Presentation on theme: "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."— Presentation transcript:
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
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.
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 * (300,000X ,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
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.
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 X X2
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
MAX: 250 X X 2 Subject to: 2 X 1 +3 X 2 < 260 1X 1 +2X 2 <140 X 1 >20 X 2 >20 LP Model
Generators Alternators Number to Make Total Profit Unit Profits $28,000 ConstraintsUsed Available Wiring Hrs Required Testing Hrs Required Electrotech Corporation
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
Generators Alternators Number to Make Total Profit Unit Profits $27,400 Constraints Used Available Wiring Hrs Required Testing Hrs Required Electrotech Corporation
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 X 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
Problem 2-20 cont., Thamer AbuDiak
Ragsdale 3.10 by Deyanira A. LP Model x1= contemporary tables x2= country tables MAX: 450x x2 } revenue Subject to: 2.0x x } router constraint 4.5x x } sander constraint 1.5x x } 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
Spread Sheet Furniture Manufacture ContemporaryCountry Number of Makes Total Revenue Unit Revenue $450 $350 $0 Constraints UsedAvailable Router Sander Polisher
Microsoft Excel 11.0 Answer Report Worksheet: [quiz 1 problems Ch 3.xls] Sheet1 Target Cell (Max) CellName Original Value Final Value $D$6Unit Revenue Total Revenue $ 227, $ 227, Adjustable Cells CellName Original Value Final Value $B$5Number of Makes Contemporary $C$5Number of Makes Country Constraints CellNameCell ValueFormulaStatusSlack $D$9Router Used $D$9<=$E$9Not Binding $D$10Sander Used2000$D$10<=$E$10Binding0 $D$11Polisher Used1500$D$11<=$E$11Binding0 $B$5Number of Makes Contemporary $B$5>=0.3Not Binding $B$5Number of Makes Contemporary $B$5>=0Not Binding $C$5Number of Makes Country $C$5>=0Not Binding $C$5Number of Makes Country $C$5>=0.2Not Binding
Optimal Solution Furniture Manufacture ContemporaryCountry Number of Makes Total Revenue Unit Revenue $450 $350 $227, Constraints UsedAvailable Router Sander Polisher
Ragsdale 3.13 by Deyanira A.Lp model. x1= bonds x2= home mortgages x3= car loans x4= personal loans Max:.10x x x x4 } total return Subject to: x } 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
Spreadsheet Bank Portfolio Amount Invested Maximum Return Bonds $0 0 10% Home Mortgages $ % Car Loans $ % Personal Loans $0 $162, % Total$ 0 Total Investment: $ 0 Total Available: $ 650,000.00
Optimal Solution Bank Portfolio Amount Invested Maximum Return Bonds $325, % Home Mortgages $162, % Car Loans $ 0 9.5% Personal Loans $162, $162, % Total$ 66, Total Investment: $ 650, Total Available: $ 650,000.00
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, M M2 10, M M2 15, M M2 5,000 M1, M2, B1, B2 0 Solved by Jose F. Lopez
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
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 am to 12 pm pm to 4 pm pm to 8 pm pm to 12 am 0125 Total Employees:640
The Solver Solved by Jose F. Lopez
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?
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 X X X 4
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 X 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
LP Model MAX:350 X X X X 4 Subject to:X 1 >0 X 2 >5 and < 15 X 3 >8 and < 22 X 4 < X X X 3 +1,000 X 4 < 40,000 X 1 +5X 2 + 8X 3 + X 4 <40
Real Estate Development Project Efficiencies 1 Bedroom 2 Bedroom 3 Bedroom Number to Make Total Profit Units to Rent $350 $450 $550 $750 $23,200 Constraints Used Available Sq. Ft Reqd ,000 33,200 40,000 Units Required
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.
Problem 3-28, Thamer AbuDiak
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 + ( )/.90 X 21 + ( )/.90 X 22 + (9.5+16)/.70 X 23 + ( )/.90 X 31 + ( )/.85 X 32 + (8.5+19)/.80 X 33 + (7.5+17)/.80 X 41 + (8.5+17)/.70 X 42 Simplifying: X X X X X X X X X 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
Ragsdale 3.41 by Dave GOAL: Minimize Net Financing Costs Jan Feb March April May June Accounts Receivable Planned Payments 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
Dielman 3.6 by Reynald VariableCoefficientStd DevT StatPValue Intercept EPS Standard error = ; R-Sq = 5.7%; R-Sq(adj) = 3.4% SourceDFSum of Squares Mean Square F StatP Value Regression Error Total a)The same regression equation relating dividends to EPS a)DIV = EPS
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 > or t < , Do Not Reject if <= t <= 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 B ( )( ) e) Construct a 95% confidence interval estimate of B ( )( )
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 = NUMBER Dielman 3.24 by Dave
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 P Constant NUMBER S = R-Sq = 91.1% R-Sq(adj) = 90.7% Analysis of Variance Source DF SS MS F P Regression Residual Error Total
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 > or if t < The test statistic: F statistic = (MSR) / (MSE). Decision: F statistic = / 8.1 = The MSR is large relative to the MSE. T = > (reject H 0 ). Conclusion: There is a significant relationship between project size and cost.
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 = Fixed Cost = b 0 +/-t n-2 s b0 = /-(2.074)4.083 = / % sure that the fixed cost is between $19.84 and $36.78
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 = = ( /-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 Error Total S = R-Sq = 98.62% R-Sq(adj) = 96.48% Individual 95% CIs For Mean