5.2.1 A plumbing problem
The fluid approximation can be seen as a plumbing problem. Stations are repre-sented by tanks connected by pipes representing the demands. Vehicles are consid-ered as a continuous fluid evolving in this network. The volume of a tank represents the capacity of a station. The length of a pipe represents the transportation time between two stations. The section of a pipe between two tanks a and b represents the demand between stations a and b, it ranges over time from 0 to the maximum demand Λt. Figure 5.1 schemes an example with 2 stations. The modeled system has no dynamic interaction with the user. Decisions are static and have to be taken before, once for all. The fluid optimization amount to setting the width of a pipe by changing the price to pass flow in it: the higher the price is, the smaller the pipe (demand) will be.
For a given policy (prices/demands) the deterministic evolution of the system is subject to different constraints. They derive from the first come first served rule that happens in practice. If a pipe (a demand) exists and there is some flow (vehicles) available in the tank (station), the flow has to pass through the pipe. If there is not enough flow to fulfill all pipes (demands), there should be somedeparture equity between them. In other words, the proportion of filling up of all pipes should be equal. However if the arrival tank of a pipe is full, it might be impossible to fulfill this pipe and respect the departure equity as it is. In this case, an arrival equity should be applied to all pipes discharging into this tank. In other words, for each pipe, if its discharging tank is full, it has the same proportion of filling up as the other pipes discharging in this tank, otherwise, it has the same proportion of filling up as the other pipes coming from its source tank. We callequity issues the problem of respecting the arrival/departure equity to model the evolution of the flow.
5.2.2 Discrete price model
The main goal of this section is to exhibit the complexity of solving the fluid model for discrete prices. Though results of this section might be interesting, the technical aspects are “hard to digest” and not useful to understand the rest of the chapter. The reader only needs to understand the contribution of each section: In Section 5.2.2.1 we propose a non-linear model that formally specifies as a mathe-matical problem the fluid approximation with discrete price. In Section 5.2.2.2 we show that the fluid dynamic with fixed prices/demands is not linear. Therefore it cannot be model as a linear program and is probably hard to solve.
5.2. THE FLUID MODEL 107
λta,b λtb,a
Ka Kb
µb,a−1
ya,b
Control
Stationa Stationb
Figure 5.1: A plumbing problem.
5.2.2.1 A non linear model
Before building a mathematical model, we recall the data and define the variables of the model as schemed in Figures5.2a and 5.2b.
Data:
N the number of vehicles available;
M the set of stations;
Ka the capacity of station a;
D the set of possible trips (=M × M);
µ−a,b1 the transportation time between station a and b λa,b(t) the demand rate from station a at timet to station b;
Pa,b(t) the set of possible prices to go from station a at timet to station b at timet+µ−a,b1; Λ(price) the function giving the demand for a given price.
Variables at timet:
pa,b(t) the price to take the trip from station a to station b;
φ+a(t) the proportion of requests accepted among those willing to leave station a;
φ−a(t) the proportion of requests accepted among those willing to arrive at station a that have been accepted to take a vehicle at their departure station;
ya,b(t) the flow leaving station a at timet and arriving at station b at timet+µ−a,b1; ya,bdep(t) the flow accepted to leave stationa but not yet accepted to arrive at station b;
ya,bref(t) the flow refused by station b “returning to station a” (one has ya,bdep(t) =ya,bref(t) +ya,b(t)), (this variable is not needed and not explicit in the model but helps the understanding) ; sa(t) the available stock (number of vehicles) at station a;
ra(t) the number of parking spots reserved at station a (flow in transit towards a).
Discrete price model
5.2. THE FLUID MODEL 109
t_ij
λta,b λta,c Ka
ya,bt
ytx,a
ya,ct ytz,a
(a) Incoming and outgoing flows: An equity issue.
(b) Variables for 2 stations.
Figure 5.2: Discrete price fluid model variables.
Remark 7. It is easy to compute the value of a solution with one price without the flow stabilization constraint. A simple iterative algorithm on the horizon works.
With the flow stabilization constraint it is not clear that looping on such iterative algorithm converges to a stable solution.
5.2.2.2 A non linear dynamic
The previous program might not be the simplest formulation of the discrete price optimization problem. However, as claims the next lemma, the discrete price dynamic is not linear and therefore there exists no linear program modeling the discrete price optimization problem.
Lemma 9. The fluid model with fixed demands is not linear.
Proof. A simple evaluation for a given price, hence a given demand λ, presents a non linear dynamic. Figure 5.3 shows an example with integer data where the instantaneous flow is an irrational number. There are 6 stations. At timet, stations a and d are not empty, c and f are not full, b is empty and e is full. All instant demands (λt) have for intensity 1. For a matter of simplicity, in the sequel, the time parameter (t) will be implicit. From the paradigm of arrival and departure equity,
we deduce the instantaneous value of the flow as follows:
(b is empty, no arrival equity) ydepa,b =ya,b =λa,b → ya,b= 1, (Departure equity in b) yb,cdep
λb,c
(Arrival equity in e) yb,e
yb,edep = yd,e in the data are rational, the flow dynamic for a given price is not linear.
5.2.3 Continuous price model
We can avoid dealing with equity issues for the continuous prices fluid evaluation.
The trick is toalways fulfill the pipes, in other words to have a flow y between two stations that is exactly equal to the demandλ for taking this trip. It is always pos-sible when assuming a continuous elastic demand,i.e. there exists a pricep(λta,b) to obtain any demandλta,b∈[0,Λta,b]. Solutions respecting this trick define the solution space of the fluid model with continuous price. More formally the “fluidification” of the state space is the following:
Continuous price fluid model solution space
A continuous space replaces the discrete one:
SF =
A continuous deterministic demand with rateλta,breplaces the discrete stochas-tic one.
A deterministic transportation time of duration µta,b−1 replaces the stochastic one.