Question

5. Suppose that (x1, X2, X3) is a feasible solution to the linear programming problem 4r, +2x2 + x3 minimize X12 3, 2a 23 2 4, subject to Let y and ybe non-negative numbers (a) Show that x1(y2y2)2(-y12) + x3y2 2 3y14y2 1 (b) Find constraints on yi and y2 so that 4x12 2 x1(y1 + 2¥2) + x2(-y1 + Y2) + x3Y2 1 at every feasible solution (xi, x2, X3) (c) Use parts (a) and (b) to find a lower bound to the optimal cost in terms of only the variables yi and y2 1 (d) Formulate the linear programming problem in the variables (y1, Y2) that determines the largest possible value for the lower bound to the optimal cost found in part (c). 1

Answer 1

Given (21, 22, 23) is a feasible solution of the linear Programming problen then Mininise a 434 +222 +236 subject to y-x273 - 7② 24 +22 thy4 0 4 M2, 37,0 Geiver y, & 2 be non-regalive - ve numbers (a) x (41 +242) + (-4, +72) + 2372 - I (24 22) & Z2 ( 2 + 2 2 +83) 71 7.3 7.4 [by & - 34 +442 4 (91 +24₂) +22 (-), +72) +234 237, they minimum value is usu +222 +22 so for 2 de 434 +222 tazy 4 (71 +242 ) + x 2 (y + 2) 8237, 38thy so we need to maximize 37 they & constrains are Y +242 24 7 - Y, & 72 ² 2 7251 JY700, 1 +2 + & the 331 4 Constremt we have lower bound of the optimal cost 34, +472. where y, 02 Salesfies the contraints (a) The LPP is maximize 34thY2 Subject to y +242 C4 - Yt Y2 42 42 41 , YY270. @ Here we see that