Pareto-scheduling of two competing agents with their own equal processing times

Highlights

• Single-machine Pareto-scheduling with two competing agents.

• The jobs of each agent have their own equal processing times.

• NP-hardness of some problems unaddressed in the literature.

• Unified frameworks in the algorithm design.

• A complete complexity classification for all the problems.

Abstract

We consider the Pareto-scheduling of two competing agents on a single machine, in which the jobs of each agent have their “own equal processing times” (shortly, OEPT). In the literature, two special versions of the OEPT model, in which the jobs have either unit or equal processing times, have been well studied, where the criteria are given by various regular objective functions without including the late work criteria. However, for equal processing times, the exact complexity of three problems is still unaddressed. Two-agent scheduling related to late work criteria is also a hot topic in recent years. This inspires our research by also including the total (weighted) late work as criteria. We show that, for equal processing times, all the problems are binary NP-hard if the criterion of one agent is the total tardiness or the total late work and the criterion of the other agent is either the total tardiness or the total late work or the weighted number of tardy jobs or the total weighted completion time. As a result, complexity classification for equal processing times is completely addressed. We further show that all the problems under the OEPT model are either polynomially solvable or ordinary NP-hard, which results in a complete complexity classification for the OEPT model.

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 $A$ and $B$, 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 $A$ and the other for company $B$. 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 $A$ and $B$ for loading their vehicles. By regarding the robot as a single machine, the two companies $A$ and $B$ 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 $A$ and $B$. For each $X\in \{A,B\}$, we use ${\mathcal{J}}^{\left(X\right)}$ to denote the set of jobs of agent $X$ and the jobs in ${\mathcal{J}}^{\left(X\right)}$ are called $X$-jobs. The term “competing agents” means that ${\mathcal{J}}^{\left(A\right)}\cap {\mathcal{J}}^{\left(B\right)}=\varnothing$.

Suppose that ${\mathcal{J}}^{\left(X\right)}=\{J_1^{\left(X\right)},J_2^{\left(X\right)},\dots ,J_{n_X}^{\left(X\right)}\}$ for $X\in \{A,B\}$. Let $n=n_A+n_B$ and $\mathcal{J}={\mathcal{J}}^{\left(A\right)}\cup {\mathcal{J}}^{\left(B\right)}$. All the jobs of $\mathcal{J}$ are available at time zero and should be scheduled on a single machine without preemption. For $X\in \{A,B\}$, each $X$-job $J_j^{\left(X\right)}\in {\mathcal{J}}^{\left(X\right)}$ has a processing time $p_j^{\left(X\right)}>0$, a due date $d_j^{\left(X\right)}\ge 0$, and a weight $w_j^{\left(X\right)}>0$. For each index $j\in \{1,2,\dots ,n_X\}$, we define

$${\mathcal{J}}_j^{\left(X\right)}=\{J_1^{\left(X\right)},J_2^{\left(X\right)},\dots ,J_j^{\left(X\right)}\}.$$

Then we have ${\mathcal{J}}_{n_X}^{\left(X\right)}={\mathcal{J}}^{\left(X\right)}$. Moreover, we use the following notation for a given schedule $\sigma$ of $\mathcal{J}$:

• $J_i{\prec }_{\sigma }{J}_j$ means that $J_i$ is scheduled before $J_j$ in $\sigma$.

• $J_{\sigma \left(j\right)}$ is the $j$th job in $\sigma$. Thus, we can write $\sigma =(J_{\sigma \left(1\right)},J_{\sigma \left(2\right)},\dots ,J_{\sigma \left(n\right)})$.

• $\sigma \left[J_j^{\left(X\right)}\right]$ is the position number of $J_j^{\left(X\right)}$ in $\sigma$. Thus, $\sigma \left[J_j^{\left(X\right)}\right]=x$ if and only if $J_j^{\left(X\right)}=J_{\sigma \left(x\right)}$.

• $S_j^{\left(X\right)}\left(\sigma \right)$ and $C_j^{\left(X\right)}\left(\sigma \right)$ are the starting time and the completion time of $J_j^{\left(X\right)}$ in $\sigma$, respectively. Then we have $C_j^{\left(X\right)}\left(\sigma \right)=S_j^{\left(X\right)}\left(\sigma \right)+p_j^{\left(X\right)}$.

• $L_j^{\left(X\right)}\left(\sigma \right)=C_j^{\left(X\right)}\left(\sigma \right)-d_j^{\left(X\right)}$ is the lateness of $J_j^{\left(X\right)}$ in $\sigma$.

• $T_j^{\left(X\right)}\left(\sigma \right)=\max \{0,C_j^{\left(X\right)}\left(\sigma \right)-d_j^{\left(X\right)}\}$ is the tardiness of $J_j^{\left(X\right)}$ in $\sigma$.

• $Y_j^{\left(X\right)}\left(\sigma \right)=\min \{p_j^{\left(X\right)},T_j^{\left(X\right)}\left(\sigma \right)\}$ is the late work of $J_j^{\left(X\right)}$ in $\sigma$.

• $U_j^{\left(X\right)}\left(\sigma \right)$ is the tardy indicator number of $J_j^{\left(X\right)}$ in $\sigma$, that is, $U_j^{\left(X\right)}\left(\sigma \right)=1$ if $C_j^{\left(X\right)}\left(\sigma \right)>d_j^{\left(X\right)}$, and $U_j^{\left(X\right)}\left(\sigma \right)=0$ if $C_j^{\left(X\right)}\left(\sigma \right)\le d_j^{\left(X\right)}$.

In this paper, we only consider non-decreasing cost functions $f_j^{\left(X\right)}(·)$, $j\in \{1,2,\dots ,n_X\}$, and define $f_j^{\left(X\right)}\left(\sigma \right)=f_j^{\left(X\right)}\left(C_j^{\left(X\right)}\left(\sigma \right)\right)$. Then $f_{\max }^{\left(X\right)}\left(\sigma \right)=\max \{f_j^{\left(X\right)}\left(\sigma \right):j=1,2,\dots ,n_X\}$ is the maximum cost of agent $X$ in $\sigma$ and $\sum f_j^{\left(X\right)}\left(\sigma \right)={\sum }_{j=1}^{n_X}{f}_j^{\left(X\right)}\left(\sigma \right)$ is the total cost of agent $X$ in $\sigma$. The criteria studied in this paper are some special choices of $f_{\max }^{\left(X\right)}\left(\sigma \right)$ and $\sum f_j^{\left(X\right)}\left(\sigma \right)$.

• $L_{\max }^{\left(X\right)}\left(\sigma \right)={\max }_{j\in \{1,2,\dots ,n_X\}}{L}_j^{\left(X\right)}\left(\sigma \right)$ is the maximum lateness of agent $X$ in $\sigma$.

• $W{C}_{\max }^{\left(X\right)}\left(\sigma \right)={\max }_{j\in \{1,2,\dots ,n_X\}}{w}_j^{\left(X\right)}{C}_j^{\left(X\right)}\left(\sigma \right)$ is the maximum weighted completion time of agent $X$ in $\sigma$.

• $\sum \left(w_j^{\left(X\right)}\right)C_j^{\left(X\right)}\left(\sigma \right)={\sum }_{j=1}^{n_X}\left(w_j^{\left(X\right)}\right)C_j^{\left(X\right)}\left(\sigma \right)$ is the total (weighted) completion time of agent $X$ in $\sigma$.

• $\sum \left(w_j^{\left(X\right)}\right)U_j^{\left(X\right)}\left(\sigma \right)={\sum }_{j=1}^{n_X}\left(w_j^{\left(X\right)}\right)U_j^{\left(X\right)}\left(\sigma \right)$ is the (weighted) number of tardy $X$-jobs in $\sigma$.

• $\sum \left(w_j^{\left(X\right)}\right)T_j^{\left(X\right)}\left(\sigma \right)={\sum }_{j=1}^{n_X}\left(w_j^{\left(X\right)}\right)T_j^{\left(X\right)}\left(\sigma \right)$ is the total (weighted) tardiness of agent $X$ in $\sigma$.

• $\sum \left(w_j^{\left(X\right)}\right)Y_j^{\left(X\right)}\left(\sigma \right)={\sum }_{j=1}^{n_X}\left(w_j^{\left(X\right)}\right)Y_j^{\left(X\right)}\left(\sigma \right)$ is the total (weighted) late work of agent $X$ in $\sigma$.

For an $X$-job $J_j^{\left(X\right)}$, we also use the following terminologies in a given schedule $\sigma$:

• $J_j^{\left(X\right)}$ is called early in $\sigma$ if $Y_j^{\left(X\right)}\left(\sigma \right)=0$, or equivalently, $C_j^{\left(X\right)}\left(\sigma \right)\le d_j^{\left(X\right)}$. In this case, $U_j^{\left(X\right)}\left(\sigma \right)=0$.

• $J_j^{\left(X\right)}$ is called strictly earlyin $\sigma$ if $C_j^{\left(X\right)}\left(\sigma \right)<d_j^{\left(X\right)}$.

• $J_j^{\left(X\right)}$ is called tardy in $\sigma$ if $Y_j^{\left(X\right)}\left(\sigma \right)>0$, or equivalently, $C_j^{\left(X\right)}\left(\sigma \right)>d_j^{\left(X\right)}$. In this case, $U_j^{\left(X\right)}\left(\sigma \right)=1$.

• $J_j^{\left(X\right)}$ is called partially earlyin $\sigma$ if $0<Y_j^{\left(X\right)}\left(\sigma \right)<p_j^{\left(X\right)}$, or equivalently, $S_j^{\left(X\right)}\left(\sigma \right)<d_j^{\left(X\right)}<C_j^{\left(X\right)}\left(\sigma \right)$.

• $J_j^{\left(X\right)}$ is called late in $\sigma$ if $Y_j^{\left(X\right)}\left(\sigma \right)=p_j^{\left(X\right)}$, or equivalently, $S_j^{\left(X\right)}\left(\sigma \right)\ge d_j^{\left(X\right)}$.

• $J_j^{\left(X\right)}$ is called non-late in $\sigma$ if $Y_j^{\left(X\right)}\left(\sigma \right)<p_j^{\left(X\right)}$ or, equivalently, $S_j^{\left(X\right)}\left(\sigma \right)<d_j^{\left(X\right)}$. 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 $\sigma$. For $X\in \{A,B\}$, let $f^{\left(X\right)}$ be the scheduling criterion of agent $X$ which only depends on the completion times of the $X$-jobs. As described in Agnetis et al. (2014), the following two types of problems are frequently studied in the single-machine two-agent scheduling:

$1\left|\beta \right|f^{\left(A\right)}:f^{\left(B\right)}\le Q^{\left(B\right)}$. The problem aims to find a feasible schedule $\sigma$ which minimizes $f^{\left(A\right)}\left(\sigma \right)$ subject to the constraint that $f^{\left(B\right)}\left(\sigma \right)\le Q^{\left(B\right)}$. For convenience, this problem is simply denoted by $1\left|\beta \right|f^{\left(A\right)}:f^{\left(B\right)}$. Problem $1\left|\beta \right|f^{\left(B\right)}:f^{\left(A\right)}$ can be similarly understood.

$1\left|\beta \right|$ ^# $(f^{\left(A\right)},f^{\left(B\right)})$. The problem aims to find all Pareto-optimal points for minimizing $f^{\left(A\right)}$ and $f^{\left(B\right)}$ 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 $1\left|\beta \right|f^{\left(A\right)}:f^{\left(B\right)}$ is (binary, unary) NP-hard, then the Pareto-scheduling problem $1\left|\beta \right|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ 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 form

$$1|p^{\left(A\right)},p^{\left(B\right)}|{}^{\#}(f^{\left(A\right)},f^{\left(B\right)}),$$

where “$p^{\left(A\right)},p^{\left(B\right)}$” in the $\beta$-field means that all the $A$-jobs have their own equal processing times $p^{\left(A\right)}$ and all the $B$-jobs have their own equal processing times $p^{\left(B\right)}$, that is, $p_j^{\left(X\right)}=p^{\left(X\right)}$ for $X\in \{A,B\}$ and $j\in \{1,2,\dots ,n_X\}$. 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 $1|p^{\left(A\right)},p^{\left(B\right)}|f^{\left(A\right)}:f^{\left(B\right)}$ is called an OEPT constrained scheduling problem. In the case where $p^{\left(A\right)}=p^{\left(B\right)}=p$, we use “$p_j=p$” to replace “$p^{\left(A\right)},p^{\left(B\right)}$”. 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 $1|p_j^{\left(A\right)}=p^{\left(A\right)}|-$ and $1|p_j^{\left(B\right)}=p^{\left(B\right)}|-$, where “$p_j^{\left(X\right)}=p^{\left(X\right)}$” in the $\beta$-field means that all the $X$-jobs have equal processing times $p^{\left(X\right)}$ and the processing times of the jobs of another agent have no restriction. For each $X\in \{A,B\}$, we assume that

$$\begin{array}{cc}& f^{\left(X\right)}\in \{f_{\max }^{\left(X\right)},\sum f_j^{\left(X\right)},\sum \left(w_j^{\left(X\right)}\right)C_j^{\left(X\right)},\sum \left(w_j^{\left(X\right)}\right)U_j^{\left(X\right)},\hfill \\ & \sum \left(w_j^{\left(X\right)}\right)T_j^{\left(X\right)},\sum \left(w_j^{\left(X\right)}\right)Y_j^{\left(X\right)}\}.\hfill \end{array}$$

Then 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 $1\parallel$^# $(\sum C_j^{\left(A\right)},\sum C_j^{\left(B\right)})$ is ordinary NP-hard and problems $1\parallel$^# $(\sum C_j^{\left(A\right)},f_{\max }^{\left(B\right)})$, $1\parallel$^# $(f_{\max }^{\left(A\right)},f_{\max }^{\left(B\right)})$, and $1\parallel$^# $(\sum U_j^{\left(A\right)},\sum U_j^{\left(B\right)})$ are polynomially solvable. For problem $1\parallel$^# $(\sum U_j^{\left(A\right)},f_{\max }^{\left(B\right)})$, 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 $1|p_j^{\left(B\right)}=1|$^# $(\sum w_j^{\left(A\right)}{C}_j^{\left(A\right)},f^{\left(B\right)})$ is unary NP-hard for $f^{\left(B\right)}\in \{L_{\max }^{\left(B\right)},W{C}_{\max }^{\left(B\right)},\sum U_j^{\left(B\right)},\sum T_j^{\left(B\right)},\sum Y_j^{\left(B\right)}\}$. 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 $Q|p_j=p|f_{\max }$ and $Q|p_j=p|\sum f_j$ 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 $Q|p_j=p|$^# $(f_{\max },g_{\max })$ and $Q|p_j=p|$^# $(\sum f_j,g_{\max })$.

Competing two-agent scheduling with equal processing times was first studied in Elvikis, Hamacher, & T’kindt (2011). They showed that the problems $Q|p_j=p|$^# $(f^{\left(A\right)},C_{\max }^{\left(B\right)})$ with $f^{\left(A\right)}\in \{\sum f_j^{\left(A\right)},f_{\max }^{\left(A\right)}\}$ are polynomially solvable, and especially for $f^{\left(A\right)}\in \{C_{\max }^{\left(A\right)},L_{\max }^{\left(A\right)},\sum T_j^{\left(A\right)},\sum U_j^{\left(A\right)},$ $\sum w_j^{\left(A\right)}{C}_j^{\left(A\right)},\sum w_j^{\left(A\right)}{U}_j^{\left(A\right)}\}$, they provided faster algorithms. Elvikis & T’kindt (2014) made a very profound study for problem $Q|p_j=p|$^# $(f_{\max }^{\left(A\right)},f_{\max }^{\left(B\right)})$ and presented an $O(n_A^2+n_B^2+n_A{n}_B\log n_B)$-time algorithm.

Oron et al. (2015) first studied the single-machine two-agent scheduling problems with unit processing times. For each agent $X\in \{A,B\}$, they studied almost all the choices with $f^{\left(X\right)}$ being from the criteria $W{C}_{\max }^{\left(X\right)},L_{\max }^{\left(X\right)},\sum C_j^{\left(X\right)},\sum U_j^{\left(X\right)},\sum T_j^{\left(X\right)},\sum w_j^{\left(X\right)}{C}_j^{\left(X\right)},\sum w_j^{\left(X\right)}{U}_j^{\left(X\right)}$, and they showed that all these problems are polynomially or pseudo-polynomially solvable. In particular, they proved that problems $1|p_j=1|\sum w_j^{\left(A\right)}{C}_j^{\left(A\right)}:\sum w_j^{\left(B\right)}{C}_j^{\left(B\right)}$ and $1|p_j=1|\sum w_j^{\left(A\right)}{U}_j^{\left(A\right)}:\sum w_j^{\left(B\right)}{U}_j^{\left(B\right)}$ are binary NP-hard. Wan et al. (2021) further showed that the two problems $1|p_j=1|\sum w_j^{\left(A\right)}{C}_j^{\left(A\right)}:\sum w_j^{\left(B\right)}{U}_j^{\left(B\right)}$ and $1|p_j=1|\sum w_j^{\left(A\right)}{U}_j^{\left(A\right)}:\sum w_j^{\left(B\right)}{C}_j^{\left(B\right)}$ 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 $1|p_j=1|$^# $(f^{\left(A\right)},f^{\left(B\right)})$, it is ordinary NP-hard if $f^{\left(X\right)}\in \{\sum w_j^{\left(X\right)}{C}_j^{\left(X\right)},\sum w_j^{\left(X\right)}{U}_j^{\left(X\right)}\}$ for $X\in \{A,B\}$, and is polynomially solvable otherwise. Dover & Shabtay (2016) showed that problems $1|r_j,p_j=1|$^# $(f^{\left(A\right)},f_{\max }^{\left(B\right)})$ with $f^{\left(A\right)}\in \{\sum f_j^{\left(A\right)},f_{\max }^{\left(A\right)}\}$ are polynomially solvable, and for some special cases, they provided faster algorithms.

Zhao & Yuan (2020) studied problems of the form $Q|p_j=p|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ 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 $f^{\left(A\right)}=\sum T_j^{\left(A\right)}$ and $f^{\left(B\right)}\in \{\sum T_j^{\left(B\right)},\sum w_j^{\left(B\right)}{U}_j^{\left(B\right)},\sum w_j^{\left(B\right)}{C}_j^{\left(B\right)}\}$, exact complexity of the problems $Q|p_j=p|$^# $(f^{\left(A\right)},f^{\left(B\right)})$, $P|p_j=p|$^# $(f^{\left(A\right)},f^{\left(B\right)})$, and $1|p_j=p|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ 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 $1|p_j^{\left(A\right)}=p^{\left(A\right)}|\sum C_j^{\left(A\right)}:\sum U_j^{\left(B\right)}$ is binary NP-hard. By Gao, Yuan, Ng, & Cheng (2021), problem $1|p_j^{\left(A\right)}=p^{\left(A\right)}|$^# $(\sum f_j^{\left(A\right)},f_{\max }^{\left(B\right)})$ is polynomially solvable. By Zhang et al. (2021), problems $1|p^{\left(A\right)},p^{\left(B\right)}|$^# $(\sum U_j^{\left(A\right)},\sum Y_j^{\left(B\right)})$ and $1|p_j^{\left(B\right)}=p^{\left(B\right)}|$^# $(\sum C_j^{\left(A\right)},f^{\left(B\right)})$ with $f^{\left(B\right)}\in \{\sum U_j^{\left(B\right)},\sum Y_j^{\left(B\right)}\}$ are polynomially solvable, and problems $1|p_j^{\left(A\right)}=p^{\left(A\right)}|$^# $(\sum w_j^{\left(A\right)}{C}_j^{\left(A\right)},\sum w_j^{\left(A\right)}{U}_j^{\left(A\right)})$ and $1|p_j^{\left(A\right)}=p^{\left(A\right)}|$^# $(\sum w_j^{\left(A\right)}{C}_j^{\left(A\right)},\sum w_j^{\left(A\right)}{Y}_j^{\left(A\right)})$ are ordinary NP-hard.

We study the OEPT (own equal processing times) model $1|p^{\left(A\right)},p^{\left(B\right)}|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ and the EPT (equal processing times) model $1|p_j=p|$^# $(f^{\left(A\right)},f^{\left(B\right)})$. We first show that some problems of the form $1|p_j=p|f^{\left(A\right)}:f^{\left(B\right)}$ are binary NP-hard. Then we present polynomial or pseudo-polynomial algorithms for all problems of the form $1|p^{\left(A\right)},p^{\left(B\right)}|$^# $(f^{\left(A\right)},f^{\left(B\right)})$. In some cases, our algorithms are designed for more general problems without limiting to the OEPT model. For each $X\in \{A,B\}$, we define

$$q^{\prime }\left(n_X\right)=\{\begin{array}{cc}O\left(n_X^2\right),\hfill & \text{if}{f}^{\left(X\right)}=f_{\max }^{\left(X\right)}\text{is}\text{arbitrary},\hfill \\ O\left(n_X\right),\hfill & \text{if}{f}^{\left(X\right)}\in \{L_{\max }^{\left(X\right)},W{C}_{\max }^{\left(X\right)}\},\hfill \end{array}$$

$$q^{″}\left(n_X\right)=\{\begin{array}{cc}O\left(n_X^3\right),\hfill & \text{if}{f}^{\left(X\right)}=\sum f_j^{\left(X\right)}\text{is}\text{arbitrary},\hfill \\ O\left(n_X\right),\hfill & \text{if}{f}^{\left(X\right)}\in \{\sum C_j^{\left(X\right)},\sum U_j^{\left(X\right)},\sum T_j^{\left(X\right)},\sum w_j^{\left(X\right)}{C}_j^{\left(X\right)}\},\hfill \\ O\left(n_X\log n_X\right),\hfill & \text{if}{f}^{\left(X\right)}=\sum w_j^{\left(X\right)}{U}_j^{\left(X\right)},\hfill \\ O\left(n_X^2\right),\hfill & \text{if}{f}^{\left(X\right)}=\sum Y_j^{\left(X\right)}.\hfill \end{array}$$

If $f^{\left(X\right)}\in \{\sum f_j^{\left(X\right)},\sum \left(w_j^{\left(X\right)}\right)C_j^{\left(X\right)},\sum \left(w_j^{\left(X\right)}\right)U_j^{\left(X\right)},\sum T_j^{\left(X\right)},\sum Y_j^{\left(X\right)}\}$, we simply write $f^{\left(X\right)}=\sum {\tilde{f}}_j^{\left(X\right)}$. If $f^{\left(X\right)}\in \{f_{\max }^{\left(X\right)},L_{\max }^{\left(X\right)},W{C}_{\max }^{\left(X\right)}\}$, we simply write $f^{\left(X\right)}={\tilde{f}}_{\max }^{\left(X\right)}$. Moreover, we simply write $f^{\left(X\right)}=\sum {\dot{f}}_j^{\left(X\right)}$ if $f^{\left(X\right)}\in \{\sum T_j^{\left(X\right)},\sum Y_j^{\left(X\right)},\sum w_j^{\left(X\right)}{C}_j^{\left(X\right)},\sum w_j^{\left(X\right)}{U}_j^{\left(X\right)}\}$. 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 $1|p^{\left(A\right)},p^{\left(B\right)}|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ and $1|p_j=p|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ have the same complexity classifications for the criteria

$$\begin{array}{cc}& f_{\max }^{\left(X\right)},\sum \left(w_j^{\left(X\right)}\right)C_j^{\left(X\right)},\sum \left(w_j^{\left(X\right)}\right)U_j^{\left(X\right)},\sum T_j^{\left(X\right)},\sum \left(w_j^{\left(X\right)}\right)Y_j^{\left(X\right)},\hfill \\ & X\in \{A,B\}.\hfill \end{array}$$

Our research also shows that complexity classifications of the two models $1|p_j=p|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ and $1|p_j=1|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ are different for some choices of $(f^{\left(A\right)},f^{\left(B\right)})$. Note that $Y_j^{\left(X\right)}=U_j^{\left(X\right)}$ if $p_j=1$. 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 $1|p^{\left(A\right)},p^{\left(B\right)}|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ and $1|p_j=p|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ are ordinary NP-hard and problem $1|p_j=1|$^# $(f^{\left(A\right)},f^{\left(B\right)})$ is polynomially solvable.

In the NP-hardness proof, we always bind the two criteria $\sum T_j^{\left(X\right)}$ and $\sum Y_j^{\left(X\right)}$ 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 $1|p^{\left(A\right)},p^{\left(B\right)}|$^# $(\sum C_j^{\left(A\right)},\sum w_j^{\left(B\right)}{C}_j^{\left(B\right)})$ imitates that for problem $1|p_j=1|$^# $(\sum C_j^{\left(A\right)},\sum w_j^{\left(B\right)}{C}_j^{\left(B\right)})$ 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.