What is sensitivity analysis? Why do we perform sensitivity analysis? How far do we like to perform sensitivity analysis? In an LP sensitivity analysis, what type of conditions or information are considered? In an LP sensitivity analysis, what type of conclusions are considered? IE 416, Chap 5:1, July 98

Shadow Price: SP page 199 Shadow price for the i th constraint of an LP is the amount by which the optimal z-value is improved if the rhs of the i th constraint is increased by 1. Reduced Cost: RC page 205 For any nonbasic variable X k, the reduced cost is the amount by which the objective function coefficient of X k must be improved before the LP will have an optimal solution in which X k is a basic variable IE 416, Chap 4:3, July 98

Terms Used in Different Books on Simplex method WinstonWinQSB System BindingTight Non-bindingLooseSlack ExcessSurplus Reduced costOpportunity cost Shadow priceDual price IE 416, Chap 4:4, July 98

Summary of LP Sensitivity Analysis, Notations O.F. Z=C BV *X BV + C NBV *X NBV S.T. a b1 *X 1 + a b2 *X 2 + … =< b b a n1 *X 1 + a n2 *X 2 + …. =<b n Where: C BV = coefficient of basic variable in Z C NBV = ….. nonbasic variable in Z X BV = basic variable in optimal solution X NBV = nonbasic variable in optimal solution b b = right hand side of binding constraint b n = right hand side of nonbinding constraint SP i = shadow price of constraint i RC i = reduced cost of X NBV C’ BV = new C BV, C’ NBV = new C NBV, b’ b = new b b, b’ n = new b n IE 416, Chap 5, Feb. 98

Summary of LP Sensitivity Analysis Summary information from output of the Quant software: Coefficient of OF. RHS. Variable RC Min Current Max Constraint s i /e i SP Min Current Max Summary sensitivity analysis:. IF.. THEN. Basis (var) bfs (value) Z (value). min C’ BV max same same Z+( C BV )X BV C’ Bv out of range change change change min C’ NBV max same same same C’ NBv out of range X NBV ->X’ BV change change min b’ b max same change Z+SP b ( b b ) b’ b out of range change change change min b’ n max same same same b’ n out of range change change change Note: y = y’ - y Note: If output of the Lindo is being used then Z SP b ( b b ); Use + if O.F. is max; Use - if O.F. is min IE 416, Chap 5, Dec. 98

Managerial Use of Shadow Price What is the maximum amount that a manager would like to pay extra for an extra unit of a resource? Consider RMC problem and material 3 in the following table. IE 416, Chap 5, Sept. 2009