Transcript Linear Programming - Texas Tech University

Linear Programming
Chapter 14 Supplement
Lecture Outline
•
•
•
•
Model Formulation
Graphical Solution Method
Linear Programming Model Solution
Solving Linear Programming Problems with
Excel
• Sensitivity Analysis
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-2
Linear Programming (LP)
• A model consisting of linear relationships
representing a firm’s objective and resource
constraints
• A mathematical modeling technique which
determines a level of operational activity in order
to achieve an objective, subject to restrictions
called constraints
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-3
Types of LP
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-4
Types of LP
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Supplement 14-5
Types of LP
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Supplement 14-6
LP Model Formulation
• Decision variables
• symbols representing levels of activity of an operation
• Objective function
• linear relationship for the objective of an operation
• most frequent business objective is to maximize profit
• most frequent objective of individual operational units
(such as a production or packaging department) is to
minimize cost
• Constraint
• linear relationship representing a restriction on
decision making
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-7
LP Model Formulation
Max/min
z = c1x1 + c2x2 + ... + cnxn
subject to:
Constraints
a11x1 + a12x2 + ... + a1nxn (≤, =, ≥) b1
a21x1 + a22x2 + ... + a2nxn (≤, =, ≥) b2
:
an1x1 + an2x2 + ... + annxn (≤, =, ≥) bn
xj = decision variables
bi = constraint levels
cj = objective function coefficients
aij = constraint coefficients
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-8
Highlands Craft Store
Resource
Requirements
Product
Bowl
Mug
Labor
(hr/unit)
1
2
Clay
(lb/unit)
4
3
Revenue
($/unit)
40
50
There are 40 hours of labor and 120 pounds of clay
available each day
Decision variables
x1 = number of bowls to produce
x2 = number of mugs to produce
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-9
Highlands Craft Store
Maximize Z = $40 x1 + 50 x2
Subject to
x1 + 2x2 40 hr (labor constraint)
4x1 + 3x2 120 lb (clay constraint)
x1 , x2 0
Solution is x1 = 24 bowls
Revenue = $1,360
Copyright 2011 John Wiley & Sons, Inc.
x2 = 8 mugs
Supplement 14-10
Graphical Solution Method
1. Plot model constraint on a set of coordinates in
a plane
2. Identify the feasible solution space on the graph
where all constraints are satisfied
simultaneously
3. Plot objective function to find the point on
boundary of this space that maximizes (or
minimizes) value of objective function
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-11
Graphical Solution Method
x2
50 –
40 –
4 x1 + 3 x2 120 lb
30 –
Objective function
Area common to
both constraints
20 –
10 –
0–
x1 + 2 x2 40 hr
|
10
Copyright 2011 John Wiley & Sons, Inc.
|
20
|
30
|
40
|
50
|
60
x1
Supplement 14-12
Computing Optimal Values
x1 + 2x2 = 40
4x1 + 3x2 = 120
x2
40 –
4 x1 + 3 x2 = 120 lb
30 –
x1 + 2 x2 = 40 hr
20 –
10 –8
0–
|
10
| 24 |
20
30
| x1
40
4x1 + 8x2 = 160
-4x1 - 3x2 = -120
5x2 =
x2 =
40
8
x1 + 2(8) =
x1
=
40
24
Z = $40(24) + $50(8) = $1,360
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-13
Extreme Corner Points
x1 = 0 bowls
x2 = 20 mugs
Z = $1,000
x2
40 –
x1 = 24 bowls
x2 = 8 mugs
Z = $1,360
30 –
20 – A
10 –
0–
x1 = 30 bowls
x2 = 0 mugs
Z = $1,200
B
|
10
|
20
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| C|
30 40
x1
Supplement 14-14
Objective Function
x240 –
4x1 + 3x2 = 120 lb
Z = 70x1 + 20x2
30 –
Optimal point:
x1 = 30 bowls
x2 = 0 mugs
Z = $2,100
A
20 –
B
10 –
0–
|
10
Copyright 2011 John Wiley & Sons, Inc.
|
20
| C
30
x1 + 2x2 = 40 hr
|
40
x1
Supplement 14-15
Minimization Problem
CHEMICAL CONTRIBUTION
Brand
Nitrogen (lb/bag)
Phosphate (lb/bag)
2
4
4
3
Gro-plus
Crop-fast
Minimize Z = $6x1 + $3x2
subject to
2x1 + 4x2  16 lb of nitrogen
4x1 + 3x2  24 lb of phosphate
x 1, x 2  0
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-16
Graphical Solution
x2 14 –
x1 = 0 bags of Gro-plus
x2 = 8 bags of Crop-fast
Z = $24
12 –
10 –
8–A
Z = 6x1 + 3x2
6–
4–
B
2–
0–
|
2
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|
4
|
6
|
8
C
|
10
|
12
|
14
x1
Supplement 14-17
Solving LP Problems with Excel
Click on “Data”
to invoke “Solver”
Objective function
=C6*B10+D6*B11
=E6-F6
=E7-F7
Decision variables
bowls (X1) = B10
mugs (x2) = B11
Copyright 2011 John Wiley & Sons, Inc.
=C7*B10+D7*B11
Supplement 14-21
Solving LP Problems with Excel
After all parameters and constraints
have been input, click on “Solve”
Objective function
Decision variables
C6*B10+D6*B11≤40
and
C7*B10+D7*B11≤120
Click on “Add” to
insert constraints
Click on “Options” to add
non-negativity and linear
conditions
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-22
LP Solution
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Supplement 14-23
Sensitivity Analysis
Sensitivity range for labor;
30 to 80 lbs.
Shadow prices – marginal
values – for labor and clay.
Copyright 2011 John Wiley & Sons, Inc.
Sensitivity range for clay;
60 to 160lbs.
Supplement 14-24
Sensitivity Range for Labor Hours
Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-25
Sensitivity Range for Profit for Bowls
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Supplement 14-26
Copyright 2011 John Wiley & Sons, Inc.
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Copyright 2011 John Wiley & Sons, Inc.
Supplement 14-27