We have presented a stochastic model to tackle the pricing optimization problem in vehicle sharing systems. This model simplifies the real-life problem, though it intends to keep important characteristics such as time-varying demands, station capacities and the reservation of parking spots at destination. In our study, we focus on the transit optimization and therefore do not consider prices explicitly.
Hence, we speak about pricing policies but they amount to considering incentive policies or simply policies regulating demand.
We proposed a formal definition for the VSS stochastic pricing problem. Al-though this formulation is compact and relatively simple, solving in general this
2.4. CONCLUSION 57
problem seems hard. We showed that even an exact measure of the VSS stochastic evaluation model is intractable for real size systems. We discussed notions of com-plexity in this stochastic framework. It allowed us to specify a frame in our search of tractable solution methods for the VSS stochastic pricing problem.
Chapter 3
Scenario-based approach
An approximate answer to the right question is worth far more than a precise answer to the wrong one.
John Tukey (1915–2000)
Chapter abstract
A direct solution method is intractable to solve the VSS stochastic pric-ing problem (defined Chapter2) for the size of systems we want to tackle.
We therefore discuss a scenario-based approach,i.e. off-line deterministic optimization problems on a given stochastic realization (scenario). This deterministic model could be used to provide heuristics and bounds for on-line stochastic optimization. This approach raises a new constraint the First Come First Served constrained flow (FCFS flow). We derive three problems based on FCFS flows: a design problem, optimizing sta-tion capacities, and two operasta-tional problems setting static prices. We show that they are all APX-Hard. We study the upper bound given by the classicalMax Flow problem and prove its poor worst case ratio.
Keywords: Scenario-based approach; Pricing; Queuing network; Com-plexity & approximation; Revenue Management; Graph vertex pricing.
59
R´esum´e du chapitre
Nous voulons r´esoudre le probl`eme stochastique de tarification dans les syst`emes de v´ehicules en libre service pr´esent´e Chapitre2. Une r´esolution directe est intractable pour la taille de syst`eme que l’on veut consid´erer.
Nous ´etudions donc une approche par sc´enario, i.e. une optimisation d´eterministe hors ligne sur une r´ealisation d’un processus stochastique (un sc´enario). Ce mod`ele d´eterministe peut ˆetre utilis´e pour fournir des heuristiques et des bornes sur le probl`eme d’optimisation en ligne. Cette approche soul`eve une nouvelle contrainte le flot premier arriv´e premier servi. Nous pr´esentons trois probl`emes bas´es sur cette contrainte : un probl`eme strat´egique, l’optimisation de la taille des stations, et deux probl`emes op´erationnels calculant des politiques tarifaires statiques. Nous montrons qu’ils sont tous trois APX-hard. Nous ´etudions une borne sup´erieure donn´ee par leFlot Max et prouvons sa faible performance dans le pire cas. Enfin nous montrons que leFlot Maxpeut donner un algorithme d’approximation de faible performance mais int´eressant d’un point de vue complexit´e.
Mots cl´es :Politiques tarifaires ; Approche par sc´enario ; R´eseau de files d’attentes ; Complexit´e & approximation ; Revenue Management ; Graph vertex pricing.
Contents
3.1 Introduction . . . 61 3.2 First Come First Served constrained flows . . . 62 3.2.1 FCFS flow in time and space network . . . 62 3.2.2 Station capacity . . . 63 3.2.3 Priced FCFS flows . . . 64 3.3 Station capacity problem . . . 65 3.4 Pricing problems . . . 68 3.4.1 FCFS Flow Trip Pricing problem . . . 68 3.4.2 FCFS Flow Station Pricing problem . . . 69 3.4.3 FCFS flow relaxation: Graph Vertex Pricing . . . 70 3.5 Connections to the Max Flow problem . . . 72 3.5.1 Max Flow upper bounds for FCFS flow problems . . . . 73 3.5.2 An approximation algorithm for FCFS Flow 0/1 Trip Pricing 75 3.6 Reservation in advance . . . 78
3.1. INTRODUCTION 61
3.6.1 No flexibilities . . . 78 3.6.2 Flexible requests . . . 78 3.7 Conclusion . . . 81 This chapter is based on the article “Vehicle Sharing System Optimization:
Scenario-based approach” (Waserhole et al., 2013b) submitted to The European Journal of Operational Research.
3.1 Introduction
In practice there is a lot of uncertainty in VSS dynamic. Dealing with human behavior, variability of user arrivals and transportation times has an important influence. In this context, stochastic optimization seems the most relevant approach to cope with randomness. In Chapter 2 we propose a stochastic model for the VSS stochastic pricing problem. For this model, a naive direct optimization with a Markov Decision Process computing the best dynamic (state dependent) policy is intractable: it can’t even scale up for systems in the order of 7 stations. This problem is known as the curse of dimensionality; the number of states of the induced Markov chain is exponential and hence exact solution techniques are not applicable. In this chapter, we study a deterministic approximation, the scenario-based approach, for the VSS stochastic pricing problem defined Chapter 2.
When dealing with stochastic problems, it is classic and natural to consider de-terministic approximations. The scenario-based approach amounts to optimizing a posteriori the system, considering that all trip requests (a scenario) are available at the beginning of the time horizon. Morency et al. (2011) show that, in Mon-treal’s BSS Bixi (2009), 68% of the trips were made by “members” and that their frequencies of use are quite stable along the week. For this context, considering deterministic requests might be a good approximation.
This approach offers two main advantages: On the one hand, the off-line deter-ministic optimization solution gives a bound for on-line stochastic optimization on a given instance; On the other hand, solving efficiently the deterministic problem on a scenario is the first step toward robust optimization methods (Bertsimaset al., 2011b), at least for models describing uncertainty by sets of scenarii.
Although this paper deals with VSS optimization, the theoretical problem ad-dressed is the optimal control of closed queuing networks with general service time and arrival rate distributions. Therefore, our results can be applied to a wider class of queuing network problems to conduce performance analysis (Bertsimaset al., 2011a) or to estimate the relevancy of robust optimization.
The remaining of this chapter is structured as follows: In Section 3.2, we describe a new type of constraint implied by the VSS scenario-based approach: the First Come First Served constrained flow (FCFS flow). In Section 3.3, we define a station capacity problem based on the FCFS flow that is shown APX-hard. In Section 3.4, we define two pricing problems based also on this constraint that are both shown APX-hard: 1) The trip pricing problem that decides a price for taking each trip and 2) The station pricing problem that decides for each station the price to take and return a vehicle. In Section3.5, we study a bound and an approximation algorithm for FCFS flow pricing problems based on the Max Flow algorithm.
Finally in Section 3.6, we study the complexity of a different deterministic problem that does not involve any FCFS flow rule: the optimization of trip reservation in advance.