Exact methods for single and multi-objective integer programming problems

In this thesis, a few new approaches and algorithms have been developed for different problems arising in the field of operations research. All results, in our opinion, are significant as either they are presenting a new approach, or computationally performing better compared to the existing results that are available in the literature. We also found that their implementation was relatively easy. In this thesis, single and multi-objective integer programming related research problems have been discussed. We have considered, single-objective, Bi-objective, Tri-objective and finally the multi-objective related research problems. We considered a single-objective minimum travelling salesman problem that is classified as NP-hard problem. We obtained a minimum travelling salesman tour with the help of an index restricted minimum spanning tree, which has, in network theory, an easy classification. We have established an equivalence between the index restricted minimum spanning tree and the minimum travelling salesman tour, i.e., the minimum spanning tree with node index less than or equal to 2 is equivalent to the minimum travelling salesman tour.  Numerical illustrations have been presented.  We have investigated the computational aspect of the knapsack and the general linear integer models. We first extended the use of a  recent reformulation of knapsack model to the general linear integer program and found that the reformulation was still providing the computational efficiency with regard to the general linear integer program, when solved by the branch and bound technique. We developed a new reformulation for the knapsack model and once again computational efficiency was improved. These ideas were extended to bi-objective case where again computational efficiency was observed. We developed two new algorithms to solve the bi-objective model using a recently developed characteristic equation approach to a single-objective linear integer program.  We obtained the whole set of non-dominated points. In the second algorithm, a concept of threshold was introduced to improve its efficiency. This second algorithm not only performed better than the first one, but also performed better than one of the existing methods in terms of CPU time and number of iterations. Tri-objective integer programming has been investigated by developing two new algorithms. The first developed algorithm was based on some ideas from two recently  published papers. The second algorithm was an improvement over the first one by applying the idea of relaxation which  was recently used in the literature. This idea is: if there are two problems and  the feasible set of one of them is included in the other, hence there is no need to solve two problems rather only one needs to be solved. The new algorithm outperformed some existing results in respect of CPU time and the number of IPs. Finally, we have developed an algorithm to find the non-dominated points for multi-objective integer programming. The algorithm enhances the performance of one recent method which is the improved recursive method. This enhancement is based on re-arranging the weighted-sum objective functions. The algorithm reduces the CPU time and the number of IPs significantly. From the computational experiments, the improvements became more significant when a greater number of objective functions were considered.