A biologically inspired model for transshipment problem

Transshipment problem is one of the basic operational research problems. In this paper, our first work is to develop a biologically inspired mathematical model for a dynamical system, which is first used to solve minimum cost flow problem. It has lower computational complexity than Physarum Solver. Second, we apply the proposed model to solve the traditional transshipment problem. Compared with the conditional methods, experiment results show the provided model is simple, effective as well as handling problem in a continuous manner. Introduction. – Physarum polycephalum is a ’smart’ true slime mold, which exhibits a various of pattern formation behaviors, such as growth, food foraging, nutrient transportation, propagation of pseudopodium and maintenance of tube shape. Recently, experiment results have shown Physarum can construct biological transport networks efficiently [1–4], solve Shortest Path (SP) problem [5–7] and Steiner tree problem with three or four food sources [13]. By extracting from the complex behaviors of Physarum, there have been some simple and distinct models developed for solving SP problem [6–9] in a maze and other optimization problems so far [2, 4]. These include a mathematical model Physarum Solver [8], Oregonator model of the Belousov-Zhabotinsky (BZ) medium [6, 10], particle-like agent model simulating the growth of Physarum [11, 12] and the cell model [2, 3]. These models have been applied to deal with some extended SP problems [21–23], Spanning tree problem [10], road navigation [14, 19], efficient transportation networks [1, 15, 26] and other problems [16, 17, 20, 24]. As mentioned above, Physarum Solver [8] is capable of finding SP in a maze where two separate food sources (FSs) are placed at two exits, respectively. Physarum polycephalum always prefers to transporting its nutrients and chemical signals in an effective manner. Nakagaki et al. show that this simple organism has the ability to find the minimum-length path of a maze [5]. In other words, it can be interpreted as transporting a certain amount of flow among the minimum-length path. As a consequence, SP problem can be regarded as single-source single-sink minimum cost flow problem. In transshipment problem, items are supplied from different sources to different destinations. It is sometimes economical if the shipment passes some transient nodes in between sources and destinations. The objective is to minimize the total cost of shipments. and thus the shipment passes through one or more intermediate nodes before it reaches its destinations [27,28]. Therefore, transshipment problem is a special multi-source multi-sink minimum cost flow problem. In this Letter, we first develop a biologically inspired mathematical model in response to local information (potential difference). This model has lower computational complexity than Physarum Solver. Second, we extend the proposed model to solve transshipment problem in an continuous manner, which is different the discrete simplex method. The paper is organized as follows. In Section 2 we present the biologically inspired mathematical model. Then, extend this model to solve transshipment problem. In Section 3 we discuss the numerical study, which illustrate the efficiency of the proposed model. Section 4 con-