A Proposed New Average Method for Solving Multi-Objective Linear Programming Problem Using Various Kinds of Mean Techniques

: In this paper, we initiated New Average Method with various kinds of mean techniques to solve multi-objective linear programming problem. In this method multi-objective functions are converted into single objective function by using different kinds of mean techniques. Also an algorithm of New Average Method is suggested for solving multi-objective linear programming problem. We illustrate numerical problem using Chandra Sen’s Method, Average Method and New Average Method. The numerical result in this paper indicates that New Average Method gives promising result than Chandra Sen’s Method and Average Method. Also we observed that, in New Average Method, Harmonic mean technique gives better result than other mean techniques like as Quadratic mean, Arithmetic mean, Identric mean, Logarithmic mean, Geometric mean techniques.


Introduction
Linear Programming problems are concerned with the efficient use or allocation of limited resources to meet desired objectives. In different sectors like design, construction, maintenance, producing planning, financial and corporate planning and engineering, decision makers have to take decisions and their ultimate goal is to minimize effort or maximize profit. Linear programming problem is formed with a cost or profit function with some constraints conditions, where a single cost or profit function need to be optimized [1]. However, in many situations, decision makers want to optimize several different objective functions at the same time under same constraints conditions. This leads to Multi-Objective concept. It is seen that if the multiple objective functions are not similar to each other, then this problem becomes more critical. There have been many methods suggested for Multi-objective linear programming problem (MOLPP).
A study of multi objective linear programming problem is introduced by Chandra Sen. In which multi-objective function are converted into single objective function with limitation that, individually each objective function optimum value must be greater than zero [2]. Using mean and median solving multi objective programming problem is studied by Sulaiman and Sadiq [3]. Sulaiman and Mustafa also used Harmonic mean to solve MOLPP [4]. A new geometric average technique is studied to solve MOLFPP by Nahar and Alim [5]. They also proposed a Statistical Average Method using Arithmetic, Geometric and Harmonic mean [6].
In order to extend this work, in this paper we propose an algorithm to solve MOLPP with New Average Method with various types of mean techniques and compare the numerical result with Chandra Sen's Method and Average Method. New Average Method gives better result than Chandra Sen's Method and Average Method. Among all New Harmonic mean technique gives the best result.

Mathematical Formulation of MOLPP
Mathematical general form of MOLPP is given as: Programming Problem Using Various Kinds of Mean Techniques Where, is n-dimensional and b is m-dimensional vectors. A is × matrix. , , … … . . , are scalars. Here, is need to be maximized for 1,2, … … … , and need to be minimized for 1, … … . , .

Chandra Sen's Method
In this method, firstly all objective functions need to be maximized or minimized individually by Simplex method. By solving each objective function of equation (1) following equations are obtained: Where, ! , ! , … … … . , ! are the optimal values of objective functions.
These values are used to form a single objective function by adding (for maximum) and subtracting (for minimum) of each result of dividing each by ! . Mathematically, Subject to the constraints are remain same as equation (1).
Then this single objective linear programming problem is optimized.

Average Method of MOLPP
In this method, initially all optimized values of each objective functions are calculated under given constraints. Then a single objective function is constructed by adding all maximization objective functions and subtracting all minimization objective functions and divided them by different kinds of means of maximization objective functions' absolute maximum values and different kinds of means of minimization objective functions' absolute minimum values respectively. Then this single objective function is optimized for the same constraints. For this method, different kinds of mean techniques are discussed.

Algorithm for Average Method of MOLPP
Step 1: Use Simplex method to find the optimal value of each of the objective function.
Step 2: Check the feasibility of step1, if it is feasible then go to step 3 otherwise use dual simplex method to remove infeasibility.

New Average Method of MOLPP
In this method firstly all objective functions are solved individually by using simplex method. Then the result of addition (maximum type) and subtraction (minimum type) of objective functions are divided by different kinds of means to build a single maximum type objective function, where means are calculated by using maximum absolute value of maximum type objective functions and minimum absolute value of minimum type objective functions. Then this single objective function is optimized for the same constraints. For this method, different kinds of mean techniques are discussed.

Algorithm for New Average Method of MOLPP
Step 1: Use Simplex method to find the optimal value of each of the objective function.
Step 2: Check the feasibility of step1, if it is feasible then go to step 3 otherwise use dual simplex method to remove infeasibility.
Step Step 6: Optimize the combined objective function using same constraints as follow: The optimal values of the each of the objective function are calculated by using simplex method and are given below: Now for first objective function,

2
Subject to:          Now using the values of ! , & different types of means are calculated and given below:   Now using Simplex method, Z is optimized for different mean techniques and result are shown below:

Conclusion
In this paper, we have used different types of method for solving MOLPP such as Chandra Sen's Method, Average Method and New Average Method. New Average Method is compared with Average Method and Chandra Sen's Method. It is seen that Chandra Sen's Method is better than some mean (Contraharmonic mean, Quadratic mean, Neuman-Sándor mean) techniques in Average Method and New Average Method is better than Chandra Sen's Method and Average Method for all mean techniques. Among all the techniques, new Harmonic mean technique gives the best result.