Development of A Heuristic to Solve the General Transportation Problem

The transportation problem is well known and has very important applications. For this well-researched model, there are very efficient approaches for solving it that are available. These approaches include formulating the transportation problem as a linear program and then using the efficient methods such as the simplex method or interior point algorithms. The Hungarian method is another efficient method for solving both the assignment model and the general transportation model. An assignment problem is a special case of the transportation model in which all supply and demand points are 1. Every transportation problem can be converted into an assignment problem since rows and columns can be split so that each supply and each demand point is 1. The transportation simplex method is another method that is also used to solve the general transportation problem. This method is also called the modified distribution method (MODI). To use this approach, a starting solution is required and the closer the starting solution to the optimal solution, the fewer the iterations that are required to reach optimality. The fourth method for transportation models is the network simplex method, which is the Fastest so far. Unfortunately, all these approaches for transportation models are serial in nature and are very difficult to parallelize, which makes it difficult to efficiently use the available massively parallel technology. There is a need for an efficient approach for the transportation problem, which is easily parallelizable. This paper presents a See-Saw approach for solving the general transportation problem. This is an extension of the See-Saw approach for solving the assignment problem. The See-Saw moves can be done independently, which makes the approach proposed in this paper more promising than the available methods for transportation model

Development of A Dummy Guided Formulation and Exact Solution Method for TSP

A traveling salesman problem (TSP) is a problem whereby the salesman starts from an origin node and returns to it in such a way that every node in the network of nodes is visited once and that the total distance travelled is minimized. An efficient algorithm for the TSP is believed not to exist. The TSP is classified as NP-hard and coming up with an efficient solution for it will imply NP=P. The paper presents a dummy guided formulation for the traveling salesman problem. To do this, all sub-tours in a traveling salesman problem (TSP) network are eliminated using the minimum number of constraints possible. Since a minimum of three nodes are required to form a sub-tour, the TSP network is partitioned by means of vertical and horizontal lines in such a way that there are no more than three nodes between either the vertical lines or horizontal lines. In this paper, a set of all nodes between any pair of vertical lines or horizontal lines is called a block. Dummy nodes are used to connect one block to the next one. The reconstructed TSP is then used to formulate the TSP as an integer linear programming problem (ILP). With branching related algorithms, there is no guarantee that numbers of sub-problems will not explode to unmanageable levels. Heuristics or approximating algorithms that are sometimes used to make quick decisions for practical TSP models have serious economic challenges. The difference between the exact solution and the approximated one in terms of money is very big for practical problems such as delivering household letters using a single vehicle in Beijing, Tokyo, Washington, etc. The TSP model has many industrial applications such as drilling of printed circuit boards (PCBs), overhauling of gas turbine engines, X-Ray crystallography, computer wiring, order-picking problem in warehouses, vehicle routing, mask plotting in PCB production, etc

Development of an Exact Method for Zero-one Linear Programming Model

The paper presents a new method for solving the 0–1 linear programming problems (LPs). The general 0–1 LPs are believed to be NP-hard and a consistent, efficient general-purpose algorithm for these models has not been found so far. Cutting planes and branch and bound approaches were the earliest exact methods for the 0–1 LP. Unfortunately, these methods on their own failed to solve the 0–1 LP model consistently and efficiently. The hybrids that are a combination of heuristics, cuts, branch and bound and pricing have been used successfully for some 0–1 models. The main challenge with these hybrids is that these hybrids cannot completely eliminate the threat of combinatorial explosion for very large practical 0–1 LPs. In this paper, a technique to reduce the complexity of 0–1 LPs is proposed. The given problem is used to generate a simpler version of the problem, which is then solved in stages in such a way that the solution obtained is tested for feasibility and improved at every stage until an optimal solution is found. The new problem generated has a coefficient matrix of 0 s and 1 s only. From this study, it can be concluded that for every 0–1 LP with a feasible optimal solution, there exists another 0–1 LP (called a double in this paper) with exactly the same optimal solution but different constraints. The constraints of the double are made up of only 0 s and 1 s. It is not easy to determine this double 0–1 LP by mere inspection but can be obtained in stages as given in the numerical illustration presented in this paper. The 0–1 integer programming models have applications in so many areas of business. These include large economic/financial models, marketing strategy models, production scheduling and labor force planning models, computer design and networking models, military operations, agriculture, wild fire fighting, vehicle routing and health care and medical model

Computational enhancement in the application of the branch and bound method for linear integer programs and related models

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

Bi-objective integer programming analysis based on the characteristic equation

In this paper, a bi-objective integer programming problem is analysed using the characteristic equation that was developed to solve a single-objective pure integer program. This equation can also provides other ranked solutions i.e. 2nd, 3rd,.. best solutions. These solutions are potential non-dominated points for a bi-objective integer program, which is being investigated in this paper. A C code is developed to solve the characteristic equation, a tool which is not available in the IBM ILOG CPLEX library. Two versions of this algorithm are developed to identify the non-dominated points for the bi-objective integer programming problem. The second version improves on the first by reducing the number of search steps. Computational experiments are carried out with respect to the two algorithms developed in this paper and comparisons have also been carried out with one of the recently developed method, the balanced box method. These computational experiments indicate that the second version of the algorithm developed in this paper performed significantly better than the first version and out performed the balanced box method with respect to both CPU time and the number of iterations