Combining Metaheuristics with Mixed Integer Linear Programming 23

It has also become very popular to combine metaheuristics with DP, LP or MIP to take advantage of both methodologies. Raidl and Puchinger [121] gave a good overview about the techniques and how they can be combined.

It is often useful to embed (meta)heuristics within an exact approach to provide the exact approach with primal bounds which can also help to keep the branch-and-bound tree smaller if the metaheuristic can provide good primal bounds to the exact approach.

Obviously, it is, however, also a trade-off because performing metaheuristic methods at every node of a branch-and-bound tree is also very time consuming. Thus, clever decisions have to be made, whether a metaheuristic should be called in a particular branch-and-bound node or not. Alternatively, if suitable also faster and/or simpler

heuristics can be called more often. In a decomposition-based exact approach like, e.g., branch-and-cut, it can also be useful to use metaheuristics to find cutsets faster and even larger cutsets which may speed up the computation of the MIP, see, e.g., Raidl et al. [123].

On the other hand, for metaheuristics, it can also be useful to guide them by relaxations of LPs or MIPs. Sometimes it can be useful to solve the LP-relaxation of a problem, and then, trying to repair infeasible solutions for providing good initial solutions to metaheuristics. Solutions providing dual bounds from, e.g., the LP-relaxation of a problem can also be exploited by metaheuristics, utilizing the primal-dual relationships.

Another possibility, which is sometimes also referred as fix-and-optimizestrategy [156], is to fix some variables and let the other variables free, so that they can be optimized by a MIP solver like CPLEX or Gurobi. This is often done within a(very) large neighborhood search(VLSN). Ahuja et al. [1] provided a survey on very-large scale neighborhood search techniques.

2.5.3 Multistage Approaches

Multistage approaches are particularly useful for (very) large real-world problems. They decompose the problem such that the original problem may not be solved at once, but only a smaller problem, or the problem can even be divided into a master and a subproblem.

In the first part of this section, we describe thecluster-first route-second methodology originally proposed by Fisher and Jaikumar [50]. This is particularly useful for vehicle routing problems (VRP), when at first the given items, which need to be visited are assigned to a particular vehicle, and in a second step for each vehicle a routing problem is solved separately. The second metaheuristic we describe, is the multilevel refinement strategy, initially proposed for combinatorial optimization by Walshaw [158], where the original, usually large, problem is coarsened until a problem size is reached which can be easily solved. After that, the problem is iteratively extended and refined (possibly by local search such as a VND) until the lowest level is reached and a solution to the original problem is retrieved.

Cluster-First Route-Second

Cluster-first route-second is a decomposition-based metaheuristic and was first proposed by Fisher and Jaikumar [50] in 1981. The basic idea is to divide the problem into a master problem and a subproblem, where the master problem is associated with the clustering and the subproblem is an independent problem, which is much easier to solve.

The union of the solutions from all subproblems corresponds to the solution of the overall problem. Note, that, e.g., for the vehicle routing problem with maximum-tour constraint an estimation of the tour length of each cluster has to be integrated into the master problem. The success of this approach is also dependent on the quality of this estimation, which results in a better assignment for the subproblem. The subproblem is usually easy to compute respectively solve.

2.5. Hybrids Algorithm 2.8: Multilevel refinement

1: l←0

2: while P_l is too large to be reasonably considered in a direct waydo

3: Pl+1 ←coarsen(Pl)

4: l←l+ 1

5: end while

6: xl{P_l} ←initialize(Pl)

7: while l >0 do

8: l←l−1

9: x⁰_l{P_l}=extend(xl+1{P_l+1}, P_l)

10: x_l{P_l}=refine(x⁰_l{P_l}, P_l)

11: end while

It is also possible to do this the other way round, which would result in the route-first cluster-second methodology, which has been shown by Prins [115], to be also a fruitful approach for VRPs. This way, a giant tour is constructed, and then, by applying splitting algorithms, multiple tours are created, such that e.g., in the VRP, each vehicle has its own route.

Multilevel Refinement

The multilevel refinement paradigm is a metaheuristic which, e.g., can be used to improve results of existing approaches. It was proposed by Walshaw [158, 159] and it was shown to be able to improve results of existing approaches. This is an intuitive paradigm as it tries to make the problem size smaller and first solve a small variant of the instance and then iteratively extend solutions to obtain a solution to the initial problem instance.

After each extension step a possible refinement can also be made, like, e.g., VND (see Section 2.3.3). The process of making the initial problem size smaller is also denoted as coarsening.

The procedure is shown in pseudo-code in Algorithm 2.8. The problem is coarsened until an instance size is reached which can be easily solved, possibly by exact methods like LP or MIP. After the coarsening is finished, the solution x_l is initialized at the highest respectively coarsest level. Then, the solution is iteratively extended to a lower level l−1 and optionally refined at the level l−1. If levell= 0 is reached, a solution to original problem instance is found. Crucial decisions in this metaheuristic are the criterion when to stop coarsening and how to most accurately coarsen the instance, as usually information is lost while coarsening. Keeping as much information as possible in the upper levels of the coarsening is crucial for the success of the algorithm.

CHAPTER 3

Balancing Bike-Sharing Systems

In this chapter, the work on the Balancing Bike Sharing Systems (BBSS) problem is presented. Essentially, we show three approaches. We present the static BBSS problem which was published in the Journal of Global Optimization [125]. Subsequently, the dynamic BBSS is discussed where we have been nominated as best paper candidate at the14th European Conference on Evolutionary Computation in Combinatorial Optimi-sation [86]. In the end, we introduce a simplified problem formulation, where we first proposed a cluster-first route-second heuristic (see Section 2.5.3), published at Interna-tional Conference on Computer Aided Systems Theory [87] which is then extended to an approach based on logic-based Benders decomposition (see Section 2.2.5) which yields optimal solutions to practical relevant problem instances for which the input data is also based on real-world data. This approach has been published inNetworks [83].

3.1 Introduction

Public bike sharing systems (PBSs) provide a modern way of shared public transport within cities. These systems, most frequently, consist of rental stations distributed in parts of a city. In state-of-the-art PBSs every station has a self-service computer terminal authenticating the customers, and ideally also used to allow instant registration for new clients. Customers have to authenticate and provide a payment method to reduce theft and vandalism. Rental stations consist ofslots which can either be empty or occupied by a bike. These slots are connected to the whole computer system allowing the operators as well as the customers to have an overview of the status of each station. If there is at least one slot occupied by a bike, customers have the opportunity to rent a bike via the terminal, and if there is at least one slot free, customers may return a bike by putting it into the free slot. To work well, a PBS has to have a reasonable density of stations in the covered region. Users can rent bikes at any station and return them at any other station. An overview of the structure of a PBS is shown in Figure 3.1. Note that user

demand can be satisfied via different stations. For instance, the demand between o1

and t₁ is satisfied by utilizing two distinct bike-sharing stations. A PBS should not be confused with classical bike rental as both have different use cases, client bases and revenue models. The major differences are that in PBS short-term usage is promoted whereas in bike rental longer rental times are not unusual, PBSs are distributed over a larger area, whereas bike rentals are more stationary with bikes usually to be returned at the same place where they have been rent [139].

PBSs are mostly implemented in public-private partnership and are financed through advertisements on the bikes, subsidies from the municipalities, and subscription fees from the users. The costs for building and operating the system have to be covered. The problem of building or extending a PBS can in principle be seen as a facility location or hub location problem with network design aspects [91] and more detailed information about it can be found in Section 4.

For continuous operation of the system, besides maintaining the bikes and stations, providers in particular have to take care ofrebalancing bikes among the stations such that users can rent and return bikes at any station with high probability. Stations should ideally neither run full nor empty, as these situations obviously significantly impact customer satisfaction.

Different approaches to achieve and maintain a reasonable balance exist. Most commonly, the PBS operator actively rebalances the stations by employing vehicles with trailers that pickup bikes at stations with excess of bikes and deliver them to stations with a lack of bikes. This is the scenario that will be considered within this work, but there are also alternative approaches in which balance should be achieved by the users themselves [53, 111]. There, the operator provides incentives for their customers to rent bikes at stations with excess and to return them at stations with a lack of bikes. These incentives can be reduced subscription fees, prizes or discounts at special partners of the PBS. Both rebalancing strategies can also be used in conjunction.

The active rebalancing of a PBS by a vehicle fleet has in the literature been referred to as acapacitated single commodity split pickup and delivery vehicle routing problem with multiple visits [125]. Diverse variants of this problem, with different objectives and constraints, have already been considered, and different algorithmic approaches have been proposed, ranging from mixed integer linear programming (MIP) methods to metaheuristics and hybrids. To our knowledge, all these approaches allow for an arbitrary number of bikes to be picked up at some stations and delivered to other stations, just limited by the vehicles’ and stations’ capacities. Observations in practice, however, indicate that in a larger well-working bike sharing system it makes rarely sense to move only few bikes for rebalancing. Drivers actually almost always pickup a full vehicle load and deliver it completely to another station. Many stations even require several visits with full load pickups or deliveries. Due to budgetary reasons, typically only just enough drivers and vehicles are employed to achieve a reasonable balance most of the time, but basically never an ideal one where single bikes play a substantial role. Drivers should use their limited working time in a best way to optimize the PBS’s overall balance as far as

3.2. Related Work