## Operation Management-QA68 Online Services

### Linear Programming and Sensitivity Analysis

** Sensitivity Analysis**

- Walling Street Investments, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of P shares of Petroleum Inc. and Q shares of Quality Steel. The annual return for Petroleum Inc is $5 per share and the annual return for Quality Steel is $8 per share. Each Petroleum Inc share sells for $30 per share and Quality Steel sells for $55 per share. The portfolio has $100,000 to be invested. The portfolio risk index (0.65 per share of Petroleum Inc and 0.30 per share for Quality Steel) has a maximum of 800. In addition, the portfolio is limited to a maximum of 800 shares of Petroleum Inc. The linear programming formulation that will maximize the total annual return of the portfolio is as follows

Max 5P + 8Q Maximize total annual return

s.t.

30P + 55Q ≤ 100,000 Funds available

0.65P + 0.30Q ≤ 800 Risk maximum

1P ≤ 800 Petroleum Inc. maximum

P, Q ≥ 0

The computer solution of this problem is shown in the following figure.

- What is the optimal solution, and what is the value of the total annual return?
- Which constraints are binding? What is your interpretation of these constraints in terms of the problem?
- What are the dual values (shadow prices) for the constraints? Interpret each.
- Would it be beneficial to increase the maximum amount invested in Petroleum Inc.? Why or why not?

- Salsa Aguilar, located in Pennsylvania, produces salsas that are sold through Heisler’s Market, a local grocery store located in McMurray, PA. Salsa Aguilar makes two types of products: Traditional Salsa and Spicy Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and onion puree.The Traditional Salsa is a blend of 50% whole tomatoes, 40% tomato sauce, and 10% chopped onions and jalapenos. The Spicy Salsa, has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% chopped onions and jalapenos. Each jar of salsa produced weighs 12 ounces. For the current production period, Salsa Aguilar can purchase up to 200 pounds of whole tomatoes, 150 pounds of tomato sauce, and 85 pounds of chopped onions and jalapenos; the price per pound for these ingredients is $0.95, $0.70, and $0.55, respectively. The cost of the spices and the other ingredients is approximately $0.15 per jar. Salsa Aguilar buys empty glass jars for $0.03 each, and labeling and filling costs are estimated to be $0.02 for each jar of salsa produced. Salsa Aguilar’s contract with Heisler’s Market results in sales revenue of $2.00 for each jar of Traditional Salsa and $2.25 for each jar of Spicy Salsa. Letting

T = jars of Traditional Salsa

S = jars of Spicy Salsa

Leads to the formulation of the linear program as follows. Note that we adjusted the weights of the ingredients to accommodate for the 12 oz jars of salsa and used their prices to find the values below

Max 2T + 2.25S

s.t.

5T + 7S ≤ 2533 Whole tomatoes

4T + 1S ≤ 1400 Tomato Sauce

1T + 2S ≤ 7333 Chopped Onions and jalapenos

T, S ≥ 0

The computer solution is shown in figure below.

- What is the optimal solution, and what are the total returns based on these optimal production quantities?
- Specify the objective function ranges
- What are the dual values (shadow prices) for each constraint? Interpret each.
- Identify each of the right-hand-side ranges.

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- Intel Investments, Inc. manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client’s needs. For a new client, Intel has been authorized to invest up to $1 million in two investment funds: a high-risk stock fund and a low-risk money market fund. Each unit of the stock fund costs $100 and provides an annual rate of return of 15%; each unit of the money market fund costs $150 and provides an annual rate of return of 4%. The client is content with minimizing risk subject to the requirement that the annual income from the investment be at least $80,000. According to Intel’s risk measurement system, each unit invested in the stock fund has a risk index of 10, and each unit invested in the money market fund has a risk index of 4. Intel’s client also specified that at least $250,000 has to be invested in the money market fund. Letting,

S = units purchased in the stock fund

M = units purchased in the money market fund

Leads to the following formulation

Min 10S + 4M

s.t.

100S + 150M ≤ 1,000,000 Funds available

15S + 4M ≥ 80,000 Annual Income

M ≥ 250,000/150 Units in money market

S,M ≥ 0

The computer solution is shown below

- Suppose the risk index for the stock fund (the value of S) increases from its current value of 10 to 12. How does the optimal solution change, if at all?
- Suppose the risk index for the money market fund (the value of M) increases from its current value of 4 to 6. How does the optimal solution change, if at all?
- Suppose S increases to 12 and M increases to 6. How does the optimal solution change, if at all?

**product code: Operation Management-QA68**

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