Processing Set Restrictions Research Articles

Approximation algorithms for batch scheduling with processing set restrictions

Approximation algorithms for batch scheduling with processing set restrictions

Auction-Based Capacity Allocation in Two Parallel Machines with Inclusive Processing Set Restrictions

We consider the scarce capacity allocation problem in two parallel machines with inclusive processing set restrictions. Our focus is to design an auction mechanism to allocate the production capacity among several selfish customers effectively and efficiently. In our iterative ascending auction mechanism, we need to design jointly two things: who wins what and who pays what. First, we propose an adaptive ascending pricing policy to ensure that bidding by truthful processing requirements and keeping on bidding until the ask prices reach his real revenue is a dominant strategy for each customer. Second, we establish a facility owner’s profit maximization model to determine the capacity allocation in two parallel machines with inclusive processing set restrictions; it is an NP hard problem; we also develop a heuristic based on the Lagrangian relaxation technology to obtain the suboptimal solutions. Our computational analysis shows that the auction mechanism can achieve more than 87% of the global system value.

Modeling and solving a real world machine scheduling problem with due windows and processing set restrictions

Motivated by a real-world problem situation faced by a large manufacturing company in the last step of their production process, we model and solve an offline, deterministic, non-preemptive, identical, parallel machine scheduling problem with processing set restrictions, availability constraints, release dates, bounded tardiness, blocking constraints and due windows. The objective is to minimize the total tardiness while penalizing the assignment of jobs to larger than necessary machines. We develop a mixed integer linear program and we propose a heuristic solution approach relying on the well known large neighborhood search paradigm which relies on tailored destroy and repair operators, embedded into a simulated annealing framework. The proposed algorithm is tested on artificial instances as well as on real-world data provided by the company and compared to optimal solutions obtained by means of CPLEX. The developed tool is now used for decision support by the company partner.

Efficient algorithms for scheduling equal-length jobs with processing set restrictions on uniform parallel batch machines.

We consider the problem of scheduling jobs with equal lengths on uniform parallel batch machines with non-identical capacities where each job can only be processed on a specified subset of machines called its processing set. For the case of equal release times, we give efficient exact algorithms for various objective functions. For the case of unequal release times, we give efficient exact algorithms for minimizing makespan.

A linear time approximation scheme for scheduling unbounded batch machines with delivery times and inclusive processing set restrictions

<abstract><p>We consider the problem of scheduling jobs with delivery times and inclusive processing set restrictions on unbounded batch machines to minimize the maximum delivery completion time, which is equivalent to minimizing the maximum lateness from the optimization viewpoint. We develop a polynomial time approximation scheme for this strongly NP-hard problem that runs in linear time for any fixed accuracy requirement.</p></abstract>

Uniform Parallel Machines Scheduling with Setup Time, Learning Effect, Machine Idle Time, and Processing Set Restrictions to Minimize Earliness/Tardiness Costs

Inspired by a real industrial case, this study deals with the problem of scheduling jobs on uniform parallel machines with past-sequence-dependent setup times to minimize the total earliness and tardiness costs. The paper contributes to the existing literature of uniform parallel machines problems by the novel idea of considering position-based learning effects along with processing set restrictions. The presented problem is formulated as a Mixed Integer linear programming (MILP) model. Then, an exact method is introduced to calculate the accurate objective function in the just-in-time (JIT) environments for a given sequence of jobs. Furthermore, three meta-heuristic approaches, (1) a genetic algorithm (GA), (2) a simulated annealing algorithm (SA), and (3) a particle swarm optimization algorithm (PSO) are proposed to solve large size problems in reasonable computational time. Finally, computational results of the proposed meta-heuristic algorithms are evaluated through extensive experiments and tested using ANOVA followed by t-tests to identify the most effective meta-heuristic.

Online Scheduling with Unit Processing Times and Processing Set Restrictions

This paper investigates online scheduling problems with processing set restrictions. There is a sequence of jobs that are revealed one by one over list, where each job has unit processing time. A job can only be processed by a subset of the machines, namely the processing set of the job. The objective is to minimize the makespan. For m machines with two different processing sets, we propose an optimal algorithm with a competitive ratio of $$\frac{3}{2}$$ . For three machines with inclusive processing sets, an optimal algorithm with competitive ratio $$\frac{5}{3}$$ is provided.

Parallel-machine group scheduling with inclusive processing set restrictions, outsourcing option and serial-batching under the effect of step-deterioration

This paper investigates a parallel-machine group scheduling problem where non-identical jobs with arbitrary sizes and inclusive processing set restrictions can be either processed on in-house parallel machines in the form of serial batch or outsourced with cost. The objective of our study is aimed at minimizing the weighted sum of the in-house makespan and the total outsourcing cost for a platform manufacturing enterprise. Some structural properties are identified for the optimal solution in some special cases of the studied problem, which contribute to the optimal solution for the studied problem. Further, based on these properties, a novel hybrid algorithm VNS–NKEA is proposed to solve the studied problem, which integrates neighborhood knowledge-based evolutionary algorithm (NKEA) and variable neighborhood search (VNS). To demonstrate the better performance including solution quality and the convergence speed of the proposed algorithm, computational experiments are conducted to evaluate its performance by comparing with other proposed algorithms. The experiment results show that the hybrid algorithm performs quite better than other compared algorithms for each instance, which reflect that the hybrid algorithm can solve the studied problem effectively.

Parallel batch scheduling with nested processing set restrictions

We study the problem of scheduling n jobs on m parallel batching machines with nested processing set restrictions. Each job has a release time before which it cannot be processed, and has a restricted set of batching machines to which it can be assigned, called its processing set. Two distinct processing sets are either nested or disjoint. Each batching machine can process up to B (B<n) jobs simultaneously as a batch, where B is known as the batch capacity. The processing time of a batch is the maximum of the processing times of the jobs belonging to it. The objective is to minimize the makespan (maximum completion time) of the schedule. When all jobs are released at the same time, we present a fast algorithm with approximation ratio 3−1/m. For the case of unequal release times, we develop a fast algorithm with approximation ratio 4−1/m, as well as a polynomial time approximation scheme (PTAS).

Approximation algorithms for scheduling jobs with release times and arbitrary sizes on batch machines with non-identical capacities

We consider the problem of scheduling jobs with release times and arbitrary sizes on batch machines with non-identical capacities to minimize makespan. A job can only be assigned to a machine whose capacity is not less than the size of the job. A machine can process a group of jobs as a batch simultaneously, as long as the total size of the jobs in this batch does not exceed the capacity of the machine. The processing time of a batch is equal to the longest processing time of all the jobs in the batch. For the case of equal release times, we obtain a fast 5-approximation algorithm, as well as an algorithm with approximation ratio arbitrarily close to 2. For the case of unequal release times and a fixed number of different batch capacities, we obtain an algorithm with approximation ratio arbitrarily close to 2. We also study the problem of scheduling jobs with equal release times, arbitrary sizes, inclusive processing set restrictions and incompatible families on batch machines with identical capacities to minimize makespan. We obtain a 8/3-approximation algorithm, as well as an algorithm with approximation ratio arbitrarily close to 2. If only a single family is considered, a 2-approximation algorithm is presented. It is likely that these results cannot be too substantially improved upon, since all of the problems studied cannot be approximated to within a ratio smaller than 2 unless P=NP.

Fast approximation algorithms for uniform machine scheduling with processing set restrictions

We consider the problem of nonpreemptively scheduling a set of independent jobs on a set of uniform machines, where each job has a set of machines to which it can be assigned. This kind of restriction is called the processing set restriction. In the literature there are many kinds of processing set restrictions that have been studied. In this paper we consider two kinds: the “inclusive processing set” and the “tree-hierarchical processing set”. Epstein and Levin (2011) have given Polynomial Time Approximation Schemes (PTAS) to solve both classes. However, the running times of their PTAS are rather high. In this paper, we give fast approximation algorithms for both cases and show that they both have a worst-case performance bound of 4/3. Moreover, we show that the bounds are achievable.

Parallel batch scheduling with inclusive processing set restrictions and non-identical capacities to minimize makespan

We consider the problem of scheduling n jobs on m parallel batching machines with inclusive processing set restrictions and non-identical capacities. The machines differ in their functionality but have the same processing speed. The inclusive processing set restriction has the following property: the machines can be linearly ordered such that a higher-indexed machine can process all those jobs that a lower-indexed machine can process. Each job is characterized by a processing time that specifies the minimum time needed to process the job, a release date before which it cannot be processed, and a machine index which is the smallest index among the machines that can process it. Each batching machine has a limited capacity and can process a batch of jobs simultaneously as long as its capacity is not violated. The capacities of the machines are non-identical. The processing time of a batch is the maximum of the processing times of the jobs belonging to it. Jobs in the same batch have a common start time and a common completion time. The goal is to find a non-preemptive schedule so as to minimize makespan (the maximum completion time). When all jobs are released at the same time, we present two fast algorithms with approximation ratios 3 and 9/4, respectively. For the general case with unequal release dates, we develop a polynomial time approximation scheme (PTAS), which is also the first PTAS even for the case with equal release dates and without processing set restrictions.

Scheduling parallel machines with inclusive processing set restrictions and job rejection

In this article, we study a parallel machine scheduling problem with inclusive processing set restrictions and the option of job rejection. In the problem, each job is compatible to a subset of machines, and machines are linearly ordered such that a higher-indexed machine can process all those jobs that a lower-indexed machine can process (but not conversely). To achieve a tight production due date, some of the jobs might be rejected at certain penalty. We first study the problem of minimizing the makespan of all accepted jobs plus the total penalty cost of all rejected jobs, where we develop a ( 5 3 + e ) -approximation algorithm with a time complexity of O ( n 2 / e 2 ) . We then study two bicriteria variants of the problem. For the variant problem of minimizing the makespan subject to a given bound on the total rejection cost, we develop a ( 5 3 + e ) -approximation algorithm with a time complexity of O ( n m e 2 · log P ) . For the variant problem of maximizing the total rejection cost of the accepted jobs subject to a given bound on the makespan, we present a 0.5-approximation algorithm with a time complexity of O ( n log n ) .© 2017 Wiley Periodicals, Inc. Naval Research Logistics, 2017

A note on scheduling jobs with equal processing times and inclusive processing set restrictions

We consider the problem of scheduling n jobs on m parallel machines with inclusive processing set restrictions. Each job has a given release date, and all jobs have equal processing times. The objective is to minimize the makespan of the schedule. Li and Li (2015) have developed an O(n2+mn log n) time algorithm for this problem. In this note, we present a modified algorithm with an improved time complexity of O(min{m, log n} ⋅ n log n).

Parallel Machine Scheduling with Nested Processing Set Restrictions and Job Delivery Times

The problem of scheduling jobs with delivery times on parallel machines is studied, where each job can only be processed on a specific subset of the machines called its processing set. Two distinct processing sets are either nested or disjoint; that is, they do not partially overlap. All jobs are available for processing at time 0. The goal is to minimize the time by which all jobs are delivered, which is equivalent to minimizing the maximum lateness from the optimization viewpoint. A list scheduling approach is analyzed and its approximation ratio of 2 is established. In addition, a polynomial time approximation scheme is derived.

Scheduling jobs with release dates, equal processing times, and inclusive processing set restrictions

We consider the problem of scheduling a given set of n jobs with equal processing times on m parallel machines so as to minimize the makespan. Each job has a given release date and is compatible to only a subset of the machines. The machines are ordered and indexed in such a way that a higher-indexed machine can process all the jobs that a lower-indexed machine can process. We present a solution procedure to solve this problem in O(n2+mnlogn) time. We also extend our results to the tree-hierarchical processing sets case and the uniform machine case.

An artificial bee colony algorithm approach for unrelated parallel machine scheduling with processing set restrictions, job sequence-dependent setup times, and due date

The problem dealt with in this study has job sequence-dependent setup times under suitable machine constraints and due date constraints. The performance criterion for this problem is to minimize the sum of makespan and total tardiness. In the literature, an application of ABC algorithm for the problems which includes all the properties that we have dealt has not been discussed. Because there is no appropriate test data, a real-life data was collected from a factory. In this study, a new approach has been proposed for the solution with meta-heuristics of unrelated parallel machine scheduling problems which is a combinatorial problem. This new neighborhood approach provides different machine assignments for every candidate job sequences. This approach is used by integrating into ABC and GA. To evaluate the performances of the algorithms, the real-life problem was solved by using ABC and GA algorithms under similar conditions. It was found that all jobs can be completed in two shifts without the need for a third shift. Computational results show that ABC algorithm has better performance than GA.

Matching with sizes (or scheduling with processing set restrictions)

Matching problems on bipartite graphs where the entities on one side may have different sizes are intimately related to scheduling problems with processing set restrictions. We survey the close relationship between these two problems, and give new approximation algorithms for the (NP-hard) variations of the problems in which the sizes of the jobs are restricted. Specifically, we give an approximation algorithm with an additive error of one when the sizes of the jobs are either 1 or 2, and generalise this to an approximation algorithm with an additive error of 2k−1 for the case where each job has a size taken from the set {1,2,4,…,2k} (for any constant integer k). We show that the above two problems become polynomial-time solvable if the processing sets are nested.

Berth allocation with time-dependent physical limitations on vessels

We consider a berth allocation problem in container terminals in which the assignment of vessels to berths is limited by water depth and tidal condition. We model the problem as a parallel-machine scheduling problem with inclusive processing set restrictions, where the time horizon is divided into two periods and the processing sets in these two periods are different. We consider both the static and dynamic cases of the problem. In the static case all of the vessels are ready for service at time zero, while in the dynamic case the vessels may have nonzero arrival times. We analyze the computational complexity and develop efficient heuristics for these two cases. Computational experiments are performed to test the effectiveness of the heuristics and to evaluate the benefits of taking tidal condition into consideration when making berth allocation decisions.

Scheduling with processing set restrictions: PTAS results for several variants

We consider multiprocessor scheduling to minimize makespan. Each job has a given processing time and in addition, a subset of machines associated with it, also called its processing set. Each job has to be assigned to one machine in its set, thus contributing to the load of this machine. We study two variants of this problem on identical machines, the case of nested processing sets, and the case of tree-hierarchical processing sets. In addition, we consider uniformly related machines with a special case of inclusive processing sets, which has a clear motivation. We design polynomial time approximation schemes for these three variants. The first case resolves one of the open problems stated in the survey by Leung and Li (2008).