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596 CHAPTER 14 MULTICRITERIA DECISIONS

Constraints for goal programming problems are handled in the same way as in an
ordinary linear programming problem. In the Nicolo Investment Advisors problem,
one constraint corresponds to the funds available. Because each share of UK Oil
costs £25 and each share of Hub Properties costs £50, the constraint representing the
funds available is:
25U þ 50H 80 000
To complete the formulation of the model, we must develop a goal equation for
each goal. Let us begin by writing the goal equation for the primary goal. Each share
of UK Oil has a risk index of 0.50 and each share of Hub Properties has a risk index
of 0.25; therefore, the portfolio risk index is 0.50U + 0.25H. Depending on the
values of U and H, the portfolio risk index may be less than, equal to, or greater
than the target value of 700. To represent these possibilities mathematically, we
create the goal equation:
þ
0:50U þ 0:25H ¼ 700 þ d d
1 1
where
þ
d ¼ the amount by which the portfolio risk index exceeds the target value of 700
1

d ¼ the amount by which the portfolio risk index is less than the target value of 700
1
þ

To achieve a goal exactly, In goal programming, d and d are called deviation variables.The purposeof
1
1
the two deviation deviation variables is to allow for the possibility of not meeting the target value
variables must both equal
zero. exactly. Consider, for example, a portfolio that consists of U ¼ 2000 shares of
UK Oil and H ¼ 0 shares of Hub Properties. The portfolio risk index is
þ
0.50(2000) + 0.25(0) ¼ 1000. In this case, d ¼ 300 reflects the fact that the port-
1
folio risk index exceeds the target value by 300 units; note also that since d þ is
1

greater than zero, the value of d must be zero. For a portfolio consisting of U ¼ 0
1
shares of UK Oil and H ¼ 1000 shares of Hub Properties, the portfolio risk index
þ

would be 0.50(0) + 0.25(1000) ¼ 250. In this case, d ¼ 450 and d ¼ 0, indicating
1 1
that the solution provides a portfolio risk index of 450 less than the target value
of 700.
In general, the letter d is used for deviation variables in a goal programming
model. A superscript of plus (+) or minus ( ) is used to indicate whether the
variable corresponds to a positive or negative deviation from the target value. If we
bring the deviation variables to the left-hand side, we can rewrite the goal equation
for the primary goal as:

þ
0:50U þ 0:25H d þ d ¼ 700
1 1
Note that the value on the right-hand side of the goal equation is the target value for
the goal. The left-hand side of the goal equation consists of two parts:
1 A function that defines the amount of goal achievement in terms of the
decision variables (e.g., 0.50U + 0.25H).
2 Deviation variables representing the difference between the target value for
the goal and the level achieved.
To develop a goal equation for the secondary goal, we begin by writing a function
representing the annual return for the investment:
Annual return ¼ 3U þ 5H
Then we define two deviation variables that represent the amount of over- or
underachievement of the goal. Doing so, we obtain:

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