Analyze the Stochastic Solid Fuzzy Transportation Problem with Mixed Constraints through Weibull Distribution
Authors: Rashi Arya  
Published: July 2024

This paper proposed a general formulation of the stochastic solid transportation problem (SSTP) with mixed constraints such as supply, demand and conveyance capacity taken as uncertain under stochastic environment, following the Weibull distribution (WD). The aim of this study is to minimize the transportation cost includes probabilistic constraints have inequalities of stochastic solid transportation problem (SSTP). SSTP with probabilistic constraints is represented as a chance constrained programming problem. Obtain alpha cut representation from cost coefficient of the fuzzy objective function. We have developed four models for stochastic solid transportation problem. The suggested models are demonstrated by taken as numerical example. A sensitivity analysis is performed to understand parameter’s sensitivity in the proposed model.
Introduction:  In system of transportation, goods are moved from various sources to destinations using different vehicles and organizational systems, involving both technology and human efforts. Efficient resource allocation in transportation system is crucial for industries and imprecision from factors like fluctuating demand, unreliable supply chains and unpredictable traffic. To address these complexities, advanced mathematical models are needed to manage stochasticity, fuzziness and mixed constraints. The study explores the stochastic solid fuzzy transportation problem with mixed constraint by utilizing the Weibull distribution to model uncertainties inherent in transportation systems. This research addresses the complexity introduced by stochastic variables and fuzzy parameters, particularly in situations where demand, supply and cost of transportation are not deterministic.    
Objectives: The aim of this study is to minimize the cost of transportation includes probabilistic constraints have inequalities of stochastic solid transportation problem (SSTP).
Methods: Obtain alpha cut representation form the cost coefficient of the fuzzy objective function and four models are developed for stochastic solid transportation problem. These models are demonstrating by using a numerical example and a sensitivity analysis is conducted to understand the sensitivity of the parameters in the propose model.
Results: Obtained optimal solutions for developed four models of SFSTPMC and sensitivity analysis shows that cost of transportation and flow of unit are sensitive to change in probabilities of demand. Improve transportation system by understanding sensitivity patterns that help decision maker choose appropriate supply availability probabilities.
Conclusions: This study presented an approach for solving the SFSTPMC using the Weibull distribution for probabilistic constraints and fuzzy objective functions for transportation cost. Developed and optimized four models, focusing on stochastic parameters. Sensitivity analysis demonstrated the impact of these parameters on transportation cost and unit flow. The results validate the model’s effectiveness in practical resource allocation and decision making under uncertainty.