Profit maximization fuzzy 4D-TP with budget constraint for breakable substitute items: a swarm based optimization approach

The concept of breakable substitute items and budget constraints is to be used in decision-making problems. For demonstration, a fixed charge multi-item four-dimensional transportation problem (4D-TP) with budget constraint as profit maximization, the problem for breakable substitute items is considered under a fuzzy environment. The items are purchased from distinct depots at different prices. The different types of breakable substitute items are supplied to separate destination points from a distinct type of supply points with a different type or capacity of vehicles via a different road. The parameters of the transportation problem like direct transportation charges, fixed charges, market prices, procuring costs, sources of origins, requirements at destination points, conveyance's volume, or size are assumed to be deterministic or imprecise. Budget restrictions are applied on-demand points where the available budget amounts are fuzzy. Requirement restrictions at destinations are on the number of items having some minimum demands for each substitutable item. The imprecise constraints are reduced to equivalent deterministic constraints using credibility measures. The reduced fuzzy optimization problem under deterministic constraints is solved by swap-based particle swarm optimization (SPSO) and credibility-based genetic algorithm (CBGA), where a comparison of fuzzy objectives is made using the credibility measure of fuzzy events. For deterministic objectives, the same SPSO algorithm is used, where a simple comparison makes a comparison of an objective of deterministic numbers. The obtained results are compared using CBGA and SPSO for 4D-TP. As a particular demonstration, the results of solid transportation problems (3D-TPs) and conventional transportation problems (2D-TPs) are also presented in this paper. Statistical analysis is demonstrated to analogize the algorithms.

Automated summary

This research utilizes the concept of breakable substitute items and budget constraints in decision-making problems. The problem of breakable substitute items is considered under a fuzzy environment in a fixed charge multi-item four-dimensional transportation problem (4D-TP) with profit maximization as the objective. The problem involves purchasing items from different depots at different prices and delivering different types of items to separate destinations using different types or capacities of vehicles. Fuzzy constraints are transformed into equivalent deterministic constraints using credibility measures, and the reduced fuzzy optimization problem is solved using swap-based particle swarm optimization (SPSO) and credibility-based genetic algorithm (CBGA). The results obtained from CBGA and SPSO for 4D-TP are compared, and the paper also presents results from solid transportation problems (3D-TPs) and conventional transportation problems (2D-TPs) for demonstration, with statistical analysis performed to compare the algorithms.