Advanced Transformation Technique to Solve Multi-Objective Optimization Problems

Multi-objective optimization problem (MOOP) is an important class of optimization problem that ensures users to model a large variety of real world applications. In this paper an advanced transformation technique has been proposed to solve MOOP. An algorithm is suggested and the computer ap-plication of algorithm has been demonstrated by a flow chart. This method is comparatively easy to calculate. Applying on different types of examples, the result indicates that the proposed method gives better solution than other methods and it is less time consuming. Physical presentation and data analysis represent the worth of the method more compactly.


Introduction
Multi-objective optimization (MOO) is an effective technique for studying optimal trade off solutions that balance several criteria. The fundamentals and applications of MOO have been already explored in great detail [1]. The main limitation of MOO is that its computational burden grows in size with the number of objectives. Various types of solution procedure have been already developed for solving MOO problems [2]- [21]. Some of them deal with theory and some of them concern with solution methods and applications.
To solve multi-objective linear programming problems, various types of methods have been proposed by various scholars, such as Mean and median method by Sulaiman and Sadiq [17], Optimal transformation technique by Sulai-man and Ameen [4], Harmonic mean by Sulaiman and Mustafa [9], New statistical average method by Nahar and Alim [12].
On the other hand, Linear fractional programming problem has been solved by different researchers by different techniques, for example, A new procedure proposed by Tantawy [3] and by Guzel [5], An improved method by Mehdi, Chergui and Abbas [7], Arithmetic average technique by Sulaiman, Sadiq and Rahim [8], A new approach presented by Akter and Modi [10], New geometric average technique proposed by Nahar and Alim [11].
Many research scholars have solved multi-objective quadratic programming problem by applying several methods. We include some of them. Optimal cutting plane procedure [19] and Arithmetic average transformation technique [20] had been used by Sulaiman and Rahim. Optimal average maximum-minimum technique and Optimal geometric average technique had been used by Sulaiman-Nawkhass [6] and by Sulaiman-Abdull [21] gradually to solve multi-objective quadratic programming problem.
The larger the size of the problem, the greater the number of inefficient solution generated and thus the slower the convergence of the algorithm. To overcome this problem, we propose an advanced transformation technique. We test the capabilities of this proposed technique drawing a comparison with other techniques.
In this paper, we focus our interest on multi-objective quadratic programming problem (MOQPP) where several quadratic objectives are to be optimized subject to a set of linear constraints and nonnegative integer variables. The optimization software package, namely AMPL has been employed in the computation.
Sen proposed a method of multi-objective programming for achieving several conflicting objectives simultaneously [1] [15]. Multi-objective linear programming problem (MOLPP) has been solved by many research scholars. Sulaiman and Sadiq had solved MOLPP by using mean and median [17]. Hamad-Amin used arithmetic average technique to solve it [18]. New statistical (arithmetic, geometric, harmonic) average techniques had been proposed by Nahar and Alim for solving MOLPP. Sulaiman and Rahim used optimal cutting plane procedure to solve MOQPP [19]

Multi-Objective Optimization Problem
Multi-objective optimization is an area of multiple criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously.
Mathematically, multi-objective decision making problems can be expressed as: The problem consists of n decision variables, m constraints and k objectives.
h g x i j h ∀ might be linear or nonlinear. Mathematically, the multi-objective linear programming problem (MOLPP) can be defined as: And mathematically the multi-objective quadratic programming problem (MOQPP) can be stated as: for maximization and 1, , i r s = + for minimization where x is an n-dimensional vector of decision variables, P is a ( )  To make the objective function dimension free, the integrated objective function was summarized by weighting each objective function by inverse of its optima. Hence the integrated objective function is formulated without facing any complex with objective functions of different dimensions.

Chandra Sen's Approach
Then the single objective optimization problem is solved by simplex method. This method is known as Chandra Sen's approach.

Harmonic Average Technique
Harmonic Mean: Harmonic mean of a set of observations is defined as the reciprocal of the arithmetic average of the reciprocal of the given values. If ∑ It provides a more reliable result when the results to be achieved are the same for the various mean adopted.
The steps to calculate the harmonic mean are as follows: Step 1: Finding the reciprocal of the given values.
Step 2: Calculating the average for the reciprocals obtained in step 1.
Step 3: Finally calculate the reciprocal of the average obtained from step 2.
Multi-objective optimization problem given in (3.1) can be translated by harmonic average technique as: Some difficulties occur when calculating with harmonic mean. The harmonic mean is greatly affected by the values of the extreme items. It can't be able to calculate if any of the item is zero. This calculation is cumbersome as it involves the calculation using reciprocals of the items.

Modified Harmonic Average Technique
According to modified harmonic average technique multi-objective optimization problem given in (3.1) can be converted into a single objective function as: Step 1: Find the minimum optimal value of the maximization problems.
Step 2: Find the minimum optimal value of the minimization problems.
Step 3: Calculate the average for the reciprocals obtained in step 1 and step 2.
Step 4: Finally calculate the reciprocal of the average obtained from step 3.
Modified harmonic average technique gives better solution than harmonic average technique.

Advanced Transformation Technique
Multi-objective optimization problem can be defined as: [ ] We require the common set of decision variables to be the best compromising optimal solution. Here we can determine the common set of decision variables from the following combined objective function.
By our proposed Advanced transformation technique, we can obtain the single objective function as follows:

Algorithm
Step 1: Find the value of each objective function which is to be maximized or minimized.
Step 2: Solve the first objective function by mathematical programming language AMPL.
Step 3: Assign a name to the optimum value of first objective function 1 z by 1 α .
Step 5: Select Step 7: Optimize the combined objective function as 1 1 Max .

Numerical Example
Consider the following Multi-Objective Quadratic Programming problem with linear constraints: After finding the value of each of individual objective functions by using AMPL, the numerical results are given in Table 1.
Using Chandra Sen's Approach, the system becomes, The system,  Table 2 summarizes the solutions of the MOQPP using different approaches.

Physical Interpretation
In this optimization problem, a process is going to search a better procedure to find maximum value of a given MOQPP.

Data Analysis
Consider, some set of numerical examples of multi-objective optimization prob-