Question

Consider the following linear programming problem Manimize $45X1 + $10X2 Subject To 15X1 + 5X2 2 1000 Constraint A 20X1 + 4X2 > 1200 Constraint B X1, X2 20 Constraint C if A and B are the two binding constraints. a) What is the range of optimality of the objective function? 3 C1/C2 s 5 b) Suppose that the unit revenues for X1 and X2 are changed to $100 and $15, respectively. Will the current optimum remain the same? NO • that because the new C1/C2 is 6.68 which is not within the range of optimality c) The range of optimality for C1 is scis d) The range of optimality for C2 is sc2 s Using Graphical Solution, e) The Shadow Price (Dual Price) for Constraint A is and the feasibility ranges is: s Constraint AS f) The Shadow Price (Dual Price) for Constraint B is and the feasibility ranges is: S Constraint B s

Answer 1

Kindly go through the solution provided below.

axit C₂ C2 = 2,750 AGO . If the line axit GX2 = 2,750 lies bo AC and BC, 2001 Then, the solution will remain optimal H. 200 s 2750 £300 and 60 < 2250 S 200 8750 26 227 60 and 29.5°2 4 2 34 225 | 13.75 > CZ > 9.167 ) and 45,83 > 0, 241.25 Roъ 200 - - - - t A 66 Xi - - As we in creaseſ the RH.S. of constraint & 1 decrease we prove harcalled to the live as shown by the up and down arrows in the figure. (R.H.S. da, may = 20 (20) = 4000 And (R.M.S.) 2, min. = 4 (200) [800] In this range, constraint 2 is Jeasible shadow price for constraint e can be calculated as Sokeunt zo Z' = 2750 - 2000 a capacity change loo Here z' is the optional value of a when - 800 Rokos of constraint 2 is changed to 800 z'= 45(0)+10(200) - Prvo [ Since, the new optional solution will be at D: ] shadow brice / Similarly, we can calculate the shadow price and range of feasiblity for constraint a 300 cm -- - T -----/ B exi CR.4.)max = 5(300) = (0,500) CR.us),, min = 15160) = 1200] shadow price = o-z 2750-2700 - Capacity change 1000 - 900 Here, I' is the new optimal value of 2 for (R.A.S), 2900 ; . The point is A (6o,o) 2!= 45(60)+1000) = 2,7 00 Shadow prite = -10,5) brce z Loo