PAZ351 MATHEMATICAL MODELING FOR BUSINESS APPLICATIONS LINEAR PROGRAMMING EXAMPLES. March 27, 2007

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1 PAZ351 MATHEMATICAL MODELING FOR BUSINESS APPLICATIONS LINEAR PROGRAMMING EXAMPLES March 27, (Production Mix) A clothier makes coats and slacks. The two resources required are wool cloth and labor. The clothier has 150 square yards of wool and 200 hours of labor available. Each coat requires 3 square yards of wool and 10 hours of labor, whereas each pair of slacks requires 5 square yards of wool and 4 hours of labor. The profit for a coat is $50, and the profit for slacks is $40. The clothier wants to determine the number of coats and pairs of slacks to make so that profit will be maximized. (a) Formulate a linear programming model for this problem. (b) Solve this model by using graphical analysis.[tay07] 2. (Production Mix) A jewelry store makes necklaces and bracelets from gold and platinum. The store has 18 ounces of gold and 20 ounces of platinum. Each necklace requires 3 ounces of gold and 2 ounces of platinum, whereas each bracelet requires 2 ounces of gold and 4 ounces of platinum. The demand for bracelet is no more than four. A necklace earns $300 in profit and a bracelet, $400. The store wants to determine the number of necklaces and bracelets to make in order to maximize profit. (a) Formulate a linear programming model for this problem. (b) Solve this model by using graphical analysis.[tay07] 3. (Blending) My diet requires that all the food I eat come from the one of the four basic food groups (chocolate cake, ice cream, soda and cheesecake). At present, the following four foods are available for consumption: brownies, chocolate ice cream, cola and pineapple cheesecake. Each brownie costs 50, each scoop of chocolate ice cream costs 20, each bottle of cola costs 30, and each piece of pineapple cheesecake costs 80. Each day, I must ingest at least 500 calories, 6 oz of chocolate, and 10 oz of sugar and 8 oz of fat. The nutritional content per unit of each food is shown in Table1. Formulate a linear programming model that can be used to satisfy my daily nutritional requirements at minimum cost[win04]. 4. (Production Mix) Giapetto s Woodcarving, Inc., manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 1

2 Calories Chocolate Sugar Fat (ounces) (ounces) (ounces) Brownie Chocolate Ice cream (1 Scoop) Cola (1 Bottle) Pineapple Cheesecake (1 Piece) Table 1: Diet Problem worth of raw materials. Each soldier that is manufactured increases Giapettos variable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapettos variable labor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing and 1 hour of carpentry labor. A train requires 1 hour of finishing and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw material but only 100 finishing hours and 80 carpentry hours. Demand for train is unlimited, but at most 40 soldiers are bought each week. Giapetto wants to maximize weekly profit (revenues costs). Formulate a mathematical model of Giapettos situation that can be used to maximize Giapettos weekly profit[win04]. Soldier Train Capacity Price $27 $21 - Raw Materials $10 $9 - Labor Cost $14 $10 - Carpentry 1 hour 1 hour 80 hours Finishing 2 hours 1 hour 100 hours Demand 40 Unlimited - Table 2: Giapetto s Woodcarving, Inc. 5. (Production Mix) A knitting machine can produce 1, 000 pants or 3, 000 shirts (or a combination of the two) each day. The finishing department can handle either 1, 500 pants or 2, 000 shirts (or a combination of the two) each day. The marketing department requires that at least 400 pants be produced each day. The company s stated objective is profit maximization[tm93]. (a) If the profit from a pair of pants is $4 and that derived from a shirt is $1.50, how many of each type should be produced? Solve graphically. (b) If the profit from a shirt is $2, what should be the minimum profit derived from selling a pair of pants that will justify production of pants? (c) Examine the solutions to (a) and (b) and interpret the difference between them. 2

3 6. (Employee Assignment) A store sells men s and ladies tennis shoes. It makes a profit of $1 a pair on the men s shoes and $1.20 a pair on ladies. It takes two minutes of a salesperson s time and two minutes of a cashier s time to sell a pair of men s shoes. It takes three minutes of a salesperson s time and one minute of a cashier s time per pair of women shoes. The store is open eight hours per day, during which time there are two salespersons and one cashier on duty. How much of the salespersons and the cashiers time should be allocated to the men s and ladies shoes should the store sell in order to maximize profit each day? What is the profit[tm93]? (a) Formulate as a linear programming problem. (b) Solve graphically 7. (Production Mix) The Flair Furniture Company produces inexpensive tables and chairs. The production process for each is similar in that both require a certain number of hours of carpentry work and a certain number of labor hours in the painting and vanishing department. Each table takes 4 hours of carpentry and 2 hours in the painting and varnishing shop. Each chair requires 3 hours of carpentry and 1 hour in the painting and varnishing. During the current production period, 240 hours of carpentry and 100 hours in the painting and varnishing time are available. Each table sold yields a profit of $7; each chair produced is sold for a $5 profit. Flair Furnitures problem is to determine the best possible combination of tables and chairs to manufacture in order to reach the maximum profit. (a) Formulate this production mix situation as an LP problem[rsh06]. 8. (Media Selection) Dorian Auto manufactures luxury cars and trucks. The company believes that its most likely customers are high-income women and men. To reach these groups, Dorian Auto has embarked on an ambitious TV advertising campaign and has decided to purchase 1-minute commercial spots on two type of programs: comedy shows and football games. Each comedy commercial is seen by 7 million high-income women and 2 million high-income men. Each football commercial is seen by 2 million high-income women and 12 million high-income men. A 1-minute comedy ad costs $50,000, and a 1-minute football ad costs $100,000. Dorian would like the commercials to be seen by at least 28 million highincome women and 24 million high-income men. Use linear programming to determine how Dorian Auto can meet its advertising requirements at minimum cost[win04]. 9. (Production Mix) An auto company manufactures cars and trucks. Each vehicle must be processed in the paint shop and body assembly shop. If the paint shop were only painting the trucks, then 40 per day could be painted. If the paint shop were only painting the cars, then 60 per day could be painted. If the body shop were only producing cars, it could process 50 per day. If the body shop were only producing trucks, it could 3

4 process 50 per day. Each truck contributes $300 to profit, and each car contributes $200 to profit. (a) Use linear programming to determine a daily production schedule that will maximize the companys profits. (b) Suppose that auto dealers require that the auto company produce at least 30 trucks and 20 cars. Find the optimal solution to the new LP[Win04]. 10. (Production Mix) Reddy Mikks produces both interior and exterior paints from two raw materials, M1 and M2. Table3 provides the basic data of the problem A market surveys restricts the maximum daily demand of interior Tons of raw material per ton of Maximum daily Exterior Paint Interior Paint availability (tons) Raw material, M Raw material, M Profit per ton($1000) 5 4 Table 3: Reddy Mikks paint to 2 tons. Additionally, the daily demand for interior demand cannot exceed that of exterior paint by more than one ton. Reddy Mikks wants to determine the optimum (best) product mix of interior and exterior paints that maximizes the total daily profit[tah97]. 11. (Blending) The Holiday Meal Turkey Ranch is considering buying two different brands of turkey feed and blending them to provide a good, lowcost diet for its turkeys. Each feed contains, in varying proportions, some or all of the three nutritional ingredients essential for fattening turkeys. Each pound of Brand 1 purchased contains 5 ounces of ingredients A, 4 ounces of ingredient B, and 1/2 ounce of ingredient C. Each pound of Brand 2 contains 10 ounces of ingredient A, 3 ounces of ingredient C. The Brand 1 feed costs the ranch 2 cents a pound, while the Brand 2 feed costs 3 cents a pound. Table4 shows the minimum monthly intake requirement for each nutritional ingredient. The owner of the ranch would like to use LP to determine the lowest- cost diet that meets the minimum monthly intake requirement for each nutritional ingredient[rsh06]. Ingredient A B C Minimum monthly requirement per turkey 90oz. 48oz. 1.5oz. Table 4: Holiday Meal Turkey Ranch 4

5 12. (Labor Scheduling) A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given in Table5. Union rules state that each full-time employee must work five consecutive days and then receive two days off. For example, an employee who works Monday to Friday must be off on Saturday and Sunday. The post office wants to meet its daily requirements using only full-time employees. Formulate an LP that the post office can use to minimize the number of full-time employees that must be hired[win04]. Day Number of Full-time Employees Required Day 1 = Monday 17 Day 2 = Tuesday 13 Day 3 = Wednesday 15 Day 4 = Thursday 19 Day 5 = Friday 14 Day 6 = Saturday 16 Day 7 = Sunday 11 Table 5: Post Office 13. (Production Mix) The Dakota Furniture Company manufactures desks, tables and chairs. The manufacture of each type of furniture requires lumber and two types of skilled labor: finishing and carpentry. The amount of each resource needed to make each type of furniture is given in Table6. Currently, 48 board feet of lumber, 20 finishing hours and 8 carpentry Resource Desk Table Chair Lumber (board ft) Finishing hours Carpentry hours Table 6: Dakota Furniture Company hours are available. A desk sells for $60, a table for $30, and a chair for $20. Dakota believes that demand for desks and chairs is unlimited, but at most 5 tables can be sold. Because the available resources have already been purchased, Dakota wants to maximize total revenue. 14. (Production) A baker has 30 oz of flour and 5 packages of yeast. Baking a loaf of bread requires 5 oz of flour and 1 package of yeast. Each loaf of bread can be sold for 30. The baker may purchase additional flour at 4 /oz or sell leftover flour at the same price. Formulate and solve an LP to help the baker maximize profits (revenues-costs). 15. (Financial) Semicond is a small electronics company that manufactures tape recorders and radios. The per-unit labor costs, raw material costs, 5

6 Tape Recorder Radio Selling Price $100 $90 Labor Cost $50 $35 Raw Material Cost $30 $40 Table 7: Semicond Price-Costs and selling price of each product are given in Table7. On December 1, 1998, Semicond has a available raw material that is sufficient to manufacture 100 tape recorders and 100 radios. On the same date, the companys balance sheet is as shown in Table8, and Semiconds asset / liability ratio (called the current ratio) is 20000/10000 = 2. Semicond must determine how many tape recorders and radios should be produced during December. Demand is large enough to ensure that all goods produced will be Assets Liabilities Cash $10,000 Accounts receivable 1 $3,000 Inventory Outstanding 2 $7,000 Bank Loan $10,000 Table 8: Semicond Price-Costs sold. All sales are on credit, however, and payment for goods produced in December will not be received until February 1, During December, Semicond will collect $2, 000 in accounts receivable, and Semicond must pay off $1, 000 of the outstanding loan and a monthly rent of $1, 000. On January 1, 1999, Semicond will receive a shipment of raw material worth $2, 000, which will be paid for the January 1, Semiconds management has decided that the cash balance on January 1, 1999 must be at least $4, 000. also, Semiconds bank requires that the current ratio at the beginning of the January be at least 2. To maximize the contribution to profit from December production, (revenues to be received) (variable production costs), what should Semicond produce during December? 16. (Media Selection) The Win Big Gambling Club promotes gambling junkets from a large Midwestern city to casinos in Bahamas. The club has budgeted up to $8, 000 per week for local advertising. The money is to be allocated among four promotional media: TV spots, newspaper ads, and two types of radio advertisements. Win Big s goal is to reach the largest possible high-potential audience through various media. Table9 presents the number of potential gamblers reached by making use of advertisement in each of four media. It also provides the cost per advertisement placed and the maximum number of ads that can be purchased per week. Win Big s contractual arrangements require that at least five radio spots be placed each week. To ensure a broad-scoped promotional campaign, management also insists that no more than $1, 800 can be spent on radio 6

7 Audience Cost Maximum Medium Reached per ad ad($) ads per week TV Spot(1 minute) 5, Daily Newspaper(full-page ad) 8, Radio Spot(30 seconds, prime time) 2, Radio Spot(1 minute, afternoon) 2, Table 9: Win Big Gambling Club advertising each week. Formulate this media selection problem by linear programming [RSH06]. 17. (Marketing Research) Management Sciences Associates (MSA) is a marketing and computer research firm based in Washington D.C., that handles consumer surveys. One of its clients is a national press service that periodically conducts political polls on issues of several requirements in order to draw statistically valid conclusions on the sensitive issue of new U.S. immigration laws: (a) Survey at least 2, 300 U.S. households in total. (b) Survey at least 1, 000 households whose heads are 30 years of age or younger. (c) Survey at least 600 households whose heads are between 31 and 50 years of age. (d) Ensure that at least 15% of those surveyed live in state that borders on Mexico. (e) Ensure that no more than 20% of those surveyed who are 51 years of age or over live in a state that borders on Mexico. MSA decides that all surveys should be conducted in person. It estimates that the costs of reaching people in each age and region category are as in Table10. MSA s goal is to meet the five sampling requirements at the Cost per Person Surveyed ($) Region Age 30 Age Age 51 State bordering Mexico $7.50 $6.80 $5.50 State not bordering Mexico $6.90 $7.25 $6.10 Table 10: Management Sciences Associates least possible cost. Formulate an LP to solve MSA s problem [RSH06]. 18. (Pricing and Marketing Strategy) The I. Kruger Paint and Wallpaper Store is a large retail distributor of the Supertrex brand of vinyl wallcoverings. Kruger will enhance its citywide image in Miami if it can outsell other retail stores in total number of rolls of Supertrex next year. It is able to estimate the demand function as follows: 7

8 Number of rolls of Supertrex sold = 20 x dollars spent on advertising x dollars spent on in-store displays + 12 x dollars invested in on-hand wallpaper inventory - 65,000 x percentage markup taken above wholesale cost of a roll The store budgets a total of $17, 000 for advertising, in-store displays, and on-hand inventory of Supertrex for next year. It decides it must spend at least $3, 000 on advertising; in addition, at least 54% of the amount invested in on-hand inventory should be devoted to displays. Markups on Supertrex seen at other local stores range from 20% to 45%. Kruger decides that its markup had the best be in this range as well. (a) Formulate as an LP problem. (b) Solve the problem. (c) What is the difficulty with the answer? (d) What constraint would you add[tay07]? 19. (Employee Scheduling) Hong Kong Bank of Commerce and Industry is a busy bank that has requirements for between 10 and 18 tellers, depending on the time of the day. The lunch time, from noon to 2P.M., is usually the heaviest. Table11 indicates the workers needed at various hours that the bank is open. Time Period Number of Tellers Required 9 A.M.-10 A.M A.M.-11 A.M A.M.-Noon 14 Noon-1 P.M P.M.-2 P.M P.M.-3 P.M P.M.-4 P.M P.M.-5 P.M. 10 Table 11: Hong Kong Bank of Commerce and Industry The bank now employs 12 full-time tellers, but many people are on its roster of available part-time employees. A part-time employee must put in exactly four hours per day but can start any time between 9A.M. and 1P.M. Part-timers are a fairly inexpensive labor pool, since no retirement or lunch benefits are provided for them. Full-timers, on the other hand, work from 9A.M. to 5P.M. but are allowed 1 hour for lunch.(half of the full timers eat at 11A.M., the other half at noon.) Full-timers thus provide 35 hours per week of productive labor time. By corporate policy, the bank limits part-time hours to a maximum of 50% of the day s total requirement. Part-timers earn $8 per hour (or $32 per day) on average, and full timers can earn $100 per day in salary and 8

9 benefits, on average. The bank would like to set a schedule that would minimize its total personnel cost. It will release one or more of its full-time tellers if it is possible to do so[rsh06]. 20. (Shipping) The Top Speed Bicycle Co. manufactures and markets a line of 10-speed bicycles nationwide. The firm has final assembly plants in two cities in which labor costs are low, New Orleans and Omaha. Its three major warehouses are located near the large market areas of New York, Chicago, and Los Angeles. 21. The sales requirements for the next year at the New York warehouse are 10,000 bicycles, at the Chicago warehouse 8,000 bicycles, and at the London warehouse 15,000 bicycles. The factory capacity at each location is limited. New Orleans can assemble and ship 20,000 bicycles; the Omaha plant can produce 15,000 bicycles per year. The cost of shipping one bicycle from each factory to each warehouse differs, and these unit shipping costs are as in Table12 The company wishes to develop a shipping schedule New York Chicago Los Angeles New Orleans $2 $3 $5 Omaha $3 $1 $4 Table 12: Top Speed Bicycle Co that will minimize its total annual transportation costs[rsh06]. References [RSH06] B. Render, R.M. Stair, and M. Hanna. Quantitative Analysis for Management. Prentice Hall, New Jersey, 9 edition, ITU Library Number: T56.R [Tah97] [Tay07] [TM93] H. Taha. Operations Research: an Introduction. Prentice Hall Inc., USA, 6 edition, B.W. Taylor. Introduction to Management Science. Prentice Hall, New Jersey, 9 edition, ITU Library Number: T56.T /T56.T E. Turban and J.R. Meredith. Fundamentals of Management Science. Plano, Tex. : Business Publications, 6 edition, ITU Library Number: HD30.23.T [Win04] W.L. Winston. Operations Research, Applications and Algorithms. Thomson Learning Inc., Canada, 4 edition,