Xqx based modeling for general integer programming problems.

摘要

We present a new way to model general integer programming (IP) problems with in- equality and equality constraints using XQX. We begin with the definition of IP problems followed by their practical applications, and then present the existing XQX based models to handle such problems. We then present our XQX model for general IP problems (including binary IP) with equality and inequality constraints, and also show how this model can be applied to problems with just inequality constraints. We then present the local optima based solution procedure for our XQX model. We also present new theorems and their proofs for our XQX model. Next, we present a detailed literature survey on the 0-1 multidimensional knapsack problem (MDKP) and apply our XQX model using our simple heuristic procedure to solve benchmark problems. The 0-1 MDKP is a binary IP problem with inequality con- straints and variables with binary values. We apply our XQX model using a heuristics we developed on 0-1 MDKP problems of various sizes and found that our model can handle any problem sizes and can provide reasonable quality results in reasonable time. Finally, we apply our XQX model developed for general integer programming problems on the general multi-dimensional knapsack problems. The general MDKP is a general IP problem with inequality constraints where the variables are positive integers. We apply our XQX model on GMDKP problems of various sizes and find that it can provide reasonable quality results in reasonable time. We also find that it can handle problems of any size and provide fea- sible and good quality solutions irrespective of the starting solutions. We conclude with a discussion of some issues related with our XQX model.