1. Integer, Goal, and Nonlinear Programming Models6
Integer, Goal, and Nonlinear Programming Models

2. LEARNING OBJECTIVES Formulate integer programming (IP) models.
Set up and solve IP models using Excel’s Solver.
Understand the difference between general integer and binary integer variables.

3. LEARNING OBJECTIVES
Understand the use of binary integer variables in formulating problems involving fixed costs.
Formulate goal programming (GP) problems and solve them using Excel’s Solver.
Formulate nonlinear programming (NLP) problems and solve them using Excel’s Solver.

4. Introduction Relax the basic assumptions Integer Programming
Fractional value
One objective
Linear equations
Integer Programming
Goal Programming
Nonlinear Programming

5. Models Integer Models Goal Models General integer variables
Binary variables
Pure IP problems
Mixed IP problems
Goal Models
More than one objective

6. Models Nonlinear Models Objective function Constraints
Maximize profit = 25X – 0.4X Y – 0.5Y 2
Constraints

7. Integer Models
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

8. General Integer Variables
Harrison Electric Company
Ornate lamps
Old-fashioned ceiling fans
Lamps Fans Hours
Wiring
Assembly
Profit $600 $700

9. Harrison Electric Decision Variables L = number of lamps
F = number of ceiling
Integer values

10. Harrison Electric Objective function Maximize profit = $600L + $700F
subject to
2L + 3F ≤ 12 (wiring hours)
6L + 5F ≤ 30 (assembly hours)
L, F ≥ 0

11. Graphical Solution + + L F 6 – 5 – 4 – 3 – 2 – 1 – – | | | | | |
| | | | | |
Figure 6.1
+ = Integer Valued Point
6L + 5F ≤ 30 +
Rounded-off IP Solution
(L = 4, F = 2, Infeasible) +
Optimal IP Solution
(L = 3.75, F = 1.50, Profit = $3,300)
2L + 3F ≤ 12
Nearest Feasible Rounded-off IP Solution
(L = 4, F = 1, Profit = $3,100)

12. Integer Solutions LAMPS (L) CEILING FANS (F) PROFIT ($600L + $700F)
0 0 $
1 0 $ 600
2 0 $1,200
3 0 $1,800
4 0 $2,400
5 0 $3,000
0 1 $ 700
1 1 $1,300
2 1 $1,900
3 1 $2,500
$3,100  Nearest feasible rounded-off solution
0 2 $1,400
1 2 $2,000
2 2 $2,600
$3,200  Optimal IP solution
0 3 $2,100
1 3 $2,700
0 4 $2,800
Table 6.1

13. Solving the Problem
Screenshot 6-1

14. Solving the Problem
Screenshot 6-1

15. Solver Options
Screenshot 6-2A

16. Solver Options
Screenshot 6-2B

17. Binary Variables Only two possible values (0, 1) Selection problems
Set covering problems

18. Simkin and Steinberg Oil stock portfolios EXPECTED COST FOR
COMPANY NAME ANNUAL RETURN BLOCK OF SHARES
(LOCATION) (IN THOUSANDS) (IN THOUSANDS)
Trans-Texas Oil (Texas) $ 50 $ 480
British Petro (Foreign) $ 80 $ 540
Dutch Shell (Foreign) $ 90 $ 680
Houston Drilling (Texas) $120 $1,000
Lone Star Petro (Texas) $110 $ 700
San Dieago Oil (California) $ 40 $ 510
California Petro (California) $ 75 $ 900
Table 6.2

19. Simkin and Steinberg Decision Variables
T = 1 if Trans-Texas Oil is included in the portfolio
= 0 if Trans-Texas Oil is not included in the portfolio
Similarly
B (British Petro),
D (Dutch Shell),
H (Houston Oil),
L (Lone Star Petro),
S (San Diego Oil), and
C (California Petro)

20. Simkin and Steinberg Objective function
Maximize ROI = $50T + $80B + $90D + $120H + $110L + $40S + $75C
subject to
$480T + $540B + $680D + $1,000H +700L + $510S + $900C ≤ $3,000 (investment limit)
T + H + L ≥ 2 (Texas co‘s)
B + D ≤ 1 (foreign co‘s)
S + C = 1 (California co‘s)
B ≤ T (Trans-Texas
and British Petro)
All variables = 0 or 1

21. Binary Requirements
Screenshot 6-3

22. Binary Requirements
Screenshot 6-3

23. Sussex County Build health care clinics Table 6.3 TO FROM A B C D E FG 15 20 35 45 40 3 50 30
Table 6.3

24. Sussex County Build health care clinics
COMMUNITY COMMUNITIES WITHIN 30 MINUTES
A A, B, C
B A, B, D
C A, C, D, G
D B, C, D, F, G
E E, F
F D, E, F
G C, D, G
Table 6.4

25. Sussex County Decision Variables
A = 1 if a clinic is located in community A
= 0 if a clinic is not located in community A
Similarly
B (community B),
C (community C),
D (community D),
E (community E),
F (community F), and
G (community G)

26. Sussex County Objective function Minimize total
number of clinics = A + B + C + D + E + F + G
subject to
A + B + C ≥ 1 (community A is covered)
A + B + D ≥ 1 (community B is covered)
A + C + D + G ≥ 1 (community C is covered)
B + C + D + F + G ≥ 1 (community D is covered)
E + F ≥ 1 (community E is covered)
D + E + F ≥ 1 (community F is covered)
C + D + G ≥ 1 (community G is covered)
All variables = 0 or 1

27. Solving the Problem
Screenshot 6-4