A Multi-objective Optimization Model for Distribution Center Location Based on Reliability of the System

Distribution centers, warehouses and other logistics facilities as a logistics network node, its location is reasonable not only directly affect the distribution center's operational performance, operational strategy and future development, but also affects the Logistics costs of suppliers upstream and downstream distributors or retailers, as well as logistics strategy and competitive strategy, its decision belongs to the medium and long term planning strategic decision. Therefore, comprehensive consideration should be given to the selection of D-center locations to achieve high efficiency, low cost, and meet the requirements of all parties involved [1].

The academic research on the location selection of D-centers has a long history, and the relevant literature is also increasing day by day. Generally speaking, it can be roughly divided into two categories: the first category is to study the location problem of D-centers separately[2-4];the second type is to combine Logistics management with inventory strategy, transportation strategy, customer service objectives, and other strategic research on the location of D-centers[5-9].Distribution center vigorous development and it will become increasingly fierce competition, so,it provides satisfactory service to customers will become the D-center to survive in the competition an important weight[10].Therefore, this article based on both the D-centers and customers through the D-center system reliability to measure the D-center service level, and combines with the traditional D-center location model Baumol-Wolfe model, then, it discusses the D-center location.

2

DESCRIPTION OF THE PROBLEM

In the traditional distribution model, the supply chain and logistics business flow without separation, each company needs to be in the establishment of small warehouse and distribution points, such as sales department to engage in commodity trading activities, and at the same time, will be responsible for goods logistics distribution, it will greatly reduce the efficiency. In order to strengthen the management of distribution, to choice some locations to set up distribution center by these distribution centers are responsible for the goods within the jurisdiction of the distribution; the distribution point is no longer dedicated to establish and manage their warehouse, but focus on specialized in commodities trading activities6, and, to some extent, promote efficiency and benefit.

We can describe the problem as follows: From a set of candidate locations select several locations as D- centers, making the point from the known resources (such as factories, enterprises, etc.), through the distribution center, shipping products to several customers, and ultimately meet the needs of customers, in this process, the key is to choose a reasonable distribution center location. Its distribution network structure shown in Figure 1[6]:

3

BUILDING DISTRIBUTION CENTER LOCATION OF THE MULTI-OBJECTIVE MODEL

3.1

The reliability of distribution system

With the development of logistics, logistics services are also to a higher level, if the logistics industry manufacturers want to attract customers and improve customer loyalty, it is necessary to change the existing concept. Distribution center can not only be satisfied with to provide services to customers, but a higher level of demand, according to the different requirements of customer service to provide the appropriate level of services to meet customer satisfaction goals. In this article, it measures the level of service which the distribution center provides to the customers by the reliability of distribution system. Only the distribution center system is in stable condition, it can provide the corresponding services for customers in time.

For system reliability, different professional combination to the professional characteristics, so, the reliability of the definition will have certain differences. But in general, these definitions are focused on the system ability which under specified conditions and within the prescribed period of time to complete the required function[11]. The reliability is a way to measure the ability.

Based on the definition above, we can define a system service reliability of distribution centers as follows: it is distribution center of internal several interconnected logistics operations unit reliability logic (series, parallel) combination. According to the scope of this study, assuming that the reliability of picking, loading and other set of goods outside delivery is 1, the reliability of system services of distribution center is the probability of product delivered which under certain conditions, upon the request time. It can be expressed using the following: (1) $R_{i j} = P (T_{w i} + T_{i j} \leq t_{w j}) = P (\frac{d_{w i}}{V_{w i}} + \frac{d_{i j}}{V_{i j}} \leq t_{w j}) = P (\frac{d_{w i} + d_{i j}}{V} \leq t_{w j}) = P (V \geq \frac{d_{w i} + d_{i j}}{t_{w j}}) = 1 - F_{v_{i j}} (\frac{d_{w i} + d_{i j}}{t_{w j}})$

In the formula (1), T_wi is the served time product from the factory k to D-center i ; T_ij is the time that product delivery from the D-center i to customers j; t_wj stands for the delivery time j given by the customer; d_wi stands for the distance between factory k and the plant's D-center i; d_ij stands for the distance between the D-center i and the customer j; V_wi is the speed for the vehicle from the factory k to the D-center i; V_ij is the speed for the vehicle from the D-center i to customer j; Assuming the turnaround time in the D-center is 0, and the vehicle performance, road conditions and nature and other factors are the same when the vehicle from the factory to the D-centers and from D-centers to customers, that is, V_wi = V_ij = V; F_{V_ij} (·) is the velocity distribution function, a vehicle from the factory k to the D-center i, then to customer j, usually, the vehicle speed is a certain statistical laws, and its distribution is some random factors comprehensive effect which are vehicle performance, traffic status, road conditions and driving level. Generally it has the characteristics of Weibull distribution or Normal distribution, and it can be calibrated by statistics methods [12].

Based on the calculation method about the reliability of D-center which serves to one customer, according to the definition, a distribution center to serve multiple customers or multiple distribution centers to serve multiple customers within their service area's D-center system reliability can be calculated as follows: (2) $R_{s y s} = \frac{\sum_{w \in W} \sum_{i \in M} \sum_{j \in N} q_{j} R_{i j}}{\sum_{j \in N} q_{j}} = \frac{\sum_{w \in W} \sum_{i \in M} \sum_{j \in N} q_{j} [1 - F_{v_{i j}} (\frac{d_{w i} + d_{i j}}{t_{w j}})]}{\sum_{j \in N} q_{j}}$

In the formula (2), R_sys stands for the system reliability for the distribution center; q_j is demand of customer j; R_ij is the system reliability of D-center i which provide logistics services for customer j[12].

3.2

Model description and basic assumptions

Already know the factory location, the customer's location and demand, then chose the location which can establish D-center from a set of candidate locations, to make the transport costs from factory to D-center and then to customer and fixed costs of the construction of distribution centers least, while make the customer's satisfaction maximum. In order to facilitate research, make the following assumptions: 1)

Since only consider single-source supply, so do not take into account capacity constraints;

2)

According to the need of the distribution center extension or new, so it does not consider the constraint of the capacity and flow, that each D-center can meet the demand of customers within the scope of radiation;

3)

If an alternate D-center is selected, the fixed cost of construction of D-centers is known;

4)

Each customer only by a distribution center to provide services;

5)

Each customer's demand is known;

In order to better reflect the actual situation, in the paper it corrects the Kuehn-Hamburger model assumption, in the Kuehn-Hamburger model it does not take into account the fixed costs to establish the distribution center. On the assumption that fixed fees are known, so, it can more close to reality.

3.3

Symbol Definition

In the whole site selection model will involve the following basic parameters:

M = {1,2,…,m}, i ∈ M, the candidates of D-centers;

N = {1,2,…,n}, j ∈ N, the number of customers

W——factory;

x_wi ——the number of products from the factory to D-centers i;

c_wi ——unit of freight from the factory to D-centers i;

q_j ——demand for products of customer j;

c_ij ——Unit freight from alternative d-Center i to customer j;

g_i —— the fixed costs of establishment and operation for D-centers i;

$\begin{array}{l} y w = & {\begin{matrix} \begin{matrix} 0 & the candidate address i does not build D-center \end{matrix} \\ \begin{matrix} 1 & the candidate address i builds D-center \end{matrix} \end{matrix} \end{array}$

$\begin{array}{l} w c = & {\begin{array}{l} \begin{matrix} 0 & the D-center i does not provide service for customers j \end{matrix} \\ \begin{matrix} 1 & the D-center i provides service for customers j \end{matrix} \end{array} \end{array}$

3.4

Build mathematical models

As the general D-center location model, it takes the minimum cost of construction and operation of D-centers as a target, but the difference is that the objective function takes into account the reliability of the system largest. Therefore, the multi-objective optimization model for selecting D-center locations is represented as follows: (3) $\min C = \sum_{i = 1}^{m} c_{w i} x_{w i} d_{w i} + \sum_{i = 1}^{m} \sum_{j = 1}^{n} c_{i j} q_{j} w c (i, j) d_{i j} + \sum g_{i} y w (i)$ (4) $R_{s y s} = \frac{\sum_{w \in W} \sum_{i \in M} \sum_{j \in N} q_{j} R_{i j}}{\sum_{j \in N} q_{j}} = \frac{\sum_{w \in W} \sum_{i \in M} \sum_{j \in N} q_{j} [1 - F_{v_{i j}} (\frac{d_{w i} + d_{i j}}{t_{w j}})]}{\sum_{j \in N} q_{j}}$ (5) $\begin{matrix} s . t . \sum_{i \in M} w c (i, j) = 1, & j \in N \end{matrix}$ (6) $\begin{matrix} x_{w i} = \sum_{j = 1}^{n} q_{j} w c (i, j), & i \in M \end{matrix}$ (7) $\begin{matrix} w c (i, j) \leq y w (i) & i \in M & j \in N \end{matrix}$ (8) $\begin{matrix} w c (i, j), y w (i) \in (0, 1), & i \in M & j \in N \end{matrix}$ (9) $\begin{matrix} x_{w i} \geq 0 & i \in M \end{matrix}$

Among them, the equation (3) is the objective function, that the costs of transport and fixed costs are minimal; (4) is the objective function too,that the largest distribution center system reliability; equation (5) is one distribution center provide services to one customer; (6) that supply-demand balance; equation (7) that if the location i did not build distribution center, it can not provide services to customers.

3.5

Solving the multi-objective optimization model

In the real world, many problems are multiobjective optimization problems, differ from the single-objective optimization, in multiobjective optimization, the objectives are conflicting and the optimal solution has an infinite number. Usually, the multi-objective optimization problem is transformed into a single objective optimization problem, and then the general solution method is applied to solve it.

In regard to this model, we adopt the method of selecting the main target. To select a main target from the minimum cost of D-centers and the maximum reliability of d-centers system. It can circumstance of the case according to the actual, if the enterprise pursuit minimum cost, it can take the minimum cost as the main goal, at the same time, take the maximum reliability of distribution centers system as a constraint to solve the model. In this paper, it takes the maximum reliability of the D-center system as the main objective as an example to solve the model. The model function is: $R_{s y s} = \frac{\sum_{w \in W} \sum_{i \in M} \sum_{j \in N} q_{j} R_{i j}}{\sum_{j \in N} q_{j}} = \frac{\sum_{w \in W} \sum_{i \in M} \sum_{j \in N} q_{j} [1 - F_{v_{i j}} (\frac{d_{w i} + d_{i j}}{t_{w j}})]}{\sum_{j \in N} q_{j}}$

Increasing constraints , $\min C = \sum_{i = 1}^{m} c_{w i} x_{w i} d_{w i} + \sum_{i = 1}^{m} \sum_{j = 1}^{n} c_{i j} q_{j} w c (i, j) d_{i j} + \sum g_{i} y w (i) < C_{0}$ , C₀ is the cost given. Namely: enterprise, to pursuit leading service strategy, and in a certain cost range, make the distribution center system reliability maximum12 .The model in this article is a mixed integer programming model with linear constraints, the general law can be used to define the branching method to solve, but because of the relatively large amount of computation, it solves the model by lingo9.0 software, which brings a lot of convenience for calculation. Calculation procedure is as follows: 1)

To select the minimum fixed costs of D-center from the known locations, there are k = 1, make the D-center service to all the customers, and calculate the total cost of this D-center and D-center system reliability;

2)

If its cost is lower than the given cost C0, then output the results of its address and the customer's assigned, transferred 3); or re-select a fixed cost of optional second small to calculate;

3)

After select a location, chose another from the rest of locations in lower fixed costs principle, there are k = 2, so that two distribution centers for all customer service, and sun up the total cost and reliability of D-centers , into 2); and so on, until the total cost above the specified cost.

4

CALCULATION EXAMPLE

The third-party logistics company agency the distribution business for a home appliance manufacturer, now it faces how to design the distribution system. Known to have a factory and eight customers, each customer's demand is qj, and there are four candidate locations, the delivery time which customer demand is t = 6 hours, try to determine the optimal location of distribution centers program.

Table 1.

Unit transportation cost from factory w to D-center i (unit: yuan / ton / km)

i	1	2	3	4
c_wi	18	16	15	20

Table 2.

the fixed costs of construction of D-centers (Unit: ten thousand Yuan)

i	1	2	3	4
g_i	87	90	92	95

Table 3.

distance from factory w to the D-center i (units: km)

i	1	2	3	4
d_wi	210	273	260	160

Table 4.

the demand of customer j (unit: tons)

j	1	2	3	4	5	6	7	8
q_j	4600	3400	2700	4000	2300	3800	2000	2300

Table 5.

Unit transportation cost from D-center i to customer j (unit: yuan / ton / km)

j	1	2	3	4	5	6	7	8
1	20	15	24	19	20	22	18	21
2	23	25	17	16	21	23	19	28
3	20	12	19	27	20	24	16	19
4	17	25	27	22	18	19	24	30

Table 6.

the distance from D-center i to customer j (unit: km)

J	1	2	3	4	5	6	7	8
1	155.9	192.5	255.6	181.3	188.1	133.5	166.3	148.6
2	99.4	138.4	146.1	89.3	122.5	200.2	150.7	102.4
3	128.6	217.3	160.3	190.3	94.6	125.6	205.6	129.4
4	178.2	198.4	280.3	290.5	210.8	161.6	181.7	185

In this model, the maximum reliability of D-center system is taken as the objective function, at the same time take the costs of distribution center lower than C₀ as a constraint. For the C₀ values is given by the company. This enterprise given the cost of C₀= 350000000 yuan. The calculation results are as follows:

The maximum value of objective function R *= 6.266. Choose the locations 2,3,4 to establish distribution centers, the distribution centers set up the locations 2 to services for customers 4; the distribution centers set up the locations 3 will services for customers 3,5; the distribution centers set up the locations 4 will services for customers 1,2,6,7,8. If the business pursuit of lower costs, the decision-makers can make an appropriate decision according to the given cost and find the right solution. Therefore, this model has a larger superiority and rationality.

5

ANALYSIS AND CONCLUSION

In real life, it is the first priority of a distribution center to provide the service level could meet customer demand. From the perspective of customers, the D-center considers maximizing the reliability of the D-center system, in order to meet customer needs in a timely manner. It is the necessary condition to achieve greater efficiency and effectiveness of the system of D-center base on both customers and D-center to balance the reliability of D-center and the total cost.

1)

In this paper, it considers both the cost of D-center and the level of customer service. So, the total cost of establish D-center is minimum and at the same time it can provide better service;

2)

It examines the level of service which the D-center provides by measuring the reliability of D-center, and it measures the reliability of D-center system by quantitative calculation. It measured a D-center can provide service to customers timeliness from one side;

3)

This article establishes a multi-objective optimization model that minimizes the total cost of the D-center and maximizes the reliability of the D-center system. And the model considered the D- center can be provided service for the timeliness from the customer's point of view. Enterprise having considered both the logistics cost and the level of customer service, to make the reliability of D-center system maximum in the case of the minimum cost. So that the D-center location will more reasonable and more realistic.

4)

There are many factors affect the D-center services, in this paper it measures the level of service to provide customers only by the reliability of the D-centers. It can choose a variety of factors to further explore to provide customers with the level of service, and how to make customers more satisfied, in order to provide a more realistic decision for D-center location.

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j	1	2	3	4	5	6	7	8
1	20	15	24	19	20	22	18	21
2	23	25	17	16	21	23	19	28
3	20	12	19	27	20	24	16	19
4	17	25	27	22	18	19	24	30

j	1	2	3	4	5	6	7	8
1	20	15	24	19	20	22	18	21
2	23	25	17	16	21	23	19	28
3	20	12	19	27	20	24	16	19
4	17	25	27	22	18	19	24	30

A Multi-objective Optimization Model for Distribution Center Location Based on Reliability of the System

Limin Wu

Pubblicato online: 17 mar 2025

Ricevuto: 05 nov 2024

Accettato: 12 feb 2025

DOI: https://doi.org/10.2478/amns-2025-0817

Parole chiaveLogistics cost, location model, system reliability, Distribution Center

© 2025 Limin Wu, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Parole chiave
Logistics cost, location model, system reliability, Distribution Center

j	1	2	3	4	5	6	7	8
1	20	15	24	19	20	22	18	21
2	23	25	17	16	21	23	19	28
3	20	12	19	27	20	24	16	19
4	17	25	27	22	18	19	24	30