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ISE 336 ART OF MATHEMATICAL
MODELLING
INTRODUCTION TO MATHEMATICAL
MODELING AND IBM ILOG CPLEX
OPTIMIZATION STUDIO
2014-2015 SPRING TERM
Dr. M. Arslan Örnek
LECTURE #4
THIS WEEK...
Linearizing Logical Forms with Binary Variables
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Using of Discrete Variables
Indicator (Binary) Variables
Introduction to the Theory of Logical Variables
Logical forms represented by Binary Variables
Linearization of compund propositions
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INTEGER VARIABLES
Indivisible (Discrete)
Quantities
Variables with whole numbers
such as aeroplanes, cars,
houses or people.
Logical Variables
Usually Binary (0-1) Variables
but...
β=0; no depot should be
built,
β=1; a depot of type A
should be built,
β=2; a depot of type B
should be built,
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LOGICAL CONDITIONS
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THE FIXED CHARGE PROBLEM
Is it a linear function ?
If no, how can you make it
linear ?
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THE FIXED CHARGE PROBLEM
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BINARY VARIABLES THEOREM
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THE FIXED CHARGE PROBLEM
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INTRODUCTION TO THE THEORY OF
LOGICAL VARIABLES
Simple Proposition :
Logical variable which can take on only the values “true” or “false”
Compound Proposition :
Modification of a simple proposition by the word “not” or connected
propositions which are connected with logical connectives such as
“and”, “or”, “if-then”, and “if and only if”.
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PROPOSITIONAL CONNECTIVES
AND CONSTRAINTS
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PROOF FOR COMPOUND
PROPOSITIONS
How could we prove this is equality ?
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CONNECTIVE TRUTH TABLES
# of rows : 2# of
propositions
# of adjacent truths : 2k  proposition number
# of propositions
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TRUTH TABLE FOR EXCLUSIVE
DISJUNCTION
?
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BOOLEAN ALGEBRA PROPERTIES
De Morgan’s
Laws
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LOGICAL FORMS REPRESENTED BY
BINARY VARIABLES
Show the following compound propositions with binary variables
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LOGICAL FORMS REPRESENTED BY
BINARY VARIABLES
 k
  n



k



1
 i
  s

 i 1
  s  k 1

 k
  n

  i  k  0      s 1  0 
 i 1
  s  k 1

k

i 1
i
k 
s  k 1

s  k 1
n

n
s
1
k
s
 k   i  1
i 1
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LOGICAL FORMS REPRESENTED BY
BINARY VARIABLES
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EXAMPLE-1
Linearize the following implication: If a company decides to produce a
certain item, then at least 10 units of it need to be produced; it is
assumed that the production capacity is 25 units.
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ANSWER-1
x : The number of units being produced, x ≥ 0
Logical Proposition :
 x  0  10  x  25
1 if x  0
 
0 if x  0
If the company decides to produce
If the company decides not to produce
x  25
x  10
x  25
x  10
 1
Do we need ? Why ?
x  0
 x 0
x  0
 0
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EXAMPLE-2
Linearize the following implication: If item A and/or item B are produced,
then at least one of the items C,D, or E have to be produced.
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ANSWER-2
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ANSWER-2
Can you linearize the same compound proposition without using an extra
binary variable ?
Is it necessary to
linearize the
compound
proposition with an
additional binary
variable ?
C   D   E   A
C   D   E   B
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EXAMPLE-3
Linearize the expression: The binary variables
1 ,  2 ,  3
satisfy
1 2   3
 3  1  1  1   2  1
or
 3  1  1   2  2 
If the first and the second binary
varibles are equal to 1, the third
binary variable has to be equal
to 1
If the third variable is equal to 1,
both the first and the second
variables have to be equal to 1
1   2   3  1
or
1   2  2 3
 3  1 ,  3   2
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EXAMPLE-4
Linearize the expression : The inequality
a x
iI
i i
b
holds if the condition X holds.
X  M
1 if X holds
 
0 otherwise


  1    ai xi  b 
 iI

a x
iI
i i
 b  M 1   
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EXAMPLE-5
A depot can be sited at any one of the positions A,B,C,D, or E. Only one
depot can be built.
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