Non-clairvoyant Scheduling Games

In a scheduling game, each player owns a job and chooses a machine to execute it. While the social cost is the maximal load over all machines (makespan), the cost (disutility) of each player is the completion time of its own job. In the game, players may follow selfish strategies to optimize their cost and therefore their behaviors do not necessarily lead the game to an equilibrium. Even in the case there is an equilibrium, its makespan might be much larger than the social optimum, and this inefficiency is measured by the price of anarchy—the worst ratio between the makespan of an equilibrium and the optimum. Coordination mechanisms aim to reduce the price of anarchy by designing scheduling policies that specify how jobs assigned to a same machine are to be scheduled. Typically these policies define the schedule according to the processing times as announced by the jobs. One could wonder if there are policies that do not require this knowledge, and still provide a good price of anarchy. This would make the processing times be private information and avoid the problem of truthfulness. In this paper we study these so-called non-clairvoyant policies. In particular, we study the RANDOM policy that schedules the jobs in a random order without preemption, and the EQUI policy that schedules the jobs in parallel using time-multiplexing, assigning each job an equal fraction of CPU time.     

Efficient Coordination Mechanisms for Unrelated Machine Scheduling

We present three new coordination mechanisms for scheduling n selfish jobs on m unrelated machines. A coordination mechanism aims to mitigate the impact of selfishness of jobs on the efficiency of schedules by defining a local scheduling policy on each machine. The scheduling policies induce a game among the jobs and each job prefers to be scheduled on a machine so that its completion time is minimum given the assignments of the other jobs. We consider the maximum completion time among all jobs as the measure of the efficiency of schedules. The approximation ratio of a coordination mechanism quantifies the efficiency of pure Nash equilibria (price of anarchy) of the induced game. Our mechanisms are deterministic, local, and preemptive in the sense that the scheduling policy does not necessarily process the jobs in an uninterrupted way and may introduce some idle time. Our first coordination mechanism has approximation ratio Θ(logm) and always guarantees that the induced game has pure Nash equilibria to which the system converges in at most n rounds. This result improves a bound of O(log² m) due to Azar, Jain, and Mirrokni and, similarly to their mechanism, our mechanism uses a global ordering of the jobs according to their distinct IDs. Next we study the intriguing scenario where jobs are anonymous, i.e., they have no IDs. In this case, coordination mechanisms can only distinguish between jobs that have different load characteristics. Our second mechanism handles anonymous jobs and has approximation ratio $O(\frac{\log m}{\log\log m})$ although the game induced is not a potential game and, hence, the existence of pure Nash equilibria is not guaranteed by potential function arguments. However, it provides evidence that the known lower bounds for non-preemptive coordination mechanisms could be beaten using preemptive scheduling policies. Our third coordination mechanism also handles anonymous jobs and has a nice “cost-revealing” potential function. We use this potential function in order, not only to prove the existence of equilibria, but also to upper-bound the price of stability of the induced game by O(logm) and the price of anarchy by O(log² m). Our third coordination mechanism is the first that handles anonymous jobs and simultaneously guarantees that the induced game is a potential game and has bounded price of anarchy. In order to obtain the above bounds, our coordination mechanisms use m as a parameter. Slight variations of these mechanisms in which this information is not necessary achieve approximation ratios of O(m ^? ), for any constant ?>0.     

带尺寸约束的二机流水车间生产运输协调博弈调度问题

研究二机流水车间生产运输协调调度问题,当工件在第1台机器加工完成后,由1台带有容量限制的运输车分批次运输到第2台机器加工,运输过程考虑工件尺寸约束,目标函数为最小化最大完工时间.考虑到源于不同客户的工件对机器及运输设备的竞争,以工件为博弈方,工件在生产运输过程中等待时间为策略,各工件完工时间为收益,建立非合作博弈模型.通过将问题转化为马尔可夫决策过程,设计线性逼近值函数的Q-learning算法求解纳什均衡调度.实验结果表明Q-learning算法求得的纳什均衡调度具有较好的全局最优性,从而能够在满足客户的利益下,提高企业的生产效率,实现客户与企业的双赢.     

Preemptive scheduling on uniformly related machines: minimizing the sum of the largest pair of job completion times

We revisit the classic problem of preemptive scheduling on m uniformly related machines. In this problem, jobs can be arbitrarily split into parts, under the constraint that every job is processed completely, and that the parts of a job are not assigned to run in parallel on different machines. We study a new objective which is motivated by fairness, where the goal is to minimize the sum of the two maximal job completion times. We design a polynomial time algorithm for computing an optimal solution. The algorithm can act on any set of machine speeds and any set of input jobs. The algorithm has several cases, many of which are very different from algorithms for makespan minimization (algorithms that minimize the maximum completion time of any job), and from algorithms that minimize the total completion time of all jobs.     

An exact algorithm for parallel machine scheduling with conflicts

We consider an extension of classic parallel machine scheduling where a set of jobs is scheduled on identical parallel machines and an undirected conflict graph is part of the input. Each node in the graph represents a job, and an edge implies that its two jobs are conflicting, meaning that they cannot be scheduled on the same machine. The goal is to find an assignment of the jobs to the machines such that the maximum completion time (makespan) is minimized. We present an exact algorithm based on branch and price that combines methods from bin packing, scheduling, and graph coloring, with appropriate modifications. The algorithm has a good computational performance even for parallel machine scheduling without conflicting jobs.     

Equilibria for two parallel links: the strong price of anarchy versus the price of anarchy

We consider a job scheduling game with two uniformly related parallel machines (or links). Jobs are atomic players, and the delay of a job is the completion time of the machine running it. The private goal of each job is to minimize its own delay and the social goal is to minimize the maximum delay of any job, that is, to minimize the makespan. We consider the well known price of anarchy as well as the strong price of anarchy, and show that for a wide range of speed ratios these two measures are very different whereas for other speed ratios these two measures give the exact same bound. We extend all our results for models of restricted assignment, where a machine may have an initial load resulting from jobs that can only be assigned to this machine, and show tight results for all variants.     

Online scheduling on two uniform machines subject to eligibility constraints

We consider the online scheduling of a set of jobs on two uniform machines with the makespan as objective. The jobs are presented in a list. We consider two different eligibility constraint set assumptions, namely (i) arbitrary eligibility constraints and (ii) Grade of Service (GoS) eligibility constraints. In the first case, we prove that the High Speed Machine First (HSF) algorithm, which assigns jobs to the eligible machine that has the highest speed, is optimal. With regard to the second case, we point out an error in [M. Liu et al., Online scheduling on two uniform machines to minimize the makespan, Theoretical Computer Science 410 (21–23) (2009) 2099–2109]; we then provide tighter lower bounds and present algorithms with worst-case analysis for various ranges of machine speeds.     

Irreducible bin packing and normality in routing open shop

The open shop is a classical scheduling problem known since 1976, which can be described as follows. A number of jobs have to be processed by a given set of machines, each machine should perform an operation on every job, and the processing times of all the operations are given. One has to construct a schedule to perform all the operations to minimize finish time also known as the makespan. The open shop problem is known to be NP-hard for three and more machines, while is polynomially solvable in the case of two machines. We consider the routing open shop problem, being a generalization of both the open shop problem and the metric traveling salesman problem. In this setting, jobs are located at nodes of a transportation network and have to be processed by mobile machines, initially located at a predefined depot. Machines have to process all the jobs and return to the depot to minimize makespan. A feasible schedule is referred to as normal if its makespan coincides with the standard lower bound. We introduce the Irreducible Bin Packing decision problem, use it to describe new sufficient conditions of normality for the two machine problem, and discuss the possibility to extend these results on the problem with three and more machines. To that end, we present two new computer-aided optima localization results.

     

A minimum-cost network flow approach to preemptive parallel-machine scheduling

We model and solve the problems of preemptive scheduling of n independent jobs with release dates on m parallel machines with machine availability and eligibility constraints to minimize the makespan and maximum lateness as the minimum-cost network flow problem. We show that the approach can be extended to solve the corresponding scheduling problems on two uniform parallel machines.     

恒速处理机的多机Flow shop最小和调度问题的启发式算法分析

本文研究有n个作业需在5个处理机中心进行加工，处理机中心i由l1个恒速机组成的非抢占式多机flow shop调度最小和问题．每个作业有s个工序，每个工序需在对应的处理机中心的任一台机器上加工处理，作业到达前不能加工，所有作业通过处理机中心的路径相同．目标是确定一个作业在每个处理机中心机器上的可行调度序列，使所有作业在最后处理机中心的加权完成时间总和最小化．在作业处理时间需求、作业权重分别为独立同分布的有界随机变量时，通过特殊flow shop调度松弛方法，我们证明该问题在作业数趋于无穷时，一个基于有效作业最短加权平均处理时间需求的启发式算法是渐近最优的．     

Minimizing makespan subject to minimum flowtime on two identical parallel machines

We consider the problem of scheduling jobs on two parallel identical machines where an optimal schedule is defined as one that gives the smallest makespan (the completion time of the last job) among the set of schedules with optimal total flowtime (the sum of the completion times of all jobs). We propose an algorithm to determine optimal schedules for the problem, and describe a modified multifit algorithm to find an approximate solution to the problem in polynomial computational time. Results of a computational study to compare the performance of the proposed algorithms with a known heuristic shows that the proposed heuristic and optimization algorithms are quite effective and efficient in solving the problem.Scope and purposeMultiple objective optimization problems are quite common in practice. However, while solving scheduling problems, optimization algorithms often consider only a single objective function. Consideration of multiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore, enumerative optimization techniques and heuristic solution procedures are required to solve multi-objective scheduling problems. This paper illustrates the development of an optimization algorithm and polynomially bounded heuristic solution procedures for the scheduling jobs on two identical parallel machines to hierarchically minimize the makespan subject to the optimality of the total flowtime.     

Preemptive online scheduling with rejection of unit jobs on two uniformly related machines

We consider preemptive online and semi-online scheduling of unit jobs on two uniformly related machines. Jobs are presented one by one to an algorithm, and each job has a rejection penalty associated with it. A new job can either be rejected, in which case the algorithm pays its rejection penalty, or it can be scheduled preemptively on the machines, in which case it may increase the maximum completion time of any machine in the schedule, also known as the makespan of the constructed schedule. The objective is to minimize the sum of the makespan of the schedule of all accepted jobs and the total penalty of all rejected jobs. We study two versions of the problem. The first one is the online problem where the jobs arrive unsorted, and the second variant is the semi-online case, where the jobs arrive sorted by a non-increasing order of penalties. We also show that the variant where the jobs arrive sorted by a non-decreasing order of penalties is equivalent to the unsorted one. We design optimal online algorithms for both cases. These algorithms have smaller competitive ratios than the optimal competitive ratio for the more general problem with arbitrary processing times (except for the case of identical machines), but larger competitive ratios than the optimal competitive ratio for preemptive scheduling of unit jobs without rejection.     

Scheduling with multiple servers

In this paper, we consider the problem of scheduling a set of jobs on a set of identical parallel machines. Before the processing of a job can start, a setup is required which has to be performed by a given set of servers. We consider the complexity of such problems for the minimization of the makespan. For the problem with equal processing times and equal setup times we give a polynomial algorithm. For the problem with unit setup times, m machines and m − 1 servers, we give a pseudopolynomial algorithm. However, the problem with fixed number of machines and servers in the case of minimizing maximum lateness is proven to be unary NP-hard. In addition, recent algorithms for some parallel machine scheduling problems with constant precessing times are generalized to the corresponding server problems for the case of constant setup times. Moreover, we perform a worst case analysis of two list scheduling algorithms for makespan minimization.     

Bounded parallel-batching scheduling with two competing agents

We consider a scheduling problem in which two agents, each with a set of non-preemptive jobs, compete to perform their jobs on a common bounded parallel-batching machine. Each of the agents wants to minimize an objective function that depends on the completion times of its own jobs. The goal is to schedule the jobs such that the overall schedule performs well with respect to the objective functions of both agents. We focus on minimizing the makespan or the total completion time of one agent, subject to an upper bound on the makespan of the other agent. We distinguish two categories of batch processing according to the compatibility of the agents. In the case where the agents are incompatible, their jobs cannot be processed in the same batch, whereas all the jobs can be processed in the same batch when the agents are compatible. We show that the makespan problem can be solved in polynomial time for the incompatible case and is NP-hard in the ordinary sense for the compatible case. Furthermore, we show that the latter admits a fully polynomial-time approximation scheme. We prove that the total completion time problem is NP-hard and is polynomially solvable for the incompatible case with a fixed number of job types.     

Minimizing the makespan on a batch machine with non-identical job sizes: an exact procedure

A batch processing machine can simultaneously process several jobs forming a batch. This paper considers the problem of scheduling jobs with non-identical capacity requirements, on a single-batch processing machine of a given capacity, to minimize the makespan. The processing time of a batch is equal to the largest processing time of any job in the batch. We present some dominance properties for a general enumeration scheme and for the makespan criterion, and provide a branch and bound method. For large-scale problems, we use this enumeration scheme as a heuristic method.Scope and purposeUsually in classical scheduling problems, a machine can perform only one job at a time. Although, one can find machines that can process several jobs simultaneously as a batch. All jobs of a same batch have common starting and ending times. Batch processing machines are encountered in many different environments, such as burn-in operations in semiconductor industries or heat treatment operations in metalworking industries. In the first case, the capacity of the machine is defined by the number of jobs it can hold. In the second case, each job has a certain capacity requirement and the total size of a batch cannot exceed the capacity of the machine. Hence, the number of jobs contained in each batch may be different. In this paper, we consider this second case (which is more difficult) and we provide an exact method for the makespan criterion (minimizing the last ending time).     

Nonpreemptive Coordination Mechanisms for Identical Machines

We focus on the problem of scheduling n weighted selfish tasks on m identical parallel machines and we study the performance of nonpreemptive coordination mechanisms. A nonpreemptive coordination mechanism consists of m local scheduling policies that decide the processing order of the tasks on each machine without delays or interruptions. We discuss the existence of Nash equilibria for this setting and show that it is not a guaranteed property of all nonpreemptive coordination mechanisms. Next, we focus on the wider class of randomized Nash equilibria and prove lower bounds on the price of anarchy. Our lower bounds are presented in comparison to the currently best known coordination mechanism (which uses delays) and lead to the conclusion that preemption or delays are required in order to improve on the currently best known solution.     

Some scheduling problems with deteriorating jobs and learning effects

Although scheduling with deteriorating jobs and learning effect has been widely investigated, scheduling research has seldom considered the two phenomena simultaneously. However, job deterioration and learning co-exist in many realistic scheduling situations. In this paper, we introduce a new scheduling model in which both job deterioration and learning exist simultaneously. The actual processing time of a job depends not only on the processing times of the jobs already processed but also on its scheduled position. For the single-machine case, we derive polynomial-time optimal solutions for the problems to minimize makespan and total completion time. In addition, we show that the problems to minimize total weighted completion time and maximum lateness are polynomially solvable under certain agreeable conditions. For the case of an m-machine permutation flowshop, we present polynomial-time optimal solutions for some special cases of the problems to minimize makespan and total completion time.     

A computer simulation model for job shop scheduling problems minimizing makespan

One of the basic and significant problems, that a shop or a factory manager is encountered, is a suitable scheduling and sequencing of jobs on machines. One type of scheduling problem is job shop scheduling. There are different machines in a shop of which a job may require some or all these machines in some specific sequence. For solving this problem, the objective may be to minimize the makespan. After optimizing the makespan, the jobs sequencing must be carried out for each machine. The above problem can be solved by a number of different methods such as branch and bound, cutting plane, heuristic methods, etc. In recent years, researches have used genetic algorithms, simulated annealing, and machine learning methods for solving such problems. In this paper, a simulation model is presented to work out job shop scheduling problems with the objective of minimizing makespan. The model has been coded by Visual SLAM which is a special simulation language. The structure of this language is based on the network modeling. After modeling the scheduling problem, the model is verified and validated. Then the computational results are presented and compared with other results reported in the literature. Finally, the model output is analyzed.     

Flowshop scheduling of deteriorating jobs on dominating machines

In this paper we consider the general, no-wait and no-idle permutation flowshop scheduling problem with deteriorating jobs, i.e., jobs whose processing times are increasing functions of their starting times. We assume a linear deterioration function with identical increasing rates for all the jobs and there are some dominating relationships between the machines. We show that the problems to minimize the makespan and the total completion time remain polynomially solvable when deterioration is considered, although these problems are more complicated than their classical counterparts without deterioration.     

A dominant class of schedules for malleable jobs in the problem to minimize the total weighted completion time

This paper is about scheduling parallel jobs, i.e. which can be executed on more than one machine at the same time. Malleable jobs is a special class of parallel jobs. The number of machines a malleable job is executed on may change during its execution.In this work, we consider the NP-hard problem of scheduling malleable jobs to minimize the total weighted completion time (or mean weighted flow time). For this problem, we introduce the class of “ascending” schedules in which, for each job, the number of machines assigned to it cannot decrease over time while this job is being processed.We prove that, under a natural assumption on the processing time functions of jobs, the set of ascending schedules is dominant for the problem. This result can be used to reduce the search space while looking for an optimal solution.