Decision Analysis: Company Pfeiffer Manages Assessment Answer

Given, the company Pfeiffer manages approximately $15 M for clients. The client can choose between growth stock fund, income fund and money market fund or a mix of the three investments. Each client has an investment objective and tolerance for risk. The company invests in each of the three funds based on the portfolio risk index. The yield of growth stock fund, income fund and money market fund are 20%, 10% and 6% respectively.

For client Hartmann, portfolio risk index = 0.05 and minimum investment in growth stock fund, income fund and money market fund must be 10% , 10% and 20% respectively. The total amount available for investment = 300000.

Let X1, X2 and X3 be the investment in growth stock fund, income fund and money market fund respectively.

Objective: Maximize yield: 20% * X1 + 10% * X2 + 6% * X3

s.t.

Fund Constraint: X1 + X2 + X3 = 300000

Risk Constraint: 0.1* X1 + 0.05* X2 + 0.01* X3 <= 0.05

Growth stock fund constraint: X1/ (X1 + X2 + X3) >= 10 % ,

Income fund and constraint: X2/ (X1 + X2 + X3) >= 10 % ,

Money market fund X3/ (X1 + X2 + X3) >= 20 % ,

X1, X2 and X3 >= 0 (Vanderbei, 2005)

Solving the linear program developed in excel we get,

X1 = 120000

X2 = 30000

X3 = 150000

Thus objective function = 20%* 120000 + 10%* 30000 + 6%* 150000 = 36000

Mr. Hartmann can earn 36000 with his investment of 300000 if the company uses linear program and optimally allocates the fund.

To find the range of yield, the company will not modify portfolio. Using the sensitivity report generated in excel, we get,

Allowance increase for growth fund = 0.4, Allowance decrease for growth fund = 0.05,

Thus range of Yield of growth fund = (0.2 – 0.05 = 0.15) to (0.2 + 0.4 = 0.6)

Similarly, Allowance increase for income fund = 0.022, Allowance decrease for income fund = e^30,

Thus range of Yield of income fund = 0 to 0.122

Allowance increase for monetary fund = 0.14, Allowance decrease for monetary fund = 0.04

Thus range of Yield of monetary fund = 0.02 to 0.2

In this range of yield, the company will not modify portfolio

To find if Hartmann’s portfolio risk increases to 0.06 from 0.05, what will be the increase in the yield. We can use shadow price of portfolio risk. Shadow price of a constraint is defined as the increase in objective function for unit increase in constraint limit. Using the sensitivity report generated in excel, we get, shadow price of portfolio Risk = 466666.67. If the risk of the portfolio increases to 0.06, the additional benefit to the objective function = 0.0 1* 466666.67 = 4666.67

If the growth fund yield decreases to 10%, the objective function will change to

New Objective function : Maximize yield: 10% * X1 + 10% * X2 + 6% * X3

Fund Constraint: X1 + X2 + X3 = 300000

Risk Constraint: 0.1* X1 + 0.05* X2 + 0.01* X3 <= 0.05

Growth stock fund constraint: X1/ (X1 + X2 + X3) >= 10 % ,

Income fund and constraint: X2/ (X1 + X2 + X3) >= 10 % ,

Money market fund X3/ (X1 + X2 + X3) >= 20 % ,

X1, X2 and X3 >= 0 (Vanderbei, 2005)

Thus solving the above linear program in excel, we get

X1 = 48000

X2 = 192000

X3 = 60000

Thus objective function = 10% * 48000+ 10% * 192000 + 6% * 60000 = 27600

To manage the portfolio of each client the company must have the information about the risk index of each client and the amount they have. Using these two input parameters and the yield of the three funds, the portfolio can be designed.

The company can use linear programming to determine the modification each week for all the clients. The company can take weighted average of the total portfolio risk for the 50 clients and based on the risk obtained, allocate funds as solved by the linear programming. After the yield is obtained, the company can provide return to each of the clients based on the amount they invested and their portfolio risk.

2: Given, the company can produces Product 1 and product 2. The profit of the Product 1 and product 2 are 30 and 15 respectively. The products have to pass through three departments A. B, C and the total labor hours available for each department is fixed. Product 1 takes 1, 0.3, 0.2 labor hours in the departments A, B, and C whereas Product 2 takes 0.35, 0.2, 0.5 labor hours the departments A, B, and C. The company wants to find the amount of Product 1 and product 2 that has to be produced to maximize its profit

Let the quantity of Product 1 and product 2 be X1 and X2 respectively.

Thus Objective function: Max 30 X1 + 15 X2

s.t. Labor Constraint:

Department A: X1 + 0.35 X2 <= 100

Department B: 0.3 X1 + 0.2 X2 <= 36

Department C: 0.2 X1 + 0.5 X2 <= 50

X1, X2 >= 0 and are integers (Solow, 2014)

Thus solving the linear program in excel, we get,

X1 = 78, X2 = 62.

Thus, No. of units of Product 1 to be produced = 78,

No. of units of Product 2 to be produced = 62  

Objective function = 30 * 78 + 15 * 62 = 3270

The company wants to find out which department should be scheduled for overtime. This can be found using the shadow price of labor hours in each department. Using the sensitivity report generated in excel, we get, shadow price of Department A = 15, shadow price of Department B = 47 and shadow price of Department C = 0. Thus increasing labor hours of department B by 1 unit will increase objective by 47 while increasing labor hours of department A will increase objective by 15. So we should increase the labor hours of department B. The price the company should be willing to pay for the overtime should be less than $47.

Given overtime cost of the three departments A, B and C is 18, 22.5 and 12 respectively. The maximum overtime that can be scheduled in the three departments A, B and C is 10, 8, 6 hours respectively.

Let Y1, Y2 and Y3 be the overtime labor hours of department A, B and C respectively

Thus the objective function will be

Max 30X1 + 15 X2 – 18 * Y1 - 22.5 * Y2 – 12 * Y3

Labor Constraint:

Department A: X1 + 0.35 X2 <= 100 + Y1

Department B: 0.3 X1 + 0.2 X2 <= 36 + Y2

Department C: 0.2 X1 + 0.5 X2 <= 50 + Y3

Overtime constraint

Y1 <= 10

Y2 <= 6

Y3 <= 8

X1, X2 >= 0 and are integers (Solow, 2014)

Solving it we get

X1 = 87, X2 = 65.

Y1 = 10

Y2 = 4

Y3 = 0

Thus, No. of units of Product 1 to be produced = 87,

No. of units of Product 2 to be produced = 65.

Overtime hours in department A = 10

Overtime hours in department B = 4

Overtime hours in department C = 0

Thus Objective function = 30 * 87 + 15 * 65 -10 * 18 - 22.5 * 4 – 0 = 3315

Increase in profit due to overtime = 3315 – 3270 = 45

3. Given, the two river oil company supplies oil to southern Ohio and has a budget of $600000. The company can purchase super tanker, regular line and econo tanker. Each of the three tankers has different capacity and cost. The monthly operating cost of Super tanker, regular line and econo tanker are 550, 425 and 350 respectively. The company wants to optimize the monthly operating cost.

Let X1, X2 and X3 be the number of super tanker, regular line and econo tanker respectively.

The objective function is to Minimize 550 * X1 + 425 * X2 + 350 * X3

s.t.

Fund Constraint: 67000 * X1 + 55000 * X2 + 46000 * X3 <= 600000

Demand Constraint: 5000 * 15 * X1 + 2500 * 20 * X2 + 1000 * 25 * X3 >= 550000

Driver constraint: (X1 + X2 + X3) <= 15