Open Access
This article is
 freely available
 reusable
Algorithms 2020, 13(1), 4; https://doi.org/10.3390/a13010004
Article
TwoMachine JobShop Scheduling Problem to Minimize the Makespan with Uncertain Job Durations
^{1}
United Institute of Informatics Problems, National Academy of Sciences of Belarus, Surganova Street 6, 220012 Minsk, Belarus
^{2}
Department of Automated Production Management Systems, Belarusian State Agrarian Technical University, Nezavisimosti Avenue 99, 220023 Minsk, Belarus
^{3}
Department of Electronic Computing Machines, Belarusian State University of Informatics and Radioelectronics, P. Brovki Street 6, 220013 Minsk, Belarus
^{*}
Author to whom correspondence should be addressed.
Received: 30 October 2019 / Accepted: 16 December 2019 / Published: 20 December 2019
Abstract
:We study twomachine shopscheduling problems provided that lower and upper bounds on durations of n jobs are given before scheduling. An exact value of the job duration remains unknown until completing the job. The objective is to minimize the makespan (schedule length). We address the issue of how to best execute a schedule if the job duration may take any real value from the given segment. Scheduling decisions may consist of two phases: an offline phase and an online phase. Using information on the lower and upper bounds for each job duration available at the offline phase, a scheduler can determine a minimal dominant set of schedules (DS) based on sufficient conditions for schedule domination. The DS optimally covers all possible realizations (scenarios) of the uncertain job durations in the sense that, for each possible scenario, there exists at least one schedule in the DS which is optimal. The DS enables a scheduler to quickly make an online scheduling decision whenever additional information on completing jobs is available. A scheduler can choose a schedule which is optimal for the most possible scenarios. We developed algorithms for testing a set of conditions for a schedule dominance. These algorithms are polynomial in the number of jobs. Their time complexity does not exceed $O\left({n}^{2}\right)$. Computational experiments have shown the effectiveness of the developed algorithms. If there were no more than 600 jobs, then all 1000 instances in each tested series were solved in one second at most. An instance with 10,000 jobs was solved in 0.4 s on average. The most instances from nine tested classes were optimally solved. If the maximum relative error of the job duration was not greater than $20\%$, then more than $80\%$ of the tested instances were optimally solved. If the maximum relative error was equal to $50\%$, then $45\%$ of the tested instances from the nine classes were optimally solved.
Keywords:
scheduling; uncertain duration; flowshop; jobshop; makespan criterion1. Introduction
A lot of reallife scheduling problems involve different forms of uncertainties. For dealing with uncertain scheduling problems, several approaches have been developed in the literature. A stochastic approach assumes that durations of the jobs are random variables with specific probability distributions known before scheduling. There are two types of stochastic scheduling problems [1], where one is on stochastic jobs and another is on stochastic machines. In the stochastic job problem, each job duration is assumed to be a random variable following a certain probability distribution. With an objective of minimizing the expected makespan, the flowshop problem was considered in References [2,3,4]. In the stochastic machine problem, each job duration is a constant, while each completion time of the job is a random variable due to the machine breakdown or nonavailability. In References [5,6,7], flowshop problems to stochastically minimize the makespan or total completion time have been considered.
If there is no information to determine a probability distribution for each random duration of the job, other approaches have to be used [8,9,10]. In the approach of seeking a robust schedule [8,11,12,13], a decision maker prefers a schedule that hedges against the worstcase scenario. A fuzzy approach [14,15,16] allows a scheduler to find best schedules with respect to fuzzy durations of the jobs. A stability approach [17,18,19,20] is based on the stability analysis of optimal schedules to possible variations of the durations. In this paper, we apply the stability approach to the twomachine jobshop scheduling problem with given segments of job durations. We have to emphasize that uncertainties of the job durations considered in this paper are due to external forces in contrast to scheduling problems with controllable durations [21,22,23], where the objective is to determine optimal durations (which are under the control of a decision maker) and to find an optimal schedule for the jobs with optimal durations.
2. Contributions and New Results
We study the twomachine jobshop scheduling problem with uncertain job durations and address the issue of how to best execute a schedule if each duration may take any value from the given segment. The main aim is to determine a minimal dominant set of schedules (DS) that would contain at least one optimal schedule for each feasible scenario of the distribution of durations of the jobs.
It is shown how an uncertain twomachine jobshop problem may be decomposed into two uncertain twomachine flowshop problems. We prove several sufficient conditions for the existence of a small dominant set of schedules. In particular, the sufficient and necessary conditions are proven for the existence of a single pair of job permutations, which is optimal for the twomachine jobshop problem with any possible scenario. We investigated properties of the optimal pairs of job permutations for the uncertain twomachine jobshop problem.
In the stability approach, scheduling decisions may consist of two phases: an offline phase and an online phase. Using information on the lower and upper bounds on each job duration available at the offline phase, a scheduler can determine a small (or minimal) dominant set of schedules based on sufficient conditions for schedule dominance. The DS optimally covers all scenarios in the sense that, for each possible scenario, there exists at least one schedule in the DS that is optimal. The DS enables a scheduler to quickly make an online scheduling decision whenever additional information on completing some jobs becomes available. The stability approach enables a scheduler to choose a schedule, which is optimal for the most possible scenarios.
In this paper, we develop algorithms for testing a set of conditions for a schedule dominance. The developed algorithms are polynomial in the number of jobs. Their asymptotic complexities do not exceed $O\left({n}^{2}\right)$, where n is a number of the jobs. Computational experiments have shown effectiveness of the developed algorithms: if there were no more than 600 jobs, then all 1000 instances in each tested series were solved in no more than one second. For the tested series of instances with 10,000 jobs, all 1000 instances of a series were solved in 344 seconds at most (on average, 0.4 s per one instance).
The paper is organized as follows. In Section 3, we present settings of the uncertain scheduling problems. The related literature and closed results are discussed in Section 4. In Section 4.2, we describe in detail the results published for the uncertain twomachine flowshop problem. These results are used in Section 5, where we investigate properties of the optimal job permutations used for processing a set of the given jobs. Some proofs of the claims are given in Appendix A. In Section 6, we develop algorithms for constructing optimal schedules if the proven dominance conditions hold. In Section 7, we report on the wide computational experiments for solving a lot of randomly generated instances. Tables with the obtained computational results are presented in Appendix B. The paper is concluded in Section 8, where several directions for further researches are outlined.
3. Problem Settings and Notations
Using the notation $\alpha \left\beta \right\gamma $ [24], the twomachine jobshop scheduling problem with minimizing the makespan is denoted as $J2{n}_{i}\le 2{C}_{max}$, where $\alpha =J2$ denotes a jobshop system with two available machines, ${n}_{i}$ is the number of stages for processing a job, and $\gamma ={C}_{max}$ denotes the criterion of minimizing the makespan. In the problem $J2{n}_{i}\le 2{C}_{max}$, the set $\mathcal{J}=\{{J}_{1},{J}_{2},\dots ,{J}_{n}\}$ of the given jobs have to be processed on machines from the set $\mathcal{M}=\{{M}_{1},{M}_{2}\}$. All jobs are available for processing from the initial time $t=0$. Let ${O}_{ij}$ denote an operation of the job ${J}_{i}\in \mathcal{J}$ processed on machine ${M}_{j}\in \mathcal{M}$. Each machine can process a job ${J}_{i}\in \mathcal{J}$ no more than once provided that preemption of each operation ${O}_{ij}$ is not allowed. Each job ${J}_{i}\in \mathcal{J}$ has its own processing order (machine route) on the machines in $\mathcal{M}$.
Let ${\mathcal{J}}_{1,2}$ denote a subset of the set $\mathcal{J}$ of the jobs with the same machine route (${M}_{1},{M}_{2}$), i.e., each job ${J}_{i}\in {\mathcal{J}}_{1,2}$ has to be processed first on machine ${M}_{1}$ and then on machine ${M}_{2}$. Let ${\mathcal{J}}_{2,1}\subseteq \mathcal{J}$ denote a subset of the jobs with the opposite machine route (${M}_{2},{M}_{1}$). Let ${\mathcal{J}}_{k}\subseteq \mathcal{J}$ denote a set of the jobs, which has to be processed only on machine ${M}_{k}\in \mathcal{M}$. The partition $\mathcal{J}={\mathcal{J}}_{1}\bigcup {\mathcal{J}}_{2}\bigcup {\mathcal{J}}_{1,2}\bigcup {\mathcal{J}}_{2,1}$ holds. We denote ${m}_{h}=\left{\mathcal{J}}_{h}\right$, where $h\in \{1;2;1,\phantom{\rule{0.166667em}{0ex}}2;2,\phantom{\rule{0.166667em}{0ex}}1\}$.
We first assume that the duration ${p}_{ij}$ of each operation ${O}_{ij}$ is fixed before scheduling. The considered criterion ${C}_{max}$ is the minimization of the makespan (schedule length) as follows:
where ${C}_{i}\left(s\right)$ denotes a completion time of the job ${J}_{i}\in \mathcal{J}$ in the schedule s and S denotes a set of semiactive schedules existing for the problem $J2{n}_{i}\le 2{C}_{max}$. A schedule is called semiactive if no job (operation) can be processed earlier without changing the processing order or violating some given constraints [1,25,26].
$${C}_{max}:=\underset{s\in S}{min}{C}_{max}\left(s\right)=\underset{s\in S}{min}\{max\{{C}_{i}\left(s\right):{J}_{i}\in \mathcal{J}\}\},$$
Jackson [27] proved that the problem $J2{n}_{i}\le 2{C}_{max}$ is polynomially solvable and that the optimal schedule for this problem may be determined as a pair $({\pi}^{\prime},{\pi}^{\u2033})$ of the job permutations (calling it a Jackson’s pair of permutations) such that ${\pi}^{\prime}=({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1})$ is a sequence of all jobs from the set ${\mathcal{J}}_{1}\bigcup {\mathcal{J}}_{1,2}\bigcup {\mathcal{J}}_{2,1}$ processed on machine ${M}_{1}$ and ${\pi}^{\u2033}=({\pi}_{2,1},{\pi}_{2},{\pi}_{1,2})$ is a sequence of all jobs from the set ${\mathcal{J}}_{2}\bigcup {\mathcal{J}}_{1,2}\bigcup {\mathcal{J}}_{2,1}$ processed on machine ${M}_{2}$. Job ${J}_{j}$ belongs to the permutation ${\pi}_{h}$ if ${J}_{j}\in {\mathcal{J}}_{h}$.
In a Jackson’s pair $({\pi}^{\prime},{\pi}^{\u2033})$ of the job permutations, the order for processing jobs from set ${\mathcal{J}}_{1}$ (from set ${\mathcal{J}}_{2}$, respectively) may be arbitrary, while for the permutation ${\pi}_{1,2}$, the following inequality holds for all indexes k and m, $1\le k<m\le {m}_{1,2}$:
(for the permutation ${\pi}_{2,1}$, the following inequality holds for all indexes k and m, $1\le k<m\le {m}_{2,1}$) [28]:
$$min\{{p}_{{i}_{k}1},{p}_{{i}_{m}2}\}\le min\{{p}_{{i}_{m}1},{p}_{{i}_{k}2}\}$$
$$min\{{p}_{{j}_{k}2},{p}_{{j}_{m}1}\}\le min\left\{{p}_{{j}_{m}2}{p}_{{j}_{k}1}\right\}$$
The aim of this paper is to investigate the uncertain twomachine jobshop scheduling problem. Therefore, we next assume that duration ${p}_{ij}$ of each operation ${O}_{ij}$ is unknown before scheduling; namely, in the realization of a schedule, a value of ${p}_{ij}$ may be equal to any real number no less than the given lower bound ${l}_{ij}$ and no greater than the given upper bound ${u}_{ij}$. Furthermore, it is assumed that probability distributions of random durations of the jobs are unknown before scheduling. Such a jobshop scheduling problem is denoted as $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. The problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ is called an uncertain scheduling problem in contrast to the deterministic scheduling problem $J2{n}_{i}\le 2{C}_{max}$. Let a set of all possible vectors $p=({p}_{1,1},{p}_{1,2},\dots ,$${p}_{n1},{p}_{n2})$ of the job durations be determined as follows: $T=\{p\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{l}_{ij}\le {p}_{ij}\le {u}_{ij},\phantom{\rule{0.277778em}{0ex}}{J}_{i}\in \mathcal{J},\phantom{\rule{0.277778em}{0ex}}{M}_{j}\in \mathcal{M}\}.$ Such a vector $p=({p}_{1,1},{p}_{1,2},\dots ,$${p}_{n1},{p}_{n2})\in T$ of the possible durations of the jobs is called a scenario.
It should be noted that the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ is mathematically incorrect. Indeed, in most cases, a single pair of job permutations which is optimal for all possible scenarios $p\in T$ for the uncertain problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ does not exist. Therefore, in the general case, one cannot find an optimal solution for this uncertain scheduling problem.
For a fixed scenario $p\in T$, the uncertain problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ turns into the deterministic problem $J2{n}_{i}\le 2{C}_{max}$ associated with scenario p. The latter deterministic problem is an individual one and we denote it as $J2p,{n}_{i}\le 2{C}_{max}$. For any fixed scenario $p\in T$, there exists a Jackson’s pair of the job permutations that is optimal for the individual deterministic problem $J2p,{n}_{i}\le 2{C}_{max}$ associated with scenario p.
Let ${S}_{1,2}$ denote a set of all permutations of ${m}_{1,2}$ jobs from the set ${\mathcal{J}}_{1,2}$, where ${S}_{1,2}={m}_{1,2}!$. Let ${S}_{2,1}$ denote a set of all permutations of ${m}_{2,1}$ jobs from the set ${\mathcal{J}}_{2,1}$, where ${S}_{2,1}={m}_{2,1}!.$ Let $S=<\phantom{\rule{0.166667em}{0ex}}{S}_{1,2},{S}_{2,1}\phantom{\rule{0.166667em}{0ex}}>$ be a subset of the Cartesian product $({S}_{1,2},{\pi}_{1},{S}_{2,1})\times ({S}_{2,1},{\pi}_{2},{S}_{1,2})$ such that each element of the set S is a pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})\in S$, where ${\pi}^{\prime}=({\pi}_{1,2}^{i},{\pi}_{1},{\pi}_{2,1}^{j})$ and ${\pi}^{\u2033}=({\pi}_{2,1}^{j},{\pi}_{2},{\pi}_{1,2}^{i})$, $1\le i\le {m}_{1,2}!$, $1\le j\le {m}_{2,1}!$. The set S determines all semiactive schedules and vice versa.
Remark 1.
As an order for processing jobs from set ${\mathcal{J}}_{1}$ (from set ${\mathcal{J}}_{2}$) may be arbitrary in the Jackson’s pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})$, in what follows, we fix both permutations ${\pi}_{1}$ and ${\pi}_{2}$ in the increasing order of the indexes of their jobs. Thus, both permutations ${\pi}_{1}$ and ${\pi}_{2}$ are now fixed, and so their upper indexes are omitted in each permutation from the pair $({\pi}^{\prime},{\pi}^{\u2033})=(({\pi}_{1,2}^{i},{\pi}_{1},{\pi}_{2,1}^{j}),({\pi}_{2,1}^{j},{\pi}_{2},{\pi}_{1,2}^{i}))$.
Due to Remark 1, the equality $\leftS\right={m}_{1,2}!\xb7{m}_{2,1}!$ holds. The following definition is used for a Jsolution for the uncertain problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$.
Definition 1.
A minimal (with respect to the inclusion) set of pairs of job permutations $S\left(T\right)\subseteq S$ is called a Jsolution for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with set $\mathcal{J}$ of the given jobs if, for each scenario $p\in T$, the set $S\left(T\right)$ contains at least one pair $({\pi}^{\prime},{\pi}^{\u2033})\in S$ of the job permutations, which is optimal for the individual deterministic problem $J2p,{n}_{i}\le 2{C}_{max}$ associated with scenario p.
From Definition 1, it follows that, for any proper subset ${S}^{\prime}$ of the set $S\left(T\right)$${S}^{\prime}\subset S\left(T\right)$, there exists at least one scenario ${p}^{\prime}\in T$ such that set ${S}^{\prime}$ does not contain an optimal pair of job permutations for the individual deterministic problem $J2{p}^{\prime},{n}_{i}\le 2{C}_{max}$ associated with scenario ${p}^{\prime}$, i.e., set $S\left(T\right)$ is a minimal (with respect to the inclusion) set possessing the property indicated in Definition 1.
The uncertain jobshop problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ is a generalization of the uncertain flowshop problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$, where all jobs from the set $\mathcal{J}$ have the same machine route. Two flowshop problems are associated with the individual jobshop problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. In one of these flowshop problems, an optimal schedule for processing jobs ${\mathcal{J}}_{1,2}$ has to be determined, i.e., ${\mathcal{J}}_{2,1}={\mathcal{J}}_{1}={\mathcal{J}}_{2}=\varnothing $. In another flowshop problem, an optimal schedule for processing jobs ${\mathcal{J}}_{2,1}$ has to be determined, i.e., ${\mathcal{J}}_{1,2}={\mathcal{J}}_{1}={\mathcal{J}}_{2}=\varnothing $. Thus, a solution of the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ may be based on solutions of the two associated problems $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and with job set ${\mathcal{J}}_{2,1}$.
The permutation ${\pi}_{1,2}$ of all jobs from set ${\mathcal{J}}_{1,2}$ (the permutation ${\pi}_{2,1}$ of all jobs from set ${\mathcal{J}}_{2,1}$, respectively) is called a Johnson’s permutation, if the inequality in Equation (1) holds for the permutation ${\pi}_{1,2}$ (the inequality in Equation (2) holds for the permutation ${\pi}_{2,1}$, respectively). As it is proven in Reference [28], a Johnson’s permutation is optimal for the deterministic problem $F2\left\right{C}_{max}$.
4. A Literature Review and Closed Results
In this section, we address uncertain shopscheduling problems if it is impossible to obtain probability distributions for random durations of the given jobs. In particular, we consider the uncertain twomachine flowshop problem with the objective of minimizing the makespan. This problem is well studied and there are a lot of results published in the literature, unlike the uncertain jobshop problem.
4.1. Uncertain ShopScheduling Problems
The stability approach was proposed in Reference [17] and developed in Reference [18,29,30,31] for the ${C}_{max}$ criterion, and in References [19,32,33,34,35] for the total completion time criterion $\sum {C}_{i}:={min}_{s\in S}{\sum}_{{J}_{i}\in \mathcal{J}}{C}_{i}\left(s\right)$. The stability approach combines a stability analysis of the optimal schedules, a multistage decision framework, and the solution concept of a minimal dominant set $S\left(T\right)$ of schedules, which optimally covers all possible scenarios. The main aim of the stability approach is to construct a schedule which remains optimal for most scenarios of the set T. The minimality of the dominant set $S\left(T\right)$ is useful for the twophase scheduling described in Reference [36].
At the offline phase, one can construct set $S\left(T\right)$, which enables a scheduler to make a quick scheduling decision at the online phase whenever additional local information becomes available. The knowledge of the minimal dominant set $S\left(T\right)$ enables a scheduler to execute best a schedule and may end up executing a schedule optimally in many cases of the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ [36]. In Reference [17], a formula for calculating the stability radius of an optimal schedule is proven, i.e., the largest value of independent variations of the job durations in a schedule such that this schedule remains optimal. In Reference [19], a stability analysis of a schedule minimizing the total completion time was exploited in the branchandbound method for solving the jobshop problem $Jm{l}_{ij}\le {p}_{ij}\le {u}_{ij}\sum {C}_{i}$ with m machines. In Reference [29], for the twomachine flowshop problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$, sufficient conditions have been identified when the transposition of two jobs minimizes the makespan.
Reference [37] addresses the total completion time objective in the flowshop problem with uncertain durations of the jobs. A geometrical algorithm has been developed for solving the flowshop problem $Fm{l}_{ij}\le {p}_{ij}\le {u}_{ij},n=2\sum {C}_{i}$ with m machines and two jobs. For this problem with two or three machines, sufficient conditions are determined such that the transposition of two jobs minimizes $\sum {C}_{i}$. Reference [38] is devoted to the case of separate setup times with the criterion of minimizing the makespan or total completion time. The job durations are fixed while each setup time is relaxed to be a distributionfree random variable within the given lower and upper bounds. Local and global dominance relations have been determined for the flowshop problem with two machines.
Since, for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ there often does not exist a single permutation of n jobs $\mathcal{J}={\mathcal{J}}_{1,2}$ which remains optimal for all possible scenarios, an additional criterion may be introduced for dealing with uncertain scheduling problems. In Reference [39], a robust solution minimizing the worstcase deviation from optimality was proposed to hedge against uncertainties. While the deterministic problem $F2\left\right{C}_{max}$ is polynomially solvable (the optimal Johnson’s permutation may be constructed for the problem $F2\left\right{C}_{max}$ in $O(nlogn)$ time), finding a job permutation minimizing the worstcase regret for the uncertain counterpart with a finite set of possible scenarios is NP hard.
In Reference [40], a binary NP hardness has been proven for finding a pair $({\pi}_{k},{\pi}_{k})\in S$ of identical job permutations that minimizes the worstcase absolute regret for the uncertain twomachine flowshop problem with the criterion ${C}_{max}$ even for two possible scenarios. Minimizing the worstcase regret implies a timeconsuming search over the set of $n!$ job permutations. In order to overcome this computational complexity in some cases, it is useful to consider a minimal dominant set of schedules $S\left(T\right)$ instead of the whole set S. To solve the flowshop problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set $\mathcal{J}$, one can restrict a search within the set $S\left(T\right)$.
4.2. Closed Results
Since each permutation ${\pi}^{\prime}$ uniquely determines a set of the earliest completion times ${C}_{i}\left({\pi}^{\prime}\right)$ of the jobs ${J}_{i}\in \mathcal{J}$ for the problem $F2\left\right{C}_{max}$, one can identify the permutation ${\pi}^{\prime}$, ($({\pi}^{\prime},{\pi}^{\prime})\in S$), with the semiactive schedule [1,25,26] determined by the permutation ${\pi}^{\prime}$. Thus, the set S becomes a set of $n!$ pairs $({\pi}^{\prime},{\pi}^{\prime})$ of identical permutations of $n={m}_{1,2}$ jobs from the set $\mathcal{J}={\mathcal{J}}_{1,2}$ since the order for processing jobs ${\mathcal{J}}_{1,2}$ on both machines may be the same in the optimal schedule [28]. Therefore, the above Definition 1 is supplemented by the following remark.
Remark 2.
For the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ considered in this section, it is assumed that a Jsolution $S\left(T\right)$ is a minimal dominant set of Johnson’s permutations of all jobs from the set ${\mathcal{J}}_{1,2}$, i.e., for each scenario $p\in T$, the set $S\left(T\right)$ contains at least one optimal pair $({\pi}_{k},{\pi}_{k})$ of identical Johnson’s permutations ${\pi}_{k}$ such that the inequality in Equation (1) holds.
In Reference [36], it is shown how to delete redundant pairs of (identical) permutations from the set S for constructing a Jsolution for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{1,2}$. The order of jobs ${J}_{v}\in {\mathcal{J}}_{1,2}$ and ${J}_{w}\in {\mathcal{J}}_{1,2}$ is fixed in the Jsolution if there exists at least one Johnson’s permutation of the form ${\pi}_{k}=({s}_{1},{J}_{v},{s}_{2},{J}_{w},{s}_{3})$ for any scenario $p\in T$. In Reference [29], the sufficient conditions are proven for fixing the order of two jobs from set $\mathcal{J}={\mathcal{J}}_{1,2}$. If one of the following conditions holds, then for each scenario $p\in T$, there exists a permutation ${\pi}_{k}=({s}_{1},{J}_{v},{s}_{2},{J}_{w},{s}_{3})$ that is a Johnson’s one for the problem $F2\leftp\right{C}_{max}$ associated with scenario p:
$${u}_{v1}\le {l}_{v2}\text{}\mathrm{and}\text{}{u}_{w2}\le {l}_{w1},$$
$${u}_{v1}\le {l}_{v2}\text{}\mathrm{and}\text{}{u}_{v1}\le {l}_{w1},$$
$${u}_{w2}\le {l}_{w1}\text{}\mathrm{and}\text{}{u}_{w2}\le {l}_{v2}.$$
If at least one condition in Inequalities (3)–(5) holds, then there exists a Jsolution $S\left(T\right)$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with fixed order ${J}_{v}\to {J}_{w}$ of jobs, i.e., job ${J}_{v}$ has to be located before job ${J}_{w}$ in any permutation ${\pi}_{i}$, $({\pi}_{i},{\pi}_{i})\in S\left(T\right)$. If both conditions in Inequalities (4) and (5) do not hold, then there is no Jsolution $S\left(T\right)$ with fixed order ${J}_{v}\to {J}_{w}$ in all permutations ${\pi}_{i}$, $({\pi}_{i},{\pi}_{i})\in S\left(T\right)$. If no analogous condition holds for the opposite order ${J}_{w}\to {J}_{v}$, then at least one permutation with job ${J}_{v}$ located before job ${J}_{w}$ or that with job ${J}_{w}$ located before job ${J}_{v}$ have to be included in any Jsolution $S\left(T\right)$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$. Theorem 1 is proven in Reference [41].
Theorem 1.
There exists a Jsolution $S\left(T\right)$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with fixed order ${J}_{v}\to {J}_{w}$ of the jobs ${J}_{v}$ and ${J}_{w}$ in all permutations ${\pi}_{k}$, $({\pi}_{k},{\pi}_{k})\in S\left(T\right)$ if and only if at least one condition of Inequalities (4) or (5) holds.
In Reference [41], the necessary and sufficient conditions have been proven for the case when a singleelement Jsolution $S\left(T\right)=\left\{({\pi}_{k},{\pi}_{k})\right\}$ exists for the problem $F2{l}_{jm}\le {p}_{jm}\le {u}_{jm}{C}_{max}$. The partition $\mathcal{J}={\mathcal{J}}^{0}\cup {\mathcal{J}}^{1}\cup {\mathcal{J}}^{2}\cup {\mathcal{J}}^{*}$ of the set $\mathcal{J}={\mathcal{J}}_{1,2}$ is considered, where
${\mathcal{J}}^{0}=\{{J}_{i}\in \mathcal{J}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{u}_{i1}\le {l}_{i2},{u}_{i2}\le {l}_{i1}\},$
${\mathcal{J}}^{1}=\{{J}_{i}\in \mathcal{J}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{u}_{i1}\le {l}_{i2},{u}_{i2}>{l}_{i1}\}=\{{J}_{i}\in \mathcal{J}\backslash {\mathcal{J}}^{0}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{u}_{i1}\le {l}_{i2}\},$
${\mathcal{J}}^{2}=\{{J}_{i}\in \mathcal{J}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{u}_{i1}>{l}_{i2},{u}_{i2}\le {l}_{i1}\}=\{{J}_{i}\in \mathcal{J}\backslash {\mathcal{J}}^{0}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{u}_{i2}\le {l}_{i1}\},$
${\mathcal{J}}^{*}=\{{J}_{i}\in \mathcal{J}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{u}_{i1}>{l}_{i2},{u}_{i2}>{l}_{i1}\}.$
For each job ${J}_{k}\in {\mathcal{J}}^{0}$, inequalities ${u}_{k1}\le {l}_{k2}$ and ${u}_{k2}\le {l}_{k1}$ imply inequalities ${l}_{k1}={u}_{k1}={l}_{k2}={u}_{k2}$. Since both segments of the possible durations of the job ${J}_{k}$ on machines ${M}_{1}$ and ${M}_{2}$ become a point, the durations ${p}_{k1}$ and ${p}_{k2}$ are fixed and equal for both machines ${M}_{1}$ and ${M}_{2}$: ${p}_{k1}={p}_{k2}=:{p}_{k}$. In Reference [41], Theorems 2 and 3 have been proven.
Theorem 2.
There exists a singleelement Jsolution $S\left(T\right)\subset S$, $\leftS\right(T\left)\right$$=1,$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ if and only if
(a) for any pair of jobs ${J}_{i}$ and ${J}_{j}$ from the set ${\mathcal{J}}^{1}$ (from the set ${\mathcal{J}}^{2}$, respectively), either ${u}_{i1}\le {l}_{j1}$ or ${u}_{j1}\le {l}_{i1}$ (either ${u}_{i2}\le {l}_{j2}$ or ${u}_{j2}\le {l}_{i2}$),
(b) ${\mathcal{J}}^{*}\le 1$; for job ${J}_{{i}^{*}}\in {\mathcal{J}}^{*}$, the inequalities ${l}_{{i}^{*}1}\ge max\{{u}_{i1}\phantom{\rule{0.277778em}{0ex}}:{J}_{i}\in {\mathcal{J}}^{1}\},\phantom{\rule{0.277778em}{0ex}}$${l}_{{i}^{*}2}\ge max\{{u}_{j2}\phantom{\rule{0.277778em}{0ex}}:{J}_{j}\in {\mathcal{J}}^{2}\}$ hold; and $max\{{l}_{{i}^{*}1},\phantom{\rule{0.277778em}{0ex}}{l}_{{i}^{*}2}\}\ge {p}_{k}$ for each job ${J}_{k}\in {\mathcal{J}}^{0}$.
Theorem 2 characterizes the simplest case of the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ when one permutation ${\pi}_{k}$ of the jobs $\mathcal{J}={\mathcal{J}}_{1,2}$ dominates all other job permutations. The hardest case of this problem is characterized by the following theorem.
Theorem 3.
If $max\{{l}_{ik}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{J}_{i}\in \mathcal{J},{M}_{k}\in \mathcal{M}\}<min\{{u}_{ik}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{J}_{i}\in \mathcal{J},{M}_{k}\in \mathcal{M}\}$, then $S\left(T\right)=S$.
The Jsolution $S\left(T\right)$ may be represented in a compact form using the dominance digraph which may be constructed in $O\left({n}^{2}\right)$ time. Let $\mathcal{J}\times \mathcal{J}$ denote the Cartesian product of two sets $\mathcal{J}$. One can construct the following binary relation ${\mathcal{A}}_{\u2aaf}\subseteq \mathcal{J}\times \mathcal{J}$ over set $\mathcal{J}={\mathcal{J}}_{1,2}$.
Definition 2.
For the two jobs ${J}_{v}\in \mathcal{J}$ and ${J}_{w}\in \mathcal{J}$, the inclusion $({J}_{v},{J}_{w})\in {\mathcal{A}}_{\u2aaf}$ holds if and only if there exists a Jsolution $S\left(T\right)$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ such that job ${J}_{v}\in \mathcal{J}$ is located before job ${J}_{w}\in \mathcal{J}$, $v\ne w$, in all permutations ${\pi}_{k}$, where $({\pi}_{k},{\pi}_{k})\in S\left(T\right)$.
The binary relation $({J}_{v},{J}_{w})\in {\mathcal{A}}_{\u2aaf}$ is represented as follows: ${J}_{v}\u2aaf{J}_{w}$. Due to Theorem 1, if for the jobs ${J}_{v}\in \mathcal{J}$ and ${J}_{w}\in \mathcal{J}$ the relation ${J}_{v}\u2aaf{J}_{w}$, $v\ne w$, holds, then for the jobs ${J}_{v}$ and ${J}_{w}$, at least one of conditions in Inequalities (4) and (5) holds. To construct the binary relation ${\mathcal{A}}_{\u2aaf}$ of the jobs on the set $\mathcal{J}$, it is sufficient to check Inequalities (4) and (5) for each pair of jobs ${J}_{v}$ and ${J}_{w}$. The binary relation ${\mathcal{A}}_{\u2aaf}$ determines the digraph $(\mathcal{J},{\mathcal{A}}_{\u2aaf})$ with vertex set $\mathcal{J}$ and arc set ${\mathcal{A}}_{\u2aaf}$. It takes $O\left({n}^{2}\right)$ time to construct the digraph $(\mathcal{J},{\mathcal{A}}_{\u2aaf})$. In the general case, the binary relation ${\mathcal{A}}_{\u2aaf}$ may be not transitive. In Reference [42], it is proven that, if the binary relation ${\mathcal{A}}_{\u2aaf}$ is not transitive, then ${\mathcal{J}}^{0}\ne \varnothing $. We next consider the case with the equality ${\mathcal{J}}^{0}=\varnothing $, i.e., $\mathcal{J}={\mathcal{J}}^{*}\cup {\mathcal{J}}^{1}\cup {\mathcal{J}}^{2}$ (the case with ${\mathcal{J}}^{0}\ne \varnothing $ has been considered in Reference [41]). For a pair of jobs ${J}_{v}\in {\mathcal{J}}^{1}$ and ${J}_{w}\in {\mathcal{J}}^{1}$ (for a pair of jobs ${J}_{v}\in {\mathcal{J}}^{2}$ and ${J}_{w}\in {\mathcal{J}}^{2}$, respectively), it may happen that there exist both Jsolution $S\left(T\right)$ with job ${J}_{v}$ located before job ${J}_{w}$ in all permutations ${\pi}_{k}$, $({\pi}_{k},{\pi}_{k})\in S\left(T\right)$ and Jsolution ${S}^{\prime}\left(T\right)$ with job ${J}_{w}$ located before job ${J}_{v}$ in all permutations ${\pi}_{l}$, $({\pi}_{l},{\pi}_{l})\in {S}^{\prime}\left(T\right)$.
In Reference [42], the following claim has been proven.
Theorem 4.
The digraph $(\mathcal{J},{\mathcal{A}}_{\u2aaf})$ has no circuits if and only if the set $\mathcal{J}={\mathcal{J}}^{*}\cup {\mathcal{J}}^{1}\cup {\mathcal{J}}^{2}$ includes no pair of jobs ${J}_{i}\in {\mathcal{J}}^{k}$ and ${J}_{j}\in {\mathcal{J}}^{k}$ with $k\in \{1,2\}$ such that ${l}_{ik}={u}_{ik}={l}_{jk}={u}_{jk}.$
The binary relation ${\mathcal{A}}_{\prec}\subset {\mathcal{A}}_{\u2aaf}\subseteq \mathcal{J}\times \mathcal{J}$ is defined as follows.
Definition 3.
For the jobs ${J}_{v}\in \mathcal{J}$ and ${J}_{w}\in \mathcal{J}$, the inclusion $({J}_{v},{J}_{w})\in {\mathcal{A}}_{\prec}$ holds if and only if ${J}_{v}\u2aaf{J}_{w}$ and ${J}_{w}\overline{)\u2aaf}{J}_{v}$, or ${J}_{v}\u2aaf{J}_{w}$ and ${J}_{w}\u2aaf{J}_{v}$ with $v<w$.
The relation $({J}_{v},{J}_{w})\in {\mathcal{A}}_{\prec}$ is represented as follows: ${J}_{v}\prec {J}_{w}$. As it is shown in Reference [42], the relation ${J}_{v}\prec {J}_{w}$ implies that ${J}_{v}\u2aaf{J}_{w}$ and that at least one condition in Inequalities (4) or (5) must hold. The relation ${J}_{v}\u2aaf{J}_{w}$ implies exactly one of the relations ${J}_{v}\prec {J}_{w}$ or ${J}_{w}\prec {J}_{v}$.
Since it is assumed that set ${\mathcal{J}}^{0}$ is empty, the binary relation ${\mathcal{A}}_{\prec}$ is an antireflective, antisymmetric, and transitive relation, i.e., the binary relation ${\mathcal{A}}_{\prec}$ is a strict order. The strict order ${\mathcal{A}}_{\prec}$ determines the digraph $\mathcal{G}=(\mathcal{J},{\mathcal{A}}_{\prec})$ with arc set ${\mathcal{A}}_{\prec}$. The digraph $\mathcal{G}=(\mathcal{J},{\mathcal{A}}_{\prec})$ has neither a circuit nor a loop. Properties of the dominance digraph $\mathcal{G}$ were studied in Reference [42]. The permutation ${\pi}_{k}=({J}_{{k}_{1}},{J}_{{k}_{2}},\dots ,{J}_{{k}_{n}})$, $({\pi}_{k},{\pi}_{k})\in S,$ may be considered as a total strict order of all jobs of the set $\mathcal{J}$. The total strict order determined by permutation ${\pi}_{k}$ is a linear extension of the partial strict order ${\mathcal{A}}_{\prec}$ if each inclusion $({J}_{{k}_{v}},{J}_{{k}_{w}})\in {\mathcal{A}}_{\prec}$ implies inequality $v<w$. Let $\mathsf{\Pi}\left(\mathcal{G}\right)$ denote a set of permutations ${\pi}_{k}\in {S}_{1,2}$ defining all linear extensions of the partial strict order ${\mathcal{A}}_{\prec}$. The cases when $\mathsf{\Pi}\left(\mathcal{G}\right)={S}_{1,2}$ and $\mathsf{\Pi}\left(\mathcal{G}\right)=\left\{{\pi}_{k}\right\}$ are characterized in Theorems 2 and 3. In the latter case, the strict order ${\mathcal{A}}_{\prec}$ over set $\mathcal{J}$ can be represented as follows: ${J}_{{k}_{1}}\prec \dots \prec {J}_{{k}_{i}}\prec {J}_{{k}_{i+1}}\prec \dots \prec {J}_{{k}_{{n}_{1,2}}}.$ In Reference [42], the following claims have been proven.
Theorem 5.
Let $\mathcal{J}={\mathcal{J}}^{*}\cup {\mathcal{J}}^{1}\cup {\mathcal{J}}^{2}$. For any scenario $p\in T$, the set $\mathsf{\Pi}\left(\mathcal{G}\right)$ contains a Johnson’s permutation for the problem $F2\leftp\right{C}_{max}$.
Corollary 1.
If $\mathcal{J}={\mathcal{J}}^{*}\cup {\mathcal{J}}^{1}\cup {\mathcal{J}}^{2}$, then there exists a Jsolution $S\left(T\right)$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ such that ${\pi}^{\prime}\in \mathsf{\Pi}\left(\mathcal{G}\right)$ for all pairs of job permutations, $\left\{({\pi}^{\prime},{\pi}^{\prime})\right\}\in S\left(T\right)$.
In Reference [42], it was studied how to construct a minimal dominant set $S\left(T\right)=\left\{({\pi}^{\prime},{\pi}^{\prime})\right\}$, ${\pi}^{\prime}\in \mathsf{\Pi}\left(\mathcal{G}\right)$. Two types of redundant permutations were examined, and the following claim was proven.
Lemma 1.
Let $\mathcal{J}={\mathcal{J}}^{*}\cup {\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}$. If permutation ${\pi}_{t}\in \mathsf{\Pi}\left(\mathcal{G}\right)$ is redundant in the set $\mathsf{\Pi}\left(\mathcal{G}\right)$, then ${\pi}_{t}$ is a redundant permutation either of type 1 or type 2.
Testing whether set $\mathsf{\Pi}\left(\mathcal{G}\right)$ contains a redundant permutation of type 1 takes $O\left({n}^{2}\right)$ time, and testing whether permutation ${\pi}_{g}\in \mathsf{\Pi}\left(\mathcal{G}\right)$ is a redundant permutation of type 2 takes $O\left(n\right)$ time. In Reference [42], it is shown how to delete all redundant permutations from the set $\mathsf{\Pi}\left(\mathcal{G}\right)$. Let ${\mathsf{\Pi}}^{*}\left(\mathcal{G}\right)$ denote a set of permutations remaining in the set $\mathsf{\Pi}\left(\mathcal{G}\right)$ after deleting all redundant permutations of type 1 and type 2.
Theorem 6.
Assume the following condition:
$$max\{{l}_{i,3k},{l}_{j,3k}\}<{l}_{ik}={u}_{ik}={l}_{jk}={u}_{jk}<min\{{u}_{i,3k},{u}_{j,3k}\}.$$
If set $\mathcal{J}={\mathcal{J}}^{*}\cup {\mathcal{J}}^{1}\cup {\mathcal{J}}^{2}$ does not contain a pair of jobs ${J}_{i}\in {\mathcal{J}}^{k}$ and ${J}_{j}\in {\mathcal{J}}^{k}$, $k\in \{1,2\}$, such that the above condition holds, then $S\left(T\right)=<{\mathsf{\Pi}}^{*}\left(\mathcal{G}\right),{\mathsf{\Pi}}^{*}\left(\mathcal{G}\right)>$.
To test conditions of Theorem 6 takes $O\left(n\right)$ time. Due to Theorem 6 and Lemma 1, if there are no jobs such that condition (6) holds, then a Jsolution can be constructed via deleting redundant permutations from set $\mathsf{\Pi}\left(\mathcal{G}\right)$. Since the set ${\mathsf{\Pi}}^{*}\left(\mathcal{G}\right)$ is uniquely determined [42], we obtain Corollary 2.
Corollary 2
([42]). If set $\mathcal{J}={\mathcal{J}}^{*}\cup {\mathcal{J}}^{1}\cup {\mathcal{J}}^{2}$ does not contain a pair of jobs ${J}_{i}$ and ${J}_{j}$ such that condition (6) holds, then the binary relation ${\mathcal{A}}_{\prec}$ determines a unique Jsolution $S\left(T\right)=<{\mathsf{\Pi}}^{*}\left(\mathcal{G}\right),{\mathsf{\Pi}}^{*}\left(\mathcal{G}\right)>$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$.
The condition of Theorem 6 is sufficient for the uniqueness of a Jsolution ${\mathsf{\Pi}}^{*}\left(\mathcal{G}\right)=S\left(T\right)$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$. Due to Theorem 1, one can construct a digraph $\mathcal{G}=(\mathcal{J},{\mathcal{A}}_{\prec})$ in $O\left({n}^{2}\right)$ time. The digraph $\mathcal{G}=(\mathcal{J},{\mathcal{A}}_{\prec})$ determines a set $S\left(T\right)$ and may be considered a condensed form of a Jsolution for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$. The results presented in this section are used in Section 5 for constructing precedence digraphs for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$.
5. Properties of the Optimal Pairs of Job Permutations
We consider the uncertain jobshop problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ and prove sufficient conditions for determining a small dominant set of schedules for this problem. In what follows, we use Definition 4 of the dominant set $DS\left(T\right)\subseteq S$ along with Definition 1 of the Jsolution $S\left(T\right)\subseteq S$.
Definition 4.
A set of the pairs of job permutations $DS\left(T\right)\subseteq S$ is called a dominant set (of schedules) for the uncertain problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ if, for each scenario $p\in T$, the set $DS\left(T\right)$ contains at least one optimal pair of job permutations for the individual deterministic problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario p.
Every Jsolution (Definition 1) is a dominant set for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. Before processing jobs of the set $\mathcal{J}$ (before the realization of a schedule $s\in S$), a scheduler does not know exact values of the job durations. Nevertheless, it is needed to choose a pair of permutations of the jobs $\mathcal{J}$, i.e., it is needed to determine orders of jobs for processing them on machine ${M}_{1}$ and machine ${M}_{2}$. When all jobs will be processed on machines $\mathcal{M}$ (a schedule will be realized) and the job durations will take on exact values ${p}_{ij}^{*}$, ${l}_{ij}\le {p}_{ij}^{*}\le {u}_{ij}$, and so a factual scenario ${p}^{*}\in T$ will be determined. A schedule s chosen for the realization should be optimal for the obtained factual scenario ${p}^{*}$. In the stability approach, one can use two phases of scheduling for solving an uncertain scheduling problem: the offline phase and the online phase. The offline phase of scheduling is finished before starting the realization of a schedule. At this phase, a scheduler knows only given segments of the job durations and the aim is to find a pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})$ which is optimal for the most scenarios $p\in T$. After constructing a small dominant set of schedules $DS\left(T\right)$, a scheduler can choose a pair of job permutations in the set $DS\left(T\right)$, which dominates the most pairs of job permutations $({\pi}^{\prime},{\pi}^{\u2033})\in S$ for the given scenarios T. Note that making a decision at the offline phase may be timeconsuming since the realization of a schedule is not started.
The online phase of scheduling can begin once the earliest job in the schedule $({\pi}^{\prime},{\pi}^{\u2033})$ starts. At this phase, a scheduler can use additional online information on the job duration since, for each operation ${O}_{ij}$, the exact value ${p}_{ij}^{*}$ becomes known at the time of the completion of this operation. At the online phase, the selection of a next job for processing should be quick.
In Section 5.1, we investigate sufficient conditions for a pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})$ such that equality $DS\left(T\right)=\left\{({\pi}^{\prime},{\pi}^{\u2033})\right\}$ holds. In Section 5.2, the sufficient conditions allowing to construct a single optimal schedule dominating all other schedules in the set S are proven. If a singleelement dominant set $DS\left(T\right)$ does not exist, then one should construct two partial strict orders ${A}_{\prec}^{1,2}$ and ${A}_{\prec}^{2,1}$ on the set ${\mathcal{J}}_{1,2}$ and on the set ${\mathcal{J}}_{2,1}$ of jobs as it is described in Section 4.2. These orders may be constructed in the form of the two precedence digraphs allowing a scheduler to reduce a size of the dominant set $DS\left(T\right)$. Section 5.4 presents Algorithm 1 for constructing a semiactive schedule, which is optimal for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ for all possible scenarios T provided that such a schedule exists. Otherwise, Algorithm 1 constructs the precedence digraphs determining a minimal dominant set of schedules for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$.
5.1. Sufficient Conditions for an Optimal Pair of Job Permutations
In the proofs of several claims, we use a notion of the main machine, which is introduced within the proof of the following theorem.
Theorem 7.
Consider the following conditions in Inequalities (7) or (8):
$$\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}}{u}_{i1}\le \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{l}_{i2}and\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}}{l}_{i2}\ge \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{u}_{i1}$$
$$\sum _{{J}_{i}\in {\mathcal{J}}_{2,1}}{u}_{i2}\le \sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{1}}{l}_{i1}and\sum _{{J}_{i}\in {\mathcal{J}}_{2,1}}{l}_{i1}\ge \sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2}}{u}_{i2}$$
If one of the above conditions holds, then any pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})\in S$ is a singleelement dominant set $DS\left(T\right)=\left\{({\pi}^{\prime},{\pi}^{\u2033})\right\}$ for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$ of the given jobs.
Proof.
Let the condition in Inequalities (7) hold. Then, we consider an arbitrary pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})\in S$ with any fixed scenario $p\in T$ and show that this pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})$ is optimal for the individual deterministic problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario p, i.e., ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})={C}_{max}$.
Let ${c}_{1}\left({\pi}^{\prime}\right)$ (${c}_{2}\left({\pi}^{\u2033}\right)$) denote a completion time of all jobs ${\mathcal{J}}_{1}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$ (jobs ${\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$) on machine ${M}_{1}$ (machine ${M}_{2}$) in the schedule $({\pi}^{\prime},{\pi}^{\u2033})$, where ${\pi}^{\prime}=({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1})$ and ${\pi}^{\u2033}=({\pi}_{2,1},{\pi}_{2},{\pi}_{1,2})$. For the problem $J2p,{n}_{i}\le 2{C}_{max}$, the maximal completion time of the jobs in schedule $({\pi}^{\prime},{\pi}^{\u2033})$ may be calculated as follows: ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})=max\{{c}_{1}\left({\pi}^{\prime}\right),{c}_{2}\left({\pi}^{\u2033}\right)\}$.
Machine ${M}_{1}$ (machine ${M}_{2}$) is called a main machine for the schedule $({\pi}^{\prime},{\pi}^{\u2033})$ if equality ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})={c}_{1}\left({\pi}^{\prime}\right)$ holds (equality ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})={c}_{2}\left({\pi}^{\u2033}\right)$ holds, respectively).
For schedule $({\pi}^{\prime},{\pi}^{\u2033})\in S$, the following equality holds:
where ${I}_{1}$ and ${I}_{2}$ denote total idle times of machine ${M}_{1}$ and machine ${M}_{2}$ in the schedule $({\pi}^{\prime},{\pi}^{\u2033})$, respectively. We next show that, if the condition in Inequalities (7) holds, then machine ${M}_{2}$ is a main machine for schedule $({\pi}^{\prime},{\pi}^{\u2033})$ and machine ${M}_{2}$ has no idle time, i.e., machine ${M}_{2}$ is completely filled in the segment $[0,{c}_{2}\left({\pi}^{\u2033}\right)]$ for processing jobs from the set ${\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}$. At the initial time $t=0$, machine ${M}_{2}$ begins to process jobs from the set ${\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}$ without idle times until the time moment ${t}_{1}={\sum}_{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2}.$
$${c}_{1}\left({\pi}^{\prime}\right)=\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{p}_{i1}+{I}_{1};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{2}\left({\pi}^{\u2033}\right)=\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2}+{I}_{2},$$
From the first inequality in (7), we obtain the following relations:
$$\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}}{p}_{i1}\le \sum _{{J}_{i}\in {\mathcal{J}}_{1,2}}{u}_{i1}\le \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{l}_{i2}\le \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2}={t}_{1}.$$
Therefore, at the time moment ${t}_{1}$, machine ${M}_{2}$ begins to process jobs from the set ${\mathcal{J}}_{1,2}$ without idle times and we obtain the following equality: ${c}_{2}\left({\pi}^{\u2033}\right)={\sum}_{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2},$ where ${I}_{2}=0$ and machine ${M}_{2}$ has no idle time. We next show that machine ${M}_{2}$ is a main machine for the schedule $({\pi}^{\prime},{\pi}^{\u2033})$. To this end, we consider the following two possible cases.
(a) Let machine ${M}_{1}$ have no idle time.
By summing Inequalities (7), we obtain the following inequality:
$$\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{u}_{i1}\le \sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{l}_{i2}.$$
Thus, the following relations hold:
$${c}_{1}\left({\pi}^{\prime}\right)=\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{p}_{i1}\le \sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{u}_{i1}\le \sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{l}_{i2}\le \sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2}={c}_{2}\left({\pi}^{\u2033}\right).$$
Hence, machine ${M}_{2}$ is a main machine for the schedule $({\pi}^{\prime},{\pi}^{\u2033})$.
(b) Let machine ${M}_{1}$ have an idle time.
An idle time of machine ${M}_{1}$ is only possible if some job ${J}_{j}$ from set ${\mathcal{J}}_{2,1}$ is processed on machine ${M}_{2}$ at the time moment ${t}_{2}$ when this job ${J}_{j}$ could be processed on machine ${M}_{1}$.
Obviously, after the time moment ${\sum}_{{J}_{i}\in {\mathcal{J}}_{2,1}}{p}_{i2}$ when machine ${M}_{2}$ completes all jobs from set ${\mathcal{J}}_{2,1}$, machine ${M}_{1}$ can process some jobs from set ${\mathcal{J}}_{2,1}$ without an idle time. Therefore, the inequality ${t}_{2}+{I}_{1}\le {\sum}_{{J}_{i}\in {\mathcal{J}}_{2,1}}{p}_{i2}$ holds and we obtain the following relations:
$${c}_{1}\left({\pi}^{\prime}\right)\le {t}_{2}+{I}_{1}+\sum _{{J}_{i}\in {\mathcal{J}}_{2,1}}{p}_{i1}\le \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}}{p}_{i2}+\sum _{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{p}_{i1}\le \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}}{p}_{i2}+\sum _{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{u}_{i1}$$
$$\le \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}}{p}_{i2}+\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}}{l}_{i2}\le \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}}{p}_{i2}+\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}}{p}_{i2}\le \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}}{p}_{i2}={c}_{2}\left({\pi}^{\u2033}\right).$$
We conclude that, in case (b), machine ${M}_{2}$ is a main machine for the schedule $({\pi}^{\prime},{\pi}^{\u2033})$. Thus, if the condition in Inequalities (7) holds, then machine ${M}_{2}$ is a main machine for the schedule $({\pi}^{\prime},{\pi}^{\u2033})$ and machine ${M}_{2}$ has no idle time, i.e., equality ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})={c}_{2}\left({\pi}^{\u2033}\right)$ holds and machine ${M}_{2}$ is completely filled in the segment $[0,{c}_{2}\left({\pi}^{\u2033}\right)]$ with processing jobs from the set ${\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}$.
Thus, the pair of permutations $({\pi}^{\prime},{\pi}^{\u2033})$ is optimal for scenario $p\in T$. Since scenario p was chosen arbitrarily in the set T, we conclude that the pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})$ is a singleton $DS\left(T\right)=\left\{({\pi}^{\prime},{\pi}^{\u2033})\right\}$ for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$ of the given jobs. As a pair of permutations $({\pi}^{\prime},{\pi}^{\u2033})$ is an arbitrary pair of job permutations in the set S, any pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})\in S$ is a singleton $DS\left(T\right)=\left\{({\pi}^{\prime},{\pi}^{\u2033})\right\}$ for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$.
The case when the condition in Inequalities (8) holds may be analyzed similarly via replacing machine ${M}_{1}$ by machine ${M}_{2}$ and vice versa. □
If conditions of Theorem 7 hold, then in the optimal pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})$ existing for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$, the orders of jobs from sets ${\mathcal{J}}_{1,2}\subseteq \mathcal{J}$ and ${\mathcal{J}}_{2,1}\subseteq \mathcal{J}$ may be chosen arbitrarily. Theorem 7 implies the following two corollaries.
Corollary 3.
If the following inequality holds:
$$\sum _{{J}_{j}\in {\mathcal{J}}_{1,2}}{u}_{i1}\le \sum _{{J}_{j}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{l}_{i2},$$
then set $<\left\{{\pi}_{1,2}\right\},{S}_{2,1}>\subseteq S$, where ${\pi}_{1,2}$ is an arbitrary permutation in set ${S}_{1,2}$, is a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$ of the given jobs.
Proof.
We consider an arbitrary vector $p\in T$ of the job durations and an arbitrary permutation ${\pi}_{1,2}$ in the set ${S}_{1,2}$. The set ${S}_{2,1}$ contains at least one Johnson’s permutation ${\pi}_{2,1}^{*}$ for the deterministic problem $F2{p}_{2,1}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$ and scenario ${p}_{2,1}$ (the components of vector ${p}_{2,1}$ are equal to the corresponding components of vector p). We consider a pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})$$=(({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1}^{*}),({\pi}_{2,1}^{*},{\pi}_{2},{\pi}_{1,2}))\in \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left\{{\pi}_{1,2}\right\},{S}_{2,1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}>\subseteq S$ and show that it is an optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$ and scenario p. Without loss of generality, both permutations ${\pi}_{1}$ and ${\pi}_{2}$ are ordered in increasing order of the indexes of their jobs.
Similar to the proof of Theorem 7, one can show that, if the condition in Inequalities (9) holds, then machine ${M}_{2}$ processes jobs without idle times and equality ${c}_{2}\left({\pi}^{\u2033}\right)={\sum}_{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2}$ holds, where the value of ${c}_{2}\left({\pi}^{\u2033}\right)$ cannot be reduced. If machine ${M}_{1}$ has no idle time, we obtain equalities
$${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})=max\{{c}_{1}\left({\pi}^{\prime}\right),{c}_{2}\left({\pi}^{\u2033}\right)\}=max\{\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{p}_{i1},\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2}\}={C}_{max}.$$
On the other hand, an idle time of machine ${M}_{1}$ is only possible if some job ${J}_{j}$ from set ${\mathcal{J}}_{2,1}$ is processed on machine ${M}_{2}$ at the time moment ${t}_{2}$ when job ${J}_{j}$ could be processed on machine ${M}_{1}$. In such a case, the value of ${c}_{1}\left({\pi}^{\prime}\right)$ is equal to the makespan ${C}_{max}\left({\pi}_{2,1}^{*}\right)$ for the problem $F2{p}_{2,1}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$ and scenario ${p}_{2,1}$. As the permutation ${\pi}_{2,1}^{*}$ is a Johnson’s permutation, the value of ${C}_{max}\left({\pi}_{2,1}^{*}\right)$ cannot be reduced and we obtain the following equalities:
$${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})=max\{{c}_{1}\left({\pi}^{\prime}\right),{c}_{2}\left({\pi}^{\u2033}\right)\}=max\{{C}_{max}\left({\pi}_{2,1}^{*}\right),\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2}\}={C}_{max}.$$
Thus, the pair of job permutation $({\pi}^{\prime},{\pi}^{\u2033})=(({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1}^{*}),({\pi}_{2,1}^{*},{\pi}_{2},{\pi}_{1,2}))\in $$\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}\left\{{\pi}_{1,2}\right\},{S}_{2,1}>\subseteq S$ is optimal for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$. The optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$ belongs to the set $<\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left\{{\pi}_{1,2}\right\},{S}_{2,1}>$. As vector p is an arbitrary vector in the set T, the set $<\left\{{\pi}_{1,2}\right\},{S}_{2,1}>$ contains an optimal pair of job permutations for all scenarios from set T. Due to Definition 4, the set $<\left\{{\pi}_{1,2}\right\},{S}_{2,1}>\subseteq S$ is a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$. □
Corollary 4.
Consider the following inequality:
$$\sum _{{J}_{j}\in {\mathcal{J}}_{2,1}}{u}_{i2}\le \sum _{{J}_{j}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{1}}{l}_{i1}.$$
If the above inequality holds, then set $<{S}_{1,2},\left\{{\pi}_{2,1}\right\}>$, where ${\pi}_{2,1}$ is an arbitrary permutation in set ${S}_{2,1}$, is a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$ of the given jobs.
This claim may be proven similar to Corollary 3. If the conditions of Corollary 3 (Corollary 4) hold, then the order for processing jobs from set ${\mathcal{J}}_{1,2}\subseteq \mathcal{J}$ (set ${\mathcal{J}}_{2,1}\subseteq \mathcal{J}$, respectively) in the optimal schedule $({\pi}^{\prime},{\pi}^{\u2033})=\left(\right({\pi}_{1,2},{\pi}_{1},$${\pi}_{2,1}),({\pi}_{2,1},{\pi}_{2},{\pi}_{1,2}))$ for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ may be arbitrary. Since the orders of jobs from the sets ${\mathcal{J}}_{1}$ and ${\mathcal{J}}_{2}$ are fixed in the optimal schedule (Remark 1), we need to determine only orders for processing jobs from set ${\mathcal{J}}_{2,1}$ (set ${\mathcal{J}}_{1,2}$, respectively). To do this, we will consider two uncertain problems $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}\subseteq \mathcal{J}$ and with the machine route $({M}_{1},{M}_{2})$ and that with job set ${\mathcal{J}}_{2,1}\subseteq \mathcal{J}$ and with the opposite machine route $({M}_{2},{M}_{1})$.
Lemma 2.
If ${S}_{1,2}^{\prime}\subseteq {S}_{1,2}$ is a set of permutations from the dominant set for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$, then $<{S}_{1,2}^{\prime},{S}_{2,1}>\subseteq S$ is a dominant set for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$.
The proof of Lemma 2 and those for other statements in this section are given in Appendix A.
Lemma 3.
Let ${S}_{2,1}^{\prime}\subseteq {S}_{2,1}$ be a set of permutations from the dominant set for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$, ${S}_{2,1}^{\prime}\subseteq {S}_{2,1}$. Then, $<{S}_{1,2},{S}_{2,1}^{\prime}>$ is a dominant set for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$.
The proof of this claim is similar to that for Lemma 2 (see Appendix A).
Theorem 8.
Let ${S}_{1,2}^{\prime}\subseteq {S}_{1,2}$ be a set of permutations from the dominant set for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$, and let ${S}_{2,1}^{\prime}\subseteq {S}_{2,1}$ be a set of permutations from the dominant set for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$. Then, $<{S}_{1,2}^{\prime},{S}_{2,1}^{\prime}>\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\subseteq S$ is a dominant set for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$.
Theorem 9.
Let a pair of identical permutations $({\pi}_{1,2},{\pi}_{1,2})$ determine a singleelement Jsolution for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$, and let a pair of identical permutations $({\pi}_{2,1},{\pi}_{2,1})$ determine a singleelement Jsolution for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$. Then, the pairs of permutations $\left\{({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1})and({\pi}_{1,2},{\pi}_{2},{\pi}_{2,1})\right\}$ are a singleelement dominant set $DS\left(T\right)$ for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$.
The following claim follows directly from Theorem 9.
Corollary 5.
If the conditions of Theorem 9 hold, then there exists a single pair of job permutations, which is an optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$ and any scenario $p\in T$.
Theorem 9 implies also the following corollary proven in Appendix A.
Corollary 6.
If the conditions of Theorem 9 hold, then there exists a single pair of job permutations which is a Jsolution for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$.
Note that the criterion for a singleelement Jsolution for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ is given in Theorem 2.
5.2. Precedence Digraphs Determining a Minimal Dominant Set of Schedules
In Section 4.2, it is assumed that ${\mathcal{J}}_{1,2}={\mathcal{J}}_{1,2}^{1}\cup {\mathcal{J}}_{1,2}^{2}\cup {\mathcal{J}}_{1,2}^{*}$ and ${\mathcal{J}}_{2,1}={\mathcal{J}}_{2,1}^{1}\cup {\mathcal{J}}_{2,1}^{2}\cup {\mathcal{J}}_{2,1}^{*}$, i.e., ${\mathcal{J}}_{1,2}^{0}={\mathcal{J}}_{2,1}^{0}=\varnothing $. Based on the results presented in Section 4.2, we can determine a binary relation ${A}_{\prec}^{1,2}$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and a binary relation ${A}_{\prec}^{2,1}$ for this problem with job set ${\mathcal{J}}_{2,1}$. For job set ${\mathcal{J}}_{1,2}$, the binary relation ${A}_{\prec}^{1,2}$ determines the digraph ${G}_{1,2}=({\mathcal{J}}_{1,2},{A}_{\prec}^{1,2})$ with the vertex set ${\mathcal{J}}_{1,2}$ and the arc set ${A}_{\prec}^{1,2}$. For job set ${\mathcal{J}}_{2,1}$, the binary relation ${A}_{\prec}^{2,1}$ determines the digraph ${G}_{2,1}=({\mathcal{J}}_{2,1},{A}_{\prec}^{2,1})$ with the vertex set ${\mathcal{J}}_{2,1}$ and the arc set ${A}_{\prec}^{2,1}$.
Let us consider the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and the corresponding digraph ${G}_{1,2}=({\mathcal{J}}_{1,2},{A}_{\prec}^{1,2})$ (the same results for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$ can be derived in a similar way).
Definition 5.
Two jobs, ${J}_{x}\in {\mathcal{J}}_{1,2}$ and ${J}_{y}\in {\mathcal{J}}_{1,2}$, $x\ne y$, are called conflict jobs if they are not in the relation ${A}_{\prec}^{1,2}$, i.e., $({J}_{x},{J}_{y})\notin {A}_{\prec}^{1,2}$ and $({J}_{y},{J}_{x})\notin {A}_{\prec}^{1,2}$.
Due to Definitions 2 and 3, for the conflict jobs ${J}_{x}\in {\mathcal{J}}_{1,2}$ and ${J}_{y}\in {\mathcal{J}}_{1,2}$, $x\ne y$, Inequalities (4) and (5) do not hold either for the case $v=x$ with $w=y$ or for the case $v=y$ with $w=x$.
Definition 6.
The subset ${\mathcal{J}}_{x}\subseteq {\mathcal{J}}_{1,2}$ is called a conflict set of jobs if, for any job ${J}_{y}\in {\mathcal{J}}_{1,2}\backslash {\mathcal{J}}_{x}$, either relation $({J}_{x},{J}_{y})\in {A}_{\prec}^{1,2}$ or relation $({J}_{y},{J}_{x})\in {A}_{\prec}^{1,2}$ holds for each job ${J}_{x}\in {\mathcal{J}}_{x}$ (provided that any proper subset of the set ${\mathcal{J}}_{x}$ does not possess such a property).
From Definition 6, it follows that the conflict set ${\mathcal{J}}_{x}$ is a minimal set (with respect to the inclusion). Obviously, there may exist several conflict sets in the set ${\mathcal{J}}_{1,2}$. (A conflict set of the jobs ${\mathcal{J}}_{x}\subseteq {\mathcal{J}}_{2,1}$ can be determined similarly.) Let the strict order ${A}_{\prec}^{1,2}$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ be represented as follows:
where all jobs between braces are conflict ones and each of these jobs is in relation ${A}_{\prec}^{1,2}$ with any job located outside the brackets in Relation (10). In such a case, an optimal order for processing jobs from the set $\{{J}_{1},{J}_{2},\dots ,{J}_{k}\}$ is determined as follows: $({J}_{1},{J}_{2},\dots ,{J}_{k})$.
$${J}_{1}\prec {J}_{2}\prec \dots \prec {J}_{k}\prec \{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}\prec {J}_{k+r+1}\prec {J}_{k+r+2}\prec \dots \prec {J}_{{m}_{1,2}},$$
Due to Theorem 5, we obtain that set $\mathsf{\Pi}\left({G}_{1,2}\right)$ of the permutations generated by the digraph ${G}_{1,2}$ contains an optimal Johnson’s permutation for each vector ${p}_{1,2}$ of the durations of jobs from the set ${\mathcal{J}}_{1,2}$. Thus, due to Definition 1, the singleton $\left\{({\pi}_{1,2},{\pi}_{1,2})\right\}$, where ${\pi}_{1,2}\in \mathsf{\Pi}\left({G}_{1,2}\right)$, is a Jsolution for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$. Analogously, the singleton $\left\{({\pi}_{2,1},{\pi}_{2,1})\right\}$, where ${\pi}_{2,1}\in \mathsf{\Pi}\left({G}_{2,1}\right)$, is a Jsolution for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$. We can determine a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$ as follows: $<\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathsf{\Pi}\left({G}_{1,2}\right),\mathsf{\Pi}\left({G}_{2,1}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\subseteq S$. The following theorems allow us to reduce a dominant set for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. We use the following notation: ${L}_{2}={\sum}_{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{l}_{i2}.$
Theorem 10.
Let the strict order ${A}_{\prec}^{1,2}$ over set ${\mathcal{J}}_{1,2}={\mathcal{J}}_{1,2}^{*}\cup {\mathcal{J}}_{1,2}^{1}\cup {\mathcal{J}}_{1,2}^{2}$ be determined as follows: ${J}_{1}\prec \dots \prec {J}_{k}\prec \{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}\prec {J}_{k+r+1}\prec \dots \prec {J}_{{m}_{1,2}}$. Consider the following inequality:
$$\sum _{i=1}^{k+r}{u}_{i1}\phantom{\rule{4pt}{0ex}}\le {L}_{2}+\sum _{i=1}^{k}{l}_{i2},$$
If the above inequality holds, then set ${S}^{\prime}=\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}<\left\{\pi \right\},\mathsf{\Pi}\left({G}_{2,1}\right)>\subset S$ with $\pi \in \mathsf{\Pi}\left({G}_{1,2}\right)$ is a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$.
Proof.
We consider an arbitrary vector $p\in T$ of the job durations and an arbitrary permutation $\pi $ from the set $\mathsf{\Pi}\left({G}_{1,2}\right)$. The set $\mathsf{\Pi}\left({G}_{2,1}\right)$ contains at least one optimal Johnson’s permutation ${\pi}_{2,1}^{*}$ for the problem $F2{p}_{2,1}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$ and vector ${p}_{2,1}$ of the job durations (components of this vector are equal to the corresponding components of the vector p).
We consider a pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})=((\pi ,{\pi}_{1},{\pi}_{2,1}^{*}),$$({\pi}_{2,1}^{*},{\pi}_{2},\pi ))\in \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{S}^{\prime}$ and show that it is an optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with set $\mathcal{J}$ of the jobs and scenario p. To this end, we show that the value of ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})=max\{{c}_{1}\left({\pi}^{\prime}\right),{c}_{2}\left({\pi}^{\u2033}\right)\}$ cannot be reduced. Indeed, an idle time for machine ${M}_{1}$ is only possible if some job ${J}_{j}$ from the set ${\mathcal{J}}_{2,1}$ is processed on machine ${M}_{2}$ at the same time when job ${J}_{j}$ could be processed on machine ${M}_{1}$. In such a case, ${c}_{1}\left({\pi}^{\prime}\right)$ is equal to the makespan ${C}_{max}\left({\pi}_{2,1}^{*}\right)$ for the problem $F2{p}_{2,1}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$ and vector ${p}_{2,1}$ of the job durations. As permutation ${\pi}_{2,1}^{*}$ is a Johnson’s permutation, the value of
cannot be reduced. In the beginning of the permutation $\pi $, the jobs of set $\{{J}_{1},{J}_{2},\dots ,{J}_{k}\}$ are arranged in the Johnson’s order. Thus, if machine ${M}_{2}$ has an idle time while processing these jobs, this idle time cannot be reduced. From Inequality (11), it follows that machine ${M}_{2}$ has no idle time while processing jobs from the conflict set.
$${c}_{1}\left({\pi}^{\prime}\right)=max\{\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{p}_{i1},{C}_{max}\left({\pi}_{2,1}^{*}\right)\}$$
In the end of the permutation $\pi $, jobs of set $\{{J}_{k+r+1},\dots ,{J}_{{m}_{1,2}}\}$ are arranged in Johnson’s order. Therefore, if machine ${M}_{2}$ has an idle time while processing these jobs, this idle time cannot be reduced. Thus, the value of ${c}_{2}\left({\pi}^{\u2033}\right)$ cannot be reduced by changing the order of jobs in the conflict set.
We obtain the qualities ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})=max\{{c}_{1}\left({\pi}^{\prime}\right),{c}_{2}\left({\pi}^{\u2033}\right)\}={C}_{max}.$ The pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})=((\pi ,{\pi}_{1},{\pi}_{2,1}^{*}),({\pi}_{2,1}^{*},{\pi}_{2},\pi ))$ is optimal for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$. Thus, set ${S}^{\prime}=<\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left\{\pi \right\},\mathsf{\Pi}\left({G}_{2,1}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}>$ contains an optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$. As vector p is an arbitrary vector in set T, set ${S}^{\prime}$ contains an optimal pair of job permutations for each vector from set T. Due to Definition 4, set ${S}^{\prime}$ is a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$. □
Theorem 11.
Let the partial strict order ${A}_{\prec}^{1,2}$ over set ${\mathcal{J}}_{1,2}={\mathcal{J}}_{1,2}^{*}\cup {\mathcal{J}}_{1,2}^{1}\cup {\mathcal{J}}_{1,2}^{2}$ be determined as follows: ${J}_{1}\prec \dots \prec {J}_{k}\prec \{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}\prec {J}_{k+r+1}\prec \dots \prec {J}_{{m}_{1,2}}$. Consider the following inequality:
If the above inequality holds for all $s\in \{1,2,\dots ,r\},$ then the set ${S}^{\prime}=<\left\{\pi \right\},{S}_{2,1}>$, where $\pi =({J}_{1},\dots ,{J}_{k1},{J}_{k},{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r},$${J}_{k+r+1},\dots ,{J}_{{m}_{1,2}})\in \mathsf{\Pi}\left({G}_{1,2}\right)$, is a dominant set for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$.
$${u}_{k+s,1}\le {L}_{2}+\sum _{i=1}^{k+s1}({l}_{i2}{u}_{i1})$$
Proof.
We consider an arbitrary scenario $p\in T$ and a pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})=((\pi ,{\pi}_{1},{\pi}_{2,1}^{*}),$ $({\pi}_{2,1}^{*},{\pi}_{2},\pi ))\in {S}^{\prime}$, where ${\pi}_{2,1}^{*}\in {S}_{2,1}$ is a Johnson’s permutation of the jobs from the set ${\mathcal{J}}_{2,1}$ with vector ${p}_{2,1}$ of the job durations (components of this vector are equal to the corresponding components of vector p). We next show that this pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})$ is optimal for the individual deterministic problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario p, i.e., ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})={C}_{max}$.
If conditions of Theorem 11 hold, then machine ${M}_{2}$ processes jobs from the conflict set $\{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}$ without idle times. At the initial time $t=0$, machine ${M}_{1}$ begins to process jobs from the permutation $\pi $ without idle times. Let a time moment ${t}_{1}$ be as follows: ${t}_{1}={\sum}_{i=1}^{k+1}{p}_{i1}.$ At the time moment ${t}_{1}$, job ${J}_{k+1}$ is ready for processing on machine ${M}_{2}$.
On the other hand, at the time $t=0$, machine ${M}_{2}$ begins to process jobs from the set ${\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}$ without idle times and then jobs from the permutation $({J}_{1},{J}_{2},\dots ,{J}_{k+1})$. Let ${t}_{2}$ denote the first time moment when machine ${M}_{2}$ is ready for processing job ${J}_{k+1}$. Obviously, the following inequality holds: ${t}_{2}\ge {L}_{2}+{\sum}_{i=1}^{k+1}{p}_{i2}.$ From the condition in Inequality (12) with $s=1$, we obtain inequality ${\sum}_{i=1}^{k+1}{u}_{i1}\le {L}_{2}+{\sum}_{i=1}^{k}{l}_{i2}.$
Therefore, the following relations hold:
$${t}_{1}=\sum _{i=1}^{k+1}{p}_{i1}\le \sum _{i=1}^{k+1}{u}_{i1}\le {L}_{2}+\sum _{i=1}^{k}{l}_{i2}\le {L}_{2}+\sum _{i=1}^{k+1}{p}_{i2}={t}_{2}.$$
Machine ${M}_{2}$ processes job ${J}_{k+1}$ without an idle time between job ${J}_{k}$ and job ${J}_{k+1}$.
Analogously, using $s\in \{2,3,\dots ,r\}$, one can show that machine ${M}_{2}$ processes jobs from the conflict set $\{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}$ without idle times between jobs ${J}_{k+1}$ and ${J}_{k+2}$, between jobs ${J}_{k+2}$ and ${J}_{k+3}$, and so on to between jobs ${J}_{k+r1}$ and ${J}_{k+r}$. To end this proof, we have to show that the value of ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})=max\{{c}_{1}\left({\pi}^{\prime}\right),{c}_{2}\left({\pi}^{\u2033}\right)\}$ cannot be reduced.
An idle time for machine ${M}_{1}$ is only possible between some jobs from the set ${\mathcal{J}}_{2,1}$. However, the permutation ${\pi}_{2,1}^{*}$ is a Johnson’s permutation of the jobs from the set ${\mathcal{J}}_{2,1}$ for the vector ${p}_{2,1}$ of the job durations. Therefore, the value of ${c}_{1}\left({\pi}^{\prime}\right)$ cannot be reduced. On the other hand, in the permutation $\pi $, all jobs ${J}_{1},{J}_{2},\dots ,{J}_{k}$ and all jobs ${J}_{k+r+1},\dots ,{J}_{{m}_{1,2}}$ are arranged in Johnson’s orders. Therefore, if machine ${M}_{2}$ has an idle time while processing these jobs, this idle time cannot be reduced. It is clear that machine ${M}_{2}$ has no idle time while processing jobs from the conflict set. Thus, the value of ${c}_{2}\left({\pi}^{\u2033}\right)$ cannot be reduced by changing the order of jobs from the conflict set. We obtain the equalities ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})=max\{{c}_{1}\left({\pi}^{\prime}\right),{c}_{2}\left({\pi}^{\u2033}\right)\}={C}_{max}.$
It is shown that the pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})=((\pi ,{\pi}_{1},{\pi}_{2,1}^{*}),({\pi}_{2,1}^{*},{\pi}_{2},\pi ))\in \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{S}^{\prime}$ is optimal for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with vector $p\in T$ of job durations. As vector p is an arbitrary one in set T, the set ${S}^{\prime}$ contains an optimal pair of job permutations for each scenario from set T. Due to Definition 4, the set ${S}^{\prime}$ is a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$. □
The proof of the following theorem is given in Appendix A.
Theorem 12.
Let the partial strict order ${A}_{\prec}^{1,2}$ over set ${\mathcal{J}}_{1,2}={\mathcal{J}}_{1,2}^{*}\cup {\mathcal{J}}_{1,2}^{1}\cup {\mathcal{J}}_{1,2}^{2}$ have the form ${J}_{1}\prec \dots \prec {J}_{k}\prec \{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}\prec {J}_{k+r+1}\prec \dots \prec {J}_{{m}_{1,2}}$. If inequalities
hold for all indexes $s\in \{1,2,\dots ,r\}$, then the set ${S}^{\prime}=\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}<\left\{\pi \right\},{S}_{2,1}>$, where $\pi =({J}_{1},\dots ,{J}_{k1},{J}_{k},$${J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r},{J}_{k+r+1},\dots ,{J}_{{m}_{1,2}})\in \mathsf{\Pi}\left({G}_{1,2}\right)$, is a dominant set of pairs of permutations for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$.
$$\sum _{i=rs+2}^{r+1}{l}_{k+i,1}\ge \sum _{j=rs+1}^{r}{u}_{k+j,2}$$
Similarly, one can prove sufficient conditions for the existence of an optimal job permutation for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$, when the partial strict order ${A}_{\prec}^{2,1}$ on the set ${\mathcal{J}}_{2,1}={\mathcal{J}}_{2,1}^{*}\cup {\mathcal{J}}_{2,1}^{1}\cup {\mathcal{J}}_{2,1}^{2}$ has the following form: ${J}_{1}\prec \dots \prec {J}_{k}\prec \{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}\prec {J}_{k+r+1}\prec \dots \prec {J}_{{m}_{2,1}}$.
To apply Theorems 11 and 12, one can construct a job permutation that satisfies the strict order ${A}_{\prec}^{1,2}$. Then, one can check the conditions of Theorems 11 and 12 for the constructed permutation. If the set of jobs $\{{J}_{1},{J}_{2},\dots ,{J}_{k}\}$ is empty in the constructed permutation, one needs to check conditions of Theorem 12. If the set of jobs $\{{J}_{k+r+1},\dots ,{J}_{{m}_{1,2}}\}$ is empty, one needs to check the conditions of Theorem 11. It is needed to construct only one permutation to check Theorem 11 and only one permutation to check Theorem 12.
5.3. Two Illustrative Examples
Example 1.
We consider the uncertain jobshop scheduling problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with lower and upper bounds of the job durations given in Table 1.
These bounds determine the set T of possible scenarios. In Example 1, jobs ${J}_{1}$, ${J}_{2}$, and ${J}_{3}$ have the machine route $({M}_{1},{M}_{2})$; jobs ${J}_{6}$, ${J}_{7}$, and ${J}_{8}$ have the machine route $({M}_{2},{M}_{1})$; and job ${J}_{4}$ (job ${J}_{5}$, respectively) has to be processed only on machine ${M}_{1}$ (on machine ${M}_{2}$, respectively). Thus, ${\mathcal{J}}_{1,2}=\{{J}_{1},{J}_{2},{J}_{3}\}$, ${\mathcal{J}}_{2,1}=\{{J}_{6},{J}_{7},{J}_{8}\}$, ${\mathcal{J}}_{1}=\left\{{J}_{4}\right\}$, ${\mathcal{J}}_{2}=\left\{{J}_{5}\right\}$.
We check the conditions of Theorem 7 for a single pair of job permutations, which is optimal for all scenarios T. For the given jobs, the condition in Inequalities (7) of Theorem 7 holds due to the following relations:
${\sum}_{{J}_{i}\in {\mathcal{J}}_{1,2}}{u}_{i1}={u}_{1,1}+{u}_{2,1}+{u}_{3,1}=7+9+9=25\le {\sum}_{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{l}_{i2}={l}_{6,2}+{l}_{7,2}+{l}_{8,2}+{l}_{5,2}=3+3+3+16=25;$
${\sum}_{{J}_{i}\in {\mathcal{J}}_{1,2}}{l}_{i2}={l}_{1,2}+{l}_{2,2}+{l}_{3,2}=6+5+5=16\ge {\sum}_{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{u}_{i1}={u}_{6,1}+{u}_{7,1}+{u}_{8,1}+{u}_{4,1}=3+3+3+3=12.$
Due to Theorem 7, the order of jobs from the set ${\mathcal{J}}_{1,2}=\{{J}_{1},{J}_{2},{J}_{3}\}$ and the order of jobs from the set ${\mathcal{J}}_{2,1}=\{{J}_{6},{J}_{7},{J}_{8}\}$ may be arbitrary in the optimal pair of job permutations for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ under consideration. Thus, any pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})\in S$ is a singleelement dominant set $DS\left(T\right)=\left\{({\pi}^{\prime},{\pi}^{\u2033})\right\}$ for Example 1.
Example 2.
Let us now consider the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with numerical input data given in Table 1 with the following two exceptions: ${l}_{5,2}=2$ and ${u}_{5,2}=3$.
We check the condition in Inequalities (7) of Theorem 7 and obtain
$$\begin{array}{c}\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}}{u}_{i1}={u}_{1,1}+{u}_{2,1}+{u}_{3,1}=7+9+9=25\nleqq \sum _{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{l}_{i2}={l}_{6,2}+{l}_{7,2}+{l}_{8,2}+{l}_{5,2}=3+3+3+2=11.\end{array}$$
Thus, the condition of Inequalities (7) does not hold for Example 2. We check the condition of Inequalities (8) of Theorem 7 and obtain
$$\begin{array}{c}\sum _{{J}_{i}\in {\mathcal{J}}_{2,1}}{u}_{i2}={u}_{6,2}+{u}_{7,2}+{u}_{8,2}=4+4+4=12\le \sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{1}}{l}_{i1}={l}_{1,1}+{l}_{2,1}+{l}_{3,1}+{l}_{4,1}=6+8+7+2=23.\end{array}$$
However, we see that the condition of Equation (8) does not hold:
${\sum}_{{J}_{i}\in {\mathcal{J}}_{2,1}}{l}_{i1}={l}_{6,1}+{l}_{7,1}+{l}_{8,1}=1+1+1=3\ngeqq {\sum}_{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2}}{u}_{i2}={u}_{1,2}+{u}_{2,2}+{u}_{3,2}+{u}_{5,2}=7+6+6+3=22.$
From Equation (14), it follows that the condition of Inequalities (9) of Corollary 3 does not hold. On the other hand, due to Equation (15), conditions of Corollary 4 hold. Thus, the order for processing jobs from set ${\mathcal{J}}_{2,1}\subseteq \mathcal{J}$ in the optimal schedule $({\pi}^{\prime},{\pi}^{\u2033})=\left(\right({\pi}_{1,2},{\pi}_{1},$${\pi}_{2,1}),({\pi}_{2,1},{\pi}_{2},{\pi}_{1,2}))$ for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ may be arbitrary. One can fix permutation ${\pi}_{2,1}$ with the increasing order of the indexes of their jobs: ${\pi}_{2,1}=({J}_{6},{J}_{7},{J}_{8})$. Since the orders of jobs from the sets ${\mathcal{J}}_{1}$ and ${\mathcal{J}}_{2}$ are fixed in the optimal schedule (Remark 1), i.e., ${\pi}_{1}=\left({J}_{4}\right)$ and ${\pi}_{2}=\left({J}_{5}\right)$, we need to determine the order for processing jobs in set ${\mathcal{J}}_{1,2}$. To this end, we consider the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$. We see that conditions of Theorem 2 do not hold for the jobs in set ${\mathcal{J}}_{1,2}$ since ${J}_{1}\in {\mathcal{J}}_{1,2}^{*}$, ${J}_{2}\in {\mathcal{J}}_{1,2}^{2}$, and ${J}_{3}\in {\mathcal{J}}_{1,2}^{2}$; however the following inequalities hold: ${u}_{2,2}>{l}_{3,2}$ and ${u}_{3,2}>{l}_{2,2}$.
We next construct the binary relation ${A}_{\prec}^{1,2}$ over set ${\mathcal{J}}_{1,2}$ based on Definition 3 and Theorem 1. Due to checking Inequalities (4) and (5), we conclude that the inequality in Equation (5) holds for the pair of jobs ${J}_{1}$ and ${J}_{2}$. We obtain the relation ${J}_{1}\prec {J}_{2}$. Analogously, we obtain the relation ${J}_{1}\prec {J}_{3}$. For the pair of jobs ${J}_{2}$ and ${J}_{3}$, neither Inequality (4) nor Inequality (5) hold. Therefore, the partial strict order ${A}_{\prec}^{1,2}$ over set ${\mathcal{J}}_{1,2}$ has the following form: ${J}_{1}\prec \{{J}_{2},{J}_{3}\}$. The job set $\{{J}_{2},{J}_{3}\}$ is a conflict set of these jobs (Definition 6).
Let us check whether the sufficient conditions given in Section 5.2 hold.
We check the conditions of Theorem 10 for the jobs from set ${\mathcal{J}}_{1,2}$. For $k=1$ and $r=2$, we obtain the following equalities: ${L}_{2}={\sum}_{{J}_{i}\in {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{l}_{i2}={l}_{6,2}+{l}_{7,2}+{l}_{8,2}+{l}_{5,2}=3+3+3+2=11.$ The condition of Theorem 10 does not hold since the following relations hold:
$$\sum _{i=1}^{k+r}{u}_{i1}={u}_{1,1}+{u}_{2,1}+{u}_{3,1}=7+9+9=25\nleqq {L}_{2}+\sum _{i=1}^{k}{l}_{i2}={L}_{2}+{l}_{1,2}=11+6=17.$$
For checking the conditions of Theorem 11, we need to check both permutations of the jobs from set ${\mathcal{J}}_{1,2}$, which satisfy the partial strict order ${A}_{\prec}^{1,2}$: $\mathsf{\Pi}\left({\mathcal{G}}_{1,2}\right)=\{{\pi}_{1,2}^{1},{\pi}_{1,2}^{2}\}$, where ${\pi}_{1,2}^{1}=\{{J}_{1},{J}_{2},{J}_{3}\}$ and ${\pi}_{1,2}^{2}=\{{J}_{1},{J}_{3},{J}_{2}\}$.
We consider permutation ${\pi}_{1,2}^{1}$. As in the previous case, ${L}_{2}=11$, $k=1$, $r=2$, and we must consider two inequalities in the condition in Equaiton (12) with $s=1$ and $s=2$. For $s=1$, we obtain the following:
$${u}_{1+1,1}={u}_{2,1}=9\le {L}_{2}+\sum _{i=1}^{1+11}({l}_{i2}{u}_{i1})={L}_{2}+\sum _{i=1}^{1}({l}_{i2}{u}_{i1})=11+({l}_{1,2}{u}_{1,1})=11+(67)=10.$$
However, for $s=2$, we obtain
$${u}_{1+2,1}={u}_{3,1}=9\nleqq {L}_{2}+\sum _{i=1}^{1+21}({l}_{i2}{u}_{i1})={L}_{2}+\sum _{i=1}^{2}({l}_{i2}{u}_{i1})$$
$$=11+({l}_{1,2}{u}_{1,1})+({l}_{2,2}{u}_{2,1})=11+(67)+(59)=6.$$
Thus, the conditions of Theorem 11 do not hold for permutation ${\pi}_{1,2}^{1}$.
We consider permutation ${\pi}_{1,2}^{2}$, where ${J}_{k+1}={J}_{3}$ and ${J}_{k+2}={J}_{2}$. Again, we must test the two inequalities in Equation (12), where either $s=1$ or $s=2$. For $s=1$, we obtain
$${u}_{k+1,1}={u}_{3,1}=9\le {L}_{2}+\sum _{i=1}^{k+11}({l}_{i2}{u}_{i1})={L}_{2}+\sum _{i=1}^{1}({l}_{i2}{u}_{i1})=11+({l}_{1,2}{u}_{1,1})=11+(67)=10.$$
However, for $s=2$, we obtain
$${u}_{k+2,1}={u}_{2,1}=9\nleqq {L}_{2}+\sum _{i=1}^{k+21}({l}_{i2}{u}_{i1})={L}_{2}+\sum _{i=1}^{k+1}({l}_{i2}{u}_{i1})=11+({l}_{1,2}{u}_{1,1})+({l}_{3,2}{u}_{3,1})$$
$$=11+(67)+(59)=6.$$
Thus, the conditions of Theorem 11 do not hold for permutation ${\pi}_{1,2}^{2}$.
Note that we do not check the conditions of Theorem 12 since the conflict set of jobs $\{{J}_{2},{J}_{3}\}$ is located at the end of the partial strict order ${A}_{\prec}^{1,2}$. We conclude that none of the proven sufficient conditions are satisfied for a schedule optimality. Thus, there does not exist a pair of permutations of the jobs in set $\mathcal{J}={\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}$ which is optimal for any scenario $p\in T$. The Jsolution $S\left(T\right)$ for Example 2 consists of the following two pairs of job permutations: $\{({\pi}_{1}^{\prime},{\pi}_{1}^{\u2033}),({\pi}_{2}^{\prime},{\pi}_{2}^{\u2033})\}=S\left(T\right)$, where
$${\pi}_{1}^{\prime}=({\pi}_{1,2}^{1},{\pi}_{1},{\pi}_{2,1})=({J}_{1},{J}_{2},{J}_{3},{J}_{4},{J}_{6},{J}_{7},{J}_{8}),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\pi}_{1}^{\u2033}=({\pi}_{2,1},{\pi}_{2},{\pi}_{1,2}^{1})=({J}_{6},{J}_{7},{J}_{8},{J}_{5},{J}_{1},{J}_{2},{J}_{3}),$$
$${\pi}_{2}^{\prime}=({\pi}_{1,2}^{2},{\pi}_{1},{\pi}_{2,1})=({J}_{1},{J}_{3},{J}_{2},{J}_{4},{J}_{6},{J}_{7},{J}_{8}),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\pi}_{2}^{\u2033}=({\pi}_{2,1},{\pi}_{2},{\pi}_{1,2}^{2})=({J}_{6},{J}_{7},{J}_{8},{J}_{5},{J}_{1},{J}_{3},{J}_{2}).$$
We next show that none of these two pairs of job permutations is optimal for all scenarios $p\in T$ using the following two scenarios: ${p}^{\prime}=(7,6,9,5,9,6,2,0,0,2,1,3,1,3,1,3)\in T$ and ${p}^{\u2033}=(7,6,9,6,9,5,2,0,0,2,1,3,1,3,1,3)\in T.$ For scenario ${p}^{\prime}$, only pair of permutations $({\pi}_{2}^{\prime},{\pi}_{2}^{\u2033})$ is optimal with ${C}_{max}({\pi}_{2}^{\prime},{\pi}_{2}^{\u2033})=30$ since ${C}_{max}({\pi}_{1}^{\prime},{\pi}_{1}^{\u2033})=31>30$. On the other hand, for scenario ${p}^{\u2033}$, only the pair of permutations $({\pi}_{1}^{\prime},{\pi}_{1}^{\u2033})$ is optimal with ${C}_{max}({\pi}_{1}^{\prime},{\pi}_{1}^{\u2033})=30$ since ${C}_{max}({\pi}_{2}^{\prime},{\pi}_{2}^{\u2033})=31>30$.
Note that the whole set S of the semiactive schedules has the cardinality $\leftS\right={m}_{1,2}!\xb7{m}_{2,1}!=3!\xb73!=6\xb76=36$. Thus, for solving Example 2, one needs to consider only two pairs of job permutations $\{({\pi}_{1}^{\prime},{\pi}_{1}^{\u2033}),({\pi}_{2}^{\prime},{\pi}_{2}^{\u2033})\}=S\left(T\right)\subset S$ instead of 36 semiactive schedules.
5.4. An Algorithm for Checking Conditions for the Existence of a SingleElement Dominant Set
We describe Algorithm 1 for checking the existence of an optimal permutation for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ if the partial strict order ${A}_{\prec}^{1,2}$ on the set ${\mathcal{J}}_{1,2}$ has the following form: ${J}_{1}\prec \dots \prec {J}_{k}\prec \{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}\prec {J}_{k+r+1}\prec \dots \prec {J}_{{m}_{1,2}}$. Algorithm 1 considers a set of conflict jobs and checks whether the sufficient conditions given in Section 5.2 hold. For a conflict set of jobs, it is needed to construct two permutations and to check the condition in Inequality (12) for the first permutation and the condition in Inequality (13) for the second one. If at least one of these conditions holds, Algorithm 1 constructs a permutation which is optimal for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with any scenario $p\in T$.
Obviously, testing the conditions of Theorems 11 and 12 takes $O\left(r\right)$, where the conflict set contains r jobs. The construction of the permutation of r jobs takes $O(rlogr)$. Therefore, the total complexity of Algorithm 1 is $O(rlogr)$.
Remark 3.
If Algorithm 1 is completed at Step 7 (STOP 1), we suggest to consider a set of conflict jobs $\{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}$ and construct a Johnson’s permutation for the deterministic problem $F2{p}^{\prime}{C}_{max}$ with job set ${\mathcal{J}}^{\prime}=\{{J}_{k+1},{J}_{k+2},\dots {J}_{k+r}\}$, where vector ${p}^{\prime}=({p}_{k+1,1}^{\prime},{p}_{k+1,2}^{\prime},\dots {p}_{k+r,1}^{\prime},{p}_{k+r,2}^{\prime})$ of the durations of conflict jobs $\{{J}_{k+1},{J}_{k+2},\dots {J}_{k+r}\}$ is calculated for each operation ${O}_{ij}$ of the conflict job ${J}_{i}\in \{{J}_{k+1},{J}_{k+2},\dots {J}_{k+r}\}$ on the corresponding machine ${M}_{j}\in \mathcal{M}$ as folows:
$${p}_{ij}^{\prime}=({u}_{ij}+{l}_{ij})/2$$
Theorem 11 and Theorem 12 imply the following claim.
Corollary 7.
Algorithm 1 constructs a permutation ${\pi}^{*}$ either satisfying conditions of Theorem 11 or Theorem 12 (such permutation ${\pi}^{*}$ is optimal for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and any scenario $p\in T$) or establishes that an optimal job permutation for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with any scenario $p\in T$ does not exist.
The set of jobs ${\mathcal{J}}_{2,1}$ for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{2,1}$ can be tested similarly to the set of jobs ${\mathcal{J}}_{1,2}$.
Algorithm 1: Checking conditions for the existence of a singleelement dominant set of schedules for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ 

6. Algorithms for Constructing a Small Dominant Set of Schedules for the Problem $\mathit{J}\mathbf{2}{\mathit{l}}_{\mathit{i}\mathit{j}}\le {\mathit{p}}_{\mathit{i}\mathit{j}}\le {\mathit{u}}_{\mathit{i}\mathit{j}},{\mathit{n}}_{\mathit{i}}\le \mathbf{2}{\mathit{C}}_{\mathit{m}\mathit{a}\mathit{x}}$
In this section, we describe Algorithm 2 for constructing a small dominant set $DS\left(T\right)$ of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. Algorithm 2 is developed for use at the offline phase of scheduling (before processing any job from the set $\mathcal{J}$). Based on the initial data, Algorithm 2 checks the conditions of Theorem 7 for a single optimal pair of job permutations for the uncertain problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. If the sufficient conditions of Theorem 7 do not hold, Algorithm 2 proceeds to consider the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$. For each of these problems, the conditions of Theorem 2 are checked. If these conditions do not hold, then strict orders of the jobs $\mathcal{J}$ based on Inequalities (4) and (5) are constructed. In this general case, Algorithm 2 constructs a partial strict order ${A}_{\prec}^{1,2}$ of the jobs from set ${\mathcal{J}}_{1,2}$ and a partial strict order ${A}_{\prec}^{2,1}$ of the jobs from set ${\mathcal{J}}_{2,1}$. Each of these partial orders may contain one or several conflict sets of jobs. For each such conflict set of jobs, Algorithm 2 checks whether the sufficient conditions given in Section 5.2 hold. Thus, if some sufficient conditions for a schedule optimality presented in Section 4 and Section 5 are satisfied, then there exists a pair of permutations of jobs from set $\mathcal{J}$ which is optimal for any scenario $p\in T$. Algorithm 2 constructs such a pair of job permutations $\left\{({\pi}^{\prime},{\pi}^{\u2033})\right\}=DS\left(T\right)$. Otherwise, the precedence digraphs determining a minimal dominant set $DS\left(T\right)$ of schedules is constructed by Algorithm 2. The more job pairs are involved in the binary relations ${A}_{\prec}^{1,2}$ and ${A}_{\prec}^{2,1}$, the more job permutations will be deleted from set S while constructing a Jsolution $S\left(T\right)\subseteq S$ for the problems $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job sets ${\mathcal{J}}_{1,2}$ and ${\mathcal{J}}_{2,1}$.
Algorithm 2: Construction of a small dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ 

Algorithm 2 may be applied for solving the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ exactly or approximately as follows. If at least one of the sufficient conditions proven in Section 5.1 hold, then Algorithm 2 constructs a pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})=(({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1}),({\pi}_{2,1},{\pi}_{2},{\pi}_{1,2}))$, which is optimal for any scenario $p\in T$ (Step 10).
It may happen that the constructed strict order on the set ${\mathcal{J}}_{1,2}$ or on the set ${\mathcal{J}}_{2,1}$ is not a linear strict order. If for at least one of the sets ${\mathcal{J}}_{1,2}$ or ${\mathcal{J}}_{2,1}$, the constructed partial strict order is not a linear one, a heuristic solution for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ is constructed similar to that for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ solved by Algorithm 1 (see Section 5.4). If Algorithm 2 is completed at Steps 1113 (STOP 1), we consider a set of conflict jobs $\{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}$ and construct a Jackson’s pair of job permutation for the deterministic problem $J2{p}^{\prime},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}=\{{J}_{k+1},{J}_{k+2},\dots {J}_{k+r}\}$, where the vector ${p}^{\prime}=({p}_{k+1,1}^{\prime},{p}_{k+1,2}^{\prime},\dots {p}_{k+r,1}^{\prime},{p}_{k+r,2}^{\prime})$ of the durations of conflict jobs $\{{J}_{k+1},{J}_{k+2},\dots {J}_{k+r}\}$ is calculated using the equality of Equation (16) for each operation ${O}_{ij}$ of the conflict job ${J}_{i}\in \{{J}_{k+1},{J}_{k+2},\dots {J}_{k+r}\}$ on the corresponding machine ${M}_{j}\in \mathcal{M}$ (Remark 3).
Algorithm 3: Construction of a strict order ${A}_{\prec}^{1,2}$ on the set ${\mathcal{J}}_{1,2}$ 

Algorithm 4 is obtained from the above Algorithm 3 by replacing the set ${\mathcal{J}}_{1,2}$ of jobs by the set ${\mathcal{J}}_{2,1}$ of jobs, machine ${M}_{1}$ by machine ${M}_{2}$, and vice versa. Obviously, testing the conditions of Theorems 11 and 12 takes $O\left(r\right)$, where conflict set contains r jobs. Construction of permutation of r jobs takes $O(rlogr)$. Therefore, the total complexity of Algorithm 1 is $O(rlogr)$.
Testing the conditions of Theorem 2 takes $O({m}_{1,2}log{m}_{1,2})$ time. A strict order ${A}_{\prec}^{1,2}$ on the set ${\mathcal{J}}_{1,2}$ is constructed by comparing no more than ${m}_{1,2}({m}_{1,2}1)$ pairs of jobs in the set ${\mathcal{J}}_{1,2}$. Thus, it takes $O\left({m}_{1,2}({m}_{1,2}1)\right)$ time. The complexity of Algorithm 1 is $O(rlogr)$ time provided that the conflict set contains r jobs, where $r\le {m}_{1,2}$. Since a strict order ${A}_{\prec}^{1,2}$ is constructed once in Algorithm 3, we conclude that a total complexity of Algorithm 3 (and Algorithm 4) is $O\left({n}^{2}\right)$ time.
In Algorithm 2, testing the condition of Theorem 7 takes $O(max\{{m}_{1,2},{m}_{2,1}\})$ time. Every Algorithm 3 or Algorithm 4 is fulfilled at most once. Therefore, the complexity of Algorithm 2 is $O\left({n}^{2}\right)$ time.
7. Computational Experiments
We describe the conducted computational experiments and discuss the results obtained for randomly generated instances of the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. In the computational experiments, each tested series consisted of 1000 randomly generated instances with the same numbers $n\in \{10,20,\dots ,100,200,\dots ,1000,2000,\dots ,10.000\}$ of jobs in the set $\mathcal{J}$ provided that a maximum relative length $\delta $ of the given segment of the possible durations of the operations ${O}_{ij}$ takes the following values: $\{5\%,10\%,$$15\%,20\%,30\%,40\%,$ and $50\%\}$. The lower bounds ${l}_{ij}$ and upper bounds ${u}_{ij}$ for possible values of the durations ${p}_{ij}$ of the operations ${O}_{ij}$, ${p}_{ij}\in [{l}_{ij},{u}_{ij}]$ using the value $\delta $ have been determined as follows. First, a value of the lower bound ${l}_{ij}$ is randomly chosen from the segment $[10,1000]$ using a uniform distribution. Then, the upper bound ${u}_{ij}$ is calculated using the following equality:
$${u}_{ij}={l}_{ij}\left(1+\frac{\delta}{100}\right)$$
For example, we assume that $\delta =5\%$. Then, for the lower bounds ${l}_{ij}=50$ and ${l}_{ij}=500$, the upper bounds ${u}_{ij}=52.5$ and ${u}_{ij}=525$ are calculated using Reference (17). If $\delta =50\%$, then based on the lower bounds ${l}_{ij}=50$ and ${l}_{ij}=500$ and on Reference (17), we obtain the upper bounds ${u}_{ij}=75$ and ${u}_{ij}=750$. Thus, rather wide ranges for the tested durations of the jobs $\mathcal{J}$ were considered.
In the experiments, the bounds ${l}_{ij}$ and ${u}_{ij}$ were decimal fractions with the maximum possible number of digits after the decimal point. For all tested instances of the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$, a strict inequality ${l}_{ij}<{u}_{ij}$ was guarantied for each job ${J}_{i}\in \mathcal{J}$ and each machine ${M}_{j}\in \mathcal{M}$.
We used Algorithms 1 – 4 described in Section 5.4 and Section 6 for solving the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. These algorithms were coded in C# and tested on a PC with Intel Core i77700 (TM) 4 Quad, 3.6 GHz, and 32.00 GB RAM. Since Algorithms 1 – 4 are polynomial in number n jobs in set $\mathcal{J}$, the calculations were carried out quickly. In the experiments, we tested 15 classes of randomly generated instances of the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with different ratios between numbers ${m}_{1}$, ${m}_{2}$, ${m}_{1,2}$, and ${m}_{2,1}$ of the jobs in subsets ${\mathcal{J}}_{1}$, ${\mathcal{J}}_{2}$, ${\mathcal{J}}_{1,2}$, and ${\mathcal{J}}_{2,1}$ of the set $\mathcal{J}$. The obtained computational results are presented in Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10, Table A11, Table A12, Table A13, Table A14 and Table A15 for 15 classes of the solved instances. Each tested class of the instances of the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ is characterized by the following ratio of the percentages of the number of jobs in the subsets ${\mathcal{J}}_{1}$, ${\mathcal{J}}_{2}$, ${\mathcal{J}}_{1,2}$, and ${\mathcal{J}}_{2,1}$ of the set $\mathcal{J}$:
$$\frac{{m}_{1}}{n}\xb7100\%:\frac{{m}_{2}}{n}\xb7100\%:\frac{{m}_{1,2}}{n}\xb7100\%:\frac{{m}_{2,1}}{n}\xb7100\%$$
Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8 and Table A9 present the computational results obtained for classes 1–9 of the tested instances characterized by the following ratios (Equation (18)):
$25\%:25\%:25\%:25\%$ (Table A1); $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}10\%:10\%:40\%:40\%$ (Table A2);
$10\%:40\%:10\%:40\%$ (Table A3); $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}10\%:30\%:10\%:50\%$ (Table A4);
$10\%:20\%:10\%:60\%$ (Table A5); $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}10\%:10\%:10\%:70\%$ (Table A6);
$5\%:20\%:5\%:70\%$ (Table A7); $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}5\%:15\%:5\%:75\%$ (Table A8);
$5\%:5\%:5\%:85\%$ (Table A9).
Note that all instances from class 1 of the instances with the ratio from Equation (18), $25\%:25\%:25\%:25\%$, were optimally solved by Algorithms 1–4 for all values of $\delta \in \{5\%,10\%,$$15\%,20\%,30\%,40\%,$ and $50\%\}$. We also tested classes 10–15 of the hard instances of the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ characterized by the following ratios (Equation (18)):
$3\%:2\%:5\%:90\%$ (Table A10); $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}2\%:3\%:5\%:90\%$ (Table A11);
$2\%:2\%:1\%:95\%$ (Table A12); $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1\%:2\%:2\%:95\%$ (Table A13);
$1\%:1\%:3\%:95\%$ (Table A14); $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1\%:1\%:1\%:97\%$ (Table A15).
All Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10, Table A11, Table A12, Table A13, Table A14 and Table A15 are organized as follows. Number n of given jobs $\mathcal{J}$ in the instances of the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ are presented in column 1. The values of $\delta $ (a maximum relative length of the given segment of the job durations) in percentages are presented in the first line of each table. For the fixed value of $\delta $, the obtained computational results are presented in four columns called $Opt$, $NC$, $SC$, and t. The column $Opt$ determines the percentage of instances from the series of 1000 randomly generated instances which were optimally solved using Algorithms 1–4. For each such instance, an optimal pair $({\pi}^{\prime},{\pi}^{\u2033})$ of the job permutations was constructed in spite of the uncertain durations of the given jobs $\mathcal{J}$. In other words, the equality ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})={C}_{max}({\pi}^{*},{\pi}^{**})$ holds, where $({\pi}^{*},{\pi}^{**})\in S$ is a pair of job permutations which is optimal for the deterministic problem $J2{p}^{*},{n}_{i}\le 2{C}_{max}$ associated with the factual scenario ${p}^{*}\in T$. The factual scenario ${p}^{*}\in T$ for the instance of the uncertain problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ is assumed to be unknown until completing the jobs $\mathcal{J}$.
Column $NC$ presents total number of conflict sets of the jobs in the partial strict orders ${A}_{\prec}^{1,2}$ on the job sets ${\mathcal{J}}_{1,2}$ and partial strict orders ${A}_{\prec}^{2,1}$ on the job sets ${\mathcal{J}}_{2,1}$ constructed by Algorithm 2. The value of $NC$ is equal to the total number of decision points, where Algorithm 2 has to select an order for processing jobs from the corresponding conflict set. To make a correct decision for such an order means to construct a permutation of all jobs from the conflict set, which is optimal for the factual scenario (which is unknown before scheduling). In particular, if all conflict sets have received correct decisions in Algorithm 2, then the constructed pair of job permutations will be optimal for the problem $J2{p}^{*},{n}_{i}\le 2{C}_{max}$, where ${p}^{*}\in T$ is the factual scenario.
Column $SC$ presents a percentage of the correct decisions made for determining optimal orders of the conflict jobs by Algorithm 2 with Algorithms 3 and 4. Column t presents a total CPU time (in seconds) for solving all 1000 instances of the corresponding series.
Average percentages of the instances which were optimally solved ($Opt$) are presented in Figure 1 for classes 1–9 of the tested instances and in Figure 2 for classes 10–15 of the hardtested instances.
Percentages of the average values of the correct decisions ($SC$) made for determining optimal orders of the conflict jobs for classes 1–9 are presented in Figure 3. Most instances from these nine classes were optimally solved (Table 2). If the values of $\delta $ were no greater than $20\%$, i.e., $\delta \in \{5\%,10\%,15\%,20\%\}$, then more than $80\%$ of the tested instances were optimally solved in spite of the data uncertainty. If the value $\delta $ is increased, the percentage of the optimally solved instances decreased. If the value $\delta $ was equal to $50\%$, then $45\%$ of the tested instances was optimally solved.
For all series of the hard instances presented in Table A10, Table A11, Table A12, Table A13, Table A14 and Table A15 (see the third line in Table 2), only a few instances were optimally solved. If $\delta =5\%$, then $70\%$ of the tested instances was optimally solved. If value $\delta $ belongs to the set $\{20\%,30\%,40\%,50\%\}$, then only $1\%$ of the tested instances was optimally solved. There were no hardtested instances optimally solved for the value of $\delta =50\%$.
Percentages of the average values of the correct decisions made for determining optimal orders of the conflict jobs by Algorithm 2, Algorithm 3 and Algorithm 4 for the hard classes 10–15 of the tested instances are presented in Figure 4. Note that there is a correlation between values of $Opt$ and $SC$ presented in Figure 1 and Figure 3 for classes 1–9 of the tested instances and those presented in Figure 2 and Figure 4 for classes 10–15 of the hardtested instances.
8. Concluding Remarks and Future Works
The uncertain flowshop scheduling problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ and its generalization the jobshop problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ attract the attention of researchers since these problems are applicable in many reallife scheduling systems. The optimal scheduling decisions for these problems allow the plant to reduce the costs of productions due to a better utilization of the available machines and other resources. In Section 5, we proved several properties of the optimal pairs $({\pi}^{\prime},{\pi}^{\u2033})$ of job permutations (Theorems 7–12). Using these properties, we derived Algorithms 1–4 for constructing optimal pairs $({\pi}^{\prime},{\pi}^{\u2033})$ of job permutations or a small dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. If it is impossible to construct a single pair $({\pi}^{\prime},{\pi}^{\u2033})$ of job permutations, which dominates all other pairs of job permutations for all possible scenarios T, then Algorithm 2 determines the partial strict order ${A}_{\prec}^{1,2}$ on the job set ${\mathcal{J}}_{1,2}$ (Algorithm 3) and the partial strict order ${A}_{\prec}^{2,1}$ on the job set ${\mathcal{J}}_{2,1}$ (Algorithm 4). The precedence digraphs $({\mathcal{J}}_{1,2},{A}_{\prec}^{1,2})$ and $({\mathcal{J}}_{2,1},{A}_{\prec}^{2,1})$ determine a minimal dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$.
From the conducted extensive computational experiments, it follows that pairs of job permutations constructed using Algorithm 2 are close to the optimal pairs of job permutations, which may be determined after completing all jobs $\mathcal{J}$ when factual operation durations become known. We tested 15 classes of the randomly generated instances $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. Most instances from tested classes 1–9 were optimally solved at the offline phase of scheduling. If the values of $\delta $ were no greater than $20\%$, i.e., $\delta \in \{5\%,10\%,15\%,20\%\}$, then more than $80\%$ of the tested instances was optimally solved in spite of the uncertainty of the input data. If $\delta =50\%$, then $45\%$ of the tested instances was optimally solved. However, less than $5\%$ of the instances with $\delta \ge 20\%$ from hard classes 10–15 were optimally solved at the offline phase of scheduling (Figure 2). There were no tested hard instances optimally solved for the value $\delta =50\%$.
In future research, the online phase of scheduling will be studied for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$. To this end, it will be useful to find sufficient conditions for existing a dominant pair of job permutations at the online phase of scheduling. The additional information on the factual value of the job duration becomes available once the processing of the job on the corresponding machine is completed. Using this additional information, a scheduler can determine a smaller dominant set DS of schedules, which is based on sufficient conditions for schedule dominance. The smaller DS enables a scheduler to quickly make an online scheduling decision whenever additional information on processing the job becomes available. To solve the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ at the online phase, a scheduler needs to use fast (better polynomial) algorithms. The investigation of the online phase of scheduling for the uncertain jobshop problem is under development.
We suggest to investigate properties of the optimality box and optimality region for a pair $({\pi}^{\prime},{\pi}^{\u2033})$ of the job permutations and to develop algorithms for constructing a pair $({\pi}^{\prime},{\pi}^{\u2033})$ of the job permutations that have the largest optimality box (or the largest optimality region). We also suggest to apply the stability approach for solving the uncertain flowshop and jobshop scheduling problems with $\left\mathcal{M}\right>2$ available machines.
Author Contributions
Methodology, Y.N.S.; software, V.D.H.; validation, Y.N.S., N.M.M. and V.D.H.; formal analysis, Y.N.S. and N.M.M.; investigation, Y.N.S. and N.M.M; writing—original draft preparation, Y.N.S. and N.M.M.; writing—review and editing, Y.N.S.; visualization, N.M.M.; supervision, Y.N.S. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Acknowledgments
We are thankful for useful remarks and suggestions provided by the editors and three anonymous reviewers on the earlier draft of our paper.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A. Proofs of the Statements
Appendix A.1. Proof of Lemma 2
We choose an arbitrary vector p in the set T, $p\in T$, and show that set $<{S}_{1,2}^{\prime},{S}_{2,1}>$ contains at least one optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$.
Let $({\pi}^{*},{\pi}^{**})=(({\pi}_{1,2}^{*},{\pi}_{1},{\pi}_{2,1}^{*}),({\pi}_{2,1}^{*},{\pi}_{2},{\pi}_{1,2}^{*}))$ be a Jackson’s pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$, i.e., ${C}_{max}({\pi}^{*},{\pi}^{**})={C}_{max}$. Without loss of generality, one can assume that jobs in both permutations ${\pi}_{1}$ and ${\pi}_{2}$ are ordered in increasing order of their indexes. It is clear that ${\pi}_{2,1}^{*}\in {S}_{2,1}$. If inclusion ${\pi}_{1,2}^{*}\in {S}_{1,2}^{\prime}$ holds as well, then $({\pi}^{*},{\pi}^{**})\in \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{S}_{1,2}^{\prime},{S}_{2,1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}>$ and set $<{S}_{1,2}^{\prime},{S}_{2,1}>$ contains an optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$. We now assume that ${\pi}_{1,2}^{*}\notin {S}_{1,2}^{\prime}$. The set ${S}_{1,2}^{\prime}$ contains at least one optimal permutation for the problem $F2{p}_{1,2}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and scenario ${p}_{1,2}$ (the components of vector ${p}_{1,2}$ are equal to the corresponding components of vector p). We denote this permutation as ${\pi}_{1,2}^{\prime}$. Remember that permutation ${\pi}_{1,2}^{\prime}$ may be not a Johnson’s permutation for the problem $F2{p}_{1,2}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and scenario ${p}_{1,2}$. We consider a pair of job permutations $({\pi}^{\prime},{\pi}^{**})=(({\pi}_{1,2}^{\prime},{\pi}_{1},{\pi}_{2,1}^{*}),$$({\pi}_{2,1}^{*},{\pi}_{2},{\pi}_{1,2}^{\prime}))\in \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{S}_{1,2}^{\prime},{S}_{2,1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}>$ and show that equality ${C}_{max}({\pi}^{\prime},{\pi}^{**})={C}_{max}$ holds. We consider the following two possible cases.
(j)${C}_{max}({\pi}^{\prime},{\pi}^{**})={c}_{1}\left({\pi}^{\prime}\right)$.
If equality ${c}_{1}\left({\pi}^{\prime}\right)={\sum}_{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{p}_{i1}$ holds, then ${c}_{1}\left({\pi}^{\prime}\right)\le {c}_{1}\left({\pi}^{*}\right)$.
We now assume that inequality ${c}_{1}\left({\pi}^{\prime}\right)>{\sum}_{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{p}_{i1}$ holds. Then, machine ${M}_{1}$ has an idle time. As it is mentioned in the proof of Theorem 7, an idle time for machine ${M}_{1}$ is only possible if some job ${J}_{j}$ from the set ${\mathcal{J}}_{2,1}$ is processed on machine ${M}_{2}$ at the time moment ${t}_{2}$ when job ${J}_{j}$ could be processed on machine ${M}_{1}$. Thus, the value of ${c}_{1}\left({\pi}^{\prime}\right)$ is equal to the makespan ${C}_{max}\left({\pi}_{2,1}^{*}\right)$ for the problem $F2{p}_{2,1}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$ and scenario ${p}_{2,1}$ (the components of vector ${p}_{2,1}$ are equal to the corresponding components of vector p). As jobs from the set ${\mathcal{J}}_{2,1}$ are processed as in the permutation ${\pi}_{2,1}^{*}$, which is a Johnson’s permutation, the value of ${c}_{1}\left({\pi}^{\prime}\right)$ cannot be reduced and so ${c}_{1}\left({\pi}^{\prime}\right)\le {c}_{1}\left({\pi}^{*}\right)$. We obtain the following relations: ${C}_{max}({\pi}^{\prime},{\pi}^{**})={c}_{1}\left({\pi}^{\prime}\right)\le {c}_{1}\left({\pi}^{*}\right)\le max\{{c}_{1}\left({\pi}^{*}\right),{c}_{2}\left({\pi}^{**}\right)\}={C}_{max}({\pi}^{*},{\pi}^{**})={C}_{max}.$ Thus, equality ${C}_{max}({\pi}^{\prime},{\pi}^{**})={C}_{max}$ holds.
(jj)${C}_{max}({\pi}^{\prime},{\pi}^{**})={c}_{2}\left({\pi}^{**}\right)$.
Similarly to case (j), we obtain the following equality:
where ${C}_{max}\left({\pi}_{1,2}^{\prime}\right)$ is the makespan for the problem $F2{p}_{1,2}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and vector ${p}_{1,2}$ of the job durations (it is assumed that ${\pi}_{1,2}^{\prime}$ is an optimal permutation for this problem). Thus, the value of ${c}_{2}\left({\pi}^{**}\right)$ cannot be reduced and equality ${C}_{max}({\pi}^{\prime},{\pi}^{**})={C}_{max}$ holds.
$${c}_{2}\left({\pi}^{**}\right)=max\{\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2},{C}_{max}\left({\pi}_{1,2}^{\prime}\right)\},$$
In both considered cases, the pair of job permutations $({\pi}^{\prime},{\pi}^{**})$ is an optimal schedule for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$. Therefore, an optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$ belongs to the set $<{S}_{1,2}^{\prime},{S}_{2,1}>$. As vector p is an arbitrary vector in set T, the set $<{S}_{1,2}^{\prime},{S}_{2,1}>$ contains an optimal pair of job permutations for each scenario from set T. Due to Definition 4, the set $<{S}_{1,2}^{\prime},{S}_{2,1}>$ is a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$.
Appendix A.2. Proof of Theorem 8
We consider an arbitrary vector $p\in T$ of the job durations from set T and relevant vectors ${p}_{1,2}$ and ${p}_{2,1}$ of the durations of jobs from set ${\mathcal{J}}_{1,2}$ and set ${\mathcal{J}}_{2,1}$, respectively. Set ${S}_{1,2}^{\prime}$ contains an optimal permutation ${\pi}_{1,2}^{\prime}$ for the problem $F2{p}_{1,2}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and with vector ${p}_{1,2}$ of the job durations. Set ${S}_{2,1}^{\prime}$ contains an optimal permutation ${\pi}_{2,1}^{\prime}$ for the problem $F2{p}_{2,1}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$ and with vector ${p}_{2,1}$ of the job durations. We next show that the pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})=(({\pi}_{1,2}^{\prime},{\pi}_{1},{\pi}_{2,1}^{\prime}),$$({\pi}_{2,1}^{\prime},{\pi}_{2},{\pi}_{1,2}^{\prime}))$ is an optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$ (the jobs in permutations ${\pi}_{1}$ and ${\pi}_{2}$ are ordered in increasing order of their indexes). From the proofs of Lemmas 2 and 3, we obtain the value of ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})=max\{{c}_{1}\left({\pi}^{\prime}\right),{c}_{2}\left({\pi}^{\u2033}\right)\}$
which cannot be reduced. Therefore, ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})={C}_{max}$. An optimal pair of job permutations for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with vector $p\in T$ of the job durations belongs to the set $<{S}_{1,2}^{\prime},{S}_{2,1}^{\prime}>$. As vector p is arbitrary in set T, the set $<{S}_{1,2}^{\prime},{S}_{2,1}^{\prime}>$ contains an optimal pair of job permutations for all vectors from set T. Due to Definition 4, the set $<{S}_{1,2}^{\prime},{S}_{2,1}^{\prime}>\subseteq S$ is a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$.
$$=max\{max\{\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{1}}{p}_{i1},{C}_{max}({\pi}_{2,1}^{\prime})\},max\{\sum _{{J}_{i}\in {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}\cup {\mathcal{J}}_{2}}{p}_{i2},{C}_{max}({\pi}_{1,2}^{\prime})\}\},$$
Appendix A.3. Proof of Theorem 9
We consider an arbitrary scenario $p\in T$. Due to Definition 1, the permutation ${\pi}_{1,2}$ is a Johnson’s permutation for the problem $F2{p}_{1,2}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and scenario ${p}_{1,2}$ (the components of this vector are equal to the corresponding components of vector p). Due to Definition 4, the singleton $\left\{({\pi}_{1,2},{\pi}_{1,2})\right\}$ is a minimal dominant set of schedules for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$.
Similarly, the singleton $\left\{({\pi}_{2,1},{\pi}_{2,1})\right\}$ is a minimal dominant set of schedules for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$. We consider permutations ${\pi}_{1}$ and ${\pi}_{2}$ of the jobs ${\mathcal{J}}_{1}$ and ${\mathcal{J}}_{2}$, respectively (due to Remark 1, the jobs in permutations ${\pi}_{1}$ and ${\pi}_{2}$ are ordered in increasing order of their indexes). Due to Theorem 8, the pair of permutations $(({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1}),({\pi}_{1,2},{\pi}_{2},{\pi}_{2,1}))$ is a singleelement dominant set (DS(T)) for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$.
Appendix A.4. Proof of Corollary 6
In the proof of Theorem 9, it is shown that the pair of job permutations $(({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1}),$$({\pi}_{1,2},{\pi}_{2},{\pi}_{2,1}))$ is a singleelement dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$. We next show that the pair of permutations $(({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1}),({\pi}_{1,2},{\pi}_{2},{\pi}_{2,1}))$ satisfies to Definition 1, i.e., this pair of permutations is a Jackson’s pair of job permutations for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$ (the minimality condition is obvious). Indeed, due to conditions of Theorem 9, the permutation ${\pi}_{1,2}$ is a Johnson’s permutation for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{1,2}$ and the permutation ${\pi}_{2,1}$ is a Johnson’s permutation for the problem $F2{l}_{ij}\le {p}_{ij}\le {u}_{ij}{C}_{max}$ with job set ${\mathcal{J}}_{2,1}$. Therefore, pair $(({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1}),({\pi}_{1,2},{\pi}_{2},{\pi}_{2,1}))$ is a Jackson’s pair of permutations for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$. Due to Definition 1, the pair of job permutations $(({\pi}_{1,2},{\pi}_{1},{\pi}_{2,1}),({\pi}_{1,2},{\pi}_{2},{\pi}_{2,1}))$ is a singleelement Jsolution for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}={\mathcal{J}}_{1}\cup {\mathcal{J}}_{2}\cup {\mathcal{J}}_{1,2}\cup {\mathcal{J}}_{2,1}$.
Appendix A.5. Proof of Theorem 12
We consider any fixed scenario $p\in T$ and a pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})=((\pi ,{\pi}_{1},{\pi}_{2,1}^{*}),({\pi}_{2,1}^{*},{\pi}_{2},\pi ))\in {S}^{\prime}$, where ${\pi}_{2,1}^{*}\in {S}_{2,1}$ is a Johnson’s permutation of the jobs from the set ${\mathcal{J}}_{2,1}$ with vector ${p}_{2,1}$ of the job durations (components of this vector are equal to the corresponding components of vector p). We next show that this pair of job permutations $({\pi}^{\prime},{\pi}^{\u2033})$ is optimal for the individual problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario p, i.e., ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})={C}_{max}$.
At time $t=0$, machine ${M}_{1}$ begins to process jobs from the permutation $\pi $ without idle times. We denote ${t}_{1}={\sum}_{i=1}^{k+r+1}{p}_{i1}.$ At time moment ${t}_{1}$, job ${J}_{k+r+1}$ is ready for processing on machine ${M}_{2}$. From the condition of Inequality (13) with $s=1$, it follows that, even if machine ${M}_{2}$ has an idle time before processing job ${J}_{k+r+1}$, machine ${M}_{2}$ is available for processing this job at time ${t}_{1}$. If in addition, the condition of Inequality (13) holds with $s\in \{2,3,\dots ,r\}$, then machine ${M}_{2}$ may also have idle times between processing jobs from the conflict set $\{{J}_{k+1},{J}_{k+2},\dots ,{J}_{k+r}\}.$ However, machine ${M}_{2}$ is available for processing job ${J}_{k+r+1}$ from the time moment ${t}_{1}={\sum}_{i=1}^{k+r+1}{p}_{i1}.$
In permutation $\pi $, jobs ${J}_{k+r+1},\dots ,{J}_{{m}_{1,2}}$ are arranged in Johnson’s order. Therefore, if machine ${M}_{2}$ has an idle time while processing these jobs, this idle time cannot be reduced.
Thus, the value of ${c}_{2}\left({\pi}^{\u2033}\right)$ cannot be reduced by changing the order of jobs from the conflict set. Note that an idle time for machine ${M}_{1}$ is only possible between some jobs from the set ${\mathcal{J}}_{2,1}$. Since the permutation ${\pi}_{2,1}^{*}$ is a Johnson’s permutation of the jobs from set ${\mathcal{J}}_{2,1}$ with scenario ${p}_{2,1}$, the value of ${c}_{1}\left({\pi}^{\prime}\right)$ cannot be reduced. Thus, we obtain ${C}_{max}({\pi}^{\prime},{\pi}^{\u2033})=max\{{c}_{1}\left({\pi}^{\prime}\right),{c}_{2}\left({\pi}^{\u2033}\right)\}={C}_{max}$ and the pair of permutations $({\pi}^{\prime},{\pi}^{\u2033})=((\pi ,{\pi}_{1},{\pi}_{2,1}^{*}),({\pi}_{2,1}^{*},{\pi}_{2},\pi ))\in \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{S}^{\prime}$ is optimal for the problem $J2p,{n}_{i}\le 2{C}_{max}$ with scenario $p\in T$. As the vector p is an arbitrary vector in the set T, set ${S}^{\prime}$ contains an optimal pair of job permutations for each vector from the set T. Due to Definition 4, set ${S}^{\prime}$ is a dominant set of schedules for the problem $J2{l}_{ij}\le {p}_{ij}\le {u}_{ij},{n}_{i}\le 2{C}_{max}$ with job set $\mathcal{J}$.
Appendix B. Tables with Computations Results
Table A1.
Computational results for randomly generated instances with ratio $25\%:25\%:25\%:25\%$ of the number of jobs in the subsets.
$\mathit{\delta}\%$  5%  10%  15%  20%  30%  40%  50%  

$\mathit{n}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$ 
20  100  6  100  0  100  19  100  0  100  35  100  0  100  70  100  0  100  139  100  0  100  250  100  0  100  339  100  0 
40  100  0    0  100  4  100  0  100  20  100  0  100  33  100  0  100  101  100  0  100  136  100  0  100  333  100  0 
50  100  7  100  0  100  3  100  0  100  16  100  0  100  8  100  0  100  50  100  0  100  114  100  0  100  224  100  0 
70  100  0    0  100  0    0  100  2  100  0  100  3  100  0  100  11  100  0  100  71  100  0  100  149  100  0 
80  100  0    0  100  0    0  100  0    0  100  3  100  0  100  0    0  100  35  100  0  100  122  100  0 
100  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  19  100  0  100  84  100  0 
200  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  5  100  0 
300  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
400  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
500  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
600  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
700  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
800  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
900  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
1000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
2000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
3000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
4000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
5000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
6000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
7000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
8000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
9000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
10,000  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0  100  0    0 
Aver.  100  0.54  100  0  100  1.08  100  0  100  3.04  100  0  100  4.88  100  0  100  12.54  100  0  100  26.04  100  0  100  52.33  100  0 
Table A2.
Computational results for randomly generated instances with ratio $10\%:10\%:40\%:40\%$ of the number of jobs in the subsets.
$\mathit{\delta}\%$  5%  10%  15%  20%  30%  40%  50%  

$\mathit{n}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$ 
10  100  71  100  0  100  153  100  0  99.6  241  98.34  0  99.4  320  98.13  0  98.1  481  96.05  0  95.8  618  93.20  0  91.9  713  88.64  0 
20  100  235  100  0  100  531  100  0  100  811  100  0  100  1032  100  0  100  1341  100  0  99.9  1450  99.93  0  99.5  1424  99.65  0 
30  100  460  100  0  100  887  100  0  100  1334  100  0  100  1643  100  0  100  1912  100  0  99.9  1893  99.95  0  100  1808  100  0 
40  100  636  100  0  100  1185  100  0  100  1659  100  0  100  2068  100  0  100  2352  100  0  100  2162  100  0  99.9  1953  99.95  0 
50  100  824  100  0  100  1496  100  0  100  2074  100  0  100  2411  100  0  100  2546  100  0  100  2211  100  0  100  2009  100  0 
60  100  893  100  0  100  1542  100  0  100  2222  100  0  100  2619  100  0  100  2758  100  0  100  2440  100  0  100  2109  100  0 
70  100  841  100  0  100  1589  100  0  100  2285  100  0  100  2775  100  0  100  2935  100  0  100  2477  100  0  100  2106  100  0 
80  100  981  100  0  100  1570  100  0  100  2342  100  0  100  2896  100  0  100  2995  100  0  100  2567  100  0  100  2249  100  0 
90  100  878  100  0  100  1660  100  0  100  2310  100  0  100  2905  100  0  100  3103  100  0  100  2598  100  0  100  2273  100  0 
100  100  826  100  0  100  1633  100  0  100  2368  100  0  100  3056  100  0  100  3114  100  0  100  2585  100  0  100  2321  100  0 
200  100  411  100  0  100  1145  100  0  100  1999  100  0  100  3065  100  0  100  3392  100  0  100  2709  100  0  100  2250  100  0 
300  100  181  100  0  100  721  100  0  100  1708  100  0  100  2888  100  0  100  3365  100  0  100  2579  100  0  100  2117  100  0 
400  100  51  100  0  100  302  100  0  100  981  100  0  100  2466  100  0  100  3263  100  0  100  2469  100  0  100  1966  100  0 
500  100  11  100  0  100  240  100  0  100  813  100  0  100  2307  100  0  100  3138  100  0  100  2362  100  0  100  1838  100  0 
600  100  0    0  100  88  100  0  100  499  100  0  100  2076  100  0  100  2951  100  0  100  2202  100  0  100  1692  100  0 
700  100  0    0  100  45  100  0  100  528  100  0  100  1894  100  0  100  2779  100  1  100  2015  100  1  100  1585  100  1 
800  100  0    0  100  36  100  0  100  294  100  0  100  1707  100  0  100  2656  100  0  100  1866  100  1  100  1485  100  1 
900  100  0    0  100  0    0  100  318  100  0  100  1442  100  0  100  2392  100  1  100  1677  100  1  100  1420  100  1 
1000  100  0    0  100  0    0  100  196  100  0  100  1275  100  0  100  2255  100  1  100  1630  100  1  100  1298  100  1 
2000  100  0    0  100  0    0  100  3  100  0  100  441  100  0  100  1452  100  3  100  1137  100  3  100  1044  100  2 
3000  100  0    0  100  0    0  100  0    0  100  160  100  0  100  1127  100  6  100  1025  100  5  100  1011  100  4 
4000  100  0    0  100  0    0  100  0    0  100  86  100  0  100  1032  100  9  100  1005  100  8  100  1000  100  7 
5000  100  0    0  100  0    0  100  0    0  100  34  100  0  100  1011  100  14  100  1000  100  12  100  1000  100  10 
6000  100  0    0  100  0    0  100  0    0  100  23  100  0  100  1002  100  21  100  1001  100  17  100  1001  100  14 
7000  100  0    0  100  0    0  100  0    0  100  8  100  0  100  1000  100  28  100  1000  100  23  100  1000  100  19 
8000  100  0    0  100  0    0  100  0    0  100  6  100  0  100  1001  100  37  100  1000  100  31  100  1000  100  25 
9000  100  0    0  100  0    0  100  0    0  100  3  100  0  100  1000  100  48  100  1000  100  39  100  1000  100  32 
10,000  100  0    0  100  0    0  100  0    0  100  4  100  1  100  1000  100  61  100  1000  100  49  100  1000  100  40 
Aver.  100  261  100  0  100  529  100  0  99.99  892  99.92  0  99.98  1486  99.93  0.04  99.93  2120  99.86  8.21  99.84  1774  99.75  6.82  99.69  1560  99.58  5.61 
Table A3.
Computational results for randomly generated instances with ratio $10\%:40\%:10\%:40\%$ of the number of jobs in the subsets.
$\mathit{\delta}\%$  5%  10%  15%  20%  30%  40%  50%  

$\mathit{n}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$ 
10  98.7  104  87.5  0  97.3  207  86.96  0  96.9  287  89.20  0  94.5  359  84.68  0  92.5  543  86.19  0  88.5  597  80.74  0  81.5  713  74.05  0 
20  99.7  466  99.36  0  99.6  830  99.52  0  99.3  1057  99.24  0  98.5  1244  98.79  0  95.3  1433  96.65  0  91  1467  93.73  0  84.6  1501  89.61  0 
30  100  1070  100  0  99.8  1597  99.87  0  99.7  1879  99.84  0  98.2  2007  99.10  0  96.7  2053  98.30  0  91.4  1945  95.58  0  82.1  1790  89.83  0 
40  100  1700  100  0  100  2355  100  0  99.8  2660  99.92  0  99.7  2628  99.89  0  97.6  2462  99.03  0  92.4  2135  96.35  0  85.5  1979  92.52  0 
50  100  2425  100  0  99.7  3174  99.91  0  99.9  3197  99.97  0  99.8  3048  99.93  0  97.8  2664  99.17  0  92.5  2283  96.54  0  82.2  2075  91.42  0 
60  100  3218  100  0  100  3808  100  0  100  3702  100  0  99.9  3394  99.97  0  98.5  2881  99.41  0  91.9  2442  96.60  0  80.5  2172  90.75  0 
70  100  3911  100  0  100  4385  100  0  99.9  4063  99.98  0  99.9  3648  99.97  0  98.7  2959  99.56  0  91.9  2517  96.62  0  78  2146  89.47  0 
80  100  4817  100  0  100  4902  100  0  99.8  4370  99.95  0  99.9  3829  99.97  0  98.5  3103  99.52  0  92.2  2627  96.99  0  77.3  2257  89.77  0 
90  100  5518  100  0  100  5398  100  0  100  4656  100  0  100  3910  100  0  97.7  3154  99.21  0  92.5  2675  97.12  0  79.5  2249  90.84  0 
100  100  6195  100  0  100  5697  100  0  99.9  4853  99.98  0  100  4047  100  0  98.2  3207  99.44  0  91.3  2706  96.78  0  75.6  2328  89.48  0 
200  100  10,620  100  0  100  7608  100  0  100  5717  100  0  100  4645  100  0  98.9  3320  99.64  0  90.8  2717  96.61  0  72.7  2281  87.90  0 
300  100  13,110  100  0  100  8259  100  0  100  6070  100  0  100  4782  100  0  99.5  3369  99.85  0  94.3  2605  97.81  0  74.5  2117  87.81  0 
400  100  14,309  100  1  100  8634  100  0  100  6113  100  0  100  4650  100  0  99.8  3247  99.94  0  94.3  2460  97.68  0  73.1  2002  86.56  0 
500  100  14,935  100  0  100  8658  100  0  100  6102  100  0  100  4630  100  0  99.9  3137  99.97  0  95.1  2297  97.87  0  78.5  1808  88.11  0 
600  100  15,780  100  0  100  8832  100  0  100  6021  100  0  100  4492  100  0  100  2911  100  0  97.2  2153  98.70  0  77.8  1705  86.98  0 
700  100  15,971  100  0  100  8753  100  0  100  5789  100  0  100  4379  100  0  100  2786  100  0  97.7  1996  98.85  0  82.4  1613  89.09  0 
800  100  16,439  100  0  100  8806  100  0  100  5793  100  0  100  4176  100  0  100  2533  100  0  98.8  1846  99.35  0  84  1487  89.24  0 
900  100  16,268  100  0  100  8574  100  1  100  5608  100  1  100  4005  100  1  100  2379  100  0  98.8  1717  99.30  0  89.1  1366  92.02  0 
1000  100  16,614  100  1  100  8419  100  1  100  5400  100  1  100  3807  100  1  100  2279  100  1  99.6  1655  99.76  1  90.9  1302  93.01  0 
2000  100  15,539  100  2  100  6906  100  2  100  3715  100  2  100  2422  100  2  100  1401  100  1  100  1135  100  1  98.4  1040  98.46  1 
3000  100  13,884  100  4  100  5259  100  4  100  2599  100  4  100  1624  100  3  100  1109  100  3  100  1021  100  3  99.8  1006  99.80  3 
4000  100  12,302  100  7  100  3911  100  7  100  1874  100  6  100  1291  100  6  100  1044  100  5  100  1004  100  4  99.8  1001  99.80  4 
5000  100  10,421  100  13  100  2935  100  11  100  1485  100  10  100  1126  100  9  100  1008  100  8  100  1000  100  6  100  1000  100  5 
6000  100  8822  100  17  100  2299  100  16  100  1262  100  14  100  1043  100  13  100  1004  100  10  100  1000  100  9  100  1000  100  8 
7000  100  7426  100  24  100  1855  100  22  100  1145  100  20  100  1026  100  17  100  1000  100  14  100  1001  100  11  100  1000  100  10 
8000  100  6346  100  33  100  1569  100  30  100  1084  100  26  100  1007  100  23  100  1000  100  18  100  1000  100  15  100  1000  100  13 
9000  100  5378  100  42  100  1362  100  38  100  1038  100  33  100  1002  100  30  100  1000  100  24  100  1000  100  19  100  1000  100  17 
10,000  100  4529  100  54  100  1237  100  48  100  1028  100  42  100  1000  100  38  100  1000  100  29  100  1000  100  24  100  1000  100  20 
Aver.  99.94  8861  99.53  7.07  99.87  4865  99.51  6.43  99.83  3520  99.57  5.68  99.66  2829  99.37  5.11  98.91  2142  99.14  4.04  95.79  1786  97.61  3.32  86.71  1569  92.38  2.89 
Table A4.
Computational results for randomly generated instances with ratio $10\%:30\%:10\%:50\%$ of the number of jobs in the subsets.
$\mathit{\delta}\%$  5%  10%  15%  20%  30%  40%  50%  

$\mathit{n}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$ 
10  99.4  194  96.91  0  97.4  334  92.22  0  95.5  474  90.51  0  94.3  566  89.93  0  88.3  749  84.38  0  79.1  849  75.03  0  74.1  934  71.95  0 
20  99.9  767  99.87  0  99.4  1232  99.51  0  98.4  1489  98.86  0  97.5  1680  98.39  0  94.7  1762  96.99  0  85.6  1770  91.64  0  75.1  1724  85.15  0 
30  99.7  1532  99.80  0  99.5  2150  99.77  0  99.2  2399  99.67  0  99  2499  99.52  0  96  2367  98.31  0  86.8  2200  93.77  0  71.7  1941  85.27  0 
40  99.8  2422  99.92  0  99.7  3151  99.90  0  99.8  3295  99.94  0  98.9  2972  99.63  0  95.2  2581  98.06  0  83  2326  92.48  0  69.5  2055  85.06  0 
50  100  3422  100  0  99.9  4013  99.98  0  99.9  3844  99.97  0  99.5  3451  99.86  0  95.3  2857  98.35  0  82.1  2486  92.76  0  64.2  2202  83.47  0 
60  100  4425  100  0  100  4681  100  0  99.9  4189  99.98  0  99.4  3750  99.84  0  94.8  2981  98.26  0  83.7  2566  93.53  0  64  2238  83.60  0 
70  100  5338  100  0  100  5181  100  0  100  4569  100  0  99.3  4027  99.83  0  94.3  3183  98.21  0  79.6  2594  92.14  0  61.1  2284  82.75  0 
80  100  6169  100  0  100  5770  100  0  99.9  4915  99.98  0  99.6  4112  99.90  0  95.6  3257  98.62  0  80.5  2625  92.42  0  58  2260  81.15  0 
90  100  6998  100  0  100  6018  100  0  100  4984  100  0  99.6  4213  99.91  0  94.6  3332  98.38  0  78.5  2680  91.87  0  52.3  2270  78.72  0 
100  100  7714  100  0  100  6298  100  0  100  5197  100  0  99.6  4358  99.91  0  94.5  3367  98.31  0  75.8  2642  90.61  0  54.6  2299  79.99  0 
200  100  12,228  100  0  100  7920  100  0  100  5951  100  0  100  4748  100  0  95.7  3330  98.71  0  73.5  2665  90.02  0  45.4  2233  75.41  0 
300  100  14,375  100  0  100  8735  100  0  100  6096  100  0  100  4723  100  0  96.4  3285  98.90  0  69.6  2464  87.66  0  43.4  2064  72.53  0 
400  100  15,022  100  0  100  8762  100  0  100  6135  100  0  100  4712  100  0  96.4  3036  98.81  0  70.5  2286  87.05  0  48.1  1820  71.43  0 
500  100  15,705  100  0  100  8823  100  0  100  5876  100  0  100  4497  100  0  97.5  2842  99.12  0  73.5  2100  87.38  0  55.1  1686  73.37  0 
600  100  16,442  100  0  100  8712  100  0  100  5753  100  0  100  4252  100  0  97.6  2674  99.10  0  74.8  1941  87.02  0  62.6  1530  75.56  0 
700  100  15,910  100  0  100  8670  100  0  100  5609  100  0  100  3976  100  0  99.1  2510  99.64  0  76.5  1776  86.77  0  69.3  1420  78.31  0 
800  100  16,215  100  1  100  8419  100  1  100  5492  100  1  100  3773  100  1  99.3  2271  99.69  1  81.9  1648  89.02  1  75.9  1319  81.73  1 
900  100  16,347  100  1  100  8268  100  1  100  5254  100  1  100  3597  100  1  99.2  2173  99.63  1  84.8  1575  90.35  1  80.7  1245  84.50  1 
1000  100  16,355  100  1  100  8133  100  1  100  5064  100  1  100  3369  100  1  99.7  1998  99.85  1  86.8  1426  90.74  1  84.9  1189  87.30  1 
2000  100  14,679  100  3  100  5955  100  3  100  3095  100  3  100  1972  100  2  100  1243  100  2  98.6  1056  98.67  2  97.7  1017  97.74  2 
3000  100  12,643  100  6  100  4207  100  6  100  2036  100  5  100  1354  100  5  100  1038  100  4  99.9  1003  99.90  4  100  1001  100  3 
4000  100  10,375  100  12  100  2927  100  11  100  1467  100  10  100  1152  100  9  100  1011  100  7  100  1000  100  6  100  1000  100  6 
5000  100  8524  100  19  100  2140  100  18  100  1205  100  15  100  1032  100  14  100  1003  100  11  100  1000  100  9  100  1000  100  8 
6000  100  6942  100  28  100  1724  100  26  100  1095  100  23  100  1014  100  20  100  1000  100  16  100  1000  100  14  100  1000  100  12 
7000  100  5463  100  40  100  1398  100  35  100  1050  100  32  100  1007  100  28  100  1002  100  23  100  1000  100  18  100  1000  100  16 
8000  100  4543  100  54  100  1240  100  48  100  1028  100  43  100  1002  100  45  100  1000  100  30  100  1000  100  24  100  1000  100  21 
9000  100  3751  100  69  100  1135  100  62  100  1005  100  55  100  1001  100  48  100  1000  100  38  100  1000  100  32  100  1000  100  26 
10,000  100  3056  100  86  100  1078  100  77  100  1003  100  68  100  1000  100  60  100  1000  100  48  100  1000  100  40  100  1000  100  33 
Aver.  99.96  8841  99.87  11.43  99.85  4896  99.69  10.32  99.74  3556  99.60  9.18  99.53  2850  99.53  8.36  97.29  2138  98.62  6.50  85.90  1774  92.89  5.43  75.28  1562  86.25  4.64 
Table A5.
Computational results for randomly generated instances with ratio $10\%:20\%:10\%:60\%$ of the number of jobs in the subsets.
$\mathit{\delta}\%$  5%  10%  15%  20%  30%  40%  50%  

$\mathit{n}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$ 
10  98.8  263  95.44  0  97.5  491  94.91  0  94.5  657  91.48  0  91  791  88.50  0  81.5  1014  81.36  0  74.9  1102  76.59  0  66  1186  70.49  0 
20  99.7  1034  99.71  0  98.8  1601  99.19  0  98.8  1904  99.32  0  97.6  1984  98.74  0  89.9  2113  95.13  0  80.4  1968  89.94  0  63  1845  79.73  0 
30  99.9  2131  99.95  0  99.8  2778  99.93  0  99  3059  99.67  0  98.3  2878  99.37  0  89.6  2608  95.82  0  74.3  2270  88.46  0  60.3  2060  80.34  0 
40  100  3224  100  0  99.9  3747  99.97  0  99.8  3698  99.95  0  98.3  3392  99.50  0  90.9  2877  96.63  0  71.8  2469  88.42  0  52  2174  77.60  0 
50  100  4370  100  0  100  4704  100  0  99.6  4174  99.90  0  99  3701  99.73  0  89.4  2988  96.32  0  66.6  2566  86.83  0  47.1  2228  75.94  0 
60  100  5473  100  0  100  5368  100  0  99.9  4608  99.98  0  98.2  3987  99.55  0  89.3  3098  96.51  0  67.2  2643  87.48  0  42.6  2279  74.59  0 
70  100  6454  100  0  100  5985  100  0  99.9  4968  99.98  0  99.4  4125  99.85  0  87.5  3214  96.02  0  62.6  2669  85.69  0  38.6  2291  72.85  0 
80  100  7498  100  0  99.9  6235  99.98  0  99.8  5194  99.94  0  98.8  4333  99.70  0  87.3  3372  96.14  0  61.4  2716  85.64  0  33.6  2260  70.40  0 
90  99.9  8281  99.99  0  100  6560  100  0  99.8  5243  99.96  0  98.9  4502  99.76  0  87.8  3388  96.40  0  61.4  2757  85.78  0  32  2361  71.03  0 
100  100  9169  100  0  100  7056  100  0  99.7  5507  99.95  0  99.2  4586  99.83  0  83.7  3360  94.97  0  58.2  2689  84.27  0  27.7  2287  68.21  0 
200  100  13,366  100  0  100  8131  100  0  99.9  6029  99.98  0  99  4814  99.79  0  83.1  3329  94.89  0  43.9  2541  77.80  0  10.3  2172  58.52  0 
300  100  14,999  100  0  100  8869  100  0  100  6010  100  0  98.9  4675  99.76  0  82  3127  94.18  0  32.2  2329  70.85  0  4.6  1870  48.66  0 
400  100  15,704  100  0  100  8848  100  0  100  6048  100  0  99.7  4490  99.93  0  82.5  2899  93.96  0  28.3  2120  66.08  0  1.3  1710  42.28  0 
500  100  15,775  100  0  100  8720  100  0  100  5825  100  0  99.7  4290  99.93  0  83.2  2638  93.63  0  21.6  1885  58.30  0  0.7  1541  35.56  0 
600  100  16,336  100  0  100  8420  100  1  100  5582  100  0  100  3938  100  0  87.7  2420  94.88  1  18  1727  52.52  0  0  1408  28.98  0 
700  100  16,298  100  1  100  8466  100  1  100  5360  100  1  100  3733  100  1  88.4  2203  94.73  1  15  1574  46.00  1  0  1282  22.00  1 
800  100  16,707  100  1  100  8030  100  1  100  5023  100  1  99.9  3479  99.97  1  88.7  2077  94.56  1  12.9  1457  40.15  1  0.2  1207  17.32  1 
900  100  16,135  100  1  100  7936  100  1  100  4808  100  1  100  3265  100  1  89.5  1934  94.52  1  10.8  1368  34.80  1  0  1172  14.68  1 
1000  100  16,015  100  1  100  7665  100  1  100  4528  100  1  100  3049  100  1  91.6  1737  95.16  1  9.6  1314  31.20  1  0  1144  12.59  1 
2000  100  13,921  100  4  100  5101  100  4  100  2549  100  4  100  1622  100  3  98.6  1138  98.77  3  1.2  1024  3.52  3  0  1002  0.20  3 
3000  100  11,344  100  9  100  3400  100  9  100  1636  100  8  100  1210  100  7  99.8  1014  99.80  6  0.4  1003  0.70  5  0  1000  0  5 
4000  100  8769  100  17  100  2283  100  16  100  1245  100  14  100  1054  100  13  100  1003  100  10  0.1  1000  0.1  9  0  1000  0  8 
5000  100  6948  100  28  100  1691  100  25  100  1102  100  27  100  1023  100  21  100  1001  100  16  0  1000  0  14  0  1000  0  11 
6000  100  5409  100  42  100  1362  100  38  100  1041  100  34  100  1003  100  30  100  1000  100  27  0  1000  0  20  0  1000  0  17 
7000  100  4121  100  59  100  1214  100  53  100  1016  100  47  100  1001  100  42  100  1000  100  34  0  1000  0  27  0  1000  0  23 
8000  100  3368  100  80  100  1093  100  71  100  1006  100  62  100  1000  100  66  100  1000  100  43  0  1000  0  36  0  1000  0  30 
9000  100  2646  100  102  100  1048  100  90  100  1000  100  80  100  1000  100  71  100  1000  100  56  0  1000  0  47  0  1000  0  39 
10,000  100  2248  100  126  100  1024  100  119  100  1002  100  100  100  1000  100  89  100  1000  100  70  0  1000  0  63  0  1000  0  48 
Aver.  99.94  8857  99.82  16.82  99.85  4923  99.79  15.36  99.67  3565  99.65  13.57  99.14  2854  99.43  12.36  91.14  2127  96.23  9.64  31.17  1757  47.90  8.14  17.14  1553  36.50  6.71 
Table A6.
Computational results for randomly generated instances with ratio $10\%:10\%:10\%:70\%$ of the number of jobs in the subsets.
$\mathit{\delta}\%$  5%  10%  15%  20%  30%  40%  50%  

$\mathit{n}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$ 
10  98.6  371  96.23  0  96.2  612  93.63  0  93.6  853  92.15  0  89.5  1015  89.06  0  82.1  1231  84.73  0  67.1  1301  73.10  0  54.3  1348  63.65  0 
20  99.8  1432  99.86  0  99.3  1988  99.65  0  97.2  2273  98.77  0  95.5  2345  98.04  0  85.9  2222  93.61  0  67.5  2119  84.29  0  50.7  1952  74.13  0 
30  99.7  2713  99.89  0  99.6  3341  99.88  0  99.2  3397  99.76  0  96.9  3204  99.03  0  85.9  2777  94.63  0  65.2  2344  84.98  0  43.5  2139  72.98  0 
40  99.9  4056  99.98  0  99.7  4439  99.93  0  99.2  4015  99.80  0  97.1  3722  99.19  0  85  2983  94.90  0  58.4  2525  82.89  0  36.5  2191  70.61  0 
50  100  5231  100  0  99.9  5130  99.98  0  99.7  4593  99.93  0  97.7  3952  99.42  0  80.4  3165  93.52  0  55  2595  82.35  0  28.6  2200  67.18  0 
60  100  6574  100  0  99.9  5804  99.98  0  99.3  4934  99.84  0  97.1  4182  99.26  0  84.1  3283  94.94  0  49.9  2656  80.80  0  25.6  2255  66.39  0 
70  99.9  7444  99.99  0  100  6365  100  0  99.4  5115  99.88  0  97.8  4328  99.49  0  80.4  3261  93.90  0  48.3  2706  80.60  0  21  2330  65.75  0 
80  100  8505  100  0  100  6737  100  0  99.7  5415  99.94  0  96.5  4422  99.21  0  75.3  3304  92.37  0  42.7  2667  78.29  0  16.8  2258  63.02  0 
90  100  9185  100  0  99.8  7333  99.97  0  99.8  5623  99.96  0  97.7  4555  99.50  0  76.4  3417  92.98  0  37.6  2696  76.34  0  13.3  2288  61.32  0 
100  100  9909  100  0  99.9  7305  99.99  0  99.6  5571  99.93  0  98.2  4546  99.60  0  74.4  3449  92.49  0  35.8  2695  75.92  0  11.8  2314  61.62  0 
200  100  13,806  100  0  100  8387  100  0  99.8  6146  99.97  0  96.2  4736  99.18  0  63.5  3261  88.75  0  16.1  2527  66.68  0  2.7  2006  51.40  0 
300  100  15,550  100  0  100  8870  100  0  99.9  6084  99.98  0  97.3  4563  99.41  0  53.6  3067  84.84  0  6.9  2215  57.97  0  0.6  1765  43.63  0 
400  100  15,856  100  0  100  8573  100  0  99.9  5852  99.98  0  96.9  4304  99.28  0  48.3  2737  81.11  0  3.4  2049  52.76  0  0  1596  37.34  0 
500  100  16,158  100  0  100  8576  100  0  100  5760  100  0  97.9  4067  99.48  1  44.9  2471  77.70  0  1.8  1727  43.14  0  0.1  1402  28.74  0 
600  100  16,216  100  1  100  8425  100  1  99.9  5416  99.98  1  98.8  3724  99.68  1  42.9  2217  74.24  1  1.6  1539  36.00  1  0  1279  21.81  1 
700  100  16,338  100  1  100  8142  100  1  100  5055  100  1  99.1  3432  99.74  1  40.4  2059  71.01  1  1.4  1420  30.56  1  0  1197  16.46  1 
800  100  16,548  100  1  100  7909  100  1  100  4744  100  1  99  3206  99.69  1  36.8  1821  65.29  1  0.4  1319  24.49  1  0  1133  11.74  1 
900  100  16,000  100  1  100  7494  100  1  100  4477  100  1  99  2929  99.66  1  30.5  1716  59.50  1  0.1  1264  20.97  1  0  1098  8.93  1 
1000  100  15,806  100  1  100  7294  100  1  100  4123  100  1  99.3  2694  99.74  1  36.1  1535  58.37  1  0.1  1177  15.12  1  0  1057  5.39  1 
2000  100  13,179  100  6  100  4424  100  5  100  2200  100  5  100  1431  100  4  24.3  1059  28.52  4  0  1007  0.70  3  0  1002  0.20  3 
3000  100  9960  100  13  100  2746  100  12  100  1407  100  11  100  1103  100  10  20.2  1007  20.75  8  0  1000  0  7  0  1001  0.10  7 
4000  100  7402  100  24  100  1843  100  22  100  1130  100  20  100  1027  100  17  16.2  1000  16.2  15  0  1000  0  12  0  1000  0  10 
5000  100  5616  100  40  100  1450  100  36  100  1042  100  31  100  1008  100  28  12.2  1000  12.2  23  0  1000  0  19  0  1000  0  15 
6000  100  4234  100  59  100  1204  100  53  100  1019  100  56  100  1000  100  42  9.8  1000  9.8  34  0  1000  0  28  0  1000  0  23 
7000  100  3236  100  82  100  1083  100  64  100  1007  100  65  100  1000  100  58  7.7  1000  7.7  47  0  1000  0  38  0  1000  0  31 
8000  100  2511  100  121  100  1040  100  98  100  1002  100  90  100  1000  100  76  7.6  1000  7.6  61  0  1000  0  51  0  1000  0  43 
9000  100  2059  100  140  100  1015  100  124  100  1001  100  110  100  1000  100  96  6.2  1000  6.2  79  0  1000  0  65  0  1000  0  52 
10,000  100  1728  100  174  100  1011  100  154  100  1001  100  159  100  1000  100  120  5  1000  5  97  0  1000  0  80  0  1000  0  65 
Aver.  99.93  8844  99.85  23.71  99.80  4948  99.75  20.46  99.51  3581  99.64  19.71  98.13  2839  99.20  16.32  47.00  2109  60.82  13.32  19.98  1734  41.00  11  10.91  1529  31.87  9.07 
Table A7.
Computational results for randomly generated instances with ratio $5\%:20\%:5\%:70\%$ of the number of jobs in the subsets.
$\mathit{\delta}\%$  5%  10%  15%  20%  30%  40%  50%  

$\mathit{n}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$  $\mathit{O}\mathit{p}\mathit{t}$  $\mathit{N}\mathit{C}$  $\mathit{S}\mathit{C}$  $\mathit{t}$ 
20  98.8  1353  99.04  0  96.7  1993  98.29  0  93.3  2325  97.08  0  86.9  2343  94.15  0  71.4  2210  86.38  0  52.1  2091  76.09  0  34.8  1941  65.33  0 
40  99.2  4101  99.80  0  98.7  4426  99.71  0  93.5  3918  98.34  0  88.8  3685  96.93  0  60.9  3007  86.63  0  33.2  2539  73.02  0  19  2189  62.27  0 
60  99.1  6628  99.86  0  98.7  5712  99.75  0  93.1  4893  98.51  0  82.5  4106  95.64  0  48.8  3176  83.44  0  21.1  2584  68.89  0  10.9  2267  60.17  0 
80  99.5  8614  99.93  0  98.3  6701  99.75  0  94.7  5335  98.93  0  79.5  4447  95.21  0  38.9  3272  80.96  0  14.5  2738  68.04  0  5.7  2254  57.59  0 
100  99.9  10,165  99.99  0  98.7  7202  99.81  0  92.6  5740  98.69  0  76.5  4634  94.80  0  29.4  3338  78.16  0  10.5  2711  66.58  0  2.7  2224  55.94  0 
200  100  13,856  100  0  98.6  8509  99.84  0  87.5  6083  97.93  0  59.5  4691  91.20  0  13.4  3333  73.69  0  4.8  2546  62.33  0  0.7  2041  51.20  0 
300  100  15,201  100  0  99.3  8705  99.92  0  82.4  6187  97.07  0  44.5  4566  87.69  0  7.3  3025  69.16  0  2.4  2287  57.24  0  0.1  1833  45.50  0 
400  99.9  15,924  99.99  0  98.4  8964  99.82  0  75.3  5888  95.77  0  32.3  4338  84.23  0  9.2  2727  66.59  0  1.9  1970  50.20  0  0.1  1592  37.25  0 
500  100  16,186  100  0  98  8588  99.76  0  71  5652  94.80  0  28.6  3987  81.89  1  12.6  2468  64.55  0  1.5  1794  45.09  0  0  1379  27.48  0 
600  100  16,531  100  1  97.5  8437  99.70  1  65.2  5391  93.51  1  26.8  3660  79.92  1  16.4  2184  61.72  1  0.6  1556  36.05  1  0  1287  22.30  1 
700  100  16,251  100  1  98.7  8282  99.84  1  63.8  4967  92.65  1  24.6  3441  78.00  1  17.1  2087  60.28  2  0.3  1452  31.34  1  0  1186  15.68  1 
800  100  16,462  100  1  98.3  7937  99.79  1  62.1  4736  91.91  1  26.5  3192  76.94  1  19.3  1806  55.26  1  0  1338  25.26  1  0  1131  11.58  1 
900  100  16,099  100  1  97  7613  99.61  1  58.8  4439  90.70  1  28.7  2885  75.22  1  23.5  1694  54.84  1  0.4  1237  19.48  1  0  1118  10.55  1 
1000  100  15,750  100  1  96.6  7157  99.52  1  59.2  4186  90.18  1  29.7  2708  74.04  1  23  1551  50.35  1  0.1  1211  17.51  1  0  1080  7.41  1 
2000  100  13,055  100  6  97.8  4521  99.51  5  65.8  2164  84.20  5  76.3  1416  83.26  5  23.5  1063  28.03  4  0  1012  1.19  3  0  1003  0.30  3 
3000  100  10,038  100  13  99.5  2766  99.82  12  86.4  1403  90.31  11  94  1109  94.59  10  17.9  1007  18.47  8  0  1000  0  7  0  1000  0  6 
4000  100  7568  100  25  99.6  1823  99.78  22  96.3  1118  96.69  20  98.5  1021  98.53  18  15.9  1001  15.98  14  0  1000  0  12  0  1000  0  10 
5000  100  5613  100  40  100  1430  100  35  97.4  1056  97.54  32  99.5  1008  99.50  29  11.4  1000  11.4  23  0  1000  0  19  0  1000  0  16 
6000  100  4157  100  59  100  1187  100  54  99.5  1014  99.51  48  99.9  1002  99.90  46  10.4  1000  10.4  34  0  1000  0  33  0  1000  0  23 
7000  100  3108  100  85  100  1076  100  74  99.7  1007  99.70  66  100  1000  100  60  8.6  1000  8.6  47  0  1000  0  39  0  1000  0  31 
8000  100  2581  100  110  100  1051  100  98  99.7  1004  99.70  88  100  1000  100  78  6.4  1000  6.4  65  0  1000  0  50  0  1000  0  41 
9000  100  2029  100  140  100  1014  100  131  100  1000  100  115  100  1000  100  99  6.7  1000  6.7  79  0  1000  0  63  0  1000  0  52 
10,000  100  1672  100  175  100  1010  100  166  100  1000  100  138  100  1000  100  122  5.9  1000  5.9  99  0  1000  0  80  0  1000  0  66 
Aver.  99.84  9693  99.94  28.61  98.71  5048  99.75  26.17  84.23  3500  95.81  22.96  68.85  2706  90.51  20.57  21.65  1954  47.13  16.48  6.23  1612  30.36  13.52  3.22  1414  23.07  11 