Non-linear Integer Programming Model and Algorithms for Connected p-facility Location Problem

1 Introduction

Given a connected graph G = (V, E) with nonnegative edge cost c_e for each edge e ∊ E, d_ij is the cost of the shortest path between vertices i and j in the graph. The Connected p-facility Location Problem (CpLP) is to choose p vertices from V such that the total shortest path cost of each pair of these p vertices is minimized.

Facility Location Problem (FL) is a classical issue which has been studied since Weber Location Problem in 1909. The problem looks for the best locations for a set of facilities that must satisfy requests of service coming from a given set of customers. We use the term “facility” here in its broadest sense. That is, it is meant to include entities such as factories, warehouse, retail outlets, hospitals, electronic switching centers and emergency warning sirens. The ubiquity of location decision-making has led to a strong interest in location analysis and modeling within the operations research and management science community. There are many facility location models, such as covering[1,2], p-center[3,4], p-dispersion[5] and p-median[3].

Cooper[6] formulated the Facility Location-Allocation (FLA) that provides a valuable method in deciding where to place facilities coupled with determining how to assign demand to the located facilities in order to utilize resources effectively. Another interesting location analysis method is to combine the vehicle routing problem with facility location, which is formulated to Location-Routing Problem (LRP). Many variations of LRPs exist, such as deterministic LRP[7], dynamic LRP[8] and planar LRP[9]. Much more work about LRP can be obtained from[10].

To optimize transportation between facilities, Gupta[11] introduce Connected Facility Location Problem (ConFL) which corresponds to the above mentioned facilities. The facilities need to be installed and connected with each other and customer nodes need to be assigned to them. The ConFL problem consists of finding an assignment of each customer to exactly one facility and connecting these facilities via a Steiner tree. Furthermore, the ConFL is transformed into the minimum Steiner arborescence problem and solved by an exact branch-and-cut method. Ten different integer programming formulations for ConFL have been presented by [12]. Several relevant special ConFLs are considered in some recent works, such as capacity constrained ConFL[13–15], hop constrained ConFL[16,17].

In this paper, a new location analysis method is presented by paying more attention to the sum distance/cost of pair-wise of these selected facilities. This problem comes from the real situation when dealing with natural disasters. In order to relieve the damage of large-scale disasters, many countries pay attention to emergency management issues. Mitigation and preparation are carefully considered in the one hand, while quick and efficient response is done on the other hand. Relief materials are very important in the whole process of disaster response. Huge demands are supplied from all over the country, sometimes outside the country from international support. Distribution center plays an important role in reserving, distributing, and transporting relief goods in order to enhance the ability of material organization. Huge demand for materials, especially for large-scale disasters, need to be transported from all other centers. The objective is to minimize the total transportation cost or time between each pair of the selected distribution centers.

The remainder of this paper is organized as follows. In Section 2, we will formulate a mathematic model for CpLP in detail and present the NP-hardness proof for this problem. Two efficient algorithms are designed for solving this problem in Section 3. The first one is based on procedure for maximum clique problem, while the other one is genetic algorithm. Computational results are shown in Section 4. In Section 5, we draw conclusions with further researches discussion.

2 Model formulation and complexity analysis

In this section, the mathematical model for CpLP will be formulated. And complexity of this problem will be analyzed.

3 Algorithms

In this section, two algorithms will be presented. The first one is a heuristic algorithm based on classical maximum clique approach, while the second one is genetic algorithm.

4 Experimental results

In this section, random method is introduced to generate the basic connected graphs. Vertices are placed randomly on a 400 × 400 square. Two vertices are adjacent when their Euclidean distance is no more than a predetermined radius. Two such graphs are shown as Figure 1.

Figure 1

Two randomly generated graph with 100 vertices and radius is 80 and 150

We have implemented the algorithm based on maximum clique procedure and genetic algorithm in Matlab 7.1 and carried out computational experiments to evaluate them. The computer used has a Inter(R) Core(TM) i5 2.30GHz CPU and Windows 7 operating system.

Parameters for genetic algorithm are set as follows. NP = 100, NG = 500, P_c = 0.6, P_m = 0.1, M = 1000, M′ = 1000000.

In the first group of instances, we will analyse the performance of our presented algorithms when the radius is fixed and number of vertices increases. p = 10 facilities need to be chosen from the vertex set. Radius is 100, and the number of vertices is increasing from 40 to 1000. In this group of instances, sparse or dense graphs are also considered. Experimental results are shown in Table 1, from which we can draw the following conclusions.

1. The objective value obtained from algorithm based on Procedure MCP is better than that of genetic algorithm. According to further analysis by comparing with the lower bound of CpLP, the produced value of algorithm based on Procedure MCP is about 3 to 7 times of lower bound while the objective value of genetic algorithm is about 5 to 2.1 × 10⁵ times of lower bound. Figure 2 shows the comparison among these values.

2. The solution output from genetic algorithm is not stability especially for large scale instances. When the number of vertices goes to 1000, there are 445 vertices been chosen. At the same time, the objective value becomes higher 2.0 × 10⁷, which is a very bad solution.

3. When the number of vertices is lower than 100, both of these two algorithms can find better solution.

4. The computational CPU time of algorithm based on Procedure MCP is much faster than that of genetic algorithm under the current parameters.

Figure 2

Comparison of objective value among lower bound, algorithm based on Procedure MCP and genetic algorithm

In the second group of instances, we will analyse the performance of our presented algorithms when the number of vertices is fixed and the radius is different. Also, p = 10 facilities need to be chosen from the vertex set. There are 100 vertices randomly placed on the square. The radius changes from 60 to 570. Obviously, the graph will be complete when radius is larger than 4002. Experimental results are shown in Table 2, from which we can draw the following conclusions.

1. The performance of algorithm based on Procedure MCP is better than that of genetic algorithm not only for the objective value but also for computational CPU time.

2. Both of the two algorithms are stability.

3. The objective values and computational CPU times of the presented algorithms do not vary too much when radius grows larger.

4. The CpLP is not sensitive for the radius when the number of vertices is fixed.

In the third group of instances, we will analyse the performance of our presented algorithms when the number of vertices and radius are all fixed while number of facilities changes. There are 100 vertices randomly placed on the square and radius is 100. The number of facilities increases from 5 to 50. Table 3 shows the computational results.

1. The algorithm based on Procedure MCP can produce better solution no matter number of facilities is small or large and the algorithm is more stability. The objective value is about 2 to 5 times of the lower bound. The genetic algorithm can also produce better solution only when the number of facilities is large, while it is not a better algorithm for small number of facilities.

2. The computational CPU time of algorithm based on Procedure MCP becomes much longer when the number of facilities increases. While the genetic algorithm is not sensitive for the facility number too much, it is because the coding method is up to total number of vertices.

From the above groups of instances, the algorithm based on Procedure MCP always produces better solution. Furthermore, other better procedure for maximum clique problem can improve the performance of our presented algorithm. On the other hand, genetic algorithm can also produce better solution if increase the size of the population and the generation of GA.

5 Conclusion

In this paper, we present a new location analysis method by considering the distance between each pair of selected vertices, which is called connected p-facilities location problem(CpLP). This problem is proved to be NP-hard and mathematical model is formulated. For solving CpLP, we present two algorithms. One is based on maximum clique procedure, while the other one is genetic method. From the computational results of three groups of instances, these two algorithms can produce better solutions by comparing with the lower bound of CpLP.

For further researches, new coding methods can improve the performance of genetic algorithm. Another direction is to design new algorithm based on relaxing the mathematical model of CpLP.