b – 1 operations management module b – linear programming © 2006 prentice hall, inc. powerpoint...

B – 1

Operations ManagementOperations ManagementModule B – Linear ProgrammingModule B – Linear Programming

© 2006 Prentice Hall, Inc.

PowerPoint presentation to accompanyPowerPoint presentation to accompany Heizer/Render Heizer/Render Principles of Operations Management, 6ePrinciples of Operations Management, 6eOperations Management, 8e Operations Management, 8e

B – 2

Lecture Outline

• Remind me what LP is

• What types of problems can we solve with LP?

• Formulating LP problems

• Example

• Sensitivity of the Answer

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Linear Programming

A model consisting of linear relationshipsrepresenting a firm’s objective and resource

constraints

LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called

constraints

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Common Elements to LP

• Decision variables– Should completely describe the decisions to be made

by the decision maker (DM)

• Objective Function (OF)– DM wants to maximize or minimize some function of

the decision variables

• Constraints– Restrictions on resources such as time, money,

labor, etc.

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LP Assumptions

• OF and constraints must be linear

• Proportionality– Contribution of each decision variable is

proportional to the value of the decision variable

• Additivity– Contribution of any variable is independent

of values of other decision variables

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LP Assumptions, cont’d.

• Divisibility– Allow both integer and non-integer (real)

numbers

• Certainty– All coefficients are known with certainty

– We are dealing with a deterministic world

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Types of Problems

• Module C– SCM: transportation models

• Chapter 13

• Aggregate Planning

• Module B– Product Mix

– Blending

– Scheduling (Production and Labor)

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Formulating LP Problems

The product-mix problem at Shader Electronics

Two products

1. Shader Walkman, a portable CD/DVD player

2. Shader Watch-TV, a wristwatch-size Internet-connected color TV

Determine the mix of products that will produce the maximum profit

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LP Model Formulation

• Data– Input to the model – given in the problem

• Decision variables– Mathematical symbols representing levels

of activity of an operation

– The quantities to be determined

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LP Model Formulation, cont’d.

• Objective function (OF)– The quantity to be optimized

– A linear relationship reflecting the objective of an operation

– Most frequent objective of business firms is to maximize profit

– Most frequent objective of individual operational units (such as a production or packaging department) is to minimize cost

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LP Model Formulation, cont’d.

• Constraint– A linear relationship representing a

restriction on decision making

– Binding relationships

– Attach a word description to each set of constraints

– Include bounds on variables

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Walkman Watch-TVs Available HoursDepartment (X1) (X2) This Week

Hours Required to Produce 1 Unit

Electronic 44 33 240240

Assembly 22 11 100100

Profit per unit $7$7 $5$5

Decision Variables:XX11 = = number of Walkmans to be produced

XX22 = = number of Watch-TVs to be produced

Table B.1Table B.1

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Objective Function:

Maximize Profit = = $7$7XX11 + + $5$5XX22

There are three types of constraints Upper limits where the amount used is ≤ the

amount of a resource Lower limits where the amount used is ≥ the

amount of the resource Equalities where the amount used is = the

amount of the resource

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Second Constraint:

22XX11 + + 11XX22 ≤ 100 ≤ 100 (hours of assembly time)

Assemblytime available

Assemblytime used is ≤ ≤

First Constraint:

44XX11 + + 33XX22 ≤ 240 ≤ 240 (hours of electronic time)

Electronictime available

Electronictime used is ≤ ≤

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XX22 >> 0 0 (non-negativity)

NoneWatch-TVsProduced is >>

Third Constraint Set:

XX11 >> 0 0 (non-negativity)

NoneWalkmansProduced is >>

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Graphical Solution

Can be used when there are two decision variables

1. Plot the constraint equations at their limits by converting each equation to an equality

2. Identify the feasible solution space

3. Create an iso-profit line based on the objective function

4. Move this line outwards until the optimal point is identified

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Graphical Solution

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Number of WalkmansNumber of Walkmans

XX11

XX22

Assembly (constraint B)Assembly (constraint B)

Electronics (constraint A)Electronics (constraint A)Feasible region

Figure B.3Figure B.3

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Graphical Solution

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XX11

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Assembly (constraint B)Assembly (constraint B)

Electronics (constraint A)Electronics (constraint A)Feasible region


Iso-Profit Line Solution Method

Choose a possible value for the objective function

$210 = 7X1 + 5X2

Solve for the axis intercepts of the function and plot the line

X2 = 42 X1 = 30

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Graphical Solution

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XX11

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(0, 42)

(30, 0)(30, 0)

$210 = $7$210 = $7XX11 + $5 + $5XX22

B – 20

Graphical Solution

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XX11

XX22


$210 = $7$210 = $7XX11 + $5 + $5XX22

$350 = $7$350 = $7XX11 + $5 + $5XX22

$420 = $7$420 = $7XX11 + $5 + $5XX22

$280 = $7$280 = $7XX11 + $5 + $5XX22

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Graphical Solution

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XX11

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$410 = $7$410 = $7XX11 + $5 + $5XX22

Maximum profit lineMaximum profit line

Optimal solution pointOptimal solution point((XX11 = 30, = 30, XX22 = 40) = 40)

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Corner-Point Method

Figure B.7Figure B.7 1

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B – 23

Corner-Point Method The optimal value will always be at a

corner point

Find the objective function value at each corner point and choose the one with the highest profit

Point 1 : (X1 = 0, X2 = 0) Profit $7(0) + $5(0) = $0

Point 2 : (X1 = 0, X2 = 80) Profit $7(0) + $5(80) = $400

Point 4 : (X1 = 50, X2 = 0) Profit $7(50) + $5(0) = $350

Point 3 : (X1 = 30, X2 = 40) Profit $7(30) + $5(40) = $410

B – 24

Sensitivity Analysis

How sensitive the results are to parameter changes Change in the value of coefficients

Change in a right-hand-side value of a constraint

Trial-and-error approach

Analytic postoptimality method

B – 25

Sensitivity Report

Program B.1Program B.1

B – 26

Changes in Resources

The right-hand-side values of constraint equations may change as resource availability changes

The shadow price of a constraint is the change in the value of the objective function resulting from a one-unit change in the right-hand-side value of the constraint

B – 27

Changes in Resources

Shadow prices are often explained as answering the question “How much would you pay for one additional unit of a resource?”

Shadow prices are only valid over a particular range of changes in right-hand-side values

Sensitivity reports provide the upper and lower limits of this range

B – 28


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00 2020 4040 6060 8080 100100 XX11

XX22

Figure B.8 (a)Figure B.8 (a)

Changed assembly constraint from Changed assembly constraint from 22XX11 + 1 + 1XX22 = 100 = 100

to to 22XX11 + 1 + 1XX22 = 110 = 110

Electronics constraint Electronics constraint is unchangedis unchanged

Corner point 3 is still optimal, but Corner point 3 is still optimal, but values at this point are now Xvalues at this point are now X11 = 45= 45, ,

XX22 = 20= 20, with a profit , with a profit = $415= $415

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B – 29


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00 2020 4040 6060 8080 100100 XX11

XX22

Figure B.8 (b)Figure B.8 (b)

Changed assembly constraint from Changed assembly constraint from 22XX11 + 1 + 1XX22 = 100 = 100

to to 22XX11 + 1 + 1XX22 = 90 = 90

Electronics constraint Electronics constraint is unchangedis unchanged

Corner point 3 is still optimal, but Corner point 3 is still optimal, but values at this point are now Xvalues at this point are now X11 = 15= 15, ,

XX22 = 60= 60, with a profit , with a profit = $405= $405

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B – 30

Changes in the Objective Function

A change in the coefficients in the objective function may cause a different corner point to become the optimal solution

The sensitivity report shows how much objective function coefficients may change without changing the optimal solution point

b – 1 operations management module b – linear programming © 2006 prentice hall, inc. powerpoint...

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