b)

B34567891011121314151617181920CDEFGHNPV ($millions)WarehouseFactoryCapital Required($millions)WarehouseFactoryLos Angeles68Los Angeles56San Francisco45San Francisco23San Diego57San Diego34CapitalSpent10<=CapitalAvailable10Build?WarehouseFactoryLos Angeles0<=0San Francisco0<=1San Diego1<=117TotalMaximumWarehousesWarehouses1<=1Total NPV ($millions) 9.8 9.9

See the articles in Interfaces.

a) Let EM = 1 if Eve does the marketing; 0 otherwise

EC = 1 if Eve does the cooking; 0 otherwise ED = 1 if Eve does the dishwashing; 0 otherwise EL = 1 if Eve does the laundry; 0 otherwise SM = 1 if Steven does the marketing; 0 otherwise SC = 1 if Steven does the cooking; 0 otherwise SD = 1 if Steven does the dishwashing; 0 otherwise SL = 1 if Steven does the laundry; 0 otherwise

Minimize Time (hours) = 4.5EM + 7.8EC + 3.6ED + 2.9EL + 4.9SM + 7.2SC + 4.3SD + 3.1SL subject to EM + EC + ED + EL = 2 (each person does 2 tasks) SM + SC + SD + SL = 2 EM + SM = 1 (each task is done by 1 person) EC + SC = 1 ED + SD = 1 EL + SL = 1

and EM, EC, ED, EL, SM, SC, SD, SL are binary variables. b)

A123456789101112BCDEFGH

Time Needed (hours)MarketingEve4.5Steven4.9Does Task?MarketingEve1Steven0Total1=1Cooking7.87.2Dishwashing3.64.3Laundry2.93.1TasksPerformed2=2=Cooking011=1Dishwashing101=1Laundry011=122Total Time (hours)18.4 9-9

9.10

a) Let x1 = 1 if invest in project 1; 0 otherwise x2 = 1 if invest in project 2; 0 otherwise x3 = 1 if invest in project 3; 0 otherwise x4 = 1 if invest in project 4; 0 otherwise x5 = 1 if invest in project 5; 0 otherwise

Maximize NPV ($million) = 1x1 + 1.8x2 + 1.6x3 + 0.8x4 + 1.4x5 subject to 6x1 + 12x2 + 10x3 + 4x4 + 8x5 ≤ 20 ($million capital available) and x1, x2, x3, x4, x5 are binary variables. b)

A12345678910BCDEFGHI

Estimated Profit($million)CapitalProject 11Project 21.8Project 31.6Project 40.8Project 51.4CapitalSpent($million)20<=CapitalAvailable($million)20Total Profit($million)3.46Capital Required for Project ($million)12104Project 20Project 31Project 418Project 1Undertake?1Project 50 c)

A121314151617181920212223BCDEFGCapitalAvailable($million)1618202224262830ABProject 1101101111CUndertake?Project 2Project 3Project 4011101001011011010100011101DEFGHProject 5001011111IJTotalProfit($million)3.42.63.23.43.844.24.85K 9.11 a)

123456789101112131415Estimated Profit($million)Capital117210Investment Opportunity3451519761379<=CapitalAvailable100Total Profit($million)41CapitalCapital Required for Investment Opportunity ($million)Spent43283448173223100Investment Opportunity1234510100<=<=only if (1 or 2)11(1 or 2)(3 or 4)11<=<=116071Undertake? 9-10

b)

A1819202122232425262728293031323334BCDEFGHICapitalAvailable($million)8090100110120130140150160170180190200AB111011111111111CUndertake Investment Opportunity?234560100001000110000100001001010100101100111010110101100111001110011100111DEF710110100111111GTotalProfit($million)4132344145485256616165656565 H9.12

12345678910111213141516171819Best TimeStrokeBackstrokeBreaststrokeButterflyCarl37.743.433.3Chris32.933.128.5David33.842.238.9Tony37.034.730.4Ken35.441.833.6Freestyle29.226.429.628.531.1AssignmentsStrokeBackstrokeBreaststrokeButterflyCarl000Chris001David100Tony010Ken000Total Assigned111===Demand111Freestyle100001=1TotalAssignments11110<=<=<=<=<=Supply11111Total Time126.2 Mutually exclusive alternatives: Each swimmer can only swim one stroke. Each stroke can only be swum by one swimmer.

9-11

9.13

A12345678910111213141516171819BCDEFGHIProduct 1Start-up Cost$50,000Marginal Revenue$70Product 2$40,000$60Product 3$70,000$90Product 4$60,000$80ResourceUsed6,000<=12,000<=ModifiedResourceAvailable6,00015,999ResourceAvailable6,0006,000Constraint 1Constraint 254Resource Used per Unit Produced364635Product 30<=00<=only if (1 or 2)1Product 22,000<=9,9991Product 40<=00<=10Units ProducedProduct 10<=Only if Setup0Setup?0Total1<=Max Products2Revenue$120,000Setup Cost$40,000Total Profit$80,000Which Constraint (0 = Constraint 1, 1 = Constraint 2): 9.14

A123456789101112131415161718BCDEFGStart-up CostUnit ProfitToy 1$50,000$10Toy 2$80,000$15ResourceUsed560<=700<=ModifiedHoursAvailable10,499700HoursAvailable500700Factory 1Factory 2Hours Used perUnit Produced0.020.0250.0250.04Toy 128,000<=99,9991Toy 20<=00Units ProducedOnly if SetupSetup?Gross Profit$280,000Setup Cost$50,000Net Profit$230,000Which Factory (0 = Factory 1, 1 = Factory 2)? ABCDE1FG 9.15

1234567891011Startup Cost ($million)Marginal net Revenue ($million)Capacity Used per PlaneMaximum OrderStart Up?Planes to ProduceMaximum Order (if start up)Customer 1322000<=0Customer 22340!2<=2Customer 300.820Q1<=5CapacityUsed100%<=CapacityAvailable100&.84.8($million)Total Startup CostTotal RevenueTotal Profit 9-12