PRIMAL-DUAL LPP

THE REDDY MIKKS COMPANY- PROBLEM Reddy

Mikks company produces both interior and exterior paints from two raw materials , M1 and M2. The following table provides the basic data of the problem:Tons of raw material per ton of Exterior paint Interior paint Maximum daily availability (tons) 24 6

Raw material, M1 Raw material, M2 Profit per ton (Rs 1000)

6 1 5

4 2 4

2

A market survey restricts the maximum daily demand of interior paint to 2 tons. Additionally, the daily demand for interior

paint cannot exceed that of exterior paint bymore than 1 ton.

Reddy Mikks wants to determine the optimum(best) product mix of interior and exterior paints that maximizes the total daily profit.

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MATHEMATICAL FORMULATIONx1 = Tons produced daily of exterior paint x2= Tons produced daily of interior paint Maximize z=5 x1+4 x2 Subject to

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SENSITIVITY ANALYSISSensitivity analysis allows us to determine how sensitive the optimal solution is to changes in data values. This includes analyzing changes in: 1. An Objective Function Coefficient (OFC) 2. A Right Hand Side (RHS) value of a constraint

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GRAPHICAL SENSITIVITY ANALYSISWe can use the graph of an LP to see what happenswhen:1. 2.

An OFC changes, or A RHS changes

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OBJECTIVE FUNCTION COEFFICIENT (OFC) CHANGESIn Reddy Mikks Problem,

What if the profit contribution for raw material of exterior paint is changed from Rs 5 to Rs 6 per ton?

6MaxX

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CHARACTERISTICS OF OFC CHANGES

There is no effect on the feasible region

The slope of the level profit line changesIf the slope changes enough, a different corner point will become optimal

There is a range for each OFC where the current optimal corner point remains optimal.

If the OFC changes beyond that range a new cornerpoint becomes optimal.9

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RHS CONSTRAINT CHANGESIn Reddy Mikks Problem,

What if the resources of raw material of exterior paint is changed from 24 ton to 25 ton?

25

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CHARACTERISTICS OF RHS CHANGES

The constraint line shifts, which could change thefeasible region

Slope of constraint line does not change Corner point locations can change The optimal solution can change

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Shadow Price The

change is the objective function value per one-

unit increase in the RHS of the constraint.Constraint RHS Changes If

the change in the RHS value is within the

allowable range, then the shadow price does not change The

change in objective function value =

(shadow price) x (RHS change) If

the RHS change goes beyond the allowable range,14

then the shadow price will change.

DUAL PROBLEM OF AN LPP Given

a LPP (called the primal problem), we shall

associate another LPP called the dual problem of theoriginal (primal) problem. We

shall see that the Optimal values of the primal

and dual are the same provided both have finitefeasible solutions. The

concept of duality is further used to develop

another method of solving LPPs and is also used in the sensitivity (or post-optimal) analysis.15

MATHEMATICAL FORMULATION OF PRIMAL DUAL PROBLEMPrimalMaximize Z=

DualMinimize W=

Subject to

Subject to

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Primal (Maximize)1) i th constraint 2) i th constraint 3) i th constraint = 4) j th variable 0 5) j th variable 0 6) j th variable unrestricted

Dual (Minimize)1) i th variable 0 2) i th variable 0 3) i th variable unrestricted 4) j th constraint 5) j th constraint 6) j th constraint =

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PROPERTIES OF PRIMAL-DUAL PAIRoThe number of dual variables is the same as the number of primal constraints. oThe number of dual constraints is the same as the number of primal variables. oThe coefficient matrix A of the primal problem is

transposed to provide the coefficient matrix of the dual

problem.oThe inequalities are reversed in direction.19

The maximization problem of the primal problembecomes a minimization problem in the dual problem.

The cost coefficients of the primal problem become the right hand sides of the dual problem. The right

hand side values of the primal become the costcoefficients in the dual problem.

The primal and dual variables both satisfy the non

negativity condition.20

ECONOMIC INTERPRETATION OF DUALVARIABLES

The primal problem represents a resource allocation

model , bi represents number of units available of resourcei and Z , a profit (in Rs).

The dual variables

yi, represent the worth per unit of

resource i and W denotes worth of resources.

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RELATIONSHIP BETWEEN THE OPTIMAL,PRIMAL AND DUAL SOLUTIONS

The dual of the dual problem is again the primalproblem.

Either of the two problems has an optimal solution if and only if the other does

If one problem is feasible but unbounded, then the

other is infeasible; if one is infeasible, then the other iseither infeasible or feasible/unbounded.22

PRIMAL - DUAL OF REDDY MIKKS PROBLEMReddy Mikks Primal Reddy Mikks Dual

MaximizeSubject to

MinimizeSubject to

Optimal solution

Optimal solution

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The optimal dual solution shows that the worth per unit of raw material , M1 is y1=0.75, whereas that of raw material,

M2 is y2=0.5.

In graphically showed that the same results hold true for the ranges (20, 36) and (4, 6.67) for resources 1 and 2.

Raw material M1 , can be increased from its present level of 24 tons to a maximum of 36 tons with a corresponding

increase in profit of 12x0.75=9.25

Similarly,

the limit on raw material M2 , can be

increased from 6 tons to a maximum of 6.67 tons ,

with The

a

corresponding

increase

in

profit

of

0.67x0.5=0.335. worth per unit for each of resources 1 and 2 are resources 3 and 4, representing the market that their associated resources are26

guaranteed only within the specified ranges. For

requirements, the dual prices are both zero, which indicates abundant. Hence, their worth per unit is zero.