Optimal Transformation Technique To Solve Multi-Objective Linear Programming Problme (MOLPP)

In this paper ,we suggested a new technique by using optimal average(OAV) for function. and an algorithm is suggested for it is solution .The MOLPP criteria of Chandra Sen and Sulaiman & Sadiq(Sen,Chandra,1983 ; Sulaiman & Sadiq, 2006, respectively) ; has been modified in this paper. The computer application of algorithm also has been demonstrated by a flow-cart and solving a numerical examples .The numerical results in(Table3) indicate that the new technique in general is promising. Introduction In (1983),Chandra Sen (Sen,1983) defined the multi-objective linear programming problem, and suggested an approach to construct that the multiobjective function under the limitation that the optimum value of individual problem is greater than zero. In (1992), Sulaiman and Mohammad (Sulaiman & Mahammad, 1992) studied the multi-objective fractional complimentary program. In(1993),Abdil-Kadir and Sulaiman (Abdul Kadir & Sulaiman, 1993) studied the multi-objective fractional programming problem .In (2006),Sulaiman and Sadiq (Sulaiman & Sadiq, 2006) studied the multiobjective function by solving the multi-objective programming problem ,using mean and mean value ;and they did try optimal solution and comparison results between Chandra Sen approach (Sen, 1983) and modified approach (Sulaiman & Sadiq, 2006). In order to extended this work we have defined a multiobjective programming problem linear and investigated the algorithm to solve linear programming problem for multi-objective functions (Sen, 1983).Irrespective of the number objectives with less computational burden and suggest a new technique by using optimal average (OAV) of objective functions; to generate the best optima solution. The computer application of our algorithm also has been discussed by solving a numerical examples. Finally we have been shown results and comparison they between the new technique and Chandra Sen approach (Sen, 1983) & Sulaiman approach (Sulaiman & Sadiq, 2006).


Introduction
,Chandra Sen (Sen,1983) defined the multi-objective linear programming problem, and suggested an approach to construct that the multiobjective function under the limitation that the optimum value of individual problem is greater than zero. In (1992), Sulaiman and Mohammad (Sulaiman & Mahammad, 1992) studied the multi-objective fractional complimentary program. In(1993),Abdil-Kadir and Sulaiman (Abdul Kadir & Sulaiman, 1993) studied the multi-objective fractional programming problem .In (2006), Sulaiman and Sadiq (Sulaiman & Sadiq, 2006) studied the multiobjective function by solving the multi-objective programming problem ,using mean and mean value ;and they did try optimal solution and comparison results between Chandra Sen approach (Sen, 1983) and modified approach (Sulaiman & Sadiq, 2006). In order to extended this work we have defined a multiobjective programming problem linear and investigated the algorithm to solve linear programming problem for multi-objective functions (Sen, 1983).Irrespective of the number objectives with less computational burden and suggest a new technique by using optimal average (O AV ) of objective functions; to generate the best optima solution. The computer application of our algorithm also has been discussed by solving a numerical examples. Finally we have been shown results and comparison they between the new technique and Chandra Sen approach (Sen, 1983) & Sulaiman approach (Sulaiman & Sadiq, 2006).

Mathimatical form of the multi-objective programming problem
A multi-objective linear programming (MOLPP) is introduced by Chandra Sen (Sen, Chandra, 1983) and suggested an approach (C A ) to construct the multi-objective function under the limitation that the optimal value of individual problem is greater than zero (Abdul Kadir & Sulaiman, 1993). He has not considered the situation when the optimum value of some of individual objective function functions may be negative or zero also (Sulaiman & Sadiq, 2006).The mathematical form of this type of problem is given as follows: Max z 1 = c 1 t .x+a 1 Max z 2 = c 2 t .x+a 2 …(2.1) Max z r = c r t .x+a r Min z r+1 = c t r+1 .x+a r+1 . . . Min z s = c t s .x+a s Subject to constraints: A.X= B …(2.2) X≥0 …(2.3) Where r is the number of objective functions to be maximized, is the number of objective functions to be max & minimized, X is an ndimensional vector of decision variables, C is n-dimensional vector of constants, B is m dimensional vector of constants, (s-r) is the number of objective functions that is to be minimized, A is a (mxn) matrix of coefficients. All vectors are assumed to be column vectors unless transposed, a i ( i=1,2,…,s) are scalar constants, C t .X +a i ; i=1,2,3….,s, are linear factors for all feasible solutions (Abdul Kadir & Sulaiman,1993). If a i =0; i= 1, 2,…, s, then the mathematical form Become:

Formulation of multiobjective functions
The same approach taken by sulaiman and Gulnar (Sulaiman & Sadiq, 2006) for multiobjective functions is folled here to for emulate the constrained objective functions given in equation (2.1).suppose we obtained a single value corresponding to each of objective functions of it being optimized individually subject to constraints (2.2) and (2.3) as follows: ,2,…,s the decision variable may not necessarily be common to all optimal solutions in the presence of conflicts among objectives (Sulaiman & Sadiq, 2006). But the common set of decision variable between object functions are necessary in order to select the best compromise solution (Azapagic, 1999). We can determine the common set of decision variable from the following combined objective function (Sen, 1983;AbdulKadir&Sulaiman,1993 andSulaiman& Sadiq,2006).

Solving the MOLPP by modified approach (M A )
We formulate the combined objective function as follows to determine the common set of decision variables ,to solving the MOLPP by modified approach (using mean and median value) (Sulaiman & Mohammad, 1992).

Algorithm
The following algorithm is to obtain the optimal solution for the multiobjective linear programming problem defined previous can be summarized as follows:-Step1: Find the value of each of individual objective functions which is to be maximized or minimized.
Step2: slove the first objective problem by simplex method.
Step3: check the feasibility of the solution in step2. if it is feasible then go to step 4, otherwise, use dual simplex methods to remove infeasibility.
Step4: assign a name to the optimum value of the first objective function Z 1 say  Ai Step5: repeat the step 2; i=1,2,3,4 for the k th objective problem, k=2, 3,---s.

Program Notation :
The following notations, which are used in computer program are defined as follows:

2-
The results higher by our optimal technique, then by modification approach, even there is only tow objective functions one is to be maximized and the other is minimized.
3-Since the results by modification is better and more optimal than the result by Chandra Sen (Sulaiman & Sadiq, 2006).Hence the result by our optimal technique is more better than the result by Chandra Sen. As indicated in table (3).
4-For all cases the introduced objective function Z is to be maximized.