MINIMIZING ABSOLUTE AND SQUARED DEVIATION OF COMPLETION TIMES FROM DUE DATES

This is a study of a single-machine scheduling problem with the objective of minimizing the sum of a function of earliness and tardiness called the earliness and tardiness (ET) problem. I will show that if priority weights of jobs are proportional to their processing times, and if earliness and tardiness cost functions are linear, the problem will be equivalent to the total weighted tardiness problem. This proves that the et problem is np -hard. In addition, I present a heuristic algorithm with worst case bound for the et problem based on the equivalence relation between the two. When earliness and tardiness cost functions are quadratic, I consider the problem for a common due date for all jobs and for different job due dates. In general, the et problem with quadratic earliness and tardiness cost functions and all job weights equal to one is np -hard. I show that in many cases, when weights of jobs are proportional to their processing times, the problem can be solved efficiently. In the published results on the et problem with quadratic earliness and tardiness cost functions other researchers have assumed a zero starting time for the schedule. I discuss the advantages of a nonzero starting time for the schedule.     

Single machine scheduling: A comparison of two solution procedures

Recent research on the single machine scheduling problem has focused on the treatment of multiple scheduling objectives. Most works have used some combination of mean flowtime, maximum tardiness, or total tardiness as scheduling criteria. Previous research has largely ignored earliness as a scheduling criterion. This paper presents a model that employs the criteria of flowtime as a measure of work-in-process (WIP) inventory and total job earliness to represent finished goods inventory. Total tardiness is used to represent customer satisfaction. The three criteria are used to form a single, weighted-sum objective function for guiding the choice of the best processing sequence. Two procedures are presented that might be used to solve this problem. The first is an enumeration scheme using bounding and dominance criteria that have been developed to aid efficient solution, and the second is a mixed integer linear programming (LP) formulation. Computational experience with the two models is also presented.     

Single-machine scheduling with periodic due dates to minimize the total earliness and tardy penalty

We consider a single-machine scheduling problem such that the due dates are assigned to each job depending on its order, and the lengths of the intervals between consecutive due dates are identical. The objective is to minimize the total penalty for the earliness and tardiness of each job. The early penalty proportionally increases according to the earliness amount, while the tardy penalty increases according to the step function. We show that the problem is strongly NP-hard, and furthermore, polynomially solvable if the two types of processing times exist.

     

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.     

Optimal Policy for a Stochastic Scheduling Problem with Applications to Surgical Scheduling

We consider the stochastic, single‐machine earliness/tardiness problem (SET), with the sequence of processing of the jobs and their due‐dates as decisions and the objective of minimizing the sum of the expected earliness and tardiness costs over all the jobs. In a recent paper, Baker ( 2014 ) shows the optimality of the Shortest‐Variance‐First (SVF) rule under the following two assumptions: (a) The processing duration of each job follows a normal distribution. (b) The earliness and tardiness cost parameters are the same for all the jobs. In this study, we consider problem SET under assumption (b). We generalize Baker's result by establishing the optimality of the SVF rule for more general distributions of the processing durations and a more general objective function. Specifically, we show that the SVF rule is optimal under the assumption of dilation ordering of the processing durations. Since convex ordering implies dilation ordering (under finite means), the SVF sequence is also optimal under convex ordering of the processing durations. We also study the effect of variability of the processing durations of the jobs on the optimal cost. An application of problem SET in surgical scheduling is discussed.     

An integer programming approach to scheduling in a contaminated area

This paper presents a time-indexed integer programming formulation for scheduling dependent jobs executed by a team of workers in an area contaminated with radio-active or chemical materials. The dynamics of the harmful factor and the norms of organism recovery imply that each work period for a job should be immediately followed by a rest period for the worker executing this job and the length of the rest period depends on the start time of the corresponding work period. The problem is modeled as an NP-hard problem of scheduling on unrelated parallel processors with start time dependent processing times and different objective functions: maximum or total completion time and maximum or total tardiness. The special case of scheduling jobs executed by a single worker is also considered. Numerical examples and some computational results are reported.     

Minmax common flow-allowance problems with convex resource allocation and position-dependent workloads

We study minmax due-date based on common flow-allowance assignment and scheduling problems on a single machine, and extend known results in scheduling theory by considering convex resource allocation. The total cost function of a given job consists of its earliness, tardiness and flow-allowance cost components. Thus, the common flow-allowance and the actual jobs’ processing times are decision variables, implying that the due-dates and actual processing times can be controlled by allocating additional resource to the job operations. Consequently, our goal is to optimize a cost function by seeking the optimal job sequence, the optimal job-dependent due-dates along with the actual processing times. In all addressed problems we aim to minimize the maximal cost among all the jobs subject to a constraint on the resource consumption. We start by analyzing and solving the problem with position-independent workloads and then proceed to position-dependent workloads. Finally, the results are generalized to the method of common due-window. For all studied problems closed form solutions are provided, leading to polynomial time solutions.

     

Single-machine scheduling with production and rejection costs to minimize the maximum earliness

In this paper, we consider the single-machine scheduling problem with production and rejection costs to minimize the maximum earliness. If a job is accepted, then this job must be processed on the machine and a corresponding production cost needs be paid. If the job is rejected, then a corresponding rejection cost has to be paid. The objective is to minimize the sum of the maximum earliness of the accepted jobs, the total production cost of the accepted jobs and the total rejection cost of the rejected jobs. We show that this problem is equivalent to a single-machine scheduling problem to minimize the maximum earliness with two distinct rejection modes. In the latter problem, rejection cost might be negative in the rejection-award mode which is different from the traditional rejection-penalty mode in the previous literatures. We show that both of two problems are NP-hard in the ordinary sense and then provide two pseudo-polynomial-time algorithms to solve them. Finally, we also show that three special cases can be solved in polynomial time.     

A two-agent single machine scheduling problem with due-window assignment and a common flow-allowance

We study a single-machine scheduling model combining two competing agents and due-date assignment. The basic setting involves two agents who need to process their own sets of jobs, and compete on the use of a common processor. Our goal is to find the joint schedule that minimizes the value of the objective function of one agent, subject to an upper bound on the value of the objective function of the second agent. The scheduling measure considered in this paper is minimum total (earliness, tardiness and due-date) cost, based on common flow allowance, i.e., due-dates are defined as linear functions of the job processing times. We introduce a simple polynomial time solution for this problem (linear in the number of jobs), as well as to its extension to a multi-agent setting. We further extend the model to that of a due-window assignment based on common flow allowance.     

Total tardiness minimization on unrelated parallel machine scheduling with auxiliary equipment constraints

This research deals with scheduling jobs on unrelated parallel machines with auxiliary equipment constraints. Each job has a due date and requires a single operation. A setup for dies is incurred if there is a switch from processing one type of job to another type. For a die type, the number of dies is limited. Due to the attributes of the machines and the fitness of dies to each, the processing time for a job depends on the machine on which the job is processed, each job being restricted to processing on certain machines. In this paper, an effective heuristic based on threshold-accepting methods, tabu lists, and improvement procedures is proposed to minimize total tardiness. An extensive experiment is conducted to evaluate the computational characteristics of the proposed heuristic. Computational experiences demonstrate that the proposed heuristic is capable of obtaining optimal solutions for small-sized problems, and significantly outperforms an ATCS procedure and a simulated annealing method for problems in larger sizes.     

混合离散差分进化算法在单机批处理调度中的应用

本文研究单机批处理调度问题,批处理机有批次容量限制,批处理时间由每个批次所含作业中的最长作业处理时间决定。每个作业具有不同的大小、处理时间、提前拖期惩罚权重,所有作业具有公共交货期,且交货期无限晚。目标函数为最小化所有作业的加权提前拖期惩罚之和。该问题已被证明为NP难题,本研究找到了其最优解具有的一些性质,在此基础上利用它们提出了一种动态规划(DP)与差分进化(DE)算法相结合的混合离散差分进化(HDDE)算法来求解该问题,通过与传统的遗传算法、模拟退火算法和迭代贪婪算法进行对比,HDDE算法显示了更加强大的全局搜索能力。     

Scheduling in a Management Context: Uncertain Processing Times and Non-Regular Performance Measures*

Decision makers often face scheduling problems in which processing times are not known with certainty. Non-regular performance measures, in which both earliness and tardiness are penalized, are also becoming more common in both manufacturing and service operations. We model a managerial environment with task processing times (which include sequenceindependent set-up times) prescribed by three-parameter lognormal distributions. Upon completion, each task derives a reward given by a particular piecewise-linear reward function. The objective is to select a sequence of tasks maximizing the expected total reward. The relative generality of the problem renders many enumerative methods inapplicable or computationally intractable. To overcome such difficulties we develop efficient priorityinduced construction (PIC) heuristics which build up a complete schedule by inserting tasks (singly from a list) into a partial sequence of tasks. In each partial and complete sequence a period of idle time is permitted prior to the first task. Performance on realistic-sized problems is very encouraging, with cost penalties averaging less than one percent.     

A fast FPTAS for single machine scheduling problem of minimizing total weighted earliness and tardiness about a large common due date

We address the single machine scheduling problem to minimize the total weighted earliness and tardiness about a nonrestrictive common due date. This is a basic problem with applications to the just-in-time manufacturing. The problem is linked to a Boolean programming problem with a quadratic objective function, known as the half-product. An approach to developing a fast fully polynomial-time approximation scheme (FPTAS) for the problem is identified and implemented. The running time matches the best known running time for an FPTAS for minimizing a half-product with no additive constant.     

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.     

A tardiness-augmented approximation scheme for rejection-allowed multiprocessor rescheduling

In the multiprocessor scheduling problem to minimize the total job completion time, an optimal schedule can be obtained by the shortest processing time rule and the completion time of each job in the schedule can be used as a guarantee for scheduling revenue. However, in practice, some jobs will not arrive at the beginning of the schedule but are delayed and their delayed arrival times are given to the decision-maker for possible rescheduling. The decision-maker can choose to reject some jobs in order to minimize the total operational cost that includes three cost components: the total rejection cost of the rejected jobs, the total completion time of the accepted jobs, and the penalty on the maximum tardiness for the accepted jobs, for which their completion times in the planned schedule are their virtual due dates. This novel rescheduling problem generalizes several classic NP-hard scheduling problems. We first design a pseudo-polynomial time dynamic programming exact algorithm and then, when the tardiness can be unbounded, we develop it into a fully polynomial time approximation scheme. The dynamic programming exact algorithm has a space complexity too high for truthful implementation; we propose an alternative to integrate the enumeration and the dynamic programming recurrences, followed by a depth-first-search walk in the reschedule space. We implemented the alternative exact algorithm in C and conducted numerical experiments to demonstrate its promising performance.

     

Minimizing tardiness on a single processor with sequence-dependent setup times: a simulated annealing approach

In today's fast-paced Just-In-Time and mass customization manufacturing in a sequence-dependent setup environment, the challenge of making production schedules to meet due-date requirements is becoming a more complex problem. Unfortunately, much of the research on operations scheduling problems has either ignored setup times or assumed that setup times on each machine are independent of the job sequence. This paper considers the problem of minimizing tardiness, a common measure of due-date performance, in a sequence-dependent setup environment. Simulated annealing was used to solve the sequencing problem, and its performance was compared with random search. Our experimental results show that the algorithm can find a good solution fairly quickly, and thus can rework schedules frequently to react to variations in the schedule. The algorithm is invaluable for ‘on-line’ production scheduling and ‘last-minute’ changes to production schedule. The results of this research also suggest ways in which more complex and realistic job shop environments, such as multiple machines with a higher number of jobs in the sequence, and other scheduling objectives can be modeled. This research also investigates computational aspects of simulated annealing in solving complex scheduling problems.     

Multi-objective scheduling of two-job families on a single machine

This paper presents a preliminary analysis of the typical scheduling environment in semiconductor manufacturing involving multiple job families, and where more than one objective such as cycle time, machine utilization and the due-date accuracy needs to be simultaneously considered. In this study, the NP-hard problem of scheduling N independent jobs on a single testing machine with due dates and sequence-dependent setup times is addressed, where the multiple objectives are to minimize average cycle time, to minimize average tardiness, and to maximize machine utilization. A Pareto optimal solution, which is not inferior to any other feasible solutions in terms of all objectives, is generated combining the analytically optimal and conjunctive simulated scheduling approach. First, the machine-scheduling problem is modeled using the discrete event simulation approach and the problem is divided into simulation clock based lot selection sub-problems. Then, a Pareto optimal lot is selected using the compromise programming technique for multiobjective optimization at each decision instant in simulated time. With the help of a broad experimental design, this developed solution is then compared with common heuristic-dispatching rules such as SPT and EDD, which show better results for all the objectives over a wide range of problems. The developed scheduling method shows approximately 16.7% reduction in average cycle time, 25.6% reduction in average tardiness, and 21.6% improvement in machine utilization over the common dispatching rules, SPT and EDD.     

Scheduling two job families on a single machine with two competitive agents

We consider the scheduling problems arising when two agents, each with a family of jobs, compete to perform their respective jobs on a single machine. A setup time is needed for a job if it is the first job to be processed on the machine or its processing on the machine follows a job that belongs to another family. Each agent wants to minimize a certain cost function, which depends on the completion times of its jobs only. The aim is to find a schedule for all the jobs of the two agents that minimizes the objective of one agent while keeping the objective of the other agent being bounded by a fixed value \(Q\). Polynomial-time and pseudo-polynomial-time algorithms are designed to solve the problem involving various combinations of regular scheduling objective functions.     

Single-machine batch delivery scheduling with an assignable common due window

This paper addresses a batch delivery single-machine scheduling problem in which jobs have an assignable common due window. Each job will incur an early (tardy) penalty if it is early (tardy) with respect to the common due window under a given schedule. There is no capacity limit on each delivery batch, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. The objective is to find the optimal size and location of the window, the optimal dispatch date for each job, as well as an optimal job sequence to minimize a cost function based on earliness, tardiness, holding time, window location, window size, and batch delivery. We show that the problem can be optimally solved in O(n⁸)$O(n^8)$ time by a dynamic programming algorithm under a reasonable assumption on the relationships among the cost parameters. A computational experiment is also conducted to evaluate the performance of the proposed algorithm. We also show that some special cases of the problem can be optimally solved by lower order algorithms.     

COMMON DUE-WINDOW SCHEDULING

In this paper, we solve common due-window scheduling problems within the just-in-time window concept, i.e., scheduling problems including both earliness and tardiness penalties. We assume that jobs share the same due window and incur no penalty as long as they are completed within the due window. We further assume that the earliness and tardiness penalty factors are constant and that the size of the window is a given parameter. For cases where the location of the due window is a decision variable, we provide a polynomial algorithm with complexity O(n * log (n)) to solve the problem. For cases where the location of the due window is a given parameter, we use dynamic programming with pseudopolynomial complexity to solve the problem.