Srikumar Acharya

Solving multi-choice linear programming problems by interpolating polynomials

Multi-choice programming solves some optimization problems where multiple information exists for a parameter. The aim of this paper is to select an appropriate parameter from a set of multiple choices, which optimizes the objective function. We consider a linear programming problem where the right hand side parameters are multi-choice in nature. In this paper, the multiple choices of a parameter are considered as functional values of an affine function at some non-negative integer nodes. An interpolating polynomial is formulated using functional values at non-negative integer nodes to take care of any multi-choice parameter. After establishing interpolating polynomials of all multi-choice parameters, a mathematical programming problem is formulated. The formulated problem is treated as a nonlinear programming problem involving mixed integer type variables. It can be solved by using standard nonlinear programming software. Finally, a numerical example is presented to illustrate the solution procedure.

Multi-choice multi-objective linear programming problem

Abstract We consider a multi-objective linear programming problem where some of the right hand side parameters of the constraints are multi-choice in nature. For some right hand side parameters of the constraints, there may exist multiple choices, out of which exactly one is to be chosen. The selection from the sets should be in such a manner that the combination of choices for each set should provide best compromise solution. In order to solve the proposed multi-choice multi-objective linear programming problem, this paper proposes an equivalent mathematical model, which can be solved with the help of existing non-linear programming method. The proposed model can accommodate a maximum of sixteen choices for a single parameter. An illustrative example is presented in support of the proposed model.

Computation of a multi-objective fuzzy stochastic transportation problem

This paper is concerned with the solution methodology of a multi-objective transportation problem where fuzziness and randomness occur under one roof. In the present transportation problem, supplies and demands are considered as fuzzy random variable. In the first step of the solution procedure, fuzziness is removed by using alpha-cut technique to obtain multi-objective stochastic transportation problem. By using the chance constrained technique, the multi-objective stochastic transportation problem is transformed to equivalent crisp multi-objective transportation problem. Then, introducing the concept of membership function, multi-objective deterministic transportation problem is converted into single objective mathematical programming problem. Finally, it is solved with the help of existing technique. A numerical example and a case study are provided in order to illustrate the methodology.

Multi-choice multi-objective mathematical programming model for integrated production planning: a case study

This article develops a multi-choice multi-objective linear programming model in order to solve an integrated production planning problem of a steel plant. The aim of the integrated production planning problem is to integrate the planning sub-functions into a single planning operation. The sub-functions are formulated by considering the capacity of different units of the plant, cost of raw materials from various territories, demands of customers in different geographical locations, time constraint for delivery the products, production cost and production rate at different stages of production process. Departure cost is also considered in the formulation of mathematical programming model. Some of the parameters are decided from a set of possible choices, therefore such parameters are considered as multi-choice type. Multi-choice mathematical programming problem cannot be solved directly. Therefore an equivalent multi-objective mathematical programming model is established in order to find the optimal solution of the problem. Computation of the mathematical programming model is performed with the practical production data of a plant to study the methodology.

Fuzzy Stochastic Genetic Algorithm for Obtaining Optimum Crops Pattern and Water Balance in a Farm

This paper is concerned with multi-objective fuzzy stochastic model for determination of optimum cropping patterns with water balance for the next crop season. The objective functions of the model is to study the effect of various cropping patterns on crop production subject to total water supply in a small farm. The decision variables are the cultivated area of different crops at the farm. The water requirement of the crops follows fuzzy uniform distribution and yields in the objective functions are taken as a fuzzy numbers. The model is solved by using fuzzy stochastic simulation based genetic algorithm without deriving the deterministic equivalents.

Multi-choice goal programming approach to solve multi-objective probabilistic programming problem

Abstract Stochastic Programming is an art of modeling optimization problems in an environment, where randomness occurs. In this manuscript, we present a multi-objective probabilistic programming problem, where the random parameter follow logistic distribution. We transform the probabilistic programming model to an equivalent deterministic mathematical model by using chance constrained technique. Multiple number of aspiration levels are allocated to the objective function by Decision maker, the main aim is to obtain such a decision. After allocating several aspiration levels to the objective function, which will provide minimum deviation from objective function and aspiration level. Such minimization of deviation is possible by using multi-choice goal programming technique. Multi-choice parameters are handled by three different techniques viz; binary variable approach, Vandermonde’s interpolating polynomial approach and linear least square approximation approach. To illustrate the methodology, a numerical example is presented.

Solving multi-objective fuzzy probabilistic programming problem

Most of the real world decision making problems involve uncertainty, which arise due to incomplete information or linguistic information on data. Stochastic programming and fuzzy programming are two powerful techniques to solve such type of problems. Fuzzy stochastic programming is concerned with optimization problems in which some or all parameters are treated as fuzzy random variables in order to capture randomness and fuzziness under one roof. A method for solving multi-objective fuzzy probabilistic programming problem is proposed in this paper. The uncertain parameters are considered as fuzzy log-normal random variables. Since the existing methods are not enough to solve fuzzy probabilistic programming problem directly, therefore the mathematical programming model is transformed to an equivalent multi-objective crisp model. Finally, a fuzzy programming technique is used to solve the multi-objective crisp model. The resulting model is then solved by standard non-linear programming methods. In order to illustrate the methodology a numerical example is provided.

Application of Multi-Choice Fuzzy Linear Programming Problem to a Garment Manufacture Company

Abstract In this paper, we consider a fuzzy multi-choice linear programming problem where some of the parameters and decision variables are trapezoidal type fuzzy numbers. In order to defuzzify a general fuzzy quantity the concept of nearest trapezoidal fuzzy number is introduced. By assuming all the decision variables as trapezoidal fuzzy number, the objective function and the left hand side of constraints are approximated to their nearest trapezoidal fuzzy number. Interpolating polynomials are formulated for all multi-choice type parameters. Multi-choice type parameters are replaced by polynomials with integer variables. Then an equivalent multi-objective non linear programming problem is established. First and second objective functions represent left and right modal values of the trapezoidal type fuzzy number, where as the third and fourth objective functions represent the left and right spreads of the trapezoidal type fuzzy number. By applying lexicographic method optimal solution is obtained. In addition, a case study on a garment manufacture company is presented to demonstrate the solution procedure.

Some modifications on sequential linear goal programming

Abstract Most of the decision making problems have multiple objectives, which cannot be optimized simultaneously due to the conflicting nature of the objectives. Such problems can be solved by various methods to obtain the “best-compromise” solutions. Sequential Linear Goal Programming (SLGP) method can be used to solve Multiple Objective Decision Making (MODM) problems, which involve the trade-off decisions. This paper proposes some modifications in traditional SLGP methods for MODM problems. It uses existing single objective Linear Programming (LP) techniques. The information required in each iteration is taken from the previous iteration. The proposed method is illustrated by some numerical examples. The method provides “best-compromise” solution.

Computation of Multi-choice Multi-objective Fuzzy Probabilistic Transportation Problem

This paper presents a methodology for solving multi-choice multi-objective fuzzy stochastic transportation problem, where the uncertain parameter presents in the supply constraint. In this case fuzzy random variable is assumed to be fuzzy Laplace random variable. The parameters which are present in demand constraint are multi-choice in nature. Fuzziness, randomness, and multi-choiceness are present under one roof. First fuzziness is removed by using alpha-cut technique. In second step randomness is removed by using chance constraint method. In third step multi-choice parameters are handled using interpolating polynomial approaches. Then multi-objective is transformed into a single-objective mathematical model by using weighting mean method. The deterministic equivalent of the model is a mixed integer nonlinear programming problem, which is solved by standard mathematical programming tool and technique. A numerical example is presented to demonstrate the usefulness of the proposed methodology.