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

Problems Assigned Ragsdale Dielman 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

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 Objective function Constraints 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

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

Solver

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 = generator X2 =alternator Objective Function: How can the Electrotech Corporation get the maximum income? MAX: 250 X1 +150 X2

Constraints 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 X1 +3 X2 < 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. 1X1 +2X2 <140 Electrotech decides it needs to make at least 20 generators and 20 alternators. X1>20 X2>20

LP Model MAX: 250 X1 +150 X2 Subject to: 2 X1 +3 X2 < 260

Solver Parameters

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

Summary 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.

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

Problem 2-20, Thamer AbuDiak Decision Variables: X1: Number of hours that Mine1 worked X2: Number of hours that Mine2 worked Objective functions: MIN: 200X1+160X2 Constrains: 6X1+2X2 >= 12 2X1+2X2 >= 8 4X1+8X2 >= 24 X1 >=0 X2 >=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: 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

Spread Sheet Furniture Manufacture Contemporary Country               Furniture Manufacture                 Contemporary Country     Number of Makes     Total Revenue   Unit Revenue $450 $350 $0   Constraints     Used Available Router 2 1.5 0 1000 Sander 4.5 3 0 2000 Polisher 1.5 2.5 0 1500         

Microsoft Excel 11.0 Answer Report Worksheet: [quiz 1 problems 10-13 Ch 3.xls] Sheet1 Target Cell (Max) Cell Name Original Value Final Value $D$6 Unit Revenue Total Revenue $ 227,777.78 $ 227,777.78 Adjustable Cells Cell Name Original Value Final Value $B$5 Number of Makes Contemporary 74.07407407 74.07407407 $C$5 Number of Makes Country 555.5555556 555.5555556 Constraints Cell Name Cell Value Formula Status Slack $D$9 Router Used 981.4814815 $D$9<=$E$9 Not Binding 18.51851852 $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 74.07407407 $B$5>=0.3 Not Binding 73.77407407 $B$5 Number of Makes Contemporary 74.07407407 $B$5>=0 Not Binding 73.77407407 $C$5 Number of Makes Country 555.5555556 $C$5>=0 Not Binding 555.3555556 $C$5 Number of Makes Country 555.5555556 $C$5>=0.2 Not Binding 555.3555556

Optimal Solution Furniture Manufacture Contemporary Country             Contemporary Country     Number of Makes 74.07407407 555.5555556 Total Revenue   Unit Revenue $450 $350 $227,777.78   Constraints     Used Available Router 2 1.5 981.481 1000 Sander 4.5 3 2000 2000 Polisher 1.5 2.5 1500 1500

Ragsdale 3.13 by Deyanira 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

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

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

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

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

Ragsdale 3.16 by Jose Decision Variables Objective Function 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

Ragsdale 3.16 Electric Gas Number to Model - Make 30,000 10,000 - Buy 5,000 Cost to - Make $55 $85 Total Cost - Buy $67 $95 $2,975,000 Hours Required Used Available - Production 0.2 0.4 10,000 - Assembly 0.3 0.5 14,000 15,000 - Packaging 0.1 4,000 5000 # Available 30,000 15,000 # Needed 15000

The Solver Solved by Jose F. Lopez

Ragsdale 2.21 by Jose Decision Variables Objective Function 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   Employees Scheduled To Minimum Start At Time Available In Time Period Period Needed 12 am to 4 am 90 4 am to 8 am 250 340 215 8 am to 12 pm 12 pm to 4 pm 175 165 4 pm to 8 pm 125 300 8 pm to 12 am Total Employees: 640 Solved by Jose F. Lopez

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?

Objective Function: How can the developer get the maximum income? Decision variables: how many of each type of apartment should the developer produce? X1 = efficiencies X2 = one bedroom apartments X3 = two bedroom apartments X4 = three bedroom apartments Objective Function: How can the developer get the maximum income? MAX: 350 X1 +450 X2 +550 X3 +750 X4

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: X1 >0 X2 >5and< 15 X3 >8 and < 22 X4 < 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 X1 +700 X2 +800 X3 +1,000 X4 < 40,000 Zoning restrictions only allow 40 or less units per location X1 +5X2 + 8X3 + X4<40

LP Model MAX: 350 X1 +450 X2 +550 X3 +750 X4 Subject to: X1>0 X2 >5 and < 15 X3 >8 and < 22 X4 < 10 500 X1 +700 X2 +800 X3 +1,000 X4 < 40,000 X1 +5X2 + 8X3 + X4<40

Solver Parameters

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

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

Before 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 + (9.75+16)/.90 X21 + (12.25+16)/.90 X22 + (9.5+16)/.70 X23 + (4.75+19)/.90 X31 + (7.75+19)/.85 X32+ (8.5+19)/.80 X33+ (7.5+17)/.80 X41+ (8.5+17)/.70 X42 Simplifying: 25.29 X11 + 32.5 X12 + 28.61 X21 + 31.39 X22 + 36.43 X23 + 26.39 X31 + 31.47 X32+ 34.38 X33+ 30.63 X41+ 36.43 X42 Constrains: X11 + X21 + X31 = 500, Newsprint that the company needs to produce. X12 + X22 + X32 + X42 = 600, Packaging that the company needs to produce. X13 + X23 + X33 = 300, Print Stock that the company needs to produce. X11/.85 + X12/.80 <= 600, Maximum Newspaper available. X21/.90 + X22/.80 + X23/.70 <= 500, Maximum Mixed Paper available. X31/.90 + X32/.85 + X33/.80 <= 300, Maximum White Office Paper to available. X41/.80 + X42/.70 <= 400, Maximum Cardboard available. X11, X12, X21, X22, X23, X31, X32, X33, X41, X42 >= 0, Non Negativity Constrain. 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

GOAL: Minimize Net Financing Costs Ragsdale 3.41 by Dave Eagle's Beach Wear and Gift Shop GOAL: Minimize Net Financing Costs Jan Feb March April May June Accounts Receivable 1.50 1.00 1.40 2.30 2.00 1.00 Planned Payments 1.80 1.60 2.20 1.20 0.80 1.20 Beginning Cash Balance = $400,000? Desired Monthly Balance >=$25,000? Finance Option Available for month Loan term Finance Charge A) Delay Pymt 1,2,3,4,5,6 1 month 2.0% B) Borrow 75% A/R 1,2,3,4,5,6 1 month 1.5% C) Short Term Loan 1 6 months 1.0% / 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.

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

Ragsdale 3.44

Ragsdale 3.45

Dielman 3.6 by Reynald Variable Coefficient Std Dev T Stat PValue Intercept 2.0336 0.5405 3.76 0.001 EPS 0.3740 0.2395 1.56 0.126 Standard error = 1.84975; R-Sq = 5.7%; R-Sq(adj) = 3.4% Source DF Sum of Squares Mean Square F Stat P Value Regression 1 8.345 2.44 0.126 Error 40 136.864 3.422 Total 41 145.208 The same regression equation relating dividends to EPS DIV = 0.3740EPS + 2.0336

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

Dielman 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 NUMBER The regression equation is COST = 28.3 + 2.15 NUMBER

Predictor 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) = 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

Ha: 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 > 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 H0). Conclusion: There is a significant relationship between project size and cost.

d. 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 = $28.31 Point estimate = sample 95% confidence mean Y t24-2 = 2.074 Fixed Cost = b0+/-tn-2sb0= 28.311+/-(2.074)4.083 = 28.311+/-8.468 95% sure that the fixed cost is between $19.84 and $36.78

e. 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 = 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

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