Advances and Novel Approaches in Discrete Optimization Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Frank Werner Edited by Advances and Novel Approaches in Discrete Optimization Advances and Novel Approaches in Discrete Optimization Editor Frank Werner MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editor Frank Werner Otto-von-Guericke-Universit ̈ at Magdeburg, Germany Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/Advance Novel Approaches Discrete Optimization). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. 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Contents About the Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface to ”Advances and Novel Approaches in Discrete Optimization” . . . . . . . . . . . . . ix Lili Zuo, Zhenxia Sun, Lingfa Lu and Liqi Zhang Single-Machine Scheduling with Rejection and an Operator Non-Availability Interval Reprinted from: Mathematics 2019 , 7 , 668, doi:10.3390/math7080668 . . . . . . . . . . . . . . . . . 1 Hongjun Wei, Jinjiang Yuan and Yuan Gao Transportation and Batching Scheduling for Minimizing Total Weighted Completion Time Reprinted from: Mathematics 2019 , 7 , 819, doi:10.3390/math7090819 . . . . . . . . . . . . . . . . . 9 Wenhua Li, Weina Zhai and Xing Chai Online Bi-Criteria Scheduling on Batch Machines with Machine Costs Reprinted from: Mathematics 2019 , 7 , 960, doi:10.3390/math7100960 . . . . . . . . . . . . . . . . . 19 Nodari Vakhania Dynamic Restructuring Framework for Scheduling with Release Times and Due-Dates Reprinted from: Mathematics 2019 , 7 , 1104, doi:10.3390/math7111104 . . . . . . . . . . . . . . . . 31 Wenhua Li , Libo Wang, Xing Chai and Hang Yuan Online Batch Scheduling of Simple Linear Deteriorating Jobs with Incompatible Families Reprinted from: Mathematics 2020 , 8 , 170, doi:10.3390/math8020170 . . . . . . . . . . . . . . . . . 73 Alexander A. Lazarev, Nikolay Pravdivets and Frank Werner On the Dual and Inverse Problems of Scheduling Jobs to Minimize the Maximum Penalty Reprinted from: Mathematics 2020 , 8 , 1131, doi:10.3390/math8071131 . . . . . . . . . . . . . . . . 85 Yuri N. Sotskov, Natalja M. Matsveichuk and Vadzim D. Hatsura Schedule Execution for Two-Machine Job-Shop to Minimize Makespan with Uncertain Processing Times Reprinted from: Mathematics 2020 , 8 , 1314, doi:10.3390/math8081314 . . . . . . . . . . . . . . . . 101 Haidar Ali, Muhammad Ahsan Binyamin, Muhammad Kashif Shafiq and Wei Gao On the Degree-Based Topological Indices of Some Derived Networks Reprinted from: Mathematics 2019 , 7 , 612, doi:10.3390/math7070612 . . . . . . . . . . . . . . . . . 153 Bin Yang, Vinayak V. Manjalapur, Sharanu P. Sajjan, Madhura M. Mathai and Jia-Bao Liu On Extended Adjacency Index with Respect to Acyclic, Unicyclic and Bicyclic Graphs Reprinted from: Mathematics 2019 , 7 , 652, doi:10.3390/math7070652 . . . . . . . . . . . . . . . . . 171 Michal Staˇ s Join Products K 2 , 3 + C n Reprinted from: Mathematics 2020 , 8 , 925, doi:10.3390/math8060925 . . . . . . . . . . . . . . . . . 181 Yuriy Shablya, Dmitry Kruchinin and Vladimir Kruchinin Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application Reprinted from: Mathematics 2020 , 8 , 962, doi:10.3390/math8060962 . . . . . . . . . . . . . . . . . 191 v Urmila Pyakurel, Hari Nandan Nath, Stephan Dempe and Tanka Nath Dhamala Efficient Dynamic Flow Algorithms for Evacuation Planning Problems with Partial Lane Reversal Reprinted from: Mathematics 2019 , 7 , 993, doi:10.3390/math7100993 . . . . . . . . . . . . . . . . . 213 Shenshen Gu and Yue Yang A Deep Learning Algorithm for the Max-Cut Problem Based on Pointer Network Structure with Supervised Learning and Reinforcement Learning Strategies Reprinted from: Mathematics 2020 , 8 , 298, doi:10.3390/math8020298 . . . . . . . . . . . . . . . . . 243 Yajaira Cardona-Vald ́ es, Samuel Nucamendi-Guill ́ en, Rodrigo E. Peimbert-Garc ́ ıa, Gustavo Macedo-Barrag ́ an and Eduardo D ́ ıaz-Medina A New Formulation for the Capacitated Lot Sizing Problem with Batch Ordering Allowing Shortages Reprinted from: Mathematics 2020 , 8 , 878, doi:10.3390/math8060878 . . . . . . . . . . . . . . . . . 263 Alexander Pankratov, Tatiana Romanova and Igor Litvinchev Packing Oblique 3D Objects Reprinted from: Mathematics 2020 , 8 , 1130, doi:10.3390/math8071130 . . . . . . . . . . . . . . . . 279 Krishan Arora, Ashok Kumar, Vikram Kumar Kamboj, Deepak Prashar, Sudan Jha, Bhanu Shrestha and Gyanendra Prasad Joshi Optimization Methodologies and Testing on Standard Benchmark Functions of Load Frequency Control for Interconnected Multi Area Power System in Smart Grids Reprinted from: Mathematics 2020 , 8 , 980, doi:10.3390/math8060980 . . . . . . . . . . . . . . . . . 295 Peter Drahoˇ s, Michal Koc ́ ur, Oto Haffner, Erik Kuˇ cera and Alena Koz ́ akov ́ a RISC Conversions for LNS Arithmetic in Embedded Systems Reprinted from: Mathematics 2020 , 8 , 1208, doi:10.3390/math8081208 . . . . . . . . . . . . . . . . 319 vi About the Editor Frank Werner studied Mathematics from 1975–1980 and graduated from the Technical University Magdeburg (Germany) with honors. He defended his Ph.D. thesis on the solution of special scheduling problems in 1984 with ‘summa cum laude’ and his habilitation thesis in 1989. In 1992, he received a grant from the Alexander von Humboldt Foundation. Currently, he works as an extraordinary professor at the Faculty of Mathematics of the Otto von Guericke University Magdeburg (Germany). He is the author or editor of seven books, among them the textbooks ‘Mathematics of Economics and Business’ and ‘A Refresher Course in Mathematics’, and he has published more than 280 papers in international journals. He serves on the Editorial Board of 17 journals, in particular, he is the Editor-in-Chief of Algorithms and an Associate Editor of the International Journal of Production Research and Journal of Scheduling . He has served as a member of the Program Committee of more than 80 international conferences. His research interests are operations research, combinatorial optimization, and scheduling. vii Preface to ”Advances and Novel Approaches in Discrete Optimization” Discrete optimization is an important area of applied mathematics which lies at the intersection of several disciplines and covers both theoretical and practical aspects. This book is the result of a Special Issue entitled ‘Advances and Novel Approaches in Discrete Optimization’. In the call for papers for this issue, I asked for submissions presenting new theoretical results, structural investigations, new models, and algorithmic approaches as well as new applications of discrete optimization problems. Among the possible subjects were integer programming, combinatorial optimization, optimization problems of graphs and networks, scheduling, logistics, and transportation, to name but a few. In response to the call for papers, 43 submissions were received. All submissions have been reviewed, as a rule, by at least three experts in the discrete optimization area. Finally, 17 papers were accepted for this Special Issue, all of which are of high quality and reflect the great interest in the area of discrete optimization. This corresponds to an acceptance rate of 39.5%. The authors of these publications represent 13 different countries: China, Pakistan, India, Nepal, Germany, Mexico, USA, Australia, Slovakia, Russia, Korea, Ukraine, and Belarus. This book contains both theoretical works and practical applications in the field of discrete optimization. Although many different aspects of discrete optimization have been addressed by the submissions, among the accepted papers, a major part deals with scheduling problems as well as graphs and networks. We hope that researchers and practitioners will find much inspiration for their future work in the exciting area of discrete optimization. Next, all published articles in this book are briefly surveyed in the order of their sequence in the book. The first seven articles deal with scheduling problems. In the first article, Zuo et al. consider two single-machine scheduling problems with possible job rejection and a non-availability interval of the operator simultaneously. The objective is to minimize the sum of either the makespan or total weighted completion time of the accepted jobs and the total cost for the rejected jobs. The authors suggest a pseudo-polynomial solution algorithm as well as a fully polynomial-time approximation scheme. In the next article, Wei et al. deal with the problem of transportation and batching scheduling. A single vehicle is considered, and the goal is to minimize total weighted completion time. The main results of this paper are the proof that the problem is N P -hard in the strong sense for any batch capacity of at least 3 as well as a polynomial-time 3-approximation algorithm for the case of a batch capacity of at least 2. In the third article, Li et al. consider a bi-criteria online scheduling problem on parallel batch machines. The batch capacity is unbounded, the processing times of all jobs and batches are equal to one, and the objective is to minimize the maximum machine cost subject to a minimum makespan. The authors consider two types of cost functions and present two best possible online algorithms for the problem under consideration. Vakhania investigates a single-machine scheduling problem with given release dates, due dates, and divisible processing times. The objective is to minimize maximum lateness. He suggests a general method which also leads to useful structural properties of this problem and helps to identify polynomially solvable cases. In particular, for the case of mutually divisible job processing times, a polynomial-time algorithm results, and this case turns out to be maximal polynomially solvable ix one of this problem with nonarbitrary processing times. Li et al. consider an online scheduling problem with parallel batch machines and linearly deteriorating jobs. The batch capacity is unbounded, and the objective is to minimize the makespan. For the special case of m = 1 , a best possible online algorithm with a competitive ratio of (1 + α max ) f is given, where f denotes the number of job families and α max gives the maximal deterioration rate of a job. Furthermore, for m = f ≥ 1 , a best possible online algorithm with a competitive ratio of 1 + α max is also derived. Then Lazarev et al. consider the single-machine problem with given release dates and the objective of minimizing the maximum job penalty. While this problem is N P -hard in the strong sense, they introduce a dual and an inverse problem, which can both be polynomially solved. The optimal function value of the dual problem is incorporated as a lower bound into a branch and bound algorithm for the original problem. The authors present computational results with this enumerative algorithm for hard benchmark instances with up to 20 jobs. Most of the instances considered can be solved very fast by the proposed algorithm. In the last scheduling article, Sotskov et al. consider the two-machine job-shop scheduling problem with interval processing times and makespan minimization. The focus of this paper is how to execute a schedule in a best way. The paper uses the concept of a minimal dominant set of schedules. An online algorithm of the complexity O ( n 2 ) has been developed, where n denotes the number of jobs. Detailed numerical results are given for instances with up to 100 jobs and different maximal percentage errors in the processing times. The next articles deal with graph-theoretic subjects and applications of graphs. Ali et al. consider degree-based topological indices and some derived graphs. The goal of this paper is to investigate the chemical behavior of these graphs by means of the topological indices. In particular, the authors find the exact results for the Forgotten index, the Balaban index, the reclassified Zagreb indices, the ABC 4 index, and the GA 5 index of Hex-derived networks of type 3. Then, Yang et al. consider the extended adjacency index of a molecular graph. In particular, the authors show some graph transformations which increase or decrease this index. Then, they derive the extremal acyclic, unicyclic, and bicyclic graphs with a minimum and a maximum extended adjacency index, respectively. Stas investigates a graph-theoretic subject, namely the crossing number of a graph which is the minimum number of edge crossings over all drawings of the graph in the plane. In particular, he presents this number for the join product K 2 , 3 + C n , where K 2 , 3 is the complete bipartite graph and C n is a cycle on n vertices. The methods applied by the author use several combinatorial properties on cyclic permutations. Then, Shablya et al. look for new combinatorial generation algorithms. They give basic general methods and investigate one of them based on AND/OR trees. They apply the method of compositae from the theory of generating functions. To show the effectiveness of the suggested modifications, they also derive new ranking and unranking algorithms for several combinatorial sets. The remaining articles deal with interesting applications of discrete optimization in several research fields. Pyakurel et al. present efficient algorithms to solve dynamic flow problems with constant attributes as well as generalized problems with partial contraflow reconfiguration in the context of evacuation planning. In particular, a strongly polynomial time algorithm for calculating an approximate solution of the quickest partial contraflow problem on two terminal networks is derived. Numerical results are given for the road network of Kathmandu (Nepal) as the evacuation network. x Then, Gu and Yang consider the max-cut problem. They develop a unique method combining a pointer network and two deep learning strategies, namely supervised learning and reinforcement learning. The pointer network model includes a long short-term memory network and an encoder–decoder. It turns out that their model can be used to solve large-scale max-cut problems heuristically, where for high-dimensional cases, reinforcement learning turned out to be superior to supervised learning. Cordona-Valdes et al. deal with the multi-product, multi-period capacitated lot sizing problem. Determining the optimal lot size allows shortages resulting in a penalty cost. Two mixed-integer formulations are developed: one model allows shortages, and the other one enforces the fulfillment of the demand. The developed models have been applied to a Mexican fashion retail company within a case study. Both formulations significantly reduced the final inventory costs. Pankratov et al. deal with packing problems of irregular 3D objects. By using the phi-function technique, the problem is reduced to the solution of a nonlinear programming model and solved by a multi-start strategy with finding local extreme points. The algorithm has been tested on some benchmark instances. Then, Arora et al. present an optimized analysis and planning for power generation and management. They describe several optimization methodologies. In particular, binary variations of the moth flame optimizer and the Harris hawks optimizer are analyzed and tested on 23 benchmark functions, e.g., unimodal, multi-modal ones and functions with fixed dimension. The comparison and simulation results demonstrate that their implemented algorithm delivered better results towards the load frequency control problem of a smart grid arrangement compared to earlier methods. In the last article, Drahos et al. present a method for a conversion between the logarithmic number system (LNS) and floating point (FLP) representations using reduced instruction set computing (RISC). After giving an overview on FLP and LNS number representations, two algorithms of the RISC conversion between both systems using the ‘looping in sectors’ procedure are presented. The proposed algorithms deliver a very small maximum relative conversion error, and the authors mention also some interesting applications such as camera systems or car control units. Finally, I would like to thank all authors for submitting their work to this Special Issue and also all referees for their support by giving timely and insightful reports. My special thanks go to the staff of the journal Mathematics for their skilled and pleasant cooperation during the preparation of this issue. Frank Werner Editor xi Preface to ”Advances and Novel Approaches in Discrete Optimization” Discrete optimization is an important area of applied mathematics which lies at the intersection of several disciplines and covers both theoretical and practical aspects. This book is the result of a Special Issue entitled ‘Advances and Novel Approaches in Discrete Optimization’. In the call for papers for this issue, I asked for submissions presenting new theoretical results, structural investigations, new models, and algorithmic approaches as well as new applications of discrete optimization problems. Among the possible subjects were integer programming, combinatorial optimization, optimization problems of graphs and networks, scheduling, logistics, and transportation, to name but a few. In response to the call for papers, 43 submissions were received. All submissions have been reviewed, as a rule, by at least three experts in the discrete optimization area. Finally, 17 papers were accepted for this Special Issue, all of which are of high quality and reflect the great interest in the area of discrete optimization. This corresponds to an acceptance rate of 39.5%. The authors of these publications represent 13 different countries: China, Pakistan, India, Nepal, Germany, Mexico, USA, Australia, Slovakia, Russia, Korea, Ukraine, and Belarus. This book contains both theoretical works and practical applications in the field of discrete optimization. Although many different aspects of discrete optimization have been addressed by the submissions, among the accepted papers, a major part deals with scheduling problems as well as graphs and networks. We hope that researchers and practitioners will find much inspiration for their future work in the exciting area of discrete optimization. Next, all published articles in this book are briefly surveyed in the order of their sequence in the book. The first seven articles deal with scheduling problems. In the first article, Zuo et al. consider two single-machine scheduling problems with possible job rejection and a non-availability interval of the operator simultaneously. The objective is to minimize the sum of either the makespan or total weighted completion time of the accepted jobs and the total cost for the rejected jobs. The authors suggest a pseudo-polynomial solution algorithm as well as a fully polynomial-time approximation scheme. In the next article, Wei et al. deal with the problem of transportation and batching scheduling. A single vehicle is considered, and the goal is to minimize total weighted completion time. The main results of this paper are the proof that the problem is N P -hard in the strong sense for any batch capacity of at least 3 as well as a polynomial-time 3-approximation algorithm for the case of a batch capacity of at least 2. In the third article, Li et al. consider a bi-criteria online scheduling problem on parallel batch machines. The batch capacity is unbounded, the processing times of all jobs and batches are equal to one, and the objective is to minimize the maximum machine cost subject to a minimum makespan. The authors consider two types of cost functions and present two best possible online algorithms for the problem under consideration. Vakhania investigates a single-machine scheduling problem with given release dates, due dates, and divisible processing times. The objective is to minimize maximum lateness. He suggests a general method which also leads to useful structural properties of this problem and helps to identify polynomially solvable cases. In particular, for the case of mutually divisible job processing times, a polynomial-time algorithm results, and this case turns out to be maximal polynomially solvable one of this problem with nonarbitrary processing times. ix Li et al. consider an online scheduling problem with parallel batch machines and linearly deteriorating jobs. The batch capacity is unbounded, and the objective is to minimize the makespan. For the special case of m = 1 , a best possible online algorithm with a competitive ratio of (1 + α max ) f is given, where f denotes the number of job families and α max gives the maximal deterioration rate of a job. Furthermore, for m = f ≥ 1 , a best possible online algorithm with a competitive ratio of 1 + α max is also derived. Then Lazarev et al. consider the single-machine problem with given release dates and the objective of minimizing the maximum job penalty. While this problem is N P -hard in the strong sense, they introduce a dual and an inverse problem, which can both be polynomially solved. The optimal function value of the dual problem is incorporated as a lower bound into a branch and bound algorithm for the original problem. The authors present computational results with this enumerative algorithm for hard benchmark instances with up to 20 jobs. Most of the instances considered can be solved very fast by the proposed algorithm. In the last scheduling article, Sotskov et al. consider the two-machine job-shop scheduling problem with interval processing times and makespan minimization. The focus of this paper is how to execute a schedule in a best way. The paper uses the concept of a minimal dominant set of schedules. An online algorithm of the complexity O ( n 2 ) has been developed, where n denotes the number of jobs. Detailed numerical results are given for instances with up to 100 jobs and different maximal percentage errors in the processing times. The next articles deal with graph-theoretic subjects and applications of graphs. Ali et al. consider degree-based topological indices and some derived graphs. The goal of this paper is to investigate the chemical behavior of these graphs by means of the topological indices. In particular, the authors find the exact results for the Forgotten index, the Balaban index, the reclassified Zagreb indices, the ABC 4 index, and the GA 5 index of Hex-derived networks of type 3. Then, Yang et al. consider the extended adjacency index of a molecular graph. In particular, the authors show some graph transformations which increase or decrease this index. Then, they derive the extremal acyclic, unicyclic, and bicyclic graphs with a minimum and a maximum extended adjacency index, respectively. Stas investigates a graph-theoretic subject, namely the crossing number of a graph which is the minimum number of edge crossings over all drawings of the graph in the plane. In particular, he presents this number for the join product K 2 , 3 + C n , where K 2 , 3 is the complete bipartite graph and C n is a cycle on n vertices. The methods applied by the author use several combinatorial properties on cyclic permutations. Then, Shablya et al. look for new combinatorial generation algorithms. They give basic general methods and investigate one of them based on AND/OR trees. They apply the method of compositae from the theory of generating functions. To show the effectiveness of the suggested modifications, they also derive new ranking and unranking algorithms for several combinatorial sets. The remaining articles deal with interesting applications of discrete optimization in several research fields. Pyakurel et al. present efficient algorithms to solve dynamic flow problems with constant attributes as well as generalized problems with partial contraflow reconfiguration in the context of evacuation planning. In particular, a strongly polynomial time algorithm for calculating an approximate solution of the quickest partial contraflow problem on two terminal networks is derived. Numerical results are given for the road network of Kathmandu (Nepal) as the evacuation network. Then, Gu and Yang consider the max-cut problem. They develop a unique method combining a x pointer network and two deep learning strategies, namely supervised learning and reinforcement learning. The pointer network model includes a long short-term memory network and an encoder–decoder. It turns out that their model can be used to solve large-scale max-cut problems heuristically, where for high-dimensional cases, reinforcement learning turned out to be superior to supervised learning. Cordona-Valdes et al. deal with the multi-product, multi-period capacitated lot sizing problem. Determining the optimal lot size allows shortages resulting in a penalty cost. Two mixed-integer formulations are developed: one model allows shortages, and the other one enforces the fulfillment of the demand. The developed models have been applied to a Mexican fashion retail company within a case study. Both formulations significantly reduced the final inventory costs. Pankratov et al. deal with packing problems of irregular 3D objects. By using the phi-function technique, the problem is reduced to the solution of a nonlinear programming model and solved by a multi-start strategy with finding local extreme points. The algorithm has been tested on some benchmark instances. Then, Arora et al. present an optimized analysis and planning for power generation and management. They describe several optimization methodologies. In particular, binary variations of the moth flame optimizer and the Harris hawks optimizer are analyzed and tested on 23 benchmark functions, e.g., unimodal, multi-modal ones and functions with fixed dimension. The comparison and simulation results demonstrate that their implemented algorithm delivered better results towards the load frequency control problem of a smart grid arrangement compared to earlier methods. In the last article, Drahos et al. present a method for a conversion between the logarithmic number system (LNS) and floating point (FLP) representations using reduced instruction set computing (RISC). After giving an overview on FLP and LNS number representations, two algorithms of the RISC conversion between both systems using the ‘looping in sectors’ procedure are presented. The proposed algorithms deliver a very small maximum relative conversion error, and the authors mention also some interesting applications such as camera systems or car control units. Finally, I would like to thank all authors for submitting their work to this Special Issue and also all referees for their support by giving timely and insightful reports. My special thanks go to the staff of the journal Mathematics for their skilled and pleasant cooperation during the preparation of this issue. Frank Werner Editor xi mathematics Article Single-Machine Scheduling with Rejection and an Operator Non-Availability Interval Lili Zuo 1 , Zhenxia Sun 1 , Lingfa Lu 1, * and Liqi Zhang 2 1 School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China 2 College of Information and Management Science, Henan Agricultural University, Zhengzhou 450002, China * Correspondence: lulingfa@zzu.edu.cn Received: 8 July 2019; Accepted: 24 July 2019; Published: 26 July 2019 Abstract: In this paper, we study two scheduling problems on a single machine with rejection and an operator non-availability interval. In the operator non-availability interval, no job can be started or be completed. However, a crossover job is allowed such that it can be started before this interval and completed after this interval. Furthermore, we also assume that job rejection is allowed. That is, each job is either accepted and processed in-house, or is rejected by paying a rejection cost. Our task is to minimize the sum of the makespan (or the total weighted completion time) of accepted jobs and the total rejection cost of rejected jobs. For two scheduling problems with different objective functions, by borrowing the previous algorithms in the literature, we propose a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme (FPTAS), respectively. Keywords: scheduling with rejection; machine non-availability; operator non-availability; dynamic programming; FPTAS 1. Introduction In this section, we introduce some models on scheduling with (machine or operator) non-availability intervals, scheduling with rejection, and scheduling with rejection and non-availability intervals, respectively. 1.1. Scheduling with Non-Availability Intervals In most scheduling problems, it is assumed that the machines are available at all times. However, in some industrial settings, the assumption might not be true. Some machines or the operator might be unavailable in some time intervals. Recently, some researchers have studied some scheduling problems with the non-availability intervals. Two models of the non-availability interval were studied mainly: one is the machine non-availability (MNA) intervals due to the machine maintenances and the other is the operator non-availability (ONA) intervals because the operator is resting from work. The difference between MNA intervals and ONA intervals is that a crossover job can exist in the ONA interval. However, no job can be processed in the MNA interval. To the best of our knowledge, the earliest scheduling problem with MNA intervals was studied by Schmidt [ 1 ]. He considered a parallel-machine scheduling problem in which each machine has different MNA intervals. The task is to find a feasible preemptive schedule if it exists. A polynomial-time algorithm is presented for the above problem. We first introduce some single-machine scheduling problems with an MNA interval ( a , b ) . The corresponding problem can be denoted by 1 | MNA ( a , b ) | f , where “MNA ( a , b ) ” means that there is an MNA interval ( a , b ) and f is the objective function to be minimized. For problem 1 | MNA ( a , b ) | ∑ C j , Adiri et al. [ 2 ] proved that the problem is NP-hard and then presented a 5 4 -approximation algorithm. For problem 1 | MNA ( a , b ) | C max , Lee [ 3 ] showed that this problem is binary NP-hard and then provided a 4 3 -approximation algorithm. For problem Mathematics 2019 , 7 , 668; doi:10.3390/math7080668 www.mdpi.com/journal/mathematics 1 Mathematics 2019 , 7 , 668 1 | MNA ( a , b ) | L max , Kacem [ 4 ] designed a 3 2 -approximation algorithm and a fully polynomial-time approximation scheme (FPTAS). If there are k ≥ 2 MNA intervals ( a 1 , b 1 ) , ( a 2 , b 2 ) , · · · , ( a k , b k ) on the machine, the corresponding problem 1 | MNA ( a i , b i ) | C max is strongly NP-hard (see [ 3 ]) when k is arbitrary. Breit et al. [ 5 ] showed that, for any ρ ≥ 1 and k ≥ 2, there is no ρ -approximation algorithm for problem 1 | MNA ( a i , b i ) | C max unless P = NP. When there are m ≥ 2 parallel machines M 1 , · · · , M m and each machine M i has an MNA interval ( a i , b i ) , the corresponding problem is denoted by Pm | MNA ( a i , b i ) | f . For problem Pm | MNA ( 0, b i ) | C max , i.e., each machine M i has a machine release time b j , Lee [ 6 ] provided a modified LPT (MLPT) algorithm with a tight approximation ratio 4 3 . Kellerer [ 7 ] improved this bound 4 3 to 5 4 by a dual approximation algorithm using a bin packing approach. For problem Pm | MNA ( 0, b i ) | ∑ C j , Schmidt [8] showed that the SPT rule is optimal. Lee [ 6 ] also studied the problem Pm | MNA ( a i , b i ) | C max with the assumption that one machine is always available. He showed that the approximation ratios of LS (List Scheduling) and LPT are m and m + 1 2 , respectively. Furthermore, Liao et al. [ 9 ] considered the same problem with m = 2 and developed exact algorithms based on the TMO algorithm for problem P 2 || C max Aggoune [ 10 ] studied the flow-shop scheduling problem with several MNA intervals on each machine. A heuristic algorithm is provided to approximately solve this problem. Burdett and Kozan [ 11 ] also addressed some MNA intervals in railway scenarios. They introduced new fixed jobs for the MNA intervals. Some constructive heuristics and meta-heuristic algorithms were proposed in this paper. For more new models and results about this topic, the reader is referred to the survey by Ma et al. [12]. Brauner et al. [ 13 ] first studied the scheduling problems with an ONA interval. Similarly, this scheduling model can be denoted by 1 | ONA ( a , b ) | f For problem 1 | ONA ( a , b ) | C max , Brauner et al. [ 13 ] proved that it is binary NP-hard and provided an FPTAS. For problem 1 | ONA ( a , b ) | L max , Kacem et al. [ 14 ] proposed an FPTAS by borrowing the FPTAS for problem 1 | MNA ( a , b ) | L max . Chen et al. [ 15 ] considered the problem 1 | ONA ( a , b ) | ∑ C j and presented a 20 17 -approximation algorithm. Wan and Yuan [ 16 ] further considered the problem 1 | ONA ( a , b ) | ∑ w j C j They designed a pseudo-polynomial-time dynamic programming (DP) algorithm and then converted the DP algorithm into an FPTAS. Burdett et al. [ 17 ] considered the flexible job shop scheduling with operators (FJSOP) for coal export terminals. A hybrid meta-heuristic and a lot of numerical testings were designed for the above problem. 1.2. Scheduling with Rejection In many practical cases, processing all jobs may occur high inventory or tardiness costs. However, rejecting some jobs can save time and reduce costs. When a job is rejected, a corresponding rejection cost is required. The decision maker needs to determine which jobs should be accepted (and a feasible schedule for accepted jobs), and which jobs should be rejected, such that the production cost and the total rejection cost are minimized. Thus, both from the practical and theoretical point of view, scheduling models with rejection are very interesting. In addition, an important application also occurs in scheduling with outsourcing. If the outsourcing cost is treated as the rejection cost, scheduling with rejection and scheduling with outsourcing are in fact equivalent. Scheduling models with rejection were first introduced by Bartal et al. [ 18 ]. They considered several off-line and on-line scheduling problems on m parallel machines. The task is to minimize the sum of the makespan of accepted jobs and the total rejection cost of rejected jobs. For the on-line version, they designed an on-line algorithm with the best-possible competitive ratio of 2.618. For the off-line version, they provided an FPTAS when m is fixed, and a PTAS when m is arbitrary. Next, we only introduce some results on the single-machine scheduling with rejection. The corresponding problem can be denoted by 1 | rej | f + e ( R ) , where f is the objective function on the set A of accepted jobs and e ( R ) is the total rejection cost on the set R of rejected jobs. For problem 1 | r j , rej | C max + e ( R ) , Cao and Zhang [ 19 ] proved that this problem is NP-hard and designed a PTAS. However, they also pointed out that it is open whether this problem is ordinary or 2 Mathematics 2019 , 7 , 668 strongly NP-hard. Zhang et al. [ 20 ] showed that this problem is binary NP-hard by providing two different pseudo-polynomial-time algorithms. Finally, they also provided a 2-approximation algorithm and an FPTAS for the above problem. For problem 1 | rej | L max + e ( R ) , Sengupta [ 21 ] proved that this problem is binary NP-hard. He also proposed two dynamic programming algorithms and converted one of the two algorithms into an FPTAS. Engels et al. [ 22 ] studied the problem 1 | rej | ∑ w j C j + e ( R ) They showed that this problem is binary NP-hard and then provided an FPTAS. They also showed that, when w j = 1, the problem 1 | rej | ∑ C j + e ( R ) is polynomial-time solvable. Recently, Shabtay et al. [ 23 ] presented a comprehensive survey for the off-line scheduling problems with rejection. For other models and results on scheduling with rejection, the reader is referred to the survey by Shabtay et al. [23]. 1.3. Scheduling with Rejection and Non-Availability Intervals There are only two articles which considered “scheduling with rejection” and “machine non-availability intervals” together. Zhong et al. [ 24 ], and Zhao and Tang [ 25 ] considered the problems 1 | MNA ( a , b ) , rej | C max + e ( R ) and 1 | MNA ( a , b ) , rej | ∑ w j C j + e ( R ) , respectively. Both of them presented a pseudo-polynomial dynamic programming algorithm and an FPTAS for the corresponding problem. In addition, Li and Chen [ 26 ] investigated several scheduling problems with rejection and a deteriorating maintenance activity on a single machine. In their model, the starting time of the maintenance activity (non-availability intervals) is not fixed and the duration is a linear increasing function of its starting time. Some (pseudo-)polynomial-time algorithms are presented for some different objective functions. However, to the best of our knowledge, no article considered “scheduling with rejection” and “operator non-availability intervals” simultaneously. In this paper, we are the first to consider scheduling with rejection and an operator non-availability interval. 2. Problem Formulation The single-machine scheduling with rejection and an operator non-availability interval can be stated formally as follows. There are n jobs J 1 , J 2 , · · · , J n and a single machine. Each job J j is available at time 0 and has a processing time p j , a weight w j and a rejection cost e j . Each job J j is either rejected and the rejection cost e j has to be paid, or accepted and then processed non-preemptively on the machine. There is an operator non-availability interval ( a , b ) on the machine, where we assume that 0 < a < b This implies that, in any feasible schedule π , no accepted job J j can be started or be completed in the interval ( a , b ) . However, a crossover job is allowed such that it can start before this interval and complete after this interval. Without loss of generality, we assume that all the parameters a , b , p j , w j and e j are non-negative integers. Let A and R be the sets of accepted jobs and rejected jobs, respectively. Denote by C max = max { C j : J j ∈ A } , ∑ w j C j = ∑ J j ∈ A w j C j and e ( R ) = ∑ J j ∈ R e j the makespan of accepted jobs, the total weighted completion time of accepted jobs and the total rejection cost of rejected jobs, respectively. Our task is to find a feasible schedule such that C max + e ( R ) or ∑ w j C j + e ( R ) is minimized. By using the notation for scheduling problems, the corresponding problems are denoted by 1 | ONA ( a , b ) , rej | C max + e ( R ) and 1 | ONA ( a , b ) , rej | ∑ w j C j + e ( R ) , respectively. Two similar problems related to the above problems are 1 | MNA ( a , b ) , rej | C max + e ( R ) and 1 | MNA ( a , b ) , rej | ∑ w j C j + e ( R ) Zhong et al. [ 24 ] and Zhao and Tang [ 25 ] presented a pseudo-p