A genetic algorithm based approach to solve multi-resource multi-objective knapsack problem for vegetable wholesalers in fuzzy environment

Vegetable wholesaling problem has a vital role in the business system. In this problem, a vegetable wholesaler is supposed to supply raw, fresh vegetables to supermarkets in an efficient way by minimizing time but maximizing profit. In this paper, we have presented a multi-resource multi-objective knapsack problem (MRKP) for vegetable wholesalers. This model is beneficial for a vegetable wholesaler who collects different types of vegetables (objects) from different villages (resources) to a market for selling the vegetables (objects). MRKP is an extension of the classical concept of 0–1 multi-dimensional knapsack problem (KP). In this model, precisely a wholesaler has a limited capacity van/trolley by which he/she collects a set of vegetables from different villages/vegetable fields. In this model, we have assumed that all the vegetables are available for each resource. Each type of vegetable is associated with a weight, a corresponding profit, and a collection time (for a particular resource). The profit, weight, and collection time of objects is different for different resources. Here a time slice is considered for each object to collect it from different villages to a particular destination/market. Also, the profit and time are the two objectives. MRKP aims to find the amount of an object and the corresponding resource name from which it is collected. We have solved the proposed problem in the fuzzy environment. In this paper, we have explained two defuzzification techniques, namely fuzzy expectation and total λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}-integral value method to solve the proposed problem. We have explained a modified multi-objective genetic algorithm (NSGA-II by Deb et al. in IEEE Trans Evol Comput 6(2):192–197, 2002) that is to maximise the profit and minimise the time to collect the objects. We have considered a multi-objective benchmark test function to show the effectiveness of the proposed Genetic Algorithm. Modification is made by introducing refinement operation. An extensive computational experimentation has been executed that generates interesting results to establish the effectiveness of the proposed model.