Cyclical schedules for one-warehouse,multi-retailer systems with dynamic demands

In a one-warehouse, multi-retailer system, a cyclical schedule is characterised by fixed order intervals at the warehouse and retailers. Cyclical schedules offer advantages in terms of simplicity and ease of control, but under dynamic demand conditions they are higher in cost than non-cyclical schedules. This study presents a mathematical formulation of the cyclical problem, and then outlines an algorithm to obtain optimal solutions. An experiment is then conducted to investigate the additional cost of cyclical schedules as a function of problem characteristics relevant to one-warehouse, multi-retailer systems. Results show that coefficient of demand variation has the strongest effect on the additional cost of cyclical scheduling. However, even at relatively high levels of demand variation, the additional costs of cyclical scheduling are quite moderate.     

A simulated annealing algorithm for resource constrained project scheduling problems

This paper presents a simulated annealing algorithm for resource constrained project scheduling problems with the objective of minimising makespan. In the search algorithm, a solution is represented with a priority list, a vector of numbers each of which denotes the priority of each activity. In the algorithm, a priority scheduling method is used for making a complete schedule from a given priority list (and hence a project schedule is defined by a priority list). The search algorithm is applied to find a priority list which corresponds to a good project schedule. Unlike most of priority scheduling methods, in the suggested algorithm some activities are delayed on purpose so as to extend search space. Solutions can be further improved by delaying certain activities, since non-delay schedules are not dominant in the problem (the set of non-delay schedules does not always include an optimal solution). The suggested algorithm is flexible in that it can be easily applied to problems with an objective function of a general form and/or complex constraints. The performance of the simulated annealing algorithm is compared with existing heuristics on problems prepared by Patterson and randomly generated test problems. Computational results showed that the suggested algorithm outperformed existing ones.     

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.     

Cyclic and non-cyclic scheduling of 12 h shift nurses by network programming

In this paper, we present a mathematical model for cyclic and non-cyclic scheduling of 12 h shift nurses. The model exploits the fact that a nurse's schedule is made up of an alternating sequence of work-stretch and ‘off-stretch’ patterns. We introduce a concept called a stint, which is a pattern characterized by a start date, a length, a `cost' and the shifts worked. Using the stints as nodes in a network, we construct an acyclic graph on which the nurse's schedules can be defined. The resulting model is essentially a shortest-path problem with side constraints. The model is quite flexible and can accommodate a variety of constraints. With a minor modification, the network is used to define both the cyclic and non-cyclic scheduling problems. The models are illustrated on sample data from a local hospital and solved using CPLEX optimization software on an IBM RISC6000/340 workstation.     

Near-optimal MIP solutions for preference based self-scheduling

Making a high quality staff schedule is both difficult and time consuming for any company that has employees working on irregular schedules. We formulate a mixed integer program (MIP) to find a feasible schedule that satisfies all hard constraints while minimizing the soft constraint violations as well as satisfying as many of the employees’ requests as possible. We present the MIP model and show the result from four real world companies and institutions. We also compare the results with those of a local search based algorithm that is designed to emulate the solution strategies when the schedules are created manually. The results show that using near-optimal solutions from the MIP model, with a relative MIP gap of around 0.01–0.1, instead of finding the optimal solution, allows us to find very good solutions in a reasonable amount of time that compare favorably with both the manual solutions and the solutions found by the local search based algorithm.     

Optimal scheduling on parallel machines for a new order class

This paper addresses the problem of scheduling n unit length tasks on m identical machines under certain precedence constraints. The aim is to compute minimal length nonpreemptive schedules. We introduce a new order class which contains properly two rich families of precedence graphs: interval orders and a subclass of the class of series parallel orders. We present a linear time algorithm to find an optimal schedule for this new order class on any number of machines.     

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.     

A decomposed branch-and-price procedure for integrating demand planning in personnel staffing problems

The personnel staffing problem calculates the required workforce size and is determined by constructing a baseline personnel roster that assigns personnel members to duties in order to cover certain staffing requirements. In this research, we incorporate the planning of the duty demand in the staff scheduling problem in order to lower the staffing costs. More specifically, the demand originates from a project scheduling problem with discrete time/resource trade-offs, which embodies additional flexibility as activities can be executed in different modes. In order to tackle this integrated problem, we propose a decomposed branch-and-price procedure. A tight lower and upper bound are calculated using a problem formulation that models the project scheduling constraints and the time-related resource scheduling constraints implicitly in the decision variables. Based upon these bounds, the strategic problem is decomposed into multiple tactical subproblems with a fixed workforce size and an optimal solution is searched for each subproblem via branch-and-price. Fixing the workforce size in a subproblem facilitates the definition of resource capacity cuts, which limit the set of eligible project schedules, decreasing the size of the branching tree. In addition, in order to find the optimal integer solution, we propose a specific search strategy based upon the lower bound and dedicated rules to branch upon the workload generated by a project schedule. The computational results show that applying the proposed search space decomposition and the inclusion of resource capacity cuts lead to a well-performing procedure outperforming different other heuristic and exact methodologies.     

Retail store scheduling for profit

In spite of its tremendous economic significance, the problem of sales staff schedule optimization for retail stores has received relatively scant attention. Current approaches typically attempt to minimize payroll costs by closely fitting a staffing curve derived from exogenous sales forecasts, oblivious to the ability of additional staff to (sometimes) positively impact sales. In contrast, this paper frames the retail scheduling problem in terms of operating profit maximization, explicitly recognizing the dual role of sales employees as sources of revenues as well as generators of operating costs. We introduce a flexible stochastic model of retail store sales, estimated from store-specific historical data, that can account for the impact of all known sales drivers, including the number of scheduled staff, and provide an accurate sales forecast at a high intra-day resolution. We also present solution techniques based on mixed-integer (MIP) and constraint programming (CP) to efficiently solve the complex mixed integer non-linear scheduling (MINLP) problem with a profit-maximization objective. The proposed approach allows solving full weekly schedules to optimality, or near-optimality with a very small gap. On a case-study with a medium-sized retail chain, this integrated forecasting–scheduling methodology yields significant projected net profit increases on the order of 2–3% compared to baseline schedules.     

A generalized linear programming model for nurse scheduling

This paper presents a 0–1 column generation model with a resource constrained shortest path auxiliary problem for nurse scheduling. The master problem finds a configuration of individual schedules to satisfy the demand coverage constraints while minimizing salary costs and maximizing both employee preferences and team balance. A feasible solution of the auxiliary problem is an acceptable schedule for a given nurse, with respect to collective agreement requirements such as seniority, workload, rotations and days off. We define a new resource structure in the auxiliary problem in order to take into account the complex collective agreement rules specific to the nurse scheduling problem. This model generalizes further the previous formulations discussed in the literature and can be viewed as a general scheme for complex personnel scheduling problems, especially in the context of organizations which operate around the clock. Solution methods and preliminary test results are discussed.     

基于改进混合遗传算法安排生产调度

研究了某工厂生产调度问题,建立了数学模型.针对这一实际问题,通过引入小生境技术、最优保存策略、近优淘汰策略、自适应调整交叉概率和变异概率,设计了用于求解多个最优顺序的混合遗传算法,用所设计的混合遗传算法对该模型进行了计算,获得了许多最优顺序,这就使得生产调度安排灵活机动,便于智能调度,同时生产量比原来大幅度提高.这表明使用混合遗传算法安排生产调度是非常有效的.     

Multi-parallel work centers scheduling optimization with shared or dedicated resources in low-volume low-variety production systems

A new methodology for modeling large-scale scheduling problems in low-volume low-variety production systems is proposed through this paper. Such scheduling problems are constrained by limited time and resources, where each work center is assigned a unique statement of work, to be completed on-time with the budgeted number of resources. Products assembled in low-volume low-variety production systems are processed through a series of stations referred to as work centers, where varying levels and classifications of resources are deployed onto the product. Aircraft, heavy aero-structures, and heavy military equipment are examples of products assembled in low-volume low-variety production systems. To ensure products are delivered on-time and on-budget, it is crucial to execute to a detailed schedule, such that all precedence, resource, zonal, and other constraints and characteristics inherent in such production systems are successfully satisfied. Despite the criticality of detailed schedules in delivering products on-time and on-budget, limited research is reported on mixed-integer programming approaches for scheduling optimization of activities in low-volume low-variety production systems. The discrete-time linear mixed-integer mathematical programming model developed in this paper fills the gap in the current literature with a direct impact on the organizations’ service levels and bottom line. The proposed mathematical programming models are validated through a real-world case-study of the assembly process of a narrow body aircraft to ensure compatibility in the modeling of large-scale industrial problems.     

Comparison and hybridization of crossover operators for the nurse scheduling problem

In this paper, we present a hybrid genetic algorithm for the well-known nurse scheduling problem (NSP). The NSP involves the construction of roster schedules for nursing staff in order to maximize the quality of the roster schedule subject to various hard constraints. In the literature, several genetic algorithms have been proposed to solve the NSP under various assumptions. The contribution of this paper is twofold. First, we extensively compare the various crossover operators and test them on a standard dataset in a solitary approach. Second, we propose several options to hybridize the various crossover operators.     

A mixed integer programming model for advanced planning and scheduling (APS)

This paper presents a mixed integer programming (MIP) model which succeeds in a system integration of the production planning and shop floor scheduling problems. The proposed advanced planning and scheduling (APS) model explicitly considers capacity constraints, operation sequences, lead times and due dates in a multi-order environment. The objective of the model is to seek the minimum cost of both production idle time and tardiness or earliness penalty of an order. The output of the model is operation schedules with order starting time and finish time. Numerical result shows that the suggested APS model can favorably produce optimal schedules.     

Timetabling for Greek high schools

The timetabling process and the resulting weekly schedules are important components for the daily operation of any school. This paper presents an efficient solution to the timetabling problem for the secondary educational system in Greece. Such a problem involves scheduling a large number of classes, teachers, courses, and classrooms to a number of time-periods. The development of the basic structure and the modelling of the problem as an integer mathematical program allows for the generation of constraints necessary for the satisfaction of all the school system rules and regulations. The integer programming approach and the commercial tools available for this class of problems facilitated the process of locating the optimal solution for the problem. The model is flexible and modular allowing for adaptations to satisfy the local characteristics of each school by changing the parameters of the model and adding or replacing constraints. A fully defined timetabling problem for a typical Greek high school is presented and optimally solved in order to demonstrate the effectiveness of the model in satisfying both the hard and the soft operational rules of the problem. Implementation of the new methodology for regular use for high schools is currently being attempted.     

Combining integer programming and the randomization method to schedule employees

We describe a method to find low cost shift schedules with a time-varying service level that is always above a specified minimum. Most previous approaches used a two-step procedure: (1) determine staffing requirements and (2) find a minimum cost schedule that provides the required staffing in every period. Approximations in the first step sometimes cause the two-step approach to find infeasible or suboptimal solutions. Our method iterates between a schedule evaluator and a schedule generator. The schedule evaluator calculates transient service levels using the randomization method and identifies infeasible intervals, where the service level is lower than desired. The schedule generator solves a series of integer programs to produce improved schedules, by adding constraints for every infeasible interval, in an attempt to eliminate infeasibility without eliminating the optimal solution. We present computational results for several test problems and discuss factors that make our approach more likely to outperform previous approaches.     

A stochastic programming model for scheduling call centers with global Service Level Agreements

We consider the issue of call center scheduling in an environment where arrivals rates are highly variable, aggregate volumes are uncertain, and the call center is subject to a global service level constraint. This paper is motivated by work with a provider of outsourced technical support services where call volumes exhibit significant variability and uncertainty. The outsourcing contract specifies a Service Level Agreement that must be satisfied over an extended period of a week or month. We formulate the problem as a mixed-integer stochastic program. Our model has two distinctive features. Firstly, we combine the server sizing and staff scheduling steps into a single optimization program. Secondly, we explicitly recognize the uncertainty in period-by-period arrival rates. We show that the stochastic formulation, in general, calculates a higher cost optimal schedule than a model which ignores variability, but that the expected cost of this schedule is lower. We conduct extensive experimentation to compare the solutions of the stochastic program with the deterministic programs, based on mean valued arrivals. We find that, in general, the stochastic model provides a significant reduction in the expected cost of operation. The stochastic model also allows the manager to make informed risk management decisions by evaluating the probability that the Service Level Agreement will be achieved.     

Supply chain scheduling: Sequence coordination

A critical issue in supply chain management is coordinating the decisions made by decision makers at different stages, for example a supplier and one or several manufacturers. We model this issue by assuming that both the supplier and each manufacturer have an ideal schedule, determined by their own costs and constraints. An interchange cost is incurred by the supplier or a manufacturer whenever the relative order of two jobs in its actual schedule is different from that in its ideal schedule. An intermediate storage buffer is available to resequence the jobs between the two stages. We consider the problems of finding an optimal supplier's schedule, an optimal manufacturer's schedule, and optimal schedules for both. The objective functions we consider are the minimization of total interchange cost, and of total interchange plus buffer storage cost. We describe efficient algorithms for all the supplier's and manufacturers’ problems, as well as for a special case of the joint scheduling problem. The running time of these algorithms is polynomial in both the number of jobs and the number of manufacturers. Finally, we identify conditions under which cooperation between the supplier and a manufacturer reduces their total cost.     

Preventive maintenance scheduling of multi-cogeneration plants using integer programming

Maintenance scheduling of cogeneration plants, which produce both electric power and desalinated water, is a typical complex process with long-term operations and planning problems. The plants' maintenance scheduling process has to determine the appropriate schedule for preventive maintenance, while satisfying all the system constraints and maintaining adequate system availability. It is an optimization problem and the maintenance and system constraints include the crew constraint, maintenance window constraint and time limitation constraint. In this paper, an integer linear-programming model, which has been developed, is described which schedules the preventive maintenance tasks in a multi-cogeneration plant. Results of a test example of such a plant situated in Kuwait are presented to show the applicability of the approach.     

Multiobjective evolutionary algorithms to identify highly autocorrelated areas: the case of spatial distribution in financially compromised farms

In this paper, we address the optimal batch sizing and just-in-time scheduling problem where upper and lower bounds on the size of the batches are imposed. The objective is to find a feasible schedule that minimizes the sum of the weighted earliness and tardiness penalties as well as the setup costs, which involves the cost of creating a new batch. We present some structural properties of the optimal schedules and describe solving algorithms for the single machine problem.