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QA-presentation - SELE test SELE test GENERAL LINEAR...

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Unformatted text preview: SELE test SELE test GENERAL LINEAR PROGRAMMING NOTATION :3! Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $0.30 per litre for regular gasoline and $0.50 per litre for premium gasoline. Each litre of regular gasoline contains 0.3 litre of grade A crude oil and each litre of premium gasoline contains 0.6 litre of grade A crude oil. For the next production period, Southern has 18,000 litres of grade A crude oil available. The refinery used to produce the gasolines has a production capacity of 50,000 litres for the next production period. Southern Oil’s distributors indicated that demand for the premium gasoline for the next production period will be at most 20,000 litres. Formulate a linear programming model that can be used to determine the number of litres of regular gasoline and the number of litres of premium gasoline that should be produced in order to maximize total profit contribution. 3: What is the optimal solution? What are the values and interpretations of the slack variables? d. What are the binding constraints? Does the following linear programme involve infeasibility, unbounded and/or _ alternative optimal solutions? Explain. Max 4x1 +8X2 S.t. 2X1 +2X2 S10 -1X1 +1X2 Z 8 x1, x2 2 0 Does the following linear programme involve infeasibility, unbounded and/or alternative optimal solutions? Explain. .3.ch Edith-'3? Max 1x1+1x2 SI. 8X1 + 6X2 2 24 4x1 + 6x2 :3 --12 2x2 2 4 X19X220 34 The manager of a small independent grocery store is trying to determine the best use of her shelf space for soft drinks. The store carries national and generic brands and currently has 200 square metres of shelf space available. The manager wants to allocate at least 60% of the space to the national brands and, regardless of the profitability. allocate at least 10% of the space to the generic brands. How many square metres of space should the manager allocate to the national brands and the generic brands if The national brands are more profitable than the generic brands? Both brands are equally profitable? The generic brands are more profitable than the national brands? :35 Discuss what happens to the M&D Chemicals problem (see Section 2.5) if the cost per litre for product A is increased to €3.00 per litre. What would you recommend? Explain. tire For the M&D Chemicals problem in Section 2.5, discuss the effect of management’s requiring total production of 500 litres for the two products. List two or three actions Identify all the extreme points of the feasible region. AN INTRODUCTION TO LINEAR PROGRAMMING b. Suppose RMC discovers a way to increase the profit of its solvent base to €60 per kilo. Does this profit increase change the optimal solution? If so, how? 0. Suppose the profit for the solvent base is €50 per kilo. What is the optimal solution now? Comment on any special characteristics that may exist with this profit for the solvent base. 38 Reconsider the HMO situation in Problem 16. Suppose that management adds the requirements that at least 30 kilos of fuel additive and at least 15 kilos of solvent base must be produced. a. Graph the constraints for this revised RMC problem. What happens to the feasible region? Explain. b. If no feasible solutions can be found, explain what is needed to produce 30 kilos of fuel additive and 15 kilos of solvent base. 39 PharmaPlus operates a chain of 30 pharmacies. The pharmacies are staffed by licensed pharmacists and pharmacy technicians. The company currently employs 85 full-time equivalent pharmacists (combination of full time and part time) and 175 full- time equivalent technicians. Each spring management reviews current staffing levels and makes hiring plans for the year. A recent forecast of the prescription load for the next year shows that at least 250 full-time equivalent employees (pharmacists and technicians) will be required to staff the pharmacies. The personnel department expects 10 pharmacists and 30 technicians to leave over the next year. To accommodate the expected attrition and prepare for future growth, management stated that at least 15 new pharmacists must be hired. In addition, PharmaPlus’s new service quality guidelines specify no more than two technicians per licensed pharmacist. The average salary for licensed pharmacists is €40 per hour and the average salary for technicians is €10 per hour. 3. Determine a minimum-cost staffing plan for PharmaPlus. How many pharmacists and technicians are needed? b. Given current staffing levels and expected attrition, how many new hires (if any) must be made to reach the level recommended in part (a)? What will be the impact on the payroll? 40 Expedition Outfitters manufactures a variety of speciality clothing for hiking, skiing and mountain climbing. They decided to begin production on two new parkas designed for use in extremely cold weather. The names selected for the two models are the Mount Everest Parka and the Fiocky Mountain Parka. Their manufacturing plant has 120 hours of cutting time and 120 hours of sewing time available for producing these two parkas. Each Mount Everest Parka requires 30 minutes of cutting time and 45 minutes of sewing time, and each Flocky Mountain Parka requires 20 minutes of cutting time and 15 minutes of sewing time. The labour and material cost is €150 for each Mount Everest Parka and €50 for each Flocky Mountain Parka, and the retail prices through the firm’s internet catalogue are €250 for the Mount Everest Parka and €200 for the Flocky Mountain Parka. Because management believes that the Mount Everest Parka is a unique coat that will enhance the image of the firm, they have Specified that at least 20% of the total production must consist of this model. Assuming that Expedition Outfitters can sell as many coats of each type as they can produce, how many units of each model should they manufacture to maximize the total profit contribution? 41 English Motors, Ltd. (EML), developed a new all-wheel-drive sports utility vehicle. As part of the marketing campaign, EML developed a video sales presentation to send to both owners of current EML four-wheel-drive vehicles as well as to owners of four- wheel-drive sports utility vehicles offered by competitors; EML refers to these two GENERAL LINEAR PROGRAMMING NOTATION target markets as the current customer market and the new customer market. Individuals who receive the new promotion video will also receive a coupon for a test drive of the new EML model for one weekend. A key factor in the success of the new promotion is the response rate, the percentage of individuals who receive the new promotion and test drive the new model. EML estimates that the response rate for the current customer market is 25% and the response rate for the new customer market is 20%. For the customers who test drive the new model the sales rate is the percentage of individuals who make a purchase. Marketing research studies indicate that the sales rate is 12% for the current customer market and 20% for the new customer market. The cost for each promotion, excluding the test drive costs, are £4 for each promotion sent to the current customer market and £6 for each promotion sent to the new customer market. Management also specified that a minimum of 30,000 current customers should test drive the new model and a minimum of 10,000 new customers should test drive the new model. In addition, the number of current customers who test drive the new vehicle must be at least twice the number of new customers who test drive the new vehicle. If the marketing budget, excluding test drive costs, is £1,200,000, how many promotions should be sent to each group of customers in order to maximize total sales? Creative Sports Design (CSD)manufactures a standard-size tennis racket and an oversize tennis racket. The firm’s rackets are extremely light due to the use of a magnesium-graphite alloy that was invented by the firm’s founder. Each standard-size racket uses 0.125 kilograms of the alloy and each oversize racket uses 0.4 kilograms; over the next two-week production period only 80 kilograms of the alloy are available. Each standard-size racket uses 10 minutes of manufacturing time and each oversize racket uses 12 minutes. The profit contributions are €1 0 for each standard-size racket and €15 for each oversize racket, and 40 hours of manufacturing time are available each week. Management specified that at least 20% of the total production must be the standard-size racket. How many tennis rackets of each type should CSD manufacture over the next two weeks to maximize the total profit contribution? Assume that because of the unique nature of their products, CSD can sell as many rackets as they can produce. riff Management of High Tech Services (HTS)would like to develop a model that will help allocate their technicians” time between service calls to regular contract customers and new customers. A maximum of 80 hours of technician time is available over the two-week planning period. To satisfy cash flow requirements, at least €800 in revenue (per technician) must be generated during the two-week period. Technician time for regular customers generates €25 per hour. However, technician time for new customers only generates an average of €8 per hour because in many cases a new customer contact does not provide billable services. To ensure that new customer contacts are being maintained, the technician time spent on new customer contacts must be at least 60% of the time spent on regular customer contacts. Given these revenue and policy requirements, HTS would like to determine how to allocate technician time between regular customers and new customers so that the total number of customers contacted during the two-week period will be maximized. Technicians require an average of 50 minutes for each regular customer contact and 1 hour for each new customer contact. a. DeveIOp a linear programming model that will enable HTS to allocate technician time between regular customers and new customers. 1.1. Find the optimal solution. Jackson Hole Manufacturing is a small manufacturer of plastic products used in the automotive and computer industries. One of its major contracts is with a large 82 CHAPTER 2 AN INTRODUCTION TO LINEAR PROGRAMMING computer company and involves the production of plastic printer cases for the computer company’s portable printers. The printer cases are produced on two injection molding machines. The M-100 machine has a production capacity of 25 printer cases per hour, and the M200 machine has a production capacity of 40 cases per hour. Both machines use the same chemical material to produce the printer cases; the M-100 uses 40 kilos of the raw material per hour and the M200 uses 50 kilos per hour. The computer company asked Jackson Hole to produce as many of the cases during the upcoming week as possible and has said that it will pay €18 for each case Jackson Hole can deliver. However, next week is a regularly scheduled holiday period for most of Jackson Hole's production employees; during this time, annual maintenance is performed for all equipment in the plant. Because of the downtime for maintenance. the M100 will be available for no more than 15 hours. and the M200 will be available for no more than 10 hours. However, because of the high setup cost involved with both machines, management has a requirement that. if production is scheduled on either machine, the machine must be Operated for at least 5 hours. The supplier of the chemical material used in the production process informed Jackson Hole that a maximum of 1000 kilos of the chemical material will be available for next week’s production; the cost for this raw material is as per kilo. In addition to the raw material cost, Jackson Hole estimates that the hourly cost of operating the M-100 and the M-200 are €50 and €75. respectively. a. Formulate a linear programming model that can be used to maximize the contribution to profit. b. Find the optimal solution. CASE PROBLEM 1 Workload Balancing igital Imaging (DI) produces photo printers for both the professional and consumer markets. The DI consumer division recently introduced two photo printers that provide colour prints rivalling those produced by a professional processing lab. The Dl—910 model can produce a 10 cm x 15 cm borderless print in approximately 37 seconds. The more sephisticated and faster Dl—950 can even pro- duce a 35 cm x 50 cm borderless print. Financial projections show profit contributions of $42 for each Dl-910 and $87 for each Dl-950. The printers are assembled, tested, and pack- aged at Dl’s plant located in Shanghai. This plant is highly automated and uses two manufacturing lines to produce the printers. Line 1 performs the assembly operation with times of 3 minutes per DI- 910 printer and 6 minutes per Dl-950 printer. Line 2 performs both the testing and packaging Operations. Times are 4 minutes per Dl-910 printer and 2 minutes per Dl-950 printer. The shorter time for the Dl—950 printer is a result of its faster print speed. Both manufacturing lines are in operation one 8—hour shift per day. Managerial Report Perform an analysis for Digital Imaging in order to determine how many units of each printer to produce. Prepare a report to Dl’s president presenting your find- ings and recommendations. Include (but do not limit your discussion to) a consideration of the following: 1 The recommended number of units of each printer to produce to maximize the total contribution to profit for an 8-hour shift. What reasons might management have for not implementing your recommendation? 2 Suppose that management also states that the number of Dl-91 0 printers produced must be at least as great as the number of Dl-950 units produced. Assuming that the objective is to maximize the total contribution to profit for an 8-hour shift, how many units of each printer should be produced? 3 Does the solution you developed in part (2) balance the total time Spent on line 1 and the total time spent on line 2? Why might this balance or lack of it be a concern to management? 4 Management requested an expansion of the model in part (2) that would provide a better balance between the total time on line 1 and the total time on line 2. Management wants to limit the difference between the total time on line 1 and the total time on line 2 to 30 minutes or less. If the objective is still to maximize the total contribution to profit, how many units of each printer should be produced? What effect does this workload balancing have on total profit in part (2)? GENERAL LINEAR PROGRAMMING NOTATION 5 Suppose that in part (1) management specified the objective of maximizing the total number of printers produced each shift rather than total profit contribution. With this objective, how many units of each printer should be produced per shift? What effect does this objective have on total profit and workload balancing? For each solution that you develop include a copy of your linear programming model and graphical solu- tion in the appendix to your report. CASE PROBLEM 2 Production Strategy B etter Fitness, Inc. (BFI), manufactures exercise equipment at its plant in Munich, Germany. It recently designed two universal weight machines for the home exercise market. Both machines use BFl—patented technology that provides the user with an extremely wide range of motion capability for each type of exercise performed. Until now, such capabilities have been available only on expensive weight machines used primarily by physical thera— pists. At a recent trade show, demonstrations of the machines resulted in significant dealer interest. In fact, the number of orders that BFI received at the trade show far exceeded its manufacturing capabil- ities for the current production period. As a result, management decided to begin production of the two machines. The two machines, which BFI named the BodyPlus 100 and the BodyPlus 200, require differ- ent amounts of resources to produce. The BodyPlus 100 consists of a frame unit, a press station and a pec-dec station. Each frame produced uses 4 hours of machining and welding time and 2 hours of painting and finishing time. Each press sta- tion requires 2 hours of machining and welding time and 1 hour of painting and finishing time, and each pec—dec station uses 2 hours of machining and weld- ing time and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling, testing and packaging each BodyPlus 100. The raw material costs are €450 for each frame, €300 for each press station and €250 for each pec-dec station; packaging costs are estimated to be €50 per unit. The BodyPlus 200 consists of a frame unit, a press station, a pec—dec station and a leg-press station. Each frame produced uses 5 hours of machining and welding time and 4 hours of painting and finishing time. Each press station requires 3 hours machining and welding time and 2 hours of painting and finishing time, each pec-dec station uses 2 hours of machining and welding time and 2 hours of painting and finishing time, and each leg- press station requires 2 hours of machining and welding time and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling, test- ing and packaging each BodyPlus 200. The raw material costs are €650 for each frame, €400 for each press station, €250 for each pec-dec station and €200 for each leg-press station; packaging costs are estimated to be €75 per unit. For the next production period, management estimates that 600 hours of machining and welding time, 450 hours of painting and finishing time and 140 hours of assembly, testing and packaging time will be available. Current labour costs are €20 per hour for machining and welding time, €15 per hour for painting and finishing time and €12 per hour for assembly, testing and packaging time. The market in which the two machines must compete suggests a retail price of €2400 for the BodyPlus 100 and €3500 for the BodyPlus 200, although some flexibil- ity may be available to BFl because of the unique capabilities of the new machines. Authorized BFl dealers can purchase machines for 70 per cent of the suggested retail price. BFl's president believes that the unique capabil- ities of the BodyPlus 200 can help position BFI as one of the leaders in high-end exercise equipment. Consequently, he has stated that the number of units of the BodyPlus 200 produced must be at least 25 per cent of the total production. 83 ...
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QA-presentation - SELE test SELE test GENERAL LINEAR...

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