sales man algorithm)
and compared the results to the solution resulted from optimal branch and bound
algorithm. However, their exact solution has the limitation of four depots and 27
demand points.
Albareta-Sambola et al [36] defined an auxiliary network for a two layer
capacitated facility allowing a single vehicle allocation to each potential facility. By
using a taboo search algorithm, they compute the upper bound of LRP as well as the
lower bound of the problem. Combining variable neighborhood search (VNS) and
taboo search (TS) principles, Melechovsky [37] presented a meta-heuristics to solve
the capacitated facility capacitated vehicle LRP with non linear costs. In addition to
considering regular cost of LRP which consists of fixed cost per open depot, fixed
cost per vehicle, and traveling cost, he considers a non linear cost per open depot
increasing with the total demand handled at this depot.
16
Wang et al [38] presented a two phase hybrid algorithm for the capacitated
vehicle multi depot LRP (MDLRP). In the first phase, a taboo search algorithm is
applied on location variables (LAP) to identify a good configuration of open facilities.
In the second phase, an ant colony algorithm is run on the routing (VRP) of the given
configuration. Bouhafs et al [39] proposed a two phase algorithm for capacitated LRP
(CLRP). In the first stage a simulated annealing is used to find a good configuration
of distribution centers while in the second stage an ant colony system is applied to
fine good routing corresponding to this configuration. They tested their approach on
some instances taken from the literature and reported that the computational results
are reasonable.
Yang and Li-Jun [40] decomposed a capacitated facility and vehicle LRP to
two sub- problems; location allocation problem (LAP) and vehicle routing problem
(VRP). They modified the classic LRP objective function by adding a penalty term
associated with infeasible solutions. Each sub problem is iteratively and sequentially
solved by the particle swarm optimization method (PSO). The same problem holding
the same assumption as Yang and Li-Jun was solved by Prins et al [41]. They refer to
LRP with both capacitated depots and vehicles as general LRP (CLRP). An
evolutionary algorithm called memetic algorithm with population management
(MA|PM) is presented. This approach is a genetic algorithm hybridized with local
search technique and a distance measure allows controlling the diversification of the
solution. They could get good results for the problems with maximum 200 customers.
Moreover, Prins et al [42] proposed another heuristics algorithm for CLRP. Their two
stage algorithm begins with a greedy randomized adaptive search procedure (GRASP)
following path re-linking algorithm in the second phase. Their results indicate that the
proposed algorithm can also be used in case of un-capacitated depots. In her PhD
17
dissertation, Prodhon [43] presented four heuristic algorithms; GRASP, MA|PM,
cooperative approach, and polyhedral method for CLRP and compared the results to
exact solution. She reported good results when using MA|PM and polyhedral
approach. Barreto et al [44] proposed a clustering method for the two layer
capacitated facility; capacitated vehicle LRP (CLRP). They classified customers
against six proximity measures and solved the problem using four algorithms of
which two are sequential and two are iterative. They compared their results with 19
CLRP instances adopted from the literature.
A variant case of LRP allowing multiple use of vehicle subjected to vehicle's
time is presented by Lin and Kwok [45]. In their multi objective function, they
considered both load and time capacity among the homogenous vehicles. Two
sequential approaches; simulated annealing (SA) and taboo search (TS) are proposed
to solve the defined LRP before and after multi route assignment. Computational
results on real data show that area characteristics play an important role in the two
proposed heuristics. Since real data do not include the traveling time, the traveling
times are estimated through a geographic information system (GIS). Moreover, they
use the regression estimation for unavailable delivery time data.
Lashin et al [46] presented a three layer supply chain model which is a variant
of LRP where customer multiple visiting is not allowed. While at the first tier the
predefined numbers of plants are located, locations of warehouses at the second level
are selected to serve retailers directly. Formulating their model as a mixed integer
linear programming problem, they used Lagrangian relaxation method to get an initial
solution and then improved it by sub-gradient search approach.
A bi-level programming formulation of the capacitated facility and vehicle
LRP is presented by Marinkis and Marinki [47] where two decision makers at two
18
levels decide on the locations and routings using a genetic algorithm in an iterative
manner. While at the first level the leader (first decision maker) selects the location, at
the second level the follower (second decision maker) calculates the routing costs.
Then, the leader optimizes the locations based on what he receives from the follower.
When a pre-specified number of generations are reached or the genetic convergence is
achieved, the algorithm stops. To test the effectiveness of their algorithm, they
formulated and solved a real problem and compared it to a set of benchmark
problems.
Caballero et al [48] presented a TS solution for a multi objective, capacitated
vehicle LRP subjected to time windows. The model consisted of weekly
transportation of specific risk material (SRM) from some slaughterhouses to a set of
incineration plants passing through some towns located at the road network. Applying
the multi-objective meta-heuristic using an adaptive memory procedure (MOAMP), it
is shown that the results are quite promising.
2.2.3 Dynamic and Stochastic LRP
When demand in an LRP is not fixed, it is dynamic or stochastic (also called
probabilistic). In dynamic case, although the demand is not fixed, the values of
demand over different period of time are available. In case of stochastic, however, the
value of demand over time is not available. In such a case demand is defined by a
probabilistic distribution such as Normal, Poisson, etc. Dynamic LRP and stochastic
LRP have not been studied as much as static one. Like in the static demand case,
stochastic LRP solutions can be classified into exact and heuristic. However, most
algorithms on stochastic LRP are presented as exact algorithms. One of the first works
19
on stochastic LRP was by Berman and Simchi-levi [49] in which the goal was to
locate a single facility service unit on a network so as to minimize the expected length
of travel by the traveling salesman. This work assumed that the daily demand of the
customers is presented by a probability distribution. It is shown that the optimal
solution is located on the nodes (not curve) of the network. Single facility LRP which
is also called traveling salesman location problem may be the first known LRP in the
literature. The problem considered a situation in which a subset of customers request
service from a single server in each time interval. As Nagy and Salhi [7] described,
there are two models of traveling salesman location problem in the literature. In the
first model, the server first receives the list of requests at the beginning of the time
interval and assigns a route to the customers on the list. This type of service is called
TSLP and Simchi-Levi and Berman [50] and Bertsimas [52] have studied such
models. The second model deals with determining the routing for all customers
regardless of their daily requests. Once the request is updated at the beginning of each
time interval, the server begins his tripe on the pre-determined rout where he ignores
un-requested customers. This type of service is called probabilistic traveling salesman
location problem (PTSLP). Bertsimas [52] has studied such a model.
McDiarmid [53] proposed locating a single facility at an arbitrary demand
point placed on a tree network to minimize the expected traveling salesman distance
tour. To simplify the model, he assumed that the demand on arbitrary demand points
is represented by a Binomial distribution. Simchi-Levi [54] designed an analytical
model to represent probabilistic distribution systems. In his model, he assumes that
only a sub set of all potential customers request a service daily based on a known
distribution system. He presents a three stage hierarchical approach to solve the
model; location stage (numbers and locations), customer allocation, and routing
20
strategy. Moreover, it is assumed that the demand can be covered by split deliveries.
A probabilistic model of sales-delivery man problem is proposed by Averbakh and
Berman [55]. In this model a single server is located on a tree network so as to
minimize the expected waiting time subject to expected traveling distance length. A
fixed priory routing for each set of potential customer demands identifies the optimal
traveling path and based on the serving request customers are visited.
A version of stochastic LRP is studied by Laporte et al [56]. A primary
decision about the fleet size, facilities locations, and routings is made without
considering actual demand. Then, vehicles begin their services on the selected
routings until the first customer violates the vehicle capacity constraint. Returning to
its depot, the vehicle is re-loaded and re-starts its trip on the same route. However, a
penalty is considered due to returning to the depot. The objective is to minimize the
expected penalty plus fixed depot opening cost (establishing cost) and routing cost.
The problem is modeled as an integer programming problem and an optimal solution
is found.
Chan et al [57] studied the stochastic LRP as well. Their model considers a
network consisting of depots and factories where raw materials are stored in depots
and transported by vehicles to factories to be processed. They assume that the demand
for requirement of raw materials follows a stochastic distribution. Using the stochastic
process, an initial estimation of demand is computed to form an asymmetric2 static
formulation for multi- depot, multiple- vehicle routing location problem
(MDMVRLP). A stochastic formulation is also presented. While the static model is
solvable by any mixed integer programming solution and also an extended Clarke
2
The distance between two points i and j is asymmetric if Cij#Cji where C indicates the distance.
21
Wright3 heuristic approach, the stochastic one is solved by three- dimensional spacefilling curve algorithm. The advantage of their formulation is that it allows load
splitting, frequency assignment, and fleet assignment.
Albareda-Sambola et al [59] presented an un-capacitated SLRP in which
uncertainty is modeled using a vector of independent Bernoulli random variable as the
demand vector. In other words, their model allows customers to have an order in a
time horizon. The model is handled in two stages; while in the first stage a priority
decision identifies the locations of open facilities assigning customers to them, the
actual tour will skip those customers who do not require service in the second stage
(posteriori decision) without any changes in the locations of open facilities. They
presented a heuristics to solve the model.
Laporte and Dejax [60] studied a dynamic location routing problem (DLRP);
consisting of three layers, facilities at two layers and demand points at the third layer.
They assumed that the facilities located at the first level and the assigned vehicles to
them are fixed and will not change over time. In contrast, secondary facility locations
and the size of vehicle fleets have to be determined periodically and may change over
time. Moreover, the number and location of customers (third layer) may change over
time. The problem dimension is also extended to more than three layers. To model
such problems, the initial problem was transformed to the Hamiltonian circuits4 and a
3
It applies to problems for which the number of vehicles is not fixed (it is a decision variable). For
further explanation please see [58].
4
A Hamiltonian circuit is a cycle in an undirected graph which visits each vertex exactly once and also
returns to the starting vertex. For further explanation please see [4] & [61].
22
single TSP over n+m-1 cities were solved where n and m were the number of
customers and depots respectively. An alternate estimation solution was also
presented for DLRP enabling the algorithm to be applied to large scale problems.
Averbakh and Berman [62] presented a dynamic programming model to locate both
single and multiple delivery man problems (DMP) on tree networks. They solved
their problem for two different objectives; minimizing the waiting time of customers
and minimizing the distance the delivery man travels. They mention that their
approach is not efficient. Nambiar et al [63] formulated a real world problem in
rubber industry as a dynamic plant location and vehicle routing. They proposed
heuristics to solve the problem.
Gorr et al [64] designed a dynamic spatial decision support system (SDSS) to
solve home-delivery (HD) service problem. They modeled this problem as an LRP
subject to time window (there is a limited time to deliver orders at customers' home).
They tried to improve the delivery network by selecting appropriate meal production
locations and efficient routing in the city network. They used location-allocation first,
route second approach to solve the proposed un-capacitated vehicle, capacitated
facility HD problem.
Yi and Ozdamar [17] proposed an application of the dynamic facility
capacitated LRP in a disaster situation. Their goal was to find the best location of
temporary centers and shelters in affected areas to speed up medical care for less
heavily wounded survivors considering the maximum coverage of the area. Their
mixed integer, multi commodity, network flow model considered capacitated vehicle
and multi commodity products pickup delivery assumptions. Moreover,
heterogeneous fleet vehicles may multi visit to the same customer in their route. In
contrast to most LRPs, it is not necessary for a vehicle to come back to the origin once
23
it visits the last customer. A two stage algorithm is proposed to solve the problem and
a real world earthquake situation verifies solvability of the algorithm.
Ago et al [65] allocated storage to raw materials and assigned them beltconveyor as transportation system. They assumed that requirements for raw materials
are not fixed over time but are determined. Although their work was not referred to as
LRP, it was in fact formulated as a dynamic in-factory-LRP. In other words, the
logistics system inside the factory was being improved while it considered the
location of raw material in the storage and transportation of them to the production
line simultaneously. They solved the problem by Lagrangian decomposition
approach.
2.2.4 LRP and Inventory
Since location, routing, and inventory decisions are inter dependent decisions,
the combination of inventory and LRP in the same model makes improvement on the
logistic system. However, most of studies concentrate on dual combination of
location, routing, and inventory rather than considering all three simultaneously. A
body of research is available on integrated routing- inventory, and locating- inventory
models. A review of the routing- inventory literature can be found in Kleywgt et al
[66] & [67]. Proposing a Markov decision formulation for inventory- routing
problem, Kleywgt et al [66] & [67] suggested an approximation method to solve the
model in a reasonable computational effort. They also reviewed inventory- routing
literature in terms of demand, vehicles, horizon, delivery, and inventory
characteristics.
Integrating location, routing, and inventory can be divided into two groups: direct
delivery and multiple delivery.
24
1.
Direct delivery: These studies assume that products are directly transported
from depots to customers. In fact, each individual customer's requirement is equal or
greater than one truck load. Works of Novick and Turnquist [68] & [69], Erlebacher
and Meller [70], Daskin et al [71], Shen et al [72], Shu et al [73], Miranda and
Garrido [74] & [75], and Shen and Qi [76] belong to this category. All works referred
in this part assume demand follows a probabilistic distribution.
Erlebacher and Meller [70] developed an analytical model for a stylized
version of a three layer supply chain system with capacitated plants. They considered
non linear inventory cost for the proposed continuous review policy. Moreover, the
rectilinear distance is assumed from plants to DCs and from DCs to customers. The
proposed algorithm takes 117 hours on a Sun Ultra Sparcsstation on a 600 node
problem to solve.
Daskin et al [71] formulated a non linear integer programming for a three
layer supply chain system (facility, DC, and retailer). They converted the model to the
Lagrangian relaxation sub problem and proposed a number of heuristics to solve it.
They used (Q, r) model with type I service5 as the inventory policy and approximated
Q* using EOQ model. Presenting the set-covering integer programming model, Shen
[72] proposed the Column generation method for the same model proposed by
Daskin et al [71] except that he assumed a single plant assumption in his work. It is
noticeable that Daskin et al[71] and Shen[72] assume that the variance of demand of
customer i, σi, is proportional to the mean of demand of customer i (µi). Relaxing this
5
Type I service assumes that service level at (Q, r) is approximately equal to the probability of
demand during lead time being less than reorder point (r).
25
condition, Shu et al[73] proposed an efficient heuristic algorithm for the same model
leading to solutions for problems up to 500 retailers efficiently.
Miranda and Garrido [74] considered the (Q, r) inventory policy for their three
layer single plant model and ignored shortage cost. Presenting the model in a non
linear mixed- integer formulation, they applied Lagrangian relaxation and subgradient method to solve the model. Later, Miranda and Garrido [75] extended their
model by adding homogenous warehouse constraint and solved it by the same
method. Shen and Qi [76] studied a two layer location-direct routing- inventory
problem considering stochastic demands. At the first level, there are some potential
DCs holding certain amount of safety stock to fill a predefined service level. In other
words, DCs use the (Q, r) policy subject to service level constraint. The objective is to
minimize total cost including DCs location cost, inventory cost at DCs, and direct
routing cost. Moreover, the uncertainty is illustrated in their model by considering
variations in the lead time. Formulating the problem using non-linear integer
programming model and solving it by Lagrangian relaxation method, reasonable
results are achieved for the problem up to 320 customers.
2.
Multiple- delivery: This is a case where multiple visits are allowed. Research
in this area has not been extensive. Perl [10], and Perl and Daskin [11] presented the
modified warehouse LRP (MWLRP) in which they introduced routing in multi depot
location problem. They assumed a variable cost for potential depots per unit of
throughput in addition to the fixed depot costs. Extending works of Perl [10] and Perl
and Daskin [11], Hensen et al [79] simplified the integer linear programming of
WLRP to the mixed integer programming (MILP) where by introducing a set of flow
variables, the numbers of constraint are reduced. Moreover, a sequential heuristic
algorithm is presented to solve the problem.
26
Ambrosino and Scutella [77] studied a complex distribution network design
considering capacitated facility location, warehousing, capacitated transportation load,
and inventory levels. Proposing a four tier logistic system consisting of plants, central
depots, regional facilities, and customers, they assumed two types of regional
facilities; regional depots (hold inventory) and transit points (do not hold inventory).
Moreover, they considered two types of customers; regular customers, and big
customers (big demand). They allowed big customers to be served by central depots.
In contrast to most of the studies on LRP, their work assumed non- homogenous
vehicle fleet. Mathematical formulations are presented for both static and dynamic
models. Although they arrived at exact solution for some small instances using
CPLEX, exact solution can not be achieved for large instances.
Burns et al [78] studied two different distribution models. In one model
products from a supplier are transported directly to customers, in the other model
peddling (multiple visits) is allowed. Both models require spatial density6 of
customers rather than the precise locations of every customer. They showed that the
optimal order quantity is attainable by calculating EOQ for direct delivery and full
truck for the peddling one.
Liu and Lee [80] proposed a multi depot LRP (MDLRP) model which
considers the inventory decisions. The single product continuous review system is
selected for the inventory policy. Adding inventory costs to the objective, it is
assumed that demands follow stochastic distributions over time. They proposed a two
stage heuristics methodology for the proposed model. In the first stage a route first,
6
Special density indicates the weight of customers in a region but does not identify the exact location of
demand points like LRP assumption.
27
location-allocation second algorithm generates the initial solution and then in the
second phase the initial solution is improved. Computational results indicate that the
algorithm achieves better solution than other algorithms that do not take into account
the inventory cost. However, the proposed model is only limited to a single-product
case. Moreover, adopting the fixed order system as their inventory policy may
increase the transportation cost dramatically. Liu and Lin [81] proposed a modified
solution for the same model proposed by Liu and Lee [80]. TS and SA are used in the
improving phase. Computational results indicate better solution than what it was
proposed by Liu and Lee [80].
2.2.5 Other Types of LRP
By relaxing basic assumptions of LRP, new models are formed. Although
these models do not hold major LRP characteristics, they belong to LRP category in
the literature. For example, work of Schwardt and Dethloff [82] is one of the
mentioned works. They studied a continuous single facility vehicle capacitated LRP
in which facilities can be located in planar spatial rather than potential nodes as
regular LRP assumption. The self organizing map (SOM) approach is used on the
Weber problem (SOMW) to classify customers and assign routes to them. It is
reported that their iterative algorithm leads to a better solution than sequential method
in terms of minimal cost. Later, Salhi and Nagi [83] proposed an iterative heuristic for
continuous un-capacitated multi facility LRP. They solved their model by using the
Weizfeld procedure7. The idea behind the algorithm is to transfer the LRP problem to
7
Weiszfeld procedure is a method using partial derivatives for solving weighted Weber problem. For
more detail please see [84].
28
a pure location problem and then solve this problem. They assumed that the number
of the facilities is predefined. Moreover, the fixed costs of the facilities are ignored
and there is limitation on the traveling distance.
Doerner et al [85] presented an application of LRP in healthcare management.
In their case a single mobile medical facility serves a set of population in a region. In
contrast to regular LRP assumption in which the location of the facility is fixed, this
problem assumes that the single facility moves around the region to give medical
help. Moreover, it ignores to serve all population but maximizes the tour coverage.
The mobile facility performs its closed tour in a time interval called period. Tours are
evaluated according to three measures; tour length, average distance to the nearest
tour stop, and coverage criterion measuring the percentage of the population unable to
reach the tour stops. They proposed two approaches to solve the model: simultaneous
location-routing approach which uses an ant colony algorithm and location by multiobjective genetic algorithm with routing as a sub-procedure. It is indicated that a
combination of two method leads to promising results.
Another branch of LRP exists in computer science, networking,
telecommunication, and internet application. Two of such works have been studied by
Lee et all [86] and Billionnet et al [87]. Lee et al [86] used LRP in the fiber- optic
communications. In order to design an optical internet network, a two layer LRP
model is presented where in the first layer a set of potential gateway nodes8 plays the
role of plants and in the second layer remote nodes indicate the customer demands.
The goal is locating gateways, allocating remote nodes to gateways so as the fixed
8
A gateway node is a special node for processing the demand of several access nodes.
29
cost of wavelength division multiplexing9 (WDM) at gateways, access node10 cost
(AN) at gateways, link cost (routing cost) from gateways to remote nodes, and line
card (modular electronic circuit) cost are minimized. Two taboo search (TS) heuristics
are proposed and their effectiveness is compared.
Billionnet et al [87] applied modified warehousing LRP (see section 4-4-2b) in
designing the synchronous digital networking (SDN11) network. The network is
consisting of two types of equipment: antennae (BTS12) and concentrator (BSC13).
The location and traffic demand of BTS are predetermined but the locations of BSCs
are to be identified. The goal is finding the best topological order of the network by
locating BSCs, identifying their type (BSCs are distinguished by their capacity), and
assigning BTSs to BSCs so as to minimize the BSC opening cost, BSC equipment
cost, fixed assigning cost, and variable link cost. The problem is modeled like the
modified warehousing LRP (MWLRP). A three phase heuristic approach is used to
solve the problem.
9
A technology which multiplexes multiple optical carrier signals on a single optical fiber by using
different wavelengths (colors) of laser light to carry different signals [88].
10
Access node (AN) is a network element of a gateway node supporting different access network
architecture.
11
SDN is a method for communicating digital information using lasers or light-emitting diodes (LEDs)
over optical fiber [89].
12
Base transceiver station
13
Base station controller
30
CHAPTER 3
MATHEMATICAL MODELING AND SOLUTION METHODOLOGY
3.1.
Problem Definition
The problem under consideration is a three layer location- routing problem
(LRP) consisting of plants (first layer), depots (second layer), and customers (third
layer) integrated with the inventory model under fixed interval policy. In such a
network, products are transferred from plants to depots and from there to customers. It
should be noted that depots not only play their role as transition points but they also
hold inventory; therefore, they act as distribution centers (DCs). Located at the predetermined locations, customers demand their requirements stochastically. The goal is
to select locations for DCs among sets of potential locations, allocate customers to
DCs, find routes from DCs to customers, and determine the order intervals and
maximum inventory at depots so as to minimize total network cost.
The presented work is divided in two parts. In the first step, we come up with
the LRP model and then in the second part, an appropriate inventory strategy model is
selected and integrated with the presented LRP.
3.1.1. Presented LRP
The presented model is an extension of the work by Perl & Daskin [11], and
Liu & Lee [80]. Recalling LRP characteristics presented by Nagy and Salhi [7], our
presented LRP model considers the following characteristics as illustrated in Figure 31: three layer supply network with multiple capacitated plants at the first level,
multiple capacitated depots (warehouses) at the second level, and customers at the
third level (3/R/T), discrete solution space, stochastic demand, multiple-
31
homogeneous capacitated vehicle, unspecified time/distance windows (no limitation
on time and distance), multiple products, integrated with inventory.
While routes from plants to depots are directed, customer multiple visits are
allowed for routes from depots to demand points.
Layer 1: Potential
Capacitated Plants
1
2
...
k
(One or more should be
Layer 2: Potential
1
2
…
m
Capacitated Depots
(One or more should be
Layer 3: Customers
Figure 3-1: The pictorial of presented LRP (3/R/T)
A common limitation which exists in real life is depot (warehouse) capacity.
Capacity limitation can be added to the constraint of the model. In practice,
sometimes it is possible to rent space from a third party in case of capacity constraint.
This third party is called third party logistics (3PL). To capture the real situation in
our model, depots are assumed to be capacitated while the use of a 3PL is allowed if
needed.
The objective is to locate depots, allocate depots to plants and customers to
depots, and routes from depots to customers simultaneously in such a way that the
32
summation of a) depots fixed establishing costs, b) plants and depots operating costs,
c) transportation costs d) inventory costs and e) 3PL costs are minimized. The
decisions which should be made in this problem are as follows:
1)
Select plants locations from a set of candidates location,
2)
Select depots locations from a set of candidates location,
3)
Determine the composition of products from plants to depots,
4)
Assign customers to depots,
5)
Construct optimal delivery routes from depots to customers,
6)
Inventory levels at selected depots,
7)
Routes order interval
Assumptions and limitations of the presented model are as follows:
1)
The locations of customers are predefined. However, demands are
assumed stochastic following any distribution.
2)
Plants at the first level of logistic network are considered capacitated.
3)
The capacities of depots are limited. Therefore, depots hold a third
party logistics contracts which can be used in the case of space limitation.
4)
The less than truck load (LTT) property is assumed which indicates
that the demand of each customer is less than a truck load. Therefore, each
vehicle may visit more than one customer in each trip.
5)
Each route starts and ends at the same depot.
6)
All of the service requirements of a customer is met by only one
vehicle, therefore, it is not allowed that a customer receives its demand from
two different depots (demand split is not allowable).
7)
Capacity of the homogenous vehicle (s) is limited.
33
8)
Vehicle routes pass through only one depot. This means that there is no
connection among depots.
3.1.2. Inventory Strategy
The model in this dissertation assumes that demand during lead time14 is
stochastic. However, we restrict our attention to the case where average demand
remains approximately constant with time. We first assume a general form of
stochastic demand during lead time and then propose the Poisson distribution for it.
Variation can be originated from demand and/or lead time. It is assumed that variation
is originated only from demand meaning that the lead time is known and constant. To
absorb the effects of the demand variability, safety stocks (also referred to as buffer
stocks or fluctuation stocks) should be considered.
Silver et al [90] indicate key issues to be solved systematically by a control
system under probabilistic demand as follows:
1. How important is the item?
2. Can, or should, the stock status be reviewed continuously or periodically?
3. What form should the inventory policy take?
4. What specific cost or service objectives should be set?
As it was mentioned previously, the presented inventory system considers multiproduct, multi- customer system. To answer question one, it is assumed that all items
have equal importance; therefore, all products are classified in the same category (not
A, B, C category). Question two deals with the strategic decision of the inventory.
The important point in combining the inventory and LRP is that the fixed vehicle cost
14
Lead time is defined as the time from releasing an order to receiving it. It is assumed here that the
fixed order period is longer than lead time period
34
is in fact a part of order cost. This means that by increasing the number of orders, the
cost of routing increases. However, at the same time the cost of holding inventory will
decrease and vice versa. Therefore, there is a trade off between the inventory
decisions and routing cost. In such a situation selecting the continuous review method
(known also as Q system; fixed order system; transaction reporting; two bin system)
will increase the order (vehicle) costs because, each time the system receives an order
it has to send the requested item (product) by a vehicle. Moreover, under this
circumstance, there may be lots of direct deliveries which have conflicts with our
basic assumption (multiple visits). In contrast, periodic review policy (known as T
system; fixed interval system) fits the problem better. It is possible to review orders at
the predefined intervals for all customers who are located in the same route. In such
systems, the order interval T and predetermined total inventory position (Imax)
completely define the system. If the minimum number of reviews is determined, it is
possible to save on order (vehicle) costs. Moreover, periodic review policy is usually
easier to administer than continuous review policy [92] especially when the inventory
works under multiple products assumption.
Once the stock reviewing status is selected, the proper order policy should be
chosen. There are different types of operating doctrine of which two are more popular
in both practice and literature; (T, Imax), and (T, r, Imax). In the former model which is
called "order up to Imax", at predetermined intervals, the inventory is reviewed and an
order is placed. The size of the order is determined by subtracting the amount on hand
from a predetermined total (Imax). In the later model, the reorder point (r) is taken into
consideration. The idea is that every T units of time the inventory level is checked and
if it is less than reorder point, an order is placed to raise the inventory level to Imax.
However, if the inventory level is above reorder point, no order is placed until at least
35
the next review. Since (T, r, Imax) system deals with three parameters and needs more
computational efforts, we simplify the model by adopting (T, Imax) system for our
model. We assume a safety stock based on minimizing cost rather than customer
service (question four).
Integrating inventory to the LRP model will increase the complexity by adding
two new decisions:
1)
Determining the best service interval time of every route (T), and
2)
Determining the maximum inventory (I) of every product to minimize
cost.
The characteristics of the presented LRP defined in detail in section 3.1.1. Moreover,
in section 3.1.2 the characteristics of an appropriate inventory policy described. In the
next section, an approach for the modeling of the integrated LRP and inventory is
presented. It also provides a solution method for the model.
3.2.
This chapter will fit a mathematical model to the presented LRP in the
previous section. An appropriate inventory model is first developed which is then
integrated with the presented LRP.
3.2.1. Inventory Model
In this section, the inventory strategy presented in Section 3.1 will be
developed in the form of a mathematical model and then integrated with the presented
LRP model. The presented inventory model is an extension of the work by Tersin
[91]. To facilitate the presentation of the presented inventory model, Tersin’s model is
described in Section 3.2.1.1 followed by the presented model.
36
Mathematical Mo
3.2.1.1.
Tersin’s Single Product- Single Customer Inventory Model
Tersine [91] proposes a fixed interval inventory model (also known as Tsystem) in which the annual cost of a single product, single customer inventory under
stochastic demand is considered. In this model, demand during order interval plus
lead time follows a statistical distribution. The model assumes that the total annual
cost of the inventory system is equal to the sum of purchasing cost, ordering cost,
holding cost, safety cost, and shortage cost. The fixed interval system is fully defined
by two parameters: the order interval, T, and the maximum inventory, I. It is assumed
that fixed order interval includes lead time. The following notations are defined and
used in this section:
L: Lead time; the time from releasing an order to receiving it
T: Order interval in years; it is assumed that L<T
T*: Optimal order interval in years; it is assumed that L<T
A: Fixed order cost
C: Processing cost
H: Holding cost per unit per year
K: Stock- out cost per unit
TIC: Total inventory cost
TIC*: Optimal total inventory cost
d: Demand during order interval– random variable
D: Expected annual demand
µ: Expected demand during order interval time
I: Maximum inventory level
E (d > I): Expected stock out during order interval
P (d > I): Probability of stock out during order interval
37
In this model the lost sale stock out is allowed and there is a cost associated
with per unit of stock out. Moreover, customer demand is not static and follows a
probabilistic distribution. In such a system, orders are placed every T year. This
means that once an order is placed at time t, the next order will be placed at time t+T
and this second order will be filled at t+T+L. Therefore, the safety stock should cover
the demand during time T+L.
Tersine [91] indicates that the annual inventory cost of a system can be
computed from equation (3.1) below. As shown, there are three terms in the cost
function. The first term indicates the annual purchasing cost. The second term
stipulates the ordering cost plus shortage cost for a cycle multiplying the number of
orders per year (1/T). The last term shows the holding cost which is the average of
order quantity plus safety stock and the expected number of units short.
TIC = DC +
1
DT
[A + KE(d > I)] + H[
+ (I − µ) + E(d > I)]
T
2
(3.1)
Therefore, the optimal time interval (T*) and optimal probability of a stock out during
order interval time (P (M> I)) can be computed from (3.2) and (3.3) respectively.
dTIC
= 0 ⇒ T* =
dT
2(A + KE (d > I) )
HD
(3.2)
dTIC
HT
= 0 ⇒ P( d > I ) =
dI
K + HT
(3.3)
Plugging (3.2) in (3.1) will result in the optimal annual cost (3.4):
TIC * = DC + 2HD (A + KE (d > I ) + H (( I − µ ) + E (d > I) ) (3.4)
Note that equation s (3.2) and (3.3) indicate that T and I are dependent. Referring to
Tersine [91], it is possible to use the following iterative procedure to calculate T* until
38
it converges. Once convergence occurs, the optimal solution will be reached. The
procedure proposed by Tersine [91] is as follows:
1. Ignoring KE (d> I) in 3.2, compute T*.
2. Using computed T*, compute P (d> I) and I.
3. Using computed I, obtain E (d> I).
4. Re-compute T* with computed E (d> I) in step 3.
5. Repeat steps 2, 3 and 4 until T* converges.
If the shortage cost value, KE (d>I), approaches or exceed the order cost, A, the
assumption of independency between T and I is not an appropriate assumption. As it
is indicated, in this dissertation it is assumed that the independency is held.
Values of E (d> I) and I-µ in equations 3.1 and 3.4 are computable for any
distribution. For example, in a case where demand follows Normal distribution with
mean µ and standard deviation σ, the following equations hold [90]:
E(d > I) = σE( Z)
(3.5)
I − µ = Zσ
(3.6)
Winston [92] indicates that in a fixed interval system, the expected order
quantity can be computed as the subtraction of expected on hand inventory from
maximum inventory:
Expected order quantity= maximum inventory- expected on hand inventory
On the other hand, he assumes that the expected on hand inventory is in fact the
average number of expected on hand inventory right before and right after an order
arrives:
Expected on hand inventory =1/2(expected on hand inventory right before an order
arrives+ expected on hand inventory right after an order arrives)
39
Expected on hand inventory right before an order arrives is the maximum inventory
minus the demand in order interval plus lead time (I- µ). The expected on hand
inventory right after an order arrives is equal to the maximum inventory minus the
demand in order interval plus lead time plus the received order (I- µ+DT).
Therefore; Winston [92] formulates the expected on hand inventory and expected
order quantity as (3.7) and (3.8) respectively.
Expected on hand inventory= I − µ + D
T
2
(3.7)
and,
Expected order quantity= µ − D
T
2
(3.8)
In the next section, this model will be extended for multi products and multi
customers.
3.2.1.2.
Multi Product- Multi Customer Combined With Transportation Cost
As it was implied in the previous section, Tersin’s model considers only
single- product, single customer situation. These assumptions may not be held in the
real world where multi- customers demand multi- products. Moreover, the Tersin’s
model ignores transportation cost limiting the usage of the model. To remove these
drawbacks, this section extends Tersin’s model to a multi- product inventory model
considering transportation cost. Similar to the previous section, it is assumed that
order interval includes lead time where lead time is less than order interval.
Notations used in this section are defined as follows:
TIC: Total inventory cost
TIC*: Optimal total inventory cost
40
T: Order interval (in years)
T*: Optimal order interval (in years)
Ap: Order cost of product p
Hp: Holding cost of product p per unit per year
C: Transportation cost per unit of distance
Csj: Cost of direct shipment from plant s to depot j per unit of product
Cp: Processing (purchasing) cost of product p
Cpj: Processing (purchasing) cost of product p at depot j
FV: Dispatching cost per order
Lv: Total distance traveled among customers
Kp: Stock out cost per unit of product p
dip: Demand of product p for customer i during order interval– random
variable
Dip: Expected annual demand of customer i for product p
µip : Expected demand of product p for customer i during order interval
Iip: Maximum inventory level of product p for customer i
E (dip > Iip): Expected stock-out during order interval plus lead time
P (dip> Iip): Probability of stock-out of product p for customer i during order interval
To extend Tersin’s model consider the ordering cost. It can be assumed that
transportation cost is a part of ordering cost as follows:
Total order cost= order cost+ vehicle dispatching cost+ traveling cost
(3.9)
Using the defined notations, (3.9) can be rewritten as follows:
Total order cost for product p= Ap+ FV+ C*Lv
(3.10)
As equation (3.10) indicates, the order cost is dedicated to each product. This
means that once a customer requests p product, the summation of A1to AP would
41
imply the total order cost for p products. In contrast, since all products demanded by a
customer are supposed to be shipped by the same vehicle, it is not required to dedicate
fixed vehicle dispatching cost to each product. Moreover, a variable transportation
cost which depends to the length of the trip should be added to the order cost (C* Lv).
Substituting (3.10) in equations (3.1), (3.2), and (3.4) presented in Section 3.2.1.1:
I
P
TIC = ∑
∑D
i =1
ip
Cp +
p =1
1 i
∑
T i =1
P
∑
p =1
i
[Ap + FV + CLv + K p E(dip > Iip )] + ∑
j=1
P
∑
Hp [
DipT
2
p =1
+ (Iip − µi, ) + E(dip > Iip )]
(3.11)
∑ (A
I
T* =
P
2∑
i =1
p
+ FV + CLv + K pE(dip > Iip ) )
I
P
i =1
p =1
∑ ∑
I
TIC* = ∑
i =1
(3.12)
p =1
P
∑
p =1
I
DipCp + 2∑
i =1
H p Dip
P
∑
p =1
I
Hp Dip ∑
i =1
∑ (A
P
+ FV + CLv + K p E(dip > Iip ) + ∑
I
p
p =1
i =1
∑ H ((I
P
p
ip
− µip, ) + E(dip > Iip )
p =1
(3.13)
3.2.2. LRP Model Integrated With Inventory
Following notations are used in this section:
TIC: Total inventory cost
Ap: Order cost of product p
Hp: Holding cost of product p per unit per unit time
H′: Holding cost at the third party per unit of product (fixed for all products) per unit
time
Lghv: Total distance traveled from point (node) g to point (node) h on route v
m: Transportation cost per unit distance
Csj: Cost of direct shipment of one unit of product from plant s to depot j
Cpj: Processing (purchasing) cost of product p at depot j
FV: Fixed dispatching cost per order
42
)
Kp: Stock out cost per unit of product p
dhvp: Demand of point h on route v for product p during order interval time (stochastic
variable)
µ hvp : Expected demand of point h on route v for product p during order interval time
Dip: annual demand of product p for customer i
Dhvp: Annual demand of point h on rout v for product p
Ihvp: Maximum inventory level of point h on route v for product p
E (dhvp> I hvp): Expected shortage of point h on rout v for product p during order
interval time
P (dhvp> I hvp): Probability of shortage of point h on route v for product p during order
interval time
Fj: Fixed depot opening cost of depot j
Ksp: Capacity
space for product p at plant s (number of units)
Kjp: Capacity space of product p at depot j (number of units)
Kv: Capacity of vehicle v for all products. The limitation is on the total number of
products serving route v not on individual product.
Tghv: Order interval for point g to h on route v in year. Connection gh is served on
route v every Tghv year.
Nv: Number of customers assigned to route v
The following decision variables are used in the model.
To develop the mathematical model of the presented problem, the following sets are
defined:
I= {I; i=1, 2… n}: Set of customers (third layer)
J= {j; j= n+1, i+2… n+m}: Set of depots (second layer)
43
S= {s; s=n+m+1…n+m+k}: Set of plants (first layer)
V= {v; v=1… v}: Set of vehicles
P= {p; p=1… p}: Set of products
H= I U J = {h; h=1, 2… n+m }: Set of all depots and customers
Table 3-1 defines the five decision variables used in the presented model. As
shown, Zghv is a three index, binary variable implying the routing decision. If in route
v node g precedes node h (g and h ∈ G), the value of the routing variable is equal to
one; otherwise it is equal to zero. Another binary variable, Xjp, is related to the
location of depots. If a potential depot j is determined to be opened, its value, Xj,
would be equal to one and zero otherwise. Yij is another decision variable which
assigns customer I to depots j. This variable indicates that if the customer i ∈ I is
allocated to the depot j ∈ J; the value of the variable is equal to one and zero
otherwise. Wsjp indicates the quantity of product p shipped from plants to a depot
(Wsjp). In this variable indexes s, j and p belong to the sets of plants (S), depots (J),
and products (P) respectively. Such a variable holds non-negative value. The last
decision variable, Bjp, relates to the space capacity which is a linear, non- negative
variable. If there is a space limitation of the capacity of depot j for product p, the cost
of renting a third party logistics (3PL) should be added to the objective function;
otherwise, no charge is added to the system cost.
Table 3-1: Variables of the Mathematical Model
Variable
Zghv
Xj
Yij
Wsjp
Bjp
Description
is 1 if g immediately precedes h on route k; otherwise, 0
is 1 if a facility is located at depot j for product p; otherwise,
0
is 1 if customer i is assigned to depots j; otherwise, 0
quantity of product p shipped from plant s to depot j
quantity of product p which should use 3PL due to space
limitation of depot j
44
Type
routing
location
allocation
quantity
3PL
1.1
As it was mentioned previously, the objective in the LRP model is to minimize
the system’s cost. In the classical LRP model, the system cost is the sum of fixed
depot cost, variable warehousing cost and delivery cost. Adopting the classical LRP
objective function, the system cost is modified based on the presented network
characteristics. The objective function and sets of constraint are presented in the
formulation format as follows:
n+m
Min
∑F X
j
j= n +1
+
n +m+k
n+m
p
∑ ∑C ∑W
+
j
sj
s = n + m +1 j= n +1

p =1
∑ ∑ ∑  2∑ H D ∑ (A
h ∈H g ∈H v∈V
h ≠g h ≠g
P
p

hvp
p =1
p
sjp
+
n+m
p
n
∑ ∑∑D
ip
C pj Yij +
j= n +1 p =1 i =1
n+m
p
∑ ∑ H ′B
jp
j= n +1 p =1
P
P

+ FV + mLghv + K p E(d hvp > I hvp ) ) + ∑ H p (I hvp − µ hvp ) + ∑ H p E(d hvp > I hvp )Zghv
p =1
p =1

(3-15)
Subject to:
∑∑ Z
=1
ihv
i∈I
(3-16)
v ∈ V, h ∈ H
(3-17)
v∈V h∈H
∑Z
hgv
−
g∈H ,g ≠ h
∑Z
ghv
=0
g∈H , g ≠ h
∑∑Z
jiv
≤1
v∈V
Z ghv ≥ 1
∀( R, R )
(3-18)
i∈I j∈J
∑∑∑
R ⊂ G, J ⊆ R
(3-19)
g∈R h∈R v∈V
− Yij +
∑ (Z
v ∈ V, i ∈ I, j ∈ J
(3-20)
− K jp X jp ≤B jp
p ∈ P, j ∈ J
(3-21)
− K sp X sp ≤ 0
p ∈ P, s ∈ S
(3-22)
ihv
+ Z jhv ) ≤ 1
h∈H
∑W
sjp
s∈S
m
∑W
sjp
j= n +1
45
∑W
s∈S
sjp
− ∑ D ip Yij = 0
p ∈ P, j ∈ J
(3-23)
i∈I
∑ ∑ (∑ µ
∑D
hvp
−
hvp
Tghv
p∈P
2
h∈H g∈H p∈P
g≠h g≠h
) Z ghv ≤ K v
v∈V
(3-24)
The objective of the LRP model is to minimize the system’s total annual cost.
In the classical LRP model, the system’s cost is the sum of fixed depot cost, variable
warehousing cost and delivery cost. Adopting the classical LRP objective function,
the system cost is modified based on the presented network characteristics. Equation
(3-15) indicates the objective function which contains five terms. The first term (
n+m
∑FX
j
j
) shows the fixed depot establishing cost in case a new depot(s) is to be
j= n +1
opened. The second term, the direct transportation cost (
n +m+k
n +m
p
∑ ∑C ∑W
sj
s = n + m +1 j= n +1
sjp
), is the
p =1
shipping cost of products from plants to depots. This cost depends on the distance
from the plant to the depot. The third term represents the total annual inventory cost
consisting of three sub-terms: purchasing (processing) cost (
n+m
p
n
∑ ∑∑D
ip
C pj Yij ), third
j= n +1 p =1 i =1
party logistics cost (
n+m
p
∑ ∑ H′ B
jp
) and inventory cost (last term in equation 1). The
j= n +1 p =1
third party logistics cost is a linear function of the quantity of product p which should
use 3PL due to space limitation of depot j. The last term indicates the combined
routing and inventory cost. The inventory policy assumed is the fixed order interval
policy adopted from [90] and [91]. The inventory cost (last term in the objective
function, 3.15) is described in Section 3.2.1 indicated by equation (3.13).
There are also nine sets of constraints in the model. The constraint sets (3-16)
to (3-24) force only one depot to be assigned to a route. Constraint (3-16) assigns one
46
and only one route to any customer. Constraint (3-17), known as flow conservation
constraints [12], shows that any node belonging to the set of depots and customers
should be entered and departed by the same vehicle. Constraint (3-18) indicates that
any vehicle on the network can depart a depot only once. This prevents the vehicle to
travel more than once for the customers located on the same routings. Moreover, it
does not allow the vehicle to pass other depots. Constraint (3-19) illustrates that there
is at least one connection from sets of depots and any customer(s) to the rest of the
customers. These constraints are known as connectivity constraints. Constraint (3-20)
connects the routing decision to the allocation decision. If there is a route from a
customer to a depot, the customer should be assigned to that depot. Constraint (3-21)
represents the 3PL constraint. If there is a limitation on the capacity of a depot for a
product, the corresponding cost should be included in the total cost. For per unit
limitation of product p at depot j a cost of H' is charged on the network cost.
Constraint (3-22) implies the capacity limitation of a plant. It ensures that the total
units of product p delivered to all open depots are less than the space limitation of the
plants. The constraint set (3-23) indicates that for any selected depot, the sum of the
products shipped from all plants should be equal to the sum of the product demanded
by customers. The constraint set (3-24) implies that for any route, the total quantity of
all products should be less than the capacity of the vehicle.
To form the equations of the presented LRP model, it is helpful to follow the
steps in Figure 3-2. As the flowchart shows, in the first step the inventory parameters
should be computed using corresponding equations. Then, the coefficient of Zghv is
computed. Note that this coefficient in fact indicates the inventory costs in the
objective function and is computable for any two pairs of nodes belonging to the set
G = I U J (see definition of sets). Computing the inventory parameters, it is possible
47
to form the integrated model equation (3.15 to 3.24) and then apply an appropriate
solution procedure.
48
step1
Define Sets I, J, S, V,
P, T, G
Calculate the inventory
parameter such as: T*ghv, I hvp ,
step2
step3
P (dhvp> I hvp), , E (dhvp> I hvp) and Dhvp)
for any two nodes belonging to
set H
Calculate the coefficient of Zghv
(last term of (3.15))
step4
Calculate other parameters in
objective function and constraints
step5
Form the mathematical model
(equation 3.15 to 3.24)
step6
Apply solution procedure
Figure 3-2: The Solution Procedure to Form and Solve the Integrated LRP and
Inventory Model
49
3.2.3. Sample Problem
To clarify the presented model, a small size sample problem is developed and
solved in this section. Consider a network with two potential plants at the first level,
one distribution center (DC) at the second level, and three customers at the third level.
One truck ships two products from DC to customers. Assume that demand during lead
time plus order interval time follows a Normal distribution. Inventory costs are as
follows:
L=1day
K1=K2=$7.5
A1=A2=$5
H1=H2=$25
FV=$20
H′=$100
C=$1 per mile
It is also assumed that there are two homogenous vehicles in the network with the
capacity of 400 units of products. There is a single depot in the network 5000 unit of
products capacity per product and $150000 fixed opening cost. The transportation
costs from the depot to the plants are $0.15 and 0.95$. The purchasing costs (Cpj) are
$30 and $50 for product 1 and product 2 respectively. Other required data are
provided at Tables 3-2 to 3-6:
Table 3-2: Capacity of Plant s for Product p (Ksp)
Plant
Product
1
2
4
12000
5000
50
5
10000
18000
Table 3-3: Distance Matrix
1
2
40
1
2
3
3
50
30
Table 3-4: Annual Demand of Customer i for Product P (Dip)
Product
1
2
Customer
1
480
360
2
740
600
Table 3-5: Demand During Order Interval Time
Product
1
2
Customer
1
100
40
2
50
120
Table 3-6: Variance of Demand During Order Interval Time
Product
1
2
Customer
1
64
81
2
9
36
Solution:
The solution procedure presented in Figure 4 is followed here:
Step 1: Define sets
Customers=I= {1, 2}
Depot=J= {3}
Potential Plants=S= {4, 5}
51
Products=P= {1, 2}
Step 2: Compute T*, P (M>I), I, and E (M>I) for any two pairs of nodes belonging to
set G. T*12, T*21, T*31, and T*32 can be calculated using equation (3.14). Note that
for connections 13, and 23 only the transportation cost should be considered (no
inventory cost). Referring to Tersine [91] procedure, described in Section 3.2.1.2:
1.
Ignoring E (d> I) in 4.11, compute T*ghv:
T *121 =
2.
2(5 + 5 + 20 + 1 * 40)
25 * 740 + 25 * 600
Using computed T*, compute P (d> I) and I:
P(d1211 > I1211 ) =
H1T *
I − 50
= 0.178 → 1211
= 0.92 → I1211=52.76
*
K1 + H1T
9
P(M1212 > I1212 ) =
3.
T*121=0.065 year=23.73 day
H 2 T*
I − 120
= 0.178 → Φ( 1212
) = 0.92 → I1212=125.52
*
K 2 + H2T
36
Using computed I, obtain E (M> I):
E(d1211 > I1211) = E(d1211 > 52.76) = σ1211E(Z1211) = 9 * E(0.92) = 3*0.0968= 0.29
E(d1212 > I1212 ) = E(d1212 > 125.52) = σ1212E(Z1212 ) = 36 * E(0.92) = 6 * 0.0968= 0.58
4.
Re-compute T* considering the computed E (d> I) in step 3.
T*121 =
2(5 + 5 + 20 + 1 * 40 + (15 * 0.29 + 15 * 0.58))
= 0.07
25 * 740 + 25 * 600
T*121=0.07 year=25.7 day
5.
Repeat step 2, 3 and 4 until T* converges:
Since the values of T*121in steps 1 and 2 are close enough (compare 0.065 with 0.07),
it is assumed that T*121 converges. However, if the calculated values are not close
enough, the algorithm should be repeated until T* values in steps 2 and 3 converge.
52
Similarly,
Connection 21: Initial T*211=0.082 yr=29.8 days, P (d2111>I2111) =P (d2112>I2112) =0.215,
I2111=106.32, I2112=47.11
E (d2111>106.32) =8*0.1223=0.98
E (d2112>47.11) =9*0.1223=1.10
Final T*211:
T*211 =
2(5 + 5 + 20 + 1 * 40 + (15 * 0.98 + 15 *1.1))
= 0.098 yr = 35.83day
25 * 480 + 25 * 360
Connection 31:
Initial T*311=0.087 yr=31.9 days, P (d3111>I3111) =P (d3112>I3112) =0.225, I3111=106.08,
I3112=46.84
E (d3111>106.32) =8*0.1289=1.031
E (d3112>47.11) =9*0.1289=1.16
T*311 =
2(5 + 5 + 20 + 1* 50 + (15 *1.031 + 15 *1.16 ))
= 0.104 yr = 37.84day
25 * 480 + 25 * 360
Connection 32:
T*321=0.06 yr=21.85 days, P (d3211>I3211) =P (d3212>I3212) =0.167, I3211=125.82, I3212=52.91
E (d3211>125.82) =3*0.08819=0.265
E (d3212>52.91) =6*0.08819=0.529
T*321 =
2(5 + 5 + 20 + 1* 30 + (15 * 0.265 + 15 * 0.529 ))
= 0.066 yr = 23.92day
25 * 740 + 25 * 600
Step 3: Calculate the coefficient of Zghv (last term of (3.15))
Coefficient of Z121=7094
Coefficient of Z211=5451
53
Coefficient of Z311=5454
Coefficient of Z321=7136
Steps 4, 5: Calculate other parameters in the objective function and constraints and
form the mathematical model (equation 3.16 to 3.24): since the decomposed form of
the objective function and constraints are too long, step 4 is ignored here. However,
the compact form of the problem is shown as follows:
5
3
2
3
2
2
5
2
MinF3X 3 + ∑ ∑ C sj ∑ Wsjp + ∑ ∑ ∑ D ip C pj Yij + ∑ ∑ H ′B jp
s = 4 j= 3
p =1
j = 3 p =1 i =1
s = 4 p =1
P
P
P


+ ∑ ∑ ∑  2 ∑ H p D hvp ∑ (A p + FV + C l L ij + K p E (d hvp > I hvp ) ) + ∑ H p (I hvp − µ hvp ) + ∑ H p E (d hvp > I hvp ) Z ghv
h =1 g =1 v =1 
p =1
p =1
 p =1

3
3
1
Subject to:
1
3
∑∑Z
ihv
=1
i∈ I
v =1 h =1
3
3
g =1
g =1
∑ Z hgv − ∑ Z ghv = 0
2
3
v ∈ V, h ∈ G
∑∑Z
jiv
≤1
v∈V
∑∑Z
ghv
≥1
R ⊂ G, J ⊆ R
i =1 j= 3
g ∈S h∈S
3
− Yij + ∑ ( Z ihv + Z jhv ) ≤ 1
v ∈ V, i ∈ I, j ∈ J
h =1
∑W
sjp
p ∈ P, j ∈ J
− K jp X jp ≤B jp
s∈S
3
∑W
sjp
p ∈ P, s ∈ S
− K sp ≤ 0
j= 3
54
∀(R , R )
5
2
s=4
i =1
∑ Wsjp − ∑ D ip Yij = 0
p ∈ P, j ∈ J
2
3
3
2
∑ ∑ (∑ µ
h =1 h =1
∑D
hvp
−
hvp
Thv
p =1
p ∈ P, v ∈ V
) Z ghv ≤ K v
2
p =1
Step 6: The model is programmed and solved by Gams. The results are shown in
Table 3-7.
Table 3-7: Optimal Solution of the Example
Variables
Routing
Allocation
Quantity
Value
Z131=Z211=Z321=1
Y13=Y23=1
W431=7200, W432=5000
W531=0, W532=4600
Location
3PL
Total Cost
X3=1
B13=2200
$1,252,433
4
5
3
1
2
Figure 3-3: Network of the Example
Table 7 indicates the optimal values for variables. Based on these values, Figure 3-3
shows the network. The solution in the table reveals that the network works at its
minimum cost when there is a routing from depot 3 to customer 2 and then customer
1 respectively (Z131=Z211=Z321=1). It indicates that while plant 4 fills the demands for
product 1 (7200) and product 2 (5000), plant 5 only fill the demand for product 2
55
(4600). The products are directly shipped from the plants to the only depot, X3=1,
The allocation variables, Y13 and Y23, decided to be 1 meaning that customers 1 and
customer2 should be served by depot 3. The solution also indicates that B13=2200
which means depot 3 has space limitation for product 1 as much as 2200 units forcing
us to rent a 3PL space for $100 per unit. However, the capacity of depot 3 for product
2 (10000) is greater than the quantity shipped to the depot (9600); therefore, the 3PL
is not required for product 2 (B23=0).
3.2.4. Mathematical Model Verification
In this section the presented mathematical model in Section 3.2.2 is verified.
Objective function, equation 3.15, is consists of five items. The first three are
presented in [12] for a single product network. Therefore, if the number of product is
considered one in the presented model, the equations should be converted to the form
of presented in [12]:
p=1 therefore, the first three items of objective function will be:
n+m
∑ Fj X j +
j= n +1
n+m+k
∑
n+m
∑ C sj Wsj +
s = n + m +1 j = n +1
n+m
n
∑∑
D i C j Yij
j= n +1 i =1
Which shows the same form of equations presented in [11] and [12].
The forth item is obvious and does not need verification. It simply shows the 3PL
holding cost.
The last item of the equation 3.15 has already proved in Section 3.2.1.2.
The first four constraints, equation 3.16 to equation 3.20 are the same equation
presented in [12].
56
Equation 3.21( ∑ Wsjp − K jp X jp ≤B jp ) implies the 3PL constraint. This can be
s∈S
verified by considering two extreme points; when the capacity of the depot j for
product p (Kjp) limits to zero and infinite:
When Kjp →0 then ∑ This means that when a depot has no capacity for a product, the summation of the
product which is transported from all plants (∑Wsjp) should be placed in the 3PL place
(Bjp). This conclusion is true. On the other hand,
When Kjp →∞ then KjpXjp →∞ and therefore; KjpXjp>> Bjp which means that KjpXjp is
a very large number compare to Bjp . Ignoring Bjpvalue, KjpXjp -Bjp≈ KjpXjp and
therefore (3.21) is converted to:
∑W
sjp
≤ K jp X jp
s∈S
This means that when the capacity of a depot for a product is infinite, the summation
of product which is transferred from all plants (∑Wsjp) should be placed in the depot,
not in 3PL space. This is a true conclusion.
Equation 3.22 is obvious and self verified. It simply implies that the total input
to any depot for any product should be less than or equal to the capacity of the depot
for that product.
Equation 3.23 is a general form of the balancing input-to-depot, output-tocustomer equation that has been proposed in [12] which considers multi products
rather than single product network.
Equation 3.24 proposes the capacity limitation on routes. Recalling Section
3.2.1.1, equation 3.8 presented for a single product, the expected order quantity for a
multiple- product, multiple- customer network can be presented by left side of
57
equation 3.24. As shown, for a single product, single customer network, 3.24 will be
converted to equation 3.8.
58
3.3.
Solution Methodology
Since the mathematical model belongs to the NP-hard class problems, it is not
possible to solve the model by exact methods or using any regular commercial
software for even mid size problems. Therefore, a heuristic algorithm based on
simulated annealing is presented as the solution methodology. The algorithm is
composed of two phases. In the first step an initial solution is generated, and in the
second step the initial solution is improved through an iterative process in which two
simulated annealing (SA) modules are embedded.
3.3.1. Initial Solution
A preliminary solution is generated in this section. The algorithm is adopted
from S.Liu and S.Lee [80]. Figure 3-4 describes the initial procedure. The algorithm
begins with randomly selecting a candidate customer. If the demand of the candidate
customer is less than or equal to the capacity of the selected vehicle, the customer is
assigned to the vehicle. Then, the assigned customer is removed from the set of
customers and the next closest customer to the previously selected customer is picked
as the next candidate. If the candidate has a demand less than or equal the remaining
vehicle capacity, it is assigned to the vehicle. Otherwise, the candidate is temporarily
ignored from consideration and the next closest customer is selected as the new
candidate. The procedure is repeated until the entire capacity of the vehicle is
satisfied. The method iterates until all customers are assigned to the vehicles. On the
other hand, if the demand of the candidate customer is greater than the remained
capacity of the selected vehicle and it is the last un- assigned customer, the candidate
is assigned to a new vehicle. However, if the candidate is not the only un-assigned
customer, it is placed in an unqualified candidate list (UCL) and the next closest
59
customer to the last assigned customer is introduced as the next candidate. It is
notable that the demand of each customer for all products is less than the capacity of
the vehicle otherwise it is not possible to assigned more than one customer to a
vehicle. Therefore, the method forces to assign all possible customers (based on their
demand) to a vehicle first and then consider the next vehicle. There are two major
assumptions in the presented algorithm. First, it is assumed that one unit of all
products occupies the same space. The next assumption implies that the vehicle
capacity is considered for all products not product by product. For example if vehicle
capacity is 400 units for 5 products, this means that the total capacity of the vehicle is
400 not 400*5. Figure 3-4 indicates the initial algorithm.
60
Randomly select a candidate customer
Check the availability of vehicle v
No
Assigned the selected
customer to the next
vehicle
Does the
vehicle have
capacity?
No
Yes
Make the unqualified
candidate list (UCL)
empty
Yes
Put the selected candidate
in the unqualified candidate
list (UCL)
Assign the candidate
customer to the vehicle
Remove the customer from
the set of customers
No
Yes
Are all customers
assigned to
vehicles?
Yes
Select a candidate customer
closest to the last assigned
customer on vehicle v
Is any other
customer in the
set of customers?
Assigned vehicles to the
closest weighted depot
using eq. (3.25)
Figure 3-4: Initial Solution Algorithm
61
Is any customer
assigned to the
vehicle v?
No
Once routes are identified for the network, they should be assigned to the
potential depots. Applying equation 3.25 which is a modified version of the formula
proposed by [80],, it is possible to assign a coordinate for every route. The equation
identifies the weighted-average
average coordinate of a route. Using this coordinate values,
the average distances from a given route to all depots are calculated and then the
shortest path is chosen.
∑
∑ ∑
∑ Where
and
and ∑
∑ ∑
∑ indicate the coordinate of route v,
(3.25)
shows the coordinate of
customer i, and Nn is the number of customers assigned to the route v. Since there is
no limitation on the capacity of the depots, the total demand assigned to a depot can
exceed the depot capacity. However, in such a case a penalty cost which is the 3PL
cost is charged on the network cots.
3.3.2.
Improvement
Figure 3-55 illustrates a high level picture of the solution methodology. To
improve the initial solution, the problem is decomposed into two sub-problems,
sub problems,
location allocation problem and vehicle routing problem. Each sub-problem
sub
em applies a
SA module as its solution methodology. A brief summary of SA is presented in
Section 3.3.2.31. The initial solution is used as an input to the location allocation
problem (also known as depot improvement). The location allocation problem is
improved
proved by applying a SA until a stopping criterion is satisfied. The improved
location allocation problem is then input to the vehicle routing module (also known as
route improvement). The vehicle routing is then improved by applying the SA module
62
until a stopping criterion is satisfied. The output is fed into the location problem. After
applying routing SA, a conditional saving algorithm called Clark-Wright method
(known as the best heuristic algorithm for the vehicle routing problem [94]) is applied
to eligible routes. The improvement algorithm iterates until a termination condition is
satisfied. The condition is explained in Section 3.3.3.2.
Initial Solution
Apply SA module for location
allocation problem (depot
improvement)
N
Depot Stopping
Criterion?
Y
Apply SA Module for vehicle
routing problem (route
Improvement)
Route Stopping
Criterion?
Y
Apply Clark-Wright Algorithm
for Qualified Routes
Improvement
termination?
Y
Stop
63
N
Figure 3-5 Improvement phase algorithm
In the next two sections depot and route improvement algorithms are
discussed in details. Both depot and routing improvement algorithms apply a
simulated annealing module to find better solutions. The depot improvement
algorithm uses three sub modules called exchange, switch, and drop. At each iteration
of depot improvement, one of the modules is randomly selected to
to improve the depot
configuration. The routing improvement algorithm applies two sub modules called
insert and swap and one conditional algorithm known as Clark-Wright
Clark Wright saving method.
Similarly, at each iteration of routing improvement module, insert or swap
swap module is
randomly chosen to improve the network route.
3.3.2.1.
Depot Improvement Algorithm
Figure 3-6 shows the depot improvement algorithm. A simulated annealing
approach adopted from B. Golden and C. Skiscim [95], and [97] improves the depot
configuration. The algorithm begins with setting the SA parameters; starting
starting
temperature, final temperature, cooling rate, epoch, and equilibrium measure (Section
3.2.3). At each temperature of SA algorithm a set of candidate changess are made to
the depot configuration. The change is made by randomly applying one of the
exchange,
ange, switch, or drop procedures as defined below. Then, the candidate network
cost (CNC), which is the cost of the network after the change is made, is compared to
the initial network cost (INC) which is the output of initial solution. If CNC is less
than INC, the candidate change is accepted. Otherwise, the random nature of SA let
the algorithm take a second chance to the candidate change to be accepted: If the
value of
is less than a randomly generated number from the interval (0,1), the
64
candidate change is accepted; otherwise, another candidate change should be
generated again. The algorithm requires generating a pre defined number of accepted
candidate change. Once thee number of accepted candidate is satisfied, it is determined
that whether the set of generated candidate network costs in average are sufficiently
close to the initial network cost. Such a decision is made by comparing the value of
with a predefined
redefined value,
.
Set SA parameters
Randomly select a candidate change
from exchange, switch, or drop
No
No
Is CNC < INC?
Is
(0,1)
Y
Y
Accept candidate change
Are the no. of accepted
candidate=parameter epoch?
No
Y
No
Is
<ε
€ (0,1)
Y
Is T=Final T
T=T-cooling rate
N
Yes
Stop
Figure 3-6: Depot Improvement Phase
65
Remove the set of candidate
change from consideration
If the value of the fraction is greater than
(the inequality holds), the SA is in
equilibrium at temperature T meaning that the generated candidates are good enough
solutions at T. Therefore, the temperature of SA should be dropped
dropped as much as the pre
determined cooling rate value and repeated the algorithm if the final temperature is
not still achieved. However, if the inequality does not hold, a new set of candidate is
needed to regenerate.
The procedures which are used in the depot improvement phase are defined as
follows:
Procedure Exchange
a.
Randomly select an open depot.
b.
Find a closed depot nearest to the selected open depot
c.
Randomly select a route from the selected open depot
d.
Remove the selected route in step c and assign it to the selected closed depot
forcing it to open.
e.
If there is no more routes in the selected open depot, close the depot.
Otherwise; leave the depot open.
f.
Apply saving algorithm proposed at [94] to modify the customer sequence.
Note that the saving algorithm is applied at each iteration of the exchange procedure.
g.
A candidate change is generated by exchange procedure.
Procedure Switch
a.
Randomly select two open depots.
b.
Randomly select two routes, one from each selected open depots.
c.
Exchange the routes.
d.
Apply saving algorithm proposed at [93] to modify the customer sequence.
quence.
e.
A candidate change is generated by switch procedure.
66
Procedure Drop:
a.
Select an open depot randomly.
b.
Select a route on the selected open depot randomly.
c.
Select another open depot randomly.
d.
Assign the selected route at step b to the depot chosen at step c forcing the
depot to open.
e.
Apply saving algorithm proposed by [93] to modify the customer sequence.
A candidate change is generated by drop procedure.
The saving algorithm proposed by [93] can be described as follows. The
algorithm is required to modify the customer sequence for the change which is made
to the depot. For example, suppose that in the depot improvement phase the exchange
algorithm leads to assigning a route called V to a depot called D. The problem is how
to add the depot D to the route V. It is a common way to substitute the selected depot
by the dropped one; however, this may not be necessarily a good choice because
replacing the new depot by the current may worsen the cost of the network. Consider
Figure 3-7 where the route V1 with three customers is assigned to the depot D1. The
distance matrix which shows the distance between any two points in the network is
shown in Table 3-8.
Table 3-8: Distance Matrix of Figure 3-7
Cus.1
Cus.1
Cus.2
Cus.3
D1
D2
Cus.2
50
Cus.3
30
10
67
D1
50
70
20
D2
70
30
40
50
Suppose that as a result of applying the exchange algorithm, depot D2 is replaced by
depot D1 (Figure 3-8). If the total distance from D1 to customer 1 plus customer 3 to
D1 in the current network is less than that of the new network, this means that the
total distance will be increased by such a change. To avoid this drawback, the first
and the last customers in the routes need to be changed with a new pair of customers.
To find such customers, the saving algorithm proposed by [93] is used. Based on this
algorithm a saving matrix is formed by replacing the selected depot among all pairs of
customers and then determining the distances. The minimum distance in the saving
matrix identifies the best starting and ending customers in the route without big
changes in the route configuration. Therefore, to form the saving matrix in the given
example it is required to measure the total distance for pairs of customer 1& 2, 1 & 3,
and 2 & 3 as the starting and ending customers to the depot D2 and then select the
minimum pair distance as the new network configuration. Table 3-9 indicates the
saving matrix which is calculated based on Table 3-8 distances. As shown the best
connection between customers and D2 is made through customer 1 and customer 2
which means that if customer1 and customer2 are the first (last) and last (first)
customers to be served respectively, the minimum cost as much as 50 unit of distance
is charged on the network cost because:
Saving of 1-2=70+30-50=50
It is obvious that when such a number is negative it means that the change will
decrease the network cost.
D1
D2
3
1
2
68
Figure 3-7: Current Route Sequence
D1
3
1
D2
2
Figure 3-8: Candidate Route sequence
Table 3-9: Saving Matrix
Points
1-2
1-3
2-3
3.3.2.2.
Saving
50
60
80
Route Improvement:
Similar to the depot improvement, the routing improvement uses the SA to
improve the routing configuration. The SA follows the same parameters and
sequences used in the depot section except that it uses two procedures insert and
swap. The procedures are defined at the end of this section. Once the SA is complete,
a conditional procedure, Clark-Wright algorithm [94], is applied if specific conditions
are satisfied. Figure 3-9 indicates how the Clark-Wright algorithm is applied on the
output of route improvement. As it is seen, once the route improvement algorithm is
finished, the sequence of customers before and after applying route improvement
69
algorithm are compared. If the number of changes in customer sequences is greater
than a predefined number, then the number of customers on the route (s) is
considered. If the numbers of customers are greater than a predefined threshold, then
it is allowed to apply the Clark-Wright procedure. Otherwise, it is not needed to apply
the procedure. The two conditions eliminate many unnecessary routing changes from
consideration. The first condition indicates that the routes holding modifications less
than the pre defined value are not considered by Clark-Wright algorithm. The reason
is that by applying few numbers of modifications on a route sequence, the network
cost probably will not substantially be affected. Similarly, the latter condition implies
that when the number of customers on a route is not large enough, the route is not an
appropriate candidate for Clark-Wright algorithm. The reason is that a route sequence
holding only few numbers of customers does not play an important role in the
network cost.
Output of SA route improvement
Compare improved routes
before and after applying
route improvement module
Are the differences on
selected route (s) >
predefined threshold?
No
Yes
Do not apply Clark-Wright algorithm
Are the number of
customers on selected
route (s) > predefined
threshold?
No
70
Yes
Apply Clark-Wright algorithm
Figure 3-9: Conditions on Applying Clark-Wright Algorithm
The Clark-Wright [94] algorithm can be described as follows. Once the
improvement of routes is done, the new network route configuration is compared to
the initial network (output of depot improvement) and if the number of changes in the
improved routes are greater than a predefined value15, here four, for those routes in
which the number of customers are greater than a predefined number16, here four, the
Clark-Wright algorithm is applied [94] . In other words, if the following two
conditions are satisfied, the Clark-Wright algorithm is applied. First, the number of
customers on the route is greater than four customers and second the number of
changes on the route is greater than four changes. The first value implies that when
the number of customers on a route are not big enough, the route is not an appropriate
candidate for Clark-Wrigth algorithm. The reason is that a route holding only few
numbers of customers does not play an important role in the network cost. Similarly,
the latter value indicates that the routes holding modifications less than the pre
defined value are not considered by Clark-Wright algorithm because applying few
numbers of modifications on a route probably will not substantially affect the network
cost. Such thresholds eliminate many unnecessary routing changes from
consideration. The Clark-Wright algorithm is the best-known [94] heuristic saving
method for the vehicle routing problem. The Clark-Wright algorithm for a single
vehicle network can be described as follows. Suppose it is going to assign n
customers to the vehicle V. For any two customers i and j in this network, s (i,j) is
calculated from equation 3.26. The equation indicates the distance saving when
15
16
The threshold value is set intuitively; however, it highly depends on the problem size.
The predefined value is set intuitively; however, it highly depends on the problem size.
71
customer i and j are placed by the same route (visited by the same vehicle) instead of
being served directly by vehicle V.
, 2 !, " 2 !, # $ !, " , " !, % !, " !, #
, (3.26)
Where d indicates the distance, D shows the depot, and i & j imply the customers.
Once the s (i, j) is calculated for any two points, it is needed to sort them decreasingly
to form saving list. Starting from top to bottom of the saving list, the links (i,j)
indicate the routing priority. For complete description of the Clark-Wright algorithm
see [94].
The insert and swap procedures are described as follows:
Procedure Insert
a.
Randomly choose two routes 1 and 2.
b.
Randomly choose one customer from route 1.
c.
If there is enough capacity on route 2, insert the selected customer to route 2
using the saving algorithm proposed by [93]; otherwise, go back to step a.
d.
A candidate change is generated by insert procedure.
Procedure Swap
a.
Randomly choose two routes 1 and 2.
b.
Randomly choose a customer on route 1
c.
Find the nearest customer on route 2 to the selected customer to route 1
d.
Exchange the position of selected customers on route 1 and 2.
e.
Apply saving algorithm proposed by [93].
f.
A candidate change is generated by swap procedure.
72
3.3.2.3.
Simulated annealing module
3.3.2.3.1.
Simulated annealing in general
Simulated annealing (SA) uses the analogy between cooling a solid and optimizing a
combinatorial problem. In annealing a solid, the goal is to cool a solid gradually to
reach its low energy ground. The process begins at a high temperature. Setting the
starting temperature depends on the material and technical issues. The solid stays for a
while at this temperature until the equilibrium is achieved. The equilibrium at a
temperature is reached when the solid structure cannot be further improved. Then the
temperature is decreased at a defined rate and the process is repeated. Decreasing the
temperature finally stops where the structural behaviour of the solid is no more
sensitive to changes in the temperature. Although there are different types of the
annealing in the industry, all approaches have four common elements: starting
temperature, final temperature, decreasing rate, and duration of the process at each
temperature. Likewise, in optimizing a combinatorial problem the goal is to minimize
the objective function which is generally a cost function. The SA algorithm begins at
an initial solution. The new solutions are generated at this level (temperature) until it
is recognized that the current solution cannot be further improved. At this point
equilibrium is achieved. The equilibrium is also known as the length of Markov chain
in the literature. The process is decreased to the next level and the procedure iterates.
The algorithm is repeated until the current solution is not sensitive to changing the
levels (temperature). Figure 3-8 illustrates the general procedure of a SA. As shown
any generated (current) solution (CS) with a cost less than initial solution (IS), is
considered as an accepted solution; however, if the CS is greater than the IS it is not
rejected yet. In this condition, there is still a chance to accept the generated solution.
73
Each SA has a random feature which allows accepting generated solutions with higher
objective function value than current solution. Such a random feature compares two
values; one generated randomly and another generated from current and candidate
solutions. Laarhoven and Aarts [96] suggest generating the number using exp-(CSIS/T) where T indicates the current temperature. In this study, if the value of exp-(IS/
CS) is greater than a randomly generated value taken from (0,1), the generated
solution is considered as acceptable solution. This random feature allows preventing
the solution to be stuck in the local minimal. In fact, the feature lets the algorithm
randomly select greater generated cost function than the initial cost to escape from
probable local optimal minimum points.
Four parameters need to be set for any SA algorithm including the initial value
of control parameter, the final value of control parameter, the length of Markov chain
(equilibrium), and the decrement of the control parameter. Setting these parameters is
referred to as cooling schedule in the literature. Based on the type of problem, there
are different approaches to set the cooling schedule. Besides, the optimal setting of
the parameters is somehow trial and error practice and highly dependent on the
experience.
74
Initial solution (IS) at temperature T
Find another solution (CS)
No
No
Is CS<IS
Is exp-(CS-IS) >
random (0,1)
Yes
Yes
Accept CS
No
Stopping
criteria?
Yes
Decrease temperature
No
Final
temperature?
Yes
Stop
Figure 3-10: General simulated annealing procedure
3.3.2.3.2.
Presented simulated annealing
As it was previously mentioned, the presented SA is adopted from the work of
Golden and Skiscim [95]. In this approach the initial value of control parameter is
randomly set to a high temperature. This number is then divided into some equal
75
intervals. For example, if the initial value is randomly set to 300 and it is decided to
have 30 intervals, the cooling schedules (parameters) are set as follows:
Initial temperature=300
Finial temperature=0
Decrement temperature=10
Therefore the cooling schedule would be 300, 290… 10, 0. To set the fourth
parameter which is the length of Markov chain, the presented approach defines the
concept of epoch. Since length of Markov chain indicates the point at which system is
in equilibrium, an epoch can be used to decide whether the system is in equilibrium or
not. An epoch at temperature T is defined as a set including more than one element at
that temperature. Every element in the set indicates a solution which is as the result of
a single change in the initial solution. Therefore, an epoch with the length of five
indicates five solutions each of which is the result of one change to the initial solution.
It is also obvious that for each solution there is a corresponding objective function
value. As a result, the average cost of an epoch can be defined as the average
objective function values of its solutions. If the absolute difference between the
average of the epoch cost at temperature T and the average of the network cost before
temperature T is less than a defined number taken from (0, 1) interval, the equilibrium
is achieved at the temperature T and therefore the temperature should be decreased by
the amount of the decrement temperature. Otherwise, a new epoch should be
generated. The mathematical representation of equilibrium condition at temperature T
is indicated by equation 3 [96] .
&'()*(+) -. )-/0 /-1 (1 23()*(+) -. 4)15-*6 /-1 7).-*) 2
()*(+) -. 4)15-*6 /-1 7).-*) 2
Where 9 0,1.
76
89
(3)
3.3.2.3.3.
Setting the SA parameter
As it was mentioned, setting SA parameters follows an intuitive attempt rather
than a logical method. The goal of this section is to determine parameters of the
presented SA including starting temperature, final temperature, decreasing rate, length
of epoch, and equilibrium condition value (9). The parameters are set in two steps.
First, to get some initial understanding of parameters, some random values have been
considered and then the program has been run. To evaluate the parameters, two
criteria have been used: running time (second) and improvement of initial cost
(percentage). While less values of the former are desirable, the greater values of the
later are more satisfactory. Since the starting value does not have any effect neither
on the quality of the network cost nor the running time, it is randomly set to 200.
Doing some experiments, an interval has been estimated for the decrement of
temperature (3-20) and final temperature (20-100). Any number beyond the estimated
intervals either does not improve initial cost substantially or takes too long to run the
algorithm. To find the optimal decrement of temperature and the final temperature
three levels of each are considered. Table 3-10 indicates the levels. As shown in the
table, the considered decrement temperature levels are 5, 10, and 15 while the final
temperature levels is chosen to be 30, 60, and 90. Moreover, the length of the epoch is
considered in three levels: 5, 10, and 20. It is intuitively revealed that the level of 20
will lead to too long running time while the cost of the objective function is not
satisfactory improved. Therefore, by removing epoch level 20 from consideration,
two levels have been tested.
77
Table 3-10: Initial Levels of Presented SA Parameters
Parameter
Initial temperature
Decrement temperature
Length of epoch
Final temperature
Level
200
(5, 10, 15)
(5, 10, 20)
(30, 60, 90)
Once the initial values of the parameters are determined, it is needed to set a good
combination of parameters. In this step, two general factor analyses, one for small size
problems and the other for large ones, each including 18 instances are designed. The
effect of three factors is examined on the percentage of cost improvement and running
time. The factors are the decrement temperature in three levels, 5, 10, and 15, the
length of epoch in two levels, 5, and 10, and the final temperature in three levels, 30,
60, and 90. By running the algorithm for each combination of parameters, the
responses are observed and gathered in Tables 3-11 and 3-12 for small and large size
problems respectively. The small size problem considers a network with 1 plant, 10
depots, 20 products, and 50 customers, while the large one takes into account a
network with 1 plant, 10 depots, 20 products, and 350 customers.
Table 3-11: Four Factor Design of Experiment Considering Two Responses; time and
% improvement (small)
Run
Final Temp.
Cooling Rate No. of epoch % improvement Time (s)
1
90
5
5
38.86
335.35
2
60
5
5
43.87
785.19
3
30
5
5
36.00
654.95
4
90
10
5
39.68
230.55
5
60
10
5
35.65
198.23
6
30
10
5
45.56
247.53
7
90
15
5
44.52
200.88
78
8
9
10
11
12
13
14
15
16
17
18
60
30
90
60
30
90
60
30
90
60
30
15
15
5
5
5
10
10
10
15
15
15
5
5
10
10
10
10
10
10
10
10
10
43.79
37.87
30.22
44.39
35.15
50.70
42.88
44.39
39.84
39.02
43.22
213.28
166.34
1540.70
3287.70
5896.40
3004.20
1474.60
1459.00
256.43
1343.90
1826.70
Table 3-12: Four Factor Design of Experiment Considering Two Responses; time and
% improvement (large)
Run
Final Temp. Cooling Rate No. of Epoch % improvement Time (s)
1
90
5
5
8.48
464.12
2
60
5
5
9.12
541.16
3
30
5
5
7.26
724.21
4
90
10
5
6.21
258.95
5
60
10
5
12.11
239.89
6
30
10
5
0.21
377.76
7
90
15
5
0.00
138.85
8
60
15
5
9.61
169.68
9
30
15
5
5.73
220.45
10
90
5
10
6.94
1203.60
11
60
5
10
11.69
1673.80
12
30
5
10
11.50
2372.70
13
90
10
10
5.65
458.91
14
60
10
10
7.35
634.28
15
30
10
10
12.16
849.72
16
90
15
10
2.37
284.99
17
60
15
10
4.40
389.21
18
30
15
10
13.94
628.15
The results are then analyzed using Stat-ease. Table 3-13 to 3-17 indicate the analysis
of variance (ANOVA) of data taken from Tables 3-11 and 3-12 respectively. The
analysis has been done to determine whether the defined factors or any of their
interactions have effect on the running time and the percentage of the cost
79
improvement. The analysis begins by removing bottom-up interactions (three factors
interactions-ABC). As shown in Table 3-13, for small size problem when the response
is the cost improvement percentage (IP), neither individual factors nor their
interactions are statistically important. Moreover, when the response is the running
time the only important factor is the number of epoch (Table 3-14). Removing two
factor interactions reveals the same result (Table 3-15).
Table 3-13: Analysis of Variance against Percentage of Cost Improvement (small)
Source
Model
A-Temperature
B-Rate
C-Epoch
AB
AC
BC
Residual
Total
Sum of Squares
0.03062
0.0005
0.00793
0.00009
0.01537
0.00033
0.0064
0.00938
0.04
df
13
2
2
1
4
2
2
4
17
Mean Square
0.00236
0.00025
0.00396
0.00009
0.00384
0.00017
0.0032
0.00235
Value
1.0042
0.10739
1.6899
0.03843
1.63796
0.07066
1.36424
Prob > F
0.56
0.90
0.29
0.85
0.32
0.93
0.35
Table 3-14: Analysis of Variance against Running Time (small)
Source
Model
A-Temperature
B-Rate
C-Epoch
AB
AC
BC
Residual
Total
Sum of Squares
34052640.87
1868283.055
6309363.759
16164017.35
5025332.21
1536985.075
3148659.428
4158476.551
38211117.43
df
13
2
2
1
4
2
2
4
17
80
Mean Square
2619433.91
934141.53
3154681.88
16164017.35
1256333.05
768492.54
1574329.71
1039619.14
Value
2.52
0.9
3.03
15.55
1.21
0.74
1.51
Prob > F
0.193
0.476
0.158
0.017
0.429
0.533
0.324
Table 3-15: Analysis of Variance against Running Time without Interactions (small)
Source
Sum of Squares df Mean Square Value Prob > F
Model
24341664.16
5 4868332.83 4.21
0.0192
A-Temperature 1868283.055
2 934141.53
0.81
0.4685
B-Rate
6309363.759
2 3154681.88 2.73
0.1054
C-Epoch
16164017.35
1 16164017.35 13.99 0.0028
Residual
13869453.26
12 1155787.77
Total
38211117.43
17
Table 3-16: Analysis of Variance against Percentage of Cost Improvement (large)
Source
Sum of Squares df Mean Square
F
p-value
Model
0.01612
13
0.00124
0.42
0.896
A-Temperature
0.00027
2
0.00013
0.05
0.9562
B-Rate
0.0015
2
0.00075
0.25
0.7884
C-Epoch
0.00042
1
0.00042
0.14
0.7253
AB
0.00794
4
0.00199
0.67
0.6468
AC
0.00281
2
0.00141
0.47
0.6535
BC
0.00317
2
0.00159
0.53
0.6227
Residual
0.01187
4
0.00297
Total
0.028
17
Table 3-17: Analysis of Variance against Running Time (large)
Source
Model
A-Temperature
B-Rate
C-Epoch
AB
AC
BC
Residual
Total
Sum of Squares
3351471
184102
980088.7
477814.6
236190.4
727240.9
746034.3
2372595
5724066
df
13
2
2
1
4
2
2
4
17
Mean Square
257805.5
92051
490044.4
477814.6
59047.6
363620.5
373017.2
593148.7
F
0.43
0.16
0.83
0.81
0.1
0.61
0.63
p-value
0.8861
0.8612
0.5008
0.4202
0.9769
0.5858
0.5788
The same results can be concluded from Tables 3-16 and 3-17 for the large size
problem. Such results verify the results gained by [93] where Wu et al discovered that
the algorithmic parameters do not have a significant effect on the solution values. As
a result, it is concluded that none of the factors plays a role in the quality of the cost
81
improvement percentage neither for the small size nor the large size problem. The
only important factor is the number of epoch for only the small size problem and not
for large one. Therefore, the size of epoch is set to be five to make running time short.
The analysis also reveals that when the size of the problem changes, the effect of
factors may change on the running time only a little bit. For this reason, parameters
for all problem sizes are considered the same level although they may not represent
the same effect on the running time. The important conclusion from the analysis is
that the results allow considering any level of factors for setting SA parameters.
Therefore the parameters are set as follows:
Starting temperature=200
Final temperature=60
Cooling rate=15
Epoch=5
The last parameter is the equilibrium value (9) where it’s value should be taken from
the interval [0,1]. For a given problem, six values are set and the algorithm is run. The
percentage of cost improvement (IP) and the running time (Time) are shown in Table
3-18. The results reveal that when the value of 9 increases from 0.1 to 0.9 the running
time decreases from a very large number (infinite) to 61.97 second. This result is
reasonable because small values of 9 show that the generated initial and generated
solutions should be quite close. This means that many of the generated solutions
which are not too much close to the initial solution are not allowed to be accepted. As
a result, a lot of iterations are needed to get the acceptable solutions. However, by
increasing the 9 value it is seen that the running time dramatically decreases. On the
other hand, the desirable high values of IP are observed not in the two extreme values
of . In fact, when 9 gets values around 0.5, the IP gets its highest value. Further
82
investigation reveals that middle values of 9 result in reliable solutions (low range)
with the highest IP while extreme values return unreliable results (wide range) of IP.
Therefore, the value of 0.5 is determined for the parameter 9 allowing high value of
IP in a reasonable running time.
Table 3-18: Setting the Equilibrium Value ()
0.1
0.3
0.5
0.7
IP
38.63%
48.31%
33.19%
Time (s)
>40000
282.1752
220.7532
119.6977
0.9
32.37%
61.9671
83
CHAPTER 4
IMPLEMENTATION AND ANALYSIS
This chapter discusses the implementation and analysis of the SA algorithm
presented in the previous chapter. The implementation considers the coding and
verification while the analysis discusses the run tests and results, as well as two case
studies indicating how the model can be applied in the real world application.
4.1. Implementation
The implementation section composes of two sub sections; coding and
verification. While the coding implies the machine language implementation, the
verification shows that the program is an exact translation of the algorithm. Moreover,
it indicates that the program can be run without any bugs and problems.
4.1.1. Coding
The developed algorithm in the previous section is programmed using Matlab
7.5.0 (R2007b). The reason for selecting Matlab as the machine code is that it uses
pre-programmed functions which simplify the programming of the procedures.
Moreover, the structure of Matlab is highly compatible with matrix; therefore, if a
problem is converted to the matrix form, it works effectively with Matlab. Codes are
written in the following six main modules:
a. Data
The data module consists of two separate Matlab files (M-file). In the first file,
called datagen, the range of mathematical parameters are determined and then
generated. For example, the parameters of inventory model such as holding cost,
order cost and also the parameters of mathematical model such as the number of
84
customers, products, depots, plants are generated here. All data are randomly
generated and it is possible to determine a desirable range for any parameter. For
instance, if the number of products are set to five with different holding costs, it is
possible to determine a range [a,b] from which holding cost are randomly taken. Once
the parameters are determined and set, another M-file, called dataread, would read
the data.
b. Initial
By running the initial M-file the initial solution is generated. The data from
dataread are input to this file and the initial algorithm is run. The output of this file
will be an initial designed network that is a primary network configuration including
selected depot (s), assigned customers and routes to depots, and assigned depot (s) to
plant (s).
c. Algorithm
There are three sub modules in this section. First, one M-file handles the depot
improvement algorithm. The code has three algorithms in it; exchange, switch, and
drop which are randomly selected in every iteration of the algorithm. Another M-file
is developed to cope with the routing improvement part. The file is composed of two
algorithms; swap and insert which are randomly selected when routing improvement
phase is run. The last M-file is developed to run Clark-Wright algorithm if needed.
d. Cost
For every algorithm’s module there should be one corresponding cost module.
Since there are four modules which handle algorithm implementation (initial, depot,
routes, Clark-Wright), four corresponding M-files should be developed to cope with
the net work cost. While the first M-file code computes the network cost of initial
solution, the rest calculate the network cost after applying the depot improvement,
85
routing improvement, and Clark-Wright improvement respectively. To shorten the
running time and to make the programming codes more compact, a sensitivity
analysis concept is considered to calculate the cost of the network. In sensitivity
analysis when some value of parameters are changed, to re-compute the cost of the
system, only the effect of those parameters on the network cost are taken into account.
Therefore, it is not required to consider the entire network again to compute the cost.
This method is more efficient when the numbers of parameters are too much or the
problem is to solve repeatedly. Therefore, once the network configuration is changed
by depot, route, or Clark-Wright algorithm, it is not required to compute the entire
network cost. It is only required to modify the initial cost by applying corresponding
changes to the depot and route configuration.
e. Main
An M-file called Main is developed to manage the entire algorithm. The
function of this file is to call and run other files. Once the file is run, the data from
other files are called and then run. Since the improvement should be iteratively done
between depot and routing parts, this program should be set to a number showing the
termination condition of the improvement.
f. Analysis
The nature of this heuristic algorithm is to search the feasible solution
randomly. Therefore, for problems with large feasible solutions if a heuristic
algorithm is run more than once, the results will not definitely be the same. To
weaken this variation, it is required to repeat the entire process more than once which
is handled by the analysis M-file. Once the file is run, the entire process from start to
finish end is repeated for a pre determined number of iterations. Then the average
86
improvement of the network cost and the average running time are computed. The
best network configuration is also recorded.
4.1.2. Program Verification
To ensure that the machine code is a correct translation of the presented
algorithm and that the code can be run without any problems, the program was
verified in the following ways.
4.1.2.1.
Input-output matching
Regardless of the problem size and parameters, the input data should match
the output. For example, if there are 10 potential vehicles each holding the capacity of
400 in the input file, the total unassigned space plus assigned space for every vehicle
should add up to 400 in the output file as well. Therefore, if 350 space units are
assigned to vehicle1, there should be only 50 units left on that vehicle. This matching
should also exist for other outputs such as plant capacity, depot capacity, number of
customers, total demand, and so on. Fifty problems have been generated and solved
using the program. For each problem the input and output data are compared. The
results show that for all problems the input and output numbers match accordingly.
4.1.2.2.
Manual versus computerized calculation
A small size problem (referred as Problem 2) has been solved both manually and
using the developed program and the results were compared. To make the manual
calculation easier, it is assumed that the demand is not stochastic. The data for
Problem 2 and the manual as well as computerized solutions are discussed in
Appendix C1. Results from the manual and computerized calculation indicate that the
87
total system costs are quite close; $241930 versus $242093.47. The $163.47
difference is interpreted as rounding error.
4.1.2.3.
Different network configuration, same cost
For any given problem, if demand for all products for all customers are
considered equal, the capacity of the potential depots are the same, and if locations of
the customers and depots are very close to each other, the total network cost of the
system in different runs should not deviate too much because the routing and
transportation cost for any network configuration is almost the same. However, it is
notable that the deviation increases when the number of nodes (plants, depots,
customers) increases. A problem (referred as Problem 3) is generated and then solved
six times. The data for Problem 3 and corresponding solutions at each run are shown
in Appendix C2. It is shown that although the network configurations are not the same
at different runs, the system costs are almost the same for all configurations.
4.1.2.4.
Deterministic condition
Recalling from section 3.2.1.1, the Tersin’s model, it is expected to observe the same
value for both initial and final value of the fixed interval service time (T*) if the
demand is assumed to be deterministic. Table 4-1 defines parameters of example
problem 4. Running the developed program T* values for initial and final solutions
have been recorded and shown in Table 4-2. As it is seen in Table 4-2, the solution
reveals that seven out of ten potential vehicles are selected to serve the customers.
The initial and final values of the fixed interval service time are indicated in second
and third columns of the table. As expected both values for all vehicles are equal.
88
Table 4-1: Model Parameters for a LRP Integrated with Inventory Problem (Problem 4)
Parameter
number of customers
number of products
number of vehicles
number of depots
number of plants
lead time+order interval demand
annual demand
vehicle capacity
depot capacity
plant capacity
order cost/product
holding cost/product
shortage cost/product
dispatching cost
indirect transportation cost
purchasing cost/depot/product
direct transportation cost
depot fixed cost
3PL holding cost
customers, depots, plants coordinate
Unit
unit less
unit less
unit less
unit less
unit less
unit
unit
unit
unit
unit
$/order
$/year
$/year
$/order
$/mile
$
$/mile
$
$/year
unit less
Range
20
2
10
10
2
50
350
350
U (9000,10000)
U (12000,13000)
10
11
0
300
2
U (0,0.005)
1
100000
13
U (-100,100)
Table 4-2: Initial versus Final Value of Fixed Interval Service Time (Problem 4)
Vehicle no.
1
2
3
4
5
6
7
8
9
10
T*1(year)
0.1649
0.1695
0.1609
0.159
0.1659
0.1951
0.1909
-
89
T*2(year)
0.1649
0.1695
0.1609
0.159
0.1659
0.1951
0.1909
-
4.1.2.5.
Equal vehicle’s order interval
If annual demand for all customers and all products are equal, lead time plus
order interval demand for all customers deviates only slightly, depots, and customers
are located at the same area very close together, it is expected to observe almost the
same values of interval service time under deterministic and stochastic demand for
any of two assigned vehicle. Table 4-3 indicates the data for a problem with low
deviation in the lead time plus order interval demand. As shown the lead time plus
order interval demand slightly deviates from customer to customer and product to
product (40, 45). Moreover, customers, depots, and plants are randomly distributed in
a very close area. Coordinates are randomly taken from the interval (-1, 1).
Table 4-3: Model Parameters for a LRP Integrated with Inventory Problem (Problem 5)
Parameter
number of customers
number of products
number of vehicles
number of depots
number of plants
lead time+order interval demand
annual demand
vehicle capacity
depot capacity
plant capacity
order cost/product
holding cost/product
shortage cost/product
dispatching cost
indirect transportation cost
purchasing cost/depot/product
direct transportation cost
depot fixed cost
3PL holding cost
customers, depots, plants coordinate
Unit
unit less
unit less
unit less
unit less
unit less
unit
unit
unit
unit
unit
$/order
$/year
$/year
$/order
$/mile
$
$/mile
$
$/year
unit less
90
Range
10
2
6
10
3
40+ U(0,5)
350
200
U (9000,10000)
U (12000,13000)
10
11
10
150
2
U (0,0.005)
1
100000
13
(-1,1)
Running the program, the fixed order intervals for selected vehicles are reported
under both deterministic and stochastic conditions. Table 4-4 indicates the results. It is
shown that under deterministic condition (initial-T*1) all selected vehicles (v1-v5)
indicates very close values for T*1. The largest difference interval is related to vehicle
2 with 0.1064 year and vehicle 4 with 0.1077 year which is still close: 0.4745 day < 1
day. The same conclusions can be made for stochastic intervals (final- T*2). The
largest difference intervals belong to vehicle 2 with 0.1074 year and vehicle 5 with
0.1091 which is ignorable: 0.1091-0.1074=0.6205 day < 1 day.
Table 4-4: Initial versus Final Value of Fixed Interval Service Time (Problem 5)
Vehicle no.
1
2
3
4
5
6
4.1.2.6.
T*1(year)
0.1071
0.1064
0.1074
0.1077
0.1073
-
T*2(year)
0.1082
0.1074
0.1089
0.1089
0.1091
-
Depot improvement match
As mentioned the depot algorithm uses three sub algorithms to apply
exchange, switch, and drop. Suppose for a given problem the output shows that only
one depot is selected as active depot. Knowing the exchange algorithm considers only
one open depot while the other two consider two open depots for the improvement. It
is expected to observe that the depot improvement has applied only the exchange
algorithm not switch or drop. The observation from running of any problem supports
the expectation.
4.1.2.7.
Comparing to optimal solution
91
In this section the optimal solution of a small size problem is compared to the solution
obtained from the algorithm. A network including one plant, two potential depots,
and three customers considering two products is considered. The parameter of the
model are shown in Table 4-4b to 4-4d. The optimal solution is gained by
enumeration method (counting all possible solution). Results show that the
developed algorithm leads to the same network gained from optimal solution. This
verifies that the developed algorithm generate optimal solutions at least for small size
problems. It is assumed that demand during order interval follows a Poisson
distribution.
Table 4-4b: Comparing Developed Algorithm Solution to Optimal Solution
Parameter
order interval demand
vehicle capacity
plant capacity
order cost/product
holding cost/product
shortage cost/product (product1, product2)
dispatching cost
indirect transportation cost
direct transportation cost (product1, product2)
depot fixed cost
3PL holding cost
customers, depots, plants coordinate
Table 4-4c: Annual and Order Interval Demand
Customer
Annual Demand
Unit
unit
unit
unit
$/order
$/year
$/year
$/order
$/mile
$/mile
$
$/year
unit less
Range
40+ U(0,5)
200
160,000
200
2
22, 24
200
3
0.0043, 0.0032
135000
4
(-100,100)
Table 4-4d: Depot Capacity
Order Interval
Demand
Product
1
2
1
2
Depot
1
2
1
1527
1478
22
24
1
20317
16728
2
1498
1486
21
25
2
22848
18365
3
1474
1467
24
22
92
4.2.
Results and Analysis
The objectives of this section are three folds: To prove that LRP integrated with
inventory model leads to less network cost than that of individual LRP and inventory
models, to study the effect of different parameters such as customers, depots, vehicles,
etc. on the system cost and finally, to discuss the application of the developed
mathematical model and algorithm in practice.
4.2.1. Comparing Network Cost Under Integrated and Individual Decision
Although comparing the network cost between routing and inventory has been
presented in the literature, only few studies address the comparison between LRP
integrated with inventory and individual one [80]. An analysis is done here to show
how combining LRP and the presented fixed order interval inventory strategy can
save the system cost. To examine the interaction, a set of problems were generated,
run, and the system costs were then analyzed. The sizes of the problems are different
but their parameters are generated from the same range as shown in Tables 4-5 and 46. It is shown that in the first and second rows of Table 4-5, 28 problems including
seven levels of customers and four levels of products are generated.
Table 4-5: Test Problem Level
Parameter
number of customers
number of products
number of depots
number of plants
Range
50, 100, 150, 200, 250, 300, 350
10, 20, 30, 40
30
1
93
Table 4-6 shows the data for the test problems. Some parameters are selected
from a uniform distribution interval such as depot fixed cost while others are chosen
as a fix number such as dispatching cost. It is assumed that demand order interval
time follows Poisson distribution with the mean taken from U [10, 20] for all problem
sizes.
Table 4-617: Test Problems Data
Parameter
demand during lead time
annual demand
vehicle capacity
depot capacity
plant capacity
order cost/product
holding cost/product
shortage cost/product
dispatching cost
indirect transportation cost
direct transportation cost
depot fixed cost
3PL holding cost
customers, depots, plants coordinate
Unit
month
unit
unit
unit
unit
$ /order
$/year
$/year
$/vehicle
$/mile
$/mile
$
$/year
unit less
Range
U (10, 20)
U (1450, 1550)
2550
U (14000, 25000)
1600000
51
2
U (1,6)
101
U (0.000,0.009)
4
135000
4
U (-100,100)
Once the problems are generated, they are run and the system cost under different
circumstances is calculated as shown in Table 4-7. For example, consider the first
problem which represents a network consisting of 50 customers and10 products. For
this network, if the LRP and inventory decisions are separately made, the total
network cost adds up to $1,730,600. However, when LRP is integrated with
inventory, the cost dropped to $1,607,689. As seen, for all test problems the
17
U stands for Uniform Distribution
94
integrated system cost is less than individual one. The percentage of cost
improvement holds a [3.59%- 11.07%] interval. To observe whether the difference
system cost under integrated and separated conditions are statistically significant or
not a hypothesis test run as follows. Consider networks in which the difference
between separated and integrated inventory costs is the least value in Table 4-7. As
seen the least improvement belongs to the test 14 with 250 customers and 20
products.
Table 4-7: Integrated and Individual Inventory Cost Comparison
Test
Customer
Product
Separated
Integrated
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
50
50
50
100
100
100
150
150
150
200
200
200
250
250
250
300
300
300
350
350
350
10
20
40
10
20
40
10
20
40
10
20
40
10
20
40
10
20
40
10
20
40
$ 1,730,600
$ 1,878,800
$ 2,412,900
$ 3,426,900
$ 4,700,400
$ 7,463,800
$ 4,946,900
$ 7,713,000
$ 9,925,800
$ 6,781,400
$ 7,938,300
$11,938,000
$ 9,530,500
$12,458,000
$24,206,000
$12,526,000
$15,733,000
$28,587,000
$14,349,000
$24,950,000
$40,322,000
$ 1,607,689
$ 1,689,104
$ 2,167,744
$ 3,199,299
$ 4,370,569
$ 6,932,675
$ 4,529,350
$ 7,154,213
$ 9,114,660
$ 6,388,846
$ 7,439,170
$ 11,170,755
$ 9,100,954
$ 12,010,771
$ 23,160,659
$ 11,139,239
$ 14,779,546
$ 27,310,337
$ 13,737,725
$ 23,426,378
$ 38,435,769
MIN
MAX
MEAN
95
Improvement
Percentage
7.10%
10.10%
10.16%
6.64%
7.02%
7.12%
8.44%
7.24%
8.17%
5.79%
6.29%
6.43%
4.51%
3.59%
4.32%
11.07%
6.06%
4.47%
4.26%
6.11%
4.68%
3.59%
11.07%
6.65%
Table 4-8b: Test Samples Results for 350 Customers, 30 Products Network
Sample Customer Product
Separate Cost
Integrated Cost
1
2
3
4
5
6
7
8
9
10
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
250
250
250
250
250
250
250
250
250
250
20
20
20
20
20
20
20
20
20
20
13,159,000
16,211,000
14,223,000
14,419,000
13,056,000
13,428,000
13,529,000
15,209,000
13,666,000
12,372,000
12,638,274
15,362,939
13,757,913
14,048,460
12,567,721
13,216,043
12,970,230
14,900,906
13,391,590
11,072,478
Improvement
Percentage
3.96%
5.23%
3.27%
2.57%
3.74%
1.58%
4.13%
2.03%
2.01%
10.50%
Mean (µ1-µ2)
d=SeparatedIntegrated
$
$
$
$
$
$
$
$
$
$
$
$
520,726
848,061
465,087
370,540
488,279
211,957
558,770
308,094
274,410
1,299,522
534,545
323374
Sigma (µ1-µ2)
Ten such sample networks are generated and run by the algorithm. For each sample
the cost under separated and integrated inventory are identified. The results are shown
in Table 4-8b. Since the network configurations for both separated and integrated
conditions are the same, the matched pairs hypothesis [98] needs to be run for the
statistical test. The test is a two-tailed test under 99.99% of confidence interval:
H0: mean of separated network = mean of integrated network (µ1=µ2)
H1: mean of separated network = mean of integrated network (µ1#µ2)
α= 0.001%
n= 10
ν= 10-1=9
t α/2, ν=t0.0005, 9=4.781
test statistic=
=
>/√A
BCD,BDB
=5.23
CEC,CFD/√GH
α= 0.025
96
α= 0.0005
-4.781
4.781 5.23
Figure 4.1: Rejection Area for t Test
Since the test statistics (5.23) is greater than t α/2, ν (4.781), it is concluded that at
99.99% of confidence level, H0 is rejected (Figure 4.1). Therefore, the mean of
separated network is greater than that of integrated network. This conclusion shows
that the integrated network significantly decreases the network cost.
4.2.2.
Tests
It is desirable to evaluate the developed algorithm from different angles. There
is no benchmark problem in the literature by which the presented algorithm can be
evaluated. Therefore, to evaluate performance of the developed algorithm, two
measures are used: “percentage improvement” and “run time”. Percentage
improvement is defined as the percentage of cost improvement of the initial solution
that is attained through the application of the algorithm. Moreover, the run time and
its relation to the size of the problem should be reasonable. While some of the
parameters of the mathematical model are set at fixed values (such as dispatching
cost), others are selected at random from the intervals shown in Table 4-8 based on
the Uniform distributions (such as depot fixed cost). The annual demand for any
customer and any product is randomly selected from the uniform distribution; U (350,
97
450). Demand during lead time is assumed to follow a Poisson distribution for every
product, every customer with the mean randomly taken from U (10, 20) interval.
Moreover, in the process of assigning customers to a vehicle, it is required to
check the total demand of the route that is the summation of the demands for all
products, for all customers. If the number of customers and products are large enough
(greater than 25), the total demand distribution can be estimated by Normal
distribution based on the Central Limit Theorem. However, there are some cases in
which the number of customers on a route is less than 25. Under this condition, using
the Central Limit Theorem will result in an inaccurate estimation. To remove this
drawback, the Poisson distribution is estimated by the Normal distribution. The
location of customers, depots, and plants are randomly generated from a square with
the side of 200 which is (-100, 100) on the x and y axes. It is notable that the
purchasing cost is removed from the objective function because it does not have any
effect on the optimality.
Table 4-8c18: Mathematical Network Parameters and Their Values/Range for Test
Problems
Range
Parameter
U (10, 20)
Unit
Per customer per product
demand during order interval
U (350, 450)
Per customer per product
annual demand
U (14000,15000)
Number of product p
depot capacity /product
160001
Number of product p
plant capacity for any product any plant
U (400, 410)
$/order
order cost/product
U (1, 1.5)
$/unit of product p/year
holding cost/product
U (40, 45)
$/unit of product p
shortage cost/product
300
$/order
dispatching cost
2
$/mile
indirect transportation cost
U (0.005,0.01)
$/mile
direct transportation cost
18
U in the Table 4-8 stands for Uniform distribution
98
U (125000, 135000)
$
depot fixed cost
U (2, 3)
$/unit of product p/year
3PL holding cost
U (-100,100)
unit less
customers, depots, plants coordinate
The major network parameters have been considered as variables and then the
percentage of the improvement and running time are measured as response for any
combination of the predefined parameters level. The parameters and their
corresponding levels are indicated in Table 4-9. Any full combination of the
mentioned levels will generate a set of parameters. For example, the combination of
(1, 30, 10, 100, 1900) indicates a test problem that is a network with 1 plant, 30 depot,
10 product, 100 customer, and vehicle capacity of 1900 unit of products. It is seen that
the total of 378 of such problems are considered. The total running time for all
problems is about 15 hours.
Table 4-9: Major Network Parameters and Their Levels for Test Problems
Parameter
plant
depot
product
customer
vehicle capacity
total test problems
No. of levels
3
3
3
7
2
378
Level
1, 3, 5
10, 20, 30
10, 20, 40
50, 100, 150, 200, 250, 300, 350
low=1900, high=4500
To ensure timely termination of the algorithm with good results, the following
termination criterion is selected. When the percentage of improvement in total cost of
the system from one iteration to the next is very small, repeating the algorithm will
not improve the objective function significantly and hence the algorithm will
terminate. In this study, the termination condition is set to 1% difference between
costs of the network in two consecutive iterations. Table 1 in Appendix D indicates
99
the results of running the test problems in a table format. In this table for each
combination of the parameters the average initial cost, final cost, percentage of
improvement (ip), running time, and variance for five solutions are shown. To analyze
the data “excel pivot chart generator” is applied. The data in the appendix are
analyzed in the following three categories:
4.2.2.1.
Cost improvement
Cost improvement can be calculated by comparing the initial with the final
cost of the network. For example, if the initial and final costs of a network are 100
and 75 respectively, the percentage of cost improvement would be: I
GHH3FB
GHH
J% 25%. It was mentioned in the previous section that each problem is run five times.
Therefore, the average percentage of cost improvement is the average among five
percentages of cost improvement numbers. Figure 4-2 shows the average percentage
of cost improvement (y axes) against the number of customers in the network and the
capacity of the vehicle (x axes). For any customer level, there are two levels of
vehicle capacity: low level indicated by 1900, and high level shown by 4500. Please
note that the figure illustrates the average cost improvement; it includes the impact of
all depots, plants, and products level at the same time.
As seen, no matter of the vehicle capacity, the most improvement is achieved
when the number of customers in the network is minimum (50). As the number of
customers increases, the percentage of the improvement decreases. In a network with
350 customers, the percentage of the improvement is at its minimum. Since the size of
the problem is small, the difficulty (complexity) which exists in the problem is less
than that of large problem where the search region converges to infinite. Therefore,
for small problems a large percentage of the feasible region is searched resulting in
100
ovement on the initial solution. However, when feasible solution is
making big improvement
too large, less feasible area is searched leading to increasing the possibility of
obtaining less improvement on the initial solution. Another reason can be the increase
of total network cost
ost when the customer size becomes larger. Since the cost of a
network with 350 customers is larger than that of 50, a huge amount of saving should
be gained at the same percentage of improvement. For example, suppose that after
applying the solution on two
o networks one small and the other large, the cost of a
small network size has decreases from $100 to $90 ($10 improved) while the large
network cost has decreased from $1000 to $950 ($50 improved).
). Although the cost
has improved $40 more in larger network (compare $10 and $50), the percentage
improvements are 10% and 5% in small and large networks respectively.
Figure 4-2: Comparing the Average
A
Cost Improvement against Number of Customer
C
and Type of Vehicles.
The figure also illustrates that the improvement gap between the low and high
vehicle capacity decreases when the number of the customers increases. At the same
level of customers, fifty, it is needed to use more low capacity vehicles in the network
than that
hat of high capacity one. Therefore, there is more chance to improve the routing
cost when low capacity vehicles are used. For instance, suppose a network using ten
101
low capacity vehicles against a network using three high capacity vehicles. The
feasible region is larger in the low capacity network than high capacity one. As a
result, there are more feasible solutions that can be examined by the algorithm.
However, when the number of customers increases, this property begins to weaken
because the feasible region becomes very large at which there is no difference
between low and large capacity vehicles.
It is also desirable to consider the individual effect of depots, plants, and
products on the improvement cost. Results indicate that the effect of the depots and
plant follow the same pattern as in Figure 4-2. The corresponding figures are shown
in Appendix E, Figure 1 to Figure 6. However, the pattern is different when the effect
of products is considered. Figures 4-3 to 4-5 illustrate the average of cost
improvement when the number of products is considered 10, 20, and 40 respectively.
When the number of products is 10, Figure 4-3, the improvement gap between the low
and high vehicle capacity is very high at any number of customers. For example, at
customer number size of 50, while the improvement level is less than 50% for low
vehicle capacity, it is less than 5% for high level capacity. The pattern of
improvement highly depends on the number of products and vehicles. As seen in
Figure 4-3, when the number of product is 10 (low enough for this network), the
percentage of improvement is very low for high capacity vehicle. The reason is that at
level of product10, only few high capacity vehicles can serve the customers. This
limits the room for improvement. Since the capacity of the large vehicle is high
enough, as the number of customer increases, the pattern do not change dramatically.
In contrast, for a network running under low capacity vehicle, the number of vehicles
is large enough allowing e room for improvement. This explains the large difference
between percentages of improvement between high and low capacity vehicles.
102
However, when the number of product increases (Figure 4-4),
4 4), the number of high
capacity vehicles also increases. Therefore, there is en
enough room to make
improvements. Besides, any change in the network configuration may considerably
affect the network cost when the number of product is large enough. For example,
consider exchanging two customers’ route, each requesting 20 products. Such a
change can substantially effects on the routing and inventory cost; or consider a
customer holding large enough products re-assigned
re
to a new depot in improvement
process. This change can substantially affect location and inventory cost. As seen in
Figure 4-55 as network runs under more products, the improvement percentage for
both low and high capacity vehicles get smaller. In general, it can be concluded
concluded that
when the number of product is large enough, the percentage of improvement for low
and high capacity vehicles converges to the same value.
Figure 4-3: Comparing the Average Cost Improvement against Customer and Vehicle
Levels at Product Level
L
When the Number of Products is set to 10..
103
Figure 4-4: Comparing the Average of Cost Improvement against Customer and
Vehicle Levels When the Number of Product
P
is set to 20.
Figure 4-5:: Comparing the Average of Cost Improvement against Customer
ustomer and
Vehicle Levels When the Number of Product is set to 40.
Note that this conclusion does not mean that the network under low capacity
vehicle and customer necessarily makes better solutions. For example, consider a
network applying low vehicle capacity
capacity and including 50 customers and 10 products
with initial and final costs of $7000 and $4500 respectively. Consider that for the
same network high capacity vehicles are applied and network cost drops from $4200
to $4000. Although the percentage impr
improvement
ovement is dropped 35% for low capacity
vehicle and only 4.7% for the high capacity one, the higher capacity vehicle makes a
104
lower network cost. In summary, it can be concluded that the improvement in the
network cost for low capacity vehicle is not as sensitive as high capacity vehicle when
the number of the products change. Moreover, the improvement follows an increasing
trend for the high capacity vehicles when the number of product increases.
The cost improvement measure can be used in practice when there is only one
choice between high and low capacity vehicle should be selected as the network
vehicle. Results show that low capacity vehicle is a more robust decision than high
capacity one since the percentage of improvement is not sensitive to the number of
products and customers. Therefore, low capacity vehicle is a more precautious
decision. On the other hand, if the number of product is low enough, running the
network under high capacity vehicle probably leads to substantially decrease in
percentage of improvement. In such cases it is suggested not to select high capacity
vehicle. However, for a large enough product network, high capacity vehicle may be
selected as vehicle solution since it results in the lower number of vehicles increasing
manageability and control of the network.
The practitioner can also use the result of such analysis to evaluate how well
their current system can be improved. For a network running under few products and
high capacity vehicle, it is not expected to receive large amount of saving on the
network cost while for a system running under many products there is potentially
huge amount of improvement on the system cost.
The evaluated measure (percentage of cost improvement) does not reveal
which network combination will result to minimizing the cost. Such a metrics is
considered in the next section.
4.2.2.2.
Network Cost
105
The average network cost for any combination of parameters can be considered to
make decisions on the optimal network parameters for the model.. It is obvious that
tha
when the number of customers and/
and or products increases, the cost of the network is
expected to increase as well. However, it is important to know which vehicle fleet,
low or high, is a better decision when all other parameters in the model are the same.
Figure 4-6 indicates that on average the cost of the network is lower when the high
vehicle capacity vehicle is used than that of low one. This result is reasonable because
the number of required vehicles is less when the high capacity vehicles serve the
customers compare to low capacity vehicles. The statistical analysis indicates that the
cost difference between low and high capacity vehicle is significant.
Figure 4-6: Effect of changes in Vehicle Capacity and Number of Customers
ustomers on Total
Network Cost
It is also desirable to consider the effect of other parameters such as number of
depots, plants, and products on the cost difference between low and high capacity
vehicle scenarios. Further analyses indicate that number of plants and depots do not
have a significant effect on the cost difference. Since this conclusion is quite obvious,
the corresponding figures are not shown here. However, the cost difference is
significantly dependent on the number of products. Figure 4-77 compares the network
106
cost under two different conditions; when network applies low vehicle capacity versus
when it uses high vehicle capacity. The costs under two conditions are divided to
form high/low cost fraction (y axis). The fraction is computed at different levels of
customers and products (x axis). For example, the fraction 25% for a network
including 50 customers and 20 products indicates that if high capacity vehicle is
chosen as vehicle system, the cost of the system is 25% less as compared to that of
low capacity vehicle. It is seen that as the number of products increases the ratio of
high capacity vehicle network cost over low capacity network cost at any customer
level also increases. For example, with 100 customers, the ratio (high/low) is about
12.29% ($741322 against $650250). This fraction increases to 18.42% and 29.64%
Vehicle Level Fraction
for product levels at 20 and 40, respectively.
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
10 40 20 10 40 20 10 40 20 10 40
50
100
150
200
250
300
350
Product(upper) and Customer (lower) Level
Figure 4-7: Ratio of High to low Vehicle Network Cost against Different Levels of
Products & Customers
In summary, from the analysis it can be concluded that using the high capacity vehicle
fleet for the network leads to saving more money than that of low capacity vehicle.
Besides, the saving will increase when the number of products increases. It is
noticeable that since there are large number of variables and parameters involved in
107
the model, the conclusion may or may not be held for any network in general.
However, such an analysis can be developed for any network and appropriate
interpretation can be made based on the observed pattern.
4.2.2.3.
Run time
It is important to measure the run time of the presented algorithm to observe
whether it is short enough. However, the run time is highly dependent on the
computer on which the program is run. In this study, test problems are run on Pentium
(R) CPU 3.00 GHz. It is expected to observe an increase in the run time when the
numbers of customers and/or products increase. Figure 4-8 shows the average run
time in second (y axes) against different product and customer levels (x axes). It is
shown that as the number of customer increases the running time increases as well. It
is also observed that at any customer level, once the number of product increases, the
running time is also increases. This result is quite expectable. The only unusual
observation is the run time for a problem with 100 customers and 40 products which
is not expected to be this high. As indicated the run time for such a network is even
greater than networks with 200 and 250 customers. Further analysis reveals that the
unusual increase in the run time is related to the high capacity vehicle. Figure 4-9
shows the average run time against different levels of product, vehicle capacity, and
customer. As seen, the only unexpected run time belongs to a network with 100
customers, 40 products, and 4500 vehicle capacity. No explanation could be given for
this observation.
108
Average of Time (second)
250
200
150
100
50
0
10 20 40 10 20 40 10 20 40 10 20 40 10 20 40 10 20 40 10 20 40
50
100
150
200
250
300
350
Depots (upper); Customer (lower)
Figure 4-8: Average Run time (second) against Product
P
and Customer L
Levels
level and Customer
Figure 4-9: Average Run Time (second) against Product, Vehicle level,
Another important issue is to observe how big a problem the presented algorithm can
solve and how the change in run time is related to problem size (Figure 5-9).
5 9). There is
no limitation on the size of the problem that the algorithm is able to handle. However,
in this analysis the largest problem size which is considered contains 350 customers.
As shown, the average run time follows a linear pattern as the number of customers
increase.
109
Figure 4-10:: Relation between Run Time and Problem Size
4.2.3. Case study
Two case studies are developed in this section to show how the presented
model and algorithm can be applied in practice. Basically, there are two conditions
under which a network is evaluated: designing
desi
a network or re-designing an existing
one. With the design of a new network, the location of customers and plants may be
pre-determined
determined while the locations of potential depots are to be determined. In the
case where a network already exists, one or more
more parameters of the network can be
changed to identify their effects on the system. Both cases are studied in this section.
4.2.3.1.
Case Study I: Designing a network
Consider the following problem. Beverage Company which produces different
types of beverages is in the process of developing its physical transportation system.
The company currently produces 20 types of beveragess all in a single plant. The
production capacities for all products are high enough to fill the market demand. The
coordinate of the location of the plant is (5.67, 3.15). There are 102 active retailers
(customers) demanding beverage products. The annual demands (in boxes) are fixed
and identified in Appendix F,, Table 1. Since there is no warehouse in the plant, the
produced beverages should be retransferred to depots and then from there to retailers.
110
The inventory system at depots runs under the fixed order interval meaning that
customers are regularly served in pre defined intervals. Lead time is fixed and equal
to 1 month for any product. However, lead time plus fixed order interval demand in
boxes follows Poisson distribution for all products, all customers. The demand and
corresponding coordinates of the customers are indicated in Appendix F, Table 3.
There are 30 potential depots in the network of which one or more will be opened as
active depot (s). The potential depot coordinates and their corresponding capacities in
boxes are shown in Appendix F, Table 4. Every depot has a third logistic party letting
the depot rent its space in the case of capacity limitation at the price of $3 per unit of
product. It is assumed that the unit cost of 3PL is fixed throughout the network. The
inventory costs are indicated in Appendix F, Table 5. Furthermore, there are two
possible types of trucks in the network known as low and high capacity vehicles.
Since using one type of truck (low or high) will save maintenance activities, the
manager has decided to select only one type in the system. A low capacity truck
handles 3300 boxes of products while the high capacity one carries 5000 boxes of
products. There is a $300 dispatching charge for any types of truck per route. Products
occupy the same space in the vehicle; therefore, they are considered the same weight
and volume. The transportation cost from any depots to customers is $1 per mile, per
product. However, the direct transportation cost from plant to depots is not fixed and
is shown in Appendix F, Table 5. Since the demand of any customer during lead time
plus order interval is less than the capacity of the vehicle, any type of truck can serve
more than one customer in every trip. The manager can use the network to make
important decisions that minimize total network cost. Some of these decisions
include:
a. How much is the total annual cost of the system?
111
b. Which potential depot (s) should be active?
c. Which vehicle types should be selected?
d. How many trucks should be chosen?
e. What is the route for each selected vehicle?
f. How often should the truck serve customers?
g. Should the 3PL be utilized? If so, how much?
h. What are the inventory levels at selected depots?
As mentioned before, the purchasing cost is ignored from the objective function
because it has no effect on the optimization. To answer the questions in this sample
case study, the data are entered in the dataread file, and the program is run twice
under two conditions; low and high capacity vehicle. The output reveals that the total
annual network cost will be $2,142,626 for high and $2,189,856 for low capacity
vehicle which indicates that the high capacity vehicle should be used. The total cost
includes inventory and routing cost, fixed depot opening cost, direct transportation
cost from plant to depot, and 3PL cost.
The output in Table 4-10 indicates the depot selection and how much of each product
should be assigned to them. For example, 39330 boxes of product 6 occupy depot 3.
The unselected depots are identified with zero such as depot 1.
Table 4-10: Depot-product Allocation for Case Study I
Depot
1
2
3
4
5
6
7
8
9
10
11
1
0
0
39772
19955
0
0
0
0
0
0
0
2
0
0
40381
20234
0
0
0
0
0
0
0
3
0
0
39883
20310
0
0
0
0
0
0
0
4
0
0
40132
19670
0
0
0
0
0
0
0
Product
5
6
0
0
0
0
40385
39330
19386
20055
0
0
0
0
0
0
0
0
0
0
0
0
0
0
112
7
0
0
39837
20445
0
0
0
0
0
0
0
8
0
0
38218
20113
0
0
0
0
0
0
0
9
0
0
39663
19747
0
0
0
0
0
0
0
10
0
0
39835
20344
0
0
0
0
0
0
0
12
13
14
15
16
17
18
19
20
total
Depot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
total
0
0
0
19285
29415
20147
38984
0
0
167558
0
0
0
20465
29890
19506
39920
0
0
170396
0
0
0
19960
29733
20031
38872
0
0
168789
0
0
0
20232
30194
19390
40114
0
0
169732
11
0
0
39881
19331
0
0
0
0
0
0
0
0
0
0
19504
29937
19849
39108
0
0
167610
12
0
0
39456
20311
0
0
0
0
0
0
0
0
0
0
20437
29350
19900
39055
0
0
168509
13
0
0
39553
19107
0
0
0
0
0
0
0
0
0
0
20309
30497
19219
39601
0
0
168286
14
0
0
39652
19494
0
0
0
0
0
0
0
0
0
0
20205
29767
19925
38719
0
0
167762
0
0
0
0
0
0
19245
19548
30274
29458
19922
19558
39608
39514
0
0
0
0
168820 167463
Product
15
16
0
0
0
0
40287
39253
19812
19660
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20278
20099
30008
30164
20078
19712
39990
39574
0
0
0
0
170453 168462
0
0
0
20299
29809
19932
39404
0
0
169726
0
0
0
19808
28976
19673
39584
0
0
166372
0
0
0
19969
29922
19955
39586
0
0
168842
0
0
0
19683
29053
19809
39487
0
0
168211
17
0
0
39642
19517
0
0
0
0
0
0
0
0
0
0
19826
29664
20486
39818
0
0
168953
18
0
0
39760
19900
0
0
0
0
0
0
0
0
0
0
19302
29313
19769
40094
0
0
168138
19
0
0
38629
19988
0
0
0
0
0
0
0
0
0
0
19468
29595
20235
39691
0
0
167606
20
0
0
39856
19915
0
0
0
0
0
0
0
0
0
0
20679
29768
19578
39260
0
0
169056
The vehicle-depot assignment is shown in Table 4-11. 9 out of 30 vehicles
should be active. The assigned vehicle-depot is illustrated by 1 in the table. It is seen
that some depots are assigned by more than a single vehicle such as depot 3 to which
vehicles 5, and 7 are assigned. Besides, there are some depots with no vehicle
assigned to them meaning that the depot should not be opened such as depot 1, depot
2, and so on. The output from Table 4-11 should be coincident with the result coming
from Table 40. For example, there are no products assigned to depot 1 and depot 2 in
Table 4-10 indicated with zeros; therefore, there should be no vehicle assigned to
these depots in Table 4-11.
Table 4-11: Vehicle-depot Allocation for Case Study I
113
Vehicle
1
2
3
4
5
6
7
8
9
Depot
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1
1
1
1
1
1
1
1
1
The assigned customers on the vehicle and their corresponding routes are
indicated in the Table 4-12. The table only indicates the active vehicles in the first
row and obviously there is no customers assigned to the unassigned vehicles.
Numbers in cells indicate the customer number. In every column, upper cells show
the higher routing priority than that of lower ones. For example, the first customer in
the route 6 (V6) is customer 11, and the second one is customer 40 and so on.
Obviously, every route is started from a depot determined in Table 4-11 and finished
with the same depot.
Table 4-12: Customer Priority on the Vehicles for Case Study I
V1
32
75
79
25
52
61
7
22
64
99
88
47
V2
5
34
63
18
3
69
28
62
41
30
90
100
V3
15
98
35
95
20
83
59
76
86
70
101
102
V4
65
48
14
29
37
39
92
36
60
81
53
74
V5
89
82
45
77
43
26
73
54
84
16
71
12
114
V6
11
40
44
94
93
66
87
51
23
27
49
68
V7
50
97
103
31
46
9
78
24
4
57
80
21
V8
67
13
58
72
56
8
6
33
2
19
17
96
V9
10
85
91
55
42
38
The output of the program is indicated in Table 4-13. The intervals shown in
the second column vary from 50 days to 72 days. The total space occupied by vehicles
plus their remained space are indicated in the last two columns of the table.
Table 4-13: Fixed Service Interval for Case Study I
Vehicle
V1
V2
V3
V4
V5
V6
V7
V8
V9
Service interval (day)
50
50
52
53
52
51
53
53
72
Occupied space
4086
4556
4926
4490
4975
4067
4881
4954
4911
Remained space
914
444
74
510
25
933
119
46
89
To decide whether 3PL cost is charged on the network cost, it is required to
compare the occupied space shown in Table 4-10 with available space (Appendix F,
Table 5) for every product, every opened depot. Table 4-14 shows such a comparison.
Negative numbers indicate the space limitation. It is seen that depot 3 and depot 18
have space limitation shown products. For example, depot 3 has space limitation for
all products but product 3. The total required 3PL is 152334 boxes for all products.
Therefore, the total cost will be:
Total 3PL cost=152334 *$3/Box=$457002
Table 4-14: Comparison between Required and Available Space for Case Study I
Product
1
2
3
3
-4103
-5887
57
Depot
4
15
16
12036 17139 7006
14953 14600 7012
12034 13937 7846
115
17
9892
16688
14712
18
-324
-3024
-4645
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
required 3PL
-5476
-10082
-2812
-1696
-4381
-7399
-1153
-1199
424
-3996
-411
-8419
-550
-4994
-4486
-2054
-1130
-70228
16361
12069
15885
15145
12295
15931
14184
15197
10791
16387
17716
19557
12081
17404
16407
17453
17401
0
17939
12428
18893
14809
15497
14660
15712
15891
11643
15444
17299
14028
15651
14755
10884
19260
16717
0
6505
824
4633
4261
4721
6537
3897
3013
9144
7058
5281
5083
4378
2984
9365
10085
2947
0
15966
12563
12963
10654
12586
19693
15745
15705
11822
14513
10150
13942
11988
14640
18702
10875
20017
0
-8144
-1825
-508
-9293
-8051
-2290
-851
-472
900
-2795
-3601
-9199
-5903
-5353
-10027
-3973
-1846
-82106
The level of inventories for vehicles is indicated in the Table 4-15. The levels
are rounded up and considered to each route. However, Table 4-11 shows that there
are some open depots to which multiple vehicles are assigned; therefore, it is required
to add up the assigned inventories of the vehicles with the same depots. Table 4-16
shows such a mixed table for every opened depot.
Table 4-15: Vehicles Inventory Levels for Case Study I
Product
1
2
3
4
5
6
7
8
9
V1
278
290
285
271
297
262
289
295
269
V2
280
285
298
282
277
277
307
309
301
V3
304
284
294
271
293
297
276
290
297
Vehicle
V4 V5 V6
284 301 280
294 299 274
289 258 280
289 290 293
280 282 292
305 294 277
271 303 286
271 291 277
285 268 290
116
V7
306
299
300
284
273
284
285
287
280
V8
289
273
301
300
271
272
295
300
291
V9
157
153
154
161
150
147
143
162
150
10
11
12
13
14
15
16
17
18
19
20
308
286
293
295
291
275
300
286
302
283
285
276
301
298
299
286
294
307
275
310
291
278
298
284
276
294
271
276
281
296
296
282
299
285
287
285
285
280
274
307
281
280
299
281
300
290
290
289
303
282
302
280
292
303
294
290
275
307
285
277
279
280
282
297
297
293
287
268
271
297
266
275
283
284
298
279
289
287
277
271
295
307
291
301
294
294
267
291
Table 4-16: Depots Inventory Levels for Case Study I
Product
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
3
607
598
558
574
555
577
588
578
547
588
558
561
586
569
557
585
565
590
582
583
4
289
273
301
300
271
272
295
300
291
287
277
271
295
307
291
301
294
294
267
291
Depot
15 16
278 437
290 438
285 453
271 442
297 426
262 425
289 450
295 471
269 450
308 431
286 453
293 456
295 442
291 434
275 448
300 457
286 419
302 457
283 456
285 431
117
17
304
284
294
271
293
297
276
290
297
298
284
276
294
271
276
281
296
296
282
299
18
563
568
569
582
571
583
557
548
575
575
562
592
571
557
553
587
564
577
595
574
156
152
159
143
148
154
150
144
147
165
152
4.2.3.2.
Case Study II: Modifying (Re-designing) an existing network
The model can be used to modify the existing network. In this section it is
going to consider changes which are made in an existence network without rerunning
the program for the entire network. In fact, it is desirable to consider only the effect of
the change (s) on the network. For example, suppose that five new customers are to be
added to an existing network. The manager wants to assign these new customers to
the open and/or closed depots. However, he is satisfied with the current network
configuration and does not want to design the entire network again only to add five
customers.
There are three major steps to handle a redesigning case as indicated in Figure
4-11.
a. Fix those parts of the problem which will not be affected by the change (s) in the
network. The simplest scenario is to temporarily eliminating those parts of the
network from consideration. For example, if there are three open depot in a
network and it is desired to observe the effect of closing the second depot on the
network, it is required to fix the first and the third depot by assuming that the
available capacity of these depots is equal to their normal capacity minus the
annual demand that has already been assigned to them. This means that the
primary available capacities are removed from the analysis.
b. Run the program only for the change (s) which will be made. This step is in fact to
solve a sub network of the original network.
c.
Reconstruct the network accordingly if required. Under some conditions after
applying the network modification (step 2), it is required to apply the second
modification. For example, suppose that it is requested to add five new customers to a
given network. Assume that the new solution reveals that four out of five customers
118
will be assigned to a new depot while the fifth one should be assigned to an existing
route. Under this condition, it is required to reconstruct the route to decide on the
priority of the new customer in the route. By re-running the program for this route or
applying the saving algorithm (described in Chapter 3) it is possible to reconstruct an
existing route. Applying the reconstruction step depends on the network configuration
obtained from previous step and may not be applicable for some cases.
Fix no-changes part (s)
Run the algorithm for the
modified (s) part (s)
Reconstruct the network
Figure 4-11: Redesigning Process
Consider the problem presented in the case study I. The manager has decided
to close depot 18 because he believes that the depot is not efficient anymore for some
reasons; first, the cost of the direct transportation from the plant to this depot is
relatively expensive (see Appendix F, Table 5). Moreover, the 3PL charge is the
largest 3PL cost among all other open depots ($82106) and the cost will still increase
in the next fiscal year because the third party is going to increase the unit space cost
from $3 to $4. Besides, the manager thinks that the two vehicles assigned to this depot
are not efficient enough because a large part of their available space is not used (Table
4-13) when they serve the customers. He wants to make a good decision to cut the
cost. The redesigning process is applied on such a case and discussed as follows.
119
First, it is required to fix the current network configuration for all depots but
depot 18 and its corresponding customers. By subtracting the used capacity of the
open depots indicated in Table 4-10 from their annual demands (Appendix F, Table 1)
the new capacities for the network are achieved (Table 4-17). For example, the depot
15 which is currently open has a capacity of 36424 for product 1. On the other hand,
the total annual demand for the product 1 at this depot is 19285; therefore, the
available space for the product 1 is 17139 boxes. It is noticeable that since depot 3
and depot 18 have space limitation (Table 4-14), their available space has been
assumed zero. The depots which are not shown in the Table 4-17 are in fact the
potential depots and their capacities are remained unchanged. See Appendix F, Table
4 to observe the primary list of depots’ capacity. Once the current solution is fixed,
the program should be run only for the modified network. The modified part can be
assumed as a new but smaller problem; that is, a problem with 12 customers
(customers on vehicle 4 and vehicle 6 assigned to depot 18), 20 products, 18 depots
(depot 18 is closed and depot 3 is ignored because it has 3PL cost) out of which 4
depots have modified capacities (Table 4-17). The program is run and results shows
that two depots should be open to serve the customers that had previously assigned to
the depot 18; depots 11 and depot 13. However, the routing and the combination of
the customers on vehicle 4 and 6 have dramatically changed. To have a better
understanding of the current and future condition of the network, two network
configurations have been compared in Table 4-18. It is seen that the customers and
their priorities on the vehicles have been completely changed. For example, customer
10 who is currently assigned to depot 18 and is the first customer to be served on the
vehicle 6 will be assigned to depot 13 and will be visited by vehicle 4 at the third
priority to the last in the future configuration.
120
Table 4-17: Modified depot Capacity for Redesigning Problem (Case Study II)
Product
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
12036
14953
12034
16361
12069
15885
15145
12295
15931
14184
15197
10791
16387
17716
19557
12081
17404
16407
17453
17401
Depot
15
16
17139 7006
14600 7012
13937 7846
17939 6505
12428 824
18893 4633
14809 4261
15497 4721
14660 6537
15712 3897
15891 3013
11643 9144
15444 7058
17299 5281
14028 5083
15651 4378
14755 2984
10884 9365
19260 10085
16717 2947
17
9892
16688
14712
15966
12563
12963
10654
12586
19693
15745
15705
11822
14513
10150
13942
11988
14640
18702
10875
20017
18
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Table 4-18: Comparison of the Current and Modified (future) Network Configuration
Configuration
Depot
Current
3, 4, 15, 16, 17, 18
Route on V4
64-47-13-28-36-38-91-35-59-80-52-73
Route on V6
10-39-43-93-92-65-86-50-22-26-48-67
3PL Depots
3, 18
Modified
3, 4, 11, 13, 15, 16, 17
Depot 13:36-28-22-59-86-26-92-43-3510-50-67
Depot 11:91-64-47-13-93-52-73-80-3948-38-65
3
However, the solution is not acceptable unless as the result of this change the cost of
the network decreases. Table 4-19 compares the cost of the changes. The first column
of the table indicates the network costs in detail. The second column shows the
121
current cost of the network for depot 18. The third column implies the cost when the
3PL increases the space cost from $3 to $4 resulting in increasing the 3PL costs for
depot 18 from $246318 to $328424. The last column indicates the new solution when
depot 18 is closed. From Table 4-19, it is observed that once the depot 18 is closed,
depot 11 and depot 13 should be opened. This change decreases the cost of the
inventory and routing, direct transportation, and 3PL while it increases the fixed depot
opening cost. The table indicates that the total network cost will be decreased from
$759308 to $424469 if the change is applied.
Table 4-19: Comparison of the Current and Modified Network Cost
Cost Type
Inventory & Routing
Direct Transportation
Fixed depot opening
3PL
Total change
Current
196230
211470
105290
246318
759308
Cost
Current respect to new condition
196230
211470
105290
328424
841414
Modified
190530
28409
205530
0
424469
The total cost of the network will be:
Total cost of the network= current total cost+ cost of the change
Recalling case study I, the total modified cost is $2142626. The cost of the change is
in fact the subtraction of total modified cost from total current cost which is shown in
Table 4-19:
Cost of the change=424469-759308= -334839
Therefore;
Total cost of the network=2142626-334839=1807787
From the analysis it is concluded that the manager should close the depot 18 and open
depot 11 and depot 13 to cut the costs. Under this condition he will be charged less for
122
inventory & routing, direct transportation from the depots to the plant, and 3PLcost.
However, he needs to pay more money for opening two depots. The reconstruction
step is not applicable for this case because the modified routes are not required for
further reorder (modification).
123
CHAPTER 5
CONCLUSIONS
A summary of what has been discussed in this dissertation is presented in the
next two sections and then a direction for the further research is suggested.
5.1.
Conclusions
A supply chain can be defined all activities which are accomplished to fill
customer demands. Such activities involve wide variety of functions from daily
processes such as transportation and inventory operations to long run activities such
as locating of depots and warehouses. Researches revealed that by integrating the
activities along the supply network it is possible to save huge amount of money on
supply network costs. However, most studies in the literature ignore to integrate
supply chain decisions allowing attaining non optimal network cost. On the other
hand, to simplify the mathematical model most studies in supply chain area ignore
realistic assumption such as multiple products network, third party logistics
availability, stochastic customer demand, and size of the network. In this dissertation
a mathematical model is represented to integrate the location routing problem, known
as LRP, with the inventory. Basically, when decision on the location of depots and
routing among customers are made at the same time LRP is formed. Such a network
can be consists of three levels; plants, potential depots, and customers at the first to
third level respectively. The goal of this study is to make five decisions
simultaneously; selecting of the depot (s), assigning customers to selected depot (s),
determining the routing among assigned customers, identifying the inventory levels at
selected depots, and finally finding out the fixed order interval period for determined
routes in a way to minimize the network cost. It is assumed that customers demand
124
their multi- products requests stochastically during lead time. It is also assumed that
the inventory is run under fixed order interval policy at potential depots.
Since the model belongs to Np-hard problem, a simulated annealing (SA)
algorithm which is a heuristic method is applied to solve the model in two phases;
initial solution and improvement. The largest network which is designed by the
presented algorithm is consists of 350 customers and 40 products.
Some tests are solved using the presented algorithm and the results are then
analyzed. Furthermore, the effect of simultaneous decisions on LRP and inventory on
the network cost is shown and concluded that when those decisions are made at the
same model, the cost of the network is less than that of separate models. Two case
studies also presented to indicate how to apply the model in a real world situation.
5.2.
Contributions
Although the integrated LRP with the inventory has already been examined
recently before this research, this study extends the previous observations and
contributes in the literature in the following sense:
•
Considering the 3PL has not ever been studied in the literature.
•
Customers are assumed to make their multiple- products request rather than
single- request which is common in the literature but may not be realistic.
•
It is assumed that the depots follow a fixed order interval system as the inventory
policy.
•
The size of the network that can be handled by the presented algorithm is larger
than what is proposed in the literature.
5.3.
Future Research
125
It is suggested to extend this study in terms of modeling and solution to
capture real situation better while generating more promising results. For example, it
is desirable to integrate other inventory policies with LRP and then compare them to
observe the best inventory strategy. Moreover, it is recommended to relax some of the
network limitation to have a more general model. For instance, if the limitation of
direct delivery from plants to customers and/or depot to depot is removed more
general networks are generated. It is also suggested to integrate the inventory with
more complex network. Such a network can be generated by adding more levels of
distribution centers. For example, the distribution centers can be classified in two
levels; warehouses and depots. In terms of solution, other heuristic methods can be
examined. Genetic algorithm may generate good results for integrated LRP with
inventory provided that a good chromosome representation is proposed to define the
solution.
Another branch of study is to apply heterogeneous vehicle fleet for the
presented model. It is also possible to save more money by integrating other supply
chain activities such as packing with the presented model.
126
REFERENCES
127
LIST OF REFERENCE
[1]
Chopra, S., and Meindl, P., “Supply chain management: strategy, planning,
and operation”, 3rd ed., Upper Saddle River, New Jersey, 2007
[2]
Blanchard, D.,"Supply chain management best practices," John Wiley & Sons,
Inc., Hoboken, NJ, 2007.
[3]
http://people.brunel.ac.uk/~mastjjb/jeb/or/facloc.html
[4]
http://neo.lcc.uma.es/radi-aeb/WebVRP/
[5]
Simchi-Levi, D., and Berman, O. "A heuristic algorithm for the traveling
salesman location problem on the networks," Operations Research, Vol.36,
No.3, 1988, 478-84.
[6]
Min, H., Jayaraman, V., and Srivastava, R., "Combined location routing
problems: A synthesis and future research directions," European Journal of
Operational Research, Vol.108, 1998, pp.1-15.
[7]
Nagi, G., Salhi, S. "Location- routing: Issues, models and methods," European
Journal of Operational Research, Vol.177, 2007, pp.649-672.
[8]
Burns, A.D., "Zweistufige standortplanung unter berucksichtigung von
tourenplanungsaspekten- Primale heuristiken und locale suchverfahren," Ph.D.
Dissertation, Sankt Gallen University, 1998.
[9]
Sirvastava, S. "Alternate solution procedures for location routing problem,"
OMEGA Internatinal Journal of Management Science, Vol.21, No.4, 1993,
497-506.
[10]
Perl, J., "Unified warehouse location routing analysis," Ph.D. Dissertation,
UMI Dissertation Information Service, Northwestern University, 1983.
[11]
Perl, J., and Daskin, M.S., "A warehouse location routing problem,"
Transportation Research B, Vol.19, 1985, pp.381-396.
[12]
Laporte, G. "Location routing problems," In : Golden, B.L.; Assad, A.A. (Ed.),
Vehicle routing: methods and studies. North-Holland Publishing, Amsterdam,
Holland, 1988, pp.163-98.
[13]
Balakrishnan, A., "Integrated facility location and vehicle routing models:
recent work and future peospects," American Journal of Mathematical and
Management Sciences, Vol.7, No.1&2, 1987, pp.35-61.
[14]
Burks, Jr, R.E., "An adaptive taboo search heuristic for the location routing
pickup and delivery problem with time windows with a theater distribution
application," Ph.D. Dissertation, Air force institute of technology, 2006.
128
[15]
Bowerman, R., Hall, B., and Calamai, P., "A multi objective optimization
approach to urban school bus routing: formulation and solution method,"
Transportation Research, Part A, Vol.22, No.2, 1995, pp.107-123.
[16]
Gunnarsson, H., Ronnqvist, M., and Carlsson, D., "A combined location and
ship routing problem," Journal of the Operational Research Society, Vo.57,
2006, 928-938.
[17]
Yi, W., and Ozdamar, L., "A daynamic logistic coordination model for
evacuation and support in disaster response activities," European Journal of
Operational Research, Vol.179, 2007, pp.1177-1193.
[18]
Laporte G., and Nobert Y., "An exact algorithm for minimizing routing and
operating costs in depot location," European journal of operational research,
Vol.6, 1981, pp.224-226.
[19]
Laporte, G., Norbert, Y., and Aprin, D., "An exact algorithm for solving a
capacitated location routing problem," Annals of Operation Research, Vol.6,
1986, pp.293-310.
[20]
Laporte, G., Nobert, Y., and Taillefer, S."Solving a family of multi depot
vehicle routing and location routing problem," Transportation Science,
Vol.22, 1988, pp.161-72.
[21]
Cox, D.W., "An air lift hub-and-spoke location routing model with time
windows: case study of conus-to-Korea air lift problem", Master Thesis,
Graduate School of Engineering, Air Force Institute of Technology, Air
University, March 1998.
[22]
Zografos, K.G., and Samara, S., "Combined location routing model for
hazardous waste transportation and disposal," Transportation Research
Record, No.1245, 1989, 52-59.
[23]
List, G., and Mirchandani, P., "An integrated network/planar multi objective
model for routing and sitting for hazardous material and waste,"
Transportation Science, Vol.25, No.2, 1991, pp.146-156.
[24]
ReVelle, C., Cohon, J., and Shobrys, D., "Simultaneous sitting and routing in
the disposal hazardous waste," Transportation Science, Vol.25, No.2, 1991,
pp.138-145.
[25]
Giannikos, I. "A multi objective programming model for locating treatment
sites and routing hazardous wastes," European Journal of Operational
Research, Vol.104, 1998, 333-342.
[26]
Alumur, S. and Kara, B.Y. "A new model for hazardous location routing
problem," Computer and operation Research, Vol.34, 2007, pp.1406-1423.
129
[27]
Singh, N. and Shah, J., "Managing tendupatta leaf logistics: an integrated
approach," International Transaction in Operation Research, Vol.11, 2004,
pp.683-699.
[28]
Belenguer, J.-M., Benavent, E., Prins, C., Prodhon, C., and Wolfer-Cavlo, R.
"A branch and cut method for the capacitated location routing problem,"
International Conference on Service System and Service Management, Vol.2,
2006, pp.1541-1546.
[29]
Berger, R.T., Caullard, C.R., and Daskin, M.S., "Location routing problem
with distance constraint," Transportation Science, Vol.41, No.1, 2007, pp.2943.
[30]
Jakobsen, S.K., and Madsen, O.B.G., "A comparative study of heuristics for
two level location routing problem," European Journal of Operational
Research, Vol.5, 1980, pp.378-87.
[31]
Sirvastava, R., and Benton, W.C., The location routing problem:
considerations in phisycal distribution system design," Computers and
Operations Research, Vol.17, No.5, 1990, pp.427.
[32]
Chien, T.W., "Heuristics procedures for practical-sized un-capacitated
location, capacitated routing problem," Decision Science, Vol. 24, No.5, 1993,
995-1021.
[33]
Nagi, G., Salhi, S. "A nested location routing heuristic using route length
estimation," Studies in Locational Analysis, Vol.10, 1996, pp.109-127.
[34]
Tuzun, D., and Burke, L., "A two phase taboo search approach to the location
routing problem," European Journal of Operational Research, Vol.116, 1999,
pp.87-99.
[35]
Lin, C.K.Y, Chow, C.K., and Chon, A. "A location routing loading problem
for bill delivery," Computer and Industrial Engineering, Vol.41, No.1-2,
2002, pp.5-25.
[36]
Albareda-Sambola, M., Diaz, J.A., and Fernandez, E., "A compact model and
tight bounds for a combined location routing problem," Computer and
Operation Research, Vol.32, 2005, pp.407-428.
[37]
Melechovsky, J., "A meta-heuristics to solve location routing problem with
non linear costs," Journal of Heuristics, Vol.11, 2005, pp.375-391.
[38]
Wang, X., Sun, X., and Fang, Y. "A two phase hybrid heuristic search
approach to the location routing problem," IEEE International Conference on
Systems, Man, and Cybernetics, Vol.4, 2005, pp.3338-3343.
130
[39]
Bouhafs, L., Hajam, A., and Koukam, A. "A combination of simulated
annealing and ant colony system for the capacitated location routing problem,"
Lecture Notes in Computer Science, No.4251, 2006, pp.409-416.
[40]
Yang, P., and Li-Jun, B., "An integrated optimization problem in logistics and
PSO solution," International Conference on Service System and Service
Management Vol.2, 2006, pp.965-970.
[41]
Prins, C., Prodhon, C., and Calvo, R.W., "A memetic algorithm with
population management (MA|PM) for the capacitated location routing
problem," Lecture Notes in Computer Science, No.3906, 2006, pp.183-194.
[42]
Prins, C., Prodhon, C., and Calvo, R.W., "Solving the capacitated location
routing problem by a GRASP complemented by a learning process and a path
relinking," A Quarterly Journal of Operations Research, Vol.4, No.3,
September 2006, pp.221-238.
[43]
Prodhon, C., "Solving the capacitated location routing," Ph.D. Dissertation,
Universite de Technologyie de Troyes, France, 2006.
[44]
Barreto, S., Ferreiera, C., Paixia, J., and Santos, B.S., "Using clustering
analysis in a capacitated location routing problem," European Journal of
Operational Research, Vol.179, 2007, pp.968-977.
[45]
Lin, C.K.Y., and Kwok, R.C.W, "Multi objective meta-heuristics for a
location routing with multiple use of vehicle on real data and simulated data,"
European Journal of Operational Research, Vol.175, 2006, pp.1833-1849.
[46]
Lashine, S.H., Fattouh, M., and Issa, A. "Location/allocation and routing
decisions in supply chain network design," Journal of Modeling in
Management, Vol.1, No.2, 2006, pp.173-183.
[47]
Marinkis, Y. and Marinki, M. "A bi-level genetic algorithm for a real life
location routing problem," International Journal of Logistics: Research and
Application, Vol.00, No.0, 2007, pp.1-17.
[48]
Caballero, R., Gonzalez, M., Guerrero, F.M., Molina, J., and Paralera, C.
"Solving a multi-objective location routing problem with a meta-heuristic
based on taboo search: Application to a real case in Andalusia," European
Journal of Operational Research, Vol.177, 2007, pp.1751-1763.
[49]
Berman, O., and Simchi-levi, D. "Minisum location of traveling salesman," An
International Journal of Networks, Vol.16, 1986, pp.239-54
[50]
Simchi-Levi, D., and Beramn, O. "A heuristic algorithm for the traveling
salesman location problem on the plane," Operations Research Letters, Vol.6,
No.5, 1987, 243-248.
131
[51]
Simchi-Levi, D., and Berman, O. "A heuristic algorithm for the traveling
salesman location problem on the networks," Operations Research, Vol.36,
No.3, 1988, 478-84.
[52]
Bertimas, D. "Traveling salesman facility location problem," Transportation
Science, Vol.23, 1989, pp.184-91.
[53]
McDiarmid, C., "Probability modeling and optimal location of traveling
salesman problem," The Journal of Operational Research Society, Vol.43,
No.5, 1992, pp.533-538.
[54]
Simchi-Levi, D. "Hierarchical planning for probabilistic distribution system in
Euclidean spaces," Management Science, Vol.38, No.2, 1992, pp.198-211.
[55]
Averbakh, I., and Berman, O. "Probabilistic sales-delivery man and salesdelivery facility location problems on a tree," Transportation Science, Vol.29,
No.2, 1995, 184-197.
[56]
Laporte, G.; Louveaux, F.; and Mercure, H. "Models and exact solutions for a
class of stochastic location routing problems," European Journal of
Operational Research, Vol.39, 1989, pp.71-78.
[57]
Chan, Y., Carter, W.B., and Burnes, M.D., "A multi-depot, multiple-vehicle,
location routing problem with stochastically processed demands," Computers
and Operations Research, Vol.28, 2001, pp.803-826.
[58]
Clark, G., and Wright, J.V., “Scheduling of vehicles from a central depot to a
number of delivery points,” Operation Research, Vol 12, 1964, pp.568-581.
[59]
Albareda-Sambola, M., Fernandez, E., and Laporte, G., "Heuristics and lower
bound for a stochastic location routing problem," European Journal of
Operational Research, Vol.175, 2007, 1833-1849.
[60]
Laporte, G., and Dejax, P.J., "Dynamic location routing problems," Journal of
Operational Research Society, Vol. 40, No.5, 1989, 471-82.
[61]
http://en.wikipedia.org/wiki/Hamiltonian_path
[62]
Averbakh, I., and Berman, O. "Routing and location routing p-delivery men
problems on a path," Transportation Science, Vol. 28, 1994, pp.162-166.
[63]
Nambiar, M., Gelders, L.F., and Wassenhove, L.N.V. "Plant location and
vehicle routing in the Malaysian rubber smallholder sector: a case study,"
European Journal of Operational Research, Vol.38, 1989, pp.14-26.
[64]
Gorr, W., Johnson, M., and Roehrig, S. "Spatial decision support system for
home-delivered services," Geographical Systems, Vol.3, 2001, pp.181-197.
132
[65]
Ago, M., Nishi, T., and Konishi, M. "Simultaneous optimization of storage
allocation and routing problems for Belt-conveyer transportation," Journal of
Advanced mechanical Design, Systems, and Manufacturing, Vol.1, No.2,
2007, 250-261.
[66]
Kleywegt, A., Nori, V.S., and Savelsbergh, M.W.P. "The stochastic inventory
routing problem with direct deliveries," Transportation Science, Vol.36, No.1,
2002, pp.94-118.
[67]
Kleywegt, A., Nori, V.S., and Savelsbergh, M.W.P. "Dynamic programming
approximation for a stochastic inventory routing problem," Transportation
Science, Vol.38, No.1, 2004, pp.42-70.
[68]
Novick, L.K., and Turnquist, M.,"Integrating inventory impacts into a fixed
charge model for location distribution centers," Transportation Research Part
E: Logistics and Transportation Review, Vol.34, No.3, 1998, pp.173-186.
[69]
Novick, L.K., and Turnquist, M., "Inventory, transportation, service quality,
and the location of distribution centers," European Journal of Operational
Research, Vol.129, No. 2, March 2001, pp.362-371.
[70]
Erlebacher, S.J., and Meller, R.D., "The interaction of location and inventory
in designing distribution system," IIE Transaction, Vol. 32, 2000, pp.155-166.
[71]
Daskin, M., Coullard, C., and Shen, Z.J., "An inventory location model:
formulation, solution algorithm, and computational results," Annals of
Operations Research, Vol.110, 2002, 83-106.
[72]
Shen, Z.J.M., Coullard, C., Daskin, M.S., "A joint inventory location model,"
Transportation Science, Vol.37, No.1, 2003, pp. 40-55.
[73]
Shu, J., Teo, C-P., and Shen, Z-J. M., "Stochastic transportation inventory
network design problem," Operations Research, Vol.53, No.1, 2005, pp. 4860.
[74]
Miranda, P.A., Garrido, R.A., "Incorporating inventory control decisions into
a strategic distribution network design model with stochastic demand,"
Transportation Research Part E: Logistics and Transportation Review,
Vol.40, No.3, 2004, pp.183-207.
[75]
Miranda, P.A., Garrido, R.A., "A simultaneous inventory control and facility
location model with stochastic capacity constraints," Networks and Spatial
Economics, Vol.6, No.1, 2006, pp.39-53.
[76]
Shen, Z.J.M., and Qi, L., "Incorporating inventory and routing costs in
strategic location models," European Journal of Operational Research,
Vol.179, 2007, pp.372-389.
133
[77]
Ambrosino, D., Scutella, M.G., "Distribution network design: new problems
and related models," European Journal of Operational Research, Vol.165,
2005, pp.610-624.
[78]
Burns, L.D., Hall, R.W., Blumenfeld, D.E., and Daganzo, C.F., "Distribution
strategies that minimize transportation and inventory costs," Operations
Research, Vol.33, No.3, May-Jun 1985, pp.469-490.
[79]
Hansen, B.H., Hegedahl, B., Wjortkjaer, S., and Obel, B., "A heuristic
solution to the warehouse location routing problem," European Journal of
Operational Research, Vol.76, 1994, pp.111-127.
[80]
Liu, S.C., and Lee, S.B., "A two phase heuristic method for the multi depot
location routing problem taking inventory control decision into consideration,"
International Journal of Advanced Manufacturing Technology, Vol.22, 2003,
941-950.
[81]
Liu, S.C., and Lin, S.B., "A heuristic method for the combined location
routing and inventory problem," International Journal of Advanced
Manufacturing Technology, Vol.26, 2005, pp.372-381.
[82]
Schwardt, M., and Dethloff, J. "Solving a continuous location routing problem
by use of self-organizing map, "International Journal of Physical Distribution
and Logistics Management, Vol.32, No.6, 2005, pp.390-408.
[83]
Salhi,S., and Nagi, G., “Local improvement in planar facility location using
vehicle routing,” Annals of Operations Research, Online July 2007.
[84]
Drezner, Z., and Hamacher, H.W., "Facility location: applications and theory",
Springer-Verlag Berlin, pp.9, 2002.
[85]
Doerner, K., Focke, A.; and Gutjahr, W.J., "Multi-criteria tour planning for
mobile healthcare facilities in a developing country," European Journal of
Operational Research, Vol.179, 2007, pp.1078-1096.
[86]
Lee, Y., Kim, S., Lee, S., and Kang, K., "A location routing problem in
designing optical internet access with WDM system," Network
Communication, Vol. 6, No.2, 2003, pp.151-160.
[87]
Billionnet, A., Elloumi, S., and Djerbi, L.G., "Designing radio mobile access
networks based on synchronous digital hierarchy rings," Computers and
Operation Research, Vol.32, 2005, pp.379-394.
[88]
http://en.wikipedia.org/wiki/Wavelength-division_multiplexing
[89]
http://en.wikipedia.org/wiki/Synchronous_optical_networking
134
[90]
Silver, E.A., Pyke, D., and Peterson, R. "Inventory management and
production planning and scheduling," 3rd ed., John Wiley & Sons, New York,
NY, 1998.
[91]
Tersine, Richard J., “Principles of inventory and materials management”, 4th
ed., PTR Prentice-Hall, New Jersey, 1994.
[92]
Winston, Wayne L., “Operations research: applications and algorithms”, 4th
ed., Brooks/Cole-Thomson Learning, Canada, 2004.
[93]
Wu, T., Low, C., and Bai, J., “Heuristic solutions to multi depot location
routing problems”, Computers and Operation Research, Vol.29, 2002, pp.
1393-1415.
[94]
Larsen, R.C., and Odoni, A.R, “Urban operation research: logistical and
transportation planning methods”, 2nd ed., Dynamic Ideas, 2007.
[95]
Golden, B.L, and Skiscim, C.C., “Using simulated annealing to solve routing
and location problems”, Naval Research Logistics Quarterly, Vol.33, 1986,
pp. 261-279.
[96]
Laarhoven, P.J.M., and Aarts, E.H.L., “Simulated Annealing: theory and
applications”, Reidel Publishing Company, Dordrecht, Netherlands, 1987
[97]
Skiscim, C., and Golden B.L. “Optimization by simulated annealing: A
preliminary computational study for TSP, Winter Simulation Conference,
Vol.2, 1983, pp.523-535.
[98]
Mendenhall, W. and Sincich, T. “ Statistics for Engineering and Science”,
Prentice Hall, Upper Saddle River, New Jersey, 4th ed., 1995.
135
APPENDICES
136
1
2
3
Reference #
[1
8]
deter. & heuristics
4
5
single
deter. & heuristics
[3
2]
zografos &
samara
1989
[9
]
[2
2]
list &
mirchandani
1991
6
7
8
[2
3]
deter. & exact deter. & exact
Row
deter. & exact
deter. & exact
deter. & exact
deter. & exact
Category
1993
1993
1981
1983
stowers and
palekar
1993
single
single
single
single
single
single
single
stage(single/multiple)
det.
det.
det.
det.
det.
det.
det.
det.
det. / stoch.
multiple
multiple
multiple
single
single
multiple
single
single
# of facility
srisvastava
chien
laporte & nobert
laporte
ghosh, et al
Author
1981
year published
single
single
multiple
homo
multiple
single
single
multiple
single
single
multiple
multiple
multiple
single
single
# of vehicle
uncap.
uncap.
uncap.
cap.
uncap.
uncap.
uncap.
uncap.
vehicle (capacitated, uncap.)
uncap.
cap.
uncap.
uncap.
uncap.
uncap.
uncap.
uncap.
facility (capacitated, uncap.)
137
pri.
pri.
sec.
pri.
sec.
sec.
pri.
pri.
primary/secondary facility layer
static
static
static
static
static
static
static
static
static/dynamic
no
no
no
no
no
no
no
no
time windows
multiple
multiple
single
single
single
single
multiple
single
rw
hp
hp
hp
hp
hp
hp
hp
single/multiple objective
data (hp=hypotethical, rw=real
world)
hold
hold
network
discrete
hierarchical structure
solution space
index
3
APPENDIX A
homo
vehicle:homo/hetero
nonrectilinear
round trip
distance
location
Focus
exactnonlinear
programmi
ng
continuou
s
obnoxious
locationrouting
Solution
Method
exactnonlinear
programmi
ng
symmetric
traveling
salesman
tourlocation
exact
integer
programmi
ng
exact-two
branching
strategy
warehous
e
locationrouting
hazardou
s waste
LRP
heuristicinsertion
method
exact
mixed
integer
goal
programmi
ng
hazardou
s waste
LRP
heuristicsaving
method &
clustering
exactmultiobjecti
ve rout
generation
method
single
single
single
single
det.
det.
det.
det.
multiple
multiple
multiple
multiple
homo
multiple
multiple
multiple
multiple
multiple
multiple
cap.
cap.
cap.
uncap.
cap.
pri.
deter. & exact
[6
0]
1990
dyn. & stoch.
9
1981
deter. & heuristics
dyn. & stoch.
1976
averbakh &
berman
1994
single
single
single
single
single
det.
det.
det.
det.
det.
multiple
multiple
multiple
single
multiple
multiple
homo
homo
multiple
multiple
multiple
homo
multiple
multiple
multiple
multiple
multiple
multiple
multiple
multiple
multiple
cap.
cap.
cap.
cap.
cap.
uncap.
uncap.
uncap.
cap.
cap.
uncap.
uncap.
uncap.
uncap.
uncap.
uncap.
uncap.
sec.
sec.
pri.
pri.
sec.
sec.
sec.
pri.
pri.
sec.
static
static
static
static
static
static
static
static
dynamic
no
no
yes
yes
yes
no
no
no
no
single
single
single
single
static
TW+distance
limitation
single
static
no
single
single
single
single
multiple
single
hp
no comput.
results
rw
hp
laporte et al
138
hp
rw
rw
rw
hp
rw
not hold
hold
hold
discrete
discrete
discrete
2
jacobsen and
madsen
1978
[2
4]
srivastava & benton
deter. & heuristics
#
nambiar et al
jacobsen and
madsen
1980
[6
2]
deter. & heuristics
#
#
#
deter. & heuristics
[3
0]
multiple
deter. & heuristics
#
det.
[3
1]
single
det.
#
single
#
#
perl &
daskin
1984
[1
9]
#
#
1986
perl &
daskin
1985
deter. & exact
rw
gillett & johnson
revelle
laporte & dejax
1991
1989
path network
3
heuristicmulti
terminal
sweep
algorithm
exactdynamic
programmi
ng
newspape
r transfer
point LRP
terminal
LRP
delivery
men
problem
LRP
heuristic
newspape
r
distributio
n
hazardou
s waste
LRP
heuristic
exact
integer
programmi
ng &
partitioning
algorithm
exact
integer,
shortest
path,
weighting
method
heuristic(sa
ving and
clustering)
warehous
e
locationrouting
warehous
e
locationrouting
generic
capacitate
d LRP
implicatio
n of
environm
ental
factor in
LRP:
ANOVA
table
heuristic
heuristic
heuristic
exact
integer
programmi
ng
1989
1979
1987
simchi-levi berman et al jamil et al
1991
1995
1994
#
1979
daskin
[2
0]
harrison
inventory lrp
burness
hansen et al
& white
1976
1994
#
laporte et al
[7
9]
or
dyn. & stoch.
simchi &
berenan
1988
#
dyn. & stoch.
averbakh &
berman
1995
#
[5
1]
[5
5]
[5
6]
dyn. & stoch.
#
#
#
#
#
#
#
#
#
deter. & heuristics
deter. & exact
1988
bookbinde
r
1988
laporte et al
139
multiple
single
single
single
single
single
single
single
single
single
single
single
single
det.
stoch.
stoch.
stoch.
stoch.
stoch.
stoch.
stoch.
stoch.
stoch.
det.
det.
det.
multiple
multiple
multiple
single
single
single
single
single
single
single
multiple
multiple
multiple
multiple
homo
multiple
N/A
multiple
multiple
single
single
single
single
homo
homo
multiple
multiple
multiple
multiple
single
multiple
multiple
single
single
single
single
multiple
multiple
multiple
cap.
cap.
uncap.
uncap.
uncap.
cap.
uncap.
uncap.
uncap.
uncap.
cap.
cap.
cap.
uncap.
uncap.
uncap.
uncap.
uncap.
uncap.
uncap.
uncap.
uncap.
uncap.
cap.
cap.
cap.
sec.
sec.
sec.
pri.
pri.
sec.
pri.
pri.
pri.
pri.
pri.
sec.
sec.
static
static
static
static
static
static
static
static
static
static
static
static
static
no
no
no
no
no
no
no
no
no
no
no
yes
no
single
single
single
single
multiple
single
single
single
single
single
single
single
single
rw
no comput.
results
hp
no comput.
results
hp
hp
hp
hp
hp
hp
hp
rw
hp
hold
hold
hold
hold
discrete
tree
discrete
discrete
3
3
2
heuristic(fib
onaci
search)
heuristic(to
ur
approximati
on)
exactiterative
procedure
heuristic(sa
ving )
exact- B&B
heuristic
traveling
sales man
location
repair
vehicle
service
LRP
traveling
sales man
location
warehous
e
locationrouting
asymmetri
c LRP
warehous
e
locationrouting
salesdelivery
LRP
exact(nonli
near p.) &
heuristic(sp
acefilling
curve)
exactnonlinear
programmi
ng
warehous
e
locationrouting
heuristic
heuristic
stochastic
modeling
probabilist
ic
traveling
salesman
location
probabilist
ic
traveling
salesman
location
emergenc
y vehicle
LRP
exact-b&b
blood
bank LRP
exact- B&B
heuristic
#
#
#
#
#
#
#
#
[4
9]
dyn. & stoch.
simchi-levi &
berman
1987
#
1989
[5
0]
[5
2]
bertsimas
#
#
dyn. & stoch.
dyn. & stoch.
berman &
simchi-levi
1986
TSFLP + PTSFLP
single
single
stoch.
stoch.
stoch.
single
single
deter. & heuristics
deter. & heuristics
billionnet at al
labbe et al
2005
2004
det.
det.
deter. & heuristics
deter. & exact
ghiani &
averbakh & berman
laporte
2002
1999
deter. & exact
deter. & exact
kolen
chan &Francis
chan & hearn
min
1985
1976
1977
1996
single
single
det.
det.
det.
det.
det.
multiple
det.
single
multiple
multiple
single
single
multiple
multiple
140
static
single
single
single
single
multiple
cap.
cap.
cap.
unidentified
cap.
cap.
cap.
uncap.
uncap.
cap.
pri.
pri.
pri.
static
static
no
no
no
no
single
single
single
single
not identified
hp
hp
rw
planar
tree
static
static
static
static
N/A
static
sec.
static
static
static
unidentified
tree network +
cycle network
discrete
pure
location
problem
tree network
minmax
location
problem
tree network
1
exact
integer &
heuristic
exact
terminal
LRP
round trip
p-center
problem,
rectilinear
distance:
variant of
pick updelivery
arc
routing
problem
TSLP
combined
location &
network
design
problem
heuristics
Traveling
SLP on
Plane
plant
cycle
location
problem
heuristics
Probabilis
tic
Traveling
Salesman
Facility
location
Problem
nagy & salhi
2001
1996b
single
single
single
single
det.
stoch.
det.
stoch.
multiple
multiple
multiple
multiple
homo
homo
homo
[5
3]
inventory lrp
chan et al
liu & lee
2003
#
homo
#
multiple
#
multiple
#
det.
#
multiple
det.
[8
0]
single
deter. & heuristics
#
2007
nambiar
et al
1989
[6
3]
2004
#
2006
#
berger
salhi &
nagy
1999
[3
3]
wasner
dyn. & stoch.
ambrosino
& scutella
2005
#
[5
7]
[2
9]
deter. & exact
peng & bai
#
#
#
#
#
[4
0]
deter. & heuristics
dyn. & stoch.
averbakh averbakh & simich-levi
mosheiov
et al
berman & berman
1994
1994
1988
1995
stoch.
stoch.
mcdiarmid
1992
single
single
single
stoch.
stoch.
stoch.
single
single
single
N/A
141
multiple
single
multiple
multiple
multiple
multiple
single
single
single
cap.
cap.
uncap.
unidentified
cap.
cap.
uncap.
cap.
uncap.
cap.
uncap.
uncap.
unidentified
uncap.
uncap.
uncap.
uncap.
uncap.
pri.
sec.
pri.
pri.
pri.
pri.
pri.
pri.
pri.
static
static
yes
single
single
static
distance
limitation
single
dynamic
no
static
TW+distance
limitation
single
hp
rw
hp
dynamic
no
single
dynamic dynamic
rw
static
dynamic
dynamic
dynamic
no
static
static
no
no
no
single
single
single
single
hp
nothing solved
hp
exact:
branch &
price
pick updelivery
system
set
partitionin
g; cutting
plane
traveling
sales man
location
heuristics
probability
theory
heuristics
proposing
a new
solution
method(P
SO)
pick updelivery
location
problem
3
heuristics
3
PTSLP
2
heuristic
3
N/A
tree network
inventory
cost in the
model
3
N/A
tree
network
customer
clustering
3
near LRP
N/A
network
fixed
facilities
in all
periods
hold
discrete
nested
heuristics
hold
discrete
route
length
estimation
using
nested
LRP
hold
discrete
clarkewright
formulatio
n
hold
discrete
clustering
heuristic
hold
discrete
distributio
n network
design
problem
hold
discrete
#
[7
7]
[4
4]
deter. & heuristics
#
[3
7]
deter. & heuristics
#
[4
7]
[8
2]
deter. & heuristics other type
#
#
[8
1]
deter. & heuristics inventory lrp
#
#
[4
2]
deter. & exact
#
[7
6]
[2
8]
#
albaredasambola
2007
#
#
[5
9]
dyn. & stoch. inventory lrp
deter. & heuristics inventory lrp
max shen
belenguer
prins
liu & lin
Ozyurt
Schwardt
Marinakis
Melechovsky
Barreto et al.
Amrosino
2007
2006
2006
2005
not identified
2005
2007
2005
2007
2005
single
single
single
single
single
single
single
single
single
single
single
stoch.
stoch.
det.
det.
det.
det.
det.
det.
det.
det.
det.
multiple
multiple
multiple
multiple
multiple
multiple
single
multiple
multiple
multiple
multiple
homo
homo
homo
homo
homo
homo
homo
heter
homo
142
multiple
single
multiple
single
multiple
single
single
multiple
single
single
multiple
uncap.
cap.
cap.
cap.
cap.
cap.
cap.
cap.
cap.
cap.
cap.
cap.
uncap.
cap.
cap.
uncap.
uncap.
uncap.
cap.
cap.
cap.
cap.
pri.
sec.
sec.
pri.
pri.
pri.
pri.
pri.
pri.
pri.
sec.
dynamic
static
static
static
static
static
static
static
static
static
both
no
no
no
no
no
yes
no
no
no
no
no
single
single
single
single
single
single
single
single
single
single
single
hp
hp
hp
hp
hp
not identified
hp
rw
hp
hp
hp
hold
hold
hold
hold
hold
hold
hold
hold
discrete
discrete
discrete
continuous
discrete
discrete
discrete
discrete
3
2
3
3
3
3
3
3
3
3
4
lagrangian
relaxation
exact:
branch &
price
GRASP
heuristics:
combined
TS& SA
lagrangian
relaxation
NN
heuristics:
bi-level GA
heuristics
heuristics
exactCPLEX
continuou
s
locationrouting
introducin
g
benchmar
k
problems
proposing
a new
solution
method
proposing
a
clustering
analysis
integrated
inventory
warehousi
ng LRP
inventory
cost in the
model(CL
RIP)
hold
discrete
non linear
inventory
costs
design
hold
discrete
PTSLP:pr
obabilistic
traveling
salesman
problem
hold
discrete
deter. & heuristics
#
dyn. & stoch.
[3
8]
[1
5]
[6
4]
deter. & exact
#
#
#
#
deter. & heuristics
[2
1]
[3
4]
[1
7]
dyn. & stoch.
#
#
#
#
#
deter. & heuristics
deter. & heuristics
wu, low, bai
mayer, wagner
yi
tuzun & burke
cox
amaya
gorr
burks
bowerman
wang, sun
2002
2002
hub and spoke
system
2007
1999
1998
2007
2001
2006
1995
2005
multiple
single
single
single
single
multiple
multiple
single
det.
dynamic
det.
det.
det.
det.
det.
det.
det.
multiple
multiple
multiple
multiple
multiple
multiple
multiple
multiple
multiple
heter
homo
multiple
multiple
multiple
single
multiple
multiple
single
single
single
cap.
cap.
cap.
cap.
cap.
uncap.
cap.
cap.
cap.
uncap.
cap.
uncap.
uncap.
cap.
cap.
uncap.
uncap.
pri.
pri.
pri.
pri.
pri.
pri.
pri.
pri.
pri.
static
dynamic
static
static
static
dynamic
static
static
static
no
no
no
yes
yes
yes
no
no
single
single
single
single
multiple
single
multiple
single
hp
rw
hp
rw
rw
rw
rw
hp
single
homo
143
rw
homo
3
2
3
3
3
3
proposing
DS for LRP
tabou
search
clustering
heuristic
hybrid
heuristic
proposing
GIS; DSS
proposing
LPDPTW
(pickup &
delivery)
proposing
USBRP;
multicommodit
y
proposing
new
solution
for classic
LRP
arc
routing
problem
3
hub-spike
LRP with
time
window
3
CPLEX
hold
discrete
TS
hold
discrete
heuristics
hold
discrete
heuristics
hold
discrete
new
model:
applicatio
n of LRP
(disaster);
multi
commoditi
es;
customer
multi visit
is allowed
hold
discrete
hub and
spoke
system
hold
discrete
hub and
spoke
system
hold
discrete
fleet may
have
different
capacities
hold
discrete
heuristics:
A
hold
discrete
#
#
[4
6]
[4
5]
deter. & exact
#
[2
6]
deter. & exact
#
[2
7]
deter. & heuristics
#
[3
6]
deter. & heuristics
#
#
deter. & heuristics
[4
1]
[3
9]
[3
5]
deter. & exact
#
#
#
deter. & heuristics
deter. & heuristics
alumur
lin, kwok
lashine
zhang et al
2004
2007
2006
2006
single
single
single
single
single
2006
MANET
system
det.
det.
det.
det.
det.
multiple
multiple
multiple
multiple
multiple
multiple
homo
homo
homo
homo
homo
homo
aykin
lin& chow
bouhafs
prins & prodhon
albareda-sambola singh and shah
1995
2002
2006
2006
2005
single
single
single
single
det.
det.
det.
det.
2
multiple
multiple
homo
144
single
multiple
single
single route
single route
multiple
multiple
multiple
multiple
uncap.
cap.
cap.
cap.
cap.
cap.
cap.
cap.
cap.
uncap.
cap.
cap.
cap.
cap.
uncap.
cap.
cap.
cap.
don't know
pri.
pri.
pri.
pri.
sec.
sec.
pri.
sec.
static
static
static
static
static
static
static
static
static
no
yes
no
no
no
yes
no
yes
no
single
single
single
single
single
single
multiple
multiple
single
hp
rw
from literature
hp
hp
rw
rw
hp
hp
hold
hold
hold
hold
hold
hold
hold
hold
discrete
discrete
discrete
discrete
discrete
discrete
discrete
discrete
4
unidentified
unidentified
unidentified
3
3
3
unidentified
3
heuristics
4heuristics
+exact
SA & Ant
Colony
memetic
algorithm
population
manageme
nt
TS
CPLEX
exact:CPL
EX
SA
heuristics:l
agrangian
relaxation;s
ubgradient
search
introducin
g new
model:hu
b interrelation
LRP+Loa
ding
constraint
(max use
of vehicle)
ant colony
introducin
g dummy
arc
applicatio
n of LRP
hazardou
s waste
LRP
designing
of scm+
capacitate
d
warehous
e
new
objective:
work load
balance+
min cost;
multiple
use of
vehicle is
allowed
no
discrete
salhi & nagy
doerner et al
2007
2007
single
single
det.
det.
multiple:pre-defined
[2
5]
deter. & heuristics
gunnarsso
n
2006
#
[4
8]
[1
6]
[8
5]
[8
3]
deter. & heuristics
#
#
#
#
deter. & heuristics
deter. & exact
145
caballero et al
giannikos
2007
1998
multiple
single
single
det.
det.
det.
single
multiple
multiple
multiple
homo
homo
heter
homo
homo
multiple
single
multiple
multiple
multiple
cap.
cap.
cap.
cap.
uncap.
uncap.
cap.
uncap.
uncap.
cap.
pri.
pri.
pri.
pri.
sec.
static
static
static
static
static
distance limitation
distance limitation
no
yes
no
single
multiple
single
multiple
multiple
hp
rw
rw
rw
hp
not hold
hold
n/a
discrete
discrete
3
2
4
2
ant colony
heuristics
non-linear
integer
programin
g
formulatio
n
a mobile
single
facility:ap
plication
in health
care+use
s GPS
goal
programmi
ng
not hold
multiobjecti
ve heuristic
hold
continous
hazardou
s waste
LRP
waste
disposal
picked up
from
different
customers
and
deliver to
one place
ship
routing;
multi
product
APPENDIX C1
Data for Problem 2 are given in Table 1. Some parameters shown in the table
are given in a uniformly distributed interval. The data for such parameters is randomly
selected from the given range.
Table 1: Model Parameters for a LRP Integrated with Inventory Problem (Problem 2)
Parameter
number of customers
number of products
number of vehicles
number of depots
number of plants
order interval demand
annual demand
vehicle capacity
depot capacity
plant capacity
order cost/product
holding cost/product
shortage cost/product
dispatching cost
indirect transportation cost
purchasing cost/depot/product
direct transportation cost
depot fixed cost
3PL holding cost
customers, depots, plants coordinate
Unit
unit less
unit less
unit less
unit less
unit less
unit
unit
unit
unit
unit
$
$
$
$
$
$
$
unit
$
unit less
Range
4
2
10
5
3
50
U (350,450)
290
U (9000,10000)
U (12000,13000)
10
11
0
25
2
2
1
U (98000,99000)
14
U (-100,100)
The results from solving the problem using the coded program are shown in Table 2.
Customers 2 & 4 are assigned to vehicle 1 while customers 1 & 3 are allocated to
vehicle 2. The routes are sequenced as follows:
Vehicle 1: depot 2, customer 2, customer 4, depot 2
Vehicle 2: depot 2, customer 3, customer 1, depot 2
146
The computer solution indicates that the total cost for such a network is $241930. It is
notable that since the purchasing cost does not play a role in optimization, it is
ignored from the objective function.
The distances among customers and selected depot and plant are indicated in Table 3.
Table 2: Algorithm Solution of Problem 2
Customer
1
2
3
4
Vehicle
2
1
2
1
Route
2
1
1
2
Depot
2
2
2
2
Plant
1
1
1
1
Table 3: Distance among Customers and Selected Depot and Plant for Problem 2
Distance
customer 1
customer 2
customer 3
customer 4
depot 6
plant 2
customer 1
0.00
0.00
0.00
0.00
0.00
0.00
customer 2
75.24
0.00
0.00
0.00
0.00
0.00
customer 3
99.73
114.37
0.00
0.00
0.00
0.00
customer 4
151.41
123.73
75.39
0.00
0.00
0.00
depot 6
133.94
122.50
46.34
31.20
0.00
0.00
plant 2
138.23
147.56
38.61
69.73
186.89
0.00
Now, the network cost should be manually calculated. Recalling section 3.1, the
objective function has been presented as follows:
n+m
Min
∑ Fj X j +
j= n +1
+

n +m+k
∑
n+m
p
∑ C sj ∑ Wsjp +
s = n + m +1 j= n +1
p =1
∑ ∑ ∑  2∑ H D ∑ (A
h∈H g∈H v∈V
h ≠g h ≠g

P
p
p =1
hvp
p
n+m
∑
p
n
∑ ∑ D ip C pj Yij +
j= n +1 p =1 i =1
n+m
p
∑ ∑ H ′B
jp
j= n +1 p =1
P
P

+ FV + mLghv + K p E(d hvp > I hvp ) ) + ∑ H p (I hvp − µ hvp ) + ∑ H p E(d hvp > I hvp )Zghv
p =1
p =1

The network cost is calculated item by item as follows:
Since the open depot, coming from computer solution, is depot 2, the fixed depot
opening cost will be considered for depot 2:
n+m
∑FX
j
j
= 98234
j= n +1
147
Computer solution reveals that products should be transferred from plant 1 to depot 2.
Recalling direct transportation from Table 6, the total direct transportation is:
n+m+k
∑
n+m
p
∑ C sj ∑ Wsjp =1*(400+440+446+413+425+435+406+366)
s = n + m +1 j= n +1
p =1
=1*40.3672*3331=134463.15
The purchasing cost has been ignored from the objective function because it does not
have any effect on the optimality:
n+m
p
n
∑ ∑∑D
ip
C pj Yij =0
j= n +1 p =1 i =1
The computer solution indicates that the 3PL cost is zero. This result is reasonable
because the total annual demand for product 1 and product 2 are less than the capacity
of depot 2. Referring to Table 6:
Total annual demand for product 1=400+446+425+406=1677 < Capacity of depot 2
for product 1=9538 and,
Total annual demand for product 2=440+413+435+366=1654 < Capacity of depot 2
for product 2=9572
Therefore,
n+m
p
∑ ∑ H ′B
jp
=0
j= n +1 p =1
Inventory cost=
+

∑ ∑ ∑  2∑ H D ∑ (A
h∈H g∈H v∈V
h ≠g h ≠g

P
p
hvp
p =1
p
P
P

+ FV + mLghv + K p E(d hvp > I hvp ) ) + ∑ H p (I hvp − µ hvp ) + ∑ H p E(d hvp > I hvp )Zghv
p =1
p =1

since it is assumed that the model is not stochastic, the equation can be summarized as
follows:
Inventory cost=
9
9
5

P
∑ ∑ ∑  2∑ H
h =1 g =1 v =1

p =1
p

D ghvp ∑ (A p + Fv + C l L ij )Z ghv

Considering the two active vehicles 1 and 2 and using the data from Table 6:
148
Inventory cost for vehicle 1=
M 2 N 11 N
446 " 406 " 413 " 366 N 10 " 10 " 25 " 2 N 122.50 " 123.73 " 31.2
S2 N 11 N 1631 N 45 " 2 N 277.43 4639.42
Inventory cost for vehicle 2=
S2 N 11 N 400 " 425 " 440 " 435 N 10 " 10 " 25 " 2 N 46.34 " 99.73 " 133.94
=S2 N 11 N 1700 N 45 " 2 N 280.01 4756.9
Total inventory cost=4639.42+4756.9=9396.32
By adding the calculated cost, the total network cost will be:
98234+134463.15+9396.32=242093.47
As mentioned the computer solution indicates that the total cost of the network is
$241930 which is quite close to the $242093.47. There is $163.47 difference which is
due to rounding error.
149
APPENDIX C2
Consider a network with the given data in Table 1. The problem has been
solved 6 times. The solution of each running is shown in Table 2 to Table 7. As
shown the solutions are different. For example, consider solution 1 in Table 26.
Vehicle one serves customers 6-1-5-4 in the same sequence while vehicle 2 serves
customers 9-8-10-2 and vehicle 3 meet customer 3 and then customer 7 respectively.
Although the final solutions do not follow the same network configuration, the
network cost for all six solutions is $1114800.
Table 1: Model Parameters for a LRP Integrated with Inventory Problem (Problem 3)
Parameter
number of customers
number of products
number of vehicles
number of depots
number of plants
order interval demand
annual demand
vehicle capacity
depot capacity
plant capacity
order cost/product
holding cost/product
shortage cost/product
dispatching cost
indirect transportation cost
purchasing cost/depot/product
direct transportation cost
depot fixed cost
3PL holding cost
customers, depots, plants coordinate
150
Unit
unit less
unit less
unit less
unit less
unit less
unit
unit
unit
unit
unit
$
$
$
$
$
$
$
unit
$
unit less
Range
10
3
5
5
1
25
350
301
10000
100001
(400,410)
(1.5,2)
(40,45)
300
(0,0.005)
0
1
100000
3
(-1,1)
Table 2:Solution of the Problem 3, Run # 1
Customer
1
2
3
4
5
6
7
8
9
10
Vehicle 1
2
0
0
4
3
1
0
0
0
0
Vehicle 2
0
4
0
0
0
0
0
2
1
3
Vehicle 3
0
0
1
0
0
0
2
0
0
0
Vehicle 4
0
0
0
0
0
0
0
0
0
0
Vehicle 5
0
0
0
0
0
0
0
0
0
0
Table 3 : Solution of the Problem 3, Run # 2
Customer
1
2
3
4
5
6
7
8
9
10
Vehicle 1
3
0
4
2
0
1
0
0
0
0
Vehicle 2
0
0
0
0
1
0
0
4
3
2
Vehicle 3
0
1
0
0
0
0
2
0
0
0
Vehicle 4
0
0
0
0
0
0
0
0
0
0
Vehicle 5
0
0
0
0
0
0
0
0
0
0
Table 4: Solution of the Problem 3, Run # 3
Customer Vehicle 1 Vehicle 2 Vehicle 3 Vehicle 4 Vehicle 5
1
1
0
0
0
0
2
0
0
2
0
0
3
3
0
0
0
0
4
0
2
0
0
0
5
0
0
1
0
0
6
2
0
0
0
0
7
0
1
0
0
0
8
4
0
0
0
0
9
0
3
0
0
0
10
0
4
0
0
0
151
Table 5: Solution of the Problem 3, Run # 4
Customer
1
2
3
4
5
6
7
8
9
10
Vehicle 1
1
0
4
0
0
2
0
3
0
0
Vehicle 2
0
0
0
3
0
0
4
0
2
1
Vehicle 3
0
1
0
0
2
0
0
0
0
0
Vehicle 4
0
0
0
0
0
0
0
0
0
0
Vehicle 5
0
0
0
0
0
0
0
0
0
0
Table 6: Solution of the Problem 3, Run # 5
Customer
1
2
3
4
5
6
7
8
9
10
Vehicle 1 Vehicle 2 Vehicle 3 Vehicle 4
2
0
0
0
0
0
1
0
3
0
0
0
1
0
0
0
0
0
2
0
4
0
0
0
0
4
0
0
0
3
0
0
0
1
0
0
0
2
0
0
Table 7: Solution of the Problem 3, Run # 6
Vehicle 5
0
0
0
0
0
0
0
0
0
0
Customer
1
2
3
4
5
6
7
8
9
10
Vehicle 1
4
0
2
0
3
1
0
0
0
0
Vehicle 5
0
0
0
0
0
0
0
0
0
0
Vehicle 2
0
0
0
2
0
0
1
3
4
0
Vehicle 3
0
2
0
0
0
0
0
0
0
1
152
Vehicle 4
0
0
0
0
0
0
0
0
0
0
APPENDIX D
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
40
40
40
40
40
40
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
773434
360107
1203205
757762
1560683
1236373
1884614
1368427
2286278
1972739
2489319
2499809
3182366
2806626
945002
568882
1679682
1183697
2402633
1659736
2888196
2074955
3271382
2685845
4697897
3239448
7615631
5945672
1957151
1341217
3122720
2200985
3860840
2780125
153
394637
346670
717340
701486
1125045
1108870
1676870
1366050
2084298
1934277
2332711
2449073
3070340
2728886
653105
458808
1134707
894643
1893798
1326213
2460480
1800939
3098124
2509378
4437866
3153736
7228129
5891308
1215945
807225
2269633
1550876
3326112
2311990
48.77
3.61
40.18
7.29
27.50
9.82
11.06
0.18
8.63
1.83
6.24
2.04
3.32
2.58
30.43
18.51
32.25
23.90
21.18
19.99
14.85
13.22
5.30
6.52
5.52
2.64
5.06
0.90
37.90
39.73
27.26
29.54
13.80
16.86
58.85
46.33
78.72
64.72
108.31
87.44
152.98
96.53
162.95
127.33
196.71
153.49
198.19
152.41
44.08
49.62
85.51
97.15
111.84
113.23
125.49
129.14
142.53
147.39
158.55
147.23
168.16
182.90
68.62
71.59
120.51
248.15
123.43
163.37
variance
time(s)
ip(%)
final cost
initial cost
vehicle
capacity
customer
product
depot
plant
problem
Table 1: Test Problems Results
15.11
10.25
16.13
23.76
54.37
27.99
14.13
0.11
32.67
5.82
14.35
17.17
19.91
14.36
63.92
99.86
36.65
52.51
19.50
6.69
8.26
0.52
2.81
2.51
10.93
4.60
3.83
1.27
12.93
7.43
22.96
42.50
18.58
7.80
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
40
40
40
40
40
40
40
40
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
40
40
40
40
40
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
5516949
3420572
8275258
5745566
11772571
8987849
14341916
10285980
768970
341524
1307119
670056
1810702
1043124
2087419
1302734
2843671
1569342
2985448
1954557
3174789
2468575
1511026
774510
2186484
1341645
3399489
1839336
3648497
2355621
4308662
2805916
4897574
3282183
5706357
4693742
2510179
1497986
4197549
2531048
5460666
154
5074046
3089816
7755968
5374140
11537810
8674518
14068098
10120264
401608
339995
754699
602700
1168486
882203
1432107
1299796
2182092
1568839
2321317
1924407
2613724
2352614
644649
481972
1231865
906367
1958841
1350463
2325807
1841214
3246017
2349405
3804866
2799447
4933303
4338809
1148956
794065
2522249
1508050
3648537
8.02
9.65
6.27
6.42
1.99
3.48
1.91
1.60
47.48
0.45
41.77
10.05
35.33
15.22
30.86
0.19
23.28
0.03
22.09
1.50
17.59
4.64
57.33
37.43
43.37
32.29
42.41
25.95
36.20
21.81
24.66
16.07
22.21
14.55
13.49
7.55
54.11
46.89
39.99
40.28
33.22
131.73
185.15
155.59
164.49
157.45
179.51
206.37
186.27
43.25
48.96
58.35
67.03
75.45
86.89
91.85
105.93
137.97
128.09
122.92
146.02
145.80
177.79
51.84
89.87
79.03
88.08
89.45
112.06
112.41
119.35
122.01
143.75
149.65
162.42
184.24
250.68
72.96
78.04
96.95
182.89
105.81
7.48
11.86
5.65
4.90
1.99
1.58
0.34
1.17
21.49
0.13
76.38
0.16
12.06
19.45
70.79
0.25
7.05
0.00
76.39
5.02
27.58
6.74
9.55
30.53
99.47
17.94
15.03
66.38
4.64
30.22
1.84
24.70
13.62
20.42
16.52
7.11
6.39
7.63
36.29
19.21
16.32
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
20
20
20
20
20
20
20
20
20
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
40
40
40
40
40
40
40
40
40
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
40
40
40
40
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
3528731
6256492
4306703
6985078
4851390
8960912
5948043
9824778
6896729
820443
338467
1382983
667116
1962122
1025404
2603668
1239697
2989995
1599777
3436551
1926909
3669417
2265218
1514598
726295
2677783
1295484
3715439
1929672
4633831
2490885
4871122
3047507
5458384
3376361
6283904
3855573
2818204
1569492
4424762
2807216
155
2307331
4983791
3033362
5991881
3760883
7667792
5274369
8930852
5985034
407420
334599
735510
600556
1168416
890404
1633983
1236115
1893397
1596942
2377123
1871911
2818571
2206912
673504
461118
1238534
931287
1994417
1355642
2575787
1853836
2948498
2319819
3836763
2798895
4945576
3334222
1219218
781085
2280415
1521201
34.56
20.30
29.55
14.22
22.47
14.42
11.21
9.09
13.27
50.33
1.13
46.79
9.98
40.47
13.17
37.04
0.27
36.52
0.17
30.76
2.86
23.20
2.44
55.29
36.51
53.33
27.62
45.96
29.69
44.40
25.35
39.47
23.68
29.65
17.10
21.26
13.46
56.72
50.23
48.43
45.68
166.11
132.48
215.84
153.25
181.83
174.38
192.43
174.87
202.38
49.36
53.56
69.46
68.29
79.32
90.49
99.24
108.34
117.99
125.13
142.74
147.17
159.44
158.54
52.41
57.99
126.38
92.61
101.15
113.29
109.66
133.00
119.53
140.25
157.48
154.38
185.34
178.69
86.43
85.27
123.80
183.41
5.04
12.68
4.64
19.31
6.84
21.65
19.97
7.67
8.49
1.12
2.67
13.96
0.20
13.54
0.03
31.41
0.21
15.19
0.01
22.40
6.41
7.98
8.62
37.47
0.05
60.39
61.96
72.57
1.23
8.61
23.93
6.37
25.28
15.00
40.09
18.90
15.06
6.56
0.28
6.24
12.80
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
1
1
1
1
1
1
1
1
1
1
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
30
30
30
30
30
30
30
30
30
30
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
40
40
40
40
40
40
40
40
40
40
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
40
40
40
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
6110713
3779893
7142499
4585023
7780637
5302275
9563954
6309953
10245248
7130127
763435
366929
1168796
690822
1415010
973321
1755388
1413168
2085248
2059935
2354785
2300805
3003727
3048749
905658
633958
1731637
1244759
2306019
1685623
2754297
2264196
2958339
2758892
4154982
3693315
6904041
6183646
1873637
1471040
2991978
156
3572588
2275708
4610986
2976470
5908100
3760620
7499970
4729597
8654110
5581919
394095
347913
713261
652036
1037686
937838
1553168
1404485
1916848
1990156
2237725
2248763
2839204
2934510
587566
477349
1142287
960797
1690771
1422612
2427172
2127905
2823904
2646582
3913018
3575187
6667381
6022448
1210510
940056
2223745
41.52
39.64
35.44
34.77
24.08
28.98
21.59
24.90
15.53
21.80
48.03
5.12
38.68
5.47
26.61
3.01
11.35
0.58
7.97
3.29
4.97
2.21
5.42
3.75
34.86
24.70
33.66
22.45
26.67
15.58
11.83
5.85
4.50
3.89
5.83
3.16
3.42
2.60
35.46
36.00
25.71
119.42
184.86
142.22
177.70
300.12
192.66
157.15
206.55
186.83
208.87
38.60
46.00
56.09
103.30
73.24
100.39
92.22
115.33
115.70
132.40
188.58
157.36
162.80
176.91
55.73
55.28
70.02
96.18
127.78
127.90
120.20
141.17
166.51
154.10
137.28
176.92
158.94
199.18
117.45
98.74
195.97
10.93
9.63
1.75
36.10
19.40
15.80
11.09
77.76
6.69
18.51
25.23
7.88
30.86
23.29
13.43
105.86
85.07
0.72
31.14
7.14
10.08
15.95
22.47
8.97
17.77
4.34
65.02
41.75
5.67
6.43
17.96
21.43
19.31
8.55
0.71
6.17
0.59
3.34
37.00
22.77
6.34
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
40
40
40
40
40
40
40
40
40
40
40
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
40
40
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
2401957
3864775
3049736
5164522
4158135
7243232
5906865
11454376
10071133
12700917
11111659
743617
375475
1354551
699877
1836311
1138526
2063827
1428608
2615429
1759960
2669259
2089048
3225813
2567890
1484347
740968
2283069
1481672
3198823
1998642
3462178
2509576
4132934
3025427
4470882
3646077
5175780
4077055
2321239
1529709
157
1542596
3441850
2546259
4751938
3691982
6956133
5736822
11090659
9747182
12539605
10915624
394798
353011
727389
631970
1046159
996501
1429977
1416332
1816996
1734881
2127444
2042515
2490216
2534146
595413
470544
1179295
1029502
1708861
1542642
2284832
2025350
2976042
2645827
3516450
3251056
4431134
3737768
1130790
878288
35.87
10.94
16.50
8.05
11.15
3.96
2.76
3.17
3.24
1.26
1.75
46.42
5.82
46.05
9.70
42.80
12.47
30.41
0.88
30.36
1.37
20.13
2.23
22.36
1.17
59.86
36.07
48.22
30.38
46.38
22.87
34.02
18.94
27.99
12.48
21.38
10.83
14.38
8.32
51.14
42.37
280.73
176.40
202.94
219.54
196.26
215.47
190.88
253.32
205.78
270.27
227.26
56.31
54.66
63.02
71.64
96.54
92.83
122.30
116.21
144.44
134.94
155.80
156.45
177.59
179.93
53.82
65.25
86.50
96.31
120.87
122.94
125.74
142.26
153.08
162.28
181.25
191.38
184.00
197.39
98.02
133.58
26.09
9.09
7.99
5.58
17.58
1.10
7.26
1.14
0.99
1.28
2.62
37.75
17.87
74.04
0.25
40.21
0.25
45.11
1.37
19.55
6.00
15.12
4.45
47.05
4.84
5.34
38.73
30.25
17.33
54.99
83.12
14.20
80.63
8.78
24.76
11.79
2.01
13.17
2.32
12.63
26.32
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
20
20
20
20
20
20
20
20
20
20
20
20
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
40
40
40
40
40
40
40
40
40
40
40
40
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
40
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4118023
3027168
5197958
3931852
5963680
4762063
6961640
5283445
7866750
6303504
9278127
7349605
802756
384861
1434241
800255
1918738
1092080
2399192
1372556
3080724
1835648
3169422
1993507
3659481
2433051
1473898
748144
2641540
1434759
3470249
2138593
4477555
2842798
4876050
3378438
5249160
3786440
5986767
4416862
2814995
158
2289024
1894339
3413843
2578556
4706560
3505245
5751272
4471997
7277086
5336140
8411689
6646200
398586
361633
812395
707922
1047274
930086
1516802
1367955
2007977
1796300
2034698
1936686
2601824
2281274
636770
464328
1256192
1002188
1882362
1468582
2596778
2193224
2769092
2594149
3849457
3204079
4296572
3737330
1151620
44.39
37.47
34.28
34.40
21.11
26.33
17.39
15.23
7.49
15.40
9.33
9.56
50.33
5.98
43.31
11.52
45.16
14.64
36.52
0.32
34.89
2.15
35.74
2.84
28.67
6.26
56.78
37.93
52.02
29.62
45.57
31.23
41.87
22.85
43.13
23.14
26.63
15.30
28.28
15.37
58.96
126.23
206.18
158.36
198.82
181.78
202.24
186.59
208.52
213.16
228.98
222.12
235.70
52.97
62.95
73.87
72.26
90.49
100.73
118.49
116.25
134.59
144.79
149.19
157.25
168.06
176.37
68.66
65.66
89.45
99.04
98.28
117.33
120.05
128.17
140.23
154.43
164.07
172.87
212.54
185.15
118.69
14.37
20.59
10.37
2.91
9.72
17.46
11.63
82.20
7.82
7.96
3.75
16.29
1.30
7.95
43.15
1.36
37.20
15.31
53.35
0.16
9.96
6.94
19.70
10.36
71.10
0.72
19.07
0.57
58.32
70.92
18.45
34.56
25.75
4.88
14.48
21.46
19.08
16.88
16.19
16.77
14.24
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
3
3
3
3
3
3
3
3
3
3
3
3
3
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
30
30
30
30
30
30
30
30
30
30
30
30
30
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
40
40
40
40
40
40
40
40
40
40
40
40
40
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1693554
4452563
2920393
5849196
4412876
7115658
5050842
7469010
5457833
9132094
6994948
9874691
7517648
697155
377257
1137957
683909
1333077
1096890
1698854
1401731
2148587
1980359
2441938
2393850
2953619
2774785
908627
542325
1656003
1257754
2330693
1785032
2611476
2299836
3098678
2751041
4056180
3636540
6897435
6171892
159
907661
2310610
1687295
3544152
2711008
4886665
3473524
5391717
4144406
7127843
5415189
8056503
6188194
403974
358499
664399
634451
1026905
1013780
1559406
1394173
1959278
1951888
2254588
2333723
2785695
2677161
576744
459538
1150226
981220
1770633
1530717
2272526
2030822
2999995
2546764
3914279
3523693
6642055
6069237
46.41
48.06
42.17
39.43
38.30
31.28
30.82
27.82
24.03
21.94
22.57
18.40
17.65
41.79
4.84
40.93
7.16
22.71
6.29
8.18
0.52
8.81
1.44
7.66
2.25
5.64
3.34
36.28
13.90
30.48
21.91
24.03
13.93
12.94
11.62
3.18
7.41
3.46
3.13
3.70
1.66
104.82
130.42
245.49
186.61
185.37
153.42
182.00
182.37
236.17
186.31
210.52
201.10
238.88
38.92
44.61
56.40
61.13
75.47
82.73
94.92
107.19
111.59
135.58
141.98
141.12
160.55
176.31
57.88
58.14
73.25
98.53
92.77
128.45
140.48
135.41
156.90
162.83
201.49
164.44
186.94
175.40
2.38
2.89
6.37
7.73
36.64
44.74
37.27
5.17
13.81
10.64
19.18
4.53
17.01
17.64
26.37
57.07
8.89
55.03
79.15
46.07
0.18
7.77
5.36
7.92
11.25
7.22
15.91
17.96
141.38
16.93
28.40
3.65
43.21
8.42
18.74
10.13
2.06
5.89
3.96
3.49
0.87
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
40
40
40
40
40
40
40
40
40
40
40
40
40
40
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
20
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
1879531
1446522
2886672
2353435
3881053
3033268
5116527
3963512
7511848
5925198
11153376
9501158
12666393
10748506
749480
348782
1306909
731558
1746972
1020486
2120356
1401811
2688556
1678645
2788683
2074883
3173068
2509973
1462149
754985
2240072
1404647
3201714
1984615
3504496
2505365
3969385
2917887
4433627
3544986
5262331
160
1143023
850667
2104456
1538202
3525512
2614498
4920146
3663935
7312020
5599407
10861413
9223143
12481376
10609351
384051
337544
779067
660160
1057616
909833
1524588
1373412
1868894
1674389
2148998
1986392
2589873
2421240
592384
458617
1201417
1039734
1690062
1435085
2299453
2099064
2997632
2526120
3409521
3109908
4594214
39.19
40.98
27.11
34.67
9.17
13.69
3.85
7.58
2.66
5.52
2.61
2.91
1.46
1.30
48.46
3.21
39.73
9.77
39.33
10.63
27.98
1.84
30.45
0.25
22.51
4.29
18.39
3.49
59.44
39.26
46.27
25.62
47.10
27.66
34.15
16.22
24.46
13.41
23.01
12.16
12.66
130.27
104.45
259.17
326.85
166.11
207.26
159.55
198.10
228.83
205.07
279.88
224.24
220.37
239.09
45.92
60.04
63.65
68.92
132.88
93.43
160.84
118.40
199.05
124.57
194.34
143.57
263.49
166.41
82.71
65.47
108.00
84.43
139.92
126.54
144.36
134.03
150.20
151.23
187.37
166.39
223.53
42.01
33.47
6.28
3.52
5.50
36.11
1.39
6.95
3.00
1.83
2.43
2.36
0.32
0.21
21.90
1.21
91.23
3.92
16.49
31.90
31.24
16.92
8.79
0.10
108.76
0.37
8.56
8.05
2.97
0.27
19.66
56.71
15.44
14.78
29.32
14.85
92.62
1.30
25.11
9.61
14.84
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
20
40
40
40
40
40
40
40
40
40
40
40
40
40
40
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
20
20
20
20
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
50
50
100
100
150
150
200
200
250
250
300
300
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
4250024
2331711
1624896
3894947
2853344
5161643
3798494
5902620
4638035
6854899
5260017
7905208
5924119
8850006
7125328
773500
370785
1366074
724149
1983117
1073810
2434832
1300824
2875864
1640085
3348598
2077508
3428969
2552700
1464430
761835
2566011
1489686
3546941
2170512
4549658
2810072
4706841
3229051
5125938
3529537
161
4028948
1193358
860291
2281659
1676443
3305957
2582482
4654080
3354150
5428145
4168687
7017220
5266576
8057968
6397557
394683
350039
767838
660966
1153319
935269
1506399
1295281
1853561
1635191
2280453
1983251
2645849
2421634
655430
474115
1193587
1006240
1912170
1492797
2376716
2100210
2971730
2589737
3635909
2996911
5.22
48.82
46.77
41.42
41.18
35.94
31.91
21.13
27.73
20.82
20.71
11.23
11.03
8.95
10.16
48.75
5.59
43.56
8.74
41.66
12.40
38.02
0.40
35.54
0.30
31.79
4.54
22.73
4.99
55.31
37.76
53.30
32.22
46.06
31.23
47.79
25.11
36.88
19.60
28.94
15.02
180.47
123.25
107.64
164.38
226.55
191.78
185.93
168.38
210.27
176.75
220.52
249.61
235.08
216.83
240.84
56.99
53.06
72.22
70.60
95.07
93.02
112.71
115.59
126.92
128.44
171.24
150.59
170.75
171.53
66.01
66.59
100.01
97.93
102.68
120.34
119.46
131.43
202.80
148.63
159.49
178.81
3.69
2.95
32.21
8.86
13.71
8.01
18.42
79.09
9.15
4.99
4.97
10.53
14.86
0.43
17.37
16.02
0.52
46.20
3.99
38.16
65.80
44.80
0.62
14.54
0.12
32.16
0.10
29.44
11.90
10.81
0.67
54.55
51.19
26.57
0.67
2.33
23.45
27.78
30.51
65.29
11.61
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
20
20
40
40
40
40
40
40
40
40
40
40
40
40
40
40
350
350
50
50
100
100
150
150
200
200
250
250
300
300
350
350
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
5468968
4151912
2762442
1614594
4360746
2794147
5610403
4087188
6912590
4802963
7718047
5728722
8830279
6814825
9783577
7453410
162
4098308
3515322
1094896
932339
2274436
1544904
3580440
2496332
4534235
3367017
5460475
4367651
6745180
5149875
7950669
5866271
24.99
15.00
60.23
42.15
47.86
44.70
36.17
38.82
34.37
29.57
29.25
23.74
23.62
24.34
18.74
21.27
183.13
178.89
111.31
144.62
140.59
318.66
156.56
273.49
179.18
263.91
231.19
252.14
229.53
302.27
276.82
241.52
43.09
30.78
13.51
62.95
7.70
0.98
15.01
17.17
9.31
70.37
7.86
26.54
3.20
55.25
5.06
6.61
50
40
30
20
10
50
100
150
200
250
300
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
0
1900
Average Cost Improvement (%)
APPENDIX E
350
Vehicle (upper) and Customer (lower) Level at
Plant#1
50
100
150
200
250
300
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
50
45
40
35
30
25
20
15
10
5
0
1900
Average Cost Improvement (%)
Figure 1: Comparing the Average of Cost Improvement against no. of Customer and
Type of Vehicles at Plant Level 1.
1
350
Vehicle (upper) and Customer (lower) Level at Plant# 3
Figure 2: Comparing the Average of Cost Improvement against no. of Customer and
Type of Vehicles at Plant Level 3.
3
163
50
100
150
200
250
300
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
Average Cost Improvement (%)
50
45
40
35
30
25
20
15
10
5
0
350
Vehicle (upper) and Customer (lower) Level at Plant#5
40
35
30
25
20
15
10
5
0
Total
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
Average Cost Improvement (%)
Figure 3: Comparing the Average of Cost Improvement against no. of Customer and
Type of Vehicles at Plant Level 5.
5
50
100
150
200
250
300
350
Vehicle (upper) and Customer (lower) Level at Depot#10
Figure 4: Comparing the Average of Cost Improvement against no. of Customer and
Type of Vehicles at Depot Level 10.
164
50
40
30
20
10
50
100
150
200
250
300
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
0
1900
Average Cost Improvement (%)
60
350
Vehicle (upper) and Customer (lower) Level at Depot #20
60
50
40
30
20
10
50
100
150
200
250
300
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
1900
4500
0
1900
Average Cost Improvement (%)
Figure 5: Comparing the Average of Cost Improvement against no. of Customer and
Type of Vehicles at Depot Level
Le 20.
350
Vehicle (upper) and Customer (lower) Level at Depot#30
Figure 6: Comparing the Average of Cost Improvement against no. of Customer and
Type of Vehicles at Depot Level 30.
165
APPENDIX F
Table 1: Annual Demand for Case Study I: 102customers & 20 products
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
1625
1761
1755
1616
1528
1690
1646
1769
1773
1554
1756
1722
1535
1520
1570
1610
1761
1633
1575
1692
2
1754
1653
1636
1568
1526
1507
1608
1769
1771
1561
1688
1569
1594
1593
1686
1550
1627
1650
1576
1624
3
1511
1625
1706
1743
1767
1538
1677
1555
1679
1792
1623
1652
1598
1649
1777
1746
1783
1549
1641
1572
4
1535
1787
1564
1776
1641
1647
1603
1532
1682
1780
1778
1628
1586
1562
1746
1686
1657
1764
1534
1555
5
1768
1541
1603
1590
1695
1649
1555
1632
1718
1734
1736
1627
1514
1552
1697
1521
1703
1776
1572
6
1706
1734
1637
1526
1665
1794
1598
1551
1611
1506
1794
1527
1674
1753
1697
1765
1718
1701
1571
1588
1729
7
1624
1740
1672
1579
1528
1589
1588
1620
1736
1717
1665
1750
1672
1519
1781
1636
1661
1784
1725
1676
8
1587
1787
1742
1585
1650
1611
1795
1752
1639
1645
1730
1684
1510
1584
1715
1561
1740
1613
1513
1782
9
1760
1546
1776
1771
1599
1545
1510
1504
1588
1657
1660
1578
1706
1597
1558
1530
1588
1625
1652
1541
10
1575
1667
1550
1739
1702
1673
1641
1720
1650
1666
1686
1604
1526
1746
1719
1715
1760
1557
1624
1565
11
1711
1579
1672
1583
1719
1736
1758
1674
1692
1667
1716
1653
1718
1557
1684
1747
1574
1587
1619
1555
12
1699
1774
1798
1588
1621
1745
1568
1789
1781
1697
1538
1619
1590
1731
1658
1733
1677
1569
1647
1694
13
1627
1731
1527
1514
1554
1690
1661
1768
1672
1522
1749
1567
1630
1671
1524
1652
1586
1767
1684
1527
14
1506
1624
1617
1566
1784
1623
1597
1714
1738
1546
1670
1563
1797
1619
1789
1580
1618
1708
1754
1518
15
1684
1528
1696
1616
1688
1724
1756
1616
1652
1528
1551
1596
1645
1596
1711
1694
1538
1753
1769
1659
16
1645
1637
1603
1780
1523
1577
1729
1683
1521
1765
1601
1717
1532
1695
1524
1593
1595
1738
1678
1759
17
1598
1628
1632
1732
1791
1640
1605
1570
1610
1752
1678
1511
1771
1592
1594
1762
1565
1502
1721
1678
18
1538
1721
1563
1511
1547
1698
1730
1782
1588
1798
1567
1531
1678
1533
1648
1683
1648
1658
1582
1732
19
1655
1695
1623
1569
1659
1653
1707
1728
1633
1697
1567
1550
1635
1647
1725
1777
1768
1613
1735
1674
20
1622
1738
1744
1661
1734
1521
1663
1583
1769
1658
1571
1721
1743
1644
1764
1525
1548
1584
1566
1698
21
1628
1700
1693
1602
1559
1758
1669
1731
1782
1609
1547
1646
1540
1644
1651
1604
1547
1775
1614
1763
22
1645
1687
1631
1680
1661
1643
1618
1531
1703
1735
1740
1636
1759
1747
1713
1703
1788
1796
1790
1710
23
1738
1589
1515
1705
1781
1593
1596
1617
1508
1711
1693
1527
1587
1735
1658
1593
1782
1599
1608
1714
24
1601
1666
1586
1639
1572
1755
1798
1672
1604
1530
1701
1762
1664
1791
1654
1711
1647
1549
1514
1785
25
1757
1796
1568
1643
1748
1662
1629
1610
1501
1736
1695
1643
1563
1800
1531
1551
1712
1651
1603
1563
26
1516
1706
1610
1668
1520
1608
1575
1590
1665
1563
1728
1508
1702
1641
1609
1764
1538
1719
1745
1704
27
1501
1572
1592
1564
1528
1569
1657
1518
1779
1755
1738
1659
1759
1667
1508
1777
1788
1689
1708
1704
28
1621
1725
1654
1733
1651
1623
1671
1538
1550
1574
1563
1776
1774
1518
1782
1794
1714
1690
1679
1523
29
1668
1503
1557
1683
1672
1798
1715
1504
1732
1783
1717
1742
1502
1761
1701
1589
1763
1536
1720
1636
30
1683
1680
1602
1635
1514
1679
1577
1624
1733
1549
1517
1618
1748
1501
1748
1781
1591
1747
1756
1712
31
1747
1612
1575
1774
1656
1688
1584
1611
1599
1503
1559
1559
1752
1564
1760
1685
1658
1665
1651
1707
32
1629
1748
1752
1671
1635
1762
1690
1547
1606
1785
1572
1662
1527
1671
1756
1526
1524
1721
1648
1798
33
1609
1556
1739
1579
1634
1533
1558
1534
1705
1665
1697
1582
1743
1658
1760
1648
1771
1506
1711
1583
34
1629
1563
1661
1589
1554
1519
1644
1631
1549
1782
1649
1672
1645
1710
1676
1514
1663
1556
1777
1551
35
1740
1700
1637
1730
1526
1712
1783
1742
1643
1553
1521
1691
1800
1525
1782
1567
1580
1541
1720
1524
36
1530
1771
1616
1518
1684
1543
1553
1587
1689
1711
1735
1661
1555
1535
1708
1634
1775
1568
1609
1755
37
1646
1685
1796
1739
1679
1622
1773
1707
1547
1746
1517
1784
1694
1601
1654
1748
1682
1728
1534
1645
38
1643
1592
1526
1661
1657
1740
1747
1551
1584
1510
1630
1718
1762
1631
1670
1787
1586
1649
1675
1614
39
1643
1716
1550
1551
1614
1741
1772
1763
1698
1532
1739
1546
1654
1578
1749
1668
1566
1671
1599
1656
40
1671
1599
1534
1752
1701
1554
1712
1658
1798
1719
1772
1668
1526
1608
1708
1585
1691
1662
1792
1761
41
1504
1732
1555
1738
1612
1599
1682
1560
1640
1543
1652
1684
1633
1629
1792
1779
1729
1695
1515
1782
42
1525
1640
1581
1666
1662
1682
1675
1531
1789
1631
1773
1638
1755
1781
1643
1563
1695
1787
1521
1526
43
1599
1539
1767
1739
1520
1765
1591
1594
1598
1510
1585
1779
1602
1719
1573
1543
1683
1749
1517
1522
44
1615
1543
1668
1581
1712
1530
1543
1559
1655
1527
1800
1581
1722
1506
1505
1566
1629
1653
1591
1697
45
1573
1563
1726
1764
1656
1672
1669
1556
1650
1643
1532
1612
1570
1626
1650
1686
1665
1576
1669
1710
46
1691
1747
1770
1766
1765
1517
1620
1515
1745
1787
1641
1715
1758
1752
1555
1767
1567
1524
1566
1540
47
1784
1757
1586
1591
1770
1657
1570
1644
1666
1688
1538
1517
1501
1799
1649
1732
1528
1554
1601
1794
48
1559
1584
1771
1749
1772
1768
1787
1712
1757
1741
1740
1575
1518
1504
1548
1686
1767
1560
1570
1530
49
1631
1737
1550
1636
1653
1723
1759
1540
1699
1527
1569
1681
1586
1791
1725
1599
1797
1617
1675
1614
50
1518
1762
1572
1681
1604
1713
1591
1553
1606
1795
1511
1714
1721
1549
1624
1622
1732
1765
1774
1695
51
1684
1706
1724
1767
1534
1655
1695
1731
1709
1794
1679
1763
1707
1616
1608
1760
1666
1512
1705
1786
52
1678
1737
1744
1765
1775
1526
1605
1733
1518
1614
1588
1743
1743
1544
1577
1523
1615
1757
1549
1708
166
53
1761
1724
1744
1761
1517
1750
1730
1623
1545
1597
1647
1612
1556
1626
1780
1528
1539
1748
1538
1685
54
1766
1571
1750
1653
1665
1727
1799
1725
1635
1773
1725
1707
1668
1705
1649
1722
1564
1728
1618
1725
55
1740
1734
1536
1773
1587
1638
1744
1562
1636
1738
1543
1755
1713
1569
1695
1544
1634
1506
1705
1575
56
1745
1671
1630
1584
1799
1785
1682
1613
1662
1799
1724
1529
1540
1718
1784
1673
1708
1526
1724
1623
57
1516
1732
1779
1734
1563
1653
1800
1696
1679
1514
1698
1793
1628
1727
1610
1784
1541
1648
1780
1520
58
1739
1777
1723
1732
1729
1614
1736
1608
1674
1667
1711
1796
1676
1595
1674
1513
1715
1522
1583
1772
59
1526
1795
1730
1730
1797
1539
1558
1501
1649
1765
1664
1622
1787
1608
1653
1523
1759
1716
1662
1538
60
1543
1794
1566
1553
1590
1626
1703
1531
1594
1567
1648
1696
1575
1794
1746
1604
1719
1594
1695
1769
61
1622
1774
1602
1790
1681
1771
1754
1588
1797
1548
1571
1605
1716
1697
1798
1514
1565
1770
1673
1755
62
1565
1745
1677
1606
1756
1629
1726
1534
1619
1624
1654
1517
1727
1520
1663
1664
1693
1546
1628
1685
63
1524
1710
1675
1578
1572
1787
1645
1659
1680
1585
1650
1642
1714
1790
1786
1684
1730
1776
1625
1729
64
1594
1760
1761
1710
1564
1515
1575
1615
1592
1799
1631
1788
1538
1701
1532
1781
1690
1565
1535
1632
65
1733
1626
1559
1731
1731
1581
1734
1613
1523
1638
1668
1636
1704
1501
1698
1530
1794
1542
1664
1551
66
1683
1774
1705
1799
1782
1636
1788
1564
1626
1516
1594
1543
1793
1673
1577
1778
1601
1748
1502
1781
67
1570
1510
1676
1750
1527
1539
1680
1799
1730
1692
1525
1692
1683
1601
1569
1741
1544
1642
1777
1713
68
1792
1689
1635
1681
1785
1534
1543
1587
1713
1568
1527
1631
1727
1562
1518
1778
1635
1503
1551
1639
69
1667
1649
1799
1663
1775
1649
1519
1629
1723
1591
1613
1510
1525
1538
1693
1621
1695
1760
1669
1529
70
1766
1750
1554
1541
1547
1731
1655
1573
1754
1763
1684
1639
1736
1791
1745
1715
1617
1598
1796
1542
71
1669
1581
1708
1651
1635
1553
1660
1722
1682
1546
1546
1699
1541
1663
1634
1519
1581
1794
1571
1778
72
1729
1670
1671
1617
1696
1736
1785
1654
1687
1741
1748
1601
1522
1759
1786
1519
1623
1753
1727
1775
73
1691
1652
1741
1620
1575
1743
1527
1504
1723
1577
1516
1598
1604
1602
1770
1641
1604
1732
1790
1678
74
1567
1735
1632
1566
1636
1657
1684
1522
1672
1545
1562
1615
1704
1641
1648
1512
1760
1567
1531
1524
75
1568
1501
1582
1602
1753
1574
1780
1683
1578
1623
1599
1694
1698
1758
1596
1647
1695
1674
1634
1515
76
1626
1658
1640
1797
1718
1693
1514
1665
1620
1646
1703
1757
1570
1766
1722
1611
1798
1730
1521
1702
77
1650
1587
1634
1796
1653
1561
1566
1638
1544
1537
1768
1698
1552
1564
1749
1781
1688
1668
1597
1573
78
1532
1689
1628
1542
1581
1673
1744
1781
1787
1785
1610
1672
1635
1628
1669
1793
1763
1749
1506
1774
79
1724
1743
1739
1633
1572
1604
1567
1591
1674
1565
1728
1704
1769
1785
1769
1596
1595
1531
1590
1551
80
1632
1519
1601
1746
1642
1502
1715
1772
1626
1621
1534
1616
1729
1514
1681
1688
1509
1687
1782
1746
81
1778
1743
1547
1670
1703
1725
1620
1632
1578
1622
1602
1614
1678
1752
1623
1738
1502
1584
1599
1725
82
1735
1624
1775
1759
1750
1684
1646
1758
1739
1613
1524
1664
1565
1655
1562
1692
1731
1588
1713
1709
83
1536
1663
1700
1784
1800
1655
1794
1556
1746
1587
1588
1768
1655
1723
1568
1657
1510
1625
1511
1792
84
1733
1701
1541
1773
1785
1773
1611
1590
1562
1619
1711
1517
1713
1736
1619
1793
1598
1530
1535
1542
85
1749
1651
1734
1582
1626
1715
1653
1743
1621
1722
1784
1784
1502
1730
1626
1698
1745
1800
1504
1667
86
1515
1608
1544
1598
1676
1759
1581
1748
1580
1647
1610
1612
1605
1574
1762
1569
1778
1735
1626
1708
87
1788
1785
1599
1799
1643
1690
1622
1562
1525
1741
1545
1750
1693
1677
1672
1679
1504
1608
1658
1657
88
1745
1653
1786
1502
1677
1643
1758
1683
1647
1680
1526
1764
1510
1548
1733
1558
1639
1680
1766
1518
89
1554
1598
1663
1529
1702
1539
1539
1627
1510
1582
1685
1623
1511
1761
1665
1764
1554
1611
1657
1553
90
1669
1688
1616
1553
1533
1719
1552
1612
1798
1554
1779
1597
1735
1796
1693
1566
1786
1641
1784
1773
91
1537
1778
1541
1777
1690
1505
1750
1522
1746
1718
1584
1749
1752
1526
1643
1651
1760
1643
1608
1791
92
1523
1785
1777
1512
1706
1505
1720
1567
1663
1742
1527
1516
1721
1568
1658
1543
1600
1735
1625
1747
93
1662
1644
1614
1759
1591
1555
1653
1598
1534
1649
1604
1745
1678
1670
1520
1667
1664
1557
1595
1622
94
1581
1630
1720
1701
1725
1531
1797
1725
1725
1732
1590
1516
1561
1601
1721
1733
1594
1688
1509
1718
95
1544
1659
1548
1718
1711
1658
1770
1601
1539
1536
1645
1761
1690
1601
1736
1626
1582
1567
1584
1753
96
1674
1515
1667
1710
1758
1569
1713
1750
1654
1618
1522
1619
1657
1657
1616
1570
1751
1686
1555
1662
97
1794
1539
1796
1509
1595
1602
1600
1556
1612
1530
1519
1565
1530
1711
1514
1690
1779
1614
1664
1567
98
1693
1715
1760
1798
1722
1526
1798
1716
1574
1611
1794
1737
1669
1655
1665
1573
1631
1513
1626
1524
99
1524
1582
1790
1634
1586
1782
1557
1746
1772
1544
1780
1761
1577
1607
1749
1532
1620
1663
1795
1591
100
1775
1541
1558
1591
1583
1516
1582
1503
1744
1543
1708
1635
1535
1729
1760
1689
1620
1681
1781
1626
101
1544
1728
1552
1529
1698
1647
1768
1711
1615
1543
1791
1538
1739
1568
1762
1649
1650
1641
1740
1659
102
1729
1737
1760
1759
1606
1556
1579
1526
1615
1783
1708
1620
1762
1553
1675
1571
1641
1730
1532
1659
167
18
19
20
169076
17
167625
16
168156
15
168970
14
168478
13
167776
12
168299
11
168521
10
167621
9
168221
8
168851
7
166380
6
169733
5
167469
4
168825
3
169736
167559
Total
annual
demand
2
168792
1
170398
Product
170468
Table 2: Total annual Demand of Case Study I (summary of Table 1)
Table 3: Fixed Order Interval Demand and Customers’ Coordinates for Case Study I
Product
Customer
Coordinate
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
X
Y
1
18
16
20
21
23
16
18
22
24
17
22
25
24
21
22
18
25
19
17
16
-18.17
76.762
2
16
21
21
21
18
21
20
25
19
23
21
23
23
19
25
18
23
25
17
17
-7.358
57.641
3
22
20
24
21
16
23
20
23
16
22
16
22
22
22
24
21
25
25
23
20
-67.49
23.225
4
22
25
20
19
16
20
24
21
17
16
23
21
19
16
22
24
16
21
25
24
38.657
51.807
5
25
23
23
23
17
16
25
20
20
23
18
19
22
24
18
24
25
19
16
19
-93.07
75.642
6
19
21
24
17
23
20
20
22
16
21
20
25
22
24
17
22
16
21
20
19
38.551
82.265
7
21
24
23
17
18
21
16
22
22
18
17
17
22
19
16
21
20
16
24
24
-65.65
24.608
8
24
22
24
20
16
23
21
20
21
20
16
17
18
17
17
23
17
24
22
23
-49.86
18.734
9
16
24
23
24
20
16
16
20
24
19
24
23
16
23
17
25
23
22
25
18
89.406
60.759
10
16
17
23
24
20
20
23
20
24
21
25
22
25
22
19
24
16
21
20
24
48.686
22.049
11
21
18
16
21
21
25
21
23
23
23
21
16
18
20
21
23
22
18
25
23
64.018
29.575
12
22
23
24
23
20
20
24
21
22
24
16
23
22
21
25
18
25
18
23
24
-42.26
59.565
13
16
17
21
19
17
19
19
18
24
18
22
17
21
21
18
25
17
17
19
24
-33.47
87.386
14
19
16
20
17
18
20
18
22
24
20
20
24
19
20
16
18
21
18
18
20
31.11
96.187
15
16
18
16
21
24
23
21
18
20
19
21
22
17
22
16
19
16
17
24
17
67.007
18.379
16
20
16
23
19
22
18
20
23
17
16
24
19
20
25
24
18
21
23
16
21
-15.14
92.317
17
22
20
20
21
21
17
20
21
19
16
20
19
17
22
24
23
19
24
18
21
-11.72
33.744
18
21
16
22
21
22
19
25
25
19
18
22
18
21
18
25
25
20
23
16
19
-13.71
72.752
19
25
23
22
20
19
25
25
18
22
21
18
19
21
16
21
23
19
16
24
23
-88.19
49.861
20
24
25
17
18
16
25
24
20
17
24
20
16
20
22
17
20
20
17
16
25
-23.37
-5.454
21
21
18
25
19
25
18
24
21
16
20
18
25
24
19
20
22
21
18
25
18
28.857
68.229
22
16
17
24
19
21
16
19
23
17
24
25
23
16
16
24
20
25
23
24
17
3.6022
74.15
23
22
19
24
20
16
17
17
25
20
18
18
19
19
17
25
17
22
21
24
18
-58.22
42.953
24
20
24
21
20
24
21
25
22
23
17
25
23
17
22
21
17
21
19
21
24
47.566
92.05
25
23
22
17
19
16
17
22
24
18
19
20
24
18
20
21
24
16
24
23
24
37.77
85.604
26
25
17
20
18
19
21
20
21
19
25
17
24
24
21
20
21
18
19
25
20
2.664
58.032
27
23
22
25
16
20
17
25
18
25
21
25
17
25
23
23
25
19
21
20
24
-26.6
62.825
28
18
19
18
19
21
24
22
17
22
19
24
19
24
19
19
24
23
23
18
22
0.4596
91.739
29
16
19
20
18
23
19
24
24
22
20
18
25
25
19
17
18
21
25
20
16
-75.26
52.428
168
30
23
20
19
22
22
20
21
18
25
22
16
19
24
19
21
22
21
25
17
23
-16.88
36.529
31
19
17
21
22
21
19
21
22
19
25
23
19
23
16
19
25
18
24
19
20
62.647
61.449
32
22
21
24
21
16
23
17
16
24
20
16
16
24
21
24
21
17
23
16
25
-94.18
30.283
33
23
24
20
23
17
21
25
25
23
16
25
21
20
20
25
24
17
22
18
19
8.1111
31.121
34
25
21
20
23
25
24
24
18
20
20
22
23
22
17
16
16
19
24
19
23
-76.81
45.317
35
19
25
24
25
17
23
17
24
16
21
21
24
22
19
19
17
24
22
24
17
46.577
36.598
36
18
21
19
20
19
24
16
18
20
23
17
20
24
22
20
25
18
23
24
24
26.406
95.455
37
25
17
18
25
22
22
19
25
20
24
17
24
20
21
24
23
17
19
25
19
9.9365
96.028
38
23
25
25
19
22
19
17
21
20
17
24
22
17
24
24
24
20
21
24
22
2.1563
54.333
39
23
20
17
20
17
25
16
17
20
17
21
20
20
21
18
16
22
20
20
17
41.364
32.436
40
22
17
22
21
21
22
21
20
23
22
22
19
25
25
23
21
21
24
24
18
2.4183
55.757
41
16
21
22
22
25
19
16
23
25
24
21
19
20
16
24
19
16
20
23
25
44.19
85.78
42
22
18
16
20
19
21
21
20
19
21
21
18
19
20
20
25
20
22
20
16
95.129
87.651
43
24
20
16
22
24
20
24
17
17
22
17
25
21
17
23
17
19
24
20
21
41.06
57.247
44
18
20
23
25
25
20
21
23
17
25
25
22
18
25
20
18
16
21
21
25
89.295
42.302
45
20
25
23
17
18
18
22
16
16
16
22
23
24
21
21
21
20
18
22
24
-23.56
30.248
46
23
17
19
22
18
16
23
20
25
25
18
17
25
22
21
24
17
21
22
21
26.033
23.078
47
20
17
20
19
20
20
22
19
20
17
19
23
17
17
16
21
18
20
25
18
-63.13
63.768
48
18
19
22
21
24
20
16
18
18
25
19
18
20
20
20
16
20
24
23
16
-2.251
48.132
49
18
22
22
21
24
18
17
23
25
21
24
22
22
16
23
23
19
20
16
21
-39.17
48.563
50
19
17
18
24
18
17
22
19
24
18
18
24
18
20
18
23
25
18
22
22
37.368
6.7058
51
17
22
24
16
23
18
19
19
17
23
17
19
20
25
19
23
24
20
17
17
40.047
95.142
52
20
22
23
18
22
24
16
16
17
21
23
22
23
20
17
16
19
16
24
16
69.887
43.565
53
24
24
24
22
20
21
17
21
17
24
22
17
16
24
23
24
20
22
22
19
-1.948
-3.255
54
25
18
16
21
18
22
25
20
19
24
18
19
19
25
22
16
24
16
19
21
60.102
86.969
55
16
20
16
25
22
24
18
18
21
22
24
17
17
21
22
25
25
21
18
19
-64.15
93.533
56
25
19
23
18
17
18
19
17
20
23
16
18
24
17
19
18
19
22
20
18
-10.04
17.414
57
21
18
22
25
18
23
21
25
21
22
17
17
18
24
17
20
21
22
22
16
-53.2
77.097
58
21
16
18
18
17
18
17
24
20
19
17
18
23
17
17
22
21
23
18
23
-95.31
30.864
59
19
25
18
24
21
19
16
20
19
21
19
23
20
22
20
23
17
22
17
16
5.5378
73.856
60
25
20
16
22
16
16
22
19
19
22
23
16
23
22
21
25
21
25
23
18
74.36
98.689
61
20
17
19
20
17
20
23
24
22
19
23
20
23
25
17
22
20
18
21
23
-32.21
61.631
62
18
25
25
24
21
18
23
17
20
24
22
25
21
20
25
25
23
23
22
19
-8.256
37.607
63
16
20
25
16
20
23
16
17
17
20
21
18
22
17
20
17
16
16
16
19
34.922
54.846
64
25
17
19
17
16
20
21
16
22
25
19
17
23
17
16
24
18
16
24
19
-62.47
2.0149
65
16
24
18
23
22
16
25
17
25
18
17
22
16
18
17
18
17
16
18
25
19.347
63.589
66
21
16
22
23
19
18
25
20
22
25
18
16
22
25
21
20
19
25
18
22
-28.57
55.917
67
19
17
18
20
24
22
20
19
20
17
17
24
19
21
25
25
19
18
23
22
64.187
8.7427
68
18
16
23
21
17
20
21
24
24
20
18
17
20
24
17
23
18
23
23
19
-15.38
61.621
69
25
19
23
23
25
17
24
20
24
24
22
17
20
16
19
19
17
18
25
17
-89.63
63.389
70
21
18
20
25
16
25
25
20
23
25
20
24
21
25
19
17
24
18
23
21
69.003
28.933
169
71
20
17
18
22
16
17
22
22
19
19
24
25
19
25
19
25
18
23
23
21
-72.26
78.729
72
25
23
17
18
20
23
23
19
16
21
21
16
24
20
17
18
24
18
17
22
84.802
-1.014
73
16
20
16
23
23
25
21
16
20
21
19
16
17
18
19
21
23
20
18
22
71.212
44.105
74
23
19
17
17
25
17
20
24
22
24
20
24
19
20
19
19
25
25
16
24
64.027
72.186
75
23
20
25
16
18
22
17
24
22
25
22
17
19
20
22
18
18
25
24
20
-95.55
29.646
76
24
24
18
20
19
16
25
25
22
22
24
24
24
24
22
24
18
24
23
23
94.982
63.245
77
17
25
24
16
23
19
21
25
23
18
23
18
16
17
16
18
21
18
18
17
-44.88
38.646
78
16
22
17
17
20
18
18
23
19
25
24
19
20
19
19
22
23
23
20
19
47.19
83.53
79
22
18
19
25
23
21
21
17
19
21
18
21
23
23
16
19
21
22
18
16
-80.17
91.368
80
24
22
22
23
22
24
22
21
19
17
18
22
18
24
21
21
20
20
22
20
65.809
27.896
81
22
25
17
23
22
24
22
16
19
18
17
16
25
22
20
20
19
20
24
24
56.435
51.195
82
23
18
24
20
23
23
16
24
22
24
21
25
25
23
18
22
24
18
16
25
-97.53
40.637
83
23
24
17
16
16
19
21
21
17
17
16
24
25
22
18
22
20
21
22
21
59.476
14.136
84
25
20
25
21
18
24
22
24
18
17
25
20
23
19
20
16
16
24
22
21
21.069
23.171
85
24
21
18
21
18
22
22
18
22
24
23
16
19
17
22
24
25
24
20
21
-96.55
29.781
86
18
22
18
16
21
16
20
24
24
22
17
17
23
22
19
19
25
25
17
25
9.4381
73.336
87
19
24
17
21
22
16
21
21
19
22
17
25
17
25
16
21
21
25
23
23
25.365
42.144
88
19
23
18
19
22
17
21
19
17
24
20
24
22
17
25
25
25
25
16
17
-94.19
83.749
89
90
21
21
20
25
22
22
18
19
19
18
23
16
20
17
24
21
22
16
22
19
23
18
22
24
21
18
16
17
17
19
23
23
20
21
17
18
20
21
21
20
-37.6
15.408
35.851
49.422
91
24
22
23
22
18
21
21
25
25
24
20
18
18
17
25
23
24
19
17
20
-35.88
32.49
92
20
20
22
22
22
25
23
20
19
22
16
23
21
17
19
20
20
23
22
21
30.324
58.766
93
24
24
23
23
17
18
16
21
21
17
25
21
21
22
17
20
16
23
20
21
62.858
86.785
94
18
22
22
21
23
18
16
23
19
16
17
18
23
22
25
18
21
23
16
17
47.126
3.2165
95
20
23
22
18
17
16
21
24
18
22
18
18
22
21
17
23
17
20
18
23
47.411
59.003
96
21
25
16
24
16
22
19
19
18
22
20
16
25
19
16
18
21
20
19
21
37.313
61.817
97
20
25
21
19
22
23
16
17
24
20
23
20
23
20
25
22
25
22
22
19
-74.05
38.495
98
18
24
18
22
17
20
18
23
17
21
18
20
21
19
23
20
22
22
19
21
50.846
15.599
99
17
18
19
19
25
18
17
23
22
16
18
25
17
16
18
18
18
23
21
16
-36.76
35.419
100
17
21
22
16
20
23
19
24
16
19
18
16
17
24
17
19
21
24
23
25
-99.36
79.972
101
21
21
17
17
22
19
21
16
20
24
19
21
21
20
18
19
24
18
16
23
-79.18
88.163
102
24
17
23
21
25
18
21
23
19
19
18
19
18
17
20
22
18
23
23
21
13.88
38.064
170
Table 4: Depots’ Capacity and Coordinates for Case Study I
Depot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Depot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
31077
32813
35669
31991
31137
39019
38502
36106
30069
38279
32248
30197
38537
38904
36424
36421
30039
38660
32358
30943
2
32487
38613
34494
35187
35950
34858
37541
30055
32605
31664
38699
35536
38258
39755
35065
36902
36194
36896
30264
32712
3
31816
34209
39940
32344
31377
34216
31921
31023
33411
34308
36803
34615
33108
36326
33897
37579
34743
34227
36433
35662
4
30698
34329
34656
36031
36160
34752
39723
32315
31517
35149
34311
39789
39910
34256
38171
36699
35356
31970
33035
37823
Product
5
39384
31948
30303
31455
33022
37310
33328
34879
39882
37648
34449
30282
36596
36569
31673
31098
32485
37783
35276
34441
10
35705
30057
38682
34528
34736
34392
33451
36730
30200
35520
33130
34075
37826
34957
35395
11
30941
30057
38682
34528
34736
34392
33451
36730
30200
35520
33130
34075
37826
34957
35395
12
31522
38664
39880
31102
32741
30145
35800
38704
36199
30935
38041
39097
39789
37579
32080
13
37725
32311
35557
35494
33141
30894
35558
31988
33990
39298
31332
34948
30996
33627
35753
14
39249
35529
39241
37210
36665
36996
30251
37847
32588
38586
31324
37901
31203
34615
37504
6
35279
31542
36518
35940
31434
38350
39126
36023
37074
31996
30636
34879
33800
37002
38441
34091
32521
39006
32361
31749
Product
15
32806
34340
31868
39369
39278
37790
34323
34944
30727
30804
36260
39451
30385
37079
34306
171
7
37662
39239
38141
35590
35489
34863
30413
35326
37825
35935
39156
31583
30374
37186
35108
34070
30586
30111
30685
30587
8
31101
39772
33837
32408
30192
33617
34187
38208
39405
33147
35440
33338
38966
36457
35305
33697
32259
31533
39135
30467
9
32217
36079
32264
35678
31833
34168
31836
36333
37803
36963
30387
32458
35394
38315
34629
36459
39648
37296
39086
33349
Coordinate
X
Y
-78.2 19.32
-16.6 -19.9
48.82 48.63
-35.3 96.09
31.94 -30.8
0.762 -23.9
1.361 92.58
79.7
65.81
-64.8 -80.1
-9.39 -73.5
-46.4 -79.1
-18.2 57.1
-9.07 58.85
90.87 56.02
91.78 -24
-3.55 -29.8
-65.2 -72
37.01 -51.3
3.158 -80.6
81.31 -95.1
16
32473
33722
38703
31741
34824
30960
30742
39551
32605
33891
31945
36110
39880
31973
35750
17
33446
30976
34648
36921
39630
32205
36467
31769
39043
38113
38484
32462
30103
35679
34581
18
33715
39138
35274
36307
30338
39758
30649
33535
35416
38479
34494
39612
34562
38583
30186
19
30082
36269
36575
37441
36396
32571
35987
34853
30420
31430
37536
34918
39031
32956
38728
20
34504
37752
38726
37316
37063
34109
38218
38133
38429
35244
34013
35872
34418
37363
37396
16
17
18
19
20
32950
35554
38636
33741
33340
32950
35554
38636
33741
33340
38494
31722
39955
36646
33466
37555
33732
36806
31100
34431
35048
30075
35118
36084
30170
35091
34020
30791
32684
33672
34542
31700
33671
30014
37624
32648
35126
34465
38994
30530
Table 5: Inventory Costs ($) per Product
Product
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Order
407
406
404
410
409
401
402
409
405
410
401
406
402
405
402
404
402
407
405
407
Holding
1.1289
1.2367
1.285
1.3902
1.0935
1.1165
1.0772
1.3168
1.3003
1.2428
1.0486
1.0155
1.1716
1.3925
1.4585
1.2066
1.4656
1.133
1.1227
1.2035
172
Shortage
45
41
43
43
45
44
43
45
44
43
42
44
41
41
43
43
44
44
42
44
Direct
0.0008
0.0008
0.0031
0.0027
0.0019
0.003
0.0023
0.0034
0.0036
0.0015
0.0007
0.0045
0.0005
0.0032
0.0048
0.003
0.0007
0.0031
0.0006
0.0046
38678
38471
30067
31047
33535
39680
31110
35718
31382
37189
32715
39595
37414
36867
34256