Parallel machines scheduling to minimize job tardiness and machine deteriorating cost with deteriorating jobs

This paper studies the parallel machines bi-criteria scheduling problem (PMBSP) in a deteriorating system. Sequencing and scheduling problems (SSP) have seldom considered the two phenomena concurrently. This paper discusses the parallel machines scheduling problem with the effects of machine and job deterioration. By the machine deterioration effect, we mean that each machine deteriorates at a different rate. This deterioration is considered in terms of cost which depends on the production rate, the machine’s operating characteristics and the kind of work done by each machine. Moreover, job processing times are increasing functions of their starting times and follow a simple linear deterioration. The objective functions are minimizing total tardiness and machine deteriorating cost. The problem of total tardiness on identical parallel machines is NP-hard, thus the problem with machine deteriorating cost as an additional term is also NP-hard. We propose the LP-metric method to show the importance of our proposed multi-objective problem. A metaheuristic algorithm is developed to locate optimal or near optimal solutions based on a Tabu search mechanism. Numerical examples are presented to show the efficiency of this model.     

Scheduling deteriorating jobs with a learning effect on unrelated parallel machines

In this study we consider unrelated parallel machines scheduling problems with learning effect and deteriorating jobs, in which the actual processing time of a job is a function of joint time-dependent deterioration and position-dependent learning. The objective is to determine the jobs assigned to corresponding each machine and the corresponding optimal schedule to minimize a cost function containing total completion (waiting) time, total absolute differences in completion (waiting) times and total machine load. If the number of machines is a given constant, we show that the problems can be solved in polynomial time under the time-dependent deterioration and position-dependent learning model.     

Parallel machines scheduling with a deteriorating maintenance activity

In this paper, we consider parallel identical machines scheduling problems with a deteriorating maintenance activity. In this model, each machine has a deteriorating maintenance activity, that is, delaying the maintenance increases the time required to perform it. We need to make a decision on when to schedule the deteriorating maintenance activities and the sequence of jobs to minimize total completion time. We provide a polynomial time algorithm to solve the total completion time minimization problem.     

Parallel machine scheduling with a deteriorating maintenance activity and total absolute differences penalties

In this paper we consider identical parallel machines scheduling problems with a deteriorating maintenance activity. In this model, each machine has a deteriorating maintenance activity, that is, delaying the maintenance increases the time required to perform it. We need to make a decision on when to schedule the rate-modifying activities and the sequence of jobs to minimize some objective function. We concentrate on two goals separately, namely, minimizing the total absolute differences in completion times (TADC) and the total absolute differences in waiting times (TADW). We show that the problems remain polynomially solvable under the proposed model.     

Optimal maintenance of two stochastically deteriorating machines with an intermediate buffer

We consider a manufacturing system in which an input generating installation transfers a raw material to a subsequent production unit. Both machines deteriorate stochastically with usage and may fail. For each machine the deteriorating process is described by some known transition probabilities between different degrees of deterioration. A buffer has been built between the two machines in order to cope with unexpected failures of the installation. A discrete-time Markov decision model is formulated for the optimal preventive maintenance of both machines. The maintenance times are geometrically distributed and the cost structure includes operating costs, storage costs, maintenance costs and costs due to the lost production. It is proved that for fixed buffer content and for fixed deterioration degree of one machine, the average-cost optimal policy initiates a preventive maintenance of the other machine if and only if its degree of deterioration exceeds some critical level. We study, by means of numerical results, the effect of the variation of some parameters on the optimal policy and on the minimum average cost. For the case in which the maintenance times follow continuous distributions, an approximate discrete-time Markov decision model is proposed.     

Single-machine scheduling with maintenance and repair rate-modifying activities

Most papers in the scheduling field are based on the assumption that machines are always available at constant speed. However, in industry applications, it is very common for a machine to be in subnormal condition after running for a certain period of time. Motivated by a problem commonly found in the surface-mount technology of electronic assembly lines, this paper deals with scheduling problems involving repair and maintenance rate-modifying activities. When a machine is running at less than an efficient speed, a production planner can decide to stop the machine and maintain it or wait and maintain it later. If the choice is made to continue running the machine without fixing it, it is possible that the machine will break down and repair will be required immediately. Both maintenance and repair activities can change the machine speed from a sub-normal production rate to a normal one. Hence, we call them rate-modifying activities. Our purpose here is to simultaneously sequence jobs and schedule maintenance activity to optimize regular performance measures. In this paper, we assume that processing time is deterministic, while machine break down is a random process following certain distributions. We consider two types of processing cases: resumable and nonresumable. We study problems with objective functions such as expected makespan, total expected completion time, maximum expected lateness, and expected maximum lateness, respectively. Several interesting results are obtained, especially for the nonresumable case.     

Two-agent single-machine scheduling problems under increasing linear deterioration

A new scheduling model in which both two-agent and increasing linear deterioration exist simultaneously is investigated in this paper. The processing time of a job is defined as an increasing linear function of its starting time. Two agents compete to perform their respective jobs on a common single machine and each agent has his own criterion to optimize. We introduce an increasing linear deterioration model into the two-agent single-machine scheduling, where the goal is to minimize the objective function of the first agent with the restriction that the objective function of the second agent cannot exceed a given upper bound. We study two scheduling problems with the different combinations of two agents’ objective functions: makespan, maximum lateness, maximum cost and total completion time. We propose the optimal properties and present the optimal polynomial time algorithms to solve the scheduling problems, respectively.     

Single-machine scheduling problems with both start-time dependent learning and position dependent aging effects under deteriorating maintenance consideration

In this paper we introduce a new model of joint start-time dependent learning and position dependent aging effects into single-machine scheduling problems. The machine may need maintenance to improve its production efficiency. The objectives are to find jointly the optimal maintenance position and the optimal sequence such that the makespan, the total completion time, and the total absolute deviation of completion times (TADC) are minimized. We also aim to determine jointly the optimal maintenance position, the optimal due-window size and location, and the optimal sequence to minimize the sum of earliness, tardiness and due-window related costs function. We show that all the studied problems can be optimally solved by polynomial time algorithms.     

Two-parallel machines scheduling with rate-modifying activities to minimize total completion time

This paper considers the two-parallel machines scheduling problem with rate-modifying activities. In this model, each machine has a rate-modifying activity that can change the processing rate of machine under consideration. Hence the actual processing times of jobs vary depending on whether the job is scheduled before or after the rate-modifying activity. We need to make a decision on when to schedule the rate-modifying activities and the sequence of jobs to minimize some objective function. We provide polynomial and pseudo-polynomial time algorithms to solve the total completion time minimization problem and total weighted completion time minimization problem under agreeable ratio condition.     

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.     

A note on parallel-machine scheduling with deteriorating jobs

We consider a parallel-machine scheduling problem of minimizing the total completion time. The processing time of a job is a linear function of its starting time and deterioration rate. This problem is known to be NP-hard, even for the case with two machines. In this note, we generalize an existing lower bound for the two-machine case to the general case with an arbitrary number of machines. Despite the generalization concerning machine number, our bound has one extra term that makes our bound tighter than the existing one.     

带两个服务等级的三台机最优在线算法

研究了带服务等级约束的三台平行机在线排序问题.每台机器和每个工件的服务等级为1或者2,工件只能在等级不高于它的机器上加工,即等级为1的工件只能在等级为1的机器上加工,等级为2的工件可在所有机器上加工.每个工件的加工时间为一个单位,目标是极小化所有工件的总完工时间.考虑两种情形:当一台机器等级为1,两台机器等级为2时,给出了竞争比为17/14的最优在线算法;当两台机器等级为1,一台机器等级为2时,给出了竞争比为43/36的最优在线算法.     

Single machine scheduling with time-dependent linear deterioration and rate-modifying maintenance

We study single machine scheduling problems with linear time-dependent deterioration effects and maintenance activities. Maintenance periods (MPs) are included into the schedule, so that the machine, that gets worse during the processing, can be restored to a better state. We deal with a job-independent version of the deterioration effects, that is, all jobs share a common deterioration rate. However, we introduce a novel extension to such models and allow the deterioration rates to change after every MP. We study several versions of this generalized problem and design a range of polynomial-time solution algorithms that enable the decision-maker to determine possible sequences of jobs and MPs in the schedule, so that the makespan objective can be minimized. We show that all problems reduce to a linear assignment problem with a product matrix and can be solved by methods very similar to those used for solving problems with positional effects.     

A hybrid genetic algorithm for an identical parallel-machine problem with maintenance activity

The scheduling of maintenance activities has been extensively studied, with most studies focusing on single-machine problems. In real-world applications, however, multiple machines or assembly lines process numerous jobs simultaneously. In this paper, we study a parallel-machine scheduling problem in which the objective is to minimize the total tardiness given that there is a maintenance activity on each machine. We develop a branch-and-bound algorithm to solve the problem with a small problem size. In addition, we propose a hybrid genetic algorithm to obtain the approximate solutions when the number of jobs is large. The performance of the proposed algorithms is evaluated based mainly on computational results.     

Minimizing total flow time in the single-machine scheduling problem with periodic maintenance

This paper deals with a single-machine scheduling problem with limited machine availability. The limited availability of machine results from periodic maintenance activities. In our research, a periodic maintenance schedule consists of several maintenance periods. Each maintenance period is scheduled after a periodic time interval. The objective is to find a schedule that minimizes the total flow time subject to periodic maintenance and nonresumable jobs. Some important theorems are proved for the problem. A branch-and-bound algorithm that utilizes several theorems is proposed to find the optimal schedule. We also develop a heuristic to solve large sized problems. In this paper, computational results show that the proposed heuristic is highly accurate and efficient.     

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.     

Machine scheduling with an availability constraint

Most literature in scheduling assumes that machines are available simultaneously at all times. However, this availability may not be true in real industry settings. In this paper, we assume that the machine may not always be available. This happens often in the industry due to a machine breakdown (stochastic) or preventive maintenance (deterministic) during the scheduling period. We study the scheduling problem under this general situation and for the deterministic case.We discuss various performance measures and various machine environments. In each case, we either provide a polynomial optimal algorithm to solve the problem, or prove that the problem is NP-hard. In the latter case, we develop pseudo-polynomial dynamic programming models to solve the problem optimally and/or provide heuristics with an error bound analysis.This research was supported in part by NSF grant DDM 9201627     

Two-machine flowshop scheduling with availability constraints

The majority of the scheduling literature carries a common assumption that machines are available all the time. However, this availability assumption may not be true in real industry settings, since a machine may become unavailable during certain periods of time when, for instance, a machine breakdown or a preventive maintenance activity is scheduled. Although the problem is realistic and important, it is relatively new and unstudied. In this paper, we study the two-machine flowshop problem under the assumption that the unavailable time is known in advance. We assume that if a job cannot be finished before the next down period of a machine then the job will have to partially restart when the machine has become available again. We call our model semiresumable. Our model contains two important special cases: resumable where the job can be continued without any penalty and nonresumable where the job needs to totally restart. We study the problem where an availability constraint is imposed only on one machine as well as on both machines. We provide complexity analysis, develop a pseudo-polynomial dynamic programming algorithm to solve the problem optimally and also propose heuristic algorithms with an error bound analysis.     

Optimal on-line flow time with resource augmentation

We study the problem of scheduling n jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has ? machines. We design an algorithm of competitive ratio , where Δ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant ?. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only Δ and n are known. This gives a trade-off between the resource augmentation and the competitive ratio.We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has m machines and the on-line algorithm has ?m machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines, and scheduling so as to minimize the total completion time.     

工件带简单线性恶化函数和共同交货期单机排序问题

本文研究单机排序问题,其中工件加工时间具有简单线性恶化函数.同时,所有工件均具有一个给定共同交货期.目标函数为最小化提前有奖延误受罚之和.在逆一致性条件下,给出了求解该排序问题的一个伪多项式时间动态规划算法.同时借助于几何舍入技巧,对求解这类排序问题给出了一个充分多项式时间的近似算法(FPTAS)。