A User-Friendly Interactive Algorithm of the Multicriteria Linear Integer Programming

The interactive algorithms are often used [2] to solve multicriteria linear integer programming problems (MCIP). These algorithms [3, 6, 7, 11, 13] are modifications of interactive approaches solving multicriteria linear problems that include the integrality constraints. Linear integer programming problems are used as scalarizing problems in these interactive algorithms. These problems are NP-difficult problems [4]. Moreover, finding a feasible integer solution can be as difficult as finding an optimal solution. That is why in the interactive algorithms solving MCIP the time to solve the scalarizing problem plays a significant role. For this reason an effort is made to reduce the number of the integer problems solved; continuous problems (instead of integer problems) are solved and continuous (weak) nondominated solutions obtained are presented to the DM for evaluation (especially in the DM’s learning phase). Some of the interactive algorithms work with the aspiration levels of the criteria, others use weight to denote the relative significance of the criteria. Many show one while others show several (weak) nondominated solutions to the DM for evaluation at each iteration. In the paper a learning-oriented [5] interactive algorithm are suggested. The main features of the algorithm proposed, which improves the dialogue with the DM, are: ­ they reduce the number of the integer problems solved because in most of the iterations the solutions of single criterion linear problems with continuous variables (which are easy to solve) are presented to the DM for evaluation. This is used under the assumption [10, 11] that the criteria values for the scalarizing problems with continuous variables differ relatively little from the solutions with integer variables and under the assumption that the DM prefers to work in the criteria rather than in the variable space.; ­ at every iteration the DM provides his/her local preferences in terms of the desired changes in the criteria values of some of the criteria, the desired directions of change of the other criteria and permitted deterioration with or without set limiting БЪЛГАРСКА АКАДЕМИЯ НА НАУКИТЕ . BULGARIAN ACADEMY OF SCIENCES