TECHNICAL NOTE: A SIMPLE MODEL FOR OPTIMIZING THE SINGLE MACHINE EARLY/TARDY PROBLEM WITH SEQUENCE-DEPENDENT SETUPS

A simple mixed integer programming model for the N job/single machine scheduling problem with possibly sequence-dependent setup times, differing earliness/tardiness cost penalties, and variable due dates is proposed and evaluated for computational efficiency. Results indicated that the computational effort required to reach optimality rose with the number of jobs to be scheduled and with decreased variance in due dates. Though computational effort was significant for the largest problems solved, the model remained viable for optimizing research scale problems.     

The use of demand information in selecting production plans and schedules

The strategic problem of selecting a production plan for a given planning horizon is usually treated as independent of the tactical problem of scheduling the production plan. This paper approaches both selecting the production plan and scheduling it as one problem. The problem is formulated as a zero-one integer program. The formulation accommodates many real-life considerations. The integer program is solved using a branch and bound procedure which provides the optimal production plan and schedule as well as the importance indices of the orders, a concept which is introduced and used in this study to rank the available orders within the planning horizon according to their importance to the firm. The integer program and the search procedure can be used as a decision supporting tool to respond to any changes in the demand information, capacity of the firm, or its operating strategy, and it guarantees the selection of feasible production plan(s) and optimal schedules.     

Heuristic for scheduling in a two-machine bicriteria dynamic flowshop with setup and processing times separated

This paper addresses the two-machine bicriteria dynamic flowshop problem where setup time of a job is separated from its processing time and is sequenced independently. The performance considered is the simultaneous minimization of total flowtime and makespan, which is more effective in reducing the total scheduling cost compared to the single objective. A frozen-event procedure is first proposed to transform a dynamic scheduling problem into a static one. To solve the transformed static scheduling problem, an integer programming model with N ² + 5N variables and 7N constraints is formulated. Because the problem is known to be NP-complete, a heuristic algorithm with the complexity of O (N ³) is provided. A decision index is developed as the basis for the heuristic. Experimental results show that the proposed heuristic algorithm is effective and efficient. The average solution quality of the heuristic algorithm is above 99%. A 15-job case requires only 0.0235 s, on average, to obtain a near or even optimal solution.     

A flight scheduling model for Taiwan airlines under market competitions

In this research, we develop a short-term flight scheduling model with variable market shares in order to help a Taiwan airline to solve for better fleet routes and flight schedules in today's competitive markets. The model is formulated as a non-linear mixed integer program, characterized as an NP-hard problem, which is more difficult to solve than the traditional fixed market share flight scheduling problems, often formulated as integer/mixed integer linear programs. We develop a heuristic method to efficiently solve the model. The test results, mainly using the data from a major Taiwan airline's operations, show the good performance of the model and the solution algorithm.     

A FRAMEWORK FOR MACHINE SCHEDULING PROBLEMS WITH CONTROLLABLE PROCESSING TIMES

This is a study of single and parallel machine scheduling problems with controllable processing time for each job. The processing time for job j depends on the position of the job in the schedule and is a function of the number of resource units allocated to its processing. Processing time functions and processing cost functions are allowed to be nonlinear. The scheduling problems considered here have important applications in industry and include many of the existing scheduling models as special cases. For the single machine problem, the objective is minimization of total compression costs plus a scheduling measure. The scheduling measures include makespan, total flow time, total differences in completion times, total differences in waiting times, and total earliness and tardiness with a common due date for all jobs. Except when the total earliness and tardiness measure is involved, each case the problem is solved efficiently. Under an assumption typically satisfied in just-in-time systems, the problem with total earliness and tardiness measure is also solved efficiently. Finally, for a large class of processing time functions; parallel machine problems with total flow time and total earliness and tardiness measures are solved efficiently. In each case we reduce the problem to a transportation problem.     

Distributed Decision‐Making in Supply Chains and Private E‐Marketplaces

This paper develops a distributed decision‐making framework for the players in a supply chain or a private e‐marketplace to collaboratively arrive at a global Pareto‐optimal solution. In this model, no player has complete knowledge about all the costs and constraints of the other players. The decision‐making framework employs an iterative procedure, based on the Integer L‐shaped method, in which a master problem is solved to propose global solutions, and each player uses his local problems to construct feasibility and optimality cuts on the master problem. The master problem is modeled as a mixed‐integer program, and the players' local problems are formulated as linear programs. Collaborative planning scenarios in private e‐marketplaces and in supply chains were formulated and solved for test data. The results show that this distributed model is able to achieve near‐optimal solutions considerably faster than the traditional centralized approach.     

Appointment Overbooking in Health Care Clinics to Improve Patient Service and Clinic Performance

The problem of no‐shows (patients who do not arrive for scheduled appointments) is particularly significant for health care clinics, with reported no‐show rates varying widely from 3% to 80%. No‐shows reduce revenues and provider productivity, increase costs, and limit patient access by reducing effective clinic capacity. In this article, we construct a flexible appointment scheduling model to mitigate the detrimental effects of patient no‐shows, and develop a fast and effective solution procedure that constructs near‐optimal overbooked appointment schedules that balance the benefits of serving additional patients with the potential costs of patient waiting and clinic overtime. Computational results demonstrate the efficacy of our model and solution procedure, and connect our work to prior research in health care appointment scheduling.     

A Model for Master Production Scheduling in Uncertain Environments

In uncertain environments, the master production schedule (MPS) is usually developed using a rolling schedule. When utilizing a rolling schedule, the MPS is replanned periodically and a portion of the MPS is frozen in each planning cycle. The cost performance of a rolling schedule depends on three decisions: the choice of the replanning interval (R), which determines how often the MPS should be replanned; the choice of the frozen interval (F), which determines how many periods the MPS should be frozen in each planning cycle; and the choice of the forecast window (T), which is the time interval over which the MPS is determined using newly updated forecast data. This paper uses an analytical approach to study the master production scheduling process in uncertain environments without capacity constraints, where the MPS is developed using a rolling schedule. It focuses on the choices of F, R, and T for the MPS. A conceptual framework that includes all important MPS time intervals is described. The effects of F, R, and T on system costs, which include the forecast error, MPS change, setup, and inventory holding costs, are also explored. Finally, a mathematical model for the MPS is presented. This model approximates the average system cost as a function of F, R, T, and several environmental factors. It can be used to estimate the associated system costs for any combination of F, R, and T.     

Scheduling professionals in the retail pharmacy chain

To control operating costs, a zero-one integer programming model is developed to assist pharmacy staff scheduling decisions. Variable scheduling needs are met by the assignment of relief (mobile) pharmacists to help or temporarily replace full-time pharmacists. Assignments of relief pharmacists over a two-week planning horizon are determined with consideration given to variations in wage rates and travel costs together with the underlying corporate, contractual and operating constraints. The developed model has been applied with considerable success using data collected from a business district in the US located in northern Louisiana related to a national retail chain pharmacy. Forecasting the number of chain retail outlets in the near future has been also performed and the results obtained argue in favor of adopting the model by the entire chain.     

Priority Rules for Multi‐Task Due‐Date Scheduling under Varying Processing Costs

We study the scheduling of multiple tasks under varying processing costs and derive a priority rule for optimal scheduling policies. Each task has a due date, and a non‐completion penalty cost is incurred if the task is not completely processed before its due date. We assume that the task arrival process is stochastic and the processing rate is capacitated. Our work is motivated by both traditional and emerging application domains, such as construction industry and freelance consulting industry. We establish the optimality of Shorter Slack time and Longer remaining Processing time (SSLP) principle that determines the priority among active tasks. Based on the derived structural properties, we also propose an effective cost‐balancing heuristic policy and demonstrate the efficacy of the proposed policy through extensive numerical experiments. We believe our results provide operators/managers valuable insights on how to devise effective service scheduling policies under varying costs.     

Discrete parallel machine makespan ScheLoc problem

Scheduling–Location (ScheLoc) problems integrate the separate fields of scheduling and location problems. In ScheLoc problems the objective is to find locations for the machines and a schedule for each machine subject to some production and location constraints such that some scheduling objective is minimized. In this paper we consider the discrete parallel machine makespan ScheLoc problem where the set of possible machine locations is discrete and a set of n jobs has to be taken to the machines and processed such that the makespan is minimized. Since the separate location and scheduling problem are both \(\mathcal {NP}\)-hard, so is the corresponding ScheLoc problem. Therefore, we propose an integer programming formulation and different versions of clustering heuristics, where jobs are split into clusters and each cluster is assigned to one of the possible machine locations. Since the IP formulation can only be solved for small scale instances we propose several lower bounds to measure the quality of the clustering heuristics. Extensive computational tests show the efficiency of the heuristics.     

Bi-objective multi-resource scheduling problem for emergency relief operations

Abstract

Resource scheduling for emergency relief operations is complex as it has many constraints. However, an effective allocation and sequencing of resources are crucial for the minimization of the completion times in emergency relief operations. Despite the importance of such decisions, only a few mathematical models of emergency relief operations have been studied. This article presents a bi-objective mixed integer programming (MIP) that helps to minimize both the total weighted time of completion of the demand points and the makespan of the total emergency relief operation. A two-phase method is developed to solve the bi-objective MIP problem. Additionally, a case study of hospital network in the Melbourne metropolitan area is used to evaluate the model. The results indicate that the model can successfully support the decisions required in the optimal resource scheduling of emergency relief operations.     

Robust Resource Allocation Decisions in Resource‐Constrained Projects*

The well‐known deterministic resource‐constrained project scheduling problem involves the determination of a predictive schedule (baseline schedule or pre‐schedule) of the project activities that satisfies the finish–start precedence relations and the renewable resource constraints under the objective of minimizing the project duration. This baseline schedule serves as a baseline for the execution of the project. During execution, however, the project can be subject to several types of disruptions that may disturb the baseline schedule. Management must then rely on a reactive scheduling procedure for revising or reoptimizing the baseline schedule. The objective of our research is to develop procedures for allocating resources to the activities of a given baseline schedule in order to maximize its stability in the presence of activity duration variability. We propose three integer programming–based heuristics and one constructive procedure for resource allocation. We derive lower bounds for schedule stability and report on computational results obtained on a set of benchmark problems.     

ALLOCATING AND SCHEDULING MOBILE DIAGNOSTIC IMAGING EQUIPMENT AMONG HOSPITALS

The allocation and weekly scheduling of mobile magnetic resonance imaging (MRI) units leased to a group of hospitals that share the equipment can be a complex problem. Similar problems occur in other domains where expensive equipment or facilities such as video conference facilities, aircraft, and supercomputers are leased. The crux of the problem was determining the number of days and which days of the week various types of equipment types should be leased to hospitals, so as to maximize the rental revenues and satisfy client preferences for days of the week and equipment types. We found rental revenues were a decreasing function of the number of days allocated to a hospital. We considered two sub-problems linked by a set of variables to model the problem. We show that one of these subproblems is a minimum cost network flow problem and the other is an integer multi-commodity transportation problem. We developed a procedure for solving the latter problem by exploiting earlier results for specialized networks. We conducted a computational study to evaluate the performance of this procedure and showed that it generally provides near-optimal integer solutions. We describe the development and implementation of a spreadsheet-based decision support system based on this model. This system was successfully implemented by a small firm with no expertise or prior experience using models.     

Intelligent scheduling model and algorithm for manufacturing

This paper puts forward an intelligent scheduling model based on Hopfield neural network and a unified algorithm for manufacturing. The energy computation function and its dynamic state equation are derived and discussed in detail about their coefficients (parameters) and steps (Delta t) in iteration towards convergence. The unified model is focused on the structure of the above function and equation, in which the goal and penalty items must be involved and meet different schedule models. The applications to different schedule mode including jobshop static scheduling, scheduling with due-date constraint or priority constraint, dynamic scheduling, and JIT (just in time) scheduling are discussed, and a series of examples with Gantt charts are illustrated.     

A simple linear time approximation algorithm for multi-processor job scheduling on four processors

Multiprocessor job scheduling problem has become increasingly interesting, for both theoretical study and practical applications. Theoretical study of the problem has made significant progress recently, which, however, seems not to imply practical algorithms for the problem, yet. Practical algorithms have been developed only for systems with three processors and the techniques seem difficult to extend to systems with more than three processors. This paper offers new observations and introduces new techniques for the multiprocessor job scheduling problem on systems with four processors. A very simple and practical linear time approximation algorithm of ratio bounded by 1.5 is developed for the multi-processor job scheduling problem P ₄|fix|C _max, which significantly improves previous results. Our techniques are also useful for multiprocessor job scheduling problems on systems with more than four processors.     

On lazy bureaucrat scheduling with common deadlines

Lazy bureaucrat scheduling is a new class of scheduling problems introduced by Arkin et al. (Inf. Comput. 184:129–146, 2003). In this paper we focus on the case where all the jobs share a common deadline. Such a problem is denoted as CD-LBSP, which has been considered by Esfahbod et al. (Algorithms and data structures. Lecture notes in computer science, vol. 2748, pp. 59–66, 2003). We first show that the worst-case ratio of the algorithm SJF (Shortest Job First) is two under the objective function [min-time-spent], and thus answer an open question posed in (Esfahbod, et al. in Algorithms and data structures. Lecture notes in computer science, vol. 2748, pp. 59–66, 2003). We further present two approximation schemes A _k and B _k both having worst-case ratio of , for any given integer k>0, under the objective functions [min-makespan] and [min-time-spent], respectively. Finally, we prove that the problem CD-LBSP remains NP-hard under several objective functions, even if all jobs share the same release time. A preliminary version of the paper appeared in Proceedings of the 7th Latin American Symposium on Theoretical Informatics, pp 515–523, 2006. Research of G. Zhang supported in part by NSFC (60573020).     

Job scheduling with dual criteria and sequence-dependent setups: mathematical versus genetic programming

Flexibility, speed, and efficiency are major challenges for operations managers in today's knowledge-intensive organizations. Such requirements are converted into three production scheduling criteria: (a) minimize the impact of setup times in flexible production lines when moving from one product to another, (b) minimize number of tardy jobs, and (c) minimize overall production time, or makespan, for a given set of products or services. There is a wide range of solution methodologies for such NP-hard scheduling problems. While mathematical programming models provide optimal solutions, they become too complex to model for large scheduling problems. Simultaneously, heuristic approaches are simpler and very often independent of the problem size, but provide “good” rather than optimal solutions. This paper proposes and compares two alternative solutions: 0-1 mixed integer linear programming and genetic programming. It also provides guidelines that can be used by practitioners in the process of selecting the appropriate scheduling methodology.     

A bicriteria approach to scheduling a single machine with job rejection and positional penalties

Single machine scheduling problems have been extensively studied in the literature under the assumption that all jobs have to be processed. However, in many practical cases, one may wish to reject the processing of some jobs in the shop, which results in a rejection cost. A solution for a scheduling problem with rejection is given by partitioning the jobs into a set of accepted and a set of rejected jobs, and by scheduling the set of accepted jobs among the machines. The quality of a solution is measured by two criteria: a scheduling criterion, F1, which is dependent on the completion times of the accepted jobs, and the total rejection cost, F2. Problems of scheduling with rejection have been previously studied, but usually within a narrow framework—focusing on one scheduling criterion at a time. This paper provides a robust unified bicriteria analysis of a large set of single machine problems sharing a common property, namely, all problems can be represented by or reduced to a scheduling problem with a scheduling criterion which includes positional penalties. Among these problems are the minimization of the makespan, the sum of completion times, the sum and variation of completion times, and the total earliness plus tardiness costs where the due dates are assignable. Four different problem variations for dealing with the two criteria are studied. The variation of minimizing F1+F2 is shown to be solvable in polynomial time, while all other three variations are shown to be $\mathcal{NP}$ -hard. For those hard problems we provide a pseudo polynomial time algorithm. An FPTAS for obtaining an approximate efficient schedule is provided as well. In addition, we present some interesting special cases which are solvable in polynomial time.