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On the complexity of assembly line balancing problems
Computers and Operations Research 108 (2019) 182–186
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Technical Note
On the complexity of assembly line balancing problems Eduardo Álvarez-Miranda a, Jordi Pereira b,∗ a b
Department of Industrial Engineering, Universidad de Talca, Curicó, Chile Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Viña del Mar, Chile
a r t i c l e
i n f o
Article history: Received 29 March 2019 Revised 3 April 2019 Accepted 4 April 2019 Available online 11 April 2019 Keywords: Line balancing Complexity Bin packing
a b s t r a c t Assembly line balancing is a family of combinatorial optimization problems that has been widely studied in the literature due to its simplicity and industrial applicability. Most line balancing problems are NPhard as they subsume the bin packing problem as a special case. Nevertheless, it is common in the line balancing literature to cite [A. Gutjahr and G. Nemhauser, An algorithm for the line balancing problem, Management Science 11 (1964) 308–315] in order to assess the computational complexity of the problem. Such an assessment is not correct since the work of Gutjahr and Nemhauser predates the concept of NPhardness. This work points at over 50 publications since 1995 with the aforesaid error.
1. Introduction Assembly line balancing is a classical problem that has been widely studied in the literature. The problem seeks an efficient assignment of tasks to workstations in an assembly line subject to operational constraints. The most basic formulation is known as type-1 simple assembly line balancing problem (SALBP-1) and tries to maximize efficiency by minimizing the number of required workstations under cumulative, knapsack-like, constraints that limit the total amount of work performed in every station and under precedence constraints among tasks that ensure that some tasks are performed before others, i.e. if some task precedes another task, the first task must be assigned to the same workstation than the second task or to a succeeding one. It is common for some papers dealing with line balancing to point at the complexity status of the problem without a proof or a reference. Some others refer to Gutjahr and Nemhauser (1964) for a proof, yet their work predates the development of computational complexity theory. For the SALBP-1 the NP-hard status is easily verified as one can reduce the bin packing problem reduces to it. Note that in Gutjahr and Nemhauser (1964) the authors provide a shortest path (i.e. dynamic programming) method to solve the SALBP-1 without covering the complexity of the problem. We believe that the error arises from the first sentences of their work, which read: “There is a class of combinatorial optimization problems which has defied solution by efficient algorithms. The travel-
∗ Corresponding author at: Av. Pedro Hurtado 750, Office A-215, Viña del Mar, Chile. E-mail address: [email protected] (J. Pereira).
https://doi.org/10.1016/j.cor.2019.04.005 0305-0548/© 2019 Elsevier Ltd. All rights reserved.
© 2019 Elsevier Ltd. All rights reserved.
ling salesman problem, job shop scheduling and line balancing all fall into this class” (op.cit. p. 308). While citing (Gutjahr and Nemhauser, 1964) to justify the NPhardness of a problem does not affect the validity of the main results of many of the citing works, the statement is still misleading and heads to false assumptions on what makes a problem NP-hard. This work investigates the issue by reviewing publications that cite (Gutjahr and Nemhauser, 1964) since 1995, and lists over 50 references where such a claim is made. We also briefly discuss the available complexity results for assembly line balancing problems. 2. Review and analysis of citing documents In order to highlight how this misunderstanding has percolated throughout the line balancing literature to become part of its folklore, the list of works that cite (Gutjahr and Nemhauser, 1964) according to its digital object identifier (doi: 10.1287/mnsc.11.2.308) was downloaded. According to this list, there were 152 documents citing (Gutjahr and Nemhauser, 1964) as of January 7, 2019. Among these references, we reviewed all works from 1995 onwards that were published in journals indexed in the InCites Journal Citation Reports (JCR). We chose to limit the search to publications from 1995 onwards as well as to journals indexed in the JCR publications (1) to access a copy of every document (older documents as well as those published in some books and conference proceedings were difficult to track), (2) to remove articles written in other languages than English, and (3) to consider only peerreviewed publications (we were not sure about the peer-reviewing requisite of every conference or journal). While we acknowledge that the selection is arbitrary, we think that it should not introduce any significant bias into the conclusions.
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Table 1 Publications grouped according to year of publication. Period
Number
Publication
1995 to 20 0 0 2001 to 2005 2006 to 2010
5 2 9
2011 to 2015
18
2016 onwards
23
Ben-Arieh (1995); Gen et al. (1996); Kim et al. (1998); Lucertini et al. (1998); Ponnambalam et al. (1999) Bukchin et al. (2002); Khan and Day (2002) Andrés et al. (2008); Baykasoglu and Dereli (2008); Gao et al. (2009); Guo et al. (2009, 2008); Lapierre et al. (2006); Miralles et al. (2008); Özcan et al. (2010); Özcan and Toklu (2009) Alavidoost et al. (2015); Fathi and Ghobakhloo (2014); Hamta et al. (2011, 2013); Hosseini and Tavakkoli-Moghaddam (2013); Manavizadeh et al. (2013); Mosadegh et al. (2012a,b); Nilakantan et al. (2015a); Özbakıir and Tapkan (2011); Rabbani et al. (2015); Rong et al. (2011); Seyed-Alagheband et al. (2011); Wei and Chao (2011); Yang et al. (2013); Yoosefelahi et al. (2012); Zheng et al. (2013) Alavidoost et al. (2017); Aydog˘ an et al. (2016); Babazadeh et al. (2018); Babazadeh and Javadian (2018); Delice et al. (2016, 2017a, 2018, 2017b); Lai et al. (2019, 2016); Li et al. (2016); Nilakantan and Ponnambalam (2016); Nilakantan et al. (2016); Özcan (2018); Roshani and Ghazi Nezami (2017); Roshani and Giglio (2017); S¸ ahin and Kellegöz (2017); Tapkan et al. (2016); Tóth et al. (2018); Triki et al. (2016, 2017); Zhou and Wu (2018); Zhou and Kang (2018)
Table 2 Publications grouped according to journal. Journals are grouped according to the number of publications. Number
Journal name
8 6 5 4 3
Computers & Industrial Engineering The international Journal of Advanced Manufacturing Technology International Journal of Production Research Journal of Intelligent Manufacturing Applied Soft Computing, Assembly Automation, Computers & Operations Research, European Journal of Operational Research and International Journal of Computer Integrated Manufacturing Expert Systems with Applications and International Journal of Production Economics 4OR, Advances in Mechanical Engineering, Applied Mathematical Modelling, Computer Integrated Manufacturing Systems, Discrete Applied Mathematics, Engineering Computations, Engineering Optimization, IEEE Transactions on Systems, Man, and Cybernetics, Part C, Journal of Cleaner Production, Journal of ¯ a¯ Manufacturing Systems, Memetic Computing, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Sadhan and Soft Computing
2 1
After filtering, the list of references was reduced to 92 references. One of them, Salehi et al. (2018) includes the cite in its reference list but never cites it within the text. Among the remaining 91 papers, 35 papers cite (Gutjahr and Nemhauser, 1964) •
•
•
•
•
as one of the pioneering works in the area (2 references, McGovern and Gupta, 2007; 2015), as a general line balancing reference (4 references, Amen, 2006; Kimms, 20 0 0; Naderi et al., 2018; Tsujimura et al., 1995) for the instances introduced in this work (1 reference, Brown and Sumichrast, 2005), for its use of the proposed mathematical formulation (1 reference, Yeh and Kao, 2009), for its resolution method (26 references, Altekin and Akkan, 2012; Battaïa and Dolgui, 2013; Bautista and Pereira, 2009; 2011; Benzer et al., 2007; Dolgui et al., 20 06; 20 08a; 2012; 20 08b; 20 09; Erel and Gökçen, 1999; Gökçen et al., 2005; Gökçen and Erel, 1998; Guerriero and Miltenburg, 2003; Hezer and Kara, 2015; Kazemi et al., 2011; Maenhout and Vanhoucke, 2016; Nicosia et al., 2002; Nilakantan et al., 2015b; Rekiek et al., 20 02; 20 0 0; Rosenblatt and Lee, 1996; Sarin et al., 1999; Scholl and Becker, 2006; Sivasankaran and Shahabudeen, 2014; Ug˘ urdag˘ et al., 1997; Van Hop, 2006).
That leaves us with 56 papers that cite (Gutjahr and Nemhauser, 1964) with regard to computational complexity issues (namely, Alavidoost et al., 2015; 2017; Andrés et al., 2008; Aydog˘ an et al., 2016; Babazadeh et al., 2018; Babazadeh and Javadian, 2018; Baykasoglu and Dereli, 2008; Ben-Arieh, 1995; Bukchin et al., 2002; Delice et al., 2016; 2017a; 2018; 2017b; Fathi and Ghobakhloo, 2014; Gao et al., 2009; Gen et al., 1996; Guo et al., 20 09; 20 08; Hamta et al., 2011; 2013; Hosseini and TavakkoliMoghaddam, 2013; Khan and Day, 2002; Kim et al., 1998; Lai et al., 2019; 2016; Lapierre et al., 2006; Li et al., 2016; Lucertini et al., 1998; Manavizadeh et al., 2013; Miralles et al., 2008; Mosadegh et al., 2012a; 2012b; Nilakantan et al., 2015a; Nilakantan and Ponnambalam, 2016; Nilakantan et al., 2016; Özbakıir and Tapkan, 2011; Özcan, 2018; Özcan et al., 2010; Özcan and Toklu,
2009; Ponnambalam et al., 1999; Rabbani et al., 2015; Rong et al., 2011; Roshani and Ghazi Nezami, 2017; Roshani and Giglio, 2017; S¸ ahin and Kellegöz, 2017; Seyed-Alagheband et al., 2011; Tapkan et al., 2016; Tóth et al., 2018; Triki et al., 2016; 2017; Wei and Chao, 2011; Yang et al., 2013; Yoosefelahi et al., 2012; Zheng et al., 2013; Zhou and Wu, 2018; Zhou and Kang, 2018). In order to fully comprehend how this misconception has permeated through the area, these works are grouped according to the year of publication, Table 1, and journal of publication, Table 2. When the year of publication is considered, see Table 1, a significant, positive trend is observed. The trend should be attributed both to the increased interest in line balancing and to the growing number of scientific publications. Moreover, given the large number of journals from different disciplines (25 different journals as shown in Table 2), it is clear that the misconception transcends a specific area or group of researchers. Also note that while most of the references belong to production, manufacturing and industrial engineering topics, there is also a significant number of papers in applied mathematics, artificial intelligence and operations research journals. Please note that by no means we assert that the full list of 56 papers are on the same page. The statements that lead to cite (Gutjahr and Nemhauser, 1964) vary greatly. We give some examples selected within the papers published in 2017 and onwards. •
•
•
The cite properly identifies the source of complexity but gives a misleading reference (i.e. “The deterministic counterpart of the problem SALBP-E is NP-hard (Gutjahr & Nemhauser, 1964; Wee & Magazine, 1982) since the NP-hard bin-packing problem is a special case of the problem SALBP-E, where the precedence digraph G = (V, A) has no arcs.”, Lai et al., 2019, p. 468) Complexity is given as a fact and the reference is used to support the statement (i.e. “In terms of mathematical complexity, ALBP is NP-hard in nature (Gutjahr and Nemhauser 1964)”. Özcan, 2018, p. 109), A wrong justification for the complexity is given (“For driving an optimum solution, considering ALBP with J number of tasks and P number of constraints, it is needed for searching j!/2P of
184
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solutions. Therefore, assembly line problems can be classified as NP-hard problem (Gutjahr & Nemhauser, 1964; Ajenblit & Wainwright, 1998).”, Babazadeh et al., 2018, p. 191). Or they state that a proof is given in the cited paper (“The ALBPs were proven to be NP-hard by Gutjahr and Nemhauser (1964) and Ajenblit and Wainwright (1998).”, Alavidoost et al., 2017, p. 314).
While some of these errors are clearly more severe than others, a reader may inadvertently reach a wrong conclusion after reading any of these excerpts. We would also like to mention that some of these 56 papers give some additional references to state the complexity of the problem. For instance, Wee and Magazine (1982) is given as a secondary reference in Lai et al. (2019, 2016), yet Wee and Magazine (1982) does not deal with the complexity by itself but with the use of a BPP heuristic for line balancing. Other authors (Alavidoost et al., 2015; Babazadeh et al., 2018; Fathi and Ghobakhloo, 2014) supplementary refer to Ajenblit and Wainwright (1998), although their work does not cover complexity by itself and, in turn, cites (Karp, 1972) for the matter. Again, Karp (1972) does not cover line balancing (nor bin packing) by itself. Still, these references and their content hint that active researchers on line balancing (correctly) considered these problems to be NP-hard without proof. Among the possible sources for such a common citing error, we give two. First, as these problems are usually NP-hard, authors may have taken as correct the references of previous authors without reading the original source to check its applicability and correctness. Second, line balancing is tackled in areas in which authors and reviewers may not be knowledgable on complexity theory and what makes a problem to be NP-hard. As in the first case, the large amount of cites to Gutjahr and Nemhauser (1964) may be sufficient to rely on this cite to justify the complexity status. 3. On the complexity of line balancing Given the negative results given in Section 2, this section gives some alternative solutions to properly state the source of complexity of line balancing problems. The simplest method to assess the NP-hardness status of a problem is to reduce problem BIN PACKING (problem SR1 Garey and Johnson, 1979, p. 226) to the feasibility version of the line balancing problem under study (i.e. for the SALBP, the feasibility version is the SALBP-F in which we are given both the cycle time and the number of stations and the question is whether we can find an assignment of tasks to stations with the given number of stations while fulfilling the cycle time constraints). A more elaborate explanation in this direction is given in Scholl (1999, pp. 34–42). Note that Scholl (1999) also discusses an altogether different topic related to the metrics that one can use to assess the expected difficulty of a certain instance. Both topics should not be confused. An alternative description including additional approximation results is given in Queyranne (1985). The approximation results make use of instances with precedence constraints in order to find tighter bounds than those available for BPP heuristics. We would like to conclude this note by offering alternatives to bin packing-based proofs. For instance, there are some line balancing problems under heterogeneity conditions (e.g. Pereira et al., 2018) in which it is possible to derive a proof based on the precedence constraints and the relationship of the problem with the partially ordered knapsack (POK) (Johnson and Niemi, 1983). These results shed light on which special characteristics of a certain problem lead to its complexity class. Consequently, we strongly recommend researchers to check the literature on the complexity of basic problems that their line balancing problem subsumes.
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Technical Note
On the complexity of assembly line balancing problems Eduardo Álvarez-Miranda a, Jordi Pereira b,∗ a b
Department of Industrial Engineering, Universidad de Talca, Curicó, Chile Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Viña del Mar, Chile
a r t i c l e
i n f o
Article history: Received 29 March 2019 Revised 3 April 2019 Accepted 4 April 2019 Available online 11 April 2019 Keywords: Line balancing Complexity Bin packing
a b s t r a c t Assembly line balancing is a family of combinatorial optimization problems that has been widely studied in the literature due to its simplicity and industrial applicability. Most line balancing problems are NPhard as they subsume the bin packing problem as a special case. Nevertheless, it is common in the line balancing literature to cite [A. Gutjahr and G. Nemhauser, An algorithm for the line balancing problem, Management Science 11 (1964) 308–315] in order to assess the computational complexity of the problem. Such an assessment is not correct since the work of Gutjahr and Nemhauser predates the concept of NPhardness. This work points at over 50 publications since 1995 with the aforesaid error.
1. Introduction Assembly line balancing is a classical problem that has been widely studied in the literature. The problem seeks an efficient assignment of tasks to workstations in an assembly line subject to operational constraints. The most basic formulation is known as type-1 simple assembly line balancing problem (SALBP-1) and tries to maximize efficiency by minimizing the number of required workstations under cumulative, knapsack-like, constraints that limit the total amount of work performed in every station and under precedence constraints among tasks that ensure that some tasks are performed before others, i.e. if some task precedes another task, the first task must be assigned to the same workstation than the second task or to a succeeding one. It is common for some papers dealing with line balancing to point at the complexity status of the problem without a proof or a reference. Some others refer to Gutjahr and Nemhauser (1964) for a proof, yet their work predates the development of computational complexity theory. For the SALBP-1 the NP-hard status is easily verified as one can reduce the bin packing problem reduces to it. Note that in Gutjahr and Nemhauser (1964) the authors provide a shortest path (i.e. dynamic programming) method to solve the SALBP-1 without covering the complexity of the problem. We believe that the error arises from the first sentences of their work, which read: “There is a class of combinatorial optimization problems which has defied solution by efficient algorithms. The travel-
∗ Corresponding author at: Av. Pedro Hurtado 750, Office A-215, Viña del Mar, Chile. E-mail address: [email protected] (J. Pereira).
https://doi.org/10.1016/j.cor.2019.04.005 0305-0548/© 2019 Elsevier Ltd. All rights reserved.
© 2019 Elsevier Ltd. All rights reserved.
ling salesman problem, job shop scheduling and line balancing all fall into this class” (op.cit. p. 308). While citing (Gutjahr and Nemhauser, 1964) to justify the NPhardness of a problem does not affect the validity of the main results of many of the citing works, the statement is still misleading and heads to false assumptions on what makes a problem NP-hard. This work investigates the issue by reviewing publications that cite (Gutjahr and Nemhauser, 1964) since 1995, and lists over 50 references where such a claim is made. We also briefly discuss the available complexity results for assembly line balancing problems. 2. Review and analysis of citing documents In order to highlight how this misunderstanding has percolated throughout the line balancing literature to become part of its folklore, the list of works that cite (Gutjahr and Nemhauser, 1964) according to its digital object identifier (doi: 10.1287/mnsc.11.2.308) was downloaded. According to this list, there were 152 documents citing (Gutjahr and Nemhauser, 1964) as of January 7, 2019. Among these references, we reviewed all works from 1995 onwards that were published in journals indexed in the InCites Journal Citation Reports (JCR). We chose to limit the search to publications from 1995 onwards as well as to journals indexed in the JCR publications (1) to access a copy of every document (older documents as well as those published in some books and conference proceedings were difficult to track), (2) to remove articles written in other languages than English, and (3) to consider only peerreviewed publications (we were not sure about the peer-reviewing requisite of every conference or journal). While we acknowledge that the selection is arbitrary, we think that it should not introduce any significant bias into the conclusions.
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183
Table 1 Publications grouped according to year of publication. Period
Number
Publication
1995 to 20 0 0 2001 to 2005 2006 to 2010
5 2 9
2011 to 2015
18
2016 onwards
23
Ben-Arieh (1995); Gen et al. (1996); Kim et al. (1998); Lucertini et al. (1998); Ponnambalam et al. (1999) Bukchin et al. (2002); Khan and Day (2002) Andrés et al. (2008); Baykasoglu and Dereli (2008); Gao et al. (2009); Guo et al. (2009, 2008); Lapierre et al. (2006); Miralles et al. (2008); Özcan et al. (2010); Özcan and Toklu (2009) Alavidoost et al. (2015); Fathi and Ghobakhloo (2014); Hamta et al. (2011, 2013); Hosseini and Tavakkoli-Moghaddam (2013); Manavizadeh et al. (2013); Mosadegh et al. (2012a,b); Nilakantan et al. (2015a); Özbakıir and Tapkan (2011); Rabbani et al. (2015); Rong et al. (2011); Seyed-Alagheband et al. (2011); Wei and Chao (2011); Yang et al. (2013); Yoosefelahi et al. (2012); Zheng et al. (2013) Alavidoost et al. (2017); Aydog˘ an et al. (2016); Babazadeh et al. (2018); Babazadeh and Javadian (2018); Delice et al. (2016, 2017a, 2018, 2017b); Lai et al. (2019, 2016); Li et al. (2016); Nilakantan and Ponnambalam (2016); Nilakantan et al. (2016); Özcan (2018); Roshani and Ghazi Nezami (2017); Roshani and Giglio (2017); S¸ ahin and Kellegöz (2017); Tapkan et al. (2016); Tóth et al. (2018); Triki et al. (2016, 2017); Zhou and Wu (2018); Zhou and Kang (2018)
Table 2 Publications grouped according to journal. Journals are grouped according to the number of publications. Number
Journal name
8 6 5 4 3
Computers & Industrial Engineering The international Journal of Advanced Manufacturing Technology International Journal of Production Research Journal of Intelligent Manufacturing Applied Soft Computing, Assembly Automation, Computers & Operations Research, European Journal of Operational Research and International Journal of Computer Integrated Manufacturing Expert Systems with Applications and International Journal of Production Economics 4OR, Advances in Mechanical Engineering, Applied Mathematical Modelling, Computer Integrated Manufacturing Systems, Discrete Applied Mathematics, Engineering Computations, Engineering Optimization, IEEE Transactions on Systems, Man, and Cybernetics, Part C, Journal of Cleaner Production, Journal of ¯ a¯ Manufacturing Systems, Memetic Computing, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Sadhan and Soft Computing
2 1
After filtering, the list of references was reduced to 92 references. One of them, Salehi et al. (2018) includes the cite in its reference list but never cites it within the text. Among the remaining 91 papers, 35 papers cite (Gutjahr and Nemhauser, 1964) •
•
•
•
•
as one of the pioneering works in the area (2 references, McGovern and Gupta, 2007; 2015), as a general line balancing reference (4 references, Amen, 2006; Kimms, 20 0 0; Naderi et al., 2018; Tsujimura et al., 1995) for the instances introduced in this work (1 reference, Brown and Sumichrast, 2005), for its use of the proposed mathematical formulation (1 reference, Yeh and Kao, 2009), for its resolution method (26 references, Altekin and Akkan, 2012; Battaïa and Dolgui, 2013; Bautista and Pereira, 2009; 2011; Benzer et al., 2007; Dolgui et al., 20 06; 20 08a; 2012; 20 08b; 20 09; Erel and Gökçen, 1999; Gökçen et al., 2005; Gökçen and Erel, 1998; Guerriero and Miltenburg, 2003; Hezer and Kara, 2015; Kazemi et al., 2011; Maenhout and Vanhoucke, 2016; Nicosia et al., 2002; Nilakantan et al., 2015b; Rekiek et al., 20 02; 20 0 0; Rosenblatt and Lee, 1996; Sarin et al., 1999; Scholl and Becker, 2006; Sivasankaran and Shahabudeen, 2014; Ug˘ urdag˘ et al., 1997; Van Hop, 2006).
That leaves us with 56 papers that cite (Gutjahr and Nemhauser, 1964) with regard to computational complexity issues (namely, Alavidoost et al., 2015; 2017; Andrés et al., 2008; Aydog˘ an et al., 2016; Babazadeh et al., 2018; Babazadeh and Javadian, 2018; Baykasoglu and Dereli, 2008; Ben-Arieh, 1995; Bukchin et al., 2002; Delice et al., 2016; 2017a; 2018; 2017b; Fathi and Ghobakhloo, 2014; Gao et al., 2009; Gen et al., 1996; Guo et al., 20 09; 20 08; Hamta et al., 2011; 2013; Hosseini and TavakkoliMoghaddam, 2013; Khan and Day, 2002; Kim et al., 1998; Lai et al., 2019; 2016; Lapierre et al., 2006; Li et al., 2016; Lucertini et al., 1998; Manavizadeh et al., 2013; Miralles et al., 2008; Mosadegh et al., 2012a; 2012b; Nilakantan et al., 2015a; Nilakantan and Ponnambalam, 2016; Nilakantan et al., 2016; Özbakıir and Tapkan, 2011; Özcan, 2018; Özcan et al., 2010; Özcan and Toklu,
2009; Ponnambalam et al., 1999; Rabbani et al., 2015; Rong et al., 2011; Roshani and Ghazi Nezami, 2017; Roshani and Giglio, 2017; S¸ ahin and Kellegöz, 2017; Seyed-Alagheband et al., 2011; Tapkan et al., 2016; Tóth et al., 2018; Triki et al., 2016; 2017; Wei and Chao, 2011; Yang et al., 2013; Yoosefelahi et al., 2012; Zheng et al., 2013; Zhou and Wu, 2018; Zhou and Kang, 2018). In order to fully comprehend how this misconception has permeated through the area, these works are grouped according to the year of publication, Table 1, and journal of publication, Table 2. When the year of publication is considered, see Table 1, a significant, positive trend is observed. The trend should be attributed both to the increased interest in line balancing and to the growing number of scientific publications. Moreover, given the large number of journals from different disciplines (25 different journals as shown in Table 2), it is clear that the misconception transcends a specific area or group of researchers. Also note that while most of the references belong to production, manufacturing and industrial engineering topics, there is also a significant number of papers in applied mathematics, artificial intelligence and operations research journals. Please note that by no means we assert that the full list of 56 papers are on the same page. The statements that lead to cite (Gutjahr and Nemhauser, 1964) vary greatly. We give some examples selected within the papers published in 2017 and onwards. •
•
•
The cite properly identifies the source of complexity but gives a misleading reference (i.e. “The deterministic counterpart of the problem SALBP-E is NP-hard (Gutjahr & Nemhauser, 1964; Wee & Magazine, 1982) since the NP-hard bin-packing problem is a special case of the problem SALBP-E, where the precedence digraph G = (V, A) has no arcs.”, Lai et al., 2019, p. 468) Complexity is given as a fact and the reference is used to support the statement (i.e. “In terms of mathematical complexity, ALBP is NP-hard in nature (Gutjahr and Nemhauser 1964)”. Özcan, 2018, p. 109), A wrong justification for the complexity is given (“For driving an optimum solution, considering ALBP with J number of tasks and P number of constraints, it is needed for searching j!/2P of
184
•
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solutions. Therefore, assembly line problems can be classified as NP-hard problem (Gutjahr & Nemhauser, 1964; Ajenblit & Wainwright, 1998).”, Babazadeh et al., 2018, p. 191). Or they state that a proof is given in the cited paper (“The ALBPs were proven to be NP-hard by Gutjahr and Nemhauser (1964) and Ajenblit and Wainwright (1998).”, Alavidoost et al., 2017, p. 314).
While some of these errors are clearly more severe than others, a reader may inadvertently reach a wrong conclusion after reading any of these excerpts. We would also like to mention that some of these 56 papers give some additional references to state the complexity of the problem. For instance, Wee and Magazine (1982) is given as a secondary reference in Lai et al. (2019, 2016), yet Wee and Magazine (1982) does not deal with the complexity by itself but with the use of a BPP heuristic for line balancing. Other authors (Alavidoost et al., 2015; Babazadeh et al., 2018; Fathi and Ghobakhloo, 2014) supplementary refer to Ajenblit and Wainwright (1998), although their work does not cover complexity by itself and, in turn, cites (Karp, 1972) for the matter. Again, Karp (1972) does not cover line balancing (nor bin packing) by itself. Still, these references and their content hint that active researchers on line balancing (correctly) considered these problems to be NP-hard without proof. Among the possible sources for such a common citing error, we give two. First, as these problems are usually NP-hard, authors may have taken as correct the references of previous authors without reading the original source to check its applicability and correctness. Second, line balancing is tackled in areas in which authors and reviewers may not be knowledgable on complexity theory and what makes a problem to be NP-hard. As in the first case, the large amount of cites to Gutjahr and Nemhauser (1964) may be sufficient to rely on this cite to justify the complexity status. 3. On the complexity of line balancing Given the negative results given in Section 2, this section gives some alternative solutions to properly state the source of complexity of line balancing problems. The simplest method to assess the NP-hardness status of a problem is to reduce problem BIN PACKING (problem SR1 Garey and Johnson, 1979, p. 226) to the feasibility version of the line balancing problem under study (i.e. for the SALBP, the feasibility version is the SALBP-F in which we are given both the cycle time and the number of stations and the question is whether we can find an assignment of tasks to stations with the given number of stations while fulfilling the cycle time constraints). A more elaborate explanation in this direction is given in Scholl (1999, pp. 34–42). Note that Scholl (1999) also discusses an altogether different topic related to the metrics that one can use to assess the expected difficulty of a certain instance. Both topics should not be confused. An alternative description including additional approximation results is given in Queyranne (1985). The approximation results make use of instances with precedence constraints in order to find tighter bounds than those available for BPP heuristics. We would like to conclude this note by offering alternatives to bin packing-based proofs. For instance, there are some line balancing problems under heterogeneity conditions (e.g. Pereira et al., 2018) in which it is possible to derive a proof based on the precedence constraints and the relationship of the problem with the partially ordered knapsack (POK) (Johnson and Niemi, 1983). These results shed light on which special characteristics of a certain problem lead to its complexity class. Consequently, we strongly recommend researchers to check the literature on the complexity of basic problems that their line balancing problem subsumes.
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