Michael Bögl

The school bus routing and scheduling problem with transfers

In this article, we study the school bus routing and scheduling problem with transfers arising in the field of nonperiodic public transportation systems. It deals with the transportation of pupils from home to their school in the morning taking the possibility that pupils may change buses into account. Allowing transfers has several consequences. On the one hand, it allows more flexibility in the bus network structure and can, therefore, help to reduce operating costs. On the other hand, transfers have an impact on the service level: the perceived service quality is lower due to the existence of transfers; however, at the same time, user ride times may be reduced and, thus, transfers may also have a positive impact on service quality. The main objective is the minimization of the total operating costs. We develop a heuristic solution framework to solve this problem and compare it with two solution concepts that do not consider transfers. The impact of transfers on the service level in terms of time loss (or user ride time) and the number of transfers is analyzed. Our results show that allowing transfers reduces total operating costs significantly while average and maximum user ride times are comparable to solutions without transfers.

Computational study of neighborhood operator performance on the Traveling Salesman Problem with Time Windows in neighborhood search based frameworks (RTS, VNS)

In this work we analyze the performance of different neighborhood operators in terms of solution quality in different neighborhood search based frameworks, namely Reactive Tabu Search and Variable Neighborhood Search, on the Traveling Salesman Problem with Time Windows. We compare the impact of the two different search concepts on the solution quality by embedding the same operators and hence be able to precisely state the impact of the basic concept. Additionally, we analyze the improvement of those search strategies compared to a simple local search procedure. The neighborhood operators under consideration are 1-shift, nodeexchange, lexicographic 2-opt, lexicographic 3-opt and or-opt.

Two heuristic solution concepts for the vehicle selection problem in line haul transports

In this article we will develop a mathematical model for a cost-efficient selection of vehicles with varying capacities for line haul transports with leasing options. For this integer optimization problem, which is a variant of the generalized assignment problem known as NP-hard, we will compare two heuristic solution concepts and try to answer the question in which cases a user should choose an exact or approximate solution concept depending on different data instances of the problem.

Metaheuristics Based on Solution Construction

In section 3.1.1 we saw how randomization can make constructive search methods better than pure greedy methods, because the danger of being too eager is dampened a little. We called this approach randomized adaptive search. Now we want to explain a metaheuristic which is based on this idea. It is called Greedy Randomized Adaptive Search Procedure (GRASP, cf [70, 170]). As it is a constructive search method, it starts with an empty solution and adds solution elements to a partial solution until it is complete. It may be argued, this is exactly the way a greedy construction method works, but there are subtle differences, which are pointed out now. Greedy methods do not perform search, they construct a single solution in an iterative fashion by evaluating all remaining solution elements and according to their performance add them to the partial solution. Elements are added as long as the solution is improved. If this is not the case anymore, the construction is stopped and the final solution is returned. Hence greedy methods do not perform search.

Metaheuristics in Vehicle Routing

The metaheuristics previously explained are now applied to a transportation problem. In modern economy goods are sourced worldwide an transportation and the organization of transportation has gained more and more attention. Transportation not only causes cost but also increases traffic, environment pollution, noise, and inefficient transportation organization may cause a decrease of the service level. There are plenty of reasons to do optimization of vehicle routes. Transportation optimization is a frequently studied problem in the context of logistics. It not only arises in collection and distribution of goods but also in service applications like taxi driving, package delivery, school bus routing, repairman routing, and so on (cf. [74]).

Metaheuristics in Machine Scheduling

Shop-based machine scheduling involves finding a sequence in which a set of work orders or jobs is processed on each of multiple machines. In this context, the order of production steps, also referred to as the process routing of each job has to be considered. In the job shop case, process routings are not required to be identical like in flow lines, hence they may be predefined individually for each job. In scientific literature, a production step is commonly referred to as an operation and the process routings are specified by so called precedence constraints.

Metaheuristics Based on Solution Modification

Section 3.2 introduced the principle of repeatedly modifying solutions such as to (finally) obtain better ones. In the following we will reconsider this principle in two ways: On the one hand we bring this principle in line with scientific literature, precisely speaking, with the field of local search methods. The reason for this is that the metaheuristics described in the subsequent Sections 6.2, 6.3 and 6.4 are commonly assigned to this class of methods. However, it has to be pointed out that the principle of search by solution modification as presented in this book is not necessarily limited to local search only. Rather is it generic enough to consider other metaheuristics in this context, as shown for example in Section 8.2. On the other hand we identify typical patterns and processing schemes of local search methods, though on a rather abstract level by concentrating on the most general aspects.

The Knapsack Problem and Straightforward Optimization Methods

In the previous chapter we gave some examples for optimization problems in the application area of production and logistics. Recall the cargo-loading problem we described at last which consists in choosing an optimal subset of available products for shipping. In the theory of optimization this task is categorized under a special class of problems, called packing problems. Precisely speaking, we are facing a subclass of packing problems, called knapsack problems. The basic idea of optimally packing items into a single object, i.e. a knapsack in the simplest case, serves as an abstract model for a broad spectrum of packing, loading, cutting, capital budgeting or even scheduling problems. In order to provide a general basis for the subsequent chapters, we will first introduce an example knapsack optimization problem and then discuss various different approaches to solve it.

Summarizing Remarks on Metaheuristics

In the preceding chapters we have discussed metaheuristics from a general point of view, concentrating on the meta-level, the associated concepts and their relation to the lower, problem-specific level. This chapter has two main goals: First it should give an overall summarizing view on the characteristics of the metaheuristic methods covered in this book. This is accomplished by finalizing the systematization approach stated in the introduction of Chapter 3. Second, as we only described the most common algorithms, we briefly sketch further selected metaheuristic approaches and relate them to our classification.

Metaheuristics in General

In Chapter 3, we pointed out different ways of conducting a heuristic search within a solution space based on the knapsack problem. We also pointed out that the basic principles of search heuristics which we identified, are generic, meaning that they can be applied to a whole variety of different optimization problems. Likewise, we can consider the strategies which build upon the basic principles as problem-independent. The actual realization of the basic principle itself, however, is a problem-specific issue, since it has to be defined how a solution may be constructed, modified or recombined. Figure 4.1 gives an illustration of these interrelationships.