Performance Guarantees of Local Search for Multiprocessor Scheduling

Increasing interest has recently been shown in analyzing the worst-case behavior of local search algorithms. In particular, the quality of local optima and the time needed to find the local optima by the simplest form of local search has been studied. This paper deals with worst-case performance of local search algorithms for makespan minimization on parallel machines. We analyze the quality of the local optima obtained by iterative improvement over the jump, swap, multi-exchange, and the newly defined push neighborhoods. Finally, for the jump neighborhood we provide bounds on the number of local search steps required to find a local optimum.operations research and management science;

A Bicriteria Simulated Annealing Algorithm for Scheduling Jobs on Parallel Machines with Sequence Dependent Setup Times

The study considers the scheduling problem of identical parallel machines subject to minimization of the maximum completion time and the maximum tardiness expressed in a linear convex objective function. The maximum completion time or makespan is the date when the last job to be completed leaves the system. The maximum tardiness is indicated by the job that is completed with the longest delay relative its due date. Minimizing both criteria can help assuring a high utilization of the production system as well as a high level of service towards the client. Due to the complexity of the problem, a Simulated Annealing (SA) heuristic has been implemented to be able to obtain an efficient solution in a reasonable running time. A set of n jobs is assigned, to one of the m identical parallel machines. Each job is processed in only one operation before its completion after which it leaves the system. Constraints, such as due dates for each job and setup times for the machines, are considered. The resolution procedure consists of two phases and begins with an initial solution generator. Then a SA heuristic is applied for further improvement of the solution. 4 generators are used to create an initial solution and 3 to generate neighbour solutions. To test and verify the performance of the proposed resolution procedure, a computational experimentation has been realized on a set of test problems generated ad-hoc

Order fulfillment in online retailing : what goes where

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2005.Includes bibliographical references (p. 139-146).We present three problems motivated by order fulfillment in online retailing. First, we focus on one warehouse or fulfillment center. To optimize the storage space and labor, an e-tailer splits the warehouse into two regions with different storage densities. One is for picking customer orders and the other to hold a reserve stock that replenishes the picking area. Consequently, the warehouse is a two-stage serial system. We investigate an inventory system where demand is stochastic by minimizing the total expected inventory- related costs subject to a space constraint. We develop an approximate model for a periodic review, nested ordering policy. Furthermore, we extend the formulation to account for shipping delays and advance order information. We report on tests of the model with data from a major e-tailer. Second, we focus on the entire network of warehouses and customers. When a customer order occurs, the e-tailer assigns the order to one or more of its warehouses and/or drop- shippers, so as to minimize procurement and transportation costs, based on the available current information. However, this assignment is necessarily myopic as it cannot account for any subsequent customer orders or future inventory replenishments.(cont.) We examine the benefits from periodically re-evaluating these real-time assignments. We construct near- optimal heuristics for the re-assignment for a large set of customer orders by minimizing the total number of shipments. Finally, we present saving opportunities by testing the heuristics on order data from a major e-tailer. Third, we focus on the inventory allocation among warehouses for low-demand SKUs. A large e-tailer strategically stocks inventory for SKUs with low demand. The motivations are to provide a wide range of selections and faster customer fulfillment service. We assume the e-tailer has the technological capability to manage and control the inventory globally: all warehouses act as one to serve the global demand simultaneously. The e-tailer will utilize its entire inventory, regardless of location, to serve demand. Given we stock certain units of system inventory, we allocate inventory to warehouses by minimizing outbound transportation costs. We analyze a few simple cases and present a methodology for more general problems.by Ping Josephine Xu.Ph.D

A Bicriteria Simulated Annealing Algorithm for Scheduling Jobs on Parallel Machines with Sequence Dependent Setup Times

The study considers the scheduling problem of identical parallel machines subject to minimization of the maximum completion time and the maximum tardiness expressed in a linear convex objective function. The maximum completion time or makespan is the date when the last job to be completed leaves the system. The maximum tardiness is indicated by the job that is completed with the longest delay relative its due date. Minimizing both criteria can help assuring a high utilization of the production system as well as a high level of service towards the client. Due to the complexity of the problem, a Simulated Annealing (SA) heuristic has been implemented to be able to obtain an efficient solution in a reasonable running time. A set of n jobs is assigned, to one of the m identical parallel machines. Each job is processed in only one operation before its completion after which it leaves the system. Constraints, such as due dates for each job and setup times for the machines, are considered. The resolution procedure consists of two phases and begins with an initial solution generator. Then a SA heuristic is applied for further improvement of the solution. 4 generators are used to create an initial solution and 3 to generate neighbour solutions. To test and verify the performance of the proposed resolution procedure, a computational experimentation has been realized on a set of test problems generated ad-hoc

Vehicle routing with multi-dimensional loading constraints

Zwei der wichtigsten Problemstellungen in der Transportlogistik behandeln einerseits das Verladen von Produkten auf LKWs und andererseits die ressourceneffiziente Belieferung der Kunden auf dem gegebenen Straßennetz. Bis dato wurden diese zwei Probleme mit Hilfe von kombinatorischer Optimierung getrennt von einander behandelt und es existieren zahlreiche Publikationen zu beiden Themen in den einschlägigen Fachzeitschriften. Erst in den letzten drei Jahren wurde einem integrierten Ansatz, der beide Problemstellungen zu einem Optimierungsproblem vereint betrachtet. Somit werden die Bestellungen einzelner Kunden nicht bloß über ihre Gewichte, sondern auch über ihre Abmessungen definiert. Der klare Vorteil dieses Ansatzes liegt darin, dass die einzelnen LKW Routen auch tatsächlich so gefahren werden können, da die tatsächliche Beladung auch berücksichtigt wurde. Andererseits steigt die kombinatorische Komplexität drastisch, weil das kapazitierte Vehicle Routing Problem (CVRP) mit Bin Packing Problemen (BPP) kombiniert wird und beide Probleme für sich alleine NP schwer sind. Diese Dissertation präsentiert drei verschiedene Probleme, die sich neben der Frage welches Fahrzeug beliefert welchen Kunden auch der Frage widmet, wie die bestellten Produkte auf den LKW geladen werden können. - Das Multi-Pile Vehicle Routing Problem (MP-VRP) bindet in das klassische CVRP eine Beladekomponente ein, die zwischen eindimensionalem und zweidimensionalem Bin Packing Problem angesiedelt ist. Die Problemstellungen wurden durch einen österreichischen Holzzulieferer motiviert. - Beim kapazitierten Vehicle Routing Problem mit zweidimensionalen Beladenebenbedingungen (2L-CVRP) bestellt jeder Kunden rechteckige Objekte, welche auf der rechteckigen Beladefläche des LKWs untergebracht werden müssen. - Das allgemeinste Beladeproblem stellt das dreidimensionale Bin Packing Problem dar. Hier bestellt jeder Kunde dreidimensionale Objekte, welche auf dem dreidimensionalen Laderaum des LKWs untergebracht werden müssen. Das klassische dreidimensionale Bin Packing Problem wird durch zusätzliche Beladenebenbedingungen erweitert. Momentan gibt es zu diesen kombinierten Problemen nur wenige Publikationen. Exakte Ansätze gibt es momentan nur zwei, einen für das MP-VRP (hier können Probleme bis zu 50 Kunden gelöst werden) und für das 2L-CVRP (hier können Probleme bis zu 30 Kunden exakt gelöst werden). Für Realweltanwendungen müssen jedoch Heuristiken gefunden werden, welche größere Probleminstanzen lösen können. In dieser Arbeit wird für alle drei Problemstellungen ein Ameisenalgorithmus verwendet und mit bestehenden Lösungsansätzen aus dem Bereich der Tabu-Suche (TS) verglichen. Es wird gezeigt, dass der präsentierte Ameisenansatz für die zur Verfügung stehenden Benchmarkinstanzen die besten Ergebnisse liefert. Darüber hinaus wird der Einfluss verschiedener Beladenebenbedingungen auf die Lösungsgüte untersucht, was eine wichtige Entscheidungsgrundlage für Unternehmen darstellt.Two of the most important problems in distribution logistics concern the loading of the freight into the vehicles, and the successive routing of the vehicles along the road network, with the aim of satisfying the demands of the clients. In the combinatorial optimization field, these two loading and routing problems have been studied intensively but separately yielding a large number of publications either for routing or packing problems. Only in recent years some attention has been brought to their combined optimization. The obvious advantage is that, by considering the information on the freight to be loaded, one can construct more appropriate routes for the vehicles. The counterpart is that the combinatorial difficulty of the problem increases consistently. One must not forget that both the vehicle routing problem and the bin packing problem are NP hard problems! This thesis presents three different problems concerning the combination of routing and loading (packing) problems. - The Multi-Pile Vehicle Routing Problem (MP-VRP) incorporates an interesting loading problem situated between one dimensional and two dimensional bin packing. This problem has been inspired by a real world application of an Austrian wood distributing company. - The Capacitated Vehicle Routing Problem with Two-Dimensional Loading Constraints (2L-CVRP) augments the classical Capacitated Vehicle Routing Problem by requiring the satisfaction of general two dimensional loading constraints. This means that customers order items represented by rectangles that have to be feasibly placed on the rectangular shaped loading surface of the used vehicles. - The most general packing problem to be integrated is the Three Dimensional Bin Packing Problem (3DBPP) resulting in the Capacitated Vehicle Routing Problem with Three-Dimensional Loading Constraints (3L-CVRP). Here the order of each customer consists of cuboid shaped items that have to be feasibly placed on the loading space of the vehicle. A feasible placement is influenced by additional constraints that extend the classical 3DBPP. Concerning the literature solving these problems with exact methods it becomes clear that this is only possible to some very limited extent (e.g.: the MP-VRP can be solved up to 50, the 2L-CVRP can be solved exact up to 30 customers, for the 3L-CVRP no exact approach exists). Nevertheless for real world applications the problem instances are much larger which justifies the use of (meta-)heuristics. The rank-based Ant System was modified and extended to solve the combined problem by integrating different packing routines. The designed methods outperform the existing techniques for the three different problem classes. The influence of different loading constraints on the objective value is investigated/is intensively studied to support the decision makers of companies facing similar problems

Identical parallel machine scheduling problems: structural patterns, bounding techniques and solution procedures

The work is about fundamental parallel machine scheduling problems which occur in manufacturing systems where a set of jobs with individual processing times has to be assigned to a set of machines with respect to several workload objective functions like makespan minimization, machine covering or workload balancing. In the first chapter of the work an up-to-date survey on the most relevant literature for these problems is given, since the last review dealing with these problems has been published almost 20 years ago. We also give an insight into the relevant literature contributed by the Artificial Intelligence community, where the problem is known as number partitioning. The core of the work is a universally valid characterization of optimal makespan and machine-covering solutions where schedules are evaluated independently from the processing times of the jobs. Based on these novel structural insights we derive several strong dominance criteria. Implemented in a branch-and-bound algorithm these criteria have proved to be effective in limiting the solution space, particularly in the case of small ratios of the number of jobs to the number of machines. Further, we provide a counter-example to a central result by Ho et al. (2009) who proved that a schedule which minimizes the normalized sum of squared workload deviations is necessarily a makespan-optimal one. We explain why their proof is incorrect and present computational results revealing the difference between workload balancing and makespan minimization. The last chapter of the work is about the minimum cardinality bin covering problem which is a dual problem of machine-covering with respect to bounding techniques. We discuss reduction criteria, derive several lower bound arguments and propose construction heuristics as well as a subset sum-based improvement algorithm. Moreover, we present a tailored branch-and-bound method which is able to solve instances with up to 20 bins