2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
Modelling a merge-in supply network with K suppliers
and one (1) client with Coxian-2 replenishment times
1
Vidalis Michael, 2Marinagi Catherine, 3Vrisagotis Vassilios
1
Department of Business Administration University of Aegean
2
Department of Logistics, TEI of Chalkis, Thiva, Greece
3
Department of Business Administration University of Aegean1
[email protected], [email protected], [email protected]
1
Abstract
This paper deals with the analytical modelling of a dynamic supply system with two
stages (supplier client). The system has an arborescent structure. The behaviour of
the system involves inputs, processing and outputs of material, information, or data.
The replenishment times between the members are random and follow a Coxian-2
phase type distribution. The members of the supply system are functionally related.
Client and suppliers are assumed saturated. We use continuous time Markov
process with discrete space to model supply chain systems. The structure of the
transition matrices is examined. A computational algorithm is developed, which
allows the calculation of performance measures from the derivation of the steady
state probabilities. Among the metrics of performance of the system are metrics of
achieving customer service targets. These performance measures and the average
inventory (WIP) of the supply chain are examined as a function of system
characteristics. The contribution of this paper is the presentation of an exact
evaluative model able to calculate performance measures of the system. The basic
purpose of generative models, which are also optimization models, is the
determination of an optimal solution to the system parameters, given an overall
system structure and an objective function to be optimised.
Keywords: Two-echelon Supply network, performance measures, merge-in system,
Markov analysis.
1. Introduction
The complexity of the supply chain makes global formulation of the supply
chain problem very difficult. Therefore, for many years the research
community used to model smaller sections of the supply chain, such as
material procurement, production, marketing, or distribution. Later on, supply
chain planning models have been developed that integrate single components
into the overall supply chain (Pyke and Cohen, 1993; Nagar and Jain,
2008).Further, the traditional serial supply chains have been replaced by
complicated
networks
of
cargo
flows
strongly
resembled
to
telecommunications networks. Consequently the research in logistics field has
altered its focus from serial supply systems to non serial supply chains. On
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
this framework, we present an arborescent supply network as shown in the
following figure.
Figure 1 : An arborescent supply network
Basic feature of our model is the fact all the suppliers’ and distribution
centre operations are subjected to breakdowns which result in the
development of contingency plans.
In our paper, attention has been focused on two-echelon dynamic supply
systems with several (up to 3) suppliers and one buyer with no buffer An
arborescent system’s structure has been assumed, where the final stage is
connected with a number of downstream members. The replenishment lead
times between the members are random and follow a Coxian-2 phase type
distribution as in (Vidalis and Papadopoulos, 1999). Coxian-2 is suitable for
modelling machine breakdowns. The system is modelled as continuous time
Markov process with discrete space. The structure of the transition matrices of
these specific systems is examined and a computational algorithm is
developed to generate it for different values of system characteristics.
In the following we give the outline of our paper. In section 2 a literature
review is presented, in section 3 we describe the algorithm development and
the solution process, in section 4 we give the numerical results and finally in
section 5 we conclude our study and we propose further research topics.
2. Prior Research
The globalization of modern supply chains draws the interest of the
research community to multi-source inventory models. This allows considering
different options for inventory replenishments.
Supply chain systems are also distinguished to deterministic and
stochastic. Stochastic systems are used to model real-world supply networks,
since they involve many uncertainties, like demand, processing time, lead
times, price, etc. These uncertainties influence the performance of dynamic
supply chains. In order to control dynamic supply chain systems and their
uncertainties, there is a need to develop evaluative models to calculate
performance measures and optimization models to determine optimal
solutions.
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
Stochastic models have been proposed for uncertainties of dynamic
supply chain systems. For example, stochastic models for demand
uncertainty have been proposed in (Gupta and Maranas, 2003; Nagar and
Jain 2008; Bernstein and deCroix, 2006; Bogataj and Horvat, 1996). Other
researchers propose Markov chains to model supply chains. Pyke and Cohen
(1993) have developed a Markov chain model of a three-level productiondistribution system (a single station, a finished goods stockpile, a single
retailer). Further, a distribution-based methodology is used to reduce
computational complexity. Nagar and Jain (2008) have developed a two-stage
stochastic programming model, which is then extended to a multi-stage
stochastic programming model. They use scenario approach to address
supply chain planning under uncertain demand environment. Researchers
have also developed a sequential approach for obtaining state distribution for
random variables that determine system performance. Markovian analysis of
production lines where the service times at each station of the line follow the
Coxian-2 distribution has been proposed in (Vidalis and Papadopoulos, 1999).
Their proposed algorithm uses Coxian-2 distribution service times to calculate
the throughput rate of the production lines.
Our research involves stochastic replenishment lead times, which may
due to unpredictable congestion and disruption. Nowadays extensive global
chains are apt to such events (Sheffi, 2005). Kaplan (1970) concerned with
characterizing optimal policies for a dynamic inventory model with stochastic
lead times. Heijden, Diks, Kok (1999) focus on stochastic lead times in a
multi-echelon divergent system for inventory control. They have developed a
heuristic algorithm to compute control parameters i.e. the order-up-to-levels
and the parameters of the Balance Stock rationing rule defined in (Heijden,
Diks and Kok, 1997).They have measured the performance and accuracy of
this algorithm using numerical experiments. Boucherie, Heidevelt and van
Houtum (2007) consider stochastic lead times, which are independent and
identically distributed. Their model represents a multi-echelon network
consisting of a central warehouse and several local warehouses. Song and
Zipkin (2009) consider an inventory system with multiple supply sources and
Poisson demand, where the replenishment lead times for each source are
stochastic.
Arts and Kiesmuller (2010) state that, although situations with one buyer
and several supply options have become increasingly common in modern
supply networks, quantitative modelling approaches to analyze these supply
networks are limited.
The present paper is related to the work of Vidalis and Papadopoulos,
(1999) as far as methodology is concerned but it is an extension of the latter
paper since on our paper we deal with arborescent and not serial supply
network. The contribution of this paper is the presentation of an exact
evaluative model able to calculate performance measures of the system such
as throughput, the average inventory WIP, fill rate. The performance
measures of the supply network are examined as a function of system
characteristics such as number of suppliers (N), replenishment times(μ1,μ2),
probability that the shipment will arrive with no delay(d1) and probability that
the shipment will arrive with delay(d2).
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
3. Research Methodology
3.1 The model’s variables
The presented model refers to a production- inventory model with many (N=3)
suppliers – manufacturers and one (1) distribution center. The distribution
center receives shipments by the suppliers – manufacturers. The shipments
time intervals are random variables with high variability which are subjected to
unexpected events (such as strikes, adverse weather conditions,
transportation breakdowns etc.). In order to cope with these unexpected
events the administration of the distribution center has developed contingency
plans (the so called “plans B”). A fraction of the shipments routed to
distribution center d1, (0≤ d1≤1) needs a random time to be served,
exponentially distributed with rate μ1. The rest of the fraction, shipments are
executed according to contingency plans, d2 (0 ≤d2=1-d1≤1) faces an
additional time of delay, also exponentially distributed with rate μ2. Thus, the
total shipment time follows the Coxian distribution with two phases (Coxian-2)
(Vidalis et al, 2012).
Not only the distribution center face risk in shipments but the suppliers –
manufacturers face risk in their manufacturing operations as well .The
suppliers assembly the products which shipped to distribution center in their
premises. The assembly time intervals are random also variables with high
variability which are subjected to machines breakdowns. As a matter of fact, a
fraction of assembly operations dsupplier1 (0≤ dsupplier1≤1) needs a random time
to be fully completed exponentially distributed with rate μsupplier1 . The rest of
the fraction, assembly operations are subjected to breakdowns , dsupplier2 (0 ≤
dsupplier2=1- dsupplier1≤1) faces an additional time of delay, also exponentially
distributed with rate μsupplier2. Thus the total assembly time follows the Coxian
distribution with two phases. (Coxian 2).
The suppliers immediately after the completion ship the products to the
distribution center. Moreover, the suppliers receive raw materials at infinite
quantities. Consequently, the under study
manufacturing system is
characterized as push with arbores cent structure since we have more than
one suppliers (N>1).
In this paper the following notation has been adopted:
Ν : Νumber of suppliers
K1,2 : number of phases of Coxian phase type distribution
μ1 : mean shipment rate from the manufacturer to the distribution center
during the first phase
μ2 : mean replenishment rate from the manufacturer to the distribution center
during the second phase
d1: probability that the shipment will arrive having only one phase of delay,
which is the usual case
d2: probability that the shipment will arrive having two phases of delay, which
happens more rarely
μN1 : mean assembly rate of the manufacturer- supplier during the first phase
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
μN2 : mean assembly rate of the manufacturer-supplier during the second
phase
dN1: probability that the assembly operation will be completed having only one
phase of delay, which is the usual case
dN2: probability that the assembly operation will be completed having two
phases of delay, which happens more rarely.
FR: fill rate at the retailer
WIP: average inventory in the entire system
THR: average output rate of the system
The effectiveness of the system, the fill rate of the distribution center, is a
function of N, μ1,μ2, d1, d2, μN1, μN2, dN1, dN2. Our goal is to express fill rate ,
throughput , WIP and cycle time as functions of the abovementioned variables
.Thinking our algorithm as a system we give the below shape referring to
inputs, processes and outputs
Figure 2. The algorithm as a system
INPUTS PROCESSES OUTPUTS Ν
Queuing theory Transition Matrix μ1
Markov chain processes μ2
d1: Matrix Analytical Methods Vector of stationary possibilities d2
K1,2
μN1 μN2
dN1, dN1
WIP FILL RATE THROUGHPUT NUMERICAL RESULTS Our production and inventory network is analyzed as a continuous time
Markov process with finite number of states. The state of the system is
defined by the vector (supplier, phase, distribution center, phase) or
(pt,Nt,DCt)
pt : number of phases of shipment and assembly process at time t
Nt : Number of supplier manufacturing the products
DCt : Number of distribution center receiving shipments of products.
The vector of (pt,Nt,DCt) defines a continuous time Markov process on the
state space. {1,2,3,…,N}X{1,2} X {1}X{1,2}.
The state of the system alters when the following events took place;
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
•
Shipment arrival at distribution center that has faced no delay It reduce
by one the entities which are in the suppliers premises and it increases
by one the entities which are in the distribution center. The rate of the
shipments arrive with no delay is d1μ1
• Shipment arrival at distribution center that has executed according to
“plan B” DCt remains stable. The rate of the contingency plan is d2μ1
• Assembly completion at manufacturer N that has faced no delay It
reduce by one the entities which are in the suppliers premises and it
increases by one the entities which are in the distribution center. The
rate of the shipments arrive with no delay is dsupplier1μsupplier1
• Assembly completion at manufacturer N that has faced breakdown to
Nt remains stable. The rate of the contingency plan is
dsupplier2μsupplier1
• If assembly operation at manufacturer N is not completed and a new
entity arrives then the system is blocked. Assembly operation is
blocked with rate dsupplier1μsupplier1 since assembly operation faces
no delays
• Assembly operation at manufacturer N can also be blocked with rate
μsupplier2 if assembly operation is subjected to breakdowns.
The total number of states of the transition rate table is
SN,0,1 = (K1+1)N+1- Σ(K1+1)N-i2i-1
In case of two suppliers and one distribution center, each supplier sends its
finished entities one by one to the distribution center, which receives
automatically the entities with the modes regular and delayed one. In this
case the number of states is:
S2,0,1= (2+1)2+1-20(2+1)-21=22 (see Table 1 below).
(since we deal with Coxian with two phases we assume k1 constant and
k1=2).
Table 1. States for the model with two (2) suppliers and one (1) distribution centre
Supplier 1
Supplier 2
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
0
0
1
1
2
2
2
0
0
1
1
1
2
2
2
0
0
1
1
Distribution
center
0
1
2
0
1
2
1
2
0
1
2
0
1
2
1
2
1
2
State
Physical description
110
busy 1st phase,busy 1st
phase, idle
Busy1,busy1, busy1
Busy1,busy1,busy2
Busy1,busy1,idle
Busy1,busy1,busy1
Busy1,busy1, busy2
Busy1,blocked, busy1
Busy1,blocked, busy2
Busy2,busy1,idle
Busy2,busy1,busy1
Busy2,busy1,busy2
Busy2,busy2,idle
Busy2,busy2,busy1
Busy2,busy2,busy2
Busy2,blocked,busy1
Busy2,blocked,busy2
Blocked, busy1, busy1
Blocked,busy1, busy2
111
112
120
121
122
101
102
210
211
212
220
221
222
201
202
011
012
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
0
0
0
0
2
2
0
0
1
2
1
2
021
022
001
002
Blocked,busy2, busy1
Blocked,busy2,busy2
Blocked,blocked,busy1
Blocked,blocked,busy2
The stationary probabilities of the above mentioned Markov chain must
be calculated. The distribution of the stationary probabilities is the basis for
the system’s performance evaluation.
3.2. Structure of Transition Matrix
The structure of the transition matrix is affected by the number of suppliers (N)
and the phases of interval times of shipment for the distribution center (k1).
This matrix is a tri-diagonal matrix and consists of three sets of sub-matrices:
(i) the set of sub-matrices in the main diagonal, denoted by ∆κ, (ii) the set of
sub-matrices under the main diagonal, denoted by Κκ and (iii) the set of submatrices above the main diagonal, denoted by Ακ.
Figure 3. The general structure of the transition matrix
∆1
Α1
Α2
Κ1
∆2
Α3
Κ2
∆3
The dimensions of the submatrices:
•
Submatrices ∆1,∆2, : the dimensions is
2aN-1+ k2(k2+1)N-2 x2xaN-1+ k2(k2+1)N-2
•
Submatrix ∆3 : The dimensions is k2 (k2+1)N-1 x k2 (k2+1)N-1
•
Submatrix K1 : the dimensions is
2aN-1+ k2(k2+1)N-2 x2aN-1+ k2(k2+1)N-2
•
Submatrix K2 : The dimensions is k2 (k2+1)N-1 x 2aN-1+ k2(k2+1)N-2
•
Submatrix A1 : 2aN-1+ k2(k2+1)N-2 x2aN-1+ k2(k2+1)N-2
•
Submatrix A2 , A3 : The dimensions are
2xaN-1+ k2(k2+1)N-2 x k2 (k2+1)N-1
where aN-1 : the states of the transition matrix for the model with N-1 suppliers
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
All the submatrices (∆1,∆2,∆3,Α1,Α2,Α3,Κ1,Κ2) for the model with two
suppliers and one distribution center have the following structure forms:
Figure 4. Submatrix ∆1
-μ21-μ11
d1μ1
d11μ11
-μ21-μ11-μ1
μ2
d22μ21
d2μ1
-μ21-μ11μ2
d22μ21
d22μ21
-μ11μ22
μ22
d21μ21
d21μ21
d11μ11
-μ21-μ11μ1
d2μ1
-μ21-μ11μ2
d1μ1
d21μ21
μ22
-μ11-μ1
μ2
Figure 5. Submatrix ∆2
μ21μ12
d1μ
1
d11μ11
-μ21-μ12μ1
μ2
d22μ21
d2μ1
-μ21μ12-μ2
d11μ11
-μ21μ11-μ1
d2μ1
-μ21μ11-μ2
μ2
Figure 6. Submatrix ∆3
d2μ1
-μ21μ2
μ22
μ22
-μ11μ1
d1μ1
-μ21μ1
d21μ2
1
d22μ21
-μ12μ22
μ22
d21μ2
1
d22μ21
d22μ21
d21μ21
d22μ21
-μ22μ1
d2μ1
-μ22μ1
d21μ21
μ22
μ22
d2μ1
-μ11μ2
μ22
d2μ1
-μ11μ2
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
-μ1
d2μ1
-μ2
Figure 7. Submatrix A1
d12μ1
1
d12μ11
d12μ11
d12μ1
1
d12μ11
d12μ11
d12μ1
1
d12μ1
1
Figure 8. Submatrix A2
d11μ11
d11μ11
d11μ11
d11μ11
d11μ11
d11μ11
Figure 9. Sumbmatrix A3
μ12
μ12
μ12
μ12
μ12
μ12
Figure 10. Submatrix K1
μ12
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
μ12
Figure 11. Submatrix K2
d1μ1
μ2
d1μ1
μ2
d1μ1
μ2
The steps of the solution method for solving the system under
consideration are similar to those applied in (Papadopoulos, 1989;
Papadopoulos and O’Kelly, 1989; Papadopoulos, Heavey and O’Kelly, 1989a;
1989b; Heavey, Papadopoulos and Browne, 1993; Vidalis, 1998; Vidalis and
Papadopoulos, 1999; Vidalis and Papadopoulos, 2001). These steps are
described below:
Figure 12. Steps of the performance evaluation algorithm
Step 1: Calculate the dimension of the transition matrix.
Step 2: Generate the transition matrix:
Step 2.1: Generation of the sub-matrix ∆1,∆2 (Rule 0).
Step 2.2: Generation of the sub-matrices D3 (Rule 1).
Step 2.3: Generation of the sub-matrices Κ1, Κ2, (Rule 2).
Step 2.4: Generation of the sub-matrices Α1, Α2,Α3 (Rule 3).
Step 3: Calculation of the steady-state probability vector and then
computation of the selected performance measures (Rule 4).
3.3. Model equations
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
The number of model equations are affected by the number of suppliers (N),
the phases k1,k2 which are thought to be constant and equal to two (2). For
the case of two suppliers and k1=k2=2, the linear equations are given in the
following table:
Table 2. Linear equations for the state possibilities’ calculation model with two (2)
suppliers and one (1) distribution centre
State
Transition equation
No
equation
110
-(μ11+μ21)*π110+ d1μ1*π111+ μ2*π112=0
1
111
d11μ11*π110 -(μ11+μ21+μ1)*π111+ μ22*π120+ d1μ1*π101 + μ2*π102
2
+μ12*π011+ d1μ1*π 011+ μ2*π012=0
112
d2μ1*π 111 -(μ11+μ21+μ1)*π111 =0
3
120
d22μ21*π110 -(μ11+μ22)*π120 + d1μ1*π 121+ μ2*π122=0-
4
121
d22μ21*π110 +d11μ11*π120 –( μ11+μ21+μ1)*π121 + μ12*π220 +
5
d1μ1*π021+μ2 *π022=0
122
d22μ21*π111 + d2μ1*π 121 – (μ11+μ21+μ2)*π122 =0
6
101
d21μ21*π110 +d21μ21*π111 + μ22*π121-(-μ11+μ1)*π101 + d1μ1*π001+μ2
7
*π002=0
102
d21μ21*π112 +μ22*π101+ d2μ1*π 101 –(μ11+μ2)*π102 =0
8
210
d12μ11π110-(μ12+μ21)π211 +d1μ1*π211 + μ2*π212 =0
9
211
d12μ11π111+d21μ21*π211 –(μ12+μ21+μ1)*π211 +μ22*π220 + d1μ1*π201
10
+μ2*π202 =0
212
d12μ11*π112+d2μ1*π211 –(μ12+μ21+μ2)π212=0
11
220
d12μ11*π120 + d22μ21*π210 –(μ12+μ22)*π220 + d1μ1*π221 + μ2*π202 =0
12
221
d12μ11*π121 + d22μ21*π211 –(μ12+μ22+μ1)*π221 =0
13
222
d12μ11*π122 + d22μ21*π212 +d2μ1*π221 –(μ12+μ22+μ2)*π222 =0
14
201
d12μ11*π101 + d21μ21*π211 +μ22*π221 –(μ12+μ1)*π201 =0
15
202
d12μ11 *π102 + d21μ21*π212 +μ22*π222 +d2μ1*π201 –(μ12+μ2)*π202 =0
16
011
d11μ11*π111 + μ12 *π211– (μ21+μ1)*π011 =0
17
012
d11μ11*π112 + μ12*π212 +d2μ1*π011 –(μ21+μ2)*π012 =0
18
021
d11μ11*π121 + μ12*π221 +d22μ21*π011 –(μ22+μ1)*π021 =0
19
022
d11μ11*π122 + μ12*π222 + d22μ21*π012 +d2μ1*π021 -(μ21+μ2)*π022=0
20
001
d11μ11*π101 + μ12*π201 + μ22*π021 –μ1*π001 =0
21
002
d11μ11*π102 + μ12*π202 + μ22*π022 + d2μ1*π001 –μ22*π002 =0
22
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
3.4. Performance measures
The most important performance measures are the average invetory, the
average flow time, the fill rate and the mean output rate or throughput of the
system.
The average inventory (or cycle inventory) is the mean number of units
in the supply network and is calculated according t the following formula:
SN,0,1
SN,0,1
WIP = NX Σ1 πο +(Ν+1) (1 - Σ1 πο)
where N : number of suppliers
Σπο : sum of state probabilities where the Distribution center is idle
SN,0,1 : number of states for a supply network with N suppliers and 0 buffer
and one (1) customers
Example : for N=2 the WIP is calculated as following:
WIP System = 2(π110 +π120 +π210+π220) +3[1- (π110 +π120 +π210+π220)]
Further, the fill rate is calculated according to the following formula:
FR = d1 (state probabilities corresponding to the phase 1 of customer) +(state
probabilities corresponding to the phase 2 of customer
Alternatively the fill rate can also be calculated:
FRN = (FiII RateN-1 for probabilities 1 to 2an-1 + k2(k2+1)N-2 ) + (FiII RateN-1 for
probabilities 2an-1 + k2(k2+1)N-2 to k1(2an-1 + k2(k2+1)N-2) +
d1 Σ1 k2 (k2+1)N-1 πphase1 + Σ 1 k2 (k2+1)N-1 πphase2
Example
Fill rate 101 = d1(π11 +π21) + (π12 +π22+) + d1π01 + π02
Fill rate 201 = Fill rate 101 + Fill Rate101 + d1(π011+π021 +π001) + (π012+π022
+π002)
The throughput is calculated according to the below mentioned
THROUGHPUT = d1μ1 + μ2
Last but not least the average flow time or cycle time is given according to the
known formula:
CYCLE TIME = WIPsystem / THROUGHPUT
4. Analysis
On this section we present an number of numerical results on which the
performance measures of the supply network are examined as a function of
system characteristics. In more detail, we examine:
• The behavior of throughput in relation with replenishment times with
μ1,μ2 and d1,d2.
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•
•
•
•
4.1
The behavior of WIP (Average Inventory) in relation with the number of
suppliers (N)
The behavior of WIP (Average Inventory) in relation with replenishment
times with μ1,μ2 and d1,d2.
The behavior of Fill rate in relation with the number of suppliers (N)
The behavior of Fill rate in relation with replenishment times with μ1,μ2
and d1,d2
Throughput in relation d1,d2
From the below diagram we can argue that as d1 increases the throughput is
also increases linearly. This finding is rational since the rate of outbound
shipments the distribution centre (throughput) will increase as the inbound
shipments arrive in this with no delay.
Figure 13. Throughput and d1
4
3.9
3.8
Throughput
3.7
3.6
3.5
3.4
3.3
3.2
3.1
0.1
4.2
0.2
0.3
0.4
0.5
0.6
d1 Customer
0.7
0.8
0.9
1
Throughput in relation μ1,μ2
From the below diagram we can argue that as μ1, μ2 increases the throughput
is also increases linearly. This finding is rational since the rate of outbound
shipments the distribution centre (throughput) will increases as the
replenishment times of inbound shipments arriving with or no delay increases.
Figure 14. Throughput and μ1 μ2
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6
5.5
5
Throughput
4.5
4
3.5
3
2.5
2
1.5
1
1
2
3
4
5
6
6
5
3
2
1
m1 Customer
m2 Customer
4.3.
4
WIP (average inventory) in relation to N (number of suppliers)
From the below diagram it is clear that as the number of suppliers increases
the average inventory increases. The argument is thoroughly rational since as
the suppliers who ship their products to the distribution centre increase the
quantity of inventory in the distribution centre increases linearly on our
example.
Figure 15. WIP and number of suppliers (N)
4
3.5
WIP
3
2.5
2
1.5
1
1.2
1.4
1.6
1.8
2
2.2
Number of Suppliers
2.4
4.4. WIP (average inventory) in relation to μ1 ,μ2
2.6
2.8
3
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
From the below figure it is easily derived that as the replenishment times
with and no delay μ1 μ2 decrease the average inventory (WIP) smoothly
increases .
Figure 16. WIP and μ1, μ2
4.02
4
3.98
WIP
3.96
3.94
3.92
3.9
3.88
1
2
3
4
5
6
4
6
m1 Customer
m2 Customer
4.5.
0
2
Fill rate in relation to N (Number of suppliers)
From the below diagram it is clear that as the number of suppliers increases
the fill rate increases. The argument is rational since as the suppliers who
ship their products to the distribution centre increase the percentage of
demand orders covered by the distribution centre increases almost linearly,
on our model.
Figure 17. Fill rate and N (number of suppliers)
0.85
0.8
0.75
Fillrate
0.7
0.65
0.6
0.55
0.5
1
1.2
1.4
1.6
1.8
2
2.2
Number of Suppliers
2.4
2.6
2.8
3
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
4.6.
Fill rate in relation to μ1 μ2
From the below diagram it is clear that as μ1 decreases and ,μ2 increases fill
rate increases and reaches its maximum.
Figure 18. Fill rate and μ1 μ2
0.9
0.85
Fillarte
0.8
0.75
0.7
0.65
1
2
3
4
5
6
0
2
4
6
m2 Customer
m1 Customer
4.7.
Fill rate in relation to d1
From the below figure we can see that as the fraction of replenishment orders
with no delay increase the percentage of demand covered increases. This is
rational because the distribution centre will rarely be out of stock since the
orders wiil arrive on time and with no delay.
Figure 19. Fill rate and d1
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
1
0.95
0.9
0.85
Fillrate
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.1
0.2
0.3
0.4
0.5
0.6
d1 Customer
0.7
0.8
0.9
1
5. Conclusions and Further Research
Concluding, on our paper we presented an arborescent supply network. In
order to examine its operation we modeled the system as queuing network
subjected to breakdowns and we used Coxian-2 phase type distribution. Also
we proceeded with the configuration of transition matrices and with the
calculation of steady state probabilities’ vector. Last we presented the
behavior of system characteristics – performance measures in relation to a
variety of variables. For further research, we propose the development of
arborescent supply network with buffer or the development of a model with
many suppliers, buffer and with many customers.
References
Arts J.J., and Kiesmuller G.P. (2010), “Analysis of a two-echelon inventory system
with two supply modes”. (BETA publicatie : working papers, No. 339).
Technische
Universiteit
Eindhoven,
available
at:
http://cms.ieis.tue.nl/Beta/Files/WorkingPapers/ wp_339.pdf
Bernstein F., and deCroix G.A. (2006), “Inventory Policies in a Decentralized
Assembly System Operations Research”, Vol. 54, No. 2, pp. 324-336.
Bogataj L., and Horvsat L. (1996), “Stochastic considerations of GrubbstromMolinder model of MRP, input-output and multi-echelon inventory systems”,
International Journal of Production Economics, Vol. 45, pp. 329-336.
Boucherie, R.B., Heidevelt M.C., and van Houtum G. (2007), “A product-form solution
for two-echelon spare parts networks with emergency shipments”, Working
paper, Technische Universiteit Eindhoven, Eindhoven, NL.
Gupta A., and Maranas C.D. (2003), “Managing demand uncertainty in supply chain
planning”, Computers and Chemical Engineering, Vol. 27, No. 8, pp. 1219-27.
2nd INTERNATIONAL CONFERENCE ON SUPPLY CHAINS
Heavey C., Papadopoulos H.T., and Browne J. (1993), “The Throughput Rate of
Multi station Unreliable Production Lines”, European Journal of Operational
Research, Vol. 68, pp. 69-89.
Heijden M.C., Van der Diks E.B., and de Kok A.G. (1999), “Inventory control in multiechelon divergent systems with random lead times”, Operational Research
(OR) Spektrum 21, pp. 331-359.
Heijden M.C., van der, Diks E.B., de Kok A.G. (1997), “Allocation policies in general
multi-echelon distribution systems with (R; S) order-up-to policies”,
International Journal of Production Economics, Vol. 49, No. 2, pp,157–174.
Kaplan R. (1970), “Dynamic Inventory Model with Stochastic Lead Times”,
Management Science, Vol. 16, No. 7, Theory Series, pp. 491-507.
Nagar L., and Jain K. (2008), “Supply chain planning using multi-stage stochastic
programming”, Supply Chain Management: An International Journal, Vol. 13,
Iss. 3, pp. 251- 256.
Papadopoulos H.T. (1989), “Mathematical Modelling of Reliable Production Lines”,
Ph.D. Thesis in Industrial Engineering/Operations Research, Department of
Industrial Engineering, National University of Ireland, Galway, Ireland.
Papadopoulos, H.T., and O’ Kelly M.E.J. (1989), “A Recursive Algorithm for
Generating the Transition Matrix of Multistation Series Production Lines”,
Computers in Industry, Vol. 12, pp. 227-240.
Papadopoulos H.T., Heavey C. and O’ Kelly M.E.J. (1989a), “Throughput Rate of
Multistation Reliable Production Lines with inter station buffers (I) Exponential
Case”, Computers in Industry, Vol. 13, pp. 229-244.
Papadopoulos H.T., Heavey C., and O’ Kelly M.E.J. (1989b), “Throughput Rate of
Multistation Reliable Production Lines with inter station buffers (II) Erlang
Case”, Computers in Industry, Vol. 13, pp. 317-335.
Papadopoulos H.T., Heavey C., and Browne L. (1993), Queuing Theory in
Manufacturing Systems Analysis and Design, Chapman and Hall.
Pyke D.,E., and Cohen M.A., (1993), “Performance characteristics of stochastic
integrated production-distribution systems”, European Journal of Operational
Research, Vol. 68, pp. 23-48.
Sheffi Y. (2005), The Resilient Enterprise, MIT Press, Cambridge, MA.
Song J.S., and Zipkin P. (2009), “Inventories with multiple supply sources and
networks of queues with overflow bypasses”, Management Science, Vol. 55,
No. 3, pp.362-372.
Vidalis M.I. (1998), “Performance Evaluation and Optimal Buffer Allocation in Serial
Production Lines”, Ph.D. Thesis in Operations Research, Department of
Mathematics, University of the Aegean, Samos island, Greece.
Vidalis M.I., and Papadopoulos H.T. (1999), “Markovian Analysis of production Lines
with Coxian-2 service times”, International Transactions in Οperations
Research, Vol. 6, pp. 495-524.
Vidalis M.I., and Papadopoulos H.T. (2001), “A Recursive Algorithm for Generating
the Transition Matrices of Multistation Multiserver Exponential Reliable
Queueing Networks”, Computers & Operations Research, Vol. 28, No 9,
pp.853–883.