6-1

CHAPTER 6Integer, Goal, and

Nonlinear Programming Models

6-2

LEARNING OBJECTIVES

1. Formulate integer programming (IP) models.

2. Set up and solve IP models using Excel’s Solver.

3. Understand the difference between general integer and binary integer variables.

6-3

LEARNING OBJECTIVES

4. Understand the use of binary integer variables in formulating problems involving fixed costs.

5. Formulate goal programming (GP) problems and solve them using Excel’s Solver.

6. Formulate nonlinear programming (NLP) problems and solve them using Excel’s Solver.

6-4

Introduction

• Relax the basic assumptions• Fractional value

• One objective

• Linear equations

• Integer Programming

• Goal Programming

• Nonlinear Programming

6-5

Models

• Integer Models• General integer variables

• Binary variables

• Pure IP problems

• Mixed IP problems

• Goal Models• More than one objective

6-6

Models

• Nonlinear Models• Objective function

Maximize profit = 25X – 0.4X 2 + 30Y – 0.5Y 2

• Constraints

6-7

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

6-8

General Integer Variables

• Harrison Electric Company• Ornate lamps

• Old-fashioned ceiling fans

Lamps Fans Hours

Wiring 2 3 12

Assembly 6 5 30

Profit $600 $700

6-9

Harrison Electric

• Decision Variables

L = number of lamps

F = number of ceiling

Integer values

6-10

Harrison Electric

Objective function

Maximize profit = $600L + $700F

subject to

2L + 3F ≤ 12 (wiring hours)

6L + 5F ≤ 30 (assembly hours)

L, F ≥ 0

6-11

Graphical Solution

+ = Integer Valued Point

6L + 5F ≤ 30

Rounded-off IP Solution(L = 4, F = 2, Infeasible)

+

2L + 3F ≤ 12+

+

+

++

+

+

+ Optimal IP Solution(L = 3.75, F = 1.50, Profit = $3,300)

F

L

6 –

5 –

4 –

3 –

2 –

1 –

–| | | | | |

0 1 2 3 4 5 6

Nearest Feasible Rounded-off IP Solution(L = 4, F = 1, Profit = $3,100)

Figure 6.1

6-12

Integer SolutionsLAMPS (L) CEILING FANS (F) PROFIT ($600L + $700F)

0 0 $ 01 0 $ 6002 0 $1,2003 0 $1,8004 0 $2,4005 0 $3,0000 1 $ 7001 1 $1,3002 1 $1,9003 1 $2,5004 1 $3,100 Nearest feasible rounded-off solution

0 2 $1,4001 2 $2,0002 2 $2,6003 2 $3,200 Optimal IP solution

0 3 $2,1001 3 $2,7000 4 $2,800 Table 6.1

6-13

Solving the Problem

Screenshot 6-1

6-14

Solving the Problem

Screenshot 6-1

6-15

Solver Options

Screenshot 6-2A

6-16

Solver Options

Screenshot 6-2B

6-17

Binary Variables

• Only two possible values (0, 1)

• Selection problems

• Set covering problems

6-18

Simkin and Steinberg

• Oil stock portfolios

EXPECTED COST FORCOMPANY NAME ANNUAL RETURN BLOCK OF SHARES (LOCATION) (IN THOUSANDS) (IN THOUSANDS)

Trans-Texas Oil (Texas) $ 50 $ 480British Petro (Foreign) $ 80 $ 540Dutch Shell (Foreign) $ 90 $ 680Houston Drilling (Texas) $120 $1,000Lone Star Petro (Texas) $110 $ 700San Dieago Oil (California) $ 40 $ 510California Petro (California) $ 75 $ 900

Table 6.2

6-19

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

SimilarlyB (British Petro), D (Dutch Shell), H (Houston Oil), L (Lone Star Petro), S (San Diego Oil), and C (California Petro)

6-20

Simkin and SteinbergObjective 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

6-21

Binary Requirements

Screenshot 6-3

6-22

Binary Requirements

Screenshot 6-3

6-23

Sussex County

• Build health care clinics

Table 6.3

TO

FROM A B C D E F G

A 0 15 20 35 35 45 40

B 15 0 35 20 35 40 40

C 20 3 50 15 50 45 30

D 35 20 15 0 35 20 20

E 35 35 50 35 0 15 40

F 45 40 45 20 15 0 35

G 40 40 30 20 40 35 0

6-24

Sussex County

• Build health care clinics

Table 6.4

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

6-25

Sussex County

• Decision Variables

A = 1 if a clinic is located in community A= 0 if a clinic is not located in community A

SimilarlyB (community B), C (community C), D (community D), E (community E), F (community F), and G (community G)

6-26

Sussex CountyObjective function

Minimize totalnumber of clinics = A + B + C + D + E + F + G

subject toA + 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

6-27

Solving the Problem

Screenshot 6-4