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Balancing a robotic spot welding manufacturing line: An industrial case study
Accepted Manuscript
Balancing a Robotic Spot Welding Manufacturing Line: an Industrial Case Study Thiago Cantos Lopes, Celso Gustavo Stall Sikora, Rafael Gobbi Molina, Daniel Schibelbain, Luiz Carlos de Abreu Rodrigues, Leandro Magatao ˜ PII: DOI: Reference:
S0377-2217(17)30518-0 10.1016/j.ejor.2017.06.001 EOR 14482
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
8 November 2016 22 March 2017 1 June 2017
Please cite this article as: Thiago Cantos Lopes, Celso Gustavo Stall Sikora, Rafael Gobbi Molina, Daniel Schibelbain, Luiz Carlos de Abreu Rodrigues, Leandro Magatao, ˜ Balancing a Robotic Spot Welding Manufacturing Line: an Industrial Case Study, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.06.001
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ACCEPTED MANUSCRIPT Highlights • A model for Robotic Spot Welding Manufacturing Line Balancing is presented. • The model takes in account accessibility and movement times between welding points. • Multiple robots per station define interference constraints in the model. • The models predictions are validated by comparisons with empirical data.
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• Up to 6.6% productivity gain achieved when compared to the as-is configurations.
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Balancing a Robotic Spot Welding Manufacturing Line: an Industrial Case Study Thiago Cantos Lopesa , Celso Gustavo Stall Sikoraa , Rafael Gobbi Molinab , Daniel Schibelbainc , Luiz Carlos de Abreu Rodriguesb , Leandro Magat˜aoa∗ a: Graduate Program in Electrical and Computer Engineering (CPGEI) Federal University of Technology - Paran´ a (UTFPR), Curitiba, Brazil, 80230-901 b: Department of Mechanical Engineering - UTFPR
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c: Renault do Brasil, S˜ ao Jos´ e dos Pinhais, Brazil, 83070-900
Abstract
Resistance spot welding is an important procedure used in the automobile manufacturing industry. After stamping, sheets of metal are mostly welded together to build the car’s body. Many of
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the industrial robots found in assembly lines are spot welders. Process characteristics include accessibility limitations, the need to move between welding points and (when there are multiple robots per station) potential interferences between robots. The combination of such characteristics is not present in classical models for the assembly line balancing problem. In this paper, the balancing problem of robotic spot welding manufacturing lines is presented and modeled based on
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a real-world car factory on the outskirts of Curitiba, Brazil. The studied line has 42 robots that perform over 700 welding points on the later stages of the body-shop. The model was developed with Mixed Integer Linear Programming (MILP) techniques, validated with empirical data, and
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solved with a universal solver. The optimized balancing achieved a cycle time reduction of up to 6.6% compared to the as-is configurations. The total movement time was observed not to
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be necessarily minimized during the optimization process, implying that trade-offs exist between movement times and the number of welding points performed. Keywords: Combinatorial Optimization, Assembly Line Balancing Problem, Robotic Welding
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Manufacturing Line, Mixed Integer Linear Programming
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1. Introduction
The automotive industry strongly relies on flow shop arrangements for the mass production of
vehicles. The flow shop or product oriented system brings as advantages a high capacity utilization, simple flow of materials, and the specialization of workers as a result of the division of labor (Scholl, 1999). The workpieces are moved along a serial arrangement of workstations, which are responsible for specific tasks in the production process. Such functional characteristics propitiate the use of automatic systems or even robotic workers. ∗ Corresponding
author Email address: [email protected] (Leandro Magat˜ aoa )
Preprint submitted to European Journal of Operational Research
June 7, 2017
ACCEPTED MANUSCRIPT In the industry, the flow shop arrangements are well represented in the assembly lines used to transform simpler components in assembled products. According to Michalos et al. (2010), the automotive processes can be roughly divided in four parts: stamping, body shop, painting, and final assembly. A product must go through all the stages, that should all be planned for an aimed throughput rate. Therefore, the cycle time is a key factor for all stages of the industrial production. Due to the simple flow of workpieces, the workstation that contains the longest processing operations is the system’s bottleneck. Hence, one of the main objectives of planning a production system is to evenly divide the tasks among the workstations to increase their efficiency. Obtaining
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the best distribution of the operations or tasks is the objective of the Assembly Line Balancing Problem (ALBP), first proposed by Salveson (1955). The base problem and its simplifications were defined by Baybars (1986) resulting in the classification of Simple Assembly Line Balancing Problem (SALBP) and General Assembly Line Balancing Problem (GALBP). According to Wee & Magazine (1982), ALBP are NP-Hard once they are a generalization of the Bin Packing Problem. Most of the assembly operations in the automotive industry are performed in the body shop
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and final assembly stages. The body shop is usually highly automated, while the final assembly is performed by manual, robotic, or hybrid systems (Michalos et al., 2010). Although it is not commonly stated in the articles, most of the academic developments on ALBP (recently reviewed in Batta¨ıa & Dolgui (2013)) relates to the final assembly. In the less reported stage of the process, the body shop, sheets of metal are transformed into the car body (Body-in-White) with welding
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procedures. The welding procedures must also be evenly divided among the workstations, however, welding tasks present differences compared to the final assembly operations. The welding itself
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occurs in less than a second, however, the welding tool movement and positioning account for the major part of the cycle time. Moreover, the accessibility of welding tasks and the path that link the operation are important to correctly measure the station processing time.
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It is common that robotic welding workstations contain more than one welding worker or robot. According to Dimitriadis (2006), multi-manned workstations, that is stations with multiple (robotic) workers, can bring advantages such as fewer workstations, shorter line length, and the
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smaller costs of product handling and station equipment. The balancing of robotic welding lines can then be described as a combination of problems. The correct balancing of welding points
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consists of an assignment problem of welding procedures to stations and/or robots, a path routing problem for each robot, and a robot coordination problem, once they must not interfere with each other.
Some previous works treated the combination of the problems or parts of them. Segeborn et al.
(2013) developed a heuristic and simulation based method for the balancing of spot welding points in robotic workstations with multiple robots. Their approach consists of a greedy algorithm for the tasks distribution, followed by a Simulated Annealing improvement procedure. In order to reduce the need for robot coordination, a separation procedure is applied sampling pairs of points at random and swapping assignments if the separation can be improved. In order to calculate the cycle time, the distances between two welding points are calculated with approximation methods.
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ACCEPTED MANUSCRIPT Segeborn et al. (2013) states that it is impractical to calculate all the distances between points considering the geometry of the pieces and the robot. Therefore, two simplified distance matrices are used: one that do not consider the geometry of the piece and robots and one that represent the robot as simple lines. The path planning is then performed by a Traveling Salesman Algorithm using the approximate distances. Finally, the coordination of robots is performed via simulation. The speed of each movement can be controlled or wait times can be inserted to avoid collisions, increasing the cycle time. The coordination simulation is also simplified: the flexible parts such as cables and hoses are not considered. Segeborn et al. (2013) tested their approach on a line with 3
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workstations with 10 robots that performed 201 spot welding procedures.
Akturk et al. (2011) present a MILP model for the balancing of robotic spot welding lines considering tool change costs. Both the cycle time and the tool costs are minimized in a costoriented model. Welding points are assigned to robots, whose tool life is measured throughout the production phases and can be changed in pauses or shift changes. In the modeling, similar points are gathered in groups. Each group of welding points is considered a task, whose processing time
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is a constant representing the time to reach the first point plus the time required to perform the welding operations. The points are not assigned individually, all the welding point group must be assigned to a single robot. Therefore, the movement time is simplified and integrates the task time. The model is developed for single robots in each station, so that no path planning and no robot coordination is presented.
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Another work that presents itself as a variation of an ALBP formulation is due to Nilakantan et al. (2015). They present a MILP model and a Particle Swarm Optimization method for the
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minimization of cycle time and energy consumption in robotic assembly lines. There are different models of robots that can perform tasks at different rates with different energy requirements. A multi-objective approach presents the concurring effects of production rate and running costs. The
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authors state that the formulations are adequate to robotic lines for the body shop. However, tasks are consider as ‘classical’ ALBP operations, the same precedence diagrams used for the final assembly are used in the computational tests. Moreover, no movement time between procedures,
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path planning, or robot coordination is considered. The literature related to path planning is strongly related to General Traveling Salesman Prob-
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lem (GTSP). Saha et al. (2006) consider a given set of points assigned to a single robot. The authors developed a heuristic method for the path planning considering obstacles caused by the piece accessibility. Their procedure consists of a TSP and a path calculation algorithm. The authors state that calculating all distances between welding points consumes more CPU time than solving the TSP problem itself. Therefore, the proposed procedure sparingly calculates the distance between points by delaying the calls of the path algorithm to only when they are needed. Similar approaches are presented by Carlson et al. (2014) and Wurll et al. (1999). Both works calculates a path between given points considering obstacles for stations with only one robot. As Saha et al. (2006), the authors also use techniques to reduce the number of distance calculations between welding points.
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ACCEPTED MANUSCRIPT Xie & Hsieh (2002) present a genetic algorithm to sequence procedures in spot welding stations. The authors aim for the minimization of the cycle time and the piece’s geometric deformation. The presented algorithm is applied to welding points aligned linearly, so that the travel distance calculation is obvious. No obstacles are considered, as well as no scheduling and no coordination between multiple robots are necessary. The robotic coordination of multiple workers in a cell is considered by Spensieri et al. (2013). This work considers a given assignment of points and an operation sequence for each robot. The problem consists in determining a path for each robot so that the robots do not collide. An MILP
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model and a Branch-and-Bound procedure were developed to solve the problem by adding wait times in the scheduling of robots. Therefore, for the given assignment of points and sequences, the model minimizes the necessary wait time.
Chakraborty et al. (2010) treat a related problem: the robotic laser drilling of electrical components. The authors present an heuristic procedure (greedy plus improvement algorithm) for the task assignment, path planning, and robot coordination considering 2 or 4 robots. The electric
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components are composed of planar plates in which drilling operations must be performed. The tasks are assigned among the robots, whose paths are calculated with a TSP algorithm considering the interference between robots. Once the planar plates do not contain obstacles, the distance between two points can be determined by euclidean distance.
As the previous works have shown, the balancing of robotic spot welding lines is composed of
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several problems that are treated mostly with heuristic procedures. For the best of the authors knowledge, the largest spot welding line reported as case study contained 10 robots allocated in 3
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stations performing 201 welding points.
This paper proposes a model to treat such combinations of problems related to the balancing of robotic spot welding assembly lines. An MILP formulation is presented for the assignment
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problem of spot welding procedures among multi-manned robotic workstations. The modeling approach simplifies the path planning and robot coordination by distributing the welding points considering such aspects. Restrictions are provided to measure in approximative form the robots’
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movement time and occupation time in the vehicle regions. The final path planning can then be determined via simulation.
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The paper is structured as follows: Section 2 presents the production context on robotic body shop lines. The section also introduces the studied problem, describing its characteristics and presents a literature review discussing ALBP prior studies that relate to the problem characteristics. Section 3 describes the developed model from its basic concepts to its sets, variables and constraints. Section 4 describes real-world industrial case studies, the problems’ size, achieved results and discusses some practical insights. Section 5 summarizes the main results and contributions of this paper.
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ACCEPTED MANUSCRIPT 2. Balancing robotic spot welding manufacturing lines 2.1. Body-in-white production context The body-in-white is a stage in automobile manufacturing in which sheet metal components are welded together. The process starts after the stamping of the sheets, pieces are assembled together forming frames of the vehicle (front, back, base, left, right, and roof parts). The frames are then mounted together, resulting in the body of the car. The assembly process is generally divided along assembly or manufacturing lines. The welding
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of pieces can be performed either by manual labor, robots, or dedicated welding machines (Aslanlar et al., 2008). The piece handling between stations and lines can be performed manually or with the use of robotic handlers, conveyors, or tow trains.
Among other possible methods, the most used assembly procedure in the automotive industry is Resistance Spot Welding (RSW) due to its high reliability, low cost, manufacturing robustness, and automation potential (Barnes & Pashby, 2000; Gould, 2012). With RSW, two or three metal
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sheets can be united by electrical current. The heat provided from the metal resistance locally melts the sheets, welding the components together. The welding procedure takes less than a second (Aslanlar et al., 2008). The spot welding is performed by two electrodes that are positioned in a welding gun. The gun must access both sides of the piece, therefore, the accessibility of points is a disadvantage of the RSW (Barnes & Pashby, 2000). Furthermore, the quality of the welding
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depends on the surface conditions of sheets and electrodes, applied force, and size and contour of the electrodes (Aslanlar et al., 2008). Therefore, multiple types of electrodes and guns may be necessary for the welding procedures.
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Each welding point can be seen as a task to be performed in the body assemblage. Automobiles require from 3000 to 7000 welding points based on their size (Hamidinejad et al., 2012). For the joining of sheets, pieces must be positioned respecting geometric tolerances. Parts are clamped
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in place, usually by pneumatic or electric actuators until pieces are properly united. According to Sikora et al. (2017b), welding points can be divided in two types of tasks: piece joining and
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reinforcement procedures. The piece joining tasks assure the cohesion of pieces for further handling. They require the actuators to hold the piece in the correct position. The reinforcement welding
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points are needed for the final product resistance, however, they do not require actuators anymore: the joining points are sufficient to assure the geometric tolerances and the piece can be handled to other stations. The balancing of spot welding manufacturing lines consists in the distribution of welding points
along the workstations or lines. There are, however, some specific considerations related to the nature of spot welding procedures: • Accessibility: due to quality requirements, the position of pieces and the reach of the welding guns, not all points can be performed by all electrodes. Furthermore, for spot welding, both sides of the sheets must be accessible for the gun. New sheets block the access of inner
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ACCEPTED MANUSCRIPT layers. Although there are theoretically no precedence relations between two welding points, the assembly of parts force a sequencing of points based on station-wise accessibility windows. • Processing time: in the balancing of assembly lines, operations must be made within a cycle or takt time. The required time for a spot welding varies between a small interval (less than one second), but the time for positioning and the movement between points is relevant and must be also accounted for. • Multi-operators: the automobile industry frequently uses multiple workers or multiple robots
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in the same station. Although there are usually no precedence relations between welding points, space disputes or interferences may be observed (due to the sizes and shapes of the vehicle and robots). Precedence relations might relate welding points to piece joining, but provided these operations are fixed the precedence relations can be converted to station-wise accessibility windows (see Appendix A for details).
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2.2. Problem statement
Otto & Otto (2014) showed that the automotive industry organizes its operations based on modules. The lines are organized to be responsible for a module, whose precedence relations are easier to identify and map. After the operations of a line, there is not much opportunity to perform changes or extra operations in further modules: the interaction between modules is seldom observed.
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During the welding procedures, pieces vary from smaller and simpler sheets into compound structures that are then welded together in the body-in-white along several assembly lines. Due to
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the progressive addition of layers of metal sheets, much of the welding points lose their accessibility throughout the process. It is very difficult to map the entire welding operation (from 3000 to 7000 points) for all the types of electrodes and guns (which have different accessibilities) considering
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every combination of subassembly (the order in which sheets are welded together). Therefore, the lines are usually organized in functional subassemblies: different stations build specific sub-
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products. There are also defined modules for the left, right, top, base, front, and back part of the car. Therefore, it is reasonable (and sometimes is the only possible form) to model each line separately.
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Even considering a single assembly line, an optimization approach could consider the balancing
of operations or the project and design of the line itself. Reassigning welding operations to workers and robots is relatively easy: orienting the worker or programming the robot may be sufficient. Changing a subassembly order or location requires changes in the layout: pieces handling systems and actuators must also be reallocated. Therefore, in this paper, the balancing of a welding line refers to the allocation of points regarding a pre-defined layout. Altering or choosing the layout and subassembly order refers to a design problem. According to Falkenauer (2005), the re-balancing of assembly lines is much more common than the project of a new one. Changes in the product, demand, and the inclusion of another product variant are reasons for the re-balancing of assembly lines. 7
ACCEPTED MANUSCRIPT After the presented considerations, the research problem treated in this paper refers to the balancing of robotic spot welding lines used in automotive industry. The proposed problem is defined based on the following hypotheses and characteristics: C0 - Occurrence constraint - a set of welding points must be performed; C1 - Assignment restrictions - the assignment of points must respect the accessibility constraints associated with the robot’s tools and position; C2 - Robot-wise dependent parameters - Not all robots are equally fast due to their model and
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to the size of their spot welding tools; C3 - Cycle time is decisively influenced by movement time - Each robot’s cycle time is not determined by the simple sum of time of each performed “task”: time of handling of pieces, welding, moving and positioning welding guns must be considered;
C4 - Multiple positions for welding - some points may be accessible from more than one direction; C5 - Multi-manned stations with interference constraints - stations can contain more than one
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robot, which must not collide with each other as they perform tasks;
C6 - Given welding parameters - welding times and accessibility are known; C7 - Given layout - the number, position, and welding guns of robots are known and fixed for a given layout;
C8 - No precedence relations - there are no precedence relations defined between any two weld-
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ing points (although subassemblies might produce station-wise accessibility windows, whose
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explanation and modeling treatment are given in Appendix A).
2.3. Literature review on ALBP formulations for the problem characteristics For the definition of assembly line balancing problems, the spot welding points can be seen as
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tasks whose processing times are similar and short when compared to the cycle time. In principle, balancing might seem simple: simply divide the number of welding points roughly evenly between
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robots. There are, however, a few difficulties: robotic tools must be moved between welding points and the movement time is not usually known between each pair of points - and it is not necessarily proportional to euclidean distance between points, either. Such time cannot be easily determined
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as optimizing robotic routes is a hard problem on its own. Furthermore, stations often have more than one robot. Robots in the same station are subject to potential interference problems if they perform tasks (welding points) at the same regions or cross the robotic arms. In manual stations, this might not be such a serious concern as workers can communicate, coordinate and adapt in real time, but the studied robots blindly follow a predefined set of instructions. The specific tools and fixed positions of robots impose accessibility constraints, meaning that each robot will only be able to access and perform a subset of the welding points. While the characteristic C0 (described in Section 2.2) is simple and a basic part of nearly all assembly line models (Bowman, 1960; White, 1961; Thangavelu & Shetty, 1971; Patterson & Albracht, 1975; Scholl, 1999; Sikora et al., 2017b), it is made slightly more complicated by the 8
ACCEPTED MANUSCRIPT characteristic C1, as explained hereafter. Figures 1 and 2 illustrate why the problem is assignment bounded, as each robot can access only a subset of welding points, due to geometrical aspects. These assignment constraints can be seen as station-type restrictions in the classification schemes proposed by Boysen et al. (2007, 2008); Batta¨ıa & Dolgui (2013). In Figure 1, the robot can reach the welding point without colliding with the vehicle. In Figure 2, the robot will collide with the
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vehicle, if it attempts to reach the welding point.
Figure 2: Inaccessible welding point.
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Figure 1: Accessible welding point.
There are models (Scholl et al., 2010) and algorithms that deal with various assignment restrictions (Dar-El & Rubinovitch, 1979; Pastor & Corominas, 2000; Lapierre & Ruiz, 2004; Bautista & Pereira, 2007) and, thus, could be used to deal with C1. C2 is a pretty common part of robotic balancing problems (first defined by Rubinovitz & Bukchin (1991)). The problem would probably
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be treatable by assignment bounded or adapted RALBP algorithms (Kim & Park, 1995; Levetin et al., 2006; Daoud et al., 2014), if C1 and C2 were the problem’s most challenging characteristics. But they are not. C3 implies on constraints that make this problem harder to translate into a
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SALBP-based model, as it violates a core hypothesis of such models: a station’s (robot’s) time is not equal to the sum of the task’s time performed in it. This fact happens because the welding time
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is comparable to the movement’s time between points. Figures 3 and 4 show how relatively close welding points might require absolutely different configurations for the robot: this fact implies that
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movements are not only a matter of Euclidean distance.
Figure 3: Robot performing a welding point in the Figure 4: Robot performing a welding point back of the vehicle, before moving to a nearby weld- close to the first one (Figure 3). Notice how ing point.
the robot’s position has changed significantly.
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ACCEPTED MANUSCRIPT There are set-up models (described for instance by Scholl et al. (2013)) and algorithms (Andr´es et al., 2008; Scholl et al., 2008; Martino & Pastor, 2010) that could attempt to describe C3 by comparing movement times to setup times. But these models and algorithms require a matrix of known set-up times between tasks. Such data is not available as the time to move between each pair of welding spots is rather difficult to determine, especially given the large number of welding points involved. One could attempt to use an approximation for such movement times, but such times would have to be robot-dependent. The line has robots of different models with different speeds and tools of different sizes, aka C2. This might not be a promising direction though.
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Attempting to adapt the problem at hand to the model described by Scholl et al. (2013), the number of variables and constraints would tend to be too large: In the studied scenarios, there were 42 robots. Assuming each of the roughly 700 welding points can be accessed by 10% of the robots (rounding down to 4) and that the points can be followed or preceded by about a third of the other points (rounding down to 200), the sequence variables (yij and wij , see Scholl et al. (2013) for more details) would amount to 280.000 binaries and would require nearly half a million constraints to be
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properly controlled. Such a MILP problem is almost certain to be computationally un-treatable. And that is setting aside the fact that the set-up times are robot-dependent. Successfully operating some simplifications, there are heuristics and algorithms that could, maybe, be able to provide good answers, were it not for the other characteristics of this problem.
The feature C3 is made even harder by the fact that some welding points can be performed in
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different manners (C4), by different accesses for the robotic arms. The manner a welding point is accessed affects the movement time the robot will take to reach other points. Two points may seem
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really close and there might be a direct route between them but only if they are performed with the robotic arm positioned in a way that such route is feasible. These characteristics are illustrated by Figures 5 and 6, where the robot is accessing the same welding point from two different directions:
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suppose the robot is to move to a welding point between the doors. Such point is reachable for the robot in Figure 6 with a simple spin of its tool. The robot in Figure 5, in the other hand, must first leave the windshield region, then spin while moving inside the front door (without colliding
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with the vehicle), and only then it reaches the welding point. A symmetrical situation occurs if the robot were to move to a welding point in the central lower part of the windshield region: this
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time would be easier for the robot in Figure 5 move between the points. This means that a setup time approximation to movements between welding points would not only be robot-dependent but also access-dependent, making the use of a set-up based models and algorithms even more challenging. Parallels with assembly lines with processing alternatives models (Pinto et al., 1983; Scholl et al., 2009) can be drawn, but the nature of the alternatives is, in this case, rather different. The processing alternatives presented in ALBP literature relate to different forms of performing operations resulting in different subgraphs. The direct effect is multiple values of task processing time and precedence relations. In the spot welding lines, the different forms of accessing a point are mainly related to the movement times and interferences. But even if it were possible to overcome this process alternative aspect and treat C0, C1, C2
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Figure 5: Welding point accessed through the wind- Figure 6: Welding Point accessed through the front shield.
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and C3 using set-up based assignment bounded robotic balancing model, there is C5: multiple robots can be assigned to the same station, so that robots must not collide with one another while
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performing the welding points. Multi-manned workstations are commonly used for the production of large products because such special workstations can reduce the line length and, consequently, some costs related to piece handling systems, tools, and support systems (energy and pressure inlets, gas exhaustion, etc.). However, multi-manned stations bring as disadvantages the necessity to coordinate the worker operations (Dimitriadis, 2006).
In the ALBP literature, two approaches for the simultaneous processing of a piece by more than
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one worker are employed: the 2-sided (or N-sided) assembly lines and the multi-manned assembly lines. The 2-sided assembly line balancing was firstly introduced by Bartholdi (1993). The studied
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assembly line contained workstations on both the left and the right side of the produced vehicle. Some tasks must be performed by a specific worker: on the left or on the right side. Other tasks could be performed by the worker of any side. Once precedence relations are expected between
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tasks, it is necessary to schedule the operations within the workstation to assure its feasibility. Scholl et al. (2009) generalizes the problem presenting the Assembly Line Balancing Problem with
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Variable Workplaces (VWALBP). The authors considers up to 24 divisions of a vehicle. Multiple workers can perform operations in the same station, but they cannot work in the same regions of the vehicle.
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The second approach that considers multiple worker in the same station was introduced by
Dimitriadis (2006) with a heuristic solution procedure. The Multi-Manned Assembly Line Balancing Problem (MALBP) is based on the hypothesis that the product is large enough to have workers performing tasks without blocking each other. In each workstation, however, the precedence relations must be respected. Therefore, the scheduling of operations is necessary. Both approaches (VWALBP and MALBP) consider a maximal number of possible workers in a station empirically. The authors describe that, in the automotive industry, stations can be divided in up to 4 or 5 workplaces. An MILP model for the MALBP was presented by Fattahi et al. (2011), along with an Ant Colony Optimization procedure with the objective of minimizing the number of workstations. Further recent developments proposed the inclusion of other factors in the objective function, such
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ACCEPTED MANUSCRIPT as the number of workers (Kelleg¨oz, 2016) or the load balance (Yilmaz & Yilmaz, 2015). Roshani & Giglio (2016) considered the MALBP type-II, in which the station length is fixed and the number of workers and the cycle time is minimized. Among the presented papers, MILP formulations are developed, however only small and medium instances can be solved with commercial solvers (Fattahi et al., 2011; Kelleg¨oz, 2016; Yilmaz & Yilmaz, 2015). Note that MALBP does not concern the physical location in which a task is performed. Once the precedence relations are respected, the solution is said feasible. Sikora et al. (2017a) also presents a model in which multiple workers perform tasks at the same station. The multiple workers, however, only perform together tasks
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that require more than one worker. Putting these common tasks aside, each worker performs tasks in separate stations.
The body-shop environment present fewer or no precedence relations between the spot welding points. In fact, an ordering is implied based on the subassemblies, but they are due to the accessibility of metal sheets and not the direct effect of a welding point (see Appendix A). The interference between robots is due to the space competition, as it is exemplified in Figure 8: two
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robots cannot perform tasks in the same regions at the same time. Therefore, the approach of MALBP models is not adequate to deal with such restrictions. On the other hand, the definition of VWALBP states that only one worker can access a vehicle region in a workstation. Although that is enough to avoid the interference of robots, this hypothesis is too strict. There is also the possibility of robots accessing the same region, but at different moments within the cycle time (e.g.
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one robot starts at the front door, and moves to another region while a second robot enters the front door after it becomes available). The available time the robots can spend on a region can be
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seen as a scarce resource. However, the amount of available time depends on the number of robots that accesses the region: the entering and exiting time must also be considered. Another interference (C5) example lies in robots crossing arms (Figure 7). It is possible for
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either the front robot perform welding points in the back of the car, or the back robot perform welding points in the front of the car. However, if they do so simultaneously the chances of collision of the robotic arms are great, as depicted by the Figure 7, where both robots perform accessible
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welds, but the station’s configuration is undesirable.
Figure 7: Arm Crossing Interference.
Figure 8: Space Dispute Interference.
These interference characteristics are very difficult to translate into adapted features of the 12
ACCEPTED MANUSCRIPT classification schemes of Becker & Scholl (2006), Boysen et al. (2007) and Batta¨ıa & Dolgui (2013). Consequently, the combination of all the problem’s features give rise to a very complex ALB problem that is even more difficult to adapt into previously described models. Prior to the work described in this paper, the studied manufacturing line’s balancing was performed manually, based almost exclusively on the engineer’s and operator’s experiences. To the best of our knowledge, there is no literature model that can describe all these characteristics in a single model. This means that balancing such manufacturing line requires a new model.
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3. Model 3.1. General concepts
The employed model operates some simplifications in order to approximate the real-world problem. The main simplification is that no sequence for points performed by each robot is established, but the rather the number of points are counted along with the number of movements (classified
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as: very small, small, medium and large) required to perform said points. This simplification is based on two observed facts: Firstly, most of the welding points were distributed in logical paths such that the robot’s time spent on that path was proportional to the number of points performed. Secondly, when the robot moved from one path to another, the time spent on such movement could usually be classified into one of four classes, which are further discussed at the end of this section. Robots are presumed to be fixed to a station, with fixed given properties and parameters (such
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as speed, tools, position and capabilities). Two parameters of interest were the relative velocities of each robot and the fixed time required per welding point. This robot-wise fixed time is associated
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with opening and closing the welding claws. In general, newer robots tend to allow higher speeds due to better technological features and robots with bigger tools (often required to access some welding points) tend to have lower speeds and longer fixed times due to slower operation. All
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stations have a fixed constant time associated with the entrance and exit of each vehicle of said station, time during which it is assumed that no welding point can be performed.
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Welding points are grouped in regions, as depicted by the Figure 9 in which welding points (red dots) are boxed in accordance with physical characteristics and the accessibility constraints.
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Regions have the following characteristics: • All welding points in a region take the same amount of time to be performed. This time is added to fixed time per point of the robot that performs it. The movement time between points of the same region can be regarded as part of the time required to perform a welding point; • The time spent by a robot in a region is proportional to the number of points performed by said robot at said region; • If a robot can access the region, it can access all points in it; • Each region r has a fixed and given number of points Nr ; 13
ACCEPTED MANUSCRIPT • In a first approximation, the movement time is considered constant between regions. That is: if a robot works at three regions, it will move twice between regions and have a movement time twice as great as a robot that only works at two regions (and thus only moves once between regions); The region hypothesis was based on a problem’s characteristic: welding points were mostly lined up in logical geometrical paths and locations, and these logical paths could be divided in
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accordance to how the robots could access them.
Figure 9: Example on how welding points were grouped in regions (identified by different colors), adapted from the case study.
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Welding points can be performed from different accesses: front door, back door, luggage compartment, windshield. Some regions can be reached by multiple accesses (Figures 5 and 6). In order to better describe movements, accesses must be taken in account. If a robot is accessing
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regions within the same access, then the movement time tend to be smaller than when the robot accesses regions from different accesses. The concept of accesses adds a different kind of movement
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to the problem. Movements between accesses are analogous to movements between regions and can be analogously counted. Figure 10 illustrates the different regions reachable by accesses: some points can only be performed with the robotic arm positioned in a certain access, for instance green
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indicate regions accessible through the back door, while blue indicates regions accessible through
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the front door.
Figure 10: Accesses for the regions and adjacencies.
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ACCEPTED MANUSCRIPT Further movement time refinements are provided by the adjacency concept: Two regions that are next to one another (one is connected to the other) are deemed adjacent, meaning that in some cases the movement between them can be ignored or reduced. For instance, the regions that surround the front door can be seen as adjacent (as shown in Figure 10). The same applies to accesses. The movement time between the front and back doors is naturally smaller than the movement time between the front door and the luggage compartment. In many cases a large physical region must be divided in several adjacent regions because of accessibility constraints: if a robot can access the region, it can access all the welding points in it. Therefore, regions were
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modeled according to both accessibility and movements. Adjacencies were used to better describe movement times. For instance, in Figure 9, the front door’s cycle of regions is composed of adjacent regions, but they must be divided in order to account for the accessibility capacities of different robots.
Some constraints required grouping regions in sets that share some particular characteristics. There are two types of groups of regions that must be taken in consideration: Macro-regions (used
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to deal with interference constraints) and cycles (used to prevent incorrect counting of the number of adjacencies a robot has benefited from). The particular way in which these groups of regions behave is described in the Section 3.3. Figure 11 illustrates a pair of possible crossing constraint macro-regions: if, in the same station, the front robot accesses the back macro-region and the back robot accesses the front region they are likely to collide their robotic arms, as illustrated by the
Figure 11: Macro-regions defined for interference constraints.
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Figure 7.
As previously mentioned, the number of movements is counted. Movements between regions
within the same access are deemed “small” (or “very small” if the regions are adjacent). Movements between accesses are deemed “large”, unless there is an adjacency between the accesses (in which case they are “medium” movements). Here are examples of each movement type: Small - Moving from the left to the right side of the front door. Medium - Moving from the front door to the back door. Large - moving from the front door to the luggage compartment. Movements are counted by adding the number of regions and accesses a robot performs welding points, and subtracting the time-wise difference linked to the number of allowable adjacencies. The restrictions that control these operations will be described in Section 3.3. 15
ACCEPTED MANUSCRIPT Table 1: Parameter Tuple sets: They are employed throughout the model from the variable definitions to the constraint formulations. Sets
Tuple
S
s
Description of each range and meaning of the tuples on each set Range for all stations
R
r
Range for all regions
W
w
Range for all robots
WS
(w, s)
W RA
(w, r, a)
WA
(w, a)
Indicates that the robot w can use the access a
RRadj
(r1 , r2 )
Indicates that the regions r1 and r2 are adjacent
W RRadj
(w, r1 , r2 , a)
W AAadj
(w, a1 , a2 )
Indicates that the robot w is allocated at the station s Indicates that the robot w can reach the region r via the access a
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Indicates that the robot w can benefit from the adjacency between r1 and r2 via the access a Indicates that the robot w can benefit from the adjacency between a1 and a2
CR
(c, r)
Indicates that the region r belongs to the region-cycle c
CA
(c, a)
Indicates that the access a belongs to the access-cycle c
W CR
(w, cr)
Indicates that the robot w can perform the region-cycle cr
(w, ca)
Indicates that the robot w can perform the access-cycle ca
(w, em)
Indicates that the robot w can access the exclusion macro-region em
W CM
(w, cm)
Indicates that the robot w can access the crossing macro-region cm
EM R
(em, r)
Indicates that the region r belongs to the exclusion macro-region em
CM R
(cm, r)
Indicates that the region r belongs to the crossing macro-region cm
WMWM
(w1 , m1 , w2 , m2 )
EA
ea
Indicates that the access a defines a exclusion interference constraint
EM
em
List of all the macro regions em that define exclusion constraints
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W CA W EM
Indicates that the robot w1 cannot access the crossing macro-region m1 at the same time that
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robot w2 accesses the crossing macro-region m2
3.2. Sets and variables
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The problem is modeled around its decision variables, that are all defined based on parameter sets. In order to differentiate sets of parameters and sets of variables, the later will start with a lower-case letter (v for integer and real variables and b for binary variables). The tuple sets that
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are employed to define the variables are a discrete collection of tuples, whose interpretations are summarized at the Table 1. The problem’s variables, their types, tuple sets and interpretation are
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listed and summarized at the Table 2. Lastly, some constraints require some additional parameters that are either vectors or constants. These parameters are listed by the Table 3. Each variable set is based on a domain (tuple sets) and has one variable for each element of
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said set. For example, the variable set bW A has one binary variable for each (w, a) tuple in the tuple set W A, namely bW A(w,a) . The tuple and variable sets mostly follow a concept-based letter
orientation. For example, R stands for regions, W for robots, A for accesses, S for stations. This
pattern seeks to simplify interpretation: for instance, the binary variable bW A(w,a) is set to true when the robot w performs welding points by the access a. All variables in the proposed model are strictly non-negative. In our case the main decision variable is an integer number of welding points, performed by the robot w at the region r via the access a. The variable set vW RA contains one integer variable for each element of the tuple set W RA of accessibility (that contains the tuples (w, r, a) of regions r accessible by the robot w via the access a). This is a heavily assignment bounded problem: in the studied cases only about 10%
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ACCEPTED MANUSCRIPT Table 2: Variables sets, their types, the sets and tuples they are defined by and their meaning Variable Set
Tuple Set
Tuple
vCT
-
-
Line’s cycle time (Time Units)
Description and meaning of the variable from the set
Robot w’s cycle time (Time Units)
vCT W
W
w
vW RA
W RA
(w, r, a)
Number of welding points performed by robot w on region r via the access a
bW RA
W RA
(w, r, a)
Whether or not the robot w performs points on region r via the access a
vN M R
W
w
Number of movements between regions performed by the robot w
vN M A
W
w
Number of movements between accesses performed by the robot w
bW RR
W RRadj
(w, r1 , r2 , a)
bW AA
W AAadj
(w, a1 , a2 )
bW A
WA
(w, a)
Whether or not the robot w uses the access a
bW EM
W EM
(w, m)
Whether or not the robot w performs points on the exclusion macro-region m
bW CM
W CM
(w, m)
Whether or not the robot w performs points on the crossing macro-region m
Whether or not the robot w benefits from the adjacency between the regions r1 and r2 via the access a
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Whether or not the robot w benefits from the adjacency between the accesses a1 and a2
Table 3: Parameters employed for occurrence, time controlling and interference constraints Set
Tuple
Ts
-
-
Meaning
Fixed time associated to preparation at stations (Time Units)
Nr
R
r
Number of points in the region r
Tr
R
r
Duration per point in the region r (Time Units)
Tw
W
w
Duration per point for the robot w (Time Units)
Vw
W
w
Relative Speed of the robot w
TmovR
-
-
Movement time between regions (Time Units)
TmovA
-
-
Movement time between accesses (Time Units)
TadjR
-
-
Time gained per adjacency between regions (Time Units)
TadjA
-
-
Time gained per adjacency between accesses (Time Units)
Up
-
-
Upper bound parameter for exclusion constraints
Dp
-
-
Decrease per robot parameter for exclusion constraints
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Parameters
of the possible (w, r, a) combinations were feasible.
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There are many decision variables that follow vW RA, the first one is the binary bW RA. Each bW RA(w,r,a) that is true when its corresponding vW RA(w,r,a) ≥ 1, and false otherwise. The variables in the set bW A are true if the worker w uses the access a (W A is another accessibility
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set).
Movement variables are defined for each w in W : vN M R counts the number of movements
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between regions and vN M A counts the number of movements between accesses. The adjacency variables bW RR (bW AA) are true when the robot w benefits from adjacencies between the regions r1 and r2 (accesses a1 and a2 ) and are defined for each tuple (w, r1 , r2 ) in W RRadj ((w, a1 , a2 )
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in W AAadj ) of feasible adjacency. The last parameter sets for movements are CR and CA, whose tuples (c, r) (and (c, a))
indicate that the region r (the access a) is part of the region cycle c (the access cycle c). Their main purpose is to limit the number of adjacencies a robot can benefit from. They are defined when there are adjacencies that form a cycle, in which case it is possible to have more adjacencies than movements. As shown by Figure 12, a cycle with 4 adjacent regions has 4 adjacencies and 3 movements. This is why cycles are part of the logical constraint of adjacencies between regions and accesses. The set W CR (W CA) indicates that the robot w can perform welding points in all
regions (accesses) of the region (access) cycle cr (ca), and is a set that can be determined by the
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ACCEPTED MANUSCRIPT accessibility sets W RA and W A.
Region
Region
Region
Region
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Cycle
Figure 12: Illustration of the reason cycles are employed to correctly count the number of movements and adjacencies: If a robot displaces through regions that form an adjacency cycle, there will be one more adjacency than the number of movements, this implies that the adjacency time benefits could be higher than the time spent on the required
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movements. Cycle constraints (Inequalities 12 and 13) prevent such wrong consideration.
Finally, the interference constraints require two sets of macro-regions: exclusion and crossing macro-regions. Their parameters are sets EM R and CM R, with tuples (m, r) that list regions r that belong to each macro-region m. The variable set bW EM (bW CM ) contains binary variables for each tuple (w, m) in the set W EM (W CM ) with macro-regions m where the robot w
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can perform welding points. These variables indicate whether or not the robot performs welding points in the macro-region. The crossing constraints also require crossing parameter tuples
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(w1 , m1 , w2 , m2 ) (set W M W M ) that inform that if the robot w1 accesses the macro-region m1 and the robot w2 accesses the macro-region m2 they may collide, therefore, leading to an unfeasible solution. Another variant of the exclusion constraint formulation is the access-wise interference
access set EA.
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constraint, defined based on the number of robots that employ the same access a in the exclusion
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The last variables within the proposed model are the cycle time of the manufacturing line (vCT ) and the cycle time of each individual robot w (vCT Ww ).
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3.3. Constraints
There are three kinds of constraints in this model. First, the occurrence constraint requires that
all welding points in all regions be performed. Second, there are the time measuring constraints that seek to determine the cycle time of each robot and of the line as a whole. Lastly, the interference constraints seek to ensure that robots do not collide during the performance of their tasks. 3.3.1. Occurrence constraint The occurrence constraint states that the sum of welding points performed by all robots by all accesses in each region equals the number of welding points in the region, as stated by the Equation 1. This constraint is defined based on the assignment binding W RA set that allows only robots that can access a region to perform points in it. 18
ACCEPTED MANUSCRIPT
X
vW RA(w,
r, a)
∀r∈R
= Nr
(w, r, a)∈W RA
(1)
3.3.2. Time controlling constraints The cycle time (vCT ) is determined by the line’s bottleneck station, worker or, in this case, robot (vCT Ww ). This is stated in the Inequality 2.
vCT ≥ vCT Ww
∀w∈W
(2)
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The cycle time of each robot w (vCT Ww ) is determined by the Equation 3. The total time of a robot is given by the time required to perform the welding points assigned to it added to the movement time and to the preparation time Ts of its station. The movement time is estimated as the sum of movements between regions (multiplied by the time between regions TmovR ), added to the sum of movements between accesses (idem with TmovA ) and subtracted by the sum of adjacency
vCT Ww =
X
vW RA(w,
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time gains between regions and between accesses (with times TadjR and TadjA ).
r, a)
(w, r, a)∈W RA
+ TmovR /Vw · vN M Rw
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+ TmovA /Vw · vN M Aw X − TadjR /Vw ·
· (Tw + Tr )
bW RR(w,
r1 , r2 , a)
(w, r1 , r2 , a)∈W RRadj
X
bW AA(w,
a1 , a 2 )
+ −
(3)
−
+ Ts
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− TadjA /Vw ·
+
(w, a1 , a2 )∈W AAadj
∀w∈W
The model’s variables that control each part of the robot’s movement approximation are de-
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scribed in the following equations. Firstly, the variables in the sets bW RA and bW A are controlled by the Inequalities 4 and 5. These variables determine in which regions and accesses each robot operates. These constraints establish a fixed charge aspect to the problem: the movement costs
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of accessing regions. This occurs both in regard to the number of points in a region and in regard to the number of regions in an access (Hillier & Lieberman, 2015). As such, it is expected that
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good solutions tend to have fewer movements, concentrating each robot’s work in areas closer to one another.
bW RA(w,
r, a)
bW A(w,
a)
≥ vW RA(w,
r, a) /Nr
≥ bW RA(w,
r, a)
∀ (w, r, a) ∈ W RA ∀ (w, r, a) ∈ W RA
(4) (5)
The first movement variables defined are the movement counters: the number of movements between regions (vN M Rw ) or accesses (vN M Aw ) is the number of regions or accesses minus 1 (unless the robot works in no region (or access) in which case the number of movements is zero). This information is codified in the Inequalities 6 and 7 (as stated in Section 3.2, all variables are strictly non-negative). 19
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vN M Rw ≥ −1 +
X
bW RA(w,
∀w ∈ W
r, a)
(w, r, a)∈W RA
vN M Aw ≥ −1 +
X
bW A(w,
(6)
∀w ∈ W
a)
(w, a)∈W A
(7)
The adjacency variables can only be set as true if the robot w works at both regions r1 , r2 (accesses a1 , a2 ) on the set W RRadj (W AAadj ). This restriction is stated by the Inequalities 8a and 8b (Inequalities 9a and 9b). The adjacency gain between regions can only happen if both
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regions are accessed by the same access a.
bW RR(w,
r1 , r2 , a)
≤ bW RA(w,
r1 , a)
∀ (w, r1 , r2 , a) ∈ W RRadj
(8a)
bW RR(w,
r1 , r2 , a)
≤ bW RA(w,
r2 , a)
∀ (w, r1 , r2 , a) ∈ W RRadj
(8b)
a 1 , a2 )
≤ bW A(w,
a1 )
∀ (w, a1 , a2 ) ∈ W AAadj
(9a)
bW AA(w,
a1 , a 2 )
≤ bW A(w,
a2 )
∀ (w, a1 , a2 ) ∈ W AAadj
(9b)
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bW AA(w,
Region-to-region adjacencies require some limitations: Each adjacency between two regions r1 and r2 (listed by (r1 , r2 ) tuples in RRadj ) can only be used by one robot (Inequality 10). Further-
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more, each robot w can only benefit from two adjacencies tied to a given region r (Inequality 11). These constraints are required to prevent mathematical gains with no real-world counterpart. X
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bW RR(w,
r1 , r 2 )
(w, r1 , r2 )∈W RRadj
X
bW RR(w,
≤ 1
∀ (r1 , r2 ) ∈ RRadj
r, r1 ) +
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(w, r, r1 )∈W RRadj
+
X
(10)
(11)
bW RR(w,
r2 , r)
(w, r2 , r)∈W RRadj
≤ 2
∀(w, r, a) ∈ W RA
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Lastly, the cycle constraints limit adjacencies region-wise (CR contains tuples (c, r) of adjacent regions r that form a cycle c) and access-wise (CA, idem as regions with (c, a) for accesses that
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form a cycle), as stated by the Inequality 12 (Inequality 13). These constraints are relevant for robots w that can access all regions r in the cycle c: even if it performs welding points in all regions of said cycle, it cannot benefit from all adjacencies.
X
bW RR(w,
r1 , r2 , a)
(c, r1 ) ∈ CR (c, r2 ) ∈ CR (w, r1 , r2 , a) ∈ W RRadj
X
(c, a1 ) ∈ CA (c, a2 ) ∈ CA (w, a1 , a2 ) ∈ W AAadj
bW AA(w,
a 1 , a2 )
≤ −1 +
≤ −1 +
20
X
1
∀ (w, c) ∈ W CR
(12)
X
1
∀ (w, c) ∈ W CA
(13)
(c, r) ∈ CR
(c, a) ∈ CA
ACCEPTED MANUSCRIPT 3.3.3. Interference constraints Four kinds of interference constraints were considered: region station-wise exclusivity, crossingmacro regions, exclusion macro-regions, and access-wise exclusion. They seek to describe problems that may happen as robots “compete” for the same physical space. The region station-wise exclusivity states that at each station s, at most one robot w can access each region r. This constraint is stated by the Inequality 14. Note that this constraint prevents a robot from accessing the same region from two different accesses. From a practical standpoint, this
the same number of points via only one access. X
bW RA(w,
r, a)
(w, s)∈W S (w, r, a)∈W RA
≤ 1
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is a desirable feature as that solution would be dominated by the one in which the robot performs
∀ s ∈ S, r ∈ R
(14)
The macro-region allocation variables control the two types of macro-region interference constraints. A robot is allocated to a macro-region if it performs welding points in one of the regions
bW EM(w,m)
≥ bW RA(w,
r, a)
bW CM(w,m)
≥ bW RA(w,
r, a)
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of said macro-region. This is stated by the Inequalities 15a and 15b.
∀ (w, r, a) ∈ W RA, (m, r) ∈ EM R
(15a)
∀ (w, r, a) ∈ W RA, (m, r) ∈ CM R
(15b)
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The crossing macro-region constraints seek to forcefully ban two different robots r1 and r2 from occupying conflicting physical space defined by the macro-regions m1 and m2 (notice that m1 and
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m2 might be the same, but the robots must be different). In essence at most either r1 occupies m1 or r2 occupies m2 . This constraint is codified by the Inequality 16. For instance, suppose that wf is the front robot and wb is the back robot, while mf and mb are the front and back door
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macro-regions. In this case (wf , mb , wb , mf ) impedes the crossing of the robot arm illustrated by the Figure 7: It allows either the front robot to work in the back of the car or (exclusive or)
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the back robot to work at the front of the car.
bW CM(w1 ,
m1 )
+ bW CM(w2 ,
m2 )
≤1
∀ (w1 , m1 , w2 , m2 ) ∈ W M W M
(16)
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Ideally, one might want each overall region of the product to be occupied by at most one robot
per station (and, thus, employ crossing macro-regions for all possible interference zones). If that is the case one could use crossing macro-region constraint with the same macro-region m for two
or more robots. In this case (wf , mf , wb , mf ) would prevent two robots from simultaneously accessing the front door, as in Figure 8. But in some cases, due to accessibility constraints, this might simply be impossible or lead to a poor solution. Exclusion macro-regions are less restrictive interference constraints that seek to allow two robots to be assigned to the same zone in a way that they will together spend less than the cycle time in said zone. In this case, because the cycle time correlates very well with the number of welding points performed, it is stated that the sum of points performed will be limited by an upper-bound value (parameter Up ) that decreases (by 21
ACCEPTED MANUSCRIPT the parameter Dp ) for each robot that works at the same exclusion macro-region. This decrease happens because robots take time to enter and leave the access. This is stated by the Inequality 17. This constraint that can be seen as an adaptation of resource constraints, as described on Section 1. X
vW RA(w,
r, a)
(w, r, a)∈W RA (w, s)∈W S (m, r)∈EM R
≤ Up −Dp · −1 +
X
bW EM(w,
(w, s) ∈ W S (w, m)∈W EM
m)
∀ s ∈ S, m ∈ EM (17)
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The goal of Inequality 17 is to limit the number of points performed in a macro region by a decreasing limit: the higher the number of robots in the same station performing welding points in the same macro-region, the smaller the total number of points they can perform. For instance, in the case studies of section 4, the upper bound value of number of the points (Up ) was 24. If two robots work at the same station in the same macro-region this limit decreases to 20 (therefore, Dp equals 4), as time will be required for one robot to leave the macro-region and for the other robot
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to enter it. These parameters were defined based on observation.
The access exclusion constraint is defined by the Inequality 18. This constraint is very similar to the macro-region exclusion constraint defined by Inequality 17. The main change is that the number of points performed on a same station by different robots via the same access (rather than macro-region) decreases the more robots simultaneously access said access-region.
(w, r, a)∈W RA (w, s)∈W S
≤ Up − Dp · −1 +
X
(w, a)∈W A
bW A(w,
a)
∀ a ∈ EA, s ∈ S (18)
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3.4. Objective function
r, a)
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vW RA(w,
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X
For the presented case study, the total number of robots and the necessary spot welding points are given. Therefore, the objective of the model is to minimize the cycle time of the line, as
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described in Expression 19. Other objective functions are also possible, such as the minimization
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of the total number of robots (an ALBP-1 variation).
Minimize: vCT
(19)
4. Case studies The optimization models were developed based on one of the robotic spot welding lines of Renault’s facility in the outskirts of Curitiba in Brazil. The line is depicted in a simplified manner by the Figure 13. It is composed of 42 robots distributed symmetrically (left and right sides of the vehicle) amongst 13 stations. Each stations had either two, four or six robots. When this model was developed, there were four car models (vehicle types) that passed through this manufacturing line, each of them with over 700 welding points. This was the last part of the car 22
ACCEPTED MANUSCRIPT body manufacturing line, the body-in-white stage: the pieces were all assembled and, therefore, there were no precedence relations to deal with.
IN
OUT
Figure 13: Studied line’s simplified layout, with 42 robots allocated to 13 stations
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The line was a mixed-model one with asynchronous pace, however, the total workload of each vehicle was very similar and their individual cycle times were not significantly different. This was partly due to the company’s previous divisions of welding points, which were performed in such a way that left this line with a rather similar workload for all car models. Although in other parts of the manufacturing line sequencing could be a significant problem (see Boysen et al. (2009) for a mixed-model sequencing survey), in this part each vehicle could be balanced individually, allowing
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four (simpler) single model optimizations to replace a much more complex mixed-model one. This reduction of a Mixed-Model ALBP into P-Single-Model ALBPs could have downsides in manual lines due to the lack of specialization effects and set-up times (Becker & Scholl, 2006). The robots do not display learning curves and can easily alternate between products, allowing us to set these considerations aside. The main mixed-model aspect remaining was the accessibility limitations, defined based on the geometrical features of both: the robot and the vehicle models. However,
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this was considered an input rather than a variable. The goal was set to the minimization of cycle
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time of each product model. Therefore, one main instance was created for each vehicle. 4.1. Instances definition and data calibration The process of defining instances was not trivial: Firstly, data was collected as for where were
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the welding points. Secondly, the model’s parameters (such as movement times, fixed times per robots) were determined by filming the robot’s activities and carefully observing the videos. The
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model was designed to be slightly conservative, and, therefore, when a parameter (say movement time between non-adjacent regions of the same access) was observed with different values in different cases, the worst case was picked.
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In order to assure that estimated task times are close to real times, a calibration must be
performed. Figure 14 shows a comparison between the model’s predictions and the robot’s real times after the due data calibration for one instance. The model’s approximation on the station workload (measured in time units) is higher than the real workload, indicating that any solution obtained by the proposed model will be an upper bound to the real solution. Once the parameters had been defined, points were grouped in regions, based on accessibility data. Such data was not initially available and was determined using a robotic simulation software owned by the company. This software had 3D models for each vehicle and robot, allowing to verify before implementation if each of the welding points in a solution could, indeed, be reached. Each region was assigned accesses from which robots could reach it. Adjacency, macro-regions and 23
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Figure 14: Comparison between the model’s approximation to the empirically measured time: Notice that the
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approximations are slightly conservative
cycle parameter tuples were also defined based on the observed geometrical characteristics. The instances’ data were adapted in order to preserve both: the reproducibility and the company’s private information.
The model and data from each product variant allowed the definition of Mixed Integer Linear Programming models for each vehicle. In order to evaluate the computational burden in regard
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to the problem’s size, instances were generated with various percentages of the total number of welding points. Each region’s number of welding points was multiplied by a reducing factor and
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rounded down to an integer. These factors were chosen, in a manner that instances would have, approximately, 10%, 20%, ..., 90% and 100% of the total number of welding points. The 100% instances represent the practical cases. Data on the Supporting Information present the tuple and
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parameter sets required to define each problem instance. Model files (.lp format) are also provided, along with the best solution files for each instance, containing the value of the objective function
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and of each variable presented in the model file. Thus, reproducibility is assured by the provided detailed information.
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4.2. Results
The computational testes were conducted using a Core i7 (2.3 GHz) computer with 8.0 GB of
RAM and a universal MILP solver was employed to solve all instances. The computational time limit of each run was set to one hour. Table 4 illustrates the solution of the computational and practical case studies presented in Section 4.1, each variant representing a different vehicle. Four vehicle models are analyzed, in instances that vary from 10% to 100% of the total number of welding points presented in the real line. The table contains the results of the best cycle time found in 3,600 seconds of solution time and the lower bound for each instance. The computational tests indicate that the number of points to be distributed influences the computational load. However, this number is not the only factor to be considered: Optimality was more difficult for Variant 4 problems
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ACCEPTED MANUSCRIPT Table 4: Number of welding points (N), upper bounds (UB) and lower bounds (LB) on cycle time, and computation times for each instance of the case studies presented in Section 4.1. Each variant represent a product type. The 100% instances represent the practical case for each product. Variant 1
Variant 2
N
UB
LB
time
N
UB
LB
time
10%
73
1,026.0
1,026.0
0s
68
972.0
972.0
74s
20%
146
1,439.5
1,439.5
2s
146
1,452.0
1,452.0
2s
30%
219
1,578.0
1,578.0
3s
221
1,564.4
1,564.4
4s
40%
287
1,745.5
1,745.5
3s
295
1,782.0
1,782.0
4s
50%
366
1,956.0
1,956.0
7s
372
1,980.0
1,980.0
4s
60%
459
2,208.0
2,208.0
34s
426
2,074.4
2,074.4
25s
70%
516
2,357.5
2,357.5
4s
518
2,334.0
2,334.0
19s
80%
591
2,561.5
2,561.5
13s
591
2,515.6
2,515.6
53s
90%
642
2,712.0
2,712.0
9s
640
2,664.0
2,664.0
5s
100%
733
2,867.5
2,867.5
45s
740
2,868.0
2,838.0
3600s
% Points
N
UB
LB
time
N
UB
LB
time
10%
71
1,075.6
1,075.6
2s
70
1,075.6
1,075.6
2s
20%
142
1,404.0
1,404.0
4s
142
1,422.0
1,422.0
6s
30%
217
1,620.0
1,620.0
9s
216
1,620.0
1,620.0
10s
40%
276
1,680.0
1,680.0
456s
273
1,713.3
1,713.3
247s
50%
348
1,986.0
1,986.0
48s
347
1,968.0
1,968.0
242s
60%
440
2,124.0
2,118.0
3600s
438
2,164.0
2,133.3
3600s
70%
495
2,334.0
2,334.0
1379s
492
2,290.0
2,256.0
3600s
80%
568
2,458.0
2,448.0
3600s
564
2,514.0
2,381.0
3600s
90%
599
2,640.0
2,640.0
159s
597
2,538.0
2,448.0
3600s
100%
708
2,874.0
2,736.0
3600s
706
2,868.0
2,723.5
3600s
Variant 4
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Variant 3
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% Points
than for Variant 1, despite the number of welding points. This means that other instance-specific
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factors such as accessibility and interferences might significantly affect the instance’s computational difficulty.
For the practical cases, the company’s mid-term productivity goal was a cycle time of 2880 T.U.
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One of the practical instances could be solved to optimality (relative gap = 0%), while the others couldn’t (up to 5% relative gap). However, all of them allowed the cycle time to reach the company’s
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goal (the value associated with the target productivity, measured in time units per vehicles). The model’s gain means that the studied part of the manufacturing line has the capacity to produce around 6.6% more vehicles without employing a single new robot. Figure 15 illustrates the station-
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wise gain in cycle time. The vehicle corresponding to the model that was solved to optimality was also the one that was produced the most, meaning that it tends to dictate the line’s overall cycle time. Furthermore, according to Table 4, the obtained cycle time difference between models for the real-world case is smaller than 1% (2,868.0 versus 2,874.0) corroborating with the hypothesis
that they could be treated separately rather than on a mixed-model (or multi-model) manner. All models did reach the main cycle time goal (2880 T.U.) and idle times in some of the robots can be used, for instance, to perform welding points that were not performed in this part of the manufacturing line. Figure 15 shows a comparison between the initial workload distribution between stations and the optimized configuration. By computing the trade-off between movements and number of per-
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Figure 15: Comparison between the model’s optimized solution for one of the instances and the empirically measured
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time: The dashed line highlights the smaller largest value for cycle time amongst stations.
formed points, the model is able to better distribute the workload and reduce the flow-shop’s cycle time: In some cases, the sum of robot times (heavily linked to movements) increased by the optimization process, but the line’s cycle time decreased. Notice that this has been achieved under the conservative parameter framework made clear by Figure 14 validation. The obtained solutions to the practical instances were afterwards internally translated back
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from mathematical to tangible practical information and were verified to provide feasible solutions
5. Conclusion
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to the real-world problem by the company’s specialists employing simulation tools.
Balancing robotic spot welding manufacturing lines is a challenging problem. The time required
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to welding the points is short compared to cycle time. However, robots have to move between them and the movement time must be taken in account. Furthermore, geometrical aspects impose assign-
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ment constraints and multiple robots per station impose interference constraints. The combination of these restrictions lead to the understanding that previously described models could not properly describe the problem. A new (MILP) model is proposed and applied to solve a real-world scenario
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from Renault factory in the outskirts of Curitiba, Brazil. Practical case studies were conducted for four vehicle models on a large-size real-world line (42
robots and over 700 welding points for each model). Each of the models was optimized separately, and optimality was not reached for all instances. However, the obtained answers offered cycle time reductions of around 6.6%, allowing the company to reach its mid-term goal of productivity for that part of the factory without the need to buy new robots. Computational studies have shown that the number of welding points is a relevant factor to the model tractability, but not the only one. Some small instances were more difficult than larger counter-parts (see Table 4). This indicates that instance-specific characteristics, such as accessibility and interference, are also relevant factors for computational tractability. 26
ACCEPTED MANUSCRIPT Instances were generated to describe each of the vehicle type produced in the factory. Parameters were set according to empirical observations. The line was a mixed-model one, however in this particular part of the factory all models could individually reach similar cycle times. This verified the hypothesis about the optimization process being able to set aside sequencing considerations and balance each model individually. The results obtained for the case studies were validated by the company specialists through simulation techniques. The insights provided by the model allowed identifying new operational conditions, with smaller cycle times, without the need of investing in new equipment; the model was used to aid in the managerial and operational decision-making
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process.
The new Mixed Integer Linear Programming model provides a framework to optimize robotic welding lines by offering a formulation to simultaneously deal with: Robot-wise variations on parameters, Assignment Restrictions, Movement Times and Interference Constraints. The proposed formulation is limited to lines without precedence relations between welding points. Any ordering of tasks within a station would require the scheduling of operations between the robots. If the
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precedence are station related, such as by adding new layers of sheets, the approach presented in Appendix A can be used.
As future development, this basic framework can be adapted to incorporate other features such as feeder lines and parallel stations. Furthermore, in an assignment line design context, the presented model could be adapted to minimize the number of robots required or optimize a cost-
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oriented function relaxing some of the hypotheses of given robot locations and line layout. This
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cost-oriented approach can be of great value to aid managerial decisions.
Appendix A. Piece joining tasks on defined subassemblies The proposed case studies presented in Section 4 represent the final manufacturing spot weld-
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ing line of the body shop. In this line, the structure of the vehicle is already assembled, so that only reinforcement procedures are necessary. No precedence relations or subassemblies are ob-
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served. However, the proposed model can also be used to balance a robotic spot welding line when subassemblies are known and fixed, along with the accessibility of each task. Suppose Figure A.16 represents a spot welding line consisting of a main line (top row) and a
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sub-product line (bottom row). The third station of the main line represents a subassembly: the product piece of the feeder line is assembled with the main product. The welding procedures can be divided in 6 types of tasks, represented by the precedence diagram on the right. The blue tasks represent the piece joining procedures: the welding points that are necessary to assure the cohesion of the assembled parts. Some welding points may lose their accessibility after the superposition of the two pieces of the assemblage. The green points represent the points that should be performed before the subassembly at the main line, while the orange points should be performed in the sub-product line. The red node represents the procedures that need to be performed after the assemblage: these tasks represent the reinforcement welding points that assure the mechanical resistance of the assembled parts. The red tasks can be performed in the joining station or after 27
ACCEPTED MANUSCRIPT it. Finally, there are some points that are independent on the subassembly. The purple node represent the independent welding points that have to be performed in the main product. The yellow tasks represent the independent welding points that must be performed in the piece that starts at sub-product. Workstations and Accessibility
Assembly-Related Precedence Relations
Main Product Welding Line
Main product welding points Independent of assembly Main product welding points obstructed by assembly
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Welding points associated to assembly or piece joining Sub-Product Welding Line
Sub-product welding points obstructed by assembly
Welding points accessible only after assembly
Sub-product welding points Independent of assembly
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Figure A.16: Example of station-wise accessibility windows for a subassembly.
Based on the type of task and the given precedence diagram, the precedence relations can be written as station-wise accessibility windows. The line’s diagram on the left side of Figure A.16 shows which stations can perform which tasks by colored squares in the workstations. Note that the green and orange tasks must be performed before the subassembly. The blue tasks must be performed at the lines’ intersection. The red tasks must be assigned to the overlap or stations
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after it. The independent purple welding points can be performed in any station of the main line. Finally, the independent yellow points can be performed in any station of the feeder line and any
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station of the main line after the sub-product line piece has been transfered to the main line. With the station-wise accessibility window concept, precedence relations that resulted from
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known and fixed subassemblies can be modeled as robotic accessibilities.
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Acknowledgments
We would like to acknowledge the financial support from Funda¸c˜ao Arauc´aria (Agreement 141/2015) and CNPq (Grant 305405/2012-8). In addition, we would also like to show our appre-
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ciation to Renault for the opportunity of working with them and their manufacturing lines.
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Balancing a Robotic Spot Welding Manufacturing Line: an Industrial Case Study Thiago Cantos Lopes, Celso Gustavo Stall Sikora, Rafael Gobbi Molina, Daniel Schibelbain, Luiz Carlos de Abreu Rodrigues, Leandro Magatao ˜ PII: DOI: Reference:
S0377-2217(17)30518-0 10.1016/j.ejor.2017.06.001 EOR 14482
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
8 November 2016 22 March 2017 1 June 2017
Please cite this article as: Thiago Cantos Lopes, Celso Gustavo Stall Sikora, Rafael Gobbi Molina, Daniel Schibelbain, Luiz Carlos de Abreu Rodrigues, Leandro Magatao, ˜ Balancing a Robotic Spot Welding Manufacturing Line: an Industrial Case Study, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.06.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Highlights • A model for Robotic Spot Welding Manufacturing Line Balancing is presented. • The model takes in account accessibility and movement times between welding points. • Multiple robots per station define interference constraints in the model. • The models predictions are validated by comparisons with empirical data.
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• Up to 6.6% productivity gain achieved when compared to the as-is configurations.
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Balancing a Robotic Spot Welding Manufacturing Line: an Industrial Case Study Thiago Cantos Lopesa , Celso Gustavo Stall Sikoraa , Rafael Gobbi Molinab , Daniel Schibelbainc , Luiz Carlos de Abreu Rodriguesb , Leandro Magat˜aoa∗ a: Graduate Program in Electrical and Computer Engineering (CPGEI) Federal University of Technology - Paran´ a (UTFPR), Curitiba, Brazil, 80230-901 b: Department of Mechanical Engineering - UTFPR
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c: Renault do Brasil, S˜ ao Jos´ e dos Pinhais, Brazil, 83070-900
Abstract
Resistance spot welding is an important procedure used in the automobile manufacturing industry. After stamping, sheets of metal are mostly welded together to build the car’s body. Many of
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the industrial robots found in assembly lines are spot welders. Process characteristics include accessibility limitations, the need to move between welding points and (when there are multiple robots per station) potential interferences between robots. The combination of such characteristics is not present in classical models for the assembly line balancing problem. In this paper, the balancing problem of robotic spot welding manufacturing lines is presented and modeled based on
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a real-world car factory on the outskirts of Curitiba, Brazil. The studied line has 42 robots that perform over 700 welding points on the later stages of the body-shop. The model was developed with Mixed Integer Linear Programming (MILP) techniques, validated with empirical data, and
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solved with a universal solver. The optimized balancing achieved a cycle time reduction of up to 6.6% compared to the as-is configurations. The total movement time was observed not to
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be necessarily minimized during the optimization process, implying that trade-offs exist between movement times and the number of welding points performed. Keywords: Combinatorial Optimization, Assembly Line Balancing Problem, Robotic Welding
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Manufacturing Line, Mixed Integer Linear Programming
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1. Introduction
The automotive industry strongly relies on flow shop arrangements for the mass production of
vehicles. The flow shop or product oriented system brings as advantages a high capacity utilization, simple flow of materials, and the specialization of workers as a result of the division of labor (Scholl, 1999). The workpieces are moved along a serial arrangement of workstations, which are responsible for specific tasks in the production process. Such functional characteristics propitiate the use of automatic systems or even robotic workers. ∗ Corresponding
author Email address: [email protected] (Leandro Magat˜ aoa )
Preprint submitted to European Journal of Operational Research
June 7, 2017
ACCEPTED MANUSCRIPT In the industry, the flow shop arrangements are well represented in the assembly lines used to transform simpler components in assembled products. According to Michalos et al. (2010), the automotive processes can be roughly divided in four parts: stamping, body shop, painting, and final assembly. A product must go through all the stages, that should all be planned for an aimed throughput rate. Therefore, the cycle time is a key factor for all stages of the industrial production. Due to the simple flow of workpieces, the workstation that contains the longest processing operations is the system’s bottleneck. Hence, one of the main objectives of planning a production system is to evenly divide the tasks among the workstations to increase their efficiency. Obtaining
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the best distribution of the operations or tasks is the objective of the Assembly Line Balancing Problem (ALBP), first proposed by Salveson (1955). The base problem and its simplifications were defined by Baybars (1986) resulting in the classification of Simple Assembly Line Balancing Problem (SALBP) and General Assembly Line Balancing Problem (GALBP). According to Wee & Magazine (1982), ALBP are NP-Hard once they are a generalization of the Bin Packing Problem. Most of the assembly operations in the automotive industry are performed in the body shop
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and final assembly stages. The body shop is usually highly automated, while the final assembly is performed by manual, robotic, or hybrid systems (Michalos et al., 2010). Although it is not commonly stated in the articles, most of the academic developments on ALBP (recently reviewed in Batta¨ıa & Dolgui (2013)) relates to the final assembly. In the less reported stage of the process, the body shop, sheets of metal are transformed into the car body (Body-in-White) with welding
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procedures. The welding procedures must also be evenly divided among the workstations, however, welding tasks present differences compared to the final assembly operations. The welding itself
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occurs in less than a second, however, the welding tool movement and positioning account for the major part of the cycle time. Moreover, the accessibility of welding tasks and the path that link the operation are important to correctly measure the station processing time.
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It is common that robotic welding workstations contain more than one welding worker or robot. According to Dimitriadis (2006), multi-manned workstations, that is stations with multiple (robotic) workers, can bring advantages such as fewer workstations, shorter line length, and the
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smaller costs of product handling and station equipment. The balancing of robotic welding lines can then be described as a combination of problems. The correct balancing of welding points
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consists of an assignment problem of welding procedures to stations and/or robots, a path routing problem for each robot, and a robot coordination problem, once they must not interfere with each other.
Some previous works treated the combination of the problems or parts of them. Segeborn et al.
(2013) developed a heuristic and simulation based method for the balancing of spot welding points in robotic workstations with multiple robots. Their approach consists of a greedy algorithm for the tasks distribution, followed by a Simulated Annealing improvement procedure. In order to reduce the need for robot coordination, a separation procedure is applied sampling pairs of points at random and swapping assignments if the separation can be improved. In order to calculate the cycle time, the distances between two welding points are calculated with approximation methods.
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ACCEPTED MANUSCRIPT Segeborn et al. (2013) states that it is impractical to calculate all the distances between points considering the geometry of the pieces and the robot. Therefore, two simplified distance matrices are used: one that do not consider the geometry of the piece and robots and one that represent the robot as simple lines. The path planning is then performed by a Traveling Salesman Algorithm using the approximate distances. Finally, the coordination of robots is performed via simulation. The speed of each movement can be controlled or wait times can be inserted to avoid collisions, increasing the cycle time. The coordination simulation is also simplified: the flexible parts such as cables and hoses are not considered. Segeborn et al. (2013) tested their approach on a line with 3
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workstations with 10 robots that performed 201 spot welding procedures.
Akturk et al. (2011) present a MILP model for the balancing of robotic spot welding lines considering tool change costs. Both the cycle time and the tool costs are minimized in a costoriented model. Welding points are assigned to robots, whose tool life is measured throughout the production phases and can be changed in pauses or shift changes. In the modeling, similar points are gathered in groups. Each group of welding points is considered a task, whose processing time
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is a constant representing the time to reach the first point plus the time required to perform the welding operations. The points are not assigned individually, all the welding point group must be assigned to a single robot. Therefore, the movement time is simplified and integrates the task time. The model is developed for single robots in each station, so that no path planning and no robot coordination is presented.
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Another work that presents itself as a variation of an ALBP formulation is due to Nilakantan et al. (2015). They present a MILP model and a Particle Swarm Optimization method for the
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minimization of cycle time and energy consumption in robotic assembly lines. There are different models of robots that can perform tasks at different rates with different energy requirements. A multi-objective approach presents the concurring effects of production rate and running costs. The
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authors state that the formulations are adequate to robotic lines for the body shop. However, tasks are consider as ‘classical’ ALBP operations, the same precedence diagrams used for the final assembly are used in the computational tests. Moreover, no movement time between procedures,
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path planning, or robot coordination is considered. The literature related to path planning is strongly related to General Traveling Salesman Prob-
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lem (GTSP). Saha et al. (2006) consider a given set of points assigned to a single robot. The authors developed a heuristic method for the path planning considering obstacles caused by the piece accessibility. Their procedure consists of a TSP and a path calculation algorithm. The authors state that calculating all distances between welding points consumes more CPU time than solving the TSP problem itself. Therefore, the proposed procedure sparingly calculates the distance between points by delaying the calls of the path algorithm to only when they are needed. Similar approaches are presented by Carlson et al. (2014) and Wurll et al. (1999). Both works calculates a path between given points considering obstacles for stations with only one robot. As Saha et al. (2006), the authors also use techniques to reduce the number of distance calculations between welding points.
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ACCEPTED MANUSCRIPT Xie & Hsieh (2002) present a genetic algorithm to sequence procedures in spot welding stations. The authors aim for the minimization of the cycle time and the piece’s geometric deformation. The presented algorithm is applied to welding points aligned linearly, so that the travel distance calculation is obvious. No obstacles are considered, as well as no scheduling and no coordination between multiple robots are necessary. The robotic coordination of multiple workers in a cell is considered by Spensieri et al. (2013). This work considers a given assignment of points and an operation sequence for each robot. The problem consists in determining a path for each robot so that the robots do not collide. An MILP
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model and a Branch-and-Bound procedure were developed to solve the problem by adding wait times in the scheduling of robots. Therefore, for the given assignment of points and sequences, the model minimizes the necessary wait time.
Chakraborty et al. (2010) treat a related problem: the robotic laser drilling of electrical components. The authors present an heuristic procedure (greedy plus improvement algorithm) for the task assignment, path planning, and robot coordination considering 2 or 4 robots. The electric
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components are composed of planar plates in which drilling operations must be performed. The tasks are assigned among the robots, whose paths are calculated with a TSP algorithm considering the interference between robots. Once the planar plates do not contain obstacles, the distance between two points can be determined by euclidean distance.
As the previous works have shown, the balancing of robotic spot welding lines is composed of
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several problems that are treated mostly with heuristic procedures. For the best of the authors knowledge, the largest spot welding line reported as case study contained 10 robots allocated in 3
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stations performing 201 welding points.
This paper proposes a model to treat such combinations of problems related to the balancing of robotic spot welding assembly lines. An MILP formulation is presented for the assignment
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problem of spot welding procedures among multi-manned robotic workstations. The modeling approach simplifies the path planning and robot coordination by distributing the welding points considering such aspects. Restrictions are provided to measure in approximative form the robots’
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movement time and occupation time in the vehicle regions. The final path planning can then be determined via simulation.
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The paper is structured as follows: Section 2 presents the production context on robotic body shop lines. The section also introduces the studied problem, describing its characteristics and presents a literature review discussing ALBP prior studies that relate to the problem characteristics. Section 3 describes the developed model from its basic concepts to its sets, variables and constraints. Section 4 describes real-world industrial case studies, the problems’ size, achieved results and discusses some practical insights. Section 5 summarizes the main results and contributions of this paper.
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ACCEPTED MANUSCRIPT 2. Balancing robotic spot welding manufacturing lines 2.1. Body-in-white production context The body-in-white is a stage in automobile manufacturing in which sheet metal components are welded together. The process starts after the stamping of the sheets, pieces are assembled together forming frames of the vehicle (front, back, base, left, right, and roof parts). The frames are then mounted together, resulting in the body of the car. The assembly process is generally divided along assembly or manufacturing lines. The welding
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of pieces can be performed either by manual labor, robots, or dedicated welding machines (Aslanlar et al., 2008). The piece handling between stations and lines can be performed manually or with the use of robotic handlers, conveyors, or tow trains.
Among other possible methods, the most used assembly procedure in the automotive industry is Resistance Spot Welding (RSW) due to its high reliability, low cost, manufacturing robustness, and automation potential (Barnes & Pashby, 2000; Gould, 2012). With RSW, two or three metal
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sheets can be united by electrical current. The heat provided from the metal resistance locally melts the sheets, welding the components together. The welding procedure takes less than a second (Aslanlar et al., 2008). The spot welding is performed by two electrodes that are positioned in a welding gun. The gun must access both sides of the piece, therefore, the accessibility of points is a disadvantage of the RSW (Barnes & Pashby, 2000). Furthermore, the quality of the welding
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depends on the surface conditions of sheets and electrodes, applied force, and size and contour of the electrodes (Aslanlar et al., 2008). Therefore, multiple types of electrodes and guns may be necessary for the welding procedures.
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Each welding point can be seen as a task to be performed in the body assemblage. Automobiles require from 3000 to 7000 welding points based on their size (Hamidinejad et al., 2012). For the joining of sheets, pieces must be positioned respecting geometric tolerances. Parts are clamped
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in place, usually by pneumatic or electric actuators until pieces are properly united. According to Sikora et al. (2017b), welding points can be divided in two types of tasks: piece joining and
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reinforcement procedures. The piece joining tasks assure the cohesion of pieces for further handling. They require the actuators to hold the piece in the correct position. The reinforcement welding
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points are needed for the final product resistance, however, they do not require actuators anymore: the joining points are sufficient to assure the geometric tolerances and the piece can be handled to other stations. The balancing of spot welding manufacturing lines consists in the distribution of welding points
along the workstations or lines. There are, however, some specific considerations related to the nature of spot welding procedures: • Accessibility: due to quality requirements, the position of pieces and the reach of the welding guns, not all points can be performed by all electrodes. Furthermore, for spot welding, both sides of the sheets must be accessible for the gun. New sheets block the access of inner
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ACCEPTED MANUSCRIPT layers. Although there are theoretically no precedence relations between two welding points, the assembly of parts force a sequencing of points based on station-wise accessibility windows. • Processing time: in the balancing of assembly lines, operations must be made within a cycle or takt time. The required time for a spot welding varies between a small interval (less than one second), but the time for positioning and the movement between points is relevant and must be also accounted for. • Multi-operators: the automobile industry frequently uses multiple workers or multiple robots
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in the same station. Although there are usually no precedence relations between welding points, space disputes or interferences may be observed (due to the sizes and shapes of the vehicle and robots). Precedence relations might relate welding points to piece joining, but provided these operations are fixed the precedence relations can be converted to station-wise accessibility windows (see Appendix A for details).
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2.2. Problem statement
Otto & Otto (2014) showed that the automotive industry organizes its operations based on modules. The lines are organized to be responsible for a module, whose precedence relations are easier to identify and map. After the operations of a line, there is not much opportunity to perform changes or extra operations in further modules: the interaction between modules is seldom observed.
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During the welding procedures, pieces vary from smaller and simpler sheets into compound structures that are then welded together in the body-in-white along several assembly lines. Due to
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the progressive addition of layers of metal sheets, much of the welding points lose their accessibility throughout the process. It is very difficult to map the entire welding operation (from 3000 to 7000 points) for all the types of electrodes and guns (which have different accessibilities) considering
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every combination of subassembly (the order in which sheets are welded together). Therefore, the lines are usually organized in functional subassemblies: different stations build specific sub-
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products. There are also defined modules for the left, right, top, base, front, and back part of the car. Therefore, it is reasonable (and sometimes is the only possible form) to model each line separately.
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Even considering a single assembly line, an optimization approach could consider the balancing
of operations or the project and design of the line itself. Reassigning welding operations to workers and robots is relatively easy: orienting the worker or programming the robot may be sufficient. Changing a subassembly order or location requires changes in the layout: pieces handling systems and actuators must also be reallocated. Therefore, in this paper, the balancing of a welding line refers to the allocation of points regarding a pre-defined layout. Altering or choosing the layout and subassembly order refers to a design problem. According to Falkenauer (2005), the re-balancing of assembly lines is much more common than the project of a new one. Changes in the product, demand, and the inclusion of another product variant are reasons for the re-balancing of assembly lines. 7
ACCEPTED MANUSCRIPT After the presented considerations, the research problem treated in this paper refers to the balancing of robotic spot welding lines used in automotive industry. The proposed problem is defined based on the following hypotheses and characteristics: C0 - Occurrence constraint - a set of welding points must be performed; C1 - Assignment restrictions - the assignment of points must respect the accessibility constraints associated with the robot’s tools and position; C2 - Robot-wise dependent parameters - Not all robots are equally fast due to their model and
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to the size of their spot welding tools; C3 - Cycle time is decisively influenced by movement time - Each robot’s cycle time is not determined by the simple sum of time of each performed “task”: time of handling of pieces, welding, moving and positioning welding guns must be considered;
C4 - Multiple positions for welding - some points may be accessible from more than one direction; C5 - Multi-manned stations with interference constraints - stations can contain more than one
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robot, which must not collide with each other as they perform tasks;
C6 - Given welding parameters - welding times and accessibility are known; C7 - Given layout - the number, position, and welding guns of robots are known and fixed for a given layout;
C8 - No precedence relations - there are no precedence relations defined between any two weld-
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ing points (although subassemblies might produce station-wise accessibility windows, whose
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explanation and modeling treatment are given in Appendix A).
2.3. Literature review on ALBP formulations for the problem characteristics For the definition of assembly line balancing problems, the spot welding points can be seen as
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tasks whose processing times are similar and short when compared to the cycle time. In principle, balancing might seem simple: simply divide the number of welding points roughly evenly between
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robots. There are, however, a few difficulties: robotic tools must be moved between welding points and the movement time is not usually known between each pair of points - and it is not necessarily proportional to euclidean distance between points, either. Such time cannot be easily determined
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as optimizing robotic routes is a hard problem on its own. Furthermore, stations often have more than one robot. Robots in the same station are subject to potential interference problems if they perform tasks (welding points) at the same regions or cross the robotic arms. In manual stations, this might not be such a serious concern as workers can communicate, coordinate and adapt in real time, but the studied robots blindly follow a predefined set of instructions. The specific tools and fixed positions of robots impose accessibility constraints, meaning that each robot will only be able to access and perform a subset of the welding points. While the characteristic C0 (described in Section 2.2) is simple and a basic part of nearly all assembly line models (Bowman, 1960; White, 1961; Thangavelu & Shetty, 1971; Patterson & Albracht, 1975; Scholl, 1999; Sikora et al., 2017b), it is made slightly more complicated by the 8
ACCEPTED MANUSCRIPT characteristic C1, as explained hereafter. Figures 1 and 2 illustrate why the problem is assignment bounded, as each robot can access only a subset of welding points, due to geometrical aspects. These assignment constraints can be seen as station-type restrictions in the classification schemes proposed by Boysen et al. (2007, 2008); Batta¨ıa & Dolgui (2013). In Figure 1, the robot can reach the welding point without colliding with the vehicle. In Figure 2, the robot will collide with the
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vehicle, if it attempts to reach the welding point.
Figure 2: Inaccessible welding point.
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Figure 1: Accessible welding point.
There are models (Scholl et al., 2010) and algorithms that deal with various assignment restrictions (Dar-El & Rubinovitch, 1979; Pastor & Corominas, 2000; Lapierre & Ruiz, 2004; Bautista & Pereira, 2007) and, thus, could be used to deal with C1. C2 is a pretty common part of robotic balancing problems (first defined by Rubinovitz & Bukchin (1991)). The problem would probably
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be treatable by assignment bounded or adapted RALBP algorithms (Kim & Park, 1995; Levetin et al., 2006; Daoud et al., 2014), if C1 and C2 were the problem’s most challenging characteristics. But they are not. C3 implies on constraints that make this problem harder to translate into a
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SALBP-based model, as it violates a core hypothesis of such models: a station’s (robot’s) time is not equal to the sum of the task’s time performed in it. This fact happens because the welding time
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is comparable to the movement’s time between points. Figures 3 and 4 show how relatively close welding points might require absolutely different configurations for the robot: this fact implies that
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movements are not only a matter of Euclidean distance.
Figure 3: Robot performing a welding point in the Figure 4: Robot performing a welding point back of the vehicle, before moving to a nearby weld- close to the first one (Figure 3). Notice how ing point.
the robot’s position has changed significantly.
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ACCEPTED MANUSCRIPT There are set-up models (described for instance by Scholl et al. (2013)) and algorithms (Andr´es et al., 2008; Scholl et al., 2008; Martino & Pastor, 2010) that could attempt to describe C3 by comparing movement times to setup times. But these models and algorithms require a matrix of known set-up times between tasks. Such data is not available as the time to move between each pair of welding spots is rather difficult to determine, especially given the large number of welding points involved. One could attempt to use an approximation for such movement times, but such times would have to be robot-dependent. The line has robots of different models with different speeds and tools of different sizes, aka C2. This might not be a promising direction though.
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Attempting to adapt the problem at hand to the model described by Scholl et al. (2013), the number of variables and constraints would tend to be too large: In the studied scenarios, there were 42 robots. Assuming each of the roughly 700 welding points can be accessed by 10% of the robots (rounding down to 4) and that the points can be followed or preceded by about a third of the other points (rounding down to 200), the sequence variables (yij and wij , see Scholl et al. (2013) for more details) would amount to 280.000 binaries and would require nearly half a million constraints to be
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properly controlled. Such a MILP problem is almost certain to be computationally un-treatable. And that is setting aside the fact that the set-up times are robot-dependent. Successfully operating some simplifications, there are heuristics and algorithms that could, maybe, be able to provide good answers, were it not for the other characteristics of this problem.
The feature C3 is made even harder by the fact that some welding points can be performed in
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different manners (C4), by different accesses for the robotic arms. The manner a welding point is accessed affects the movement time the robot will take to reach other points. Two points may seem
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really close and there might be a direct route between them but only if they are performed with the robotic arm positioned in a way that such route is feasible. These characteristics are illustrated by Figures 5 and 6, where the robot is accessing the same welding point from two different directions:
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suppose the robot is to move to a welding point between the doors. Such point is reachable for the robot in Figure 6 with a simple spin of its tool. The robot in Figure 5, in the other hand, must first leave the windshield region, then spin while moving inside the front door (without colliding
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with the vehicle), and only then it reaches the welding point. A symmetrical situation occurs if the robot were to move to a welding point in the central lower part of the windshield region: this
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time would be easier for the robot in Figure 5 move between the points. This means that a setup time approximation to movements between welding points would not only be robot-dependent but also access-dependent, making the use of a set-up based models and algorithms even more challenging. Parallels with assembly lines with processing alternatives models (Pinto et al., 1983; Scholl et al., 2009) can be drawn, but the nature of the alternatives is, in this case, rather different. The processing alternatives presented in ALBP literature relate to different forms of performing operations resulting in different subgraphs. The direct effect is multiple values of task processing time and precedence relations. In the spot welding lines, the different forms of accessing a point are mainly related to the movement times and interferences. But even if it were possible to overcome this process alternative aspect and treat C0, C1, C2
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Figure 5: Welding point accessed through the wind- Figure 6: Welding Point accessed through the front shield.
door.
and C3 using set-up based assignment bounded robotic balancing model, there is C5: multiple robots can be assigned to the same station, so that robots must not collide with one another while
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performing the welding points. Multi-manned workstations are commonly used for the production of large products because such special workstations can reduce the line length and, consequently, some costs related to piece handling systems, tools, and support systems (energy and pressure inlets, gas exhaustion, etc.). However, multi-manned stations bring as disadvantages the necessity to coordinate the worker operations (Dimitriadis, 2006).
In the ALBP literature, two approaches for the simultaneous processing of a piece by more than
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one worker are employed: the 2-sided (or N-sided) assembly lines and the multi-manned assembly lines. The 2-sided assembly line balancing was firstly introduced by Bartholdi (1993). The studied
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assembly line contained workstations on both the left and the right side of the produced vehicle. Some tasks must be performed by a specific worker: on the left or on the right side. Other tasks could be performed by the worker of any side. Once precedence relations are expected between
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tasks, it is necessary to schedule the operations within the workstation to assure its feasibility. Scholl et al. (2009) generalizes the problem presenting the Assembly Line Balancing Problem with
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Variable Workplaces (VWALBP). The authors considers up to 24 divisions of a vehicle. Multiple workers can perform operations in the same station, but they cannot work in the same regions of the vehicle.
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The second approach that considers multiple worker in the same station was introduced by
Dimitriadis (2006) with a heuristic solution procedure. The Multi-Manned Assembly Line Balancing Problem (MALBP) is based on the hypothesis that the product is large enough to have workers performing tasks without blocking each other. In each workstation, however, the precedence relations must be respected. Therefore, the scheduling of operations is necessary. Both approaches (VWALBP and MALBP) consider a maximal number of possible workers in a station empirically. The authors describe that, in the automotive industry, stations can be divided in up to 4 or 5 workplaces. An MILP model for the MALBP was presented by Fattahi et al. (2011), along with an Ant Colony Optimization procedure with the objective of minimizing the number of workstations. Further recent developments proposed the inclusion of other factors in the objective function, such
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ACCEPTED MANUSCRIPT as the number of workers (Kelleg¨oz, 2016) or the load balance (Yilmaz & Yilmaz, 2015). Roshani & Giglio (2016) considered the MALBP type-II, in which the station length is fixed and the number of workers and the cycle time is minimized. Among the presented papers, MILP formulations are developed, however only small and medium instances can be solved with commercial solvers (Fattahi et al., 2011; Kelleg¨oz, 2016; Yilmaz & Yilmaz, 2015). Note that MALBP does not concern the physical location in which a task is performed. Once the precedence relations are respected, the solution is said feasible. Sikora et al. (2017a) also presents a model in which multiple workers perform tasks at the same station. The multiple workers, however, only perform together tasks
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that require more than one worker. Putting these common tasks aside, each worker performs tasks in separate stations.
The body-shop environment present fewer or no precedence relations between the spot welding points. In fact, an ordering is implied based on the subassemblies, but they are due to the accessibility of metal sheets and not the direct effect of a welding point (see Appendix A). The interference between robots is due to the space competition, as it is exemplified in Figure 8: two
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robots cannot perform tasks in the same regions at the same time. Therefore, the approach of MALBP models is not adequate to deal with such restrictions. On the other hand, the definition of VWALBP states that only one worker can access a vehicle region in a workstation. Although that is enough to avoid the interference of robots, this hypothesis is too strict. There is also the possibility of robots accessing the same region, but at different moments within the cycle time (e.g.
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one robot starts at the front door, and moves to another region while a second robot enters the front door after it becomes available). The available time the robots can spend on a region can be
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seen as a scarce resource. However, the amount of available time depends on the number of robots that accesses the region: the entering and exiting time must also be considered. Another interference (C5) example lies in robots crossing arms (Figure 7). It is possible for
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either the front robot perform welding points in the back of the car, or the back robot perform welding points in the front of the car. However, if they do so simultaneously the chances of collision of the robotic arms are great, as depicted by the Figure 7, where both robots perform accessible
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welds, but the station’s configuration is undesirable.
Figure 7: Arm Crossing Interference.
Figure 8: Space Dispute Interference.
These interference characteristics are very difficult to translate into adapted features of the 12
ACCEPTED MANUSCRIPT classification schemes of Becker & Scholl (2006), Boysen et al. (2007) and Batta¨ıa & Dolgui (2013). Consequently, the combination of all the problem’s features give rise to a very complex ALB problem that is even more difficult to adapt into previously described models. Prior to the work described in this paper, the studied manufacturing line’s balancing was performed manually, based almost exclusively on the engineer’s and operator’s experiences. To the best of our knowledge, there is no literature model that can describe all these characteristics in a single model. This means that balancing such manufacturing line requires a new model.
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3. Model 3.1. General concepts
The employed model operates some simplifications in order to approximate the real-world problem. The main simplification is that no sequence for points performed by each robot is established, but the rather the number of points are counted along with the number of movements (classified
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as: very small, small, medium and large) required to perform said points. This simplification is based on two observed facts: Firstly, most of the welding points were distributed in logical paths such that the robot’s time spent on that path was proportional to the number of points performed. Secondly, when the robot moved from one path to another, the time spent on such movement could usually be classified into one of four classes, which are further discussed at the end of this section. Robots are presumed to be fixed to a station, with fixed given properties and parameters (such
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as speed, tools, position and capabilities). Two parameters of interest were the relative velocities of each robot and the fixed time required per welding point. This robot-wise fixed time is associated
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with opening and closing the welding claws. In general, newer robots tend to allow higher speeds due to better technological features and robots with bigger tools (often required to access some welding points) tend to have lower speeds and longer fixed times due to slower operation. All
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stations have a fixed constant time associated with the entrance and exit of each vehicle of said station, time during which it is assumed that no welding point can be performed.
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Welding points are grouped in regions, as depicted by the Figure 9 in which welding points (red dots) are boxed in accordance with physical characteristics and the accessibility constraints.
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Regions have the following characteristics: • All welding points in a region take the same amount of time to be performed. This time is added to fixed time per point of the robot that performs it. The movement time between points of the same region can be regarded as part of the time required to perform a welding point; • The time spent by a robot in a region is proportional to the number of points performed by said robot at said region; • If a robot can access the region, it can access all points in it; • Each region r has a fixed and given number of points Nr ; 13
ACCEPTED MANUSCRIPT • In a first approximation, the movement time is considered constant between regions. That is: if a robot works at three regions, it will move twice between regions and have a movement time twice as great as a robot that only works at two regions (and thus only moves once between regions); The region hypothesis was based on a problem’s characteristic: welding points were mostly lined up in logical geometrical paths and locations, and these logical paths could be divided in
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accordance to how the robots could access them.
Figure 9: Example on how welding points were grouped in regions (identified by different colors), adapted from the case study.
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Welding points can be performed from different accesses: front door, back door, luggage compartment, windshield. Some regions can be reached by multiple accesses (Figures 5 and 6). In order to better describe movements, accesses must be taken in account. If a robot is accessing
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regions within the same access, then the movement time tend to be smaller than when the robot accesses regions from different accesses. The concept of accesses adds a different kind of movement
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to the problem. Movements between accesses are analogous to movements between regions and can be analogously counted. Figure 10 illustrates the different regions reachable by accesses: some points can only be performed with the robotic arm positioned in a certain access, for instance green
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indicate regions accessible through the back door, while blue indicates regions accessible through
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the front door.
Figure 10: Accesses for the regions and adjacencies.
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ACCEPTED MANUSCRIPT Further movement time refinements are provided by the adjacency concept: Two regions that are next to one another (one is connected to the other) are deemed adjacent, meaning that in some cases the movement between them can be ignored or reduced. For instance, the regions that surround the front door can be seen as adjacent (as shown in Figure 10). The same applies to accesses. The movement time between the front and back doors is naturally smaller than the movement time between the front door and the luggage compartment. In many cases a large physical region must be divided in several adjacent regions because of accessibility constraints: if a robot can access the region, it can access all the welding points in it. Therefore, regions were
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modeled according to both accessibility and movements. Adjacencies were used to better describe movement times. For instance, in Figure 9, the front door’s cycle of regions is composed of adjacent regions, but they must be divided in order to account for the accessibility capacities of different robots.
Some constraints required grouping regions in sets that share some particular characteristics. There are two types of groups of regions that must be taken in consideration: Macro-regions (used
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to deal with interference constraints) and cycles (used to prevent incorrect counting of the number of adjacencies a robot has benefited from). The particular way in which these groups of regions behave is described in the Section 3.3. Figure 11 illustrates a pair of possible crossing constraint macro-regions: if, in the same station, the front robot accesses the back macro-region and the back robot accesses the front region they are likely to collide their robotic arms, as illustrated by the
Figure 11: Macro-regions defined for interference constraints.
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Figure 7.
As previously mentioned, the number of movements is counted. Movements between regions
within the same access are deemed “small” (or “very small” if the regions are adjacent). Movements between accesses are deemed “large”, unless there is an adjacency between the accesses (in which case they are “medium” movements). Here are examples of each movement type: Small - Moving from the left to the right side of the front door. Medium - Moving from the front door to the back door. Large - moving from the front door to the luggage compartment. Movements are counted by adding the number of regions and accesses a robot performs welding points, and subtracting the time-wise difference linked to the number of allowable adjacencies. The restrictions that control these operations will be described in Section 3.3. 15
ACCEPTED MANUSCRIPT Table 1: Parameter Tuple sets: They are employed throughout the model from the variable definitions to the constraint formulations. Sets
Tuple
S
s
Description of each range and meaning of the tuples on each set Range for all stations
R
r
Range for all regions
W
w
Range for all robots
WS
(w, s)
W RA
(w, r, a)
WA
(w, a)
Indicates that the robot w can use the access a
RRadj
(r1 , r2 )
Indicates that the regions r1 and r2 are adjacent
W RRadj
(w, r1 , r2 , a)
W AAadj
(w, a1 , a2 )
Indicates that the robot w is allocated at the station s Indicates that the robot w can reach the region r via the access a
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Indicates that the robot w can benefit from the adjacency between r1 and r2 via the access a Indicates that the robot w can benefit from the adjacency between a1 and a2
CR
(c, r)
Indicates that the region r belongs to the region-cycle c
CA
(c, a)
Indicates that the access a belongs to the access-cycle c
W CR
(w, cr)
Indicates that the robot w can perform the region-cycle cr
(w, ca)
Indicates that the robot w can perform the access-cycle ca
(w, em)
Indicates that the robot w can access the exclusion macro-region em
W CM
(w, cm)
Indicates that the robot w can access the crossing macro-region cm
EM R
(em, r)
Indicates that the region r belongs to the exclusion macro-region em
CM R
(cm, r)
Indicates that the region r belongs to the crossing macro-region cm
WMWM
(w1 , m1 , w2 , m2 )
EA
ea
Indicates that the access a defines a exclusion interference constraint
EM
em
List of all the macro regions em that define exclusion constraints
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W CA W EM
Indicates that the robot w1 cannot access the crossing macro-region m1 at the same time that
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robot w2 accesses the crossing macro-region m2
3.2. Sets and variables
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The problem is modeled around its decision variables, that are all defined based on parameter sets. In order to differentiate sets of parameters and sets of variables, the later will start with a lower-case letter (v for integer and real variables and b for binary variables). The tuple sets that
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are employed to define the variables are a discrete collection of tuples, whose interpretations are summarized at the Table 1. The problem’s variables, their types, tuple sets and interpretation are
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listed and summarized at the Table 2. Lastly, some constraints require some additional parameters that are either vectors or constants. These parameters are listed by the Table 3. Each variable set is based on a domain (tuple sets) and has one variable for each element of
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said set. For example, the variable set bW A has one binary variable for each (w, a) tuple in the tuple set W A, namely bW A(w,a) . The tuple and variable sets mostly follow a concept-based letter
orientation. For example, R stands for regions, W for robots, A for accesses, S for stations. This
pattern seeks to simplify interpretation: for instance, the binary variable bW A(w,a) is set to true when the robot w performs welding points by the access a. All variables in the proposed model are strictly non-negative. In our case the main decision variable is an integer number of welding points, performed by the robot w at the region r via the access a. The variable set vW RA contains one integer variable for each element of the tuple set W RA of accessibility (that contains the tuples (w, r, a) of regions r accessible by the robot w via the access a). This is a heavily assignment bounded problem: in the studied cases only about 10%
16
ACCEPTED MANUSCRIPT Table 2: Variables sets, their types, the sets and tuples they are defined by and their meaning Variable Set
Tuple Set
Tuple
vCT
-
-
Line’s cycle time (Time Units)
Description and meaning of the variable from the set
Robot w’s cycle time (Time Units)
vCT W
W
w
vW RA
W RA
(w, r, a)
Number of welding points performed by robot w on region r via the access a
bW RA
W RA
(w, r, a)
Whether or not the robot w performs points on region r via the access a
vN M R
W
w
Number of movements between regions performed by the robot w
vN M A
W
w
Number of movements between accesses performed by the robot w
bW RR
W RRadj
(w, r1 , r2 , a)
bW AA
W AAadj
(w, a1 , a2 )
bW A
WA
(w, a)
Whether or not the robot w uses the access a
bW EM
W EM
(w, m)
Whether or not the robot w performs points on the exclusion macro-region m
bW CM
W CM
(w, m)
Whether or not the robot w performs points on the crossing macro-region m
Whether or not the robot w benefits from the adjacency between the regions r1 and r2 via the access a
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Whether or not the robot w benefits from the adjacency between the accesses a1 and a2
Table 3: Parameters employed for occurrence, time controlling and interference constraints Set
Tuple
Ts
-
-
Meaning
Fixed time associated to preparation at stations (Time Units)
Nr
R
r
Number of points in the region r
Tr
R
r
Duration per point in the region r (Time Units)
Tw
W
w
Duration per point for the robot w (Time Units)
Vw
W
w
Relative Speed of the robot w
TmovR
-
-
Movement time between regions (Time Units)
TmovA
-
-
Movement time between accesses (Time Units)
TadjR
-
-
Time gained per adjacency between regions (Time Units)
TadjA
-
-
Time gained per adjacency between accesses (Time Units)
Up
-
-
Upper bound parameter for exclusion constraints
Dp
-
-
Decrease per robot parameter for exclusion constraints
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Parameters
of the possible (w, r, a) combinations were feasible.
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There are many decision variables that follow vW RA, the first one is the binary bW RA. Each bW RA(w,r,a) that is true when its corresponding vW RA(w,r,a) ≥ 1, and false otherwise. The variables in the set bW A are true if the worker w uses the access a (W A is another accessibility
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set).
Movement variables are defined for each w in W : vN M R counts the number of movements
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between regions and vN M A counts the number of movements between accesses. The adjacency variables bW RR (bW AA) are true when the robot w benefits from adjacencies between the regions r1 and r2 (accesses a1 and a2 ) and are defined for each tuple (w, r1 , r2 ) in W RRadj ((w, a1 , a2 )
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in W AAadj ) of feasible adjacency. The last parameter sets for movements are CR and CA, whose tuples (c, r) (and (c, a))
indicate that the region r (the access a) is part of the region cycle c (the access cycle c). Their main purpose is to limit the number of adjacencies a robot can benefit from. They are defined when there are adjacencies that form a cycle, in which case it is possible to have more adjacencies than movements. As shown by Figure 12, a cycle with 4 adjacent regions has 4 adjacencies and 3 movements. This is why cycles are part of the logical constraint of adjacencies between regions and accesses. The set W CR (W CA) indicates that the robot w can perform welding points in all
regions (accesses) of the region (access) cycle cr (ca), and is a set that can be determined by the
17
ACCEPTED MANUSCRIPT accessibility sets W RA and W A.
Region
Region
Region
Region
CR IP T
Cycle
Figure 12: Illustration of the reason cycles are employed to correctly count the number of movements and adjacencies: If a robot displaces through regions that form an adjacency cycle, there will be one more adjacency than the number of movements, this implies that the adjacency time benefits could be higher than the time spent on the required
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movements. Cycle constraints (Inequalities 12 and 13) prevent such wrong consideration.
Finally, the interference constraints require two sets of macro-regions: exclusion and crossing macro-regions. Their parameters are sets EM R and CM R, with tuples (m, r) that list regions r that belong to each macro-region m. The variable set bW EM (bW CM ) contains binary variables for each tuple (w, m) in the set W EM (W CM ) with macro-regions m where the robot w
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can perform welding points. These variables indicate whether or not the robot performs welding points in the macro-region. The crossing constraints also require crossing parameter tuples
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(w1 , m1 , w2 , m2 ) (set W M W M ) that inform that if the robot w1 accesses the macro-region m1 and the robot w2 accesses the macro-region m2 they may collide, therefore, leading to an unfeasible solution. Another variant of the exclusion constraint formulation is the access-wise interference
access set EA.
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constraint, defined based on the number of robots that employ the same access a in the exclusion
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The last variables within the proposed model are the cycle time of the manufacturing line (vCT ) and the cycle time of each individual robot w (vCT Ww ).
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3.3. Constraints
There are three kinds of constraints in this model. First, the occurrence constraint requires that
all welding points in all regions be performed. Second, there are the time measuring constraints that seek to determine the cycle time of each robot and of the line as a whole. Lastly, the interference constraints seek to ensure that robots do not collide during the performance of their tasks. 3.3.1. Occurrence constraint The occurrence constraint states that the sum of welding points performed by all robots by all accesses in each region equals the number of welding points in the region, as stated by the Equation 1. This constraint is defined based on the assignment binding W RA set that allows only robots that can access a region to perform points in it. 18
ACCEPTED MANUSCRIPT
X
vW RA(w,
r, a)
∀r∈R
= Nr
(w, r, a)∈W RA
(1)
3.3.2. Time controlling constraints The cycle time (vCT ) is determined by the line’s bottleneck station, worker or, in this case, robot (vCT Ww ). This is stated in the Inequality 2.
vCT ≥ vCT Ww
∀w∈W
(2)
CR IP T
The cycle time of each robot w (vCT Ww ) is determined by the Equation 3. The total time of a robot is given by the time required to perform the welding points assigned to it added to the movement time and to the preparation time Ts of its station. The movement time is estimated as the sum of movements between regions (multiplied by the time between regions TmovR ), added to the sum of movements between accesses (idem with TmovA ) and subtracted by the sum of adjacency
vCT Ww =
X
vW RA(w,
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time gains between regions and between accesses (with times TadjR and TadjA ).
r, a)
(w, r, a)∈W RA
+ TmovR /Vw · vN M Rw
M
+ TmovA /Vw · vN M Aw X − TadjR /Vw ·
· (Tw + Tr )
bW RR(w,
r1 , r2 , a)
(w, r1 , r2 , a)∈W RRadj
X
bW AA(w,
a1 , a 2 )
+ −
(3)
−
+ Ts
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− TadjA /Vw ·
+
(w, a1 , a2 )∈W AAadj
∀w∈W
The model’s variables that control each part of the robot’s movement approximation are de-
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scribed in the following equations. Firstly, the variables in the sets bW RA and bW A are controlled by the Inequalities 4 and 5. These variables determine in which regions and accesses each robot operates. These constraints establish a fixed charge aspect to the problem: the movement costs
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of accessing regions. This occurs both in regard to the number of points in a region and in regard to the number of regions in an access (Hillier & Lieberman, 2015). As such, it is expected that
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good solutions tend to have fewer movements, concentrating each robot’s work in areas closer to one another.
bW RA(w,
r, a)
bW A(w,
a)
≥ vW RA(w,
r, a) /Nr
≥ bW RA(w,
r, a)
∀ (w, r, a) ∈ W RA ∀ (w, r, a) ∈ W RA
(4) (5)
The first movement variables defined are the movement counters: the number of movements between regions (vN M Rw ) or accesses (vN M Aw ) is the number of regions or accesses minus 1 (unless the robot works in no region (or access) in which case the number of movements is zero). This information is codified in the Inequalities 6 and 7 (as stated in Section 3.2, all variables are strictly non-negative). 19
ACCEPTED MANUSCRIPT
vN M Rw ≥ −1 +
X
bW RA(w,
∀w ∈ W
r, a)
(w, r, a)∈W RA
vN M Aw ≥ −1 +
X
bW A(w,
(6)
∀w ∈ W
a)
(w, a)∈W A
(7)
The adjacency variables can only be set as true if the robot w works at both regions r1 , r2 (accesses a1 , a2 ) on the set W RRadj (W AAadj ). This restriction is stated by the Inequalities 8a and 8b (Inequalities 9a and 9b). The adjacency gain between regions can only happen if both
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regions are accessed by the same access a.
bW RR(w,
r1 , r2 , a)
≤ bW RA(w,
r1 , a)
∀ (w, r1 , r2 , a) ∈ W RRadj
(8a)
bW RR(w,
r1 , r2 , a)
≤ bW RA(w,
r2 , a)
∀ (w, r1 , r2 , a) ∈ W RRadj
(8b)
a 1 , a2 )
≤ bW A(w,
a1 )
∀ (w, a1 , a2 ) ∈ W AAadj
(9a)
bW AA(w,
a1 , a 2 )
≤ bW A(w,
a2 )
∀ (w, a1 , a2 ) ∈ W AAadj
(9b)
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bW AA(w,
Region-to-region adjacencies require some limitations: Each adjacency between two regions r1 and r2 (listed by (r1 , r2 ) tuples in RRadj ) can only be used by one robot (Inequality 10). Further-
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more, each robot w can only benefit from two adjacencies tied to a given region r (Inequality 11). These constraints are required to prevent mathematical gains with no real-world counterpart. X
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bW RR(w,
r1 , r 2 )
(w, r1 , r2 )∈W RRadj
X
bW RR(w,
≤ 1
∀ (r1 , r2 ) ∈ RRadj
r, r1 ) +
PT
(w, r, r1 )∈W RRadj
+
X
(10)
(11)
bW RR(w,
r2 , r)
(w, r2 , r)∈W RRadj
≤ 2
∀(w, r, a) ∈ W RA
CE
Lastly, the cycle constraints limit adjacencies region-wise (CR contains tuples (c, r) of adjacent regions r that form a cycle c) and access-wise (CA, idem as regions with (c, a) for accesses that
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form a cycle), as stated by the Inequality 12 (Inequality 13). These constraints are relevant for robots w that can access all regions r in the cycle c: even if it performs welding points in all regions of said cycle, it cannot benefit from all adjacencies.
X
bW RR(w,
r1 , r2 , a)
(c, r1 ) ∈ CR (c, r2 ) ∈ CR (w, r1 , r2 , a) ∈ W RRadj
X
(c, a1 ) ∈ CA (c, a2 ) ∈ CA (w, a1 , a2 ) ∈ W AAadj
bW AA(w,
a 1 , a2 )
≤ −1 +
≤ −1 +
20
X
1
∀ (w, c) ∈ W CR
(12)
X
1
∀ (w, c) ∈ W CA
(13)
(c, r) ∈ CR
(c, a) ∈ CA
ACCEPTED MANUSCRIPT 3.3.3. Interference constraints Four kinds of interference constraints were considered: region station-wise exclusivity, crossingmacro regions, exclusion macro-regions, and access-wise exclusion. They seek to describe problems that may happen as robots “compete” for the same physical space. The region station-wise exclusivity states that at each station s, at most one robot w can access each region r. This constraint is stated by the Inequality 14. Note that this constraint prevents a robot from accessing the same region from two different accesses. From a practical standpoint, this
the same number of points via only one access. X
bW RA(w,
r, a)
(w, s)∈W S (w, r, a)∈W RA
≤ 1
CR IP T
is a desirable feature as that solution would be dominated by the one in which the robot performs
∀ s ∈ S, r ∈ R
(14)
The macro-region allocation variables control the two types of macro-region interference constraints. A robot is allocated to a macro-region if it performs welding points in one of the regions
bW EM(w,m)
≥ bW RA(w,
r, a)
bW CM(w,m)
≥ bW RA(w,
r, a)
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of said macro-region. This is stated by the Inequalities 15a and 15b.
∀ (w, r, a) ∈ W RA, (m, r) ∈ EM R
(15a)
∀ (w, r, a) ∈ W RA, (m, r) ∈ CM R
(15b)
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The crossing macro-region constraints seek to forcefully ban two different robots r1 and r2 from occupying conflicting physical space defined by the macro-regions m1 and m2 (notice that m1 and
ED
m2 might be the same, but the robots must be different). In essence at most either r1 occupies m1 or r2 occupies m2 . This constraint is codified by the Inequality 16. For instance, suppose that wf is the front robot and wb is the back robot, while mf and mb are the front and back door
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macro-regions. In this case (wf , mb , wb , mf ) impedes the crossing of the robot arm illustrated by the Figure 7: It allows either the front robot to work in the back of the car or (exclusive or)
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the back robot to work at the front of the car.
bW CM(w1 ,
m1 )
+ bW CM(w2 ,
m2 )
≤1
∀ (w1 , m1 , w2 , m2 ) ∈ W M W M
(16)
AC
Ideally, one might want each overall region of the product to be occupied by at most one robot
per station (and, thus, employ crossing macro-regions for all possible interference zones). If that is the case one could use crossing macro-region constraint with the same macro-region m for two
or more robots. In this case (wf , mf , wb , mf ) would prevent two robots from simultaneously accessing the front door, as in Figure 8. But in some cases, due to accessibility constraints, this might simply be impossible or lead to a poor solution. Exclusion macro-regions are less restrictive interference constraints that seek to allow two robots to be assigned to the same zone in a way that they will together spend less than the cycle time in said zone. In this case, because the cycle time correlates very well with the number of welding points performed, it is stated that the sum of points performed will be limited by an upper-bound value (parameter Up ) that decreases (by 21
ACCEPTED MANUSCRIPT the parameter Dp ) for each robot that works at the same exclusion macro-region. This decrease happens because robots take time to enter and leave the access. This is stated by the Inequality 17. This constraint that can be seen as an adaptation of resource constraints, as described on Section 1. X
vW RA(w,
r, a)
(w, r, a)∈W RA (w, s)∈W S (m, r)∈EM R
≤ Up −Dp · −1 +
X
bW EM(w,
(w, s) ∈ W S (w, m)∈W EM
m)
∀ s ∈ S, m ∈ EM (17)
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The goal of Inequality 17 is to limit the number of points performed in a macro region by a decreasing limit: the higher the number of robots in the same station performing welding points in the same macro-region, the smaller the total number of points they can perform. For instance, in the case studies of section 4, the upper bound value of number of the points (Up ) was 24. If two robots work at the same station in the same macro-region this limit decreases to 20 (therefore, Dp equals 4), as time will be required for one robot to leave the macro-region and for the other robot
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to enter it. These parameters were defined based on observation.
The access exclusion constraint is defined by the Inequality 18. This constraint is very similar to the macro-region exclusion constraint defined by Inequality 17. The main change is that the number of points performed on a same station by different robots via the same access (rather than macro-region) decreases the more robots simultaneously access said access-region.
(w, r, a)∈W RA (w, s)∈W S
≤ Up − Dp · −1 +
X
(w, a)∈W A
bW A(w,
a)
∀ a ∈ EA, s ∈ S (18)
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3.4. Objective function
r, a)
M
vW RA(w,
ED
X
For the presented case study, the total number of robots and the necessary spot welding points are given. Therefore, the objective of the model is to minimize the cycle time of the line, as
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described in Expression 19. Other objective functions are also possible, such as the minimization
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of the total number of robots (an ALBP-1 variation).
Minimize: vCT
(19)
4. Case studies The optimization models were developed based on one of the robotic spot welding lines of Renault’s facility in the outskirts of Curitiba in Brazil. The line is depicted in a simplified manner by the Figure 13. It is composed of 42 robots distributed symmetrically (left and right sides of the vehicle) amongst 13 stations. Each stations had either two, four or six robots. When this model was developed, there were four car models (vehicle types) that passed through this manufacturing line, each of them with over 700 welding points. This was the last part of the car 22
ACCEPTED MANUSCRIPT body manufacturing line, the body-in-white stage: the pieces were all assembled and, therefore, there were no precedence relations to deal with.
IN
OUT
Figure 13: Studied line’s simplified layout, with 42 robots allocated to 13 stations
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The line was a mixed-model one with asynchronous pace, however, the total workload of each vehicle was very similar and their individual cycle times were not significantly different. This was partly due to the company’s previous divisions of welding points, which were performed in such a way that left this line with a rather similar workload for all car models. Although in other parts of the manufacturing line sequencing could be a significant problem (see Boysen et al. (2009) for a mixed-model sequencing survey), in this part each vehicle could be balanced individually, allowing
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four (simpler) single model optimizations to replace a much more complex mixed-model one. This reduction of a Mixed-Model ALBP into P-Single-Model ALBPs could have downsides in manual lines due to the lack of specialization effects and set-up times (Becker & Scholl, 2006). The robots do not display learning curves and can easily alternate between products, allowing us to set these considerations aside. The main mixed-model aspect remaining was the accessibility limitations, defined based on the geometrical features of both: the robot and the vehicle models. However,
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this was considered an input rather than a variable. The goal was set to the minimization of cycle
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time of each product model. Therefore, one main instance was created for each vehicle. 4.1. Instances definition and data calibration The process of defining instances was not trivial: Firstly, data was collected as for where were
PT
the welding points. Secondly, the model’s parameters (such as movement times, fixed times per robots) were determined by filming the robot’s activities and carefully observing the videos. The
CE
model was designed to be slightly conservative, and, therefore, when a parameter (say movement time between non-adjacent regions of the same access) was observed with different values in different cases, the worst case was picked.
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In order to assure that estimated task times are close to real times, a calibration must be
performed. Figure 14 shows a comparison between the model’s predictions and the robot’s real times after the due data calibration for one instance. The model’s approximation on the station workload (measured in time units) is higher than the real workload, indicating that any solution obtained by the proposed model will be an upper bound to the real solution. Once the parameters had been defined, points were grouped in regions, based on accessibility data. Such data was not initially available and was determined using a robotic simulation software owned by the company. This software had 3D models for each vehicle and robot, allowing to verify before implementation if each of the welding points in a solution could, indeed, be reached. Each region was assigned accesses from which robots could reach it. Adjacency, macro-regions and 23
CR IP T
ACCEPTED MANUSCRIPT
Figure 14: Comparison between the model’s approximation to the empirically measured time: Notice that the
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approximations are slightly conservative
cycle parameter tuples were also defined based on the observed geometrical characteristics. The instances’ data were adapted in order to preserve both: the reproducibility and the company’s private information.
The model and data from each product variant allowed the definition of Mixed Integer Linear Programming models for each vehicle. In order to evaluate the computational burden in regard
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to the problem’s size, instances were generated with various percentages of the total number of welding points. Each region’s number of welding points was multiplied by a reducing factor and
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rounded down to an integer. These factors were chosen, in a manner that instances would have, approximately, 10%, 20%, ..., 90% and 100% of the total number of welding points. The 100% instances represent the practical cases. Data on the Supporting Information present the tuple and
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parameter sets required to define each problem instance. Model files (.lp format) are also provided, along with the best solution files for each instance, containing the value of the objective function
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and of each variable presented in the model file. Thus, reproducibility is assured by the provided detailed information.
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4.2. Results
The computational testes were conducted using a Core i7 (2.3 GHz) computer with 8.0 GB of
RAM and a universal MILP solver was employed to solve all instances. The computational time limit of each run was set to one hour. Table 4 illustrates the solution of the computational and practical case studies presented in Section 4.1, each variant representing a different vehicle. Four vehicle models are analyzed, in instances that vary from 10% to 100% of the total number of welding points presented in the real line. The table contains the results of the best cycle time found in 3,600 seconds of solution time and the lower bound for each instance. The computational tests indicate that the number of points to be distributed influences the computational load. However, this number is not the only factor to be considered: Optimality was more difficult for Variant 4 problems
24
ACCEPTED MANUSCRIPT Table 4: Number of welding points (N), upper bounds (UB) and lower bounds (LB) on cycle time, and computation times for each instance of the case studies presented in Section 4.1. Each variant represent a product type. The 100% instances represent the practical case for each product. Variant 1
Variant 2
N
UB
LB
time
N
UB
LB
time
10%
73
1,026.0
1,026.0
0s
68
972.0
972.0
74s
20%
146
1,439.5
1,439.5
2s
146
1,452.0
1,452.0
2s
30%
219
1,578.0
1,578.0
3s
221
1,564.4
1,564.4
4s
40%
287
1,745.5
1,745.5
3s
295
1,782.0
1,782.0
4s
50%
366
1,956.0
1,956.0
7s
372
1,980.0
1,980.0
4s
60%
459
2,208.0
2,208.0
34s
426
2,074.4
2,074.4
25s
70%
516
2,357.5
2,357.5
4s
518
2,334.0
2,334.0
19s
80%
591
2,561.5
2,561.5
13s
591
2,515.6
2,515.6
53s
90%
642
2,712.0
2,712.0
9s
640
2,664.0
2,664.0
5s
100%
733
2,867.5
2,867.5
45s
740
2,868.0
2,838.0
3600s
% Points
N
UB
LB
time
N
UB
LB
time
10%
71
1,075.6
1,075.6
2s
70
1,075.6
1,075.6
2s
20%
142
1,404.0
1,404.0
4s
142
1,422.0
1,422.0
6s
30%
217
1,620.0
1,620.0
9s
216
1,620.0
1,620.0
10s
40%
276
1,680.0
1,680.0
456s
273
1,713.3
1,713.3
247s
50%
348
1,986.0
1,986.0
48s
347
1,968.0
1,968.0
242s
60%
440
2,124.0
2,118.0
3600s
438
2,164.0
2,133.3
3600s
70%
495
2,334.0
2,334.0
1379s
492
2,290.0
2,256.0
3600s
80%
568
2,458.0
2,448.0
3600s
564
2,514.0
2,381.0
3600s
90%
599
2,640.0
2,640.0
159s
597
2,538.0
2,448.0
3600s
100%
708
2,874.0
2,736.0
3600s
706
2,868.0
2,723.5
3600s
Variant 4
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Variant 3
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% Points
than for Variant 1, despite the number of welding points. This means that other instance-specific
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factors such as accessibility and interferences might significantly affect the instance’s computational difficulty.
For the practical cases, the company’s mid-term productivity goal was a cycle time of 2880 T.U.
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One of the practical instances could be solved to optimality (relative gap = 0%), while the others couldn’t (up to 5% relative gap). However, all of them allowed the cycle time to reach the company’s
CE
goal (the value associated with the target productivity, measured in time units per vehicles). The model’s gain means that the studied part of the manufacturing line has the capacity to produce around 6.6% more vehicles without employing a single new robot. Figure 15 illustrates the station-
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wise gain in cycle time. The vehicle corresponding to the model that was solved to optimality was also the one that was produced the most, meaning that it tends to dictate the line’s overall cycle time. Furthermore, according to Table 4, the obtained cycle time difference between models for the real-world case is smaller than 1% (2,868.0 versus 2,874.0) corroborating with the hypothesis
that they could be treated separately rather than on a mixed-model (or multi-model) manner. All models did reach the main cycle time goal (2880 T.U.) and idle times in some of the robots can be used, for instance, to perform welding points that were not performed in this part of the manufacturing line. Figure 15 shows a comparison between the initial workload distribution between stations and the optimized configuration. By computing the trade-off between movements and number of per-
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Figure 15: Comparison between the model’s optimized solution for one of the instances and the empirically measured
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time: The dashed line highlights the smaller largest value for cycle time amongst stations.
formed points, the model is able to better distribute the workload and reduce the flow-shop’s cycle time: In some cases, the sum of robot times (heavily linked to movements) increased by the optimization process, but the line’s cycle time decreased. Notice that this has been achieved under the conservative parameter framework made clear by Figure 14 validation. The obtained solutions to the practical instances were afterwards internally translated back
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from mathematical to tangible practical information and were verified to provide feasible solutions
5. Conclusion
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to the real-world problem by the company’s specialists employing simulation tools.
Balancing robotic spot welding manufacturing lines is a challenging problem. The time required
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to welding the points is short compared to cycle time. However, robots have to move between them and the movement time must be taken in account. Furthermore, geometrical aspects impose assign-
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ment constraints and multiple robots per station impose interference constraints. The combination of these restrictions lead to the understanding that previously described models could not properly describe the problem. A new (MILP) model is proposed and applied to solve a real-world scenario
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from Renault factory in the outskirts of Curitiba, Brazil. Practical case studies were conducted for four vehicle models on a large-size real-world line (42
robots and over 700 welding points for each model). Each of the models was optimized separately, and optimality was not reached for all instances. However, the obtained answers offered cycle time reductions of around 6.6%, allowing the company to reach its mid-term goal of productivity for that part of the factory without the need to buy new robots. Computational studies have shown that the number of welding points is a relevant factor to the model tractability, but not the only one. Some small instances were more difficult than larger counter-parts (see Table 4). This indicates that instance-specific characteristics, such as accessibility and interference, are also relevant factors for computational tractability. 26
ACCEPTED MANUSCRIPT Instances were generated to describe each of the vehicle type produced in the factory. Parameters were set according to empirical observations. The line was a mixed-model one, however in this particular part of the factory all models could individually reach similar cycle times. This verified the hypothesis about the optimization process being able to set aside sequencing considerations and balance each model individually. The results obtained for the case studies were validated by the company specialists through simulation techniques. The insights provided by the model allowed identifying new operational conditions, with smaller cycle times, without the need of investing in new equipment; the model was used to aid in the managerial and operational decision-making
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process.
The new Mixed Integer Linear Programming model provides a framework to optimize robotic welding lines by offering a formulation to simultaneously deal with: Robot-wise variations on parameters, Assignment Restrictions, Movement Times and Interference Constraints. The proposed formulation is limited to lines without precedence relations between welding points. Any ordering of tasks within a station would require the scheduling of operations between the robots. If the
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precedence are station related, such as by adding new layers of sheets, the approach presented in Appendix A can be used.
As future development, this basic framework can be adapted to incorporate other features such as feeder lines and parallel stations. Furthermore, in an assignment line design context, the presented model could be adapted to minimize the number of robots required or optimize a cost-
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oriented function relaxing some of the hypotheses of given robot locations and line layout. This
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cost-oriented approach can be of great value to aid managerial decisions.
Appendix A. Piece joining tasks on defined subassemblies The proposed case studies presented in Section 4 represent the final manufacturing spot weld-
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ing line of the body shop. In this line, the structure of the vehicle is already assembled, so that only reinforcement procedures are necessary. No precedence relations or subassemblies are ob-
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served. However, the proposed model can also be used to balance a robotic spot welding line when subassemblies are known and fixed, along with the accessibility of each task. Suppose Figure A.16 represents a spot welding line consisting of a main line (top row) and a
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sub-product line (bottom row). The third station of the main line represents a subassembly: the product piece of the feeder line is assembled with the main product. The welding procedures can be divided in 6 types of tasks, represented by the precedence diagram on the right. The blue tasks represent the piece joining procedures: the welding points that are necessary to assure the cohesion of the assembled parts. Some welding points may lose their accessibility after the superposition of the two pieces of the assemblage. The green points represent the points that should be performed before the subassembly at the main line, while the orange points should be performed in the sub-product line. The red node represents the procedures that need to be performed after the assemblage: these tasks represent the reinforcement welding points that assure the mechanical resistance of the assembled parts. The red tasks can be performed in the joining station or after 27
ACCEPTED MANUSCRIPT it. Finally, there are some points that are independent on the subassembly. The purple node represent the independent welding points that have to be performed in the main product. The yellow tasks represent the independent welding points that must be performed in the piece that starts at sub-product. Workstations and Accessibility
Assembly-Related Precedence Relations
Main Product Welding Line
Main product welding points Independent of assembly Main product welding points obstructed by assembly
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Welding points associated to assembly or piece joining Sub-Product Welding Line
Sub-product welding points obstructed by assembly
Welding points accessible only after assembly
Sub-product welding points Independent of assembly
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Figure A.16: Example of station-wise accessibility windows for a subassembly.
Based on the type of task and the given precedence diagram, the precedence relations can be written as station-wise accessibility windows. The line’s diagram on the left side of Figure A.16 shows which stations can perform which tasks by colored squares in the workstations. Note that the green and orange tasks must be performed before the subassembly. The blue tasks must be performed at the lines’ intersection. The red tasks must be assigned to the overlap or stations
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after it. The independent purple welding points can be performed in any station of the main line. Finally, the independent yellow points can be performed in any station of the feeder line and any
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station of the main line after the sub-product line piece has been transfered to the main line. With the station-wise accessibility window concept, precedence relations that resulted from
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known and fixed subassemblies can be modeled as robotic accessibilities.
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Acknowledgments
We would like to acknowledge the financial support from Funda¸c˜ao Arauc´aria (Agreement 141/2015) and CNPq (Grant 305405/2012-8). In addition, we would also like to show our appre-
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ciation to Renault for the opportunity of working with them and their manufacturing lines.
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