Unbounded Parallel-batch Machine Research Articles

Bicriteria scheduling on an unbounded parallel-batch machine for minimizing makespan and maximum cost

This paper studies the bicriteria problem of scheduling n jobs on an unbounded parallel-batch machine. The goal is to minimize makespan and maximum cost simultaneously. When the jobs have arbitrary processing times and equal release dates, we obtain an O(n3)-time algorithm, improving the previously best known O(n4)-time algorithm. When the jobs have equal processing times and arbitrary release dates with a precedence relation, we also obtain an O(n3)-time algorithm, which is the first polynomial time algorithm for this problem.

Bicriteria Scheduling on an Unbounded Parallel-Batch Machine for Minimizing Makespan and Maximum Cost

Bicriteria Scheduling on an Unbounded Parallel-Batch Machine for Minimizing Makespan and Maximum Cost

Min–Max Scheduling of Batch or Drop-Line Jobs Under Agreeable Release and Processing Times

We study the Pareto optimization scheduling on an unbounded parallel-batch machine with jobs having agreeable release dates and processing times for minimizing makespan and maximum cost simultaneously. The jobs considered in this paper are of two types: batch jobs and drop-line jobs. For batch jobs, the completion time of a job is given by the completion time of the batch containing this job. For drop-line jobs, the completion time of a job is given by the starting time of the batch containing this job plus the processing time of this job. For both of batch jobs and drop-line jobs, we present polynomial-time algorithms for finding all Pareto optimal points.

A best possible online algorithm for parallel batch scheduling with delivery times and limited restart

We consider the online scheduling on an unbounded (drop-line) parallel batch machine to minimize the time by which all jobs have been delivered. In this paper, all jobs arrive over time and the running batches are allowed limited restart. Here limited restart means that a running batch which contains restarted jobs cannot be restarted again. A drop-line parallel batch machine can process several jobs simultaneously as a batch, and all jobs in a batch start at the same time, and the completion time of a job equals the sum of its starting time and its processing time. Here we consider the restricted model: all jobs have agreeable processing times and delivery times. We provide a best possible online algorithm H with a competitive ratio of $$\frac{3}{2}$$ for the problem on an unbounded parallel batch machine and the corresponding problem on an unbounded drop-line parallel batch machine, respectively.

Online Algorithms for Scheduling Unit Length Jobs on Unbounded Parallel-Batch Machines with Linearly Lookahead

In this paper, we propose a new online scheduling model with linear lookahead intervals, which has the character that at any time [Formula: see text], one can foresee the jobs that will coming in the time interval [Formula: see text] in which [Formula: see text]. In this new lookahead model, the length of the lookahead intervals are variable as the time going on and the number of jobs increasing, and has the tend of steady growth. In this paper, we consider online scheduling of unit length jobs on [Formula: see text] identical parallel-batch machines under this new lookahead model to minimize makespan. The batch capacity is unbounded, that is [Formula: see text]. We present an optimal online algorithm for [Formula: see text], and provide a best possible online algorithm of competitive ratio [Formula: see text] for [Formula: see text], where [Formula: see text] is the positive root of [Formula: see text].

Online Scheduling of Incompatible Family Jobs with Equal Length on an Unbounded Parallel-Batch Machine with Job Delivery

In this paper, we consider the online scheduling of incompatible family jobs with equal length on an unbounded parallel-batch machine with job delivery. The jobs arrive online over time and belong to [Formula: see text] incompatible job families, where [Formula: see text] is known in advance. The jobs are first processed in batches on an unbounded parallel-batch machine and then the completed jobs are delivered in batches by a vehicle with infinite capacity to their customers. The jobs from distinct families cannot be processed and delivered in the same batch. The objective is to minimize the maximum delivery completion time of the jobs. For this problem, we present an online algorithm with the best competitive ratio of [Formula: see text].

Online Scheduling on Two Uniform Unbounded Parallel-Batch Machines to Minimize Makespan

We study an online (over time) scheduling problem on two uniform unbounded parallel-batch machines with processing speed 1 and $$v (0<v\leqslant 1$$ ), respectively, to minimize the makespan. We first show that no online algorithm can have a competitive ratio less than $$1+ \alpha _L$$ , where $$\alpha _L= (\sqrt{5}-1)/2\approx 0.618$$ if $$0<v\leqslant 1/2$$ , and $$\alpha _L= \alpha _2(v)$$ is the positive root of $$\alpha ^3 +3 \alpha ^2+(3-1/v)\alpha -1/v=0$$ if $$1/2 < v \leqslant 1$$ . This lower bound is still valid when all jobs have the same processing times. Then, we provide an online algorithm with a competitive ratio at most $$\min \{(\sqrt{5}+1)/2,\; \sqrt{2}/v\}$$ . When the jobs have the same processing times, we present the best possible online algorithm with a competitive ratio $$1+ \alpha _L$$ .

Scheduling family jobs on an unbounded parallel-batch machine to minimize makespan and maximum flow time

This paper investigates the scheduling of family jobs with release dates on an unbounded parallel-batch machine. The involved objective functions are makespan and maximum flow time. It was reported in the literature that the single-criterion problem for minimizing makespan is strongly NP-hard when the number of families is arbitrary, and is polynomially solvable when the number of families is fixed. We first show in this paper that the single-criterion problem for minimizing maximum flow time is also strongly NP-hard when the number of families is arbitrary. We further show that the Pareto optimization problem (also called bicriteria problem) for minimizing makespan and maximum flow time is polynomially solvable when the number of families is fixed, by enumerating all Pareto optimal points in polynomial time. This implies that the single-criterion problem for minimizing maximum flow time is also polynomially solvable when the number of families is fixed.

Competitive Project Scheduling on Two Unbounded Parallel Batch Machines

This paper considers competitive project scheduling on two unbounded parallel batch machines. There are two competing firms, and each firm has an unbounded parallel batch machine. All projects must be performed in batches by Firms 1 and 2 on their machines, respectively. The profit that each firm obtains from each project depends on whether the firm finishes the job before or after its competitor. In the first problem, given a feasible schedule for Firm 1, the objective is to find an optimal schedule to maximize the total reward for Firm 2 under the given schedule for Firm 1. The corresponding total reward for Firm 1 is called the worst-case total reward of the given schedule for Firm 1. In the second problem, the objective is to find an optimal schedule for Firm 1 to maximize the worst-case total reward. We provide optimal algorithms for the two problems, respectively.

Online algorithms for scheduling on batch processing machines with interval graph compatibilities between jobs

We consider the online (over time) scheduling problem of minimizing the makespan on m unbounded parallel-batch machines, in which jobs in the same batch have to be pairwise compatible. Compatibility is a symmetric binary relation, which is represented by an interval compatibility graph. The processing time of a batch is equal to the maximum processing time of the jobs in it, and all jobs in the same batch start and finish at the same time. For this problem, firstly, we show that there exists no online algorithm with a competitive ratio less than 2. Then we provide an online algorithm with a competitive ratio 2+m−1m+1, which is optimal for the case m=1. When all jobs have the same processing times, we also give an optimal online algorithm.

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 note on unbounded parallel-batch scheduling

This paper revisits the scheduling problem on an unbounded parallel batch machine for minimizing a maximum cost fmax. It was reported in the literature that the decision version of the problem is solvable in O(n2+nlog⁡P) time, where n is the number of jobs and P is the total processing time of jobs. This implies that the optimization version for minimizing fmax can be solved in weakly polynomial time. But a strongly polynomial-time algorithm has not been provided for this problem. In this paper, we present an O(n4)-time algorithm for the Pareto optimization problem for minimizing Cmax and fmax, where Cmax is the maximum completion time of jobs. Consequently, the problem for minimizing fmax can also be solved in O(n4) time.

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 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.

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.

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 .

Unbounded parallel-batch scheduling with family jobs and delivery coordination

We study a scheduling problem that integrates parallel-batch production with family jobs and job delivery at the same time. The jobs are first processed on an unbounded parallel-batch machine and then delivered in batches to their specified customers by a transportation vehicle. We assume that jobs from different families (customers) cannot be processed together by the batch machine and also transported together by the vehicle. The objective is to minimize the time when the vehicle finishes delivering the last delivery batch to its customer and returns to the machine. We first show that the problem is NP -hard, and then propose for it a heuristic algorithm with a worst-case performance ratio of 3/2.

Online scheduling on unbounded parallel-batch machines with incompatible job families

We consider online scheduling on m parallel-batch machines where the batch capacity is unbounded and the jobs belong to m incompatible job families. By incompatible job families, we mean that jobs from different families cannot be processed together in the same batch. The processing time of a job becomes known only upon its arrival. The objective is to minimize the makespan. The problem is difficult to solve so we consider the case where the number of families is equal to the number of machines. We give a lower bound 5 + 1 2 ≈ 1.618 on the competitive ratio of any online algorithm for this restricted problem. We also provide an online algorithm H m ( θ ) , where θ ∈ ( 0 , 1 ) is a parameter, and show that its competitive ratio is no less than 1 + 10 5 ≈ 1.632 . When m = 2 or under the condition that jobs belonging to the same family have identical processing times, we show that H m ( α ) , where α = 5 − 1 2 , is a best possible online algorithm. When m ≥ 3 , we prove that H m ( β ) , where β = 2 − 1 , has a competitive ratio no greater than 1 + 1 β + 1 ≈ 1.707 .

Best semi-online algorithms for unbounded parallel batch scheduling

We consider semi-online scheduling of an unbounded parallel batch machine to minimize the makespan where, at the present time instant t , information on the first longest job arriving after t is known. In this paper online means that jobs arrive over time, J ∗ ( t ) denotes the first longest job arriving after t , and p ∗ ( t ) and r ∗ ( t ) denote the processing time and arrival time of J ∗ ( t ) , respectively. Given information p ∗ ( t ) , we present an online algorithm with a competitive ratio ( 5 − 5 ) / 2 ≈ 1.382 , and show that the algorithm is the best possible; furthermore, this algorithm generates at most two batches. This algorithm is also the best possible given information J ∗ ( t ) . Given information r ∗ ( t ) , we present an online algorithm with a competitive ratio 3 / 2 , and show that any online algorithm cannot have a competitive ratio less than 3 3 ≈ 1.442 ; furthermore, this algorithm generates at most three batches. Given information r ∗ ( t ) with the restriction that an online algorithm generates at most two batches, we present an online algorithm with a competitive ratio ( 5 + 1 ) / 2 ≈ 1.618 , and show that the algorithm is the best possible.

Online scheduling on unbounded parallel-batch machines to minimize the makespan

We consider online scheduling on m parallel-batch machines to minimize the makespan where the batch capacity is unbounded. The processing time of a job becomes known only upon its arrival. We give an improved lower bound 1 + α m on the competitive ratio, where α m is the positive solution of the equation α m 2 + m α m − 1 = 0 , and establish a best possible online algorithm matching this lower bound. We also provide a new lower bound 3/2 on the competitive ratio for dense-algorithms, and present a best possible dense-algorithm matching this lower bound.