Given two permutations of n elements, a pair of intervals of these permutations consisting of the same set of elements is called a common interval. 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(n 2 ) time algorithm (LHP), whose expected running time becomes O(n) for two randomly generated permutations, (ii) a practically fast O(n 2 ) time algorithm (MNG) using the reverse Monge property, and (iii) an O(n + K ) time algorithm (RC), where K (≤ n 2 ) 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 = 10 6 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 (n 2 ) time, but MNG runs in o(n 2 ) time and is faster than both LHP and RC.

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 automatic mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions. Computational comparisons on benchmark instances of small sizes show that the method obtain solutions that are optimal or that deviate by at most 0.16% from the best known solutions. For instances of larger sizes, our method obtains best solutions among all heuristics tested.

Abstract: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 non-convex 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. It 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 then 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 such 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.

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