# Assignment marks will be adjusted for sloppiness, poor grammar and spelling, as well as for technical errors.

Problem 1 (15 points)The price of oil has been dropping significantly recently. This urges fund managers to adjust their investment portfolios. In particular, many fund managers in Canada used to have a significant portion of their investments in the energy sector, but now they need to look into otherinvestment opportunity. The fund manager of XYZ investment corp. estimates that in the past the rate-of-return for the investment made in the energy sector is about 3%, i.e. 100 dollars of investment can generate 3 dollars of profit, but given the trend of oil price the manager alsopredicts that the rate of return might drop to -5% in the worst-case scenario. The manager read the market reports of the other sectors and learned that in average the past rate-of-return for nonenergy sectors is around 1.5%. Given that investing in other sectors appears to be less risky, the manager predicts that even in the worst-case scenario the rate-of-return for non-energy sectors can still be around 1%. The manager needs to determine how much money she should invest in the energy sector and how much she should invest in the non-energy sectors so that the total profit estimated based on the past rate-of-return can be maximized. However, she cannot invest more than 5 million dollarsand has to make sure that the overall investment cannot lose more than \$25,000 in the worst-case scenario estimated based on the predicted rate-of-return.(a) Formulate algebraically a linear programming model for the above problem. Define the decision variables, objective function, and constraints. (5 points)(b) Draw the feasible region for the linear programming model. (2 points)(c) Find the optimal solution(s) and optimal value of the objective function for the linear programing model. Justify why the solution is optimal. Describe also verbally how the manager should invest. (3 points)

(d) Suppose that the manager can also invest in bonds, which in the past had rate-of-returnaround 0.5%. The rate-of-return for bonds is known to be stable, so the manager predicts thateven in the worst-case scenario the rate-of-return remains the same, i.e. 0.5%.Formulate a linear programming model on a spreadsheet to find out how much the managershould invest in the energy sector, non-energy sectors, and bonds with the objective andrequirements specified above. SOLVE using Excel solver (Provide a printout of thecorresponding “Excel Spreadsheet” and the “Answer Report”). Describe verbally how themanager should invest. (5 points)

Problem 2 (6 points)Consider the following linear programming model:Minimize x+0.5ySubject to2y – x<=1

y<=1.5

x+2y >=22x – y<=8

2x+3y<=12

x >=1x >=0, y>=0(a) Draw the feasible region and objective function for the model. Show the optimal solution andoptimal value of the objective function. Justify why the solution is optimal. (4 points)(b) What other method can you choose to find the optimal solution without drawing the objectivefunction? Considering the structure of the feasible region, which method is better? Justify youranswer. (2 points)

Problem 3 (7 points)Consider the following linear programming model:Maximize 3x + ySubject to2x + y<=14

x – y<=1

x +12y<=108

x >=3, y >=0(a) Draw the feasible region and objective function for the model. Report what you find aboutthe optimal solution(s) and the optimal value of the objective function. Justify your finding.(4 points)(b) Is there any redundant constraint? Which one(s) and why? (1 point)(c) Is there any constraint(s) that can be removed without changing the optimal solution(s) youobtained in (a) and why? (2 points)Problem 4 (6 points)Consider the following linear programming model:Maximize –2x + 0.5ySubject toy – 4x<=1

4x + y>=4y<=6

y>=0x >=0.5(a) Draw the feasible region and objective function for the model. Report what you find aboutthe optimal solution(s) and the optimal value of the objective function. Justify your finding.(4 points)(b) Change the objective function in the LP model to “2x-0.5y”. Report what you find about theoptimal solution(s) and the optimal value. Justify your finding. (2 points)

Problem 5 (6 points)Consider the following linear programming model:Maximize 2x – ySubject tox+5y<=12

x – 2y<=6

x – y>=0x+2y<=8

x+3y<=9

x>=0, y>=0(a) Draw the feasible region for the model, however DO NOT draw the objective function.Without graphing the objective function (i.e. use the corner point method), find the optimalsolution(s) and the optimal value of the objective function. Justify your method and why thesolution(s) you obtain is (are) optimal. (4 points)(b) Add the constraint “x+5y>=12” to the linear programming model. Is the optimal solution thesame as the one in (a)? If yes, justify your answer by highlighting the new feasible region. Ifnot, provide the new optimal solution also by highlighting the new feasible region. In bothcases, justify why the solution is optimal. (2points)