Sensitivity and Duality.pptx

8/10/2019 Sensitivity and Duality.pptx

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Decisions variables:

X1

= Weekly production level of Space Rays (in dozens)

X2= Weekly production level of Zappers (in dozens).

Objective Function:

Weekly profit, to be maximized

The Galaxy LP Model


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Max 8X1+ 5X2 (Weekly profit)

subject to

2X1+ 1X21000 (Plastic)

3X1+ 4X22400 (Production Time)

X1+ X2

700 (Total production)X1 - X2 350 (Mix)

Xj> = 0, j = 1,2 (Nonnegativity)

The Galaxy Linear Programming Model


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1000

500

Feasible

X2

Infeasible

ProductionTime

3X1+4X2 2400

Total production constraint:

X1+X2 700 (redundant)500

700

Production mix

constraint:

X1-X2 350

The Plastic constraint

2X1+X2 1000

X1

700

Graphical Analysis the Feasible Region

There are three types of feasible points

Interior points. Boundary points. Extreme points.


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Reduced cost

Assuming there are no other changes to the input

parameters, the reduced cost for a variable Xjthat has a value of 0at the optimal solution is:

the amount the variable's objective function coefficient

would have to improve (increase for maximization

problems, decrease for minimization problems) before thisvariable could assume a positive value.

RANGE OF INSIGNIFICANCE

Complementary slacknessAt the optimal solution, either the value of a variable is

zero, or its reduced cost is 0.


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In sensitivity analysis of right-hand sides of constraints we are

interested in the following questions:

Keeping all other factors the same, how much would the

optimal value of the objective function (for example, the

profit) change if the right-hand side of a constraint

changed by one unit? (SHADOW PRICE/DUAL PRICE)

For how many additional or fewer units will this per unitchange be valid? (RANGE OF FEASIBILITY)

Sensitivity Analysis of

Right-Hand Side Values


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Any change to the right hand side of a

binding constraint will change the

optimal solution.

Any change to the right-hand side of a

non-binding constraint that is less than

its slack or surplus, will cause no change

in the optimal solution.

Sensitivity Analysis of

Right-Hand Side Values


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Shadow Prices Assuming there are no other changes to the input

parameters, the change to the objective function

value per unit increase to a right hand side of a

constraint is called the Shadow Price

The shadow price for a nonbinding constraint (one

in which there is positive slack or surplus when

evaluated at the optimal solution) is 0.


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Dual Price

A dual price for a right-hand side (or resource

limit) is the amount the objective function will

improve per unit increase in the right-hand side

value of a constraint.

For maximization problems, dual prices and

shadow prices are the same.

For minimization problems, shadow prices are the

negative of dual prices.


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1000

500

X2

X1

500

Whenmoreplastic becomes available(the plastic constraint is relaxed), theright hand side of the plastic constraintincreases.

Production timeconstraint

Maximum profit = $4360

Maximum profit = $4363.4

Shadow price =4363.40 4360.00 = 3.40

Shadow Price

graphical demonstration -

The Plasticconstraint


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Range of Feasibility

Assuming there are no other changes to theinput parameters, the range of feasibility is

The range of values for a right hand side of a

constraint, in which the shadow prices for the

constraints remain unchanged.

In the range of feasibility the objective function

value changes as follows:

value]sidehandrighttheinngeprice][Cha[Shadow

valueobjectiveinChange


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1000

500

X2

X1

500

Increasing the amount of

plastic is only effective until a

new constraint becomes active.


This is an infeasible solutionProduction time

constraint

Production mix

constraint

X1+ X2 700

A new activeconstraint


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1000

500

X2

X1

500


Production timeconstraint

Note how the profit increasesas the amount of plasticincreases.


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1000

500

X2

X1

500

Lessplastic becomes available(the plastic constraint is more

restrictive).The profit decreases

A new activeconstraint

Infeasiblesolution


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Using Excel Solver Answer ReportMicrosoft Excel 9.0 Answer Report

Worksheet: [Galaxy.xls]Galaxy

Report Created: 11/12/2001 8:02:06 PM

Target Cell (Max)

Cell Name Original Value Final Value

$D$6 Profit Total 4360 4360

Adjustable Cells

Cell Name Original Value Final Value

$B$4 Dozens Space Rays 320 320

$C$4 Dozens Zappers 360 360

Constraints

Cell Name Cell Value Formula Status Slack

$D$7 Plastic Total 1000 $D$7


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Using Excel Solver Sensitivity ReporMicrosoft Excel Sensitivity Report

Worksheet: [Galaxy.xls]Sheet1

Report Created:

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$4 Dozens Space Rays 320 0 8 2 4.25

$C$4 Dozens Zappers 360 0 5 5.666666667 1

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$D$7 Plastic Total 1000 3.4 1000 100 400

$D$8 Prod. Time Total 2400 0.4 2400 100 650

$D$9 Total Total 680 0 700 1E+30 20

$D$10 Mix Total -40 0 350 1E+30 390

Allowable I ncrease/ Decreaseentries indicate how much a given

decision variables Objective Coefficient may change, holding all the

other data constant, and still have the same LP solution.


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When a coefficient is changed by less than the allowable

amounts, the current optimal solution remains the unique

optimal solution.

For a Max model, when a coefficient is increased by its

allowable amount exactly, there will be an alternative optimalcorner solution with a larger optimal value for the

distinguished variable.

For a MIN model, increasing a coefficient by the allowableamount exactly will produce an alternative optimum

corner with a lower optimal value for the distinguished

variable.


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Solver Infeasible Model


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Solver Unbounded solution


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Solver does not alert the user to the existence

of alternate optimal solutions.

Many times alternate optimal solutions existwhen the allowable increase or allowable

decrease is equal to zero.

Solver Alternate Optimal Solution

Li P i D lit


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Linear Programming Duality

Theory

Every LP has an associated dual problem. The Dual is

essentially the inverse of the Primal (original

problem).

The optimal dual solution produces the dual priceor

shadow price reported in the Lindo/Solver output

reports.


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Duality

The Dual

An alternate formulation of a linear programming

problem as either the original problem or its

mirror image, the dual, which can be solved to

obtain the optimal solution.

Its variables have a different economic

interpretation than the original formulation of the

linear programming problem (the primal).It can be easily used to determine if the addition of

another variable to a problem will change the

optimal.


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Formulation of a Dual

DualThe number of decision variables in the primal is

equal to the number of constraints in the dual.

The number of decision variables in the dual isequal to the number of constraints in the primal.

Since it is computationally easier to solve problems

with less constraints in comparison to solvingproblems with less variables, the dual gives us the

f lexibil i ty to choose which problem to solve.


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Example

A comparison of these two versions of the

problem will reveal why the dual might be

termed the mirror image of the primal.


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Economic Interpretation of Dual

Economic interpretation of dual solutionresults

Analysis enables a manager to evaluate the

potential impact of a new product.Analysis can determine the marginal values of

resources (i.e., constraints) to determine how much

profit one unit of each resource is equivalent to.

Analysis helps the manager to decide which of

several alternative uses of resources is the most

profitable.

E ample: Primal and Its D al


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Primal ProblemGross profit:

Max GP = 50X + 30Y

st. 5X + 2Y 220

3X + 2Y 180

X, Y 0

Dual Problem*Total opportunity cost:

Min C = 220a + 180p

st. 5a + 3p 50

2a + 2p 30

a, p 0

Production Data for the Making of PCs and Printers by A-1 Clone

Assembly time Packaging time

per unit (hr.) per unit (hr.) Number of items Profit

(site 1) (site 2) to be made per unit

PCs 5 3 X $50Printers 2 2 Y $30

Labor available 220 hours 180 hours

for production

Example: Primal and Its Dual

*John von Neumann proved the Duality Theorem

C ffi i i f d l bl i


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Coefficient matrix of a dual problem is a

transpose of the primals coefficient matrix

a = opportunity cost of using an additional unit of labor

for the assembly of PCs and printersp =opportunity cost of using an additional unit of labor

for the packaging of PCs and printers

C180220

3022

5035

GP3050

18023

22025

X Y RHS a p RHS


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The solution for the above problem

is:

GP = 2800, X = 20 and Y = 60

a = 2.50 and p = 12.50

Sensitivity and Duality.pptx

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