Set Of Identical Machines Research Articles

Empowering the configuration-IP: new PTAS results for scheduling with setup times

Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and, therefore, be solved efficiently. As an application, we consider scheduling problems with setup times in which a set of jobs has to be scheduled on a set of identical machines with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed, an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time f(1/varepsilon )cdot mathrm {poly}(|I|). Previously, only constant factor approximations of 5/3 and 4/3 + varepsilon , respectively, were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.

An Efficient Convolution Algorithm for the Non-Markovian Two-Node Cyclic Network

Consider a closed cyclic queueing model that consists of two nodes and a total of M customers. Each node buffer can accommodate all M customers. Node 1 has N ≤ M servers, each having an exponential service time with rate λ . The second node consists of a single server with a general service time distribution function B . . The well-known machine repair model with spares, where a set of identical machines, N , is served by a single repair person, is a key application of this model. This model has several other applications in performance evaluation, manufacturing, computer networks, and in reliability studies as it can be easily used to compute system availability. In this article, we give an efficient algorithm to derive an exact solution for the steady state system size probabilities. Our approach is based on developing an efficient polynomial convolution method to compute the transition probabilities of a birth process over node 2 service times and solving an imbedded Markov chain at node 2 service completion epochs. This is a significant improvement over the exponential algorithm given in an earlier paper. Numerical examples are given to demonstrate the performance of our method.

Scheduling Stochastic Jobs - Complexity and Approximation Algorithms

Uncertainty is an omnipresent issue in real-world optimization problems. This paper studies a fundamental problem concerning uncertainty, known as the beta-robust scheduling problem. Given a set of identical machines and a set of jobs whose processing times follow a normal distribution, the goal is to assign jobs to machines such that the probability that all the jobs are completed by a given common due date is maximized. We give the first systematic study on the complexity and algorithms for this problem. A strong negative result is shown by ruling out the existence of any polynomial-time algorithm with a constant approximation ratio for the general problem unless P=NP. On the positive side, we provide the first FPT-AS (fixed parameter tractable approximation scheme) parameterized by the number of different kinds of jobs, which is a common parameter in scheduling problems. It returns a solution arbitrarily close to the optimal solution, provided that the job processing times follow a few different types of distributions. We further complement the theoretical results by implementing our algorithm. The experiments demonstrate that by choosing an appropriate approximation ratio, the algorithm can efficiently compute a near-optimal solution.

Coordination mechanisms for scheduling games with machine modification

AbstractThis paper studies scheduling games with machine modification. A set of jobs is to be processed on a set of identical machines. An initial schedule before the modification of machines is given as a prior. Then some machines are removed and some new machines are added. Each job has the right either to stay on its original machine if the machine is not removed, or to move to another machine. The choices of all jobs after the machine modification constitute a new schedule. The individual cost of each job is its completion time, which is determined by the scheduling policy that the machines follow. We present properties of the Nash equilibriums under generalizations of the classical scheduling policies such as LPT, SPT and Makespan. For the social cost of minimizing the makespan, we establish the Price of Anarchy and Price of Stability of the games, and majorities of the parametric bounds are tight for each combination of the number of final machines, the number of added machines and the number of removed machines.

Efficient Approximation Algorithms for the Bounded Flexible Scheduling Problem in Clouds

Clouds, such as Amazon Infrastructure-as-a-Service (IaaS) clouds and EMC Hybrid Cloud, impose growing requirements of resource-efficiency scheduling. The bounded flexible scheduling (BFS) problem is one of the problems proposed to meet such requirements. In BFS, we are given a set of identical machines and a set of jobs, each of which is with a value, a workload, a deadline and a parallelism degree, i.e., the maximum number of machines on which the job can execute concurrently. The problem is to compute an assignment of the given jobs to the machines, such that the total value of the jobs successfully completed by their deadlines is maximized. This paper presents a factor- $\frac{C-k}{C}$ approximation algorithm for BFS, where $k$ is the maximum parallelism degree and $C$ is the capacity of the system (i.e., the number of machines). Since $C\gg k$ in BFS, our result significantly improves the known best approximation ratio of $(\frac{C-k}{2C-k})(1-\epsilon)$ for tight deadlines [17] , and $\frac{C-k}{C}\cdot \frac{s-1}{s}$ for loose deadlines [18] on a slackness ratio $s\geq 1$ that is the maximum ratio between a job’s earliest actual finish time and its deadline. We first propose feasibility condition to determine whether an instance of BFS is feasible, i.e., whether there exists a scheduling according to which all jobs can finish before their deadlines, which is the key to achieve the ratio improvement of our algorithm. To prove the correctness of the feasibility condition, we give a simple linear program (LP) for a weaker version of BFS, and show that it is with an integral polyhedron and hence the version of BFS is polynomial-time solvable. Then we present a greedy algorithm and its equivalent primal-dual algorithm for the complementary problem of BFS. Both algorithms have an approximation ratio of $\frac{C-k}{C}$ , and time complexity $O(n^{2}+nT)$ , where $n$ is the number of jobs and $T$ is the number of time slots. As a by-product, we show that the BFS admits a polynomial-time approximation scheme (PTAS) when $T$ is fixed.

Analysis of a two-node closed queueing cyclic network with one non-exponential node

Consider a two-node closed cyclic queueing network with a total of M customers. The first node consists of N⩽M parallel identical servers having exponential service times. Node 2 consists of a single server having a general service time distribution. Waiting space at each node is sufficient to accommodate all M customers.We derive an exact solution for the steady state system size probabilities. Our approach is based on developing and solving an imbedded Markov chain at Node 2 service completion epochs coupled with a birth process over Node 1 service times. In that sense, our methodology is intuitively appealing and easy to apply. We focus on analytically and numerically investigating the effect of Node 2 service time variability on system performance. We find that closed queueing networks, like the one considered here, are less vulnerable to variability than open networks. However, the effect of variability is still significant to merit considerable attention. Numerical results are presented to illustrate the viability of our method for many service time distributions. As by-products, our work yield new results on birth processes and combinatorial identities, which can be useful in other contexts.A primary application of this model is the well-known machine repair model where a set of identical machines are attended by a single repairman. Other applications include performance evaluation of manufacturing and computer networks, as well as reliability studies where our model can be easily used to compute system availability.

An efficient algorithm for semi-online multiprocessor scheduling with given total processing time

We consider a semi-online multiprocessor scheduling problem with a given a set of identical machines and a sequence of jobs, the sum of whose processing times is known in advance. The jobs are to be assigned online to one of the machines and the objective is to minimize the makespan. The best known algorithm for this problem achieves a competitive ratio 1.6 (Cheng et al. in Theor Comput Sci 337:134---146, 2005). The best known lower bound is approximately 1.585 (Albers and Hellwig in Theor Comput Sci 443:1---9, 2012) if the number of machines tends to infinity. We present an elementary algorithm with competitive ratio equal to this lower bound. Thus, the algorithm is best possible if the number of machines tends to infinity.

Heuristic and Exact Algorithms for the Identical Parallel Machine Scheduling Problem

Given a set of jobs with associated processing times, and a set of identical machines, each of which can process at most one job at a time, the parallel machine scheduling problem is to assign each job to exactly one machine so as to minimize the maximum completion time of a job. The problem is strongly NP-hard and has been intensively studied since the 1960s. We present a metaheuristic and an exact algorithm and analyze their average behavior on a large set of test instances from the literature. The metaheuristic algorithm, which is based on a scatter search paradigm, computationally proves to be highly effective and capable of solving to optimality a very high percentage of the publicly available test instances. The exact algorithm, which is based on a specialized binary search and a branch-and-price scheme, was able to quickly solve to optimality all remaining instances.

Semi-on-line multiprocessor scheduling with given total processing time

We are given a set of identical machines and a sequence of jobs, the sum of whose weights is known in advance. The jobs are to be assigned on-line to one of the machines and the objective is to minimize the makespan. An algorithm with performance ratio 1.6 and a lower bound of 1.5 is presented. These results improve on the recent results by Azar and Regev, who proposed an algorithm with performance ratio 1.625 for the less general problem that the optimal makespan is known in advance.

Lower Bounds for the Head-Body-Tail Problem on Parallel Machines: A Computational Study of the Multiprocessor Flow Shop

The multiprocessor flow-shop is the generalization of the flow-shop in which each machine is replaced by a set of identical machines. As finding a minimum-length schedule is NP-hard, we set out to find good lower and upper bounds. The lower bounds are based on relaxation of the capacities of all machine sets except one. This results in a parallel-machine scheduling problem with release dates and delivery times, for which we derive a number of lower bounds. We pay special attention to the time complexity of algorithms for computing these bounds. To obtain the upper bounds a constructive algorithm in subsequent stages is used. We present an experimental comparison of the various lower and upper bounds for the multiprocessor flow-shop problem.

Scheduling Problems on Two Sets of Identical Machines

In this paper we investigate the following scheduling problem: We have two sets of identical machines, the jobs have two processing times one for each set of machines. We consider two different objective functions, in the first model the goal is to minimize the maximum of the makespans on the sets, in the second model we minimize the sum of the makespans. We consider the online, semi online and offline versions of these problems.

Scheduling and Sequencing Jobs on Machines by Way of Simulated Annealing-Based Heuristics

The following paper addresses the problem of scheduling and sequencing a number of jobs to be processed on a set of identical machines. To find solutions to this problem, a set of models is studied with the purpose of maximizing the operational margin of autonomy of the scheduled jobs in the proposed solutions. The article proposes two methods to solve the problem. The first is based on an integer programming model; the second, on a simulated annealing method supported by a neighborhood heuristic in which two variants are studied: one with a coherence verification mechanism, the other without.

Scheduling no-wait production with time windows and flexible processing times

This paper presents a low-complexity algorithm for online job scheduling at workcenters along a given route in a manufacturing system. At each workcenter, the job has to be processed by any one of a given set of identical machines. Each machine has a preset schedule of operations, leaving out time-windows during which the job's processing must be scheduled. The manufacturing system has no internal buffers and the job cannot wait between two consecutive operations. There is some flexibility in the job's processing times, which must be confined to given time intervals. The scheduling algorithm minimizes the job's completion time, and it is executable in real time whenever the job requirement is generated.

Locating sets of identical machines in a linear layout

The assignment of M unique machines to M locations with the objective of minimizing the total machine-to-machine material transportation cost in a flow line may be formulated as a quadratic assignment problem (QAP). Instead of having M unique machines, if an application involves one or more sets of identical machines, the location problem becomes a tertiary assignment problem (TAP). Solving a large problem of this kind is extremely difficult because of its combinatorial nature. When machine-to-machine flow is fixed, the TAP may be specialized to a QAP for which the unique machine problem is a special case. Obtaining an optimum solution to this problem when M is large is also computationally intractable. However, this problem may be solved by identifying sets of identical machines which may be partitioned into individual, "unique" machines. Properties of a special type of matrix called the amoebic matrix are used in the partitioned problems to provide approximate solutions, which are relabeled to prescribe a solution to the original problem. Results are demonstrated along with suggestions for further research.

A Little's Result Approach to the Service Constrained Spares Provisioning Problem for Repairable Items

A set of identical machines are deployed to meet a known and constant demand. If a machine fails, a replacement part must be available before repair of the machine may be initiated. If the part is currently out of stock, it must be ordered. Once repaired at one of a finite number of repair stations, the machine serves as an “operational ready” standby if demand is currently being met; otherwise, the machine is immediately deployed. Machine time-to-failure, ordering leadtimes, and repair times are all assumed to be exponentially distributed. The objective is to determine the number of machines and repair channels that minimize a cost function subject to the service constraint; i.e., on average, the number of machines operating should be at least some fraction of the demand. We present an algorithm that efficiently generates all the boundary points of the feasible region, from which the optimal solution is readily identified, for the special cases in which there is either zero or infinite stock. Use of the algorithm, which is based upon Little's result and first-passage time analysis, requires negligible storage and computational effort. In fact, for problems of moderate size, computation of optimal solutions via a nonprogrammable pocket calculator is feasible.

Capacity optimization of flexible manufacturing systems under budget constraints

The capacity of a flexible manufacturing system (FMS) is optimized with the objective to maximize the system's throughput, while a budget constraint is considered. Decisions are performed on the capacity of machine groups (sets of identical machines), the transportation system and, in case of a significant cost impact, the number of pallets in the system. Throughput evaluation is achieved either by an open finite queueing network or by a closed queueing network if the number of pallets is included in the decision process. For both cases the solution procedure is based on the marginal allocation scheme.

A modification of Hochbaum and Shmoys' algorithm for scheduling problems

The dual approximation algorithm of Hochbaum and Shmoys for solving the minimum makespan problem on a set of identical machines is modified in such a way that it turns into an optimal algorithm when the relative error is chosen properly.

Cyclical job sequencing on multiple sets of identical machines

AbstractThe problem posed in this paper is to sequence or route n jobs, each originating at a particular location or machine, undergoing r−1 operations or repairs, and terminating at the location or machine from which it originated. The problem is formulated as a 0‐1 integer program, with block diagonal structure, comprised of r assignment subproblems; and a joint set of constraints to insure cyclical squences. To obtain integer results the solutions to each subproblem are ranked as required and combinations thereof are implicitly enumerated. The procedure may be terminated at any step to obtain an approximate solution. Some limited computational results are presented.