Another heuristic for minimizing total average cycle stock subject to practical constraints

This paper considers the setting of reorder intervals of a population of items for minimizing the total average cycle stock subject to a limit on the total number of replenishments per unit time, and a restricted set of possible intervals. Silver and Moon have investigated the problem with the use of dynamic programming, and they also proposed a heuristic for solving it. This paper presents a new 0-1 linear programming approach to the problem. Based upon the solution of the relaxed 0-1 linear programming formulation, a simple heuristic is proposed to solve the reorder problem. Limited numerical results using realistic test examples indicate that the new heuristic performed very well for each example.     

A greedy look-ahead heuristic for the vehicle routing problem with time windows

In this paper we consider the problem of physically distributing finished goods from a central facility to geographically dispersed customers, which pose daily demands for items produced in the facility and act as sales points for consumers. The management of the facility is responsible for satisfying all demand, and promises deliveries to the customers within fixed time intervals that represent the earliest and latest times during the day that a delivery can take place. We formulate a comprehensive mathematical model to capture all aspects of the problem, and incorporate in the model all critical practical concerns such as vehicle capacity, delivery time intervals and all relevant costs. The model, which is a case of the vehicle routing problem with time windows, is solved using a new heuristic technique. Our solution method, which is based upon Atkinson's greedy look-ahead heuristic, enhances traditional vehicle routing approaches, and provides surprisingly good performance results with respect to a set of standard test problems from the literature. The approach is used to determine the vehicle fleet size and the daily route of each vehicle in an industrial example from the food industry. This actual problem, with approximately two thousand customers, is presented and solved by our heuristic, using an interface to a Geographical Information System to determine inter-customer and depot–customer distances. The results indicate that the method is well suited for determining the required number of vehicles and the delivery schedules on a daily basis, in real life applications.     

Estimation for nonhomogeneous Poisson processes from aggregated data

A well-known heuristic for estimating the rate function or cumulative rate function of a nonhomogeneous Poisson process assumes that the rate function is piecewise constant on a set of data-independent intervals. We investigate the asymptotic (as the amount of data grows) behavior of this estimator in the case of equal interval widths, and show that it can be transformed into a consistent estimator if the interval lengths shrink at an appropriate rate as the amount of data grows.     

On the recoverable robust traveling salesman problem

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal values. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared.     

Predetermined intervals for start times of activities in?the?stochastic project scheduling problem

This paper proposes a new methodology to schedule activities in projects with stochastic activity durations. The main idea is to determine for each activity an interval in which the activity is allowed to start its processing. Deviations from these intervals result in penalty costs. We employ the Cross-Entropy methodology to set the intervals so as to minimize the sum of the expected penalty costs. The paper describes the implementation of the method, compares its results to other heuristic methods and provides some insights towards actual applications.     

Minimizing the total completion time in a two-machine flowshop with sequence-independent setup times

We consider the problem of minimizing the sum of completion times in a two-machine permutation flowshop subject to setup times. We propose a new priority rule, several constructive heuristics, local search procedures, as well as an effective multiple crossover genetic algorithm. Computational experiments carried out on a large set of randomly generated instances provide evidence that a constructive heuristic based on newly derived priority rule dominates all the proposed constructive heuristics. More specifically, we show that one of our proposed constructive heuristics outperforms the best constructive heuristic in the literature in terms of both error and computational time. Furthermore, we show that one of our proposed local search-based heuristics outperforms the best local search heuristic in the literature in terms of again both error and computational time. We also show that, in terms of quality-to-CPU time ratio, the multiple crossover genetic algorithm performs consistently well.     

The open vehicle routing problem with time windows

In this paper, we consider the open vehicle routing problem with time windows (OVRPTW). The OVRPTW seeks to find a set of non-depot returning vehicle routes, for a fleet of capacitated vehicles, to satisfy customers’ requirements, within fixed time intervals that represent the earliest and latest times during the day that customers’ service can take place. We formulate a comprehensive mathematical model to capture all aspects of the problem, and incorporate in the model all critical practical concerns. The model is solved using a greedy look-ahead route construction heuristic algorithm, which utilizes time windows related information via composite customer selection and route-insertion criteria. These criteria exploit the interrelationships between customers, introduced by time windows, that dictate the sequence in which vehicles must visit customers. Computational results on a set of benchmark problems from the literature provide very good results and indicate the applicability of the methodology in real-life routing applications.     

ZI round,a MIP rounding heuristic

We introduce a new pure integer rounding heuristic, ZI Round, and compare this heuristic to recent extremely fast pure integer rounding heuristic Simple Rounding. Simple Rounding was introduced in the non-commercial code SCIP. ZI Round attempts to round each fractional variable while using row slacks to maintain primal feasibility. We use the MIPLIB 2003 library for the test set. The average time in our run per instance for both Simple Rounding and ZI Round was 0.8 milliseconds, but ZI Round found more feasible solutions with a the same or better objective value. Also the average time to solve the lp relaxation per instance was 2.2 seconds, so these two rounding heuristics are several magnitudes faster than other heuristics which must use the lp solver, including diving heuristics. We also show that ZI Round performs well on a set covering class and a railway crew scheduling class.     

A New Efficient Heuristic for the Minimum Set Covering Problem

We solve approximately the minimum set covering problem by developing a new heuristic, which is essentially based on the flow algorithm originally developed by Ford and Fulkerson. We perform a comparative study of the performances (concerning solution qualities and execution times) of the flow algorithm as well as of the natural greedy heuristic for set covering originally studied by Johnson and Lovász.     

Asymptotic Analysis of a Greedy Heuristic for the Multi-Period Single-Sourcing Problem: The Acyclic Case

The multi-period single-sourcing problem that we address in this paper can be used as a tactical tool for evaluating logistics network designs in a dynamic environment. In particular, our objective is to find an assignment of customers to facilities, as well as the location, timing and size of production and inventory levels, that minimizes total assignment, production, and inventory costs. We propose a greedy heuristic, and prove that this greedy heuristic is asymptotically optimal in a probabilistic sense for the subclass of problems where the assignment of customers to facilities is allowed to vary over time. In addition, we prove a similar result for the subclass of problems where each customer needs to be assigned to the same facility over the planning horizon, and where the demand for each customer exhibits the same seasonality pattern. We illustrate the behavior of the greedy heuristic, as well as some improvements where the greedy heuristic is used as the starting point of a local interchange procedure, on a set of randomly generated test problems. These results suggest that the greedy heuristic may be asymptotically optimal even for the cases that we were unable to analyze theoretically.     

The Orienteering Problem with Time Windows

The orienteering problem with time windows, denoted by OPTW, belongs to a class of routeing and scheduling problems that arise in physical distribution. It may be modelled as a problem on a graph. It considers a set of nodes (customers), each with an associated profit and service duration (time window), and a set of arcs, each with an associated travel time. The objective of the problem is to construct an acyclic path beginning at a specified origin and ending at a specified destination that maximizes the total profit while observing time window constraints on all nodes and not exceeding a designated time limit. The problem is classified as NP-hard and, thus, an exact algorithm that executes in reasonable computational time is unlikely to exist. Since the problem is highly-constrained, we were able to develop a heuristic (referred to as the ‘tree’ heuristic) based upon an exhaustive search of the feasible solution space. The tree heuristic systematically generates a list of feasible paths and then selects the most profitable path from the list. In comparison with an insertion heuristic, the tree heuristic was found to produce improved values of total profit for heavily-constrained, modest-sized problems with reasonable computational effort.     

A heuristic multi-start decomposition approach for optimal design of serial machining lines

We study an optimal design problem for serial machining lines. Such lines consist of a sequence of stations. At every station, the operations to manufacture a product are grouped into blocks. The operations within each block are performed simultaneously by the same spindle head and the blocks of the same station are executed sequentially. The inclusion and exclusion constraints for combining operations into blocks and stations as well as the precedence constraints on the set of operations are given. The problem is to group the operations into blocks and stations minimizing the total line cost. A feasible solution must respect the given cycle time and all given constraints. In this paper, a heuristic multi-start decomposition approach is proposed. It utilizes a decomposition of the initial problem into several sub-problems on the basis of a heuristic solution. Then each obtained sub-problem is solved by an exact algorithm. This procedure is repeated many times, each time it starts with a new heuristic solution. Computational tests show that the proposed approach outperforms simple heuristic algorithms for large-scale problems.     

A combined Lagrangian, linear programming, and implication heuristic for large-scale set partitioning problems

Given a finite ground set, a set of subsets, and costs on the subsets, the set partitioning problem is to find a minimum cost partition of the ground set. Many combinatorial optimization problems can be formulated as set partitioning problems. We present an approximation algorithm that produces high-quality solutions in an acceptable amount of computation time. The algorithm is iterative and combines problem size-reduction techniques, such as logical implications derived from feasibility and optimality conditions and reduced cost fixing, with a primal heuristic based on cost perturbations embedded in a Lagrangian dual framework, and cutting planes. Computational experiments illustrate the effectiveness of the approximation algorithm.     

Numerical Methods for Single Machine Scheduling with Non-linear Cost Functions to Minimize Total Cost

In this paper we are concerned with the problem of sequencing a given set of jobs without preemption on a single machine so as to minimize total cost, where associated with each job is a processing time and a differentiable cost function defined on the completion time of the job. The problem, in general, is NP-complete and, therefore, there is unlikely to be an algorithm to solve the problem in reasonable time, thus a heuristic algorithm is desirable. We present two heuristic algorithms to solve the problem. The first algorithm is based on the differential of the cost functions, and the second algorithm is based on the least square approximation of the cost functions. Computational experiences for the case of quadratic, cubic, and exponential cost functions are presented.     

Scheduling a single batch-processing machine with arbitrary job sizes and incompatible job families: An ant colony framework

This paper investigates the first attempt on the batch-processing machine scheduling problem, where the machine can process multiple jobs simultaneously, using an ant colony optimization metaheuristic. We consider the scheduling problem of a single batch-processing machine with incompatible job families and the performance measure of minimizing total weighted completion time. Jobs of a given family have an identical processing time and are characterized by arbitrary sizes and weights. Based on a number of developed heuristic approaches, we propose an ant colony framework (ACF) in two versions, which are distinguished by the type of embedded heuristic information. Each version is also investigated in two formats, that is the pure ACF and the hybridized ACF. To verify the performance of our framework, comparisons are made based on using a set of well-known existing heuristic and meta-heuristic algorithms taken from the literature, on a diverse set of artificially generated test problem instances. Computational results show the high performance of the proposed framework and signify its ability to outperform the comparator algorithms in most cases as the problem size increases.     

一种带时间窗和容量约束的车辆路线问题及其Tabu Search算法

本文提出一种带时间窗和容量约束的车辆路线问题（CVRPTW），并利用Tabu Search快速启式算法，针对Solomon提出的几个标准问题，快捷地得到了优良的数值结果。     

Scheduling multi-stage parallel-processor services to minimize average response time

The problem of scheduling on a multi-stage parallel-processor architecture in computer centres is addressed with the objective of minimizing average completion time of a set of requests. The problem is modelled as a flexible flowshop problem for which few heuristics exist in the flowshop scheduling literature. A new three-phase heuristic is proposed in this paper. An extensive computational experiment has been conducted to compare the performance of the existing heuristics and the proposed heuristic. The results indicate that the proposed heuristic significantly outperforms the existing ones. More specifically, the overall average error of the best existing heuristic is about five times that of the proposed heuristic while the overall average CPU time of the proposed heuristic is about half of the best existing one. More importantly, as the number of requests increases, the CPU time of the proposed heuristic decreases considerably (compared to the best existing heuristic) while the ratio of the error (of the best existing to the proposed heuristic) of about five times remains almost the same.     

Optimal techniques for class-dependent attribute discretization

Preprocessing of raw data has been shown to improve performance of knowledge discovery processes. Discretization of quantitative attributes is a key component of preprocessing and has the potential to greatly impact the efficiency of the process and the quality of its outcomes. In attribute discretization, the value domain of an attribute is partitioned into a finite set of intervals so that the attribute can be described using a small number of discrete representations. Discretization therefore involves two decisions, on the number of intervals and the placement of interval boundaries. Previous approaches for quantitative attribute discretization have used heuristic algorithms to identify partitions of the attribute value domain. Therefore, these approaches cannot be guaranteed to provide the optimal solution for the given discretization criterion and number of intervals. In this paper, we use linear programming (LP) methods to formulate the attribute discretization problem. The LP formulation allows the discretization criterion and the number of intervals to be integral considerations of the problem. We conduct experiments and identify optimal solutions for various discretization criteria and numbers of intervals.     

Heuristics for a project management problem with incompatibility and assignment costs

Consider a project which consists of a set of jobs to be performed, assuming each job has a duration of at most one time period. We assume that the project manager provides a set of possible durations (in time periods) for the whole project. When a job is assigned to a specific time period, an assignment cost is encountered. In addition, for some pairs of jobs, an incompatibility cost is encountered if they are performed at the same time period. Both types of cost depend on the duration of the whole project, which also has to be determined. The goal is to assign a time period to each job while minimizing the costs. We propose a tabu search heuristic, as well as an adaptive memory algorithm, and compare them with other heuristics on large instances, and with an exact method on small instances. Variations of the problems are also discussed     

Solving large-scale maximum expected covering location problems by genetic algorithms: A comparative study

This paper compares the performance of genetic algorithms (GAs) on large-scale maximum expected coverage problems to other heuristic approaches. We focus our attention on a particular formulation with a nonlinear objective function to be optimized over a convex set. The solutions obtained by the best genetic algorithm are compared to Daskin's heuristic and the optimal or best solutions obtained by solving the corresponding integer linear programming (ILP) problems. We show that at least one of the GAs yields optimal or near-optimal solutions in a reasonable amount of time.