Question one is completed I need help with question 2 (project management). I added Question one answers because they may help with Question 2.

Investment Municipal bonds Real estate stocks Pharmaceutical stocks Energy stocks Domestic automobile stocks Projected Rate of Return (%) 5.5 11 8.2 4.8 8.4 Risk Low High Moderate Low Moderate QUESTION 1 Topic: an LP application in Portfolio selection (14 marks) A brokerage firm has been tasked with investing $500,000 for a new client with the following guidelines: At least 22% of the total amount invested should be in municipal bonds The return from moderate-risk investments must be at least 35% of the return from high-risk investment. At least 15% in energy stocks At least 16% in domestic automobile stocks No more than 60% of the total amount invested should be in real estate stock and domestic automobile stock. The client's goal is to maximize the return on investments. The broker has prepared a list of high-quality stocks and bonds and their corresponding rates of return as shown in the table above. 1) Formulate a linear programming model for this problem. Clearly define the variables, objective function, and constraints. [5] Answer: Decision variables: Let M=the amount of money($) invested in Municipal bonds Let R=the amount of money($) invested in Real estate stocks Let P=the amount of money($) invested in Pharmaceutical stocks Let E=the amount of money($) invested in Energy stocks Let D=the amount of money($) invested in Domestic automobile stocks Objective function: Maximize the return on investments: 0.055M+0.11R+0.082P+0.048E+0.084D subject to: 0.78M-0.22R-0.22P-0.22E-0.22D >= 0 (Before simplified: M>=0.22(M+R+P+E+D)) 0.082P+0.084D-0.0385R >= 0 (Before simplified: 0.082P+0.084D >= 0.35*0.11R) 0.85E-0.15M-0.15R-0.15P-0.15D >= 0 (Before simplified: E >=0.15(M+R+P+E+D)) 0.84D-0.16M-0.16R-0.16P-0.16E >=0 (Before simplified: D>= 0.16(M+R+P+E+D)) M+R+P+E+D = 0 2) Solve the problem using Excel or Excel QM and answer the following questions: a) What is the optimal solution to this LP problem? [7] Answer: The optimal solution: M= $110,000 (Municipal bonds), R= $215,755 (Real estate stocks), P= $15,000 (Pharmaceutical stocks), E= $75,000 (Energy stocks), D= $84,245 (Domestic automobile stocks) The maximum profit is $41,689.6 ($41,689.633) b) Would the optimal solution change if the rate of return from Energy stocks investment increases to 5% and all other rates of return are unchanged? Explain. [2] Answer: According to the Sensitivity Report 1, the Allowable Increase of Objective Coefficient for Variable E is 5.14% (0.051404082). If the rate of return from Energy stock increases from 4.8% to 5%, the amount of increase is 0.05-0.048=0.002 (0.2%); therefore, it is within the allowable increase, and there is no change in the optimal solution. In addition, it is confirmed through Excel QM that the optimal solution does not change when we input 5% (0.05) for the Variable E instead of 4.8% (0.048); only the optimal value of the objective function increase slightly ($41,839.6 ) due to the increase. Activity Immediate Predecessor Optimistic (Weeks) 4 2 Most Likely (Weeks) 7 Pessimistic (Weeks) 10 A B A 8 20 C A 8 12 16 D B 1 2 E 6 8 22 D C C F 3 4 2 2 G F 2 2 H F 6 8 10 8 1 J E, G, H I 4 1 12 3 2 QUESTION 2 Topic: Project Management (11 marks) The Laurenster Corporation needs to perform the following activities with times given in weeks. (Please see the table above) 1) Using Excel QM, draw a project network for this problem. (You can draw the network manually and then cut and paste the image into Excel or word document.) [4] 2) Use Excel QM to find the expected project completion time? [3] 3) What is the probability that the project will be completed within 42 weeks? (Note: keep three decimal places for the standard deviation of the project time when you do this question.) [4]