Finding an efficient solution to linear bilevel programming problem: An effective approach

Multilevel programming is developed to solve the decentralized problem in which decision makers (DMs) are often arranged within a hierarchical administrative structure. The linear bilevel programming (BLP) problem, i.e., a special case of multilevel programming problems with a two level structure, is a set of nested linear optimization problems over polyhedral set of constraints. Two DMs are located at the different hierarchical levels, both controlling one set of decision variables independently, with different and perhaps conflicting objective functions. One of the interesting features of the linear BLP problem is that its solution may not be Paretooptimal. There may exist a feasible solution where one or both levels may increase their objective values without decreasing the objective value of any level. The result from such a system may be economically inadmissible. If the decision makers of the two levels are willing to find an efficient compromise solution, we propose a solution procedure which can generate effcient solutions, without finding the optimal solution in advance. When the near-optimal solution of the BLP problem is used as the reference point for finding the efficient solution, the result can be easily found during the decision process.     

Pricing policies for substitutable products in a supply chain with Internet and traditional channels

This study considers pricing policies in a supply chain with one manufacturer, who sells a product to an independent retailer and directly to consumers through an Internet channel. In addition to the manufacturer’s product, the retailer sells a substitute product produced by another manufacturer. Given the wholesale prices of the two substitute products, the manufacturer decides the retail price of the Internet channel, and the retailer decides the retail prices of the two substitute products. Both the manufacturer and the retailer choose their own decision variables to maximize their respective profits. This work formulates the price competition, using the settings of Nash and Stackelberg games, and derives the corresponding existence and uniqueness conditions for equilibrium solutions. A sensitivity analysis of an equilibrium solution is then conducted for the model parameters, and the profits are compared for two game settings. The findings show that improving brand loyalty is profitable for both of the manufacturer and retailer, and that an increased service value may alleviate the threat of the Internet channel for the retailer and increase the manufacturer’s profit. The study also derives some conditions under which the manufacturer and the retailer mutually prefer the Stackelberg game. Based on these results, this study proposes an appropriate cooperation strategy for the manufacturer and retailer.     

Type II sensitivity analysis of cost coefficients in the degenerate transportation problem

This paper focuses on sensitivity analysis of the degenerate transportation problem (DTP) when perturbation occurs on one cost coefficient. The conventional Type I sensitivity analysis of the transportation problem (TP) determines the perturbation ranges for the invariant optimal basis. Due to different degenerate optimal basic solutions yielding different Type I ranges, the Type I range is misleading for the DTP. Type II sensitivity analysis, which determines the perturbation ranges for the invariant shipping pattern, is more practical for the DTP. However, it is too tedious to obtain Type II ranges by enumerating all optimal basic solutions and all primal optimal basic solutions while getting the union of each corresponding Type I ranges. Here, we propose two labeling algorithms to determine the Type II ranges of the cost coefficient. Besides, three lemmas are provided for obtaining the upper bound or lower bound of the Type II ranges of the cost coefficient directly under specific conditions of the DTP. A numerical example is given to demonstrate the procedure of the proposed labeling algorithms and computational results have been provided.     

A review of Hopfield neural networks for solving mathematical programming problems

The Hopfield neural network (HNN) is one major neural network (NN) for solving optimization or mathematical programming (MP) problems. The major advantage of HNN is in its structure can be realized on an electronic circuit, possibly on a VLSI (very large-scale integration) circuit, for an on-line solver with a parallel-distributed process. The structure of HNN utilizes three common methods, penalty functions, Lagrange multipliers, and primal and dual methods to construct an energy function. When the function reaches a steady state, an approximate solution of the problem is obtained. Under the classes of these methods, we further organize HNNs by three types of MP problems: linear, non-linear, and mixed-integer. The essentials of each method are also discussed in details. Some remarks for utilizing HNN and difficulties are then addressed for the benefit of successive investigations. Finally, conclusions are drawn and directions for future study are provided.     

Linear Bi-level Programming Problems — A Review

Journal of the Operational Research Society - Multi-level programming is characterized as mathematical programming to solve decentralized planning problems. The decision variables are partitioned...