Solution-investment Lec 10

Solution to Investment Problem 1. a. Final value of MB is 0 as its associated reduced cost is non-zero. Allowable increase is –reduced cost = 0.045, allowable decrease is infinity. b. Reduced cost of CD is -0.015 (note the relationship between the reduced cost and the allowable increase for a non-basic decision variable in a max problem). c. Reduced cost of TB is 0 as this is a basic decision variable. d. Final value of Fund LHS is 70,000 as this is binding constraint (with a non-zero shadow price). e. Final value of MB LHS is 0. For a non-binding constraint like this one, the allowable decrease = RHS – LHS. As we know the allowable decrease and RHS are both 14000, LHS has to be 0. Allowable increase is infinity. f. Shadow price of CD & others is 0 as this is a nonbinding constraint. Allowable increase is infinity, and allowable decrease is 70,000. g. Shadow price of CD & TB is 0 as this is a nonbinding constraint. Allowable increase is 17181.818 and allowable decrease is infinity. i. Final value of Ratio LHS is 0 as this is a binding constraint. 2. MB coefficient 0.085 Æ 0.0875, increases by 0.0025, within the allowable increase 0.045, no change in the optimal solution or the OFV as this is a non-basic decision variable. 3. CD coefficient 0.05 Æ 0.0525, increases by 0.0025, within the allowable increase 0.015, no change in the optimal solution or the OFV as this is a non-basic decision variable. 4. TB coefficient 0.065 Æ 0.06 decreases by 0.005, within the allowable decrease limit 0.015, optimal solution remains optimal, but OFV will change as this is a basic decision. New OFV = 38181.82*0.06+31818.18*0.13 = 6427.27. 5. GS coefficient 0.13 Æ 0.135, increases by 0.005, within allowable increase infinity, optimal solution remains optimal, new OFV = 38181.82*0.065 + 31818.18*0.135 = 6777.27. 6. MB constraint RHS decreases by 7,000, within the allowable decrease of 14000. As this is a nonbinding constraint, no change in the optimal solution or OFV. 7. Ratio CD&TB vs MB&GS, RHS increases by 5000, within allowable increase 70,000, shadow price of -0.0295455 is valid. + OFV = 5000*(-0.0295455) = -147.73. So, new OFV = 6618.18 – 147.73 = 6470.45 8. New decision variable, must re-solve. 9. Removing a non-binding constraint, no change in the optimal solution or OFV. 10. Change of left-hand side coefficients, must resolve. 11. RHS value of fund constraint decreases by 500, allowable decrease is 31500, within the allowable decrease, so the shadow price of 0.094545 is valid. + OFV = (-500)*0.094545 = -47.27. So the new OFV = 6618.18-47.27 = 6570.91 12. New constraint MB+TB < 60%*70,000 Æ MB + TB < 42,000. Plugging in the current optimal solution, you will find this constraint is satisfied, so there is no change in the optimal solution or the OFV.