1. Chapter 6 Integer and Goal Programming Models
Jason C. H. Chen, Ph.D.
Professor of MIS
School of Business Administration
Gonzaga University
Spokane, WA 99223

2. Variations of Basic Linear Programming
Integer Programming
Goal Programming
Nonlinear Programming (skip)

3. Integer Programming (IP)
Where some or all decision variables are required to be whole numbers.
General Integer Variables (0,1,2,3,etc.)
Values that count how many
Binary Integer Variables (0 or 1)
Usually represent a Yes/No decision

4. General Integer Example: Harrison Electric Co.
Produce 2 products (lamps and ceiling fans) using 2 limited resources
Decision: How many of each product to make? (must be integers)
Objective: Maximize profit

5. Decision Variables L = number of lamps to make
F = number of ceiling fans to make
Lamps
(per lamp)
Fans
(per fan)
Hours Available
Profit Contribution
$600
$700
Wiring Hours
2 hrs
3 hrs 12
Assembly Hours
6 hrs
5 hr 30

6. LP Model Summary Max 600 L + 700 F ($ of profit)
Subject to the constraints:
2L + 3F < 12 (wiring hours)
6L + 5F < 30 (assembly hours)
L, F > 0

7. Graphical Solution

8. Properties of Integer Solutions
Rounding off the LP solution might not yield the optimal IP solution
The IP objective function value is usually worse than the LP value
IP solutions are usually not at corner points

9. Using Solver for IP
IP models are formulated in Excel in the same way as LP models
The additional integer restriction is entered like an additional constraint
int - Means general integer variables
bin - Means binary variables
Go to file 6-1.xls

10. Harrison Electric (General Integer)F
Lamps
Fans
Number of units
3.00
2.00
Profit
$600
$700
$3,200.00
Constraints:
Wiring hours 2 3
12.00
<= 12
Assembly hours 6 5
28.00 30
LHS
Sign
RHS
Go to file 6-1.xls

11. Binary Integer Example: Portfolio Selection
Choosing stocks to include in portfolio
Decision: Which of 7 stocks to include?
Objective: Maximize expected annual return (in $1000’s)

12. Stock Data

13. Decision Variables Use the first letter of each stock’s name
Example for Trans-Texas Oil:
T = 1 if Trans-Texas Oil is included
T = 0 if not included

14. Restrictions Invest up to $3 million
Include at least 2 Texas companies
Include no more than 1 foreign company
Include exactly 1 California company
If British Petro is included, then
Trans-Texas Oil must also be included

15. Objective Function (in $1000’s return)
Max 50T + 80B + 90D + 120H + 110L S + 75C
Subject to the constraints:
Invest up to $3 Million
480T + 540B + 680D H
+ 700L + 510S + 900C < 3000

16. Include At Least 2 Texas Companies
T + H + L > 2
Include No More Than 1 Foreign Company
B + D < 1
Include Exactly 1 California Company
S + C = 1

17. If British Petro is included (B=1), then
Trans-Texas Oil must also be included (T=1)
T=0
T=1
B=0 ok
B=1
not ok
Combinations
of B and T
allows the 3 acceptable combinations and prevents the unacceptable one
B < T

18. IP Model for Portfolio Selection
Max $50T + $80B + $90D + $120H + $110L + $40S + $75C
Subject to the constraints:
480T + 540B + 680D H
+ 700L + 510S + 900C < (investment limit)
T + H + L > (Texas companies)
B + D < (foreign companies)
S + C = (California companies)
B < T (Trans-Texas and British petro)
All variables = 0 or 1
Go to file 6-3.xls

19. Go to file 6-3.xls Simkin and Steinberg (Binary) T B D H L S C
Trans-Texas Oil
British Petro
Dutch Shell
Houston Oil
Lone Star Petro
San Diego Oil
Calif Petro
Invest? (1 = Yes, 0 = No) 1
Exp annual return ('000)
$50
$80
$90
$120
$110
$40
$75
$360
Constraints:
Investment limit
480
540
680
1000
700
510
900
2890
<=
3000
Foreign companies
British & Trans-Texas
Texas companies 2
>=
California companies =
LHS
Sign
RHS
Go to file 6-3.xls

20. Goal Programming Models
Permit multiple objectives
Try to “satisfy” goals rather than optimize
Objective is to minimize underachievement of goals

21. Goal Programming Example: Wilson Doors Co.
Makes 3 types of doors from 3 limited resources
Decision: How many of each of 3 types of doors to make?
Objective: Minimize total underachievement of goals

22. Data

23. LP Model Maximize $70E+ $110I + $110C St.
4E + 3I + 7 C < 9,000 (steel usage)
2E + 4I + 3C < 6,000 (forming time)
2E + 3I + 4C < 5,200 (assembly time)
E, I, C > 0
Go to file 6-6.xls

24. E: 1400, I=800, and C=0 with a total sales of $186,000
LP Solution (File: 6-6.xls)
Wilson Doors (LP) E I C
Exterior doors
Interior doors
Comm doors
Number of units
800.00
0.00
Revenue
$70
$110
$186,000.00
Constraints:
Steel usage 4 3 7
<=
9000
Forming time 2
6000
Assembly time
5200
LHS
Sign
RHS
E: 1400, I=800, and C=0 with a total sales of $186,000

25. Goals Total sales at least $180,000
Exterior door sales at least $70,000
Interior door sales at lest $60,000
Commercial door sales at least $35,000

26. Regular Decision Variables
E = number of exterior doors made
I = number of interior doors made
C = number of commercial doors made
Deviation Variables
di+ = amount by which goal i is overachieved
di- = amount by which goal i is underachieved

27. Goal Constraints Goal 1: Total sales at least $180,000
70E + 110I + 110C + dT- - dT+ = 180,000
Goal 2: Exterior door sales at least $70,000
70E + dE- - dE+ = 70,000
Note: Each highlighted deviation variable measures goal underachievement

28. Goal 3: Interior door sales at least $60,000
110 I + dI- - dI+ = 60,000
Goal 4: Commercial door sales at least $35,000
110C + dC- - dC+ = 35,000

29. Goals Total sales at least $180,000
Exterior door sales at least $70,000
Interior door sales at lest $60,000
Commercial door sales at least $35,000
Goal 1: 70E + 110I + 110C + dT- - dT+ = 180,000
Goal 2: 70E + dE- - dE+ = 70,000
Goal 3: 110 I + dI- - dI+ = 60,000
Goal 4: 110C + dC- - dC+ = 35,000

30. Objective Function Minimize total goal underachievement
Min dT- + dE- + dI- + dC-
Subject to the constraints:
The 4 goal constraints
The “regular” constraints (3 limited resources)
nonnegativity

31. Objective Function Minimize dT- + dE- + dI- + dC-
Subject to the constraints:
70E + 110I + 110C + dT- - dT+ = 180,000 (total sales goal)
70E + dE- - dE = 70,000 (exterior door sales goal)
110 I + dI- - dI = 60,000 (interior door sales goal)
110C + dC- - dC = 35,000 (comm. door sales goal)
4E + 3I + 7 C < 9,000 (steel usage)
2E + 4I + 3C < 6,000 (forming time)
2E + 3I + 4C < 5,200 (assembly time)
E, I, C, dT-, dT+, dE- , dE+, dI- , dI+, dC- , dC+ > 0
Go to file 6-6.xls

32. Go to file 6-6.xls, sheet:6-6A
Weighted Goals
When goals have different priorities, weights can be used
Suppose that Goal 1 is 5 times more important than each of the others
Objective Function
Min 5dT- + dE- + dI- + dC-
Go to file 6-6.xls, sheet:6-6A

33. Go to file 6-6.xls, sheet:6-6A GP#1
Wilson Doors (Weighted GP #1) E I C
dT-
dT+
dE-
dE+
dI-
dI+
dC-
dC+
Exterior doors
Interior doors
Comm doors
Under ach total sales
Over ach total sales
Under ach exter doors
Over ach exter doors
Under ach inter doors
Over ach inter doors
Under ach comm doors
Over ach comm doors
Solution value
800.00
200.00
0.00
Goal weights 5 1
Constraints:
Achieved
Total sales goal 70
110 -1 =
180000
Exterior doors goal
70000
Interior doors goal
60000
Comm doors goal
35000
Steel usage 4 3 7
<=
9000
Forming time 2
6000
Assembly time
5200
LHS
Sign
RHS
Go to file 6-6.xls, sheet:6-6A GP#1

34. Properties of Weighted Goals
Solution may differ depending on the weights used
Appropriate only if goals are measured in the same units
What if Goal 1 is only 2.5 times important than each of the others?
Objective Function
Min 2.5dT- + dE- + dI- + dC-
Go to file 6-6.xls, sheet:6-6B GP#2, 6-6B IP

35. Go to file 6-6.xls, sheet:6-6B GP#2
Wilson Doors (Weighted GP #2) E I C
dT-
dT+
dE-
dE+
dI-
dI+
dC-
dC+
Exterior doors
Interior doors
Comm doors
Under ach total sales
Over ach total sales
Under ach exter doors
Over ach exter doors
Under ach inter doors
Over ach inter doors
Under ach comm doors
Over ach comm doors
Solution value
642.42
318.18
0.00
0.0
Goal weights
2.5 1
Constraints:
Achieved
Total sales goal 70
110 -1 =
180000
Exterior doors goal
70000
Interior doors goal
60000
Comm doors goal
35000
Steel usage 4 3 7
<=
9000
Forming time 2
6000
Assembly time
5200
LHS
Sign
RHS
Go to file 6-6.xls, sheet:6-6B GP#2

36. Go to file 6-6.xls, sheet:6-6B IP
Wilson Doors (Weighted GP #2 - IP) E I C
dT-
dT+
dE-
dE+
dI-
dI+
dC-
dC+
Exterior doors
Interior doors
Comm doors
Under ach total sales
Over ach total sales
Under ach exter doors
Over ach exter doors
Under ach inter doors
Over ach inter doors
Under ach comm doors
Over ach comm doors
Solution value
644.00
317.00
0.00
130.00
0.0
Goal weights
2.5 1
Constraints:
Achieved
Total sales goal 70
110 -1 =
180000
Exterior doors goal
70000
Interior doors goal
60000
Comm doors goal
35000
Steel usage 4 3 7
<=
9000
Forming time 2
6000
Assembly time
5200
LHS
Sign
RHS
Go to file 6-6.xls, sheet:6-6B IP

37. Ranked Goals
Lower ranked goals are considered only if all higher ranked goals are achieved
Suppose they added a 5th goal
Goal 5: Steel usage as close to 9000 lb as possible
4E + 3I + 7C + dS- = 9000 (lbs steel)
(no dS+ is needed because we cannot exceed 9000 pounds)

38. Rank R1: Goal 1
Rank R2: Goal 5
Rank R3: Goals 2, 3, and 4
A series of LP models must be solved
Solve for the R1 goal while ignoring the other goals
Objective Function: Min dT-

39. Objective Function Objective Function: Min dT-
Subject to the constraints:
70E + 110I + 110C + dT- - dT+ = 180,000 (total sales goal)
4E + 3I + 7C + dS = 9000 (steel usage goal)
70E + dE- - dE = 70,000 (exterior door sales goal)
110 I + dI- - dI = 60,000 (interior door sales goal)
110C + dC- - dC = 35,000 (comm. door sales goal)
4E + 3I + 7 C < 9,000 (steel usage)
2E + 4I + 3C < 6,000 (forming time)
2E + 3I + 4C < 5,200 (assembly time)
E, I, C, dT-, dT+, dE- , dE+, dI- , dI+, dC- , dC+ > 0
Go to file 6-7.xls

40. Go to file 6-7A R1.xls Wilson Doors (Rank R1 Goals Only) E I C dT- dT+E I C
dT-
dT+
dS-
dE-
dE+
dI-
dI+
dC-
dC+
Exterior doors
Interior doors
Comm doors
Under ach total sales
Over ach total sales
Under ach steel usage
Under ach exter doors
Over ach exter doors
Under ach inter doors
Over ach inter doors
Under ach comm doors
Over ach comm doors
Solution value
800.00
200.00
0.00
Objective coeff 1
Constraints:
Achieved
Total sales goal 70
110 -1 =
180000
Steel usage goal 4 3 7
9000
Exterior doors goal
70000
Interior doors goal
60000
Comm doors goal
35000
Forming time 2
<=
6000
Assembly time
5200
LHS
Sign
RHS
Go to file 6-7A R1.xls

41. Objective Function: Min dS-
2) If the R1 goal can be achieved (dT- = 0), then this is added as a constraint and we attempt to satisfy the R2 goal (Goal 5)
Objective Function: Min dS-
3) If the R2 goal can be achieved (dS- = 0), then this is added as a constraint and we solve for the R3 goals (Goals 2, 3, and 4)
Objective Function: Min dE- + dI- + dC-
Go to file 6-7.xls

42. Go to file 6-7B R2.xls Wilson Doors (Rank R2 Goals Only) E I C dT- dT+E I C
dT-
dT+
dS-
dE-
dE+
dI-
dI+
dC-
dC+
Exterior doors
Interior doors
Comm doors
Under ach total sales
Over ach total sales
Under ach steel usage
Under ach exter doors
Over ach exter doors
Under ach inter doors
Over ach inter doors
Under ach comm doors
Over ach comm doors
Solution value
466.67
0.00
Objective coeff 1
Constraints:
Achieved
Total sales goal 70
110 -1 =
180000
Steel usage goal 4 3 7
9000
Exterior doors goal
70000
Interior doors goal
60000
Comm doors goal
35000
Forming time 2
<=
6000
Assembly time
5200
LHS
Sign
RHS
Go to file 6-7B R2.xls

43. Go to file 6-7B R2 IP.xls Wilson Doors (Rank R2 Goals Only - IP) E I CE I C
dT-
dT+
dS-
dE-
dE+
dI-
dI+
dC-
dC+
Exterior doors
Interior doors
Comm doors
Under ach total sales
Over ach total sales
Under ach steel usage
Under ach exter doors
Over ach exter doors
Under ach inter doors
Over ach inter doors
Under ach comm doors
Over ach comm doors
Solution value
422.00
134.00
0.00
90.00
Objective coeff 1
Constraints:
Achieved
Total sales goal 70
110 -1 =
180000
Steel usage goal 4 3 7
9000
Exterior doors goal
70000
Interior doors goal
60000
Comm doors goal
35000
Forming time 2
<=
6000
Assembly time
5200
LHS
Sign
RHS
Go to file 6-7B R2 IP.xls

44. Go to file 6-7C R3.xls Wilson Doors (Rank R3 Goals Only) E I C dT- dT+E I C
dT-
dT+
dS-
dE-
dE+
dI-
dI+
dC-
dC+
Exterior doors
Interior doors
Comm doors
Under ach total sales
Over ach total sales
Under ach steel usage
Under ach exter doors
Over ach exter doors
Under ach inter doors
Over ach inter doors
Under ach comm doors
Over ach comm doors
Solution value
421.05
136.84
0.00
Objective coeff 1
Constraints:
Achieved
Total sales goal 70
110 -1 =
180000
Steel usage goal 4 3 7
9000
Exterior doors goal
70000
Interior doors goal
60000
Comm doors goal
35000
Forming time 2
<=
6000
Assembly time
5200
LHS
Sign
RHS
Go to file 6-7C R3.xls

45. Go to file 6-7C R3 IP.xls Wilson Doors (Rank R3 Goals Only - IP) E I CE I C
dT-
dT+
dS-
dE-
dE+
dI-
dI+
dC-
dC+
Exterior doors
Interior doors
Comm doors
Under ach total sales
Over ach total sales
Under ach steel usage
Under ach exter doors
Over ach exter doors
Under ach inter doors
Over ach inter doors
Under ach comm doors
Over ach comm doors
Solution value
422.00
134.00
0.00
90.00
Objective coeff 1
Constraints:
Achieved
Total sales goal 70
110 -1 =
180000
Steel usage goal 4 3 7
9000
Exterior doors goal
70000
Interior doors goal
60000
Comm doors goal
35000
Forming time 2
<=
6000
Assembly time
5200
LHS
Sign
RHS
Go to file 6-7C R3 IP.xls