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business
operations research an introduction
Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions
Identify which of Case apply to the following LP: max z = -x + 3x2 s.t. x1 x24 x + 2x = 4 x1 x1, x20
Identify which of Case apply to the following LP: max z s.t. 3x1 + x2 2x1 + x2 6 x + 3x2 9 x1, x20
8 Graphically find all optimal solutions to the following LP min z = x - x2 s.t. x1 + x2 6 x1 x2 3 x2-x = 3 X1, X20
9 Graphically determine two optimal solutions to the following LP: min z=3x+5x2 s.t. 3x+2x236 3x + 5x2 = 45
10 Money manager Boris Milkem deals with French currency (the franc) and American currency (the dollar). At 12 midnight, he can buy francs by paying .25 dollars per franc and dollars by paying 3 francs per dollar. Let x1 number of dollars bought (by paying francs) and x2 number of francs bought
My diet requires that all the food I eat come from 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¢,
1 There are three factories on the Momiss River (1, 2, and 3). Each emits two types of pollutants (1 and 2) into the river. If the waste from each factory is processed, the pollution in the river can be reduced. It costs $15 to process a ton of factory 1 waste, and each ton processed reduces the
2 U.S. Labs manufactures mechanical heart valves from the heart valves of pigs. Different heart operations require valves of different sizes. U.S. Labs purchases pig valves from three different suppliers. The cost and size mix of the valves purchased from each supplier are given in Table 3.Each
3 Peg and Al Fundy have a limited food budget, so Peg is trying to feed the family as cheaply as possible. However, she still wants to make sure her family members meet their daily nutritional requirements. Peg can buy two foods. Food1 sells for $7 per pound, and each pound contains 3 units of
4 Goldilocks needs to find at least 12 lb of gold and at least 18 lb of silver to pay the monthly rent. There are two mines in which Goldilocks can find gold and silver. Each day that Goldilocks spends in mine 1, she finds 2 lb of gold and 2 lb of silver. Each day that Goldilocks spends in mine 2,
6 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 Table 4. Union rules state that each full-time employee must work five consecutive days and then receive two days off. For example, an
1 In the post office example, suppose that each full-time employee works 8 hours per day. Thus, Monday’s requirement of 17 workers may be viewed as a requirement of 8(17) 136 hours. The post office may meet its daily labor requirements by using both full-time and part-time employees. During
2 During each 4-hour period, the Smalltown police force requires the following number of on-duty police officers: 12 midnight to 4 A.M.—8; 4 to 8 A.M.—7; 8 A.M. to 12 noon—6; 12 noon to 4 P.M.—6; 4 to 8 P.M.—5; 8 P.M. to 12 midnight—4. Each police officer works two consecutive 4-hour
3 Suppose that the post office can force employees to work one day of overtime each week. For example, an employee whose regular shift is Monday to Friday can also be required to work on Saturday. Each employee is paid $50 a day for each of the first five days worked during a week and $62 for the
4 Suppose the post office had 25 full-time employees and was not allowed to hire or fire any employees. Formulate an LP that could be used to schedule the employees in order to maximize the number of weekend days off received by the employees.
5 Each day, workers at the Gotham City Police Department work two 6-hour shifts chosen from 12 A.M. to 6 A.M., 6 A.M.to 12 P.M., 12 P.M. to 6 P.M., and 6 P.M. to 12 A.M. The following number of workers are needed during each shift:12 A.M. to 6 A.M.—15 workers; 6 A.M. to 12 P.M.—5 workers;12
6 During each 6-hour period of the day, the Bloomington Police Department needs at least the number of policemen shown in Table 5. Policemen can be hired to work either 12 consecutive hours or 18 consecutive hours. Policemen are paid $4 per hour for each of the first 12 hours a day they work and
7 Each hour from 10 A.M. to 7 P.M., Bank One receives checks and must process them. Its goal is to process all the checks the same day they are received. The bank has 13 check-processing machines, each of which can process up to 500 checks per hour. It takes one worker to operate each machine. Bank
For a discount rate of 15%, consider a project with the cash flows shown in Figure 8.a Compute project NPV if cash flows are at the end of the year.b Compute project NPV if cash flows are at the beginning of the year.c Compute project NPV if cash flows are at the middle of the year. D E F G H A B C
Suppose on April 8, 2001, we paid out $900. Then we receive $300 on 8/15/01 $400 on 1/15/02 $200 on 6/25/02 $100 on 7/03/03.If the annual discount rate is 10%, what is the NPV of these cash flows?
Star Oil Company is considering five different investment opportunities. The cash outflows and net present values (in millions of dollars) are given in Table 7. Star Oil has $40 million available for investment now (time 0); it estimates that one year from now (time 1) $20 million will be available
1 Show that if r 0.02, investment 2 has a larger NPV than investment 1.
2 Two investments with varying cash flows (in thousands of dollars) are available, as shown in Table 8. At time 0, $10,000 is available for investment, and at time 1, $7,000 is available. Assuming that r 0.10, set up an LP whose solution maximizes the NPV obtained from these investments.
3 Suppose that r, the annual interest rate, is 0.20, and that all money in the bank earns 20% interest each year (that is, after being in the bank for one year, $1 will increase to$1.20). If we place $100 in the bank for one year, what is the NPV of this transaction?
4 A company has nine projects under consideration. The NPV added by each project and the capital required by each project during the next two years is given in Table 9. All figures are in millions. For example, Project 1 will add $14 million in NPV and require expenditures of $12 million during
5 Finco must determine how much investment and debt to undertake during the next year. Each dollar invested reduces the NPV of the company by 10¢, and each dollar of debt increases the NPV by 50¢ (due to deductibility of interest payments). Finco can invest at most $1 million during the coming
Semicond is a small electronics company that manufactures tape recorders and radios. The per-unit labor costs, raw material costs, and selling price of each product are given in Table 10. On December 1, 2002, Semicond has available raw material that is sufficient to manufacture 100 tape recorders
1 Graphically solve the Semicond problem.
2 Suppose that the January 1 inventory shipment had been valued at $7,000. Show that Semicond’s LP is now infeasible
Sunco Oil manufactures three types of gasoline (gas 1, gas 2, and gas 3). Each type is produced by blending three types of crude oil (crude 1, crude 2, and crude 3). The sales price per barrel of gasoline and the purchase price per barrel of crude oil are given in Table 12. Sunco can purchase up to
1 You have decided to enter the candy business. You are considering producing two types of candies: Slugger Candy and Easy Out Candy, both of which consist solely of sugar, nuts, and chocolate. At present, you have in stock 100 oz of sugar, 20 oz of nuts, and 30 oz of chocolate. The mixture used to
2 O.J. Juice Company sells bags of oranges and cartons of orange juice. O.J. grades oranges on a scale of 1 (poor)to 10 (excellent). O.J. now has on hand 100,000 lb of grade 9 oranges and 120,000 lb of grade 6 oranges. The average quality of oranges sold in bags must be at least 7, and the average
3 A bank is attempting to determine where its assets should be invested during the current year. At present,$500,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual rate of return on each type of investment is known to be: bonds, 10%; home loans, 16%;
4 Young MBA Erica Cudahy may invest up to $1,000. She can invest her money in stocks and loans. Each dollar invested in stocks yields 10¢ profit, and each dollar invested in a loan yields 15¢ profit. At least 30% of all money invested must be in stocks, and at least $400 must be in
5 Chandler Oil Company has 5,000 barrels of oil 1 and 10,000 barrels of oil 2. The company sells two products:gasoline and heating oil. Both products are produced by combining oil 1 and oil 2. The quality level of each oil is as follows: oil 1—10; oil 2—5. Gasoline must have an average quality
6 Bullco blends silicon and nitrogen to produce two types of fertilizers. Fertilizer 1 must be at least 40% nitrogen and sells for $70/lb. Fertilizer 2 must be at least 70% silicon and sells for $40/lb. Bullco can purchase up to 80 lb of nitrogen at $15/lb and up to 100 lb of silicon at $10/lb.
7 Eli Daisy uses chemicals 1 and 2 to produce two drugs.Drug 1 must be at least 70% chemical 1, and drug 2 must be at least 60% chemical 2. Up to 40 oz of drug 1 can be sold at $6 per oz; up to 30 oz of drug 2 can be sold at $5 per oz. Up to 45 oz of chemical 1 can be purchased at $6 per oz, and up
8 Highland’s TV-Radio Store must determine how many TVs and radios to keep in stock. A TV requires 10 sq ft of floorspace, whereas a radio requires 4 sq ft; 200 sq ft of floorspace is available. A TV will earn Highland $60 in profits, and a radio will earn $20. The store stocks only TVs and
9 Linear programming models are used by many Wall Street firms to select a desirable bond portfolio. The following is a simplified version of such a model. Solodrex is considering investing in four bonds; $1,000,000 is available for investment. The expected annual return, the worst-case annual
10 Coalco produces coal at three mines and ships it to four customers. The cost per ton of producing coal, the ash and sulfur content (per ton) of the coal, and the production capacity (in tons) for each mine are given in Table 16. The number of tons of coal demanded by each customer are given in
11 Eli Daisy produces the drug Rozac from four chemicals. Today they must produce 1,000 lb of the drug.The three active ingredients in Rozac are A, B, and C. By weight, at least 8% of Rozac must consist of A, at least 4%of B, and at least 2% of C. The cost per pound of each chemical and the amount
12 (A spreadsheet might be helpful on this problem.) The risk index of an investment can be obtained from return on investment (ROI) by taking the percentage of change in the value of the investment (in absolute terms) for each year, and averaging them.Suppose you are trying to determine what
13 The owner of Sunco does not believe that our LP optimal solution will maximize daily profit. He reasons,“We have 14,000 barrels of daily refinery capacity, but your optimal solution produces only 13,500 barrels. Therefore, it cannot be maximizing profit.” How would you respond?
14 Oilco produces two products: regular and premium gasoline. Each product contains .15 gram of lead per liter.The two products are produced from six inputs: reformate, fluid catalytic cracker gasoline (FCG), isomerate (ISO), polymer (POL), MTBE (MTB), and butane (BUT). Each input has four
Rylon Corporation manufactures Brute and Chanelle perfumes. The raw material needed to manufacture each type of perfume can be purchased for $3 per pound. Processing 1 lb of raw material requires 1 hour of laboratory time. Each pound of processed raw material yields 3 oz of Regular Brute Perfume
Sunco Oil has three different processes that can be used to manufacture various types of gasoline. Each process involves blending oils in the company’s catalytic cracker.Running process 1 for an hour costs $5 and requires 2 barrels of crude oil 1 and 3 barrels of crude oil 2. The output from
2 Furnco manufactures tables and chairs. A table requires 40 board ft of wood, and a chair requires 30 board ft of wood. Wood may be purchased at a cost of $1 per board ft, and 40,000 board ft of wood are available for purchase. It takes 2 hours of skilled labor to manufacture an unfinished table
3 Suppose that in Example 11, 1 lb of raw material could be used to produce either 3 oz of Brute or 4 oz of Chanelle.How would this change the formulation?
4 Chemco produces three products: 1, 2, and 3. Each pound of raw material costs $25. It undergoes processing and yields 3 oz of product 1 and 1 oz of product 2. It costs$1 and takes 2 hours of labor to process each pound of raw material. Each ounce of product 1 can be used in one of three ways.It
5 A company produces A, B, and C and can sell these products in unlimited quantities at the following unit prices:A, $10; B, $56; C, $100. Producing a unit of A requires 1 hour of labor; a unit of B, 2 hours of labor plus 2 units of A; and a unit of C, 3 hours of labor plus 1 unit of B. Any A that
6 Daisy Drugs manufactures two drugs: 1 and 2. The drugs are produced by blending two chemicals: 1 and 2. By weight, drug 1 must contain at least 65% chemical 1, and drug 2 must contain at least 55% chemical 1. Drug 1 sells for $6/oz, and drug 2 sells for $4/oz. Chemicals 1 and 2 can be produced by
7 Lizzie’s Dairy produces cream cheese and cottage cheese. Milk and cream are blended to produce these two products. Both high-fat and low-fat milk can be used to produce cream cheese and cottage cheese. High-fat milk is 60% fat; low-fat milk is 30% fat. The milk used to produce cream cheese must
8 A company produces six products in the following fashion. Each unit of raw material purchased yields four units of product 1, two units of product 2, and one unit of product 3. Up to 1,200 units of product 1 can be sold, and up to 300 units of product 2 can be sold. Each unit of product 1 can be
9 Each week Chemco can purchase unlimited quantities of raw material at $6/lb. Each pound of purchased raw material can be used to produce either input 1 or input 2.Each pound of raw material can yield 2 oz of input 1, requiring 2 hours of processing time and incurring $2 in processing costs. Each
10 LIMECO owns a lime factory and sells six grades of lime (grades 1 through 6). The sales price per pound is given in Table 25. Lime is produced by kilns. If a kiln is run for an 8-hour shift, the amounts (in pounds) of each gradeof lime given in Table 26 are produced. It costs $150 to run a kiln
11 Chemco produces three products: A, B, and C. They can sell up to 30 pounds of each product at the following prices (per pound): product A, $10; product B, $12; product C, $20. Chemco purchases raw material at $5/lb. Each pound of raw material can be used to produce either 1 lb of A or 1 lb of B.
12 Chemco produces 3 chemicals: B, C, and D. They begin by purchasing chemical A for a cost of $6/100 liters. For anadditional cost of $3 and the use of 3 hours of skilled labor, 100 liters of A can be transformed into 40 liters of C and 60 liters of B. Chemical C can either be sold or processed
13 Carrington Oil produces two types of gasoline, gas 1 and gas 2, from two types of crude oil, crude 1 and crude 2. Gas 1 is allowed to contain up to 4% impurities, and gas 2 is allowed to contain up to 3% impurities. Gas 1 sells for$8 per barrel, whereas gas 2 sells for $12 per barrel. Up to
14 You have been put in charge of the Melrose oil refinery.The refinery produces gas and heating oil from crude oil.Gas sells for $8 per barrel and must have an average “grade level” of at least 9. Heating oil sells for $6 a barrel and musthave an average grade level of at least 7. At most,
Sailco Corporation must determine how many sailboats should be produced during each of the next four quarters (one quarter three months). The demand during each of the next four quarters is as follows: first quarter, 40 sailboats; second quarter, 60 sailboats;third quarter, 75 sailboats; fourth
1 A customer requires during the next four months, respectively, 50, 65, 100, and 70 units of a commodity (no backlogging is allowed). Production costs are $5, $8, $4, and $7 per unit during these months. The storage cost from one month to the next is $2 per unit (assessed on ending inventory). It
2 A company faces the following demands during the next three periods: period 1, 20 units; period 2, 10 units; period 3, 15 units. The unit production cost during each period is as follows: period 1—$13; period 2—$14; period 3—$15.A holding cost of $2 per unit is assessed against each
3 James Beerd bakes cheesecakes and Black Forest cakes.During any month, he can bake at most 65 cakes. The costs per cake and the demands for cakes, which must be met on time, are listed in Table 33. It costs 50¢ to hold a cheesecake, and 40¢ to hold a Black Forest cake, in inventory for a month.
4 A manufacturing company produces two types of products: A and B. The company has agreed to deliver the products on the schedule shown in Table 34. The company has two assembly lines, 1 and 2, with the available production hours shown in Table 35. The production rates for each assembly line and
5 During the next two months, General Cars must meet(on time) the following demands for trucks and cars: month 1—400 trucks, 800 cars; month 2—300 trucks, 300 cars.During each month, at most 1,000 vehicles can be produced.Each truck uses 2 tons of steel, and each car uses 1 ton of steel. During
6 Gandhi Clothing Company produces shirts and pants.Each shirt requires 2 sq yd of cloth, each pair of pants, 3.During the next two months, the following demands for shirts and pants must be met (on time): month 1—10 shirts, 15 pairs of pants; month 2—12 shirts, 14 pairs of pants.During each
7 Each year, Paynothing Shoes faces demands (which must be met on time) for pairs of shoes as shown in Table 37. Workers work three consecutive quarters and then receive one quarter off. For example, a worker may work during quarters 3 and 4 of one year and quarter 1 of the next year.During a
8 A company must meet (on time) the following demands:quarter 1—30 units; quarter 2—20 units; quarter 3—40 units. Each quarter, up to 27 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlimited number of units can be produced with overtime
9 Donovan Enterprises produces electric mixers. During the next four quarters, the following demands for mixers must be met on time: quarter 1—4,000; quarter 2—2,000;quarter 3—3,000; quarter 4—10,000. Each of Donovan’s workers works three quarters of the year and gets one quarter off.
Finco Investment Corporation must determine investment strategy for the firm during the next three years. Currently (time 0), $100,000 is available for investment. Investments A, B, C, D, and E are available. The cash flow associated with investing $1 in each investment is given in Table 38.For
1 A consultant to Finco claims that Finco’s cash on hand at time 3 is the sum of the cash inflows from all investments, not just those investments yielding a cash inflow at time 3.Thus, the consultant claims that Finco’s objective function should beExplain why the consultant is incorrect. max z
2 Show that Finco’s objective function may also be written as max z=> 100,000 +0.54 + 0.5B+ 0.2C+0.9D + 0.5E + 0.08S+ 0.08S + 0.0852
3 At time 0, we have $10,000. Investments A and B are available; their cash flows are shown in Table 39. Assume that any money not invested in A or B earns no interest.Formulate an LP that will maximize cash on hand at time 3. Can you guess the optimal solution to this problem?
4 Broker Steve Johnson is currently trying to maximize his profit in the bond market. Four bonds are available for purchase and sale, with the bid and ask price of each bond as shown in Table 40. Steve can buy up to 1,000 units of each bond at the ask price or sell up to 1,000 units of each bond at
5 A small toy store, Toyco projects the monthly cash flows(in thousands of dollars) in Table 42 during the year 2003.A negative cash flow means that cash outflows exceed cash inflows to the business. To pay its bills, Toyco will need to borrow money early in the year. Money can be borrowed in two
6 Consider Problem 5 with the following modification:Each month Toyco can delay payments on some or all of the cash owed for the current month. This is called “stretching payments.” Payments may be stretched for only one month, and a 1% penalty is charged on the amount stretched. Thus, if it
7 Suppose we are borrowing $1,000 at 12% annual interest with 60 monthly payments. Assume equal payments are made at the end of month 1, month 2, . . . month 60. We know that entering into Excel the function PMT(.01, 60, 1,000)would yield the monthly payment ($22.24).It is instructive to use LP to
8 You are a CFA (chartered financial analyst). Madonna has come to you because she needs help paying off her credit card bills. She owes the amounts on her credit cards shown in Table 43. Madonna is willing to allocate up to $5,000 per month to pay off these credit cards. All cards must be paid off
9 Winstonco is considering investing in three projects. If we fully invest in a project, the realized cash flows (in millions of dollars) will be as shown in Table 44. For example, project 1 requires cash outflow of $3 million todayand returns $5.5 million 3 years from now. Today we have $2 million
CSL is a chain of computer service stores. The number of hours of skilled repair time that CSL requires during the next five months is as follows:Month 1 (January): 6,000 hours Month 2 (February): 7,000 hours Month 3 (March): 8,000 hours Month 4 (April): 9,500 hours Month 5 (May): 11,000 hours At
1 If y1= 38, then what would be the optimal solution to CSL’s problem?
2 An insurance company believes that it will require the following numbers of personal computers during the next six months: January, 9; February, 5; March, 7; April, 9;May, 10; June, 5. Computers can be rented for a period of one, two, or three months at the following unit rates: onemonth rate,
3 The IRS has determined that during each of the next 12 months it will need the number of supercomputers given in Table 45. To meet these requirements, the IRS rentsvsupercomputers for a period of one, two, or three months.It costs $100 to rent a supercomputer for one month, $180 for two months,
4 You own a wheat warehouse with a capacity of 20,000 bushels. At the beginning of month 1, you have 6,000 bushelsof wheat. Each month, wheat can be bought and sold at the price per 1000 bushels given in Table 46.The sequence of events during each month is as follows:a You observe your initial
1 Bloomington Breweries produces beer and ale. Beer sells for $5 per barrel, and ale sells for $2 per barrel.Producing a barrel of beer requires 5 lb of corn and 2 lb of hops. Producing a barrel of ale requires 2 lb of corn and 1 lb of hops. Sixty pounds of corn and 25 lb of hops are available.
2 Farmer Jones bakes two types of cake (chocolate and vanilla) to supplement his income. Each chocolate cake can be sold for $1, and each vanilla cake can be sold for 50¢.Each chocolate cake requires 20 minutes of baking time and uses 4 eggs. Each vanilla cake requires 40 minutes of baking time
3 I now have $100. The following investments are available during the next three years:Investment A Every dollar invested now yields $0.10 a year from now and $1.30 three years from now.Investment B Every dollar invested now yields $0.20 a year from now and $1.10 two years from now.Investment C
4 Sunco processes oil into aviation fuel and heating oil. It costs $40 to purchase each 1,000 barrels of oil, which is then distilled and yields 500 barrels of aviation fuel and 500 barrels of heating oil. Output from the distillation may be sold directly or processed in the catalytic cracker. If
5 Finco has the following investments available:Investment A For each dollar invested at time 0, we receive$0.10 at time 1 and $1.30 at time 2. (Time 0 now;time 1 one year from now; and so on.)Investment B For each dollar invested at time 1, we receive$1.60 at time 2.Investment C For each
6 All steel manufactured by Steelco must meet the following requirements: 3.2–3.5% carbon; 1.8–2.5% silicon;0.9–1.2% nickel; tensile strength of at least 45,000 pounds per square inch (psi). Steelco manufactures steel by combining two alloys. The cost and properties of each alloy are given in
7 Steelco manufactures two types of steel at three different steel mills. During a given month, each steel mill has 200 hours of blast furnace time available. Because of differences in the furnaces at each mill, the time and cost to produce a ton of steel differs for each mill. The time and cost
8 Walnut Orchard has two farms that grow wheat and corn. Because of differing soil conditions, there are differences in the yields and costs of growing crops on the two farms. The yields and costs are shown in Table 49. Each farm has 100 acres available for cultivation; 11,000 bushels of wheat and
9 Candy Kane Cosmetics (CKC) produces Leslie Perfume, which requires chemicals and labor. Two production processes are available: Process 1 transforms 1 unit of labor and 2 units of chemicals into 3 oz of perfume. Process 2 transforms 2 units of labor and 3 units of chemicals into 5 oz of perfume.
10 Carco has a $150,000 advertising budget. To increase automobile sales, the firm is considering advertising in newspapers and on television. The more Carco uses a particular medium, the less effective is each additional ad.Table 50 shows the number of new customers reached by each ad. Each
11 Sunco Oil has refineries in Los Angeles and Chicago.The Los Angeles refinery can refine up to 2 million barrels of oil per year, and the Chicago refinery up to 3 million.Once refined, oil is shipped to two distribution points:Houston and New York City. Sunco estimates that each distribution
12 For a telephone survey, a marketing research group needs to contact at least 150 wives, 120 husbands, 100 single adult males, and 110 single adult females. It costs $2 to make a daytime call and (because of higher labor costs)$5 to make an evening call. Table 52 lists the results. Because of
13 Feedco produces two types of cattle feed, both consisting totally of wheat and alfalfa. Feed 1 must contain at least 80% wheat, and feed 2 must contain at least 60%alfalfa. Feed 1 sells for $1.50/lb, and feed 2 sells for $1.30/lb.Feedco can purchase up to 1,000 lb of wheat at 50¢/lb and up to
Feedco (see Problem 13) has decided to give its customer (assume it has only one customer) a quantity discount. If the customer purchases more than 300 lb of feed 1, each pound over the first 300 lb will sell for only$1.25/lb. Similarly, if the customer purchases more than 300 pounds of feed 2,
15 Chemco produces two chemicals: A and B. These chemicals are produced via two manufacturing processes.Process 1 requires 2 hours of labor and 1 lb of raw material to produce 2 oz of A and 1 oz of B. Process 2 requires 3 hours of labor and 2 lb of raw material to produce 3 oz of A and 2 oz of B.
16 Suppose that in the CSL computer example of Section 3.12, it takes two months to train a technician and that during the second month of training, each trainee requires 10 hours of experienced technician time. Modify the formulation in the text to account for these changes.
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