Workforce scheduling with multiple objectives

In workforce scheduling, the optimal schedule has traditionally been determined by minimizing the cost of labor subject to an acceptable service level, which is defined as the percentage of customers served within a predetermined time interval. We propose an alternative multidimensional paradigm, where cost minimization and service level maximization are considered simultaneously, together with other, complementary criteria. The ultimate goal of the proposed approach is to open a broader workforce scheduling paradigm that incorporates service quality into the analysis and provides the possibility to study the interaction between cost and service quality. Furthermore, the approach enables us to avoid strong assumptions. An example with real-world, empirical demand data is provided.     

Mutual exclusion scheduling with interval graphs or related classes: Complexity and algorithms

This note summarizes the main results presented in the author's Ph.D. thesis, supervised by Professor Michel Van Caneghem and defended on 14th June 2005 at University of Aix-Marseille II, France. The thesis, written in French, is available at http: //www.lif-sud.univ-mrs.fr/Rapports/25-2005.html. The mutual exclusion scheduling problem has an elegant graph-theoretic formulation: given an undirected graph G and an integer k, find a minimum coloring of G such that each color appears at most k times. When G is an interval graph, this problem has some applications in workforce planning. Then, the object of the thesis is to study the complexity of mutual exclusion scheduling problem for interval graphs and related classes. Received: August 2005 / Revised version: September 2005 Frédéric Gardi: On leave from Laboratoire d'Informatique Fondamentale - CNRS UMR 6166, Parc Scientifique et Technologique de Luminy, Marseille, France.     

Fast approximation algorithms for bi-criteria scheduling with machine assignment costs

We consider parallel machine scheduling problems where the processing of the jobs on the machines involves two types of objectives. The first type is one of two classical objective functions in scheduling theory: either the total completion time or the makespan. The second type involves an actual cost associated with the processing of a specific job on a given machine; each job-machine combination may have a different cost. Two bi-criteria scheduling problems are considered: (1) minimize the maximum machine cost subject to the total completion time being at its minimum, and (2) minimize the total machine cost subject to the makespan being at its minimum. Since both problems are strongly NP-hard, we propose fast heuristics and establish their worst-case performance bounds.     

An approximation algorithm for scheduling two parallel machines with capacity constraints

We consider the problem of scheduling n independent jobs on two identical parallel machines, with a limit on the number of jobs that can be assigned to each single machine, so as to minimize the total weighted completion time of the jobs. We study a semidefinite programming-based approximation algorithm for solving this problem and prove that the algorithm has a worst case ratio at most 1.1626.     

Generalized p-Center problems: Complexity results and approximation algorithms

In an earlier paper, two alternative p-Center problems, where the centers serving costumers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected, were shown to be NP-complete. This paper considers a generalized version of these problems, in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range. We refer to these problems as the ‘Set’ p-Center problems. We establish that the triangle inequality (Δ-inequality) versions of these problems, in which the edge weights are assumed to satisfy the triangle inequality, are also NP-complete. We also provide a polynomial time approximation algorithm for the two Δ-inequality Set p-Center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee, unless P = NP. For the special case ‘alternative’ p-Center problems, which we refer to as the ‘Pair’ p-Center problems, we extend the previous results in several ways. For example, the results mentioned above for the Set p-Center problems also apply to the Pair p-Center problems. Furthermore, we establish and exploit a correspondence between satisfiability and the dominating set type of problems that naturally arise when considering the decision versions of the Pair p-Center problems.     

堆取料机调度问题的一个近似算法

研究源自煤炭港口料场管理的堆取料机调度问题.该问题中,堆取料机从长为L的料场作业区最左端开始空机或载料运行,两者的速度之比为s：1（s ≥ 1）.决策者需确定储料堆在作业区的分布以及堆取料机处理它们的先后次序,目的是极小化一批煤料运输船的在港服务时间.考虑堆取料机处理完毕需要回到料场终端的作业模型,证明了存在最坏情况界不超过1+1/4s的近似算法.     

List scheduling algorithms to minimize the makespan on identical parallel machines

In this paper we present constructive algorithms for the classical deterministic scheduling problem of minimizing the makespan on identical machines. Since the problem is known to beNP-hard in the strong sense, the approximate algorithms play a relevant role when solving this problem. The proposed algorithms are based on list scheduling procedures, but the assignment rule is not the same for the full set of jobs. Computational results show that these algorithms perform very well. This research has been partially supported by the Research Project H015/2000, Universidad de Alcalá. The authors are indebted to Joaquín Pérez and the referees for their helpful remarks and comments. We also wish to thank Paul Alexander Ayres for his help in the correct use of English.     

Exact and heuristic methodologies for scheduling in hospitals: problems, formulations and algorithms

This text summarizes the PhD thesis defended by the author in January 2006 under the supervision of Professor Erik Demeulemeester at the Katholieke Universiteit Leuven. The thesis is written in English and is available from the author’s website (http://www.econ.kuleuven.be/jeroen.belien). In this research we propose a number of exact and heuristic algorithms for various scheduling problems encountered in hospitals. The emphasis lies on the design of new methodologies as well as on the applicability of the algorithms in real-life environments. The main contributions include a new decomposition approach for a particular class of staff scheduling problems, an extensive study of master surgery scheduling algorithms that aim at leveling the resultant bed occupancy and an innovative method for integrating nurse and surgery scheduling.      

A railway scheduling problem

It is discussed hown railway routes arriving regularly at some station should be scheduled to minimize the maximum waiting time for passengers changing trains.

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Zusammenfassung In der Arbeit wird untersucht, wie man fürn Eisenbahnlinien, die in regelmä\igen Abständen in einem Bahnhof eintreffen, die optimale Reihenfolge erhält, so da\ die maximale Wartezeit für die Reisenden, die in den Zug einer anderen Linie umsteigen wollen, minimiert wird.

     

A linear time approximation algorithm for multiprocessor scheduling

Givenn jobs andm identical processors anO(n) approximation algorithm is presented which tries to determine a nonpreemptive schedule with minimum finish time. Ifr is the number of jobs placed onto the processor with maximum finish time, then the worst case ratio of the new algorithm's finish time to the optimal solution is shown to be less thanrm/(rm–m+1). Extensive empirical results show that the new algorithm is competitive with the LPT algorithm in terms of quality of solution and faster in terms of computing time.     

A dynamic edge covering and scheduling problem: complexity results and approximation algorithms

We consider a new dynamic edge covering and scheduling problem that focuses on assigning resources to nodes in a network to minimize the amount of time required to process all edges in it. Resources need to be co-located at the endpoints of an edge for it to be processed and, therefore, this problem contains both edge covering and scheduling decisions. These new problems have motivating applications in traffic systems and military intelligence operations. We provide complexity results for the dynamic edge covering and scheduling problem over different types of networks. We then show that existing approximation algorithms for parallel machine scheduling problems can be leveraged to provide approximation algorithms for this new class of problems over certain types of networks.     

Optimal receiver scheduling algorithms for a multicast problem

This paper studies a multicast problem arising in wavelength division multiplexing single-hop lightwave networks, as well as in Video-on-Demand systems. In this problem, the same message of duration Δ has to be transmitted to a set of n receivers which are not all available simultaneously. The receivers can be partitioned into subsets, each served by a different transmission, with the objective of minimizing their overall waiting cost. When there is a single data channel available for transmission, a dynamic programming algorithm is devised which finds an optimal solution in O(nlogn+min{n²,nΔ²}) time, improving over a previously known O(n³) time algorithm. When multiple data channels are available for transmission, an optimal O(n) time algorithm is proposed which finds an optimal solution if the message has constant transmission duration, whereas an NP-completeness proof is given if the message has arbitrary transmission duration.     

The project scheduling problem with production and consumption of resources: A list-scheduling based algorithm

The paper deals with algorithms for applying classical list scheduling to a project scheduling problem where the units of resources are produced or consumed at the occurrence of precedence-related events. It is shown that the feasibility variant of the project scheduling problem is NP-complete. Moreover, polynomial-time scheduling algorithms are devised for the three cases where the occurrence time sequence of all events or the consuming events or the producing events is given in advance. By enumerating these sequences (called linear orders), one obtains a list-scheduling based algorithm for minimizing the makespan of a project scheduling problem with production and consumption of resources.     

A school bus scheduling problem

This paper introduces a school bus scheduling problem wherein trips for each school are given. A trip consists of a sequence of bus stops and their designated school. Each school has its fixed time window within which trips should be completed. A school bus can serve multiple trips for multiple schools. The school bus scheduling problem seeks to optimize bus schedules to serve all the given trips considering the school time windows. We first model the problem as a vehicle routing problem with time windows (VRPTW) by treating a trip as a virtual stop. Two assignment problem based exact approaches are then proposed for special cases and a heuristic algorithm is proposed for more general cases. Benchmark problems and computational experiments are presented. Computational experiments show the effectiveness of the proposed approaches.     

优先序约束的排序问题：基于最大匹配的近似算法

本文研究具有加工次序约束的单位工件开放作业和流水作业排序问题,目标函数为极小化工件最大完工时间。工件之间的加工次序约束关系可以用一个被称为优先图的有向无圈图来刻画。当机器数作为输入时,两类问题在一般优先图上都是强NP-困难的,而在入树的优先图上都是可解的。我们利用工件之间的许可对数获得了问题的新下界,并基于许可工件之间的最大匹配设计近似算法,其中匹配的许可工件对均能同时在不同机器上加工。对于一般优先图的开放作业问题和脊柱型优先图的流水作业问题,我们在理论上证明了算法的近似比为$2-frac 2m$,其中$m$是机器数目。     

优先序约束的排序问题：基于最大匹配的近似算法

本文研究具有加工次序约束的单位工件开放作业和流水作业排序问题,目标函数为极小化工件最大完工时间。工件之间的加工次序约束关系可以用一个被称为优先图的有向无圈图来刻画。当机器数作为输入时,两类问题在一般优先图上都是强NP-困难的,而在入树的优先图上都是可解的。我们利用工件之间的许可对数获得了问题的新下界,并基于许可工件之间的最大匹配设计近似算法,其中匹配的许可工件对均能同时在不同机器上加工。对于一般优先图的开放作业问题和脊柱型优先图的流水作业问题,我们在理论上证明了算法的近似比为$2-\frac 2m$,其中$m$是机器数目。     

Optimal appointment scheduling in continuous time: The lag order approximation method

We study appointment scheduling problems in continuous time. A finite number of clients are scheduled such that a function of the waiting time of clients, the idle time of the server, and the lateness of the schedule is minimized. The optimal schedule is notoriously hard to derive within reasonable computation times. Therefore, we develop the lag order approximation method, that sets the client’s optimal appointment time based on only a part of his predecessors. We show that a lag order of two, i.e., taking two predecessors into account, results in nearly optimal schedules within reasonable computation times. We illustrate our approximation method with an appointment scheduling problem in a CT-scan area.     

一类无干涉作业的码头起重机调度问题的近似算法研究

集装箱港口上的大型货轮通常是由从船头到船尾纵向分布的集装箱船舱构成,而码头起重机主要负责装载或卸载集装箱.如何调度码头起重机在很大程度上影响着集装箱货轮的运输效率.该文主要研究一类无干涉作业的起重机调度问题,目标是极小化装(卸)载总耗时.对三台,四台起重机情形设计了新型调度算法,并给出了最坏情况分析,改进了文献中的已有结果.     

Models and algorithms for a staff scheduling problem

We present mathematical models and solution algorithms for a family of staff scheduling problems arising in real life applications. In these problems, the daily assignments to be performed are given and the durations (in days) of the working and rest periods for each employee in the planning horizon are specified in advance, whereas the sequence in which these working and rest periods occur, as well as the daily assignment for each working period, have to be determined. The main objective is the minimization of the number of employees needed to perform all daily assignments in the horizon. We decompose the problem into two steps: the definition of the sequence of working and rest periods (called pattern) for each employee, and the definition of the daily assignment to be performed in each working period by each employee. The first step is formulated as a covering problem for which we present alternative ILP models and exact enumerative algorithms based on these models. Practical experience shows that the best approach is based on the model in which variables are associated with feasible patterns and generated either by dynamic programming or by solving another ILP. The second step is stated as a feasibility problem solved heuristically through a sequence of transportation problems. Although in general this procedure may not find a solution (even if one exists), we present sufficient conditions under which our approach is guaranteed to succeed. We also propose an iterative heuristic algorithm to handle the case in which no feasible solution is found in the second step. We present computational results on real life instances associated with an emergency call center. The proposed approach is able to determine the optimal solution of instances involving up to several hundred employees and a working period of up to 6 months. Mathematics Subject Classification (2000): 90B70, 90C10, 90C27, 90C39, 90C57, 90C59