In this paper, a reformulation that was proposed for a knapsack problem has been extended to single and bi-objective linear integer programs. A further reformulation by adding an upper bound constraint for a knapsack problem is also proposed and extended to the bi-objective case. These reformulations significantly reduce the number of branch and bound iterations with respect to these models. Numerical illustrations have been presented and computational experiments have been carried out to compare the behaviour before and after the reformulation. For this purpose, Tora software was used.

A need for an optimal solution for a given mathematical model is well known and many solution approaches have been developed to identify efficiently an optimal solution for a given situation. For example, one such class of mathematical models with industrial applications have been classified as mathematical programming models (MPM). The main idea behind these models is to find the optimal solution described by those models and interpret the solution back with respect to the given industrial situation. However, the same is not true for a ‘K’number of ranked optimal solutions, where K ≥ 2. Mathematically, the Kth best solution, K ≥ 2, deals with determination of the 2nd, 3rd, 4th or in general the Kth best solution. This Kth best solution, K ≥ 2, suddenly becomes much more demanding with respect to computational requirements, which increases with the increase in the value of K. This paper first identifies difficulties associated with determination of ranked solutions and later develops a random search method to find ranked optimal solutions in the case of an assignment problem. We test the efficiency of the proposed approach by executing the random search method on a number of different size assignment problems.

In this paper, we developed a new algorithm to find the set of a non-dominated points for a multi-objective integer programming problem. The algorithm is an enhancement on the improved recursive method where the authors have used a lexicographic method for analysis. In this approach a sum of two objectives is considered as one weighted sum objective for each iteration. Computational results show that the proposed approach outperforms the currently available results obtained by the improved recursive method with respect to CPU time and the number of integer problems solved to identify all non-dominated points. Many problems such as assignment, knapsack and travelling salesman have been investigated on different sized problems. The benefit of this approach becomes more visible with the increase in the number of objective functions.

For any single-objective mathematical programming model, rank-based optimal solutions are computationally difficult to find compared to an optimal solution to the same single-objective mathematical programming model. In this paper, several methods have been presented to find these rank-based optimal solutions and based on them a new rank-based solution method (RBSM) is outlined to identify non-dominated points set of a multi-objective integer programming model. Each method is illustrated by a numerical example, and for each approach, we have discussed its limitations, advantages and computational complexity.