Linear Programming, Sensitivity Analysis

Case Problem..1 Product Mix

Par, Inc., is a small manufacturer of golf equipment and supplies whose management has decided to move into the market for medium and high priced golf bags. Par’s distributor is enthusiastic about the new product line and has agreed to buy all the golf bags Par produces over the next three months. After thorough investigation of the steps involved in manufacturing a golf bag, management determined that each golf bag produced will require the following operations: 1. Cutting and dyeing the material 2. Sewing 3. Finishing (inserting umbrella holder, club separators, etc.) 4. Inspection and packaging The director of manufacturing analyzed each of the operations and concluded that if the company produces a medium priced standard model, each bag will require 7/10 hour in cutting and dyeing department, ½ hour in the sewing department, 1 hour in finishing department and 1/10 hour in the inspection and packaging department. The more expensive deluxe model will require 1 hour for cutting and dyeing, 5/6 hour for sewing, 2/3 hour for finishing and ¼ hour for inspection and packaging.

Par’s production is constrained by a limited number of hours available in each department. After studying departmental workload projections, the director of manufacturing estimates that 630 hours for cutting and dyeing, 600 hours for sewing, 708 hours for finishing and 135 hours for inspection and packaging will be available for the production of golf bags during the next three months. The accounting department analyzed the production data, assigned all relevant variable costs and arrived at prices for both bags that will result in a profit contribution of $10 for every standard bag and $9 for every deluxe bag produced. Determine the number of bags of each type should be produced in order to maximize total profit contribution.

Suppose the management is also considering producing a light weight model designed specifically for golfers who prefer to carry their bags. The design department estimates that each new lightweight model will require 0.8 hours for cutting and dyeing, 1 hour for sewing, 1 hour for finishing and 0.25 hours for inspection and packaging. Because of the unique capabilities designed into the new model, Par’s management feels they will realize a profit contribution of $12.85 for each lightweight model produced during the current production period. What is the effect of the new product? Linear Programming Minimization Problems

1. Ashok Chemicals Company manufactures two chemicals A and B which are sold to the manufacturers of soaps and detergents. On the basis of the next month’s demand, the management has decided that the total production for chemicals A and B should be at least 350 kilograms. Moreover, a major customer’s order for 125 kilogram of product A must also be supplied. Product A requires 2 hours of processing time per kilogram and product B requires one hour of processing time per kilogram. For the coming month, 600 hours of processing time are available.

The company wants to meet the above requirements at minimum total production cost. The production costs are Rs.2 per kilogram for product A and Rs.3 per kilogram for product B. Ashok Chemicals Company wants to determine its optimal product mix and the total minimum cost relevant thereto.

2. Bluegrass Farms, located in Lexington, Kentucky has been experimenting with a special diet for its race horses. The feed components available for the diet are a standard horse feed product, a vitamin enriched oat product and a new vitamin and mineral feed additive. The nutritional values in units per pound of the standard feed components are summarized below. For example, each pound of the standard feed component contains 0.8 unit of ingredient A, 1 unit of ingredient B and 0.1 unit of ingredient C. The minimum daily diet requirements for each horse are three units of ingredient A, six units of ingredient B and four units of ingredient C. In addition, to control the weight of the horses, the total daily feed for a horse should not exceed 6 pounds. Bluegrass Farms would like to determine the minimum-cost mix that will satisfy the daily diet requirements.

Feed Component Standard Enriched Additive Oat Ingredient A 0.8 0.2 0.0 Ingredient B 1.0 1.5 3.0 Ingredient C 0.1 0.6 2.0 Cost per pound 0.25 0.50 3.00 ($)

3. National Insurance Associates carries an investment portfolio of stocks, bonds and other investment alternatives. Currently $200,000 of funds are available and must be considered for new investment opportunities. The four stock options National is considering and the relevant financial data are as follows: Stock

A B C D Price per share ($) 100 50 80 40 Annual rate of return 0.12 0.08 0.06 0.10 Risk measure per dollar invested 0.10 0.07 0.05 0.08

The risk measure indicates the relative uncertainty associated with the stock in terms of its realizing the projected annual return; higher values indicate greater risk. The risk measures are provided by the firm’s top financial advisor. National’s top management has stipulated the following investment guidelines: the annual rate of return for the portfolio must be at least 9% and no one stock can account for more than 50% of the total dollar investment.

a) Use linear programming to develop an investment portfolio that minimizes risk. b) If the firm ignores risk and uses a maximum return-on-investment strategy, what is the investment portfolio? c) What is the dollar difference between the portfolio in parts (a) and (b)? Why might the company prefer the solution in part (a)?

Case Problem.. 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% to 40% of the total portfolio value. Similar percentages for the other two funds stipulate that between 20% to 50% of the total portfolio must be in the income fund and 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 evaluation 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, William’s is currently forecasting annual yields of 18% for the growth fund, 12.5% for the income fund and 7.5% fir the money market fund. Based on the information provided, how should the new client be advised to allocate $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: a. 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 change? b. Assume that the client’s risk index could be increased to 0.55. How much would the yield increase and how would the investment recommendation change? c. Refer again to the original situation where the client’s risk index was assessed to be 0.05. How would your investment recommendation change if the annual yield for the growth fund were revised downward to 16% or even to 14%? d. 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? e. The asset allocation model you developed may be useful in modifying the portfolios for all 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? Case Workforce Assignment:

McCormick Manufacturing Company produces two products with contributions to profit per unit of $10 and $9, respectively. The labour requirements per unit produced and the total hours of labour available from personnel assigned to each of four departments are shown in the table. Assuming that the number of hours available in each department is fixed, find the optimal solution. Labour-Hours per Unit

Department Product 1 Product 2 Total Hours Available 1 0.65 0.95 6500 2 0.45 0.85 6000 3 1.00 0.70 7000 4 0.15 0.30 1400

Suppose that McCormick has a cross-training program that enables some employees to be transferred between departments. By taking advantage of the cross-training skills, a limited number of employees and labour hours may be transferred as shown in the table below.

Cross-Training Transfers Permitted to Department From Maximum Hours Department 1 2 3 4 Transferable 1 …. yes yes …. 400 2 …. …. yes yes 800 3 …. …. …. yes 100 4 yes yes …. …. 200

Row 1 of this table shows that some employees assigned to department 1 have cross-training skills that permit them to be transferred to department 2 or 3. The right hand column shows that, for the current production planning period, a maximum of 400 hours can be transferred from department 1. Similar cross-training transfer capabilities and capacities are shown for departments 2, 3, 4.

If the production manager has the flexibility to assign personnel to different departments, reduced workforce idle time, improved workforce utilization, how will the solution improve?