Operations
Management
Linear Programming
Module B
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Outline
Requirements of a Linear Programming
Problem
Formulating Linear Programming Problems

Shader Electronics example
Graphical Solution to a Linear Programming
Problem
Graphical representation of Constraints
 Iso-Profit Line Solution Method
 Corner-Point Solution Method

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Outline - continued
Sensitivity Analysis
Solving Minimization Problems
Linear Programming Applications
Production Mix Example
 Diet Problem Example
 Production Scheduling Example
 Labor Scheduling Example

The Simplex Method of LP
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Learning Objectives
When you complete this chapter, you should
be able to :
 Identify or Define:
Objective function
 Constraints
 Feasible region
 Iso-profit/iso-cost methods
 Corner-point solution
 Shadow price

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Learning Objectives - continued
When you complete this chapter, you should
be able to :
 Describe or Explain:



How to formulate linear models
Graphical method of linear programming
How to interpret sensitivity analysis
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What is Linear Programming?
 Mathematical technique

Not computer programming
 Allocates scarce resources to achieve an
objective
 Pioneered by George Dantzig in World War II

Developed workable solution in 1947
 Called Simplex Method
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Examples of Successful LP
Applications
 Scheduling school busses to minimize total
distance traveled when carrying students
 Allocating police patrol units to high crime
areas in order to minimize response time to
911 calls
 Scheduling tellers at banks to that needs are
met during each hour of the day while
minimizing the total cost of labor
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Examples of Successful LP
Applications - continued
 Picking blends of raw materials in feed mills
to produce finished feed combinations at
minimum costs
 Selecting the product mix in a factory to
make best use of machine- and labor-hours
available while maximizing the firm’s profit
 Allocating space for a tenant mix in a new
shopping mall so as to maximize revenues
to the leasing company
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Requirements of a Linear
Programming Problem
1 Must seek to maximize or minimize some
quantity (the objective function)
2 Presence of restrictions or constraints limits ability to achieve objective
3 Must be alternative courses of action from
which to choose
4 Objectives and constraints must be
expressible as linear equations or
inequalities
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Formulating Linear Programming
Problems
Assume:
You wish to produce two products (1) Walkman
AM/FM/Cassette and (2) Watch-TV
 Walkman takes 4 hours of electronic work and 2
hours assembly
 Watch-TV takes 3 hours electronic work and 1 hour
assembly
 There are 240 hours of electronic work time and
100 hours of assembly time available
 Profit on a Walkman is $7; profit on a Watch-TV $5

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Formulating Linear Programming
Problems - continued
Let:
X1 = number of Walkmans
 X2 = number of Watch-TVs

Then:
4X1 + 3X2  240
 2X1 + 1X2  100
 7X1 + 5X2 = profit

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electronics constraint
assembly constraint
maximize profit
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Graphical Solution Method
 Draw graph with vertical & horizontal axes
(1st quadrant only)
 Plot constraints as lines, then as planes

Use (X1,0), (0,X2) for line
 Find feasible region
 Find optimal solution


Corner point method
Iso-profit line method
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Shader Electronic Company Problem
Department
Electronic
Hours Required to
Produce 1 Unit
X1
X2
Walkmans Watch-TV’s
4
3
Assembly
2
1
Profit/unit
$7
$5
Available Hours
This Week
240
100
Constraints: 4x1 + 3x2  240 (Hours of Electronic Time)
2x1 + 1x2  100 (Hours of Assembly Time)
Objective:
Maximize: 7x1 + 5x2
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Number of Watch-TVs (X2)
Shader Electronic Company
Constraints
Electronics
120
(Constraint A)
100
Assembly
(Constraint B)
80
60
40
20
0
0
10
20
30
40
50
60
70
80
Number of Walkmans (X1)
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Number of Watch-TVs (X2)
Shader Electronic Company
Feasible Region
Electronics
(Constraint A)
120
100
Assembly
(Constraint B)
80
60
40
Feasible
Region
20
0
0
10
20
30
40
50
60
70
80
Number of Walkmans (X1)
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Number of Watch-TVs (X2)
Shader Electronic Company
Iso-Profit Lines
Electronics
(Constraint A)
Assembly
(Constraint B)
120
100
80
60
40
20
0
0
10
20
30
40
50
60
70
80
Number of Walkmans (X1)
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Number of Watch-TVs (X2)
Shader Electronic Company Solution
Electronics
(Constraint A)
Assembly
(Constraint B)
120
ISO-Profit Line
100
80
Solution Point
(X1=30, X2=40)
60
40
20
0
0
10
20
30
40
50
60
70
80
Number of Walkmans (X1)
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Number of Watch-TVs (X2)
Shader Electronic Company Solution
Corner Point Solution
Electronics
(Constraint A)
Assembly
(Constraint B)
120
100
Possible Corner
Point Solution
80
60
Optimal solution
40
20
0
0
10
20
30
40
50
60
70
80
Number of Walkmans (X1)
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Formulation of Solution
 Decision variables


X1 = tons of BW chemical produced
X2 = tons of color chemical produced
 Objective

Minimize Z = 2500X1 + 3000X2
 Constraints



X1 30 (BW); X2 20 (Color)
X1 + X2 60 (Total tonnage)
X1  0; X2  0 (Non-negativity)
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Simplex Steps for Maximization
1 Choose the variable with the greatest positive
Cj- Zj to enter the solution
2 Determine the row to be replaced by selecting
that one with the smallest (non-negative)
quantity-to-pivot column ratio
3 Calculate the new values for the pivot row
4 Calculate the new values for the other row(s)
5 Calculate the Cj and Cj-Zj values for this tableau.
If there are any Cj-Zj numbers greater than zero,
return to step 1.
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Sensitivity Analysis
Projects how much a solution might change
if there were changes in variables or input
data.
Shadow price (dual) - value of one additional
unit of a resource
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Minimization Example
BW: $2,500
You’re an analyst for a division of
manufacturing cost
Kodak, which makes BW & color
per month
chemicals. At least 30 tons of BW
and at least 20 tons of color must
be made each month. The total
chemicals made must be at least 60
tons. How many tons of each
chemical should be made to
minimize costs?
Color: $ 3,000 manufacturing
© 1995 Corel Corp.
cost per month
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Graphical Solution
80
Find values for
X1 + X2 60.
BW
60
X1  30, X2  20.
Total
Feasible
Region
Tons, Color
Chemical 40
(X2)
20
Color
0
0
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20
40
60
Tons, BW Chemical (X1)
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80
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Optimal Solution:
Corner Point Method
80
Find corner
points.
BW
60
Total
Tons, Color
Chemical 40
Feasible
Region
B
20
Color
A
0
0
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20
40
60
Tons, BW Chemical
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80
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Simplex Steps for Minimization
1 Choose the variable with the greatest negative Cj- Zj
to enter the solution
2 Determine the row to be replaced by selecting that
one with the smallest (non-negative) quantity-topivot column ratio
3 Calculate the new values for the pivot row
4 Calculate the new values for the other row(s)
5 Calculate the Cj and Cj-Zj values for this tableau. If
there are any Cj-Zj numbers less than zero, return to
step 1.
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