P RIMAL -D UAL LPP

T HE R EDDY M IKKS C OMPANY - PROBLEM Reddy Mikks company produces both interior and exterior paints from two raw materials, M 1 and M 2. The following table provides the basic data of the problem: 2 Tons of raw material per ton of Maximum daily availability (tons) Exterior paintInterior paint Raw material, M 1 Raw material, M Profit per ton (Rs 1000) 54

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 by more than 1 ton. Reddy Mikks wants to determine the optimum (best) product mix of interior and exterior paints that maximizes the total daily profit. 3

M ATHEMATICAL FORMULATION x 1 = Tons produced daily of exterior paint x 2 = Tons produced daily of interior paint Maximize z =5 x 1 +4 x 2 Subject to 4

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S ENSITIVITY ANALYSIS Sensitivity 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 6

G RAPHICAL S ENSITIVITY A NALYSIS We can use the graph of an LP to see what happens when: 1. An OFC changes, or 2. A RHS changes 7

O BJECTIVE F UNCTION C OEFFICIENT (OFC) C HANGES In Reddy Mikks Problem, What if the profit contribution for raw material of exterior paint is changed from Rs 5 to Rs 6 per ton? 6 Max X 8

There is no effect on the feasible region The slope of the level profit line changes If 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 corner point becomes optimal. C HARACTERISTICS OF OFC C HANGES 9

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RHS C ONSTRAINT C HANGES In Reddy Mikks Problem, What if the resources of raw material of exterior paint is changed from 24 ton to 25 ton? 25 X 11

C HARACTERISTICS OF RHS C HANGES The constraint line shifts, which could change the feasible region Slope of constraint line does not change Corner point locations can change The optimal solution can change 12

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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, then the shadow price will change. 14

Given a LPP (called the primal problem), we shall associate another LPP called the dual problem of the original (primal) problem. We shall see that the Optimal values of the primal and dual are the same provided both have finite feasible 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. D UAL PROBLEM OF AN LPP 15

M ATHEMATICAL FORMULATION OF PRIMAL – DUAL PROBLEM PrimalDual Maximize Z= Subject to Minimize W= Subject to 16

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Primal (Maximize)Dual (Minimize) 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 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 = 18

P ROPERTIES OF P RIMAL - DUAL PAIR 19 o The number of dual variables is the same as the number of primal constraints. o The number of dual constraints is the same as the number of primal variables. o The coefficient matrix A of the primal problem is transposed to provide the coefficient matrix of the dual problem. o The inequalities are reversed in direction.

The maximization problem of the primal problem becomes 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 cost coefficients in the dual problem. The primal and dual variables both satisfy the non negativity condition. 20

E CONOMIC INTERPRETATION OF D UAL VARIABLES The primal problem represents a resource allocation model, b i represents number of units available of resource i and Z, a profit (in Rs). The dual variables y i, represent the worth per unit of resource i and W denotes worth of resources. 21

The dual of the dual problem is again the primal problem. 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 is either infeasible or feasible/unbounded. R ELATIONSHIP BETWEEN THE OPTIMAL, PRIMAL AND DUAL SOLUTIONS 22

P RIMAL - DUAL OF R EDDY M IKKS PROBLEM Reddy Mikks PrimalReddy Mikks Dual Maximize Subject to Minimize Subject to Optimal solution 23

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The optimal dual solution shows that the worth per unit of raw material, M 1 is y 1 =0.75, whereas that of raw material, M 2 is y 2 =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 M 1, 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 M 2, can be increased from 6 tons to a maximum of 6.67 tons, with a corresponding increase in profit of 0.67x0.5= The worth per unit for each of resources 1 and 2 are guaranteed only within the specified ranges. For resources 3 and 4, representing the market requirements, the dual prices are both zero, which indicates that their associated resources are abundant. Hence, their worth per unit is zero. 26