This document presents a linear programming problem involving the production of fitness trackers by a company called Sungram. It provides the details of the problem including available resources, product requirements, profits, and formulation as a LP model. It then asks six questions requiring the use of the sensitivity report from solving the model in Excel Solver to determine how changes impact the optimal solution without resolving. The questions examine how the solution is affected by changes in available resources and product parameters.

1. THIS IS JUST AN EXAMPLE OF RELATED PROBLEMS
AND SOLUTION
b) Solve it using Excel solver. Please attach your Solver model
and output (including Answer and Sensitivity Report) as Exhibit
A.
As part of Exhibit A please attach 1) your spreadsheet model
with formulas printed. To do this click on Ctrl-~ (the tilde key).
Excel will then display all formulas. Then print your
spreadsheet. Format so that it fits on 1 page landscape or
portrait. 2) Your Solver dialog box. To do this when you are in
the Solver dialog box click on Alt-PrtSc. Then go into any
program like word and click on Paste (or Ctrl-V). Your Solver
dialog box will be pasted.
See attachments.
c) What is the maximum profit? What production quantities
should be made to achieve this profit?
The maximum profit is $4800. To achieve this profit, they
should make 40 Chairs, 180 Desks, and 40 Tables.

2. d) Which resources are economically scarce (i.e., they are
limiting our profits)?
Assembly hours, Machining hours, and Wood supply are the
economically scarce resources (they are the binding constraints
limiting our profit).
3
Microsoft Excel 16.0 Answer Report
Worksheet: [brown.xlsx]Brown
Report Created: 5/11/2019 4:51:09 PM
Result: Solver found a solution. All Constraints and optimality
conditions are satisfied.
Solver Engine
Engine: Simplex LP
Solution

3. Time: 0.031 Seconds.
Iterations: 4 Subproblems: 0
Solver Options
Max Time 100 sec, Iterations 1000, Precision 0.000001
Max Subproblems 1000, Max Integer Sols 1000, Integer
Tolerance 0%, Assume NonNegative
Objective Cell (Max)
CellNameOriginal ValueFinal Value
$E$8Profit Total04800
Variable Cells
Cell
Name
Original Value
Final ValueInteger
$B$5
Production plan C
0
40 Contin
$C$5
Production plan D
0
180 Contin
$D$5
Production plan T
0

4. 40 Contin
Constraints
CellNameCell ValueFormulaStatusSlack
$E$11 Fabrication LHS 1320 $E$11<=$G$11 Not Binding 680
$E$12 Assembly LHS 1800 $E$12<=$G$12 Binding 0 $E$13
Machining LHS 1600 $E$13<=$G$13 Binding 0 $E$14 Wood
LHS 9400 $E$14<=$G$14 Binding 0 Microsoft Excel 16.0
Sensitivity Report
Worksheet: [brown.xlsx]Brown
Report Created: 5/11/2019 4:51:09 PM
Variable Cells
Final
Reduced
ObjectiveAllowableAllowable
Cell
Name
Value
Cost
CoefficientIncreaseDecrease
$B$5
Production plan C
40

5. 01655.5
$C$5
Production plan D
180
020 2.095238095 0.476190476
$D$5
Production plan T
40
0140.37037037 1.444444444
Final
Shadow
ConstraintAllowableAllowable
Cell
Name
Value
Price
R.H. SideIncreaseDecrease
Constraints$E$11 Fabrication LHS1320020001E+30680
$E$12 Assembly LHS 1800 1.238095238 1800 360 46.66666667
$E$13 Machining LHS 1600 1.047619048 1600 360 70 $E$14
Wood LHS 9400 0.095238095 9400 155.5555556 900

6. (c) Create a “transportation model” similar to the one discussed
in the live session. Explain what your variables mean, and what
each of the constraints specify. Attach your spreadsheet
formulation as Exhibit B. Please note that your spreadsheet
must be sufficiently annotated so that it is readable in order to
receive credit. See note earlier regarding printing Solver dialog
box and solver model with formulas.
See attachments.
(d) What is the value of the objective function after
optimization?
The value of the objective after optimization is 130 distance
units.
(e) Explain in detail the movement plan to get cars from
locations with excess supply to locations that are in deficit.

7. Move 2 cars from Location 4 to 2.
Move 3 cars from Location 4 to 6.
Move 4 cars from Location 5 to 1.
Move 11 cars from Location 5 to 3.
Move 2 cars from Location 6 to 2.
Move 2 cars from Location 8 to 3.
Move 7 cars from Location 8 to 6.
(f) Stretch your thinking: Bike sharing is growing around the
world especially in countries such as China and Brazil. It is
anticipated once driverless cars become a reality (and
commonplace) car sharing would become very common. Think
about similarities between the problem you solved and the issue
of getting bikes to the locations they are needed. How might an
optimization model be used to improve efficiencies for a bike
sharing provider?
Various answers are acceptable. The key aspect is that the
imbalance in supply and demand can be resolved by efficiently
moving bikes and driverless cars around.

8. 5 MATHEMATICAL MODEL
Decision Variables
Let Xij = Number of cars moved from location i to location j.
Supply locations (i) = 4, 5, 6, 8.
Demand Locations (j) = 1, 2, 3, 7.
Objective (Minimize distance):
MIN 7X41 + 5X42 + 8X43 + 3X47 + 3X51 + 8X52 + 5X53 +
6X57 + 5X61 + 4X62 + 4X63 + 3X67 + 3X81 + 7X82 + 4X83 +
4X87
Constraints:
(supply)
X41 + X42 + X43 + X47 = 5 (supply of cars at 4) X51 + X52
+ X53 + X57 = 15 (supply of cars at 5) X61 + X62 + X63 +
X67 = 2 (supply of cars at 6)
X81 + X82 + X83 + X87 = 9 (supply of cars at 8)
(demand)
X41 + X51 + X61 + X81 = 4 (demand of cars at 1)

9. X42 + X52 + X62 + X82 = 4 (demand of cars at 2) X43 + X53
+ X63 + X83 = 13 (demand of cars at 3)
X47 + X57 + X67 + X87 = 10 (demand of cars at 7)
(Non-negativity)
X41, X42, X43, X47, X51, X52, X53, X57, X61, X62, X63,
X67, X81, X82, X83, X87 >= 0
6
A
B
C
D
E
F
G
1
American RentaCar: Tra
Parameters
Supply4

10. Locations5
6
8
Demand
Decisions
Supply4
Locations5
6
8
Demand
Objective
Total Distance
Demand Locations
123
7
Supply
2
3

11. 4
5
758
385
544
374
3
6
3
4
5
15
2
9
6

12. 7
8
9
4413
10
Supply
10
Demand Locations
123
6
11

13. 12
13
020
4011
020
002
3
0
0
7
=SUM(C13:F13)
=SUM(C14:F14)
=SUM(C15:F15)
=SUM(C16:F16)
14

14. 15
16
17
=SUM(C13:C16)=SUM(D13:D16)=SUM(E13:E16)
=SUM(F13:F16)
=SUM(G13:G16)
18
19

15. 20
=SUMPRODUCT(C5:F8,C13:F16)
American RentaCar: Transportation Problem
Parameters
Supply4
Locations5
6
8
Demand
11 Decisions
13 Supply4
14 Locations5
6
8
Demand
19 Objective
Total Distance

16. Part 1 – Proposal (LAYOUT EXAMPLE)
ASSIGNMENT TITLE
Date
Course Code
Course Name

17. Professor's Name
Student's Name
Student ID #
Topic
**DO NOT INCLUDE THESE HEADINGS IN YOUR OWN
PAPER - THEY ARE USED
FOR EXAMPLE PURPOSES ONLY**
State topic chosen from list provided and include two points for
why it was chosen. These points
should simply be personal opinion. The supportive points
should explain why you are personally
interested in the topic.

18. Current Knowledge
State three points with explanation regarding your basic current
knowledge on the topic. These
points should not be personal opinion, and should include
proper APA citation of sources both
within the assignment, as well as in a reference page.
*Hint: try to remember where you learned this information in
the first place, and use this as the
source. If the source cannot be cited appropriately, find a
reputable resource that includes
similar information. It is not acceptable to state information
without a citation.
Prediction for Future Knowledge and Awareness
State two points with explanation for what you hope to learn
about the topic and how you think
your future awareness may change surrounding the issue. This

19. is personal opinion and should be
stated as a prediction or hypothesis on what you think will be
achieved through the completion of
the assignment.
Two Questions That You Hope to Answer Through Your
Research
State two questions that you hope to be answered by the time
your research on the topic has been
completed. These questions will be restated and addressed in
your final paper.
Reference Page
This should list all sources used within the body of the proposal
following proper APA

20. formatting style (see APA Citation Style guide for details). In-
text citations must also be used
within the text of this proposal.
Individual Assignment 2
1) Sungram, an active lifestyle company, manufactures three
models of novel fitness trackers called Jump (J), Run (R), and
Walk (W). They have a limited supply of common parts---wifi
module (450 in inventory), cellular module (250 in inventory),
heart rate monitor (800 in inventory), GPS module (450 in
inventory), LCD screen (600 in inventory)---that these products
use. A Jump model requires a wifi module, 2 heart rate
monitors, a GPS module, and 2 LCD screens. A Run model
requires a wifi module, a cellular module, 2 heart rate monitors,

21. a GPS, and an LCD screen. A Walk model requires a heart rate
monitor and an LCD screen. The profit on the Jump model is
$65, the profit on the Run model is $75, and the profit on the
Walk Model is $25. The following is a linear programming
formulation of the problem.
Let
J = Number of Jump models produced
R = Number of Run models produced
W = Number of Walk models produced
We may write a model for this problem as follows. Maximize
65J + 75R + 25W subject to:
(wifi module constraint) J + R ≤ 450
(cellular module constraint) R ≤ 250
(heart rate monitor constraint) 2J + 2R + W ≤ 800
(GPS module constraint) J + R ≤ 450 (LCD screen constraint)
2J + R + W ≤ 600 (non-negativity) J, R, W ≥ 0.
Implement the above model in Solver (make sure to choose
Simplex as the solving method and to choose the option “Make
Unconstrained Variables non-negative”---do not explicitly put
in the non-negativity constraints in the model) and using the
sensitivity report only (do not resolve the problem and explain
your calculation using the sensitivity report) answer the
following questions.

22. a. Does the solution change if only 415 wifi modules are
available?
b. Is it profitable to produce the Walk model? If not, by how
much should the profit margin on the Walk model be increased
to make it profitable to produce the Walk model?
c. Because of a change in production technology the profit
margin on the Jump model has increased to $70. Should the
production plan of Sungram change? What is their new profit?
d. 100 heart rate monitors were found to be defective, making
the number of available heart rate monitors 700. What will the
profit be in this situation?
e. Another supplier is willing to sell cellular modules to
Sungram. However, their prices for a cellular module are $8
higher than what Sungram pays its regular supplier. Should
Sungram go ahead and purchase these cellular modules? If yes,
at most how many units should they purchase?
f. Sungram is considering introducing a new fitness tracker
model called the RunLite. This product uses a wifi module, a
cellular module, a heart rate monitor, and an LCD screen, and is
expected to make a profit of $50. Should Sungram produce the
RunLite? Why or Why not?
Your submission to this question should be a word or pdf file

23. with all of the answers for the parts a through f. Make sure to
explain how you used the sensitivity report to figure out your
answer. Please also attach the solver model and sensitivity
report you used for this question.
2) Braxton oil company produces four brands of oil: regular,
multigrade, supreme, and extreme. Each brand of oil is
composed of one or more of four crude stocks, each having a
different lubrication index. The relevant data concerning the
crude stocks are as follows.
Crude Stock
Lubrication Index
Cost ($/Barrel)
Daily Supply (Barrels)
1
20
7.1
1100
2
40
8.5
1100
3
30

24. 7.7
1100
4
55
9
1100
Each brand of oil must meet a standard for a lubrication index,
and each brand thus sells at a different price. The relevant data
concerning the four brands of oil are as follows.
Brand
Minimum Lubrication Index
Sales Price ($/Barrel)
Daily Demand (Barrels)
Regular
25
8.5
1500
Multigrade
32
9
1000
Supreme
42
10

25. 1000
Extreme
50
10.5
750
The task is to determine an optimal output plan for a single day,
assuming that production can be either sold or else stored at
negligible cost. The daily demand figures are subject to
alternative interpretations. Investigate the following.
(a) The daily demands represent potential sales. In other words,
the model should contain demand ceilings (upper limits). What
is the optimal profit?
(b) The daily demands are strict obligations. In other words, the
model should contain demand constraints that are met precisely.
What is the optimal profit?
(c) The daily demands represent minimum sales commitments,
but all output can be sold. In other words, the model should
permit the production to exceed the daily commitments. What is
the optimal profit?
Your submission to this question should be a spreadsheet
containing a Solver Model for each part. Please make sure to
answer the questions explicitly in words. Your Solver Model
needs to be clear so that someone who looks at it can follow

26. what you are doing without going into Solver (use colors for
decision variables, objective, constraints etc as in the
examples). If necessary, explain the equations you are modeling
mathematically. Make sure to also paste your Solver dialog box
in the spreadsheet. Explain how you change the models between
parts a, b, and c.