A single transferable reservation model for public transportation

Transcription

1 A single transferable reservation model for public transportation Claude Tardif Royal Military College of Canada PO Box Stn Forces, Kingston, ON Canada, K7K 7B4 June 21, 2014 Keywords: Public transportation, ranked ballots, interval graphs AMS 2010 Subject Classification: 05C90 Abstract We present a model for selecting public transportation routes using interval graphs and ranked ballots. 1 Introduction In this paper we propose a model of public transportation with many virtual routes rather than a smaller number of fixed routes. The users rank some virtual routes in order of preference, and the routes that actually run are decided on the basis of the preferences expressed. Public transportation websites often already have an interface where the user can input origin, destination and preferred time of arrival or departure, and be shown a list of suitable bus routes. These are fixed routes that will run independently of demand. It would be possible to modify such websites so that the output is a list of virtual routes that may run, depending on demand. The model is inspired by the problem of public transportation in lowdensity areas. Fixed bus routes need wide zig-zags to cover the area, which makes them slow and discourages their use. Smaller vehicles could instead be used to move commuters between their homes and a stop for an express bus. These vehicles would be between the taxi and the bus in terms of size, but also in terms of flexibility. Indeed the routes would not be fixed like those 1

2 of buses, but at the same time a single customer could not decide the route if the vehicle were to carry many occupants at the same time. Instead, the commuters would vote by ranking (implicitly) a set of virtual bus routes. The routes would be decided on the basis of that information. Assuming the users have portable phone or internet device, it would be possible to do this shortly before departure time, and have the relevant information communicated to the users. In this paper we will be concerned with the problem of how to aggregate user preferences of virtual routes into a selection of actual bus routes. Our approach is inspired by the single transferable vote. We assume that the virtual routes all depart from and return to a central terminal. Their schedule vary, sometimes overlapping and sometimes not. Therefore, even if the service provider has a limited fleet of N vehicules, many routes may be selected as long as no more than N of them overlap at a single time. Thus in mathematical terms our problem amounts to selecting a best possible N- colourable subgraph of a given interval graph. We will present parameters and problems that seem to be relevant to the selection procedure. The paper is structured as follows. In the next section we present the ranked ballot and give simple examples of its use in the selection of bus routes. In Section 3 we introduce the single transferable vote for buses with limited capacity. In Section 4 we introduce interval graphs and their relevance to the model. We give an example in Section 5, and conclude with open problems in Section 6 2 Ranked ballots Consider a set of commuters who wish to commute from a central origin T to peripheral destinations A through F as illustrated in Figure 1. There are six bus routes available: route r X, X {A,...,F } goes from T to X and then circles through the other destinations clockwise. Only one bus will run, and the commuters decide on its route selected by listing the choices in order of their preference, as indicated in Table 1. Note that the commuters generally rank the routes according to how fast they will get them to their destinations, with some exceptions: A few commuters going E are willing to get off at the next stop clockwise, and then presumably walk to E. Also, a few commuters from F will not take the bus at all if it means too long a commute time. In general it is not necessary to have the commuters go through the process of ranking many bus routes. Most of their preferences are implicit just from inputting origin 2

3 Figure 1: Transportation network Destination A B C D E F route preferences 6 : r A,r F,r E,r D,r C,r B 3 : r B,r A,r F,r E,r D,r C 8 : r C,r B,r A,r F,r E,r D 7 : r D,r C,r B,r A,r F,r E 6 : r E,r D,r C,r B,r A,r F 2 : r E,r D,r C,r B,r F,r A 8 : r F,r E,r D,r C,r B,r A 2 : r F,r E,r D,r C,r B Table 1: Commuter preferences and destination (along with prefered arrival or departure time if many buses run in a day). Some commuters (like two going to F) might also want to input a maximum tolerable travel time, and some commuters (like two going to E) might want to input a secondary destination, and criteria for switching to it. We next show how two electoral procedures, the instant-runoff procedure and a variation of the Borda count, aggregate these preferences into a selection of a bus route. The instant-runoff procedure proceeds in several elimination rounds as follows. At each round, the score of a route r X is the number s(r X ) of ballots on which it the prefered route amongst those that are not eliminated. The route with the lowest score is eliminated. The scores are recalculated at the next round. Thus one route eventually gets at least half the votes, and is elected. In our example, the process leads to the 3

4 selection of r C, as indicated in Table 2. round scores s(r A ),s(r B ),...,s(r F ) result 1 6, 3, 8, 7, 8, 10 r B is eliminated 2 9, 0, 8, 7, 8, 10 r D is eliminated 3 9, 0, 15, 0, 8, 10 r E is eliminated 4 9, 0, 23, 0, 0, 10 r C is elected Table 2: Instant-runoff procedure rounds With the Borda count, each ballot contributes weights to the score of each route, according to its rank in the order of preferences. For instance, suppose that having route r X ranked k-th preference of a commuter contributes 1/2 k 1 to the weight of r X. The score of a route is then its total weight. In our example the scores are given in Table 3. The route selected is the one with the highest score, namely r E. Route r A r B r C r D r E r F Score Table 3: Borda scores Thus the route selected depends on the method used. If we know the time required to traverse each edge, we can also compute the total travel time for each route, and choose the route that minimises it. For instance if each edge is traversed in one unit of time, r C minimises the total travel time. In this case it happens to be the route selected with the instant-runoff procedure, but this is a coincidence. Nonetheless, for both procedures it would be worthwhile to refine the definition of the score function by taking travel time into consideration, rather than rely only on an ordinal ranking. In particular, it seems tempting to use a variant of the Borda count where the weight given to a preference by a voter is the negative of the travel time using this route. The route selected is then r C, which minimises the travel time. However, this score function is ill defined: The two voters from E which rank r F above r A give r F an indeterminate score, and the two voters from F which do not rank r A contribute 0 or to the weight of r A (depending on the model). Thus it is probably best to stick with nonnegative weights, and seek a score function that gives the best results. Now, if two buses are to run instead of just one, the instant-runoff procedure can be adapted by selecting the last two survivors rather than just the last one. In our example, r C and r F would be selected in this way. This is 4

5 much better than selecting instead the two routes with highest Borda scores, namely r E and r F. Indeed, much of the score of r E comes from commuters who give r F an even higher score, and end up not using r E. A better way to use Borda scores is in a multi-round process, in which the weight of individual voters is modified after each selection. In our example we can use the following procedure: After r E is selected in the first round, the weight of a voter who had r E as n-th preference is reduced to 1 1 n, and the voter stops contributing to the score of routes which he ranks below r E. The new scores of the routes are as given in Table 4; route r c is therefore selected in the second round. Route r A r B r C r D r E r F Score Table 4: Second round Borda scores Perhaps the satisfactory selection procedures will combine useful features of both the instant-runoff procedure and the Borda method. The Borda method allows a commuter to spread its weight among many desirable alternatives, which is sensible in the early stages of the selection procedure. However the instant-runoff procedure links each commuter to a specific route, which is sensible at the end of the selection procedure. This is particularily the case when we take into account the limited capacity of the vehicules, as we will see in the next section. 3 The single transferable vote Buses have a maximum capacity, and it is necessary to adapt the selection procedure to reflect this. The single transferable vote is an adaptation of the instant-runoff procedure used when each candidate elected can keep only a maximum number of votes, the other votes being transfered to the next preference expressed on the ballot. In political elections with the single transferable vote, a threshold of votes a candidate can keep is set, usually at v c, where v is the number of voters, and c is the number of candidates. For our purposes, the threshold we will use is the maximum capacity of the buses. The rules for transfering votes are as follows: If one route has too many votes, then the excess votes are transfered to the next (noneliminated) preference on the ballot. Otherwise, the route with the lowest score is eliminated. It is difficult to adapt the multi-round Borda count to this situation: v+1 or c+1 5

6 In successive rounds, the weight of a voter should be decreased only if the voter ends up using one of the earlier selected routes, but this is difficult to determine before the end of the process. Therefore we will only consider the single transferable vote. In the example of the previous section, suppose that two buses with a maximum capacity of 21 passengers are available. The rule for transfering excess votes only comes to play after round 4 of Table 2, when two excess votes need to be transfered from route r C. (We will ignore the fact that the next preference of voters transfered from r C would be a clone of r C.) The other routes still running are r A and r F, with 9 and 10 votes respectively. Among the 23 votes currently with r C, most rank r A above r F ; only two belong to commuters going to E who rank r F above r A. If at least one of the latter two are transfered, then r F will keep its advantage over r A, so that r A will next be eliminated. In all other cases, r A will end up with more votes than r F which will be eliminated. The votes of r F will all be transfered to r C. Ten more votes will then be transfered from r C to r A. Thus the final selection, r C and r A or r C and r F, depends on which excess votes are transfered from r C. In political elections with the single transferable vote, the traditional way to select the excess votes to be transfered is to pick them randomly. Modern methods transfer a fraction of every vote instead. The latter method is clearly inadequate for our purposes, since a vote corresponds to a commuter s reservation in a bus. The following example shows that random selection is also inadequate. Figure 2: Random transfers Example 1 Consider six commuters selecting three taxi routes r 1,r 2,r 3, each with a maximum capacity of two passengers. They first go to their first preference as illustrated in Figure 2, commuters a,b going to r 1, c,d,e going to r 2 and f to r 3. But r 2 exceeds capacity, so c is asked to transfer to his next preference which is r 1. But then, r 1 exceeds capacity, so a is asked to transfer to her next preference which is r 2. Again, r 2 exceeds capacity, so e is asked to transfer to his next preference which is r 3. The taxis can now run, except that a and c would prefer to switch places. 6

7 This is an instance of Pareto inefficiency in the affectation of passengers to vehicules, in the sense that it is possible to improve the solution for some passengers without making it any worse for any passenger. This situation can be prevented by giving the commuters an order of priority, perhaps by giving higher priority to those who reserve earlier. The commuters with lower priority are then the first ones to go when votes are transfered from a bus route exceeding capacity. This indeeds prevents Pareto inefficient solutions like that of the previous example. For instance, c being asked first to transfer from r 2 means that c has a lower priority than e, and a being asked first to transfer from r 1 means that a has a lower priority than c. Thus the priority of a is lower than that of e, so e cannot then be asked to transfer from r 2. However, even with an order of priority amongst commuters, we can still get solutions that are Pareto inefficient when compared with solutions using virtual bus routes that have been eliminated earlier in the selection process. The following example illustrates such an instance. Example 2 Suppose that in the network of Figure 1, there are 1, 2, 3, 4, 4, 4 commuters going to A, B,..., F respectively. There are three buses with a maximum capacity of six passengers each. The routes again run clockwise, and the passengers rank them according to how fast they get them to their destination: Passengers Route preferences a 1 b 1,b 2 c 1,c 2,c 3 d 1,d 2,d 3,d 4 e 1,e 2,e 3,e 4 f 1,f 2,f 3,f 4 r A,r F,r E,r D,r C,r B r B,r A,r F,r E,r D,r C r C,r B,r A,r F,r E,r D r D,r C,r B,r A,r F,r E r E,r D,r C,r B,r A,r F r F,r E,r D,r C,r B,r A In the selection process, routes r A, r B, r C are eliminated in this order. If the order of priority of the passengers is lexicographic, a 1 through c 3 gradually bump passengers f 1 through f 4 from r F to r E and r D. The resulting affectation is the following. Route Passengers r F a 1,b 1,b 2,c 1,c 2,c 3 r E e 1,e 2,e 3,e 4,f 1,f 2 r D d 1,d 2,d 3,d 4,f 3,f 4 7

8 The problem with this solution is that all the passengers that end up in r F would prefer to be in r A, which has been eliminated earlier in the selection process. It is tempting to solve this problem by modifying the rule for transfering votes. We can stipulate that votes can never transfer to a bus that is already full and must instead go further down the list of preferences. In Example 2, this would modify the seating arrangement as follows. Route Passengers r F f 1,f 2,f 3,f 4,a 1,b 1 r E e 1,e 2,e 3,e 4,b 2,c 1 r D d 1,d 2,d 3,d 4,c 2,c 3 Clearly, no other solution would be agreed upon by all commuters, since each route selected is the first choice of at least one of its passengers. However this is not an improvement. This second solution as essentially equivalent to the first, since the average travel time is the same. The first solution at least outlines the possibility of an improvement. In voting terms, Example 2 can be seen as an instance of the spoiler effect, where r B and r C spoil the chances of r A. Given the well-known paradox of voting of Arrow [1], it is not surprising that a voting procedure may give unsatisfactory results in some circumstances. But perhaps a better analogy for Example 2 is the Braess paradox [2], where the addition of routes r B and r C among the selection of virtual routes gives worse results. Indeed the whole set of virtual routes is designed by a unique service provider. Perhaps it can be designed as to avoid the situation of Example 2 as much as possible. 4 Non-synchronous bus routes and interval graphs We will now consider the situation where virtual bus routes have varying start time and finish time. We will assume, though, that all routes start and end at the same terminal. Let s(r i ), f(r i ) be respectively the start time and finish time of virtual route r i. Consider the graph G whose vertices are the virtual bus routes and whose edges join pairs r i,r j of routes such that [s(r i ),f(r i )] [s(r j ),f(r j )]. Two routes r i,r j can be serviced by the same vehicle if they are not joined by an edge of G. Therefore, if there are N vehicles available, a set of routes that can be serviced corresponds to a N-colourable subgraph of G. 8

9 The graph G and all its subgraphs are interval graphs. Interval graphs are well understood (see for instance [4]). It is well known that the chromatic number of an interval graph is equal to its clique number. A clique in an interval graph is a set of intersecting intervals. Thus a necessary condition for a set of routes to be serviceable by N vehicles is that no more than N routes overlap at any given time. This leads to an efficient algorithm to assign routes to vehicles, that can be performed online with no backtracking: A route starting at time t can be assigned at time t to any vehicle that is at the terminal at time t. However our main problem is to select a N-colourable subgraph from a much larger interval graph. We first note that selection criterion in the single transferable vote needs to be modified: A route can be selected as soon as it does not create any (N + 1)-clique with any of the routes still running; it does not need to a number of votes reaching a certain quota. This leads to the complications illustrated in the following example. Figure 3: Virtual routes schedules Example 3 Consider the virtual routes represented in Figure 3, to be serviced by two vehicules with a capacity of six passengers each. The commuter s preferences are as follows. 9

10 Passengers Route preferences a i,i = 1,...,8 r 1 b 1 r 2,r 4 b 2 r 2,r 7 c i,i = 1,...,8 r 3 d i,i = 1,...,7 r 4,r 2,r 7 e i,i = 1,...,4 r 5,r 6 f i,i = 1,...,4 r 6,r 5 g i,i = 1,...,7 r 7,r 2,r 4 Suppose we first declare r 1,r 3 selected, since they each have eight votes. The next step would seem to be to eliminate r 2 which has the smallest number of votes. However on closer inspection, we see that r 4 and r 7 cannot both be selected. Indeed, if r 4 is selected, then r 6 must be eliminated since it creates a 3-clique with r 1 and r 4. But then its votes are transfered to r 5 which is elected with eight votes, so that r 7 is then eliminated. Similarly, if r 7 is selected, then r 4 must be eliminated. Now if at most one of r 4 and r 7 is selected, there is no reason to eliminate r 2, which would not create 3-cliques with any other routes. Perhaps graph-theoretic features like the vertex degree should be incorporated in the score function of the single-transferable vote to reflect the situation. In Example 3 above, route r 2 has only two passengers, but its vertex degree of 2 means that it interferes with only two routes. The routes r 4,r 5,r 6,r 7 have more passengers, but also higher vertex degrees. Also note that Example 3 depicts the final stages of the selection process. We may suppose that much more virtual routes will be present initially, with each route having many variants and departure times every five minutes. It is likely that there will be more virtual routes than passengers. Hence it is sensible to preselect a set of virtual routes using variants of Borda scores. For this purpose, the following result may be useful. Theorem 1 (Carlisle, Lloyd [3]) There exists a polynomial algorithm that inputs an interval graph G with a weight function on its vertices and an integer M and outputs a maximum weight M-colourable subgraph of G. Thus for any M N it is possible to shortlist a M-colourable set of virtual routes. There are many ways in which this could be done. For one thing, different weight functions could be used for different parts of the short list. Also, selecting a maximum weight M-colourable subgraph all at once is different from selecting a maximum weight M -colourable subgraph (M < M) 10

11 and then selecting a maximum weight (M M )-colourable subgraph of what remains. For instance in Figure 3 we can define the weight of a vertex to be the number of passengers that have it as first preference. The maximum weight independent set is I 1 = {r 1,r 2,r 3 }, and I 2 = {r 4,r 5 } is a maximum weight independent set in what remains. I 1 I 2 is 2-colourable, but it is different from the maximum weight 2-colourable subgraph J = {r 1,r 3,r 4,r 7 }. In a multi-round process using Borda weights, the weights are changed after a selection, thus there might be even more discrepancy between the two approaches. For that matter, a greedy selection of maximum-weight vertices might sometimes be even better than selecting maximum-weight independent sets. Thus there are many approaches to preselecting a good set of virtual routes. Perhaps a combination of many approaches would give the best results. 5 An example We will examine an example using the network of Figure 1, where two passengers arrive at the terminal T at each of the times t i,i = 0,...,8 intending to go on to one of the periferal destination A,...,F. The travel time between vertices is as indicated in Figure 4. The commuter arrivals at T are as indicated in Table 5. Figure 4: Transportation network with travel times There are two vehicules with a capacity of three passengers each available to transport the commuters from T to their destination. Note that if these were used for individual travel, they would be insufficient to meet 11

12 Time t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 Destinations C, F A, B C, D D, F A, A B, F D, E D, B A, E Table 5: Passenger arrivals at T the demand. We can then suppose that they service fixed routes r 0,AB, r 1,CD, r 2,EF,..., r 9,AB where r i,xy is a route leaving T at time t i and going through the adjacent destinations X and Y in that order. It can be checked that in this case the average wait and travel time of the commuters is 2.26 units of time. More generally, if two passengers arrive at T at each t i, bound for a random destination, then with this fixed schedule the average waiting and travel time is 2 units of time per passenger, plus the delays caused when more than 3 commuters are in the waiting line for a route. A fixed schedule of three-stop routes r i,xy Z departing every three time intervals gives worse results: Six new passengers arrive at T every three time intervals, and at most six leave. So the queue of delayed passengers can only increase over time. We will now consider a schedule with all virtual routes r i,xy, where X and Y are consecutive (clockwise or counterclockwise) destinations, along with all virtual routes r i,xy Z, where X,Y,Z are consecutive (clockwise or counterclockwise) destinations. Thus there are 24 virtual routes departing at every t i. We will use a weight function to preselect a restricted number of routes. Tentatively, the weight we will assign to a route r will be the sum of the three largest terms of the form w P t min,p /t r,p, where w P is the weight of passenger P (which is originally set at 1), t min,p is the minimum possible travel time of passenger P (which is 0.7 for all passengers in our example) and t r,p is the total wait and travel time of passenger P by using route r. The maximum weights of two-stop and three-stop routes starting at each time interval is given in Table 5. Note that at any time interval, the maximum weight of a route is achieved by a three-stop route. However the three-stop routes overlap more routes, so they are at a disadvantage in the selection of a maximum weight independent set (except at t 7,t 8 ). The maximum weight of an independent set is 8.46, achieved by the set S 0 = {r 0,FA,r 2,CD,r 4,AF,r 6,DE,r 8,EFA }. Note that r 0,FA is chosen over r 0,CD listed in Table 5. Actually r 0,FA and r 0,CD both have a weight of 1.00, but the tie is broken sensibly: When r 2,CD is selected, r 0,CD becomes less essential to C 0 than r 0,FA is essential to F 0 (where X i denotes a commuter arriving at T at time t i bound for destination 12

13 Time Two-stop route Weight Three-stop route Weight t 0 r 0,CD 1.00 r 0,CDE 1.00 t 1 r 1,BA 1.54 r 1,BAF 1.78 t 2 r 2,CD 1.80 r 2,CDE 1.80 t 3 r 3,DC 1.72 r 3,DEF 1.78 t 4 r 4,AF 2.30 r 4,AFE 2.30 t 5 r 5,BA 1.61 r 5,BAF 1.67 t 6 r 6,DE 1.73 r 6,DEF 1.78 t 7 r 7,DE 1.72 r 7,DCB 1.78 t 8 r 8,ED 1.55 r 8,EFA 1.63 Table 6: Route weights X). This is akin to decreasing the weight of C 0 in comparison to that of F 0. A systematic reweighing of passengers is indeed done before a second independent set is chosen. Every time a passenger P contributes to the weight of a route r selected in S 1, his original weight of w P = 1 is multiplied by 1 t min /(t min + t r,p ) (with t min and t r,p defined as above). The weight of the routes not in S 1 are recalculated using these weights. In general the weight of routes with an even-indexed start time decreases more than that of the routes with an odd-indexed start time. The maximum weight independent set in what remains turns out to be S 1 = {r 1,BA,r 3,DC,r 5,FA,r 7,BCD } (where the tie between r 1,BA and r 1,AB is broken randomly). It can be checked that the set of routes S 0 S 1 allows all commuters to reach their destination in an average time of 1.24 units of time, which is much better than the result obtained using fixed routes. It is possible that such a selection process of maximum weights independent sets would give satisfactory results most of the time. However, since the process does not link a commuter to a specific route, it is possible that some commuters could end up with more than one suitable choice while others get only bad choices or no choice at all. Therefore the final affectation of commuters to route will be done using the single transferable vote. First we need to enlarge the short list of vitual routes by selecting a third independent set S 2. We will change the selection process, to account for the possibility that the process used to select S 0 and S 1 discriminates against potentially good candidates. Thus we will select an independent set greedily, using the original weights. Discarting the routes 13

14 already in S 0 and S 1, we get S 2 = {r 1,ABC,r 4,AB,r 6,DEF } (where the tie between r 1,ABC and r 1,BAF is broken randomly). The single transferable vote will then be used to select a relatively large subset of S 0 S 1 S 2, compared to the number of voters. We will assign weight to voters as follows: If t 1 is the time taken by a passenger P with her prefered route and t 2 the time taken with her second choice, then P s weight will be t 2 /t 1. This weighing scheme makes ties less likely, but it is also chosen to alleviate the disadvantage of having a voter s weight concentrated on her first preference: if the first preference is eliminated while the second preference is still running, then there were no ill effects in having the weight concentated on the first preference. But if the second preference is eliminated, then the voter s weight and the likelihood of having her first preference selected will increase accordingly. Note that a voter s weight is at least 1, with equality only if the voter is indifferent between two or more routes as first preference. In the latter case, her weight of 1 is split equally among her first preferences. A voter s weight is infinite if there is only one route available for her. The score of a route r is the sum of the contributions of the voters that have it as first preference, plus two values derived from graph-theoretic considerations: Let d r and k r denote respectively the degree of r and the size of the largest clique containing r in the interval graph G induced by the non-eliminated routes. Then the values 1/(d r 1) and 1/(w r 2) are added to the score of r. This gives an advantage to the routes that have smaller degree and are in smaller cliques, that is, the routes that interfere less with other routes running in the election. In particular 1/(w r 2) becomes infinite when r is not in a 3-clique, hence there are no inconvenients in selecting it. The passenger weights and the scores of the routes in S 0 S 1 S 2 is given in Table 5. Passengers D 6 and E 6 split their weight equally between r 6,DE and r 6,DEF. But since r 6,DEF has higher degree and is in a larger clique, it ends up with the overall lowest score, with r 6,DE tied with r 4,AB for second lowest. Eliminating r 6,DEF will increase the weight of D 6 and E 6 and transfer these weights solely to r 6,DE. More significantly, r 6,DE will not be in any 3-clique anymore, so that its score will be infinite. Thus r 4,AB will next be eliminated, followed by r 1,ABC. Thus the single transferable vote confirms the selection of S 0 S 1 as the set of routes that will run, while assigning passengers to routes. Note that in this small example we did not need to worry about transfering excess passengers from vehicles that exceed capacity. Things might be more complex with larger examples. 14

15 Time t 0 t 1 t 2 t 3 t 4 Passengers C 0,F 0 A 1,B 1 C 2,D 2 D 3,F 3 A 4,A 4 Weights 1.07, , , , , 1.00 S 0 routes r 0,FA r 2,CD r 4,AF Voters F 0 C 0,C 2,D 2 F 3,A 4,A 4 Score S 1 routes r 1,BA r 3,DC Voters B 1 D 3 Score S 2 routes r 1,ABC r 4,AB Voters A 1 A 4,A 4 Score Time t 5 t 6 t 7 t 8 Passengers B,F D,E D,B A,E Weights, , 1.00,, S 0 routes r 6,DE r 8,EFA Voters D 6,E 6 A 8,E 8 Score 2.50 S 1 routes r 5,FA r 7,BCD Voters F 5 B 5,B 7,D 7 Score 4.62 S 2 routes r 6,DEF Voters D 6,E 6 Score 1.83 Table 7: Weights of passengers and scores of routes 6 Concluding comments We have presented a single-transferable reservation model for public transportation, in which user input influences the selection of the routes that run. We highlighted problems that might arise in the selection of routes. However, there seems to be much leeway in the design of a selection process, so perhaps a satisfactory process can be found. It is likely that real life data would be more effective than theoretical examples for this purpose. There are a few additionnal complexities of the model that could be considered: Perhaps the selection process should work on line rather than all at 15

16 once, with earlier routes selected while commuters are still making reservations for later routes. Also, since commuters mostly go from their homes to the terminal in the morning, and from the terminal to their homes in the evening, their order of preferences are usually from later departures to earlier departures in the morning and from earlier arrivals to later arrivals in the evening. It is possible that the selection process that is best suited for the morning would not be the same as the selection process for the evening. At times when the system is used both for users going from the terminal to the home and from the home to the terminal, the occupancy of a vehicle is not a single parameter, but rather the chromatic number of an interval graph with all (or most) of its vertices either in the initial clique or in the final clique. Since the chromatic number of an interval graph is easy to determine, it would be possible to adapt the model to reflect this situation. Vehicules could have different sizes, possibly with mostly mid-sized vehicules, along with a few smaller vehicules to pick up users that fall through the cracks. Preferably the selection process could account for that possibility. Perhaps it is possible to consider models with more than one terminal as well: With two terminals T 1 and T 2, we can consider routes starting and ending at the same terminal, but also routes with different start and end points. The interference graph of the virtual routes is not an interval graph anymore. If we allow a fixed time for empty vehicules to transfer between terminals, each virtual route r is modeled by a pair I 1,r,I 2,r of intersecting intervals. Two routes r,r interfere with each other if and only if I 1,r I 1,r and I 2,r I 2,r are both nonempty. It is not known whether this variation of interval graphs admits efficient algorithms for the relevant graph-theoretic parameters. References [1] K. J. Arrow, Social Choice and Individual Values, 1951, 2nd ed., Yale University Press. [2] D. Braess, Über ein Paradoxon aus der Verkehrsplanung, Unternehmensforschung 12 (1968),

17 English translation in [D. Braess, A. Nagurney, T, Wakolbinger, On a paradox of traffic planning, Transportation Science 39 (2005), ]. [3] M. Carlisle, E. Lloyd, On the k-coloring of intervals, Discrete Appl. Math. 59 (1995), [4] D. B. West, Introduction to graph theory, Prentice Hall, Inc., Upper Saddle River, NJ,