Linear Programming Models in Services

Stereo Warehouse Let x = number of receivers to stock y = number of speakers to stock Maximize 50x + 20y gross profit Subject to 2x + 4y 400 floor space 100x + 50y 8000 budget x 60 sales limit x, y 0

Graphical Solution Stereo Warehouse Z=2000 Z=3000 Z=3600 Z=3800 A B C D E Optimal solution ( x = 60, y = 40)

Model in Standard Form Let s 1 = square feet of floor space not used s 2 = dollars of budget not allocated s 3 = number of receivers that could have been sold Maximize Z = 50x + 20y subject to 2x + 4y + s 1 = 400 (constraint 1) 100x + 50y + s 2 = 8000 (constraint 2) x + s 3 = 60 ( constraint 3) x, y, s 1, s 2, s 3 0

Stereo Warehouse Extreme-Point Solutions Extreme Nonbasic Basic Variable Objective-function point variables variables value value Z A x, y s s s 3 60 B s 3, y s s x 60 C s 3, s 2 s y 40 x 60 D s 1, s 2 s y 80 x 40 E s 1, x s y 100 s

Sensitivity Analysis Objective-Function Coefficients z = 50x + 20y (constraint 3 ) (constraint 1) (constraint 2) A B C D

Sensitivity Analysis Right-Hand-Side Ranging (constraint 3 ) (constraint 2) A BI C D H