0 Price

Demand

λ ←− potential demand

Λ ←− generous price policy

y^{λ} ←− satisfied demand

y^{Λ} ←− baseline

Figure 1.5: Can pricing improve on the transit of the generous policy?

⇔ ∃? pricing policyλ such that P

a,by^{λ}_{a,b}>P

a,by^{Λ}_{a,b}.

### 1.4 VSS pricing optimization

### 1.4.1 Revenue management in vehicle rental system

The origin of Revenue Management (RM) lies in the airline industry. It started in the 1970s and 1980s with the deregulation of the market in the United States.

In the early 1990s RM techniques were then applied to improve the efficiency of round-trip Vehicle Rental Systems (VRS); see Carroll and Grimes (1995) and Geraghty and Johnson (1997). One-way rental is now offered in many VRS but usually remains much more expensive than round-trip rental. One-way VRS RM literature is recent. The closest paper we refer to isHaensela et al.(2011) who model a network of round trip car VRS with the possibility of transferring cars between rental sites for a fixed cost.

For trucks rental, companies such asRentn’Dropin France orBudget Truck Rental in the United States are specialized in one-way rental, offering dynamic pricing. This problem is tackled byGuerriero et al.(2012) who consider the optimal management of a fleet of trucks rented by a logistic operator. The logistic operator has to decide whether to accept or reject a booking request and which type of truck should be used to address it.

Results for one-way VRS are not directly applicable to VSS, because they differ on several points: 1) Renting are by the day in VRS and by the minute in VSS with a change of scale in the temporal flexibility; 2) One-way rental is the core of VSS; for instance round trip rental represents only 5% of sold trips in Bixi (Morency et al., 2011), whereas it is classically the opposite in car VRS; 3) There is usually no

reservation in advance in VSS, it is a “first come first served rule”, whereas usually trips are planned several days in advance in VRS. At the best of our knowledge there are no results in the RM literature dedicated to pricing in VSS.

### 1.4.2 Pricing policies classification

We distinguish between different class of pricing policies. Classic pricing poli-cies take into account only the rental length and traveled distance. But apart from adjusting the offer and demand to optimize an overall revenue, they give no opera-tional leverage on the system. To drive the system towards its most profitable state, we need to influence the users by considering their diversities. With a parking spot reservation protocol, the system knows exactly which trip a user wants to take. We define two types of pricing policy, by station or by trip.

Definition 1(Station Pricing). A station pricing policy sets for each station a price
to take a vehicle and a price to return it. More formally, the pricep_{a,b}(t_{a}, t_{b}) to take
a trip from stationa at time ta to station b at timetb is a function P of the classical
price c_{a,b}(t_{a}, t_{b}) to take such trip and the incentives tak_{a}(t_{a}) to take a vehicle at
station a at time ta and retb(tb) to return it at station b at time tb:

pa,b(ta, tb) =P(taka(ta), retb(tb), ca,b(ta, tb)).

Definition 2 (Trip Pricing). A trip pricing policy sets a price to take each trip.

More formally, the pricepa,b(ta, tb) to take a trip from stationa at time ta to station
b at time t_{b} is a function P of the classical price c_{a,b}(t_{a}, t_{b})to take such trip and the
incentive tripa,b(ta, tb) to take the trip (a, b) between ta and tb:

pa,b(ta, tb) =P(tripa,b(ta, tb), ca,b(ta, tb)).

Incentives can depend or not on the system’s state, i.e. the vehicle distribution among the stations and the trip routes. It defines two types of policies: static and dynamic.

Definition 3 (Static Pricing). A static pricing policy sets (in advance) a price to take every trip at any time independently of the system’s state.

Definition 4 (Dynamic Pricing). A dynamic pricing policy can set a price to take a trip that depends on the current state of the system.

Operators have to consider the KISS principle (Keep It Simple and Stupid) in order to have their optimized policies accepted by users. They might be interested

1.4. VSS PRICING OPTIMIZATION 31

in a simple class of policies, easier to conceptualize for them, for the user as well as for the optimizer. We define a simple class of dynamic policies in which prices depend only on the current state of the pick up and return stations, and a simple class of static policies in which prices do not change along the day.

Definition 5 (Locally Dynamic Pricing). A station state dependent pricing policy can set the price to take a trip from a station a to a station b in function of the current states of stations a and b (parking filling and number of vehicles in transit toward them).

Definition 6 (Fully Static Pricing). A fully static pricing policy is a static policy that is constant over time, i.e. prices depend neither on the system’s state nor on the time the trip is taken.

When studying pricing policies, the demand is usually considered elastic in func-tion of the proposed price. It means roughly that higher the price is lower the demand will be. In theory, with perfectly rational users we would have a bijec-tive function linking prices and demands and therefore consider only continuous prices. However, because of psychological or marketing issues, we can be interested in studying discrete prices.

Definition 7 (Continuous Pricing). The pricep to take each trip has to be selected in a range: p∈[pmin, pmax].

Definition 8 (Discrete Pricing). The price p to take each trip has to be selected in discrete values: p∈ {p1, . . . , pk}.

With such formal definitions, we are now enable to categorize the studies done in the literature.

### 1.4.3 Classified literature review

Only few studies have been conduced in the literature to study the influence of pricing in VSS. Papanikolaou (2011) proposes ideas for locally dynamic station pricing based on the system dynamics framework. His model can be used as an educational tool for studying the behavior of VSS and exploring the impact of pricing policies on transit optimization. Pricing concepts are developed but no experimental results are provided.

Chemla et al.(2013) implement a simple locally dynamic continuous station pric-ing heuristic based on Monge’s transportation problem. Prices are updated regu-larly and aim at deterring users from parking at stations that are already nearly

full. Users have incentives to park at other stations that have a greater number of available parking places. Taking a sample of cities generated randomly, they show that adding a pricing regulation improves significantly the level of service.

Fricker and Gast (2012) study the impact of a simple dynamic policy in
homo-geneous cities. It is based on the power of two choices paradigm: the user indicates
randomly two destination stations and he is routed to the least loaded one. They
show that it improves dramatically the situation: with an optimal sizing the
prob-ability for a station to be empty or full decreases from _{K}^{2}_{+1} to 2^{−K} with K the
uniform station capacity. However, their results concern perfectly balanced cities
(homogeneous) and studies on more realistic cites are not investigated yet.

Pfrommer et al. (2013) consider a combination of intelligent repositioning de-cisions and dynamic pricing. Based on model predictive control principles, they develop a locally dynamic discrete station pricing with 10 different prices for the destination station (no incentive for the pick up one). They want to encourage cus-tomers to change their destination station in exchange of a payment to improve the overall service level.

### 1.4.4 Pricing in practice

V´elib’has developed a fully static station pricing. Some “stations +” are charac-terized by their high elevation such as Montmartre (hill). To reduce the gravitation phenomenon, when a user returns a bike at a “station +”, he receives in exchange 15 minutes of free ride that he can accumulate to execute a longer journey without paying (recall that the first half an hour is free).

Figure 1.1, page 19, gives an example of what could produce a discrete pricing per station. A color code is used to represent the different discrete prices to take and to return a vehicle. The green, the yellow and the red triangles represent cheap, normal and expensive stations respectively. The black triangles represent unavailable stations for taking on the left part, or returning on the right part. In this example, the user has finally chosen to pay for a “green taking” and a “yellow returning”.

Figure 1.6 shows a graphical user interface of a discrete pricing per station for GPS based system (in this case it could have been named a discrete pricing per location). The colors on the area represent different price levels.

1.4. VSS PRICING OPTIMIZATION 33

Figure 1.6: Price heated map. Source MIT Media Lab (Papanikolaou, 2011).

### 1.4.5 Digressions on pricing psychological impact

For station pricing policies, at least two types of price function exist to compute the incentives: (1)additive incentive adding a fixed fee to the classical renting price and (2)multiplicative incentivemultiplying the classical renting price by an incentive factor.

We conjecture that considering only additive incentives will impact mainly small distance trips. For instance, if a trip from a to b has a negative incentive of +5e to take a vehicle in station a, and no incentive (+0e) to return it in station b, this overall incentive of +5e can be considered as substantial for a 15etrip but barely unnoticeable for a 70e trip.

On the contrary, considering only multiplicative incentives will impact mainly the long distance trips. For instance, for a trip from a to b which has a negative incentive of +10% to take the vehicle in stationa, and no incentive (+0%) to return it in station b: this overall incentive of +10% has probably almost no effect on a 15e trip (1.5e) whereas on a 70e trip, this 7e difference might represent more for the potential user.

Therefore, to tackle both short and long trips, the best pricing would be a com-bination of multiplicative (α) and additive (β) incentives. Hence, a reasonable set of discrete station prices to take/return a vehicle could be the following couples (α, β):

(+0e,+0%), (+2e,+3%) and (+5e ,+5%).

Finally, a user going from a stationa, with incentive (α^{a}_{tak}, β_{tak}^{a} ) to take a vehicle,
to station b, with incentive (α^{b}_{ret}, β_{ret}^{b} ) to return a vehicle and for a renting time T
costingc(T), will have to pay:

p(a, b, T) =α^{a}_{tak}+α^{b}_{ret}+ (β_{tak}^{a} +β_{ret}^{b} )×c(T).

In the end, these digressions are related to the economical and psychology field.

In the mathematical models developed in this study, we assume that users are per-fectly rational and do not consider explicitly the economical incentives.