Operations management by stevenson 9th ch6s

6S

Linear
Programming

McGraw-Hill/Irwin

Copyright © 2007 by The McGraw-Hill Companies, Inc. All


Learning Objectives
 Describe the type of problem tha would lend itself
to solution using linear programming
 Formulate a linear programming model from a
description of a problem
 Solve linear programming problems using the
graphical method
 Interpret computer solutions of linear programming
problems
 Do sensitivity analysis on the solution of a linear
progrmming problem

6S-2


Linear Programming
 Used to obtain optimal solutions to problems
that involve restrictions or limitations, such
as:
 Materials
 Budgets

 Labor
 Machine time

6S-3


Linear Programming
 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

6S-4


Linear Programming Model
 Objective Function: mathematical statement
of profit or cost for a given solution

 Decision variables: amounts of either inputs
or outputs

 Feasible solution space: the set of all

feasible combinations of decision variables as
defined by the constraints

 Constraints: limitations that restrict the
available alternatives

 Parameters: numerical values
6S-5


Linear Programming
Assumptions
 Linearity: the impact of decision variables is
linear in constraints and 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

6S-6


Graphical Linear Programming
Graphical method for finding optimal
solutions to two-variable problems
1. Set up objective function and constraints
in mathematical format
2. Plot the constraints
3. Identify the feasible solution space
4. Plot the objective function
5. Determine the optimum solution

6S-7


Linear Programming Example
 Objective - profit
Maximize Z=60X1 + 50X2
 Subject to
Assembly

4X1 + 10X2 <= 100 hours

Inspection

2X1 + 1X2 <= 22 hours

Storage

3X1 + 3X2 <= 39 cubic feet

X1, X2 >= 0

6S-8


Linear Programming Example

Product X2

Assembly Constraint
4X1 +10X2 = 100

12
10
8
6
4
2
0
Product X1

6S-9


Linear Programming Example
Add Inspection Constraint
2X1 + 1X2 = 22

Product X2

25
20
15
10
5
0
Product X1

6S-10


Linear Programming Example

Add Storage Constraint
3X1 + 3X2 = 39

Product X2

25
20
15

Inspection
Storage

10

Assembly

5
0

Feasible solution space

Product X1

6S-11


Linear Programming Example
Add Profit Lines

Product X2

25
20

Z=900

15
10
5
0

Z=300

Z=600

Product X1

6S-12


Solution
 The intersection of inspection and storage
 Solve two equations in two unknowns
2X1 + 1X2 = 22
3X1 + 3X2 = 39
X1 = 9
X2 = 4
Z = $740

6S-13

Constraints
 Redundant constraint: a constraint that
does not form a unique boundary of the
feasible solution space

 Binding constraint: a constraint that forms
the optimal corner point of the feasible
solution space

6S-14


Solutions and Corner Points
 Feasible solution space is usually a polygon
 Solution will be at one of the corner points
 Enumeration approach: Substituting the
coordinates of each corner point into the objective
function to determine which corner point is optimal.

6S-15


Slack and Surplus
 Surplus: when the optimal values of
decision variables are substituted into a
greater than or equal to constraint and the
resulting value exceeds the right side value
 Slack: when the optimal values of decision

variables are substituted into a less than or
equal to constraint and the resulting value is
less than the right side value

6S-16


Simplex Method
 Simplex: a linear-programming algorithm
that can solve problems having more than
two decision variables

6S-17


MS Excel Worksheet for
Microcomputer
Problem
Figure 6S.15

6S-18


MS Excel Worksheet Solution
Figure 6S.17

6S-19


Sensitivity Analysis

 Range of optimality: the range of values for

which the solution quantities of the decision
variables remains the same
 Range of feasibility: the range of values for

the fight-hand side of a constraint over which
the shadow price remains the same
 Shadow prices: negative values indicating

how much a one-unit decrease in the original
amount of a constraint would decrease the
final value of the objective function
6S-20