LPX-1

Linear Programming Models in Services

Linear Programming Models in Services

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

LPX-3

Describe the features of constrained optimization models.

Formulate LP models for computer solution.

Solve two-variable models using graphics.

Explain the nature of sensitivity analysis.

Solve LP models with Excel Add-in Solver and interpret the results.

Formulate a goal programming model.

Learning ObjectivesLearning Objectives

LPX-4

Let x = number of receivers to stock

y = number of speakers to stock

Maximize 50x + 20y gross profit

Subject to 2x + 4y 400 floor space

100x + 50y 8000 budget

x 60 sales limit

x, y 0

Stereo WarehouseStereo Warehouse

LPX-5

Let E = units of egg custard base in the shake

C = units of ice cream in the shake

S = units of butterscotch syrup in the shake

Minimize

Subject to

cholesterol

fat

protein

calories

015 0 25 010. . .E C S 50 150 90 175E C S

100 50 150C S 70 10E C 20030 80 200 100E C S

E C S, , 0

Diet ProblemLakeview Hospital

Diet ProblemLakeview Hospital

LPX-6

Let xi = number of officers reporting at period i for i =1, 2, 3, 4, 5, 6

Minimize

x1 + x6 6 period 1

x1 + x2 4 period 2

x2 + x3 14 period 3

x3 + x4 8 period 4

x4 + x5 12 period 5

x5 + x6 16 period 6

x x x x x x1 2 3 4 5 6

Shift-Scheduling ProblemGotham City Police PatrolShift-Scheduling ProblemGotham City Police Patrol

LPX-7

Let Tt = number of trainees hired at the beginning of period t

for t = 1,2,3,4,5,6

At = number of tellers available at the beginning of period t

for t = 1,2,3,4,5,6

Minimize

subject to

A1 = 12

For t = 2,3,4,5,6

At , Tt 0 and integer for t = 1,2,3,4,5,6

( )600 3001

6

A Ttt

t

160 80 1500

160 80 1800

160 80 1600

160 80 2000

160 80 1800

160 80 2200

1 1

2 2

3 3

4 4

5 5

6 6

A T

A T

A T

A T

A T

A T

0 9 01 1. A T At t t

Workforce-Planning ProblemLast National Drive-in Bank

Workforce-Planning ProblemLast National Drive-in Bank

LPX-8

Let xij = number of cars sent from city i to city j for i = 1,2,3 and j = 1,2,3,4

Minimize 439x11 + 396 x12 + . . . +479x33 + 0x34

subject to x11 + x12 + x13 + x14 = 26

x21 + x22 + x23 + x24 = 43

x31 + x32 + x33 + x34 = 31

x11 + x21 + x31 = 32

x12 + x22 + x32 = 28

x13 + x23 + x33 = 26

x14 + x24 + x34 = 14

xij 0 for all i , j

Transportation ProblemLease-a-Lemon Car Rental

Transportation ProblemLease-a-Lemon Car Rental

LPX-9

0

50

100

150

200

0 50 100 150 200

Z=2000

Z=3000

Z=3600

Z=3800

A B

C

D

EOptimal solution( x = 60, y = 40)

Graphical SolutionStereo WarehouseGraphical SolutionStereo Warehouse

LPX-10

Let s1 = square feet of floor space not used

s2 = dollars of budget not allocated

s3 = number of receivers that could have been sold

Maximize Z = 50x + 20y

subject to 2x + 4y + s1 = 400 (constraint 1)

100x + 50y + s2 = 8000 (constraint 2)

x + s3 = 60 ( constraint 3)

x, y, s1, s2, s3 0

Model in Standard FormModel in Standard Form

LPX-11

Extreme Nonbasic Basic Variable Objective-function

point variables variables value value Z

A x, y s1 400 0

s2 8000

s3 60

B s3, y s1 280 3000

s2 2000

x 60

C s3, s2 s1 120 3800

y 40

x 60

D s1, s2 s3 20 3600

y 80

x 40

E s1, x s3 60 2000

y 100

s2 3000

Stereo WarehouseExtreme-Point Solutions

Stereo WarehouseExtreme-Point Solutions

LPX-12

0

50

100

150

200

0 50 100 150 200

z = 50x + 20y

x 60 (constraint 3 )

2 4 400x y (constraint 1)

100 50 8000x y (constraint 2)

A B

C

D

Sensitivity AnalysisObjective-Function Coefficients

Sensitivity AnalysisObjective-Function Coefficients

LPX-13

0

50

100

150

200

0 50 100 150 200

x 60 (constraint 3 )

100 50 8000x y (constraint 2)

A B I

C

DH

Sensitivity AnalysisRight-Hand-Side Ranging

Sensitivity AnalysisRight-Hand-Side Ranging

LPX-14

Let x = number of receivers to stock

y = number of speakers to stock

d

d

d

d

d

d

d

d

Pk

1

1

2

2

3

3

4

4

= amount by which profit falls short of $99,999= amount by which profit exceeds $99,999= amount by which floor space used falls short of 400 square feet= amount by which floor space used exceeds 400 square feet= amount by which budget falls short of $8000= amount by which budget exceeds $8000= amount by which sales of receivers fall short of 60= amount by which sales of receivers exceed 60= priority level with rank k

Minimizesubject to

Z Pd P d d P d d 1 4 2 1 3 3 1 22 2 2( ) ( )

50 20 99 999

2 4 400

100 50 8000

1 1

2 2

3 3

x y d d

x y d d

x y d d

,

x d d 4 4 60

x y d d d d d d d d, , , , , , , , ,1 1 2 2 3 3 4 4 0

profit goal

floor-space goal

budget goal

sales-limit goal

Goal ProgrammingStereo Warehouse Example

Goal ProgrammingStereo Warehouse Example

LPX-15

Topics for DiscussionTopics for Discussion

How can the validity of LP models be evaluated?

Interpret the meaning of the opportunity cost for a nonbasic decision variable that did not appear in the LP solution.

Explain graphically what has happened when a degenerate solution occurs in an LP problem.

Is LP a special case of goal programming? Explain.

What are some limitations to the use of LP?