**3.5 Connections to the Max Flow problem**

**3.6.2 Flexible requests**

We consider now flexible requests where users are ready to change their origin and/or their destination stations, delay or advance the date of their trip. A user request can be satisfied by several station-to-station trip alternatives with possibly different gains. Each request can be arbitrarily accepted, i.e. served with one of its alternative, or refused. There is no consideration of a first come first served rule.

The problem is to find the set of requests to serve in order to maximize the overall gain.

3.6. RESERVATION IN ADVANCE 79

Max Flow With Alternative

Instane: A set of stations S with capacities Ks for s ∈ S, a number N
of vehicles with their distribution among the stations at the beginning of the
horizon, a set R = {(s^{k,r}_{o} , t^{k,r}_{o} , s^{k,r}_{d} , t^{k,r}_{d} , p^{k,r}), k ∈ K, r ∈ R} of trip requests
with|K| alternatives.

Solution: The set of requests R^{′} to serve with the alternative k chosen.

Measure: The generated gain of the flow R^{′}:
X

(r,k)∈R^{′}

p^{k,r}.

Theorem 6. Max Flow With Alternative is NP-hard even with requests of unitary price.

Proof. We reduce the NP-hard problem 3-SAT to Max Flow With
Alterna-tive. We use a gadget called the “k-choices”. It directs a flow of k vehicles from
a station to exactly one station out of two. Figure 3.10 schemes an example for
k = 3. The general construction is the following. There are k vehicles at station a
to go either all to station b or c. At time step 0, there are k trip requests with no
alternative to go from station a to stations s_{0}. . . s_{k}_{−}_{1} at time step 1. At time step
1, we can have a vehicle in each station s0. . . sk−1. Then, there are k trip requests
(r_{i}, i ∈ {0. . . k−1}) with two alternatives: (1) to go from stations^{1,i}_{o} =s_{i}to station
s^{1,i}_{d} =c or (2) to go from station s^{2,i}_{o} =si+1 mod k to station s^{2,i}_{d} = b, arriving both
at time step 2. The only possibility to serve all 2k trip requests is to accept either
all trip alternatives (1) going to stationb or all trip alternatives (2) going to station
c. All other policies incur a loss of at least two trip requests.

We consider now a 3-SATinstance with m clauses andn literals. Each literal l is represented by 3 stations: ˙l when the literal is unassigned, l when it is set to true and ¯l when it is set to false. At the beginning of the horizon, there are m vehicles available at every station ˙l. At time step 0, there is a “k-choice” gadget with k=m to direct a flow ofm vehicles either to station l or ¯l. We create a station r to store the number of clauses satisfied (represented as the number of vehicles in station r at the end of the horizon). For each clause i (i=1 to m), there is a trip request at time step i with three alternatives. For clause a∨b∨c the three alternative trips are to go from station a to station r, b to r or ctor.

The 3-SATinstance is satisfiable if and only if theMax Flow With Alter-nativeinstance serves 2mn+m demands: 2mfor each of the n literal assignments (through ak-choice gadget) and m to satisfy all clauses.

(1,0)

Figure 3.10: k-choices gadget, example with k = 3. On the upper part of the figure, the compact representation of the gadget.

+m

Figure 3.11: 3-SAT reduction as a Max Flow With Alternative. Two clauses are represented: a∨¯b∨c and a∨b∨¯c.

3.7. CONCLUSION 81

### 3.7 Conclusion

In this chapter, we have investigated a scenario-based approach for the VSS stochastic pricing problem. Its principle is to work a posteriori on a realization of the stochastic process: a scenario. Optimizing on a scenario provides heuristics and bounds for the stochastic problem. In this context, such approximation raises deter-ministic problems with a new constraint: the First Come First Served constrained flow (FCFS flow). We presented three such problems: 1) a system design problem, optimizing station capacity (FCFS Flow Station Capacities) and two opera-tional problems setting static prices, 2) on the trips (FCFS Flow Trip Pricing), or 3) on the stations (FCFS Flow Station Pricing).

We showed that all three problems are APX-hard, i.e. inapproximable in poly-nomial time within a constant ratio. Therefore, we investigated a bound and an approximation algorithm using theMax Flowalgorithm (hence relaxing the FCFS flow constraint). The theoretical guaranty (worst case) for the bound provided by the Max Flow algorithm on a scenario is exponential in the number of stations.

Nevertheless, it is competitive in practice. We use Max Flow With
Reserva-tion to compute upper bounds in Chapter 6 devoted to the simulation. Moreover,
from a theoretical point of view, it can be used to build a _{N((M}^{1}_{+2)!)}-approximation
algorithm for the FCFS Flow Trip Pricing problem with unitary prices; with
N the number of vehicles and M the number of stations.

We conjecture that the inapproximability ratios ofFCFS Flow Trip/Station Pricing and FCFS Flow Station Capacities are greater than a factor linked to the number of stations. One can hence be satisfied to have an approximation algorithm that does not depend on the number of trip requests |R|. However, in current VSS, the number of trips sold in one day is in the order of M (or N).

Therefore, an approximation algorithm in |R| might be more useful.

Finally, giving good and usable heuristic solutions using scenario-based opti-mization, studying metaheuristic approaches might be interesting. However, it is not sure that they can explore such large space and provide good solutions within a reasonable time. Indeed, the evaluation cost of amovement on a static policy seems important, at first sight basically in the order of computing again the whole FCFS flow.

## Chapter 4

## Queuing Network Optimization with product forms

The art of doing mathematics consists in finding that special case which contains all the germs of generality.

David Hilbert (1862–1943)

Chapter abstract

This chapter proposes an approximation algorithm to solve a sim-pler stochastic VSS pricing problem than the general one presented in Chapter 2. In order to provide exact formulas and analytical insights:

transportation times are assumed to be null, stations have infinite ca-pacities and the demand is Markovian stationary over time. We propose a heuristic based on computing a Maximum Circulation on the de-mand 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). Hence, whenever the number of vehicles is large compared to the number of stations, the performance of this approximation is very good.

Keywords: Closed Queuing Networks; Pricing; Product forms; Continuous-time Markov decision process; Stochastic optimization; Approximation algorithms.

83

R´esum´e du chapitre

Ce chapitre propose un algorithme d’approximation pour r´esoudre un probl`eme stochastique de tarification dans les syst`emes de v´ehicules en libre service. Ce probl`eme est simplifi´e par rapport `a celui pr´esent´e Chapitre 2. De mani`ere `a obtenir des formules exactes et des r´esultats analytiques, les temps de transports sont consid´er´es nulle, les stations ont des capacit´es infinis et la demande est markovienne stationnaire.

Nous proposons une heuristique bas´ee sur le calcul d’une Circulation Maximum sur le graphe des demandes coupl´e `a un programme entier convexe r´esolu optimalement par un algorithme glouton. PourM stations et N v´ehicules, le ratio de performance de cette heuristique est prouv´e ˆetre exactement N/(N +M −1). Par cons´equent, lorsque le nombre de v´ehicules est grand devant le nombre de stations, la performance de cette approximation est tr`es bonne.

Mots cl´es : R´eseau de files d’attentes ferm´e ; Tarification ; Forme pro-duit ; Processus de d´ecision Markovien `a temps continu ; Optimisation stochastique ; Approximation.

### Contents

4.1 Introduction . . . 85
4.2 Simplified stochastic framework . . . 85
4.2.1 Simplified protocol . . . 85
4.2.2 Simplified VSS stochastic evaluation model . . . 86
4.2.3 Simplified VSS stochastic pricing problem . . . 88
4.3 ^{Maximum} ^{Cirulation} approximation . . . 90
4.3.1 Maximum Circulation Upper Bound . . . 90
4.3.2 Maximum Circulation static policy . . . 91
4.3.3 Performance evaluation . . . 96
4.4 Conclusion . . . 101
This chapter is based on the article “Pricing in Vehicle Sharing Systems: Queuing
Network Optimization with product forms” (Waserhole and Jost,2013a) submitted
to the special issue on shared mobility systems in EURO Journal on transportation
and logistics.

4.1. INTRODUCTION 85

### 4.1 Introduction

In Chapter 2, Section 2.3.3, we discussed the properties of optimal dynamic
and static policies. An optimal dynamic policy can be computed with an action
decomposable Markov decision process. However the number of states of the MDP
grows roughly as N^{M}, where N is the number of vehicles and M is the number of
stations considered. This chapter proposes an approximation algorithm to solve a
simpler stochastic VSS pricing problem than the general one presented in Chapter2.

In order to provide exact formulas and analytical insights: we investigate simplified stochastic models allowing an analytic formula for the performance evaluation of the system.

In Section 4.2, we define the simplified model we are going to restrain to.

We consider VSS with stationary O-D demands and infinite station capacities, as in George and Xia (2011), but we also assume null transportation times. Under these assumptions, the VSS can be modeled as a closed queuing network of BCMP type. Its performance can therefore be computed analytically. We define static and dynamic stochastic pricing problems on such queuing networks.

In Section4.3we study a static heuristic policy provided by the Maximum Cir-culation on the demand graph. When theMaximum Circulation disconnects the city, vehicles have to be spread among the connected components. The vehicle distribution problem amounts to maximizing a separable concave function under linear and integrality constraints. It can be solved optimally by a greedy algorithm.

The exact guaranty of performance of our heuristic on dynamic and static policies
is proved to be _{N}_{+M}^{N}_{−}_{1}.