This document discusses linear programming techniques. It provides an overview of linear programming concepts like objectives, decision variables, constraints, and parameters in linear programming models. It also covers graphical linear programming including setting up and solving an example problem. The assumptions of linear programming like linearity and nonnegativity are outlined. Methods for solving linear programs are introduced, including the graphical method and Simplex method. Sensitivity analysis concepts like range of optimality and shadow prices are also defined.

1. 6s-1 Linear Programming
William J. Stevenson
Operations Management
8th edition

2. 6s-2 Linear Programming
CHAPTER
6s
Linear
Programming
McGraw-Hill/Irwin
Operations Management, Eighth Edition, by William J. Stevenson
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

3. 6s-3 Linear Programming
 Used to obtain optimal solutions to problems
that involve restrictions or limitations, such
as:
 Materials
 Budgets
 Labor
 Machine time
Linear Programming

4. 6s-4 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
Linear Programming

5. 6s-5 Linear Programming
 Objective: the goal of an LP model is maximization or
minimization
 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
Linear Programming Model

6. 6s-6 Linear Programming
 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
Linear Programming Assumptions

7. 6s-7 Linear Programming
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
Graphical Linear Programming

8. 6s-8 Linear Programming
 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
Linear Programming Example

9. 6s-9 Linear Programming
Assembly Constraint
4X1 +10X2 = 100
0
2
4
6
8
10
12
0 2 4 6 8 10 12 14 16 18 20 22 24
Product X1
ProductX2
Linear Programming Example

10. 6s-10 Linear Programming
Linear Programming Example
Add Inspection Constraint
2X1 + 1X2 = 22
0
5
10
15
20
25
0
2
4
6
8
10
12
14
16
18
20
22
24
Product X1
ProductX2

11. 6s-11 Linear Programming
Add Storage Constraint
3X1 + 3X2 = 39
0
5
10
15
20
25
0
2
4
6
8
10
12
14
16
18
20
22
24
Product X1
ProductX2
Assembly
Storage
Inspection
Feasible solution space
Linear Programming Example

12. 6s-12 Linear Programming
Add Profit Lines
0
5
10
15
20
25
0
2
4
6
8
10
12
14
16
18
20
22
24
Product X1
ProductX2
Z=300
Z=900
Z=600
Linear Programming Example

13. 6s-13 Linear Programming
 The intersection of inspection and storage
 Solve two equations in two unknowns
2X1 + 1X2 = 22
3X1 + 3X2 = 39
X1 = 9
X2 = 4
Z = $740
Solution

14. 6s-14 Linear Programming
 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
Constraints

15. 6s-15 Linear Programming
 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
Slack and Surplus

16. 6s-16 Linear Programming
 Simplex: a linear-programming algorithm
that can solve problems having more than
two decision variables
Simplex Method

17. 6s-17 Linear Programming
Figure 6S.15
MS Excel Worksheet for
Microcomputer Problem

18. 6s-18 Linear Programming
Figure 6S.17
MS Excel Worksheet Solution

19. 6s-19 Linear Programming
 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
Sensitivity Analysis