Mutsunori Yagiura

Fast Algorithms to Enumerate All Common Intervals of Two Permutations

Abstract. Given two permutations of n elements, a pair of intervals of these permutations consisting of the same set of elements is called a commoninterval . Some genetic algorithms based on such common intervals have been proposed for sequencing problems and have exhibited good prospects. In this paper we propose three types of fast algorithms to enumerate all common intervals: (i) a simple O(n2) time algorithm (LHP), whose expected running time becomes O(n) for two randomly generated permutations, (ii) a practically fast O(n2) time algorithm (MNG) using the reverse Monge property, and (iii) an O(n+K) time algorithm (RC), where K

Effective Local Search Algorithms for Routing and Scheduling Problems with General Time-Window Constraints

(\leq {n \choose 2})

An Ejection Chain Approach for the Generalized Assignment Problem

is the number of common intervals. It will also be shown that the expected number of common intervals for two random permutations is O(1) . This result gives a reason for the phenomenon that the expected time complexity O(n) of the algorithm LHP is independent of K . Among the proposed algorithms, RC is most desirable from the theoretical point of view; however, it is quite complicated compared with LHP and MNG. Therefore, it is possible that RC is slower than the other two algorithms in some cases. For this reason, computational experiments for various types of problems with up to n=106 are conducted. The results indicate that (i) LHP and MNG are much faster than RC for two randomly generated permutations, and (ii) MNG is rather slower than LHP for random inputs; however, there are cases in which LHP requires Ω(n2) time, but MNG runs in o(n2) time and is faster than both LHP and RC.

On metaheuristic algorithms for combinatorial optimization problems

We propose local search algorithms for the vehicle routing problem with soft time-window constraints. The time-window constraint for each customer is treated as a penalty function, which is very general in the sense that it can be nonconvex and discontinuous as long as it is piecewise linear. In our algorithm, we use local search to assign customers to vehicles and to find orders of customers for vehicles to visit. Our algorithm employs an advanced neighborhood, called the cyclic-exchange neighborhood, in addition to standard neighborhoods for the vehicle routing problem. After fixing the order of customers for a vehicle to visit, we must determine the optimal start times of processing at customers so that the total penalty is minimized. We show that this problem can be efficiently solved by using dynamic programming, which is then incorporated in our algorithm. We report computational results for various benchmark instances of the vehicle routing problem. The generality of time-window constraints allows us to handle a wide variety of scheduling problems. As an example, we mention in this paper an application to a production scheduling problem with inventory cost, and report computational results for real-world instances.

An iterated local search algorithm for the time-dependent vehicle routing problem with time windows

We propose a tabu search algorithm for the generalized assignment problem, which is one of the representative combinatorial optimization problems known to be NP-hard. The algorithm features an ejection chain approach, which is embedded in a neighborhood construction to create more complex and powerful moves. We also incorporate an adaptive mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions. Computational results on benchmark instances of small sizes show that the method obtains solutions that are optimal or that deviate by at most 0.16% from the best known solutions. Comparisons with other approaches from the literature show that, for instances of larger sizes, our method obtains the best solutions among all heuristics tested.

One-dimensional cutting stock problem to minimize the number of different patterns

Metaheuristic algorithms are widely recognized as one of the most practical approaches for combinatorial optimization problems. Among representative metaheuristics are genetic algorithms, simulated annealing, tabu search, and so on. In this paper, we explain essential ideas used in such metaheuristic algorithms within a generalized framework of local search. We then conduct numerical experiments of metaheuristic algorithms using rather simple implementations, to observe general tendencies of their performance. From these results, we propose a few recommendations about the use of metaheuristics as simple optimization tools. We also mention some advanced techniques to enhance the ability of metaheuristics. Finally, we summarize some theoretical results on metaheuristic algorithms.

A path relinking approach with ejection chains for the generalized assignment problem

We generalize the standard vehicle routing problem with time windows by allowing both traveling times and traveling costs to be time-dependent functions. In our algorithm, we use a local search to determine routes of the vehicles. When we evaluate a neighborhood solution, we must compute an optimal time schedule for each route. We show that this subproblem can be efficiently solved by dynamic programming, which is incorporated in the local search algorithm. The neighborhood of our local search consists of slight modifications of the standard neighborhoods called 2- opt^*, cross exchange and Or-opt. We propose an algorithm that evaluates solutions in these neighborhoods more efficiently than the ones computing the dynamic programming from scratch by utilizing the information from the past dynamic programming recursion used to evaluate the current solution. We further propose a filtering method that restricts the search space in the neighborhoods to avoid many solutions having no prospect of improvement. We then develop an iterated local search algorithm that incorporates all the above ingredients. Finally we report computational results of our iterated local search algorithm compared against existing methods, and confirm the effectiveness of the restriction of the neighborhoods and the benefits of the proposed generalization.

The vehicle routing problem with flexible time windows and traveling times

Abstract As the cost associated with the change of cutting patterns become more important in recent industry, we consider 1D-CSP in which the number of different cutting patterns is constrained within a given bound. The proposed approach is based on metaheuristics, and incorporates an adaptive pattern generation technique. According to our computational experiments, it is observed that the proposed algorithm provides comparable solutions to other existing heuristic approaches for 1D-CSP.

Exact algorithms for the two-dimensional strip packing problem with and without rotations

The generalized assignment problem is a classical combinatorial optimization problem known to be NP-hard. It can model a variety of real world applications in location, allocation, machine assignment, and supply chains. The problem has been studied since the late 1960s, and computer codes for practical applications emerged in the early 1970s. We propose a new algorithm for this problem that proves to be more effective than previously existing methods. The algorithm features a path relinking approach, which is a mechanism for generating new solutions by combining two or more reference solutions. It also features an ejection chain approach, which is embedded in a neighborhood construction to create more complex and powerful moves. Computational comparisons on benchmark instances show that the method is not only effective in general, but is especially effective for types D and E instances, which are known to be very difficult.

An iterated local search algorithm based on nonlinear programming for the irregular strip packing problem

We generalize the standard vehicle routing problem by allowing soft time window and soft traveling time constraints, where both constraints are treated as cost functions. With the proposed generalization, the problem becomes very general. In our algorithm, we use local search to determine the routes of vehicles. After fixing the route of each vehicle, we must determine the optimal start times of services at visited customers. We show that this subproblem is NP-hard when cost functions are general, but can be efficiently solved with dynamic programming when traveling time cost functions are convex even if time window cost functions are non-convex. We deal with the latter situation in the developed iterated local search algorithm. Finally we report computational results on benchmark instances, and confirm the benefits of the proposed generalization.