Discrete OptimizationPareto-scheduling of two competing agents with their own equal processing times
Introduction
Baker & Smith (2003) and Agnetis, Mirchandani, Pacciarelli, & Pacifici (2004) were the pioneers that introduced the two-agent concept to the scheduling field. In the past decade, various two- and multi-agent scheduling models have received increasing attention from researchers, see Agnetis, Billaut, Gawiejnowicz, Pacciarelli, & Soukhal (2014) and Perez-Gonzalez & Framinan (2014) for details. In particular, the competing multi-agent scheduling model plays an important role and it appears in many application fields. For example, in the universal mobile telecommunication system Arbib, Smriglio, & Servilio (2004), in preventive maintenance scheduling (Leung, Pinedo, & Wan, 2010), in flowshops (Ahmadi-Darani, Moslehi, Reisi-Nafchi, 2018, Fan, Cheng, 2016), in parallel-batching scheduling (Gao, Yuan, Ng, Cheng, 2019, Tang, Zhao, Liu, Leung, 2017, Wang, Kang, Shiau, Wu, Hsu, 2017a), in the warehouse management (Park, Min, & Choi, 2018), in fairness issues (Agnetis, Chen, Nicosia, & Pacifici, 2019).
In practical application, a common phenomenon is that the jobs belonging to each agent may have equal processing times. As an example, let us consider a scheduling problem arising in the vehicle loading process in logistic industry. In this problem, we have two logistic companies and , each with its own fixed customer base consisting of some retailers. Each company has sufficiently many vehicles so that the goods of each retailer of him can be sent by a single vehicle. Each day, all the goods ordered from the retailers are stored at two different places in a given warehouse, one for company and the other for company . One robot is used to load the goods onto the vehicles to be dispatched such that the loading times of the vehicles belonging to the same company are equal (or almost equal). Each company wants to have a good performance in the vehicle loading process so that its customers can obtain satisfactory service. This means that the robot is the common resource to be used by two competing companies and for loading their vehicles. By regarding the robot as a single machine, the two companies and as two competing agents, and the vehicles as jobs, the above loading vehicle process can be formulated as the scheduling problem with two competing agents on a single machine in which the jobs of each agent have their own equal processing times (shortly, OEPT). This motivates the research in this paper.
Our research is also motivated by the theoretical progress in competing two-agent Pareto-scheduling on a single machine. In the literature, when the jobs have either unit or equal processing times, the competing two-agent scheduling model has been well studied, where the criteria are given by maximum cost, total (weighted) completion time, (weighted) number of tardy jobs, and total tardiness. For unit processing times, complexity classification of all the problems has been completely addressed by the works in Oron, Shabtay, & Steiner (2015) and Wan, Mei, & Du (2021). For equal processing times, Zhao & Yuan (2020) studied all the problems again and presented polynomial or pseudo-polynomial algorithm for each problem. However, the exact complexity (polynomially solvable or ordinary NP-hard) of three problems is still unaddressed. This motivates us to address the exact complexity of these remaining problems. Moreover, it is theoretically meaningful to consider the more general version in which the jobs of each agent have their own equal processing times. Notice that scheduling with late work criteria is a hot topic in recent years, see Chen, Yuan, & Gao (2019); Chen, Chau, Xie, Sterna, & Blazewicz (2017); Chen, Sterna, Han, & Blazewicz (2016); Gerstl, Mor, & Mosheiov (2019); He & Yuan (2020); Li, Gajpal, & Bector (2020); Mosheiov, Oron, & Shabtay (2021); Sterna & Czerniachowska (2017); Wang, Fan, Zhang, Zhang, & Leung (2017b); Yin, Xu, Cheng, Wu, & Wang (2016); Yin, Y., Wang, & Cheng (2017); Zhang & Wang (2017); Zhang & Yuan (2019), and Zhang, Geng, & Yuan (2020); Zhang, Yuan, Ng, & Cheng (2021). Among them, competing two-agent scheduling accounts for a large proportion. This inspires our research by also including the total (weighted) late work as criteria.
Consider the single-machine scheduling with two competing agents and . For each , we use to denote the set of jobs of agent and the jobs in are called -jobs. The term “competing agents” means that .
Suppose that for . Let and . All the jobs of are available at time zero and should be scheduled on a single machine without preemption. For , each -job has a processing time , a due date , and a weight . For each index , we defineThen we have . Moreover, we use the following notation for a given schedule of :
- •
means that is scheduled before in .
- •
is the th job in . Thus, we can write .
- •
is the position number of in . Thus, if and only if .
- •
and are the starting time and the completion time of in , respectively. Then we have .
- •
is the lateness of in .
- •
is the tardiness of in .
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is the late work of in .
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is the tardy indicator number of in , that is, if , and if .
In this paper, we only consider non-decreasing cost functions , , and define . Then is the maximum cost of agent in and is the total cost of agent in . The criteria studied in this paper are some special choices of and .
- •
is the maximum lateness of agent in .
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is the maximum weighted completion time of agent in .
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is the total (weighted) completion time of agent in .
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is the (weighted) number of tardy -jobs in .
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is the total (weighted) tardiness of agent in .
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is the total (weighted) late work of agent in .
For an -job , we also use the following terminologies in a given schedule :
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is called early in if , or equivalently, . In this case, .
- •
is called strictly earlyin if .
- •
is called tardy in if , or equivalently, . In this case, .
- •
is called partially earlyin if , or equivalently, .
- •
is called late in if , or equivalently, .
- •
is called non-late in if or, equivalently, . Clearly, a job is non-late if and only if it is either an early job or a partially early job.
Without causing confusion, we can omit the notation . For , let be the scheduling criterion of agent which only depends on the completion times of the -jobs. As described in Agnetis et al. (2014), the following two types of problems are frequently studied in the single-machine two-agent scheduling:
. The problem aims to find a feasible schedule which minimizes subject to the constraint that . For convenience, this problem is simply denoted by . Problem can be similarly understood.
# . The problem aims to find all Pareto-optimal points for minimizing and and, for each Pareto-optimal point, a corresponding Pareto-optimal schedule.
It is implied in Hoogeveen (2005) and T’kindt & Billaut (2006) that, if the constrained scheduling problem is (binary, unary) NP-hard, then the Pareto-scheduling problem # is also (binary, unary) NP-hard.
In this paper, we focus on the research for the single-machine competing two-agent Pareto-scheduling problems of the formwhere “” in the -field means that all the -jobs have their own equal processing times and all the -jobs have their own equal processing times , that is, for and . For convenience, we use OEPT to abbreviate “own equal processing times”. Then each problem of the form (1) is called an OEPT Pareto-scheduling problem. Correspondingly, each constrained scheduling problem of the form is called an OEPT constrained scheduling problem. In the case where , we use “” to replace “”. Since all the jobs have equal processing times (EPT), the corresponding OEPT problems are called EPT problems. For technical reasons, we also study problems of the forms and , where “” in the -field means that all the -jobs have equal processing times and the processing times of the jobs of another agent have no restriction. For each , we assume thatThen the complexity classifications for the EPT model and OEPT model are studied.
In the following review, the complexity of each problem is classified by stating that the problem is polynomially solvable, ordinary NP-hard, or unary NP-hard, where a problem is ordinary NP-hard if it is binary NP-hard and is also pseudo-polynomially solvable.
Agnetis et al. (2004) showed that problem # is ordinary NP-hard and problems # , # , and # are polynomially solvable. For problem # , Agnetis et al. (2004) presented a weakly polynomial algorithm and Wan, Yuan, & Wei (2016) further presented a strongly polynomial algorithm. Leung et al. (2010) presented a comprehensive research for the competing two-agent scheduling with release dates and preemption. Choi & Chung (2014) studied the competing two-agent scheduling problems with just-in-time jobs. Perez-Gonzalez & Framinan (2014) introduced and studied some new models of two-agent scheduling. Agnetis et al. (2014) collected all the achievements in this field before 2014. In particular, it is implied in Agnetis et al. (2014) that problem # is unary NP-hard for . Recent research on two-agent scheduling can be found in Chen et al. (2019); Cheng, Li, Ying, & Liu (2019); Gao & Yuan (2017); He & Yuan (2020); Hermelin, Kubitza, Shabtay, Talmon, & Woeginger (2019); Li et al. (2020); Mor & Mosheiov (2010); Yuan (2018); Yuan, Ng, & Cheng (2020); Zhang et al. (2020), and Zhao & Yuan (2020), among many others. Nowadays, the two-agent scheduling has become a popular topic in scheduling research. For our purpose, we only review the known results related to competing two-agent scheduling or single-agent scheduling with equal processing times.
Single-criterion scheduling with equal processing times was first studied in Dessouky, Lageweg, Lenstra, & de Vel (1990). Under the uniform-machine environment, the authors showed that all the problems of the forms and are polynomially solvable. Bicriteria scheduling with equal processing times on uniform machines was studied in Tuzikov, Makhaniok, & Manner (1998). They presented polynomial algorithms for the general problems # and # .
Competing two-agent scheduling with equal processing times was first studied in Elvikis, Hamacher, & T’kindt (2011). They showed that the problems # with are polynomially solvable, and especially for , they provided faster algorithms. Elvikis & T’kindt (2014) made a very profound study for problem # and presented an -time algorithm.
Oron et al. (2015) first studied the single-machine two-agent scheduling problems with unit processing times. For each agent , they studied almost all the choices with being from the criteria , and they showed that all these problems are polynomially or pseudo-polynomially solvable. In particular, they proved that problems and are binary NP-hard. Wan et al. (2021) further showed that the two problems and are also binary NP-hard. Consequently, the complexity classification of all the problems studied in Oron et al. (2015) has been completely addressed. For problem # , it is ordinary NP-hard if for , and is polynomially solvable otherwise. Dover & Shabtay (2016) showed that problems # with are polynomially solvable, and for some special cases, they provided faster algorithms.
Zhao & Yuan (2020) studied problems of the form # for all the criteria studied in Oron et al. (2015). They showed that all the problems are polynomially or pseudo-polynomially solvable. In the case where and , exact complexity of the problems # , # , and # are still unaddressed. This also shows that the model with equal processing times is in fact different from the model with unit processing times. More research achievements on scheduling with equal processing times can be found in Gerstl, Mosheiov, 2013, Gerstl, Mosheiov, 2017; Sadi & Soukhal (2017), and Sarin & Prakash (2004).
To our knowledge, there are a few works related to the OEPT two-agent scheduling. By Chen et al. (2019), problem is binary NP-hard. By Gao, Yuan, Ng, & Cheng (2021), problem # is polynomially solvable. By Zhang et al. (2021), problems # and # with are polynomially solvable, and problems # and # are ordinary NP-hard.
We study the OEPT (own equal processing times) model # and the EPT (equal processing times) model # . We first show that some problems of the form are binary NP-hard. Then we present polynomial or pseudo-polynomial algorithms for all problems of the form # . In some cases, our algorithms are designed for more general problems without limiting to the OEPT model. For each , we define
If , we simply write . If , we simply write . Moreover, we simply write if . Then the contributions of this paper can be described in Table 1.
From Table 1 and combining the known results in the literature, we can obtain that the two models # and # have the same complexity classifications for the criteriaOur research also shows that complexity classifications of the two models # and # are different for some choices of . Note that if . We use Table 2 to display and compare the complexity classifications of the three models, where “P” indicates “polynomially solvable”, “ONP” indicates “ordinary NP-hard”, and “ONP” indicates that the two problems # and # are ordinary NP-hard and problem # is polynomially solvable.
In the NP-hardness proof, we always bind the two criteria and together so that the seven NP-hardness results (in the sense of symmetry) can be proved in three theorems. In our algorithm design, some ideas are learned from Agnetis et al. (2014, 2004); Elvikis et al. (2011); Gao et al. (2021); Lawler (1976); Oron et al. (2015); T’kindt & Billaut (2006); Zhang et al. (2021), and Zhao & Yuan (2020). In particular, our solution for problem # imitates that for problem # in Oron et al. (2015). Moreover, for problems with similar properties, we design algorithms for solving them under unified frameworks.
This paper is organized as follows: In Section 2, we present some NP-hardness results. In Section 3, we present some preliminaries for algorithm design. In Section 4, we provide polynomial algorithms. In Section 5, we provide pseudo-polynomial algorithms. We conclude the paper and suggest future research topics in the last section.
Section snippets
NP-hardness results
In our NP-hardness proof, the two criteria and take the same value, and so, have the same role. Let . Then for . We use the following lemma to identify the two criteria and with under a certain condition. Lemma 2.1 Consider problem and suppose that for some . Let be a schedule such that . Then we have every -job is non-late in , and so, for every -job
Pareto optimization
Consider two-dimensional vectors (shortly, 2-vectors, or vectors). We say that vector dominates vector if and . A vector set is called well sorted if the vectors of are sorted in the non-decreasing order of their first components. A vector is called non-dominated if no vector in dominates . The set of all non-dominated vectors of is denoted by . In the case where , we call a non-dominated vector set. The
Problems of the form #
The -constraint method was used in Gao et al. (2021) to solve problem # . The associated algorithm for solving problem can be simply described as follows in our own sentences. Algorithm 4.1 For solving problem .
Implementation: The algorithm consists of two stages. Stage 1 schedules the -jobs and Stage 2 schedules the -jobs.
- Stage 1:
Schedule the -jobs from time backwards in the following way. Initially set . If some
Pseudo-polynomial algorithms
From Oron et al. (2015) and Wan et al. (2021) and from Theorems 2.1–2.3, all the problems # with for are binary NP-hard. In the following, we present pseudo-polynomial algorithms for solving these 15 problems (in the sense of symmetry) in a unified framework.
Conclusion
In this paper, we study competing two-agent Pareto-scheduling on a single machine, where the jobs of each agent have their own equal processing times , but not limited. We show that problems # and # are binary NP-hard even when , where . Moreover, we present polynomial or pseudo-polynomial algorithms for some versions of problem # . Combining the
Acknowledgments
We sincerely thank the Associate Editor (AE) and three anonymous referees for their constructive comments and helpful suggestions on an early version of our paper. This research was supported in part by the National Natural Science Foundation of China under grant numbers 12101567, 12071442, and 11771406.
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