International journal of computer integrated manufacturing , tập 24, số 1, 2011

International Journal of Computer Integrated Manufacturing Vol 24, No 1, January 2011, 1–31 Computer-aided process planning – A critical review of recent developments and future trends Xun Xua*, Lihui Wangb and Stephen T Newmanc a Department of Mechanical Engineering, School of Engineering, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand; bVirtual Systems Research Centre, University of Sko¨vde, P.O Box 408, 541 28 Sko¨vde, Sweden; cDepartment of Mechanical Engineering, University of Bath, Bath, BA2 7AY, United Kingdom (Received 28 January 2010; final version received 22 August 2010) For the past three decades, computer-aided process planning (CAPP) has attracted a large amount of research interest A huge volume of literature has been published on this subject Today, CAPP research faces new challenges owing to the dynamic markets and business globalisation Thus, there is an urgent need to ascertain the current status and identify future trends of CAPP Covering articles published on the subjects of CAPP in the past 10 years or so, this article aims to provide an up-to-date review of the CAPP research works, a critical analysis of journals that publish CAPP research works, and an understanding of the future direction in the field First, general information is provided on CAPP The past reviews are summarised Discussions about the recent CAPP research are presented in a number of categories, i.e feature-based technologies, knowledge-based systems, artificial neural networks, genetic algorithms, fuzzy set theory and fuzzy logic, Petri nets, agent-based technology, Internet-based technology, STEP-compliant CAPP and other emerging technologies Research on some specific aspects of CAPP is also provided Discussions and analysis of the methods are then presented based on the data gathered from the Elsevier’s Scopus abstract and citation database The concepts of ‘Subject Strength’ of a journal and ‘technology impact factor’ are introduced and used for discussions based on the publication data The former is used to gauge the level of focus of a journal on a particular research subject/domain, whereas the latter is used to assess the level of impact of a particular technology, in terms of citation counts Finally, a discussion on the future development is presented Keywords: CAPP; machining process; features; planning; operation Introduction Design and manufacturing is a critical phase of product development process The pivotal link between design and manufacturing is process planning Process planning deals with the selection of necessary manufacturing processes and determination of their sequences to ‘transform’ a designer’s ideas (namely the designed part) into a physical component economically and competitively In the domain of machining processes, which is the focus of this article, the major process planning activities may include interpretation of design data, selection of machining operations, machine tools, cutting tools, datum and fixture as well as calculation of cost and production time No doubt, this is a complex engineering problem The traditional approach to solving process-planning problems in a manufacturing company is to leave it in the hands of manufacturing experts These domain experts use their experience and knowledge to generate instructions for the manufacture of products based on design specifications and available facilities Different process planners often come up *Corresponding author Email: x.xu@auckland.ac.nz ISSN 0951-192X print/ISSN 1362-3052 online Ó 2011 Taylor & Francis DOI: 10.1080/0951192X.2010.518632 http://www.informaworld.com with different plans for the same problem, adding inconsistency to the already complicated problem Since Niebel (1965) first discussed the use of computers to assist process-planning tasks, more than 40 years have elapsed In comparison with computer-aided design (CAD) and computer-aided manufacturing (CAM), computer-aided process planning (CAPP) has been lagging behind in terms of providing practical, matured, professional and commercialised solutions to the manufacturing industry This is though not attributed to the lack of research effort On the contrary, there is a prolonged and prolific history of research and publications In general, there are two approaches in CAPP, variant and generative Variant approach follows the principle that similar parts require similar plans It requires a human operator to classify a part, input part information, retrieve a similar process plan from a database and make the necessary modifications This approach is suited to enterprises involving stable manufacturing processes and manufactured products that vary little The advantage of this approach is the ease of maintenance, but the shortcoming is the lack of X Xu et al an on-time calculation of manufacturing process, and quality of the process plan still depends on the knowledge of a process planner Manual input is still required to establish the mass data of manufacturing processes In a generative approach, process plans are generated with little human intervention It generates new process plans by means of decision logic and process knowledge Earlier developments of this approach adopted decision-making to determine the manufacturing process and use group technology code or special descriptive languages to define workpieces Later development (since the mid 1980s) focused on the use of features to define product models The bottleneck of this approach is the difficulty in obtaining useable features, and the difficulty in representing, managing and utilising human expertise This is the main reason that both feature technology and knowledge-based techniques have been heavily researched in association with CAPP Some of the desirable characteristics of an effective CAPP system are to Be interconnected with up- and down-stream activities, i.e design and manufacturing, in such a way that a CAPP system can take design data as it is and generate output that can be fed into a CAM and later a CNC system; Be extendible, adaptable and customisable for individual enterprises and to new processes; Provide effective knowledge acquisition, representation and manipulation mechanisms as well as the means to check the completeness and consistency of that knowledge; Involve users in some parts of the decisionmaking process, provide heuristics as needed and supplement the system’s abilities; and Come with a user-friendly interface in support of effective interaction by facilitating inputs, producing outputs and reports, and displaying the results graphically Because the concept of CAPP was suggested, there have been numerous research publications as well as technical surveys It is also evident that the trend of CAPP research has also undergone drastic changes To the authors’ knowledge, there has not been a comprehensive review of the CAPP research for machining since 1998 Therefore, the aim of this article is to provide a comprehensive review on CAPP technologies developed for machining since the late 1990s but mostly after 2000 It is also to be pointed out the ontology and architecture matters are not included in this review There are seven sections in the article Section provides an excerpt for the past review publications on CAPP Section is the main section describing various technologies developed or implemented These include the feature-based technologies, knowledge-based systems, neural network, genetic algorithm (GA), fuzzy set theory/logic, Petri net (PN), agent-based technologies, Internet-enabled CAPP, STEP-compliant CAPP as well as some emerging technologies Section comments on the research work on some specific areas of CAPP, e.g tool section, setup planning, operation selection and sequencing, decision models and integrated process planning and knowledge representation Discussions and future trends in CAPP are presented in Sections and 6, respectively Section concludes the article Previous reviews in CAPP The idea of developing process plans using computers was first presented in 1965 by Neibel (1965), and the first CAPP system was developed in 1976 under the sponsorship of Computer Aided Manufacturing International (CAM-I) (Cay 1997) Since then, there has been a plethora of research work in the area of CAPP; so too are a significant number of surveys This section provides a snap-shot of the past surveys in the field One of the first review articles was written by Steudel in 1984 The author discussed the approaches and strategies for structuring manufacturing methods and data for the development of a generative type, automated planning system An overview of the information needed for such a task was provided This article also outlined the anticipated development of a ‘common language of geometry’ to relate a part to the process, and development of CAD/CAM systems that incorporated CAPP In the following year, Eversheim and Schulz (1985) presented a survey based on questionnaires sent to the CAPP developers and endusers in Europe, North America and several Asian countries during 1983 and 1984 From the survey, it was apparent that the CAPP development and applications were still relatively new In 1988, Ham and Lu (1988) presented an evaluation of the status of CAPP at the time, and correctly stated that the direction of future research lies in the integration between design and manufacturing, and the use of artificial intelligence (AI) technologies Following on from that is probably the most significant survey of the time, by Alting and Zhang in 1989 In this survey, the authors reviewed over 200 published works and featured 14 well-known CAPP systems One hundred and fifty-six existing systems were listed in a table format The survey indicated the difficulty in integrating CAD with CAPP due to a lack of common methods to represent geometric entities The authors also suggested the interfacing issues between CAPP and CAM and other computerised production systems International Journal of Computer Integrated Manufacturing such as NC tool path, MRP, production simulation, etc They recognised AI technologies as a crucial technology in the development of an effective process planning system In addition, the importance of learning systems was pointed out, and an ideal approach identified to integrate all the information involved in production of a part into a single database In the same year, Gouda and Taraman (1989) published a survey of the 128 CAPP systems at the time Four types of CAPP systems were highlighted, variant, semi-generative, generative and expert process-planning systems The next survey was compiled by the CAPP Working Group of the CIRP in 1993 (ElMaraghy et al 1993) Aspects covered by this article include the major development thrust in CAPP, the industry perspectives of CAPP, evolving trends and challenges, and integration of design, CAPP and production planning Issues of quality and evolving standards were also addressed In the same year, Eversheim and Schneewind (1993) provided a broad but concise review of CAPP in real industrial environments It suggested that the future of CAPP development was an extension to assembly planning, function integration with NC programming, use of AI methods in decisionmaking, and use of database sharing for data integration with CAD In 1995, Kamrani et al presented an overview of the techniques and the role of process planning It also discussed the critical issues and the characteristics associated with evaluation and selection of a CAPP system These issues are the range of product support, classification / coding / graphic capabilities, work instruction creation, process planning approaches, time analysis capabilities, machining parameters, material and tooling databases, system requirements, cost, commercial availability and user friendliness, vendor qualification and support The next comprehensive review was written by Leung (1996), where he compiled and annotated about 200 publications in CAPP from 1989 to 1996 The author observed that solid modelling in CAPP systems was not as adequate as anticipated, hence the revitalisation of variant process planning systems Leung believed that it was logical that future process planning systems be built on intelligent system architecture with AI techniques Following on from Leung was the review by Cay and Chassapis (1997), which covers the research work on CAPP from 1990 to 1997 It gave an overview of manufacturing features and feature recognition techniques with CAPP research Cay maintained that a fully automated environment was not far from a reality, should effective integration of design and manufacturing systems be achieved The last general review on CAPP in the 1990s is by Marri et al (1998) This review covered the literature from 1989 to 1996 The advantages and disadvantages of these systems were discussed with the generative approach highlighted Aside from general CAPP reviews, there are also surveys in a more specific area, such as CAD and feature-based process planning (Shah 1991, Shah et al 1991), neural network-based process planning (Yue et al 2002), expert system-based process planning (Gupta and Ghosh 1989, Kiritsis 1995, Metaxiotis et al 2002a; Liao 2005), and virtual reality-based process planning (Peng et al 2000) More recently, Shen et al (2006a,b) presented a state-of-the-art survey on agent-based, distributed manufacturing process planning and scheduling They addressed the complexity of manufacturing process-planning and scheduling problems, and reviewed the literature in process planning, scheduling, and their integration, with a focus on agent-based approaches Zhang and Xie (2007) provided a review on agent technology for collaborative process planning Key issues in developing an agent-based process planning system were explored, including agent and system architecture, communication standards and protocols, and applications Current status of CAPP Centred on the CAPP technologies and systems, this section consists of 10 sub-sections, each representing a category to which the related technologies belong The categories are feature-based technologies, knowledgebased systems, artificial neural networks, GAs, fuzzy set theory and fuzzy logic, PNs, agent-based technology, Internet-based technology, STEP-compliant CAPP and other emerging technologies 3.1 Feature-based technologies Feature technology has been the central topic for CAD/CAM integrations for years, so has it been for CAPP This is because almost all CAPP systems function on the basis of features, or require features to be the input data There are two approaches to obtaining features: feature recognition and design by features (Shah 1991, Shah et al 1991) The feature recognition approach examines the topology and geometry of a part and determines the existence and definitions of features To achieve this, a geometric model of lower-level entities (lines, points, etc.) is converted into a feature-based model in terms of higher-level entities (holes, pockets, etc.) Feature recognition has been adapted in various approaches, such as rule-based approach, volume decomposition approach, expert system, and graph-based approach The design-by-feature approach builds a part from predefined features stored in a feature library X Xu et al Geometry of these features is defined but their dimensions are left as variables to be instantiated when a feature is used in the modelling process There are two distinct methodologies for the design-byfeature approach The first is destruction by machining features and the other is synthesis by design features Destruction by machining features method starts with the model of the raw stock from which a part is to be machined The design model is then generated by subtracting depression features corresponding to the material to be removed by machining operations from the stock Synthesis by the design features method is built by both adding and subtracting features In a recent survey by Babic et al (2008), three major feature recognition problems were identified, (i) extraction of geometric primitives from a CAD model; (ii) defining a suitable part representation for form feature identification; and (iii) feature pattern matching/recognition The review has a focus on the rulebased methods It is perhaps fair to say that with some exceptions, much of the focus of the feature recognition research has been on finding all possible features, leaving the task of manufacturability analysis to process planners Xu and Hinduja (1997, 1998) recognised this issue and have developed methods for recognising features specifically for machining operations such as roughing, semi-finishing and finishing operations Han and Han (1999) proposed to integrate feature recognition with process planning They used feature recognition for manufacturing and setup minimisation, feature dependence construction, and generation of an optimal feature-based machining sequence The system interacts with the tool database and does manufacturability analysis together with feature dependency construction In the same year, Khoshevis et al (1999) published an integrated process-planning system, using feature reasoning and space search-based optimisation The process-planning system consists of feature completion module, process selection module and process sequencing module They used graph-based methods and volume-based methods The features are recognised by an objectorientated feature finder The feature completion module creates a feature precedence network It also allows the process planning system to accept the input from some other feature recognizers Lee et al (2007) developed a projective feature recognition algorithm that outputs features that can be directly used for process planning Process planning is based on the topological sorting and breadth-first search of graphs A great deal of the research work in the area of machining feature recognition is limited to 2½ and axis milling features (Sridharan and Shah 2004, Ranjan et al 2005) In the work of Huang and YipHoi (2002), they suggested a methodology to extract user-specific features from generic features This is achieved by specifying patterns for these specific features High-level features are recognised and are more meaningful to process planning Sadaiah et al (2002) developed a generative CAPP system for prismatic parts The proposed system is divided into three modules The first module is concerned with feature extraction The second and third modules deal with planning the set-up, machine selection, cutting tool selection, cutting parameter selection, and generation of process plan sheet Woo et al (2005) developed a hybrid feature recognizer for machining process planning It is the integration of three distinct feature recognition methods, i.e graph matching, cellbased maximal volume decomposition, and negative feature decomposition Hou and Faddis (2006) investigated the integration of CAD/CAPP/CAM based on machining features In the system, machining features are utilised to carry machining geometry information from CAPP to CAM systems for the preparation of tool paths Hwang and Miller (1997) described a process-planning model using mixed-type reasoning designed for processing prismatic parts on CNC machine tools in a batch-manufacturing environment The mixed-type reasoning handles feature interactions by combining forward chaining for feature sequencing and backward chaining for the construction of a process plan, allowing the human problemsolving strategies to be decoupled from the tools for analysis and sorting algorithms Clearly, feature-based approaches have been widely adopted by both CAD and CAPP systems (Patil and Pande 2002) In the research by Markus et al (1997), a feature-based process planning system was developed, where planning of the framework is accomplished via retrieving and adapting previous part family related plans The developed method can generate feasible matching of different parts Another crucial issue in process planning is the sequence of machining processes Using STEP AP224 features (ISO 10303–22 1999), Gonzalez and Rosado (2004) defined an internal feature model for process planning This way, all the information is represented around the machining feature for process planning without the use of geometric entities In the past, operation sequencing received more attention and has been studied deeply in various aspects Wang et al (2006) presents a different approach as part of their distributed process planning (DPP) system As a two-layer hierarchy is considered to separate the generic data from those that are machine-specific in DPP, machining process sequencing is treated as machining feature sequencing within the context The advantage of their approach is that both manufacturing interactions and geometric interactions are handled during feature sequencing International Journal of Computer Integrated Manufacturing One of the major hurdles in the development of a comprehensive CAPP system is that each CAPP domain is quite unique in terms of the analytical models and knowledge bases it utilises As a result, the field of CAPP has become highly fragmented (Yuen et al 2003) A unifying theme seems to be needed to reverse this trend One way of achieving this goal is to adopt the paradigm of feature-orientation as the unifying theme Yuen et al (2003) created a generic CAPP support system (GCAPPSS) that can act as the ‘front end’ for all domain-specific CAPP systems The GCAPPSS invokes a set of algorithms that enable feature extraction, recognition, coding, classification and decomposition The output from this system enables a multi-layered hierarchical part representation that can facilitate interpretation of feature relations and the object itself 3.2 Knowledge-based systems The art of process planning relies heavily on the knowledge of experienced workers or domain experts Hence, it is a knowledge intensive exercise An expert system (also known as knowledge-based system) as Welband (1983) stated in his work, ‘is a program which has a wide base of knowledge in a restricted domain, and uses complex inferential reasoning to perform tasks which a human expert could do’ Undoubtedly, knowledgebased systems have found numerous applications in process planning An expert system usually consists of three main components: the knowledge base, the inference engine and the user interface Expert systems have advantages over traditional computer systems, since they organise knowledge in rules and control strategies which allow users to modify a program with ease, and they are able to organise knowledge in such a way that they can reason intelligently Thus, expert systems are able to deal with more complicated problems such as process planning Also, export systems can be designed so that they accumulate knowledge as time passes, in the form of separate facts, production rules, objects, etc The inference mechanism of an expert system makes it possible to perform operations on the knowledge base of analysed elements (Grabowik and Knosala 2003) Park (2003) discussed the knowledge capturing methodology in process planning To identify the knowledge elements, three sub-models were suggested: object model, functional model and dynamic model, based on which three knowledge elements for process planning were derived: facts (from the object model), constraints (from the functional model), and way of thinking and rules (from the dynamic model) In process planning, an organised relationship between design and manufacturing knowledge is important The knowledge should be structured in such a way that it allows for easier reasoning in the generation of a sound process plan In the research by Sormaz and Khoshnevis (1995), a knowledge representation scheme that recognised both geometric and feature-based representation of parts was proposed The developed system can connect feature and process knowledge with part geometry, and use an objectoriented approach for a detailed presentation of machining knowledge Jia et al (2003) also adopted the object-oriented technology to represent setup process information, process decision knowledge and decision procedure control knowledge In the same year, Grabowik and Knosala (2003) presented a method of knowledge representation in the form of a set of objects Based on the set, the hierarchical structure of classes was prepared A more recent publication by Denkena et al (2007) described a holistic process-planning model based on an integrated approach combining technological and business considerations Halevi and Wang (2007) argued that instead of making decisions engineers should be engaged in the development of knowledge-based ‘road map’ The road map method can introduce flexibility and dynamics in the manufacturing process and thus simplifies the decision-making process in production planning Liu and Wang (2007) used a hybrid approach whereby knowledge-based rules and geometric reasoning rules are combined to sort out the sequence of interacting prismatic machining features Knowledge-based CAPP systems remain to be a popular branch of CAPP research since the 1980s Anwer and Chep (1999) created an Intelligent Process Planning Assistant (IPPA), which supports knowledgeassisted planning of machining operations and presented an opportunity for CAD/CAM integration In the same year, Jiang et al (1999) created an automatic process planning system (APPS) for the generation of manufacturing process plans directly from CAD drawings The APPS uses knowledge such as machine limitations, tooling availability and other processrelated manufacturing information In CAPP, selection of cutting tools and determination of machining conditions require a considerable amount of experience and knowledge The objectives may include selecting the best tool holders and inserts from an available cutting tool stock, and determining the optimum cutting conditions Arezoo et al (2000) developed a knowledge-based system EXCATS (expert computer aided cutting tool selection) for selection of cutting tools (including tool holders and inserts) and conditions of turning operations such as feed, speed and depth of cut The system demonstrates the key role of a knowledge-based system in achieving maximum flexibility in process planning automation Zhao et al (2002) further extended EXCATS by integrating it X Xu et al with a CAD system Their system is capable of processing CAD data and automatically generating the component representation file for EXCATS Pham and Gologlu (2001) designed a hybrid CAPP system called ProPlanner, to facilitate concurrent product development In their work, a hybrid knowledge representation scheme, and objects were used to store domain-related declarative knowledge and production rules used to codify procedural knowledge Gologlu (2004) extended the ProPlanner, and presented an efficient heuristic algorithm for finding near-optimal operation sequences from all available process plans in a machining set-up In the adopted approach, a fourlevel hierarchy was used: feature level, machining scheme level, operation level and tool level This enabled the problem of operation sequencing to be systematically addressed 3.3 Neural networks Neural networks are the techniques developed by simulating the human neuron function and using weights distributed among their neurons to perform implicit inferences (Ming et al 1999) The function of a neural network-based system is determined by four parameters: net topology, training or learning rules, input node characteristics and output node characteristics (Prabhakar and Henderson 1992) Use of neural networks can give a process planning system an adaptive and learning capability Neural network has several advantages over other methods used in CAPP (Yue et al 2002) It can tolerate slight errors from input It is usually faster because the process is limited to simple mathematical computations and does not use either a search or a logical rule to parse information A neural network also has the ability to derive rules or knowledge through training with examples and can allow exceptions and irregularities in the knowledge/ rule base Using a neural network, one can easily consider multiple constraints in parallel The pioneering research on a neural network approach using perception for feature recognition was proposed by Hwang and Henderson (1992) Perception is a pattern classifier for only linearly separable patterns, with supervised training The network, trained to recognise intermediate features such as a pocket, a slot and a through-hole, was able to recognise partial features presented but failed to recognise more complex features, such as a cross-slot In the article by Onwubolu (1999), a back propagation neural network was applied to the problem of feature recognition Devireddy and Ghosh (1999) presented a methodology of integrating design with process planning using neural networks The system can be trained to handle new types of components Because it is easy to modularise a neural network solution for a particular problem, neural networks are found to be combined with a number of other methods A hybrid intelligent inference model for CAPP was developed by Ming et al (1999) Their model combines the advantages of both expert systems and neural networks The methodology provided an effective means to administrate, control and coordinate the CAPP functions It also enhanced the adaptability and flexibility of the CAPP system to cope with the dynamic nature of the manufacturing environment Chang and Chang (2000) developed a system that integrates the variant and generative forms of CAPP It consists of process planning expert system modules and a dynamic learning recognition mechanism, through integration of fuzzy logic rules, artificial neural networks and expert systems It is able to decide whether to use the variant or generative procedures Fuzzy logic rules and neural networks enable process planning to have a dynamic adaptive learning ability Ben Yahia et al (2002) presented a feed-forward neural network based system for CAPP The methodology can cater for some difficult problems Ming and Mak (2000a) formulated the problem of selecting exactly one representative from a set of alternative process plans for each part, to minimise, for all the parts to be manufactured, the sum of both the costs of the selected process plans and the dissimilarities in their manufacturing resource requirements They combined a Hopfield neural network (Hopfield and Tank 1985) and GAs to solve the above problem Later, the same authors used the Hopfield neural network to solve the manufacturing operation selection problem (Ming and Mak 2001) Similarly, Deb et al (2006) used the back-propagation neural network method for the selection of all possible operations for machining rotationally symmetrical components This was done by pre-structuring the neural network with prior domain knowledge in the form of heuristic or thumb rules Ding et al (2005) presented an optimisation strategy for process sequencing based on multiobjective fitness: minimum manufacturing cost, shortest manufacturing time and best satisfaction of manufacturing sequence rules They used an artificial neural network to allocate the relative weights for the three main evaluating factors for process sequencing, and applied an analytical hierarchical process to evaluate the satisfaction degree of the manufacturing sequence rules for process sequencing Amaitik and Kilic (2007) developed a process planning system for prismatic parts where several neural network models were developed The main neural network model is utilised to select proper cutting tool(s) for each machining feature The idea is that for each machining feature and machining operation combination there is a corresponding cutting tool to be used to create that feature The neural network International Journal of Computer Integrated Manufacturing is trained based on this criterion For each cutting tool, a neural network was designed and trained to select the proper tool geometry Selection of a machine tool on which the machining operations can be performed to produce the given part is also implemented by a neural network The input vector of the neural network includes machining part characteristics and machining operation characteristics, and the output vector of the neural network contains recommended specifications of the machine tool to be used to perform the task These recommended specifications are used to search in an available machine tool database for a proper machine tool Sunil and Pande (2008) proposed a 12-node vector scheme to represent machining feature families having variations in topology and geometry The data of the recognised features was then post-processed and linked to a feature-based CAPP system for CNC machining To solve setup planning problems, Ming and Mak (2000b) used Kohonen self-organising neural networks and Hopfield neural networks Kohonen self-organising neural networks were specially developed to solve the setup generation problem, by considering fixtures/jigs constraint, approach direction constraint, feature precedence relationship constraint, and position tolerance relationship constraint Hopfield neural network was adopted to solve the operation sequence problem and the setup sequence problem These two problems are NP-complete problems, and can be mapped onto the travelling salesman problem by numerating the constraints, in the operation sequence and the setup sequence, into the distance among operations and among setups Park et al (2000, 2001) developed a methodology for incremental supervised learning of cutting conditions for process planning It enabled the model for generating cutting conditions to be enhanced while the system is in continual use The methodology of the fuzzy neural network is applied to model the process of learning and enhancing cutting condition, where it is capable of online and offline supervised learning in response to arbitrary sequences of analogue and binary input / target vector pairs The issue of intelligent toolpath generation was addressed by Balic and Korosec (2002) They presented a discussion to show that artificial neural network is able to establish a desirable milling tool-path strategy/sequence for free-surface machining Joo et al (2001) presented a dynamic planning model for determining cutting parameters Using neural networks, they developed a dynamic planning model for determining cutting parameters and this model is executed by a shop-floor controller to determine the cutting parameters for the removal feature based on current shop-floor status Devireddy et al (2002) proposed a three-layer, back-propagation neural network for selection of machining operations for all the features at a time, by taking into consideration the global sequencing of operations across all the features of a part This approach is able to overcome some limitations of decision trees and expert system-based approaches Korosec et al (2005) reported a neural-fuzzy model that uses the concept of ‘feature manufacturability’ to identify and recognise the degree of difficulty in machining The model was created by means of constructing parametric fuzzy membership functions, based on the neural networks learning process A three-layer, feed-forward architecture was used The system created was successfully implemented by Balic and Korosec (2002) for intelligent tool-path generation for free-form surface machining 3.4 Genetic algorithms A GA is an intelligent search method requiring domain-specific knowledge to solve a problem It has been successfully applied to various optimisation problems since the mid-1960s GAs are in the category of post-collation optimisation approach (Qiao et al 2000) By mimicking the evolutionary process of nature, such algorithms have been employed as global search and optimisation techniques for various scientific and engineering problems GAs search from a population of points, unlike the enumerative techniques where the objective function is calculated at each point in a search space, one point at a time GAs mimic the process of natural evolution by combining the survival of the fittest among solution structures with a structured, yet randomised, information exchange and offspring creation The offspring displaces weak solutions during each generation Therefore, GAs are very simple, straight forward, yet powerful methods for global search and optimisation of multimodal functions (Singh et al 2003) The main advantage that the GA-based approach has over other CAPP approaches is in the task of concurrently considering machine tools, cutting tools, tool access directions for each operation, and the sequence among the operations Therefore, the resulting process plan model retains the entire solution space This makes it possible to find a globally optimal process plan for a part The CAPP model for machined parts to be made in a job shop environment as reported in Zhang et al (1997) is such an example Similar work has been done by Rocha et al (1999), with the use of GA in a CAPP system to generate the sequence of operations and to select machine tools and cutting tools that minimise machining time In the system developed by Dereli and Filiz (1999), a reward/penalty matrix called REPMAX for each setup was determined based on the selected criterion, such as safety or X Xu et al minimum tool change The objective of optimisation was to gain the least total penalty or largest total reward Bo et al (2006) reconstructed GAs based on the analysis of various constraints in process route sequencing, including the establishment of coding strategy, evaluation operators and fitness function, to meet the requirement of sequencing work The mentioned constraints are used as the control strategy for GAs in the searching process to direct the calculation of GAs, and to find the optimal result that can satisfy the constraints Li et al (2000a) used GAs to optimize machining datum selection and machining tolerance allocation in process planning Much like neural network, GA is also found to be combined with other methods to solve optimisation problems in CAPP The use of a GA in conjunction with a neural network is mentioned earlier in the work of Ming and Mak (2000b) The weakness of the Hopfield neural network that leads to a local optimal solution to the problem was eliminated by the use of GA Genetic operations, such as individual evaluation, parent selection, reproduction and mutation, are performed to obtain the best individual as the solution Therefore, the combination approach of the hybrid Hopfield network is able to obtain the near-global optimal solution to the process plan selection problem The combined approach is also adopted by Ding et al (2005) They presented an optimisation strategy for process sequencing based on multi-objective fitness: minimum manufacturing cost, shortest manufacturing time and best satisfaction of manufacturing sequence rules They proposed to incorporate the GA, neural network and analytical hierarchical process (AHP) for process sequencing A globally optimised fitness function is then defined including the evaluation of manufacturing rules using AHP, calculation of cost and time, and determination of relative weights using neural network techniques Li et al (2002) developed a hybrid GA-SA (simulated annealing) approach to solve the optimisation problem of process planning for prismatic parts In this approach, the assignment of machining resources, selection of set-up plans and sequencing of machining operations are considered concurrently The advantage of this hybrid GA-SA approach is that it can generate multiple optimal or near-optimal process plans, with acceptable computation efficiency based on a combined machining cost criterion with weights Based on the multiple process plans, process planners can make a more accurate and flexible decision according to the actual conditions This approach can conveniently simulate a practical and dynamic workshop environment, considering the unavailability of a machine or tool in bottleneck (competition) usage or breakdown, change of machining cost evaluation strategy, and substitution of machines or tools in another shop floor Qin et al (2005) introduced a fuzzy inference system for choosing appropriate machines In addition, the load for each machine is balanced by using the GA based on the capability information, which is measured by a reliability index For the most reliable machine, the load given will be more than the unreliable one The load on each machine is measured by the machine utilisation, i.e the percentage of time the machine is being utilised Bhaskara Reddy et al (1999) proposed a quick identification of (near) optimal operation sequences in a dynamic planning environment using a GA They identified the feasible sequences based on a Feature Precedence Graph and used minimum production cost as the objective function A precedence cost matrix was generated for any pair of features based on the relative costs corresponding to the number of tasks that needed to be performed In the following year, Qiao et al (2000) proposed a GA-based operation sequencing method that provides a potential for finding ‘good’ machining operation sequences A fitness function is developed for this purpose, considering multiple process planning rules simultaneously and flexibly The value of the fitness function is a criterion to evaluate the degree of satisfaction of a searched operation sequence to generate a feasible operation sequence and, eventually, a near-optimal solution In the work of Shunmugam et al (2000, 2002), they considered a face-milling operation for selection of machining parameters such as number of passes, depth of cut in each pass, speed and feed, which were obtained using a GA, to yield minimum total production cost while considering technological constraints such as allowable speed and feed, dimensional accuracy, surface finish, tool wear and machine tool capabilities From experiments, the method proposed always yields less production cost than, or equal to, that by other methods Chiung et al (1998) proposed a GA-based evolutionary approach to solve the sequencing problem by simultaneously considering the operation flexibility, realistic shop factors and transportation time of an AGV (automated-guided vehicle) system It was formulated as a bi-criteria mathematical model minimising the total processing and transportation time and minimising the load variation between machines Li et al (2005) presented a GA to search for an optimal process plan for a single manufacturing system as well as distributed manufacturing systems according to prescribed criteria such as minimising processing time By applying a GA, their CAPP system could generate optimal or near-optimal process plans based on the chosen criterion They claimed that the algorithm adopted a crossover operator described by Bhaskara International Journal of Computer Integrated Manufacturing Reddy et al (1999) The developed technique is comparative or better in dealing with process planning in a single manufacturing system or factory Salehi and Tavakkoli-Moghaddam (2009) applied GAs for process planning in both the preliminary and detailed planning stages In the preliminary stage, feasible sequences of operations are generated based on the analysis of constraints In the detailed planning stage, the GA prunes the initial feasible sequences to give an optimised operation sequence and optimised selection of machine, cutting tool, and tool approach direction for each operation 3.5 Fuzzy set theory/logic Much of the decision-making in the real world such as process planning takes place in an environment in which goals and constraints are fuzzy, i.e not known precisely This generates the need of approximation to obtain a reasonable model of a real system Fuzzy theory deals with this type of problems by transforming human knowledge to mathematical formulae (Beg and Shunmugam 2003) and puts it into engineering systems together with other information like mathematical models and sensory measurements (Wang 1997), where goals and constraints can be modelled by fuzzy sets Xu and Hinduja (1997) developed a fuzzy decision system to evaluate the tolerances in a design model for a decision whether only a roughing operation will suffice or a semi-finishing and/or finishing operation are needed Fuzzy logic is often combined with other methodologies to solve process-planning problems Chang and Chang (2000) proposed a system that integrates the variant and generative CAPP It consists of process planning expert system modules and a dynamic learning recognition mechanism Fuzzy logic rules, artificial neural networks and expert systems are used Fuzzy logic and neural network techniques are used for dynamic and adaptive learning, where fuzzy set theory provides a suitable tool to deal with uncertainty or ambiguity problems Wong et al (2003) created a prototype process planning system that uses a hybrid of fuzzy and genetic approaches for solving the process-sequencing problem The costtolerance relationship is developed by fuzzy linguistic variables and fuzzy ‘if-then’ rules are established The imprecise manufacturing information, such as tool setup cost, is expressed in fuzzy numbers Zhao et al (2004) introduced a fuzzy inference system for the purpose of choosing appropriate machines, as an alternative way to integrate the production capability during scheduling In addition, based on the capability information, the load for each machine is balanced by using a GA This is used to overcome the problem that if a machine is unreliable it is not being utilised at all Park et al (2000) presented a methodology that enabled the model of generating cutting conditions to be enhanced while the system was in continual use Wang et al (2001) developed a method that used grey relational analysis and fuzzy clustering to form part families efficiently, based on factors such as processing time, lot size, and operation sequence In grey relational analysis, black represents having no information and white represents having all information A grey system has a level of information between black and white In the developed system, grey relational analysis is used to relate important product process factors and obtain a similarity matrix, and fuzzy clustering is used to form part families by truncating the transitive closure of the similarity matrix 3.6 Petri nets Nowadays, process planning is required to be flexible enough to meet the requirements in dynamic manufacturing Recent research on alternative process planning represents another trend (Wu et al 2002) PNs have the ability to represent and analyse concurrency and synchronisation phenomena in an easy way, such as concurrent evolutions, where various processes that evolve simultaneously are partially independent (Li et al 2000b) Furthermore, PN approach can be easily combined with other techniques and theories such as object-oriented programming, fuzzy theory, neural networks, etc These modified PNs have also found applications in process planning PNs have an inherent quality in representing logic in an intuitive and visual way Based on the basic PNs, fuzzy PNs (FPNs) have been developed to address issues characterised by uncertainty, imprecision and ambiguity Kiritsis et al (1999) considered cost estimation of operation sequencing in nonlinear process planning, i.e taking into consideration of processing alternatives To determine overall costs for a feasible process plan, they took into account the costs caused by machine, setup and tool changing in addition to the pure operation cost They developed two PN techniques for process planning cost estimation; both are based on a new PN model (PP-net: Process Planning net) that allows the modelling of partial process plans The first technique is based on building a complex PN called PPC-system (Process Planning Cost system) by integrating the PP-net and separate PNs describing the costs of machine, setup and tool changing The second method proceeds with cost calculation by attaching a specific data structure to each PP-net transition that describes the associated machine, setup and tool for the operation modelled by that transition International Journal of Computer Integrated Manufacturing 79 Dgdim1 ¼ þðgmax dim À gN Þ ¼ ð2:1064 À 1:2702Þ ffi þ0:84 mm Dgdim2 ¼ ÀðgN À gmin dim Þ ¼ Àð1:2702 À 0:4909Þ ffi À0:78 mm ð12Þ Although operating on a simplified problem (geometrical tolerances are neglected), the manual computation of the gap boundary values provides a useful support for the quantitative comparison of the five methods, at least when they are applied by considering dimensional tolerances only The manually obtained, extreme gap values will be used as reference values later on, then the results of the five methods will be discussed 3.2 Jacobian model solution 3.2.1 Dimensional tolerances only The Jacobian model of the case study is made under the simplified hypothesis to consider as fixed at 908 the orientation of the four sides of the box This simplification is needed to avoid the network in the assembly Indeed it is to observe (see Figure 3) that the functional requirement g has to be measured between the top side of the box and the second disk The second disk being assembled with the sub-assembly box-disk 1, firstly the assembly between part and part has to be solved Therefore, the assembly between the part and the sub-assembly box-disk has to be solved Once indicated with x1 and x2 the dimensions of the box, with x3, and x4 the diameter of the two disks and with U1, U2, U3, U4 the assembly variables (see Figure 5), the simplification adopted makes it possible to directly solve the assembly problem as: U1 ¼ x U2 ¼ x U ¼ x À x4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U4 ¼ x3 þ ðx3 þ x4 Þ2 Àðx1 À x3 À x4 Þ2 T24 60 ¼6 40 0 0 0 ð13Þ Figure Functional requirement and the functional elements pairs of the case study pair is associated with the points G and O and it is an internal one The second functional elements pair is associated with the points O and O1, while the third functional elements pair is associated with the points O1 and O2; they are both externals The last internal functional elements pair is associated with the points O2 and H The required functional requirement g is correspondent to the functional elements pair associated with the points G and H, and it is evaluated as the chain of the four functional elements pairs just identified At this point, it is possible for each FE to locate the virtual joints and the reference frames and to evaluate the transformation matrices T10 ; T20 ; T30 ; :::; T24 with the Equations (1) and (2) The matrix of the total transformation is equal to: 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi7 25 Àx2 þ x3 þ x4 þ ðx3 þ x4 Þ Àðx1 À x3 À x4 Þ Therefore, the functional requirement g and the functional elements pairs of the case study are identified (Figure 5) The first functional elements ð14Þ Therefore, the Jacobian matrix of the functional requirement is evaluated using Equations (4) and (5): 80 M Marziale and W Polini dz0 ¼ Àdx2 0 0 À1 0 07 dz6 ¼ dx3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 7 dz12 ¼ r ðx3 þ x4 Þ2 Àðx1 À x3 À x4 Þ2 0 17 x1 À x 6 Àh 07 ¼ À0:2582 Á dx1 þ 1:2910 Á dx3 þ 1:2910 Á dx4 h x4 À x 1 07 7 0 0 07 dz18 ¼ dx4 ð18Þ À1 0 0 6 0 0 07 and then is: 7 x À x À x 0 1 6 h À x3 07 Dg ¼ 0:2582 Á dx1 þ dx2 À 2:2910 Á dx3 À 2:2910 Á dx4 h À x3 Àðx1 À x3 À x4 Þ T J ¼6 ð19Þ 0 07 6 À1 0 07 6 0 0 07 Once obtained the required stack-up function, it can be 6 0 0 17 solved with the usual methods of the literature For 6 Àx4 07 example, for the worst case approach (Whitney 2004): 6 x4 0 07 X 0 0 07 ¼ Æ Dg jSi j Á ti ¼ Æð0:2582 Á 0:20 þ 0:50 WC À1 0 07 i¼1 ð20Þ 0 0 07 þ2:2910 Á 0:05 þ 2:2910 Á 0:05Þ 0 0 ¼ Æ0:7807 ffi Æ0:78 mm 0 05 0 0 while for statistical case approach: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð15Þ u uX qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DgSC ¼ Æt ðSi Á ti Þ2 ¼ Æ0:5281 ffi Æ0:53 mm where h ¼ x3 þ x4 þ ðx3 þ x4 Þ2 Àðx1 À x3 À x4 Þ2 i¼1 Once calculated the Jacobian matrix of the funcð21Þ tional requirement pair, the stack-up function may be formalised considering that the requirement Dg must 3.2.2 Dimensional and geometrical tolerances be evaluated as the translation of point H long the 7Z0 axis (Figure 5), and then as the third component If the considered case study included the dimensional of Equation (3) It is: and the geometrical tolerances, nothing changes as regards to the previous case, since it has adopted the simplification to consider fixed the angles in order to Dg ¼ Àdz24 ¼ Àdz0 þ dfz5 Á ðx1 À x4 Þ À dz6 avoid the network In fact, to solve the stack-up þ dfz11 Á ðx1 À x3 À x4 Þ À dz12 À dz18 ð16Þ function, it is needed to relate the virtual joints displacements to the tolerances assigned on the where dzi (i ¼ 0, 1, 2, 6, 7, 8, 12, 13, 14, 18, 19, 20) is components However, the form tolerances (the planar one applied on the bottom side of the box and the two the translation along the ith axis and dfi (i ¼ 3, 4, 5, 9, circularities applied on the circles) not produce any 10, 11, 15, 16, 17, 21, 22, 23) is the rotation around the effect because in the Jacobian model the features are ith axis This equation relates the functional requireconsidered with nominal shape; the other ones (the ment Dg to the virtual joints displacements assigned to perpendicularity applied on the left side of the box, the functional requirement pairs; now it is necessary to and the two parallelisms applied on the other sides of relate these virtual joints displacements to the tolerthe box) cannot produce any orientation deviation, ances assigned on the components since the angles of the box are considered fixed This step is the critical task of the Jacobian model Therefore, the simplification to consider fixed the However considering the simplification adopted (anangles of the box, due to the need to avoid network, gles fixed) is df5 ¼ df11 ¼ and then: causes the assigned geometrical tolerances not to Dg ¼ Àdz0 À dz6 À dz12 À dz18 ð17Þ produce any effects and the results are the same as the previous case where only dimensional tolerances are considered Moreover, the application of the Moreover, with reference to Figure of the four Envelope Principle or of the Independence Principle functional requirement pairs and to Equations (13), does not produce any effect for the Jacobian model and considering the nominal dimensions: 81 International Journal of Computer Integrated Manufacturing 3.3 Torsor model solution 3.3.1 Dimensional tolerances only The required functional characteristic g has to be evaluated by considering that there is a network among the components of the assembly At the state of the art, the solution of parallel chains through the torsor method is not completely developed Therefore, the simplification to consider fixed the angles of the box has been used in order to avoid the network This simplification may solve the assembly problem as showed in Equations (13) The first step of the method is to identify the elements of the parts (see Figure 6), and the relations among them; these information are reported in the surfaces graph of the case study (see Figure 7) Considering the angles of the box as fixed, the network can be solved and, therefore, the surfaces graph is simplified as shown in Figure The cumulative torsor G is expressed as: < a uH = b vH ð22Þ GH ¼ ÀT1;1 þ T1=3 þ T3;1 ¼ : ; g Dg R H Therefore, the functional requirement Dg is expressed by the translation component along the global Z axis The next step is to evaluate the components of the torsors (indeed it is enough to evaluate the third components due to translation) For the T1,1 torsor, with reference to Figure and Table and considering that the case study is a 2D problem on x7z plane (i.e a ¼ 0, g ¼ 0, v ¼ 0), it is: À = < À À b ð23Þ T1;1 ¼ ; : 1;1 À wM1;1 R1;1 M where M is the median point of the feature Considering that the point of interest is H, from Equation (8): 9 À b1;1 > < uH > = > < À > = À À ¼ þ4 À vH > > > > : ; : ; Àb1;1 À wM1;1 wH 9 > > = > < À1:2702b1;1 > = < À ð24Þ ¼ Á À > > > ; : ; > : w À 5b À1:2703 M1;1 1;1 and then from Equation (7): where T1,1 is the torsor of feature of part (the box); T3,1 is the torsor of feature of part (the circle 2); and T1/3 is the torsor of the link between part and part Figure study T1;1 < À b ¼ : 1;1 À M À1:2702b1;1 = À ; wM1;1 À 5b1;1 R ð25Þ 1;1 Torsor model: elements and parts of the case Figure Torsor model: surface graph of the case study 82 M Marziale and W Polini This torsor (note that only three of its components are not null) is expressed in the local frame of the feature and it has to be expressed in the global reference frame R (Figure 9) In this simple case it is possible to note that the local y-axis coincides with the global Y-axis, therefore, the angle b1,1 is the same too The local x and z axes are the inverse of the global X and Z axes, respectively, therefore, the correspondent translation needs to be inverted It is: À1:2702b1;1 = < À b1;1 À T1;1 ¼ ð26Þ À Á; : À À wM1;1 À 5b1;1 H R Considering the simplification of considering fixed the angles of the box, b1,1 ¼ and therefore: À = ¼ h À À Á À Á i   > > H: À À t1;3 þ t3 2; þ t1;3 þ t3 ; R ð31Þ 83 International Journal of Computer Integrated Manufacturing The second term (T2/3) is the torsor of the link between part þ and part With reference to Figure it is: T2=3 > H: À evaluate the results due to a statistical approach, since the torsor’s components are considered the extreme possible intervals of the small displacements, and this Á À Á i h À > À t1;4 þ t1 þ t3 þ t4 2; þ t1;4 þ t1 þ t3 þ t4 = À h i> Àð0:2582 Á t1 þ 1:2910 Á t3 þ 1:2910 Á t4 Þ=2; þð0:2582 Á t1 þ 1:2910 Á t3 þ 1:2910 Á t4 Þ=2 ; R where t1,2, t1,3, and t1,4 are the thicknesses of the tolerance zones of S1,2, S1,3 and S1,4; and t1, t3, and t4 are the tolerances on the dimensions x1, x3 and x4 Therefore, the functional requirement is: is not compatible with the statistical approach where a probability density function is to each parameter 3.3.2 Dg ¼ Æð0:2582 Á t1 þ t2 þ 2:2910 Á t3 þ 2:2910 Á t4 þ t11 þ t13 Þ=2 ð33Þ Now, it is necessary to relate the thicknesses of each tolerance zone assigned to each feature to the tolerances required on the components This is another critical step of the torsor model However, under the simplified hypothesis adopted (i.e fixed angles of the box) and by considering only dimensional tolerances, it may have: t11 ¼ t12 ¼ t13 ¼ t14 ¼ mm t1 ¼ 0:40 mm t2 ¼ 1:00 mm t3 ¼ 0:10 mm ð34Þ And, therefore, the functional requirement in the worst case approach is: DgWC ¼ þð0:2582 Á 0:40 þ 1:00 þ 2:2910 Á 0:10 þ 2:2910 Á 0:10Þ=2 ¼ Æ0:7807 ffi Æ0:78 mm ð35Þ as obtained with the Jacobian approach It may be added that the torsor method does not allow to Dimensional and geometrical tolerances If the considered case study included the dimensional and the geometrical tolerances, Equation (30) is still valid (under the hypothesis of fixed angles of the box), and it is always needed to relate the thickness of the tolerance zones to the tolerances required on the components Moreover, by using the simplification to consider fixed the angles of the box, none changes as regards the case of only dimensional tolerances Therefore, the simplification to consider fixed the angles of the box causes the geometrical tolerances not to have effects on the results of the case study Moreover, the application of the Envelope Principle or the Independence Principle does not produce any effect for the torsor model too 3.4 t4 ¼ 0:10 mm Table Comparison Table shows the results due to the application of the two considered models to the same case study If only the dimensional tolerances are applied, the worst case approach gives values that are under the estimated geometrical exact solution of about 4% This is probably due to the same way the dimensional tolerances are schematised (i.e the first datum is nominal, the variability due to the dimensional tolerance is considered applied only on one of the two features delimiting the dimension) If the geometrical tolerances are applied too, the worst case Case study results Tolerance case Dg Method Approach Exact geometric solution Worst case Jacobian Worst case Statistical case Worst case Statistical case Torsor ð32Þ Only dimensional dim & geo þ0.84 mm 70.78 mm +0.78 mm +0.53 mm +0.78 mm – – +0.78 mm +0.53 mm +0.78 mm – 84 M Marziale and W Polini approach gives the same result, since the simplification to consider fixed the angles of the box has been adopted in order to avoid the network In fact, to solve the stack-up function, it is needed to relate the virtual joints displacements to the tolerances assigned on the components However, the form tolerances (the planar one applied on the bottom side of the box and the two circularities applied on the circles) not produce any effect because in the Jacobian model the features are considered with nominal shape; the other ones (the perpendicularity applied on the left side of the box, and the two parallelisms applied on the other sides of the box) cannot produce any orientation deviation, since the angles of the box are considered fixed The statistical approach gives similar results, when only dimensional tolerances or both dimensional and geometrical tolerances are applied In this case the variability range is smaller than that of the worst case approach, as it is foreseen The Jacobian model has the advantage, as regards the torsor model, to be able to perform the analysis by both the worst case and the statistical approaches Moreover, the Jacobian model allows to evaluate the Jacobian matrix from the nominal conditions and, therefore, it is possible to directly relate the displacements of the functional requirements to the virtual joints displacements Another advantage of the Jacobian model is that it uses the usual algebraic rules to evaluate the displacements of the functional requirements, while the torsor model needs intervals algebraic rules which are much complex; this aspect is fundamental to approach the solution of the network functions which need to be developed for both the models Despite these advantages of the Jacobian model, its virtual joints displacements are difficult to relate to the tolerances applied on the components This step is critical for the torsor model too, but it seems easier to approach, despite with this second model the computation of the displacement components of the points of interest in the same global datum reference frame is very difficult and it has to be developed completely yet The torsor model allows to easily evaluate the variability ranges of the small displacements from the tolerances applied on the components, but it is very difficult to relate these ranges to the variability ranges of the functional requirements of the assembly In the last years the idea of the unified Jacobiantorsor model has been presented in order to evaluate the virtual joint displacements from the tolerances applied on the components by the torsors and, therefore, to relate the displacements of the functional requirements to the virtual joint displacements by the Jacobian matrix (Laperrie`re et al 2002, Desrochers et al 2003); it is theoretically possible since the deviations are usually small and the equations may be linearised The proposed unified model expands the functionalities of the Jacobian model under two important aspects (Ghie et al 2003) First, the punctual small displacement variables of the former Jacobian formulation are now considered as intervals formulated and solved using interval-base arithmetic The equations describing the bounds within which the feature is permitted to move, which are the constraint equations of the torsor formulation, are applied on the unified model Second, some of the small displacement variables used in the model are eliminated due to the invariant nature of the movements they generate with respect to the toleranced feature This standard result of the torsor formulation is applied to the unified model The effect of this is to significantly reduce the unified model size This new model enables to perform tolerance analysis and tolerance synthesis (Laperrie`re et al 2002) or to redesign the assembly tolerances (Ghie et al 2007) The unified Jacobian-torsor model has been developed for deterministic (worst-case) computer aidedtolerancing Recently, the same set of interval-based deterministic equations has been applied to a statistical context (Ghie et al 2010) and the model has been used to develop a method for obtaining the functional requirement cost for product (Ghie 2009) Unified Jacobian-torsor model As just underlined, the Jacobian model takes its best advantage in the simplicity to evaluate the Jacobian matrix from the nominal conditions; Equations (4) and (5) This makes it possible to directly relate the displacements of the functional requirements to the virtual joints displacements; equation (3) The solution of the network functions seems easier to approach than the torsor one too Despite this advantage, the virtual joint displacements are difficult to relate to the tolerances applied on the components Conclusions This paper firstly makes a brief review of two state of the art tolerance analysis models for rigid-parts assembly, the Jacobian and the torsor The two models are compared in order to highlight the advantages and the weakness of each model, on the basis of the experimental results and the information available from the literature The two considered models have some common limits The first two deal with the assembly cycle: the two models have not yet developed an approach to consider functional requirement functions arranged in a network and to correctly represent the coupling with International Journal of Computer Integrated Manufacturing clearance between two parts The third deals with the representation of the tolerances applied on the assembly’s components: the two models not give a complete correspondence among the model variables (displacements) and the part’s tolerances In other words the translation of the part’s tolerances into model variables does not satisfy the standards (ASME or ISO) The last deals with the form deviations and the Independence Principle: the two models not allow to consider form deviations (the features are considered with nominal shape) and to apply the Independence and/or the Envelope Rule to different tolerances of the same parts However, the Jacobian model has the advantage to directly relate the displacements of the functional requirements to the virtual joints displacements by the evaluation of the Jacobian matrix from the nominal conditions Moreover, it can be used to perform both a worst case and a statistical case approach The torsor model has an easy evaluation of the variability ranges of the small displacements from the tolerances applied to the components The experimental 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Box 14115-179, Tehran, Iran (Received 20 January 2010; final version received 11 July 2010) This article proposes a genetic algorithm (GA) for the simultaneous lotsizing and sequencing problem in permutation flow shops involving sequence-dependent setups and capacity constraints To evaluate the effectiveness of the proposed GA, two lower bounds are developed and compared against the optimal solution The results of GA are compared with the selected lower bound Keywords: capacitated lotsizing problem; permutation flow shop; sequence-dependent setups; genetic algorithm Introduction Flow shop is one of the most widely investigated production and scheduling problems of the literature (Alouloua and Artigues 2010, Dugardin et al 2010) It comprises a series of machines that perform operations on a production as it progresses down the line A special case of flow shop that assumes the same order of products in all machines is called permutation flow shop Traditionally, the problem of scheduling jobs on a flow shop is decomposed into the sub-problems of lotsizing and sequencing This is an approximate way of solving the problem because, in general, the lotsizing decision is dependent on the sequencing decision Nevertheless, there have been several heuristics developed for each of the above two problems For a more detailed review of the relevant work done in this area, please refer to Sikora et al (1996) and Zhu and Wilhelm (2006) Researchers have also investigated the problem of integrating lotsizing and sequencing decisions in the flow shop Sikora et al (1996) considered a variation with limited intermediate buffer space and deadlines, and they studied the objectives of minimising Cmax and inventory holding costs They integrated the Silver– Meal lotsizing heuristic (Silver and Meal 1973), which they modified to deal with lot splitting, with Palmer’s (1965) flow shop heuristic, which they augmented with an improvement procedure They demonstrated the effectiveness of their approach by scheduling an actual assembly line In another paper, Sikora (1996) presented a GA that used separate crossover and *Corresponding author Email: mohammad_9091@yahoo.com ISSN 0951-192X print/ISSN 1362-3052 online Ó 2011 Taylor & Francis DOI: 10.1080/0951192X.2010.511654 http://www.informaworld.com mutation operators for lotsizing and sequencing decisions He compared this GA with the integrated approach presented by Sikora et al (1996) and found that the GA that used a population size of 10 prescribed much better schedules with significantly less run time than the integrated approach However, the performance of the GA was sensitive to the selection of parameter values and it was difficult to determine effective values Lee et al (1997) presented a hybrid GA to minimise Cmax in the case of a finite buffer space This hybrid GA incorporates SA in its mutation operation and pairwise-exchange improvement procedures in an attempt to avoid the local optima at which GAs frequently stop They evaluated the performance of this hybrid GA in an application to an actual assembly plant and their computational tests showed that it performs better than the pure GA, especially when the problem size is large Recently, simultaneous lotsizing and scheduling in capacitated flow shop with sequence-dependent setups have been considered by Mohammadi et al (2008 and 2009a) They proposed a mathematical formulation and mixed integer programming (MIP)-based heuristics for the problem Involving capacity constraints, setup carry over and variable lotsizes in production stages are the main features of their model Because of restriction in computation times, the quality of solutions was poor especially for large instances of the problem To solve larger instances of problem, Mohammadi et al (2009b) also proposed a new algorithmic approach based on a simplified mathematical model 88 M Mohammadi et al In this article, instead of solving a succession of smaller MIPs, they would relax all binary variables of the problem The resulting problem would be solved through a T-iteration-based algorithm In a specific iteration k, relaxed binary variables of period k would be divided into two groups where members of the first group would get value and members of the second group would get value The current article proposes a genetic algorithm (GA) for the simultaneous lotsizing and sequencing problems in permutation flow shops involving sequencedependent setups and capacity constraints Permutation flow shop is one of the most usual production systems in industry (Ribas et al 2010) The former papers in permutation flow shops considered lotsizing and sequencing decisions in two distinct phases and would lead to infeasible solutions, but this article assumes simultaneous lotsizing and sequencing in permutation flow shops This article presents an efficient solution algorithm to solve this problem The article has the following structure Section introduces a detailed description of the problem and its underlying assumptions Section deals with the development and comparison of lower bounds in detail and section provides the GA Section reports the numerical experiments and finally section discusses the concluding remarks and recommendations for future studies Problem formulation 2.1 Assumptions Several products are produced on serially arranged machines The order of products in all machines is the same Each machine is constrained in capacity When the machines are setup, sequence-dependent setup costs and times occur The setting-up of a machine must be completed in a stipulated period There must be precisely N (number of products) setups in each period on each machine, even if a setup is just from a product to itself Since a setup time (and cost) from a product to itself is zero, note that the model does not force a machine to have exactly N positive-time (and cost) setups but rather up to N such setups The remaining zero-time (cost) setups are modelling phantoms and not exist in reality (Clark and Clark 2000, Clark 2003) External demand exists for final products and is satisfied at the end of each period There are no lead times between the different production levels for transportation or cooling the products Shortages are not permitted A component cannot be produced earlier in a period than the production of its required component is finished In other words, production on a production level can only be started if a sufficient amount of the product from the previous production level is available; this is called vertical interaction To guarantee the vertical interaction, idleness between each setup and its production is defined with the help of shadow product (Fandel and Stammen-Hegene 2006, Mohammadi et al (2008, 2009a,b) There are no demand and no storage costs for shadow products At the beginning of each period, machines are not setup for each of the products The triangle inequality holds, i.e it is never faster to change over from one product to another by means of a third product In other words, a direct changeover is at least as capacity-efficient as going via another product 2.2 Mathematical model The following notations are used in the model formulation: Indices: i, j, k Index of product type n Index of setup number n0 Designation for a specific setup number m Index of production’s level t Index of planning period Parameters: T Planning horizon N Number of different products M Number of production levels/number of machines bigM A large real number Cm,t Available capacity of machine m in period t (in time units) dj,t External demand for product j at the end of period t (in units of quantity) hj,m Storage costs unit rate for product j in level m bj,m Capacity of machine m required to produce a unit of product (or shadow product) j in time units per quantity units pj,m,t Production costs to produce one unit of product j on machine m in period t (in money unit per quantity unit) Si,j,m Sequence-dependent setup time at machine m when switching from product i to j in time units; (for i 6¼ j, Si,j,m ! and for i ¼ j, Si,j,m ¼ 0) 89 International Journal of Computer Integrated Manufacturing Sequence-dependent setup cost at machine m when switching from product i to j in money units; (for i 6¼ j, Wi,j,m ! and for i ¼ j, Wi,j,m ¼ We also assume that the relation between setup costs and times can be considered as: Wi,j,m ¼ fw Á Si,j,m where fw is opportunity cost per unit of setup time The setup configuration on machines at the beginning of each period Decision variables: Ij,m,t Inventory level of product j at level m at the end of period t yni;j;t 1, if the nth setup on machines is performed at period t when switching from product i to j; 0, otherwise xnj;m;t Quantity of product j produced after nth setup on machine m at period t qnj;m;t Shadow product indicating the gap (in quantity units) between nth setup (to product j) on machine m at period t and its related production in order to ensure that direct predecessor of this product (production of product j on machine m71 at period t) has been completed In other words, it denotes the idle time (in quantity units) before production of product j on machine m in period t in order to guarantee vertical interaction Wi,j,m According to the above notation, the proposed mathematical formulation for the problem can be written as follows: Min N X N X N X M X T X wi;j;m yni;j;t þ n¼1 j¼1 i¼0 m¼1 t¼1 xnj;m;t þ N X M X T X N X N X M X T X ð1Þ subject to: dj;t ¼ Ij;M;tÀ1 þ N X xnj;M;t À Ij;M;t ; n¼1 j ¼ 1; ; N; t ¼ 1; ; T Ij;m;tÀ1 þ N X xnj;m;t ¼ Ij;m;t þ n¼1 N X xnj;mþ1;t ; ð2Þ j ¼ 1; ; N; n¼1 m ¼ 1; ; MÀ1; t ¼ 1; ; T n X N X N X yni;j;t Si;j;m þ n¼1 i¼0 j¼1 qnj;m;t þ ð3Þ n0 X N X n¼1 j¼1 n X N X bj;m n¼1 j¼1 bj;m xnj;m;t n0 X N X N X n¼1 i¼0 j¼1 yni;j;t bj;mþ1 n0 À1 X N X bj;mþ1 n¼1 j¼1 n0 ¼ 1; ; N; m ¼ 1; ; MÀ1; t ¼ 1; ; T xnj;mþ1;t ; ð4Þ N X N X N X yni;j;t Á Si;j;m þ n¼1 i¼0 j¼1 qnj;m;t xnj;m;t Cm;t ; À N X N X n bj;m xj;m;t þ n¼1 j¼1 N X N X bj;m n¼1 j¼1 m ¼ 1; ; M; t ¼ 1; ; T N X  Á Cm;t bj;m ð5Þ yni;j;t ; n ¼ 1; ; N; i¼0;i6¼jðfor n>1Þ j ¼ 1; ; N; m ¼ 1; ; M; t ¼ 1; ; T qnj;m;t À N X  Á Cm;t bj;m yni;j;t ; ð6Þ n ¼ 1; ; N; i¼0 j ¼ 1; ; N; m ¼ 1; ; M; t ¼ 1; ; T y1j;i;1 ¼ 0; j 6¼ 0; i ¼ 1; ; N N X y10;i;1 ¼ ð7Þ ð8Þ ð9Þ i¼1 N X ynj;i;t ¼ j¼0 N X ynþ1 i;k;t ; n ¼ 1; ; NÀ1; k¼1 i ¼ 1; ; N; t ¼ 1; ; T Ij;m;0 ¼ 0; j¼1 m¼1 t¼1 qnj;mþ1;t þ n¼1 j¼1 pj;m;t n¼1 j¼1 m¼1 t¼1 hj;m Ij;m;t Si;j;mþ1 þ n0 X N X ð10Þ yni;j;t ¼ or ð11Þ Ij;m;t ; xnj;m;t ; qnj;m;t ! ð12Þ j ¼ 1; ; N; m ¼ 1; ; M ð13Þ In this model, Equation (1) represents the objective function which minimises the sum of the sequencedependent setup costs, the storage costs and the production costs Equation (2) ensures the demand supply in each period Equation (3) shows that in a network, the total of in-flows to each node is equal to out-flows from that node Equation (4) guarantees that within one period each typical product j on machine m is produced before its direct successor (product j on machine m þ 1) Equation (5) represents the capacity constraints of machines during periods Equation (6) indicates that setup is considered in the production process Equation (7) indicates the relationship between shadow products and setups Equations (8) and (9) guarantee that for each machine, the first setup at the beginning of the planning horizon is from a defined product Equation (10) represents the relationship 90 M Mohammadi et al between successive setups Equations (11) and (12) represent the type of variables Equation (13) indicates that at the beginning of planning horizon there is no on-hand inventory Development of lower bounds The formulation presented in the previous section is not a practical approach to solve large instances of the problem In this section, we obtain two lower bounds on the problem The first lower bound is achieved by solving model M1 that is obtained from the initial model by relaxing all binary variables The second lower bound is obtained by solving a new model M2 that is derived from M1, adding the following equation: N X y1i;j;t þ i¼0 N X N X yni;j;t ¼ aj;t ð14Þ i¼1;i6¼j n¼2 where aj,t is a binary variable Equation (14) PN is similar ton the right hand side of y Equation (6), i¼0;i6¼jðfor n>1Þ i;j;t In Equation (14), we P n y aggregate N i¼0;i6¼jðfor n>1Þ i;j;t by summing over all n Lemma Equation (14) is valid to M1 Proof Suppose that Equation (14) is not valid to M1, it means there at least one couple PN that PN is P N yn y (j,t), where þ i;j;t i¼1;i6 n¼2 i;j;t > Suppose P PN i¼0 P ¼j N n À1 n ynk;j;t y y þ ¼ 1, that i¼0 i;j;t i¼1;i6¼j n¼2 i;j;t ¼ and (k ¼ j) For n ¼ n0 to N, all j would be changed to k PN y By this modification, for all couple (j,t), i¼0 i;j;t þ PN PN n y i¼1;i6¼j n¼2 i;j;t ¼ or According to the fact that setup costs from a product to itself is zero and considering the triangle inequality: Á À n0 þ1 yk;k;t : Wk;k;m þ ynk;i;t : Wk;i;m Á À n0 yk;j;t : Wk;j;m þ ynj;i;tþ1 : Wj;i;m Therefore, by assuming that Equation (14) is not valid to M1, there would be a solution better than or equal to the optimum of M1 and it is impossible to intractable problems (Goren et al in press) GAs are probabilistic search optimisation algorithms that were inspired by the process of natural evolution and the principles of ‘survival of the fittest’ (Holland 1975, McWilliams 2009) A GA starts with an initial random population of feasible solutions; when applied to scheduling, each individual in the population corresponds to one possible solution of the scheduling problem The algorithm generates a new candidate pool of solutions iteratively from the presently available solutions and replaces some or all of the existing members of the current solution tool with the newly created feasible solutions The expanding role of GAs to solve complicated lotsizing problems has been studied by Jans and Degraeve (2007) and Goren et al (in press) Components of GA which are used in this article are described as follows Genetic algorithm GAs are one of the most popular heuristic algorithms that represent a powerful and robust approach for developing heuristics for complex and large-scale combinatorial optimisation problems They have been used by many to overcome the limitations of the mathematical models, finding acceptable solutions 4.1 Chromosome representation In this research, chromosomes are represented in the form of matrices with T N dimensions to show the value of binary variables In Figure 1, a sample chromosome for a specific iteration of a problem with T ¼ and N ¼ is depicted The chromosome indicates the sequence of products in all periods According to this chromosome: y10;1;1 ¼ y21;3;1 ¼ y33;2;1 ¼ y10;3;2 ¼ y23;1;1 ¼ y31;1;2 ¼ 1, and other binary variables would get value 4.2 Initial population Selecting an appropriate initial population can significantly increase the efficiency of a GA For this reason, a simple and effective heuristic has been used The procedure is used for t ¼ to T and would be described as follows: (1) The products P are sorted in the decreasing order of Wj;m ¼ N i¼1 Wi;j;m ; j ¼ 1; ; N: (2) Let [i] indicate the ith product in an ordered sequence in this heuristic For [i] ¼ to N: (a) Consider inserting product [i] into every position (b) Calculate the sum of setup costs for all products scheduled so far using the actual setup costs (c) Place product i in the position with the lowest resultant sum of setup costs Figure A sample chromosome 91 International Journal of Computer Integrated Manufacturing M different initial populations have been made through this method (for m ¼ 1, ., M), the remaining initial populations have been generated randomly Table Comparison of lower bounds and exact optimal solutions in problem size N ¼ 3, M ¼ and T ¼ No 4.3 Fitness function The fitness value of each chromosome has been obtained by solving the corresponding problem 4.4 Selection operator The tournament selection approach and roulette wheel (Gen and Cheng 1997) have been used to undergo crossover operation 4.5 4.6 Mutation operator In order to produce small perturbations on chromosomes to promote diversity of the population, a shift mutation operator has been used in this article 4.7 Population replacement Chromosomes for the next generation are selected from the enlarged population The best pop_size chromosomes of the enlarged population have been selected for the next generation 4.8 Termination criterion The termination criterion determines when GA will stop We stop GA, if maximum number of generations, max_gen, has been executed or the pre-set number of generations without improvement in the last best solution, max_no_improve, is reached 3107.06 (12.71) 3496.35 (10.17) 3205.10 (17.22) 3333.55 (13.41) 3381.63 (6.34) Crossover operation Recombining the features of two randomly selected parents with the aim of producing better offspring is the main purpose of crossover operation Several crossover operators have been proposed (Nagano et al 2008) To maintain the blocks of products and improve the algorithm, similar block order crossover (SBOX), similar job two point crossover (SJ2OX) and similar block two point crossover (SB2OX) have been used in this research Numerical experiments In order to ascertain the accuracy of the mentioned lower bounds, we performed some numerical tests Tables and 2, respectively show the results of such tests in some instances of the problem with (N ¼ 3, M ¼ 2, T ¼ 3) and (N ¼ , M ¼ 3, T ¼ 3) To apply Original problem First lower bound Second lower bound 2799.56 9.90% (0.06) 2907.08 16.86% (0.08) 2737.97 14.58% (0.13) 2811.88 15.65% (0.09) 2813.06 16.81% (0.06) 2990.33 3.76% (0.21) 3341.76 4.42% (0.31) 3056.67 4.63% (0.29) 3231.77 3.05% (0.19) 3158.12 6.61% (0.31) Note: The values inside the brackets are the computational time in seconds and the percentage values are the difference between the objective values of the lower bound against the original model Table Comparison of lower bounds and exact optimal solutions in problem size N ¼ 3, M ¼ and T ¼ No Original problem 4550.98 (131.41) 4998.95 (149.77) 5143.65 (116.53) 4766.01 (186.51) 5310.58 (188.13) First lower bound Second lower bound 3788.75 16.75% (0.11) 4301.39 13.95% (0.19) 4279.41 16.80% (0.16) 4074.29 14.51% (0.22) 4496.89 15.32% (0.23) 4373.22 3.91% (0.65) 4770.74 4.57% (0.41) 4993.17 2.93% (0.49) 4610.72 3.26% (0.78) 5007.38 5.71% (0.59) Note: The values inside the brackets are the computation times in seconds and the percentage values are the difference between the objective values of the lower bound against the original model the exact model and lower bounds, GAMS models are provided using GAMS IDE (ver 2.0.19.0) and solved using OSL GAMS models are run on a personal computer with a Pentium processor running at 3.4 GHZ From the results of the second columns of Tables and 2, it is evident that computation time grows exponentially by increasing the dimension of the problem According to Table 1, the average computation time for problems with (N ¼ 3, M ¼ 2, T ¼ 3) is 11.97 s According to Table 2, the average computation time for problems with (N ¼ 3, M ¼ 3, T ¼ 3) is 92 M Mohammadi et al Table The dimension of problems used in calibration of parameters 36363 56565 76767 154.47 s It means that by increasing one level to production levels, the average computation time increases by more than 12 times Tables and confirm the advantages of the second lower bound, and therefore it has been used to compare heuristics Now, let us see the behaviour of the different parameters of the proposed GA All different combinations of the aforementioned factors and parameters yield many alternative GAs In order to calibrate the algorithms, we have chosen a full factorial design in which all possible combinations of the following factors are tested: Selection type: levels (Tournament and Roulette wheel) Crossover type: levels (SBOX, SJ2OX, SB2OX) Crossover probability (pc): levels (0.5, 0.6, 0.7, 0.8) Mutation probability (pm): levels (0.1, 0.15, 0.2, 0.25) Population size (psize) : levels (3M, 4M, 5M) All the cited factors result in 6 6 ¼ 288 different combinations and thus, 288 different GAs Every algorithm is tested with a set of problems presented in Table There is one replicate for each combination; therefore, 288 ¼ 1440 problems have been solved The response variable is based on the following performance measure: % increase over the second lower bound: ,  X Heusol i À LBi  100 LBi i¼1 where Heusoli is the solution obtained by a specific problem and LBi is the second lower bound for this specific problem By following the same procedure for all combinations and all parameters we get the following ‘best case’ calibrations of the proposed algorithm: Selection type: Tournament Crossover type: SB2OX Crossover probability: 0.7 Mutation probability: 0.15 Population size: 5M Our proposed GA is coded in Matlab programming language and is run on a personal computer with 10 10 10 15 15 15 Table Comparison of the second lower bound and the proposed GA Problem size (N M T) The second lower bound Proposed GA 36363 (0.49) 56363 (11.61) 36563 (2.80) 36365 (4.23) 56565 (58.37) 76565 (149.95) 56765 (98.53) 56567 (109.21) 76767 (288.38) 10 6 (912.25) 10 (150.06) 6 10 (211.61) 10 7 (1837.74) 10 (611.34) 7 10 (958.13) 10 10 10 (4306.88) 15 10 10 47200a (208.51) 11.51% (351.93) 10.67% (289.98) 11.98% (294.13) 11.16% (918.37) 12.88% (1411.22) 11.05% (1008.13) 11.98% (1100.51) 11.15% (1836.58) 13.09% (1613.28) 12.02% (1395.15) 11.74% (1411.21) 13.31% (3807.97) 13.81% (1668.33) 11.96% (1893.12) 12.29% (6104.28) 13.19% 10 15 10 47200a b 10 10 15 47200a b 15 15 15 a b 47200 b b b b b Note: The values inside the brackets are the computation times in seconds a Finding the optimum value for the second lower bound requires more than 7200 s and the objective value at this time has been considered b Feasible solution has not been found after 7200 s of computing time The percentage values are the difference between the objective values of the heuristics against the second lower bound a Pentium processor running at 3.4 GHZ 20 Twenty problems with different sizes are selected for testing For each problem size, five problem instances are randomly generated Table shows the comparison International Journal of Computer Integrated Manufacturing between the proposed GA and the second lower bound The required parameters for all numerical experiments are extracted from the following uniform distributions: bj;m % Uð1; 5; 2Þ; dj;t % Uð0; 180Þ; hj;m % Uð0:2; 0:4Þ; pj;m;t % Uð1:5; 2Þ; Wi;j;m % Uð35; 70Þ; Si;j;m % Uð35; 70Þ; Cm;t ¼ Uða; bÞ a ¼ 200.N þ 100.(m71), b ¼ 200.N þ 200.(m71) Cm,t is calculated in accordance to satisfy the demands of each period on a just-in-time basis with average setups Concluding remarks The contribution of the article has been to derive and test one exact formulation and one GA for simultaneous lotsizing and sequencing in permutation flow shops with sequence-dependent setups To test the accuracy of the proposed GA, two lower bounds are developed and compared against the optimal solution Selected lower bound is used to test the accuracy of the proposed GA Because of the expanding role of meta-heuristic approaches to solve complicated lotsizing problems (Jans and Degraeve 2007, Defersha and Chen 2008), the application of other meta-heuristic approaches to face this complex problem is recommended as an area for future research References Aloulou, M.A and Artigues, C., 2010 Flexible solutions in disjunctive scheduling: general formulation and study of the flow-shop case Computers and Operations Research, 37 (5), 890–898 Clark, A.R., 2003 Optimization approximations for capacity constrained material requirements planning International Journal of Production Economics, 84 (2), 115–131 Clark, A.R and Clark, S.J., 2000 Rolling-horizon lot-sizing when setup times are sequence-dependent International Journal of Production Research, 38 (10), 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