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The Pennsylvania State University
The Graduate School
Harold and Inge Marcus Department of Industrial and Manufacturing Engineering
MULTIPLE CRITERIA SUPPLY CHAIN INVENTORY MODELS
WITH STOCHASTIC DEMANDS
A Thesis in
Industrial Engineering
by
Yang Jin
2013 Yang Jin
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
August 2013
ii
The thesis of Yang Jin was reviewed and approved* by the following:
A. Ravi Ravindran
Professor of Industrial Engineering, Chair-Enterprise Integration Consortium
Thesis Advisor
Chia-Jung Chang
Assistant Professor of Industrial Engineering
Thesis Reader
Paul Griffin
Professor of Industrial Engineering
Head of the Harold and Inge Marcus Department of Industrial and Manufacturing
Engineering
*Signatures are on file in the Graduate School
iii
ABSTRACT
This thesis considers the use of multiple criteria optimization methods applied to the
supply chain inventory problem with stochastic demand. The supply chain system is modeled
with a single wholesaler and multiple retailers. The demand of the customer is met by the retailers
and the retailers are independent from each other and see independent customer demands. The
wholesaler meets the demand of all the retailers.
Two types of supply chains are examined in this thesis, the de-centralized supply chain
system and the centralized supply chain system. In the de-centralized case, we assume that each
decision maker is responsible for setting the inventory policies for each company in the supply
chain system. In the centralized case, we assume that the decision maker is responsible for setting
the inventory policies for the entire supply chain system.
We study this supply chain problem with three objectives under stochastic and stationary
demand. The demands of the customer follow normal distributions. The criteria considered are
(1) capital invested in inventory; (2) annual number of orders; (3) customer service level. We first
develop the formulas for all the three objectives and then provide general models for the supply
chain system for both de-centralized and centralized cases.
Two methods are used to solve this multiple criteria mathematical programming problem.
The first method uses the weighted objective method. Interval programming is used to for
generating the weights. The second method uses the compromise programming method. Several
efficient points are generated by these two methods and presented to the decision maker. In the
de-centralized case, the retailer problems are solved first using both methods. We then solve the
wholesaler problem using both methods based on the results of the retailer problems. In the
centralized case, two scenarios are considered: common replenishment epochs and different
replenishment epochs. Both scenarios are also solved using the weighted objective method and
iv
the compromise programming method. Numerical examples are provided for each case to
illustrate the methods.
Finally, we compare the results of the common replenishment epochs with those of the
different replenishment epochs. We also compare the results of the de-centralized supply chain
system with those of the centralized supply chain system. We then draw conclusions based on the
comparisons of the results.
v
Table of Contents
List of Figures ................................................................................................................................ vii
List of Tables ................................................................................................................................ viii
Acknowledgement ........................................................................................................................... x
1 Introduction .............................................................................................................................. 1
1.1 Supply Chain System ....................................................................................................... 1
1.2 Inventory Policies ............................................................................................................ 3
1.3 Problem Statement ........................................................................................................... 4
1.4 Thesis Outline .................................................................................................................. 5
2 Literature Review ..................................................................................................................... 6
2.1 Multi Criteria Optimization ............................................................................................. 6
2.1.1 Problem Description ................................................................................................ 6
2.1.2 Definitions ................................................................................................................ 6
2.1.3 Methods of Solving .................................................................................................. 7
2.2 Inventory Control ............................................................................................................. 9
2.3 Multiple Criteria Inventory Control ............................................................................... 13
3 Problem Description .............................................................................................................. 20
3.1 Criteria ........................................................................................................................... 21
3.2 Mathematical Model ...................................................................................................... 21
3.2.1 Notations ................................................................................................................ 21
3.2.2 Demand during Cycle Time plus Lead Time at the Retailers ................................ 22
3.2.3 Capital Invested in Inventory at the Retailers ........................................................ 23
3.2.4 Capital Invested in Inventory at the Wholesaler .................................................... 24
3.2.5 Order Frequency .................................................................................................... 31
3.2.6 Service Level ......................................................................................................... 32
3.3 General Model for De-centralized Supply Chain System .............................................. 33
3.4 General Model for Centralized Supply Chain System ................................................... 34
3.5 Solution Methods ........................................................................................................... 35
3.5.1 Weighted Objective Method .................................................................................. 35
3.5.2 Compromise Programming Method (Zeleny, 1982) .............................................. 38
4 Single Wholesaler, Two Retailers De-centralized Supply Chain .......................................... 42
vi
4.1 Supply Chain Model ...................................................................................................... 43
4.1.1 Assumptions ........................................................................................................... 43
4.1.2 Notations ................................................................................................................ 43
4.1.3 General Model of Decentralized Supply Chain ..................................................... 44
4.2 Retailer Problem ............................................................................................................ 46
4.2.1 Solution Procedure ................................................................................................. 46
4.2.2 Numerical Example for Retailer Problems ............................................................ 48
4.3 Wholesaler Problem ....................................................................................................... 56
4.3.1 Solution Procedure ................................................................................................. 57
4.3.2 Numerical Example for the Wholesaler Problem .................................................. 58
4.4 More Than Two Retailers .............................................................................................. 65
5 Single Wholesaler, Two Retailers Centralized System Problem ........................................... 66
5.1 Common Replenishment Epochs ................................................................................... 67
5.1.1 Assumptions ........................................................................................................... 68
5.1.2 Solution Procedure ................................................................................................. 70
5.1.3 Numerical Example for Common Replenishment Epochs .................................... 70
5.2 Different Replenishment Epochs ................................................................................... 77
5.2.1 Assumptions ........................................................................................................... 78
5.2.2 Solution Procedure ................................................................................................. 79
5.2.3 Numerical Example for Different Replenishment Epochs ..................................... 80
6 Conclusions ............................................................................................................................ 88
6.1 Policy Comparisons ....................................................................................................... 89
6.1.1 Common Replenishment Epochs vs. Different Replenishment Epochs ................ 89
6.1.2 Decentralized vs. Centralized ................................................................................ 91
6.2 Future Work ................................................................................................................... 93
References ...................................................................................................................................... 94
vii
List of Figures
Figure 1.1 Serial Supply Chain with Five Members ........................................................................ 1
Figure 1.2 Supply Chain Network ................................................................................................... 2
Figure 1.3 Two-stage Supply Chain Network ................................................................................. 5
Figure 3.1 Two-stage Supply Chain Network ............................................................................... 21
Figure 3.2 Inventory Position for the Retailers .............................................................................. 23
Figure 3.3 Inventory Position for Retailers and Wholesaler in the Example ................................. 26
Figure 3.4 Inventory Level at the Wholesaler in the example ....................................................... 30
Figure 3.5 Parametric Space for Three Criteria ............................................................................. 37
Figure 4.1 Supply Chain System of a Single Wholesaler and Two Retailers ................................ 42
viii
List of Tables
Table 2.1 Comparison of Literature in Multi-criteria Inventory Control ....................................... 19
Table 3.1 Initial Weights for the Objectives Using Interval Programming ................................... 37
Table 3.2 An Example Illustrating the Outlier Problem when p=∞ .............................................. 40
Table 4.1 Efficient Points Generated by Weighted Objective Method .......................................... 47
Table 4.2 Efficient Points Generated by Compromise Programming Method .............................. 48
Table 4.3 Retailer Data Used in the Numerical Example .............................................................. 48
Table 4.4 Weights Used in the Weighted Objective Method......................................................... 49
Table 4.5 Values of Each Objective Using Weighted Objective Method for Retailer 1 ............... 50
Table 4.6 Values of Decision Variables using Weighted Objective Method for Retailer 1 .......... 51
Table 4.7 Ideal Solutions for Retailer 1 ......................................................................................... 52
Table 4.8 Weights for Each Objective for Retailer 1 ..................................................................... 52
Table 4.9 Values of Each Objective for Retailer 1 ........................................................................ 53
Table 4.10 Values of Decision Variables for Retailer 1 ................................................................ 53
Table 4.11 Ten Efficient Points Presented to the Decision Maker for Retailer 1 .......................... 53
Table 4.12 Values of the Decision Variables for Retailer 1 .......................................................... 54
Table 4.13 Weights for Each Objective for Retailer 2 ................................................................... 54
Table 4.14 Values of Each Objective for Retailer 2 ...................................................................... 55
Table 4.15 Values of Decision Variables for Retailer 2 ................................................................ 55
Table 4.16 Values Used in the Numerical Example of Wholesaler Problem ................................ 58
Table 4.17 Solutions to the Retailer Problem Chosen by the Decision Makers ............................ 59
Table 4.18 Objective Weights for the Weighted Objective Method .............................................. 60
Table 4.19 Values of Each Objective using Weighted Objective Method for the Wholesaler ...... 61
Table 4.20 Values of Decision Variables Using Weighted Objective Method for the
Wholesaler .................................................................................................................. 61
Table 4.21 Ideal Solutions for the Wholesaler ............................................................................... 62
Table 4.22 Weights for Each Objective for the Wholesaler .......................................................... 63
Table 4.23 Values of Each Objective in the Wholesaler Problem ................................................. 63
Table 4.24 Values of Decision Variables in the Wholesaler Problem ........................................... 64
Table 4.25 Seven Efficient Points Presented to the Decision Maker for the Wholesaler .............. 64
Table 4.26 Values of the Decision Variables for the Wholesaler .................................................. 65
Table 5.1 Retailers Data Used in the Numerical Example of Centralized Supply Chain .............. 71
ix
Table 5.2 Wholesaler Data Used in the Numerical Example of Centralized Supply Chain .......... 71
Table 5.3 Values of Weights Using the Weighted Objective Method ........................................... 71
Table 5.4 Values of Each Objective Using Weighted Objective Method for the Supply Chain ... 73
Table 5.5 Values of Decision Variables using Weighted Objective Method for the
Supply Chain ................................................................................................................. 73
Table 5.6 Ideal Solutions for Each Objective in the Entire Supply Chain ..................................... 74
Table 5.7 Weights for Each Objective for the Supply Chain ......................................................... 74
Table 5.8 Values of Each Objective Using Common Replenishment Epochs............................... 75
Table 5.9 Values of Decision Variables Using Common Replenishment Epochs ......................... 76
Table 5.10 Ten Efficient Points Presented to the Decision Maker for the Supply Chain .............. 76
Table 5.11 Values of the Decision Variables for the Supply Chain .............................................. 77
Table 5.12 Retailers Data Used in the Numerical Example of Centralized Supply Chain ............ 80
Table 5.13 Wholesaler Data Used in the Numerical Example of Centralized Supply Chain ........ 80
Table 5.14 Weights for the Objectives Using Weighted Objectives Method ................................ 80
Table 5.15 Values of Each Objective Using Weighted Objective Method for the Supply Chain . 82
Table 5.16 Values of Decision Variables using Weighted Objective Method for the
Supply Chain .............................................................................................................. 82
Table 5.17 Ideal Solutions for Each Objective for Different Replenishment Epochs ................... 83
Table 5.18 Weights for Each Objective for the Supply Chain ....................................................... 84
Table 5.19 Values of Each Objective Using Different Replenishment Epochs ............................. 85
Table 5.20 Values of Decision Variables Using Different Replenishment Epochs ....................... 85
Table 5.21 Nine Efficient Points Presented to the Decision Maker for the Supply Chain ............ 86
Table 5.22 Values of the Decision Variables for the Supply Chain .............................................. 87
Table 6.1 Values of Each Objective Using Common Replenishment Epochs............................... 89
Table 6.2 Values of Each Objective Using Different Replenishment Epochs ............................... 90
Table 6.3 Objective Values for Each Company and for the Entire Supply Chain in
Decentralized Case ........................................................................................................ 92
Table 6.4 Comparison of Decentralized System with Centralized System ................................... 92
x
Acknowledgement
I would like to thank Dr. Ravindran for his excellent guidance, patience and advice throughout
the writing of my thesis. Through many discussions with him, I have learned a lot. I also would
like to thank Dr. Chang for her valuable advice.
I would like to thank my family for their support in my graduate studies. I also would like to
thank Xiran. She encourages me all the time.
I would like to thank Yangshen He, Yining Chen and Xuebin Huang for their friendship and
support that they have given to me throughout the writing of this thesis.
1
1 Introduction
A supply chain system is involved in activities related to delivering products from
supplier to customer. High efficiency and high responsiveness are two conflicting objectives for a
supply chain system. Efficiency is closely related to the minimization of the cost in the supply
chain. Common efficiency measures include: manufacturing costs, distribution costs, inventory
holding costs, transportation costs, and shortage costs. Responsiveness is related to the ability of
satisfying customer demand. Common measures of responsiveness include: reliability of fulfilling
customer orders, delivery time and percent of demand filled from inventory. Inventory is
necessary for industry today because it helps provide the customers with continuous supply and
increase the responsiveness of the supply chain. However, higher inventory level also means
higher cost due to the capital invested in the inventory. Therefore, the balance between efficiency
and responsiveness is highly related to inventory control and policies to control inventory become
essential to achieve success in business.
1.1 Supply Chain System
A supply chain system is comprised of several companies that are necessary to move
products to the customers. If the companies in a supply chain are in series, the supply chain can
be considered as a serial supply chain. Figure 1.1 is an example of a typical serial supply chain
with five members (stages).
Supplier Manufacturer Distributor Wholesaler Retailer
Figure 1.1 Serial Supply Chain with Five Members
The raw materials are provided by the supplier and are delivered to the manufacturer.
After the manufacturing process, the finished product leaves the manufacturer and is delivered to
the distributor, and then to the wholesaler. Finally the product goes to the retailer for sale to the
2
customer. In a supply chain network, there can be more than one company in each stage. Figure
1.2 is an example of a supply chain network.
Supplier
Distributor
Wholesaler
Retailer
Manufacturer Distributor
Wholesaler
Retailer
Manufacturer
Wholesaler
Retailer
Retailer
Figure 1.2 Supply Chain Network
In both serial supply chain and supply chain network, the demand is driven by the
customer. The flows of the products follow the arrows in the above figures.
A supply chain can also be classified as centralized or de-centralized. In a centralized
supply chain, one company controls the entire supply chain and makes decisions such that the
entire supply chain will be optimized. A centralized supply chain is present where one company
owns the whole supply chain network including the transportation between stages.
In a de-centralized supply chain, which is more common in practice, the companies at
each stage have their own objectives and the decision maker in each company will make
decisions in order to achieve them. The decisions of each company are independent from other
members in the supply chain. Often the objectives of different companies conflict with one
another. Hence, the optimization of the entire supply chain is difficult when different companies
are just optimizing their own objectives.
3
1.2 Inventory Policies
In the area of inventory control, the most basic model is the economic order quantity
(EOQ) model. The decision variable of the EOQ model is the order quantity and the objective is
to minimize the total cost for a company where the demand is deterministic and shortage is not
allowed. Based on the basic EOQ model, models considering shortages or production period have
also been proposed and studied.
When the demand is stochastic, there are two main types of inventory control policies:
continuous review and periodic review policies. In a continuous review policy, reorder point-
order quantity policy is widely used and is called model. In this model, when the inventory
level is below , an order will be placed and the order quantity is . Therefore, the time intervals
between orders are not always the same while the order quantity is always the same. In a periodic
review policy, policy is widely used in practice. It is a reorder point – order-up-to
system under periodic review with period length . Every time units, review the inventory
position , and if , order a sufficient amount to bring the inventory level back to the order-
up-to level . The order quantity . One variation of the periodic review policy is called
the policy. This policy considers the case where the ordering cost is sufficiently small that
we can essentially be assured that the optimal behavior is to order at each review period
(Ravindran and Warsing, 2012). The orders are made at a fixed time interval and is the order-
up-to level. Therefore, in the policy, the time interval between orders is always the same,
while the order quantity can vary every time based on the on-hand inventory level at the time of
review. In this thesis, the policy will be used.
Multi-criteria optimization problems are very common in many industries. In practice,
often the objectives that we are trying to achieve are conflicting with each other. As mentioned
above, the efficiency and the responsiveness of a supply chain are two conflicting objectives.
Having larger inventory of several different products will result in the increase of customer
4
responsiveness, but this will also increase the inventory cost and reduce the efficiency. Similarly,
fewer warehouses can reduce the facility costs and the overall inventory level while the
responsiveness of the supply chain decreases due to the longer transportation time. Ravindran and
Warsing (2012) point out that supply chain optimization problems are generally multiple criteria
optimization models. Multi-criteria optimization methods should be used to solve this type of
problem.
There are three types of approaches that can be used to solve the multi-criteria
mathematical programming problem. The first approach does not need to use the preference
information from the decision maker. The second approach needs the input of preferences from
the decision maker. The third approach is known as the interactive method. Hence, the preference
information from the decision maker is progressively used and the decision maker is involved in
every step. This approach is more preferred in practice.
1.3 Problem Statement
In multi-echelon supply chain inventory control, the research that has been done so far
mainly focused on the centralized supply chain with deterministic demand. In practice, de-
centralized supply chain is more common. Every member of the supply chain system has its own
decision maker and makes its own decisions. Every decision maker only focuses on the
optimization of their own criteria. Furthermore, the stochastic demand is closer to real-world
scenario compared with deterministic demand. This thesis will consider the supply chain system
with stochastic demand while studying both de-centralized and centralized cases. Figure 1.3
illustrates the two-stage supply chain network considered in this thesis.
5
Wholesaler
Retailer
Retailer
Retailer
.
.
.
Figure 1.3 Two-stage Supply Chain Network
We will consider a two-stage supply chain problem in series with one wholesaler and
multiple retailers. The demand of the customers is stochastic and stationary. There are three
conflicting objectives: 1) capital invested in inventory, 2) number of orders, and 3) customer
service level. inventory policy will be used in this thesis and a multi-criteria optimization
model will be presented to determine the optimal order-up-to level and order interval for all
members in the supply chain system.
1.4 Thesis Outline
Chapter 2 consists of the literature review of inventory control, multi-criteria inventory control
and multi-criteria optimization. Chapter 3 introduces the general multi-criteria optimization
models for both de-centralized and centralized systems. Chapter 4 presents an optimization model
and the solution procedure for a de-centralized system with one wholesaler and two retailers. We
will then extend the model to multiple retailers. Chapter 5 presents an optimization model and the
solution procedure for the centralized system with one wholesaler and two retailers. Two types of
the replenish time of the retailers will be included. Chapter 6 consists of conclusions and future
research.
6
2 Literature Review
2.1 Multi Criteria Optimization
Multiple criteria optimization methods are used to solve problems involving multiple
conflicting objectives. The objective is to find the best compromise solution that maximizes the
decision maker’s preference. A review of problem description, definitions and solution methods
are presented below.
2.1.1 Problem Description
The multi-criteria mathematical programming problem (MCMP) can be written as
follows:
{ }
where is an n-dimensional vector of decision variable, is the decision space and is a
vector of real-value functions (Shin and Ravindran, 1991).
2.1.2 Definitions
Ideal Solution
The optimal solution that can maximize all objectives at the same time is not achievable
in the multi-criteria optimization problem since the criteria are conflicting with each other. When
the problem is a maximization problem, the ideal solution is the individual maxima (minimum for
minimization problem) for each objective while ignoring other objectives.
Let be the solution by solving the maximization problem.
Then the solution
is called the ideal solution.
7
Efficient/Non-dominated Solution
An efficient solution is a feasible solution where one objective cannot be improved
without sacrificing on at least one other objective.
is said to be properly efficient of the multi-criteria problem if it is efficient and there
exists a value such that
for some such that whenever and
.
Dominated Solution
A dominated solution is a feasible solution where at least one objective can be improved
without sacrificing on any other objectives.
2.1.3 Methods of Solving
There exist several efficient solutions for a multiple criteria decision making (MCDM)
problem. The set of all efficient solutions is called the efficient frontier. The decision maker
needs to choose the best efficient solution according to his/her preference. There are many
different methods to solve a multiple criteria optimization problem.
Best Compromise Solution
Let the decision maker’s preference function be . Then the multi-criteria optimization
problem reduces to
( )
The best compromise solution is the efficient solution that maximizes the decision
maker’s utility function, , which is not explicitly known.
8
Keeny and Raiffa (1976) present the methods to model the utility functions. However, it
is difficult to map the utility function correctly. By asking about the preference between different
points on the efficient frontier, we are able to help the decision maker to find the best compromise
solution. There are three different types of approaches to solve the MCMP problem. The
difference is when and what type of preference information is asked.
Prior Articulation of Preferences
This approach uses pre-specified preference information from the decision maker. The
priorities of the conflicting objectives will be specified according to the preference information.
The solution to the problem will be an efficient point and it will be the decision maker’s most
preferred solution.
Goal Programming is one of the important methods in this category. Lee (1972) provides
the formulation and solution of goal programming problems. In this method, decision maker will
be asked to set goals for each objective. The decision maker will then need to give preference
information on achieving these goals. The objective of this problem is to determine an optimal
solution that comes as close as possible to achieving the goals with given preference information.
Post Articulation of Preferences
If the efficient frontier is known before selecting the most preferred solution, these types
of solution methods are in this category. When the efficient frontier only consists of a finite
number of points, such methods could be used.
Interactive Methods
Methods that rely on the progressive articulation of preference from the decision maker
during the solving procedure are called interactive methods. Shin and Ravindran (1991), have a
survey of numerous interactive methods to solve MCMP problems. They present four typical
interaction styles: binary pairwise comparison; pairwise comparison; vector comparison; precise
local tradeoff ratio. Methods covered in this paper include: feasible region reduction methods;
9
feasible direction methods; criterion weight space methods; tradeoff cutting plane methods;
Lagrange multiplier methods; visual interactive methods; branch and bound methods. The basic
idea of these methods is to find a few efficient solutions and ask decision maker to provide
preference information among them. The decision space is reduced based on the response of the
decision maker. This is repeated until the best compromise solution is achieved.
Sadagopan and Ravindran (1982), presents an interactive method to solve the bi-criteria
mathematical programming problem and it is called the Paired Comparison Method. Decision
maker is presented with two intermediate efficient points and the decision maker chooses which
solution is preferred. By doing this, the efficient set is reduced and finally the best compromise
solution is reduced to a small interval.
2.2 Inventory Control
Extensive research has been conducted in the area of inventory control. Handley and
Whitin (1963) provides a good introduction of the techniques of constructing and analyzing
mathematical models of inventory systems. Clark (1972) provides an overview of research in
multi-echelon inventory theory, covering published results through 1971. The author presents
important distinguishing features among various models. Deterministic – Stochastic; Single-
Product – Multi-Product; Stationary – Non-Stationary; Continuous Review – Periodic Review;
Consumable Product – Reparable Product; Backlog – No Backlog. In a deterministic model,
demands are known in advance with certainty. In the stochastic model, the demands are based on
probability distributions. In a stationary model, parameters used to define the external demands
are assumed to be independent of time. Parameters may vary over time in a non-stationary model.
Two types of review policies can be followed. In a continuous review model, opportunities to
review the stock of the system and to implement ordering policies occur continuously. In a
periodic review model, such opportunities exist only at discrete points in time, normally with a
10
given periodicity. The model assumes backlogging of demand if unsatisfied demands are satisfied
from later resupply. In the non-backlog assumption, unsatisfied demands are lost sales.
Evans (1958) provides a stationary model considering transportation and production in
order to minimize the total cost of maintaining the system, which is one of the earliest
deterministic papers. Zangwill (1966) analyzes a deterministic multi-period production and
inventory model. It permits backlogging of unsatisfied demand. An efficient dynamic
programming algorithm is also presented to calculate the production schedule with minimum
cost. Veinott (1969) extends this by providing algorithms whose computational effort increases
algebraically, instead of exponentially, with the size of the problem. Kalymon (1970) develops an
algorithm that can decompose the problem which has been studied by Veinott and Zangwill into
single-stage problems and the computational effort of this decomposition algorithm might
increase exponentially with the number of facilities but it will only increase linearly with the
number of the facilities at the lowest echelon in the system.
Muckstadt (1979) develops a mathematical model for a particular type of three-echelon
inventory system with stochastic and stationary demand. The items considered in the paper are
recoverable items, which mean the items can be repaired when they fail. The model is used to
determine the stock levels at each location for each item and for a given amount of investment,
the optimum performance of the inventory system can be achieved.
Zheng (1992) points out that the application of the stochastic model has been limited
because of the absence of insightful analytical results. The paper compares the stochastic order
quantity/reorder point model with a corresponding deterministic EOQ model. The paper
concludes that the relative increase in costs incurred by using the ordering quantity from EOQ
model instead of the optimal quantity from the stochastic model is no more than 1/8 and vanishes
when ordering costs are significant relative to other costs.
Clark and Scarf (1960) are the first to discuss the concept of echelon stock. They
characterize optimal policies in a two-echelon inventory model. A centralized inventory system is
11
considered. Federgruen and Zipkin (1984) extend the results to the infinite-horizon case. They
show that the computation for the infinite-horizon case is much easier compared with the finite-
horizon case. Cases of multiple locations in the lower echelon are also considered.
De Bodt and Graves (1985) present a continuous-review inventory control policy that
minimizes the expected average costs in a multi-stage, serial inventory system with stochastic and
stationary demand. A fixed ordering cost, an inventory holding cost for each echelon and a
backordering cost are included in this model. This paper considers a centralized inventory system.
An approximate cost model is presented and their approximation is similar to that for the single-
item inventory model with continuous review using reorder point and reorder quantity policy.
Svonoros and Zipkin (1988) consider a two-echelon inventory system with multiple
retailers and one warehouse. Backorders are allowed in the model. The demand at the retailers is
stochastic and a continuous review policy is used in the model with a reorder point and a constant
order quantity. They aim to estimate the long-run average backorders at the retailers and average
inventory level at all facilities. The estimates are then used within a cost-minimization framework
to choose the best inventory policies. The cost-minimization framework considers a centralized
inventory system.
Axsater (1992) considers a two-stage inventory system with one warehouse and multiple
retailers. The demand is stochastic and the lead time is a constant in the model. A continuous
review policy is used and a centralized system is considered. The paper presents the method to
evaluate the exact holding and shortage cost for two retailers and extend to multiple retailers
using the same methodology.
Axsater and Juntti (1997) points out that a multi-level inventory system is often
controlled by two reorder point policies – an installation stock policy and an echelon stock policy.
In an installation stock policy, ordering decisions at each installation (i.e. stage) are based only on
the inventory at that installation. In the echelon stock policy, ordering decisions at each
12
installation are based on the sum of the installation inventory at that installation and all its down-
stream installations.
Axsater (1997) considers the same problem in Axsater’s (1992) paper. The difference is
that instead of an installation stock policy, an echelon stock policy is used in this paper. The
paper points out that installation stock policy and echelon stock policy may perform better than
each other in different situations. The inventory system is still a centralized system and a
continuous review policy is used. A simple method to exactly evaluate the holding and shortage
costs is presented.
Axsater and Zhang (1999) present a joint replenishment policy to deal with a two-level
inventory system with a central warehouse and several identical retailers. A centralized inventory
system is considered in this paper. Demands from the customers are stochastic and stationary and
a continuous-review policy is used. The joint replenishment policy means that when the sum of
the inventory positions of retailers declines to a certain “joint” reorder point, a batch quantity will
be ordered from the retailer with the lowest inventory position.
Cachon (2001) presents a model that deals with a two-echelon supply chain system with
stochastic and discrete demand. So a periodic review policy is used. For each facility, average
inventory, backorders and fill rate are evaluated exactly and for the lower echelon, safety stock is
also evaluated exactly. A centralized supply chain system is considered when calculating the
cost. The paper concludes that the cost of the supply chain system increase substantially if the
upper echelon carry zero inventory and a high fill rate is also required for the upper echelon. The
paper also indicates that in a periodic review environment, continuous review policy gives a poor
performance.
13
2.3 Multiple Criteria Inventory Control
Starr and Miller (1962) solve the inventory problem with a single facility, multi-item as a
bi-criteria problem. The objectives are cycle stock investment and number of orders and
Lagrangian relaxation technique is used to solve this problem. They point out that the product of
the cycle stock investment to the number of orders is a constant. They plot the efficient curve and
show that inventory policy could be chosen based on the curve. They are also the first to use the
efficient curves.
Brown (1967) introduces the concept of exchange curve between two criteria. In the
deterministic demand case, capital invested in inventory and number of orders are the two criteria
considered, while in the stochastic demand case, capital invested in inventory and the customer
service level are considered as the two criteria. By generating the exchange curve, the decision
maker can choose the preferred solution from the efficient set.
Gardner and Dannenbring (1979) point out that in the traditional inventory models, the
assumed marginal ordering cost, holding cost and shortage cost are actually very difficult to
calculate in practice. For the ordering cost, they cite the conclusion from Ziegler’s survey that the
approaches used in the literature to determine the ordering cost result in average cost instead of
marginal costs. The holding cost is also hard to calculate because it mostly related to the cost of
capital and it is highly subjective. Shortage cost has not been widely used by most practitioners.
They also point out another gap between theory and practice that the traditional inventory model
emphasized models with single item and these models lack the insights for multi-item inventory
management. Therefore, the paper deals with policy tradeoffs on three objectives: aggregate
customer service, workload and investment in cycle and safety stock in a centralized supply chain
inventory system. By using these three objectives, they generate a response surface which is
essentially the efficient frontier. They show that any optimal decision must be a point located on
the response surface. The computational results also show that the models considered in the paper
are efficient and can make improvements in the inventory system.
14
Buffa and Munn (1989) present a recursive algorithm that can minimize the total logistics
cost by determining the order intervals. The transportation cost is modeled as a function of
shipping distance and weight. A relaxation procedure is used to find a good initial approximation
of the optimal order intervals. The optimal result will then be achieved through a series of
recursive steps. They also indicate that this recursive algorithm converges to the optimal solution
in a very few steps.
Bookbinder and Chen (1992) used Multiple Criteria Decision Making (MCDM)
methodology in a two-echelon centralized inventory system with both deterministic and
probabilistic demands. In the deterministic case, they use integer-ratio policy, which is that the lot
size at a given echelon is an integral multiple of the lot size at its immediate successor echelon
(Taha and Skeith, 1970). Two objectives are considered in this case. The first one is the
minimization of annual inventory, holding and ordering cost. The second one is the minimization
of annual transportation costs. Two types of probabilistic cases are examined. In the first type, the
marginal costs are known. The objective of transportation costs in the previous case is replaced
by the minimization of the mean annual number of stock outs. An exchange curve is obtained as
results in this case. In the second type, the marginal costs are not available. The three objectives
are capital invested in inventory, expected annual number of stock out and transportation costs. A
non-dominated policy surface of the trade-offs among three criteria is obtained. The results of this
paper are MCDM generalizations of Starr and Miller’s optimal policy curve and Gardner and
Dannenbring’s optimal policy surface.
Crowston, Wagner and Williams (1973) consider the optimal lot size in multi-stage
assembly systems. In the system, each facility can have multiple predecessors but only a single
successor. They prove that in this system, the optimal lot size is an integer multiple of the lot size
at successor facility. Based on this, they construct a dynamic programming algorithm to calculate
the optimal lot sizes and they use the concept of echelon stock in this algorithm. The costs at each
facility include a fixed cost for each lot and a linear holding cost.
15
Lenard and Roy (1994) propose another approach for the inventory policies based on the
notion of efficient policy surfaces. Multi-criteria and multi item inventory problem are
considered. The three criteria they used are inventory level, customer service and workload. In
the multi item case, they define a structure of attributes in order to group items and three different
levels of attributes are identified. By using grouping techniques, they group the items into
coherent families. They then propose to have one aggregate item to represent the different items
in a family in order to let the decision maker make the decisions on the basis of a multi-criteria
single-item model.
Agrell (1995) solves a multi-criteria problem for a single item with stochastic demand.
The three criteria used are total annual cost, expected number of stock out occurrences annually
and the expected annual number of items stocked out. The paper also presents an interactive
multi-criteria framework for an inventory control decision support system. It is implemented as
FORTRAN modules and Excel spreadsheet macros such that the decision maker can use the
interface to interact with the models and find the preferred efficient solution.
Ettl et al (1996) develop a supply network model that discusses the trade-offs between
capital invested in inventory and customer service levels. A centralized supply chain system is
considered in this paper. They consider each facility in the supply chain as an inventory queue
and they obtain the performance and service measures for each facility by using a set of
approximations. A set of inventory policies will be generated if the desired service level for each
facility is specified. The trade-offs between the two criteria can then be shown to the decision
maker and the decision maker can find the best preferred solution. They conclude that this supply
chain network model can be used to support the decision making in large and complex supply
chains.
Puerto and Fernandez (1998) solve the inventory problem with backorders in both
deterministic and stochastic cases for a single company. The paper uses the trade-off analysis in
the context of vector optimization theory. The two criteria considered in the model are the
16
ordering cost and holding plus backlogging cost. They first obtain the Pareto-optimal solution set
for the unconstrained inventory model and then add constraints to solve the constrained inventory
model.
Thirumalai (2001) provides an algorithm for serial supply chain inventory problem with
both deterministic and stochastic demands. It is a multi-criteria problem with multiple decision
makers, so it considers a de-centralized supply chain inventory system. The system has three
stages arranged in series - one manufacturer, one warehouse and one retailer. In the deterministic
case, the two criteria are average capital invested in inventory and the annual number of orders.
An interactive method is proposed for this bi-criteria problem with one single decision maker. In
the stochastic demand case, three criteria are used in the paper - capital invested in inventory,
number of orders and customer service. A continuous review policy is used when solving the
cases with stochastic demand. Then, he extends the problem to a supply chain system and uses
the case of three companies arranged in series. Multiple decision makers are considered and the
decision of each company is made by an independent decision maker. A collaborative and
interactive optimization algorithm is provided to solve the problem. He shows that by using the
interactive algorithm, a compromise solution that is efficient not only for the entire supply chain
system but also for each company can be achieved.
DiFilippo (2003) provides a multi-criteria, two-stage supply chain inventory model with
deterministic demand. Transportation cost is also incorporated in the model. The three criteria
used in the paper are capital invested in inventory, number of orders and transportation cost.
Both de-centralized and centralized supply chain systems are examined. Freight rate is used in
this paper to calculate the transportation cost. In the de-centralized case, the multi-criteria retailer
problems are solved first, then the solutions are used to solve the warehouse problem. Two types
of order policy of the warehouse are considered. A stationary policy in which the warehouse will
place an order of same quantity at a fixed time interval, while a non-stationary policy in which the
quantity ordered by the warehouse can be different every time. Two order policies are used and
17
compared for the warehouse problem because it is not always optimal to follow a stationary
policy when optimizing a single cost objective without transportation cost when there are
multiple retailers in the system. In the centralized case, two different cases are considered for the
retailers. In the case with common replenishment time, the retailers place an order at the same
time and the problem will be the same as a single warehouse, single retailer problem. In the case
of different replenishment times, the replenishment time for the retailers can be different. The
results show that the case of different replenishment performs better in the numerical example.
Natarajan (2007) presents three multi-criteria models to deal with the inventory planning
problem for a de-centralized supply chain system. There is a single warehouse and multiple
retailers in the system. The first model considers deterministic demand and lead time and
marginal costs are all known to the decision maker. A new coordination scheme is proposed such
that the warehouse can lower its inventory level and satisfy the demand from the retailers at the
same time. The second model assumes that the marginal costs are unknown and freight rate
functions are used to calculate the transportation costs. Three criteria used in the second model
are capital invested in inventory, annual number of orders and annual transportation costs. The
third model considers stochastic demand and random lead time. The fourth criterion is the fill
rate. Several efficient solutions are generated after solving the models and the value path method
is used to show the trade-offs among the efficient solutions of each company in the system to the
decision makers. Then, the models are tested with real world data.
Many of the previous studies focused on the centralized supply chain inventory system
with only one decision maker. However, in practice, de-centralized supply chain systems are
more common. Table 2.1 gives a comparison of the literature in multi-criteria inventory control.
It shows that the field of multi-criteria inventory problem has been broadly studied in the past
four decades. Models that deal with both deterministic and stochastic demand have been
developed. Problems with both single and multiple facilities under stochastic demand have also
been studied. Also, in the multiple facilities case, most models use a continuous review policy
18
instead of a periodic review policy. The periodic review policy is commonly used in practice.
Therefore, the focus of this thesis is to consider an inventory model of a two-stage supply chain
system with stochastic demand using a periodic review policy, for both centralized and de-
centralized supply chains.
20
3 Problem Description
A supply chain has been defined as an integrated process, with both a forward flow of
materials and a backward flow of information, that involves suppliers, manufacturers,
distributors, and retailers working together to acquire raw materials, convert them into final
products, and deliver the final products to retailers (Beamon, 1998). There are mainly two types
of supply chain. One is a de-centralized supply chain, where each supplier, manufacturer,
distributor and retailer has its own objectives and need to make decisions in order to achieve
those objectives. In this case, each stage of the supply chain is generally owned by a different
company. Optimizing one member of the supply chain will not necessarily optimize the entire
supply chain (DeFilippo, 2003). The other one is centralized supply chain where a single decision
maker takes charge of the entire supply chain and makes decisions for each member of the supply
chain such that the entire supply chain will be optimized. In a centralized supply chain, all stages
are owned by the same company. This thesis aims to develop multi-criteria inventory models with
stochastic demand and lead time for both decentralized and centralized supply chains.
DeFilippo (2003) presents some inventory policies for multi-criteria supply chain
inventory models with transportation costs. A nested inventory policy is one where if one member
of the supply chain places an order, all downstream stages also place an order at the same time.
Under the nested policy, if the time between orders at the warehouse is constant and the order
quantity is fixed, this policy is called a stationary policy. If the order quantity and the time
between orders are not necessarily the same, the policy is called a non-stationary policy.
21
3.1 Criteria
Three criteria will be included in the multi-criteria model developed in this thesis. The
three criteria are: 1) the total capital invested in inventory; 2) number of orders; 3) service level.
The number of orders should be minimized because the number of orders is related to the
workload.
This research will be limited to a two-stage supply chain with a wholesaler and multiple
retailers with stochastic and stationary demand. Figure 3.1 shows the supply chain considered in
this thesis. There are retailers and one wholesaler, denoted as in the supply
chain.
Wholesaler
Retailer
Retailer
Retailer
.
.
.
1
2
m
m+1
Figure 3.1 Two-stage Supply Chain Network
3.2 Mathematical Model
3.2.1 Notations
The notation is for a supply chain with retailers and a single wholesaler, with
stochastic demand and deterministic lead time. The demands of the customers follow a normal
22
distribution. The daily demands seen by the retailer are independent. We assume an order-up-to
policy for all companies.
Decision variables:
cycle time or order interval for company (in days),
safety factor for company
order-up-to level for company
safety stock for company
Data:
Here, we assume that the daily demands at retailers follow normal distributions.
mean daily demand seen by company
standard deviation of the daily demand seen by company
unit cost of the product for company
lead time for company (in days),
3.2.2 Demand during Cycle Time plus Lead Time at the Retailers
Let denote the mean demand during cycle time plus lead time for company
Let
denote the variance of demand during cycle time plus lead time for
company
Recall that the daily demands seen by the retailer are independent, so
is given as,
23
3.2.3 Capital Invested in Inventory at the Retailers
The product of the average cycle inventory plus safety stock and cost of the product is
defined as the inventory capital. Figure 3.2 shows the inventory position for the retailers.
T
ss
Lead Time
S
Figure 3.2 Inventory Position for the Retailers
In the (T, S) inventory policy, the safety stock for company is,
Note that the value of is a decision variable in this model because the customer service level is
an objective function.
From Figure 3.2, we can find that the order-up-to level for company is the sum of the
safety stock and the expected demand during cycle time plus lead time Therefore, the order-
up-to level for company is,
The expected inventory level for retailers is the sum of expected average cycle inventory
and safety stock.
24
Let denote the mean demand during cycle time for company
Let denote the mean demand during lead time for company
From Figure 3.2, the expected inventory level when the order arrives is .
Because the mean rate of the daily demand is a constant, the cycle inventory level will vary
linearly between and . Therefore, the expected average cycle inventory is,
Because
, this can be further simplified to,
Recall that the safety stock for company is,
Therefore, the expected inventory level for retailer is,
Therefore, the expected capital invested in inventory for company is,
(
)
3.2.4 Capital Invested in Inventory at the Wholesaler
The order-up-to level of the wholesaler depends on the order quantities of the retailers.
The retailers will often order at different time periods in both de-centralized and centralized
supply chain system. We also assume that the lead time for the wholesaler is strictly shorter than
25
one day, so that no order from a retailer will arrive at the wholesaler during its lead time. Abdul-
Jalbar et al (2003) presents a near optimal solution method to solve the single wholesaler and
multiple retailers problem in order to minimize inventory holding cost and ordering cost. The
inventory holding cost and ordering cost are proportional to the capital invested in the inventory
and the number of orders in this model, respectively. Therefore, Abdul-Jalbar’s method can be
utilized in this model. In Abdul-Jalbar’s method (using our notation),
where is an integer and is the least common multiple (LCM) of
The time between orders at the warehouse is shown as , where and
is an integer. Therefore,
An example will be presented below to illustrate the relationship between the order
intervals of the retailers and the order interval of the wholesaler and their inventory levels.
In the example, we assume a single wholesaler, two retailers system. The order interval
of retailer 1 is 2 days and the order interval for retailer 2 is 3 days. For simplicity, we assume that
the lead time is zero for both retailers and wholesaler. Let and . We also assume
that the retailers and wholesaler have no cycle inventory at the beginning. For the example, the
least common multiple of and is 6. We assume that , so , i.e., the wholesaler
orders every 6 days.
Let denote the order quantity of retailer 1 and denote the order quantity of retailer 2.
26
Because the demand is stochastic, the inventory level at the time of review will change every
time. Hence, can be different every time and is a random variable.
Figure 3.3 shows the inventory position of the retailers and the wholesaler for this example.
3 days
ss1
S1
3 days
Retailer 1
2 days
ss2
S2
2 days2 days
Retailer 2
S3
I1
ss
6 days
I2
2 days
1 day
I3
1 day
2 days
Wholesaler
Figure 3.3 Inventory Position for Retailers and Wholesaler in the Example
27
Let denote the order quantity for retailer at th time. Let denote the inventory
level at the wholesaler at th time. Let denote the order-up-to level of the wholesaler.
At time zero, the wholesaler gets the replenishment from its supplier and the inventory level at
the wholesaler reaches its order-up-to level. At the same time, the two retailers place their orders
and an amount of goods corresponding to the order is removed from the wholesaler’s inventory.
The inventory level for the wholesaler then drops to
At the end of day 2, retailer 2 places the second order. The inventory level for the wholesaler then
drop to
At the end of day 3, retailer 1 places the second order. The inventory level for the wholesaler then
drop to
At the end of day 4, retailer 2 places the third order. In the example, we assume that the inventory
level for the wholesaler then drop to the level of safety stock.
On day 6, a new cycle begins and what we described above will be repeated.
In the (T, S) inventory policy, the safety stock for the wholesaler is,
The wholesaler is responsible for meeting the demands from the retailers and the
retailers’ demands follow normal distributions. Recall that,
28
The variation of the wholesaler’s demand can be considered as times the sum of the variation of
demands viewed from retailer times . Therefore, the variation of the wholesaler’s demand can
be written as,
∑
Then the standard deviation of wholesaler’s demand can be derived.
√ ∑
Therefore, the safety stock for wholesaler is,
√ ∑
Let denote the expected daily demand for wholesaler. The expected daily demand for
wholesaler is,
∑
And recall the assumption that no retailer order will arrive at the wholesaler during its lead time.
Again, we recall that,
The expected order quantity for retailer is,
29
The expected demand for the wholesaler during the order interval is times the sum of the
expected order from the retailer times .
Thus, the expected demand during is,
∑
∑
The order-up-to level for wholesaler is the sum of the safety stock and the expected demand
during . Therefore, the order-up-to level can be written as,
∑
∑
√ ∑
The average inventory level for the wholesaler also consists of average cycle inventory and safety
stock.
For the average cycle inventory, we recall the example illustrated in Figure 3.3. In the example,
we know that the order quantities of the retailer will vary every time. As mentioned above, the
expected order quantity for retailer is,
Then we can use the expected value to represent the order quantity because we are considering a
long-run performance of the supply chain. So in the example,
Recall that
∑
and ∑ is the expected demand during . Therefore, the expected inventory level
when all of the orders from the retailers are satisfied is the level of safety stock.
30
S
I1
ss
6 days
I2
2 days
1 day
I3
1 day
2 days
Wholesaler
Figure 3.4 Inventory Level at the Wholesaler in the example
Figure 3.4 is the inventory level at the wholesaler shown in Figure 3.3. From this figure,
we can derive the expression for the average cycle inventory level at the wholesaler. The average
cycle inventory level is the area above the safety stock level divided by the length of the order
interval. Let denote the level of safety stock at the wholesaler.
The area is calculated as,
The expression can then be simplified to,
The length of the order interval is 6, so the average cycle inventory level is,
From the graph, we can see that,
31
Thus the average cycle inventory level can be written as,
Recall that
Therefore, in the example, the average cycle inventory can be written as,
∑ ( )
This expression can then be generalized to multiple retailers and is shown as,
∑ ( )
Therefore, the average inventory level for the wholesaler is,
∑ ( )
∑
The expected inventory capital for wholesaler is,
( ∑
)
The in-transit inventory capital is the value of inventory when it is in-transit between its
origin and destination. We assume the shipments are FOB origin, which means the buyer owns
the goods when they leave the seller. The expected in-transit inventory capital for company is,
3.2.5 Order Frequency
Order frequency is the number of times the company places an order per year. The order
interval for each company is:
32
The annual number of orders is:
3.2.6 Service Level
In-stock probability is used as a customer service measure in this model. It represents the
probability of stocking out of items in the selling season (Ravindran and Warsing, 2012). The
stock out probability for company is:
∫
The in-stock probability for company is:
∫
where represents the normal distribution of the demand during cycle time and lead time
with mean and variance
for retailer , and with mean ∑ and variance
∑
for the wholesaler.
Recall that for retailers,
And for wholesaler,
∑
√ ∑
So the stock out probability for company can be simplified to,
33
∫
∫
where represents standard normal distribution.
We can give the in-stock probability for company :
∫
∫
In this model, the objective is to maximize the in-stock probability for the retailers and the
wholesaler.
3.3 General Model for De-centralized Supply Chain System
The general multi-criteria optimization model in a decentralized two-stage supply chain
is presented below.
For the retailers:
(
)
∫
For the wholesaler:
(
∑
)
35
3.5 Solution Methods
We have presented the general model for both de-centralized and centralized supply
chain systems. The model will be solved using the weighted objective method and the
compromise programming method (Ravindran, 2008; Zeleny, 1982).
3.5.1 Weighted Objective Method
Consider the general MCMP problem given below:
where S is the feasible region and are the criteria functions.
The weighted objective problem, also known as the problem, is formulated as follows
(Ravindran, 2008),
∑
∑
where is the weight assigned to objective .
36
Theorem 3.1 (Geoffrion, 1968): Let for all be specified. If is optimal for the
problem, then is an efficient solution to the MCMP problem. Note that Theorem 3.1 is only
a sufficient condition. If the constraint region is a closed convex set and ’s are convex
functions on , then, is efficient if and only if is an optimal solution to the problem with
for all .
The general solution procedure for the weighted objective method is given below.
Step 1
In this thesis, for the weighted objective method, we use Steuer’s (Steuer, 1986) interval
programming method to generate the initial set of positive weights and solve the problem.
This method searches the -space, define by { | ∑ }, which is also known as
the parametric or weight space.
Using interval programming, we generate efficient points for problems with
criteria. The weights for each objective assigned to each efficient point are shown below. Let
denote the th efficient point . Note that is a very small positive number.
37
(
)
where
Here, and there are total 7 efficient points that will be generated initially as shown
in Table 3.1. The initial weights are chosen in such a way that they are evenly spread in the
parametric space, so that we can concentrate in the area where the decision maker’s optimal
weights are.
Table 3.1 Initial Weights for the Objectives Using Interval Programming
Efficient Point Number
1 1
2 1
3 1
4 ⁄ ⁄ ⁄
5 ⁄ ⁄ ⁄
6 ⁄ ⁄ ⁄
7 ⁄ ⁄ ⁄
Figure 3.5 illustrates the graph of the parametric space.
(1,ε,ε)
(ε,1,ε) (ε,ε,1)
(4/9,1/9,4/9)
(1/9,4/9,4/9)
(4/9,4/9,1/9)
(1/3,1/3,1/3)
Figure 3.5 Parametric Space for Three Criteria
38
Step 2
Seven efficient points are then generated by solving the single objective problem (
problem) using the weights shown in Table 3.1.
Note that there is no guarantee that each set of weights would produce a different
efficient solution.
3.5.2 Compromise Programming Method (Zeleny, 1982; Ravindran, 2008)
The compromise programming method uses the ideal values of each objective and
weighted metric to define the solution closest to the ideal value.
Consider the MCMP problem stated in section 3.5.1. Let be the ideal value for .
is determined by minimizing over the feasible region and ignoring all other objectives and is
given by:
The compromise programming problem is then formulated as follows:
(∑[
]
)
∑
39
where ’s are weights assigned to objective .
In compromise programming, given ∑ and , the optimal
solution to Problem P is called a compromise solution and it would be an efficient solution. The
value for has to be assessed subjectively by the decision maker. For example, setting
,
the objectives can be scaled by the ideal value. The constraint ∑ , serves as a
normalization constraint in the compromise program formulation. Without loss of generality, this
constraint can be omitted and the result will still hold as long as for all (DeFilippo,
2003).
The value of can be any integer between 1 and infinity. In this thesis, three values of
will be chosen for analysis. and infinity.
For
∑[ ]
For
(∑[
]
)
For
⌊ ⌋
40
When we minimize the maximum deviation from the ideal value for all objectives.
When this problem can also be formulated as follows:
When using in the compromise programming method, outliers are given more
emphasis and the solution may not always be preferred by the decision maker. Table 4.1
illustrates the anomaly. Here, the ideal value is 100. The problem will choose solution (1)
because the maximum deviation is 4 and the maximum deviation in solution (2) is 5. However, a
decision maker would prefer solution (2) because it achieves the ideal for 3 of the 4 objectives
by sacrificing a small amount in .
Table 3.2 An Example Illustrating the Outlier Problem when p=∞
Objective
(Min)
Ideal
Solution
Solution
(1)
Solution
(2)
Z1 100 102 105
Z2 100 104 100
Z3 100 103 100
Z4 100 101 100
The solution procedure for compromise programming method is shown below.
41
Step 1
Get the ideal solution for each objective.
Step 2
The weight of each objective needs to be assigned by the decision maker. Let denote the
normalized weight for objective . In order to calculate decision maker needs to give a rating
in a scale of 1 to 10 for each of the objective. We then use the following equations to get the
weights for objective .
Step 3
After deciding the weights, compromise programming method can then be used to solve the
multi-criteria problem for different values of . Three different values will be chosen for
, the compromise programming method will generate three efficient points.
Recall that seven efficient points are generated by using the weighted objective method.
Thus, 10 efficient points will be generated and presented to the decision maker for evaluation.
The best compromise solution will be achieved if the decision maker is satisfied with one of the
solutions. If the decision maker is not satisfied with any of the solutions presented, additional
weights can be generated to each objective and new efficient solutions can be generated by the
weighted objective method. We can use interval programming to generate additional weights
around the efficient solution which the decision maker prefers. In addition, more efficient
solution can be obtained by solving the compromise programming problem by using
etc.
42
4 Single Wholesaler, Two Retailers De-centralized Supply Chain
In a de-centralized supply chain, each company has its own decision makers. The
decision makers optimize their own objectives independently. In this chapter, the single
wholesaler, two retailers de-centralized supply chain under stochastic demand and fixed lead time
will be analyzed.
The retailer problem is always the same no matter how many retailers are in the system
and the ordering policy determined by the wholesaler is largely dependent on the results of the
retailer problem. The model and solution procedure of the retailer problem will be presented first.
Then a numerical example will be provided for the retailer problem. The model and solution
procedure of the wholesaler problem will then be presented. Based on the numerical results of the
retailer problem, the wholesaler problem will also be solved using a numerical example.
The supply chain system discussed in this chapter consists of a single wholesaler that
provides goods for the two retailers. The retailers need to meet the stochastic and stationary
demand from the customer and the retailers operate independently of each other and of the
wholesaler. The supply chain system of a single wholesaler and two retailers is shown in Figure
4.1, where and are the average retailer demands.
Wholesaler
Retailer 1
Retailer 2
µ1
µ2
Figure 4.1Supply Chain System of a Single Wholesaler and Two Retailers
43
4.1 Supply Chain Model
The model of the de-centralized case will be presented. In this case, the retailer problems
and the wholesaler problem are not solved at the same time.
4.1.1 Assumptions
Several assumptions will be made for this supply chain system. The demand seen by the
retailer is stochastic, stationary and independent. The lead time for each company is a constant. In
addition, we assume that the lead time for the wholesaler is negligible. Hence, the expected
number of backorders of the wholesaler equals 0 and the retailers’ order will always be satisfied
on the average. The decision maker for each company will use a periodic review policy by
reviewing the inventory level at a fixed period of time and order the quantity which brings the
inventory to the order-up-to level. We assume that the supplier for the wholesaler has unlimited
supply of goods. The cost of the products at the wholesaler is always smaller than that at any
retailers.
4.1.2 Notations
The notation below is for the retailers and the wholesaler in the de-centralized case.
Order-up-to level for retailer (1)
Order-up-to level for retailer (2)
Order-up-to level for the wholesaler
Order interval for retailer (1) (days)
44
Order interval for retailer (2) (days)
Order interval for the wholesaler (days)
Safety factor for retailer (1)
Safety factor for retailer (2)
Safety factor for the wholesaler
Cost per unit for retailer (1)
Cost per unit for retailer (2)
Cost per unit for the wholesaler
Mean daily demand for retailer (1)
Mean daily demand for retailer (2)
Standard deviation of daily demand for retailer (1)
Standard deviation of daily demand for retailer (2)
4.1.3 General Model of Decentralized Supply Chain
Recall the general model developed in Chapter 3, section 3.3.
For the retailers,
(
)
( ∫
)
45
For the wholesaler,
(
∑
)
( ∫
)
∑
Note that in this model, both the order interval and safety factor have upper and lower bounds.
Let denote the maximum order interval for company
Let denote the minimum order interval for company
Let denote the maximum safety factor for company
Let denote the minimum safety factor for company
Note that the third objective is different from the model given in section 3.3. In a supply
chain system, the customer service level is measured in terms of the “in-stock probability” and it
46
should be maximized. Because the other two objectives are to minimize, a negative sign has been
added to this objective so that service level also becomes a minimization objective.
4.2 Retailer Problem
First, we need to solve the problems for each retailer. Based on the results, the inventory
control policy for the wholesaler will be determined.
As discussed in section 3.5, to solve this multi-criteria optimization problem for each
retailer, both weighted objective method and compromising programming method will be used.
4.2.1 Solution Procedure
Weighted Objective Method
The general procedure for solving the multi-criteria retailer problem using the weighted
objective method has been presented in section 3.5.1.
The formulation of the weighted objective method for the retailers is shown below,
((
) )
[ ( ∫
)]
is the weight assigned to the jth objective for company i.
The objective value of the seven efficient points generated using the weighted objective
method are shown in Table 4.1.
47
Table 4.1 Efficient Points Generated by Weighted Objective Method
Objective 1 Objective 2 Objective 3
Efficient Point 1 Z 1,0,0
Z 1,0,0
Z 1,0,0
Efficient Point 2 Z 0,1,0
Z 0,1,0
Z 0,1,0
Efficient Point 3 Z 0,0,1
Z 0,0,1
Z 0,0,1
Efficient Point 4 Z 1/9,4/9,4/9
Z 1/9,4/9,4/9
Z 1/9,4/9,4/9
Efficient Point 5 Z 4/9,1/9,4/9
Z 4/9,1/9,4/9
Z 4/9,1/9,4/9
Efficient Point 6 Z 4/9,4/9,1/9
Z 4/9,4/9,1/9
Z 4/9,4/9,1/9
Efficient Point 7 Z 1/3,1/3,1/3
Z 1/3,1/3,1/3
Z 1/3,1/3,1/3
Compromise Programming Method
The general procedure for solving the multi-criteria retailer problem using the
compromise programming method has been presented in section 3.5.2.
The formulation of the compromise programming problem for the retailers is shown
below,
((
)
((
) )
(
)
(
)
(
)
[ ( ∫
) ]
)
is the ideal value of the j
th objective for company i.
48
The three efficient points generated using the compromise programming method is
shown in Table 4.2.
Table 4.2 Efficient Points Generated by Compromise Programming Method
Objective 1 Objective 2 Objective 3
Efficient Point 8
Efficient Point 9
Efficient Point 10
4.2.2 Numerical Example for Retailer Problems
A numerical example for the retailer problem will be presented in this section. Table 4.3
includes the mean monthly demand, the standard deviation of the monthly demand, the costs and
the lead time for each retailer.
Table 4.3 Retailer Data Used in the Numerical Example
Mean
Daily
Demand
Variation
of Daily
Demand
Cost
Lead
Time
(Days)
Retailer 1 180 100 50 15 30 120 1.645 2.33
Retailer 2 120 64 60 12 30 150 1.28 2.33
The multi-criteria problems for the two retailers are given as:
For retailer 1,
(
√ )
∫
49
For retailer 2,
(
√ )
∫
For the retailer problem, we first focus on solving the problem of retailer 1.
Weighted Objective Method
We first solve the problem of retailer 1 using the weighted objective method. The weights
used to solve this problem are shown in Table 4.4.
Table 4.4 Weights Used in the Weighted Objective Method
Efficient Point Number
1 1
2 1
3 1
4 ⁄ ⁄ ⁄
5 ⁄ ⁄ ⁄
6 ⁄ ⁄ ⁄
7 ⁄ ⁄ ⁄
In the weighted objective method, simple scaling is used to scale the objectives and the
new weights are
. In the example, for retailer 1, we use
to scale objective 1,
to scale objective 2 and
to scale objective 3.
50
Using the equal weights given in Table 4.4, the scaled weighted objective problem for
retailer 1 will become,
((
√ ) )
[ ( ∫
)]
LINGO is then used to solve the single objective optimization problem. The objective
values of the seven efficient points for retailer 1, generated using the weighted objective method,
are shown in Table 4.5. The values of order interval and safety factor for retailer 1 are
presented in Table 4.9. Note that in Table 4.5, for objective 3, we use the “in-stock” probability to
measure the service level.
Table 4.5 Values of Each Objective Using Weighted Objective Method for Retailer 1
Retailer 1 Weights Objective 1
Capital invested in inventory
Objective 2
No. of orders
Objective 3
Service level
Efficient Point 1 (1, , ) 275517 12.17 95%
Efficient Point 2 ( ,1, ) 684557 3.04 95%
Efficient Point 3 ( , ,1) 428789 5.79 99%
Efficient Point 4 (1/9,4/9,4/9) 643025 3.32 99%
Efficient Point 5 (4/9,1/9,4/9) 277815 12.17 99%
Efficient Point 6 (4/9,4/9,1/9) 389382 6.64 95%
Efficient Point 7 (1/3,1/3,1/3) 392163 6.64 98.9%
Sensitivity Analysis on the Weights
Note that the first three efficient points have been generated using maximum weight on
one of the objectives. Hence, the best values of the objectives are reached for each objective. For
51
example, for efficient point 1, objective 1 (capital invested in inventory) is the lowest. As the
weights are changed, the achievements of the objectives also change. When two of the objectives
are given higher weights relative to the third one, for example, for efficient point 4, the
achievements of objective two and three are better compared with that of objective 1. When the
weights are equal, for example, for efficient point 7, the achievements of the objectives are even.
The decision variables for the seven efficient points are shown in Table 4.6.
Table 4.6 Values of Decision Variables using Weighted Objective Method for Retailer 1
Retailer 1 Weights Order Interval (Days) Safety Factor
Efficient Point 1 (1, , ) 30 1.645
Efficient Point 2 ( ,1, ) 120 1.645
Efficient Point 3 ( , ,1) 55 2.33
Efficient Point 4 (1/9,4/9,4/9) 110 2.33
Efficient Point 5 (4/9,1/9,4/9) 30 2.33
Efficient Point 6 (4/9,4/9,1/9) 55 1.645
Efficient Point 7 (1/3,1/3,1/3) 55 2.31
Compromise Programming Method
Step 1: First, we get the ideals values for each objective. From Table 4.5, we can see that
Efficient Point 1 corresponds to the ideal value for objective 1 and Efficient Point 2 and 3
correspond to the ideal value for objective 2 and 3, respectively. Table 4.7 includes the ideals
solutions to each objective for retailer 1. The ideal solutions will then be used to calculate the
compromise solution for each retailer with different value of
52
Table 4.7 Ideal Solutions for Retailer 1
Retailer 1
30
275517
1.645
120
3.04
1.645
2.33
99%
55
Step 2: The decision maker for retailer 1 will then choose the weights for each objective.
In this numerical example, we assume that the decision maker chooses the ratings (relative
importance) in a scale of 1-10, of the three objectives. The ratings and the normalized weights are
presented in Table 4.8.
Table 4.8 Weights for Each Objective for Retailer 1
Retailer 1
Ratings Normalized Weights
Objective 1 7 0.5
Objective 2 2 0.143
Objective 3 5 0.357
The compromise programming problem for retailer 1 will become:
(
(
)
(( √ ) )
(
)
(
)
(
)
( ( ∫
) )
)
The compromise programming problem is then solved for three different values of . The
values for each objective are presented in Table 4.9 and the values of the order interval and
safety factor for retailer 1are presented in Table 4.10.
53
Table 4.9 Values of Each Objective for Retailer 1
Retailer 1
Objectives p=1 p=2 p=
Capital invested in inventory 346524 366631 375732
Annual number of orders 8.11 7.3 7.02
Customer service level 0.99 0.95 0.95
Table 4.10 Values of Decision Variables for Retailer 1
Retailer 1
Variables p=1 p=2 p=
Order interval (days) 45 50 52
Safety factor 2.33 1.645 1.645
Thus, 10 efficient points will be presented to the decision maker as shown in Table 4.11.
The first seven are generated using weighted objective method and the last three are generated
using compromise programming method. Note that in Table 4.11, for objective 3, we use the “in-
stock” probability to measure the service level.
Table 4.11 Ten Efficient Points Presented to the Decision Maker for Retailer 1
Retailer 1 Weights Objective 1
Capital invested in inventory
Objective 2
No. of orders
Objective 3
Service Level
Efficient Point 1 (1, , ) 275517 12.17 95%
Efficient Point 2 ( ,1, ) 684557 3.04 95%
Efficient Point 3 ( , ,1) 428789 5.79 99%
Efficient Point 4 (1/9,4/9,4/9) 643025 3.32 99%
Efficient Point 5 (4/9,1/9,4/9) 277815 12.17 99%
Efficient Point 6 (4/9,4/9,1/9) 389382 6.64 95%
Efficient Point 7 (1/3,1/3,1/3) 392163 6.64 98.9%
Efficient Point 8 p=1 346524 8.11 99%
Efficient Point 9 p=2 366631 7.3 95%
Efficient Point 10 p= 375732 7.02 95%
Ideal Solution 275517 3.04 99%
54
The decision variables for all ten efficient points are shown in Table 4.12.
Table 4.12 Values of the Decision Variables for Retailer 1
Retailer 1 Weights Order Interval (Days) Safety Factor
Efficient Point 1 (1, , ) 30 1.645
Efficient Point 2 ( ,1, ) 120 2.33
Efficient Point 3 ( , ,1) 55 2.33
Efficient Point 4 (1/9,4/9,4/9) 110 2.33
Efficient Point 5 (4/9,1/9,4/9) 30 2.33
Efficient Point 6 (4/9,4/9,1/9) 55 1.645
Efficient Point 7 (1/3,1/3,1/3) 55 2.31
Efficient Point 8 p=1 45 2.33
Efficient Point 9 p=2 50 1.645
Efficient Point 10 p= 52 1.645
For retailer 2, we repeat the same procedure used for solving the numerical problem for
retailer 1.
In the weighted objective method, for retailer 2, we use to scale objective
1, to scale objective 2 and
to scale objective 3.
The weights used in compromise programming method for retailer 2 are shown in Table 4.13.
Table 4.13 Weights for Each Objective for Retailer 2
Retailer 2
Ratings Normalized Weights
Objective 1 4 0.267
Objective 2 3 0.2
Objective 3 8 0.533
The objective values of the seven efficient points generated by the weighted objective
method and three efficient points generated by the compromise programming method for retailer
55
2 are shown in Table 4.14. The values of order interval and safety factor for retailer 2 are
also presented in Table 4.15.
Table 4.14 Values of Each Objective for Retailer 2
Retailer 2 Weights Objective 1
Capital invested in inventory
Objective 2
No. of orders
Objective 3
Service Level
Efficient Point 1 (1, , ) 198381 12.17 90%
Efficient Point 2 ( ,1, ) 639535 2.43 98.4%
Efficient Point 3 ( , ,1) 359511 5 99%
Efficient Point 4 (1/9,4/9,4/9) 626058 2.5 99%
Efficient Point 5 (4/9,1/9,4/9) 223749 10.14 99%
Efficient Point 6 (4/9,4/9,1/9) 356281 5 94.5%
Efficient Point 7 (1/3,1/3,1/3) 359423 5 98.9%
Efficient Point 8 p=1 377810 4.68 99%
Efficient Point 9 p=2 363653 4.87 94.7%
Efficient Point 10 p= 354864 5 90%
Ideal Solution 198381 2.43 99%
Table 4.15 Values of Decision Variables for Retailer 2
Retailer 1 Weights Order Interval (Days) Safety Factor
Efficient Point 1 (1, , ) 30 1.28
Efficient Point 2 ( ,1, ) 150 2.15
Efficient Point 3 ( , ,1) 73 2.33
Efficient Point 4 (1/9,4/9,4/9) 146 2.33
Efficient Point 5 (4/9,1/9,4/9) 36 2.33
Efficient Point 6 (4/9,4/9,1/9) 73 1.6
Efficient Point 7 (1/3,1/3,1/3) 73 2.31
Efficient Point 8 p=1 78 2.33
Efficient Point 9 p=2 75 1.62
Efficient Point 10 p= 73 1.28
56
4.3 Wholesaler Problem
The order interval of the retailer’s problem will be used in the wholesaler problem to
determine the order interval of the wholesaler.
The wholesaler problem is given by:
(
∑
)
( ∫
)
∑
where denote the maximum safety factor for the wholesaler and
denote the minimum
safety factor for the wholesaler. Let denote the maximum value of .
The value of needs to be determined. The number of possible values for will be
small because in practice, many restrictions and requirements will be placed on the wholesaler.
For instance, the wholesaler may have to order at least every half year due to the capacity limit of
the transportation mode used in delivering goods from the factory to the wholesaler. Moreover,
57
the order interval of the wholesaler is largely dependent on the order interval of the retailers.
Assuming a single wholesaler, single retailer supply chain system and the order interval of the
retailer as two months, the retailer orders 6 times a year. Then the number of orders of the
wholesaler will vary between 6 and 1 as goes from 1 to 6. Because it is impractical that the
wholesaler could order less than once a year, the value of is highly unlikely to be greater than 6
in this case.
Thus, the value of will be relatively small. In order to decrease the computational
complexity of the numerical example, simple enumeration of will be sufficient.
4.3.1 Solution Procedure
The solution procedure of both weighted objective method and compromise
programming method for the wholesaler problem is similar to those of the retailers. One step is
added to both solution procedures at the beginning.
Before solving the wholesaler problem, we need to determine the value of .
From the solution to the retailers’ problem, we obtain the order interval for each retailer. Recall
that
The equation above represents a linear equations system with two equations and three
variables. In order to assure that the are integers, we first compute
Then,
58
Modifications are needed for the first two objectives shown in the general model above.
For the first objective,
(
∑
)
We substitute with equation 3.2 in Chapter 3 and substitute with equation 3.1 in Chapter 3.
Therefore, the first objective can be written as,
(
∑
∑
√ ∑
)
For the second objective,
Therefore, the second objective can be written as,
4.3.2 Numerical Example for the Wholesaler Problem
A numerical example for the wholesaler problem will be presented in this section. Table
4.16 includes the costs and the lead time for the wholesaler.
Table 4.16 Values Used in the Numerical Example of Wholesaler Problem
Cost Lead Time (Days)
Wholesaler 30 10 1.645 2.33
59
The wholesaler problem is dependent on the solutions to the retailer problems. Hence, the
results of the retailer problems will be used as input.
In this numerical example, we assume that the decision maker for retailer 1 chooses the
solution corresponding to efficient point 9 in Table 4.12 and the decision maker for retailer 2
chooses the solution corresponding to efficient point 9 in Table 4.15. Therefore, the values of the
decision variables of the two retailers are shown in Table 4.17
Table 4.17 Solutions to the Retailer Problem Chosen by the Decision Makers
Order Interval Safety Factor
Retailer 1 50 1.645
Retailer 2 75 1.62
Because it will become impractical if the wholesaler orders less than once a year, we then let
Weighted Objective Method
We first solve the wholesaler problem using the weighted objective method. The weights
used are shown in Table 4.18.
60
Table 4.18 Objective Weights for the Weighted Objective Method
Efficient Point Number
1 1
2 1
3 1
4 ⁄ ⁄ ⁄
5 ⁄ ⁄ ⁄
6 ⁄ ⁄ ⁄
7 ⁄ ⁄ ⁄
In the weighted objective method, simple scaling is used for the objectives and the new
weights are
. For the wholesaler, we use
to scale objective 1, to scale
objective 2 and to scale objective 3.
Using the equal weights given in Table 4.18, the scaled weighted objective problem for
the wholesaler will become,
(
√ )
[ ( ∫
)]
LINGO is then used to solve the single objective optimization problem. The objective
values of the seven efficient points for the wholesaler, generated using the weighted objective
method, are shown in Table 4.19. The values of order interval and safety factor for the
wholesaler are presented in Table 4.20.
61
Enumeration will be used to determine the value of . We can determine which value of
to pick up by comparing the objective values generated from different values of and choose
the one with the smallest objective value since the objective function is to minimize. For
example, for efficient point 1, when , the objective value is 2.518 and when , the
objective value is 5.911. Therefore, we choose the solution when .
Table 4.19 Values of Each Objective using Weighted Objective Method for the Wholesaler
Wholesaler Weights Objective 1 Objective 2 Objective 3
Efficient Point 1 (1, , ) 503636 2.43 95%
Efficient Point 2 ( ,1, ) 1184701 1.22 97.6%
Efficient Point 3 ( , ,1) 507233 2.43 99%
Efficient Point 4 (1/9,4/9,4/9) 507233 2.43 99%
Efficient Point 5 (4/9,1/9,4/9) 506445 2.43 98.6%
Efficient Point 6 (4/9,4/9,1/9) 503636 2.43 95%
Efficient Point 7 (1/3,1/3,1/3) 506445 2.43 98.6%
Table 4.20 Values of Decision Variables Using Weighted Objective Method for the
Wholesaler
Wholesaler Weights Order Interval (Days) Safety Factor
Efficient Point 1 (1, , ) 1 150 1.645
Efficient Point 2 ( ,1, ) 2 300 1.98
Efficient Point 3 ( , ,1) 1 150 2.33
Efficient Point 4 (1/9,4/9,4/9) 1 150 2.33
Efficient Point 5 (4/9,1/9,4/9) 1 150 2.18
Efficient Point 6 (4/9,4/9,1/9) 1 150 1.645
Efficient Point 7 (1/3,1/3,1/3) 1 150 2.18
62
Note that efficient point 3 is the same as efficient point 4, efficient point 1 is the same as
efficient point 9 and efficient point 5 is the same as efficient point 7. Therefore, the weighted
objective method generates only four efficient points for the wholesaler problem.
Compromise Programming Method
Step 1: First, we get the ideals values for each objective. From Table 4.19, we can see
that Efficient Point 1 corresponds to the ideal value for objective 1 and Efficient Point 2 and 3
correspond to the ideal value for objective 2 and 3, respectively. Table 4.21 gives the ideals
solutions to each objective for the wholesaler. The ideal solutions will then be used to calculate
the compromise solution for the wholesaler with different value of
Table 4.21 Ideal Solutions for the Wholesaler
Wholesaler
1
503636 1.645
2
1.22
1.98
1
99%
2.33
Step 2: The decision maker for the wholesaler will then choose the weights for each
objective. In this numerical example, we assume that the decision maker chooses the ratings
(relative importance) in a scale of 1-10, of the three objectives. The ratings and the normalized
weights are presented in Table 4.22.
63
Table 4.22 Weights for Each Objective for the Wholesaler
Wholesaler
Ratings Normalized Weights
Objective 1 8 0.533
Objective 2 2 0.133
Objective 3 5 0.333
For the compromise programming problem, the compromise programming problem for
the wholesaler will be,
(
(
)
(( √ ) )
(
)
(
)
(
)
( ( ∫
) )
)
As mentioned above, the value of is small, so enumeration of will be sufficient when
calculating the best compromise solution. The procedure of picking up the value of is also
discussed above.
The values for each objective are presented in Table 4.23 and the values of , namely the
order interval , and safety factor for the wholesaler are presented in Table 4.24.
Table 4.23 Values of Each Objective in the Wholesaler Problem
Wholesaler
Objectives p=1 p=2 p=
Capital invested in inventory 507233 506498 503636
Annual number of orders 2.43 2.43 2.43
Customer service level 0.99 0.986 0.95
64
Table 4.24 Values of Decision Variables in the Wholesaler Problem
Wholesaler
Variables p=1 p=2 p=
1 1 1
Order interval (days) 150 150 150
Safety factor 2.33 2.19 1.645
We can see that two solutions generated by the compromise programming method when
p=1 and p= are the same as efficient point 3 and efficient point 1 in Table 4.18. Therefore, only
one new efficient point is generated by the compromise programming method.
Thus, 5 efficient points will be presented to the decision maker as shown in Table 4.25.
The first 4 are generated using weighted objective method and the last one is generated using
compromise programming method. Note that in Table 4.10, for objective 3, we use the “in-stock”
probability to measure the service level.
Table 4.25 Seven Efficient Points Presented to the Decision Maker for the Wholesaler
Wholesaler Weights Objective 1
Capital invested in inventory
Objective 2
No. of orders
Objective 3
Service Level
Efficient Point 1 (1, , ) 503636 2.43 95%
Efficient Point 2 ( ,1, ) 1184701 1.22 97.6%
Efficient Point 3 ( , ,1) 507233 2.43 99%
Efficient Point 4 (4/9,1/9,4/9) 506445 2.43 98.5%
Efficient Point 5 p=2 506498 2.43 98.6%
Ideal Solution 503636 1.22 99%
65
The decision variables for all ten efficient points are shown in Table 4.26.
Table 4.26 Values of the Decision Variables for the Wholesaler
Wholesaler Weights Order Interval (Days) Safety Factor
Efficient Point 1 (1, , ) 150 1.645
Efficient Point 2 ( ,1, ) 300 1.98
Efficient Point 3 ( , ,1) 150 2.33
Efficient Point 4 (4/9,1/9,4/9) 150 2.18
Efficient Point 5 p=2 150 2.19
4.4 More Than Two Retailers
There are usually more than two retailers in most supply chain systems. In the case of
decentralized supply chain, the number of retailers does not have an impact on the complexity of
solving the retailer problems. Every single retailer problem will always be solved in the same way
and independently. For the wholesaler problem, the number of retailers may have an impact.
Recall that the order interval of the wholesaler needs to be an integer multiple of all retailers’
order intervals. Thus, with the number of retailers increase, the order interval of the wholesaler
may become very large. This can make the ordering policy of the wholesaler impractical when
the wholesaler order frequency is less than once a year.
66
5 Single Wholesaler, Two Retailers Centralized System Problem
A centralized supply chain is not very common in practice. In real world, most supply
chains consist of many independent companies. These companies are involved with the
production process and the transportation of goods from companies in upstream to downstream
companies. However, a multi-criteria centralized supply chain system is still worthy to be studied.
In the centralized supply chain system, all the companies are controlled by a single decision
maker and it is the responsibility for that decision maker to decide the ordering policy for each
member in the supply chain. Therefore, the main objective becomes the overall performance of
the entire supply chain system and the decision will be the most preferred solution to the entire
supply chain. In a centralized supply chain, the inventory policies of all companies in the system
will be solved simultaneously. Abdul-Jalbar et al (2003) point out that there are two possible
scenarios in a centralized supply chain system: the first scenario is common retailer
replenishment time and the second one is different retailer replenishment time. These two
scenarios are also called common replenishment epochs and different replenishment epochs,
respectively. Same as the previous chapter, the problem is restricted to two retailers. After
providing the solution to the problem of two retailers, case of multiple retailers will also be
discussed.
The model for the centralized supply chain system is similar to that for decentralized
supply chain system. The main difference is that, in decentralized case, the retailer problem has
been solved first for each retailer. With the results of the retailer problem, we then solve the
67
wholesaler problem. However, in the centralized case, all the decision variables will be
determined at the same time for both the retailers and the wholesaler.
Recall the general model for centralized supply chain developed in Chapter 3, section 3.4.
The general model for the single wholesaler, two retailers centralized supply chain system is,
[(
) (
)
( ∑
) ]
[ ( ∫
) ( ∫
)
( ∫
)]
∑
is an integer
5.1 Common Replenishment Epochs
We will first present the special case where all the retailers will place their order at the
same time.
68
5.1.1 Assumptions
Several assumptions will be made for this supply chain system. The demand seen by the
retailer is stochastic, stationary and independent and follows a normal distribution. The lead time
for each company is a constant. We also assume that the lead time for the wholesaler is
negligible. Hence, the expected number of backorders of the wholesaler equals 0 and the
retailers’ order will always be satisfied on the average. The decision maker for each company will
use a periodic review policy by reviewing the inventory level at a fixed period of time and
ordering an amount which brings the inventory to the order-up-to level. In addition, we assume
that the supplier for the wholesaler has unlimited supply of goods. The unit cost of the product at
the wholesaler is again smaller than that at any retailer. The notations used in this chapter are the
same as those listed in section 4.1.2.
In the first objective,
(
) (
)
( ∑
)
We substitute with equation 3.2 in Chapter 3 and substitute with equation 3.1 in Chapter 3.
Then the first objective is given as,
(
) (
)
(
∑
∑
√ ∑
)
69
In the second objective,
The third objective remains the same and is presented as,
( ∫
) ( ∫
) ( ∫
)
In the case of common replenishment epochs, and is not needed because the
orders of the retailer are placed at the same time.
Let denote the common order interval for both retailers.
Therefore, the model for the single wholesaler, multiple retailers centralized supply chain
system with common replenishment epochs will be as follows:
(
) (
)
(
∑
∑
√ ∑
)
( ∫
) ( ∫
)
( ∫
)
70
is an integer
where we let denote the maximum order interval for retailers.
Let denote the minimum order interval for retailers.
Let denote the maximum safety factor for company
Let denote the minimum safety factor for company
Let denote the maximum value of .
The determination of the value of follows the same procedure discussed in section
4.3. In order to decrease the computational complexity of the numerical example, simple
enumeration of will be sufficient. The process of picking up the value of follows the same
procedure discussed in section 4.3.2.
5.1.2 Solution Procedure
The solution procedure of both weighted objective method and compromise programming
method for the centralized supply chain with common replenishment epochs are the same as those
for the retailer problem in Chapter 4.
5.1.3 Numerical Example for Common Replenishment Epochs
We will use the same data for the retailers and wholesaler problems given in chapter 4. Table
4.4 is reprinted below as Table 5.1 and Table 4.12 is reprinted below as Table 5.2.
71
Table 5.1 Retailers Data Used in the Numerical Example of Centralized Supply Chain
Mean
Daily
Demand
Variation
of Daily
Demand
Cost
Lead
Time
(Days)
Retailer 1 180 100 50 15 1.645 2.33
Retailer 2 120 64 60 12 1.28 2.33
Table 5.2 Wholesaler Data Used in the Numerical Example of Centralized Supply Chain
Cost Lead Time (Days)
Wholesaler 30 10 1.645 2.33
For the upper bound and lower bound of the safety factor, we still use the same value
given in Chapter 4. For the upper bound and lower bound of the order interval of the retailers, we
set and
. We also set .
Weighted Objective Method
We first solve the problem for the entire supply chain using the weighted objective
method. The weights used to solve this problem are shown in Table 5.3, where is a small
positive number.
Table 5.3 Values of Weights Using the Weighted Objective Method
Efficient Point Number
1 1
2 1
3 1
4 ⁄ ⁄ ⁄
5 ⁄ ⁄ ⁄
6 ⁄ ⁄ ⁄
7 ⁄ ⁄ ⁄
72
In the weighted objective method, simple scaling is used to scale the objectives and the
new weights are
. In this example, we use
to scale objective 1, to
scale objective 2 and to scale objective 3.
For example, using the equal weights given in the last row of Table 5.5, the scaled
weighted objective problem for the entire supply chain will become,
((
√ )
(
√ )
√ [ ] )
(
)
[ ( ∫
) ( ∫
) ( ∫
)]
and is an integer
The model is then solved using LINGO. The objective values of the seven efficient points
for the entire supply chain, generated using the weighted objective method, are shown in Table
5.4. The values of order interval and safety factor for each company in the supply chain are
presented in Table 5.5. Note that in Table 5.4, for objective 3, we use the “in-stock” probability to
measure the service level.
73
Table 5.4 Values of Each Objective Using Weighted Objective Method for the Supply Chain
Supply Chain Weights
Objective 1
Inventory
Capitals
Objective 2
No. of
orders
Objective 3
Service Level
Retailer 1 Retailer 2 Wholesaler
Efficient Point 1 (1, , ) 732918 21.9 95% 90% 95%
Efficient Point 2 ( ,1, ) 2048401 8.52 95% 90% 95%
Efficient Point 3 ( , ,1) 1155280 10.95 99% 99% 99%
Efficient Point 4 (1/9,4/9,4/9) 1609236 9.125 99% 99% 99%
Efficient Point 5 (4/9,1/9,4/9) 741161 21.9 98.7% 98.8% 99%
Efficient Point 6 (4/9,4/9,1/9) 1127413 11.17 95% 91.1% 95%
Efficient Point 7 (1/3,1/3,1/3) 1128102 11.29 98.3% 98.4% 98.7%
Table 5.5 Values of Decision Variables using Weighted Objective Method for the Supply
Chain
Supply Chain Weights Order Interval (Days) Safety Factor
Retailers Wholesaler Retailer 1 Retailer 2 Wholesaler
Efficient Point 1 (1, , ) 1 50 50 1.645 1.28 1.645
Efficient Point 2 ( ,1, ) 3 100 300 1.645 1.28 1.645
Efficient Point 3 ( , ,1) 1 100 100 2.33 2.33 2.33
Efficient Point 4 (1/9,4/9,4/9) 2 100 200 2.33 2.33 2.33
Efficient Point 5 (4/9,1/9,4/9) 1 50 50 2.24 2.27 2.33
Efficient Point 6 (4/9,4/9,1/9) 1 98 98 1.645 1.35 1.645
Efficient Point 7 (1/3,1/3,1/3) 1 97 97 2.12 2.14 2.24
Compromise Programming Method
Step 1: First we get the ideals values for each objective. From Table 5.4, we can see that
Efficient Point 1 corresponds to the ideal value for objective 1 and Efficient Point 2 and 3
correspond to the ideal value for objective 2 and 3, respectively. Table 5.6 includes the ideals
74
solutions to each objective for the entire supply chain. The ideal solutions will then be used to
calculate the compromise solution for the entire supply chain with different value of
Table 5.6 Ideal Solutions for Each Objective in the Entire Supply Chain
Supply chain
50
732918
1.645
1.28
1.645
1
100
8.52
1.645
1.28
1.645
3
100
-2.97
2.33
2.33
2.33
1
Step 2: The decision maker for the entire supply chain will then choose the weights for
each objective. In this numerical example, we assume that the decision maker chooses the rating
(relative importance) in a scale of 1-10, of the three objectives. The ratings and the normalized
weights are presented in Table 5.7.
Table 5.7 Weights for Each Objective for the Supply Chain
Centralized Supply Chain
Actual Weight Normalized Weight
Objective 1 9 0.5
Objective 2 4 0.222
Objective 3 5 0.278
75
The compromise programming problem for the centralized supply chain will become:
[ {(
)
(( √ )
( √ )
√ [ ] }
{(
)
(
)
}
{(
)
( ∫
∫
∫
)
} ]
and is an integer
The compromise programming problem is then solved for three different values of . The
values for each objective are presented in Table 5.8 and the values of the order interval and
safety factor for all the members in the supply chain are presented in Table 5.9.
Table 5.8 Values of Each Objective Using Common Replenishment Epochs
Centralized Supply Chain
Objectives p=1 p=2 p=
Total capital invested in inventory 914878 938390 954809
Sum of annual number of orders 15.42 14.6 14.22
Customer service level for Retailer 1 98.6% 95% 95%
Customer service level for Retailer 2 98.8% 90% 90%
Customer service level for Wholesaler 99% 95% 95%
Average customer service level 98.8% 93.3% 93.3%
76
Table 5.9 Values of Decision Variables Using Common Replenishment Epochs
Retailer 1
Variables p=1 p=2 p=
Order interval (days) 71 75 77
Safety factor 2.22 1.645 1.645
Retailer 2
Variables p=1 p=2 p=∞
Order interval (days) 71 75 77
Safety factor 2.25 1.28 1.28
Wholesaler
Variables p=1 p=2 p=∞
1 1 1
Order interval (days) 71 75 77
Safety factor 2.33 1.645 1.645
Thus, 10 efficient points will be presented to the decision maker as shown in Table 5.10.
The first seven are generated using the weighted objective method and the last three are generated
using the compromise programming method. Note that in Table 5.10, for objective 3, we use the
“in-stock” probability to measure the service level.
Table 5.10 Ten Efficient Points Presented to the Decision Maker for the Supply Chain
Supply Chain Weights
Objective 1
Invested
Captial
Objective 2
No. of
orders
Objective 3
Service Level
Retailer
1
Retailer
2
Whole-
saler Average
Efficient Point 1 (1, , ) 732918 21.9 95% 90% 95% 93.3%
Efficient Point 2 ( ,1, ) 2048401 8.52 95% 90% 95% 93.3%
Efficient Point 3 ( , ,1) 1155280 10.95 99% 99% 99% 99%
Efficient Point 4 (1/9,4/9,4/9) 1609236 9.125 99% 99% 99% 99%
Efficient Point 5 (4/9,1/9,4/9) 741161 21.9 98.7% 98.8% 99% 98.8%
Efficient Point 6 (4/9,4/9,1/9) 1127413 11.17 95% 91.1% 95% 93.7%
Efficient Point 7 (1/3,1/3,1/3) 1128102 11.29 98.3% 98.4% 98.7% 98.5%
Efficient Point 8 p=1 914878 15.42 98.6% 98.8% 99% 98.8%
Efficient Point 9 p=2 938390 14.6 95% 90% 95% 93.3%
Efficient Point 10 p=∞ 954809 14.22 95% 90% 95% 93.3%
Ideal Solution 732918 8.52 99% 99% 99% 99%
77
The decision variables for all ten efficient points are shown in Table 5.11.
Table 5.11 Values of the Decision Variables for the Supply Chain
Supply Chain Weights Order Interval (Days) Safety Factor
Retailers Wholesaler Retailer 1 Retailer 2 Wholesaler
Efficient Point 1 (1, , ) 1 50 50 1.645 1.28 1.645
Efficient Point 2 ( ,1, ) 3 100 300 1.645 1.28 1.645
Efficient Point 3 ( , ,1) 1 100 100 2.33 2.33 2.33
Efficient Point 4 (1/9,4/9,4/9) 2 100 200 2.33 2.33 2.33
Efficient Point 5 (4/9,1/9,4/9) 1 50 50 2.24 2.27 2.33
Efficient Point 6 (4/9,4/9,1/9) 1 98 98 1.645 1.35 1.645
Efficient Point 7 (1/3,1/3,1/3) 1 97 97 2.12 2.14 2.24
Efficient Point 8 p=1 1 71 71 2.22 2.25 2.33
Efficient Point 9 p=2 1 75 75 1.645 1.28 1.645
Efficient Point 10 p=∞ 1 77 77 1.645 1.28 1.645
The ten efficient points will be presented to the decision maker. The best compromise
solution will be achieved if the decision maker is satisfied with one of the solutions. If the
decision maker is not satisfied with any of the solutions presented, additional weights can be
generated and new efficient solutions can be generated. We can generate additional weights
around the efficient solution which the decision maker prefers using interval programming. These
steps will be repeated until the best compromise solution is found.
5.2 Different Replenishment Epochs
In this case, the supply chain system is still with one decision maker, who is responsible
for determining the inventory policy for all retailers and for the wholesaler. However, the
replenishment times for the retailers do not need to be the same.
78
5.2.1 Assumptions
The assumptions for this model are similar to that of the centralized supply chain model
using common replenishment epochs. The demand seen by the retailer is stochastic, stationary
and independent and follows a normal distribution. The lead time for each company is a constant.
We also assume that the lead time for the wholesaler is negligible. Hence, the expected
number of backorders of the wholesaler equals 0 and the retailers’ order will always be satisfied
on the average. The decision maker for each company will use a periodic review policy by
reviewing the inventory level at a fixed period of time and order the quantity which brings the
inventory to the order-up-to level. In addition, we assume that the supplier for the wholesaler has
unlimited supply of goods. The cost of the product at the wholesaler is again smaller than that at
any retailers. The notations are still the same as those listed in section 4.1.2.
The model for the centralized supply chain system using different replenishment epochs is
presented below:
[(
) (
)
(
∑
∑
√ ∑
)
]
[ ( ∫
) ( ∫
)
( ∫
)]
79
are integers
where we let denote the maximum order interval for retailer
Let denote the minimum order interval for retailer
Let denote the maximum safety factor for company
Let denote the minimum safety factor for company
Let denote the maximum value of .
Let denote the upper bounds of
Because the order intervals of the retailers are not necessarily the same, ’s are needed
in the model. The upper bounds of ’s also need to be determined by the decision maker. The
constraint must be satisfied, without the upper bounds of ’s, ’s may become
very large which may lead to the impractical order interval for the wholesaler. Because there are
three integers in this model, enumeration will no longer be used in solving this model.
5.2.2 Solution Procedure
The solution procedure of both weighted objective method and compromise
programming method for the centralized supply chain with different replenishment epochs are the
same as those for the centralized supply chain with common replenishment epochs.
80
5.2.3 Numerical Example for Different Replenishment Epochs
We will use the same data used in Section 5.1.3. Table 5.3 is reprinted below as Table
5.12 and Table 5.4 is reprinted below as Table 5.13.
Table 5.12 Retailers Data Used in the Numerical Example of Centralized Supply Chain
Mean
Daily
Demand
Variation
of Daily
Demand
Cost
Lead
Time
(Days)
Retailer 1 180 100 50 15 30 120 1.645 2.33
Retailer 2 120 64 60 12 30 150 1.28 2.33
Table 5.13 Wholesaler Data Used in the Numerical Example of Centralized Supply Chain
Cost Lead Time (Days)
Wholesaler 30 10 1.645 2.33
For the upper bound and lower bound of the safety factor and the order intervals, we still
use the same values given in Chapter 4. For the upper bounds of ’s, we set and
. For the upper bound of , we set .
Weighted Objective Method
We first solve the problem for the entire supply chain with different replenishment
epochs using the weighted objective method. The weights used to solve this problem are shown in
Table 5.14.
Table 5.14 Weights for the Objectives Using Weighted Objectives Method
Efficient Point Number
1 1
2 1
3 1
4 ⁄ ⁄ ⁄
5 ⁄ ⁄ ⁄
6 ⁄ ⁄ ⁄
7 ⁄ ⁄ ⁄
81
In the weighted objective method, simple scaling is used to scale the objectives and the
new weight are
. In this example, we use
to scale objective 1, to
scale objective 2 and to scale objective 3.
Using the equal weights in the last row of Table 5.5, the weighted objective method for
the entire supply chain will become,
((
√ )
(
√ )
( [ ]
)
√ ( ) )
(
)
[ ( ∫
) ( ∫
) ( ∫
)]
are integers
82
The model is then solved using LINGO. The objective values of seven efficient points for
the entire supply chain, generated using the weighted objective method, are shown in Table 5.15.
The values of order interval and safety factor for each company in the supply chain are
presented in Table 5.16.
Table 5.15 Values of Each Objective Using Weighted Objective Method for the Supply
Chain
Supply Chain Weights
Objective 1
Inventory
Capitals
Objective 2
No. of
Orders
Objective 3
Service Level (In-Stock Probability)
Retailer 1 Retailer 2 Wholesaler
Efficient Point 1 (1, , ) 568083 36.5 95% 90% 95%
Efficient Point 2 ( ,1, ) 4571438 6.42 97.7% 96.5% 97.7%
Efficient Point 3 ( , ,1) 575389 36.5 99% 99% 99%
Efficient Point 4 (1/9,4/9,4/9) 1320667 7.10 97.8% 97.9% 96.9%
Efficient Point 5 (4/9,1/9,4/9) 661673 21.29 98.7% 98.8% 98.2%
Efficient Point 6 (4/9,4/9,1/9) 992329 10.51 95% 90.7% 95%
Efficient Point 7 (1/3,1/3,1/3) 992028 10.65 98.2% 98.3% 97.5%
Table 5.16 Values of Decision Variables using Weighted Objective Method for the Supply
Chain
Supply
Chain Weights
Order Interval (Days) Safety Factor
Retailer Wholesaler
Retailer Wholesaler
1 2 1 2
Efficient
Point 1 (1, , ) 1 1 1 30 30 30 1.645 1.28 1.645
Efficient
Point 2 ( ,1, ) 3 3 3 120 120 1080 1.988 1.81 1.988
Efficient
Point 3 ( , ,1) 1 1 1 30 30 30 2.33 2.33 2.33
Efficient
Point 4 (1/9,4/9,4/9) 1 1 3 120 120 360 2.01 2.03 1.87
Efficient
Point 5 (4/9,1/9,4/9) 1 1 3 40 40 120 2.22 2.25 2.10
Efficient
Point 6 (4/9,4/9,1/9) 1 1 3 81 81 243 1.645 1.32 1.645
Efficient
Point 7 (1/3,1/3,1/3) 1 1 3 80 80 240 2.10 2.12 1.96
83
Compromise Programming Method
Step 1: First we get the ideals values for each objective. From Table 5.15, we can see that
Efficient Point 1 corresponds to the ideal value for objective 1 and Efficient Point 2 and 3
correspond to the ideal value for objective 2 and 3, respectively. Table 5.17 includes the ideals
solutions to each objective for the entire supply chain. The ideal solutions will then be used to
calculate the compromise solution for the entire supply chain with different value of .
Table 5.17 Ideal Solutions for Each Objective for Different Replenishment Epochs
Supply chain
30
568083
30
1.645
1.28
1.645
1
1
1
120
6.42
120
1.988
1.81
1.988
3
3
3
30
-2.97
30
2.33
2.33
2.33
1
1
1
84
Step 2: The decision maker for the entire supply chain will then choose the weights for
each objective. In this numerical example, we assume that the decision maker chooses the rating
(relative importance) in a scale of 1-10, of the three objectives. The ratings and the normalized
weights are presented in Table 5.18.
Table 5.18 Weights for Each Objective for the Supply Chain
Centralized Supply Chain
Actual Weight Normalized Weight
Objective 1 9 0.5
Objective 2 4 0.222
Objective 3 5 0.278
The compromise programming problem for the centralized supply chain will become:
[ {(
)
(( √ )
( √ ) [ ]
√ [ ] }
{(
)
(
)
}
{(
)
( ∫
∫
∫
)
} ]
85
are integers
The compromise programming problem is then solved for the three different values of .
The values for each objective are presented in Table 5.19 and the values of the order interval
and safety factor for all the members in the supply chain are presented in Table 5.20.
Table 5.19 Values of Each Objective Using Different Replenishment Epochs
Centralized Supply Chain
Objectives p=1 p=2 p=
Total capital invested in inventory 859945 843880 843880
Sum of annual number of orders 13.3 13.5 13.5
Customer service level for Retailer 1 98.3% 95% 95%
Customer service level for Retailer 2 98.4% 90% 90%
Customer service level for Wholesaler 97.7% 95% 95%
Average customer service level 98.1% 93.3% 93.3%
Table 5.20 Values of Decision Variables Using Different Replenishment Epochs
Retailer 1
Variables p=1 p=2 p=
Order interval (days) 64 63 63
Safety factor 2.13 1.645 1.645
Retailer 2
Variables p=1 p=2 p=∞
Order interval (days) 64 63 63
Safety factor 2.15 1.28 1.28
Wholesaler
Variables p=1 p=2 p=∞
3 3 3
Order interval (days) 192 189 189
Safety factor 1.99 1.645 1.645
86
Note that the solution of p=2 is the same as that of p= . Therefore, the compromise
programming method generates two new efficient solutions.
Therefore, 9 efficient points are presented to the decision maker as shown in Table 5.21.
The first seven are generated using the weighted objective method and the last two are generated
using the compromise programming method. Note that in Table 5.21, for objective 3, we use the
“in-stock” probability to measure the service level.
Table 5.21 Nine Efficient Points Presented to the Decision Maker for the Supply Chain
Supply Chain Weights
Objective
1
Invested
Capital
Objective
2
No. of
Orders
Objective 3
Service Level
Retailer
1
Retailer
2
Whole-
saler Average
Efficient Point 1 (1, , ) 568083 36.5 95% 90% 95% 93.3%
Efficient Point 2 ( ,1, ) 4571438 6.42 97.7% 96.5% 97.7% 97.3%
Efficient Point 3 ( , ,1) 575389 36.5 99% 99% 99% 99%
Efficient Point 4 (1/9,4/9,4/9) 1320667 7.10 97.8% 97.9% 96.9% 97.5%
Efficient Point 5 (4/9,1/9,4/9) 661673 21.29 98.7% 98.8% 98.2% 98.6%
Efficient Point 6 (4/9,4/9,1/9) 992329 10.51 95% 90.7% 95% 93.6%
Efficient Point 7 (1/3,1/3,1/3) 992028 10.65 98.2% 98.3% 97.5% 98%
Efficient Point 8 p=1 859945 13.3 98.3% 98.4% 97.7% 98.1%
Efficient Point 9 p=2 843880 13.5 95% 90% 95% 93.3%
Ideal Solution 568083 6.42 99% 99% 99% 99%
87
The decision variables for all nine efficient points are shown in Table 5.22.
Table 5.22 Values of the Decision Variables for the Supply Chain
Supply
Chain Weights
Order Interval (Days) Safety Factor
Retailer Wholesaler
Retailer Wholesaler
1 2 1 2
Efficient
Point 1 (1, , ) 1 1 1 30 30 30 1.645 1.28 1.645
Efficient
Point 2 ( ,1, ) 3 3 3 120 120 1080 1.988 1.81 1.988
Efficient
Point 3 ( , ,1) 1 1 1 30 30 30 2.33 2.33 2.33
Efficient
Point 4 (1/9,4/9,4/9) 1 1 3 120 120 360 2.01 2.03 1.87
Efficient
Point 5 (4/9,1/9,4/9) 1 1 3 40 40 120 2.22 2.25 2.10
Efficient
Point 6 (4/9,4/9,1/9) 1 1 3 81 81 243 1.645 1.32 1.645
Efficient
Point 7 (1/3,1/3,1/3) 1 1 3 80 80 240 2.10 2.12 1.96
Efficient
Point 8 p=1 1 1 3 64 64 192 2.13 2.15 1.99
Efficient
Point 9 p=2 1 1 3 63 63 189 1.645 1.28 1.645
The nine efficient points will be presented to the decision maker. The best compromise
solution will be achieved if the decision maker is satisfied with one of the solutions. If the
decision maker is not satisfied with any of the solutions presented, additional weights can be
generated and new efficient solutions can be generated. We can generate additional weights
around the efficient solution which the decision maker prefers using interval programming. These
steps will be repeated until the best compromise solution is found.
88
6 Conclusions
The use of periodic review policy in supply chain system is limited in the past literature
even though the periodic review policy is widely used in practice. In addition, many of the
previous studies focused on the centralized supply chain system with only one decision maker.
But de-centralized supply chain system with multiple decision makers is more common.
This thesis has built multi-criteria inventory models with stochastic demand for both de-
centralized and centralized supply chain systems. Three criteria are considered: 1.capital invested
in inventory; 2. annual number of orders; 3. customer service level. After the completion of the
model, weighted objective method and Zeleny’s compromise programming method are used and
steps for the solution procedure are presented. For all models, numerical examples are provided in
order to give clear illustration of how to solve the problem and also give a deeper insight into the
multi-criteria solutions. Solving the decentralized supply chain system problem can help us
further study the problems with conflicting objectives and multiple decision makers. The best
compromise solution can also be obtained for each member in the supply chain based on the
preference of each decision maker. The solution procedure of solving the centralized supply chain
system problem with only one decision maker can lead us to the best compromise solution which
is best for the entire supply chain system based on the decision maker’s preference.
89
6.1 Policy Comparisons
6.1.1 Common Replenishment Epochs vs. Different Replenishment Epochs
In the centralized supply chain system, two different cases are examined in the thesis, the
common replenishment epochs and different replenishment epochs. The weighted objective
method and the compromise programming method use the same weights in both cases, so we
compare the results of these two cases. Table 5.10 is reprinted as Table 6.1 below and Table 5.16
is reprinted as Table 6.2 below.
Table 6.1 Values of Each Objective Using Common Replenishment Epochs
Supply Chain Weights
Objective
1
Invested
Captial
Objective
2
No. of
orders
Objective 3
Service Level
Retailer
1
Retailer
2
Whole-
saler Average
Efficient Point
1 (1, , ) 732918 21.9 95% 90% 95% 93.3%
Efficient Point
2 ( ,1, ) 2048401 8.52 95% 90% 95% 93.3%
Efficient Point
3 ( , ,1) 1155280 10.95 99% 99% 99% 99%
Efficient Point
4 (1/9,4/9,4/9) 1609236 9.125 99% 99% 99% 99%
Efficient Point
5 (4/9,1/9,4/9) 741161 21.9 98.7% 98.8% 99% 98.8%
Efficient Point
6 (4/9,4/9,1/9) 1127413 11.17 95% 91.1% 95% 93.7%
Efficient Point
7 (1/3,1/3,1/3) 1128102 11.29 98.3% 98.4% 98.7% 98.5%
Efficient Point
8 p=1 914878 15.42 98.6% 98.8% 99% 98.8%
Efficient Point
9 p=2 938390 14.6 95% 90% 95% 93.3%
Efficient Point
10 p=∞ 954809 14.22 95% 90% 95% 93.3%
Ideal Solution 732918 8.52 99% 99% 99% 99%
90
Table 6.2 Values of Each Objective Using Different Replenishment Epochs
Supply Chain Weights
Objective
1
Invested
Capital
Objective
2
No. of
Orders
Objective 3
Service Level
Retailer
1
Retailer
2
Whole-
saler Average
Efficient Point 1 (1, , ) 568083 36.5 95% 90% 95% 93.3%
Efficient Point 2 ( ,1, ) 4571438 6.42 97.7% 96.5% 97.7% 97.3%
Efficient Point 3 ( , ,1) 575389 36.5 99% 99% 99% 99%
Efficient Point 4 (1/9,4/9,4/9) 1320667 7.10 97.8% 97.9% 96.9% 97.5%
Efficient Point 5 (4/9,1/9,4/9) 661673 21.29 98.7% 98.8% 98.2% 98.6%
Efficient Point 6 (4/9,4/9,1/9) 992329 10.51 95% 90.7% 95% 93.6%
Efficient Point 7 (1/3,1/3,1/3) 992028 10.65 98.2% 98.3% 97.5% 98%
Efficient Point 8 p=1 859945 13.3 98.3% 98.4% 97.7% 98.1%
Efficient Point 9 p=2 843880 13.5 95% 90% 95% 93.3%
Ideal Solution 568083 6.42 99% 99% 99% 99%
All the solutions provided in the two tables above are efficient solutions. Assume that the
decision maker is satisfied with Efficient Point 7 for both cases, we can see that when using
common replenishment epochs, the total capital invested inventory is 13.7% higher than that
using different replenishment epochs. The sum of annual number of orders is 6% higher than that
using different replenishment epochs. However, the average customer level of the entire supply
chain is 98.8% using common replenishment epochs while it is 98% when using different
common replenishment epochs.
The inventory policy with different replenishment epochs will intuitively result in a better
solution than the policy with common replenishment epochs because the policy with different
replenishment epochs allows retailers to order at different times and has more flexibility. We can
also find that the solutions to the case of common replenishment epochs are also feasible
91
solutions to the case of different replenishment epochs. From the numerical examples, we can
also say that the solutions to the case of common replenishment epochs are also efficient
solutions to the case of different replenishment epochs.
6.1.2 Decentralized vs. Centralized
Intuitively, the centralized supply chain should outperform the de-centralized supply
chain when considering the performance of the entire supply chain. That is because in a
decentralized supply chain system, each member only focuses on their own objectives and their
own benefits and the performance of the entire supply chain is likely to suffer from these
decisions.
Consider the numerical examples we used in Chapters 4 and 5. For the decentralized
system, we assume that retailers 1 and 2 both choose the Efficient Point 9 from Table 4.10 and
Table 4.13, respectively. The wholesaler choose the Efficient Point 3 from Table 4.24 based on
the retailers’ results, the objective values for each company and the objective values for the entire
supply chain are shown in Table 6.3. Here the objective values for the first two objectives of the
entire supply chain are the sum of that for all companies and the customer service level of the
entire supply chain is the average value of that for all companies.
92
Table 6.3 Objective Values for Each Company and for the Entire Supply Chain in
Decentralized Case
Capital invested in
inventory
Annual number of
orders
Customer Service
Level
Retailer 1
(Efficient Point 9) 366631 7.3 95%
Retailer 2
(Efficient Point 9) 363653 4.87 94.7%
Wholesaler
(Efficient Point 3) 507233 2.43 99%
Entire Supply Chain 1237517 14.6 96.2%
For the centralized system using different replenishment epochs, the objective values are
shown in Table 6.2. The reason we choose the case of different replenishment epochs is that in a
decentralized supply chain system, the order intervals of the retailers are more likely to be
different from each other, and it is more similar to the case of different replenishment epochs in a
centralized system. Table 6.4 is the comparison between the decentralized supply chain system
and the centralized system using different replenishment epochs. We assume that the decision
maker is satisfied with Efficient Point 8 in Table 6.2 for the centralized case.
Table 6.4 Comparison of Decentralized System with Centralized System
Capital invested in
inventory
Annual number of
orders
Average Customer
Service Level
Decentralized system 1237517 14.6 96.2%
Centralized system with
different replenishment
epochs
859945 13.3 98.1%
From Table 6.4, we can see that for the three criteria, the solution to the centralized
system outperforms that of the decentralized system. Thus, we can say that in this numerical
example, the solution to the decentralized system is dominated by the solution to the centralized
system with different replenishment epochs. However, this may not always be true when
93
extended to general cases because the solutions generated are largely dependent on the weights
given by the decision maker. Also, the solutions are dependent on which solution the decision
maker is satisfied with. For example, in the numerical example of the decentralized system, if the
decision maker of retailer 1chooses another efficient point, then the wholesaler’s problem is
affected and needs to be solved again, so the objective values of the entire supply chain will
change and the solution to the decentralized system may not be dominated by that to the
centralized system with different replenishment epochs.
6.2 Future Work
This thesis focuses on the case where the demand is stochastic and lead time is
deterministic. A possible extension for further study would be looking at the case of stochastic
lead time. In the case of stochastic demand and lead time, the calculation of the mean demand and
variation during lead time will be affected.
Transportation cost is also a part of the total cost of the entire supply chain. Therefore, a
new study of the supply chain system could include the transportation costs between companies
of different stages.
There is more than one wholesaler in many cases. Multiple wholesalers, multiple retailers
could be used in future studies and also the model may not only limited to a two-stage supply
chain system. In addition, building a model with multiple types of goods would also enhance the
study.
94
References
Abdul-Jalbar, B., Guiterrez, J., Puerto J. and Sicilia J., 2003, “Policies for inventory/distribution
systems: The effect of centralization vs. decentralization”, International Journal of Production
Economics, 81-82, 281-293.
Agrell, P.J., 1995, “A multicriteria framework for inventory control”, International Journal of
Production Economics, 41, 59-70.
Axsater, S., 1992, “Evaluation of (R,Q)-policies for two-level inventory systems with Poisson
demand”, Operations Research, 46, 135-145.
Axsater, S. and Juntti, L., 1997, “Comparison of echelon stock and installation stock policies for
two-level inventory systems”, International Journal of Production Economics, 45, 303-310.
Axsater, S., 1997, “Simple evaluation of echelon stock (R,Q) policies for two-level inventory
systems”, IIE Transactions, 29, 661-669.
Axsater, S. and Zhang, W., 1999, “A joint replenishment policy for multi-echelon inventory
control”, International Journal of Production Economics, 59, 243-250.
Beamon, B.M., 1998, “Supply chain design and analysis: Models and methods”, International
Journal of Production Economics, 55, 281-294.
Bookbinder, J.H. and Chen, V.Y.X., 1992, “Multicriteria trade-offs in a warehouse/retailer
system”, Journal of the Operational Research Society, 43, 707-720.
Buffa, F.P. and Munn, J.R., 1989, “A recursive algorithm for order cycle-time that minimizes
logistics cost”, The Journal of Operational Research Society, 40, 367-377.
Cachon, G.P., 2001, “Exact evaluation of batch-ordering inventory policies in two-echelon supply
chains with periodic review”, Operations Research, 49, 79-98.
95
Clark, A.J. and Scarf, H., 1960, “Optimal policies for multi-echelon inventory problem”,
Management Science, 6, 470-475.
Clark, A.J., 1972, “An informal survey of multi-echelon inventory theory”, Naval Research
Logistics, 19, 271-275.
Crowston, W.B., Wagner, M. and Williams, J.F., 1973, “Economic lot size determination in
multi-stage assembly systems”, Management Science, 19, 517-527.
De Bodt, M.A. and Graves, S.C., 1985, “Continuous-review policies for a multi-echelon
inventory problem with stochastic demand”, Management Science, 31, 1286-1299.
DiFilippo, A.M., 2003, “Multi criteria supply chain inventory models with transportation costs”,
MS thesis Harold and Inge Marcus Department of Industrial and Manufacturing Engineering.
State College: The Pennsylvania State University.
Ettl, M., Feigin, G.E., Lin, G.Y. and Yao, D.D., 1996, “A supply network model with base-stock
control and service requirements”, IBM Research Report.
Evans II, G.W., 1958, “A transportation and production model”, Naval Research Logistics
Quarterly, 5, 137-154.
Federgruen, A. and Zipkin, P., 1984, “Computational issues in an infinite-horizon, multi-echelon
inventory model”, Operations Research, 32, 818-836.
Jr. Gardner, E.S. and Dannenbring, D.G., 1979, “Using optimal policy surfaces to analyze
aggregate inventory tradeoffs”, Management Science, 25, 709-720.
Geoffrion, A.M., 1968, “Proper efficiency and the theory of vector maximization”, Journal of
Mathematical Analysis and Applications, 22, 618-630.
Handley, G. and Whitin, T.M., 1963, Analysis of inventory systems, Englewood Cliffs, N.J.
96
Kalymon, B.A., 1970, “A decomposition algorithm for arborescence inventory systems”,
Operations Research, 20, 860-874.
Keeny, R. and Raiffa, H., 1993, Decisions with multiple objectives: preferences and value
tradeoffs, Cambridge, England: Cambridge University Press.
Lee, S.M., 1972, Goal programming for decision analysis, Philadelphia, PA: Auerbach
Publishers.
Lenard, J.D. and Roy, B., 1994, “Multi-item inventory control: A multicriteria view”, European
Journal of Operations Research, 87, 685-692.
Muckstadt, J.A., 1979, “A three echelon, multi item model for recoverable items”, Naval
Research Logistics Quarterly, 26, 199-222.
Natarajan, A., 2007, “Multi-criteria supply chain inventory models with transportation costs”,
PhD thesis Harold and Inge Marcus Department of Industrial and Manufacturing Engineering.
State College: The Pennsylvania State University.
Puerto, J. and Fernndez, F.R., 1998, “Pareto-optimality in classical inventory problems”, Naval
Research Logistics, 45, 83-98.
Ravindran, A., 2008, Operations research and management science handbook, CRC Press, Boca
Raton, FL.
Ravindran, A. and Warsing, D.P., 2012, Supply chain engineering: Models and applications,
Taylor & Francis, Boca Raton, FL.
Sadagopan, S. and Ravindran, A., 1982, “Interactive solution of bicriteria mathematical
programs”, Naval Research Logistics, 29, 442-459.
Shin, W.S. and Ravindran, A., 1991, “Interactive multiple objective optimization: Survey I –
Continuous Case”, Computers and Operations Research, 18, 97-114.
97
Starr, M.K. and Miller, D.W., 1962, Inventory Control: Theory and Practice , Prentice Hall,
Englewood Cliffs, N.J.
Steuer, R.E., 1986, Multiple criteria optimization: theory, computation and application, Wiley,
New York.
Svoronos, A. and Zipkin, P., 1988, “Estimating the performance of multi-level inventory
systems”, Operations Research, 36, 57-72.
Taha, H.A. and Skeith, R.W., 1970, “The economic lot size in multistage production system”,
AIIE Trans, 2, 157-162.
Thirumalai, R., 2001, “Multi criteria-multi decision maker inventory models for serial supply
chains”, PhD thesis Harold and Inge Marcus Department of Industrial and Manufacturing
Engineering. State College: The Pennsylvania State University.
Jr. Veinott, A.F., 1969, “Minimum concave cost solution of Leontief substitution models of
multi-facility inventory systems”, Operations Research, 17, 262-291.
Zangwill, W.I., 1966, “A deterministic multi-period production scheduling model with
backlogging”, Management Science, 13, 105-119.
Zeleny, M., 1982, Multiple criteria decision making, McGraw-Hill, New York.
Zheng, Y., 1992, “On properties of stochastic inventory systems”, Management Science, 38, 87-
103