Sankar Kumar Roy

Solving multi-choice multi-objective transportation problem: a utility function approach

This paper explores the study of multi-choice multi-objective transportation problem (MCMTP) under the environment of utility function approach. MCMTP is converted to multi-objective transportation problems (MOTP) by transforming the multi-choice parameters like cost, demand, and supply to real-valued parameters. A general transformation procedure using binary variables is illustrated to reduce MCMTP into MOTP. Most of the MOTP are solved by goal programming (GP) approach. Using GP, the solution of MOTP may not be satisfied all the time by the decision maker (DM) when the proposed problem contains interval-valued aspiration level. To overcome this difficulty, here we propose the approaches of revised multi-choice goal programming (RMCGP) and utility function into the MOTP and then compared the solution between them. Finally, numerical examples are presented to show the feasibility and usefulness of our paper.

Solving a multi-objective transportation problem with nonlinear cost and multi-choice demand

This study develops a mathematical model for a transportation problem consisting of a multi-objective environment with nonlinear cost and multi-choice demand. The objective functions of the proposed transportation problem are non-commensurable and conflict with each other. The focus of the paper is on objective functions of nonlinear type, which occur due to the extra cost of supplying goods remaining at their points of origin to various destinations, and on demand parameters that are considered to be of multi-choice type. Thus, the mathematical model is formulated by considering nonlinear cost and multi-choice demand. Multi-choice programming models cannot be solved directly. A general transformation technique is developed to make multi-choice demand tractable with the help of binary variables. Therefore, an equivalent multi-objective decision making model is established in order to find the optimal solution of the problem. The outcome from a numerical example demonstrates the feasibility of the proposed method.

Multi-choice stochastic transportation problem involving Weibull distribution

This paper presents a multi-choice stochastic transportation problem (TP) where the supply and demand parameters of the constraints follow Weibull distribution. The cost coefficients of the objective function associated with TP are multi-choice type. At first, all the stochastic constraints are transformed into deterministic constraints using stochastic programming approach. Multi-choice type cost coefficients are tractabled by introducing binary variables in the multi-choice programming. Secondly, the transformed problem is considered as a deterministic multi-choice transportation problem. Finally, a numerical example is presented to illustrate the methodology.

Conic scalarization approach to solve multi-choice multi-objective transportation problem with interval goal

This paper explores the study of multi-choice multi-objective transportation problem (MCMTP) under the light of conic scalarizing function. MCMTP is a multi-objective transportation problem (MOTP) where the parameters such as cost, demand and supply are treated as multi-choice parameters. A general transformation procedure using binary variables is illustrated to reduce MCMTP into MOTP. Most of the MOTPs are solved by goal programming (GP) approach, but the solution of MOTP may not be satisfied all times by the decision maker when the objective functions of the proposed problem contains interval-valued aspiration levels. To overcome this difficulty, here we propose the approaches of revised multi-choice goal programming (RMCGP) and conic scalarizing function into the MOTP, and then we compare among the solutions. Two numerical examples are presented to show the feasibility and usefulness of our paper. The paper ends with a conclusion and an outlook on future studies.

Solving matrix game with rough payoffs using genetic algorithm

In this paper, we analyze a matrix game using a rough programming approach. The combination of a matrix game and a rough programming approach represents a new class defined as a rough matrix game. The pay-off elements are characterized by rough variables, and the uncertainties of the rough variables are measured using a measure known as trust. Based on this trust measure, we defined trust equilibrium strategies and a rough expected value. We derived a series of optimal solutions to a rough matrix game using a genetic algorithm. We present a numerical example that illustrates the effectiveness of our rough matrix game.

Analysis of Prey-Predator Three Species Fishery Model with Harvesting Including Prey Refuge and Migration

In this article, a prey-predator system with Holling type II functional response for the predator population including prey refuge region has been analyzed. Also a harvesting effort has been considered for the predator population. The density-dependent mortality rate for the prey, predator and super predator has been considered. The equilibria of the proposed system have been determined. Local and global stabilities for the system have been discussed. We have used the analytic approach to derive the global asymptotic stabilities of the system. The maximal predator per capita consumption rate has been considered as a bifurcation parameter to evaluate Hopf bifurcation in the neighborhood of interior equilibrium point. Also, we have used fishing effort to harvest predator population of the system as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent is earned from the resource. Finally, we have presented some numerical simulations to verify the analytic results and the system has been analyzed through graphical illustrations.

Multi-objective two-stage grey transportation problem using utility function with goals

Multi-Objective Goal Programming is applied to solve problems in many application areas of real-life decision making problems. We formulate the mathematical model of Two-Stage Multi-Objective Transportation Problem (MOTP) where we design the feasibility space based on the selection of goal values. Considering the uncertainty in real-life situations, we incorporate grey parameters for supply and demands into the Two-Stage MOTP, and a procedure is applied to reduce the grey numbers into real numbers. Thereafter, we present a solution procedure to the proposed problem by introducing an algorithm and using the approach of Revised Multi-Choice Goal Programming. In the proposed algorithm, we introduce a utility function for selecting the goals of the objective functions. A numerical example is encountered to justify the reality and feasibility of our proposed study. Finally, the paper ends with a conclusion and an outlook to future investigations of the study.

Rough set approach to bi-matrix game

The paper attempts to solve the bi-matrix game under rough set environment. The elements of the bi-matrix game are characterised by rough variables and defined as rough bi-matrix game (RBG). To handle RBG, we introduce the rough measurable function denoted as trust in it. Based on trust measure and rough expected operator, trust equilibrium strategies are defined for RBG. Then using rough set theory and game theory, the RBG converts into crisp quadratic programming problem (QPP) which depends upon the confidence level. Finally, using Wolfes modified simplex method, the solution of RBG is derived. The numerical examples are presented to illustrate the methodology.

Solving Solid Transportation Problem with Multi-Choice Cost and Stochastic Supply and Demand

This paper proposes a new approach to analyze the solid transportation problem (STP). This new approach considers the multi-choice programming into the cost coefficients of objective function and stochastic programming which is incorporated in three constraints namely sources, destinations and capacities constraints followed by Cauchys distribution for solid transportation problem. The multi-choice programming and stochastic programming are combined into solid transportation problem and this new problem is called multi-choice stochastic solid transportation problem (MCSSTP). The solution concepts behind the MCSSTP are based on a new transformation technique which will select an appropriate choice from a set of multi-choice which optimizes the objective function. The stochastic constraints of STP convert into deterministic constraints by stochastic programming approach. Finally, the authors have constructed a non-linear programming problem for MCSSTP and have derived an optimal solution of the specified problem. A realistic example on STP is considered to illustrate the methodology.

Solving Single-Sink, Fixed-Charge, Multi-Objective, Multi-Index Stochastic Transportation Problem

SYNOPTIC ABSTRACT The aim of this article is to present a fuzzy programming approach to a single-sink, fixed-charge, multiobjective, multi-index stochastic transportation problem (SSMISTP). This article focuses on the minimization of the transportation cost, deterioration rate, and underused capacity for transportation of raw materials from different sources to the ”Single-Sink” by different transportation modes. The parameters of the proposed problem are transportation cost, fixed-charge, deterioration rate, and underused capacity. These parameters are to be treated here as random variables. Because of the globalization of the market, assume that the “Sink” demand is an interval representing the inexact demand component for the SSMISTP. By considering the uncertainties of these parameters, we formulate the mathematical model of the proposed problem. Using a stochastic programming approach, all the stochastic objective functions are converted into deterministic objective functions. Again using the interval concept, the proposed interval-valued sink constraint decomposes into two deterministic constraints. Finally, using all deterministic objective functions and constraints, we design a multiobjective transportation problem(MOTP). The optimal compromise solution to the MOTP has been obtained by using a fuzzy programming technique. We demonstrate the feasibility of the proposed problem using a real-life practical example.