Since all CJ–ZJ � 0, this tableau is optimal. The optimal solution to the quadraticprogramming problem is X1 � 1000, X2 � 4500. Substituting these values into theobjective function gives us the optimal objective function value of $810,000.

P r o b l e m s

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1. Fulton Oil Company (Fuloco) has a truck at the docksthat, after loading its assigned cargo, has space availableto legally transport an additional 650 pounds. It hasdecided to purchase oil from the leftover supplies of amajor oil importer that also keeps a ship at the dock.The importer has enough light crude to fill six barrels,enough medium crude to fill four barrels, and enoughheavy crude to fill three barrels. Fuloco’s usual profit perbarrel and the oil’s weight per barrel are as follows:

Availability Weight Profit Oil (Barrels) per Barrel per Barrel

Light 6 100 $4Medium 4 120 $5Heavy 3 160 $6

If Fuloco must purchase full barrels from the importer,use a dynamic programming approach to determine thenumber of barrels of light, medium, and heavy crude oilFuloco should purchase from the importer.

2. On December 8, 1992, U.S. troops landed in Somaliafor a huge humanitarian relief effort. Their objective wasto maximize the number of lives saved per day. Initially,five troop convoys were sent to various areas around thecountry: Chisimayu in the south; Mogadishu, the capital,in the central region; and Hargeisa in the northernmountain area, near the Ethiopian border. Expertsconstructed the following table, estimating the numberof lives saved in each region daily based on the numberof troop convoys dispersed to that region:

Lives Saved Daily

Number of Troop Convoys

0 1 2 3 4 5

Chisimayu 0 360 560 720 840 960Region Mogadishu 0 400 560 640 800 1040

Hargeisa 0 160 360 600 880 1200

Formulate and solve this problem as a dynamic program.Give the recommended distribution of troop convoysand the overall expected number of lives saved daily.

3. Sometimes minor changes to a problem can alter thesolution approach as well as the optimal solution.Consider the situation faced by Granite Industries ofHelena, Montana, which has $6 million to invest incapital improvements to one or more of its four plants.Investments will be made in $1 million increments, and

no plant will receive more than $5 million. Initialprojections for the increase in annual profits for an $Xmillion investment at each plant are as follows:

Plant

1 2 3 4

Increase in profits 7X 3X 4X 5X

a. Formulate and solve the problem as an integer linearprogram.

b. Suppose the estimates are revised as follows:

Plant

1 2 3 4

Increase in profits 7X � 2 3X � 9 4X � 8 5X � 6

Formulate and solve this problem using a dynamicprogramming approach.

4. Atlantic Recycling has an aluminum recycling plant inWest Orange, New Jersey; its four machines recyclealuminum to different purity specifications. Eachmachine fills different-sized containers, which are thenshipped to Continental Aluminum, where the recycledaluminum is used to make various products, from doorhinges to cans and foil. The profit Atlantic makes oneach container, the container size, and the dailyproduction capacity are summarized in the followingtable:

Maximum Container

Container Production Profit per Machine Purity Size per Day Container

Masher 95% 1.0 tons 3 $240Smasher 96% 1.2 tons 2 $300Crusher 90% .7 tons 4 $180Pounder 98% 1.5 tons 2 $380

Atlantic uses a 5-ton truck to make daily deliveries ofrecycled aluminum to Continental. However,Continental also contracts with Atlantic to ship otherproducts from the West Orange area in the same truck.Thus the daily mix of containers varies, depending onthe space available after the other products have beenloaded. On December 13, Continental contracted withAtlantic to deliver 1.8 tons of other materials, so the

P r o b l e m s CD-201

capacity that can be allocated to recycled aluminum is3.2 tons.a. Given that only full containers are delivered, use

dynamic programming to determine the productionschedule that will maximize Atlantic’s profit onDecember 13.

b. Suggest a solution approach (other than dynamicprogramming) for solving this problem.

5. It is the beginning of June, and the productiondepartment at the Mann Company must plan itsstaffing strategy for the next three months (June, July, and August). Currently, the company employs 20 people, but it must implement a 5% cut (to 19employees) at the beginning of September (threemonths from now).

In any given month the firm may hire two employees,hire one employee, retain the same level as the monthbefore, lay off one employee, or lay off two employees.Production quotas for the summer require the followingminimum staffing levels:

June 19July 21August 20

The cost to hire an employee is $5000, and the costto lay off a worker is $8000. If more workers areemployed than are indicated by the above staffing levels,each inactive worker costs the company $3000 in idletime. Use a dynamic programming approach todetermine an optimal staffing policy for the summerproduction at the Mann Company.

6. Equipment failure on a Global Airlines B767 has putin jeopardy its Oriental Eagle service flight from LosAngeles to Taipei, which is scheduled to leave in 24hours. Management has determined that it cantemporarily reschedule a B767 that it has been usingon selected flights from New York to Miami for thisflight and use a smaller B757 aircraft for the New Yorkto Miami route.

The B767 is currently in New York. Rather than fly the plane empty from New York to Los Angeles the company would like to transport some muchneeded supplies and equipment from New York tovarious U.S. cities served by the airline. Accordingly,the plane will be loaded in New York and make stopsat one mid-Atlantic city (Cleveland, Cincinnati, orAtlanta), one midwestern city (Chicago, St. Louis, or Dallas), and one western city (Denver, Las Vegas, or Phoenix) before reaching Los Angeles. The accompanying figure gives the airline mileagebetween the cities.a. Formulate and solve a dynamic program to determine

the minimum total distance flown by the aircraftbefore reaching Los Angeles.

b. The average flying speed of the aircraft betweenany two cities (including takeoff and landing times)is 420 miles per hour. If it takes two hours to loadthe plane in New York with supplies and two hours

to unload supplies in each city visited (includingLos Angeles), using the solution from part a,determine the amount of time Global Airlines willhave to prepare the aircraft for the Oriental Eagleservice.

Airline Mileage for Problem 6

868

311 411

592

765

731

487

811

1021

484

596

1056645358

222

781

1372

308

254

1262

908

834

1521

1445

L.A.

PHX

L.V.

DEN

DAL

ST.L.

CHI CLEVE

CIN

N.Y.

ATL

7. Mueller’s Nautical of Minneapolis, Minnesota,manufactures a variety of boats that are sold to vacationrental operators. It is currently planning production ofits Family Cruise houseboat for the upcoming four-month production run. Due to commitments during thisperiod to other boat products that Mueller’smanufactures, the unit production cost per houseboat,the maximum monthly production level, the monthlystorage costs, and the capacities vary as shown in thefollowing table. One houseboat is in inventory at thebeginning of April.

Production Maximum Maximum InventoryMonth Orders Cost/Boat Production Storage Cost/Boat

April 1 $20,000 4 5 $ 800

May 3 $21,600 3 5 $ 800

June 3 $24,000 3 3 $1600

July 4 $23,200 3 0 $ —

a. Develop and solve a dynamic programming model tominimize total production and inventory costs, giventhat all orders are to be filled each month.

b. Suppose that Mueller’s offers a $2000/monthdiscount to its customers for each houseboat notdelivered on time. Assume that no orders are lost ifthe delivery of a houseboat is delayed past itsintended order date. Revise your formulation in parta to take these factors into account. Solve for the newoptimal solution and compare the results with thosedetermined in part (a).

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8. Many models in management science can be solved by a variety of solution techniques. Considerthe simple assignment problem in which fourcontractors have submitted bids for four differentprojects to be performed this summer in Framingham,Massachusetts. The city must select a differentcontractor to perform each of the four projects so that they are all completed by the end of the summer.The bids follow (in $1000s):

Refurbish MakeRenovate High School Improvements Repave

Child Football in Gerontology Tennis Care Center Field Center Courts

Crane Co. $300 $ 70 $150 $65Parker Ind. $400 $ 95 $200 $75Clanton Ent. $285 $ 90 $175 $70Tish Inc. $375 $110 $195 $75

Determine the least cost selection of contractors for theprojects using:a. Complete enumeration of all contractor–project

possibilitiesb. A linear programming modelc. A transportation modeld. An assignment modele. A shortest path model

(Hint: Draw a network with a single node at stage 0;then four nodes representing the four possiblecontractors that can be assigned the first project(Child Care Center Renovation); then three nodesconnecting each of the previous nodes signifying theremaining contractors available for the secondproject; then two nodes for each of these nodesdenoting the remaining contractors available for thethird project; and, finally, one node connecting eachof the previous nodes denoting the contractorremaining to do the final project.)

f. A dynamic programming model[Hint: Use the network generated in part (e).

9. Kohl Industries has just won a seven-year contract toproduce a component for a new submarine for the U.S.Navy. Production of the component requires the use of alarge, very precise drill press, which the company willpurchase for $100,000 at the beginning of year 1. At thestart of each year thereafter, Kohl must decide whetheror not to keep the drill press or trade it in for a new one.The following tables give relevant information. Costsare in $1000s.

Purchase Price (in $1000s) of a New Machine in Year N

N Price N Price N Price N Price

1 $100 3 $111 5 $122 7 $1362 $105 4 $116 6 $130

Yearly Costs of an N-Year Old Machine

Operating Trade-in Salvage Cost of an Value of an Value of an

N-Year N-Year N-Year Age N Old Machine Old Machine Old Machine

0 $ 20 — —1 $ 26 $65 $502 $ 40 $45 $353 $ 65 $25 $204 $100 $15 $ 55 $150 $10 $ 06 $210 $ 0 $ 07 — — $ 0

Formulate a dynamic programming model to determinean optimal equipment replacement policy for KohlIndustries. Rigorously define each of the eightcomponents of the dynamic programming approach.Hints: (1) there are two possible values for the decisionvariable for each state at each stage: keep or trade (thecost consequences are different for each); (2) use thesalvage value as the boundary condition at the end ofyear 7.

10. Kellycomp is producing its new KC10 and KC20computers. The profit and production requirements forthe current production run are as follows:

Inspection/Unit Production Packaging Drives

Profit Time Time Hard Floppy CD

KC10 $200 .8 hr. .2 hr. 1 1 0KC20 $600 1 hr. .5 hr. 1 2 1

A total of 2000 hard drives, 2500 floppy drives, and 500CD drives are available for this production run;additional drives meeting Kellycomp’s specifications willnot be available until the next production run.

How should Kellycomp schedule production for thisrun in light of the following goals for the production run?

Priority 11. Meet commitment for 800 KC10 models.2. Meet commitment for 400 KC20 models.

(These goals are equally important.)

Priority 23. Earn a total profit of at least $480,000.

Priority 34. Use no more than 1200 production time hours.5. Use no more than 500 inspection/packaging hours.

(Underachieving the hours in goal 5 is twice asdetrimental as underachieving the hours in goal 4.)

11. Gresham Medical Supplies currently employs 20 salesrepresentatives, each earning a base salary of $1500 permonth plus commissions. The following table detailsthe average number of contacts and the average gross

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profit dollars to Gresham per contact for each salesrepresentative.

Sales Contacts/Gross Profit Data

Physicians Clinics Hospitals

Average Number of Contracts 60 24 16per Representative per Month

Average Gross Profit $200 $500 $1,000per Contact

Thus the average monthly gross profit to the companyper sales representative is 60($200) � 24($500) �16($1000) � $40,000.

Recently, however, Gresham has been experimentingwith advertising on the World Wide Web. The followingtable summarizes additional contacts and sales estimated forevery $10,000 used to support the website activities. (Thesenumbers are valid for web support up to $50,000 monthly.)

Web Site Sales/Profit Data

Physicians Clinics Hospitals

Average Number of Sales per 250 50 100$10,000 Website Support per Month

Average Gross Profit per Sale $80 $200 $1,200

Thus the average monthly additional gross profit per$10,000 in website support is 250($80) � 50($200) �100($1200) � $150,000.

Management at Gresham has already decided it willnot lay off more than two sales representatives and it willspend at most $50,000 monthly to pay the base salariesof its sales representatives and support websiteoperations. Determine the number of salesrepresentatives the company should retain and howmuch it should spend on World Wide Web operations ifthe company has the following prioritized goals:

Priority 1: Spend at least $10,000 on website operations.Priority 2: Achieve at least $1 million in monthly grossprofit.Priority 3: Maintain a total of (1) at least 2000 physiciancontacts; (2) at least 600 clinic contacts; and (3) at least500 hospital contacts. The relative importance of failingto meet each of these goals is 1�2�4, respectively, peroccurrence.

12. Bernie Collins is an investment counselor for PaulaSmith, a soon to be retired legal secretary. Paula has justreceived an inheritance of $100,000, which she wouldlike to invest in two Wilson mutual funds: the WilsonIncome Fund and the Wilson Aggressive Growth Fund.The projected annual yield for the year and the riskindex for each fund are as follows.

Projected Estimated Annual Yield Risk Factor

Wilson Income Fund .08 10Wilson Aggressive Growth Fund .20 80

Paula is concerned primarily about security; thus shedesires a portfolio with a maximum risk factor of 25.However, she would also like to supplement herretirement by $15,000 annually from her investment andinvest at least $25,000 in the aggressive growth fund.a. Show (graphically) that Paula’s three goals are

inconsistent; that is, they cannot all be metsimultaneously.

b. Determine and solve a goal programming model forPaula with two prioritized goals: (i) Attaining a riskfactor for the portfolio of no more than 25; and (ii)attaining an expected return from the portfolio of atleast $15,000 and investing at least $25,000 in theaggressive growth fund. Dollar for dollar, failure toachieve the $15,000 is weighted five times moreimportant than failure to meet the $25,000 investedin the aggressive growth fund.

13. KarKleen is a new product from the KarKare Companywhich sells for $20 per bottle. The company plans tomarket KarKleen by phone solicitation and door-to-door canvassing and has the following goals:Priority 11. Achieve $20,000 in sales per week.2. Spend no more than $10,000 weekly on employee

salaries.(These goals are of equal importance.)

Priority 23. Reach 6000 potential customers per week.Priority 34. Assign at least 10 employees to work the phones.5. Assign at least 10 employees to canvass door to door.

(These goals are of equal importance.)Relevant data are given in the following table:

Number of % Successful Employee Status Salary Contacts Sales

Phone contacts $240/wk. 400 6%Door to door $300/wk. 150 20%

Given its prioritized goals, how many employees shouldKarKare assign to phone solicitation and how many todoor-to-door canvassing?

14. The Milwaukee Theatre Guild has just received ananonymous $250,000 donation to be used as follows:block grants of $7500 each to children’s theater groups;grants of $6000 each for new playwrights; and grants fortheater scholarships of $5000 each. The donor hasstipulated that at least eight children’s theater groupsmust be funded. Other than that, the Guild may awardthe donation any way it sees fit. The Guild hasdetermined the following goals.a. Award at least 40 block grants.b. Give at least 60% of the total funding to education

(children’s theater and scholarships).c. Keep the number of scholarships from exceeding the

number of new playwright awards by more than five.d. Award at least 15 of each type of grant.

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Formulate a nonpreemptive goal program with theweight of 4 for each detrimental deviation from goal 1, aweight of 3 for every $1000 deviation from goal 2, aweight of 2 for each detrimental deviation from goal 3,and a weight of 1 for each deviation from goal 4. Solvefor the recommended allocation of the funds.

15. With seven days to go before the Iowa presidentialcaucuses, Larry Adler, the campaign manager for PaulPowell, is trying to organize his staff and volunteers tocontact potential voters by (1) phone, (2) householdvisits, and (3) personal contacts at local businessestablishments (restaurants, strip malls, etc.). A group of10 full-time experienced staff members and 200volunteers are trying to convince likely caucus votersthat “Paul is the man!”

Full-time staff members work 12 hours per day, whilevolunteers average 5 hours per day. The following tablesummarizes Larry’s analysis of the situation.

Personal ContactHours

Phone Household By Contacts Visits Total Staff

Staff member 5/hr. 3/hr.Volunteer 8/hr. 4/hr.Minimum required 15,000 6000 1500 200Target goal 35,000 25,000 5000 700

Larry has decided to assign at least two experiencedstaff members to each of the three modes of votercontact. In addition, he feels that (1) each staffpersonal-contact hour below target is twice asdetrimental as each total personal contact hour belowtarget; (2) each total personal-contact hour belowtarget is 100 times more detrimental than eachhousehold visit below target; and (3) each householdvisit below target is three times more detrimental thaneach phone contact hour below target.a. Formulate and solve a nonpreemptive goal program

for the allocation of experienced staff workers andvolunteers during the 7 days.

b. Formulate and solve a nonpreemptive goal programfor this problem if the two goals concerning personalcontacts are considered priority 1 level goals and thetwo goals concerning phone and household visits arepriority 2 level goals.

16. Larry Adler, campaign manager for Paul Powell (seeproblem 15), is also planning a television ad blitzduring the seven days before the Iowa caucuses. Larryhas produced two commercials for the Powellcampaign. One is a 30-second upbeat positive adshowing Paul and his family on a picnic discussingissues that are important to Iowa voters. The second isa one-minute, negative attack against Paul’s principalopponent in the caucuses.

Larry has identified four possible televisionadvertising options, determined ad costs and audienceexposure, and estimated the maximum number of

minutes available to run ads during the upcoming week,as shown in the following table.

Minutes Exposures Television Type/Time Available per Ad Cost per Ad

Commercial/Daytime 70 45,000 60 sec.-$10,00030 sec.-$ 7,000

Commercial/Evening 35 250,000 60 sec.-$40,00030 sec.-$25,000

Cable/Evening 140 40,000 60 sec.-$ 6,00030 sec.-$ 5,000

Late night 200 8000 60 sec.-$ 2,00030 sec.-$ 1,500

“Exposures per Ad” indicates the number of likely voterswho will see the ad (whether or not for the first time),not the number of new voters reached with each ad.Larry has determined the following goals for thetelevision advertising:

Priority 1Goal 1: At least 50% of the ads should be positive ads.Priority 2Goal 2: The campaign should spend less than $1 millionon television advertising.Priority 3Goal 3: The ads should generate at least 5 millionexposures.Priority 4Goal 4: The campaign should run at least 100 ads.Goal 5: The campaign should run at least five positiveads in each of the four type/time slots.Goal 6: The campaign should run at least 10 total adsduring each of the four type/time slots.Goal 7: The campaign should run at least 35 evening ads.(Failing to meet goals 4 and 7 is twice as detrimental perad as failing to meet goals 5 and 6.)

Formulate and solve a goal programming model todetermine how Larry should advertise on television.

17. The pairings for the first round of the Southern Sectionhigh school basketball tournament are to be held onWednesday night, and tournament officials must assignreferee crews to officiate the games. The SouthernSection is divided into four regions: (1) Los AngelesCounty, (2) Orange County, (3) Riverside County, and(4) San Diego County.

In the first round, teams play within their own area,but the officiating crew must be from another area; thatis, a Los Angeles crew cannot referee a Los Angelesplayoff game. The following table gives the number ofplayoff-qualified, two-man officiating crews available,the number of games to be played in each area, and anestimate of the average driving distance between areas.Playoff officials are paid $50 each for a playoff game (afixed expense); in addition, the Southern Section paysone of the officials (the “driving official”) 25 cents a mile.Each county is guaranteed a minimum of 12 playoff crewassignments.

P r o b l e m s CD-205

Average Driving Distances

Number ofLos Angeles Orange Riverside San Diego Qualified Crews

Los Angeles xxx 40 50 120 42Orange 40 xxx 35 90 24Riverside 50 35 xxx 60 22San Diego 120 90 60 xxx 30Total first round games 26 18 16 20

The Southern Section has established the followingpriorities concerning the playoff official assignments:

Priority 1. Assign no more than 50% of the games in acounty to officials from any one other countyPriority 2. Assign at least 50% of the qualified officialcrews from each countyPriority 3. Do not exceed more than $600 in total travelexpenses.

a. What is your recommendation for game assignmentsif each goal is treated as a separate priority level?

b. Will your recommendation in part (a) change if thefirst two priorities are reversed?

c. How sensitive is your recommendation to changes inthe maximum amount allowed for travel expenses?

18. Pyramid Printers is about to introduce its new Q100color printer. The marketing department has indicatedthat demand for the printer is linearly related to theprice. It has forecast that if the printers sell for $1000,demand will be 2000 per month, whereas if they sell for$300, demand will be 9000 per month. Production costsare $200 per printer. Fixed operating expenses amountto approximately $1 million per month.a. Determine a linear function that relates monthly

demand, X, for the Q100 to its price, P.b. Use the result of part a to form a profit function in

terms of the one variable, X.c. Using calculus, solve for the optimal monthly

production quantity of Q100 printers. What shouldbe the selling price P? What is the optimal monthlyprofit?

d. Suppose the production costs to Pyramid increasedby $20, to $220. How much of this increase is passedalong to the customer if Pyramid wishes to continueto maximize its total monthly profit?

19. Consider the problem faced by Pyramid Printers inproblem 18(c). What are the optimal monthlyproduction quantity and price in each of the followingcircumstances:a. The price must be no more than $500.b. Monthly production cannot exceed 6000.c. Monthly production must be at least 6000.d. Monthly production cannot exceed 2500.

20. Consider the problem faced by Pyramid Printers inproblem 18(c). Suppose that in addition to the Q100,Pyramid produces the Q20, a similar but noncolorversion of the Q100. The production cost of this modelis $170; forecasts indicate that if its price were $1000,

monthly demand would be only 100, but if the pricewere $300, monthly demand would be 5100. The totalmonthly profit is reduced by .01 times the product of theproduction rates of the Q100 and the Q20.a. Assuming demand for Q20 printers is linearly related

to its price, formulate an unconstrained objectivefunction and solve for the total monthly productionof each product.

b. What should the price of each be?c. What is the total monthly profit?

21. Consider the two-product production problem ofPyramid Printers in problem 20. Production facilitieslimit production to a maximum of 4000 of eitherproduct. Each Q100 requires .5 labor-hour ofproduction time, and each Q20 requires .4 labor-hour ofproduction time; 2800 labor-hours are availablemonthly.a. Formulate this problem as a constrained nonlinear

programming problem.b. Write the Kuhn–Tucker conditions for this problem.c. Solve for the optimal production quantities using

Excel Solver.d. Given the result to (c), verify that the Kuhn–Tucker

conditions hold at the optimal solution.e. What is the instantaneous value of an extra labor-

hour?22. MS Investments uses a crude mathematical model for

the return on two types of investments. Let X1 representthe amount invested in Pyramid Corporation and X2 theamount in Kar Kare Industries. The return per dollar isgiven by

R(X1,X2) � � .00032X1X2� .8X1 � 1.6X2

MS has received authorization to invest up to $10,000 ineither one or both of the investments, with therestriction that no more than $7500 be invested in eitherinvestment.a. Formulate this problem as a nonlinear programming

model.b. Using the Kuhn–Tucker conditions, show that the

optimal solution is to invest $5000 in each investment.c. What is the return if there is an extra dollar to invest?

23. Crater Crates, Inc. of Medford, Oregon, designs andmanufactures crates that meet the specifications of itscustomers. One customer, Pets R Us, has asked Craterto design and manufacture a rectangular “doggie box”with the following characteristics: (1) The box is to have

�.00016X21 � .00024X22

CD-206 C H A P T E R 1 3 N o n l i n e a r M o d e l s : D y n a m i c , G o a l , a n d N o n l i n e a r P r o g r a m m i n g

a volume of at least 15 cubic feet; (2) the height of thebox is not to exceed 2 feet; (3) the length is not to exceedthe depth of the box by more than 6 inches (�.5 foot);(4) each dimension must be at least 1 foot; and (5) thetop, bottom, sides, and back of the box are to be moldedplastic. The front of the box is to be made of a wire meshthat allows an enclosed animal to see and breathe.

The sturdy molded plastic Crater uses costs $0.30 persquare foot, and the wire mesh costs $0.25 per squarefoot. Since the doggie box may be in high demand,Crater is interested in producing the product to meet theabove specifications at minimum cost.a. Let X1 � the length of the box; X2 � the depth of the

box; and X3 � the height of the box. Develop anonlinear programming model for Crater Crates.

b. Write the Kuhn–Tucker conditions for yourformulation in part a.

c. Show that the optimal dimensions of the box are not3 feet by 2.5 feet by 2 feet by demonstrating thatthese values do not satisfy the Kuhn–Tuckerconditions.

d. Solve for the optimal dimensions using Excel Solver.e. Interpret the Lagrange multiplier (shadow price) for

each constraint.24. Caramel Heads is an up-and-coming rock group that is

scheduled to play a concert in the University of Missourifootball stadium. The concert promoter, a graduate ofthe University of Missouri business school, hasdeveloped a regression model predicting that ticketdemand, X, is related to ticket price (in dollars), P, by

X � 60,000 � 3000P

a. Develop a total sales revenue function.b. The total sales revenue function is concave. Using

this fact, determine which ticket price will maximizetotal sales revenues for the concert.

c. What attendance is expected, given this ticket price?d. What is the expected sales revenue from ticket prices?

25. A more sophisticated regression model for relating ticketsales and price for the Caramel Heads (problem 24)gives the following relationship between sales and ticketprices:

X � 64,000 � 3200P � 10P3

a. Develop a total sales revenue function.b. The total sales revenue function is concave. Using

this fact, determine the ticket price that will maximizetotal sales revenues for the concert.

c. What attendance is expected, given this ticket price?d. What is the expected revenue from ticket sales given

this ticket price?26. According to a multiple regression model, the

relationship between sales, X, ticket prices, P, andadvertising (in $100’s), A, for the Caramel Heads concert(problem 24) is given by

X � 28,000 � 5000P � 1000A � 5A2

a. If all costs other than the advertising costs are fixed at$100,000, develop a net profit function resulting from

revenues from ticket sales less the variable cost ofadvertising, A, and the fixed costs of $100,000.

b. Develop the necessary conditions for optimality.c. Show that a ticket price of $7.80 and an advertising

expenditure of $9871.78 approximately satisfy thenecessary conditions. Use Excel Solver to verify these quantities. What is the net profit, given thisticket price and advertising expenditure?

27. KarKare Products is introducing a new antitheft device,the AT40, which costs $60 per unit to produce. It plansto sell the device for $100 each. To determine anappropriate production quantity and advertisingstrategy, KarKare test-marketed the product in threesimilar market environments. By spending differentamounts on weekly advertising it determined the effectof advertising on sales. The results are given in thefollowing table:

Weekly Advertising Expenditures ($1000s) AT40 Weekly Sales (Units)

1 1003 1505 175

a. As can be seen from the table, the relationship betweenadvertising expenditures and sales of the AT40 is notlinear. Given the above three points, it is estimated thatthe relationship of sales to advertising expenditures canbe given by a quadratic relationship of the form:

S � aX2 � bX � c

where S denotes the weekly sales and X is the amountspent weekly on advertising. Based on the above datadetermine the values for a, b, and c. [Hint: By lettingX � 1, S � 100, then X � 3, S � 150, and finally X � 5, S � 175, find three linear relationshipsbetween the coefficients a, b, and c and solve thesethree equations in three unknowns.]

b. Based on your answer to part (a), develop and solve amodel for KarKare’s optimal weekly production ofthe AT40 antitheft devices in terms of its advertisingexpenditures X. [Hint: Profit � Revenue � All Costs � $100(S) � $60(S) � $1000(X). The last termis the advertising cost.]

i) What should be the weekly production ofAT40s?

ii) How much should be spent weekly foradvertising?

iii) What is the optimal weekly profit?c. How will your answer to part (b) change if the

maximum weekly advertising expenditure is limitedto (i) $2500? (ii) $1500?

28. Consider the KarKare production problem (problem27). Suppose KarKare is considering introducing twoadditional models of the antitheft device, the AT20 andthe AT50. The AT20 costs only $30 to produce and sellsfor $50; the AT50 costs $150 to produce and sells for$200. Test-market results for these products aresummarized in the following tables:

C a s e S t u d i e s CD-207

Weekly Advertising Expenditures ($1000s) AT20 Weekly Sales (Units)

1 4003 8005 1000

Weekly Advertising Expenditures ($1000s) AT50 Weekly Sales (Units)

1 403 1005 120

a. Develop a quadratic relationship between sales ofAT20 models and advertising expenditures (X2).

b. Develop a quadratic relationship between sales ofAT50 models and advertising expenditures (X3).

c. Sales of these products are projected to beindependent, so KarKare’s total weekly profit simplyequals the sum of the weekly profits for each of theproducts. Develop a total weekly profit function interms of the three products (including the AT40s inproblem 27) and solve for the optimal advertisingexpenditures (X1, X2, and X3). What should be theweekly production quantities (S1, S2, and S3)? What isthe optimal weekly profit?

d. Use a quadratic programming model to solve for theoptimal advertising model if only $7500 can be spentweekly for advertising. What are the optimalproduction quantities?

29. Consider the KarKare production problem of threeproducts (problems 27, 28).a. Suppose that, in addition to the limit on advertising

expenditures, a weekly limit of 40 hours are availablefor production of the antitheft devices. Each AT40requires .15 production hours, each AT20 .12production hours, and each AT50 .16 productionhours. Using the expressions for sales (production)developed in part a of problems 27 and 28, write anonlinear constraint expressing the production timeconstraint in terms of X1, X2, and X3.

b. Write the complete nonlinear model for total weeklyprofit constrained by the maximum limits on weekly

advertising expenditures and production hours. Writethe Kuhn–Tucker conditions for optimality.

c. Solve the problem using a nonlinear programmingmodule. Determine: (i) the optimal weeklyadvertising expenditures for each product; (ii) theweekly production quantities for each product; (iii)the total weekly profit.

d. Show that the Kuhn–Tucker conditions are satisfiedby the answer in part (c).

e. What is the instantaneous increase in profit for: (i) anextra dollar spent on advertising; and (ii) an extraproduction hour?

30. An investor wishes to invest $5000 in two stocks:Microworld and Delphi. Historical data indicate thatMicroworld has an expected annual return of 20% andDelphi 16%, but these are by no means certain andinvolve risk. The risk is measured by the variance of thetotal return, given by

where X1 is the amount invested in Microworld (in$1000s) and X2 is the amount invested in Delphi (in$1000s). Note that risk increases with the totalinvestment in each stock.

One approach to maximizing return whileminimizing risk is to maximize:

Return � Risk

If the coefficients in the expression for risk are consistentwith those in the return function, this model can beexpressed as

MAX 20X1 � 16X2 �subject to the $5000 total investment (and nonnegativityof investment in each stock.)a. Write the Kuhn–Tucker conditions for optimality for

this model.b. Show that the objective function is concave.c. Why are the conditions in part a also sufficient

conditions for this model?d. Show that the optimal solution is to invest $2333.33

thousand in Microworld and $2666.67 thousand inDelphi.

e. What is the value of the dual variable for the $5000constraint? Give an interpretation of this value.

[2X21 � X22 � (X1 � X2)2]

2X21 � X22 � (X1 � X2)2

CASE STUDIES

CASE 1: L e g e n d s , I n c .

Legends, Inc. assembles collections of records from the1950s and 1960s, secures release rights, and then marketsa double CD of approximately 44 classic rock-and-rollrecords through late-night infomercials. In the past, it hasreleased several collections under such names as “The

Doo-Wop Sound,” “Soul Express,” and “Beach BlanketBonanza.” From experience, it has found that, with propermarketing, demand for a double CD grows during thefirst three months of commercials, and then experiences adownward turn in the fourth month. By the fifth month,

CD-208 C H A P T E R 1 3 N o n l i n e a r M o d e l s : D y n a m i c , G o a l , a n d N o n l i n e a r P r o g r a m m i n g

demand has usually fallen so low that continued produc-tion is unprofitable. Thus company policy is to producethe CD and sell it for four months for $19.95 each (plus$4.95 shipping and handling), and then move on to an-other project. Since the revenue per CD is fixed, Legends,Inc. concentrates on minimizing its expenses.

Forecasts of demand, unit manufacturing and advertis-ing costs, and the maximum number of production runs(of 10,000 CDs each) for its latest project, “The Haight-Ashbury Era,” are summarized in the following table.Legends pays $0.60 to store a double CD from one monthto the next. Company policy restricts inventory to at most30,000 double CDs so that if demand is overestimatedduring the production cycle, it will not be stuck with alarge amount of unsold product.

Sales Forecasts and Production Limits for “The Haight-Ashbury Era”

MaximumForecasted Unit Production Number ofMonthly and Promotion Production

Month Demand Costs Runs

1 20,000 $2.40 62 40,000 $3.60 73 60,000 $3.60 44 30,000 $5.00 2

Prepare a detailed report recommending an initial four-month production and inventory policy for Legends, Inc.for “The Haight-Ashbury Era” project.

CASE 2: C i t y o f S t . F r a n c i s

Last year, after fixed expenses were taken off the top, theCity of St. Francis had the following budget for its em-ployees:

AverageEmployees Salary Total

1. AdministrationSupervisors 3 $65,000 $ 195,000Staff 18 $35,000 $ 630,000

21 $ 820,0002. Fire/Paramedic

Supervisors 3 $90,000 $ 270,000Staff 12 $42,000 $ 504,000

15 $ 774,0003. Human Services

Supervisors 3 $60,000 $ 180,000Staff 20 $30,000 $ 600,000

23 $ 780,0004. Library

Supervisors 2 $52,000 $ 104,000Staff 15 $28,000 $ 420,000

17 $ 524,0005. Police Services

Supervisors 3 $85,000 $ 255,000Staff 22 $45,000 $ 990,000

25 $1,245,0006. Public Works

Supervisors 1 $80,000 $ 80,000Staff 10 $34,000 $ 340,000

11 $ 420,0007. Recreation

Supervisors 1 $48,000 $ 48,000Staff 7 $27,000 $ 189,000

8 $ 237,000Miscellaneous

Supervisors 2 $54,000 $ 108,000Staff 15 $30,000 $ 450,000

17 $ 558,000

TOTALS Supervisors 18 $1,240,000Staff 119 $4,123,000

127 $5,363,000

Given current fiscal conditions, funding for employeesalaries will be cut by 20%. All seven departments listed inthe table must be funded and have at least one supervisor.The city wants to determine its employee structure, giventhe following set of priorities:

Priority 1Goal 1: Increase fire/paramedic staff by at least 1.Goal 2: Increase police staff by at least 2.Priority 2Goal 3: Reduce the number of supervisorial positions by

at least 50%.Goal 4: Keep the total number of layoffs to a maximum of

30.Goal 5: Cut no department’s staff by more than 50%.Priority 3Goal 6: Retain at least $250,000 in “miscellaneous” funds

for supervisors or staff.Goal 7: Reduce the overall budget by at least 25% to keep

a “reserve” for next year.

Both priority 1 goals are equally important. For priority 2goals, both goals 4 and 5 are considered twice as impor-tant as goal 3. Goal 6 is twice as important as goal 7.

Prepare a report that details a recommended fundingstrategy for the City of St. Francis. Consider other goalsand priorities you feel may be relevant and incorporatetheir analysis into a “what-if” section of the report. Dis-cuss other factors that should be considered and othermeasures that could be taken, such as reductions insalaries in lieu of layoffs.

Return to Additional ChaptersChapter 1313.1 Introduction to Nonlinear Prpgramming13.2 Dynamic Programming13.3 Computational Properties of Dynamic Programming13.4 Dynamic Programming Examples13.5 Goal Programming13.6 Nonlinear Programming Concepts13.7 Unconstrained Nonlinear Programming13.8 Constrained Nonlinear Programming Problems13.9 SummaryAppendixAppendix 13.1Appendix 13.2Appendix 13.3Appendix 13.4

ProblemsCase StudiesCase 1 Legends Inc.Case 2 City of St. Francis