Bus Line Optimisation Using Autonomous Minibuses

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Oscar Bergqvist and Mikaela ˚Astrand

Abstract—Due to demographic changes, the transportation demand is predicted to increase significantly in the next decades.

Considering the transport sector’s impact on society and the environment, the development of a sustainable transport system is of great importance. Two possible building blocks in such a system are connectivity and automation, and this project aims to study a way of combining these two.

The purpose of this project is to investigate how the introduction of autonomous minibuses to a pre-existing bus system would affect its operational cost and environmental impact. This is done using a linear programming model that finds the optimal combination of conventional buses and autonomous minibuses with respect to cost. The model is implemented in the modelling system GAMS for bus lines 1–4 in Stockholm using data on travel demand. Two scenarios are analysed; the first allowing an arbitrary number of minibuses, and the second being more realistic and restricting the number of minibuses. The solutions are then compared to the corresponding solutions using only conventional buses.

In both cases, the results indicate that considerable savings can be obtained while maintaining or even improving availability.

From this, we draw the conclusion that when such technology is truly available, it would be advisable to investigate if these savings can weigh up the costs related to necessary investments.

I. INTRODUCTION

THE United Nations predicts an increase in the world’s total population from 7.3 billion people in 2015 to 9.7 billion in 2050 [1]. Simultaneously, from 2014 to 2050, the global urbanisation is projected to increase from 54 % to 66 % [2]. The total number of vehicles will double from approximately 1 billion now to 2 billions in 2030. Traffic congestion, noise and air pollution are already major problems in many cities, and the situation is predicted to deteriorate [3].

All these changes make efficient transportation a necessity in our future society.

When improving a transportation system, investing in public transport is usually the most cost-efficient way to do it.

Moreover, compared to an increased use of private vehicles, public transport can provide better accessibility and safety, and decrease environmental impact and congestion [4].

Connectivity and autonomous vehicles could both contribute to an improved public transport system. By connecting vehicles to a central traffic manager as well as other vehicles, connectivity allows for the collection of real-time data on travel demand and traffic conditions. This can be used for more efficient transport planning, leading to reduced emissions, congestion, and accident rates [3]. Some of the anticipated benefits of autonomous vehicles in contrast to ordinary vehicles are:

better traffic flow, increased road capacity, improved safety, decreased energy consumption and pollution, and reduced

driver costs [5], [6]. In addition to this and unlike conventional buses, autonomous buses do not depend on the schedule being made in advance, making them more flexible.

If connectivity and automation were to be combined, public transport providers would be able to offer a system more adaptive to changes in traffic flow and travel demand. This could lead to higher user friendliness; less waiting time, less crowding and cheaper tickets. Such improvements would make people more prone to choose public transport over the use of private vehicles, increasing its share of the total transportation.

Traditionally, our public transport systems are made of different kinds of vehicles running on a fixed schedule, each with a separate driver. The driver’s salary often amounts to a major part of the total operating cost, making it more profitable to use large vehicles. However, this leads to unnecessarily high fuel consumption and environmental impact during off-peak hours when the passenger flow is low, as well as low tour frequency due to profitability requirements. In Sweden, buses are by far the most common means of public transport, representing more than 50 % of total public transport boardings [7]. Hence, improvements in the bus system can have a great impact on the total public transport performance.

A future element in an improved bus system could be autonomous minibuses. Such buses are currently being developed by several different companies and have during 2016 and 2017 undergone trials in for instance Sweden [8], Finland [9], Singapore [10], and the US [11].

The purpose of this project is to study how the introduction of autonomous minibuses would affect the cost efficiency and environmental impact of bus lines 1–4 in Stockholm. As explained above, it is hoped that a public transport system including such buses would in the long term improve user friendliness and public well-being, and decrease congestion and environmental impact.

II. LINEAR PROGRAMMING

Linear programming is a method of finding the optimal value of an objective function, involving a linear combination of variables, subject to a set of linear constraints. Together, the objective function and the constraints form a set of linear equations and inequalities, which is referred to as a linear program. A linear integer program is a linear program where the domain of the variables is a set of integers [12].

Some famous linear integer programming problems related to this project include the travelling salesman problem (TSP) and the vehicle routing problem (VRP). The TSP is the problem of finding the shortest way to visit a set of cities

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exactly once and then come back to the city where the route started, given the distances between each pair of cities in the system. The VRP is a more general version of the TSP where several vehicles and routes can be used to visit all cities [12]. If the cities are generalised and replaced by other types of nodes, these problems have applications in many different fields.

A few examples are X-ray crystallography, order-picking in warehouses, and drilling of printed circuit boards [13].

In this project, the routes are already defined and the linear integer optimisation problem searches for the optimal combination of two bus types that minimises the total operational cost. This is formulated as an integer program as described in Section III.

In our implementation, the branch and cut algorithm is used to solve the integer program. The branch and cut algorithm solves a sequence of linear programs (branches) using the simplex algorithm, and removes (cuts) non integer solutions from the feasible set of each branch [14]. The simplex algorithm is a commonly used method to solve linear programs. It uses the fact that the maximum value of the objective function lies on one of the vertices of the defined region in the euclidean space [12]. If the simplex algorithm returns a non-integer solution, the branch and cut algorithm finds and adds more linear constraints that are satisfied by integer solutions but not by this non-integer solution. The simplex algorithm is again used to solve the resulting linear program and the process is iterated. The iteration continues until an integer solution is reached [14].

III. OPTIMISATION MODEL

The following equations are the mathematical formulation of the optimisation problem of this project. The objective function is Equation 1 and all variables and parameters are as in Tables I and II. The values of the parameters and their derivation can be found in Section V.

min

N

X

i=1

D_i(c_mx_m,i+ c_cx_c) (1)

i−M

X

j=1

aj−

i−1

X

j=M +1

rj ≥ ri ∀i = 1...N (2)

N

X

j=1

(a_j− r_j) = 0 (3)

µ_mm_mx_m,i+ µ_cm_cx_c≥ di ∀i = 1...N (4)

xm,i+ xc≥ f ∀i = 1...N (5)

x_m,i≤ L ∀i = 1...N (6)

x_m,i=

i

X

j=1

(a_j− r_j) ∀i = 1...N (7)

The objective function, Equation 1, gives the total operational cost per hour for the bus line. As mentioned in Section II the objective function is the equation being optimised, in this case minimised. The first constraint, Equation 2, ensures that the minibuses always visit a minimum of M consecutive bus stops: For some i, the first sum in the equation represents

TABLE I: Variables

Variable Description

xc= 0, 1, 2, ... The number of conventional buses per hour operating the bus line.

ai= 0, 1, 2, ... The number of new minibuses added at bus stop i per hour.

ri= 0, 1, 2, ... The number of minibuses removed at bus stop i per hour.

xm,i∈ Z The number of minibuses leaving bus stop i per hour.

TABLE II: Parameters

Parameter Description

Di The distance in km from bus stop i to bus stop i + 1.

cm The operational cost for a minibus in SEK/km.

cc The operational cost for a conventional bus in SEK/km.

M The minimum number of consecutive bus stops a minibus must visit.

N The total number of bus stops in the examined bus line.

µm Maximum average occupancy rate in % for a minibus.

mm The capacity of a minibus in persons/bus.

µc Maximum average occupancy rate in % for a conventional bus.

mc The capacity of a conventional bus in persons/bus.

di The number of people per hour wanting to go from bus stop i to bus stop i + 1.

f Minimum bus frequency in buses/hour.

L Maximum number of minibuses per hour at a bus stop.

the number of minibuses added to the bus line up to bus stop i − M . The second sum represents the number of buses removed from the bus stop up to bus stop i − 1. Thus the left side of Equation 2 represents the number of minibuses added up to bus stop i − M , that still have not been removed at bus stop i. Forcing a minibus to visit more than M consecutive bus stops is logically equivalent to restricting the number of minibuses being removed at bus stop i to the number given by the left side of Equation 2.

Equation 3 is a constraint making sure that all minibuses added to the bus line get removed again, by forcing the sum of added buses minus the sum of removed buses to be equal to zero. Equation 4 guarantees that the demand at each bus stop is satisfied, taking into account the maximum average seat occupancies µmand µc. Equation 5 prevents the frequency of buses to fall below the value f for any bus stop, ensuring high user friendliness. Equation 6 limits the number of minibuses visiting a given bus stop per hour, thereby avoiding congestion at the bus stop. Finally, Equation 7 is a constraint ensuring that xm,igives the number of minibuses per hour leaving bus stop i. This is achieved by taking the difference between the total number of minibuses added and the total number of minibuses removed up to bus stop i.

IV. THE TWO BUS TYPES

Most autonomous minibuses under development are electric, so we decided to compare two electric buses; the autonomous minibus Navya Arma and the conventional bus Eurabus 2.0.

Thereby, the comparison is made as fair as possible concerning

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Fig. 1: The autonomous minibus Arma by Navya. (Navya)

Fig. 2: The electric 12 m bus Eurabus 2.0. (Eurabus)

their environmental impact and the investments needed in infrastructure.

A. The autonomous minibus

There are not yet any autonomous minibuses for use on public roads on the market, but there are some similar bus types for use in closed sites, such as amusement parks and airports. We choose such a bus type, Arma by Navya (see Figure 1), as the minibus in our analysis. It is a completely electric and driverless vehicle launched in 2015 [15].

B. The conventional bus

As to the conventional buses, there are many different types with different fuels and different capacities. In this project, an electric bus is being considered: Eurabus 2.0 (see Figure 2). It runs on a battery and is charged in the garage like Arma, and is currently used in several different places in Sweden [16].

V. PARAMETERS

In this section, the values of all the parameters mentioned in Section III are derived.

A. Operational costs cm and cc

In this model, we only consider the costs related to the electricity consumption and the driver. According to [17], the electricity cost for charging an electric car in Sweden is approximately 1.2 SEK/kWh.

For the minibus there is no driver, so the cost is only related to the electricity consumption. Arma can be equipped with a battery of capacity 16–64 kWh [15] and has a range of up to 145 km [11]. Assuming that this range is for a battery of the highest capacity 64 kWh, this gives an electricity consumption of approximately Em= 0.44 kWh/km, which we round up to 0.5 kWh/km. Using the electricity price of 1.2 SEK/kWh, the operational cost of the minibus is found to be (in SEK/km):

cm= 0.6 (8)

The conventional bus Eurabus 2.0 has an approximate electricity consumption of Ec= 1.1 kWh/km [18], implying a cost of 1.3 SEK/km. According to [19], the average salary of a Swedish bus driver is 26,868 SEK/month, corresponding to a total cost for the employer of 37,683 SEK/month [20]. The average speed of a bus in city traffic in Stockholm is 12 km/h [21], and assuming an actual driving time of 35/40 h/week, the cost per kilometre is given by:

37, 683 SEK/month (35 · 52)/12 h/month· 1

12 km/h ≈ 20.7 SEK/km (9) Hence (in SEK/km):

c_c = 1.3 + 20.7 = 22.0 (10)

B. Capacitiesmmand mc

The capacity of Arma is 15 people, whereof four standing [15]. According to [22], the standard capacity of a 12 m bus is 70 people, whereof 35 standing. Hence (in persons/bus):

mm= 15 (11)

mc = 70 (12)

C. Maximum average occupancy ratesµm andµc

According to [22], on average maximum 20 % of the standing capacity should be used if the bus line is to be classified as having ”good standard”. For the conventional bus this corresponds to 7 people standing and for the minibuses 0.8 people standing. This gives the maximum average occupancy rate:

µm= 11 + 0.8

15 & 0.78 (13)

µc= 35 + 7

70 = 0.6 (14)

D. Number of bus stopsN and distances D_i between them A total of eight bus lines are being analysed; bus lines 1–4 in both directions, here denoted ”a” and ”b”. They have the following number of bus stops each:

N1a = 31 N2a = 23 N3a = 25 N4a= 31 N_1b= 32 N_2b = 24 N_3b= 26 N_4b= 30 The distances Dibetween all bus stops in the different lines can be found in Appendix D.

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E. Demand at each bus stop d_i

From SL (Storstockholms Lokaltrafik), the organisation responsible for all land based public transport in Stockholm, we obtain the data on travel demand used in our analysis.

This data consists of tables on the average demand between each bus stop on bus lines 1–4 in both directions during the four time intervals 06:00–08:59, 09:00–14:59, 15:00–17:59, and 18:00–05:59. All tables can be found in Appendix C.

F. Minimum bus frequency f

As a constraint on user friendliness, the maximum waiting time is set to 20 minutes, corresponding to the minimum bus frequency (in buses/h):

f = 3 (15)

G. Minimum number of bus stops visited by a busM Since most people travel several bus stops, it is reasonable that no bus should be added to the system unless it visits a minimum of four bus stops in a row. Therefore:

M = 4 (16)

H. Maximum number of minibuses per hourL

The unit of L is buses/h. Two different values of L are used, corresponding to two different sets of solutions. First the number of minibuses is unlimited, that is:

L = ∞ (17)

Then, one minibus every two minutes is allowed, that is:

L = 30. (18)

VI. IMPLEMENTATION

To solve the linear problem of Section III, we use the com- puter software GAMS. GAMS stands for General Algebraic Modelling System and is a high-level modelling system, used to solve mixed integer and real value linear programs. We import the demand data and the model parameters from Excel sheets to GAMS, and formulate the model equations in the GAMS IDE using its built-in scripting language.

The GAMS algorithm solves the integer program for each time interval using the branch and cut algorithm described in Section II. This is done for each of the bus lines 1–4 using a separate GAMS algorithm, see Appendix E. The resulting Excel sheets are finally imported into MATLAB, where the data is plotted in a presentable way, see Appendix F.

VII. RESULTS

For the model described in Section III, together with the parameters described in Section V, the optimal number of conventional buses is zero for all bus lines and all time intervals. Figures 3 and 5 show the corresponding numbers of minibuses for bus lines 1a and 4b.

When limiting the number of minibuses to 30 per hour, the optimal number of conventional buses for all bus lines and time intervals are as in Table III. The resulting minibus fleets for bus lines 1a and 4b can be seen in Figures 4 and 6.

Similar plots showing the number of minibuses for the remaining six bus lines can be found in Appendix B

TABLE III: The number of conventional buses operating the bus lines when a maximum of 30 minibuses can be used.

1a 1b 2a 2b 3a 3b 4a 4b

06:00 - 08:59 0 5 0 3 1 6 2 8

09:00 - 14:59 0 0 0 0 0 0 3 2

15:00 - 17:59 4 3 3 2 11 2 8 6

18:00 - 05:59 0 0 0 0 0 0 0 0

VIII. DISCUSSION

In this section we discuss our results in the context of how the solutions found would affect the cost and the environmental impact of the analysed bus lines. Finally, future work necessary to generalise our model and make it more useful for future traffic planning is considered.

A. Resulting bus fleets with unlimited number of minibuses When using the parameters specified in Section V and L =

∞, all the resulting fleets consist only of minibuses. This is a result of the big difference in operational cost per person and km between a conventional bus and a minibus. For a minibus and a conventional bus respectively, the cost can be calculated

as: cm

µmmm

= 0.05 SEK/(person · km) (19) c_c

µcmc

= 0.52 SEK/(person · km). (20) Thus, according to the model it is about ten times more profitable to use minibuses than conventional buses. Moreover, the minibuses can be distributed freely over the bus line and more effectively cover the demand. These things make the optimal solutions exclude the usage of conventional buses.

B. Practical issues with the usage of too many minibuses While using only minibuses is the optimal solution according to our model, it is clear that it causes some practical issues.

As can be seen in Figures 3 and 5, the number of minibuses can be very high during peak hours. For example in Figure 5, we see that during one time interval almost 60 minibuses per hour are needed, at two of the bus stops in the line. Having too many minibuses per hour will increase the traffic congestion without notably increasing the convenience of having a higher frequency of buses, and unnecessarily large bus stops will be needed. Moreover, the model does not account for the purchasing costs of the vehicles, which for an electric bus can amount to more than 65 % of its total life time cost [23].

Hence, we decided to limit the minibus frequency to one every two minutes. This is done by setting the model parameter L to 30, see Section III.

C. Resulting bus fleets when limiting the number of minibuses When limiting the number of minibuses but keeping the rest of the model and the parameters the same, the resulting minibus fleets for bus lines 1a and 4b are as in Figures 4 and 6. Table III displays the number of conventional buses for all bus lines and time intervals.

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5 10 15 20 25 30 Bus stop index

0 10 20 30 40 50

Minibusesperhour

06:00-08:59 09:00-14:59 15:00-17:59 18:00-05:59

Fig. 3: Minibuses per hour for bus line 1a when an unlimited number of minibuses can be used.

5 10 15 20 25 30

Bus stop index 0

10 20 30 40 50 60

Minibusesperhour

06:00-08:59 09:00-14:59 15:00-17:59 18:00-05:59

Fig. 5: Minibuses per hour for bus line 4b when an unlimited number of minibuses can be used.

Examining the results, we see that only minibuses are used when the demand is lower than the effective capacity of 30 minibuses. However, when the demand exceeds this value, two different behaviours of the solutions can be seen. Either, as for bus line 1a at 15:00–17:59 (Figure 4), the number of minibuses is maximised to 30 throughout the whole bus line and conventional buses are only used to cover the additional demand. Or, like in bus line 4b at 06:00–08:59 (Figure 6), we notice that the number of conventional buses and minibuses are balanced to cover the demand, and that the minibuses are distributed with a varying frequency over the bus line.

The remaining bus lines all show different versions of the behaviours seen in 1a and 4b, and the total number of solutions where conventional buses are used is 16 out of 32, see Table III. Out of these there are six solutions where the number of minibuses is maximised (like 1a) and ten solutions where the number of minibuses varies over the line (like 4b).

5 10 15 20 25 30

Bus stop index 0

10 20 30 40 50

Minibusesperhour

06:00-08:59 09:00-14:59 15:00-17:59 18:00-05:59

Fig. 4: Minibuses per hour for bus line 1a when a maximum of 30 minibuses per hour can be used.

5 10 15 20 25 30

Bus stop index 0

10 20 30 40 50 60

Minibusesperhour

06:00-08:59 09:00-14:59 15:00-17:59 18:00-05:59

Fig. 6: Minibuses per hour for bus line 4b when a maximum of 30 minibuses per hour can be used.

We see that the big difference in operational cost between the minibuses and the conventional buses tends to favour the minibuses, but the routing flexibility of the minibuses is in most cases an essential feature in optimising the solutions.

D. Bus line costs

In order to compare the solutions obtained to similar tra- ditional solutions, the implementation is repeated with the constraint of using only conventional buses, setting L = 0 (see Section III). In Tables IV and V, the total resulting bus line costs corresponding to the solutions with unlimited and limited number of minibuses, are presented. The numbers show the costs in percentage of the costs when using only conventional buses. All absolute costs can be found in Appendix A.

We note that the costs corresponding to the solutions using only minibuses are significantly lower than the ones also having conventional buses, typically in the range 3–7 %. For the solutions using a combination of buses, we still see a

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TABLE IV: Bus line costs using an unlimited number of minibuses, in % of the costs using only conventional buses.

1a 1b 2a 2b 3a 3b 4a 4b

06:00 - 08:59 5.9 5.8 5.8 4.7 4.6 4.4 6.0 6.4 09:00 - 14:59 6.5 6.1 7.0 5.5 6.3 6.5 6.7 6.4 15:00 - 17:59 6.3 6.1 6.3 5.7 4.9 5.5 6.7 6.7 18:00 - 05:59 5.3 4.0 4.9 3.9 4.5 3.5 5.0 5.5

TABLE V: Bus line costs using a maximum of 30 minibuses, in % of the costs using only conventional buses.

1a 1b 2a 2b 3a 3b 4a 4b

06:00 - 08:59 5.9 44.8 5.8 29.8 14.6 43.8 22.4 51.8 09:00 - 14:59 6.5 6.1 7.0 5.5 6.3 6.5 34.7 22.8 15:00 - 17:59 37.1 30.8 31.8 23.8 62.2 23.6 51.8 45.8 18:00 - 05:59 5.3 4.0 4.9 3.9 4.5 3.5 5.0 5.5

major gain, with values raging between 15–62 %. Adding autonomous minibuses to a public transportation system thus implies considerable savings in fuel and driver employment costs.

E. Environmental impact

To compare the environmental impact of the different bus types, we use the data on energy consumption from Section V. For a minibus this value is Em= 0.44 kWh/km and for a conventional bus it is Ec = 1.1 kWh/km. From these values, the effective energy consumption per person and km for a minibus and a conventional bus respectively is given by

E_m µmmm

= 0.038 kWh/(person · km) (21) Ec

µcmc

= 0.026 kWh/(person · km) (22) These numbers imply that the minibuses consume 46 % more energy per person and km than the conventional buses.

However, according to [24] the energy consumption of an electric bus is about one third of the energy consumption of a diesel bus and one quarter of a bus driven by gas. Using the same number of passengers as for the electric bus above (see Section V) and values on emission from [24], the CO2- eq emissions per person and kilometre for buses run on diesel or gas are as in Table VI. When using electricity as fuel, the CO2-eq emission depends entirely on the energy source; if using electricity from renewable sources the CO2-eq emission is negligible but usually this is not the case. In Sweden, the standard value used is the average CO2-eq emission from Nordic electricity (”Nordisk Elmix”), which in the time span 2005–2009 was 131.2 g/kWh [25]. The CO₂-eq emissions per person and kilometre for the two bus types considered in this project can also be found in Table VI.

It is clear that the difference in CO2-eq emission between these two bus types is negligible compared to the difference between electric buses and buses using fossil fuels. Yet another benefit of using buses running on electricity instead of fossil fuels is the improved local environment, since they impose less noise and particle emission [16].

TABLE VI: CO2-eq emissions for different bus types and fuels.

Bus type Fuel CO2-eq emission/(person · km)

Minibus Electricity 5 g

12 m bus Electricity 3 g

12 m bus Diesel 32 g

12 m bus Natural gas 35 g

12 m bus Biogas 7 g

F. Summary of possible consequences

From Section VIII-D it is clear that adding autonomous minibuses to a bus line significantly decreases its costs related to fuel consumption and driver employment. The minibuses are more flexible than conventional buses and follow the variations in demand at the different bus stops throughout the line as seen in VIII-A and VIII-C. Hence, they can decrease the cost while keeping the seat occupancy at a desired level. Also, constraints on the frequency of buses can be fulfilled without implying unnecessary high capacity during hours when the demand is low. This decreases the cost and the emissions of the bus line, though what would make the main difference for the greenhouse gas emissions is to replace fossil fuels with electricity (see Section VIII-E).

Introducing minibuses in a real world situation, at a smaller or larger scale, might thus be profitable. However, when considering such implementation it is important to also consider costs and constraints not included in our model. Improvements that could be made to our model in order to increase the reliability and usefulness of this project’s results are discussed in Section VIII-G.

G. Future work

In this study, it is not possible to include acquisition and maintenance costs since these data are not available for the minibuses. Future work needs to be done on refining the model and include these parameters, in order to get more reliable results. It would also be interesting to study several bus lines simultaneously, making the minibuses flexible to switch between bus lines and take other routes than the predefined.

Other related topics worth studying are other combinations of vehicle types, and technologies that can be used to predict travel demand in real time.

IX. CONCLUSIONS

From the results presented in this report, we can draw the following conclusions:

• Adding autonomous minibuses with flexible routing would decrease the costs related to fuel and driver employment for bus lines 1–4 in Stockholm.

• When such buses are accessible, it would therefore be interesting to investigate if these gains can weigh up the related investment costs.

• To minimise the environmental impact of the bus system, focus should be put on the choice of fuel rather than the choice of bus type.

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ACKNOWLEDGMENTS

First of all, we would like to thank our supervisor Othmane Mazhar for his dedication to our project, for teaching us essential skills in linear programming, and for his good advice and feedback on our work throughout the project. We would also like to thank Tomas Lindroth and Lisa Svenneberg at Trafikförvaltningen for providing us with real world data on buses in Stockholm. Special acknowledgements to Sofia Rahlén, also at Trafikförvaltningen, for taking the time to compile the right bus line data for our project.

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