Abstract - This paper addresses a distribution center selection problem in a three-echelon supply chain including suppliers, distribution centers (DCs) and retailers. There is a set of DCs, in which some of them should be selected in each period. Due to the imperative role of the time value of money (TVM), we consider this decisive factor in the problem. Besides it is assumed that the setup cost of each DC can be decreased in period t based on its number of the previous periods that it has been selected. This phenomenon, which shows the loyalty of a customer, is named as loyal customer (LC) effect. The aim of the problem is to minimize the total costs, including transportation and setup costs. A non-linear mathematical model is developed for the mentioned problem and it has been solved by the GAMS optimization software to obtain the optimum set of DCs.

Keywords – distribution center selection, supply chain,

time value of money, LC effect.

I. INTRODUCTION

The problem of distribution centers (DCs) location in supply chains have been an interesting subject for researchers to study. Finding suitable locations for DCs can help the chain to improve its performance and reduce the total cost. Sometimes there is a set of potential DCs to be selected for distributing products in the supply chain network. Each DC has its own characteristics and it is required to choose best DCs for achieving goals. On the other hand, it is critical to consider the time value of money (TVM) criterion, which shows the effect of time on the value of money, to have proper plans for different industries. It is worth mentioning that neglecting TVM may make companies far from their goals and it can have the undesired results for them.

Furthermore, in a competitive market, some ways are required to attract customers. Valuing the old customers is one of them which reduces some costs for them. This phenomenon, which shows the loyalty of a customer, is named as loyal customer (LC) effect in our given problem. As a result, investigating a DCs selection problem in a multi-period supply chain with the TVM and LC effect is the purpose of this paper. In the next section, the literature review is given. In Section III, we define the considered problem, and Section IV presents some numerical examples and their results. Finally, in Section V, there are some conclusions.

II. LITERARUE REVIEW

Most of the studies focus on finding optimum location of DCs. Kou [1] considered an international DC (IDC) to find an optimal location for the IDC. He has developed a new hybrid method by combining the concepts of fuzzy DEMATEL and a novel fuzzy multiple criteria decision making to solve the problem. Also, Li et al. [2] proposed an axiomatic fuzzy set (AFS) clustering method and a TOPSIS-based approach to select the location of logistics center.

Chen [3] studied location selection of a DC problem in a fuzzy environment. Yang et al. [4] considered a DC location problem from a potential set under fuzzy environment. Awasthi et al. [5] discussed about urban DCs problem under uncertainty and presented a multi-criteria decision-making approach for location planning of them. Liu et al. [6] combined rough set methods and fuzzy logic to study about DCs location problem. Huijun et al. [7] considered a bi-level DCs location problem to minimize planners’ costs in the up-level DCs and customers’ costs in the low-level ones. Nozick and Turnquist [8] investigated a real case in an automotive manufacturer for the optimal location of DCs by considering facility costs, transportation costs, inventory costs and customer responsiveness, simultaneously.

On the other hand, in supply chain management, selection problems have been assumed in selecting suppliers and vendor management. For example, Wu [9] has presented a hybrid model, including data envelopment analysis (DEA), decision tree and neural network, in order to assess supplier performance. Moreover, there are such problems in [10-13]. Also, DCs selection problem has been considered by Ou and Chou [14] who examined six different factors of a DC from a foreign market perspective and proposed a weighted fuzzy factor rating system to select and international DC. In the best of our knowledge, DCs selection problem has been neglected in the literature and as a result, this paper introduces a DCs selection problem by taking into account TVM. Consequently, this paper discusses a DCs selection problem with the TVM criterion and the LC effect.

Selection of Distribution Centers with the Time Value of Money and the Loyal Customer Effect

A. Amini 1, R. Tavakkoli-Moghaddam 1, Armand Baboli 2

1 Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran 2 Université de Lyon, INSA-Lyon, DISP Laboratory, F-69621, France

(alirezaamini@ut.ac.ir, tavakoli@ut.ac.ir, armand.baboli@insa-lyon.fr)

978-1-4577-0739-1/11/$26.00 ©2011 IEEE 289

III. MODEL DEFINITION AND FORMULATION

A. Model definition

Undoubtedly, DCs have an essential role in large-

scaled supply chain networks. This paper addresses a DCs selection problem, in which there is a set of DCs to choose the required number of them in each period. As it is obvious in Figure 1, a three-echelon supply chain is supposed that consists of suppliers, DCs and retailers. In this chain, each retailer has its own demand in each period and suppliers should satisfy retailers’ demands. They prepared the required amount of goods and sent them to retailers by DCs.

All DCs have their own setup cost in each period, which it can be affected by the LC effect. It means that if a supplier selects a DC in k periods of t periods, there is a decrease in a setup cost of the mentioned DC based on k. It can be on account of achieving goals in a competitive world for different companies. It is a kind of attracting other companies to cooperate that it helps the company to increase the benefit and decrease the total costs. Furthermore, there are transportation costs for transporting products from suppliers to DCs and form DCs to retailers. This cost depends on distances between nodes and the value of the transportation costs are been changed in different periods. As the matter of fact, the TVM criterion makes this difference in the problem.

 Fig. 1. Three-echelon supply chain

 

As a result, the problem is to select the best DCs in each period and in a three-echelon supply chain in order to minimize the setup and transportation costs with the TVM criterion and the LC effect.  

B. Notations and mathematical model

To develop a mathematical programming model, we

introduce necessary notations in Table I.

TABLE I

NOTATIONS

Indices index of suppliers index of DCs index of retailers index of period

Parameters

setup cost of jth DC in period t demand of retailer r in period t distance between supplier i and jth DC

′ distance between the jth DC and retailer r unit transportation cost in each unit of distance

from supplier i to the jth DC ′ unit transportation cost in each unit of distance

from the jth DC to retailer r interest rate loyal customer coefficient

Decision variables

number of transported units from supplier i to the jth DC in period t

′ number of transported units from the jth DC to retailer r in period t

a binary variable which is equal 1 if the jth DC is selected in period t

According to Table I, the mathematical model is presented as shown below.

Min

∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑ 1

∑ ∑ ∑ 1

                                              (1) 

 s.t.

∑ ∑ ,                           (2) 

∑ ,                           (3) 

∑ 1                                              (4) 

∑ ,                                             (5) 

The objective function (1) minimizes the total

transportation and setup costs. Constraint (2) has a

 

Sup

plie

rs

Set

of

DC

s Ret

aile

rs

290

balanced approach, in which the number of received products to the jth DC should be equal to the number of sent item form that DC. Constraint (3) makes the model to provide all demands of retailers. Also, Constraint (4) represents that at least one DC should be selected in each period. Finally, Constraint (5) counts the number of selecting of the jth before period t.

IV. NUMERICAL EXAMPLE To illustrate the performance of the model, some

examples concluding a set of five potential DCs (A, B, C, D and E) are generated based on the pattern mentioned in Table II.

TABLE II

DATA GENERATION PATTERN

Setup cost DU (100, 1000) Demand DU (10, 1000) Distance DU (10, 100) Unit traveling cost DU (10, 100) Interest rate 0.1, 0.2 LC effect -0.1, -0.2 Number of suppliers 5 Number of DCs 5 Number of retailers 5 Number of periods 5

DU represents a discreet uniform distribution. Four examples are solved by the GAMS optimisation software on a desktop Core 2 Duo PC, 2.67 GHz with 4 MB RAM and the associated results are illustrated in Tables III and IV.

There are selected DCs in each period for two different levels of the interest rate and the LC effect rates in Table III. There are four cases in Table III:

Case 1: 0.1, 0.1 Case 2: 0.1, 0.2 Case 3: 0.2, 0.1 Case 4: 0.2, 0.2

In the second row of Table III, it is obvious that the

supply chain prefers to choose A as a DC in four of five periods and also E is selected for all periods. In addition, in the last case, there is a same pattern for A and it is clear that E is selected for all periods too. In these mentioned case, the LC effect shows its impact because of having a bigger rate than case 1 and 3. The total cost of the problem is shown in Table VI. As it is palpable, increasing the level of the interest rate and power of LC effect simultaneously (case 1 versus case 4) helps the chain to reduce the amount of the total cost that demonstrates the LC effect. Shifting LC effect from -0.1 to -0.2 with constant interest rate resulted in selecting more number of DCs and decreasing total cost. It demonstrates the power of LC effect in the supply chain.

TABLE III

SELECTED DCS IN EACH PERIOD FOR DIFFERENT CASES

  1  2  3  4  5 0.1

0.1          

0.10.2          

0.20.1          

0.20.2          

TABLE IV

TOTAL COST FOR DIFFERENT CASES

  Total cost 0.1

0.1 3.36E+07

0.10.2 2.47E+07

0.20.1 4.45E+07

0.20.2 3.22E+07

V. CONCLUSION This paper has addressed a distribution center (DC)

selection problem with considering the time value of money (TVM) and introducing a novel concept which has been called LC effect in this paper. It is obvious that the problem should be built by considering TVM and LC effect simultaneously to obtain the best solution which can be practical in the real world. There is a set of potential DCs with their own setup costs for each period, in which some of them should be chosen in each period to use as centers of distributing. A mathematical model has been designed, and four numerical examples have been generated to exemplify the performance of the model. It is shown that the LC effect can be used to attract the customers and reduce the total cost of the chain. On the other hand neglecting TVM criterion may be dangerous for organisations and the whole supply chain. For future research, considering some assumptions, such as uncertain demands, transportation vehicles restrictions and budget constraint when TVM criterion and LC effect are considered can be useful.

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