has possibly become too central, and the area has now a ... · depot costs related to the 848 Genoa...

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 amethodology 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 (causingpossible inefficiencies in terms of land use);- the growing demand for public transport over time (causing ineficiencies in exploitingeconomies 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 bydetermining the optimal combination of number, size and location of depots, given the urbantransportation network and the overall number of buses.

A case study is then developed for the city of Genoa. The present scenario is analyzedwith respect to several decision constraints, (geography, feasible depot sizes, resourceallocation, bus types, time of investment and budget restrictions). Then, the shortest pathbetween any pair of possible bus stops is computed together with the average number oftrips 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 basisof the average daily traffic data in the city of Genoa. The model is solved using anoptimization software library on a personal computer.

1 Purpose of the paper

This papers aims to point out a methodology for optimizing size and locationof bus depots in an urban transit company.

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

Transactions on the Built Environment vol 30, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509

94 Urban Transport and the Environment for the 21st Century

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

Besides, the potential opposition of resident population makes it politicallydifficult to move a depot toward any new site. So that, as a result, the previouslocation is more acceptable and eventually very stable, despite of its possibleinefficiency.

It should also be said that land values of nearby areas are negativelyaffected by the new depot, while land values of previous neighbourhood willbe positively affected. Both changes will be included only in future prices, andresult in unfair damages for present landowners. Thus, the overall number ofrelocation should be reduced to the minimum.

The lack of satisfactory methods to find out the optimal solution in terms ofnumber/size/location of depots often causes urban planning to be almostcompletely 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 ofbuses, the busline network, and the frequency on each line, are given (sincedepot supply must be optimized with respect to service supply, and not theopposite). It could be objected that location and size of depots are tipical longterm choices. But on the other side, since the transit network depends ondemand, no reliable forecasts can be developed on future demand over thewhole life of the depot (at least 50 years). Obvoiusly, some overcapacity ofdepots with respect to present number of bus should be considered, but thisjust means a different overall capacity, and does not implies adjustments inmethodology.

We have assumed that costs to be minimized are the costs of the transitcompany. No external costs (like pollution of empty-journeys from/to thedepot) have been considered, since:- transit companies deficit are presently so relevant that this should beconsidered a priority goal;- given the network and the frequency (which means, given the level ofefficacy), any further gaps between social and company costs are basiclyconnected to congestion and pollution, which are implicitly minimized sincethe proposed method sets up a trade off between minimum empty journeys andeconomies 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 isexpected to be located);- the costs of depot (the cost of depot management and of all functions andactivities taking place inside the depot);- the costs of empty journeys (from/to the depot: mainly staff costs and fuelcosts);without reducing the performance of the network (i.e.: same network, samenumber 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 thedemand for alternative use of the space, and is influenced by features such ascentral location, accessibility, environmental quality, etc.;- an "actual" value criterium, where land value is strongly affected by urbanplanning regulations concerning the area.

Quite often, planners follow the latter, and just confirm the present locationof depots, thus keeping low the value of those areas. Clearly, decisionsconcerning location of depots, instead of being influenced by present location,should come first, and should only be influenced by the potential value of theareas, determined by alternative land uses.

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

So, the objective function to minimize is a cost function including the costsof depots and the costs of journey to/from depots. A trade off is clearly to bestated between possible economies of scale in depots management and costs ofjourney 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.



The analysis of depot costs for the transit company of the city of Genoashows the existence of economies of scale up to a size of 180-200 vehicles perdepot (the total number of vehicles is 848). For bigger depots, average costsappear more or less constant, without relevant diseconomies of scale. Theseresults are confirmed by a productivity index (namely: number of vehicles perstaff unit), which increases until the depot sizes reach the above mentionednumber of buses.

Obviously, total depot costs can be minimized only if the number of depotsis minor or equal to the ratio between the total number of buses and theminimum optimal size, and if no depot is below that size. For example, thedepot costs related to the 848 Genoa buses are properly minimized with 1,2,3or 4 depots (provided that none of them is smaller than 180-200 units), whilecosts 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 toreach the terminus of the correspondind bus line from the depots, andviceversa. Note that during this journey buses cannot offer any transportationto the users, so that only pure costs can be considered in this case to be addedto 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 betweendepots and bus termina, that is denoted as "empty path". The "empty path" isusually as shorter as greater is the number of depots (and then as smaller istheir 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 iscloser to the depot.

Moreover, the journey from/to depots cost are also a function of thenumber, say F, of input/out daily entries of each bus. It can be observed thatusually buses with low frequency are in service the whole day, such that theytravel 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/todepots about two/three times a day In our case study we consider that about50% of buses moves only once from their depots within 24 hours.

The overall daily journey from/to depot cost for each depot, denoted byCA/, 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 onan empty path by any of the N, buses running on the y'-th bus line, consideringN be the total number of buses running daily.

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



An empiric investigation performed during the data collection phase relatedto our case study has allowed us to evaluate the gasoline cost Q: to be about8647 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 travellingspeed Vm on the empty path to be constant within the day and independent onthe 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 pathsplay a crucial role in the determination of the best depot location. For thisreason, a relevant part of our work has been devoted to the selection of theoptimal area for the bus depots for which the distance between bus termina anddepots is the minimum one.

The first step in looking for optimal terminus-depot paths in an urbantransportation network is to derive the corresponding mathematical model,reflecting all the possible features of the selected geographical area withrespect to the problem under consideration.

In order to derive the model related to the public tranportation network inthe city of Genoa we performed the following four steps, that are usuallyperformed in any transportation planning project (Note: for recent advances inmethods and models in urban transportation planning interested readers arereferred to [2]. See also [1] for basic definition and notation for transportationmodels 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 busdepots. In our case, we have considered the only 10 areas within the urbanterritory which could allow in a very next future the location of bus depotsalso satisfying sizing and environmental constraints.

2. Then, the selected area is split into many zones; each of them, characterizedby a relevant location, is identified by the so-called centroid, in such a wayonly 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 ofGenova, that is East side, West side and Center, that are represented in totalby 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, andA is the set of arcs (/,/) connecting node / to node/ V /,/ e N', A hence givesthe possible road connections between the selected centroids. Starting fromthe 146 nodes belonging to TV that have been identified at step 2, we haveidentified the set C ci TV consisting of 82 nodes representing the termina ofthe bus lines that have been taken into account in this study, the set R a Nconsisting of the 10 depots previously identified, and the set T c Nconsisting of 54 nodes, denoted transition nodes, that correspond to



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

4. Each arc (ij) e A has a weight associated with it giving the cost required totravel from / toy e Nin the urban transportation network. According to theproblem under consideration, we assign as arc weight the distance dybetween / and y, thus expressing our cost function as a function of thekilometers travelled in the journeys from/to depots, as it has been explainedin Section 3. Note that the distances asociated with the arcs in G have beencomputed starting from the map reflecting the transportation network usinga 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 busterminus, for each of the 1 0 areas derived in step 1 , we had to compute thedistances to be travelled by buses related to the shortest paths between anynode 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 ina graph; that is, given a digraph G = (N,A), we have to compute, among allthe feasible paths P connecting an origin node, say node O e N, to adestination node, say node D e N, the path with the minimum total cost withrespect to the objective function under consideration, that in our case is thetotal travelling distance, given by the sum of the weights dy associated with thearcs (ij) belonging to the path between the selected origin node e C and thedestination node e R. Our objective function is given in (2), where Xy is aBoolean 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 ingraph G, besides a better representation of the urban area under consideration,allows a better flexibility in the model when considering, for instance, theanalysis 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 theseselected relevant locations (nodes in 7), thus allowing to reduce theinproductive distance travelled by buses without people on board.

In order to compute the required shortest paths a graphic-interactiveimplementation of the Dijkstra algorithm running on a PC has been used (see[3] for further details about the main shortest path algorithm presented in therecent literature).

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



determine the so called shortest path tree, that is a spanning tree on G rootedat node /, in which each branch from the root node / to the leave node y, Vy eR, 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 ofthe m nodes belonging toR(m = 10); such an arc, denoted by &%, V / e C, Vye R, is weighted with the lenght y^ of the shortest path between / andy, that isthe value z of the objective function reported in (2) where nodes / and y areconsidered, respectively, as the origin and destination node of the problem.

By looking at Figure 7, considering all outcoming arcs from each node inC 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 ispossible to locate a bus depot is computed, the next step is to determine theoptimal set of areas that minimize the total costs reated to the journey to/fromdepots. In this preliminary work, our analysis is restricted to consider all thepossible combinations of 4 and 3, respectively, depots.

In the first case we have considered 2 depots located at the east side and theother one located at the west side of the city. In particular, we have considered



all the possible combinations of pairs of areas among all the selected ones(respectively 4 and 6), computing, for each combination, the shortest path andthe corresponding lenght, as it is described in Section 4, and associatingconsequently with each bus terminus the closest depot. Then, once the totalnumber of buses circulating in a given day is known, for each bus line it ispossible to compute the total number of kilometers travelled on empty pathswith respect to all the considered bus lines.

In particular, in order to determine the best combination of 4 bus depotsamong the 10 feasible ones, and then assigning to each bus line thecorresponding depot, a Boolean Linear Programming model has been derived.Another interested approach for determining the optimal location and sizing ofbus depot is given in [4]. In this proposed model, we consider as objectivefunction the minimization of the sum of the distances with respect to allcombinations of areas and depots identified in Section 4, where the costcoefficient is given by the optimal value y,j associated with the shortest pathtree 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 aredirect 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 pathfrom 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 eachnode in C to have one and only one outgoing arc; therefore, such set ofconstraints establishes a link between each bus terminus and the correspondingdepot. Relations (5) and (6) give the upper and lower bounds to the number ofingoing arcs of each depot. In particular, Boolean variables My, V / E R, forceeach 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 amongthe m feasible ones (in this case k = 4).Finally, (7) and (8) are the definition of the variables of our problem that canhave only a {0,1} value.



For solving model (3) - (8) we use the software environment LINDO [5],thus deriving the total kilometers travelled in the empty paths given by thevalue of (3) with respect to all the possible combinations of areas located at thewest and east side of Genoa.

In Table 1 we report the total kilometers travelled in the empty paths for allthe combinations of 4 depots among the 10 considered. Note that thecombination of the 4 areas denoted by M-B and SP-TG gives the minimumvalue with respect to model (3) - (8) and then is the optimal combination as faras the journey from/to depot costs is concerned. However, such solutioncannot be considered as the optimal solution of our localization problem sincethe capacity constraint of the depots, for which the optimal size in terms ofoperational cost minimization is in the range of 100-300 buses, is violated.Thus, by taking into account this further constraint and the trade-off betweenoperational costs and journey from/to depot costs we can consider thecombination SP-CA and A-M as the optimal solution of the problem.

A-CSA-MA-TEA-BA-QCS-MCS-TECS-BCS-QM-TM-BM-QTE-BTE-QB-Q

CA-CO2277,422250,432313,452567,662871,712319,572356,693602,412348,832308,162124.652331,232242,052423,772863,17

CA-SP2214,742189.022240.532436.782731,942258,162294,012265,832286,152246,752263,242269,822189.132350,852732,29

CA-TG2319.482243,902310,892655,592963,742313,042398,752170,572390,892301,632118,122324.702239,492421,212951,10

CO-SP2185.042159.322210,832407,082702.242228,462264.312036.132256,452217,052033,542240.122139.432321.152702.59

CO-TG2164.762137,772200.792455.002750.742206.912244.032015.852236.172195.502011.992218.572129.392311.112750,51

SP-TG2141.842116.122167.632363.882659.042185.262221.111992.932213,252173,851990,342196.922096.232277.952659,39

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 haveconsidered 2 depots located at the eastern side of Genoa an the other onelocated at the western side of the city The same procedure as before has beenused, solving model (3)-(8) with k = 3 in (5). The results are reported in Table2.

We can see that the minimum value corresponds to the combination of theareas M-B and SP, but, as in the case of 4 depots, this solution gives rise to anumber of buses in depot B less than the minimum allowed (100), and thus theoptimal combination is given by the 3 areas A-M and SP.

Finally, the total number of kilometers associated with each depot has beendetermined by multiplying the total kilometers related to each bus lines times



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

DepotsKm

C781

SP M190 9591014 959

A177177

A-CSA-MA-TEA-BA-QCS-MCS-TECS-BCS-QM-TM-BM-QTE-BTE-QB-Q

CA2388,542361,382426,582711,863005,902430,522467,812239,632459,952419,112235,602442,182355,182536,903007,37

CO2319,272292,352355,302609,512908,992361,492398,542170,362390,682350,082166,572373,152283,902465,622905,02

SP2257,782232,102283,612479,862775,022301,242337,052108,872329,192289,832106,322312,902212,212393,932775,37

TG2611,582584,422649,622963,863267,752653,562690,852462,672682,992642,152458,642665,222578,222759,943259,37

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 depotscases the overall handling costs as the sum of the costs associated with eachdepot, by considering the function cost (1).

The yearly costs have been computed starting from the daily coststhroughout a suitable multiplicative factor (about 78% of 365), taking into aproper account the effective running period of each bus in the different seasonsof 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 thehandling costs also the operational and sizing costs. A work is in progress fordeveloping an optimization model which considers these last costs directly ascoefficients of the objective function.

6 Some conclusions

The proposed model makes it clear that decisions concerning number, locationand size of bus depot are strictly connected.

Given the number of buses, few big depots allow greater economies of scalein operational costs, but imply higher costs for journey to/from depots. Andvice versa.



It must be said that in an urban area the number of sites potentially suitablefor a bus depot are usually not so many: so it is probably better starting from aselection of the suitable sites, and then try to minimize the objective functionfor 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 isestablished.

It should also be stressed that phase n.5, which is obviously not included inthe model, should not be underestimated. Economic, social and political costsof relocation can heavily affect the final decision, and a non-optimal solutionmight be preferred, for example, if the total number of relocations is lower.

Acknowledgement

This work has been partially supported by the Italian National ResearchCouncil (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 ditrasporto (in Italian), CEDAM, Padova, 1990.

[2] Florian M, Recent advances in methods and models in urbantransportation planning, Proceedings TRISTAN 2 Conference, pp. 7-9,1994.

[3] Gallo G, Pallottino S., Shortest paths algorithms, Annals of Operationscarc/?, N. 13, pp. 3-79, 1988.

[4] Maze T.H. et al., Application of a bus garage location and sizingoptimization, Transportation Research 17A, pp.65-72, 1982.

[5] Schrage L Lindo, An optimization modeling system. The Scientific Press,San Francisco, 1992.


has possibly become too central, and the area has now a ... · depot costs related to the 848 Genoa...

Documents

Transcript of has possibly become too central, and the area has now a ... · depot costs related to the 848 Genoa...