WORKSHEET 2 (LINEAR PROGRAMMING FORMULATIONS)
1. (Product Mix) The Beaver Creek Pottery Company is a small crafts operation run by a Native American tribal council. The company employs skilled artisans to produce clay bowls and mugs with authentic Native American designs and colors. The two primary resources used by the company are special pottery clay and skilled labor. Given these limited resources, the company desires to know how many bowls and mugs to produce each day in order to maximize profit. The two products have the following resource requirements for production and profit per item produced.
Product Bowl Mug
Resource Requirements Labor Clay (hr/unit) (lb/unit) 1 4 2 3
Profit ($/unit) 40 50
There are 40 hours of labor and 120 pounds of clay available each day for production. Define the decision variables, formulate and solve the problem.
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(Product mix) A small company in the office furnishings field produces desks and chairs. There are 3 stages of production: assembly, finishing and inspection. A desk requires 1 man-hour and a chair requires 2 manhours of assembly time. The finishing time needed is 4 man-hours for a desk and 6 man-hours for a chair. Finally a desk requires 8 man-hours and a chair requires 4 man-hours per unit of inspection time. For the coming week, this company has available 10 man-hours in assembly, 36 man-hours in finishing and 40 manhours in inspection. The manager has determined that each desk can be sold for a profit of $400 and each chair for a profit of $500. Define the decision variables and formulate the linear programming model for this problem.
3. (Product mix) Gillian’s Restaurant has an ice-cream counter where it sells two main products, ice cream and frozen yogurt, each in a variety of flavors. The restaurant makes one order for ice cream and yogurt each week, and the store has enough freezer space for 115 gallons of both products. A gallon of frozen yogurt costs $0.75; a gallon of ice cream costs $0.93 and the restaurant budgets $90 each week for these products. The manager estimates that each week the restaurant sells at least twice as much ice cream as frozen yogurt. Profit per gallon of ice cream is $4.15 and profit per gallon of yogurt is $3.60. Formulate the LP model for this problem. 4. (Product mix) The manager of an airlines company is going to buy some planes. There are 3 models of planes. The prices of these are $33,500,000 for model A, $25,000,000 for model B and $17,500,000 for model C. The board of directors have decided to spare $750,000,000 for purchasing the planes. The capacity requirement is enough to permit purchasing as many planes as possible within the budget limit and other limitations. The annual profit figures of Models A, B and C are $2,100,000, $1,500,000 and $1,100.000 respectively. There are enough pilots to run as many as 30 planes. If only model A planes are bought, the maintenance facilities are able to serve at most 40 planes. In terms of maintenance requirements, model B planes are equivalent to 4/3 model A planes and model C planes are equivalent to 5/3 model A planes. Define the decision variables and formulate the problem
5. (Product mix) Shirtstop makes T-shirts with logos and sells them in its chain of retail stores. They contract with two different plants- one in Puerto Rico and one in the Bahamas. The shirts from the plant in Puerto Rico cost $0.46 apiece and 9% of them are defective and can’t be sold. The shirts from the Bahamas cost only $0.35 each but they have an 18% defective rate. Shirtstop needs 3,500 shirts. To retain their relationship with the two plants they want to order at least 1,000 shirts from each. They would also like at least 88% of the shirts they receive to be salable. Formulate LP model. Now suppose Shirtstop decided they wanted to minimize the defective shirts while keeping costs below $2,000. Reformulate the problem with these changes
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6. (Product mix) A travel agency arranges tours to various regions. It rents automobiles, minibuses or buses for this purpose. The passenger capacity of them are 4, 14 and 45 respectively excluding only the drivers. There are 42 tourist guides and only 8 of them have driving licence. In each of the buses there should be 2, in each of the minibuses there should be 1 tourist guide. In each of the automobiles there should be one tourist guide having driving licence. There are 38 drivers (excluding the tourist guides having licence). The tourist guides who have licences are not assigned to buses or minibuses. The demand is estimated to be at most 680 tourists for each tour. The rental cost for buses is $154,000,000, for minibuses is $105,000,000 and for automobiles $33,000,000 each. Define the decision variables and formulate the problem.
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(Product mix) A total of $600,000,000,000 of capital have been procured, $400,000,000,000 of which is foreign capital, to be used in a tourism investment project. There will be 4 (A, B, C, and D) kinds of houses in the holiday village. These houses will be of 80, 90, 110 and 150 m2 respectively. The company have purchased a land of 10,000 m2 at a cost of $110,000,000. 3,000 m2 of the total land will be devoted to roads, green area, etc. The unit construction costs will be $110,000,000/m2; $ 110,000,000/m2, $120,000,000/ m2 and $120,000,000/ m2 respectively. The number of Type D houses will be at most 2 times the total number of Type B and Type C houses. The company has estimated that after the holiday village begins to operate, the following annual revenues and costs will be realized. Units 1 2 3 4
Revenues $6,000,000 $8,000,000 $9,000,000 $13,000,000
Costs $3,000,000 $4,000,000 $5,000,000 $6,000,000
Define the decision variables and formulate the model.
8. (Product mix) Turkish airlines aims to load the cargo plane (flying on the Ankara-Hamburg route) in the best way. There are four sections of the plane: one main cabin, two wing cabins and one bottom cabin. The freight costs of each of the cabins differ according to the properties of cabins. A kg. of load costs $800 in the main cabin, $600 in the wing cabins and $500 in the bottom cabin. The loading capacity of the plane is 50 tons. In order to maintain flight safety, the loads of the wing cabins should be equal to each other. The load of the main cabin should be at least equal to the total of the loads of the wing cabins; and should be at least 2 times of the load of the bottom cabin. The plane should fly fully loaded. Define the decision variables and formulate the problem.
9. (Product mix) The Weinberger Electronics Corporation manufactures four highly technical products that it supplies to aerospace firms that hold NASA contracts. Each of the products must pass through the following departments before they are shipped: wiring, drilling, assembly and inspection. The time requirement in hours for each unit produced and its corresponding profit value are summarized in the following table: Product XJ201 XM897 TR29 BR788
Wiring 0.5 1.5 1.5 1
Department Drilling Assembly 0.3 0.2 1 4 2 1 3 2
Inspection 0.5 1 0.5 0.5
Unit profit 9 12 15 11
The production available in each department each month, and the minimum monthly production requirement to fulfill contracts, are as follows: Capacity (hours) Minimum Prod. Department Product Level Wiring 15,000 XJ201 150 Drilling 17,000 XM897 100 Assembly 26,000 TR29 300 Inspection 12,000 BR788 400
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The production manager has the responsibility of specifying production levels for each product for the coming month.
10. (Product mix) Androgynous Bicycle Company (ABC) has the hottest new products on the upscale toy market – boys’ and girls’ bikes in bright fashion colors with oversized hubs and axles; shell design safety tires; a strong padded frame; chrome-plated chains, brackets and valves; and a nonslip handlebar. Due to the seller’s market for high-quality toys for the newest baby boomers, ABC can sell all the bicycles it manufactures at the following prices: Boys’ bikes- $220, girls’ bikes- $175 per bike. . The firm’s accountant has determined the direct labor costs will be 45% of the price ABC receives for the boys’ model and 40% of the price received for the girls’ model. Production costs other than labor, but excluding painting and packaging are $44 per boys’ bicycle and $30 per girls’ bicycle. Painting and packaging are $20 per bike, regardless of the model. The Orlando plant’s overall production capacity is 390 bicycles per day. Each boy’s bike requires 2.5 labor hours to complete and each girl’s model 2.4 hours. ABC currently employs 120 workers, who each put in an 8-hour day. The firm has no desire to hire or fire to affect labor availability, for it believes its stable workforce is one of its biggest assets. Using a graphical approach, determine the best product mix for ABC.
11. (Fertilizer blending) The Sweet Smell Fertilizer Company markets bags of manure labeled “not less than 60 pounds dry weight”. The packaged manure is a combination of compost and sewage wastes. To provide goodquality fertilizer, each bag should contain at least 30 pounds of compost but no more than 40 pounds of sewage. Each pound of compost costs Sweet Smell 5 cents and each each pound of sewage costs 4 cents.
12. *(Feed mix) A livestock supplement is to be mixed to contain exactly 25 pounds of vitamin A, at least 15 pounds of B and at least 40 pounds of C. The supplement is made from two commercial feeds. Each pound of feed 1 contains 2 ounces of A, 6 ounces of B, 8 ounces of C and costs $5. A pound of feed 2 contains 4 ounces of A, 1 ounce of B, 3 ounces of C and costs $3. Define the decision variables and formulate the problem.
13. (Drug mix)The Elixer Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics in different proportions. One gram of ingredient 1 contributes 3 units, and ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required, and the ingredients each contribute 1 unit per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is $80 and the cost for a gram of ingredient 2 is $50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at minimum cost
14. (Portfolio selection) An investor has to allocate $100,000 among four investment opportunities. Two common stock funds- one high risk and one low risk and a corporate bond fund are available. The respective expected annual rates of return from these three funds are 20%, 15% and 12%. Any money not invested in these funds will be deposited in fixed interest securities, yielding a 10% annual rate of return. The investor wants to invest no more than $30,000 in the high-risk common stock fund and also requires that the total amount invested in the two common stock funds not exceed that invested in the corporate bond fund by more than $20,000. Define the decision variables and formulate the problem
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(Media selection) The advertising agency promoting the new Breem dishwashing detergents wants to get the best exposure possible for the product within the $ 100,000 advertising budget ceiling placed upon it. To do so, the agency needs to decide how much of the budget to spend on each of its two most effective media: (1) television spots during the afternoon hours and (2) large ads in the city’s Sunday Newspaper. Each television spot costs $3,000; each Sunday newspaper ad cost $1,250. The expected exposure, based on industry ratings, is 35,000 viewers for each television commercial and 20,000 readers for each newspaper advertisement. The agency director, Mavis Early, knows from experience that it is important to use both media in order to reach the broadest spectrum of potential Breem customers. She decides that at least 5 but no more than 25 television spots should be ordered and that at least 10 newspaper ads should be contracted. How many times should each of the two media be used to obtain maximum exposure while staying within the budget.
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16. (Advertising mix) A sports club intends to advertise an introductory offer. It faces the problem, however, that many people take up such offers and then immediately discontinue membership, leading to promotional losses. Advertisements can be placed both in newspapers and in magazines. The club has calculated that each thousand dollars spent on newspaper advertisements generates 280 customers who take up the offer and continue membership, and 200 who take up the offer but immediately discontinue membership. Each thousand dollars spent on magazine advertisements produces 300 customers taking up the offer and subsequently retaining membership, and 250 customers taking up the offer and immediately discontinuing membership. The club has a maximum of $50,000 to spend on advertising and feels that it wants no more than 12,000 responses from people who will accept the introductory offer but immediately discontinue membership. Define the decision variables and formulate the problem.
17. (Course planning) The dean of the Western College of Business must plan the school’s course offerings for the fall semester. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught costs the college an average of $2,500 in faculty wages, and each graduate course costs $ 3,000.
18. (Workforce scheduling) A large travel agency employs telephone operators who work 8-hour shifts, either from 8:00 a.m. to 4:00 p.m., or from 2:00 p.m. to 10:00 p.m. Those working the earlier shift are paid $40 per day, while those on the later shift are paid $45 per day. The manager of this agency has determined that the minimum numbers of operators that must be available at various times of the day are: Time 8 a.m. – 10 a.m 10 a.m.- 2 p.m. 2 p.m.- 4 p.m. 4 p.m.- 8 p.m. 8 p.m. – 10 p.m.
Minimum Number of Operators 3 4 12 5 2
Define the decision variables and formulate the problem if the objective is to meet these requirements at the lowest possible cost.
19. (Determination of work force size) Universal Claims Processors processes insurance claims for large national insurance companies. Most claim processing is done by a large pool of computer operators, some of whom are permanent and some temporary. A permanent operator can process 16 claims per day, whereas a temporary operator can process 12 per day, and on average the company processes at least 450 claims each day. The company has 40 computer workstations. A permanent operator will generate about 0.5 claims with errors each day, whereas a temporary operator averages about 1.4 defective claims per day. The company wants to limit claims with errors to 25 per day. A permanent operator is paid $64 per day and a temporary operator is paid $42 per day. The company wants to determine the number of permanent and temporary operators to hire. Develop the LP model.
20. (Production planning) The Copperfield Mining Company owns two mines, each of which produces three grades of ore- high, medium, and low. The company has a contract to supply a smelting company with at least 12 tons of high-grade ore, 8 tons of medium-grade ore, and 24 tons of low-grade ore. Each mine produces a certain amount of each type of ore during each hour that it operates. Mine 1 produces 6 tons of high-grade ore, 2 tons of medium-grade ore, and 4 tons of low-grade ore per hour. Mine 2 produces 2, 2, and 12 tons, respectively of high-, medium-, and low-grade ore per hour. It costs Copperfield $200 per hour to mine each ton of ore from mine 1 and $160 per hour to mine a ton of ore from mine 2. The company wants to determine the number of hours it needs to operate each mine so that its contractual obligations can be met at the lowest cost. Formulate the LP problem.
21. (Transportation) A manufacturer of television sets has two plants and two distribution centers. For the coming week, distribution center A requires 300 sets, and center B 250 sets. At most, 275 sets will be available at plant 1 and at most 325 sets will be available at plant 2. Television sets can be shipped from either plant to either distribution center. However, unit shipping costs differ along the four routes. It costs $10 per set for shipments from plant 1 to center A, $12 per set for shipments from plant 1 to center B, $14 per set for
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shipments from plant 2 to center A and $11 per set for shipments from plant 2 to center B. Demand at the two centers is to be fully met. Define the decision variables and formulate the problem
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