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7/21/2019 OR I- Chapter V
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Operations Research I-
Chapter V: Duality TheoryM. NACEUR AZAIEZ, Professor
Tunis Business School
Tunis University
https://sites.google.com/site/naceurazaiez/
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Content Introduction & Motivation
Definition of the dual of LP
Primal-Dual Relationships
Results on duality theory
Dual of LP in non-canonical form
Economic interpretations
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Introduction & Motivation The theory of duality is a very elegant
and important concept within the field
of operations research. This theory was first developed in
relation to linear programming.
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Introduction & Motivation Next, it was found out that it has many
applications, and perhaps even a more
natural and intuitive interpretation, inseveral related areas such as
nonlinear programming, networks and
game theory
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Introduction & Motivation The notion of duality within LP asserts that
every LP has associated with it a related LP
called its dual. The original problem in relation to its dual is
termed the primal.
It is the relationship between the primal andits dual, both on a mathematical and economic
level, that is truly the essence of duality theory.
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Example 1 There is a small company in Melbourne
which has recently become engaged in the
production of office furniture. The company manufactures tables, desks and
chairs.
The production of a table requires 8 Kgs ofwood and 5 Kgs of metal and is sold for $80.
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Example 1-continued A desk uses 6 Kgs of wood and 4 Kgs of
metal and is sold for $60; and a chair requires
4 Kgs of both metal and wood and is sold for$50.
We would like to determine the revenue
maximizing strategy for this company,given that their resources are limited to 100
Kgs of wood and 60 Kgs of metal.
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Formulation-example 1max
x Z x x x 80 60 501 2 3
8 6 4 100
5 4 4 60
0
1 2 3
1 2 3
1 2 3
x x x
x x x
x x x
, ,
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Duality-example 1 Now consider that there is a much bigger
company in Melbourne which has been the lone
producer of this type of furniture for manyyears.
They don't appreciate the competition from
this new company; so they have decided totender an offer to buy all of their competitor's
resources and therefore put them out of
business.
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Formulating the dual LP
The challenge for this large company then is
to develop a linear program which will
determine the appropriate amount of moneythat should be offered for a unit of each type
of resource, such that the offer will be
acceptable to the smaller company while
minimizing the expenditures of the larger
company.
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Problem D1
8 5 80
6 4 60
4 4 50
0
1 2
1 2
1 2
1 2
y y
y y
y y
y y
,
min y
w y y 100 601 2
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Example 2-continued
Given that a kg of steak costs $10 and provides
80 units of protein, 20 units of carbohydrates
and 30 units of fat, and that a kg of potatoescosts $2 and provides 40 units of protein, 50
units of carbohydrates and 20 units of fat, he
would like to find the minimum cost diet
which satisfies his nutritional requirements.
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Formulating example 2
80 40 400
20 50 200
30 20 100
0
1 2
1 2
1 2
1 2
x x
x x
x x
x x
,
min x
Z x x 10 21 2
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The Dual of example 2
Now consider a chemical company which
hopes to attract this individual away from
his present diet by offering him syntheticnutrients in the form of pills.
This company would like to determine
prices per unit for their synthetic nutrientswhich will bring them the highest possible
revenue while still providing an acceptable
dietary alternative to the individual.
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Dual Problem of example 2
max
y
w y y y 400 200 1001 2 3
80 20 30 101 2 3 y y y
40 50 20 21 2 3 y y y
y y y1 2 3 0, ,
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Comments
Each of the two examples describes some kind
of competition between two decision makers.
The notion of “competition” could beinvestigated more formally in “Game Theory”.
We shall investigate the economic
interpretation of the primal/dual relationshiplater in this chapter.
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Canonical form of the Primal
Problem
a x a x a x ba x a x a x b
a x a x a x b
x x x
n n
n n
m m mn n m
n
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
1 2 0
. . .. ..
. .. .. . . . . .. .
. .. .. . . . . .. .. ..
, , .. .,
max x
j j j
n Z c x
1
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Canonical form of the Dual
Problem
a y a y a y ca y a y a y c
a y a y a y c
y y y
m m
m m
n n mn m n
m
11 1 21 2 1 1
12 1 22 2 2 2
1 1 2 2
1 2 0
.. .. . .
. .. .. . . . . .. .
. .. .. . . . . .. .
. ..
, , .. . ,
min y
ii
m
iw b y 1
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Definition
z Z cx
s t
Ax b x
x*: max
. .
0
w* : min x w yb s.t .
yA c
y 0
Primal Problem Dual Problem
b is not assumed to be non-negative
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21
Max Z = 10X1 + 5X2
2X1 + 3X2 10
6X1 + X2 15
4X1
– 5X2
35
X1, X2 0
Example
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22
Max ZX= X1 + X2
X1 X2
X1 X2
X1 X2
X1, X2 0
Min ZY = Y1 Y2 Y3
Y1 Y2 Y3
Y1 Y2 Y3
Y1
,Y2
,Y3
0
Y1
Y2
Y3
1st constraint 2nd constraint
10
15
35
10 5
2
6
4
3
+
– 5
+ +
+ +
+
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Example
3 8 9 15 20
18 5 8 4 12 30
0
1 2 4 5
1 2 3 4 5
1 2 3 4 5
x x x x
x x x x x
x x x x x
, , , ,
max x Z x x x x x
5 3 8 0 121 2 3 4 5
Primal
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min
y
w y y 20 301 2
3 18 5
8 5 3
8 8
9 4 0
15 12 12
0
1 2
1 2
2
1 2
1 2
1 2
y y
y y
y
y y
y y
y y
,
Dual
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Primal-Dual relationship
x1 0 x2 0 xn 0 w =
y1 0 a11 a12 a n1 b1D u a l y
20 a
21 a
22 a
n2 b
2(m i n w ) . . . . . . . . . . . . . . .
ym 0 am1 am2 amn bn
Z = c1 c2 cn
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Example
5 18 5 158 12 8 8
12 4 8 10
2 5 5
0
1 2 3
1 2 3
1 2 3
1 3
1 2 3
x x x x x x
x x x
x x
x x x
, ,
max x
Z x x x 4 10 91 2 3
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x1 0 x2 0 x3 0 w=
1 0 5 - 18 5 15Dual y2 0 8 12 0 8
(min w) 3 0 12 - 4 8 10 y4 0 2 0 - 5 5
Z= 4 10 - 9
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Dual
5 8 12 2 418 12 4 10
5 8 5 9
0
1 2 3 4
1 2 3
1 3 4
1 2 3 4
y y y y y y y
y y y
y y y y
, , ,
min y
w y y y y 15 8 10 51 2 3 4
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FINDING THE DUAL OF NON-
CANONICAL LP
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Handling variables unrestricted in
sign Replace the variable unrestricted in sign, , by
the difference of two nonnegative variables.
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Example
2 4
3 4 5
2 3
0
2 3
1 2 3
1 2
1 2 3
x x
x x x
x x
x x x
, ; urs:= unrestricted sign
max x
Z x x x 1 2 3
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Conversion
Multiply through the greater-than-or-equal-to
inequality constraint by -1
Use the approach described above to convertthe equality constraint to a pair of inequality
constraints.
Replace the variable unrestricted in sign, , bythe difference of two nonnegative variables.
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Dual
y2
y3 y4
1
2 y13 y2 3 y3 2 y4 1
y1 4 y2 4 y3 1
y1 4 y2 4 y3 1
y1, y2, y3, y4 0
min y
w y y y y 4 5 5 31 2 3 4
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Streamlining the conversion ...
An equality constraint in the primal
generates a dual variable that is
unrestricted in sign. An unrestricted in sign variable in the
primal generates an equality constraint in
the dual.
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Example (Continued)
min y
w y y y y 4 5 5 31 2 3 4
y2
y3 y4
1
2 y13 y2 3 y3 2 y4 1
y1 4 y2 4 y3 1
y1 4 y2 4 y3 1
y1, y2, y3, y4 0
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y y
y y y
y y
y y y
2 3
1 2 3
1 2
1 3 2
1
2 3 2 1
4 1
0
, ,
, , ,
, ,
, , ,, ;
urs
min, , , , y
w y y y 4 5 31 2 3
+
correction
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Primal-Dual relationship
Primal Problem
opt=max
Constraint i :
<= form
= form
Variable j:
x j >= 0
x j urs
opt=min
Dual Problem
Variable i :
yi >= 0
yi urs
Constraint j:
>= form
= form
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Example
3 8 66 5
8 100
1 2
1 2
1
2 1
x x x x
x x x
; urs
max x
Z x x 5 41 2
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Equivalent non-standard form
3 8 6
6 5
8 10
0
1 2
1 2
1
2 1
x x
x x
x
x x; urs
max x
Z x x 5 41 2
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Dual from the recipe
3 8 5
8 6 4
0
1 2 3
1 2
1 2 3
y y y
y y
y y y; , urs
min y
w y y y 6 5 101 2 3
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What about opt=min ?
Can use the usual trick of multiplying the
objective function by -1 (remembering to
undo this when the dual is constructed.) It is instructive to use this method to construct
the dual of the dual of the standard form.
i.e, what is the dual of the dual of thestandard primal problem?
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What is the dual of
w* :min x w yb
s.t .
yA c
y 0
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max y
w yb
s.t .
yA c
y 0
max y
w yb
s.t .
yA c
y 0
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min
. .
x
Z cx
s t
Ax b
x
0
max
. .
x Z cx
s t Ax b
x
0
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Important Observation
FOR ANY PRIMAL LINEAR
PROGRAM, THE DUAL OF THEDUAL IS THE PRIMAL
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Primal-Dual Relationship
Primal or Dual
opt=max opt=min
Dual or Primal
Variable i :
yi >= 0
yi urs
Constraint j:
>= form
= form
Constraint i :
<= form
= form
Variable j:
x j >= 0
x j urs
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Example
3 5 122 8
5 100
1 2
1 2
1 2
1 1
x x x x
x x x x
,
min x
Z x x 6 41 2
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Equivalent form
min x
Z x x 6 41 2
3 5 122 8
5 100
1 2
1 2
1 2
1 2
x x x x
x x x x
,
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Dual
3 5 6
5 2 4
0
1 2 3
1 2 3
1 3 2
y y y
y y y
y y y
, ; urs
max y
w y y y 12 8 101 2 3
Maximization Minimization
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52
Number of constraints Number of variables
constraint Variable positive or zero
constraint Variable negative or zero
constraint = Unconstrained Variable
Number of variables Number of constraints
Variable positive or zero constraint
Variable negative or zero constraint
Unconstrained Variable constraint =
Coefficient of the jth variable
In the objective functionRHS of the jth constraint
RHS of the ith constraintCoefficient of the ith variable
In the objective function
Technological coefficient of the
jth variable in the ith constraint
Technological coefficient of the
ith variable in the jth constraint
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Example
Primal Dual
Max ZX = 5X1 + 4X2 –
2X3
2X1 + 3X2 20
X1 – 4X3 5
X1 + X2 + X3 = 30
X1, X2 0
Min ZY = 20Y1 + 5Y2 + 30Y3
2Y1 + Y2 + Y3
3Y1 + Y3
- 4Y2 + Y3
Y1 0
5
4
= - 2
, Y2 0, Y3 IR
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Primal Dual
Min ZX = 15X1 + 2X2
X1 – X2 10
7X1 + 2X2 50
- X1 + 3X2 2
X1 0, X2 IR
Max ZY = 10Y1+50Y2+2Y3+120Y4
Y1 + 7Y2 - Y3 + 9Y4
-Y1 + 2Y2 +3Y3 + Y4
Y1 0
9X1 + X2 = 120
15
= 2
, Y2 0, Y3 0, Y4 IR
Note that the dual of the dual is the primal
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Results
Consider the primal and dual LP:
55
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Theorem
T
. The dual of the dual is the primal.
The dual problem LP2 is equivalent to the following problem:
Max -
s. t. -
0.By definition, the dual of th
Theore
is pro
m
Proof
blem is
Min c
:
T
T
b y
A y c
y
T
x
s. t. ( A x b
x 0,
or equivalently,
Max
s. t.
0.
which is the primal problem.
T c x
Ax b
x
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57
Various theorems related to duality
(without proof)
If x is primal feasible and y is dual feasible,then cTx ≤ bTy (weak duality theorem).
If x* is primal feasible and y* is dual feasible
and cTx* ≥ bTy* then x* and y* are optimal If one of a pair of primal and dual problems
has an optimal solution, then the other alsohas an optimal solution and the optimal values
of their objective functions are equal.
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Interpretation of Duality
The dual variable y* measures the change in optimal
profit due to a unit change in resource i.
The fact that profit would increase by yi* if an
additional unit of resourcei were available imputes a
value or price to resource i.
This value or price is called a shadow price. Thus,
the shadow price is the amount of contribution of anadditional unit of a resource to total profit.
More will be said later in this context.
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Discussion-continued
Thus, the weak duality theorem states that the
profit is less than or equal to the worth of
resources. That is, the resources are not exploited
according to the best allocation except at
optimality.
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Discussion-continued
The yi's are the shadow prices of the resources.
A shadow price can be interpreted as the
additional unit profit that could be made byacquiring additional units of resource i.
In particular, when a resource is not totally
consumed at optimal exploitation the relatedconstraint is inactive and the corresponding
shadow price must be zero.
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Discussion-continued
Note that in that case, the corresponding slack
variable is basic
If however the constraint is active (i.e., theresource is entirely consumed at optimal
exploitation), then the corresponding shadow
price is positive and therefore it will be
profitable to acquire additional units of that
resource.
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Example
Consider the following LP and its dual :
Max ZX = 5X1 + 2X2 + 7X3
X1 + X2 + X3 50
2X1 + 4X2 + 3X3 75
X1, X2, X3 0
Min ZY = 50Y1 + 75Y2
Y1 + 2Y2
Y1 + 4Y2
Y1 + 3Y2
Y1 0
5
2
≥ 7
, Y2 0
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65
CJ 5 2 7 0 0
Basis X1 X2 X3 S1 S2 bI
0
5
S1
X1
0
1
-1
2
-1/2
3/2
1
0
-1/2
1/2
12,5
37,5
J 0 -8 -1/2 0 -5/2 187,5
The optimal tableau is given by
DualSolution
E1 E2 E3 Y1 Y2 ZY
0 8 1/2 0 5/2 187,5
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Economic Interpretation of thedual problem• Assume that in the context of the previous LP, 3
products P1, P2 et P3 are to be manufactured.
• The production process requires the use of 2
resources R1 & R2 , which are available in limited
quantities of 50 et 75 units respectively.
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Let Xi be the quantity of Pi to produce, i=1,3. As
the producer seeks to maximize his profit, his LP
can be written as:
Max ZX = 5X1 + 2X2 + 7X3
X1 + X2 + X3 50
2X1 + 4X2 + 3X3 75
X1, X2, X3 0
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• However, when the producer sells his resources,he will renounce to produce P1, P2 & P3.
• Hence, he will lose the corresponding profit.
• Consequently, the producer will not accept the
offer unless he gets sufficient compensation for his
lost profit.
• The manufacturing of P1 requires one unit
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• The manufacturing of P1 requires one unit
of R1 and 2 units of R2.
• Hence, the producer is willing to sell oneunit of R1 and two units of R2 for an
amount of at least 5 dinars (unit profit of
P1).
• As Yi is the amount of money proposed
against one unit of Ri, i =1,2,then the sum
received against the amounts of resources
necessary for manufacturing 1 unit of P1 is
Y1+2Y2.
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• We must ensure that:
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• We must ensure that:
Y1 + 2Y2 5
• Similarly, the sale of 1 unit of R1 and 4 unitsof R2 must provide at least a profit of 2
dinars.
• In addition, the sale of 1 unit of R1 and 3units of R2 must provide a profit of at least 7
dinars. This leads to the following two
constraints:Y1 + 4Y2 2
Y1 + 3Y2 771
Fi ll i th t Y &Y t f
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• Finally, given that Y1 &Y2 are amounts of money,
they cannot be negative.
• It follows that the problem is formulated as:
Min ZY = 50Y1 + 75Y2
Y1 + 2Y2 5
Y1 + 4Y2 2
Y1 + 3Y
2 7
Y1, Y2 0
• Clearly, this LP is the dual of the original LP.
• We Call the marginal value of a resource the
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We Call the marginal value of a resource the
amount of change (increase or decrease) in
the value of Z due to the use of an additional
unit of the resource.
• If the problem is a maximization of profit, the
marginal value of a resource is also called
marginal gain or shadow price (introduced
above)
• If the problem is a minimization of cost, the
marginal value is also called marginal cost
or reduced cost.
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Reconsider the following example
Max ZX = 5X1 + 2X2 + 7X3
X1 + X2 + X3 50 (R1)
2X1 + 4X2 + 3X3 75 (R2)
X1, X2, X3 0
CJ 5 2 7 0 0
Basis X1 X
2 X
3 S
1 S
2 b
i
0
5
S1
X1
0
1
-1
2
-1/2
3/2
1
0
-1/2
1/2
12,5
37,5
CJ-ZJ 0 -8 -1/2 0 -5/2 187,5
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• From the corresponding optimal Tableau,
the marginal value or shadow price of R1 is 0 and the shadow price of R2 is 2.5.
• That is, an additional unit of R1 has no
effect on the value of Z, whereas anadditional unit of R2 will increase Z by 2.5
dinars.
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• The producer will not be ready to cede a
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The producer will not be ready to cede a
unit of R1 (respectively R2) unless he
obtains against it at least 0 (respectively2.5 dinars) additional gain.
• Equivalently, in order to acquire one more
unit of R1 (respectively R
2), the producer
will be willing to pay additional cost of 0
(respectively 2.5 dinars).
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The CJ-ZJ of a decision variable
• CJ-ZJ is called the reduced cost of thecorresponding decision variable
• It is interpreted as the deficit for the
decision variable to become basic
• In the last tableau, C2-Z2 =-8. This says
that the unit profit of P2 must increase at
least by 8 dinars for P2 to becomeprofitable.
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The CJ-ZJ of a decision variable
• Equivalently, by forcing one unit of P2 to beproduced, the profit will be reduced by 8
dinars.
• Similarly, the unit profit of P3 must increaseat least by 0.5 dinars for P3 to become
profitable