Heuristic Solution Approaches to the Solid Assignment Problem

The 3-dimensional assignment problem, also known as the Solid Assignment Problem (SAP), is a challenging problem in combinatorial optimisation. While the ordinary or 2-dimensional assignment problem is in the P-class, SAP which is an extension of it, is NP-hard. SAP is the problem of allocating n jobs to n machines in n factories such that exactly one job is allocated to one machine in one factory. The objective is to minimise the total cost of getting these n jobs done. The problem is commonly solved using exact methods of integer programming such as Branch-and-Bound B&B. As it is intractable, only approximate solutions are found in reasonable time for large instances. Here, we suggest a number of approximate solution approaches, one of them the Diagonals Method (DM), relies on the Kuhn-Tucker Munkres algorithm, also known as the Hungarian Assignment Method. The approach was discussed, hybridised, presented and compared with other heuristic approaches such as the Average Method, the Addition Method, the Multiplication Method and the Genetic Algorithm. Moreover, a special case of SAP which involves Monge-type matrices is also considered. We have shown that in this case DM finds the exact solution efficiently. We sought to provide illustrations of the models and approaches presented whenever appropriate. Extensive experimental results are included and discussed. The thesis ends with a conclusions and some suggestions for further work on the same and related topics

Maximum {Supplies, Demands} Method to Find the Initial Transportation Problem

In this paper, we have developed an additional method using the Maximum {Supplies, Demands} and combining both of them with the minimum cost to find an initial solution which is very close to the optimal or at most it is the optimum solution. The transportation algorithm follows the exact steps of the simplex method. However, instead of using the regular simplex tableau, we take advantage of the special structure of the transportation model to organize the computation in a more convenient form. There are several methods for finding the initial basic feasible solution (BFS) of Transportation Problem (TP). But, there is no suitable answer to the question: Which method is the best on