Pricing in vehicle sharing systems: optimization in queuing networks with product forms

Abstract

One-way vehicle sharing systems (VSS) such as Vélib’ Paris are flourishing. The usefulness of VSS for users is highly impacted by the availability of vehicles and parking spots. Most existing systems are ruled by the trips of users. We study the potential interest of influencing the users to improve the performance of the system. We assume that each user is associated with a pair origin–destination (O–D) of stations, and only interacts with the system if his O–D trip is available. We consider leverage that can influence the rate of user requests for each pair O–D, such as a price that will be prohibitive for a prescribed proportion of users. We focus on optimizing the number of trips taken in the system. To provide exact formulas and analytical insights, transportation times are assumed to be null, stations to have infinite capacities and the demand to be stationary over time. In other words, VSS are modelled as closed queuing networks with infinite buffer capacity and Markovian demands. We propose a heuristic based on computing a Maximum Circulation on the demand graph together with a convex integer program solved optimally by a greedy algorithm. For $M$ stations and $N$ vehicles, the performance ratio of this heuristic is proved to be exactly $N/(N+M-1)$. We discuss our understanding on the possibility of extending this result to more realistic models in the perspectives. The complexity of computing optimum policies remains open. Insights on this issue are provided in the Appendix. The Appendix also contains an example showing that VSS can have poor performances without regulation.

Appendix

1.1 Toward computing optimal policies

In this Appendix, we discuss structures of optimal policies to develop tractable stochastic models to optimize a VSS through pricing. We discuss in “Structures of optimal dynamic policies” in Appendix the problem of characterizing dynamic optimal policies and in “Suboptimal classes of static policies” in Appendix the problem of characterizing static ones. Simple classes of policies, easier to optimize, are shown suboptimal.

1.2 Markov decision process: the curse of dimensionality

Computing optimal dynamic policies: The continuous-time Markov chain formulation of the VSS stochastic evaluation model leads directly to a Markov Decision Process (MDP), named the VSS MDP model. This model considers, in each state $s \in {\mathcal {S}}$, a set ${\mathcal {Q}}$ of discrete prices for each possible trip. Solving the VSS MDP model computes the optimal dynamic discrete pricing policy.

MDPs are known to be polynomially solvable in the number of states $|{\mathcal {S}}|$ and actions $|{\mathcal {A}}|$ available in each state. To solve an MDP, efficient solution methods exist such as value iteration, policy iteration algorithm or linear programming; see Puterman (1994) textbook. In each state $s \in {\mathcal {S}}$, the VSS MDP model’s action space ${\mathcal {A}}(s)$ is the Cartesian product of the available prices for each trip, i.e. ${\mathcal {A}}(s)={\mathcal {Q}}^{|{\mathcal {M}}|^2}$. The action space size is then exponential in the number of stations. However, to avoid suffering from this explosion, we can model this problem as an action decomposable Markov decision process; see Waserhole et al. (2013a). Thanks to this general framework, based on the event-based dynamic programming (Koole 1998), the complexity of solving the VSS MDP model becomes polynomial in $|{\mathcal {S}}|$ and $|{\mathcal {Q}}| |{\mathcal {M}}|^2$ (that is far less than $|{\mathcal {Q}}|^{|{\mathcal {M}}|^2}$). Nevertheless, the VSS MDP model has another problem: the explosion of its state space ${\mathcal {S}}$ with the number of vehicles and stations. This phenomenon is known as the curse of dimensionality (Bellman 1953).

1.3 Structures of optimal dynamic policies

Recall that dynamic policies have prices to take a trip that depend on the state of the system, i.e. the vehicle distribution. Unfortunately, even with homogeneous demand ($\varLambda _{a,b}=\varLambda$) optimal dynamic policies seem hard to describe.

Since the number of states is exponential, we would like to restrict to dynamic policies allowing a compact description. Capacity policies amount to specifying a virtual station capacity ${\mathcal {K}}$, and to accept a trip from station $a$ to station $b$ if only if the number of vehicles in $b$ is not exceeding ${\mathcal {K}}_b$.

We show in the next proposition that capacity policies are suboptimal among dynamic policies for the VSS stochastic pricing optimization problem.

Proposition 4

Capacities policies are suboptimal among dynamic policies, even in homogeneous cities.

Proof

Figure 6 compares the induced Markov chain (state graph) of three policies in an homogeneous city ($\varLambda =1$) with three stations and eight vehicles. An edge represents that the trip is open to its maximum in both directions, an arc indicates that it is open only in one way. Figure 6a represents the generous policy opening all trips and expects to sells 4.8 trips per time unit. Figure 6b represents the optimal dynamic capacity policy and increases the gain to $\approx 4.857$. Finally, the optimal dynamic policy is represented in Fig. 6, and increases the number of trips sold to $\approx 4.865$. $\square$

Fig. 6

Induced Markov chain of three policies evaluated in an homogeneous city with eight vehicles and three stations. ($\circ$) reachable state, ($\bullet$) unreachable state, ($-$) trip between two states open in both directions, ($\rightarrow$) trip open in only one direction. a Policy opening all trips, value $4.8$, b optimal dynamic capacity policy, value $\approx 4.857$ and c Optimal dynamic policy, value $\approx 4.865$

Figure 6 shows that using dynamic pricing policies can increase the number of trips sold by the system even in homogeneous cities (perfectly balanced). Figure 7 represents the optimal dynamic policies in an homogeneous cities with three stations when the number of vehicles increases: from eight vehicles (as in Fig. 6b), to 14 and 30 vehicles. Only the “spikes” of the dynamic policies’ induced Markov chain are represented since, the solution is invariant under the group $S_3$ of permutation of the stations. These solutions are the unique optimum. The optimal dynamic policy is solved with the VSS (decomposed) MDP model. This model is of exponential size in $N$ and $|{\mathcal {M}}|$ but still solvable for the size of these three instances. The solution uniqueness has been checked greedily solving several decomposed MDPs. It seems hard to find a compact description of optimal solutions in general.

Fig. 7

Spikes of optimal dynamic policies’ state graph for an homogeneous city with three stations and $N=8, 14$ or 30 vehicles

1.4 Suboptimal classes of static policies

1.4.1 Generous policies/no regulation

When investigating (pricing) policies, the most important practical issue is the trade-off between the simplicity (and in particular, the readability for users) and the performance. The first practical question might always be whether “unoptimized” policies perform well.

The (static) generous policy sets all demands to their maximum value ($\lambda =\varLambda$). To the best of our understanding, the generous policy is the most natural and relevant to compare with in theoretical studies, as long as the objective function is in terms of service quality and not in terms of monetary gain.

In Proposition 5, provides an example in which the number of trips sold by the generous policy can be arbitrarily far from an optimal static policy. It contains a “gravitational” phenomenon, which occurs in particular for bike sharing systems in non-flat cities.

Proposition 5

The ratio between the number of trips sold by the (static) generous policy ($\lambda =\varLambda$) and the static optimal policy is unbounded.

Proof

Consider a complete demand graph where all trip maximum demands are equal to $1$ except the trips from a special station $z \in {\mathcal {M}}$ to any other station that are worth $L^{-1}$: $\varLambda _{a,b}=1,\,\varLambda _{z,a}=1,\,\forall a \in {\mathcal {M}},\, \forall b \in {\mathcal {M}}\setminus \{ z\}.$

For any number of vehicle, when $L \rightarrow \infty$ the expected number of trips sold $T(G)$ for the generous policy $G$ tends to 0: The stationary distribution for one vehicle is $\pi _a=\frac{1}{L+M-1},\,\forall a \in {\mathcal {M}}\setminus \{z\}$ and $\pi _z=\frac{L}{L+M-1}$, hence $\lim _{L \rightarrow \infty } \pi _a=0 ,\,\forall a \in {\mathcal {M}}\setminus \{z\}$ and $\pi _z=1$. Since for all $N$, the availability vector $A$ satisfies $A = \alpha _N \pi$ for some scalar $\alpha _N$, we have:

$$\begin{aligned} \forall N\ge 1, \quad \lim _{L \rightarrow \infty } A_a=0,\quad \forall a\in {\mathcal {M}}\setminus \{ z \} \quad \text {and} \quad \lim _{L \rightarrow \infty } A_z=1, \end{aligned}$$

hence

$$\begin{aligned} \forall N\ge 1, \quad T(G)= \sum _{a\in {\mathcal {M}}} A_a (M-1) + A_z L^{-1} (M-1) \quad \Rightarrow \quad \lim _{L \rightarrow \infty } T(G)=0. \end{aligned}$$

On the other hand, the static circulation policy $C$ closing only trips to and from station $a$ has a expected number of trips sold $T(C)>1$ that is independent of $L$:

$$\begin{aligned} \forall L>0,\, \forall N\ge 1,\quad A_b=\frac{N}{N+M-2} ,\, \quad \forall b\in {\mathcal {M}}\setminus \{ a \} \quad \text {and} \quad A_a=0, \end{aligned}$$

hence independently of $L$, and for all $N\ge 1$ and $M \ge 3$

$$\begin{aligned} T(C)=\sum _{a\in {\mathcal {M}}\setminus \{z\} } A_a (M-2) = \frac{N (M-1)(M-2)}{N+M-2} \ge 1. \end{aligned}$$

$\square$

1.4.2 Bang-bang policies

Static policies directly have a compact representation: only one price per trip needs to be set, independently of the system’s state.

However, a compact formulation does not directly lead to a polynomial optimization. When considering only two possible prices per trip, a brute force solution method still needs $2^{|{\mathcal {M}}|^2}$ calls to the stochastic evaluation model. We need to exhibit structures to design efficient algorithms.

With the continuous demand assumption, static policies optimization amounts to setting the user arrival rates $\lambda$ with $0 \le \lambda _{a,b} \le \varLambda _{a,b},\, \forall (a,b) \in {\mathcal {D}}$. We investigate bang-bang policies (all or nothing) that set each trip $(a,b)\in {\mathcal {D}}$ to be either open ($\lambda _{a,b} = \varLambda _{a,b}$), or closed ($\lambda _{a,b} = 0$). One can wonder if bang-bang policies are dominant for the transit maximization. It is true for dynamic policies: bang-bang dynamic policies optimization can be reduced to a discrete price dynamic policies optimization in which deterministic policies are dominant (classic MDP results; Puterman 1994). Nevertheless, we show that bang-bang policies are not dominant among static policies even (which is more surprising) when the number of vehicles tends to infinity.

Proposition 6

Bang-bang policies are suboptimal among static policies even when the number of vehicles tends to infinity.

Proof

Figure 8 exhibits a counter example with four stations ($a,b,c,d$) and maximum trip demands $\varLambda _{a,b}=\varLambda _{b,c}=3$, $\varLambda _{c,d}=\varLambda _{d,a}=\varLambda _{c,a}=2$, all others are equal to $0$. There are only $2$ bang-bang static policies $\lambda$ defining a strongly connected demand graph: $\lambda _{i,j}=\varLambda _{i,j},\, (i,j)\ne (c,a)$ and either $\lambda _{c,a}=0$ or $\lambda _{c,a}=2$. When the number of vehicles tends to infinity, the availability of a vehicle at station $a$ equals $\frac{\pi _a}{\max _{b \in {\mathcal {M}}}\pi _{b}}$, where $\pi$ is the stationary distribution for one vehicle (George and Xia 2011). For the $\lambda _{c,a}=0$ policy, we have $\pi _a=\pi _b=\frac{2}{10}$ and $\pi _c=\pi _d=\frac{3}{10}=\pi _{\max }$, so the expected transit when $N \rightarrow \infty$ is worth $\frac{\pi _a}{\pi _{\max }}(3+3)+\frac{\pi _c}{\pi _{\max }} (2+2)=8$. For the $\lambda _{c,a}=2$, policy we have $\pi _a=\pi _b=\frac{4}{14}$ and $\pi _c=\pi _d=\frac{3}{14}$, so the expected transit when $N \rightarrow \infty$ is worth $10.5$ which is thus the optimal bang-bang static policy. Yet, for the non bang-bang policy with $\lambda _{c,a}=1$ and still $\lambda _{i,j}=\varLambda _{i,j},\, (i,j)\ne (c,a)$, we have $\pi _a=\pi _b=\pi _c=\pi _d=\frac{1}{4}$, so the expected transit when $N \rightarrow \infty$ is worth $11 >10.5$. Hence, bang-bang policies are suboptimal even when the number of vehicles tends to infinity. $\square$

Fig. 8

Bang-bang policies are suboptimal even when the number of vehicles tends to infinity

1.4.3 Single-component policies

One may wonder whether it is useful to have a policy dividing the city.

Notice that when considering static pricing policies with more than one strongly connected component, one should explicitly consider the vehicle distribution among these components. In fact, dividing the city sometimes leads to better performances: It is a leverage to prevent the system from being in unprofitable (unbalanced) states.

Proposition 7

Static policies with one single strongly connected component are suboptimal among static policies.

Fig. 9

Static policies with a single strongly connected component are suboptimal

Proof

An example is schemed Fig. 9 with four stations and a symmetric demand matrix. For two vehicles, the optimal static policies in this case is to close the trips $(b,c)$ and $(c,b)$ and open all other trips to their maximum value, i.e. $\lambda =\varLambda$ except $\lambda _{b,c}=\lambda _{c,b}=0$. The demand graph of this policy has two strongly connected components. The optimal vehicle distribution is to put one vehicle on each of them. With such distribution it expects to sell 200 trips per time unit. The optimal static policy with a single strongly connected component opens all trips to their maximum value, $\lambda =\varLambda$. It expects to sell 160.8 trips per time unit. $\square$