Posted: May 19th, 2021

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 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

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.

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

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 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.

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 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.

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

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