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Questions & Conjectures

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5.5 Discussion

5.5.2 Questions & Conjectures

Fluid model as an asymptotic limit To the best of our understanding, Fluid Solution Space (5.1) is a fluid approximation of the VSS stochastic problem. In the literature,e.g. Maglaras(2006), it is classic to interpret this model as an asymptotic limit of as-scaled problem sequence.

s-scaled stochastic continuous pricing problem

ˆ The stochastic process evolves in a scaled discrete space:

S(s) =

ˆ The state space contains fractions instead of integers and the basic unit cor-responding to a vehicle (job) and a time step is 1/s.

ˆ Each time stept ∈ T is divided into s parts with duration τts1.

ˆ The route/transportation time from stationa to station b is represented by s servers in series with rateµt,ra,b(s) =sµta,b.

ˆ The maximum time-varying transition rates are accelerated by a factor s:

Λta,b(s) = sΛta,b.

→ A solution is a continuous control on the prices for each trip, at each time step. Any demand λta,b(s) ∈ [0,Λta,b(s)] can be obtained at a price pta,b(s) =

1

spta,b(1sλta,b(s)).

The above scaling allows the convergence of not only the rewards, but also of the state process. We do not include a mathematical study of the convergence model to the fluid model, this is beyond our scope. However, in simulation (Section 6.5.3), Fluid SCSCLP (5.2) seems to converge towards thes-scaled problem.

a b

c

λ= 1

λ= 1

λ= 1 λ= 1

λ= 1

λb,c ∼=∞

Figure 5.4: Counter example for the convergence of thes-scaled problem as s → ∞towards the discrete-price fluid model.

Conjecture 1. Static optimal policies (and their values) of the s-scaled problem converge towards optimal policies of Fluid SCSCLP (5.2) when s→ ∞.

Remark 9. For discrete prices/controls, the s-scaled problem does not converge ass tends to infinity to the discrete-price fluid model (given in Section5.2.2.1). Indeed, as shows the example of Figure 5.44, the evaluation for a given price of the s-scaled problem does not converge to the discrete fluid model. In this instance, the difference is exhibited when the demand for the trip between stations b and ctends to infinity.

In the discrete-price fluid model, given a fixed demand λ, the flow yb,c from station b to station c tends to 1 as λb,c tends to infinity. While this value differs for the s-scaled problem evaluation where it is equal to: lims→∞limλb,c→∞yb,c = 23. However this is not a counter example to Conjecture 1.

Fluid model as an upper bound One would expect that the uncertainty in sales in the stochastic problem results in lower expected revenues than in the deterministic one. It is shown in many applications as inGallego and van Ryzin(1994). However

4. We are greatful to Nicolas Gast for pointing out this problem and providing this counter example.

5.5. DISCUSSION 119

for our problem, we have only been able to prove that the fluid optimal value function gives an upper bound for stationary demands (Theorem 10).

Conjecture 2. The value of Fluid SCSCLP (5.2)optimal solution is an upper bound on dynamic policies of the s-scaled problem (∀s).

Complexity of the fluid approximation The complexity of solving the fluid approximation is open. For a stationary demand and finite station capacities, the fluid approximation for the VSS discrete pricing stochastic problem seems “hard” to compute since its dynamic is non linear for a single discrete price. For a stationary demand, the fluid approximation for the VSS stochastic continuous pricing transit maximization problem is polynomially solvable in the number of stations M and constant in the number of vehiclesN. Indeed Stable fluid LP (5.3) gives the optimal policy (that is fully static) solving this problem.

For time-varying demands, the fluid model optimal static policy, solution of Fluid SCSCLP (5.2), may have an exponential number of “price patterns” in the network size (Fleischer and Sethurama, 2005). To strike this explosion, we might restrict our research to fully static policies, where prices do not depend on time. Fully static policies have a compact formulation but they are not dominant among general static policies. Moreover, solving the fluid approximation for the VSS fully static trip/station policies transit maximization problem is APX-hard for time-varying demands. The proof can be done with the same complexity argument as given for the deterministic VSS trip/station pricing problem, arising in the scenario-based approach in Chapter 3.

Chapter 6 Simulation

Measure what is measurable, and make measurable what is not so.

Galileo Galilei (1564–1642)

Chapter abstract

We want to estimate the potential impact of pricing in VSS. In the pre-vious chapters we have formulated heuristic policies and upper bounds.

We test them on case studies. A practical case study is conduced on Capital Bikeshare historical data. A simple demand pattern is generated from these data. We show that for such demand there is no potential gain for pricing policies. It exhibits the problem of accessing the real de-mand. A simple reproducible benchmark and an experimental protocol is proposed. We exhibit that the pricing leverage needs to be consid-ered jointly with the best fleet sizing. The static fluid heuristic policy appears the best one on the simulations. It allows to increase between 10% to 30% the number of trips sold. Max Flow With Reservation provides the best upper bound. Optimization gaps for dynamic policies optimization are between 50% to 100%.

Keywords: Monte-Carlo simulation; Benchmark; Experimental pro-tocol; Pricing; Fleet sizing; SCSCLP time discretization; Optimization gap; Real-case analysis; Reservation constraint.

121

R´esum´e du chapitre

Nous voulons estimer l’impact potentiel des politiques tarifaires dans les syst`emes de v´ehicules en libre service. Dans les chapitres pr´ec´edents nous avons propos´e diff´erentes politiques heuristiques ainsi que des bornes sup´erieures sur les gains possibles d’optimisation. Nous effectuons des tests sur des cas d’´etudes. Un cas d’´etude r´eel est analys´e sur les donn´ees d’exploitation de Capital Bikeshare. Un patron de demande simple est extrapol´e. Nous montrons que pour une telle demande il n’y a pas de gain d’optimisation. Cela met en exergue la n´ecessit´e d’acc´eder `a la de-mande r´eelle. Un benchmark simple et reproductible ainsi qu’un proto-cole exp´erimental sont propos´es. Nous montrons que l’´etude des poli-tiques tarifaires doit se faire conjointement avec un dimensionnement optimal de la flotte de v´ehicules. La politique statique donn´e par l’ap-proximation fluide apparait ˆetre la meilleure dans nos simulations. Elle permet de d’am´eliorer entre 10% `a 30% le nombre de trajets vendus.

Flot Max Avec R´eservation fournit la meilleure borne sup´erieure.

Des gains d’optimisations de l’ordre de 50% `a 100% sont observ´es pour les politiques dynamiques.

Mots cl´es :Simulation de Monte-Carlo ; Benchmark ; Protocole exp´erimental ; Tarification ; Dimensionnement de flotte ; Discr´etisation temporelle d’un SCSCLP ; Gain potentiel d’optimisation ; Analyse de cas r´eel ; Contrainte de r´eservation.

Contents

6.1 Introduction . . . 123 6.1.1 How to estimate pricing interest? . . . 123 6.1.2 Instance generation – Literature review . . . 123 6.1.3 The demand estimation problem . . . 124 6.1.4 Plan of the chapter . . . 125 6.2 A real-case analysis . . . 125 6.2.1 A trivial demand generation. . . 125 6.2.2 Discussion . . . 126 6.3 A simple reproducible benchmark . . . 127 6.3.1 Origin . . . 127 6.3.2 Instances . . . 128 6.3.3 Sizing . . . 130

6.4 Is there any potential gain for pricing policies? An experimental study132

6.1. INTRODUCTION 123

6.4.1 Experimental protocol . . . 132 6.4.2 Preliminary results . . . 135 6.5 Technical discussions – Models’ feature . . . 136 6.5.1 SCSCLP uniform time discretization . . . 136 6.5.2 The reservation constraint – Computing time vs quality . 138 6.5.3 Fluid as an∞-scaled problem. . . 139 6.6 Conclusion . . . 140 Part of this chapter is based on the working paper “Vehicle Sharing Systems pricing regulation: A fluid approximation” (Waserhole and Jost, 2013b).

6.1 Introduction

Dans le document The DART-Europe E-theses Portal (Page 132-138)