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GOAL PROGRAMMING: FORMULATION AND GRAPHICAL SOLUTION 601

Problem 2 will test your Although the graphical solution procedure is a convenient method for solving goal
ability to formulate a goal programming problems involving two decision variables, the solution of larger prob-
programming model and
use the graphical solution lems requires a computer-aided approach. In Section 14.2 we illustrate how to use a
procedure to obtain a computer software package to solve more complex goal programming problems.
solution.
Goal Programming Model
As we stated, preemptive goal programming problems are solved as a sequence of
linear programmes: one linear programme for each priority level. However, notation
that permits writing a goal programming problem in one concise statement is helpful.
In writing the overall objective for the portfolio selection problem, we must write
the objective function in a way that reminds us of the preemptive priorities. We can
do so by writing the objective function as:
þ

Min P 1 ðd Þþ P 2 ðd Þ
2
1
The priority levels P 1 and P 2 are not numerical weights on the deviation variables,
but simply labels that remind us of the priority levels for the goals.
We now write the complete goal programming model as:

þ
Min P 1 ðd Þþ P 2 ðd Þ
1 2
s:t:
25U þ 50H 80 000 Funds available
þ
0:50U þ 0:25H d þ d ¼ 700 P 1 goal
1
1
þ
3U þ 5H d þ d ¼ 9 000 P 2 goal
2 2

þ
þ

U; H; d ; d ; d ; d ; 0
1 1 2 2
With the exception of the P 1 and P 2 priority levels in the objective function, this model
is a linear programming model. The solution of this linear programme involves solving
a sequence of linear programmes involving goals at decreasing priority levels.
We now summarize the procedure used to develop a goal programming model.
Step 1. Identify the goals and any constraints that reflect resource capacities or
other restrictions that may prevent achievement of the goals.
Step 2. Determine the priority level of each goal; goals with priority level P 1 are most
important, those with priority level P 2 are next most important and so on.
Step 3. Define the decision variables.
Step 4. Formulate the constraints in the usual linear programming fashion.
Step 5. For each goal, develop a goal equation, with the right-hand side specifying
þ

the target value for the goal. Deviation variables d and d are included in
1 1
each goal equation to reflect the possible deviations above or below the
target value.
Step 6. Write the objective function in terms of minimizing a prioritized function
of the deviation variables.

NOTES AND COMMENTS

1 The constraints in the general goal programming analysts call the goal equations goal constraints
model are of two types: goal equations and and the ordinary linear programming constraints
ordinary linear programming constraints. Some system constraints.

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