Random search in the one-dimensional cutting stock problem

The paper reports on a randomized approach to the one-dimensional cutting stock problem. In a sequential heuristic the next pattern to be processed is chosen according to the best outcome of 200 trials of random bin packing. The quality of the solutions is discussed and the problem of multiple occurrence of equivalent solutions with different patterns is explained by the cross over technique known from genetic algorithms. For one data instance with 27 demand lengths, reported by Haessler (1975), four different optimal solutions are given.     

Approximating the single source unsplittable min-cost flow problem

In the single source unsplittable min-cost flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs; the demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity. Moreover, the total cost must not exceed a given budget. This problem has been introduced by Kleinberg [7] and generalizes several NP-complete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling, load balancing, and virtual-circuit routing. Kolliopoulos and Stein [9] and Dinitz, Garg, and Goemans [4] developed algorithms improving the first approximation results of Kleinberg for the problem of minimizing the violation of edge capacities and for other variants. However, known techniques do not seem to be capable of providing solutions without also violating the cost constraint. We give the first approximation results with hard cost constraints. Moreover, all our results dominate the best known bicriteria approximations. Finally, we provide results on the hardness of approximation for several variants of the problem. Received: August 23, 2000 / Accepted: April 20, 2001?Published online October 2, 2001     

Routing and wavelength assignment in optical networks using bin packing based algorithms

This paper addresses the problem of routing and wavelength assignment (RWA) of static lightpath requests in wavelength routed optical networks. The objective is to minimize the number of wavelengths used. This problem has been shown to be NP-complete and several heuristic algorithms have been developed to solve it. We suggest very efficient, yet simple, heuristic algorithms for the RWA problem developed by applying classical bin packing algorithms. The heuristics were tested on a series of large random networks and compared with an efficient existing algorithm for the same problem. Results indicate that the proposed algorithms yield solutions significantly superior in quality, not only with respect to the number of wavelength used, but also with respect to the physical length of the established lightpaths. Comparison with lower bounds shows that the proposed heuristics obtain optimal or near optimal solutions in many cases.     

Space defragmentation for packing problems

One of main difficulties of multi-dimensional packing problems is the fragmentation of free space into several unusable small parts after a few items are packed. This study proposes a defragmentation technique to combine the fragmented space into a continuous usable space, which potentially allows the packing of additional items. We illustrate the effectiveness of this technique using the two- and three-dimensional bin packing problem, where the aim is to load all given items (represented by rectangular boxes) into the minimum number of identical bins. Experimental results based on well-known 2D and 3D bin packing data sets show that our defragmentation technique alone is able to produce solutions approaching the quality of considerably more complex meta-heuristic approaches for the problem. In conjunction with a bin shuffling strategy for incremental improvement, our resultant algorithm outperforms all leading meta-heuristic approaches based on the commonly used benchmark data by a significant margin.     

Packing problems and project scheduling models: an integrating perspective

This paper considers two problem classes, namely packing and project scheduling problems that are important to researchers as well as practitioners. The two problem categories are described, and a classification is given for the different kinds of packing problems and project scheduling concepts. While both problem classes are different with respect to their fields of application, similarities of their mathematical structures are examined. It is shown that all packing problems considered here are special cases of models for project scheduling. The aim is to indicate which project scheduling models can be used to capture the different types of packing problems. Finally, some implications for research on optimisation algorithms for these two problem classes are discussed, and the applicability of the results of this work in practice are pointed out.     

On the computational complexity of continuous routing

We study the problem of routing and broadcasting messages in a network, in which messages are generated at processors at arbitrary times and each message must reach its destination by a specific deadline. We present distributed and global routing algorithms for some restricted continuous routing problems on arrays of processors. We show that distributed algorithms are unlikely to exist in more general situations by giving an NP-hardness proof for their corresponding feasibility problem; i.e., the problem of determining whether all messages can be routed without violating the constraints of the network. We also present a distributed algorithm for the continuous broadcasting problem.     

Hop Constrained Network Design Problem with Partial Survivability

This paper studies the hop constrained network design problem with partial survivability, namely, given an undirected network, a set of point-to-point demands (commodities), and transmission link costs, identify two node disjoint paths for each demand (commodity) to minimize the total costs subject to the constraints that each demand is routed and traverses at most a specified number of links (or hops) on both the paths.A mathematical programming formulation of the problem is presented and an efficient solution procedure based on the linear programming relaxation is developed. Extensive computational results across a number of networks are reported. These results indicate that the solution procedure is effective for a wide range of problem sizes.     

A heuristic for solving large bin packing problems in two and three dimensions

The more-dimensional bin packing problem (BPP) considered here requires packing a set of rectangular-shaped items into a minimum number of identical rectangular-shaped bins. All items may be rotated and the guillotine cut constraint has to be respected. A straightforward heuristic is presented that is based on a method for the container loading problem following a wall-building approach and on a method for the one-dimensional BPP. 1,800 new benchmark instances are introduced for the two-dimensional and three-dimensional BPP. The instances include more than 1,500 items on average. Applied to these very large instances, the heuristic generates solutions of acceptable quality in short computation times. Moreover, the influence of different instance parameters on the solution quality is investigated by an extended computational study.     

A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing

The three-dimensional bin packing problem consists of packing a set of boxes into the minimum number of bins. In this paper we propose a new GRASP algorithm for solving three-dimensional bin packing problems which can also be directly applied to the two-dimensional case. The constructive phase is based on a maximal-space heuristic developed for the container loading problem. In the improvement phase, several new moves are designed and combined in a VND structure. The resulting hybrid GRASP/VND algorithm is simple and quite fast and the extensive computational results on test instances from the literature show that the quality of the solutions is equal to or better than that obtained by the best existing heuristic procedures.     

The selection and scheduling of telecommunication calls with time windows

We consider the bandwidth scheduling problem that consists of selecting and scheduling calls from a list of available calls to be routed on a bandwidth-capacitated telecommunication network in order to maximize profit. Each accepted call should be routed within a permissible scheduling time window for a required duration. To author's knowledge, this study represents the first work on bandwidth scheduling with time windows. We present an integer programming formulation of the problem. We also propose a solution procedure based on the well-established Lagrangean relaxation technique. The results of extensive computational experiments over a wide range of problem structures indicate that the procedure is both efficient and effective.     

Using different response-time requirements to smooth time-varying demand for service

Many service systems have demand that varies significantly by time of day, making it costly to provide sufficient capacity to be able to respond very quickly to each service request. Fortunately, however, different service requests often have very different response-time requirements. Some service requests may need immediate response, while others can tolerate substantial delays. Thus it is often possible to smooth demand by partitioning the service requests into separate priority classes according to their response-time requirements. Classes with more stringent performance requirements are given higher priority for service. Lower capacity may be required if lower-priority-class demand can be met during off-peak periods. We show how the priority classes can be defined and the resulting required fixed capacity can be determined, directly accounting for the time-dependent behavior. For this purpose, we exploit relatively simple analytical models, in particular, M_t/G/∞ and deterministic offered-load models. The analysis also provides an estimate of the capacity savings that can be obtained from partitioning time-varying demand into priority classes.     

A hybrid inexact-stochastic water management model

An inexact-stochastic water management (ISWM) model is proposed and applied to a case study of water quality management within an agricultural system. The model is based on an inexact chance-constrained programming (ICCP) method, which improves upon the existing inexact and stochastic programming approaches by allowing both distribution information in B and uncertainties in A and C to be effectively incorporated within its optimization process. In its solution process, the ICCP model (under a given pi level) is first transformed into two deterministic submodels, which correspond to the upper and lower bounds for the desired objective function value. This transformation process is based on an interactive algorithm, which is different from normal interval analysis or best/worst case analysis. Interval solutions, which are feasible and stable in the given decision space, can then be obtained by solving the two submodels sequentially. Thus, decision alternatives can be generated by adjusting decision variable values within their solution intervals. The obtained ICCP solutions are also useful for decision makers to obtain insight regarding tradeoffs between environmental and economic objectives and between increased certainties and decreased safeties (or increased risks). Results of the case study indicate that useful solutions for the planning of agricultural activities in the water quality management system have been obtained. A number of decision alternatives have been generated and analyzed based on projected applicable conditions. Generally, some alternatives can be considered when water quality objective is given priority, while the others may provide compromises between environmental and economic considerations. The above alternatives represent various options between environmental and economic tradeoffs. Willingness to accept low agricultural income will guarantee meeting the water quality objectives. A strong desire to acquire high agricultural income will run into the risk of violating water quality constraints.     

Minimizing total weighted late work in the resource-constrained project scheduling problem

This paper deals with resource-constrained project scheduling problem under the weighted late work criterion. Late work objective functions estimate the quality of a schedule based on durations of late parts of activities, not taking into account the amount of delay for fully late activities. It is assume that a project contains activities interrelated by finish-to-start type precedence relations with time lag of zero, which require one or more constrained renewable resources. The objective is to schedule each activity such that the total weighted late work is minimized. The problem has been formulated using a linear integer programming model and solved by the CPLEX. Also, a set of priority rules have been designed to quickly generate a set of initial solutions. In order to solve the problem optimally, a depth-first branch-and-bound algorithm is applied based on idea of minimal delaying alternatives. The branching order of nodes that belong to the same level of the search tree is determined on the basis of the developed priority rules. This results in generation six different versions of the branch-and-bound algorithm. Computational results on randomly generated problem sets are provided to analyze the efficiency of the priority rules and the branch-and-bound algorithm.     

一种基于区间型联系数的多指标决策方法

首先定义了区间数贴近度,并讨论了相应的性质;然后在联系数的基础上定义了区间型联系数及运算法则,并给出了区间型决策矩阵的规范化公式,提出了一种基于区间型联系数相对贴近度的决策方案排序方法.运用该方法进行多指标决策和评价,概念清楚、涵义明确,决策和评价结果准确、可信、不具有主观随意性.最后通过实例分析验证了该方法的有效性和...     

The time constrained maximal covering salesman problem

We introduce the time constrained maximal covering salesman problem (TCMCSP) which is the generalization of the covering salesman and orienting problems. In this problem, we are given a set of vertices including a central depot, customer and facility vertices where each facility can supply the demand of some customers within its pre-determined coverage distance. Starting from the depot, the goal is to maximize the total number of covered customers by constructing a length constrained Hamiltonian cycle over a subset of facilities. We propose several mathematical programming models for the studied problem followed by a heuristic algorithm. The developed algorithm takes advantage of different procedures including swap, deletion, extraction-insertion and perturbation. Finally, an integer linear programming based improvement technique is designed to try to improve the quality of the solutions. Extensive computational experiments on a set of randomly generated instances indicate the effectiveness of the algorithm.     

Reducing the bandwidth of a sparse matrix with a genetic algorithm

Abstract

The matrix bandwidth minimization problem (MBMP) consists in finding a permutation of the lines and columns of a given sparse matrix in order to keep the non-zero elements in a band that is as close as possible to the main diagonal. Equivalently in terms of graph theory, MBMP is defined as the problem of finding a labelling of the vertices of a given graph G such that its bandwidth is minimized. In this paper, we propose an improved genetic algorithm (GA)-based heuristic for solving the matrix bandwidth minimization problem, motivated by its robustness and efficiency in a wide area of optimization problems. Extensively computational results are reported for an often used set of benchmark instances. The obtained results on the different instances investigated show improvement of the quality of the solutions and demonstrate the efficiency of our GA compared to the existing methods in the literature.     

Mixed integer quadratically-constrained programming model to solve the irregular strip packing problem with continuous rotations

The irregular strip packing problem consists of cutting a set of convex and non-convex two-dimensional polygonal pieces from a board with a fixed height and infinite length. Owing to the importance of this problem, a large number of mathematical models and solution methods have been proposed. However, only few papers consider that the pieces can be rotated at any angle in order to reduce the board length used. Furthermore, the solution methods proposed in the literature are mostly heuristic. This paper proposes a novel mixed integer quadratically-constrained programming model for the irregular strip packing problem considering continuous rotations for the pieces. In the model, the pieces are allocated on the board using a reference point and its allocation is given by the translation and rotation of the pieces. To reduce the number of symmetric solutions for the model, sets of symmetry-breaking constraints are proposed. Computational experiments were performed on the model with and without symmetry-breaking constraints, showing that symmetry elimination improves the quality of solutions found by the solution methods. Tests were performed with instances from the literature. For two instances, it was possible to compare the solutions with a previous model from the literature and show that the proposed model is able to obtain numerically accurate solutions in competitive computational times.     

Mean delay analysis for a message priority-based polling scheme

We consider a multi-access communication channel such as a centrally-controlled polling system, a distributed token-based ring, or a bus network. A message priority-based polling procedure is used to control the access to the channel. This procedure requires the server to have no advance information concerning the number of messages resident at a station prior to its visit to the station. Messages arriving at each station belong to one of two priority classes: class-1 (high priority) and class-2 (low priority). Class-1 messages are served under an exhaustive service discipline, while class-2 messages are served under a limited service discipline. Class-1 messages have non-preemptive priority over class-2 messages resident at the same station. Using a fully symmetric system model, an exact expression for the sum of the mean waiting times of class-1 and class-2 messages is first derived. Upper and lower bounds for the mean message waiting times for each individual message class are then obtained.This work was supported by NFS Grant No. NCR-8914690, Pacific-Bell and MICRO Grant No. 90-135 and US West Contract No. D890701.     

Discrete-Time Geo X /G/1 Queue with Preemptive Repeat Different Priority

This paper considers a discrete-time Geo ^X /G/1 queue accepting two classes of messages with preemptive repeat different priority. Service times of messages of each priority class are i.i.d. according to a general discrete distribution function that may differ between two classes. The completion time and the stability condition for our system are investigated. By using the supplementary variable method and the generating function technique, we derive the joint system contents distributions at various observation instants and also compute the probability distribution for the unfinished work.     

A Lagrangean relaxation and decomposition algorithm for the video placement and routing problem

Video on demand (VoD) is a technology used to provide a number of programs to a number of users on request. In developing a VoD system, a fundamental problem is load balancing, which is further characterized by optimally placing videos to a number of predefined servers and routing the user program requests to available resources. In this paper, an exact solution algorithm is described to solve the video placement and routing problem. The algorithm is based on Lagrangean relaxation and decomposition. The novelty of the approach can be described as the use of integer programs to obtain feasible solutions in the algorithm. Computational experimentation reveals that for randomly generated problems with up to 100 nodes and 250 videos, the use of such integer programs help greatly in obtaining good quality solutions (typically within 5% of the optimal solution), even in the very early iterations of the algorithm.