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Quiz Number 1 Group 1 – North of Newark Thamer AbuDiak Reynald Benoit Jose Lopez Rosele Lynn Dave Neal Deyanira Pena Professor Kenneth D. Lawerence New Jersey Inst. Of Tech

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Problems Assigned Ragsdale 2.13, 2.16, 2.20 3.10, 3.13, 3.16, 3.21, 3.24, 3.28, 3.41, 3.44, 3.45 Dielman 3.6 3.24

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Ragsdale 2.13 by Reynald The marketing manager for Mountain Mist soda needs to decide how many TV spots and magazine ads to run during the next quarter.

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Initial Set up Decision Variable The number of TV spots and magazine ads to run X1 = TV Spot X2 = Magazine Ad Objective function TV spots expected to increase sales by 300,000 cans Magazine ads expected to increase sales by 500,000 cans Mountain Mist makes.05 cents a can MAX: 0.05 * (300,000X1 + 500,000X2 ) Constraints A total of $100,000 may be spent No more than $70,000 may be spent on TV spots No more than $50,000 may be spent on magazine ads 5,000X1 + 2,000X2 <= 100,000 5,000X1 <= 70,000 2,000X1 <= 50,000

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Excel Initial Settings D7 = (B7*B6+C7*C6)*C15 D10 = B10*B6 + C10*C6 D11 = B11*B6 + C11*C6 D12 = B12*B6 + C12*C6 Changing Cells B6 and C6

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Solver

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Results In order to maximize profit Mountain Mist should run 10 TV spots and 25 Magazine ads which will result in $775,000 in profit.

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Ragsdale 2.16 by Rosele Problem: What combination of generators and alternators should Electrotech Corporation manufacture in order to maximize profit? Decision variables: how many generators and alternators should the Electrotech Corporation manufacture? X1 = generatorX2 =alternator Objective Function: How can the Electrotech Corporation get the maximum income? MAX: 250 X1 +150 X2

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Each generator requires 2 hours of wiring, each alternator requires 3 hours of wiring. Electrotech can not exceed a total of 260 hours wiring time. 2 X 1 +3 X 2 < 260 Each generator requires 1 hour of testing time, each alternator requires 2 hours of testing time. Electrothech can not exceed a total of 140 hours testing time. 1X 1 +2X 2 <140 Electrotech decides it needs to make at least 20 generators and 20 alternators. X 1 >20 X 2> 20 Constraints

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MAX: 250 X 1 +150 X 2 Subject to: 2 X 1 +3 X 2 < 260 1X 1 +2X 2 <140 X 1 >20 X 2 >20 LP Model

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Solver Parameters

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Generators Alternators Number to Make 100 20Total Profit Unit Profits 250 150 $28,000 ConstraintsUsed Available Wiring Hrs Required 2 3 260 260 Testing Hrs Required 1 2 140 140 Electrotech Corporation

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If additional wiring time becomes available at a reasonable cost should Electrotech do so? Why or why not? No, Electrotech should not do so because they do not see an increased profit since they are again only making 120 units. Summary

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Generators Alternators Number to Make 100 20Total Profit Unit Profits 245 145 $27,400 Constraints Used Available Wiring Hrs Required2 3 260 500 Testing Hrs Required 1 2 140 140 Electrotech Corporation

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Problem 2-20, Thamer AbuDiak Decision Variables: X 1 : Number of hours that Mine1 worked X 2 : Number of hours that Mine2 worked Objective functions: MIN: 200X 1 +160X 2 Constrains: 6X 1 +2X 2 >= 12 2X 1 +2X 2 >= 8 4X 1 +8X 2 >= 24 X 1 >=0 X 2 >=0 Answer: 1 Hour of Operation/Day for Mine 1 3 Hour of Operation/Day for Mine 2 Before After

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Problem 2-20 cont., Thamer AbuDiak

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Ragsdale 3.10 by Deyanira A. LP Model x1= contemporary tables x2= country tables MAX: 450x1 + 350x2 } revenue Subject to: 2.0x1 + 1.5x2 1000 } router constraint 4.5x1 + 3.0x2 2000 } sander constraint 1.5x1 + 2.5x2 1500 } polisher constraint X1.30 } has to produce at least 30% X2.20 } has to produce at least 20% X1 0 ) simple lower bound 1X2 0 } simple lower bound

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Spread Sheet Furniture Manufacture ContemporaryCountry Number of Makes Total Revenue Unit Revenue $450 $350 $0 Constraints UsedAvailable Router21.501000 Sander4.5302000 Polisher1.52.501500

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Microsoft Excel 11.0 Answer Report Worksheet: [quiz 1 problems 10-13 Ch 3.xls] Sheet1 Target Cell (Max) CellName Original Value Final Value $D$6Unit Revenue Total Revenue $ 227,777.78 $ 227,777.78 Adjustable Cells CellName Original Value Final Value $B$5Number of Makes Contemporary 74.07407407 74.07407407 $C$5Number of Makes Country 555.5555556 555.5555556 Constraints CellNameCell ValueFormulaStatusSlack $D$9Router Used981.4814815$D$9<=$E$9Not Binding18.51851852 $D$10Sander Used2000$D$10<=$E$10Binding0 $D$11Polisher Used1500$D$11<=$E$11Binding0 $B$5Number of Makes Contemporary74.07407407$B$5>=0.3Not Binding73.77407407 $B$5Number of Makes Contemporary74.07407407$B$5>=0Not Binding73.77407407 $C$5Number of Makes Country 555.5555556$C$5>=0Not Binding555.3555556 $C$5Number of Makes Country 555.5555556$C$5>=0.2Not Binding555.3555556

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Optimal Solution Furniture Manufacture ContemporaryCountry Number of Makes 74.07407407555.5555556Total Revenue Unit Revenue $450 $350 $227,777.78 Constraints UsedAvailable Router21.5981.4811000 Sander4.5320002000 Polisher1.52.515001500

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Ragsdale 3.13 by Deyanira A.Lp model. x1= bonds x2= home mortgages x3= car loans x4= personal loans Max:.10x1 +.085 x2 +.095x3 +.125x4 } total return Subject to: x4 162500 } 25% of total portfolio x2 x4 } invest more on mortgages than personal loans x1 x4 } invest more on bond than personal loans x1 + x2 + x3 + x4 = $650,000 } total investment x1,x2,x3,x4 0 } no negativity conditions

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Spreadsheet Bank Portfolio Amount Invested Maximum Return Bonds $0 0 10% Home Mortgages $0 0 8.5% Car Loans $0 0 9.5% Personal Loans $0 $162,500.00 12.5% Total$ 0 Total Investment: $ 0 Total Available: $ 650,000.00

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Microsoft Excel 11.0 Limits Report Worksheet: [quiz 1 problems 10-13 ch 3.xls]Sheet2 Target CellNameValue $D$9Total Return $ 66,625.00 Adjustable LowerTargetUpperTarget CellName ValueLimitResultLimitResult $B$5Bonds Amount Invested $325,000.00 $325,000.00 $66,625.00 $325,000.00 $66,625.00 $B$6Home Mortgages Amount Invested $162,500.00 $162,500.00 $66,625.00 $162,500.00 $66,625.00 $B$7Car Loans Amount Invested $ - $ - $66,625.00 $ - $66,625.00 $B$8Personal Loans Amount Invested $162,500.00 $162,500.00 $66,625.00 $162,500.00 $66,625.00

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Excel 11.0 Answer Report Worksheet: [quiz 1 problems 13 ch 3.xls]Sheet2 Target Cell (Max) CellNameOriginal ValueFinal Value $D$9Total Return $ 66,625.00 $ 66,625.00 Adjustable Cells CellName Original ValueFinal Value $B$5Bonds Amount Invested $ 325,000.00 $ 325,000.00 $B$6Home Mortgages Amount Invested $ 162,500.00 $ 162,500.00 $B$7Car Loans Amount Invested $ - $ - $B$8Personal Loans Amount Invested $ 162,500.00 $ 162,500.00 Constraints CellName Cell ValueFormulaStatusSlack $B$11Total Investment: Amount Invested $ 650,000.00 $B$11=$B$12Binding 0 $B$5Bonds Amount Invested $ 325,000.00 $B$5>=$B$8Not Binding $162,500.00 $B$6Home Mortgages Amount Invested $ 162,500.00 $B$6>=$B$8Binding $ - $C$5Bonds Maximum 0$C$5>=$C$5Binding 0 $C$6Home Mortgages Maximum 0$C$6>=$C$6Binding 0 $C$7Car Loans Maximum 0$C$7>=$C$7Binding 0 $B$5Bonds Amount Invested $ 325,000.00 $B$5>=0Not Binding $325,000.00 $B$6Home Mortgages Amount Invested $ 162,500.00 $B$6>=0Not Binding $162,500.00 $B$7Car Loans Amount Invested $ - $B$7>=0Binding $ - $B$8Personal Loans Amount Invested $ 162,500.00 $B$8>=0Not Binding $162,500.00 $B$8Personal Loans Amount Invested $ 162,500.00 $B$8<=$C$8Binding 0

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Optimal Solution Bank Portfolio Amount Invested Maximum Return Bonds $325,000.00 0 10% Home Mortgages $162,500.00 0 8.5% Car Loans $ 0 9.5% Personal Loans $162,500.00 $162,500.00 12.5% Total$ 66,625.00 Total Investment: $ 650,000.00 Total Available: $ 650,000.00

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Ragsdale 3.16 by Jose Decision Variables M1 = Number of electric trimmers to make M2 = Number of buy trimmers to make B1 = Number of electric trimmers to buy B2 = Number of gas trimmers to buy Objective Function MIN: 55M1 + 85M2 + 67B1 + 95B2 Constraints Subject To: M1 + B1 = 30,000 M2 + B2 = 15,000 0.2M1 + 0.4M2 10,000 0.3M1 + 0.5M2 15,000 0.1M1 + 0.1M2 5,000 M1, M2, B1, B2 0 Solved by Jose F. Lopez

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Ragsdale 3.16 ElectricGas Number toModel - Make30,00010,000 - Buy05,000 Cost to - Make$55$85Total Cost - Buy$67$95$2,975,000 # Available30,00015,000 # Needed30,00015000 Hours Required UsedAvailable - Production0.20.410,000 - Assembly0.30.514,00015,000 - Packaging0.1 4,0005000

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The Solver Solved by Jose F. Lopez

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Ragsdale 2.21 by Jose Decision Variables X1 = Number of workers starting at 12 am X2 = Number of workers starting at 4 am X3 = Number of workers starting at 8 am X4 = Number of workers starting at 12 pm X5 = Number of workers starting at 4 pm X6 = Number of workers starting at 8 pm Objective Function MIN: X1 + X2 + X3 + X4 + X5 + X6 Constraints Subject To: X6 + X1 90 X1 + X2 215 X2 + X3 250 X3 + X4 165 X4 + X5 300 X5 + X6 125 Xi 0 Solved by Jose F. Lopez

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Solution Solved by Jose F. Lopez THEME PARK SCHEDULING Employees Scheduled ToEmployeesMinimum Start At TimeAvailable InEmployees Time PeriodPeriod Time PeriodNeeded 12 am to 4 am 90 4 am to 8 am 250340 215 8 am to 12 pm 0250 12 pm to 4 pm 175 165 4 pm to 8 pm 125300 8 pm to 12 am 0125 Total Employees:640

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The Solver Solved by Jose F. Lopez

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Ragsdale 2.24 by Rosele 3-24 Problem: How many of each type of apartment should the developer produce while leasing 5 one bedroom apartments and 8 two bedroom apartments to a silent partner, having a maximum of 40 units per location, and 40,000 square feet per location?

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Decision variables: how many of each type of apartment should the developer produce? X 1 = efficiencies X 2 = one bedroom apartments X 3 = two bedroom apartments X 4 = three bedroom apartments Objective Function: How can the developer get the maximum income? MAX: 350 X 1 +450 X 2 +550 X 3 +750 X 4

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Constraints: The developer can build no more than 15 one bedroom apartments, 22 two bedroom apartment and 10 three bedroom apartments. As well, the silent partner requires the developer to lease to him 5 one bedroom apartments and 8 two bedroom apartments. Upper and Lower Bounds: X 1 >0 X 2 >5and< 15 X 3 >8 and < 22 X 4 < 10 Each efficiency requires 500 square feet, each one bedroom apartment requires 700 square feet each two bedroom apartment requires 800 square feet and each three bedroom apartment requires 1000 square feet. The developer can not exceed a total of 40,000 square feet in a location. 500 X 1 +700 X 2 +800 X 3 +1,000 X 4 < 40,000 Zoning restrictions only allow 40 or less units per location X 1 +5X 2 + 8X 3 + X 4 <40

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LP Model MAX:350 X 1 +450 X 2 +550 X 3 +750 X 4 Subject to:X 1 >0 X 2 >5 and < 15 X 3 >8 and < 22 X 4 < 10 500 X 1 +700 X 2 +800 X 3 +1,000 X 4 < 40,000 X 1 +5X 2 + 8X 3 + X 4 <40

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Solver Parameters

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Real Estate Development Project Efficiencies 1 Bedroom 2 Bedroom 3 Bedroom Number to Make 0 8 22 10 Total Profit Units to Rent $350 $450 $550 $750 $23,200 Constraints Used Available Sq. Ft Reqd. 500 700 800 1,000 33,200 40,000 Units Required 1 1 1 1 40 40

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Questions C and D C. The optimal solution is to make: no – efficiencies 8 – one bedroom apartments 22 – two bedroom apartments 10- three bedroom apartments D. The number of units to make limits the builders potential income. In this example the builder maxed out at 40 units while only using 33,200 square feet.

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Problem 3-28, Thamer AbuDiak

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Decision Variables: X 11 : Newspaper to be used for Newsprint, X 12 : Newspaper to be used for Packaging. X 21 : Mixed Paper to be used for Newsprint, X 22 : Mixed Paper to be used for Packaging, X 23 : Mixed Paper to be used for Print Stock. X 31 : White Office Paper to be used for Newsprint, X 32 : White Office Paper to be used for Packaging, X 33 : White Office Paper to be used for Print Stock. X 41 : Cardboard to be used for Newsprint, X 42 : Cardboard to be used for Packaging. Objective functions: MIN (6.5+15)/.85 X 11 + (11+15)/.80 X 12 + (9.75+16)/.90 X 21 + (12.25+16)/.90 X 22 + (9.5+16)/.70 X 23 + (4.75+19)/.90 X 31 + (7.75+19)/.85 X 32 + (8.5+19)/.80 X 33 + (7.5+17)/.80 X 41 + (8.5+17)/.70 X 42 Simplifying: 25.29 X 11 + 32.5 X 12 + 28.61 X 21 + 31.39 X 22 + 36.43 X 23 + 26.39 X 31 + 31.47 X 32 + 34.38 X 33 + 30.63 X 41 + 36.43 X 42 Constrains: X 11 + X 21 + X 31 = 500, Newsprint that the company needs to produce. X 12 + X 22 + X 32 + X 42 = 600, Packaging that the company needs to produce. X 13 + X 23 + X 33 = 300, Print Stock that the company needs to produce. X 11 /.85 + X 12 /.80 <= 600, Maximum Newspaper available. X 21 /.90 + X 22 /.80 + X 23 /.70 <= 500, Maximum Mixed Paper available. X 31 /.90 + X 32 /.85 + X 33 /.80 <= 300, Maximum White Office Paper to available. X 41 /.80 + X 42 /.70 <= 400, Maximum Cardboard available. X 11, X 12, X 21, X 22, X 23, X 31, X 32, X 33, X 41, X 42 >= 0, Non Negativity Constrain. Before After Answer BOX: (After Recycling): Newsprint produced from Newspaper:499 Packaging produced from Newspaper:10 Newsprint produced from Mixed Paper:1 Packaging produced from Mixed Paper:56 Print Stock produced from Mixed Paper:300 Newsprint: produced from White Office Paper:0 Packaging produced from White Office Paper: 255 Print Stock produced from White Office Paper :0 Newsprint produced from Mixed Paper:0 Packaging produced from Mixed Paper:279

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Ragsdale 3.41 by Dave GOAL: Minimize Net Financing Costs Jan Feb March April May June Accounts Receivable1.501.001.402.302.001.00 Planned Payments1.801.602.201.200.801.20 Beginning Cash Balance = $400,000? Desired Monthly Balance >=$25,000? Finance OptionAvailable for month Loan term Finance Charge A) Delay Pymt1,2,3,4,5,6 1 month2.0% B) Borrow 75% A/R1,2,3,4,5,6 1 month1.5% C) Short Term Loan 1 6 months1.0% / month Defining the Decision Variables A1, A2, A3, A4, A5, A6 = amount (in $1,000s) financed in option A at the beginning of months 1,2,3,4,5,6, respectively. B1, B2, B3, B4, B5, B6 = amount (in $1,000s) financed in option B at the beginning of months 1,2,3,4,5,6, respectively. C1 = amount (in $1,000s) financed in option C at the beginning of month 1. Eagle's Beach Wear and Gift Shop

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3.41 cont. monthly balance = (total amount available @ beginning of month) - (payment) + A/R + (amount financed) Interest = 0.5% per month

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Ragsdale 3.44

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Ragsdale 3.45

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Dielman 3.6 by Reynald VariableCoefficientStd DevT StatPValue Intercept2.03360.54053.760.001 EPS0.37400.23951.560.126 Standard error = 1.84975; R-Sq = 5.7%; R-Sq(adj) = 3.4% SourceDFSum of Squares Mean Square F StatP Value Regression18.345 2.440.126 Error40136.8643.422 Total41145.208 a)The same regression equation relating dividends to EPS a)DIV = 0.3740EPS + 2.0336

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Cont. b) Is there a linear relationship between dividend yield and EPS? Hypothesis: Ho: B1 = 0, H1: B1 ~= 0 Decision Rule: Reject H0 if t > 2.021 or t < -2.021, Do Not Reject if -2.021 <= t <= 2.021 Test: t = 1.56 Decision: Do not reject c) There is not sufficient evidence to conclude that a linear relationship between dividend yield and EPS d) Construct a 95% confidence interval estimate of B1 0.374 +- ( 2.021 )( 0.2395 ) 0.374 +- 0.4840 e) Construct a 95% confidence interval estimate of B1 2.0336 +- ( 2.021 )( 0.5405 ) 2.0336 +- 1.0924

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Dave Neal / Group North Dielman Problem 3.24 (Dependent Variable): Y = COST is the total cost of the production run. (Independent Variable): X = NUMBER is the number of items produced during that run. Regression calculated using Minitab. Regression Analysis: COST versus NUMBER The regression equation is COST = 28.3 + 2.15 NUMBER Dielman 3.24 by Dave

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b. What percentage of the variation in Y has been explained by the regression? R-Sq = regression sum of squares(SSR)/total sum of squares(SST) = 1813.9/1991.3 = 91.1% Predictor Coef SE Coef T P Constant 28.311 4.083 6.93 0.000 NUMBER 2.1549 0.1437 15.00 0.000 S = 2.84016 R-Sq = 91.1% R-Sq(adj) = 90.7% Analysis of Variance Source DF SS MS F P Regression 1 1813.9 1813.9 224.86 0.000 Residual Error 22 177.5 8.1 Total 23 1991.3

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c. Are Y and X linearly related? Conduct a hypothesis test to answer this question and use a 5% level of significance. Hypothesis to be tested: Is the total cost of a production run linearly related to the number of items produced during that run? The hypotheses are: H 0 : 1 = 0 (Cost does not change when number of items produced increases) Ha: 1 0 (Cost does change) The decision rule: If the data fits well, Mean Square Regression (MSR) will be large compared to the Mean Square due to Error (MSE). Reject H0 if t > 2.074 or if t < -2.074 The test statistic: F statistic = (MSR) / (MSE). Decision: F statistic = 1813.9 / 8.1 = 223.9 The MSR is large relative to the MSE. T = 15.00 > 2.074 (reject H 0 ). Conclusion: There is a significant relationship between project size and cost.

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d. Estimate the fixed cost involved in the production process. Find a point and a 95% confidence interval estimate. Fixed cost is equal to the slope of the equation = $28.31 Point estimate = sample 95% confidence mean Y t 24-2 = 2.074 Fixed Cost = b 0 +/-t n-2 s b0 = 28.311+/-(2.074)4.083 = 28.311+/-8.468 95% sure that the fixed cost is between $19.84 and $36.78

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e. Estimate the variable cost involved in the production process. Find a point estimate and a 95% confidence interval estimate. Calculate using one-way ANOVA. Variable Cost = 2.1549 x NUMBER. COST Mean = 78.25 78.25 = (28.311+/-8.468) + Variable Cost 95% sure that the variable cost is between $41.47 and $58.41 One-way ANOVA: COST versus NUMBER Source DF SS MS F P NUMBER 14 1963.92 140.28 46.05 0.000 Error 9 27.42 3.05 Total 23 1991.33 S = 1.745 R-Sq = 98.62% R-Sq(adj) = 96.48% Individual 95% CIs For Mean

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Level N Mean StDev -+---------+---------+---------+-------- 20 1 74.50 * (--*--) 22 2 78.25 1.06 (-*--) 23 1 81.00 * (--*---) 24 1 80.50 * (--*--) 25 2 82.25 1.77 (--*-) 26 1 81.50 * (--*--) 27 2 83.75 1.77 (--*-) 28 2 85.75 1.06 (-*--) 29 2 86.75 1.77 (-*--) 30 3 91.83 2.02 (-*) 31 1 96.00 * (--*--) 32 2 98.25 1.77 (-*-) 33 2 101.50 2.12 (--*-) 34 1 103.00 * (--*--) 35 1 109.00 * (--*--) -+---------+---------+---------+-------- 72 84 96 108

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