A neural model for the p-median problem

• Authors: Enrique Domínguez;José Muñoz
• Venue: Computers and Operations Research
• Year: 2008

There exist several neural techniques for solving NP-hard combinatorial optimization problems. At the beginning of the 1980s, recurrent neural networks were shown to be able to solve optimization problems. Criticism of this approach includes the tendency of recurrent neural networks to produce infeasible solutions and poor local minima. This paper proposes a new technique which always provides feasible solutions and removes the tuning phase since the constraints are incorporated in the neural architecture instead of the energy function, therefore the tuning parameters are unnecessary. One of the most popular and well-known facility location problems is the p-median problem, which concerns the location of p facilities (or medians) in order to minimize the total weighted distance between the demand points and the facilities. There exist several heuristics to find optimal solutions of the problem based on the traditional formulation. In this paper a new formulation for the p-median problem based on two types of decision variables and with only n+p linear equality constraints is presented, where n is the number of demand points or customers and p is the number of facilities (medians). Also, a competitive recurrent neural network is proposed for this problem. The neural network consists of two layers (allocation layer and location layer) with 2np process units. The process units constitute n+p groups, where only one process unit per group is active at the same time and process units of the same layer are updated in parallel. Moreover, the energy function (objective function) always decreases or remains constant as the system evolves according to the dynamical rule proposed. The effectiveness and efficiency of our algorithm for different problem sizes are analyzed in comparison to conventional heuristic methods. The results show that our recurrent neural network generates good solutions with a reasonable computational effort. Scope and purpose: Geographical information systems (GIS) have occupied the attention of many researches involving a number of academic fields including geography, civil engineering, computer science, land use planning, and environmental sciences. GIS can support a wide range of spatial queries that can be used to support location studies. Model application and model development are the major impact of GIS on the field of location science. These systems are designed to store, retrieve, manipulate, analyze, and map geographical data. GIS can serve as the source of input data for a location model and it can also be used to present the model results. For example, if a p-median problem has to be solved, then a GIS executes a heuristic algorithm that reads the data from the GIS and it presents the results in real time. Location-allocation models simultaneously locate facilities and allocate demand points to them. These models also arise in a variety of public and private sector problems. The p-median is the most widely used location-allocation model. The p-median model is NP-hard and its data set became very large in real problem, so heuristic solutions are required. As the size of the data set grow, the number of feasible solutions grows and the quality of solutions and computation times from the most commonly used heuristic are deteriorated. The purpose of this paper is to develop a neural model to be integrated in GIS software. Thus, we propose a new formulation for the p-median problem based on 2np binary variables and n+p equality linear constraints. A recurrent neural model is proposed for solving the p-median problem based on this formulation without the difficulty in selecting appropriate tuning parameters, since these parameters are avoided. Moreover, an np-parallel algorithm has been developed based on this formulation. The central property of this algorithm is that the objective function always decreases (or remains constant) as the algorithm evolves according to its dynamical rule. Moreover, this algorithm can be also implemented using optical or VLSI technology.