Parallel Batch Machines Research Articles

BATCHING MACHINE SCHEDULING WITH BICRITERIA: MAXIMUM COST AND MAKESPAN

In this paper, the problem of minimizing maximum cost and makespan simultaneously on an unbounded parallel-batching machine is considered. An unbounded parallel-batching machine is a machine that can handle any number of jobs in a batch and the processing time of a batch is the largest processing time of jobs in the batch. The main goal of a multicriteria problem is to find Pareto optimal solutions. We present a polynomial-time algorithm to produce all Pareto optimal solutions of this bicriteria scheduling problem.

Online scheduling of incompatible unit-length job families with lookahead

We consider the online scheduling of incompatible unit-length job families on an unbounded parallel-batch machine with lookahead. In this paper online means that jobs arrive over time. The objective is to mininize the makespan. In the lookahead model, at a time instant t, an online algorithm can foresee the information of all jobs arriving in the time segment (t,t+β]. Incompatible job families mean that jobs which belong to different families cannot be assigned to the same batch. For this problem, we provide a best possible online algorithm of competitive ratio 1+αf for 0≤β<1, where αf is the positive root of the equation f⋅αf2+(1+β)αf+β−f=0 and f is the number of job families which is known in advance.

Online scheduling of equal length jobs on a bounded parallel batch machine with restart or limited restart

We consider the online (over time) scheduling of equal length jobs on a bounded parallel batch machine with a capacity b to minimize makespan with restart or limited restart. Let α≈0.29, β≈0.47, γ≈0.38, and φ≈0.40 be four parameters defined in the text. When b=2, we present a best possible online algorithm HL(b=2) with a competitive ratio of 1+α under the assumption of restart or limited restart. Under the assumption of limited restart with b≥3, we present a best possible online algorithm HL(b≥3) with a competitive ratio of 1+β. Under the assumption of restart, when b=3, we present a best possible online algorithm H(b=3) with a competitive ratio of 1+γ, and when b≥4, we present a best possible online algorithm H(b≥4) with a competitive ratio of 1+φ. In our online algorithms under the assumption of restart, each job is interrupted at most two times. Consequently, our results cover the setting of k-limited restart for all k≥1.

Single-machine parallel-batching scheduling with family jobs to minimize weighted number of tardy jobs

We consider the problem of scheduling n jobs in batches on a single parallel-batching machine, where the jobs are partitioned into jobs families and the jobs in each family have the same due date. The objective is to minimize the weighted number of tardy jobs. We first devise an efficient pseudo-polynomial time and a fully polynomial time approximation scheme for the weighted problem. Then we present O(n2)-time and O(nlogn)-time algorithms for the case where the jobs have the same weight and for the case where the jobs have the same processing time, respectively.

Parallel-batch scheduling and transportation coordination with waiting time constraint.

This paper addresses a parallel-batch scheduling problem that incorporates transportation of raw materials or semifinished products before processing with waiting time constraint. The orders located at the different suppliers are transported by some vehicles to a manufacturing facility for further processing. One vehicle can load only one order in one shipment. Each order arriving at the facility must be processed in the limited waiting time. The orders are processed in batches on a parallel-batch machine, where a batch contains several orders and the processing time of the batch is the largest processing time of the orders in it. The goal is to find a schedule to minimize the sum of the total flow time and the production cost. We prove that the general problem is NP-hard in the strong sense. We also demonstrate that the problem with equal processing times on the machine is NP-hard. Furthermore, a dynamic programming algorithm in pseudopolynomial time is provided to prove its ordinarily NP-hardness. An optimal algorithm in polynomial time is presented to solve a special case with equal processing times and equal transportation times for each order.

Online scheduling on an unbounded parallel-batch machine and a standard machine to minimize makespan

We consider the online scheduling on an unbounded parallel-batch machine and a standard machine to minimize makespan. In the problem, the jobs arrive online over time and to be processed on two machines M1 and M2. M1 is an unbounded parallel-batch machine and M2 is a standard machine. The objective is to minimize the makespan, i.e., the maximum completion time of all jobs. For this problem, we present an online algorithm of competitive ratio 5+54≈1.809.

Dynamic set-up rules for hybrid flow shop scheduling with parallel batching machines

An S-stage hybrid (or flexible) flow shop, with sequence-independent uniform set-up times, parallel batching machines with compatible parallel batch families (like in casting or heat treatments in furnaces, chemical or galvanic baths, painting in autoclave, etc.) has been analysed with the purpose of reducing the number of tardy jobs (and the makespan); in Graham’s notation: FPB(m1, m2, …, mS)|p-batch, . Jobs are sorted dynamically (at each new delivery); batches are closed within sliding (or rolling) time windows and processed in parallel by multiple identical machines. Computation experiments have shown the better performance on benchmarks of the two proposed heuristics based on new formulations of the critical ratio (CRsetup) considering the ratio of allowance set-up and processing time in the scheduling horizon, which improves the weighted modified operation due date rule.

Closed loop control-based real-time dispatching heuristic on parallel batch machines with incompatible job families and dynamic arrivals

In this paper, a real-time closed loop control dispatching heuristic (RCLC) algorithm is proposed to address the scheduling problem of parallel batch machines with incompatible job families, limited waiting time constraints, re-entrant flow and dynamic arrivals in the diffusion and oxidation areas of a semiconductor wafer fabrication system (SWFS), which is known to be strongly NP-hard. The basis of this algorithm is the information of lots in the buffer when the parallel batch machines are idle and available. In RCLC, if the number of any family lots is less than the maximum batch size, the dispatching heuristic can be seen as a pull–pull–push–push (P4) strategy; otherwise, a genetic algorithm (GA). A look-itself strategy, P4 strategy and GA can build a closed loop control system. The experiments are implemented on the Petri nets-based real-time scheduling simulation platform of SWFS, and demonstrate the effectiveness of our proposed method.

Heuristics for scheduling a two-stage hybrid flow shop with parallel batching machines: application at a hospital sterilisation plant

The model of a two-stage hybrid (or flexible) flow shop, with sequence-independent uniform setup times, parallel batching machines and parallel batches has been analysed with the purpose of reducing the number of tardy jobs and the makespan in a sterilisation plant. Jobs are processed in parallel batches by multiple identical parallel machines. Manual operations preceding each of the two stages have been dealt with as machine setup with standardised times and are sequence-independent. A mixed-integer model is proposed. Two heuristics have been tested on real benchmark data from an existing sterilisation plant: constrained size of parallel batches and fixed time slots. Computation experiments performed on combinations of machines and operator numbers suggest balancing the two stages by assigning operators proportionally to the setup time requirements.

Batch scheduling on parallel machines with dynamic job arrivals and incompatible job families

We study the scheduling problem of minimising weighted completion times on parallel identical batching machines with dynamic job arrivals and incompatible job families. Each job is associated with a family, weight (priority), release time, and size. Batching machines can process simultaneously up to a specified total size of the jobs of a particular family. The scheduling problem can be represented by . We present a mathematical model and heuristic algorithms employing different local search procedures individually and sequentially under a variable neighbourhood search scheme. We have shown that among local searches, repositioning the batches instead of jobs yields better results. The best-performing heuristic algorithm is capable of generating solutions within 0.6% of the best overall heuristic solution for each instance in a reasonable amount of time. When this heuristic is compared against the mathematical model, solutions that are 3.7% above optimal on average in the 15-job problem instances are possible.

A note on two-agent scheduling on an unbounded parallel-batching machine with makespan and maximum lateness objectives

This paper studies the two-agent scheduling on an unbounded parallel-batching machine. In the problem, there are two agents A and B with each having their own job sets. The jobs of a common agent can be processed in a common batch. Moreover, each agent has an objective function to be minimized. The objective function of agent A is the makespan of his jobs and the objective function of agent B is maximum lateness of his jobs. Yazdani Sabouni and Jolai [M.T. Yazdani Sabouni, F. Jolai, Optimal methods for batch processing problem with makespan and maximum lateness objectives, Appl. Math. Model. 34 (2010) 314–324] presented a polynomial-time algorithm for the problem to minimize a positive combination of the two agents’ objective functions. Unfortunately, their algorithm is incorrect. We then dwell on the problem and present a polynomial-time algorithm for finding all Pareto optimal solutions of this two-agent parallel-batching scheduling problem.

An optimal online algorithm for single parallel-batch machine scheduling with incompatible job families to minimize makespan

We consider the online scheduling of incompatible job families on an unbounded parallel-batch machine to minimize the makespan, where jobs arrive over time and the number of job families, f, is known in advance. We provide an optimal online algorithm for the problem with a competitive ratio of 1+4f2+1−12f.

An improved ant colony optimization for scheduling identical parallel batching machines with arbitrary job sizes

In this paper we consider the problem of scheduling parallel batching machines with jobs of arbitrary sizes. The machines have identical capacity of size and processing velocity. The jobs are processed in batches given that the total size of jobs in a batch cannot exceed the machine capacity. Once a batch starts processing, no interruption is allowed until all the jobs are completed. First we present a mixed integer programming model of the problem. We show the computational complexity of the problem and optimality properties. Then we propose a novel ant colony optimization method where the Metropolis Criterion is used to select the paths of ants to overcome the immature convergence. Finally, we generate different scales of instances to test the performance. The computational results show the effectiveness of the algorithm, especially for large-scale instances.

An integrated scheduling and material-handling approach for complex job shops: a computational study

We suggest an extension of the shifting bottleneck heuristic for complex job shops that takes the operations of automated material-handling systems (AMHS) into account. The heuristic is used within a rolling horizon approach. The job-shop environment contains parallel batching machines, machines with sequence-dependent setup times, and re-entrant process flows. Jobs are transported by an AMHS. Semiconductor wafer fabrication facilities (wafer fabs) are typical examples for manufacturing systems with these characteristics. Our primary performance measure is total weighted tardiness (TWT). The shifting bottleneck heuristic (SBH) uses a disjunctive graph to decompose the overall scheduling problem into scheduling problems for single machine groups and for transport operations. The scheduling algorithms for these scheduling problems are called subproblem solution procedures (SSPs). We consider SSPs based on dispatching rules. In this paper, we are also interested in how much we can gain in terms of TWT if we apply more sophisticated SSPs for scheduling the transport operations. We suggest a Variable Neighbourhood Search (VNS) based SSP for this situation. We conduct simulation experiments in a dynamic job-shop environment in order to assess the performance of the suggested algorithms. The integrated SBH outperforms common dispatching rules in many situations. Using near to optimal SSPs leads to improved results compared with dispatching based SSPs for the transport operations.

Online algorithms for scheduling unit length jobs on parallel-batch machines with lookahead

We consider online scheduling of unit length jobs on m parallel-batch machines with lookahead to minimize makespan. A parallel-batch machine can handle up to b jobs simultaneously as a batch with processing time equal to the maximum processing time of the jobs included in the batch. In the lookahead model, at a time instant t, an online algorithm can foresee all the jobs that will arrive in the time segment (t,t+β]. In this paper, we deal with two variants: the unbounded model with b=∞ and the bounded model with b<∞. For the unbounded model, we present an optimal online algorithm for β⩾1/m, and provide a best possible online algorithm of competitive ratio 1+αm for 0⩽β<1/m, where αm is the positive root of (1+α)(m+1)=α+2−β∑i=1m(1+α)i. For the bounded model, we establish a lower bound with a form of piecewise function, and provide an online algorithm with competitive ratios 1+α⁎ for 0⩽β⩽16 and 32 for β>16, respectively, where α⁎ is the positive root of α2+(β+1)α+β−1=0. The online algorithm is the best possible when 0⩽β⩽16.

An improved on-line algorithm for single parallel-batch machine scheduling with delivery times

We study an on-line single parallel-batch machine scheduling problem where each job has a processing time and a delivery time. Jobs arrive over time and the batch capacity is unbounded. Jobs can be processed in a common batch, and each job is delivered independently and immediately at its completion time on the machine. The objective is to minimize the time by which all the jobs are delivered. We provide an on-line algorithm with a competitive ratio 22−1≈1.828, which improves on a 2-competitive on-line algorithm for this problem in the literature.

Metaheuristics for scheduling jobs with incompatible families on parallel batching machines

In this paper, we discuss the scheduling of jobs with incompatible families on parallel batching machines. The performance measure is total weighted tardiness. This research is motivated by a scheduling problem found in the diffusion and oxidation areas of semiconductor wafer fabrication where the machines can be modelled as parallel batch processors. Given that this scheduling problem is NP-hard, we suggest an ant colony optimization (ACO) and a variable neighbourhood search (VNS) approach. Both metaheuristics are hybridized with a decomposition heuristic and a local search scheme. We compare the performance of the two algorithms with that of a genetic algorithm (GA) based on extensive computational experiments. The VNS approach outperforms the ACO and GA approach with respect to time and solution quality.

Unbounded parallel-batching scheduling with two competitive 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.

Bi-criteria scheduling on a single parallel-batch machine

This paper considers bi-criteria scheduling on a single parallel-batch machine to minimizing two regular scheduling criteria that are non-decreasing in the job completion times. For minimizing the total weighted completion time and the makespan simultaneously, we prove that the problem is NP-hard in the ordinary sense and possesses a fully polynomial-time approximation scheme. For minimizing the weighted total number of tardy jobs and the makespan simultaneously, we provide a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme. Furthermore, we identify that all Pareto-optimal solutions for ∑ j = 1 n w j C j , C max and ∑ j = 1 n w j U j , C max can be found in pseudo-polynomial time, respectively.

Online scheduling on unbounded parallel-batch machines to minimize maximum flow-time

We consider online scheduling on m unbounded parallel-batch machines to minimize maximum flow-time of the jobs. We show that no online algorithm can have a competitive ratio less than 1 + α m , where α m is the positive root of α 2 + ( m + 1 ) α − 1 = 0 , and this lower bound is still valid even when all jobs have the same processing times. Then we provide an online algorithm of competitive ratio 1 + 1 / m . When the jobs have the same processing times, we present a best possible online algorithm of competitive ratio 1 + α m .