1. Linear Goal Programming

2. What is Goal Programming?
Single objective in formulating LPs so far
Max profit
Min cost
Several simultaneous objectives in practice
Increase reliability
Improve employee morale
Improve company’s image
Goal programming: a way of solving multi-objective problems

3. Categorization of Goal Programming
Nonpreemptive goal programming
Goals with roughly same importance
Assign weights to goals
Construct a single objective function to minimize the weighted sum of deviations from goals
Preemptive goal programming
Hierarchy of priority levels for goals
1st priority is to satisfy goal 1
2nd priority is to satisfy goal 2, etc.
Solve sequentially

4. Three Possible Types of Goals
A lower, one-sided goal
a1 x1 + a2 x2 + … + an xn ≥ b
example: profit exceeds some threshold
An upper, one-sided goal
a1 x1 + a2 x2 + … + an xn ≤ b
example: cost less than some value
A two-sided goal
a1 x1 + a2 x2 + … + an xn = b
example: production equal to forecasted demand

5. Dewright Company Example – Nonpreemptive Goal Programming
The Dewright Company is considering three new products to replace current models that are being discontinued, so their OR department has been assigned the task of determining which mix of these products should be produced. Management wants primary consideration given to three factors: long-run profit, stability in the workforce, and the level of capital investment that would be required now for new equipment. In particular, management has established the goals of (1) achieving a long-run profit of at least $125 million from these products, (2) maintaining the current employment level of 4000 employees, and (3) holding the capital investment to less than $55 million. However, management realizes that it probably will not be possible to attain all these goals simultaneously, so it has discussed priorities with the OR department. This discussion has led to setting penalty weights of 5 for missing the profit goal (per $1 million under), 2 for going over the employment goal (per 100 employees), 4 for going under this same goal, and 3 for exceeding the capital investment goal (per $1 million over). Each new product’s contribution to profit, employment level, and capital investment level is proportional to the rate of production. These contributions per unit rate of production are shown in the following table, along with the goals and penalty weights.

6. Dewright Company Example (Cont’d)
Unit Contribution
Product
Factor 1 2 3
Goal (Units)
Penalty Weight
Long-run profit 12 9 15
≥ 125 (millions of dollars) 5
Employment level 4
= 40 (hundreds of employees)
2(+), 4(-)
Capital investment 7 8
≤ 55 (millions of dollars)

7. Dewright Company Example (Cont’d)
Decision variables
x1: production rate of product 1
x2: production rate of product 2
x3: production rate of product 3
Three goals
Profit goal: 12 x1 + 9 x x3 ≥ 125 (lower one-sided goal)
Employment goal: 5 x1 + 3 x2 + 4 x3 = 40 (two-sided goal)
Investment goal: 5 x1 + 7 x2 + 8 x3 ≤ 55 (upper one-sided goal)
Objective function:
Min Z = 5(12 x1 + 9 x x3 – 125)-
+ 2(5 x1 + 3 x2 + 4 x3 – 40)+
+ 4(5 x1 + 3 x2 + 4 x3 – 40)-
+ 3(5 x1 + 7 x2 + 8 x3 – 55)+
where
x if x ≥ if x > 0
x+ = x- =
0 if x < x = |x| if x ≤ 0

8. Dewright Company Example (Cont’d)
To put into a proper LP form
Let y1 = 12 x1 + 9 x x3 – 125
y2 = 5 x1 + 3 x2 + 4 x3 – 40
y3 = 5 x1 + 7 x2 + 8 x3 – 55
and y1 = y1+ - y1- and y1+ ≥ 0, y1- ≥ 0
y2 = y2+ - y2- and y2+ ≥ 0, y2- ≥ 0
y3 = y3+ - y3- and y3+ ≥ 0, y3- ≥ 0

9. Dewright Company Example (Cont’d)
Final LP for nonpreemptive goal programming:
Min Z = 5 y y y y3+
subject to
12 x1 + 9 x x3 – (y1+ - y1- ) = 125
5 x1 + 3 x x3 – (y2+ - y2- ) = 40
5 x1 + 7 x x3 – (y3+ - y3- ) = 55
xi, yi+, yi- ≥ 0 i=1,2,3
add any other functional constraints too

10. Dewright Company Example (Cont’d)
Optimal solution:
x1 = 25/3
x2 = 0
x3 = 5/3
y1+ = y1- = 0 → y1 = → 1st goal is satisfied
y2+ = 25/3, y2- = 0 → y2 = 25/3 → 2nd goal exceeds employment level by /3 hundred employees
y3+ = y3- = 0 → y3 = 0 → 3rd goal is satisfied

11. Modified Dewright Company Example - Preemptive Goal Programming
Priority Level
Factor
Goal
Penalty Weight
1st Priority
Employment level
≤ 40 2M
Capital investment
≤ 55 3M
2nd Priority
Long-run profit
≥ 125 5
≥ 40 4

12. Modified Dewright Company Example -Preemptive Goal Programming (Cont’d)
1st LP:
Min Z = 2M y2+ + 3M y3+
subject to
5 x1 + 3 x x3 – (y2+ - y2- ) = 40
5 x1 + 7 x x3 – (y3+ - y3- ) = 55
xi ≥ 0 i=1,2,3,
yi+, yi- ≥ 0 i=2,3
There are multiple optimal solutions with y2+ = 0 and y3+ = 0.
To get 2nd LP, drop y2+, y3+ forcing those goals to remain satisfied,
also add 2nd priority constraints

13. Modified Dewright Company Example -Preemptive Goal Programming (Cont’d)
2nd LP:
Min Z = 5 y y2-
subject to
12 x1 + 9 x x3 – (y1+ - y1- ) = 125
5 x1 + 3 x x y = 40
5 x1 + 7 x x y3- = 55
xi ≥ 0 i=1,2,3,
y1+, y1-, y2-, y3- ≥ 0
There is a unique solution: x1 = 5, x2 = 0, x3 = 15/4, y1+ = 0, y1- = 35/4, y2- = 0 and y3- = 0.

14. Dewright Company Example: Compare the two solutions
Solution by nonpreemptive goal programming:
(x1, x2, x3 ) = (25/3, 0, 5/3)
12 x1 + 9 x x3 = 125 profit
5 x1 + 3 x2 + 4 x3 = employment
5 x1 + 7 x2 + 8 x3 = 55 capital investment
Solution by preemptive goal programming:
(x1, x2, x3 ) = (5, 0, 15/4)
12 x1 + 9 x x3 = profit
5 x1 + 3 x2 + 4 x3 = 40 employment

15. Camyo Manufacturing Example
Camyo Manufacturing produces four parts that require the use of a lathe and a drill press. The two machines operate 10 hours a day. The following table provides the time in minutes required by each part:
It is desired to balance the use of the two machines by requiring the difference between their total operation times not to exceed 30 minutes. The market demand limits the number of units produced of each part to at least 10 units. Additionally, the number of units of part 1 may not exceed that of part 2.
Part
Lathe
Drill Press 1 5 3 2 6 4 7

16. Camyo Manufacturing Example (Cont’d)
Decision variables
xi = number of parts produced per day of type i, i=1, 2, 3, 4
Operation time goal:
Difference between the time on machine 1 (lathe) and time on machine 2 (drill press) is not to exceed 30 minutes:
|time1 – time2| ≤ 30
-30 ≤ time1 – time2 ≤ 30
time1 = 5 x1 + 6 x2 + 4 x3 + 7 x4 lathe
time2 = 3 x1 + 2 x2 + 6 x3 + 4 x4 drill press
-30 ≤ (5 x1 + 6 x2 + 4 x3 + 7 x4) – (3 x1 + 2 x2 + 6 x3 + 4 x4) ≤ 30
-30 ≤ 2 x1 + 4 x2 - 2 x3 + 3 x4 ≤ 30

17. Camyo Manufacturing Example (Cont’d)
Final LP for nonpreemptive goal programming:
Min Z = y1+ + y2-
subject to
5 x1 + 6 x2 + 4 x3 + 7 x ≤ 600 lathe, 600 min in a day
3 x1 + 2 x2 + 6 x3 + 4 x ≤ 600 drill press, 600 min a day
x ≥ 10
x ≥ 10
x ≥ 10
x ≥ 10
x x ≤ 0
2 x1 + 4 x2 - 2 x3 + 3 x4 – (y1+ - y1-) = 30 time1 - time2 ≤ 30
2 x1 + 4 x2 - 2 x3 + 3 x – (y2+ - y2-) = -30 time1 - time2 ≥ -30
xi ≥ 0, i=1, 2, 3, 4
yi+ ≥ 0, yi- ≥ 0, i=1, 2
demand must exceed 10 units