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Unformatted text preview: Chapter 3 Linear Programming: SensitivilyAnalysis and Interpretation of Solution 19. Better Products, Inc., manufactures three products on two machines. In a typical w 40 hours are available on each machine. The profit contribution and production time hours per unit are as follows: Wegwp""2
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Machine 1 timelunit
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.. . B .. .$$o. . E ! r @ & , MpeMe 2 t imdrmit '. 0.5 1:O' . 1 .6' 2.0  u  ired for machine 1; thus, 2 hours of lab& &"s't be scheduled for each hour of machine 1 time. Only one operator is required for machine 2. A maximum of 100 laborhours is available for assignment to the machines during the coming week. Other production requirements are that product 1 cannot account for more than 50% of the units produced and that product 3 must account for at least 20% of the units produced. a. How many units of each product should be produced to maximize the total profit contribution? What is the projected weekly profit associated with your solution? b. How many hours of production time will be scheduled on each machine? c. What is the value of an additional hour of labor? d. Assume that labor capacity can be increased to 120 hours. Would you be interested in using the additional 20 hours available for this resource? Develop the optimal product mix assuming the extra hours are made available. 20. Adirondack Savings Bank (ASB) has $1 million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are 7% for home loans, 12% for personal loans, and 9% for automobile loans. The bank's planning committee has decided that at least 40% of the new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans. a. Formulate a linear programming model that can be used to determine the amount of funds ASB should allocate to each type of loan in order to maximize the total annual return for the new funds. b. How much should be allocated to each type of loan? What is the total annual return? What is the annual percentage return? c. If the interest rate on home loans increased to 9%,would the amount allocated to each type of loan change? Explain. d. Suppose the total amount of new funds available was increased by $10,000. What effect would this have on the total annual return? Explain. e. Assume that ASB has the original $1 million in new funds available and that the planning committee has agreed to relax the requirement that at least 40% of the new funds must be allocated to home loans by 1%. How much would the annual return change? How much would the annual percentage return change? 21. Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows: Rental Class Deluxe
I Business $35 $40 ' ope 1 1 $20
.r,' , $30
>, Type I rooms do not have Internet access and are not available for the Business rental class. 1 Problems Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 130 rentals in the Super Saver class, 60 rentals in the Deluxe class, and 50 rentals in the Business class. Round Tree has 100 Type I rooms and 120 Type Il rooms. a. Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types. Is the demand by any rental class not satisfied? Explain. b. How many reservations can be accommodated in each rental class? c. Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5, should this incentive be offered? d. With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a 'Qpe I or a v p e TI room? Why? e. Could the linear programming model be modified to plan for the allocation of rental demand for the next night? What information would be needed and how would the model change? Industrial Designs has been awarded a contract to design a label for a new wine produced by Lake View Winery. The company estimates that 150 hours will be required to complete the project. The firm's three graphics designers available for assignment to this project are Lisa, a senior designer and team leader; David, a senior designer; and Sarah, a junior designer. Because Lisa has worked on several projects for Lake View Winery, management specified that Lisa must be assigned at least 40% of the total number of hours assigned to the two senior designers. To provide labeldesigning experience for Sarah, Sarah must be assigned at least 15% of the total project time. However, the number of hours assigned to Sarah must not exceed 25% of the total number of hours assigned to the two senior designers. Due to other project commitments, Lisa has a maximum of 50 hours available to work on this project. Hourly wage rates are $30 for Lisa, $25 for David, and $18 for Sarah. a. Formulate a linear program that can be used to determine the number of hours each graphic designer should be assigned to the project in order to minimize total cost. b. How many hours should each graphic designer be assigned to the project? What is the total cost? c. Suppose Lisa could be assigned more than 50 hours. What effect would this have on the optimal solution? Explain. d. If Sarah were not required to work a minimum number of hours on this project, would the optimal solution change? Explain. 23. Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times (in minutes) required on each machine are as follows: . I +Component 1
Shaper 6 Machine * Grinder 4 5 The shaper is availablk far 120 hours, and the grinder is available for 110 hours. No more than 200 units of component 3 can be sold, but up to 1000 units of each of the other components can be sold. I n fact, the company already has orders for 600 units of component 1 Chapter 3 Linear Programming:Sensitivity Arralysis .and Interpretdmof Solution that must be satisfied. The profit coatributions for oomponents 1,2, and 3 are $8, $6, $9, mpxtively. a. Formulate and solve f r the recommended p rolction quantities. o b, What are the objective coefficient ranges for the three components? Intetpret ranges for company management. . c. What are the righthandsidemges? Interpret these 24. National h urance Associates carries an investment portfolio of stocks, bonds, and o investment alternativw. ChrimtIp for new invwtmnr opporNnides. The four relevant financial data are as follows:
. . .. . Annual rate of return ' I~iskmeasure per dollar invested The risk measure indicates the relative uncertainty associated with the stock in terms of its realizing the projected annual retum; 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 returnoninvestmentstrategy, what is the investment portfolio? c. What is the dollar difference between the portfolios in parts (a) and (b)? Why might the company prefer the solution developed in part (a)? 25. Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the Southeast. Because of a large backlog of orders for oak and cherry cabinets, the company decided to contract with three smaller cabinetmakers to do the final finishing operal tion. For the three cabinetmakers, the number of hours required to complete al the oak cabinets, the number of hours required to complete all the cherry cabinets, the number of ' hours available for the final finishing operation, and the cost per hour to perform the work are shown here. Cabinetmaker 1
Hours required to complete all the oak cabinets Hours required to complete all the cherry cabinets Hours available Cost per hour
50 60 40 $36 Cabinetmaker 2
42 48 30 $42 Cabinetmaker 3
30 35 35 $55 Problems 147 For example, Cabinetmaker 1 estimates it will take 50 hours to complete a ll the oak cabinets and 60 hours to complete a ll the cherry cabinets. However, Cabinetmaker 1 only has 40 hours available for the final finishing operation. Thus, Cabinetmaker 1 can only complete 40150 = 0.80 or 80% of the oakcabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1can only complete 40f60 = 0.67 or 67% of the cherry cabinets if it worked only on cherry cabinets. a. Formulate a linear programming model that can be used to determine the percentage of the oak cabinets and the percentage of the cherry cabinets that should be given to each of the three cabinetmakers in order to minimize the total cost of completing both projects. b. Solve the model formulated in part (a). What percentage of the oak cabinets and what percentage of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects? c. If Cabinetmaker 1 has additional hours available, would the optimal solution change? Explain. d. If Cabinetmaker2 has additional hours available, would the optimal solution change? Explain. e. Suppose Cabinetmaker 2 reduced its cost to $38 per hour. What effect would this change have on the optimal solution? Explain. 26. Benson Electronics manufactures three components used to produce cell telephones and other communicationdevices. In a given production period, demand for the three components may exceed Benson's manufachuing capacity. In this case, the company meets demand by purchasing the components from another manufacturer at an increased cost per unit. Benson's manufacturing cost per unit and purchasing cost per unit for the three components are as follows: . ., , Source Manufacture Purchase Component 1
$4.50 $6.50 Component 2
$5.00 $8.80
\I Component 3
$2.75 $7.00 : . . .Lr:ii,!?i&!iT\
A ,\ I { &4ii,
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Component 1
2 1 1.5 Manufacturing times in minutes per unit for Benson's three departments are as follows: Department Praduction Assembly Testing & Packaging Component 2
3 1.5 2 Component 3
4 3 5 For instance, each unit of component 1 that Benson manufactures requires 2 minutes of production time, 1minute of assembly time, and 1.5 minutes of testing and packaging time. For the next production period, Benson has capacities of 360 hours in the production department, 250 hours in the assembly department, and 300 hours in the testing and packaging department. a. Formulate a linear programming model that can be used to determine how many units of each component to manufacture and how many units of each component to purchase. Assume that component demands that must be satisfied are 6000 units for component 1,4000 units for component 2, and 3500 units for component 3. The objective is to minimize the total manufacturing and purchasing costs. Chapter 3 Lineor Programming: Sensitivity Analysis and Interpretation of Solution b. What is the optimal solution? How many units of each component should be man factured and how many units of each component should be purchased? c. Which departments are limiting Benson's manufacturing quantities? Use the du ditional unit. els used primarily by average golfers to extra stiff models used primarily by lowhan previous orders, neither plant has sufficient capacity by itself to fill the new order. The San Diego plant can produce up to a total of 120,000 shafts and the Tampa plant can product up to a total of 180,000shafts. Because of equipment differences at each of the plants and differing labor costs, the perunit production costs vary as shown here: ow GSI should schedule ' ,4#! order to free up additional capacity for the new order. Would this option be worthwhile? Explain. d. Suppose that the cost to produce a stiff shaft in Tampa had been incorrectly computed, and that the correct cost is $5.30 per shaft. What effect, if any, would the correct cost have on the optimal solution developed in part (b)? What effect would it have on total production cost?  28. The Pfeiffer Company manages approximately $15 million for clients. For each client, i Pfeiffer chooses a m x of three investment vehicles: 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 works for Dennis Hartmann, one of Pfeiffer's clients. Based on an evaluation of Hartmann's risk tolerance, Pfei£fer 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% for each, and at least 20% must be 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. . Problems 149 a. Develop a linear programming model to select the best mix of investmentsfor Hartmann's
portfolio. Solve the model you developed in part (a). How much may the yields on the three funds vary before it will be necessary for Pfeiffer 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 the yield estimate for the growth fund downward to 0.10, how would you recommend modifying Hartmann's portfolio? f. What information must Pfeiffer maintain on each client in order to use this system to manage client portfolios? g. On a weekly basis Pfeiffer revises the yield estimates for the three funds. Suppose Pfeiffer has 50 clients. Describe how you would envision Pfeiffer making weekly modifications in each client's portfolio and allocating the total funds managed among the three investment funds. b. c. 29. La Jolla Beverage Products is considering producing a wine cooler that would be a blend of a white wine, a rosC 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% r od, and exactly 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 rosC wine can be purchased; an unlimited amount of fruit juice can be ordered. The costs for the wine are $1. O per gallon O for the white and $1.50 per gallon for the rosC; the fruit juice can be purchased for $0.50 per gallon. La Jolla Beverage Products can sell all of the wine cooler they 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 rosC 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 dual 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 dual price? f. Interpret the dual 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 dual price?
30. The program manager for Channel 10 would like 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 30minute 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 Chapter 3 Linear Programming: SensitiviiyAnalysis and Interpretation of Solution sports segment should be no longer than the total time spent on the local and nati news; and at least 20% of the time should be devoted to the weather segment. The pro tion costs per minute are $300 for local news, $200 for national news, $100 for wea and $100 for sports. a Formulate and solve a linear program that can determine how the 20 available . utes should be used to m n m z the total cost of producing the program. iiie b. Interpret the dual price for the constraint corresponding to the available time. \ advice would you give the station manager given this dual price? c. Interpret the dual price for the constraint correspondingto the requirement that at 15% of the available time should be devoted to local coverage. What advice w you give the station manager given this dual price? d. Interpret the dual price for the constraint corresponding to the requirement tha time devoted to the local and the national news must be at least 50% of the total b~ cast time. What advice would you give the station manager given this dual price e. Interpet the dual price for the constraint correspondingto the requirement that the devoted to the weather segment must be less than or equal to the time devoted c t sports segment. What advice would you give the station manager given this dual p i 31. Gulf Coast Electronics is ready to award contracts for printing their annual report. Fo past several years, the fourcolor annual report has been printed by Johnson Printing Lakeside Litho. A new f m ,Benson Printing, inquired into the possibility of doing a tion of the printing. The quality and service level provided by Lakeside Litho has beel tremely high; in fact, only 0.5% of their reports have had to be discarded because of qu problems. Johnson Printing has also had a high quality level historically, producin average of only 1% unacceptable reports. Because Gulf Coast Electronics has had no e rience with Benson Printing, they estimated their defective rate to be 1 0%. Gulf C would like to determine how many reports should be printed by each firm to obtain 7 5 acceptablequality reports. To ensure that Benson Printing will receive some of the tract, management specified that the number of reports awarded to Benson Printing be at least 10% of the volume given to Johnson Printing. In addition, the total volum signed to Benson Printing, Johnson Printing, and Lakeside Litho should not exceed 30. 50,000,and 50,000 copies, respectively. Because of the longterm relationship Lakeside Litho, management also specified that at least 30,000reports should be awa 2.45 for Benson Printing, $2.50for Johnson F to Lakeside Litho. The cost per copy is $ ing, and $2.75for Lakeside Litho. a. Formulate and solve a linear program for determining how many copies should b signed to each printing fm to minimize the total cost of obtaining 75,000accept; quality reports. b. Suppose that the quality level for Benson Printing is much better than estimated. 7 * effect, if any, would this quality level have? c. Supposethat management is willing to reconsider their requirement that Lakeside I be awarded at least 30,000reports. What effect, if any, would this consideration h 32. PhotoTech, Inc., a manufacturerof rechargeable batteries for digital cameras, signed a tract with a digital photography company to produce t hree different lithiumion ba packs for a new line of digital cameras. The contract calls for the following:
Battery Pack PT200 Production Quantity 200,000 100,000 Case Problem 1 Product Mix 151 PhotoTech can manufacture the battery packs at manufacturing plants located in the Philippines and Mexico. The unit cost of the battery packs differ at the two plants because of differences in production equipment and wage rates. The unit costs for each battery pack at each manufacturing plant are as follows: Plant I
Mexico $0.98 $1.06 $1.15
I Product PT100 PT200 PT300 Philippines $0.95 $0.98 $1.34 &cwrn~k~datim?
The PT100and PT200battery packs are produced using similar production equipment available at both plants. However, each plant has a limited capacity for the total number of PT and PT200 attery packs produced. The combined PT 00 and PT200 roduction 100 b 1 p capacities are 175,000units at the Philippines plant and 160,000units at the Mexico plant. p u The PT300 roduction capacities are 75,000 nits at the Philippines plant and 100,000units at the Mexico plant. The cost of shipping from the Philippines plant is $0.18per unit, and the cost of shipping from the Mexico plant is $0.10per unit. a Develop a linear program that PhotoTech can use to detenninehow many units of each . battery pack to produce at each plant in order to minimize the total production and shipping cost associated with the new contract. b. Solve the linear program developed in part (a) to determine the optimal production plan. C. Use sensitivity analysis to determine how much the production andlor shipping cost i per unit would have to change in order to produce additional units of the PT100n the Philippines plant. d. Use sensitivity analysis to determine how much the production andlor shipping cost per unit would have to change in order to produce additional units of the PT200in the Mexico plant. ...
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This note was uploaded on 03/07/2010 for the course MATH 2310 taught by Professor Shakroh during the Spring '09 term at Langara.
 Spring '09
 shakroh
 Math, Linear Programming

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