sensitivity

Dual formation and Sensitivity Analysis

Panchatantra Revisited

Max 0.90 L + 0.60 S

Subject to

0.2 L + (1/12) S

How much does Jataka pay?

Total payment: 3000PLoom+2400P40+480P60+11000PLS+22000PSS

where Px is the price paid per unit of resource x.

Jatakas Objective: minimize total payment.

Constraints?

Jatakas Constraints

Jataka decides to buy the resources from Panchatantra and they want to set unit prices for each of the inputs. Their objective is to minimize the total cost of procuring the set amounts of inputs.

Panchatantra would accept the deal only if the resources are valued such that it provides them with a lower bound on the unit price of each of their output.

Therefore, the constraints are

Jatakas Constraints

1. Lungi valuation (one metre):

0.2PLoom+0.06P40+0.04P60+PLS+0PSS 0.9

2. Shirting valuation (one metre):

(1/12)PLoom+0.1P40+0P60+0PLS+PSS 0.6

3. All prices 0.

Solve it.

Resource Allocation vs Resource Valuation

So far we considered resource allocation problems (primal)

Corresponding to every resource allocation problem there exists an equivalent resource valuation (dual) problem

Every linear program can be converted into the dual form

In case of linear programs, the optimal solution of a primal and the corresponding dual problem coincide

Primal Formulation

What are we trying to do? Allocate resources in an optimal manner to maximize the

revenues

Revenue maximization or resource allocation

Dual Formulation

What are we trying to do in the above formulation? Valuate the resources in an optimal manner to minimize expenditure

Primal problem involves physical quantities while dual problem involves economic values

Optimal value of a dual variable represents the shadow price of the corresponding resource

Dual of the dual is the primal.

Writing Dual of a Primal

When your primal is in the standard form, in the dual Objective function is to be minimized

All variables should be non-negative

All constraints should be of less than or equal to type

For example, the following (primal) problem is in the standard form

Max. Z=3X1+5X2 so that

X14

2X212

3X1+2X218

X10, X20

Primal-Dual Relationship for Standard Forms

Primal ProblemObjective: Max

Constraint i :

= 0

Dual ProblemObjective: Min

Variable i :

yi >= 0

Constraint j:

>= form


Primal Problem Dual Problem

,53 21 xx ,18124 321 yyy

1823 21 xx

122 2 x

41x

0,0 21 xx

522 32 yy

33 3 y1y

0y,0y,0y 321

Max

Subject to

Min

Subject to

Primal Problem Dual Problem

Max

Subject to

Min

Subject to

Flip and Switch

,5,32

1

x

xZ

18

12

4

2

2

0

3

0

1

2

1

x

x

.0

0

2

1

x

x .

0

0

0

3

2

1

y

y

y

5

3

220

301

3

2

1

y

y

y

3

2

1

18,12,4

y

y

y

W

Flip

Switch


Primal in standard form Dual in standard form

0,...,,

...

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

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

...

...

21

2211

22222121

11212111

n

mnmnmm

nn

nn

xxx

bxaxaxa

bxaxaxa

bxaxaxa

nx

ncxcxc ...

22

11Max

..ts

0,,,

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

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

21

2211

22222112

11221111

m

nmmnnn

mm

mm

yyy

cyayaya

cyayaya

cyayaya

Min

..ts

my

mbybyb ...

22

11


Primal in standard form Dual in standard form

1 2

; 1, 2,

, ,..

.

.,

,

0

j

n

jb mAx j

x x x

Tc xMax

..ts

1 2

; 1, 2,

, , , 0

.i

m

T

iy nA c i

y y y

Min

..ts

Tb y

Forming the Dual

Primal for

Panchantantra

Dual for Panchantantra

Max 0.90L + 0.60S

s.t.: 0.2L + (1/12)S

What if Primal is not in standard form?

Convert it into standard form as follows: If objective is to Minimize 5 x1 - 2 x2

Replace it by Maximize -5 x1 + 2 x2

If a constraint is 4 x1 - 7 x2 0

Replace it by 4 x1 - 7 x2 0

If a variable is negative x2 0

Define a new variable x2 = -x2 which implies x2 0, replace x2 by -x2

If x1 is unrestricted

Define new variables u1 0, u2 0 such that x1=u1-u2 , replace x1 by u1-u2

If a constraint is 3 x1 - 5 x2 = 0

Replace it with two constraints, 3 x1 - 5 x2 0, 3 x1 - 5 x2 0

Convert both to less than or equal to type 3 x1 - 5 x2 0, - 3 x1 + 5 x2 0

Write the dual

Max

..ts

0 ,0,

155672

104849

253535

04520302

4321

4321

4321

4321

4321

xfree,xxx

xxxx

xxxx

xxxx

xxxx

Primal Formulation

Primal-Dual Relationship in General

Primal ProblemObjective: Max

Constraint i :

= form

Variable j:

xj >= 0

xj free

xj = 0

yi free

yi = form

= form

Laws of Duality

Weak Law of Duality:Each feasible solution for the primal(maximization) problem has an objective valuethat is less than or equal to the objective value ofevery feasible solution to the dual (minimization)problem.

In other words: ybxc TT

Laws of Duality

Strong Law of Duality:If the primal problem has a finite optimum,then at the optimum:Objective value of Primal = Objective value of Dual

In other words:

Primal Unbounded Dual Infeasible Primal Infeasible Dual Unbounded or Infeasible

** ybxc TT

Complementary Slackness

Consider at an optimal solution to the primal problem:

Primal Constraint Corresponding Dual Variable (Shadow Price)

Non-binding (Slack0) 0

Binding (Slack=0) Non-zero

Slack Shadow Price = 0

Complementary slackness: The above conditions always holds at the optimum

Both ways implication: If x and y satisfy the above complementary slackness conditions then they are the optimal solutions

Reduced Costs

The reduced cost for a decision variable at its Lower or Upper Bound in the optimal solution indicates by how much its coefficient in the objective function must be improved before that variable enters a positive level.

Some Properties

What is the reduced cost for a decision variable already at non-zero value in an optimal solution?

Can a decision variable at zero level have a reduced cost of zero?

Is there any relation between a primal constraint and the reduced cost of the corresponding dual variable?

-> Validate these by looking at the sensitivity reports of Panchatantra and Jatakas problems.

Zero!

Yes. If it does, that indicates multiple solutions. (not vice versa.)

Yes. Slack in a constraint equals the reduced cost of the corresponding

dual variable.

Multiple Solutions

Only happens when the objective function is parallel to one of the constraints.

At optimal, the objective function will coincide with that constraint.

How much can I change the slope coefficients before the optimal product mix is disturbed?

Ans: not at all (at least in one direction)

if multiple optimal solutions exist!

Another Pointer towards multiple solution

If the dual problem has degenerate solutions (i.e. more than two constraints intersect at the optimal solution) then the primal problem has multiple solutions.

Proof is a little advanced for this class and hence skipped.

Summary: Sensitivity Analysis

Sensitivity analysis tells us the maximum amount by which we can change any of the coefficients in a linear program such that the set of constraints that determine an optimal solution does not change.

We are concerned with changing only one coefficient and keeping all others fixed.

We are bothered only about the set of constraints that define the optimal solution they should not change. But, the optimal solution can change, the objective function value can change.

sensitivity

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