A metaheuristic for evaluation of an integrated special transport service

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International Journal of Urban Sciences

ISSN: 1226-5934 (Print) 2161-6779 (Online) Journal homepage: https://www.tandfonline.com/loi/rjus20

A metaheuristic for evaluation of an integrated

special transport service

M. Posada & C. H. Häll

To cite this article: M. Posada & C. H. Häll (2020): A metaheuristic for evaluation of an integrated special transport service, International Journal of Urban Sciences, DOI: 10.1080/12265934.2019.1709533

To link to this article: https://doi.org/10.1080/12265934.2019.1709533

Published online: 03 Jan 2020.

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A metaheuristic for evaluation of an integrated special

transport service

M. Posada and C. H. Häll

Department of Science and Technology, Linköping University, Norrköping, Sweden

ABSTRACT

This work concerns evaluation of integrated demand responsive services, in which certain parts of the passengers’ door-to-door journeys are served by fixed route public transport. The purpose of combining a special transport service with afixed route service is to reduce the high, publicly subsidized, operational costs of the special transport service. To be able to recommend in which situations, or areas, it is beneficial to use an integrated service we present a metaheuristic for solving the routing problem in such a service. The metaheuristic can be implemented as part of an evaluation tool for policy makers and officials, providing insights into the effects of an integrated demand responsive service compared to a non-integrated. The metaheuristic is based on the adaptive large neighbourhood search (ALNS) framework. It is applied to a data set from a real-world, rural, demand responsive special transport service and the fixed route service available in the area. The objective used in our heuristic is to minimize the distance driven by the demand responsive vehicles. The distance driven is strongly related to the operational cost of the service. Our tests show that the distance driven by the demand responsive vehicles is reduced by 16%, giving a substantial cost reduction for the special transport service in the given area.

Highlights

. Evaluation of integrated demand responsive service

. Combining special transport with aﬁxed route service to reduce operational costs

. Metaheuristic, based on the adaptive large neighbourhood search (ALNS) framework

. Comparing integrated service to non-integrated

ARTICLE HISTORY Received 5 May 2019 Accepted 17 December 2019 KEYWORDS

Metaheuristic; public transport; integration; door-to-door; paratransit

1. Introduction

To serve the transport needs of the whole population, demand responsive services are pro-vided as a complement tofixed route public transport in many parts of the world. Such services are often aimed specifically at the elderly and/or people with disabilities since these groups often have other mobility needs than the rest of the population due to for example wheelchair use. They require a more flexible transport service either due to

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issues with driving their own vehicle, or problems using or getting to theﬁxed route public transport system. This kind of shared-ride, door-to-door service for people with impaired mobility is often called a special transport service or paratransit, and is performed with demand responsive vehicles. These demand responsive vehicles are typically smaller than other public transport vehicles, and can be adapted with specialized equipment such as wheelchair lifts and ramps. In general, these services are provided as complements to the rest of the public transport system in urban as well as rural areas.

Special transport services are often very costly to operate. As an example of a special transport service, and the costs associated with such a service, we can look at the Swedish special transport service. This is a demand responsive special transport service aimed for those with special mobility needs. In 2015, there were 318,000 people holding special transport service permits (3.5% of the total population); they per-formed roughly 11 million trips. The total cost of the service, nationwide, was 399 million €, of which 74% was covered by public funds (Trafikanalys,2016b). This can be compared to the rest of the Swedish local and regional public transport, where 52% of the total cost was covered by public funds (Trafikanalys, 2016a). The number of trips in the regular public transport service is though much higher, giving that each trip in the special transport service is approximately subsidized ten times higher than a trip in the regular transport service. The travel patterns, between different origins and destinations, are similar in both services. In practice, this often creates situ-ations where one, subsidized, vehicle from each service (e.g. one bus and one minivan) travers the same parts of the road network at the same time. This is the main reason to why integrated demand responsive services are being considered for operation of special transport services. Using the existing public transport network for certain legs of the passengers’ journeys could potentially lower the operational cost of the special transport service, since demand responsive services are typically associated with higher oper-ational costs thanfixed route services (Häll,2011).

Figure 1a and b illustrate the integration we consider in this paper. Instead of transport-ing certain passengers directly from their origin to their destination with a demand responsive vehicle, the passengers go by public transport for a leg of the trip (as illustrated inFigure 1a). A demand responsive vehicle can have an itinerary including several pick-up and drop-off locations for several different passengers, out of which some of the passengers (0, 1 or several) should be transported to, or from, a transfer location (e.g. a bus stop) to thefixed route service as illustrated in Figure 1b.

When implementing an integrated special transport service, there are several design choices which have to be made, as well as demand characteristics which have to be taken into account. There are also several possible metrics by which to evaluate the service. The design choices involved in an integrated special transport service concern how the service makes use of a public transport network. Examples of such choices are: how many, and which, public transport stops should be used as transfer locations for the integration? Which passengers are considered for integration, i.e. to use thefixed route public trans-port? How many changes between different fixed route public transport vehicles are they allowed to make? And, are passengers assumed to be capable of walking short dis-tances on their own if, for example, the public transport stop is situated close to their final destination?

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There are demand characteristics which depend on the chosen service area. Whether the chosen service area is rural or urban aﬀects both the demand density and the lengths of the passengers’ trips, as well as the characteristics of the available public trans-port network. Urban areas have both denser public transtrans-port networks with shorter head-ways as well as higher demand densities. Rural areas have lower demand densities, longer headways, and longer passenger trips. Apart from this spatio-temporal demand diﬀerences and these public transport characteristics, there are also passenger characteristics which must be taken into account, for example concerning the use of wheelchairs. A demand responsive service aimed at a segment of the population with fewer mobility impairments is more likely to be well integrated with the public transport service.

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Figure 1.(a) An illustration of an integrated special transport service trip from an origin to a destina-tion. (b) Example of routes for demand responsive vehicles (black lines) serving several requests of which one (red line) is served as an integrated trip.

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There are several possible metrics which can be used in the evaluation of an integrated special transport service. They can be sorted into two categories: those concerning oper-ational costs (which are important from the operator’s perspective) and those concerning level-of-service (which are important from the passenger’s perspective). From an oper-ator’s perspective, interesting metrics include distance driven, the number of demand responsive vehicles used as well as the capacity utilization of these vehicles. From the pas-senger’s perspective other aspects are more important, such as the number of integrated trips, how the travel times are aﬀected, and how many bus-to-bus changes are performed. Examples of design choices, demand characteristics, and evaluation metrics are listed in

Figure 2.

To attain relevant evaluation metrics of an integrated service with given design choices and demand characteristics, a planning algorithm is needed to solve the vehicle routing problem that occurs in the planning of such a service. The problem lies in designing vehicle routes for a fleet of vehicles in order to serve all demand in time. In this paper we present a metaheuristic algorithm for solving this vehicle routing problem. This algor-ithm can be implemented as part of an evaluation tool for policy makers and public trans-port officials aimed at providing insights into how large realistic effects an integrated special transport service can have, given certain design choices. The metaheuristic is an adaptation of the adaptive large neighbourhood search (ALNS) framework previously suc-cessfully applied to the planning of non-integrated demand responsive transport. A

Figure 2.Both design choices and demand characteristics are input to a planning algorithm. The sol-ution from the planning algorithm can be evaluated from both the operator’s and passenger’s perspective.

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metaheuristic algorithm is chosen over an exact method since it is hard to solve the com-binatorial optimization problem associated with an integrated service to optimality for instances of realistic sizes (Posada, Andersson and Häll,2017).

To exemplify the use of our metaheuristic as an evaluation tool, we apply it to a test case of real-world special transport service requests. The case considered is a rural area. Due to the longer travel distances in a rural area, the potential operational cost reductions are likely to be larger than in an urban area. Furthermore, as is suggested by theﬁndings in Nguyen-Hoang and Yeung (2010), economies of population density exist in special trans-port services. Nguyen-Hoang and Yeung (2010) ﬁnd that a 1% increase in population density is associated with a 0.4% decline in operating costs of the service. This indicates that decreasing operating costs of special transport services could be especially important in rural areas. Also, in many real-world applications, especially in an urban area, the majority of requests involve short trips. These trips might be infeasible to serve as inte-grated trips, due to level-of-service constraints.

An evaluation of the eﬀects of introducing an integrated special transport service necessitates a fair comparison with the non-integrated case. Thus, the metaheuristic is applied to solve the routing problems of both the integrated and non-integrated services, using the same real-world data set as well as a widely used benchmark data set from

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Cordeau and Laporte (2003). This data set is used to verify that the metaheuristic can produce reasonable solutions for the non-integrated case, which is necessary for a fair com-parison. A description of how the data sets are used, and to what, is given inFigure 3, in which DARP (the dial-a-ride problem) and IDARP (the integrated dial-a-ride problem) represent problem instances of non-integrated respectively integrated services. These terms will be described in more detail in Section 2.

By presenting the algorithm and applying it to a real-world example, this paper contrib-utes to the development of evaluation tools for integrated services and exempliﬁes the beneﬁts of using integration for rural special transport services. The paper is organized in the following way. Section 2 presents a literature review of the topic, focusing on mod-elling of the routing problem associated with both an integrated and non-integrated special transport service. The implemented metaheuristic is described in Section 3. The algorithm is applied to real-world data from a rural area in Sweden, as well as to a bench-mark case, as described in Section 4. We end the paper by presenting our conclusions and ideas for future research in Section 5.

2. Dial-a-ride planning

The problem of planning vehicle routes for a demand responsive door-to-door service is known in the literature as the dial-a-ride problem (DARP). Note that the definition of the DARP varies between different authors due to the application oriented character of the problem (Cordeau & Laporte, 2003; Parragh, Doerner, & Hartl,2010). Dial-a-ride pro-blems can arise in situations such as rural transport, patient transport, or in special trans-port services. In the DARP, a number of individual passengers, or groups of passengers, request transport from given origins to destinations, and these requests are to be served by a fleet of demand responsive vehicles. The passengers request desired departure or arrival times, which are used together with predefined policy rules to create time windows at all locations, within which the service must be performed. The vehicle routes are designed based on this information under certain constraints, typically regard-ing vehicle capacities as well as maximum route-times and maximum ride-times (the times the vehicle is in service and the time the passengers spend in the vehicles, respectively).

There are two main alternatives when it comes to the objective function in the DARP: (1) maximize level-of-service subject to constraints regarding available resources (vehicles) and other side constraints, or (2) minimize cost subject to level-of-service constraints and resource constraints. The most common cost function to minimize is the distance driven by the demand responsive vehicles, but other costs can also be included in the objective function, such as a ﬁxed cost for each extra vehicle or costs related to lowered level-of-service.

The DARP is either static or dynamic. In the static case, all requests are known in advance, prior to scheduling the vehicle itineraries. This corresponds to a demand respon-sive service where requests have to be made no later than the day before service. In the dynamic case, the requests are made over time and hence the planning must start before all requests are known. This corresponds to a service where requests can be made at any time, far in advance or right up until the time when the customer wants to be served.

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The DARP is a hard, combinatorial, problem which can be solved to optimality, but in general only for small cases. Thus, for larger cases, heuristics or metaheuristics are com-monly applied. Metaheuristics can search the solution space quicker than an exact optim-ization method, but the solutions found cannot be veriﬁed to be optimal. For further details on DARP the reader is referred to the review on both exact and heuristic solution methods for both the static and dynamic versions of the problem presented in Cordeau and Laporte (2007), as well as a recent typology and literature review for dial-a-ride pro-blems, presented in Molenbruch, Braekers, and Caris (2017). For overviews of metaheur-istics applied to vehicle routing problems and the DARP, we refer to Gendreau, Potvin, Bräumlaysy, Hasle, and Løkketangen (2008) and Cordeau and Laporte (2007), respect-ively. For a general introduction to, and overview of, the ﬁeld of metaheuristics, we refer the reader to Talbi (2009).

Some recent contributions to the application of metaheuristics to the DARP are Parragh et al. (2010), Parragh and Schmid (2013), Braekers, Caris, and Janssens (2014) and Gschwind and Drexl (2019). Parragh et al. (2010) test a competitive variable neigh-bourhood search-based heuristic with three diﬀerent types of neighneigh-bourhoods. Parragh and Schmid (2013) implement a matheuristic comprised of a hybrid column generation and large neighbourhood search algorithms. Braekers et al. (2014) propose a deterministic annealing metaheuristic. To the best of our knowledge, the current state-of-the-art is pre-sented in Gschwind and Drexl (2019). They develop a constant-time procedure (with cubic-time preprocessing) for checking the feasibility of a DARP solution and implement an adaptive large neighbourhood search metaheuristic which outperforms previous approaches. The algorithms in these papers are all applied to a set of test cases for the DARP provided in Cordeau and Laporte (2003).

The planning of services where dial-a-ride transit andfixed route transit are combined wasfirst introduced in Wilson, Weissberg, and Hauser (1976). Liaw, White, and Bander (1996) formulate a model for the integrated problem and present heuristics to solve the problem. Häll, Andersson, Lundgren, and Värbrand (2009) present an exact formulation of the integrated dial-a-ride problem (IDARP), based on the arc-flow formulation of the DARP in Cordeau (2006), and present some techniques to strengthen the formulation. The IDARP is in many aspects similar to the pickup and delivery problem with transship-ments (PTPT), described in for example Cortés, Matamala, and Contardo (2010) and the pickup and delivery problem with time windows and scheduled lines (PDPTW-SL) described in Ghilas, Demir, and Van Woensel (2016). Ghilas et al. (2016) present an ALNS metaheuristic to efficiently solve the problem, which concerns integrating short-haul passenger and freight transportation using public transport. The differences between the PDPTW-SL and IDARP lie mainly in the human level-of-service perspective inherent to dial-a-ride problems. The formulation in Häll et al. (2009) is extended in Posada et al. (2017) to the integrated dial-a-ride problem with timetables (IDARP-TT) to also handle timetables for the fixed route service, as well as some further features which aim at making the model more applicable to real-world planning situations. Hand-ling of timetabled fixed route services is essential to be able to synchronize transfers between different travel modes, or different vehicles belonging to the same mode, with the aim to minimize transfer times. Coordinated transfers between different fixed route services has been studied, and used, for several year. More recently has also the interest in coordination between fixed route and flexible route services increased. The work of

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Kim and Schonfeld (2014) studies effects on system costs that can be achieved by combin-ing coordinated timed transfers and integration of conventional and flexible services. Their results show that integration of conventional andflexible services has best potential in regions where the demand densities vary among different parts of the region, and that waiting times can be reduced by coordinated transfers.

Some work has also focused on how to useﬂexible services as feeders to ﬁxed route services. For example, the work of Chandra and Quadrifoglio (2013) presents a model to compute what cycle length a demand responsive feeder should have, based only on geometric data and demand data. The aim is to maximize the level-of-service provided to the users.

Hickman and Blume (2001) perform an analysis of the level-of-service eﬀects of an

integrated demand responsive service using a passenger scheduling heuristic. They also point out the need for a full analysis of the implications on both levels-of-service and oper-ator costs using a vehicle scheduling heuristic. Horn (2002) describes the main com-ponents of the LITRES-2 modelling system, a framework which is used in Häll, Lundgren, and Värbrand (2008) to simulate and evaluate an integrated public transport system. The results of the simulation highlight the importance of the design of the demand responsive part of the integrated service, and that the attractiveness of an inte-grated service depends strongly on the placement and number of transfer locations.

Door-to-door services offered to the general public, and not only to elderly or people with disabilities, are more common in rural areas than in urban areas, sincefixed route public transport is expensive to operate in low density areas. As the operational costs, mainly associated with personnel, stand to be diminished with the advent of autonomous vehicles, moreflexible transport services could be offered to the general public. Chebbi and Chaouachi (2016) propose a multi-objective optimization approach, minimizing empty vehicle movements and the total number of vehicles, for planning afleet of small electric automatic vehicles to move groups of passengers between stations. If this type of service for a general public in urban areas are to be integrated with fixed route services, the type of coordinated transfers and effects of them, as considered in Kim and Schonfeld (2014) will be vital to take into consideration.

When considering services for a general public, it is often more interesting to investi-gate the effects of introducing a demand responsive service as a substitution to fixed route service, in order tofind recommendations regarding when, and where, to choose which service. Such questions are e.g. addressed in Kalpakcı and Ünverdi (2016), and Li and Quadrifoglio (2010). It is also important to have a bigger understanding for the system itself, and its users, when making recommendations regarding when and where to operate a specific service. Therefore, some research, as in Alonso-González, Liu, Cats, Van Oort, and Hoogendoorn (2018), focus on creation of frameworks helping authorities to assess the mobility improvements users experience by the introduction of a demand responsive service, or as in Papanikolaou, Basbas, Mintsis, and Taxiltaris (2017) present-ing a method for categorization of demand responsive services and matchpresent-ing them to their specific public transport markets.

3. Metaheuristic algorithm

In this section a proposed metaheuristic algorithm for the IDARP and DARP is pre-sented. The metaheuristic is based on the ALNS framework, which has previously

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been applied successfully to the DARP and similar vehicle routing problems (see e.g. Gschwind & Drexl, 2019; Ropke & Pisinger, 2006). The ALNS is based on the LNS, introduced in Shaw (1998). The underlying principle of the LNS is that in each iter-ation, the currently incumbent solution is first destroyed and then repaired. In the area of vehicle routing, this corresponds to first removing a certain number of requests from their routes and then reinserting them according to some rule. An ALNS can use several different removal and insertion heuristics (equivalent to using local search with several neighbourhoods), where the probability of choosing a specific combination of removal/insertion heuristics is updated as the search ceeds. There are several different destroy and repair operators that have been pro-posed in the literature, see e.g. Häll and Peterson (2013), Parragh and Schmid (2013), and Ropke and Pisinger (2006). In contrast to the variable neighbourhood search (VNS) and LNS, in an ALNS the probabilities of using the various destroy/ repair operators are adaptively updated according to their past performance.

We begin in Section 3.1 by presenting some considerations and design choices made during the algorithm development process. In Section 3.2 we present the metaheuristic design. Sections 3.3 through 3.7 describe various aspects of the algorithm. Section 3.8 pre-sents two operators speciﬁc to the IDARP which have not previously been published in the literature.

3.1. Algorithm design considerations

Cordeau, Gendreau, Laporte, Potvin, and Semet (2002) present four critical criteria for assessing heuristics: accuracy, speed, simplicity, and flexibility. We have made our algorithm design choices with these four attributes in mind and in the following dis-cussion we address each of them, and how they apply to our solution method. Accu-racy measures how far the obtained solution is from either a proven optimal value, if it exists, or the current best known solution. Rather than designing a state-of-the-art algorithm intended to produce new best known solutions to the DARP, the aim has been to produce consistently good solutions compared to current best known sol-utions for the DARP, to provide a good basis for the comparison between the IDARP and DARP solutions. This has also allowed us to focus on algorithmic simpli-city, in the sense that there are few operations involved in the algorithm, and the par-ameters which control their behaviour have clear meanings. As is pointed out in Cordeau et al. (2002), the parameters should make sense to the end-user, and ideally be limited in number. This simplicity of design provides a basis for algorithmic flexibility in the sense that problems with different sets of constraints, such as those in the DARP and IDARP, can be solved without changing the overall design. Another measure of how flexible an algorithm is, is its ability to solve diverse problem instances, preferably without changing parameter values. Thisflexibility is exemplified in Section 4, where the algorithm is applied to two diverse data sets using the same parameter settings, producing reasonable results. The final assessment criterion for heuristics presented in Cordeau et al. (2002) is speed. In this paper, we have chosen not to focus on speed, since computational time is not critical for evaluating policy implementations of the type discussed in this paper.

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3.2. Adaptive large neighbourhood search

Diﬀerent metaheuristics could be combined with ALNS to deﬁne when it should end and to help it escape local minima (Ropke & Pisinger,2006). Here, as in Ropke and Pisinger (2006), a simulated annealing (SA) framework is chosen. Apart from accepting all improv-ing solutions, SA also accepts some solutions that impair the objective function. The selec-tion of which impairing soluselec-tions to accept is done according to a stochastic process where a current solution, solcurrent, is accepted over the incumbent solution, solincumbent, with probability ( exp (− f (solcurrent)− f (solincumbent)))/T, where T . 0 is the simulated annealing temperature, and f (·) is the objective function. Accepting solutions which impair the objective function is done to avoid getting trapped in local minima.

The algorithm has three removal operators: random removal, Shaw removal, and worst removal. The three operators are described in Section 3.3. The removed requests are reinserted with a greedy insertion heuristic, described in Section 3.4. There is also an inter-route intensification operator added to further explore promising regions of the solution space. The removal and insertion operators on their own seldom produce new best solutions, in this implementation, without further intensification, but instead fill the function of diver-sifying the search. The intensification operator is further described in Section 3.5.

The algorithm has four primary steps (seeFigure 4). The algorithm starts in Step 1 by ﬁnding an initial solution. The removal and insertion operators are then applied to produce a new current solution, Step 2. If the new request is accepted by the SA acceptance criterion, the inter-route intensiﬁcation operator is used in Step 3 to possibly improve the current solution, after which it becomes the new incumbent solution. If the current sol-ution is feasible and better than the previous best found solsol-ution, it is saved in Step 4. These four steps are performed for a pre-set number of iterations, imax, after which the algorithm terminates.

Apart from the four main steps, the simulated annealing temperature, T, as well as the number of requests to be removed in the iteration, q, are updated at the beginning of each iteration. q is then used together with the incumbent solution as input to the removal and

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insertion operators in Step 2. Which removal operator to employ in each iteration is chosen using weights based on the past performance of the operators. The weights assigned to each operator are updated at regular intervals, every iupdateiteration.

Note that the incumbent solution does not need to be feasible. Allowing intermediate infeasible solutions helps reduce the risk of the algorithm becoming trapped in local minima. Infeasible solutions are here considered to be solutions which have a larger vehicle fleet than the available vehicle fleet size. An extra operator, with the purpose of increasing the probability of finding feasible solutions, is applied each iteration if the current solution is infeasible. This operator attempts to move as many requests as possible away from the routes of the extra vehicles. An alternative way of dealing with infeasible routes and solutions would be to assign costs to relaxations of time window, ride and route times and vehicle load constraints, as is done in Cordeau and Laporte (2003). This penalized objective function approach has previously been shown to be successful. However, it leads to additional parameters and decreases the simplicity of the method.

3.3. Removal operators

The three removal operators in the algorithm are used together with the acceptance cri-terion to visit new regions of the solution space. The parameter q[ [0, n] in Step 2 sets the number of requests which are to be removed by the removal operator, i.e. the size of the neighbourhood. q is chosen randomly in the interval [qlower, qupper] in each iter-ation. This means that the number of requests removed by the removal operation at each iteration changes as the search progresses. Both the lower and upper endpoints of this interval are parameters set by the user. Which of the three removal operators that should be applied in each iteration is decided through a roulette wheel selection process, using adaptive weights based on the previous performance of each operator. The performance is measured as the number of times each operator has produced either a new solution or a new best solution after the intensiﬁcation operator has been applied.

The random removal operator is straightforward and removes q random requests from their respective routes. In the Shaw removal operator, proposed by Shaw (1998), a relat-edness measure based on the spatio-temporal distance between each pair of requests is used to select a set of requests which are ‘similar’ to be removed together. The worst removal operator attempts to select requests that appear to be misplaced, i.e. requests for which the route cost is considerably decreased if it is removed. For further details regarding all three removal operators, see Ropke and Pisinger (2006).

3.4. Greedy insertion heuristic

The initial solution heuristic as well as the removal/insertion operators and the intensi fi-cation in the algorithm (Steps 1, 2 and 3) require an insertion heuristic to function. In the insertion heuristic, all feasible insertions of a request, i.e. all possible insertion positions of the pick-up and drop-off locations corresponding to that request, are tested and the best insertion is saved. This heuristic can be applied to the route of a single vehicle, or to the routes of all vehicles in thefleet, as required. If it is applied to all routes, the best possible

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insertion of the request into the routes of theﬂeet is guaranteed. If the insertion heuristic is applied with the aim of shifting the positions of locations within a single route, of course only insertions which decrease the objective function value have to be considered, and if there are none, the current positions of the locations are kept. Also, to further speed up the insertion, feasibility checks are not performed for insertions inferior to the best insertion found.

If no feasible insertion can be found, a new empty vehicle is added, departing from the depot closest to the pick-up location of the request to be inserted. If the number of vehicles is higher than a predeﬁned ﬂeet size, the solution is considered infeasible.

3.5. Intensiﬁcation

To increase the efficiency of the method by exploring promising regions of the solution space, an intensification operator is added, Step 3. This operator iterates over the requests in the vehicle routes in random order and attempts to reinsert the locations belonging to each request in positions which decrease the objective function value (i.e. the distance tra-velled by the demand responsive vehicles). When an improving move is found, the search restarts with a new random order of the vehicles and requests. The intensification heuristic terminates when no further improving moves can be found.

3.6. Initial solution

Toﬁnd a good initial solution, the following initial heuristic is applied a hundred times, where after the best found solution is selected as the initial solution to the rest of the meta-heuristic, as in Braekers et al. (2014), in Step 1. Each initial solution is created by incre-mentally adding the locations of each request in random order at the best insertion points in any vehicle in theﬂeet. This insertion is done by the greedy insertion heuristic. Which solution is best is decided by (1) the number of (extra) vehicles, and (2) the objec-tive function value (distance driven by all vehicles).

3.7. Feasibility evaluation

The insertion heuristic needs to evaluate the feasibility of various insertions of a request into a route. It is important to note the difference between a feasible solution, a feasible route, and a feasible insertion. A solution is considered feasible if the total number of vehicles used does not exceed a predefined level; a route is considered feasible if no con-straints are violated (vehicle load, time windows, maximum route-time, and maximum ride-times); the insertion of a specific request into a specific vehicle route at a specific point is considered feasible if the route is still feasible after the insertion. All insertions in our algorithm have passed feasibility checks in the insertion heuristic, thus all routes are kept feasible at all times in the algorithm, i.e. entire solutions can be infeasible while all routes are kept feasible.

Since many possible insertions have to be evaluated in each iteration of the algorithm, typically resulting in millions of feasibility checks during an algorithm run, it is important that the feasibility checks are performed in an eﬃcient manner. To this end, we implement the constant-time feasibility checking procedure recently presented in Gschwind and

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Drexl (2019). In this procedure, the feasibility check itself can be performed inO(1) time, given that specific auxiliary data are available. Computation of these data require an O(r3₎ preprocessing step, where r is the number of stops on the route. Previously, the most com-monly implemented feasibility checking procedures have been based on theO(r2) eight-step procedure from Cordeau and Laporte (2003), which in turn incorporates concepts from Savelsbergh (1992). In our implementation, the procedure of Gschwind and Drexl (2019) was significantly faster, despite the preprocessing step, than the eight-step pro-cedure of Cordeau and Laporte (2003) with efficient preliminary feasibility checks from

Braekers et al. (2014). The reason behind this diﬀerence in eﬃciency is that while the feasi-bility check is performed frequently, the pre-processed data only have to be recomputed when a route is changed.

3.8. IDARP-speciﬁc operators

We have implemented two IDARP-speciﬁc removal operators, which are applied instead of the removal/insertion operators in certain iterations of the algorithm. They are selected using weights which adapt to the past performance of the operators, in the same way as the removal operators presented in Section 3.3.

They are constructed in a similar fashion to the Shaw and worst removal operators and are selected using the same roulette wheel selection principle, with self-updating weights, as the three removal operators, in Step 2. The first IDARP-specific operator removes a request (or rather, the locations corresponding to a request) and replaces it with two requests corresponding to the first and last legs of the passenger’s journey, essentially introducing a fixed route leg into the journey. The two resulting requests are denoted as artificial. The first artificial request has the original pick-up location as pick-up and the embarkation point to the public transport line as drop-off location. The second artifi-cial request has the disembarkation point from the public transport line as pick-up location and the original drop-off location as drop-off. The second operator removes two artificial requests corresponding to the same real request and replaces them with the original request, essentially removing the fixed route leg for a certain passenger. Changes due to the IDARP-specific operators, are kept if the current solution becomes incumbent, otherwise the algorithm returns to the incumbent solution.

The time-windows of the resulting ‘new’ locations are set using the original time-windows, service times, minimum travel times with public transport and demand respon-sive vehicles, and the public transport timetable. In the feasibility checks, certain compu-tations have to be slightly altered to compensate for the fact that there is an interdependence between the routes of two different demand responsive vehicles when a request is integrated. These changes are simplified by only allowing changes to the service times of one of the new integrated requests at a time. This simplification decreases theflexibility of the insertions but maintains the computational complexity of the feasi-bility checks.

Which requests are selected for integration by thefirst operator depends on the poten-tial difference between the direct travel distance with a demand responsive vehicle from the pickup to destination, and the length of the two new special transport service legs. The requests are sorted into a descending list according to this cost difference. As in the Shaw and worst removal operators, a determinism parameter p≥ 1 introduces

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randomness in the selection procedure to avoid cycling. The two best feasible public trans-port stops are selected for the removed request and the new artiﬁcial requests are inserted using the greedy insertion heuristic.

The second IDARP-specific operator acts in a similar fashion, but replaces the descend-ing cost difference list with a list of the requests which have been integrated. If a request is selected, both thefirst and last legs are removed, and the original door-to-door locations are reinserted into one of the routes.

It should be noted that in a pre-processing step prior to the algorithm execution, the different combinations of public transport stops that can be used to serve each request without breaking time window and ride time constraints are listed. Thus, requests whose pickup and drop-off locations are located such that no public transport line can serve them are excluded from the list of requests. This pre-processing heuristic is similar to the passenger itinerary heuristic in Hickman and Blume (2001) and makes the computations more efficient.

4. Computational experiments

To evaluate the diﬀerences between IDARP and DARP, we apply the ALNS algorithm to both benchmark instances and real-world data. This section presents the chosen par-ameter values, and some assumptions made. Section 4.1 presents the experiments based on the benchmark instances and Section 4.2 present those based on the real-world data set. The parameter values presented inTable 1were chosen based on an initial trial-and-error phase followed by individual tuning of the parameters. After this tuning, the same parameter settings are used for all instances. Each of the parameters is described below.

The number of ALNS iterations to perform before the algorithm terminates is set to 1000. The start temperature is set such that the probability of accepting a solution which is 5% (the start temperature control parameter) worse than the initial solution is accepted with probability 0.5. The temperature is cooled every iteration, by a cooling rate chosen such that the temperature at the last iteration equals 10% (the cooling rate par-ameter) of the initial temperature.

The self-adjusting weights used in the roulette-wheel selection of the removal operators are updated based on their past performance. Each operator is awarded a score, for either a new solution or a new best solution. These scores are set to 5 and 20, respectively. The search is divided into segments of 100 iterations. The weights are updated at the end of

Table 1.Chosen parameter settings.

Parameter Value

No. of ALNS iterations 1000

Start temperature control parameter 5%

Cooling rate parameter 10%

Score for new best solution 20

Score for new solution 5

Segment size 100

Reaction factor 0.1

qlower 2

qupper 0.2n

Determinism parameter of worst removal 3 Determinism parameter of Shaw removal 6 Determinism parameter of IDARP operators 3

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each such segment. A reaction factor of 0.1 is applied to the weights, if the reaction factor is zero the initial weights are not changed, if it is set to one, the performance of the operators in the most recent segment determines the weights completely. For further details regard-ing the self-adjustregard-ing weights and roulette-wheel selection, see Ropke and Pisregard-inger (2006). qlowerand qupperare thefloor and roof, respectively, for the number of requests removed in each iteration. Thefloor is set to 2 since setting it to 1 leads to increased cycling, while higher values could lead to missing new local optima close to the incumbent solution. If the roof is set too low, the neighbourhoods searched are too small to escape local minima. If it is set too high, the risk that the insertion and intensification heuristics fail to deliver high quality solutions increases, since only remnants of the previous routes remain.

Both the worst and Shaw removal operators as well as the IDARP-speciﬁc operators use determinism parameters to avoid removing the same requests each iteration. A higher value on the determinism parameter corresponds to lower randomness in the selection. The three determinism parameters are set to 3, 6 and 3, respectively.

In the computational experiments, the vehicleﬂeet is assumed homogeneous, as are the requests, but the algorithm can also be applied to cases with heterogeneous vehicleﬂeets and heterogeneous requests. The algorithm is adapted to handle some driver-related con-straints, such as maximum route duration, but not for example mandatory lunch breaks, as is considered in Parragh, Cordeau, Doerner, and Hartl (2012).

The algorithm is applied to single as well as multiple depot cases. In multiple depot cases, the vehicle fleet is divided evenly among the depots at the start of the algorithm, but as the search progresses, the distribution of the vehicles can change. We have chosen the objective of minimizing the distance travelled by the demand responsive vehicles, which is the most common objective function in dial-a-ride studies. The objective function used in the integrated case does not include thefixed route public transport as a cost, since it is considered small compared to the cost of the demand responsive vehicles and will thus not affect the vehicle routes.

4.1. Benchmark instances

It is necessary to produce reasonably good results for the DARP in order to fairly compare them to those of the IDARP. To verify that our method can produce good enough results, we use the 20 problem instances from Cordeau and Laporte (2003). The instances can be grouped into four categories according to certain characteristics. Theﬁrst half (R1a-R10a) have narrow time windows (randomly set to be between 15 and 45 time steps wide for each node) and the second half (R1b-R10b) have wider time windows (randomly set between 30 and 90 time steps for each node). Furthermore, within these two groups of requests, the sizes of the vehicle ﬂeets are set as to either provide tight constraints (R7a-R10a and R7b-R10b) or provide moderately full routes (R1a-R6a and R1b-R6b). The instances with moderately full routes could potentially be solved with fewer vehicles.

To be considered‘good enough’ we require that the algorithm consistently produce sol-utions that are at worst 5% away from the current best known solsol-utions. Gschwind and Drexl (2019) presents the current best known solutions for this data set, including compu-tational results from Braekers et al. (2014) and Chassaing, Duhamel, and Lacomme (2016).

Table 2shows the size of each problem instance from Cordeau and Laporte (2003), the current best known solution (column BKS), as well as our results (column ALNS) and

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the diﬀerence (in %) between these results. As can be seen, the algorithm produces good enough results to all instances.

4.2. Real-world instances

The real-world instances are based on historic data for requests made to a special transport service in a rural, low density, region of Sweden during two autumn weeks (weekends excluded). The customers are all capable of riding thefixed route service, however they are not able to take themselves to and from the service (e.g. to/from the bus stop). There-fore they are suitable to serve with integrated trips. The total area covers about 1160 km2 and has 47,000 inhabitants. In the area there is a main town (situated in the southern part), a small town (situated in the northern part) and a large rural area in between with a small village as a central part of this rural area. These three locations (the main town, the small town and the village) are connected viafixed route public transport, in the form of both regional buses and a regional train line. The population generates between 97 and 145 requests per day (with an average of 119) to the special transport service. Each of these requests has a known origin, destination and requested pick-up time. Time windows are set to be 15 min wide, beginning at the requested pick-up times. The time windows for the drop-off locations are created by setting the lower ends of the time windows to the first possible time the request can be dropped off. This time is calculated as the start of the pick-up time window plus the service times and the shortest travel time between the pickup and drop-off locations. The shortest travel time could be either the direct travel time using a demand responsive vehicle, or the shortest possible integrated route. The upper end of a drop-off time window is calculated by adding the service dur-ation and maximum ride-time to the upper end of the pick-up time window. The maximum ride-time is set to the largest of 30 min and 1.5 times the direct travel time. This corresponds to real-world rules in Swedish special transport services.

Table 2.Results for the 20 benchmark instances.

Instance No. req. No. Veh. BKS ALNS Diﬀerence between BKS and ALNS (%)

R1a 24 3 190.02 190.02 0 R2a 48 5 301.34 301.34 0 R3a 72 7 532.00 533.46 0.3 R4a 96 9 570.00 579.57 1.7 R5a 120 11 625.64 635.60 1.6 R6a 144 13 783.78 803.28 2.5 R7a 36 4 291.71 291.71 0 R8a 72 6 487.84 495.63 1.6 R9a 108 8 653.94 672.32 2.8 R10a 144 10 845.47 869.13 2.8 R1b 24 3 164.46 164.46 0 R2b 48 5 295.66 295.66 0 R3b 72 7 484.83 489.61 1.0 R4b 96 9 529.33 542.79 2.5 R5b 120 11 573.56 588.33 2.6 R6b 144 13 725.22 750.02 3.4 R7b 36 4 248.21 248.21 0 R8b 72 6 458.73 461.71 0.6 R9b 108 8 592.23 603.09 1.8 R10b 144 10 783.81 815.22 4.0

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Locations for the requests from one day as well as the three public transport stops are shown inFigure 5. Note that the majority of the requests lie within one of the two towns. This means that many requests have both pick-up and drop-oﬀ locations within the same town, automatically excluding them as candidates for being served as integrated trips (since theﬁxed route network only has one stop within each town). These requests still have to be planned as good as possible into the demand responsive vehicle routes.

The demand responsive vehicles are available in two different depots (one in the main town and one in the small town). Each vehicle can carry six passengers and has a maximum route time of 8 h. Each customer request has the load of one person and the service duration at each location (pick-up, drop-off and transfer) is 5 min. The travel times for demand responsive vehicles are based on the road network in the area. The travel times of the public transport are set according to a simplified version of the timetable of the regional train line and regional bus lines, which is slightly faster than the direct travel time of the demand responsive vehicles, with head-ways of 30 min.

Figure 5.Locations for theﬁrst day’s requests in the data set, as well as the three public transport stops. (The background map has been excluded, to maintain the anonymity of the special transport service passengers).

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The results of the computations, for both DARP and IDARP instances, using the evalu-ation metrics presented inFigure 2, Section 1, can be seen inTable 3(presenting metrics from the operator’s perspective) andTable 4(presenting metrics from the passenger’s per-spective). The values are averages of ten algorithm runs of each day.Table 3shows the distances (in km) driven by the demand responsive vehicles, the number of demand responsive vehicles used, as well as the demand responsive vehicle capacity utilization for both DARP and IDARP solutions. The capacity utilization is calculated as the quotient of the sum of total direct travel times for all requests divided by the total distance driven by the vehicles. The rightmost column shows the decrease in terms of distances driven for IDARP compared to DARP solutions (in %). Table 4 presents the average passenger travel time (in minutes). Note that in this table, the travel time for those requests which were selected for integration in the integrated case, and those which were not selected, are separated. This is done in both the DARP and IDARP cases, to make comparisons possible.

The results show that only a small proportion of all trips, between 2 and 15 (on average 8) out of 119 requests in the data set are integrated. All these requests have long travel distances, as could be expected, with pick-up and drop-oﬀ locations in two diﬀerent towns, or in the rural area between the towns but close to the public transport

Table 3.A comparison of the best found results for DARP and IDARP, respectively, on the same data set, from the operator’s perspective.

DARP IDARP Comparison

Total distance driven No. vehicles Capacity utilization Total distance driven No. vehicles Capacity utilization Reduction in driven distance (%) Week 1 Monday 794.2 16 0.71 679.7 16 0.83 14.42 Tuesday 814.4 11 0.71 721.4 11 0.65 11.42 Wednesday 864.0 16 0.82 723.6 16 0.66 16.25 Thursday 1306.7 19 0.80 1036.4 20 0.69 20.69 Friday 986.9 13 0.72 785.4 13 0.67 20.42 Week 2 Monday 1046.0 20 0.79 917.2 16 0.67 12.31 Tuesday 968.4 16 0.76 907.2 14 0.71 6.32 Wednesday 1018.98 15 0.80 859.2 14 0.74 15.68 Thursday 1014.76 14 0.79 807.96 14 0.72 20.38 Friday 1040.51 16 0.75 858.4 15 0.65 17.50

Table 4.A comparison of the best found results for DARP and IDARP, respectively, on the same data set, from the passenger’s perspective.

No. req. No. Int. req.

DARP IDARP Travel time (non-int.) Travel time (int.) Travel time (non-int.) Travel time (int.) Week 1 Monday 119 6 14.93 30.60 15.25 38.54 Tuesday 111 7 14.05 52.88 14.49 54.24 Wednesday 97 7 14.81 30.00 16.52 39.11 Thursday 145 15 16.83 37.69 18.35 42.30 Friday 111 7 17.38 58.83 16.99 59.50 Week 2 Monday 139 9 16.95 35.58 16.65 39.19 Tuesday 119 2 15.21 78.28 18.58 80.47 Wednesday 124 9 17.75 39.25 17.12 48.15 Thursday 110 7 18.84 55.74 17.22 58.00 Friday 117 11 16.04 39.90 15.60 48.38

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line. The travel times for these integrated requests increase (on average) by 5 min. This increase corresponds to the difference in travel time for these specific requests when being served as integrated requests in the IDARP compared to being served in the DARP (where all requests are non-integrated). It can be noted that for the remaining non-integrated requests, the average travel times show no significant difference. Even though that few requests are served as integrated requests, it still leads to (on average) a 16% reduction in travelled distance for the demand responsive vehicles. The smallest cost reduction for the service, i.e. decrease in objective function value, between the DARP solution and IDARP solution is 6.3%, while the largest is 20.7%. This indicates that while an improvement can be found for each day of the test case, the inter-day varia-bility is quite large. There is no clear correlation between the size of the specific instance (the number of requests per day) and the resulting decrease in distance driven. Also, no significant difference can be found between neither the number of vehicles used in the DARP and IDARP solutions nor the vehicle capacity utilisations.

5. Discussion and conclusions

We have proposed a metaheuristic and shown how it can be used e.g. by policy makers and public transport officials, to evaluate differences in efficiency and level-of-service between special transport services and integrated special transport services given specific design choices and demand characteristics. The algorithm has been applied to different test cases, and the results have given the following mainfindings:

. The proposed metaheuristic is ﬂexible and can be applied to problems of varying characteristics without changes to parameters.

. The algorithm produces reasonable solutions to both the 20 randomly generated instances in the benchmark data set and to the real-world instances.

. An integrated special transport service has the potential to be more eﬃcient in practice than a regular special transport service, in a rural setting, even though only a few requests are integrated.

. The travel times of the integrated requests increased on average by 5 min, while the travel times of non-integrated requests were not signiﬁcantly changed.

The results obtained from the real-world case, should not be seen as an estimation of what results to expect in other cases. The tests performed in this paper show the simplicity of the algorithm and that it can be used for this type of evaluations. When considering introducing an integrated special transport service in a new area, we therefore recommend performing this type of evaluations based on historical data from the speciﬁc region that is under consideration.

To find out general guidelines to where, when and how to operate an integrated service, future research should apply the proposed algorithm to larger test cases with different demand characteristics and service design choices. Results from such research can show if integrated dial-a-ride services are, and if so to what extent, in general more efficient than regular dial-a-ride services. In this context, a comparison between rural and urban demand distributions should also be included. Such an extensive study lay beyond the scope of the present paper. Studies to find general guidelines

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could also be conducted using more aggregate data, or even generated demand-data and networks.

An important consideration in many public transport applications is that of different, heterogenous, groups of customers. It is important to note that the results from the real-world test case presented in this paper show how much the distance travelled by the demand responsive vehicles could decrease in the case where the service rules allow and accept that all passengers can make use of the public transport system. As noted in Dick-erson et al. (2007), the group ‘older adults’ is heterogeneous – not all older adults have impaired mobility, and no single transport solution will work for all. A viable implemen-tation of an integrated special transport service should thus offer different levels-of-service to different segments of the passengers. Some passengers can make use of integrated trips, while some will require door-to-door transport without mode changes. This can, and should in future research, be incorporated into the proposed metaheuristic.

Future research should also consider new questions that arise when integrated demand responsive andfixed line public transport are offered to a general public, one such appli-cation is exemplified in Kalpakcı and Ünverdi (2016). They point out that while demand responsive services in the developed world are usually targeted at either specific segments of the population or intended to serve low-density areas, demand responsive transport ser-vices in developing countries aim at compensating for insufficiencies in the public trans-port supply. Thus, the goal of demand responsive services in developing countries is to integrate well with the public transport network, in order to shift demand from cars to public transport. If the demand responsive service is used, by the general public, to a degree where the changes in demand are large enough to affect the use, and perhaps design, of the public transport system, an interesting direction for future research is the adaptation of line planning and timetabling models, such as those discussed in Schöbel (2012) and Schmidt and Schöbel (2015).

When an integrated service is to be implemented in practice, the dynamic versions of the DARP and IDARP (as described in Section 2) should be solved, since a real-time oper-ational planning method is then required. Since the algorithm presented in this paper is used for evaluation of services, computational time has not been an important issue. However, if using the algorithm for replanning in a dynamic setting, this becomes impor-tant. The computational times on each of the benchmark instances varied between a minute or two on the smallest instances, and up to 15 min on the largest. The implemen-tation could have been speeded up by parallel computing, see for example Crainic (2015), by introducing more removal and insertion operators or by performing sensitivity analysis of the parameter setting (or by other means trim the parameter values).

Acknowledgements

This research was supported by the Swedish Governmental Agency for Innovation Systems (VINNOVA) and their assistance is gratefully acknowledged. The authors would like to thank Väst-traﬁk, responsible for public transport in western Sweden, for providing data and input to the study.

Disclosure statement

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Funding

This research was supported by the Swedish Governmental Agency for Innovation Systems (VINNOVA).

ORCID

M. Posada http://orcid.org/0000-0001-9008-6407

C. H. Häll http://orcid.org/0000-0001-6829-8219

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