Operations management 12th stevenson ch19 linear programming

Chapter 19
Linear Programming

McGraw-Hill/Irwin

Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.


Chapter 19: Learning Objectives
You should be able to:

1.

Describe the type of problem that would lend itself to solution using linear programming

2.

Formulate a linear programming model from a description of a problem

3.

Solve simple linear programming problems using the graphical method

4.

Interpret computer solutions of linear programming problems

5.

Do sensitivity analysis on the solution of a linear programming problem

Instructor Slides

19-2


Linear Programming (LP)

LP
 A powerful quantitative tool used by operations and other manages to obtain
optimal solutions to problems that involve restrictions or limitations

Applications include:
 Establishing locations for emergency equipment and personnel to minimize response time
 Developing optimal production schedules
 Developing financial plans
 Determining optimal diet plans

Instructor Slides

19-3


LP Models

 LP Models
 Mathematical representations of constrained optimization problems
 LP Model Components:
 Objective function
 A mathematical statement of profit (or cost, etc.) for a given solution

 Decision variables
 Amounts of either inputs or outputs

 Constraints
 Limitations that restrict the available alternatives

 Parameters
 Numerical constants

Instructor Slides

19-4


Linear Programming

Used to obtain optimal solutions to problems that involve
restrictions or limitations, such as:

 Materials
 Budgets
 Labor
 Machine time

Linear programming (LP) techniques consist of a sequence of steps
that will lead to an optimal solution to problems, in cases where an
optimum exists


LP Assumptions

 In order for LP models to be used effectively, certain assumptions must be
satisfied:

 Linearity
 The impact of decision variables is linear in constraints and in the objective function
 Divisibility
 Noninteger values of decision variables are acceptable
 Certainty
 Values of parameters are known and constant
 Nonnegativity
 Negative values of decision variables are unacceptable

Instructor Slides

19-6


Model Formulation
1.

List and define the decision variables (D.V.)



2.

State the objective function (O.F.)



3.

These typically represent quantities

It includes every D.V. in the model and its contribution to profit (or cost)

List the constraints



Right hand side value



Relationship symbol (≤, ≥, or =)



Left Hand Side

 The variables subject to the constraint, and their coefficients that indicate how much of
the RHS quantity one unit of the D.V. represents

4.

Non-negativity constraints

Instructor Slides

19-7


Example– LP Formulation

 x1 = Quantity of product 1 to produce

Decision Variables  x2 = Quantity of product 2 to produce
 x = Quantity of product 3 to produce
 3
Maximize

5 x1 + 8 x2 + 4 x3 (profit)

Subject to
Labor

2 x1 + 4 x2 + 8 x3 ≤ 250 hours

Material

7 x1 + 6 x2 + 5 x3 ≤ 100 pounds

Product 1

x1

(Constraints)

≤ 10 units

x1 , x2 , x3 ≥ 0

Instructor Slides

(Objective function)

(Nonnegativity constraints)

19-8


Graphical LP

 Graphical LP
 A method for finding optimal solutions to two-variable problems
 Procedure
1.

Set up the objective function and the constraints in mathematical format

2.

Plot the constraints

3.

Indentify the feasible solution space

 The set of all feasible combinations of decision variables as defined by the constraints

Instructor Slides

4.

Plot the objective function

5.

Determine the optimal solution

19-9


Linear Programming Example

Assembly
Time/Unit

Inspection
Time/Unit

Storage
Space/Unit

Model A

4

2

3

$

60

Model B

10

1

3

$

50

Available

100 hours

22 hours

39 cubic feet

Profit/Unit


Linear Programming Example

Find the quantity of each
model to produce in order to
maximize the profit


LP Example – Decision Variables

Decision Variables
A: # of model A product to be built
B: # of model B product to be built


Example– Graphical LP: Step 1

 x1 = quantity of type 1 to produce
Decision Variables 
 x2 = quantity of type 2 to produce
Maximize
60 x1 + 50x2
Subject to
Assembly

4 x1 + 10 x2 ≤ 100 hours

Inspection

2 x1 + 1x2 ≤ 22 hours

Storage

3 x1 + 3 x2 ≤ 39 cubic feet
x1 , x2 ≥ 0

Instructor Slides

19-13


LP Example – Objective Function
Profit:
Profit = profit from model A + profit from model B
(profit/model A) x (# of model A) + (profit/model B) x (# of model B)
Z = 60A + 50B

Obejective Function:

Maximize
Z = 60A + 50B

(Profit)


LP Example – Objective Function
30
25
20
60A+50B=600

B 15

60A+50B=1200

10
5
0
1

2

3

4

5

6
A

7

8

9

10

11

Example– Graphical LP: Step 2

Plotting constraints:
 Begin by placing the nonnegativity constraints on a graph

Instructor Slides

19-16


Example– Graphical LP: Step 2

Plotting constraints:

Instructor Slides

1.

Replace the inequality sign with an equal sign.

2.

Determine where the line intersects each axis

3.

Mark these intersection on the axes, and connect them with a straight line

4.

Indicate by shading, whether the inequality is greater than or less than

5.

Repeat steps 1 – 4 for each constraint

19-17


Example– Graphical LP: Step 2
4A+10B<100
12

Infeasible

10
8
B 6

Feasible

4
2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
A


Example– Graphical LP: Step 2

Example– Graphical LP: Step 2

Instructor Slides

19-20


Example– Graphical LP: Step 2

Instructor Slides

19-21


Example– Graphical LP: Step 2

Instructor Slides

19-22


Storage Space Constraint

Storage space

Storage
Space/Unit
3
3

39 cubic feet

3A + 3B < 39 cubic feet


Storage Space Constraint
3A + 3B < 39
14

Infeasible

12
10
B

8

Feasible

6
4
2
0
1

2

3

4

5

6

7

8
A

9

10

11

12

13

14


Linear Programming Formulation

Objective - profit
Maximize Z=60A + 50B

Subject to
Assembly 4A + 10B <= 100 hours

Inspection
Storage

2A + 1B <= 22 hours
3A + 3B <= 39 cubic feet

A, B >= 0

Nonnegativity Condition