An optimal online algorithm for the parallel-batch scheduling with job processing time compatibilities

We consider the online (over time) scheduling on a single unbounded parallel-batch machine with job processing time compatibilities to minimize makespan. In the problem, a constant \(\alpha >0\) is given in advance. Each job \(J_{j}\) has a normal processing time \(p_j\). Two jobs \(J_i\) and \(J_j\) are compatible if \(\max \{p_i, p_j\} \le (1+\alpha )\cdot \min \{p_i, p_j\}\). In the problem, mutually compatible jobs can form a batch being processed on the machine. The processing time of a batch is equal to the maximum normal processing time of the jobs in this batch. For this problem, we provide an optimal online algorithm with a competitive ratio of \(1+\beta _\alpha \), where \(\beta _\alpha \) is the positive root of the equation \((1+\alpha )x^{2}+\alpha x=1+\alpha \).     

A temporal investigation of the direct,interactive, and reverse relations between demand and control and affective strain

Abstract

Few studies have tested how stressors affect outcomes over time. We sought to extend the literature by means of a longitudinal study testing for direct, interactive, and causal relations between demands and control and affective strain. We extended prior work testing causal relationships for Karasek's (1979 Karasek, R.A. Jr. 1979. Job demands, job decision latitude, and mental strain: Implications for job redesign. Administrative Science Quarterly, 24: 285–307. [Crossref], [Web of Science ®] , [Google Scholar]) Job Demand-Control (JDC) model by examining both the effects of demands and control on strain and in turn the effects of strain on demand and control. We tested our hypotheses using hierarchical linear modelling with a military sample of 1539 soldiers who completed six waves of survey data at 3-month time lags. The results replicate earlier cross-sectional studies reporting effects of work characteristics on strain; however, in our study these effects did not persist past three months. The results also provide evidence for reverse causal effects such that higher strain was associated with higher subsequent work overload and lower control over a six month time period. Similar to past research, we did not find support for the interactive effects of work overload and control on strain. We discuss the implications of our findings for theory and practice (such as the optimum time for applying interventions during the management of change), especially in terms of understanding the specific time lags for different stress–strain associations and the need for additional theories to explain reverse relationships.     

A simple greedy approximation algorithm for the minimum connected $$$$-Center problem

In this paper, we consider the connected \(k\)-Center (\(CkC\)) problem, which can be seen as the classic \(k\)-Center problem with the constraint of internal connectedness, i.e., two nodes in a cluster are required to be connected by an internal path in the same cluster. \(CkC\) was first introduced by Ge et al. (ACM Trans Knowl Discov Data 2:7, 2008), in which they showed the \(NP\)-completeness for this problem and claimed a polynomial time approximation algorithm for it. However, the running time of their algorithm might not be polynomial, as one key step of their algorithm involves the computation of an \(NP\)-hard problem. We first present a simple polynomial time greedy-based \(2\)-approximation algorithm for the relaxation of \(CkC\)—the \(CkC^*\). Further, we give a \(6\)-approximation algorithm for \(CkC\).     

The cost of selfishness for maximizing the minimum load on uniformly related machines

Consider the following scheduling game. A set of jobs, each controlled by a selfish agent, are to be assigned to m uniformly related machines. The cost of a job is defined as the total load of the machine that its job is assigned to. A job is interested in minimizing its cost, while the social objective is maximizing the minimum load (the value of the cover) over the machines. This goal is different from the regular makespan minimization goal, which was extensively studied in a game theoretic context. We study the price of anarchy (poa) and the price of stability (pos) for uniformly related machines. The results are expressed in terms of s, which is the maximum speed ratio between any two machines. For uniformly related machines, we prove that the pos is unbounded for s>2, and the poa is unbounded for s≥2. For the remaining cases we show that while the poa grows to infinity as s tends to 2, the pos is at most 2 for any s≤2.     

Online scheduling with rejection and reordering: exact algorithms for unit size jobs

We study an online scheduling problem with rejection on \(m\ge 2\) identical machines, in which we deal with unit size jobs. Each arriving job has a rejection value (a rejection cost or penalty for minimization problems, and a rejection profit for maximization problems) associated with it. A buffer of size \(K\) is available to store \(K\) jobs. A job which is not stored in the buffer must be either assigned to a machine or rejected. Upon the arrival of a new job, the job can be stored in the buffer if there is a free slot (possibly created by evicting other jobs and assigning or rejecting every evicted job). At termination, the buffer must be emptied. We study four variants of the problem, as follows. We study the makespan minimization problem, where the goal is to minimize the sum of the makespan and the penalty of rejected jobs, and the \(\ell _p\) norm minimization problem, where the goal is to minimize the sum of the \(\ell _p\) norm of the vector of machine completion times and the penalty of rejected jobs. We also study two maximization problems, where the goal in the first version is to maximize the sum of the minimum machine load (the cover value of the machines) and the total rejection profit, and in the second version the goal is to maximize a function of the machine completion times (which measures the balance of machine loads) and the total rejection profit. We show that an optimal solution (an exact solution for the offline problem) can always be obtained in this environment, and determine the required buffer size. Specifically, for all four variants we present optimal algorithms with \(K=m-1\) and prove that in each case, using a buffer of size at most \(m-2\) does not allow the design of an optimal algorithm, which makes our algorithms optimal in this respect as well. The lower bounds hold even for the special case where the rejection value is equal for all input jobs.     

Lawler’s minmax cost problem under uncertainty

The well-known \(O(n^2)\) minmax cost algorithm of Lawler (Manag Sci 19(5):544–546, 1973) was developed to minimize the maximum cost of jobs processed by a single machine under precedence constraints. We first develop a fast updating algorithm to obtain optimal solutions for two neighboring instances. This method will be used throughout this article. Then we consider job cost functions that depend on the completion time and on one or more additional numerical parameters. The parameters are uncertain and take values from given intervals. Under the uncertainty, we apply the minmax regret criterion for choosing a solution. We generalize results by Brauner et al. (J Sched, 2015) for decomposable cost functions with deterministic processing times and a single uncertain parameter to general cost functions. We describe different conditions, under which minmax regret solutions can be obtained with the time complexity \(O(n^3)\) or \(O(n^2)\). Then the updating algorithm is applied to the lateness model by Kasperski (Oper Res Lett 33:431–436, 2005) where both the processing time and the due date of each job are uncertain. The original \(O(n^4)\) running time is improved to the time complexity \(O(n^3)\). Finally, we extend the cost functions with a single uncertain parameter to those with a vector of additional uncertain parameters, improve one of the complexity results by Volgenant and Duin (Comput Oper Res 37:909–915, 2010) and solve some new problems.     

Minimizing the number of tardy jobs in two-machine settings with common due date

We consider two-machine scheduling problems with job selection. We analyze first the two-machine open shop problem and provide a best possible linear time algorithm. Then, a best possible linear time algorithm is derived for the job selection problem on two unrelated parallel machines. We also show that an exact approach can be derived for both problems with complexity \(O(p(n) \times \sqrt{2}^n)\), p being a polynomial function of n.     

Total completion time minimization in online hierarchical scheduling of unit-size jobs

This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Our goal is to minimize the total completion time of all jobs. Each job has a unit processing time and a hierarchy. The job with a lower hierarchy can only be processed on the first machine and the job with a higher hierarchy can be processed on any one of m machines. We first show that the lower bound of this problem is at least \(1+\min \{\frac{1}{m}, \max \{\frac{2}{\lceil x\rceil +\frac{x}{\lceil x\rceil }+3}, \frac{2}{\lfloor x\rfloor +\frac{x}{\lfloor x\rfloor }+3}\}\), where \(x=\sqrt{2m+4}\). We then present a greedy algorithm with tight competitive ratio of \(1+\frac{2(m-1)}{m(\sqrt{4m-3}+1)}\). The competitive ratio is obtained in a way of analyzing the structure of the instance in the worst case, which is different from the most common method of competitive analysis. In particular, when \(m=2\), we propose an optimal online algorithm with competitive ratio of \(16\) \(/\) \(13\), which complements the previous result which provided an asymptotically optimal algorithm with competitive ratio of 1.1573 for the case where the number of jobs n is infinite, i.e., \(n\rightarrow \infty \).     

Semi-online scheduling for jobs with release times

In this paper we consider three semi-online scheduling problems for jobs with release times on m identical parallel machines. The worst case performance ratios of the LS algorithm are analyzed. The objective function is to minimize the maximum completion time of all machines, i.e. the makespan. If the job list has a non-decreasing release times, then $2-\frac{1}{m}$ is the tight bound of the worst case performance ratio of the LS algorithm. If the job list has non-increasing processing times, we show that $2-\frac{1}{2m}$ is an upper bound of the worst case performance ratio of the LS algorithm. Furthermore if the job list has non-decreasing release times and the job list has non-increasing processing times we prove that the LS algorithm has worst case performance ratio not greater than $\frac{3}{2} -\frac{1}{2m}$ .     

Scheduling with release times and rejection on two parallel machines

In this paper we study scheduling with release times and job rejection on two parallel machines. In our scheduling model each job is either accepted and then processed by one of the two machines at or after its release time, or it is rejected and then a rejection penalty is paid. The objective is to minimize the makespan of the accepted job plus the total penalty of all rejected jobs. The scheduling problem is NP-hard in the ordinary sense. In this paper, we develop a \(1.5+\epsilon \)-approximation algorithm for the problem, where \(\epsilon \) is any given small positive constant.     

Some extremal results on the colorful monochromatic vertex-connectivity of a graph

A path in a vertex-colored graph is called a vertex-monochromatic path if its internal vertices have the same color. A vertex-coloring of a graph is a monochromatic vertex-connection coloring (MVC-coloring for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph G, the monochromatic vertex-connection number, denoted by mvc(G), is defined to be the maximum number of colors used in an MVC-coloring of G. These concepts of vertex-version are natural generalizations of the colorful monochromatic connectivity of edge-version, introduced by Caro and Yuster (Discrete Math 311:1786–1792, 2011). In this paper, we mainly investigate the Erd?s–Gallai-type problems for the monochromatic vertex-connection number mvc(G) and completely determine the exact value. Moreover, the Nordhaus–Gaddum-type inequality for mvc(G) is also given.     

Online tradeoff scheduling on a single machine to minimize makespan and maximum lateness

In this paper, we consider the following single machine online tradeoff scheduling problem. A set of n independent jobs arrive online over time. Each job \(J_{j}\) has a release date \(r_{j}\), a processing time \(p_{j}\) and a delivery time \(q_{j}\). The characteristics of a job are unknown until it arrives. The goal is to find a schedule that minimizes the makespan \(C_{\max } = \max _{1 \le j \le n} C_{j}\) and the maximum lateness \(L_{\max } = \max _{1 \le j \le n} L_{j}\), where \(L_{j} = C_{j} + q_{j}\). For the problem, we present a nondominated \(( \rho , 1 + \displaystyle \frac{1}{\rho } )\)-competitive online algorithm for each \(\rho \) with \( 1 \le \rho \le \displaystyle \frac{\sqrt{5} + 1}{2}\).     

A coordination mechanism for a scheduling game with parallel-batching machines

In this paper we consider the scheduling problem with parallel-batching machines from a game theoretic perspective. There are m parallel-batching machines each of which can handle up to b jobs simultaneously as a batch. The processing time of a batch is the time required for processing the longest job in the batch, and all the jobs in a batch start and complete at the same time. There are n jobs. Each job is owned by a rational and selfish agent and its individual cost is the completion time of its job. The social cost is the largest completion time over all jobs, the makespan. We design a coordination mechanism for the scheduling game problem. We discuss the existence of pure Nash Equilibria and offer upper and lower bounds on the price of anarchy of the coordination mechanism. We show that the mechanism has a price of anarchy no more than \(2-\frac{2}{3b}-\frac{1}{3\max \{m,b\}}\).     

Assessing and Increasing Staff Preference for Job Tasks Using Concurrent-Chains Schedules and Probabilistic Outcomes

The assessment and improvement of staff members' subjective valuation of nonpreferred work tasks may be one way to increase the quality of staff members' work life. The Task Enjoyment Motivation Protocol (Green, Reid, Passante, &; Canipe, 2008 Green, C. W., Reid, D. H., Passante, S. and Canipe, V. 2008. Changing less-preferred duties to more-preferred: A potential strategy for improving supervisor work enjoyment. Journal of Organizational Behavior Management, 28: 90–109. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) provides a process for supervisors to identify the aversive qualities of nonpreferred job tasks. Through participative management, the process reduces these aversive qualities while increasing the appetitive properties via the pairing of these tasks with enjoyable consequences. The present study provides an extension of Green et al.'s work through utilization of a concurrent-chains schedule arrangement via the pairing of reinforcing consequences with a target job task using probabilistic outcomes to directly assess job task preferences for eight direct support staff in a human service organization.     

2-Distance vertex-distinguishing index of subcubic graphs

This paper addresses the performance of scheduling algorithms for a two-stage no-wait hybrid flowshop environment with inter-stage flexibility, where there exist several parallel machines at each stage. Each job, composed of two operations, must be processed from start to completion without any interruption either on or between the two stages. For each job, the total processing time of its two operations is fixed, and the stage-1 operation is divided into two sub-parts: an obligatory part and an optional part (which is to be determined by a solution), with a constraint that no optional part of a job can be processed in parallel with an idleness of any stage-2 machine. The objective is to minimize the makespan. We prove that even for the special case with only one machine at each stage, this problem is strongly NP-hard. For the case with one machine at stage 1 and m machines at stage 2, we propose two polynomial time approximation algorithms with worst case ratio of \(3-\frac{2}{m+1}\) and \(2-\frac{1}{m+1}\), respectively. For the case with m machines at stage 1 and one machine at stage 2, we propose a polynomial time approximation algorithm with worst case ratio of 2. We also prove that all the worst case ratios are tight.     

Zur Struktur des Marktes für Abschlussprüfungen

This comment refers to the paper ofStefani/Bleibtreu. In their model they combine two important topics of audit research: Auditor’s independence and audit market’s concentration tendencies. The objective of their paper is to demonstrate that abandoning non audit services might have negative impacts on auditors’ competitive behavior.

This comment addresses the following features of the model:

• The paper’s relevance, because it addresses aspects of the audit market structure, which have been so far neglected in the political discussion on the EU-level.
• The way of modeling different auditors’ offered product portfolios and auditors’ cost structures.
• The implications of the paper’s results, because only very few papers make use of analytical models for analyzing the audit market.

     

Building better measures of role ambiguity and role conflict: The validation of new role stressor scales

Occupational stress researchers have given considerable attention to role ambiguity and role conflict as predictors of employee health, job attitudes and behaviour. However, the validity of the Rizzo, House, and Lirtzman’s (1970 Rizzo, J. R., House, R. J., &; Lirtzman, S. I. (1970). Role conflict and ambiguity in complex organizations. Administrative Science Quarterly, 15, 150–163. doi: 10.2307/2391486[Crossref], [Web of Science ®] , [Google Scholar]) scales – the most popular role stressor measures – has been a source of disagreement among researchers. In response to the disputed validity of the Rizzo et al. scales, we developed new measures of role ambiguity and role conflict and conducted five studies to examine their psychometric qualities (Study 1 N?=?101 U.S. workers; Study 2 N?=?118 workers primarily employed in the U.S.; Study 3 N?=?135 employed U.S. MBA students; Study 4 N?=?973 members of the U.S. Air Force (USAF); Study 5 N?=?234 workers primarily employed in the U.S.). Across these five studies, we found that the new role stressor scales have desirable psychometric qualities: they displayed high levels of substantive validity, high levels of internal consistency and test–retest reliability, they produced an interpretable factor structure, and we found evidence of their construct validity. We therefore recommend that these new scales be used in future research on role stress.     

Online MapReduce scheduling problem of minimizing the makespan

MapReduce system is a popular big data processing framework, and the performance of it is closely related to the efficiency of the centralized scheduler. In practice, the centralized scheduler often has little information in advance, which means each job may be known only after being released. In this paper, hence, we consider the online MapReduce scheduling problem of minimizing the makespan, where jobs are released over time. Both preemptive and non-preemptive version of the problem are considered. In addition, we assume that reduce tasks cannot be parallelized because they are often complex and hard to be decomposed. For the non-preemptive version, we prove the lower bound is \(\frac{m+m(\Psi (m)-\Psi (k))}{k+m(\Psi (m)-\Psi (k))}\), higher than the basic online machine scheduling problem, where k is the root of the equation \(k=\big \lfloor {\frac{m-k}{1+\Psi (m)-\Psi (k)}+1 }\big \rfloor \) and m is the quantity of machines. Then we devise an \((2-\frac{1}{m})\)-competitive online algorithm called MF-LPT (Map First-Longest Processing Time) based on the LPT. For the preemptive version, we present a 1-competitive algorithm for two machines.     

Asymptotically optimal policy for stochastic job shop scheduling problem to minimize makespan

This paper studies the large-scale stochastic job shop scheduling problem with general number of similar jobs, where the processing times of the same step are independently drawn from a known probability distribution, and the objective is to minimize the makespan. For the stochastic problem, we introduce the fluid relaxation of its deterministic counterpart, and define a fluid schedule for the fluid relaxation. By tracking the fluid schedule, a policy is proposed for the stochastic job shop scheduling problem. The expected value of the gap between the solution produced by the policy and the optimal solution is proved to be O(1), which indicates the policy is asymptotically optimal in expectation.     

Optimal job insertion in the no-wait job shop

The no-wait job shop (NWJS) considered here is a version of the job shop scheduling problem where, for any two operations of a job, a fixed time lag between their starting times is given. Also, sequence-dependent set-up times between consecutive operations on a machine can be present. The NWJS problem consists in finding a schedule that minimizes the makespan. We address here the so-called optimal job insertion problem (OJI) in the NWJS. While the OJI is NP-hard in the classical job shop, it was shown by Gröflin & Klinkert to be solvable in polynomial time in the NWJS. We present a highly efficient algorithm with running time $\mathcal {O}(n^{2}\cdot\max\{n,m\})$ for this problem. The algorithm is based on a compact formulation of the NWJS problem and a characterization of all feasible insertions as the stable sets (of prescribed cardinality) in a derived comparability graph. As an application of our algorithm, we propose a heuristic for the NWJS problem based on optimal job insertion and present numerical results that compare favorably with current benchmarks.