6S
Linear
Programming
McGraw-Hill/Irwin
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
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
24
22
20
18
16
14
12
10
8
6
4
2
12
10
8
6
4
2
0
0
Product X2
Assembly Constraint
4X1 +10X2 = 100
Product X1
6S-9
Linear Programming Example
Add Inspection Constraint
2X1 + 1X2 = 22
20
15
10
5
24
22
20
18
16
14
12
10
8
6
4
2
0
0
Product X2
25
Product X1
6S-10
Linear Programming Example
Add Storage Constraint
3X1 + 3X2 = 39
Product X2
25
Inspection
20
Storage
15
Assembly
10
5
Feasible solution space
24
22
20
18
16
14
12
10
8
6
4
2
0
0
Product X1
6S-11
Linear Programming Example
Add Profit Lines
Product X2
25
20
Z=900
15
10
5
Z=300
Z=600
24
22
20
18
16
14
12
10
8
6
4
2
0
0
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