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GOAL PROGRAMMING: FORMULATION AND GRAPHICAL SOLUTION 597

þ
d ¼ the amount by which the annual return for the portfolio is greater than
2
the target value of £9000

d ¼ the amount by which the annual return for the portfolio is less
2
than the target value of £9000
Using these two deviation variables, we write the goal equation for goal 2 as:
þ
3U þ 5H ¼ 9000 þ d d
2 2
or
þ

3U þ 5H d þ d ¼ 9000
2
2
This step completes the development of the goal equations and the constraints for
the Nicolo portfolio problem. We are now ready to develop an appropriate objective
function for the problem.
Developing an Objective Function with Preemptive Priorities
The objective function in a goal programming model calls for minimizing a function
of the deviation variables. In the portfolio selection problem, the most important
goal, denoted P 1 , is to find a portfolio with a risk index of 700 or less. This problem
has only two goals, and the client is unwilling to accept a portfolio risk index greater
than 700 to achieve the secondary annual return goal. Therefore, the secondary goal
is denoted P 2 . As we stated previously, these goal priorities are referred to as
preemptive priorities because the satisfaction of a higher level goal cannot be traded
for the satisfaction of a lower level goal.
Goal programming problems with preemptive priorities are solved by treating
priority level 1 goals (P 1 ) first in an objective function. The idea is to start by
finding a solution that comes closest to satisfying the priority level 1 goals. This
solution is then modified by solving a problem with an objective function involving
only priority level 2 goals (P 2 ); however, revisions in the solution are permitted
only if they do not hinder achievement of the P 1 goals. In general, solving a goal
programming problem with preemptive priorities involves solving a sequence of
linear programmes with different objective functions; P 1 goals are considered first,
P 2 goals second, P 3 goals third and so on. At each stage of the procedure, a revision
in the solution is permitted only if it causes no reduction in the achievement of a
higher priority goal.
We must solve one linear The number of linear programmes that we must solve in sequence to develop the
programme for each solution to a goal programming problem is determined by the number of priority
priority level.
levels. One linear programme must be solved for each priority level. We will call the
first linear programme solved the priority level 1 problem, the second linear pro-
gramme solved the priority level 2 problem and so on. Each linear programme is
obtained from the one at the next higher level by changing the objective function
and adding a constraint.
We first formulate the objective function for the priority level 1 problem. The
client stated that the portfolio risk index should not exceed 700. Is underachieving
the target value of 700 a concern? Clearly, the answer is no because portfolio risk
index values of less than 700 correspond to less risk. Is overachieving the target
value of 700 a concern? The answer is yes because portfolios with a risk index
greater than 700 correspond to unacceptable levels of risk. Thus, the objective
function corresponding to the priority level 1 linear programme should minimize
þ
the value of d .
1
The goal equations and the funds available constraint have already been devel-
oped. Thus, the priority level 1 linear programme can now be stated.

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