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RAVI PRASAD K.J., profile picture
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Economic interpretations of Linear Programming Problem

The document discusses the economic interpretations of primal and dual linear programming. It explains that the primal problem seeks to maximize revenue under limited resources by allocating activities across resources. The dual variables represent the worth per unit of each resource. The dual constraints represent the "imputed cost" per unit of each resource, and the reduced cost of an activity is the imputed cost of resources needed to produce it. The optimality condition of the simplex method states that an unused activity should only increase if its reduced cost is negative, meaning its revenue exceeds its imputed costs, making it economically advantageous. An example of assembling toys illustrates how reduced cost can determine whether a toy is economically attractive to produce.

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ECONOMIC INTERPRETATIONS PRIMAL/DUAL LINEAR PROGRAMMING
ECONOMIC INTERPRETATIONS Primal Linear Problem •Primal LP problem can be viewed as •a resource allocation model that seeks to maximize the revenueunder limited resources •Primal problem •‘n’ economic activities •‘m’ resources •‘cj’ revenue per unit of activity j •Resource “i” with availability b(i) Is consumed at the rate a(ij)] units per unit activity j
ECONOMIC INTERPRETATIONS OF DUAL VARIABLES The dual variable represents the worth per unit of resource “i” [z = w ] happens only at optimum……………!
Example:
ECONOMIC INTERPRETATION OF DUAL CONSTRAINTS •Formula 2:
ECONOMIC INTERPRETATION OF DUAL CONSTRAINTS •Dual variable “y_i” represents “Imputed cost” per unit resource “i” •The imputed cost of all resources needed to produce one unit of activity “j” is given by •The quantity represents “reducedcost” of activity “j”
ECONOMIC INTERPRETATION OF DUAL CONSTRAINTS •Simplex method – the maximization optimality condition says an increase in the level of an unused (non-basic) activity “j” can improve revenue only if its reduced cost is negative The maximization optimality condition means it is economically advantageous to increase the level of an activity if its revenue exceeds its unit imputed cost.
Example : Assembling 3 types of toys
Example : Assembling 3 types of toys •Toy 1 is not to be produced as x_1 = 0 •Reduced cost for x_1 : a toy becomes attractive economically only if its unit imputed cost is strictly less than the unit revenue •Company can achieve this by increasing the unit price •It can also decrease the imputed cost of the consumed resources •( = y_1 + 3 y_2 + y3) •The reduction ratios on the operations is represented as r_1, r_2 and r_3 •Since y_1 = 1 , y_2 = 2 and y_3 =0 we have •Even 50% reduction fails to satisfy the inequality •Company should not produce this toy
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