EMGT 6412/MATH 6665
Mathematical Programming
Spring 2016
Integrated Districting, Fleet
Composition, and Inventory
Control
This is a joint work with Dr. Joe Geunes, Professor at University of
Florida, Department of Industrial and Systems Engineering
Dincer Konur
Engineering Management and Systems
Engineering
1
Outline
•
•
•
•
Introduction and Motivation
Problem Definition
Literature Review
Problem Formulation
– Districting
– Fleet Composition
– Inventory Control
• Solution Analysis
– Branch and Price
• Numerical Studies
• Conclusions
2
Outline
•
•
•
•
Introduction and Motivation
Problem Definition
Literature Review
Problem Formulation
– Districting
– Fleet Composition
– Inventory Control
• Solution Analysis
– Branch and Price
• Numerical Studies
• Conclusions
3
Introduction and Motivation
•
An air-conditioning company in Florida operates as follows. The company has a set of retailers
throughout the region, and these retailers place orders for products from multiple suppliers via the
company's distribution department. Currently, the distribution department simply passes retailer
orders to suppliers who then ship to individual retailers
Long-haul
trucking costs
Supply
Region
Retailing
Region
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Introduction and Motivation
•
Our focus lies in the area of strategic planning for a multi-unit distribution system.
Speficically, we seek the best way to partition a set of geographically dispersed
retailers within a region into districts such that the inventory and transportation
operations of the retailers within the same district are coordinated.
–
–
•
Districting can roughly be defined as partitioning a service area into districts so that the overall
system efficiency can be increased by coordinating the operations within the districts (Miranda
et al., 2011).
Fleet composition problems determine which types of vehicles, as well as the number of each
vehicle type, to acquire in order to satisfy anticipated shipment requirements (see, e.g., Etezadi
and Beasley, 1983, Loxton et al., 2012, and the references therein).
Because a fleet's composition constrains inventory replenishment decisions, and
inventory management strategy influences fleet operations costs, we take an integrated
approach to fleet composition decisions by accounting for anticipated inventory-related
costs.
5
Problem Definition
•
•
•
Which retailers should be in the same district?
What should be the fleet composition dedicated to each retailer district?
What should be the inventory control decisions of the retailers within the same district?
Fleet
Composition
for District 4
District 2
District 3
District 1
Districting
Fleet
Composition
for District 3
Fleet
Composition
for District 1
Inventory Management:
Coordinated replenishment of
the retailers in the same district
6
Literature Review
• The paper is related to three topics:
– Districting: Districting has been integrated wit h various vehicle routing problems
(see, e.g., Ouyang, 2007, Haugland et al., 2007, Bard and Jarrah, 2009, Bard et
al., 2010, Lei et al., 2012, Carlsson and Delage, 2013, and Lei et al., 2015)
– Fleet composition: analyzed in many applications, specifically, in distribution
management. TL transportation in inventory control: single-unit inventory control
problems (see, e.g., Aucamp, 1982, Lee, 1986, Hwang et al., 1990, C etinkaya
and Lee, 2002, Lee et al., 2003, Toptal et al., 2003, Zhao et al., 2004, Toptal and
C etinkaya, 2006, Mendoza and Ventura, 2008, Toptal, 2009, Zhang et al., 2009,
Konur and Toptal, 2012, Konur and Schaefer, 2014), as well as multi-unit inventory
control problems (see, e.g., Ben-Khedher and Yano, 1994, Sindhuchao et al.,
2005, Kiesmuller, 2009, Gurler et al., 2014).
– Coordinated inventory control: shipment consolidation (see, e.g., Higgison,
1995, Higgison and Bookbinder, 1995, Cetinkaya and Lee, 2000, Cetinkaya and
Bookbinder, 2003), joint replenishment problem (Khouja and Goyal, 2008)
• This is strategic planning study
7
Outline
•
•
•
•
Introduction and Motivation
Problem Definition
Literature Review
Problem Formulation
– Districting
– Fleet Composition
– Inventory Control
• Solution Analysis
– Branch and Price
• Numerical Studies
• Conclusions
8
Problem Formulation
• A single retailer’s (retailer i among n retailers) inventory
control problem without explicit transportation costs:
– Parameters:
– Decision variables (continuous):
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Problem Formulation
• Forming districts can be beneficial:
– Coordinated inventory control of the selected suppliers
– Coordination implies that the retailers within the same district will
be shipped at the same time intervals
• So, their replenishment cycles are equal, but they might have
different shipment quantities considering individual demand, holding,
and shortage costs
• Note that there can be at most n district (each retailer by
himself) and there will be at least 1 district (all retailers
within the same district). Let districts be indexed by j.
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Problem Formulation
• Each district will have a dedicated truck fleet
– A fleet should be able to carry the district’s cumulative demand
– There are m different types of trucks indexed by k=1,2,…,m
– Each truck type has different capacity,
, and cost
• The problem is then to jointly determine
– which retailers should be in which districts,
– what should be the replenishment cycle for each district
considering the inventory related costs of the retailers it has, and
– what should be the truck fleet shipping to each district considering
the shipment requirements of the district
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Problem Formulation
• The total cost:
–
–
–
–
First term is the inventory cost
Second term is the shortage cost
Third term is the setup cost
Last term if the trucking cost
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Problem Formulation
• The model:
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Outline
•
•
•
•
Introduction and Motivation
Problem Definition
Literature Review
Problem Formulation
– Districting
– Fleet Composition
– Inventory Control
• Solution Analysis
– Branch and Price
• Numerical Studies
• Conclusions
14
Solution Analysis
• Given a district x:
– What is its replenishment cycle length
– What is its truck fleet
Mixed-integer-nonlinear
NP-hard
Branch and bound?
Heuristic?
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Solution Analysis
• Given a district x: Analysis of P-x:
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Solution Analysis
• Given a district x and the total number of trucks in
the district’s fleet:
Mixed-integer-nonlinear
NP-hard
Branch and bound?
Heuristic?
• Then
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Solution Analysis
• Given a district x, and its fleet composition z
• Corollary:
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Solution Analysis
• Using the properties, we have a neighborhood
search for
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Solution Analysis
• Efficiency of Algorithm 1
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Solution Analysis
• Recall that
• So, we can use Algorithm 1 over all possible total number
of trucks (we know the upper and lower bounds for the
total number of trucks!!)
• The method to solve P-x
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Solution Analysis
• Efficiency of Algorithm 2
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Solution Analysis
• So far, we can find a good fleet composition for a
given district of retailers considering their
coordinated inventory decisions
• How about districting? What districts to form?
• A set partitioning formulation:
– A set partitioning problem is to determine a partition (a
collection of subsets) of a given set of items such that
each item is included in one subset
• Set S={1,2,3,4,5}
• A partition P={{1,3},{2,5},{4}}
• Another partition P={{1,3,4}, {2,5}}
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Solution Analysis
• Note that each district corresponds to a subset of retailers, so the
districting problem is a set partitioning problem (a retailer cannot be in
two districts)
• So each district can be defined by a n binary vector
• There are
possible subsets (districts)
• Cost of a subset (district) will be achieved by solving the optimum
fleet and inventory decisions of the subset (i.e., we will need to solve
P-x for each given subset)
• Let
cost of district e
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Solution Analysis
• Set partitioning formulation:
• A large scale binary model
– And you need to calculate the cost of each column
(variable, or the district, a subset)
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Solution Analysis
• Branch and price: use column generation within
the branch and bound search
– Note that when you try to use branch and bound for
the set partitioning problem, the setting up the linear
relaxation is computationally burdensome
• You need to calculate each column and there are
exponentially many of them
– In a branch and price scheme, you solve the linear
relaxations at the nodes of the branch and bound tree
using column generation
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Solution Analysis
• Branch and price:
– Column generation for the nodes
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Solution Analysis
• Column generation for the nodes:
–
–
– But we still cannot calculate the reduced cost for each nonconsidered district in the RM-P
– So, let’s reformulate the pricing problem:
– Let
– Then the negative reduced cost will be:
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Solution Analysis
• Column generation for the nodes:
– The pricing problem:
Mixed-integer-nonlinear
NP-hard
Branch and bound?
Heuristic?
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Solution Analysis
• Pricing problem:
– One does not need to find the column with the most negative
reduced cost! Finding a column with negative reduced cost is
enough to say that current RM-P is not optimum for the full LR-P
– So, solve PP with heuristics?
• The whole branch and price will be heuristic
• A good thing:
– When the problem is hard to solve
– You will lean towards heuristic
– Before structuring the heuristic
• Analyze the problem
• Try to characterize properties (if any) of the optimum solution
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Solution Analysis
• Pricing problem:
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Solution Analysis
• Pricing problem:
– Given T, consider the linear relaxation of the PP
Given T, xi values will be binary when you solve the
linear relaxation of the PP!  Search over T?
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Solution Analysis
• Heuristic for the pricing problem:
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Solution Analysis
• Efficiency of Algorithm 3
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Solution Analysis
• Solving the pricing problem completes the branch
and price
• Implementation details:
– Finding a starting feasible set of columns (heuristic)
– Branching rule: branch on the binary variable
• A column to be used implies that some of the retailers are
already in the district, so other districts should only consider
the remaining retailers
• Hence, the problem size reduces as we go down on the
branch and bound tree
35
Outline
•
•
•
•
Introduction and Motivation
Problem Definition
Literature Review
Problem Formulation
– Districting
– Fleet Composition
– Inventory Control
• Solution Analysis
– Branch and Price
• Numerical Studies
• Conclusions
36
Numerical Studies
• Efficiencies of each heuristic
• Note
– When you have meta-heuristics, i.e., embedded
heuristics, or a heuristic which uses heuristics within
– Check the efficiency of each heuristic starting from the
most inner one
• So, we checked the efficiencies of Algorithms 1, 2, and 3
• Now, let’s check the efficiency of the branch-and-price
• Recall that our branch and price is also heuristic
– It can be made exact when you solve the pricing problem exactly
37
Numerical Studies
• Efficiency of the branch-and-price (small problems)
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Numerical Studies
• For larger problems, BARON takes long time
• Compare with evolutionary algorithm?
– What will be the evolutionary algorithm then?
• A n-vector of integers v=[v1,v2,…,vn] where 1<=vi<=n shows
the district retailer i belongs to. But how to calculate the cost of
a given v?
• Given v, we know x, so we need to solve P-x
• We already have an efficient algorithm for that, Algorithm 2
• So pick the top B% of the chromosomes to be the parents, who
we will use to generate the next population
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Numerical Studies
– What will be the evolutionary algorithm then?
• Generates the next population
– Make sure to include randomness to achieve diversity
– The idea is to generate a good population using the good
chromosomes we have (parent chromosomes)
– There are lots of lots of lots of mutation rules and cross-over
rules, pick the one fits you the best
» Consider that when you mutate, you do not want to end up
with a infeasible chromosome
» Avoid mutations that might require feasibility check?
• You need to stop search at some point!
• Non-improving generations, time, number of generations…40
Numerical Studies
• Efficiency of the branch-and-price (large problems)
41
Outline
•
•
•
•
Introduction and Motivation
Problem Definition
Literature Review
Problem Formulation
– Districting
– Fleet Composition
– Inventory Control
• Solution Analysis
– Branch and Price
• Numerical Studies
• Conclusions
42
Conclusions
• So, we have analyzed a practical problem
– Formulated it
– Tried to solve it:
•
•
•
•
Analyzed the subproblem properties
Used them to construct efficient solution methods
Used those within a branch and price and an evolutionary algorithm
Efficient solution methods at the end! Yay!
– On the way,
• we proved some important bounds on fleet size (can be used for
managerial decisions)
• We proved polynomial solvability of a special case of the problem,
which extends another problem
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Conclusions
• Questions? Comments?
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