1. Case Study 2: Giant Motor Company
Ck Bobba
Farshad Miraftab
Matthew Rosendin
Nabeel Saleem
Sheffield Wang
11/20/2016
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2. Contents
1 Problem Statement 3
2 Formulation 3
2.1 Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Decision Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4.2 Logical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4.3 Capacity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4.4 Car Model Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4.5 Demand Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4.6 Non-negativity, Integer, and Binary Constraints . . . . . . . . . . . . 5
2.5 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5.1 Hydra Plant Fixed Costs are Inevitable . . . . . . . . . . . . . . . . 5
2.5.2 Lyra and Libra Fixed Costs . . . . . . . . . . . . . . . . . . . . . . . 6
2.5.3 Demand Diversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Solution 6
3.1 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Demand Diversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Discussion 8
4.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.1.1 Improving Diversion Conversion Rates . . . . . . . . . . . . . . . . . 8
4.1.2 Generate Additional Initial Demand . . . . . . . . . . . . . . . . . . 10
4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Appendix 11
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3. 1 Problem Statement
The goal of this study is it to maximize the overall net profit for GMC in the coming year.
The decisions that GMC needs to make include:
1. Decide which plants to operate
2. Decide the quantity of cars to produce from the plants
3. Decide the number of customers it intends to divert (unmet demand)
As a result, this study provides the optimal decisions, while satisfying capacity and demand
constraints in order to maximize profit.
2 Formulation
2.1 Indices
j: Index of manufacturing plants for j = {1, 2, 3, 4, 5} where 1..5 indicate the Lyra, Libra,
Hydra, New Lyra, and New Libra plants, respectively.
i: Index of car models for i = {1, 2, 3} where 1..3 indicate Lyra, Libra, and Hydra models
respectively.
ij: Index of car model i produced from plant j
2.2 Parameters
pij: Marginal profit for car model i from plant j (in $1000s)
cj: Fixed cost of manufacturing plants (in $millions)
aij: Demand diversion ratios
bi: Logical, demand, and capacity constraints (in 1000s) for GMC cars
2.3 Decision Variables
xj: The availability of plant j; these are a binary-valued integers
yij: The amount of car i produced from plant j (in 1000s)
zi: The unmet demand of car i (in 1000s)
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4. 2.4 Model
max 2y11 + 2.5y14 + 2.3y15 + 3y22 + 3y24 + 3.5y25 + 5y33 + 4.8y35
− 2000x1 − 2000x2 − 2600x3 + 3400x4 − 3700x5
s.t. x1 + x4 = 1 (1)
x2 + x5 = 1 (2)
x3 = 1 (3)
y11 ≤ 1000x1 (4)
y22 ≤ 800x2 (5)
y33 ≤ 900x3 (6)
y14 + y24 ≤ 1600x4 (7)
y15 + y25 + y35 ≤ 1800x5 (8)
y12, y13, y21, y23, y31, y32, y34 = 0 (9)
y11 + y14 + y15 + z1 = 1400 (10)
y22 + y24 + y25 − 0.3z1 + z2 = 1100 (11)
y33 + y35 − 0.05z1 − 0.1z2 + z3 = 800 (12)
xj, yj, zi integer ∀ i, ∀ j (13)
yj, zi ≥ 0 ∀ i, ∀ j (14)
xj ∈ {0, 1} ∀ j (15)
2.4.1 Objective Function
The objective function is to maximize profit. Included in the maximization function is
revenue from each car line less the fixed cost of the plants. We have shown the above
model with the parameter values. More generally, the objective function is shown here:
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j=1
pijyij − cjxj ∀ i = 1, 2, 3
2.4.2 Logical Constraints
Constraint (1), x1 + x4 = 1, is a logical constraint ensures that either the Lyra plant is
retooled or not. Constraint (2), x2 + x5 = 1, is a similar logical constraint which ensures
that either the Libra plant is retooled or not. Constraint (3) ensures that the Hydra plant
fixed costs are inevitable. This does not mean that the Hydra plant must produces cars, but
rather it ensures that the fixed costs are incurred, regardless of whether cars are produced
or not.
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5. 2.4.3 Capacity Constraints
If the switch variables are activated, constraints (4) through (8) ensure that the production
of cars from each plant don’t exceed the capacity. Note: if the switch variable is 0 for plant
j, then the right-hand side of the equation, capacity of plant j, is multiplied by 0 and thus
production will be forced to remain 0 for that plant.
2.4.4 Car Model Constraints
Constraint (9) reflects that certain plants can only produce a certain model of car. For
example, the Lyra plant can only produce the Lyra cars, but a Lyra retooled plant can
produce both the Libra and the Lyra cars.
2.4.5 Demand Constraints
Constraints (10), (11), and (12) ensure that the production for each car meets the demand
for that car. The formulation accounts for the original expected demand, gained demand
from diversion, and unmet demand. Recall that the variables for demand gained/lost from
diversion is zi.
2.4.6 Non-negativity, Integer, and Binary Constraints
All decision variables are integers and non-negative. The switch variables xj are binary.
2.5 Assumptions
In a few cases we have made assumptions where the case study did not explicitly supply
direction. As such, the following assumptions have guided our model.
2.5.1 Hydra Plant Fixed Costs are Inevitable
The Hydra plant fixed cost of $2.6B is inevitable to GMC regardless if the plant is operated
or not. From the case study text,
“The fixed costs are annual costs incurred by GMC, independent of the number
of cars produced by the plant. For the current plant configurations, the fixed
costs include property taxes, insurance, payments on the loan that was taken
out to construct the plant, and so on.”
As a result, given that property taxes, insurance, and payments on the loan cannot be
avoided so long that GMC owns the plant (regardless of if it is operated or not), it is
reasonable and prudent to assume those fixed costs as inevitable.
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6. 2.5.2 Lyra and Libra Fixed Costs
If the Lyra or Libra plants are retooled, then the fixed costs increase from $2B to $3.4B
and $3.7B respectively. According to the case study, the “new” fixed costs include the
previous fixed costs.
2.5.3 Demand Diversion
GMC can identify the number of customers it attempts to divert, which will then be
subject to the diversion matrix provided by the case study. As an example, we know initial
Lyra demand is 1.4 million. However, GMC can choose to attempt to divert 0.4 million
customers in which the new Lyra demand will be only 1 million (a loss of 0.4 million). As
a result, the Libra demand will increase by 0.12 million (0.4 × 0.3) and the Hydra demand
will increase by 0.02 million (0.4 × 0.05). The remaining 0.26 million customers were lost
during the attempt to divert!
3 Solution
The optimal solution indicates that Giant Motor Company should retool the Lyra plant
but not the Libra plant. The Libra plant should produce 800,000 million Libras. The
Hydra plant should produce 808,000 million Hydras. Finally, the New Lyra plant should
produce 1,300,000 million Lyras and 300,000 million Libras. The optimal solution yields a
profit of $2.59 billion. These results are summarized in the subsections and tables below.
The following discussion section provides a more in-depth analysis of these results.
3.1 Production
This subsection reflects the plants that should be operated, retooled, and thus used for
production. It will also reflect the quantity of cars produced for each car by each plant.
Table 1: Production Summary (Thousands)
Plant/Model Lyra Libra Hydra New Lyra New Libra
Lyra - - 1300
Libra 800 - 300
Hydra - 808 -
To summarize the earlier solution, GMC should make the following production decisions:
1. Retool the Lyra plant
2. Do not retool the Libra plant (keep original)
3. Have the Libra plant produce 800,000 Libra cars
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7. 4. Have the Hydra plant produce 808,000 Hydra cars
5. Have the retooled Lyra plant produce 1,300,000 Lyra cars and 300,000 Libra cars
A summary of the total number of cars produced in table 2.
Table 2: Amount of Cars Produced per Model (Thousands)
Car Model Amount
Lyra 1,300
Libra 1,100
Hydra 808
Total 3,208
3.2 Demand Diversion
As discussed earlier, another decision that management needs to make is the number
of customers it attempts to divert to a different car model. Unfortunately, through the
diversion process, the majority of customers are actually lost and thus no sale is made.
The diversion ratios identified by GMC are displayed in table 3 for reference.
Table 3: Demand Diversion Ratios
Lyra Libra Hydra
Lyra N/A 30% 5%
Libra N/A N/A 10%
Hydra N/A N/A N/A
However, the table below summarizes the number of customers for each car model GMC
should attempt to divert.
Table 4: Number of Diversion Attempts (Thousands)
Car Model Unmet Demand
Lyra 100
Libra 30
Hydra N/A
As a result, tables 3 and 4 together can provide the total number of “gained” customers
through diversion. Table 5 below reflects those values.
Table 5: Gained Customers Through Diversion (Thousands)
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8. Lyra Libra Hydra
Lyra N/A 30 5
Libra N/A N/A 3
Hydra N/A N/A N/A
Table 4 identifies that GMC should divert 100,000 Lyra customers (in effect reducing
Lyra demand). Of the 100,000 previous Lyra customers, 30,000 will become “newly”
Libra customers and 5,000 will become “newly” Hydra customers and the remaining 75,000
customers will be lost. Similarly, GMC should attempt to divert 30,000 Libra customers
(in effect reducing Libra demand). Of the 30,000 previous Libra customers, only 3,000 will
be successfully diverted and become “newly” Hydra customers and the remaining 25,000
customers will be lost.
3.3 Profit
Based on the production and demand values discussed, the overall objective value based
on the decisions above would result in:
Table 6: Profit (Millions)
Marginal Profit $10,590
Fixed Costs $8,000
Total Net Profit $2,590
As a result, the total net profit for GMC will be $2.59 billion.
4 Discussion
In this discussion section, we present the assumptions that guided our model along with
justifications, and we provide recommendations to GMC to increase total net profit.
4.1 Recommendations
In an effort to further increase maximum profit, we have provided additional recommenda-
tions. Our analysis involved sensitivity analysis in two areas of the report: (1) improving
the diversion conversion values and (2) generating additional demand.
4.1.1 Improving Diversion Conversion Rates
Table 3 earlier in the report reflects to current estimates of the diversion matrix. However,
if GMC can improve those either by hiring more skilled salesman that can better persuade
customers, or by proving training courses for existing salesman, the impacts to net profit
may be significant. The chart below reflects the sensitivity to the total net profit as a
function of the Lyra to Libra diversion ratio.
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9. Note that the current estimate in the proposed solution assumes the ratio to be 30%.
However, if GMC can improve that ratio by as little as 10% ( 33% increase), the Total
Net Profit will increase to $2.611 Billion. Similarly, improvements from the Lyra to Hydra
diversion ratio will also have positive impact to the Total Net Profit.
Note that the base case ratio used in the proposed solution was 5%. Yet, increasing the
ratio to 6% will have no impact to the Total Net Profit. However, improving the ratio to
7% will result in a jump in Total Net Profit to $2.62 Billion.
Although these charts above reflect single input changes, a combination of such changes can
have even a higher impact. We feel that such an assumption is not unreasonable. Investing
in either training courses, or in more skilled salesman may result in improvements in the
diversion ratios across the board, and not necessarily to only one type of model. To repre-
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10. sent this effect, the chart below demonstrate how improving diversion ratios simultaneously
for both Libras and Hydras can have a compounding effect.
Recall the proposed solution assumed 0.3 and 0.05 for Lyra to Libra and Lyra to Hydra re-
spectively. Consequently, investing in either more skilled salesmen or sales training courses
to improve salesmen ability to divert consumers to higher model cars may potentially result
in roughly a $100 Million increase in Total Net Profit. For these reasons, we suggest GMC
explore opportunities to improve such ratios.
4.1.2 Generate Additional Initial Demand
GMC had initially provided expected demand for each line of car prior to any unmet de-
mand calculations. However, investing in advertisements to attract more initial demand,
particularly in the more luxurious models (Libra and Hydra), can reap huge benefits to
GMC.
As a result, this section of the report aims to demonstrate the magnitude of benefits by
attracting more initial demand for all three models. For the Lyra model, the current
demand is 1,400,000 customers but an increase of 20% in demand will result in a Total Net
Profit of $3,120M (a 20% increase!). Similarly, for the Libra mode, the current demand
in the proposed solution is 1,100,000 customers. Increasing demand by 20% will result in
1,320,000 customers. Such improvement will result in a Total Net Profit of $3,264M (a
26% increase!). Lastly, increasing Hydra demand by 20% will result in a Total net Profit
of $3,330 (a 29% increase!). The figure below reflects these values.
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11. It is also important to note that these values may be even larger given the combination of
increases in demand across all three models.
4.2 Conclusion
Given the proposed estimates, constraints, and assumptions, the optimal solution discussed
in the report will result in GMC having a generous Total Net Profit of $2.590 Billion.
Furthermore, given the two recommendations provided, we advise that increasing initial
demand through advertisements should take precedence over improving the diversion ratio.
Although investing in advertisements may be potentially more expensive than investing in
training courses or skilled salesmen to improve the diversion ratios, the overall increase in
Total Net Profit is much more sensitive to the initial demand than diversion ratios. The
diversion ratio improvement scenario may result in a $2.7B Total Net Profit (best case)
whereas the increase in initial demand scenario may result in a $3.3B Total Net Profit
(best case).
5 Appendix
In this section we include the Solver parameters and output with brief explanations.
Figure 1 shows the Solver parameters, or the data we entered into Solver. Notice that in
figure 2 we have color-coded the decision variable, constraint, and the objective function
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12. Figure 1: Solver Parameters
cells. We have entered the corresponding cells (mapped by color) into the Solver param-
eters dialog box. The decision variables, constraints, and objective function inputs also
correspond to the model discussed in Section 1.4. Upon clicking “solve,” the decision vari-
able cells are optimized and the total net profit is shown as $2.59 billion. The full results
as given by Solver are shown in figure 2.
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13. Figure 2: Solver Output in Excel
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