PA.Assignment 3

Assignment 3 Q1. a. The optimal solution for the problem is 800 of US Oil and 1200 of Huber Steel. The total profit is $8400. b. The constraints in this LP problem are a risk index maximum of 700 and $80,000 in available funds for investment. These constraints are critical to the optimal solution because changes to the problem parameters that affect these constraints have a large impact on the optimal solution. To ensure a balanced portfolio in terms of risk and return, it is important to carefully consider the optimal solution to changes in the problem parameters. c. The sensitivity report shows that the shadow price for US Oil for the maximum investment is $0. Since investing more in US Oil is not recommended as US Oil additional units are $0. Q2. a. 260X1+220X2+290X3+230Y1+240Y2+310Y3 - Minimized. X1+X2+X3 <= 20 Y1+Y2+Y3 <= 20 Constraints:

X1+Y1 = 10 X2+Y2 = 15 X3+Y3 = 10 X1, X2, X3>=20 Y1+Y2+Y3<=20 b. c. The best solution is to produce 20 tons of concentrate in Eutis and supply 10 tons each to Orlando and Tallahassee, and to produce 15 tons of concentrate in Clermont and supply 10 tons to Miami and 5 tons to Orlando. We calculate the total production cost to be $8600. d. No, the optimal solution is not degenerate as the allowable may increase or decrease on any of the constraints is not zero in the sensitivity report.

b. The basic variables in this LP problem are: Basic Variables Non-Basic Variables 1 S1, S2, S3 X1, X2 2 X1, S1, S3 X2, S2 3 X1, X2, S3 S1, S2 4 X1, X2, S2 S1, S3 5 X2, S1, S2 X1, S3 6 X1, X2, S1 S2, S3 7 X1, S1, S2 X2, S3 8 X1, S2, S3 X2, S1 9 X2, S2, S3 X1, S1 10 X2, S1, S3 X1, S2 c. Values for all the variables at feasible solutions: Basic Variables Non-Basic Variables Solution X1, S1, S3 X2, S2 X1=12, X2=0, S1=20, S2=0, S3=10 X1, X2, S3 S1, S2 X1 = 2, X2 = 5, S1 = 0, S2 = 0, S3 = 5 X2, S1, S2 X1, S3 X1 = 0, X2 = 2, S1 = 4, S2 = 8, S3 = 0 X1, S1, S2 X2, S3 X1 = 2, X2= 0, S1 = 10, S2 = 10, S3 = 0 X2, S2, S3 X1, S1 X1 = 0, X2 = 4, S1 = 0, S2 = 4, S3 = 2 d. The values for the objective functions for each of the following equations are: Basic Variables Non-Basic Variables Solution Objective function value X1, S1, S3 X2, S2 X1=12, X2=0, S1=20, S2=0, S3=10 24 X1, X2, S3 S1, S2 X1 = 2, X2 = 5, S1 = 0, S2 = 0, S3 = 5 24 X2, S1, S2 X1, S3 X1 = 0, X2 = 2, S1 = 4, S2 = 8, S3 = 0 8 X1, S1, S2 X2, S3 X1=2, X2= 0, S1 = 10, S2 = 10, S3= 0 4 X2, S2, S3 X1, S1 X1 = 0, X2 = 4, S1 = 0, S2 = 4, S3 = 2 16 e. The highest objective function value is 24, which is at two solutions: (X1,X2) = (12,0) and (2,5) f. When (X1,X2) = (12,0), X1 + 2X2 <= 12 is a binding constraint since slack(left over) = 0. When (X1,X2) = (2,5), -X1 + 2X2 <=8, X1+2X2 <= 12 are binding constraints since slack(left over) = 0.