In this work we introduced new MILP formulations for solving the NDPR that utilize an exponential number of variables (and constraints). We proved several conditions and properties of optimal solutions and revised the concept of communication graphs for the NDPR to state our MILP formulations. Two BP&C algorithms have been developed. The first one is based on a multi-commodity flow formulation on an undirected communication graph whereas the second is based on a cut-set formulation on a directed communication graph. The computational study on instances from the literature and a newly created set of instances shows that the cut-set formulation on the directed communication graph has the overall best performance. The multi-commodity flow formulation on the undirected communication graph performs reasonably well, but only up to a limited number of nodes and/or commodities. Due to its excessive number of variables, the latter formulation exhibits serious memory issues that makes it less appealing for practical applications involving larger graphs or a higher number of commodities.
(a) 40N30C50L80F A (original) (b) 40N30C50L80F A (K0)
Figure 3.12: Solutions to ARLP instance 40N30C50L80F_A with different numbers of commodities. Solid lines indicate free edges and dashed lines augmenting edges. Selected relays are marked with triangles. On the right the single source is marked with a square.
We conducted a sensitivity analysis on the Cabral instances with differentdmaxrestrictions.
The results showed that our algorithms are quite robust against changes to this parameter featuring only a comparatively small increase in computation times and priced columns.
Compared to the previous attempt from the literature to develop an exact model for the NDPR by [30], the main advantage of our modeling approach is that we do not use variables corresponding to entire walks between commodities (including the decisions for placing relays). Instead, we consider only simple paths between two consecutive relays and model the decision where to place the relays through the communication graph.
That way, our pricing draws a computational advantage from the fact that augmenting edges can be simultaneously shared by multiple commodities. The positive effect of our modeling approach is striking for instances with many commodities, where the need for the simultaneous reuse of augmenting edges is even more amplified.
3.6.1 Future Work
Besides telecommunication network design, the NDPR can be used to answer strategic questions in the context of electric mobility. A company running its operations based on a fleet of electric vehicles (EVs), be it a logistics company or an e-car provider, faces a difficult decision problem of planning the underlying charging infrastructure. Due to expensive EV batteries and their limited range, a stable and robust charging infrastructure is crucial for running the business suitably. Since building and/or renting charging stations is expensive, logistic companies or e-car providers are interested in minimizing the number
of charging stations whilst enabling travel between specific locations (cf. commodities).
Furthermore, in some metropolises (including Stockholm, Gothenburg, and Singapore) congestion taxing or congestion charging mechanisms are implemented. This means that shorter distances can be traversed (e.g., via shortcuts through the inner city), but in that case a certain road toll is to be paid(see, e.g., [58, 104]). Similarly, urban freeways passing through a downtown area can be subject to compulsory electronic toll (like the case in Santiago de Chile, or some Norwegian cities). Consequently, if a company is interested in building charging stations for its fleet, it also has to gauge whether toll roads are to be used. When making strategic decisions, costs for toll roads are typically estimated over a longer planning period and considered as fixed link costs.
Two main strategic design questions arising in this complex decision process are addressed by the NDPR: Given a family of origin-destination pairs EVs need to travel, and given the existing links that can be traversed:
1. What are the optimal locations for placing the charging stations and how many of them are needed?
2. Could the available infrastructure be enhanced by including additional links (short-cuts), to reduce the travel distances?
The relevance of the NDPR for planning EVs’ charging infrastructure has not been sufficiently acknowledged in the existing literature. This is maybe due to the additional aspects that need to be taken into account when dealing with e-mobility. These include restricting the distance arising from detours necessary for vehicle recharging and the maximum number of (time-consuming) recharging stops an EV requires before reaching its final destination. Although the NDPR does not consider these additional aspects, there is no doubt that the problem plays an important role for e-mobility applications for two reasons: (1) the NDPR may appear as a subproblem (i.e., in some decomposition schemes), and (2) the proposed algorithms can be used to derive heuristic solutions in multi-phase approaches, where the complex decision process is approached step-by-step.
Hence, the NDPR provides important insights for the companies running their business with a fleet of EVs. It helps in estimating the initial set-up costs (induced by the installation of recharging stations and potential purchases of road-toll passes). Moreover, by using a correlation between the edge lengths and lengths of the trips, the routing decisions obtained through an NDPR solution implicitly help in estimating a lower bound on the number of required EVs. Similarly, assuming that all trips will be covered, an upper bound on the expected profit can be calculated.
Interesting and more difficult NDPR variants that are important directions for future work include the following aspects: (1) limiting the maximum number of recharging stops (relays) used by a single commodity, (2) limiting the maximum waiting times imposed by
recharging, or (3) limiting the overall trip length per commodity.
CHAPTER 4
Solving a Selective Dial-a-Ride Problem with Logic-based Benders Decomposition
In the previous chapter we employed column generation as solution approach. While being used for decades and belonging to the most well-known decomposition techniques for mixed integer linear programming (MILP) it is still very successful and achieves state-of-the-art results for various problems. The approach considered in this chapter also originates from a well-established technique, Benders decomposition (BD), which has been first proposed in the 1960s. BD received only moderate attention for several years before enjoying a renaissance since the 2000s, see [144]. During this time also an interesting extension, logic-based Benders decomposition (LBBD), was developed that will be considered in the following.
We propose an LBBD approach for solving a selective dial-a-ride problem (DARP). This problem considers transportation requests of people from pick-up to drop-off locations.
Users specify time windows with respect to these points. Requests are served by a given vehicle fleet with limited capacity and maximum tour duration per vehicle. Moreover, user inconvenience considerations are taken into account by limiting the travel time between origin and destination for each request. In contrast to previous work we do not focus on travel cost minimization under the assumption that all requests can be satisfied but rather consider maximization of the number of served requests with a given vehicle fleet. This appears to be particularly relevant for funded systems that have to cope with overallocation.
In our study we in particular investigate the impact of strengthening the Benders cuts.
We compare plain Benders cuts to heuristically strengthened ones, as well as two variants of theoretically strongest cuts. Moreover, we consider heuristic boosting techniques as
well as valid inequalities to speed up solving the Benders master problem. The models are implemented as LBBD and also as Branch-and-Check (BaC) algorithms and empirically compared on a diverse set of benchmark instances.
This chapter has been published in Computers & Operations Research:
M. Riedler and G. R. Raidl. Solving a selective dial-a-ride problem with logic-based Benders decomposition. Computers & Operations Research, 96:
30–54, 2018
4.1 Introduction
The DARP considers the design of vehicle routes for a set of customers who specify transportation requests from origin (pick-up) to destination (drop-off) points. Users typically impose time windows with respect to these locations. To reduce user inconve-nience the time required to go from the pick-up to the drop-off location (ride time) is limited. The available requests shall be served by a fleet of vehicles. Each vehicle has a limited capacity corresponding to the number of customers that can be transported and a maximum total travel time. The restriction on the tour duration is important in order to deal with regulations regarding driver shifts.
As done by Jaw et al. [95], Cordeau [41], and others we distinguish between outbound and inbound requests. An outbound request considers the case that a customer wants to go from some starting location to a destination. An inbound request corresponds to the opposite case, i.e., a customer who wants to return to his/her starting location.
According to the survey presented in [134] customers have different priorities with respect to the adherence to time windows. For outbound requests it is critical to stay within the time window at the drop-off location and for inbound requests it is important to adhere to the time window at the pick-up location.
In the literature several variants of the DARP have been investigated, see [42, 44, 135].
The two main variants are the static and the dynamic case. In the former it is assumed that all requests are known in advance whereas in the latter requests become known gradually over time and routes need to be adjusted accordingly. There are also mixed variants for which some requests are known in advance and some are revealed dynamically.
Moreover, there is a distinction between the single- and the multi-vehicle case. In the former variant the requests have to be served using a single vehicle and in the latter multiple vehicles are available. In the following we deal with a variant of the static multi-vehicle DARP.
4.1.1 Outline and Discussion of the Contributions
In many DARP applications it is assumed that all requests can be served and that the total travel expenses together with the user inconvenience have to be minimized. In contrast, we consider the scenario that in general not all customers can be handled with
the given fixed-size vehicle fleet and aim at maximizing the number of served requests.
This is intended to deal with situations in which dial-a-ride systems are overallocated.
In these cases serving as many customers as possible appears to be more relevant than savings due to shorter tour lengths. Of course user inconvenience considerations still have to be taken into account to provide reasonable service conditions.
We consider solution algorithms based on LBBD (see Hooker and Ottosson [91]) and BaC (see Thorsteinsson [161]). The Benders master problem focuses on the selection of requests and their assignment to vehicles. It is modeled as integer linear programming (ILP) that we enhance through valid inequalities originating from subproblem relaxations.
For solving the Benders subproblems, which correspond to the route planning tasks, we consider MILP as well as constraint programming (CP) approaches. In particular, we also present a hybrid approach that combines MILP and CP. Several strategies for constructing Benders cuts are studied. We consider cuts derived from greedily obtained minimal infeasible request subsets, the full set of all minimal infeasible request subsets, as well as the set of all minimum cardinality infeasible request subsets and compare them to the unrefined cuts that are directly obtained from the subproblem assignments.
Moreover, we consider heuristic boosting techniques to possibly speed up the solution process. To this end we terminate the master problem prematurely according to a specific termination criterion and use the suboptimal solution to derive Benders cuts. As soon as no further cuts can be obtained this way, we fall back to solving the master problem to optimality and continue with regular Benders iterations. This is necessary to obtain a provably optimal solution. As termination criterion we consider a decreasing sequence of thresholds for the optimality gap and an increasing sequence of time limits. Employing an adaptive approach we start at the first element of the sequence and move to the next one whenever no further cuts can be found with the current termination condition. A more flexible approach allows traversing the sequence in both directions, depending on whether cuts could be obtained or not. The suggested algorithms are tested extensively on a novel set of benchmark instances as well as on instances from the literature.
The remainder of the chapter is organized as follows. We first provide an overview of previous work in the area. Then, we give a formal definition of the specific problem variant, including a complexity discussion. In terms of the formal specification we provide a compact reference model that is a straightforward extension of the MILP from [41]
for the tour-length-minimization DARP. In the main part we present the details of our decomposition approaches; including important implementation details. Finally, we discuss computational results on various test instances and conclude with an outlook on future research directions.
4.1.2 Previous Work
The DARP has a rather long research history. Among the first was the work by Psaraftis [138] that deals with the static single-vehicle variant. Sexton and Bodin [158, 159] solve the problem by splitting it into a routing and a scheduling phase which they formally describe in the context of BD. The routing is done by an insertion heuristic. In [24] the
same authors use this approach to tackle the multi-vehicle case by first forming clusters of requests and then solving the single-vehicle problem for each cluster. Since they construct the clusters (grouping close customers) as well as the routes heuristically, neither method can guarantee optimal solutions. Later on, this approach for the multi-vehicle problem has been refined by using so-called “mini-clusters”, see [53, 55]. The most recent contribution by Ioachim et al. [92] relying on this technique shows the positive influence of using mathematical optimization methods to globally define the set of “mini-clusters”. The authors argue that more sophisticated techniques provide a significant advantage over simpler heuristic approaches. However, all of these algorithms are still heuristics.
Only few contributions so far do not minimize traveling costs. Wolfler Calvo and Colorni [171] maximize the number of served customers and consider a penalty term regarding user inconvenience. This term considers the relative ratio between the direct and the actual travel time. The authors consider a fast heuristic construction approach based on an auxiliary graph.
Berbeglia et al. [15] and Häme and Hakula [86] focus on feasibility checking of DARP instances. Similarly, also the large neighborhood search by Jain and van Hentenryck [93]
has been tested as feasibility checking algorithm. Although we consider an optimization problem here, we are still concerned with feasibility checking when it comes to the Benders subproblems.
For a broader overview on the DARP we refer to the surveys by Cordeau and Laporte [44, 42] and Parragh et al. [135].
An optimization problem closely related to the DARP is the pickup and delivery problem with time windows (PDPTW). The main difference between the two problems is that the PDPTW primarily deals with the transportation of goods rather than persons. As a consequence, it does not consider user inconvenience and related concerns. In this area branch-price-and-cut (BP&C) approaches have been shown to be able to provide state-of-the-art results in terms of exact solution approaches, see Ropke and Cordeau [151] and Baldacci et al. [8]. For further details consider the survey conducted in [135].
Recently, also revenue maximizing variants of the PDPTW have been considered. In Qiu and Feuerriegel [140] and Qiu et al. [141] each transportation request is assigned a profit.
The goal is then to identify a subset of requests to be served with a given heterogeneous vehicle fleet that maximizes the revenue, i.e., sum of profits minus transportation cost.
The problem is solved using a graph search algorithm as well as a set packing formulation for the case of a homogeneous vehicle fleet. A similar scenario is also considered in Gansterer et al. [65] and solved with different metaheuristic approaches.
Somewhat related are also certain variants of the team orienteering problem. A contribu-tion in this respect is from Baklagis et al. [7] who solve a variant considering pick-up and delivery with a branch-and-price (B&P) approach.
Finally, we want to review contributions that are relevant to our work from the method-ological point of view, i.e., works that apply (logic-based) BD in the context of vehicle
routing problems. Cire and Hooker [38] consider the home health care problem in which medical services need to be provided to patients. Each service is represented as a job and requires a certain minimal qualification level. The services are provided by nurses that travel to the patients. The aim is to design routes and shift plans such that all required services can be provided while minimizing the costs for the nurses’ working hours. The problem is solved using LBBD. In the master problem the jobs are assigned to the nurses and the subproblems determine the actual shift plan and route per nurse.
After solving a subproblem a cut is introduced into the master problem reflecting the cost of the assignment or prohibiting an infeasible allocation. In case of an infeasible subproblem it is often possible to strengthen the obtained cut by identifying a subset of assigned jobs that is the cause of the infeasibility. Moreover, a local search procedure is employed that tries to repair infeasible solutions by reassigning jobs to other nurses.
The authors solve the master problem only heuristically and therefore optimal solutions cannot be guaranteed. In the computational study the LBBD approach is compared to a CP model which it outperforms clearly.
The bi-level vehicle routing problem (VRP) considers the distribution of goods in two stages. The goods are first transported from the main depot to satellite depots. Starting at each satellite depot the goods are then brought to the customers. This kind of VRP arises for example in newspaper distribution. Raidl et al. [145, 146] consider a bi-level VRP with a global restriction on the time until which all customers need to receive their goods. The assignment of customers to the satellite depots is pre-specified. Deliveries are carried out with a homogeneous fleet of vehicles with restricted capacity. The goal is to perform all deliveries within the time limit at minimal routing cost. Due to the structure of the problem routing costs at the first level as well as for every satellite depot can be considered independently. However, the levels are still interlinked via the global time limit. These properties provide a promising basis for the application of LBBD. Raidl et al.
[145, 146] consider a decomposition approach in which the master problem determines the route from the main depot to the satellite depots. With the now fixed starting times at the satellite depots the corresponding routes can be computed independently as subproblems. Infeasibilities (due to the global time limit) are prevented by computing a minimal starting time for each satellite depot that guarantees the existence of a feasible route. Hence, only Benders optimality cuts are required. These cuts turn out to be quite strong here since routing costs can only be reduced given a smaller starting time at the respective depot. Raidl et al. [145] consider an exact variant of this decomposition, as well as a hybrid approach with either the master or the subproblems solved via metaheuristics, and a completely heuristic approach. In Raidl et al. [146] the hybrid approach is further refined by verifying and, if needed, correcting the heuristically added Benders cuts in a
[145, 146] consider a decomposition approach in which the master problem determines the route from the main depot to the satellite depots. With the now fixed starting times at the satellite depots the corresponding routes can be computed independently as subproblems. Infeasibilities (due to the global time limit) are prevented by computing a minimal starting time for each satellite depot that guarantees the existence of a feasible route. Hence, only Benders optimality cuts are required. These cuts turn out to be quite strong here since routing costs can only be reduced given a smaller starting time at the respective depot. Raidl et al. [145] consider an exact variant of this decomposition, as well as a hybrid approach with either the master or the subproblems solved via metaheuristics, and a completely heuristic approach. In Raidl et al. [146] the hybrid approach is further refined by verifying and, if needed, correcting the heuristically added Benders cuts in a