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programming Linear formulations
LP and Sensitivity Solutions
We return now to a set of problem visited before. For these problems you need now to answer sensitivity analysis problems. These problems are colored and appear in italic fonts. 1. Media Selection. The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year's program. Advertising alternatives include television, radio, and newspaper. Audience estimates, costs, and maximum media usage limitations are as shown.
Constraint
Audience per ad Cost per ad Maxuimum Media usage
TV Radio Newspaper 100,000 18,000 40,000 2,000 $ 300.00 $600 10 20 10
To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized. a. If the promotional budget is limited to $18,200, how many commercial messages should be run on each medium to maximize total audience contact? What is the allocation of the budget among the three media, and what is the total audience reached? b. By how much would audience contact increase if an extra $100 were allocated to the advertising budget? Excel solution 2. Investment and loans. The employee credit union at State University is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue-producing investments together with annual rates of return are as follows.
Type of loan
Automobile loans Furniture loans Other secured loans Signature loans Risk free securities
Ann rate of return
8 10 11 12 9
The credit union will have $2,000,000 available for investment during the coming year. State laws and credit union policies impose the following restrictions on the composition of the loans and investments. 1. Risk-free securities may not exceed 30% of the total funds available for investment. 2. Signature loans may not exceed 10% of the funds invested in all loans (automobile, furniture, and other secured and signature loans). 3. Furniture loans plus other secured loans may not exceed the automobile loans. 4. Other secured loans plus signature loans may not exceed the funds invested in risk-free securities. How should the $2,000,000 be allocated to each of the loan/investment alternatives to maximize total annual return? What is the projected total annual return? Excel Solution
3. Quality assurance. Hilltop Coffee manufactures a coffee product by blending three types of coffee beans. The cost per pound and the available pounds of each bean are as follows.
Bean 1 2 3 Cost per lb Available lbs 0.50 500 0.70 600 0.45 400
Consumer tests with coffee products were used to provide ratings on a 0-to-100 scale, with higher ratings indicating higher quality. Product quality standards for the blended coffee require a consumer rating for aroma to be at least 75 and a consumer rating for taste to be at least 80. The individual ratings of the aroma and taste for coffee made from 100% of each bean are as follows.
Bean 1 2 3 AromaRating TasteRating 75 86 85 88 60 75
Assume that the aroma and taste attributes of the coffee blend will be a weighted average of the attributes of the beans used in the blend. This leads to the following constraint:
so we multiply it by X1+X2+X3 in order to get rid of the denominators and reorganize the constraint in a standard form. See the resulting constraint in the Excel model provided under the solution link below. a. What is the minimum-cost blend that will meet the quality standards and provide 1000 pounds of the blended coffee product? b. b. What is the cost per pound for the coffee blend? c. Determine the aroma and taste ratings for the coffee blend. d. If one additional pound of coffee were to be produced, what would be its expected cost? e. If management decides to increase total production by 200 pounds of coffee, while increasing the amount available of bean 1by 200 pounds, by how much would the total cost change? Excel Solution
X1 X2 X3 (75 + ) (85 + ) (60 75 Of course, this is not a linear constraint, ) . X1+ X2+ X3 X1+ X2+ X3 X1+ X2+ X3
4. Make or Buy. Frandec Company manufactures, assembles, and rebuilds material hand equipment used in warehouses and distribution centers. One product, called a Liftmaster is assembled from four components: a frame, a motor, two supports, and a metal strap. Frandec's production schedule calls for 5000 Liftmasters to be made next month. Frandec purchases the motors from an outside supplier, but the frames, supports, and straps may either be manufactured by the company or purchased from an outside supplier. Manufacturing and purchase costs per unit are shown. Component Frame Support Strap Manufacturing Costs $38.00 11.50 6.50 Purchase cost $51.00 15.00 7.50
Three departments are involved in the production of these components. The time (in min per unit) required to process each component in each department is given, along with available capacity (in hours) for the three departments. Component Cutting Dept Milling Dept Shaping Dept
Frame Support Strap Capacity a.
3.5 1.3 0.8 350
2.2 1.7 420
3.1 2.6 1.7 680
Formulate and solve a linear programming model for this make-or-buy application. How many of each component should be manufactured and how many should be purchased? b. What is the total cost of the manufacturing and purchasing plan? c. How many hours of production time are used in each department? d. How much should Frandec be willing to pay for an additional hour of time in the shaping department? e. Another manufacturer has offered to sell frames to Frandec for $45.00 each. Could Frandec improve its position by pursuing this opportunity? Why or why not? Excel Solution
5. Portfolio selection The Pfeiffer Company manages approximately $15 million for clients. For each client, Pfeiffer chooses a mix of three types of investments: a growth stock fund, an income fund, and a money market fund. Each client has different investment objectives and different tolerances for risk. To accommodate these differences, Pfeiffer places limits on the percentage of each portfolio that may be invested in the three funds and assigns a portfolio risk index to each client. Here's how the system for works Dennis Hartmann, one of Pfeiffer's clients. Based on an evaluation of Hartmann's risk tolerance, Pfeiffer has assigned Hartmann's portfolio a risk index of 0.05. Furthermore, to maintain diversity, the fraction of Hartmann's portfolio invested in the growth and income funds must be at least 10% in each, and at least 20% must be invested in the money market fund. The risk ratings for the growth, income, and money market funds are 0.10, 0.05, and 0.01, respectively. A portfolio risk index is computed as a weighted average of the risk ratings for the three funds where the weights are the fraction of the portfolio invested in each of the funds. Hartmann has given Pfeiffer $300,000 to manage. Pfeiffer is currently forecasting a yield of 20% on the growth fund, 10% on the income fund, and 6% on the money market fund. a. Develop a linear programming model to select the best mix of investments for Hartmann's portfolio. b. Solve the model you developed in part (a). c. How much may the yields on the three funds vary before Pfeiffer has to modify Hartmann's portfolio? d. If Hartmann were more risk tolerant, how much of a yield increase could he expect? For instance, what if his portfolio risk index is increased to 0.06? e. If Pfeiffer revised its yield estimate for the growth fund downward to 0.10, how would you recommend modifying Hartmann's portfolio? f. If Pfeiffer revised its yield estimates for the growth fund downward to 0.17, and for the income fund upward to .11, would you recommend modifying Hartmann's portfolio? To answer this question you need to use the 100% rule. Excel Solution
6. Blending La Jolla Beverage Products is considering producing a wine cooler that would be a blend of a white wine, a rose wine, and fruit juice. To meet taste specifications, the wine cooler must consist of at least 50% white wine, at least 20% and no more than 30% rose, and 20% fruit juice. La Jolla purchases the wine from local wineries and the fruit juice from a processing plant in San Francisco. For the current production period, 10,000 gallons of white wine and 8000 gallons of rose wine can be purchased; there is no limit on the
amount of fruit juice that can be ordered. The costs for the wine are $1.00 per gallon for the white and $1.50 per gallon for the rose; the fruit juice can be purchased for $0.50 per gallon. La Jolla Beverage Products can sell all the wine cooler it can produce for $2.50 per gallon. a. Is the cost of the wine and fruit juice a sunk cost or a relevant cost in this situation? Explain. b. Formulate a linear program to determine the blend of the three ingredients that will maximize the total profit contribution. Solve the linear program to determine the number of gallons of each ingredient La Jolla should purchase and the total profit contribution they will realize from this blend. c. If La Jolla could obtain additional amounts of the white wine, should they do so? If so, how much should they be willing to pay for each additional gallon and how many additional gallons would they want to purchase? d. If La Jolla Beverage Products could obtain additional amounts of the rose wine, should they do so? If so, how much should they be willing to pay for each additional gallon and how many additional gallons would they want to purchase? e. Interpret the shadow price for the constraint corresponding to the requirement that the wine cooler must contain at least 50% white wine. What is your advice to management given this shadow price? f. Interpret the shadow price for the constraint corresponding to the requirement that the wine cooler must contain exactly 20% fruit juice. What is your advice to management given this shadow price? Excel Solution Blending - continued La Jolla is now planning to include a new cooler wine in the production plan. As for the first product, to meet taste specifications, the second wine cooler must consist of at least 50% white wine, at least 20% rose, and the rest fruit juice. The new cooler will be sold for $2.00 per gallon, and we assume (as before), that all the wine produced can be sold. a. Re-formulate the linear programming model. b. If not both products are scheduled to be produced, change the projected selling price of the one that was left out in a way that will make it more profitable. Can you create a solution that includes both products? c. Since management insists on the production of both products, it requires now that the amount of cooler 1 will not exceed by more than 50% the amount of cooler 2 produced. Add this constraint and re-solve the problem. 7. Resource allocation in Broadcasting The program manager for Channel 10 wants to determine the best way to allocate the time for the 11:0011:30 evening news broadcast. Specifically, she would like to determine the number of minutes of broadcast time to devote to local news, national news, weather, and sports. Over the 30-minute broadcast, 10 minutes are set aside for advertising. The station's broadcast policy states that at least 15% of the time available should be devoted to local news coverage; the time devoted to local news or national news must be at least 50% of the total broadcast time; the time devoted to the weather segment must be less than or equal to the time devoted to the sports segment; the time devoted to the sports segment should be no longer than the total time spent on the local and national news; and at least 20% of the time should be devoted to the weather segment. The production costs per minute are $300 for local news, $200 for national news, $100 for weather, and $100 for sports. a. Formulate and solve a linear program that can determine how the 20 available minutes should be used to minimize the total cost of producing the program. b. Interpret the shadow price for the constraint corresponding to the available time. What advice would you give the station manager given this shadow price? c. Interpret the shadow price for the constraint corresponding to the requirement that at least 15% of the available time should be devoted to local coverage. What advice would you give the station manager given this shadow price?
d.
Interpret the shadow price for the constraint corresponding to the requirement that the time devoted to the local and the national news must be at least 50% of the total broadcast time. What advice would you give the station manager given this shadow price?
Solution part a: Solution