**4.3 Indications regarding the traffic model**

**Figure 4** shows the chain of causation which determines the traffic conditions. The traffic model establishes a 'stochastic equilibrium' for the availability of

**Figure 3.** *Four-tier architecture.*

**Figure 4.** *Systemic diagram of the traffic model for a ring-format shuttle service.*

#### **Figure 5.**

*Traffic laws for a ring-format shuttle service (under N = 200).*

vehicles, taking into account the components of demand and their fluctuation along the ring route, over time and in terms of ride length. This gives us the average speed of the service, with the operating speed per vehicle, which is of interest to the operator, as well as the average usage speed, of interest to users. These speeds will also determine the average ride time and access time.

**Pattern Objective function and side**

*Volume of demand with respect to generalised cost.*

*Towards Shared Mobility Services in Ring Shape DOI: http://dx.doi.org/10.5772/intechopen.94410*

SO, system optimum <sup>P</sup>ou � <sup>P</sup><sup>o</sup> <sup>þ</sup> <sup>P</sup><sup>u</sup> with

S2 = SO under budget

balance

**Table 2.**

**Figure 7.**

**175**

**Figure 6.**

S1 = SO under min return Pou under P<sup>u</sup>*=*ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>b</sup>* <sup>P</sup><sup>o</sup> <sup>≥</sup><sup>0</sup>

*Objective function and pricing rule depending on regulation pattern.*

*Supply-demand state is the solution to a fixed-point problem.*

MO, monopoly <sup>P</sup><sup>o</sup>ð Þ� *<sup>τ</sup>*, *<sup>N</sup> <sup>τ</sup>:*Qð Þ� *<sup>τ</sup>*, *<sup>N</sup> C N*ð Þ , Qð Þ *<sup>τ</sup>*, *<sup>N</sup>*

<sup>P</sup><sup>u</sup> � <sup>Ð</sup> *<sup>Q</sup>*

**constraint**

<sup>0</sup> <sup>D</sup>ð Þ �<sup>1</sup> ð Þ*<sup>q</sup> :*d*<sup>q</sup>* � *<sup>Q</sup>:<sup>g</sup>*

Min benefit per € of public funds, *b*

**Pricing rule Notation**

*<sup>τ</sup>* <sup>¼</sup> *<sup>C</sup>*\_ *<sup>Q</sup>* <sup>þ</sup> <sup>Q</sup>*g*\_*<sup>Q</sup> <sup>τ</sup>* <sup>¼</sup> ^*τ*<sup>o</sup>

*<sup>τ</sup>* <sup>¼</sup> ^*τ*<sup>o</sup> MO

*<sup>τ</sup>* <sup>¼</sup> ^*τ*<sup>o</sup> S1

SO

S2

*<sup>τ</sup>* <sup>¼</sup> *<sup>ε</sup> <sup>C</sup>*\_ *<sup>Q</sup>* <sup>þ</sup>Q*g*\_ ð Þ *<sup>Q</sup> ε*þ1

*<sup>Q</sup>* � *gT*þ*C=<sup>Q</sup>* 1�ð Þ 1þ*ε* ð Þ 1þ*b*

*C*

Min Pou subject to P<sup>o</sup> <sup>≥</sup><sup>0</sup> *<sup>τ</sup>* <sup>≥</sup>*C N*ð Þ , *<sup>Q</sup> <sup>=</sup><sup>Q</sup> <sup>τ</sup>* <sup>¼</sup> ^*τ*<sup>o</sup>

**Figure 5** shows the variations in average access time *t*<sup>A</sup> and ride time *t*<sup>R</sup> in response to the volume of demand, for a fixed-size shuttle fleet.

#### **4.4 Indications regarding the supply-demand equilibrium**

**Figure 6** shows the variations in the volume of demand in response to price, or rather the generalised cost (the sum of the ticket price plus access and ride times, weighted by their respective 'values of time'), for a fleet of fixed size and for a demand function with constant elasticity of �2 with respect to generalised cost. The blue curve shows the 'original' demand function for exogenous traffic conditions, while the red curve illustrates the 'adjusted' demand function which takes into account the interaction between the volume of demand and the traffic conditions (simulated in the preceding tier of the model).

#### **4.5 Indications regarding optimised service management**

**Table 2** shows the objective function assigned to the production of the service, for each of the different operating regimes available as options in the strategic phase. Demand function Qð Þ *τ*, *N* represents the volume of demand in response to trip price *τ* and fleet size *N*: it summarises both the demand model and the traffic model. Cost function *C N*ð Þ , *Q* represents the cost of producing the service, on a daily basis: this depends on the number of vehicles and number of trips required. The underlying parameters, for example the capacity of the shuttles or the energy costs involved, are to be determined at the strategic level.

**Figure 7** provides a graphic representation of the solution to the production optimisation problem, in this case for a shuttle service. The primary unknown is price per ride *τ*, on the y axis. The variable *x* is a load factor: this is the key factor to resolving the traffic model. The pink function models the behaviour of demand: demanded price

#### **Figure 6.**

vehicles, taking into account the components of demand and their fluctuation along the ring route, over time and in terms of ride length. This gives us the average speed of the service, with the operating speed per vehicle, which is of interest to the operator, as well as the average usage speed, of interest to users. These speeds will

**Figure 5** shows the variations in average access time *t*<sup>A</sup> and ride time *t*<sup>R</sup> in

**Figure 6** shows the variations in the volume of demand in response to price, or rather the generalised cost (the sum of the ticket price plus access and ride times, weighted by their respective 'values of time'), for a fleet of fixed size and for a demand function with constant elasticity of �2 with respect to generalised cost. The blue curve shows the 'original' demand function for exogenous traffic conditions, while the red curve illustrates the 'adjusted' demand function which takes into account the interaction between the volume of demand and the traffic conditions

**Table 2** shows the objective function assigned to the production of the service,

**Figure 7** provides a graphic representation of the solution to the production optimisation problem, in this case for a shuttle service. The primary unknown is price per ride *τ*, on the y axis. The variable *x* is a load factor: this is the key factor to resolving the traffic model. The pink function models the behaviour of demand: demanded price

for each of the different operating regimes available as options in the strategic phase. Demand function Qð Þ *τ*, *N* represents the volume of demand in response to trip price *τ* and fleet size *N*: it summarises both the demand model and the traffic model. Cost function *C N*ð Þ , *Q* represents the cost of producing the service, on a daily basis: this depends on the number of vehicles and number of trips required. The underlying parameters, for example the capacity of the shuttles or the energy

also determine the average ride time and access time.

*Traffic laws for a ring-format shuttle service (under N = 200).*

**Figure 5.**

**174**

(simulated in the preceding tier of the model).

response to the volume of demand, for a fixed-size shuttle fleet.

*Models and Technologies for Smart, Sustainable and Safe Transportation Systems*

**4.4 Indications regarding the supply-demand equilibrium**

**4.5 Indications regarding optimised service management**

costs involved, are to be determined at the strategic level.

*Volume of demand with respect to generalised cost.*


#### **Table 2.**

*Objective function and pricing rule depending on regulation pattern.*

**Figure 7.** *Supply-demand state is the solution to a fixed-point problem.*

*<sup>τ</sup>*uð Þ *<sup>x</sup>* . Each of the other curves represents a specific pricing policy adopted by the producer: proposed price *<sup>τ</sup>*oð Þ *<sup>x</sup>* . Optimising production consists of levelling up demanded price and proposed price: for each producer behaviour, the corresponding solution is the point of intersection (x<sup>∗</sup> , τ<sup>∗</sup> ) which ensures that *<sup>τ</sup>*oð Þ¼ *<sup>x</sup> <sup>τ</sup>*uð Þ *<sup>x</sup>* .

capacity enables us to reduce the number of stops required to pick up/drop off other passengers, thus increasing the usage speed. Third, in order to satisfy a given volume of ridership, the fleet size will need to be bigger than it would if using vehicles with a larger capacity (e.g. around fifty passengers in a standard city bus); this makes it possible to increase the frequency of the service, and thus to reduce the time users wait to access it. Fourth, on an industrial level, the manufacturing of larger fleets of vehicles allows for greater economies of scale and drives down the cost price per vehicle. We might also expect to see greater flexibility in the internal logistics of the service: recharging, cleaning, maintenance and repairs. Fifth, regarding the environment, the impact of the fleet of vehicles over its whole life cycle may be reduced: this applies to the construction phase, due to the industrial efficiency gained, and the usage phase, thanks to the increased operating speed. Nevertheless, using collective vehicles of a smaller size presents at least two disadvantages. The first is that the cost of driving the vehicles is increased proportionally to 1*=K*. The second is the sacrifice of potential economies of scale for vehicles powered by combustion engines. However, these disadvantages could feasibly be swept away by the advent of self-driving vehicles and the rise of electric motors. For all of the simulation scenarios envisaged here, the common parameters are

• Benchmark volume of *Q*<sup>0</sup> ¼ 26,000 journeys per day at a generalised cost of €2 per trip. The elasticity of the volume of demand to this generalised cost is

• Values of time are set at €12/h for running time and €8/h for access time.

• For each shuttle, a daily cost price of €600 (resp. 500) for electric (resp. combustion) vehicles with drivers and €100 for self-driving electric vehicles.

• Energy cost: €0.05/km for an electric motor, or €0.15/km for a combustion

**Table 3** presents the main results for each of the 12 scenarios simulated, with one scenario per column. The scenarios are grouped by technological generation, in ascending order of technological progress. For each generation, the three regulatory

Taking all of these scenarios into consideration, the following indicators emerge. The number of trips per day varies from 455 to 8000 depending on the scenario. The size of the fleet varies from 4 vehicles, for an MO regime in the pre-platform

• For each day of activity, service in operation for H ¼ 14 hours.

as follows. The territorial aspects involve:

*Towards Shared Mobility Services in Ring Shape DOI: http://dx.doi.org/10.5772/intechopen.94410*

• Driving speed *v*<sup>0</sup> ¼ 20 km/h.

For demand:

set at �2.

engine.

**177**

**5.3 Results and discussion**

regimes are presented in the order MO – S2 – SO.

Production parameters:

• A ring route with a radius of R ¼ 2.3 km.

• An average journey distance of *L*<sup>R</sup> ¼ 3.7 km.
