Goal Programming

Linear program has multiple objectives, often conflicting in nature

Target values or goals can be set for each objective identified

Not all goals can be simultaneously obtained, resulting in a problem that would otherwise be considered infeasible

Investment Portfolio Example

An investment service company has $50,000 to use in developing a portfolio for a client that is restricted to 2 stocks shown on next slide

The company has two goals– Obtain at least 9% return– Limit investment in Key Oil to at most 60%

of the total investment ($30,000)

Stock Data

Price/Share Estimated Annual Return

AGA Products $50 6%

Key Oil $100 10%

Satisficing Solutions

Instead of optimizing the model to determine the best solution for one objective, the model is satisficed:

several objectives are simultaneously maximized to obtain minimal satisfactory levels.

GP Constraint Types

System or hard constraints:Constraints for which no flexibility in standards or

basic requirements exist (e.g. capital available, limited capacity)

Goal or soft constraints:• Constraints for which targets or goals at various

levels would be acceptable (e.g. required return or acceptable risk)

Deviation Variables

Di+ = amount by which goal i exceeds specified target value

Di- = amount by which goal i falls short of specified target value

Goal Constraints have format:Actual value - Di

+ + Di- = Target Value

Solution Techniques

Absolute Priorities: Goals are ranked in priority. Several models are solved, requiring one goal be satisfied at a time, in the order of its importance.

Weighted Variables: Preferences for deviations from goals are expressed by specifying a weight for the respective deviation variable and including this weighted variable in the objective function that is to be minimized. The model is run just once.

GP Objective Functions

Minimize sum of relevant deviations– Problem with different units ($ -vs- pounds)– Implicit trade-offs between goals hard to assess

Minimize sum of percentage deviations– (1/target)*deviation=percent deviation– Won’t work when target is 0– Implicit trade-offs between goals hard to assess

Minimize sum of weighted percentage deviations– Pick wi for each percentage deviation and use

iterative procedure to refine weights

Summary of Goal Programming1. Identify the decision variables2. Identify hard constraints3. State goals along with their target values4. Create constraints using the decision variables that

would achieve the goals exactly5. Transform soft constraints into goal constraints by

including deviational variables6. Determine which deviational variables are undesirable7. Formulate an objective that penalizes undesirable

deviations8. Identify appropriate weights for objective9. Optimize the problem10. Inspect the solution, not the objective! If

unacceptable, return to step 8.