To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Linear Programming Linear Programming Models: Graphical MethodsModels: Graphical Methods

What is Linear Programming?What is Linear Programming?

• Linear Programming (LP) is a technique that helps in resource allocation decisions.• Programming refers to modelling and solving a

problem mathematically

• Linear programming can solve two variable problems (graphical method); or more than two variable problems (simplex method)

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Requirements of a Linear Requirements of a Linear Programming ProblemProgramming Problem

• All problems seek to maximize or minimize some quantity (the objective function).

• The presence of restrictions or constraints, limits the degree to which we can pursue our objective.

• There must be alternative courses of action to choose from.

• The objective and constraints in linear programming problems must be expressed in terms of linear equations or inequalities.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Examples of Successful LP Examples of Successful LP ApplicationsApplications

1. Development of a production schedule that will satisfy

future demands for a firm’s production and at the same

time minimize total production and inventory costs

2. Selection of the product mix in a factory to make best

use of machine-hours and labor-hours available while

maximizing the firm’s products

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Examples of Successful LP Examples of Successful LP ApplicationsApplications

3. Determination of grades of petroleum products to yield the

maximum profit

4. Selection of different blends of raw materials to feed mills

to produce finished feed combinations at minimum cost

5. Determination of a distribution system that will minimize

total shipping cost from several warehouses to various

market locations

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Basic Assumptions of Linear Basic Assumptions of Linear ProgrammingProgramming

• Certainty – coefficients in the objective function and constraints are known with certainty and do not change during the period being studied.

• Proportionality-in the objective function and constraints. If one product uses 5 hours of a machine resource, then making 10 of that product uses 50 hours of machine time.

• Additivity- the total of all activitiesequals the sum of each individual activity

• Divisibility – solutions need not be integers but may also

be fractional values

• Nonnegativity – all answers and variable values are

either zero or positive.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Equations and InequalitiesEquations and Inequalities

• Solve for X1 and X2 and graph solution

1. X1 ≥ 80

2. X2 ≤ 100

3. 3X1+ 2X2 ≤ 240

4. 2X1+ 1X2 ≥ 140

5. 1X1 + 2X2 = 16

6. 2X1 - X2 ≥ 2

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Data - Flair Furniture Company Data - Table Table 7.17.1

Hours Required to Produce One Unit

Department TTables

CChairs

AvailableHours This

Week

CarpentryPainting &Varnishing

42

31

240100

Profit Amount $7 $5

Constraints: 4T + 3C 240 (Carpentry)

2T + 1C 100 (Paint & Varnishing)

Objective: Max: 7T + 5C

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Corner Point Solution MethodCorner Point Solution Method

• Find the objective function and constraints• Define the decision variables• Use the decision variables to write mathematical

expressions for the objective function and constraints• Solve for unknown decision variables in each constraint

and graph the solution• Find the value of the variables at each corner point to

form the feasible region or feasible line segment• Test corner points by substituting value of decision

variables and corresponding profit or cost.• Do simultaneous equation for corner points with

intersecting constraint lines• Select the corner point with the highest (profit) or lowest

(cost) objective coefficient values as optimal solution

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Flair Furniture Company ConstraintsConstraints

Number of Tables

120

100

80

60

40

20

0

Num

ber

of C

hair

s

20 40 60 80 100

Painting/Varnishing

Carpentry

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Feasible Flair Furniture Company Feasible RegionRegion

120

100

80

60

40

20

0

Num

ber

of C

hair

s

20 40 60 80 100Number of Tables

Painting/Varnishing

CarpentryFeasibleRegion

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Flair Furniture Company Optimal SolutionOptimal Solution

Num

ber

of C

hair

s

120

100

80

60

40

20

0

20 40 60 80 100Number of Tables

Painting/Varnishing

Carpentry

Solution(T = 30, C = 40)

Corner PointsCorner Points

1

2

3

4

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Holiday Meal Turkey Ranch p. Holiday Meal Turkey Ranch p. 290 (11290 (11thth ed) or p. 296 (10 ed) or p. 296 (10thth ed) ed)

(C)

(B)

toSubject

:Minimize

½ X

XX

A)(XX:

XX

INGREDIENTS BRAND 1 FEED BRAND 2 FEED MINIMIMUM REQUIREMENT PER TURKEY (OZ)

A 5 10 90

B 4 3 48

C 0.5 0 1.5

COST PER POUND 2 cents 3 cents

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Holiday Meal Turkey Problem Holiday Meal Turkey Problem

Corner PointsCorner Points

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

LP-GraphicalLP-Graphical

• SEATWORK

Solve 7-18 and 7-19 p319 10th and 11th ed p313

• HOMEWORK

Solve 7-27 p. 320 10th ed.; p314 for 11th edition

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Special Cases in LPSpecial Cases in LP

• Infeasibility

• Unbounded Solutions

• Redundancy

• Degeneracy

• More Than One Optimal Solution

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-17 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

A Problem with No Feasible A Problem with No Feasible SolutionSolution

X2

X1

8

6

4

2

02 4 6 8

Region Satisfying3rd Constraint

Region Satisfying First 2 Constraints

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-18 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

A Solution Region That is A Solution Region That is Unbounded to the RightUnbounded to the Right

X2

X1

15

10

5

05 10 15

Feasible Region

X1 > 5 X2 < 10

X1 + 2X2 > 10

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-19 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

A Problem with a Redundant A Problem with a Redundant ConstraintConstraint

X2

X1

30

25

20

15

10

5

05 10 15 20 25 30

Feasible Region

2X1 + X2 < 30

X1 < 25

X1 + X2 < 20

RedundantConstraint

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-20 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

An Example of Alternate Optimal An Example of Alternate Optimal SolutionsSolutions

8

7

6

5

4

3

2

1

0

1 2 3 4 5 6 7 8

Optimal Solution Consists of All Combinations of X1 and X2 Along the AB Segment

Isoprofit Line for $12Overlays Line Segment

Isoprofit Line for $8A

BAB

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-21 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Sensitivity AnalysisSensitivity Analysis

• Changes in the Objective Function

Coefficient

• Changes in Resources (RHS)

• Changes in Technological Coefficients

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-22 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Changes in the Technological Coefficients Changes in the Technological Coefficients for High Note Sound Co.for High Note Sound Co.

Ste

reo

Rec

eive

rs

X1

60

40

20

0

CD Players

20 40

X2

(a) Original Problem

3X1 + 1X2 < 60

Optimal Solution

a2X1 + 4X2 < 80

b

c

X2

(b) Change in CircledCoefficient

Still Optimal

a2X1 + 4X2 < 80

d

e

2X1 + 1X2 < 60

20 40 X130

CD Players

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-23 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Changes in the Technological Coefficients Changes in the Technological Coefficients for High Note Sound Co.for High Note Sound Co.

X1

Ste

reo

Rec

eive

rs

60

40

20

0

CD Players

20 40

X2

(a) Original Problem

3X1 + 1X2 < 60

Optimal Solution

a2X1 + 4X2 < 80

b

c

20 40

X2

X1

(c) Change in CircledCoefficient

3X1 + 1X2 < 60

Optimal Solution

f2X1 + 5X2 < 80

g

c

CD Players