chapter 6 integer and goal programming models jason c. h. chen, ph.d. professor of mis school of...

Chapter 6Integer and Goal

Programming Models

Jason C. H. Chen, Ph.D.Professor of MIS

School of Business AdministrationGonzaga UniversitySpokane, WA 99223

chen@jepson.gonzaga.edu

Dr. Chen, Decision Support Systems 2

Variations of Basic Linear Programming

• Integer Programming

• Goal Programming

• Nonlinear Programming (skip)

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 5

Decision VariablesL = 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

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 7

Graphical Solution

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 10

Harrison Electric (General Integer)

L F

 Lamp

s 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

Dr. Chen, Decision Support Systems 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)

Dr. Chen, Decision Support Systems 12

Stock Data

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 15

Objective Function (in $1000’s return)

Max 50T + 80B + 90D + 120H + 110L + 40S + 75C

Subject to the constraints:

Invest up to $3 Million

480T + 540B + 680D + 1000H

+ 700L + 510S + 900C < 3000

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 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 ok

B=1 not ok ok

allows the 3 acceptable combinations and prevents the unacceptable one

Combinationsof B and T

B < T

Dr. Chen, Decision Support Systems 18

IP Model for Portfolio Selection

Max $50T + $80B + $90D + $120H + $110L + $40S + $75C

Subject to the constraints:

480T + 540B + 680D + 1000H + 700L + 510S + 900C < 3000 (investment limit)T + H + L > 2 (Texas companies)B + D < 1 (foreign companies)S + C = 1 (California companies)B < T (Trans-Texas and British petro)

All variables = 0 or 1

Go to file 6-3.xls

Dr. Chen, Decision Support Systems 19

Simkin and Steinberg (Binary)

T B D H L S C

Trans-Texas Oil

British

Petro

Dutch

Shell

Houston Oil

Lone Star Petr

o

San Diego Oil

Calif

Petro

Invest? (1 = Yes, 0 = No) 0 0 1 1 1 1 0

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 1 1 1 <= 1

British & Trans-Texas 1 0 <= 0

Texas companies 1 1 1 2 >= 2

California companies           1 1 1 = 1

LHS Sign RHS

Go to file 6-3.xls

Dr. Chen, Decision Support Systems 20

Goal Programming Models• Permit multiple objectives

• Try to “satisfy” goals rather than optimize

• Objective is to minimize underachievement of goals

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 22

Data

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 24

Wilson Doors (LP)

  E I C

 Exterior doors

Interior doors

Comm doors

Number of units 1400.00 800.00 0.00

Revenue $70 $110 $110 $186,000.00

Constraints:            

Steel usage 4 3 7 8000.00 <= 9000

Forming time 2 4 3 6000.00 <= 6000

Assembly time 2 3 4 5200.00 <= 5200

LHS Sign RHS

LP Solution (File: 6-6.xls)

E: 1400, I=800, and C=0 with a total sales of $186,000

Dr. Chen, Decision Support Systems 25

Goals

1. Total sales at least $180,000

2. Exterior door sales at least $70,000

3. Interior door sales at lest $60,000

4. Commercial door sales at least $35,000

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 27

Goal ConstraintsGoal 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

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 29

Goals1. Total sales at least $180,000

2. Exterior door sales at least $70,000

3. Interior door sales at lest $60,000

4. 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

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 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

+ > 0Go to file 6-6.xls

Dr. Chen, Decision Support Systems 32

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

Dr. Chen, Decision Support Systems 33

Wilson Doors (Weighted GP #1)

  E I C dT- dT

+ dE- dE

+ dI- dI

+ dC- dC

+

 Exterior doors

Interior

doorsComm doors

Under

ach tota

l sale

s

Over

ach tota

l sale

s

Under

ach exter

doors

Over

ach exter

doors

Under

ach inte

r doors

Over ach inter

doors

Under ach

comm doors

Over

ach co

mm doors

Solution value1000.0

0800.0

0200.0

00.00

0.00

0.00

0.00

0.00

28000.00

13000.00

0.00

Goal weights       5   1   1   1   13000.00

Constraints:                             Achieved

Total sales goal 70 110 110 1 -1  180000.0

0 =18000

0180000.0

0

Exterior doors goal 70   1 -1   70000.00 = 70000 70000.00

Interior doors goal   110   1 -1   60000.00 = 60000 88000.00

Comm doors goal     110             1 -1 35000.00 = 35000 22000.00

Steel usage 4 3 7                 7800.00 <= 9000

Forming time 2 4 3   5800.00 <= 6000

Assembly time 2 3 4                 5200.00 <= 5200

LHSSign RHS

Go to file 6-6.xls, sheet:6-6A GP#1

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 35

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 sale

s

Under

ach exte

r door

s

Over

ach exte

r door

s

Under

ach inter door

s

Over ach inter doors

Under

ach comm

doors

Over

ach comm

doors

Solution value 1000.00 642.42 318.18 4333.33 0.00 0.00 0.00 0.00 10666.67 0.00 0.0

Goal weights       2.5   1   1   1   10833.33

Constraints:                             Achieved

Total sales goal 70 110 110 1 -1   180000.00 = 180000 175666.67

Exterior doors goal 70   1 -1   70000.00 = 70000 70000.00

Interior doors goal   110   1 -1   60000.00 = 60000 70666.67

Comm doors goal     110             1 -1 35000.00 = 35000 35000.00

Steel usage 4 3 7                 8154.55 <= 9000

Forming time 2 4 3   5524.24 <= 6000

Assembly time 2 3 4                 5200.00 <= 5200

LHS Sign RHS

Go to file 6-6.xls, sheet:6-6B GP#2

Dr. Chen, Decision Support Systems 36

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 sale

s

Under

ach exte

r door

s

Over

ach exte

r door

s

Under

ach inter door

s

Over ach inter doors

Under ach

comm doors

Over

ach comm

doors

Solution value 1000.00 644.00 317.00 4290.00 0.00 0.00 0.00 0.00 10840.00 130.00 0.0

Goal weights       2.5   1   1   1   10855.00

Constraints:                             Achieved

Total sales goal 70 110 110 1 -1   180000.00 = 180000 175710.00

Exterior doors goal 70   1 -1   70000.00 = 70000 70000.00

Interior doors goal   110   1 -1   60000.00 = 60000 70840.00

Comm doors goal     110             1 -1 35000.00 = 35000 34870.00

Steel usage 4 3 7                 8151.00 <= 9000

Forming time 2 4 3   5527.00 <= 6000

Assembly time 2 3 4                 5200.00 <= 5200

LHS Sign RHS

Go to file 6-6.xls, sheet:6-6B IP

Dr. Chen, Decision Support Systems 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)

Dr. Chen, Decision Support Systems 38

• Rank R1: Goal 1

• Rank R2: Goal 5

• Rank R3: Goals 2, 3, and 4

A series of LP models must be solved

1) Solve for the R1 goal while ignoring the other goals

Objective Function: Min dT-

Dr. Chen, Decision Support Systems 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

Dr. Chen, Decision Support Systems 40

Wilson Doors (Rank R1 Goals Only)

  E I C dT- dT

+ dS- dE

- dE+ dI

- dI+ dC

- dC+

 Exterior doors

Interior

doorsComm doors

Under

ach tota

l sale

s

Over

ach tota

l sale

s

Under ach steel

usage

Under

ach exter

doors

Over

ach exter

doors

Under

ach inte

r doors

Over ach inter doors

Under ach

comm doors

Over

ach co

mm doors

Solution value 1000.00800.0

0200.0

00.00

0.00 1200.00

0.00

0.00

0.00 28000.00 13000.00

0.00

Objective coeff       1                 0.00

Constraints:                               Achieved

Total sales goal 70 110 110 1 -1               180000.00 = 180000 180000.00

Steel usage goal 4 3 7     1             9000.00 = 9000 7800.00

Exterior doors goal 70   1 -1   70000.00 = 70000 70000.00

Interior doors goal   110   1 -1   60000.00 = 60000 88000.00

Comm doors goal     110               1 -1 35000.00 = 35000 22000.00

Forming time 2 4 3   5800.00 <= 6000

Assembly time 2 3 4                   5200.00 <= 5200

LHSSign RHS

Go to file 6-7A R1.xls

Dr. Chen, Decision Support Systems 41

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

Dr. Chen, Decision Support Systems 42

Wilson Doors (Rank R2 Goals Only)

  E I C dT- dT

+ dS- dE

- dE+ dI

- dI+ dC

- dC+

 Exterior doors

Interior doors

Comm

doors

Under

ach total sale

s

Over ach total sales

Under

ach stee

l usage

Under

ach exte

r door

s

Over ach exter doors

Under ach inter

doors

Over

ach inter door

s

Under ach comm doors

Over

ach comm

doors

Solution value 1900.00 466.67 0.00 0.00 4333.33 0.00 0.00 63000.00 8666.67 0.00 35000.00 0.00

Objective coeff           1             0.00

Constraints:                               Achieved

Total sales goal 70 110 110 1 -1               180000.00 = 180000 184333.33

Steel usage goal 4 3 7     1             9000.00 = 9000 9000.00

Exterior doors goal 70   1 -1   70000.00 = 70000 133000.00

Interior doors goal   110   1 -1   60000.00 = 60000 51333.33

Comm doors goal     110               1 -1 35000.00 = 35000 0.00

Forming time 2 4 3   5666.67 <= 6000

Assembly time 2 3 4                   5200.00 <= 5200

LHS Sign RHS

Go to file 6-7B R2.xls

Dr. Chen, Decision Support Systems 43

Wilson Doors (Rank R2 Goals Only - IP)

  E I C dT- dT

+ dS- dE

- dE+ dI

- dI+ dC

- dC+

 Exterior doors

Interior doors

Comm doors

Under

ach total sale

s

Over ach total sales

Under

ach steel usag

e

Under

ach exte

r door

sOver ach

exter doorsUnder ach inter doors

Over ach inter door

s

Under ach comm doors

Over ach comm

doors

Solution value 1699.00 422.00 134.00 0.00 90.00 0.00 0.00 48930.00 13580.00 0.00 20260.00 0.00

Objective coeff           1             0.00

Constraints:                               Achieved

Total sales goal 70 110 110 1 -1               180000.00 = 180000 180090.00

Steel usage goal 4 3 7     1             9000.00 = 9000 9000.00

Exterior doors goal 70   1 -1   70000.00 = 70000 118930.00

Interior doors goal   110   1 -1   60000.00 = 60000 46420.00

Comm doors goal     110               1 -1 35000.00 = 35000 14740.00

Forming time 2 4 3   5488.00 <= 6000

Assembly time 2 3 4                   5200.00 <= 5200

LHS Sign RHS

Go to file 6-7B R2 IP.xls

Dr. Chen, Decision Support Systems 44

Wilson Doors (Rank R3 Goals Only)

  E I C dT- dT

+ dS- dE

- dE+ dI

- dI+ dC

- dC+

 Exterior doors

Interior doors

Comm doors

Under

ach total sale

s

Over ach total sale

s

Under

ach steel usag

e

Under

ach exter door

sOver ach

exter doorsUnder ach inter doors

Over ach inter door

s

Under ach comm doors

Over ach comm

doors

Solution value 1694.74 421.05 136.84 0.00 0.00 0.00 0.00 48631.58 13684.21 0.00 19947.37 0.00

Objective coeff             1   1   1   33631.58

Constraints:                               Achieved

Total sales goal 70 110 110 1 -1               180000.00 = 180000 180000.00

Steel usage goal 4 3 7     1             9000.00 = 9000 9000.00

Exterior doors goal 70   1 -1   70000.00 = 70000 118631.58

Interior doors goal   110   1 -1   60000.00 = 60000 46315.79

Comm doors goal     110               1 -1 35000.00 = 35000 15052.63

Forming time 2 4 3   5484.21 <= 6000

Assembly time 2 3 4                   5200.00 <= 5200

LHS Sign RHS

Go to file 6-7C R3.xls

Dr. Chen, Decision Support Systems 45

Go to file 6-7C R3 IP.xls

Wilson Doors (Rank R3 Goals Only - IP)

  E I C dT- dT

+ dS- dE

- dE+ dI

- dI+ dC

- dC+

 Exterior doors

Interior doors

Comm doors

Under

ach total sale

s

Over ach total sales

Under

ach steel usage

Under

ach exte

r door

s

Over ach exter doors

Under ach inter doors

Over

ach inter door

s

Under ach comm doors

Over

ach comm

doors

Solution value 1699.00 422.00 134.00 0.00 90.00 0.00 0.00 48930.00 13580.00 0.00 20260.00 0.00

Objective coeff             1   1   1   33840.00

Constraints:                               Achieved

Total sales goal 70 110 110 1 -1               180000.00 = 180000 180090.00

Steel usage goal 4 3 7     1             9000.00 = 9000 9000.00

Exterior doors goal 70   1 -1   70000.00 = 70000 118930.00

Interior doors goal   110   1 -1   60000.00 = 60000 46420.00

Comm doors goal     110               1 -1 35000.00 = 35000 14740.00

Forming time 2 4 3   5488.00 <= 6000

Assembly time 2 3 4                   5200.00 <= 5200

LHS Sign RHS