Posted: June 29th, 2022

J. D. Williams, Inc. is an investment advisory firm that manages more than

Case 2: Investment Strategy
J. D. Williams, Inc. is an investment advisory firm that manages more than $120 million in funds
for its numerous clients. The company uses an asset allocation model that recommends the
portion of each client’s portfolio to be invested in a growth stock fund, an income fund, and a
money market fund. To maintain diversity in each client’s portfolio, the firm places limits on the
percentage of each portfolio that may be invested in each of the three funds. General guidelines
indicate that the amount invested in the growth fund must be between 20 and 40% of the total
portfolio value. Similar percentages for the other two funds stipulate that between 20 and 50%
of the total portfolio value must be in the income fund and that at least 30% of the total portfolio
value must be in the money market fund.
In addition, the company attempts to assess the risk tolerance of each client and adjust the
portfolio to meet the needs of the individual investor. For example, Williams just contracted with
a new client who has $800,000 to invest. Based on an Assessment of the client’s risk tolerance,
Williams assigned a maximum risk index of 0.05 for the client. The firm’s risk indicators show the
risk of the growth fund at 0.10, the income fund at 0.07, and the money market fund at 0.01. An
overall portfolio risk index is computed as a weighted average of the risk rating for the three
funds, where the weights are the fraction of the client’s portfolio invested in each of the funds.
Additionally, Williams is currently forecasting annual yields of 18% for the growth fund, 12.5%
for the income fund, and 7.5% for the money market fund. Based on the information provided,
how should the new client be advised to allocate the $800,000 among the growth, income, and
money market funds? Develop a linear programming model that will provide the maximum yield
for the portfolio. Use your model to develop a managerial report.
Managerial Report
1. Recommend how much of the $800,000 should be invested in each of the three funds.
What is the annual yield you anticipate for the investment recommendation?
2. Assume that the client’s risk index could be increased to 0.055. How much would the yield
increase, and how would the investment recommendation change?
3. Assume that the client expressed some concern about having too much money in the
growth fund. How would the original recommendation change if the amount invested in
the growth fund is not allowed to exceed the amount invested in the income fund?
4. The asset allocation model you developed may be useful in modifying the portfolios for
all of the firm’s clients whenever the anticipated yields for the three funds are periodically
revised. What is your recommendation as to whether use of

this model is possible?

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Sample Answer Guide:

1. Investment Recommendation To maximize the annual yield for the portfolio, we need to determine the amount of money to be invested in each fund. Let x, y, and z be the fractions of the total portfolio value invested in the growth, income, and money market funds, respectively. Then, the objective function to be maximized is:

Maximize: 0.18x + 0.125y + 0.075z

subject to the following constraints:

• x + y + z = 1 (the fractions invested in each fund must add up to 1)
• 0.2 ≤ x ≤ 0.4 (the growth fund must be between 20% and 40% of the total portfolio value)
• 0.2 ≤ y ≤ 0.5 (the income fund must be between 20% and 50% of the total portfolio value)
• z ≥ 0.3 (at least 30% of the total portfolio value must be in the money market fund)
• 0.10x + 0.07y + 0.01z ≤ 0.05 (the overall portfolio risk index must be at most 0.05)

Solving this linear programming problem yields an optimal solution of x = 0.4, y = 0.2, and z = 0.4. Therefore, we recommend investing 40% of the portfolio in the growth fund, 20% in the income fund, and 40% in the money market fund. The anticipated annual yield for this investment recommendation is:

0.18(0.4) + 0.125(0.2) + 0.075(0.4) = 0.123, or 12.3%

2. Increase in Risk Index If the client’s risk index could be increased to 0.055, we need to modify the last constraint in the linear programming problem to be:

0.10x + 0.07y + 0.01z ≤ 0.055

Solving the modified linear programming problem yields an optimal solution of x = 0.4, y = 0.24, and z = 0.36. Therefore, we recommend investing 40% of the portfolio in the growth fund, 24% in the income fund, and 36% in the money market fund. The anticipated annual yield for this investment recommendation is:

0.18(0.4) + 0.125(0.24) + 0.075(0.36) = 0.1302, or 13.02%

The increase in the risk index has led to a slightly higher allocation to the income fund, resulting in a higher anticipated yield.

3. Limit on Growth Fund Investment If the amount invested in the growth fund is not allowed to exceed the amount invested in the income fund, we need to modify the second constraint in the linear programming problem to be:

x ≤ y

Solving the modified linear programming problem yields an optimal solution of x = 0.3333, y = 0.3333, and z = 0.3333. Therefore, we recommend investing 33.33% of the portfolio in the growth fund, 33.33% in the income fund, and 33.33% in the money market fund. The anticipated annual yield for this investment recommendation is:

0.18(0.3333) + 0.125(0.3333) + 0.075(0.3333) = 0.1196, or 11.96%

Limiting the growth fund investment has led to a lower allocation to the growth fund, resulting in a lower

Case 3: Supply Chain Optimization

ABC Company is a consumer goods manufacturer with several plants that produce a range of products, including food, home goods, and personal care items. The company has four distribution centers (DCs) located across the United States. Each DC serves a specific region of the country and is responsible for distributing products to retailers in that region.

Currently, the company is experiencing some supply chain inefficiencies and is looking to optimize its distribution network. The company has identified several key areas where improvements could be made, including inventory management, transportation, and warehouse operations.

Inventory Management: The company wants to reduce the amount of inventory it holds in its warehouses while still maintaining high levels of customer service. Currently, the company’s inventory holding cost is $2.00 per unit per year, and its annual demand for each product is as follows:

Product A: 10,000 units Product B: 20,000 units Product C: 30,000 units Product D: 15,000 units

Transportation: The company wants to minimize transportation costs while ensuring that products are delivered to retailers on time. The transportation costs between each plant and DC are shown in the following table:

	DC 1	DC 2	DC 3	DC 4
Plant 1	$5	$10	$7	$12
Plant 2	$8	$6	$11	$9
Plant 3	$9	$7	$5	$8
Plant 4	$6	$9	$8	$11

Warehouse Operations: The company wants to improve warehouse operations by reducing the amount of time products spend in the warehouse and improving order fulfillment rates. Currently, the average time a product spends in the warehouse is 7 days, and the order fulfillment rate is 90%.

The company wants to develop an optimization model that will help it determine the optimal distribution network that will minimize inventory holding and transportation costs while maintaining high levels of customer service.

Managerial Report:

To optimize the distribution network of ABC Company, we need to determine the optimal amount of inventory to hold at each DC and the optimal transportation plan for delivering products from each plant to each DC. Let xi be the amount of product i (in units) held at DC i, and let xij be the amount of product i (in units) transported from Plant i to DC j. Then, the objective function to be minimized is:

Minimize: 2(10,000×1 + 20,000×2 + 30,000×3 + 15,000×4) + 5×11 + 10×12 + 7×13 + 12×14 + 8×21 + 6×22 + 11×23 + 9×24 + 9×31 + 7×32 + 5×33 + 8×34 + 6×41 + 9×42 + 8×43 + 11×44

subject to the following constraints:

x11 + x12 + x13 + x14 = 10,000 (the total demand for Product A at all DCs must be met) x21 + x22 + x23 + x24 = 20,000 (the total demand for Product B at all DCs must be met) x31 + x32 + x33 + x34 = 30,000 (the total demand for Product C at all DCs must be met) x41 + x42 + x43 + x44 = 15,000 ( ———

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