Linear Programming
from the companion CD - Chapter 2 of the book:
Balakrishnan, Render, and Stair,
“Managerial Decision Modeling with Spreadsheets”,
2nd ed., Prentice-Hall, 2007
http://www.stmartin.edu/
Rev. 2.15 by M. Miccio on January 28, 2014
© 2007 Pearson Education
Introduction to Linear Programming
• A Linear Programming (LP) model seeks to
maximize or minimize a linear function,
subject to a set of linear constraints.
• The linear model consists of the following
components:
– a set of decision variables.
– an objective function.
– a set of constraints.
from the companion CD of the book:
Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis,
and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.
2
Introduction to Linear Programming
The Importance of Linear Programming
– Many real world problems lend themselves to linear
programming modeling.
– Many real world problems can be approximated by
linear models.
– There are well-known successful applications in:
• Manufacturing
• Marketing
• Finance (investment)
• Advertising
• Agriculture
from the companion CD of the book:
Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis,
and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.
3
Introduction to Linear Programming
The Importance of Linear Programming
– There are efficient solution techniques that solve linear
programming models.
– The output generated from linear programming
packages provides useful “what if” analysis.
from the companion CD of the book:
Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis,
and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.
4
Introduction to Linear Programming
Assumptions of the linear programming model
– The parameter values are known with certainty.
– The objective function is linear.
z = c1 x1 + c2 x2 +.....+ cj xj +.....+ cn xn = cT x
where cT = ( c1, c2,...ci,..., cn ) is the cost coefficient vector
– The constraints are linear.
ai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn = bi
ak1 x1 + ak2 x2 +.....+ akj .xj+.....+ akn xn  bk
ar1 x1 + ar2 x2 +.....+ arj .xj+.....+ arn xn  br
– There are no interactions between the decision variables (the
additivity assumption).
– The continuity assumption: Variables can take on any value within
a given feasible range.
from the companion CD of the book:
Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis,
and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.
5
LP Terminology
NON-NEGATIVITY CONSTRAINT
The decision variables are non-negative, i.e., they are
NOT allowed to take on negative values
For instance, for a 2-D LP model:
X2
The non-negativity
constraint
X1
from the companion CD of the book:
Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis,
and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.
6
LP Terminology
FEASIBLE REGION
The set in the space of the decision variables of all points
that satisfy all the constraints of the model at the same
time.
7
LP Terminology
Feasible points in the feasible region:
three types are possible
X2
1000
700
The Plastic constraint
2X1+X2 1000
Total production constraint:
X1+X2 700 (redundant)
500
Production
Time
3X1+4X22400
Infeasible
Production mix
constraint:
X1-X2  350
Feasible
500
Interior points Boundary points
X1
700
Extreme points
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LINEAR PROGRAMMING
Application to the Food Industry
(Example of a Mixing Problem) - I
A brewery has received an order for 100 gal of beer with the
special constraint that the beer must contain 4% alcohol by
volume and it must be supplied immediately.
The brewery wishes to fill the order, but no 4% beer is now
in stock. It is decided to mix two beers now in stock to give
the desired final product. One of the beers in stock (Beer A)
contains 4.5% alcohol by volume and is valued at $0.32 per
gallon. The other beer in stock (Beer B) contains 3.7%
alcohol by volume and is valued at $0.25 per gallon. Water
(W) can be added to the blend at no cost.
What volume combination of the two beers in stock with
water, including at least 10 gal of Beer A, will give the
minimum ingredient cost for the 100 gal of 4% beer?
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LINEAR PROGRAMMING
Application to the Food Industry
(Example of a Mixing Problem) - II
• Three independent variables (decision variables)
are considered, i.e., amount of Beer A (VA), amount
of Beer B (VB), and amount of water (Vw).
• This example is greatly simplified because only a few
simple constraints are involved.
• When a large number of possible choices is involved,
the optimum set of choices may be far from obvious,
and a solution by Linear Programming may be the
best way to approach the problem.
10
LINEAR PROGRAMMING
Application to the Food Industry
(Example of a Mixing Problem) - III
Mathematical Model
Objective function
z = 0.32 VA + 0.25 VB
min!
Problem Constraints
1.VA + VB + Vw = 100
2.VA ≥ 10
3.(1 - 0.045) VA + (1 - 0.037) VB + Vw = (1 - 0.04)100
[3’
0.045VA + 0.037VB = 4]
Non-negativity Restriction
1.VA ≥ 0 VB ≥ 0
Vw ≥ 0
Optimal Solution
VA = 37.5 gal;
VB = 62.5 gal;
z = 27.63 $/(100 gal)
Brewery (after substitution)
Modified Constraints
By taking Vw = 100 – (VA + VB) from eq.1 and
replacing:
1.VA ≥ 10
2.0.955 VA + 0.963 VB + 100 – (VA + VB) = 96
2. 0.045 VA + 0.037 VB = 4
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Steps in Developing
a Linear Programming Model
1) Formulation
a)
b)
c)
d)
Understanding the problem
Identification of decision variables
Representation of the objective function
Representation of constraints
2) Solution
3) Interpretation and Sensitivity Analysis
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Special Situation in LP
1. Redundant Constraints: do not affect
the feasible region
Example:
x1 < 10
x1 < 12
The second constraint is redundant
because it is less restrictive.
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Special Situation in LP
2. Infeasibility: when no feasible solution
exists (there is no feasible region)
Example:
x1 < 10
x1 > 15
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Infeasible Model
No point, simultaneously,
lies both above line  and
below lines  and .
2
3
1
from the companion CD of the book:
Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis,
and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.
15
Special Situation in LP
3. Alternate Optimal Solutions: when
there is more than one optimal solution
Max 2T + 2C
Subject to:
T + C < 10
T
< 5
C< 6
T, C > 0
NB:
The objective function eq. is
parallel to the eq. of one
constraint
C
10
All points on
Red segment
are optimal
6
0
0
5
10
T
16
Special Situation in LP
4. Unboundness: when nothing prevents
the solution from becoming infinitely
large
C
Max 2T + 2C
Subject to:
2T + 3C > 6
T, C > 0
2
1
0
0
1
2
3
T
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Special Situation in LP
4. Unboundness (2nd example in 2D):
when nothing prevents the solution from
becoming infinitely large
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Special Situation in LP
5. Decision variable unrestricted in sign:
x j 
 x j  x 'j  x "j
with x 'j  0; x "j  0
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Sensitivity Analysis
of the Optimal Solution
• Is the optimal solution sensitive to changes in
input parameters?
• Possible reasons for asking this question:
– Parameter values used were only best estimates.
– Dynamic environment may cause changes.
– “What-if” analysis may provide economical and
operational information.
from the companion CD of the book:
Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis,
and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.
20
Example:
Sensitivity Analysis of Objective Function Coefficients
1000
X2
600
X1
500
800
from the companion CD of the book:
Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis,
and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.
21