Respond to the Quantitative Methods Assignment

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Christy Tallant MAT540 Quantitative Methods Week 7 August 11, 2014 Chapter 3 Homework Problem 8 (solve using the computer), 10, 11 (solve using the computer), 12, 14, 15, 16, 18, 24 8.) Solve the model formulated in Problem 7 for Southern Sporting Goods Company graphically. a. Identify the amount of unused resources (i.e., slack) at each of the graphical extreme points. A: 3(0) + 2(160) + s1 = 500 s1 = 180 4(0) + 5(160) + s2 = 800 s2 = 0 B: 3(128.5) + 2(57.2) + s1 = 500 s1 = 0 4(128.5) + 2(57.2) + s2 = 800 s2 = 0 C: 2(167) + 2(0) + s1 = 500 s1 = 0 4(167) + 5(0) + s2 = 800 s2 = 132 b. What would be the effect on the optimal solution if the profit for a basketball changed from $12 to $13? What would be the effect if the profit for a football changed from $16 to $15? Z = 12x1 + 16x2 and, x2 = Z/16 − 12x1/16 The slope of the objective function, −12/16, would have to become steeper (i.e., greater) than the slope of the constraint line 4x1 + 5x2 = 800, for the solution to change. The profit, c1, for a basketball that would change the solution point is, −4/5 = −c1/16 5c1 = 64 c1 = 12.8 Since $13 > 12.8 the solution point would change to B where x1 = 128.5, x2 = 57.2. The new Z value is $2,585.70. For a football, −4/5 = −12/c2 4c2 = 60 c2 = 15 Thus, if the profit for a football decreased to $15 or less, point B will also be optimal (i.e., multiple optimal solutions). The solution at B is x1 = 128.5, x2 = 57.2, and Z = $2,400. c. What would be the effect on the optimal solution if 500 additional pounds of rubber could be obtained? What would be the effect if 500 additional square feet of leather could be obtained? If the constraint line for rubber changes to 3x1 + 2x2 = 1,000, it moves outward, eliminating points B and C. However, since A is the optimal point, it will not change and the optimal solution remains the same, x1 = 0, x2 = 160, and Z = 2,560. There will be an increase in slack, s1, to 680 lbs. If the constraint line for leather changes to 4x1 + 5x2 = 1,300, point A will move to a new location, x1 = 0, x2 = 250, Z = $4,000. 10.) A company produces two products, A and B, which have profits of $9 and $7, respectively. Each unit of product must be processed on two assembly lines, where the required production times are as follows: Product A B Total Hours Hours/Units Line 1 12 4 60 Hours/Units Line 2 4 8 40 a. Formulate a linear programming model to determine the optimal product mix that will maximize profit. Maximize...Z = 9 x1 + 7 x 2 subject..to.. 12 x1 + 4 x 2  60 4 x1 + 8 x 2  40 x1 , x 2  0 b. Transform this model into standard form Maximize...Z = 9 x1 + 7 x 2 + 0s1 + 0s 2 subject..to.. 12 x1 + 4 x 2 + s1 = 60 4 x1 + 8 x 2 + s 2 = 40 x1 , x 2 , s1 , s 2  0 11.) Solve Problem 10 graphically a. Identify the amount of unused resources (i.e., slack) at each of the graphical extreme points b. What would be the effect on the optimal solution if the production time on line 1 was reduced to 40 hours? c. What would be the effect on the optimal solution if the profit for product B was increased from $7 to $15? To $20. 12.) For the linear programming model formulated in Problem 10 and solved graphically in Problem 11: a. Determine the sensitivity ranges for the objective function coefficients, using graphical analysis. b. Verify the sensitivity ranges determined in (a) by using the computer. c. Using the computer, determine the shadow prices for additional hours of production time on line 1 and line 2 and indicate whether the company would prefer additional line 1 or line 2 hours. 14.) Solve the model formulated in Problem 13 for Irwin Textile Mills graphically. a. How much extra cotton and processing time are left over at the optimal solution? Is the demand for corduroy met? 5.0(456) + 7.5(510) + s1 = 6,500 s1 = 6,500 − 6,105 s1 = 395 lbs. 3.0(456) + 3.2(510) + s2 = 3,000 s2 = 0 hrs. 510 + s3 = 510 s3 = 0 Therefore demand for corduroy is met. b. What is the effect on the optimal solution if the profit per yard of denim is increased from $2.25 to $3.00? What is the effect if the profit per yard of corduroy is increased from $3.10 to $4.00? In order for the optimal solution point to change from B to C the slope of the objective function must be at least as great as the slope of the constraint line, 3.0x1 + 3.2x2 = 3,000, which is −3/3.2. Thus, the profit for denim would have to be, c. −c1/3.0 = −3/3.2 d. c1 = 2.91 e. f. If the profit for denim is increased from $2.25 to $3.00 the optimal solution would change to point C, where x1 = 1,000, x2 = 0, Z = 3,000. Profit for corduroy has no upper limit that would change the optimal solution point. c. What would be the effect on the optimal solution if Irwin Mills could obtain only 6,000 pounds per cotton per month? The constraint line for cotton would move inward as shown in the following graph where point C is optimal. 15.) Solve the linear programming model formulated in Problem 13 for Irwin Mills by using the computer. Z = 2,607.000 a. If Irwin Mills can obtain additional cotton or processing time, but not both, which should it select? How much? Explain your answer. The company should select 237 additional hours of processing time, with a shadow price of $0.75 per hour. Cotton has a shadow price of $0 because there is already extra (slack) cotton available and not being used, so any more would have no marginal value. b. Identify the sensitivity ranges for the objective function coefficients and for the constraint quantity values. Then explain the sensitivity range for the demand for corduroy. 0 ≤ c1 ≤ 2.906 6,105 ≤ q1 ≤ ∞ 2.4 ≤ c2 ≤ ∞ 1,632 ≤ q2 ≤ 3,237 0 ≤ q3 ≤ 692.308 The demand for corduroy can decrease to zero or increase to 692.308 yds. without changing the current solution mix of denim and corduroy. If the demand increases beyond 692.308 yds., then denim would no longer be produced and only corduroy would be produced. 16.) United Aluminum Company of Cincinnati produces three grades (high, medium, and low) of aluminum at two mills. Each mill has a different production capacity (in tons per day) for each grade as follows: Aluminum Grade High Medium Low Mill 1 6 2 4 Mill 2 2 2 10 The company has contracted with a manufacturing firm to supply at least 12 tons of high-grade aluminum, 8 tons of medium-grade aluminum, and 5 tons of low grade aluminum. It costs United $6,000 per day to operate mill 1 and $7,000 per day to operate mill 2. The company wants to know the number of days to operate each mill in order to meet the contract at the minimum cost. Formulate a linear programming model for this problem. Let X1= No of days Mill 1 is operated X2= no of days mill 2 is operated Minimize, Z= $6000X1+$7000X2 Subject to: 6X1+2X2>=12(Demand for High grade of aluminium) 2X1+2X2>=8(Demand for High grade of aluminium) 4X1+10X2>=5(Demand for High grade of aluminium) X1, X2>=0 18.) Solve the linear programming model formulated in Problem 16 for United Aluminum Company by using the computer. a. Identify and explain the shadow prices for each of the aluminum grade contract requirements. Cost/Day No of days to run Aluminium grade High Medium Low X1 X2 $6,000 4 $7,000 0 6 2 4 2 2 10 $ 24,000 24 8 16 >= >= >= 12 8 5 The United Aluminium company should run Mill 1 for 4 days at a minimum total cost of $24,000. b. Identify the sensitivity ranges for the objective function coefficients and the constraint quantity values. 24.) Solve the linear programming model developed in Problem 22 for the Burger Doodle restaurant by using the computer. The decision variables are x1 = Number of Sausage biscuits and x2 = Number of Ham biscuits The objective function is Maximize Z = $0.60 x1 + $0.50 x2 The model constraints are 0.010 x1 + 0.024 x2 < 6 (Labor) 0.10 x1 < 30 (Sausage) 0.15 x2 < 30 (Ham) 0.04 x1 + 0.04 x2 < 16 (Flour) x1, x2 > 0 Thus, the linear programming model is Maximize Z = $0.60 x1 + $0.50 x2 Subject to 0.010 x1 + 0.024 x2 < 6 0.10 x1 < 30 0.15 x2 < 30 0.04 x1 + 0.04 x2 < 16 x1, x2 > 0 The output of the above linear programming problem using Excel Solver is given below: Target Cell (Max) Cell Name Original Value Final Value $B$14 Z = 0 230 Adjustable Cells Cell Name Original Value Final Value $B$12 x1 = 0.00 300.00 $B$13 x2 = 0.00 100.00 Constraints Cell Name $E$6 Labor Used 5.4 $E$6<=$G$6 Not Binding $E$7 Sausage Used 30 $E$7<=$G$7 Binding $E$8 Ham Used 15 $E$8<=$G$8 Not Binding $E$9 Flour Used 16 $E$9<=$G$9 Binding Thus, the optimum solution is Cell Value Formula Status Slack 0.6 0 15 0 x1 = 300, x2 = 100 and the maximum value of Z is $230. The sensitivity report of the LPP is given below: Adjustable Cells Cell Name Final Reduced Objective Allowable Allowable Value Decrease Cost Coefficient Increase $B$12 x1 = 300.00 0.00 0.6 1E+30 0.1 $B$13 x2 = 100.00 0.00 0.5 0.1 0.5 Constraints Final Shadow Constraint Allowable Value Price R.H. Side Increase Allowable Cell Name Decrease $E$6 Labor Used 5.4 0 6 $E$7 Sausage Used 30 1 30 $E$8 Ham Used 15 0 30 1E+30 15 $E$9 Flour Used 16 12.5 16 1 4 1E+30 0.6 10 4.285714286 a. Indentify and explain the shadow prices for each of the resource constraints. From the sensitivity report, the shadow price for both Labor and Ham are 0 since there are extra labor hours and ham available. This means that additional amounts of those resources would add nothing to profit. The shadow price for Sausage is $1. This means that for every additional pound of sausage that can be obtained, profit will increase by $1. The shadow price for Flour is $12.5. This means that for every additional pound of flour that can be obtained, profit will increase by $12.5. b. Which of the resources constraints profit the most? The shadow price for Flour is the highest. Thus, the resources constraint for Flour profits the most. c. Identify the sensitivity ranges for the profit of a sausage biscuit and the amount of sausage available. Explain these sensitivity ranges. For a sausage biscuit (x1), the objective function coefficient is 0.6, allowable increase is  and allowable decrease is 0.1. Thus, the sensitivity range for the profit of a sausage biscuit is 0.50 < c1 <  This sensitivity range for the profit of a sausage biscuit indicates that the optimal mix of sausage and ham biscuits will remain optimal as long as profit of a sausage biscuit does not fall below $0.50. MAT540 Quantitative Methods Week 10 August 25, 2014 Chapter 6 Homework Complete the following problems from Chapter 6: Problems 4, 6, 36, 48 4. Consider the following transportation problem: To (cost) From A B C Demand 1 $6 12 4 80 2 $9 3 8 110 3 $100 5 11 60 Supply 130 70 100 Formulate this problem as a linear programming model and solve it by using the computer. 6. Consider the following transportation problem: To (cost) From A B C Demand 1 $6 12 4 80 2 $9 3 8 110 3 $7 5 11 60 Supply 130 70 100 Solve it by using the computer. 36. World foods, Inc., imports food products such as meats, cheesed, and pastries to the United States from warehouses at ports in Hamburg, Marseilles, and Liverpool. Ships from these ports deliver the products to Norfolk, New York, and Savannah, where they are stored in company warehouses before being shipped to distribution centers in Dallas, St. Louis, and Chicago. The products are then distributed to specialty food stores and sold through catalogs. The shipping costs ($/1,000 lb.) from the European ports to the U.S. cities and the available supplies (1,000 lb.) at the European ports are provided in the following table: U.S. Cities European Port 1. Hamburg 2. Marseillies 3. Liverpool 4. Norfolk $420 510 450 5. New York $390 590 360 6. Savannah $610 470 480 Supply 55 78 37 The transportation costs (#/1,000 lb.) from each U.S. city of the three distribution centers and the demands (1,000 lb) at the distribution center are as follows: Distribution Center Warehouse 4. Norfolk 5. New York 6. Savannah Demand 7. Dallas $75 125 68 60 8. St. Louis $63 110 82 45 9. Chicago $81 95 95 50 Determine the optimal shipments between the European ports and the warehouses and the distribution centers to minimize total transportation costs. 48.The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions. Given their various previous contracts, the salespersons are able to cover the regions in different amounts of time. The amount of time (days) required by each salesperson to cover each city is shown in the following table: Region Salesperson 1 2 3 4 5 A 17 12 11 14 13 B 10 9 16 10 12 C 15 16 14 10 9 D 16 9 15 18 15 E 20 14 12 17 11 Which salesperson should be assigned to each region to minimize total time? Identify the optimal assignments and compute the total minimum time. MAT540 Quantitative Methods Week 10 August 25, 2014 Chapter 6 Homework Complete the following problems from Chapter 6: Problems 4, 6, 36, 48 4. Consider the following transportation problem: To (cost) From A B C Demand 1 $6 12 4 80 2 $9 3 8 110 3 $100 5 11 60 Supply 130 70 100 Formulate this problem as a linear programming model and solve it by using the computer. 6. Consider the following transportation problem: To (cost) From A B C Demand 1 $6 12 4 80 2 $9 3 8 110 3 $7 5 11 60 Supply 130 70 100 Solve it by using the computer. 36. World foods, Inc., imports food products such as meats, cheesed, and pastries to the United States from warehouses at ports in Hamburg, Marseilles, and Liverpool. Ships from these ports deliver the products to Norfolk, New York, and Savannah, where they are stored in company warehouses before being shipped to distribution centers in Dallas, St. Louis, and Chicago. The products are then distributed to specialty food stores and sold through catalogs. The shipping costs ($/1,000 lb.) from the European ports to the U.S. cities and the available supplies (1,000 lb.) at the European ports are provided in the following table: U.S. Cities European Port 1. Hamburg 2. Marseillies 3. Liverpool 4. Norfolk $420 510 450 5. New York $390 590 360 6. Savannah $610 470 480 Supply 55 78 37 The transportation costs (#/1,000 lb.) from each U.S. city of the three distribution centers and the demands (1,000 lb) at the distribution center are as follows: Distribution Center Warehouse 4. Norfolk 5. New York 6. Savannah Demand 7. Dallas $75 125 68 60 8. St. Louis $63 110 82 45 9. Chicago $81 95 95 50 Determine the optimal shipments between the European ports and the warehouses and the distribution centers to minimize total transportation costs. 48.The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions. Given their various previous contracts, the salespersons are able to cover the regions in different amounts of time. The amount of time (days) required by each salesperson to cover each city is shown in the following table: Region Salesperson 1 2 3 4 5 A 17 12 11 14 13 B 10 9 16 10 12 C 15 16 14 10 9 D 16 9 15 18 15 E 20 14 12 17 11 Which salesperson should be assigned to each region to minimize total time? Identify the optimal assignments and compute the total minimum time. ...
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