Scheduling problems with two competing agents to minimize minmax and minsum earliness measures

A relatively new class of scheduling problems consists of multiple agents who compete on the use of a common processor. We focus in this paper on a two-agent setting. Each of the agents has a set of jobs to be processed on the same processor, and each of the agents wants to minimize a measure which depends on the completion times of its own jobs. The goal is to schedule the jobs such that the combined schedule performs well with respect to the measures of both agents. We consider measures of minmax and minsum earliness. Specifically, we focus on minimizing maximum earliness cost or total (weighted) earliness cost of one agent, subject to an upper bound on the maximum earliness cost of the other agent. We introduce a polynomial-time solution for the minmax problem, and prove NP-hardness for the weighted minsum case. The unweighted minsum problem is shown to have a polynomial-time solution.     

Flowshop scheduling problem to minimize total completion time with random and bounded processing times

The flowshop scheduling problems with n jobs processed on two or three machines, and with two jobs processed on k machines are addressed where jobs have random and bounded processing times. The probability distributions of random processing times are unknown, and only the lower and upper bounds of processing times are given before scheduling. In such cases, there may not exist a unique schedule that remains optimal for all feasible realizations of the processing times, and therefore, a set of schedules has to be considered which dominates all other schedules for the given criterion. We obtain sufficient conditions when transposition of two jobs minimizes total completion time for the cases of two and three machines. The geometrical approach is utilized for flowshop problem with two jobs and k machines.     

Two meta-heuristics for parallel machine scheduling with job splitting to minimize total tardiness

Parallel machine scheduling is a popular research area due to its wide range of potential application areas. This paper focuses on the problem of scheduling n independent jobs to be processed on m identical parallel machines with the aim of minimizing the total tardiness of the jobs considering a job splitting property. It is assumed that a job can be split into sub-jobs and these sub-jobs can be processed independently on parallel machines. We present a mathematical model for this problem. The problem of total tardiness on identical parallel machines is NP-hard. Obtaining an optimal solution for this type of complex, large-sized problem in reasonable computational time by using an optimization solver is extremely difficult. We propose two meta-heuristics: Tabu search and simulated annealing. Computational results are compared on random generated problems with different sizes.     

Scheduling multiple orders per job in a single machine to minimize total completion time

This paper deals with a single-machine scheduling problem with multiple orders per job (MOJ) considerations. Both lot processing machines and item processing machines are also examined. There are two primary decisions that must be made in the proposed problem: (1) how to group the orders together, and (2) how to schedule the jobs once they are formed. In order to obtain the optimal solution to a scheduling problem, these two decisions should be made simultaneously. The performance measure is the total completion time of all orders. Two mixed binary integer programming models are developed to optimally solve this problem. Also, two efficient heuristics are proposed for solving large-sized problems. Computational results are provided to demonstrate the efficiency of the models and the effectiveness of the heuristics.     

Minimizing maximum tardiness and number of tardy jobs on parallel machines subject to minimum flow-time

This paper considers the problems of scheduling jobs on parallel identical machines where an optimal schedule is defined as one that gives the smallest maximum tardiness (or the minimum number of tardy jobs) among the set of schedules with optimal total flow-time (the sum of the completion times of all jobs). We show that these problems are unary NP-Hard, develop lower bounds for these two secondary criteria problems, and describe heuristic algorithms for their solution. Results of a computational study show that the proposed heuristic algorithms are quite effective and efficient in solving these hierarchical criteria scheduling problems.     

Minimizing total tardiness in an unrelated parallel-machine scheduling problem

We consider a problem of scheduling n independent jobs on m unrelated parallel machines with the objective of minimizing total tardiness. Processing times of a job on different machines may be different on unrelated parallel-machine scheduling problems. We develop several dominance properties and lower bounds for the problem, and suggest a branch and bound algorithm using them. Results of computational experiments show that the suggested algorithm gives optimal solutions for problems with up to five machines and 20 jobs in a reasonable amount of CPU time.     

Ordering the structural set of jobs,minimizing the total penalty

The problem of setting up a schedule minimizing the total penalty is solved. A job set is specified. There are no precedence conditions. Each job consists of elementary operations. Job interruptions are allowed. Interruptions of elementary operations are not allowed. The penalty for an operation of a job to be executed at instantt is. A schedule is evaluated by a penalty that is the sum of the penalties of all the elementary operations of all jobs in set. It is assumed that set is structural. The problem is to find the schedule with a start in a prescribed interval [ T₁,T₂] with the least penalty. Compatible structural schedules are introduced into consideration. The class of compatible structural schedules contains the solution of the problem. An algorithm solving the problem is proposed having complexity, in the worst case, whereK is the number of jobs in the set.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 229–264, 1979.     

Parallel-machine scheduling with non-simultaneous machine available time

We consider a problem of scheduling n independent jobs on m parallel identical machines. The jobs are available at time zero, but the machines may not be available simultaneously at time zero. We concentrate on two goals separately, namely, minimizing a cost function containing total completion time and total absolute differences in completion times; minimizing a cost function containing total waiting time and total absolute differences in waiting times. In this paper, we present polynomial time algorithm to solve this problem.     

On some properties of optimal schedules in the job shop problem with preemption and an arbitrary regular criterion

Optimal schedules in the job shop problem with preemption and with the objective of minimizing an arbitrary regular function of operation completion times are studied. It is shown that for any instance of the problem there always exists an optimal schedule that meets several remarkable properties. Firstly, each changeover date coincides with the completion time of some operation, and so, the number of changeover dates is not greater than the total number of operations, while the total number of interruptions of the operations is no more than the number of operations minus the number of jobs. Secondly, every changeover date is “super-integral”, which means that it is equal to the total processing time of some subset of operations. And thirdly, the optimal schedule with these properties can be found by a simple greedy algorithm under properly defined priorities of operations on machines. It is also shown that for any instance of the job shop problem with preemption allowed there exists a finite set of its feasible schedules which contains at least one optimal schedule for any regular objective function (from the continuum set of regular functions).     

Minimizing makespan and total completion time for parallel batch processing machines with non-identical job sizes

This paper considers the scheduling problem of parallel batch processing machines with non-identical job sizes. The jobs are processed in batches and the machines have the same capacity. The models of minimizing makespan and total completion time are given using mixed integer programming method and the computational complexity is analyzed. The bound on the number of feasible solutions is given and the properties of the optimal solutions are presented. Then a polynomial time algorithm is proposed and the worst case ratios for minimizing total completion time and makespan is proved to be 2 and (8/3–2/3 m) respectively. To test the proposed algorithm, we generate different levels of random instances. The computational results demonstrate the effectiveness of the algorithm for minimizing the two objectives.     

基于重新排序的退化工件最小化总延误时间问题

考虑了错位限制下的含有退化工件的重新排序问题,即工件的实际加工时间看作是工件开工时间的线性函数．重新排序就是在原始工件已经按照某种规则使目标函数达到最优时有一新工件集到达,新工件的安排使得原始工件重新排序进而产生错位．研究了最大序列错位和总序列错位限制下的退化工件最小化总延误时间问题,其最优排序的结构性质是使得原始工件集和新工件集中的工件是按加工率αj非减的序列排列,基于此通过分阶段排序和动态规划方法给出了两个问题的多项式时间的最优算法．     

Punctuality and idleness in just-in-time scheduling

In scheduling problems with irregular cost functions, such as deviation functions in earliness–tardiness scheduling problems, optimal solutions usually contain some idleness periods during which no activity is processed. Then, minimizing the penalties for not delivering on time and minimizing the idleness cost are two complementary criteria for a schedule. A Dynamic Programming procedure, in which the states are represented by continuous two-dimensional piecewise linear functions, is proposed to compute the cost of the Pareto optimal schedules.     

Minimizing due date related performance measures on two batch processing machines

This paper considers a scheduling problem in a two-machine flowshop of two batch processing machines. On each batch processing machine, jobs are processed in a batch, and each batch is allowed to contain jobs up to the maximum capacity of the associated machine. The scheduling problem is analyzed with respect to three due date related objectives including maximum tardiness, number of tardy jobs and total tardiness. In the analysis, several solution properties are characterized and based upon these properties, three efficient polynomial time algorithms are developed for minimizing the due date related measures.     

Due-window assignment with identical jobs on parallel uniform machines

A scheduling problem with a common due-window, earliness and tardiness costs, and identical processing time jobs is studied. We focus on the setting of both (i) job-dependent earliness/tardiness job weights and (ii) parallel uniform machines. The objective is to find the job allocation to the machines and the job schedule, such that the total weighted earliness and tardiness cost is minimized. We study both cases of a non-restrictive (i.e. sufficiently late), and a restrictive due-window. For a given number of machines, the solutions of the problems studied here are obtained in polynomial time in the number of jobs.     

Parallel machine scheduling with machine availability and eligibility constraints

In this paper we consider the problem of scheduling n independent jobs on m identical machines incorporating machine availability and eligibility constraints while minimizing the makespan. Each machine is not continuously available at all times and each job can only be processed on specified machines. A network flow approach is used to formulate this scheduling problem into a series of maximum flow problems. We propose a polynomial time binary search algorithm to either verify the infeasibility of the problem or solve it optimally if a feasible schedule exists.     

Single machine scheduling problems with controllable processing times and total absolute differences penalties

In this paper, we consider single machine scheduling problem in which job processing times are controllable variables with linear costs. We concentrate on two goals separately, namely, minimizing a cost function containing total completion time, total absolute differences in completion times and total compression cost; minimizing a cost function containing total waiting time, total absolute differences in waiting times and total compression cost. The problem is modelled as an assignment problem, and thus can be solved with the well-known algorithms. For the case where all the jobs have a common difference between normal and crash processing time and an equal unit compression penalty, we present an O(n log n) algorithm to obtain the optimal solution.     

A performance analysis of dispatching rules and a heuristic in static flowshops with missing operations of jobs

An experimental investigation of the performance of dispatching rules and a heuristic for scheduling in static flowshops with missing operations is undertaken in this study. The measure of performance is the minimization of total flow time of jobs. Permutation schedules are generated by using the heuristic for scheduling. General schedules, which can be permutation or non-permutation schedules, are obtained by using dispatching rules. Four dispatching rules, including a new dispatching rule, are considered. Two types of flowshops are studied: one with no missing operations of jobs and another with missing operations of jobs. In the latter type of flowshops, jobs with varying number of missing operations are considered. An extensive investigation of the performance of the dispatching rules and the heuristic is carried out. It is observed that the heuristic minimizes total flow time of jobs more than dispatching rules up to a certain level of missing operations of jobs in flowshops, after which dispatching rules perform better. The performance of the heuristic and the dispatching rules in terms of minimizing the makespan as a secondary measure is also reported.     

On the minimization of total weighted flow time with identical and uniform parallel machines

We consider the NP-hard problem of scheduling jobs on identical parallel machines to minimize total weighted flow time. We discuss the properties that characterize the structure of an optimal solution, present a lower bound and propose a branch and bound algorithm. The algorithm is superior to prior methods presented in the literature. We also extend the algorithm to uniform parallel machines and solve medium-sized problem instances.     

Coordination mechanisms with hybrid local policies

We study coordination mechanisms for scheduling n jobs on m parallel machines where agents representing the jobs interact to generate a schedule. Each agent acts selfishly to minimize the completion time of his/her own job, without considering the overall system performance that is measured by a central objective. The performance deterioration due to the lack of a central coordination, the so-called price of anarchy, is determined by the maximum ratio of the central objective function value of an equilibrium schedule divided by the optimal value. In the first part of the paper, we consider a mixed local policy with some machines using the SPT rule and other machines using the LPT rule. We obtain the exact price of anarchy for the problem of minimizing the makespan and some bounds for the problem of minimizing the total completion time. In the second part of the paper, we consider parallel machine scheduling subject to eligibility constraints. We devise new local policies based on the flexibilities and the processing times of the jobs. We show that the newly devised local policies outperform both the SPT and the LPT rules.     

Scheduling to minimize maximum earliness and number of tardy jobs where machine idle time is allowed

We address the bicriteria problem of minimizing the number of tardy jobs and maximum earliness on a single machine where machine idle time is allowed. We show that the problem of minimizing the number of tardy jobs while maximum earliness is kept at its minimum value of zero is polynomially solvable. We present polynomial time algorithms for the maximum earliness problem subject to no tardy jobs and the maximum earliness problem for a given set of tardy jobs. Finally, we discuss the use of the latter algorithm in generating all efficient schedules.