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We employ uncertain programming to investigate the competitive logistics distribution center location problem in uncertain environment, in which the demands of customers and the setup costs of new distribution centers are uncertain variables. This research was studied with the assumption that customers patronize the nearest distribution center to satisfy their full demands. Within the framework of uncertainty theory, we construct the expected value model to maximize the expected profit of the new distribution center. In order to seek for the optimal solution, this model can be transformed into its deterministic form by taking advantage of the operational law of uncertain variables. Then we can use mathematical software to obtain the optimal location. In addition, a numerical example is presented to illustrate the effectiveness of the presented model.

Research on distribution center location problem is a necessary component of the optimization of logistics distribution’s system. The distribution center plays the role of a bridge that links customers with suppliers so as to transport goods from suppliers to customers. A lot of researches have been devoted to the distribution center location problem. For example, Lu and Bostel [

A large part of location problems have been studied in an ideal environment, in which only a unique facility offers services or products in the market. In practice, with the rapid development of economic, the environment is growing more complex. The facility has to compete with other players for more benefits. Thus the research on competitive location problem plays an important role in location theory. Competitive location problem differs from the general location problem because we must consider the competition between the existing facilities and new facilities. The briefly explain of the competitive location problem is that some facilities have been located in the market and the new facility will be located at the optimal place so as to compete with others for their market share.

Because of the importance of competitive location problem, many scholars devoted to the related research of competitive location problem in deterministic environment, that is, all parameters are known in advance and assumed to be fixed. Hotelling [

In practical life, there are many indeterminacy factors in competitive location problem. For example, the demand of customer for a kind of product is variable because it is influenced by other things liked weather. Hence, many scholars established models by stochastic method. Leonardi and Tadei [

In the above mentioned literatures, these researches cannot be proceeded smoothly without the assumption that there are enough history data to obtain probability distribution which is closed to the real frequency. However, sometimes the lack of history data posed difficulties for applying probability theory, especially when a new product was shipped to the customer by distribution center. In this case, we have to invite some experts to give the belief degree that each event will occur. In order to deal with belief degree, uncertainty theory was found by Liu [

Within the framework of uncertainty theory, the research of uncertain facility location problem had made a great number of achievements. Gao [

This paper addresses the problem that a logistics company enters a market by locating a new distribution center where there are many existing competitors in uncertain environment. The demands of customers and setup costs of the potential distribution centers are assumed to be uncertain variables. Then we construct the expected value model with the objective of maximizing the profit of the new distribution center. In order to obtain the optimal solution, the expected value model is transformed into its crisp equivalent model. At last we can use the mathematical software to find the optimal solution.

The innovations of this paper are as follows. We investigate the competitive logistics distribution center location problem under uncertain environment instead of logistics distribution center location problem in uncertain environment. It is different form the problem which dealt with by Wu and Peng [

The remainder of this paper is organized as follows. We introduce some basic and necessary knowledge about uncertainty theory in Section 2. In Section 3, we state the competitive distribution center location problem and construct an expected value model. In Section 4, we transform the expected value model into its deterministic one. In Section 5, we give a numerical example to illustrate the modeling idea of this paper. At last, the Section 6 concludes the paper.

In order to understand the presented model of competitive location problem better, we introduce some necessary knowledge about uncertainty theory in this section.

Let us introduce the concept of uncertain measure first. Let

1) (Normality Axiom)

2) (Duality Axiom)

3) (Subadditivity Axiom) For every countable sequence of events

The triplet

4) (Product Axiom) Let

where

In order to describe the quantities with uncertainty, Liu [

Definition 1. (Liu [

Liu [

Definition 2. (Liu [

for any number x in

Definition 3. (Liu [

Definition 4. (Liu [

Example 1. If

denoted by

Definition 5. (Liu [

for any Borel sets

A real-valued function

whenever

whenever

Theorem 1. (Liu [

has an inverse uncertainty distribution

We review the important concept of the expected value, which represents the size of uncertain variable.

Definition 6. (Liu [

provided that at least one of the two integrals is finite.

Theorem 2. (Liu [

Example 2. Let

In this section, we mainly propose the expected value model for competitive distribution center location problem within the framework of uncertain programming. Uncertain programming, proposed by Liu [

This paper investigates the competitive logistics distribution center location problem in uncertain environment. That is the problem in which a logistics company enters a market by locating a new distribution center where there are many existing distribution centers. The goal of the decision maker is to choose the location of the new distribution center so as to maximize its profit. The flow diagram of logistics distribution is shown in

Before we begin to study competitive location problem with uncertain variables, we need to make some assumptions as follows (which are referred to Revelle [

1) There is one supplier and many existing distribution centers.

2) The supplier only supplies one kind of product.

3) There is no difference among the products provided by all distribution centers.

4) The location of the new distribution center can be selected from potential distribution centers.

5) The distances between the supplier and potential distribution centers, the distances between potential distribution centers and customers and the distances between existing distribution centers and customers are known in advance.

6) The allocation of customers demands is related to the distance. The full demands of customers will be assigned to the nearest distribution center.

In order to model the competitive location problem, we introduce the following indices and parameters:

Remark 1: The meaning of variable

In order to maximize the profit of the new distribution center, the decision maker must choose the appropriate site to build the new distribution center which attracts more customers. The majority of competitive location models assumed that customers will patronize the nearest distribution center. It is rationally for customers who want to sustain the less travel cost. In this paper, we consider that the customers choose the distribution center according to the distance between their sites and distribution centers rather than other conditions, such as price, service and attractiveness.

Thus we assume that the customers patronize the nearest distribution center, and this assumption which has been used by Revelle [

We note that the total profit of the new distribution center is made up of four parts. Thus the total profit is a function related to x, y and

where

and

are decision vectors, and

is an uncertain vector.

Since we know the uncertain objective function

We employ the uncertain programming model to study the competitive logistics distribution center location problem. So we can build the expected value model as follow:

where

In this expected value model, the first constraint means that the quantity supply is not exceed the demand of customer. The second constraint implies the volume of transport is less than the capacity of the new distribution center. The third one shows that we select single site to build the new distribution center. And the last one ensures the nonnegativity of decision variables

The key problem of the model is seeking for the optimal solution. Taking advantage of the operational law of uncertain variable, we can transform the expected value model (1) into its deterministic form. It is clearly that the total profit function is strictly decreasing with setup costs. According to Theorem 1 and Theorem 2, the objective function

can be converted into

Similarly, the first constraint

can be turned into

It follows from the formulas (2) and (3) that expected value model (1) can be switched to the following equivalent model:

Clearly, we note that the equivalent model (4) is a deterministic programming model. As all know, the software Lingo cannot show the integral function. Therefore, we can find the optimal solution by using the mathematical software Matlab. In addition, in uncertainty theory, the inverse distribution function is easy to calculate for us. Thus we can calculate the inverse distribution function before we use the software Lingo to find the optimal solution. We can choose one of them to solve this programming. For convenience, the following example is seeking for solution by Lingo.

In order to illustrate the modeling idea and the effectiveness of this model, we give a numerical example in this section. Suppose that there is a supplier in a city. And there are 3 distribution centers to distribute the new model of the televisions to 7 customers. A logistics company want to select an optimal site from 4 potential distribution centers after survey. The distances d_{j}, d_{jk} and D_{ik} are listed in

j | 1 | 2 | 3 | 4 |
---|---|---|---|---|

d_{j} | 150 | 200 | 210 | 170 |

i\k | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

1 | 10 | 20 | 30 | 28 | 15 | 15 | 25 |

2 | 14 | 25 | 25 | 32 | 30 | 12 | 20 |

3 | 18 | 28 | 8 | 13 | 23 | 25 | 20 |

j\k | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

1 | 15 | 25 | 18 | 12 | 20 | 8 | 18 |

2 | 22 | 15 | 16 | 10 | 15 | 21 | 6 |

3 | 21 | 8 | 24 | 20 | 16 | 25 | 21 |

4 | 18 | 12 | 20 | 14 | 12 | 20 | 12 |

j | 1 | 2 | 3 | 4 |
---|---|---|---|---|

200 | 240 | 180 | 210 | |

k | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

According to the model (4), the expected value model of this numerical example is listed in the following formula

Then we use Lingo to get the optimal solution which is listed in the following.

The solution _{2} = 233 means that the supplier must supply 233 products to the new distribution center, and y_{22} = 73, y_{24} = 68, y_{25} = 35, y_{27} = 57, state that the quantity of product are supplied form new distribution center to customers. The above solutions show that we can choose the second potential distribution center to built the new distribution center and the total profit is 644.58. Meanwhile we can know how to distribute goods for customers according to the plan which is shown in

In the process of practical logistics network optimization, uncertain factors often appear in competitive logistics distribution center location problem because of lacking of or even without historical data. This paper investigated a useful model to handle competitive logistics distribution center location problem with uncertain customers demands and uncertain setup costs. The mathematical model of this problem was established by uncertain programming based on the expected value criterion. In order to solve this model, we took advantage of the properties of uncertain variable. Then the expected value model was transformed into its crisp equivalent model, and we used mathematical software Lingo to find its optimal solution. At last, a numerical example was presented to illustrate the effectiveness of the proposed model.

This paper only considers the demands of customers and setup costs of new distribution center are uncertain variables. Indeed, other uncertain factors in competitive logistics distribution center location problem are worthy of studying. We can further focus on the uncertain utility which can be used to describe the uncertainty of customers’ patronizing behavior. Furthermore, we can seek for the expression of uncertain utility. In this paper, we only center on the static competition. It is necessary for further research to consider dynamic competition problem. Thus we can establish dynamic uncertain programming model for uncertain dynamic competitive facility location problem.

This work was supported by the Projects of the Humanity and Social Science Foundation of Ministry of Education of China (No. 13YJA630065), and the Key Project of Hubei Provincial Natural Science Foundation (No. 2012FFA065).

BingyuLan,JinPeng,LinChen, (2015) An Uncertain Programming Model for Competitive Logistics Distribution Center Location Problem. American Journal of Operations Research,05,536-547. doi: 10.4236/ajor.2015.56042