A Comparative Study of Multiple-Objective Metaheuristics on the Bi-Objective Set Covering Problem and the Pareto Memetic Algorithm

Many organizations face employee scheduling problems under conditions of variable demand for service over the course of an operating day and across a planning horizon. These organizations are concerned with the tour scheduling problem that involves assigning shifts and break times to the work days of employees and allocating days off to individual work schedules. Nowadays, organizations try to adopt various scheduling flexibility alternatives to meet the fluctuating service demand. On the other hand, they have also realized that providing employee productivity and satisfaction is as much important as meeting the service demand. Up to date, tour scheduling solution approaches have neglected considering employee preferences and tried to develop work schedules for employees in a subsequent step. This paper presents a goal programming model that implicitly represents scheduling flexibility and also incorporates information about the preferred working patterns of employees. After solving the proposed model, a work schedule will be generated for each employee without requiring a further step for the assignment of shifts, break times, and work days to employees. The model is capable of handling multiple scheduling objectives, and it can produce optimal solutions in very short computing times.     

Network flow models for intraday personnel scheduling problems

Personnel scheduling problems can be decomposed into two stages. In the first stage for each employee the working days have to be fixed. In the second stage for each day of the planning period an intraday scheduling problem has to be solved. It consists of the assignment of shifts to the employees who have to work on the day and for each working period of an employee a task assignment such that the demand of all tasks for personnel is covered. In Robinson et al. (Burke and Trick (Eds.), Proceedings of the 5th International Conference on the Practice and Theory of Automated Timetabling, 18th August–20th August 2004, Pittsburgh, PA, USA, pp. 561–566, 2005), the intraday problem has been formulated as a maximum flow problem. The assumptions are that, employees are qualified for all tasks, their shifts are given, and they are allowed to change tasks during the day. In this work, we extend the network flow model to cover the case where not all employees are qualified to perform all tasks. The model is further extended to be able to calculate shifts of employees for the given day, assuming that an earliest starting time, a latest finishing time, and a minimal working time are given. Labour cost can be also taken into account by solving a minimum cost network flow problem.     

An algorithm for single shift scheduling of hierarchical workforce

This paper presents an optimal algorithm for single shift scheduling of hierarchical workforce in seven-day-a-week industries. There are many categories of workers; their capabilities are arranged hierarchically. The objective is to arrive at the most economical mix of categories of employees satisfying the pattern of demand for employees and desired work characteristics. The demand for employees varies from one category to the other and for each category the demand during weekdays is fixed and is different from that of weekends. The desired work characteristics are: (i) each employee must be given two days off every week including at least a proportion A out of every B weekends off and (ii) each employee can have at most 5 working shifts between any two offdays. The algorithm provides a computational scheme for arriving at the minimum workforce size comprising the most economical mix of categories of employees for this demand context and a new technique for characterizing the problem to achieve feasible assignment of employees to shifts.     

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     

A constructive heuristic for staff scheduling in the glass industry

In this paper a constructive heuristic for solving the staff scheduling problem of a glass manufacture unit is proposed. Based on simple calculations and algorithms, the developed procedure assigns working shifts and days-off to teams of employees, ensuring the satisfaction of a mandatory sequence of working shifts and the balance of the workload between employees. The computational times for the experiments with the case study company, with three eight-hour working shifts and five teams of employees, fell consistently below 5 seconds for a set of different planning periods. Results are compared with the ones achieved with an optimization model (MIP), demonstrating the good performance of the heuristic, also in terms of the quality of the achieved solutions. The heuristic rarely fails to produce a feasible solution and whenever the solution is feasible then it is also optimal. When tackling problems with a large number of teams, the heuristic maintains the good performance while the MIP model is not able to find any solution within 16 hours of running time. Although it was designed for a particular problem of the glass industry, tests show that the heuristic is flexible enough to be applied to problems with different features, from other activity sectors, encouraging further extensions of this work.     

An acyclic days-off scheduling problem

This paper studies the days off scheduling problem when the demand for staffing fluctuates from day to another and when the number of total workdays is fixed in advance for each employee. The scheduling problem is then to allocate rests to employees with different days off policies: (1) two or three consecutive days off for each employee per week and (2) at least three consecutive days off for each employee per month. For each one, we propose a polynomial time algorithm to construct a solution if it exists. Received: April 2005 / Revised version: October 2005 AMS classification: 60K25, 60K30     

Selecting and adapting weekly work schedules with working time accounts: A case of a retail clothing chain

A case of selection and adaptation of weekly work schedules is presented. Weekly work schedules in two franchises of an important retail clothing chain have to be established. Working time accounts are used: each week, an employee can owe the company a certain number of hours or vice versa. Nevertheless, over a certain threshold, the hours have to be paid for by the company and the account balance returns to zero. A minimum and desired level of capacity of employees is contemplated. Hierarchically, the planned capacity must attempt to reach the minimum level; then it must fit a desired level as much as possible. At present, the task of allocation and the final adjustment of schedules is done manually, which is difficult, ineffective and often inaccurate. The procedure proposed is divided into two phases. Firstly, a work schedule, selected from a list, is assigned to each worker; a mixed linear program, followed by a local optimization process, is used. In the second phase, the work schedules are modified according to predefined rules: if there is a surplus of capacity, work schedules are reduced, and if there is a shortage, work schedules are extended. The company considers the results to be satisfactory.     

Multiple shift scheduling of workforce on four-day workweeks

This paper presents an efficient algorithm for scheduling of employees working on four-day or three-day workweeks in industries operating on multiple shift basis. Such contexts prevail in manufacturing organisations such as paper mills, steel plants and service organisations such as security or police departments and hospitals. The objective is to arrive at the minimum workforce size and to produce schedules that meet demand and achieve certain desired work characteristics. The pattern of demand for employees is such that the number of employees required for each shift during weekdays is at least as large as the number required in the corresponding shift during weekends. The desired work characteristics are parameters relating to the frequency of weekends-off in the days-off, the length of the workstretches and the transition time required when changing shifts.     

Production planning with approved vendor matrices for a hard-disk drive manufacturer

We develop an optimal production schedule for a manufacturer of hard-disk drives that offers its customers the approved vendor matrix (AVM) as a competitive advantage. An AVM allows each customer to pick and choose the various product component vendors for individual or pairs of components constituting their product. The production planning problem faced by the manufacturer is to meet customer demand as precisely as possible while observing the matrix restrictions and also the limited availability of production resources. We formulate this problem as a linear programming model with a large number of variables, and present a solution procedure based on the column generation technique. A special class of the problem is then studied, whereby the number of production setups in each period is limited and discrete. We modify our formulation into a mixed-integer problem, and proceed to develop procedures that can obtain good feasible solutions using linear programming rounding techniques.     

Weekly staff scheduling with workstation group restrictions

This paper addresses the weekly adjustment problem for staff scheduling when movement restrictions exist between workstation groups (WSGs). In practice, it is common for employees to be organized into physical or logical groups to match the layout of a facility or to facilitate managerial oversight. A complication in the problem arises when each employee is required to spend more time at his or her assigned home base during the week than at any other WSG. This conflicts with a common strategy of reassigning employees to different WSGs when idle time exists in their schedules. Ordinarily, the full problem is tackled with a two-phase approach, where optimal shifts and overtime allocations are first derived and then tasks are assigned. When movement restrictions exist in a facility, this approach is no longer practical or even possible for all but the smallest instances. Alternatively, a new model is proposed that integrates WSG restrictions with the shift scheduling and task assignment constraints. The model takes the form of a large-scale integer program and is solved with one of two decomposition heuristics. The first splits the movement restrictions network into manageable pieces; the second uses column generation to identify good individual schedules that are used to construct a set-covering-type master problem. A solution to the master problem provides a feasible solution to the original integer program. Extensive testing was done with data obtained from the U.S. Postal Service mail processing and distribution center in Dallas. The results show that good feasible solutions can be obtained in less than an hour.     

An agent-based approach to modeling insider threat

This paper describes the modeling of the potential of an organization to develop an insider threat given certain attributes of its culture. The model represents all employees of the organization and their likelihood of becoming an insider threat. Each employee is instantiated in an agent-zero construct, which accounts for affective, rational, and social behavioral influences. The main driver of behavior is the employee’s level of disgruntlement against the organization. The simulation is run over a period of 10 years and the total number of employees that exceed a certain threshold of becoming an insider threat are computed. This number is compared with survey data on work force ethics as a measure of validity of the simulation results.     

The implementor/adversary algorithm for the cyclic and robust scheduling problem in health-care

A general problem in health-care consists in allocating some scarce medical resource, such as operating rooms or medical staff, to medical specialties in order to keep the queue of patients as short as possible. A major difficulty stems from the fact that such an allocation must be established several months in advance, and the exact number of patients for each specialty is an uncertain parameter. Another problem arises for cyclic schedules, where the allocation is defined over a short period, e.g. a week, and then repeated during the time horizon. However, the demand typically varies from week to week: even if we know in advance the exact demand for each week, the weekly schedule cannot be adapted accordingly. We model both the uncertain and the cyclic allocation problem as adjustable robust scheduling problems. We develop a row and column generation algorithm to solve this problem and show that it corresponds to the implementor/adversary algorithm for robust optimization recently introduced by Bienstock for portfolio selection. We apply our general model to compute master surgery schedules for a real-life instance from a large hospital in Oslo.     

总体报酬与员工创新行为的关系研究:组织自尊和工作投入的作用

本研究基于社会信息加工理论,构建了一个以组织自尊为中介变量、工作投入为调节变量的总体报酬影响员工创新行为的两阶段调节中介模型,并采用时间滞后法获得来自多个行业与地区的140家企业470位非研发人员及其领导的多时点配对调查数据对模型进行检验,旨在揭示中国情境下总体报酬激发员工创新行为的作用机制和边界条件。实证研究发现,总体报酬与员工的创新行为、组织自尊显著正相关,组织自尊完全中介了总体报酬与创新行为的关系,工作投入正向调节了总体报酬和组织自尊的关系、以及组织自尊和员工创新行为的关系,并进一步正向调节了总体报酬通过组织自尊对创新行为的间接影响。     

Solving an integrated job-shop problem with human resource constraints

We propose two exact methods to solve an integrated employee-timetable and job-shop-scheduling problem. The problem is to find a minimum cost employee-timetable, where employees have different competences and work during shifts, so that the production, that corresponds to a job-shop with resource availability constraints, can be achieved. We introduce two new exact procedures: (1) a decomposition and cut generation approach and (2) a hybridization of a cut generation process with a branch and bound strategy. We also propose initial cuts that strongly improve these methods as well as a standard MIP approach. The computational performances of those methods on benchmark instances are compared to that of other methods from the literature.     

校车安排问题

探讨如何安排校车运行使得教师和工作人员尽量满意的问题.首先建立动态规划模型和选址规划模型,求出合理站点位置及其总距离.然后用归一法定义满意度与距离的函数关系,考虑各区域人数,建立选址规划模型.得到合理站点位置和总满意度.之后建立双目标非线性规划模型,利用量纲分析法给出权重,以此求出合理乘车位置和满意度.最后对问题进行推...     

Workforce planning at USPS mail processing and distribution centers using stochastic optimization

Service organizations that operate outside the normal 8-hour day and face wide fluctuations in demand constantly struggle to optimize the size and composition of their workforce. Recent research has shown that improved personnel scheduling methods that take demand uncertainty into account can lead to significant reductions in labor costs. This paper addresses a staff planning and scheduling problem that arises at United States Postal Service (USPS) mail processing & distribution centers (P&DCs) and develops a two-stage stochastic integer program with recourse for the analysis. In the first stage, before the demand is known, the number of full-time and part-time employees is determined for the permanent workforce. In the second stage, the demand is revealed and workers are assigned to specific shifts during the week. When necessary, overtime and casual labor are used to satisfy demand. This paper consists of two parts: (1) the analysis of the demand distribution in light of historical data, and (2) the development and analysis of the stochastic integer programming model. Using weekly demand for a three-year period, we first investigate the possibility that there exists an end-of-month effect, i.e., the week at the end of month has larger volume than the other weeks. We show that the data fail to indicate that this is the case. In the computational phase of the work, three scenarios are considered: high, medium, and low demand. The stochastic optimization problem that results is a large-scale integer program that embodies the full set of contractual agreements and labor rules governing the design of the workforce at a P&DC. The usefulness of the model is evaluated by solving a series of instances constructed from data provided by the Dallas facility. The results indicate that significant savings are likely when the recourse problem is used to help structure the workforce. This work was supported in part by the National Science Foundation under grants DMI-0218701 and DMI-0217927.     

Mixed-integer programming models for an employee scheduling problem with multiple shifts and work locations

This paper is concerned with the problem of assigning employees to gas stations owned by the Kuwait National Petroleum Corporation (KNPC), which hires a firm to prepare schedules for assigning employees to about 86 stations distributed all over Kuwait. Although similar employee scheduling problems have been addressed in the literature, certain peculiarities of the problem require novel mathematical models and algorithms to deal with the specific nature and size of this problem. The problem is modeled as a mixed-integer program, and a problem size analysis based on real data reveals that the formulation is too complex to solve directly. Hence, a two-stage approach is proposed, where the first stage assigns employees to stations, and the second stage specifies shifts and off-days for each employee. Computational results related to solving the two-stage models directly via CPLEX and by specialized heuristics are reported. The two-stage approach provides daily schedules for employees for a given time horizon in a timely fashion, taking into consideration the employees’ expressed preferences. This proposed modeling approach can be incorporated within a decision support system to replace the current manual scheduling practice that is often chaotic and has led to feelings of bias and job dissatisfaction among employees.     

Bridging the gap between self schedules and feasible schedules in staff scheduling

Every company that has employees working on irregular schedules must deal with the difficult and time consuming problem of creating feasible schedules for the employees. We introduce an algorithm that takes a partial schedule created by requests from employees and creates feasible schedule where most of the employee’s requests are unchanged, while still making sure that rules and regulations are not violated. The algorithm is based on independent modules, which can be executed in any order, and each module tries to emulate some action taken by a staff manager. Our goal is to create a transparent and fair system that creates feasible schedules of high quality, but also a system where the employees can get an explanation and justification for every change that the algorithm makes to the employee requests. By emulating the actions of staff managers, the algorithm is easily understood by staff managers and, using detailed logs of any action, make any decision easy to explain to the employees. We will present the algorithm and show results from four real world companies and institutions. The results show that a simple module based heuristic can get good results and create fair and feasible schedules that encourage employees to participate in the self-scheduling process.     

A real options approach to labour shifts planning under different service level targets

Firms that experience uncertainty in demand as well as challenging service levels face, among other things, the problem of managing employee shift numbers. Decisions regarding shift numbers often involve significant expansions or reductions in capacity, in response to changes in demand. In this paper, we quantify the impact of treating shifts in workforce expansion as investments, while considering required service level improvements. The decision to increase shifts, whether by employing temporary workers or hiring permanent employees, is one that involves significant risks. Traditional theories typically consider reversible investments, and thus do not capture the idiosyncrasies involved in shift management, in which costs are not fully reversible. In our study, by using real options theory, we quantify managers’ ability to consider this irreversibility, aiming to enable them to make shift decisions under conditions of uncertainty with the maximum level of flexibility. Our model aims to help managers make more accurate decisions with regard to shift expansion under service level targets, and to defer commitment until future uncertainties can be at least partially resolved. Overall, our investigation contributes to studies on the time required to introduce labour shift changes, while keeping the value of service level improvements in mind.     

Medium-term hydro-thermal coordination via hydro and thermal subsystem decomposition

Summary This paper addresses the medium-term hydro-thermal coordination problem in an electric energy system. That is, the problem of finding the energy production of every power plant (hydro or thermal) in every subperiod of a given planning period, so that the customer load is supplied at minimum cost. The planning horizon is typically one to two months and the first week of this planning period is modeled in detail. The solution method proposed decomposes the problem in two subproblems corresponding to the hydro and thermal subsystems. These two subproblems are coordinated using a coordinating function for every subperiod. The coordinating function of a given subperiod expresses total production cost in that subperiod as a function of the total hydro production in that subperiod. The decomposition proposed makes it possible to use specialized algorithms to solve the hydro and thermal subproblems. This results in a very efficient computational procedure. From an experimental point of view the coordinating mechanism is robust. A case study is provided. It considers 61 thermal plants, a hydro system including 8 cascaded hydro plants and a 48 subperiods planning period.