Two-agent scheduling of unit processing time jobs to minimize total weighted completion time and total weighted number of tardy jobs

We consider the single‐machine Pareto‐scheduling problem to minimize the weighted number of tardy jobs and total weighted late work simultaneously. The problem is to find the set of all the Pareto‐optimal points, that is, the Pareto frontier, and their corresponding Pareto‐optimal schedules. We consider the corresponding weighted‐sum scheduling problem and primary‐secondary scheduling problems, being subproblems of the general Pareto‐scheduling problem. The NP‐hardness of the general problem follows directly from the NP‐hardness of the two constituent single‐criterion problems. We present a pseudo‐polynomial algorithm and a fully polynomial‐time approximation scheme (FPTAS) running in weakly polynomial time to deal with the general problem. When all the jobs have a common due date, we further provide an FPTAS running in strongly polynomial time. We also study some special cases of the general problem where the jobs have equal processing times, a common due date, or a common weight, and analyze their computational complexity status.

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“…j ≤ Q is binary NP-hard and pseudo-polynomially solvable. Wan et al (2021) showed that problem 1|CO,…”

Section: Literature Reviewmentioning

confidence: 99%

Pareto‐scheduling with double‐weighted jobs to minimize the weighted number of tardy jobs and total weighted late work

Guo

Yuan

et al. 2022

Naval Research Logistics

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“…The job J 1 ∈ J is represented by a union of the path (v 1 , v 2 , v 3 ) in the digraph (V, A, ∅) and the chain (v 1 , v 2 , v 3 ) in the graph (V, ∅, E). The job J 2 ∈ J includes the set V (2) = {v 4 , v 5 , v 6 , v 7 , v 8 } of the linearly ordered operations: (v 4 , v 5 , v 6 , v 7 , v 8 ). The job J 2 ∈ J is represented by a union of the path (v 4 , v 5 , v 6 , v 7 , v 8 ) in the digraph (V, A, ∅) and the chain (v 4 , v 5 , v 6 , v 7 , v 8 ) in the graph (V, ∅, E).…”

Section: Propertymentioning

confidence: 99%

“…In our article, we focus on the above mass production, which presupposes the scheduling problems with unit processing times of the jobs to minimize either makespan C max (the schedule length) or maximal lateness L max . Scheduling models with equal (unit) processing times of the jobs are an approximation for coping with the mass-industrial productions and manufactures of similar items, particularly for job-shop manufacturing that allows a manager to personalize an individual item [2].…”

Section: Introductionmentioning

confidence: 99%

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

Sotskov

Mihova²

2021

Algorithms

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This article extends the scheduling problem with dedicated processors, unit-time tasks, and minimizing maximal lateness for integer due dates to the scheduling problem, where along with precedence constraints given on the set of the multiprocessor tasks, a subset of tasks must be processed simultaneously. Contrary to a classical shop-scheduling problem, several processors must fulfill a multiprocessor task. Furthermore, two types of the precedence constraints may be given on the task set . We prove that the extended scheduling problem with integer release times of the jobs to minimize schedule length may be solved as an optimal mixed graph coloring problem that consists of the assignment of a minimal number of colors (positive integers) to the vertices of the mixed graph such that, if two vertices and are joined by the edge , their colors have to be different. Further, if two vertices and are joined by the arc , the color of vertex has to be no greater than the color of vertex . We prove two theorems, which imply that most analytical results proved so far for optimal colorings of the mixed graphs , have analogous results, which are valid for the extended scheduling problems to minimize the schedule length or maximal lateness, and vice versa.

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“…Wan et al (2016) provided a strongly polynomial‐time algorithm for the Pareto‐scheduling problem

1 ‖ (\sum U_{j}^{(A)}, f_{\max}^{false(B false)})

. Wan et al (2020) showed that problem

1 | p_{j} = 1 | \sum w_{j}^{(A)} C_{j}^{(A)} : \sum w_{j}^{(B)} U_{j}^{(B)} \leq Q

is binary NP ‐hard. Yin et al (2016) studied the problems of minimizing the performance criterion of agent A in the form

φ d^{(A)} + \sum w_{j}^{(A)} U_{j}^{(A)}

, while keeping the objective value of

f_{\max}^{(B)}

(resp.,

\sum C_{j}^{(B)}

\sum w_{j}^{(B)} C_{j}^{(B)}

, and

\sum w_{j}^{(B)} U_{j}^{(B)}

) no greater than a given limit Q under the common due date and slack due dates, respectively.…”

Section: Literature Reviewmentioning

confidence: 99%

Pareto‐optimization of three‐agent scheduling to minimize the total weighted completion time, weighted number of tardy jobs, and total weighted late work

Zhang

Yuan

et al. 2020

Naval Research Logistics

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We consider three‐agent scheduling on a single machine in which the criteria of the three agents are to minimize the total weighted completion time, the weighted number of tardy jobs, and the total weighted late work, respectively. The problem is to find the set of all the Pareto‐optimal points, that is, the Pareto frontier, and their corresponding Pareto‐optimal schedules. Since the above problem is unary NP‐hard, we study the problem under the restriction that the jobs of the first agent have inversely agreeable processing times and weights, that is, the smaller the processing time of a job is, the greater its weight is. For this restricted problem, which is NP‐hard, we present a pseudo‐polynomial‐time algorithm to find the Pareto frontier. We also show that, for various special versions, the time complexity of solving the problem can be further reduced.

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Two-agent scheduling of unit processing time jobs to minimize total weighted completion time and total weighted number of tardy jobs

Cited by 19 publications

References 26 publications

Pareto‐scheduling with double‐weighted jobs to minimize the weighted number of tardy jobs and total weighted late work

Pareto‐scheduling with double‐weighted jobs to minimize the weighted number of tardy jobs and total weighted late work

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

Pareto‐optimization of three‐agent scheduling to minimize the total weighted completion time, weighted number of tardy jobs, and total weighted late work

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