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MD235 Spring 2012 Exam 2 VanderWerf Solution1.pdf

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Department
Business Law
Course
BSLW 1102
Professor
Pieter Vander Werf
Semester
Fall

Description
Math for Management Science MD235, Sections 6 and 7 Name_____________________________________ Exam 2 Spr12g 2022a,ch Sign this page and initial each following page on the lines provided. Closed book, closed notes. You may use something to write with, a calculator, and a straightedge. There are 4 questions, each with multiple parts. Points for each question are given in parentheses. Do all work on this test paper. If you take the pages apart, staple them with the stapler at the front of the room before you turn the test paper in. Initials ____________ 1. (27 points) Carroll Communications operates the data network represented in the diagram below. 2 Detroit 6 tbs 8 tbs 5 4 tbs 4 tbs Denver 6 tbs 1 6 tbs 3 7 Newark Louisville Los Angeles 2 tbs 6 4 tbs 6 tbs Santa Fe 4 tbs 8 tbs 4 10 tbs Birmingham The network has seven communications nodes across the U.S., and each communications line between nodes has a maximum transmission capacity measured in terabits per second (tbs), as shown in the diagram. Each line allows only one-way transmission, as shown by the arrows. Carroll used a linear programming spreadsheet model to calculate the maximum data flow that can go through the network from Newark to Los Angeles. The spreadsheet model was run with Solver, and the result is shown with the correct solution below. The Sensitivity Report from the solution is on the next page. Answer the questions after it. 2 Initials ____________ Sensitivity Report: Variable Cells        Final  Reduced Objective  Allowable  Allowable  Cell  Name  Value  Cost  Coefficient Increase  Decrease  $B$3  Line 1‐2 Thru (tbs)  6  0 0 0  1  $B$4  Line 1‐3 Thru (tbs)  6  0 0 1E+30  0  $B$5  Line 1‐4 Thru (tbs)  8  0 0 1E+30  0  $B$6  Line 2‐3 Thru (tbs)  0  0 0 0  1  $B$7  Line 2‐5 Thru (tbs)  6  0 0 1E+30  0  $B$8  Line 3‐4 Thru (tbs)  4  0 0 0  1  $B$9  Line 3‐5 Thru (tbs)  0  0 0 0  1  $B$10  Line 3‐6 Thru (tbs)  2  0 0 1E+30  0  $B$11  Line 4‐6 Thru (tbs)  2  0 0 0  1  $B$12  Line 4‐7 Thru (tbs)  10  0 0 1E+30  1  $B$13  Line 5‐7 Thru (tbs)  6  0 0 1E+30  1  $B$14  Line 6‐7 Thru (tbs)  4  0 0 1E+30  1  $B$15  Line 7‐1 Thru (tbs)  20  0 1 1E+30  1  Constraints        Final  Shadow  Constraint  Allowable  Allowable  Cell  Name  Value  Price  R.H. Side  Increase  Decrease  $H$3  Node 1 balance LHS  0  1 0 1E+30  0  $H$4  Node 2 balance LHS  0  1 0 2  0  $H$5  Node 3 balance LHS  0  1 0 2  0  $H$6  Node 4 balance LHS  0  1 0 2  0  $H$7  Node 5 balance LHS  0  1 0 2  0  $H$8  Node 6 balance LHS  0  1 0 2  0  $H$9  Node 7 balance LHS  0  0 0 0  1E+30  $H$10  Line 1‐2 capac. LHS  6  0 8 1E+30  2  $H$11  Line 1‐3 capac. LHS  6  0 6 0  2  $H$12  Line 1‐4 capac. LHS  8  0 8 0  2  $H$13  Line 2‐3 capac. LHS  0  0 4 1E+30  4  $H$14  Line 2‐5 capac. LHS  6  0 6 0  4  $H$15  Line 3‐4 capac. LHS  4  0 6 1E+30  2  $H$16  Line 3‐5 capac. LHS  0  0 4 1E+30  4  $H$17  Line 3‐6 capac. LHS  2  0 2 2  2  $H$18  Line 4‐6 capac. LHS  2  0 4 1E+30  2  $H$19  Line 4‐7 capac. LHS  10  1 10 2  0  $H$20  Line 5‐7 capac. LHS  6  1 6 2  0  $H$21  Line 6‐7 capac. LHS  4  1 4 2  0  3 Initials ____________ 1a. (3 points) Write the correct formula for cell B18 using correct Excel format in the space below. = B15 1b. (3 points) Write the correct formulas for cells F4, G4, H4, and J4 in the spaces provided below using correct Excel format. F4 G4 H4 J4 =B3 =B6 + B7 = F4 – G4 0 1c. (3 points) Write the correct formulas for cells H15, and J15 in the spaces provided below using correct Excel format. H15 J15 = B8 = C8 1d. (3 points) Does this model result in no feasible solution? Circle the correct answer and explain how you know. O N S E Y Explanation: Solver returned a solution, so there must be a feasible solution. 1e. (3 points) Does this model result in alternative optimal solutions? Circle the correct answer and explain how you know. O N S E Y Explanation: There are alternating zeros in the allowable increase and allowable decrease columns of the decision variable section of the sensitivity report. 1f. (3 points) Does this model result in an unbounded solution? Circle the correct answer and explain how you know. O N S E Y Explanation: Solver returned a bounded solution, so there cannot be an unbounded solution. 4 Initials ____________ 1g. (3 points) Carroll has the resources to increase the maximum transmission capacity along any one data line by 1.5 tbs. Which line should they do this to if they want to achieve the greatest increase in total flow through the system? Any one of the lines 4-7, 5-7, and 6-7. They all have the same shadow price (1) and an allowable increase over 1.5. 1h. (3 points) Which constraints have slack? All the ones with a LHS final value lower than the RHS limit: Capacity constraint for 1-2, 2-3, 3-4, 3-5, and 4-6. 1i. (3 points) The capacity of the line from node 2 to node 3 will drop by one-half tomorrow. How much will this reduce the maximum total flow through the system? Not at all. It’s true that a fall by on-half (half of 4 is 2) is within the allowable increase, but the shadow price is 0, so the decrease will result in zero change (0x2 = 0) in the objective function value (total flow). 5 Initials ____________ 2. (27 points) Zachary Botanicals makes two different natural shampoos. Zachary Hydration is designed mainly to moisturize the hair and Zachary Luster is designed mainly to increase its shine. Each product is a blend of three different natural ingredients: aloe, betadine, and carotene. Zachary needs to mix up 100,000 liters of Hydration and 50,000 liters of Luster in its next production run to fill demand. Key properties of the ingredients are: Cost per liter Water retention Reflectance Aloe $20.40 Betadine $0.20 4 83 Carotene $0.13 9 42 There is some flexibility in the amount of each ingredient to use. However, to achieve the desired properties of Hydration, the following conditions must be met: The average water retention of the ingredients must be at least 6.5 The average reflectance must be at least 55 Carotene must not be over 40% of the total volume of ingredients. For Luster: The average water retention must be at least 5.2 The average reflectance must be at least 75 You must formulate a model to determine how much of each ingredient to use so as to minimize the total cost of ingredients. 2a. (3 points) In the table below, provide a symbol and a definition of each decision variable. There may be more spaces than necessary. Symbol Definition A H Number of liters of ingredient A used in Hydration A Number of liters of ingredient A used in Luster L B H Number of liters of ingredient B used in Hydration B L Number of liters of ingredient B used in Luster C H Number of liters of ingredient C used in Hydration C L Number of liters of ingredient C used in Luster 2b. (4 points) In the space below, write the formula for the objective function. MIN Z = 0.4(A + H ) +L0.2(B + B H + 0L13(C + C ) H L 6 Initials ____________ 2c. (10 points) In the table below, write all constraints for the model. There may be more spaces than necessary. A H B +HC = 1H0,000 A L B +LC = L0,000 7A + 4B + 9C ≥ 6.5(A + B + C ) H H H H H H 27A H 83B + H2C ≥ 55HA + B + H ) H H C H 0.4(A +HB + H ) H 7A L 4B +L9C ≥ 5L2(A + B L C )L L 27A L 83B + L2C ≥ 7L(A + B +LC ) L L A H A L B H B L C H C L 0 2d. (10 points) One year later, Zachary has exactly the same situation with one change. There are only 36,000 liters of Carotene in stock. The company can get more at the usual cost per liter, but it will have to pay a one- time order fee of $2100 no matter how much it orders. Zachary wishes to produce the same amount of eac
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