economic interpretation of duality

7/27/2019 Economic Interpretation of Duality

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Economic Interpretation of DuShadow Price

&

Complementary Slackness TheoANKIT CHOUDH

APOORVA GU

AYESHA AH


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Primal

We are starting a business selling two FORE insigniaproducts:

Sweaters and Scarves.

Profits on each are Rs. 35 and Rs. 10 respectively.

Each has a pre-bought embroidered crest sewn on it;

We have 2000 crests in hand.

Sweaters take four skeins of yarn, while scarves only take

one, and there are 2300 skeins of yarnavailable.

Finally, we have available storage space for 1250 scarves;

we could use any of that space for sweaters, too, butsweaters take up half again as much space as scarves.

What product mix maximizes revenue?

Crest

Scarf 1

Sweater 1


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Formulating the Problem

Let x be the number of sweaters and y the number of scarves made.We want to

Max z = 35x + 10y

subject to

x + y 2000 (Crest Constraint)

4x + y 2300 (Yarn Constraint)

3x + 2y 2500 (Storage Constraint)

x; y 0 (Non Negative Constraint)


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Graphical Solution

Objective Function Value at CornePoints

z(0; 0) = 0

z(575; 0) = 20125

z(0; 1250) = 12500

z*(420; 620) = 20900

Notice one constraint is Non Bindin

We should make 420 sweaters and 620 scarve


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Initial Table

Cj 35 10 0 0 0

B.V CB X Y S1 S2 S3 SOLUTION RATIO

S1 0 1 1 1 0 0 2000 2000

S2 0 4 1 0 1 0 2300 575

S3 0 3 2 0 0 1 2500 833.3

Cj-Zj 35 10 0 0 0


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1st Iteration

Cj 35 10 0 0 0

B.V CB X Y S1 S2 S3 SOLUTION

S1 0 0 3/4 1 -1/4 0 1425

x 35 1 1/4 0 0 575

S3 0 0 5/4 0 - 1 775

Cj-Zj 0 5/4 0 -35/4 0


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Optimal Solution

Cj 35 10 0 0 0

B.V CB X Y S1 S2 S3 SOLUTION

S1 0 0 0 1 1/5 -3/5 960

X 35 1 0 0 2/5 -1/5 420

Y 10 0 1 0 -3/5 4/5 620

Cj-Zj 0 0 0 -8 -1


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What if...?

Suppose our business were suddenly given

one additional crest patch

one additional skein of yarn

one additional unit of storage space

How much would profits change?


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One Additional ResourceNon Binding Constraint

So no change in solutionWe'll make

and m

We'll make more sweatersand lesser scarves

Z=35*(0)+10*(0)=0

Additional Crest

Z=35*(0.4)+10*(-0.6)=8

Additional Yarn

Z=35*(-0

Additio


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Shadow Price

In a linear programming problem in standard form,

Shadow Price is the amount by which the optimal value of thobjective function is improved (so that it increases the z value for maximization problem and decreases the z value for a minimizatioproblem) if the right hand side of constraint is increased by one un

and all other constraints remaining constant.

Z=35*(0)+10*(0)=0 Z=35*(0.4)+10*(-0.6)=8 Z=35*(-0


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Dual

Suppose an entrepreneur wants to buy ourbusiness's resources.

What prices should be quoted for each crest?skein of yarn? Unit of storage?

Suppose the entrepreneur quotes p for each crestpatch, q for each skein of yarn, and r for eachstorage unit. Each sweater takes one patch, 4skeins, and 3 storage units, so effectively

p + 4q + 3r is bid per sweater.

Likewise, p + q + 2r is bid per scarf.

Crest

Scarf 1

Sweater 1


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Formulating the Problem

Min z1 = 2000p + 2300q + 2500r

subject to

p + 4q + 3r 35

p + q + 2r 10

p, q, r 0

Where

p= price of each crest

q= price of each yarn

r= price of unit storage space

Max z = 35x + 10ysubject tox + y 20004x + y 23003x + 2y 2500x; y 0

Wherex= Number oy= Number o

Primal ProblemDual Problem


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Initial Table

Cj 2000 2300 2500 0 0 M M

B.V CB p q r S1 S2 A1 A2 SOLUTION R

A1 M 1 4 3 -1 0 1 0 35

A2 M 1 1 2 0 -1 0 1 10

Cj-Zj2000-

2M2300-2M

2500-2M

M M 0 0


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Optimal Solution

Cj 2000 2300 2500 0 0

B.V CB p q r S1 S2 SOLUTION

q 2300 -1/5 1 0 -2/5 3/5 8

r 2500 3/5 0 1 1/5 -4/5 1

Cj-Zj 960 0 0 420 620


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Economic Interpretation

The values corresponding to p , q , r in the optimal table show the shai.e. the price that will be incurred when we use one more unit of crest

storage, respectively.

If we employ one more unit of yarn, for example, the cost of using thaunit will be Rs. 8. This price is known as the Shadow Price.

This shadow price computed from the dual is same as the value of slavariables calculated from the optimal table of the primal

Thus, shadow prices are the solutions to the dual problem.

Also, the payoff is the same in both the primal problem and the dual p


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Complementary SlacknessTheorem

The principal of complementary slackness establishes relation betweethe optimal value of a primal variable in one problem with a slack orsurplus variable of the dual problem

If at the optimal solution of the primal problem a primal constrainthas a positive value of a slack variable the corresponding resourceis not completely used up and must have a Zero Opportunity Cost.But if the value of slack variable is Zero in that constraint the entireresource is being used up and must have a positive opportunitycost. Therefore mathematical condition is

Primal Slack Variable x Dual Variable = 0


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Thank YouFOR YOUR ATTENTION

economic interpretation of duality

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