An Improved Graph Method for Linear Goal Programming

Goal programming is one of the most widely used methodologies in operations research and management science, and it encompasses most classes of multiple objective programming models. Goal programming is first brought up in 1961 by A. Charnes and W. W. Cooper (Editor Group of O.R., 1990), and then there were many scholars studying it and it developed rapidly in the past several decades. Goal programming allowed a decidemaker to put many requests within one decision. With goal programming the decide-maker should not look for an absolute optimal solution, instead, he should only look for a solution that can make himself more satisfactory than any other solutions. Because goal programming makes up some defects of linear programming, it is considered as a decision tool that is nearer to real decision process than linear programming (Hu Yunquan, 2003).


Introduction
Goal programming is one of the most widely used methodologies in operations research and management science, and it encompasses most classes of multiple objective programming models.Goal programming is first brought up in 1961 by A. Charnes and W. W. Cooper (Editor Group of O.R., 1990), and then there were many scholars studying it and it developed rapidly in the past several decades.Goal programming allowed a decidemaker to put many requests within one decision.With goal programming the decide-maker should not look for an absolute optimal solution, instead, he should only look for a solution that can make himself more satisfactory than any other solutions.Because goal programming makes up some defects of linear programming, it is considered as a decision tool that is nearer to real decision process than linear programming (Hu Yunquan, 2003).
In 1972, Lee.Sang.M put forward the graph method for linear goal programming for the first time in his monograph 'Goal Programming for Decision Analysis (Editor Group of O.R., 1990).In the last 30 years, the graph method was included as an important content of linear goal programming in many textbooks, such as in (Hu Yunquan, 2003, Ignizio, 1976. Lee and Sang M., 1972).Analogous to the graph method for liner programming, the graph method for goal programming can only solve problems with no more than two decision variables.Since there are lots of problems with no more than two decision variables and graph method can give us some help on understanding the characteristics of the optimal solution of linear programming and the satisfaction solution of linear goal programming, graph method was an important part in almost all textbooks that included goal programming.Hence it's of theoretic sense to study graph methods for goal programming.

Goal programming
We describe linear goal programming in this section.Goal programming is a programming problem with multiple goals, in which there is a priority order among the goals.We first introduce some basic notations used commonly in goal programming as follows:

Deviation variable
For each decision goal, we introduce a positive deviation variable d + and a negative deviation variable d − , while d + denotes how much the decision has exceeded the goal, and d − denotes how far the decision is from the goal.

Absolute constraint and goal constraint
If a constraint must be satisfied, we call it absolute constraint.Because it is a hard constraint, a solution is not a feasible one if it can not satisfy any one absolute constraint.Goal constraint is a special weak constraint that can only be seen in goal programming.And sometimes a feasible solution does not satisfy a goal constraint.

Priority factor and weight coefficient
Some goal is important, while some others are unimportant.If a goal is far more important than another one, we give it a priority factor P l , and we give another goal a priority factor P l+1 .Here we have P l >> P l+1 .If a goal is a little more important than another one, we can give it a bigger weight coefficient while we give them same priority factor.

Goal function
Goal function is composed of goal constraints' deviation variables, their priority factors and weight coefficient.Usually, goal programming try to minimize its deviation variables, such as min{ f (d

Satisfaction solution
If a solution satisfies all the absolute constraints, and its cost value is no bigger than any other solution, we call it a satisfaction solution.
3 Traditional graph method for linear goal programming While constraints and cost function are all linear, goal programming is linear goal programming.
In traditional graph method for linear goal programming, firstly the feasible solutions should satisfy all absolute constraints.Then we consider every goal constraint according to their priority factors.Generally, if R j is the solution region for priority factor P j , solution region R j+1 for priority factor P j+1 must be a subspace of R j , i.e.R j+1 ⊆ R j .If R j Φ, and R j+1 = Φ, there is a satisfaction solution in R j .It satisfies goal P 1 , P 2 , • • • , P j , but it can not satisfy the other goals always.
Since there is no solution that can satisfy all the goals of P j+1 , we preferentially let deviation variable be zero whose cost coefficient is larger.For example, if the goal function is 2d − 3 + 3d − 4 , we let d − 4 be zero preferentially as the coefficient 3 of d − 4 is larger than that 2 of d − 3 .So the satisfaction solution satisfies d − 4 = 0.The above described method is just the Graph Method for Linear Goal Programming.But when we look for a satisfaction solution simply by whether the coefficient is larger or smaller, we ignore the difference among the influence that the constraint functions have upon different deviation variables, and so it is not reliable.Now let us look at an example of goal programming and solve it by Graph Method: From Fig. 1, we know that the solution region R 2 for goal P 1 and P 2 is quadrangle ABCD.While considering the goal P 3 , we minimize d − 4 in priority as its cost coefficient 3 is larger than that 2 of P 3 .So the feasible solution region is reduced to quadrangle CDEF.Then we minimize d − 3 .But there is no point that satisfies d − 3 = 0 in quadrangle CDEF, so we have to try to look for a point that minimize d − 3 .The point is F(5, 2).So the feasible solution for the Goal Programming is x 1 = 5, and x 2 = 2.

Defect of tradition graph method
In fact, the cost value of point F(5, 2) about P 3 (2d − 3 + 3d −