Optimal Location of Bus Depots in an Urban Area Enrico Musso & Anna

Optimal location of bus depots in an urban area

Enrico Musso & Anna Sciomachen akg/z

Abstract

This paper deals with the so-called "plant location problem" and aims to set up a methodology for determining the optimal location and size of bus depots. Location and size of bus depots are usually not efficient, mainly because of: - the subsequent growth of the city, or significant changes in location patterns (causing possible inefficiencies in terms of land use); - the growing demand for public transport over time (causing ineficiencies in exploiting economies of scale). Nevertheless, any "present" location is usually very difficult to change, due to: - the lack of an appropriate methodology to find out better locations; - the unpopularity of any decision concerning new locations for bus depots. The poposed methodology aims to investigate the overall costs connected with depots

(land use costs + operational costs + journey-to-depot costs) and to minimize them by determining the optimal combination of number, size and location of depots, given the urban transportation network and the overall number of buses. A case study is then developed for the city of Genoa. The present scenario is analyzed with respect to several decision constraints, (geography, feasible depot sizes, resource allocation, bus types, time of investment and budget restrictions). Then, the shortest path between any pair of possible bus stops is computed together with the average number of trips for each bus line and the location, handling and managing costs for each depot. Finally. a Mixed Integer Programming (MIP) model is derived for solving the problem on the basis of the average daily traffic data in the city of Genoa. The model is solved using an optimization software library on a personal computer.

1 Purpose of the paper

This papers aims to point out a methodology for optimizing size and location

of bus depots in an urban transit company. Sizes and locations of bus depots are often a result of choices taken in the past, no longer efficient with respect to the present technological and market conditions. For example, after a significant urban growth the depot location

94 Urban Transport and the Environment for the 21st Century

has possibly become too central, and the area has now a greater economic potential. Or, the oldest depots may be too small for present need, so that the company may be induced to set up additional smaller depots.

Besides, the potential opposition of resident population makes it politically difficult to move a depot toward any new site. So that, as a result, the previous location is more acceptable and eventually very stable, despite of its possible inefficiency.

It should also be said that land values of nearby areas are negatively affected by the new depot, while land values of previous neighbourhood will be positively affected. Both changes will be included only in future prices, and result in unfair damages for present landowners. Thus, the overall number of relocation should be reduced to the minimum. The lack of satisfactory methods to find out the optimal solution in terms of number/size/location of depots often causes urban planning to be almost completely ineffective. These methods are needed in order to:

- achieve higher long-term efficiency in land use; - reduce speculations deriving from discretionary choices on depot location.

2 Some specifications

In the proposed methodology, we have assumed that the overall number of buses, the busline network, and the frequency on each line, are given (since depot supply must be optimized with respect to service supply, and not the opposite). It could be objected that location and size of depots are tipical long term choices. But on the other side, since the transit network depends on demand, no reliable forecasts can be developed on future demand over the whole life of the depot (at least 50 years). Obvoiusly, some overcapacity of depots with respect to present number of bus should be considered, but this just means a different overall capacity, and does not implies adjustments in methodology.

We have assumed that costs to be minimized are the costs of the transit company. No external costs (like pollution of empty-journeys from/to the depot) have been considered, since: - transit companies deficit are presently so relevant that this should be considered a priority goal; - given the network and the frequency (which means, given the level of efficacy), any further gaps between social and company costs are basicly connected to congestion and pollution, which are implicitly minimized since the proposed method sets up a trade off between minimum empty journeys and economies of scale in depot management.

3 The cost function

Basicly, the problem is to minimize a cost function including:

Urban Transport and the Environment for the 21st Century 95

- the costs of space (the pontential land value of the area where the depot is expected to be located); - the costs of depot (the cost of depot management and of all functions and activities taking place inside the depot); - the costs of empty journeys (from/to the depot: mainly staff costs and fuel costs); without reducing the performance of the network (i.e.: same network, same number of vehicles, same frequency).

3.1 The costs of space

We could consider two different criteria: - a "potential" value criterium, where land value is basicly determined by the demand for alternative use of the space, and is influenced by features such as central location, accessibility, environmental quality, etc.;

- an "actual" value criterium, where land value is strongly affected by urban planning regulations concerning the area. Quite often, planners follow the latter, and just confirm the present location of depots, thus keeping low the value of those areas. Clearly, decisions concerning location of depots, instead of being influenced by present location, should come first, and should only be influenced by the potential value of the areas, determined by alternative land uses.

In any application of the proposed model a given number of alternative sites will be considered, and for each of them the total cost includes the value of the objective function (costs of depot + costs of journeys to/from depots) and the specific potential value of the area. The model allows to rank the alternatives from the point of view of all costs related to depots. We have not included the cost of space in the objective function of the model, since it is easier to estimate and add the potential value of the area only for best potential locations.

So, the objective function to minimize is a cost function including the costs of depots and the costs of journey to/from depots. A trade off is clearly to be stated between possible economies of scale in depots management and costs of journey to/from depots, which are higher if the number of depots is lower.

3.2 The depot costs

The depot costs include:

- labour costs: the bigger part of the depot costs (in the study case - Genoa, Italy - they weigh over 75%); - consumption: electric power, fuel, water, telephone, etc.; - disposable and spare parts.

Cost categories rank from general and sundry expenses, contract works, bus preparation, servicing expenses, cleaning, plant servicing.

96 Urban Transport and the Environment for the 21st Century

The analysis of depot costs for the transit company of the city of Genoa shows the existence of economies of scale up to a size of 180-200 vehicles per depot (the total number of vehicles is 848). For bigger depots, average costs appear more or less constant, without relevant diseconomies of scale. These

results are confirmed by a productivity index (namely: number of vehicles per staff unit), which increases until the depot sizes reach the above mentioned number of buses. Obviously, total depot costs can be minimized only if the number of depots is minor or equal to the ratio between the total number of buses and the minimum optimal size, and if no depot is below that size. For example, the depot costs related to the 848 Genoa buses are properly minimized with 1,2,3 or 4 depots (provided that none of them is smaller than 180-200 units), while

costs increase if the number is higher (they are presently 7).

3.3 The jouney to/from depot costs

The jouney to/from depot costs are those costs that are necessary for buses to reach the terminus of the correspondind bus line from the depots, and viceversa. Note that during this journey buses cannot offer any transportation to the users, so that only pure costs can be considered in this case to be added to the overall costs connected with depots. These costs consist of: - labour costs; - fuel and consumption costs.

The above costs can be considered as a function of the distance between depots and bus termina, that is denoted as "empty path". The "empty path" is usually as shorter as greater is the number of depots (and then as smaller is their size), and as shorter as the path followed by buses is the optimal one.

Therefore, the journey from/to depots are as smaller as the bus terminus is closer to the depot. Moreover, the journey from/to depots cost are also a function of the number, say F, of input/out daily entries of each bus. It can be observed that usually buses with low frequency are in service the whole day, such that they travel on their empty path only twice a day (once from and once to the depot); on the contrary, buses that are active only during rush hours move from/to depots about two/three times a day In our case study we consider that about

50% of buses moves only once from their depots within 24 hours. The overall daily journey from/to depot cost for each depot, denoted by CA/, is given in (1), and includes all the different costs previously analyzed. CM (1) is given as a function of the average number of kilometers Knij travelled on an empty path by any of the N, buses running on the y'-th bus line, considering N be the total number of buses running daily.

- L[0.57 V x2F + 0.5Nx4F]xJ \I Q . +i - » ^j ( 1 ) TV

Urban Transport and the Environment for the 21st Century 97

An empiric investigation performed during the data collection phase related to our case study has allowed us to evaluate the gasoline cost Q: to be about

8647 Lira/km and the manpower cost Cp to be about 44200 Lira/hour It has to be noted that in our analysis we consider the average travelling speed Vm on the empty path to be constant within the day and independent on the direction of the path; it is evaluated to be about 15 km/hour.

4. The shortest path from/to depots

The computation and the subsequent analysis of the above defined empty paths play a crucial role in the determination of the best depot location. For this reason, a relevant part of our work has been devoted to the selection of the optimal area for the bus depots for which the distance between bus termina and depots is the minimum one. The first step in looking for optimal terminus-depot paths in an urban transportation network is to derive the corresponding mathematical model, reflecting all the possible features of the selected geographical area with respect to the problem under consideration. In order to derive the model related to the public tranportation network in the city of Genoa we performed the following four steps, that are usually performed in any transportation planning project (Note: for recent advances in methods and models in urban transportation planning interested readers are referred to [2]. See also [1] for basic definition and notation for transportation models and algorithmic approaches).

1. First, we have to identify within the geographic area under consideration, i.e. the city of Genoa, all the possible areas which are feasible for urban bus depots. In our case, we have considered the only 10 areas within the urban territory which could allow in a very next future the location of bus depots

also satisfying sizing and environmental constraints. 2. Then, the selected area is split into many zones; each of them, characterized by a relevant location, is identified by the so-called centroid, in such a way only journey between zones is considered as a journey between centroids.

As a result of this phase, we have identified three main areas in the city of Genova, that is East side, West side and Center, that are represented in total by 146 centroids. 3. We then model the urban public transportation network by a digraph G =

(N,A), where,Vis the set of nodes representing the selected centroids, and A is the set of arcs (/,/) connecting node / to node/ V /,/ e N', A hence gives the possible road connections between the selected centroids. Starting from the 146 nodes belonging to TV that have been identified at step 2, we have

identified the set C ci TV consisting of 82 nodes representing the termina of the bus lines that have been taken into account in this study, the set R a N consisting of the 10 depots previously identified, and the set T c N

consisting of 54 nodes, denoted transition nodes, that correspond to

98 Urban Transport and the Environment for the 21st Century

particularly relevant locations within the city, such as squares or intersection points among large roads. Note that the arcs in our digraph G have been derived by considering all the possible connections between pair of nodes taking into account the bus travel direction.

4. Each arc (ij) e A has a weight associated with it giving the cost required to travel from / toy e Nin the urban transportation network. According to the problem under consideration, we assign as arc weight the distance dy

between / and y, thus expressing our cost function as a function of the kilometers travelled in the journeys from/to depots, as it has been explained in Section 3. Note that the distances asociated with the arcs in G have been computed starting from the map reflecting the transportation network using

a 1:17.000 scale. In order to determine the closest depot to each bus terminus and consequently to minimize the empty path from/to each depot to/from each bus terminus, for each of the 1 0 areas derived in step 1 , we had to compute the

distances to be travelled by buses related to the shortest paths between any node in C, i.e. any bus terminus, and any node in R, i.e. any depot. Such distances are the solution of the well known shortest path problem in a graph; that is, given a digraph G = (N,A), we have to compute, among all

the feasible paths P connecting an origin node, say node O e N, to a destination node, say node D e N, the path with the minimum total cost with respect to the objective function under consideration, that in our case is the

total travelling distance, given by the sum of the weights dy associated with the arcs (ij) belonging to the path between the selected origin node e C and the destination node e R. Our objective function is given in (2), where Xy is a

Boolean variable associated with the direct arc (ij) e A (that is Xy = 1 if arc (ij) belongs to path f, and Xy = 0 otherwise), and dy is the distance (in meters) between / andy.

(2)

It is worth mentioning that considering the set of transition nodes T in graph G, besides a better representation of the urban area under consideration,

allows a better flexibility in the model when considering, for instance, the analysis of the empty paths not only between bus termina and depots but also, as it will be considered in a forthcoming paper, between depots and these selected relevant locations (nodes in 7), thus allowing to reduce the

inproductive distance travelled by buses without people on board. In order to compute the required shortest paths a graphic-interactive implementation of the Dijkstra algorithm running on a PC has been used (see [3] for further details about the main shortest path algorithm presented in the recent literature).

Note that by using the Dijkstra algorithm for solving the shortest path problem from each bus terminus to each depot, for each origin node / e C, representing one among the 82 bus terminus previously identified, we

Urban Transport and the Environment for the 21st Century 99

determine the so called shortest path tree, that is a spanning tree on G rooted at node /, in which each branch from the root node / to the leave node y, Vy e

R, gives the shortest path from / toy. In the following, we will denote by T(i) the shortest path tree related to the bus terminus node / e C. Once we have computed T(i) V / e C the lenghts of the shortest paths between any bus terminus to any area where it is possible to locate a bus depot are known. Figure 1 gives a representation of the the solution of this problem, where from any of n nodes belonging to C (n = 82) there is an oriented arc to each of the m nodes belonging toR(m = 10); such an arc, denoted by &%, V / e C, Vy e R, is weighted with the lenght y^ of the shortest path between / andy, that is the value z of the objective function reported in (2) where nodes / and y are considered, respectively, as the origin and destination node of the problem. By looking at Figure 7, considering all outcoming arcs from each node in C to each node in R it is possible to determine the lenghts of all paths

(branches) of T(i).

Figure 1. The shortest path tree from each bus terminus to each bus

5. Optimal depots location

Once the shortest path between all terminal nodes and the areas where it is possible to locate a bus depot is computed, the next step is to determine the optimal set of areas that minimize the total costs reated to the journey to/from

depots. In this preliminary work, our analysis is restricted to consider all the possible combinations of 4 and 3, respectively, depots. In the first case we have considered 2 depots located at the east side and the other one located at the west side of the city. In particular, we have considered

100 Urban Transport and the Environment for the 21st Century

all the possible combinations of pairs of areas among all the selected ones (respectively 4 and 6), computing, for each combination, the shortest path and the corresponding lenght, as it is described in Section 4, and associating consequently with each bus terminus the closest depot. Then, once the total number of buses circulating in a given day is known, for each bus line it is possible to compute the total number of kilometers travelled on empty paths with respect to all the considered bus lines. In particular, in order to determine the best combination of 4 bus depots among the 10 feasible ones, and then assigning to each bus line the corresponding depot, a Boolean Linear Programming model has been derived. Another interested approach for determining the optimal location and sizing of bus depot is given in [4]. In this proposed model, we consider as objective function the minimization of the sum of the distances with respect to all combinations of areas and depots identified in Section 4, where the cost coefficient is given by the optimal value y,j associated with the shortest path tree T(i) , V i E C, represented as weights of arcs (/,/) in Figure 1. The proposed model is the following:

subject to v.. - 1 Vi 4)

7-1

(n-3) >,>M Vy (5)

V/ (7)

iA. E{0,I) V/,7 (8)

In particular, (3) is our objective function, where Boolean variables v,y are direct arcs representing paths between nodes / and / as reported in Figure 1; variables v,y are then such that v,y = 1 if in the optimal solution the empty path from bus terminus / to depot 7, is travelled, and v,y = 0 otherwise. Relations (4)

- (8) define the constraints of the present optimization problem. (4) forces each node in C to have one and only one outgoing arc; therefore, such set of constraints establishes a link between each bus terminus and the corresponding depot. Relations (5) and (6) give the upper and lower bounds to the number of ingoing arcs of each depot. In particular, Boolean variables My, V / E R, force each depot node to be the destination node of at most n-3 bus lines, while, according to our definition of the problem, k depots have to be chosen among the m feasible ones (in this case k = 4).

Finally, (7) and (8) are the definition of the variables of our problem that can have only a {0,1} value.

Urban Transport and the Environment for the 21st Century 101

For solving model (3) - (8) we use the software environment LINDO [5], thus deriving the total kilometers travelled in the empty paths given by the value of (3) with respect to all the possible combinations of areas located at the west and east side of Genoa. In Table 1 we report the total kilometers travelled in the empty paths for all the combinations of 4 depots among the 10 considered. Note that the combination of the 4 areas denoted by M-B and SP-TG gives the minimum value with respect to model (3) - (8) and then is the optimal combination as far as the journey from/to depot costs is concerned. However, such solution cannot be considered as the optimal solution of our localization problem since the capacity constraint of the depots, for which the optimal size in terms of operational cost minimization is in the range of 100-300 buses, is violated. Thus, by taking into account this further constraint and the trade-off between operational costs and journey from/to depot costs we can consider the combination SP-CA and A-M as the optimal solution of the problem.

CA-CO CA-SP CA-TG CO-SP CO-TG SP-TG A-CS 2277,42 2214,74 2319.48 2185.04 2164.76 2141.84 A-M 2250,43 2189.02 2243,90 2159.32 2137,77 2116.12 A-TE 2313,45 2240.53 2310,89 2210,83 2200.79 2167.63 2655,59 2407,08 2455.00 2363.88 A-B 2567,66 2436.78 A-Q 2871,71 2731,94 2963,74 2702.24 2750.74 2659.04 CS-M 2319,57 2258,16 2313,04 2228,46 2206.91 2185.26 CS-TE 2356,69 2294,01 2398,75 2264.31 2244.03 2221.11 CS-B 3602,41 2265,83 2170,57 2036.13 2015.85 1992.93 CS-Q 2348,83 2286,15 2390,89 2256,45 2236.17 2213,25 M-T 2308,16 2246,75 2301,63 2217,05 2195.50 2173,85 M-B 2124.65 2263,24 2118,12 2033,54 2011.99 1990,34 M-Q 2331,23 2269,82 2324.70 2240.12 2218.57 2196.92 TE-B 2242,05 2189.13 2239,49 2139.43 2129.39 2096.23 TE-Q 2423,77 2350,85 2421,21 2321.15 2311.11 2277.95 2863,17 2732,29 2951,10 2702.59 2750,51 2659,39 B-Q Table 1. kilometers travelled in empty paths in the case of 4 depots

In the analysis for deriving the best combination of 3 depots, we have

considered 2 depots located at the eastern side of Genoa an the other one located at the western side of the city The same procedure as before has been used, solving model (3)-(8) with k = 3 in (5). The results are reported in Table

2. We can see that the minimum value corresponds to the combination of the areas M-B and SP, but, as in the case of 4 depots, this solution gives rise to a number of buses in depot B less than the minimum allowed (100), and thus the

optimal combination is given by the 3 areas A-M and SP. Finally, the total number of kilometers associated with each depot has been determined by multiplying the total kilometers related to each bus lines times

102 Urban Transport and the Environment for the 21st Century

the number of buses running on the same line in a working day, thus obtaining the following value for 4 and 3 depots, respectively:

Depots C SP M A

Km 781 190 959 177 1014 959 177

CA CO SP TG A-CS 2388,54 2319,27 2257,78 2611,58 A-M 2361,38 2292,35 2232,10 2584,42 A-TE 2426,58 2355,30 2283,61 2649,62 A-B 2711,86 2609,51 2479,86 2963,86

A-Q 3005,90 2908,99 2775,02 3267,75 CS-M 2430,52 2361,49 2301,24 2653,56 CS-TE 2467,81 2398,54 2337,05 2690,85 CS-B 2239,63 2170,36 2108,87 2462,67 CS-Q 2459,95 2390,68 2329,19 2682,99 M-T 2419,11 2350,08 2289,83 2642,15 M-B 2235,60 2166,57 2106,32 2458,64 M-Q 2442,18 2373,15 2312,90 2665,22 TE-B 2355,18 2283,90 2212,21 2578,22 TE-Q 2536,90 2465,62 2393,93 2759,94

B-Q 3007,37 2905,02 2775,37 3259,37 Table 2. kilometers travelled in empty paths in the case of 3 depots

With these final data it has been possible to compute in both 3 and 4 depots cases the overall handling costs as the sum of the costs associated with each depot, by considering the function cost (1). The yearly costs have been computed starting from the daily costs throughout a suitable multiplicative factor (about 78% of 365), taking into a proper account the effective running period of each bus in the different seasons of the year. Note that by solving the boolean linear programming model (3)-(8) and computing the total cost with respect to (1) we are able to include in the handling costs also the operational and sizing costs. A work is in progress for developing an optimization model which considers these last costs directly as coefficients of the objective function.

6 Some conclusions

The proposed model makes it clear that decisions concerning number, location and size of bus depot are strictly connected. Given the number of buses, few big depots allow greater economies of scale in operational costs, but imply higher costs for journey to/from depots. And vice versa.

Urban Transport and the Environment for the 21st Century 103

It must be said that in an urban area the number of sites potentially suitable for a bus depot are usually not so many: so it is probably better starting from a selection of the suitable sites, and then try to minimize the objective function for any possible combination of them. The steps are:

1. the supply model (network and sites); 2. shortest paths between lines and sites; 3. choice of optimal sites for any possible number of sites (minimization of

total empty journeys for any number of depots); 4. operational costs related to depot sizes; 5. possible benefits/costs of relocation (potential value of the present site minus potential value of new site minus costs of relocation);

6. comparison among best solution for any number of depots and choice of optimal solution. In the case study a 4-depots solution ranks first. Three depots are kept in the present locations, while four present depots are closed and a new depot is established. It should also be stressed that phase n.5, which is obviously not included in the model, should not be underestimated. Economic, social and political costs of relocation can heavily affect the final decision, and a non-optimal solution might be preferred, for example, if the total number of relocations is lower.

Acknowledgement

This work has been partially supported by the Italian National Research Council (CNR) project on Transportation PFT2, contract n.96.00112.PF74.

Special thanks to dr. Marco Benacchio for his kind co-operation.

References

[1] Cascetta E , Metodi quantitativi per la pianificazione dei sistemi di trasporto (in Italian), CEDAM, Padova, 1990. [2] Florian M, Recent advances in methods and models in urban

transportation planning, Proceedings TRISTAN 2 Conference, pp. 7-9, 1994. [3] Gallo G, Pallottino S., Shortest paths algorithms, Annals of Operations ^carc/?, N. 13, pp. 3-79, 1988.

[4] Maze T.H. et al., Application of a bus garage location and sizing optimization, Transportation Research 17A, pp.65-72, 1982. [5] Schrage L Lindo, An optimization modeling system. The Scientific Press, San Francisco, 1992.