Shadow Prices - Introduction to Operations Research - Lecture Slides, Slides of Operational Research

Download Shadow Prices - Introduction to Operations Research - Lecture Slides and more Slides Operational Research in PDF only on Docsity! Shadow Prices Economic interpretation? a x a x a x b a x a x a x b a x a x a x b x x x n n n n m m mn n m n 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 1 2 0 + + + ≤ + + + ≤ + + + ≤ ≥ . .. . .. . .. .. . . .. .. . . .. .. . . .. .. . . .. , , .. ., max x j j j n Z c x= ∑ =1 Docsity.com xj : units of activity j bi : resource level i aij : units of resource level i used per one unit of activity j cj : return/loss from unit of activity j z : total return/loss z* : optimal return/loss Docsity.com More generally,  y = cBB-1 is the “dual variable”, and for the last tableu, y is an optimal solution for the dual. Docsity.com Recipe For problems in stadard form: The shadow price of the i-th resource is equal to the optimal value of the i-th dual variable. Docsity.com warning: The recipe “assumed” that the change in the RHS value does not change the basis itself, namely the elements of the basis are assumed to be the same after the change. If this assumption is not valid, the recipe may not be valid. Docsity.com Let (x,s) be a feasible solution to the primal and (y,t) be a feasible solution to the dual.  Then a necessary and sufficient condition for optimality of both solutions is sy = 0 ; tx = 0 Observe that because all the variables are non- negative, this is equivalent to Docsity.com s y i mi i = =0 1 2, , , .. . , t x j nj j = =0 1 2, , , . . ., Docsity.com Example 7.5.2 In example 7.4.2 we have x = (12,0,0) ; s=(4,0) t = (0,4,14) ; y =(0,16) Docsity.com