Linear Programming Strategies for Civil Systems Analysis Homework

WESTERN MICHIGAN UNIVERSITY CCE-6100-100 - Civil Systems Analysis HW.2 Done by: Aws Tarawneh

1. Solve the following linear program using the graphical solution procedure: Feasible area Max at 100,50 (100*5+50*5)=750

2. Given the linear program 2.1. Write the problem in standard form Max 3A+4B+0S1+0S2+0S3 S.t -1A+2B+0S1 =8 1A+2B + 0S2 =12 2A+1B + 0S3=16 A, B, S1, S2, S3≥0 2.2. Solve the problem using the graphical solution procedure Intersection 1 Intersection 2 Feasible area

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At Intersection 1 -1A+2B=8 1A+2B=12 B=5 A=2 3A+4 B=3*2+4*5=26 At Intersection 2 1A+2B=12 2A+1B=16 A=6.6667 B=2.6667 3A+4 B=30.66 Lingo output Global optimal solution found. Objective value: 30.66667 Infeasibilities: 0.000000 Total solver iterations: 2 Model Class: LP Total variables: 2 Nonlinear variables: 0 Integer variables: 0 Total constraints: 6 Nonlinear constraints: 0 Total nonzeros: 10 Nonlinear nonzeros: 0 Variable Value Reduced Cost A 6.666667 0.000000 B 2.666667 0.000000 Row Slack or Surplus Dual Price 1 30.66667 1.000000 2 9.333333 0.000000 3 0.000000 1.666667 4 0.000000 0.6666667 5 6.666667 0.000000 6 2.666667 0.000000

2.3.What are the values of the three slack variables at the optimal solution? A= 20/3 B=8/3 S1=8+A-2B= 8+ (20/3)-2(8/3) = 28/3 S2=12-A-2B=12-(20/3)-2(8/3) = 0 S3=16-2A-B=16-2(20/3) -(8/3) =0 3. 3.1.What is the recommended investment portfolio for this client? What is the annual return for the portfolio? Max. 0.12 I + 0.09B S.t. I+B ≤ 50,000 maximum to invest I ≤ 35,000 max investment in the Internet fund. 6I+4B≤240 max risk rating I, B≥ 0 I= Internet Fund Investment. B= Blue clip Fund Investment. Using Lingo Global optimal solution found. Objective value: 5.100000 Infeasibilities: 0.000000 Total solver iterations: 2 Model Class: LP Total variables: 2 Nonlinear variables: 0 Integer variables: 0 Total constraints: 6 Nonlinear constraints: 0 Total nonzeros: 9 Nonlinear nonzeros: 0 Variable Value Reduced Cost I 20.00000 0.000000 B 30.00000 0.000000 Row Slack or Surplus Dual Price 1 5.100000 1.000000 2 0.000000 0.3000000E-01 3 15.00000 0.000000 4 0.000000 0.1500000E-01

5 20.00000 0.000000 6 30.00000 0.000000 I=20,000 B=30,000 The annual return is = 20,000 *.12 + 30,000* 0.09 = $5100 3.2.Suppose that a second client with $50,000 to invest has been classified as an aggressive investor. B&R recommends that the maximum portfolio risk rating for an aggressive investor is 320. What is the recommended investment portfolio for this aggressive investor? Discuss what happens to the portfolio under the aggressive investor strategy 6I+4B≤320 lingo output Global optimal solution found. Objective value: 5.550000 Infeasibilities: 0.000000 Total solver iterations: 2 Model Class: LP Total variables: 2 Nonlinear variables: 0 Integer variables: 0 Total constraints: 6 Nonlinear constraints: 0 Total nonzeros: 9 Nonlinear nonzeros: 0 Variable Value Reduced Cost I 35.00000 0.000000 B 15.00000 0.000000 Row Slack or Surplus Dual Price 1 5.550000 1.000000 2 0.000000 0.9000000E-01 3 0.000000 0.3000000E-01 4 50.00000 0.000000 5 35.00000 0.000000 6 15.00000 0.000000 I=35,000 B=15,000 The annual return is = 35,000 *.12 + 15,000* 0.09 = $5,500 Under the aggressive investor the annual return has increased from 5100 to 5550

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3.3.Suppose that a third client with $50,000 to invest has been classified as a conservative investor. B&R recommends that the maximum portfolio risk rating for a conservative investor is 160. Develop the recommended investment portfolio for the conservative investor. Discuss the interpretation of the slack variable for the total investment fund constraint. 6I+4B≤160 lingo output Global optimal solution found. Objective value: 3.600000 Infeasibilities: 0.000000 Total solver iterations: 1 Model Class: LP Total variables: 2 Nonlinear variables: 0 Integer variables: 0 Total constraints: 6 Nonlinear constraints: 0 Total nonzeros: 9 Nonlinear nonzeros: 0 Variable Value Reduced Cost I 0.000000 0.1500000E-01 B 40.00000 0.000000 Row Slack or Surplus Dual Price 1 3.600000 1.000000 2 10.00000 0.000000 3 35.00000 0.000000 4 0.000000 0.2250000E-01 5 0.000000 0.000000 6 40.00000 0.000000 When the risk is decreased the investor will invest in I=0 B=40,000 The annual return is = 0 *.12 + 40,000* 0.09 = $3600 The slack will be S1=10,000, S2= 35,000, S3=0

4. Consider the following linear programming problem: 4.1. Find the optimal solution using the graphical solution procedure and the value of the objective function Feasible area min at Intersection 2 when A=3, B=1 the value of the objective function is A + 2B=5 Intersection 2 Intersection 3 Intersection 1 Intersection 4

4.2. Determine the amount of slack or surplus for each constraint slack A+4B +S1=21 S1= 21-7= 14 Surplus 2A+B-S2=7 S2= 7-7= 0 Slack 3A+1.5B+S3=21 S3=21-10.5= 10.5 Surplus -2A+6B-S4=0 S4= 0 4.3.Suppose the objective function is changed to max 5A+2B. Find the optimal solution and the value of the objective function. using the graph Max will be at intersection 3 A=6 B=2 the value of the objective function is 5A+2B=30+4=34 5. Southern Oil Company 5.1.Formulate a linear programming model that can be used to determine the number of gallons of regular gasoline and the number of gallons of premium gasoline that should be produced in order to maximize total profit contribution. R= number of gallons of regular gasoline P= number of gallons of premium gasoline. Objective Function max 0.3R+0.5P Constraints: 0.3R+0.6P≤18,000 R+P≤50,000 P≤20,000 R≥0,P≥0

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Using Lingo Global optimal solution found. Objective value: 17000.00 Infeasibilities: 0.000000 Total solver iterations: 2 Model Class: LP Total variables: 2 Nonlinear variables: 0 Integer variables: 0 Total constraints: 6 Nonlinear constraints: 0 Total nonzeros: 9 Nonlinear nonzeros: 0 Variable Value Reduced Cost R 40000.00 0.000000 P 10000.00 0.000000 Row Slack or Surplus Dual Price 1 17000.00 1.000000 2 0.000000 0.6666667 3 0.000000 0.1000000 4 10000.00 0.000000 5 40000.00 0.000000 6 10000.00 0.000000 5.2.What is the optimal solution? R=40,000 P=10,000 0.3R+0.5P=0.3*40,000+0.5*10,000=17,000 5.3.What are the values and interpretations of the slack variables? Slack in the Crude Oil Constraint: S1=18,000-(0.3*40,000+0.6*10,000)=0 Slack in the Production Capacity Constraint: S2= 50,000-(40,000+10,000)=0 Slack in the Premium Gasoline Demand Constraint: S3 = 20,000-10,000=10,000 5.4.What are the binding constraints? 1. Crude Oil Constraint 2. Production Capacity Constraint Which have 0 slack S1 & S2 =0

6. Consider the following linear program: 6.1. Graph the feasible region for the problem 6.2. Is the feasible region unbounded? Explain Yes, the feasible region is unbounded. Because the value of the solution is indefinitely large without violating any of the constraints 6.3.Find the optimal solution. There is no optimal solution 6.4.Does an unbounded feasible region imply that the optimal solution to the linear program will be unbounded? Yes, If the feasible region is unbounded, the objective function (whether maximizing or minimizing) can be increased or decreased indefinitely, resulting in an unbounded optimal solution. In other words, there is no finite upper or lower limit on the objective function value, and the solution space extends infinitely in at least one direction.

7. Management of High-Tech Services 7.1. R= regular customers N= new customers Objective Function Max 1.2R + N R+N≤80 25R+8N≥800 N≥0.6R R≥0,N≥0 7.2.Optimal solution using Lingo Global optimal solution found. Objective value: 90.00000 Infeasibilities: 0.000000 Total solver iterations: 2 Model Class: LP Total variables: 2 Nonlinear variables: 0 Integer variables: 0 Total constraints: 6 Nonlinear constraints: 0 Total nonzeros: 10 Nonlinear nonzeros: 0 Variable Value Reduced Cost A 50.00000 0.000000 B 30.00000 0.000000 Row Slack or Surplus Dual Price 1 90.00000 1.000000 2 0.000000 1.125000 3 690.0000 0.000000 4 0.000000 -0.1250000 5 50.00000 0.000000 6 30.00000 0.000000 R=50 N=30 1.2R+N=60+30=90

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Case Problem. Production Strategy A=Body Plus 100 B=Body Plus 200 Objective Function: Max (0.7*2400-450-300-250-50-20*8-15*5-12*2) *A+(0.7*3500-650-400-250-200-75-20*12- 15*10-12*2)*B Max 371*A+461 B Machining and welding time constraint: 8A+12B ≤600 hours Painting and finishing time constraint: 5A+10B ≤450 hours Assembly, testing, and packaging time constraint: 2A+2B ≤140 hours Body Plus 200 production requirement: B ≥0.25*(A+B) -0.25A+0.75B ≥0 A ≥0, B ≥0 a. What is the recommended number of BodyPlus 100 and BodyPlus 200 machines to produce? Using Lingo max 371A+461B s.t. 8A+12B<600 5A+10B<450 2A+2B<140 -0.25A+0.75B>0 Global optimal solution found. Objective value: 26233.33 Infeasibilities: 0.000000 Total solver iterations: 2 Model Class: LP Total variables: 2 Nonlinear variables: 0 Integer variables: 0 Total constraints: 5 Nonlinear constraints: 0 Total nonzeros: 10 Nonlinear nonzeros: 0 Variable Value Reduced Cost A 50.00000 0.000000 B 16.66667 0.000000 Row Slack or Surplus Dual Price 1 26233.33 1.000000 2 0.000000 43.72222 3 33.33333 0.000000 4 6.666667 0.000000 5 0.000000 -84.88889

Recommended production Body Plus 100 is 50 Recommended production Body Plus 200 is 16.6667=16 cant produce 17 because it will affect the Machining and welding time constraint 8*50+12*17 ≤600 = 604<600 Max profit=371*50+461*16= $25,926 b. How does the requirement that the number of units of the BodyPlus 200 produced be at least 25% of the total production affect profits? Answers are from Lingo with frictions (A=50, B=16.67) BodyPlus 200 percentage profit 30% $25,956.52 25% $26,233.33 24% $26,290.18 20% $26,522.73 The total profit will increase because BodyPlus 200 needs more time to be produced than BodyPlus 100 ex. BodyPlus 200 needs 12 hours for Machining and welding but BodyPlus 100 needs 8 hours for Machining and welding which mean we can produce 1.5 BodyPlus 100 in the time we need to produce 1 BodyPlus 200 c. Where should efforts be expended in order to increase profits? According to the graph the intersection was between the Machining and welding time and Body Plus 200 production requirement constraints which means we can either higher the Machining and welding time or lower the 25% production of Body Plus 200 to get a higher profit

CCE 6100 HW2 edited

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