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Business Law

BSLW 1102

Pieter Vander Werf

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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