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Shadow Prices - Introduction to Operations Research - Lecture Slides, Slides of Operational Research

Central University of Jammu and Kashmir Operational Research

These are the important key points of lecture slides of Introduction to Operations Research are:Shadow Prices, Economic Interpretation, Units of Activity, Resource Level, Marginal Value of Resource, Duality Theory, Optimal Solution, Shadow Price, Optimal Value, Complementary Slackness Theorem

Typology: Slides

2012/2013

Uploaded on 01/09/2013

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Shadow Prices

Economic interpretation?

a x a x a x b

x x x

n n

m m mn n m

11 112 2 1 1

21 122 2 2 2

1 1 2 2

1 2 0

+ + + ≤

≥

...

... ... ... ...

...

, ,...,

max

xj j

Z c x=∑

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Discover Slides of Operational Research Central University of Jammu and Kashmir

Partial preview of the text

Download Shadow Prices - Introduction to Operations Research - Lecture Slides and more Slides Operational Research in PDF only on Docsity!

Shadow Prices

 Economic interpretation?

a x a x a x b a 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

x j : units of activity j

b i : resource level i

aij : units of resource level i used per one unit

of activity j

cj : return/loss from unit of activity j

z : total return/loss

z* : optimal return/loss

Clue ...

From duality theory we know that

z* = y*b

where y* is the optimal solution to the

dual problem.

Furthermore, for problems in standard

form y* is equal to the reduced costs of

the slack variables.

More generally,

 y = c B B -

is the “dual variable”, and for the last tableu, y

is an optimal solution for the dual.

warning:

The recipe “assumed” that the change in

the RHS value does not change the basis

itself, namely the elements of the basis are

assumed to be the same after the change.

 If this assumption is not valid, the recipe

may not be valid.

7.5.1 Complementary Slackness

Theorem

a x a x a x s b

x j n m

n n n n

m m mn n m m

11 1 12 2 1 1 1 21 1 22 2 2 2 2

1 1 2 2

0 1

maxx s, Z = c x 1 1 + ... + c xn n + 0 s 1 + 0 s 2 + ... + sm

Correction: Add s (^) i >= 0, i=1,2,...,m. Docsity.com

Let (x,s) be a feasible solution to the primal

and (y,t) be a feasible solution to the dual.

 Then a necessary and sufficient condition

for optimality of both solutions is

sy = 0 ; tx = 0

Observe that because all the variables are non-

negative, this is equivalent to

s yi i = 0 , i = 1 2, ,...,m

t xj j = 0 , j = 1 2, ,..., n

Clue ...

Eq. # (^) x 1 --- (^) x (^) n s 1 --- (^) s (^) m RHS ---Z (^) t --- --- (7.74) 1 ---^ tn y 1 ---^ y^ m yb

Shadow Prices - Introduction to Operations Research - Lecture Slides, Slides of Operational Research

Related documents

Partial preview of the text

Download Shadow Prices - Introduction to Operations Research - Lecture Slides and more Slides Operational Research in PDF only on Docsity!

Shadow Prices

 Economic interpretation?

x j : units of activity j

b i : resource level i

aij : units of resource level i used per one unit

of activity j

cj : return/loss from unit of activity j

z : total return/loss

z* : optimal return/loss

Clue ...

From duality theory we know that

z* = y*b

where y* is the optimal solution to the

dual problem.

Furthermore, for problems in standard

form y* is equal to the reduced costs of

the slack variables.

More generally,

 y = c B B -

is the “dual variable”, and for the last tableu, y

is an optimal solution for the dual.

warning:

The recipe “assumed” that the change in

the RHS value does not change the basis

itself, namely the elements of the basis are

assumed to be the same after the change.

 If this assumption is not valid, the recipe

may not be valid.

7.5.1 Complementary Slackness

Theorem

Let (x,s) be a feasible solution to the primal

and (y,t) be a feasible solution to the dual.

 Then a necessary and sufficient condition

for optimality of both solutions is

sy = 0 ; tx = 0

Observe that because all the variables are non-

negative, this is equivalent to