# Solving Knapsack Problem with Fuzzy Random Variable Coefficients

Conference paper
Part of the Learning and Analytics in Intelligent Systems book series (LAIS, volume 12)

## Abstract

This paper represents the knapsack problem in fuzzy and random environment. In this work, we introduce the profit and weights of the knapsack problem as the fuzzy random variable. In real-world decision-making situations, the simultaneous existence of randomness and fuzziness of the weights and the profit is a common requirement. To overcome this difficulty, these weights and profit can be considered as the fuzzy random variable. Thus a fuzzy stochastic knapsack problem is introduced. A method of finding a solution of the fuzzy stochastic knapsack problem is obtained using two-stage decision-making approach. At first stage, the problem is converted into equivalent fuzzy knapsack problem. On the second stage, using the graded mean integration representation of the fuzzy number, the equivalent deterministic knapsack problem has been obtained. An investment problem in fuzzy and random environment has been defined as a fuzzy stochastic knapsack problem and the solution procedure is given to signify the proposed approach.

## Keywords

Knapsack problem Fuzzy random variable Dynamic programming Optimal profit Investment problem

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