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

[email protected]

Dr. Chen, Decision Support Systems 2

Variations of Basic Linear Programming

• Integer Programming

• Goal Programming

• Nonlinear Programming (skip)


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


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


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


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


Graphical Solution


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


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


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


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)


Stock Data


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


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


Objective Function (in $1000’s return)

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


Invest up to $3 Million

480T + 540B + 680D + 1000H

+ 700L + 510S + 900C < 3000


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


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


IP Model for Portfolio Selection

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


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


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


Goal Programming Models• Permit multiple objectives

• Try to “satisfy” goals rather than optimize

• Objective is to minimize underachievement of goals


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


Data


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


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


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


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


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


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


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


Objective Function

Minimize total goal underachievement

Min dT- + dE

- + dI- + dC

-


• The 4 goal constraints

• The “regular” constraints (3 limited resources)

• nonnegativity


Objective Function

Minimize dT- + dE

- + dI- + dC

-


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


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


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


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


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


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



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


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


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


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



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


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)


• 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-


Objective Function

Objective Function: Min dT-


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


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


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


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



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


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



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



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



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


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


Under

ach steel usag

e

Under

ach exte

r door

sOver ach

exter doorsUnder ach inter doors

Over ach inter door

s


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



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



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



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


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


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



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


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


Under

ach steel usage

Under

ach exte

r door

s

Over ach exter doors

Under ach inter doors

Over

ach inter door

s


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


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



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

Chapter 6 Integer and Goal Programming Models Jason C. H. Chen, Ph.D. Professor of MIS School of...

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Transcript of Chapter 6 Integer and Goal Programming Models Jason C. H. Chen, Ph.D. Professor of MIS School of...