3Ragsdale 2.13 by ReynaldThe marketing manager for Mountain Mist soda needs to decide how many TV spots and magazine ads to run during the next quarter.
4Initial Set up Decision Variable Objective function Constraints The number of TV spots and magazine ads to runX1 = TV SpotX2 = Magazine AdObjective functionTV spots expected to increase sales by 300,000 cansMagazine ads expected to increase sales by 500,000 cansMountain Mist makes .05 cents a canMAX: 0.05 * (300,000X ,000X2 )ConstraintsA total of $100,000 may be spentNo more than $70,000 may be spent on TV spotsNo more than $50,000 may be spent on magazine ads5,000X1 + 2,000X2 <= 100,0005,000X1 <= 70,0002,000X1 <= 50,000
7ResultsIn order to maximize profit Mountain Mist should run 10 TV spots and 25 Magazine ads which will result in $775,000 in profit.
8Ragsdale 2.16 by RoseleProblem: 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 = generator X2 =alternatorObjective Function: How can the Electrotech Corporation get themaximum income?MAX: 250 X X2
9ConstraintsEach 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 X1 +3 X2 < 260Each 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.1X1 +2X2 <140Electrotech decides it needs to make at least 20 generators and 20 alternators.X1>20X2>20
10LP Model MAX: 250 X1 +150 X2 Subject to: 2 X1 +3 X2 < 260
12Electrotech Corporation Generators AlternatorsNumber to Make Total ProfitUnit Profits $28,000Constraints Used AvailableWiring Hrs RequiredTesting Hrs Required
13SummaryIf 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.
14Electrotech Corporation Generators AlternatorsNumber to Make Total ProfitUnit Profits $27,400Constraints Used AvailableWiring Hrs RequiredTesting Hrs Required
15Problem 2-20, Thamer AbuDiak Decision Variables:X1: Number of hours that Mine1 workedX2: Number of hours that Mine2 workedObjective functions:MIN: 200X1+160X2Constrains:6X1+2X2 >= 122X1+2X2 >= 84X1+8X2 >= 24X1 >=0X2 >=0Answer:1 Hour of Operation/Day for Mine 13 Hour of Operation/Day for Mine 2BeforeAfter
17Ragsdale 3.10 by Deyanira A. LP Model x1= contemporary tables x2= country tablesMAX: 450x x2 } revenueSubject to:2.0x x2 1000 } router constraint4.5x x2 2000 } sander constraint1.5x x2 1500 } polisher constraintX .30 } has to produce at least 30%X .20 } has to produce at least 20%X 0 ) simple lower bound1X 0 } simple lower bound
18Spread Sheet Furniture Manufacture Contemporary Country Furniture Manufacture Contemporary Country Number of Makes Total Revenue Unit Revenue $ $350 $0 Constraints Used AvailableRouterSanderPolisher
19Microsoft Excel 11.0 Answer Report Worksheet: [quiz 1 problems Ch 3.xls] Sheet1Target Cell (Max)Cell Name Original Value Final Value$D$6 Unit Revenue Total Revenue $ , $ 227,777.78Adjustable CellsCell Name Original Value Final Value$B$5 Number of Makes Contemporary$C$5 Number of Makes CountryConstraintsCell Name Cell Value Formula Status Slack$D$9 Router Used $D$9<=$E$9 Not Binding$D$10 Sander Used 2000 $D$10<=$E$10 Binding 0$D$11 Polisher Used 1500 $D$11<=$E$11 Binding 0$B$5 Number of Makes Contemporary $B$5>=0.3 Not Binding$B$5 Number of Makes Contemporary $B$5>=0 Not Binding$C$5 Number of Makes Country $C$5>=0 Not Binding$C$5 Number of Makes Country $C$5>=0.2 Not Binding
20Optimal Solution Furniture Manufacture Contemporary Country Contemporary Country Number of Makes Total Revenue Unit Revenue $ $ $227, Constraints Used AvailableRouterSanderPolisher
21Ragsdale 3.13 by Deyanira Lp model. x1= bonds x2= home mortgages x3= car loansx4= personal loansMax: .10x x x x4 } total returnSubject to:x4 } 25% of total portfoliox2 x } invest more on mortgages than personal loansx1 x } invest more on bond than personal loansx1 + x2 + x3 + x4 = $650,000 } total investmentx1,x2,x3,x4 0 } no negativity conditions
23Microsoft Excel 11.0 Limits Report Worksheet: [quiz 1 problems ch 3.xls]Sheet2TargetCell Name Value$D$9 Total Return $ 66,625.00Adjustable Lower Target Upper TargetCell Name Value Limit Result Limit Result$B$5 Bonds Amount Invested $325, $325, $66, $325, $66,625.00$B$6 Home Mortgages Amount Invested $162, $162, $66, $162, $66,625.00$B$7 Car Loans Amount Invested $ $ $66, $ $66,625.00$B$8 Personal Loans Amount Invested $162, $162, $66, $162, $66,625.00
24Excel 11.0 Answer ReportWorksheet: [quiz 1 problems 13 ch 3.xls]Sheet2Target Cell (Max)Cell Name Original Value Final Value$D$9 Total Return $ , $ 66,625.00Adjustable CellsCell Name Original Value Final Value$B$5 Bonds Amount Invested $ , $ 325,000.00$B$6 Home Mortgages Amount Invested $ , $ 162,500.00$B$7 Car Loans Amount Invested $ $$B$8 Personal Loans Amount Invested $ , $ 162,500.00ConstraintsCell Name Cell Value Formula Status Slack$B$11 Total Investment: Amount Invested $ , $B$11=$B$12 Binding$B$5 Bonds Amount Invested $ , $B$5>=$B$8 Not Binding $162,500.00$B$6 Home Mortgages Amount Invested $ , $B$6>=$B$8 Binding $$C$5 Bonds Maximum $C$5>=$C$5 Binding$C$6 Home Mortgages Maximum $C$6>=$C$6 Binding$C$7 Car Loans Maximum $C$7>=$C$7 Binding$B$5 Bonds Amount Invested $ , $B$5>=0 Not Binding $325,000.00$B$6 Home Mortgages Amount Invested $ , $B$6>=0 Not Binding $162,500.00$B$7 Car Loans Amount Invested $ $B$7>=0 Binding $$B$8 Personal Loans Amount Invested $ , $B$8>=0 Not Binding $162,500.00$B$8 Personal Loans Amount Invested $ , $B$8<=$C$8 Binding
26Ragsdale 3.16 by Jose Decision Variables Objective Function M1 = Number of electric trimmers to makeM2 = Number of buy trimmers to makeB1 = Number of electric trimmers to buyB2 = Number of gas trimmers to buyObjective FunctionMIN: 55M1 + 85M2 + 67B1 + 95B2ConstraintsSubject To: M1 + B1 = 30,000M2 + B2 = 15,0000.2M M2 10,0000.3M M2 15,0000.1M M2 5,000M1, M2, B1, B2 0Solved by Jose F. Lopez
27Ragsdale 3.16 Electric Gas Number to Model - Make 30,000 10,000 - Buy 5,000Cost to- Make$55$85Total Cost- Buy$67$95$2,975,000Hours RequiredUsedAvailable- Production0.20.410,000- Assembly0.30.514,00015,000- Packaging0.14,0005000# Available30,00015,000# Needed15000
29Ragsdale 2.21 by Jose Decision Variables Objective Function X1 = Number of workers starting at 12 amX2 = Number of workers starting at 4 amX3 = Number of workers starting at 8 amX4 = Number of workers starting at 12 pmX5 = Number of workers starting at 4 pmX6 = Number of workers starting at 8 pmObjective FunctionMIN: X1 + X2 + X3 + X4 + X5 + X6ConstraintsSubject To: X6 + X1 90X1 + X2 215X2 + X3 250X3 + X4 165X4 + X5 300X5 + X6 125Xi 0Solved by Jose F. Lopez
30Solution Solved by Jose F. Lopez THEME PARK SCHEDULING Employees EmployeesScheduled ToMinimumStart At TimeAvailable InTime PeriodPeriodNeeded12 am to 4 am904 am to 8 am2503402158 am to 12 pm12 pm to 4 pm1751654 pm to 8 pm1253008 pm to 12 amTotal Employees:640Solved by Jose F. Lopez
32Ragsdale 2.24 by Rosele3-24Problem: 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?
33Objective Function: How can the developer get the maximum income? Decision variables: how many of each type of apartment should the developer produce?X1 = efficienciesX2 = one bedroom apartmentsX3 = two bedroom apartmentsX4 = three bedroom apartmentsObjective Function: How can the developer get the maximum income?MAX: 350 X X X X4
34Constraints: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:X1 >0X2 >5and< 15X3 >8 and < 22X4 < 10Each 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 X X3 +1,000 X4 < 40,000Zoning restrictions only allow 40 or less units per locationX1 +5X2 + 8X3 + X4<40
35LP Model MAX: 350 X1 +450 X2 +550 X3 +750 X4 Subject to: X1>0 X2 >5 and < 15X3 >8 and < 22X4 < 10500 X X X3 +1,000 X4 < 40,000X1 +5X2 + 8X3 + X4<40
37Real Estate Development Project Efficiencies 1 Bedroom Bedroom 3 BedroomNumber to Make Total ProfitUnits to Rent $ $ $ $ $23,200Constraints Used AvailableSq. Ft Reqd , , ,000Units Required
38Questions C and D C. The optimal solution is to make: no – efficiencies8 – one bedroom apartments22 – two bedroom apartments10- three bedroom apartmentsD. 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.
40Before After Decision Variables: X11: Newspaper to be used for Newsprint, X12: Newspaper to be used for Packaging.X21: Mixed Paper to be used for Newsprint, X22: Mixed Paper to be used for Packaging, X23: Mixed Paper to be used for Print Stock.X31: White Office Paper to be used for Newsprint, X32: White Office Paper to be used for Packaging, X33: White Office Paper to be used for Print Stock.X41: Cardboard to be used for Newsprint, X42: Cardboard to be used for Packaging.Objective functions:MIN (6.5+15)/.85 X11 + (11+15)/.80 X12 + ( )/.90 X21 + ( )/.90 X22 + (9.5+16)/.70 X23 + ( )/.90 X31 + ( )/.85 X32+ (8.5+19)/.80 X33+ (7.5+17)/.80 X41+ (8.5+17)/.70 X42Simplifying: X X X X X X X X X X42Constrains:X11 + X21 + X = 500, Newsprint that the company needs to produce.X12 + X22 + X32 + X42 = 600, Packaging that the company needs to produce.X13 + X23 + X = 300, Print Stock that the company needs to produce.X11/.85 + X12/ <= 600, Maximum Newspaper available.X21/.90 + X22/ X23/.70 <= 500, Maximum Mixed Paper available.X31/.90 + X32/ X33/.80 <= 300, Maximum White Office Paper to available.X41/.80 + X42/ <= 400, Maximum Cardboard available.X11, X12, X21, X22, X23, X31, X32, X33, X41, X >= 0, Non Negativity Constrain.Answer BOX: (After Recycling):Newsprint produced from Newspaper: 499Packaging produced from Newspaper: 10Newsprint produced from Mixed Paper: 1Packaging produced from Mixed Paper: 56Print Stock produced from Mixed Paper: 300Newsprint: produced from White Office Paper: 0Packaging produced from White Office Paper: 255Print Stock produced from White Office Paper : 0Newsprint produced from Mixed Paper: 0Packaging produced from Mixed Paper: 279
41GOAL: Minimize Net Financing Costs Ragsdale 3.41 by DaveEagle's Beach Wear and Gift ShopGOAL: Minimize Net Financing CostsJan Feb March April May JuneAccounts ReceivablePlanned PaymentsBeginning Cash Balance = $400,000?Desired Monthly Balance >=$25,000?Finance Option Available for month Loan term Finance ChargeA) Delay Pymt 1,2,3,4,5, month 2.0%B) Borrow 75% A/R 1,2,3,4,5, month 1.5%C) Short Term Loan months 1.0% / monthDefining the Decision VariablesA1, 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.
423.41 cont.monthly balance = (total amount beginning of month) - (payment) + A/R + (amount financed)Interest = 0.5% per month
45Dielman 3.6 by ReynaldVariableCoefficientStd DevT StatPValueIntercept2.03360.54053.760.001EPS0.37400.23951.560.126Standard error = ; R-Sq = 5.7%; R-Sq(adj) = 3.4%SourceDFSum of SquaresMean SquareF StatP ValueRegression18.3452.440.126Error403.422Total41The same regression equation relating dividends to EPSDIV = EPS
46Cont.b) Is there a linear relationship between dividend yield and EPS?Hypothesis: Ho: B1 = 0, H1: B1 ~= 0Decision Rule: Reject H0 if t > or t < , Do Not Reject if <= t <= 2.021Test: t = 1.56Decision: Do not rejectc) There is not sufficient evidence to conclude that a linear relationship between dividend yield and EPSd) Construct a 95% confidence interval estimate of B1( )( )e) Construct a 95% confidence interval estimate of B1( )( )
47Dielman 3.24 by Dave 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 NUMBERThe regression equation isCOST = NUMBER
49Predictor Coef SE Coef T P 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) = / = 91.1%Predictor Coef SE Coef T PConstantNUMBERS = R-Sq = 91.1% R-Sq(adj) = 90.7%Analysis of VarianceSource DF SS MS F PRegressionResidual ErrorTotal
50Ha: 1 ≠ 0 (Cost does change) 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:H0: 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 > or if t <The test statistic: F statistic = (MSR) / (MSE).Decision: F statistic = / 8.1 = 223.9The MSR is large relative to the MSE.T = > (reject H0).Conclusion: There is a significant relationship between project size and cost.
51d. Estimate the fixed cost involved in the production process 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 = $ Point estimate = sample 95% confidence mean Y t24-2 = Fixed Cost = b0+/-tn-2sb0= /-(2.074) = / % sure that the fixed cost is between $19.84 and $36.78
52e. Estimate the variable cost involved in the production process 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 = x NUMBER.COST Mean = 78.2578.25 = ( /-8.468) + Variable Cost95% sure that the variable cost is between $41.47 and $58.41One-way ANOVA: COST versus NUMBERSource DF SS MS F PNUMBERErrorTotalS = R-Sq = 98.62% R-Sq(adj) = 96.48%Individual 95% CIs For Mean
53Level N Mean StDev -+---------+---------+---------+-------- * (--*--)(-*--)* (--*---)* (--*--)(--*-)* (--*--)(--*-)(-*--)(-*--)(-*)* (--*--)(-*-)(--*-)* (--*--)* (--*--)