Congratulations! You have recently been promoted to the position of regional manager for Hurts Car Rental Company in Florida. You need to make pricing decision and fleet size decision for Tampa branch to maximize its expected total profit in the coming month (with 4 weeks or periods), taking into account the behavior of both your customers and your competitor. Hurts has one primary competitor in Tampa with whom Hurts competes for market share. Your Market Intelligence (MI) has found that the competitor has adopted a masked price following strategy; that is, the competitors p 2 t price in period t will follow your last price p 1 t–1 within ±20% range: p 2 t = p 1 t–1 × εt , εt ∼ U[0.80, 1.20], t = 2, 3, 4. (37) Assume that the competitor begins by choosing the same price as yours in period (week) 1, p 1 1 = p 2 1 . MI also provides the weekly demand forecast for the coming 4 weeks based on the historical data. Your and your competitor’s pricing strategies, pt = (p 1 t , p 2 t ), jointly determine both the market size Nt(pt) and the market share πt(pt). The market size Nt(pt), or the total number of demands for Hurts and its competitor in week t, is determined by pt = (p 1 t , p 2 t ) via Nt(pt) = b0 + b1p 1 t + b2p 2 t , (38) where b0 = 105644.5, b1 = –377.5, and b2 = –676.25.  The market share πt(pt) of Hurts, or the probability that a unit demand choosing Hurts instead of its competitor, is determined by the Multinomial logit (MNL) model:11 πt(pt) = 1 1 + exp(–c0 – c1p 1 t – c2p 2 t ) (39) where c0 = 0.0875, c1 = –0.0220, and c2 = 0.0170. Note that 0 ≤ πt(pt) ≤ 1, and your competitor’s market share is 1 – πt(pt). Therefore, your demand Dt(pt) in week t follows Binomial distribution, Dt(pt) ∼ Bino(Nt , πt), where both the number of trials (market size Nt(pt)) and the probability of success (market share πt(pt)) are determined by the pricing strategies via (38) and (39), respectively. Since the market size is sufficiently large, as MI pointed out, by Central Limit Theorem, your demand Dt(pt) is assumed to follow normal distribution Dt(pt) ∼ N (μt , σ 2 t ), (40) where μt = Nt · πt , and σt = p Nt · πt · (1 – πt), with Nt and πt are given in (38) and (39), respectively. Each unit demand materializes as a unit sale if met by a day-car supply. Unmet demand is lost. Thus, your revenue Rt(pt , St) in week t is given by Rt(pt , St) = p 1 t × min(St , Dt(pt)), (41) 11Based on the historical data, MI estimates coefficients bi in (38) via linear regression (regress() in MATLAB); and market share coefficients ci in (39) via Multinomial logistic regression ( mnrfit(), glmfit() in MATLAB). 49 where St = 7Q is the weekly day-car supply with the fleet size Q, and Dt(pt) is Hurts weekly demand in (40). Currently the branch has a fleet of Q = 1500 cars. Total costs are comprised of three parts: maintenance, inventory and fixed costs. Each unit sale.

1. Congratulations! You have recently been promoted to the
position of regional manager for Hurts
Car Rental Company in Florida. You need to make pricing
decision and fleet size decision for Tampa
branch to maximize its expected total profit in the coming
month (with 4 weeks or periods), taking
into account the behavior of both your customers and your
competitor.
Hurts has one primary competitor in Tampa with whom Hurts
competes for market share. Your
Market Intelligence (MI) has found that the competitor has
adopted a masked price following
strategy; that is, the competitors p
2
t
price in period t will follow your last price p
1
t–1 within ±20%
range:
p

2. 2
t = p
1
t–1 × εt
, εt ∼ U[0.80, 1.20], t = 2, 3, 4. (37)
Assume that the competitor begins by choosing the same price
as yours in period (week) 1, p
1
1 = p
2
1
.
MI also provides the weekly demand forecast for the coming 4
weeks based on the historical data.
Your and your competitor’s pricing strategies, pt = (p
1
t
, p
2

3. t
), jointly determine both the market size
Nt(pt) and the market share πt(pt). The market size Nt(pt), or
the total number of demands for
Hurts and its competitor in week t, is determined by pt = (p
1
t
, p
2
t
) via
Nt(pt) = b0 + b1p
1
t + b2p
2
t
, (38)

4. where b0 = 105644.5, b1 = –377.5, and b2 = –676.25.
The market share πt(pt) of Hurts, or the probability that a unit
demand choosing Hurts instead of
its competitor, is determined by the Multinomial logit (MNL)
model:11
πt(pt) = 1
1 + exp(–c0 – c1p
1
t
– c2p
2
t
)
(39)
where c0 = 0.0875, c1 = –0.0220, and c2 = 0.0170. Note that 0
≤ πt(pt) ≤ 1, and your competitor’s
market share is 1 – πt(pt).
Therefore, your demand Dt(pt) in week t follows Binomial
distribution, Dt(pt) ∼ Bino(Nt

5. , πt),
where both the number of trials (market size Nt(pt)) and the
probability of success (market share
πt(pt)) are determined by the pricing strategies via (38) and
(39), respectively. Since the market
size is sufficiently large, as MI pointed out, by Central Limit
Theorem, your demand Dt(pt) is
assumed to follow normal distribution
Dt(pt) ∼ N (μt
, σ
2
t
), (40)
where μt = Nt
· πt
, and σt =
p
Nt
· πt

6. · (1 – πt), with Nt and πt are given in (38) and (39),
respectively.
Each unit demand materializes as a unit sale if met by a day-car
supply. Unmet demand is lost.
Thus, your revenue Rt(pt
, St) in week t is given by
Rt(pt
, St) = p
1
t × min(St
, Dt(pt)), (41)
11Based on the historical data, MI estimates coefficients bi
in (38) via linear regression (regress() in MATLAB);
and market share coefficients ci
in (39) via Multinomial logistic regression ( mnrfit(), glmfit() in

7. MATLAB).
49
where St = 7Q is the weekly day-car supply with the fleet size
Q, and Dt(pt) is Hurts weekly
demand in (40).
Currently the branch has a fleet of Q = 1500 cars. Total costs
are comprised of three parts:
maintenance, inventory and fixed costs. Each unit sale incurs
unit maintenance cost M = $13 (per
sale), the amount of work attributable to oil changes, cleaning,
and preventative maintenance. Each
car in the fleet incurs unit monthly inventory cost I = $298 (per
car). Fixed costs, K = $344978,
are the sum of all other monthly costs of running the branch and
do not vary based on sales or
inventory quantity.
Each unit demand requires one day-car supply, i.e., one car for
one day rental and returns that
car in the following day; multi-day rentals count as multi-unit
demands. For example, a request
for 4-day rental of a car is treated as 4 units demands. With
fleet size Q = 1500, you have weekly

8. day-car supply St = 7 × Q = 10500 in week t.
As the manager, you need to decide the weekly price p
1
t
, t = 1, 2, 3, 4, and monthly fleet size Q that
together maximize net monthly profit before taxes. The
performance of each strategy is measured
by two criteria, monthly total profit V , the sum of profits of
four weeks, and monthly fill rate f ,
defined by the ratio between your monthly sales and your
monthly demand.
Parameters
Q=1500 M= $13 per sale I= $298 per car K= $344978
C0=0.0875 c1=-0.0220 c2=0.0170 and
b0=105644.5 b1= -377.5 b2= -676.25
Underlying Assumptions
According to the case study, the demand follows a binomial
distribution Dt(pt) ∼(Nt, πt) where both the market size, the
market share (trials), and the market share (the number of
success) are determined by the price strategies. Our competitor
price is in plus or minus 20% than our rental price. To compute
a simulation of 1000 months, we will follow a normal

9. distribution according to our market size and market share Mu
and sigma. It is given that our fleet size is 1500 and our price
will be determined every week to give us the optimal profit.
Objective
Our objective is to maximize the weekly price P and our
monthly fleet size Q that together will maximize our net
monthly profit for the Tampa branch, taking into account the
behavior of both customers and competition. We will measure
these performances by two criteria, the monthly total profit (V)
and monthly fill rate (f) defined by the ratio between the
monthly sales and the monthly demand. Following are the
questions we intend to utilize to ensure that we optimize our
total profit.
a)
Fleet size Q = 1500 and the price P = $40 (p1=p2=p3=p4).
Simulate the operation of the Tampa branch for n = 10^3
months (sample-path with 4 observations). Compute the
expected fill rate E[f ], total expected monthly profit VQ(p 1 ),
and its 95% confidence interval.
To simulate 1000 months (sample size N) we will use the
following metrics: where the fleet size Q= 1500 and the p=$40,
maintenance cost M=$13 per sale, monthly inventory cost
I=$298 per car, and fixed costs K=$344978, we will be able to
estimate the expected fill rate and the expected monthly profit.
Then, we will be able to measure the 95% probability of these
expectations.
b)
Assume Q = 1500. Repeat question 1 for p 1 ∈ { 45, 50, . . . ,
70 }. Graph VQ(p 1 ) against p 1 and find the optimal price p
1∗ that maximizes the total profit V ∗ Q = maxp 1 VQ(p 1 ).
Using the same metrics above but using multiple rental prices
to find out the optimal price that optimizes total profit, then use

10. these finding to plot the total profit in relation to the price P1*.
c)
Now change Q = 2000, repeat questions 1 and 2. Find the
optimal price p1∗ and total profit V ∗ Q for fleet size Q =
2000.
Our new Q = 2000 while other metrics are still the same. Using
the new Q and simulating a new sample size we will be able to
generate a new expected fill rate, optimal price, and a new
expected monthly profit.
d)
Repeat questions 1 and 2, for Q = { 2500, 3000, . . . , 4000 }.
For each Q, find the optimal associated price p 1∗ and the
optimal total profit V ∗ Q. Graph V ∗ Q against Q. Find the
optimal fleet size Q∗ .
Our simulation in this part will be containing different fleet
sizes. We will use the same parameters to apply the different
costs on each Q and find the optimal price P for each Q. We can
then use our findings to graph the total profit in relation to the
fleet size.
e)
Now consider a dynamic pricing strategy. Suppose you may set
the price p 1 t for each week differently, where p 1 t ∈ { 40, 45,
. . . , 70 }. By the similar approach in questions 1-3, find the
optimal price path (p 1∗ 1, p 1∗ 2, p 1∗ 3, p 1∗ 4 ) for each
fleet size Q ∈ { 1500, 2000, . . . , 4000 } and its associated
optimal profit V ∗ Q. Graph V ∗ Q against Q. Find the optimal
Q∗ . Compare the total profit under the dynamic pricing strategy
with that under the static pricing strategy. Which one performs
better? By how much? Why?
In this part, we will consider different prices and different fleet
sizes. Do the simulation for 1000 months and calculate the
optimal price and profit. Using these findings we will be able to
plot our optimal profit in relation to the optimal fleet size.

11. f)
Assume fleet size Q=2000 and static pricing strategy:
p1=p2=p3=p4=40 , simulate the operation for n= 1000 months.
what is the expected market size and the expected market share
for the Tampa branch?
Doing this simulation using the following metrics: the market
size Nt(Pt) and the market share πt(pt) where b0 = 105644.5, b1
= –377.5, and b2 = –676.25 and c0 = 0.0875, c1 = –0.0220, and
c2 = 0.0170 we will be able to visualize our expected market
size and our expected market share.
Market share = πt(pt) = 1/1+exp(-c0-c1*p1-c2*p2)
Market size = Nt(pt) = b0 + b1p1+b2*p2