6s-1
Linear Programming
CHAPTER
6s
Linear
Programming
6s-2
Linear Programming
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

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
Linear Programming Model

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
6s-5
Linear Programming
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
Linear Programming
Graphical 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
6s-7
Linear Programming
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
Linear Programming
Linear Programming Example
Assembly Constraint
4X1 +10X2 = 100
Product X1
24
22
20
18
16
14
12
10
8
6
4
2
12
10
8
6
4
2
0
0
Product X2
6s-8
Linear Programming
Linear Programming Example
Add Inspection Constraint
2X1 + 1X2 = 22
25
20
15
10
5
Product X1
24
22
20
18
16
14
12
10
8
6
4
2
0
0
Product X2
6s-9
6s-10 Linear Programming
Linear Programming Example
Add Storage Constraint
3X1 + 3X2 = 39
Product X2
25
Inspection
20
Storage
15
Assembly
10
5
Feasible solution space
Product X1
24
22
20
18
16
14
12
10
8
6
4
2
0
0
6s-11 Linear Programming
Linear Programming Example
Add Profit Lines
Product X2
25
20
Z=900
15
10
5
Z=300
Z=600
Product X1
24
22
20
18
16
14
12
10
8
6
4
2
0
0
6s-12 Linear Programming
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 Linear Programming

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 Linear Programming
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-15 Linear Programming
MS Excel Worksheet for
Microcomputer Problem
Figure 6S.15
6s-16 Linear Programming
MS Excel Worksheet Solution
Figure 6S.17