Shinji Imahori

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

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.

The vehicle routing problem with flexible time windows and traveling times

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.

An iterated local search algorithm for the vehicle routing problem with convex time penalty functions

We propose an iterated local search algorithm for the vehicle routing problem with time window constraints. We treat the time window constraint for each customer as a penalty function, and assume that it is convex and piecewise linear. Given an order of customers each vehicle to visit, dynamic programming (DP) is used to determine the optimal start time to serve the customers so that the total time penalty is minimized. This DP algorithm is then incorporated in the iterated local search algorithm to efficiently evaluate solutions in various neighborhoods. The amortized time complexity of evaluating a solution in the neighborhoods is a logarithmic order of the input size (i.e., the total number of linear pieces that define the penalty functions). Computational comparisons on benchmark instances with up to 1000 customers show that the proposed method is quite effective, especially for large instances.

MARA: Maximum Alternative Routing Algorithm

In hop-by-hop networks, provision of multipath routes for all nodes can improve fault tolerance and performance. In this paper we study the multipath route calculation by constructing a directed acyclic graph (DAG) which includes all edges in the network. We define new DAG construction problems with the objectives of 1) maximizing the minimum connectivity, 2) maximizing the minimum max-flow, and 3) max- imizing the minimum max-flow as an extension of shortest path routing. A family of new algorithms called Maximum Alternative Routing Algorithms (MARAs) is described, proven formally to solve the problems optimally, and contrasted with existing multipath algorithms. MARAs are evaluated for the number of paths, the length of paths, the computational complexity, and the computation time, using simulations based on several real Internet Autonomous System (AS) network topologies. We show that MARAs run in sub-second times on moderate-speed processors and achieve a significant increase in the number of paths compared to existing multipath routing algorithms. These results should help further the process of deploying multipath routing in real-world networks.

Improved local search algorithms for the rectangle packing problem with general spatial costs

The rectangle packing problem with general spatial costs is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. This problem is very general, and various types of packing problems and scheduling problems can be formulated in this form. For this problem, we have previously presented local search algorithms using a pair of permutations of rectangles to represent a solution. In this paper, we propose speed-up techniques to evaluate solutions in various neighborhoods. Computational results for the rectangle packing problem and a real-world scheduling problem exhibit good prospects of the proposed techniques.

Local Search Algorithms for the Rectangle Packing Problem with General Spatial Costs

Abstract.We propose local search algorithms for the rectangle packing problem to minimize a general spatial cost associated with the locations of rectangles. The problem is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. Each rectangle has a set of modes, where each mode specifies the width and height of the rectangle and its spatial cost function. The spatial costs are general piecewise linear functions which can be non-convex and discontinuous. Various types of packing problems and scheduling problems can be formulated in this form. To represent a solution of this problem, a pair of permutations of n rectangles is used to determine their horizontal and vertical partial orders, respectively. We show that, under the constraint specified by such a pair of permutations, optimal locations of the rectangles can be efficiently determined by using dynamic programming. The search for finding good pairs of permutations is conducted by local search and metaheuristic algorithms. We report computational results on various implementations using different neighborhoods, and compare their performance. We also compare our algorithms with other existing heuristic algorithms for the rectangle packing problem and scheduling problem. These computational results exhibit good prospects of the proposed algorithms.

The best-fit heuristic for the rectangular strip packing problem: An efficient implementation and the worst-case approximation ratio

We investigate the best-fit heuristic algorithm by Burke et al. [2004. A new placement heuristic for the orthogonal stock-cutting problem. Operations Research 52, 655-671] for the rectangular strip packing problem. For its simplicity and good performance, the best-fit heuristic has become one of the most significant algorithms for the rectangular strip packing. In this paper, we propose an efficient implementation of the best-fit heuristic that requires O(n) space and O(nlogn) time, where n is the number of rectangles. We prove that this complexity is optimal, and we also show the practical usefulness of our implementation via computational experiments. Furthermore, we give the worst-case approximation ratio of the best-fit heuristic.

Solving the irregular strip packing problem via guided local search for overlap minimization

The irregular strip-packing problem (ISP) requires a given set of non-convex polygons to be placed without overlap within a rectangular container having a fixed width and a variable length, which is to be minimized. As a core sub-problem to solve ISP, we consider an overlap minimization problem (OMP) whose objective is to place all polygons into a container with given width and length so that the total amount of overlap between polygons is made as small as possible. We propose to use directional penetration depths to measure the amount of overlap between a pair of polygons, and present an efficient algorithm to find a position with the minimum overlap for each polygon when it is translated in a specified direction. Based on this, we develop a local search algorithm for OMP that translates a polygon in horizontal and vertical directions alternately. Then we incorporate it in our algorithm for OMP, which is a variant of the guided local search algorithm. Computational results show that our algorithm improves the best-known values of some well-known benchmark instances.

Recent progress of local search in handling the time window constraints of the vehicle routing problem

Vehicle routing and scheduling problems have a wide range of applications and have been intensively studied in the past half century. The condition that enforces each vehicle to start service at each customer in the period specified by the customer is called the time window constraint. This paper reviews recent results on how to handle hard and soft time window constraints, putting emphasis on its different definitions and algorithms. With these diverse time windows, the problem becomes applicable to a wide range of real-world problems.

Success guaranteed routing in almost Delaunay planar nets for wireless sensor communication

This paper proposes a routing strategy for wireless sensor networks that is valid even when the sensor nodes are distributed non-uniformly. Energy efficiency is the most important consideration in wireless sensor networks; hence, global communication should be achieved by combinations of local messages. We first propose a new underlying graph called the almost Delaunay planar net, which can be calculated from information local to the sensor nodes. We statistically analyse the efficiency of existing routing strategies in the almost Delaunay planar net. We also propose a new routing strategy that always guarantees the reachability on planar graphs, including the almost Delaunay planar net. We show that the proposed routing strategy exhibits good performance if the underlying graph in the sensor network is the almost Delaunay planar net.