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Transportation Research Procedia 24C (2017) 377–384 www.elsevier.com/locate/procedia

3rd Conference on Sustainable Urban Mobility, 3rd CSUM 2016, 26 – 27 May 2016, Volos, Greece

Online algorithm for dynamic dial a ride problem and its metrics Athanasios Loisa*, Athanasios Ziliaskopoulos a "University

"University of Thessaly, Pedion Areos, Volos 38222, Greece"

Abstract In this paper, an online regret based dial-a-ride (OR-DARP) algorithm is introduced and its performance evaluated on an actual demand responsive transit (DRT) system. The innovative part of the algorithm is the design of the optimization engine. A signal communication scheme between the trip dispatcher and the algorithm is used to improve utilization of the available idle time that can then be devoted to the optimization engine. The basic concept is as follows: a. Every trip request is treated as an emergency request demanding an immediate answer, b. The optimization engine runs continuously, thereby consuming every idle time fragment unless interrupted by a new trip request. The trip data are real, and they are sourced from a DRT system operating at a municipality in northern Greece where a static dial-a-ride algorithm was used as the optimization engine. Given the fact that these trips data provide all trip details plus the show-up time (the most important feature for our study), these data are the ideal basis for an “a posteriori” evaluation of the proposed online approach. Another contribution of this paper is the identification of the critical parameters in the trade-off between benefits gained from continuing to optimize an online system versus the losses of non-served demands. This important issue when applying online algorithms has not been studied extensively in the literature so far (to the best of our knowledge). © 2016 The Authors. Published by Elsevier B.V. © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the organizing committee of the 3rd CSUM 2016. Peer-review under responsibility of the organizing committee of the 3rd CSUM 2016. Keywords: "Online dial-a-ride; regret optimization;cost metrics; profits metrics; performance metrics" * Corresponding author. Tel.: +30-24210-74651; fax: +30-24210-74400. E-mail address: [email protected]

1. Introduction and Literature review In this paper, we introduce an online regret based dial-a-ride algorithm and we study its performance under heavy demand situations. The basic structure of our algorithm is founded on the utilization of two separate sub-algorithms for handling incoming trip requests and continuously improving the solution. The first algorithm mainly examines trip requests myopically (minimal optimization) and makes a fast yes/no decision regarding trip acceptance. The second algorithm is focused on optimization and uses all available idle time between trip requests to improve the solution. In comparison with similar algorithms proposed by other authors, our algorithm benefits from a new concept. The optimization process is not based on rigidly defined constant optimization time windows (e.g. 10 seconds, 1 minute, 5 minutes). It operates continuously to improve the solution, exploiting even small fragments of idle time between incoming trip requests. The main focus of this paper is to identify critical points in the benefits of the optimization procedures under a heavy or cascading load. The scheme was evaluated on an actual demand responsive transit (DRT) system, operating in the former municipality of Philippi in Northern Greece. The computational results indicate that

2352-1465 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the organizing committee of the 3rd CSUM 2016. 10.1016/j.trpro.2017.05.097

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there is a specific point, regarding the number of trip requests, after which it is more profitable to accept more minimally optimized trip requests than to insist on heavy optimization for existing trip requests. Existing studies in the field of online dial-a-ride problems presented are as follows. Psaraftis, H. N. (1980) presented an exact method for the dynamic DARP as an adaptation of the static version for the same problem. Although Psaraftis’ version has been limited to very small problem instances due to the combinatorial nature of the problem, it remains one of the few exact methods that help gain insight into the problem. Madsen et al. (1995) have presented an algorithm for a real-life multi-vehicle dynamic DARP consisting of up to 300 daily requests for the transportation of elderly and handicapped people in Copenhagen. The problem had many constraints such as time windows, multi-dimensional capacity restrictions, customer priorities and a heterogeneous vehicle fleet. When a new request arrives, it is inserted in a current route using an efficient insertion algorithm based on that of Jaw et al(1986). Computational results on real-life instances with up to 300 requests and 24 vehicles have shown that the algorithm was fast enough to be used in a dynamic context and that it is capable of producing good quality solutions. Teodorovic and Radivojevic (2000) developed a fuzzy logic model for the online DARP problem. They combine fuzzy logic reasoning in the insertion procedure to make the decision about which vehicle will accept the new request. The model was tested for a set of 900 trip demands and a fleet of 30 vehicles with seemingly reasonable results. Diana et al. (2006) proposed a probabilistic model that requires only the knowledge of the demand distribution over the service area, and the quality of the service. The quality of service is defined in terms of time windows associated with pickup and delivery nodes. Given a number of n trip requests in a service area, the objective is to estimate the number of vehicles needed to serve these requests. To benchmark the model, they compare it to a simulation approach that requires knowledge of the complete daily schedule. The requests were scheduled using a parallel regret insertion algorithm introduced by Diana et al. (2004). Computational results proved that the probabilistic model produced better results concerning the minimum number of vehicles required to service all trip requests. However, for the largest problem instance, the model gave worse results. Horn (2002) provides a software environment for fleet scheduling and dispatching of demand responsive services. The system can handle in advance as well as immediate requests. New incoming requests are inserted into existing routes according to least cost insertion. A steepest descent improvement phase was running periodically. Also, automated vehicle dispatching procedures, to achieve a good combination of efficient vehicle deployment and customer service, are included. The system was tested in the modeling framework LITRES-2 by Horn (2002b), using a 24-hours real-life data set of taxi operations with 4282 customer requests. Attanasio et al. (2004) presented a modified version of the static tabu search algorithm presented by Cordeau et al. (2003) to handle the online nature of the DARP. The online algorithm can be described as follows: A static solution is constructed on the basis of the requests known at the start of the planning horizon. When a new request arrives, the algorithm performs a feasibility check, i.e., it searches for a feasible solution to include the new service request. If the new request is accepted, the algorithm performs post-optimization, i.e., it tries to improve the current solution. One practical problem with this approach is the difficulty of solving the problem in a shorter time interval than the updating interval. Coslovich et al. (2006) presented an algorithm which follows a two-phase strategy for the insertion of a new request into an existing route. An off-line phase is first used to create a feasible neighborhood of the current route through a 2-opt solution improvement mechanism. An on-line phase is then used to insert the new request with the objective of minimizing user dissatisfaction. Marco Diana (2006) presented a study about the importance of information flow related to temporal attributes for a dynamic DRT system. Authors focused on three characteristics of the information flow. These characteristics were: Percentage of real time requests, interval between call-in and requested pickup time, and the length of computational time. They handled demands in batch mode. Demands were grouped and processed by algorithms in time slots of 5 minutes, 1 minute, 10 seconds. Because of this grouping and batch processing, customers have to be called back to be notified of the acceptance or rejection of their request. The authors tested two different algorithms to draw conclusions about the influence on the solution quality. The first algorithm was the dynamic version of the algorithm proposed by Jaw et al. (1986) and it is based on the best insertion method. The second algorithm was the dynamic version of an older algorithm proposed by the authors, based on the regret method. In their formulation, the quality of the schedule is given by the number of rejected requests and by the value of the objective function z. Their experiments were based on trip requests gathered by the transportation service for elderly and disabled people in Los Angeles County. In order to run specific scenarios, authors made some assumptions concerning: the number of requests known in advance, the expected value of time intervals between call-

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in and requested pickup time, the number of vehicles and the cycle time. By the term 'cycle time', authors define a specific time period where the underlying algorithm processes trips requests without interruption. Based on the presented literature, we can come to the following conclusions:    

Optimization procedures for online problems are based on time horizons. There is no clear view on how we can use the optimization procedure continuously as the algorithm runs. The response time – a critical feature – for the online DARP problems has not been studied sufficiently. No economic study was provided to define proposed system viability.

This paper comprises three sections. In section 2 we present the proposed algorithmic scheme and its description. In Section 3 we present numerical results and analysis. Section 4 contains concluding remarks and future research pointers. 2. Online regret dial-a-ride algorithm (OR-DARP) - basic concept and description Online algorithms by design are used to handle online - relevant input becomes available during operationdemands. The proposed online regret algorithm processes online demands quickly, while continuously optimizing the produced solution. To achieve this goal, we developed two sub-modules combined in the general regret online algorithm. The first module is the online insertion algorithm- a typical myopic insertion algorithm modified to handle online demands- and it is used to give an initial fast response for any incoming trip demand. The insertion algorithm complexity to successfully process a new trip request is O(n2), where n is equal to the number of assigned trip requests. The second module is the regret optimization algorithm modified to handle online demands as well. Our OR-DARP algorithm uses the original regret concept where multiple trip assignment combinations are explored, and a combination that results in higher gains replaces a previously chosen solution (maximum gain principle). Our ORDARP algorithm uses a fast heuristic to produce an initial solution. The regret process is instigated by building a regret 2-d matrix. Each row represents the route generated by the fast heuristic. Each diagonal item in the 2-d matrix represents the cost of the most expensive demand of the route described by the corresponding row. Each element of a column (except the one on the diagonal) represents the cost that will be produced if we insert the aforementioned most expensive demand to the route represented by the corresponding row. If the insertion is not feasible, then we set the insertion cost to an arbitrarily large number. By using this matrix, we define the “profit” we gain if we move demands from one route to another. This can be done by using the following rule: for every column calculate, the differences of diagonal element minus every other element. If at least one difference is positive, select the greatest one. Then move the corresponding demand to the appropriate row (route in our case). The regret optimization sub-algorithm works to optimize current solutions to all served demands and its complexity is O(rmn2 ), where n is equal to the number of assigned trip requests, m is equal to routes number, and r is equal to iterations number till further optimization is not possible. A more descriptive section of the OR-DARP algorithm follows: The algorithm reads an initial input and uses the fast insertion heuristic to produce an initial solution. The dispatcher is continuously monitoring the system and feeds the online algorithm with events (trip requests). If there is no event, then the optimization sub-module performs a one-cycle optimization. If on completion, there are still no new events, optimization continues for another cycle. If the optimization is completed (the regret sub-algorithm produces no more optimization in a full cycle) the system goes to idle state while the only activity that continues is the dispatcher looking for new events. The most important feature of the OR-DARP algorithm is that the algorithm uses every single idle time unit between events in order to optimize the solution. The proposed OR-DARP algorithm satisfies the following key features.  Fast response concerning the acceptance or denial of the incoming trip request. An online algorithm is used mainly for real applications. Prompt response is very critical to such applications. This is the main reason why the algorithm gives an answer as soon as possible. All trip requests are considered as urgent requests in order to be scheduled immediately.  Continuous improvement of the solution quality. It means that idle intervals at any time can be used by the optimization process.

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The following figures present flowcharts of OR-DARP algorithmic scheme and its sub-algorithms as well.

Read Dispatcher

NO Call Regret Optimization

YES New trip request?

Call Insertion

Figure 1 Online Regret algorithmic (OR-DARP) scheme

Sort routes in ascending order according to their current route cost

for current trip request, calculate minimum insertion cost for all routes

Assign trip request to the route with the minimum insertion cost Exit

Figure 2 Fast insertion sub-algorithm

Calculate initial solution cost

For every route, calculate the most expensive trip assignment

1. Use routes most expensive trip assignments to build regret matrix 2. Calculate new solution cost

NO Exit

YES

Initial solution cost = new solution cost

new solution cost < initial solution cost?

Figure 3 One cycle regret optimization sub-algorithm

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3. Algorithm computational study The proposed algorithm and the software dispatcher were implemented in C++ and tested on a Linux machine with a dual core processor. To evaluate its performance, we have designed a sequence of experiments based on real data gathered during a pilot testing in the municipality of Philippi. Each trip request was characterized by various factors such as earlier and late pickup time, origin and destination, maximum travel time, quantity concerning the number of persons, special trip requirements as well as the timestamp. The timestamp is the time that the trip request came to the operator for processing. We employed nine (9) different scenarios, each of different size with regard to the number of requested trip demands (94, 226, 414, 624, 803, 949, 1151, 1370, 1619). Some details about the environment of the experiments are: Maximum fleet size: 50 vehicles, vehicle capacity 12 seats. The maximum time that an online trip request is allowed to wait until processed or rejected, was 60 seconds. After that, the dispatcher, proceeds to the next request. Maximum pickup waiting time was 15 minutes. The maximum trip demand ride time equaled 1.5 times the absolute shortest path time. The charging price per passenger travel unit (Km in our case) was 0.32 currency units (Euro), and for each vehicle the cost per travel unit was 0.32 currency units as well. Work period was one day (86400 seconds). For clarity we define the following as:  OR-DARP-Opt the version of our online OR-DARP algorithm that uses both fast insertion and regret optimization sub-algorithms.  OR-DARP-NoOpt the version of our online OR-DARP algorithm that uses only the fast insertion sub-algorithm. Tables 1 and 2 present the basic behavior of the algorithmic scheme. A series of comparative results examine the performance of this online algorithm in comparison vs. profits. Table 1 presents: Results related to the number of served trip requests vs. the number of requested trips, results concerning the time required for the myopic fast insertion sub-algorithm to handle all trip requests and the time required for the regret optimization procedure. The 2nd column represents the number of requested trip demands. The 3rd column represents the number of served trips that can be handled by OR-DARP-NoOpt. The 4th column represents the number of served trips managed by OR-DARP-Opt. The 5th column represents the percentage difference between requested and served trips by OR-DARP-Opt. The 6th column represents the total time needed by fast insertion module of OR-DARP-Opt to satisfy all incoming trip requests defined in 4th column. The 7th column represents the total time needed by regret optimization sub-algorithm of OR-DARP-Opt to optimize existing solution. The 8th column represents the maximum time required by fast insertion of OR-DARP-Opt to successfully process a trip request for trip requests defined in 4th column. The 9th column represents the maximum time required for a full cycle of regret optimization sub-algorithm of OR-DARP-Opt. Table 1 shows that the OR-DARP-Opt reaches a critical limit after experiment 3 where the number of served trips (4th column) starts to decrease monotonically in comparison with the requested trip requests (2nd column). In the last experiment, we found that the number of serviced trips is 37.5% less than the requested trips. The main reason for this behavior is that during the optimization cycle, some trip requests cannot get an answer within 60 seconds of the time they arrive at the dispatcher. However, OR-DARP-NoOpt (3rd column) can to serve all requested trips. Table 1: OR-DARP algorithm (not optimized, optimized versions) test results for 1-day (86440 Seconds) work period. #

(2)Trip Requests

(3)Assigned Trips by OR-DARPNoOpt

(4)Assigned Trips by OR-DARPOpt

(5)Difference %

1 2 3 4 5 6 7

94 226 415 624 803 949 1151

94 226 415 624 803 949 1151

94 226 414 622 735 727 873

0.00% 0.00% -0.24% -0.32% -8.47% -23.39% -24.15%

(6)Total Insertion Time OR-DARPOpt (Seconds) 12.97 182.69 404.88 730.92 1862.51 2406.16 2456.5

(7)Total Regret Time OR-DARPOpt (Seconds)

(8)Max Insertion Time OR-DARP-Opt (Seconds)

(9)Max Regret Time OR-DARPOpt (Seconds)

60.41 1266.11 7330.59 15246.4 29432.5 37519.2 41219

0.41 2.57 2.82 3.51 7.16 11.41 9.2

3.3 39.2 196.8 343.3 790.9 909 768

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382 8 9

1370 1619

1370 1619

890 1012

-35.04% -37.49%

2857.13 3636.53

44882.7 48463.1

12.37 14.1

709.7 1207

It can clearly be seen that the regret optimization procedure consumes the major part of the total execution time required by OR-DARP-Opt to provide a solution. In experiment 9, the execution time needed by the regret optimization sub-algorithm is almost 48463/(48463+3636) = 93% of the total execution time. Clearly, when no optimization is used, the OR-DARP-NoOpt algorithm accepts all trip requests (3rd column). The th 8 column of Table 1 provides a good explanation why this happens. Trip request acceptance takes no more than 15 seconds while the waiting time threshold to reject it is 60 seconds. As to whether the presence of optimization procedures negatively affects profitability remains to be seen. Table 2 presents results related to profits vs. the number of requested trip demands between OR-DARP-NoOpt and OR-DARP-Opt algorithms. Column 2 represents the number of trip requests. Column 3 represents the number of requests served by OR-DARP-NoOpt. Column 4 represents the number of requests served by OR-DARP-Opt. Column 5 represents the total profits number for OR-DARP-NoOpt (OR-DARP-NoOpt serve all trip requests) solutions when applied to trip requests in column 3. Column 6 represents the total profits number for OR-DARP-Opt solutions when applied to trip requests in column 4. If the incoming trip requests were only those shown in column 4, then they would be all served by both algorithms. Additionally, column 8 represents the profit difference (in percentage) between ORDARP-Opt and OR-DARP-NoOpt solutions when both algorithms serve trip requests in column 4. Column 9 describes the number of vehicles engaged in the service of trip requests in column 4. It is the same for both algorithms. Table 2. Comparison between the OR-DARP-NoOpt and OR-DARP-Opt. #

(2)Trip Requests

(3)Assigned Trips by OR-DARPNoOpt

(4)Assigned Trips by OR-DARPOpt

1 2 3 4 5 6 7 8 9

94 226 415 624 803 949 1151 1370 1619

94 226 415 624 803 949 1151 1370 1619

94 226 414 622 735 727 873 890 1012

(5)Absolute Profits for OR-DARP NoOPT, for column 3 46€ 193€ 535€ 874€ 1180€ 1514€ 1794€ 2076€ 2521€

(6)Absolute Profits for OR-DARPOPT, for column 4 48€ 200€ 554€ 886€ 1096€ 1071€ 1263€ 1225€ 1496€

(7)Profits Difference (Euros)

(8)Profits Diff OPT vs NoOPT for column 4 3.2% 3.5% 4.4% 3.1% 2.5% 0.5% 0.7% 3.3% 2.7%

2€ 7€ 19€ 12€ -84€ -443€ -531€ -851€ -1025€

(9)Vehicles number for column 4 7 9 16 22 25 24 28 27 28

Figure 4 presents results related to profit gain or loss percentages between OR-DARP-Opt and OR-DARP-NoOpt algorithms when applied to requested trips presented in column 2 (Table 2). Horizontal axis describes the number of assigned trip requests processed by either the OR-DARP-Opt or OR-DARP-NoOpt algorithms for each experiment.

Profits (gain-loss) OR-DARP-Opt Vs OR-DARP-NoOpt

20,0% 0,0% -20,0% -40,0%

1,4% (1)94-94

4,3%

(2)226-226

3,6%

(3)414-415

3,6%

turn point

(4)622-624

(5)735-803

(6)727-949

(7)873-1151

-29,3%

(8)850-1370 (9)1012-1619

-29,6%

-7,1%

-40,7%

-41,0%

-60,0% Figure 4 OR-DARP-Opt vs. OR-DARP-NoOpt profit (gain – loss) chart

The information extracted from Table 2 is significant. By observing column 8, it is obvious that the OR-DARP-

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Opt algorithm always outperforms OR-DARP-NoOpt. The optimum percentage profit increase is approximately 4.4%. High optimization techniques such as regret optimization offer better solutions when compared to minimal (myopic) optimization techniques. However, since real time DRT systems have different response time requirements than offline systems, we should explore the behavior of optimization routines as a factor to the trip requests number more deeply. For experiments 1 to 4 (Table 2), columns 5, 6 and 7 show that the OR-DARP-Opt produces solutions with higher absolute profits than the ones produced by OR-DARP-NoOpt. However, for experiments 5 to 9, there is a loss of profits starting from 84€ (-7.1%) and reaching 1025€ (-40%) due to unserved trip requests (column 5, Table 1). Column 9 in Table 1 provides a plausible explanation. A full cycle of optimization requires an exhaustive search over all possible trip re-assignments to reach the best solution for that particular problem instance. The time needed for a full cycle of optimization is, in the worst case, 1207 seconds (experiment 9, column 9, Table 1). It is possible for a significant number of incoming trip requests to be rejected during this processing time. Experiment 5, with a profit loss of 7.1% (Figure 4) is the critical point where the optimization scheme degrades system profitability due to the required processing time to reach an optimized solution. Clearly, the inherent core optimization technique of the regret optimization sub-algorithm is insufficient for heavy demand situations. There are a few modifications that may push this critical point to a higher number of trip requests. For example, instead of a full cycle of regret optimization, we may opt for a single step regret optimization. Therefore, only a single route would be considered at a time. However, it is clear that we would degrade the optimization procedure significantly since we use only a small fragment of the available information given the fact that the full cycle optimization procedure engages all (28 in the worst case) routes. Our algorithmic approach proved that it is highly dependent on the trip requests number and when applied to real systems, has a critical point where it turns from an asset to a liability and should be replaced by the fast, myopic and minimal optimization insertion algorithm. However, even the fast insertion algorithm, needs 0.41 (experiment 1, column 8, Table 1) seconds (which is the maximum time for one trip assignment) when the number of trip requests is low (experiment 1, column 4, Table 1) and 14.1 (experiment 9, column 8, Table 1) seconds when that number is high (experiment 9, column 4, Table 1). Eventually, the fast insertion algorithm may be proved insufficient when the number of trip requests reaches a point where it is a hundred times more than the number described in experiment 9. In a few words, the valuable information extracted by the above analysis lies in the fact that for online DRT systems, we can find turn points where -high (or less high) time consuming - optimization procedures are no longer profitable. 4. Conclusions and future work In this study, we presented metrics on performance vs. profits of an online dial-a-ride algorithm. The proposed online algorithm is designed to use optimization procedures during idle processing time. Every trip request is handled as an emergency trip request, taking the highest priority for optimization tasks. This approach ensures that customers get prompt, appropriate, and final response, almost independently of the system load. The proposed online (regret based) dial-a-ride algorithm was applied to real data and produced some interesting results regarding profitability metrics. For our data set, the critical point arises in experiment 5 (Table 2) where the application of the optimization procedure is no longer a profit factor for the online DRT system. After experiment 5, it is more profitable to serve more trip requests using the myopic fast insertion algorithm (lower optimization performance) than using the regret (high optimization performance) algorithm. In short, this study suggests that online DRT systems have distinct turn points where optimization procedures as a function of trip requests number are not suitable for system profitability anymore. Such online DRT systems operated by governmental organizations or private transportation companies should carefully use a variety –from pure myopic to highly optimized- optimization engines in order to be profitable for any demand load scenario. Many other factors related to online DRT systems profitability as a parameter of optimization engines could be explored as well. We would like to propose investigation into the following issues: (a) Optimization procedures. Specifically, the way optimization into online algorithms is embedded, the execution times for one cycle (or step) optimization, and the usability of optimization in comparison with rejected requested trips (if any) are all interesting issues. (b) The forecast of the incoming demands. The utilization of historical data regarding users’ mobility habits and the duration of the forecasting period is an issue regarding online dial-a-ride algorithms. Many parameters concerning forecasting (time horizons, demand occurrence probabilities, evaluation of probabilistic demands concerning this 'importance') are all possible research subjects.

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