Optimization of product assignment to assembly lines

António Maria Ferreira de Lemos Falcão Optimization of product assignment to assembly lines Master Dissertation Master in Industrial engineering and Management Work done under the guidance of: Filipe Pereira Pinto Cunha Alvelos October 2019 ii DIREITOS DE AUTOR E CONDIÇÕES DE UTILIZAÇÃO DO TRABALHO POR TERCEIROS Este é um trabalho académico que pode ser utilizado por terceiros desde que respeitadas as regras e boas práticas internacionalmente aceites, no que concerne aos direitos de autor e direitos conexos. Assim, o presente trabalho pode ser utilizado nos termos previstos na licença abaixo indicada. Caso o utilizador necessite de permissão para poder fazer um uso do trabalho em condições não previstas no licenciamento indicado, deverá contactar o autor, através do RepositóriUM da Universidade do Minho. Licença concedida aos utilizadores deste trabalho Atribuição CC BY https://creativecommons.org/licenses/by/4.0/ iii ACKNOWLEDGMENTS I would like to express my most sincere gratitude for all those who have accompanied me in this crucial part of my academic life. I want to particularly acknowledge: Professor Filipe Alvelos, for all the constant support throughout the year. His constant availability, experience, suggestions and guidance were crucial for me to complete this work and without his support this research would not have reached this level. All of the Bosch team that welcomed me and treated me as part of the group. Thank you for all the experience shared, companionship and coffee breaks talks. My tutor in Bosch, Luís Florêncio, I must deeply thank for, besides fulfilling amazingly as a tutor, played a very important part in my life as a true friend. I could not have asked for a better person to welcome me in my first professional experience. I will never forget the advises given me and laughs we shared. Last but certainly not least, I thank my parents, brothers, friends and rest of the family for the constant support on my work and life decisions. You are the foundation of the man I have become today. “More is lost by indecision, than by wrong decision." - Tony Soprano iv DECLARAÇÃO DE INTEGRIDADE Declaro ter atuado com integridade na elaboração do presente trabalho académico e confirmo que não recorri à prática de plágio nem a qualquer forma de utilização indevida ou falsificação de informações ou resultados em nenhuma das etapas conducente à sua elaboração. Mais declaro que conheço e que respeitei o Código de Conduta Ética da Universidade do Minho. v RESUMO O trabalho apresentado nesta dissertação foi desenvolvido em contexto industrial integrado no departamento de planeamento e controlo de produção da área de inserção automática da Bosch Car Multimédia Portugal S.A - Braga. O problema abordado nesta dissertação foi encontrar a melhor distribuição de famílias de produtos a alocar nas diferentes linhas de produção de acordo com as suas restrições físicas e técnicas. Na abordagem do problema recorreu-se a técnicas de Investigação Operacional através de modelação matemática, para analisar situações complexas e obter soluções mais eficientes. Tendo como base dados da produção, previsões de necessidades e disponibilidade física da produção, foram criados modelos com diferentes conjuntos de restrições e funções objetivo por forma a apresentar soluções que melhor se adequassem à pergunta e ao problema específico do contexto produtivo presente. Através da utilização de software, foram resolvidos os modelos criados anteriormente, sendo que as soluções foram analisadas e avaliadas para a adequação às necessidades atuais da empresa e para a sua possível e viável implementação. PALAVRAS-CHAVE Afetação; Investigação Operacional; Planeamento; Otimização; Gestão Industrial: Inserção automática vi ABSTRACT The work presented in this dissertation was developed in an industrial context integrated in the production control and management department of the Bosch Car Multimedia Portugal S.A – Braga automatic insertion. The problem addressed in this dissertation was finding the best distribution of product families to assign in different lines according to the physical and technical constraints of the assembly lines. In the approach of the problem, it was used tools and techniques of the Operational Research discipline through mathematical modeling, in order to analyze complex situation and obtain more efficient solutions to help in the decision-making process. Based on production data, production needs forecasts and assembly line physical availability, models with different sets of constraints and objective functions were created to present solutions that best fit the question and the specific problem of the present production context. Through specific software that suited the problem, the previously created models were solved, and the solutions were analyzed and evaluated to suit the company’s current needs and for possible and feasible implementation of the solutions. KEYWORDS Assignment; Operation Research; Planning; Optimization; Industrial Management; Automatic Insertion vii TABLE OF CONTENTS Acknowledgments ................................................................................................................ iii Resumo................................................................................................................................ v Abstract .............................................................................................................................. vi Figure Index......................................................................................................................... ix Table Index........................................................................................................................... x List of abbreviations and acronyms........................................................................................ xi 1. Introduction ................................................................................................................ 1 1.1 Background............................................................................................................. 1 1.2 Objectives ............................................................................................................... 2 1.3 Research Methodology ............................................................................................. 2 1.4 Dissertation Structure .............................................................................................. 2 2. Literature review.......................................................................................................... 4 2.1 Printed circuit boards assembly system ..................................................................... 4 2.2 PCB production management ................................................................................... 5 2.2.1 Setup Strategy Selection ................................................................................... 5 2.2.2 PCB line assignment ........................................................................................ 6 2.2.3 PCB grouping................................................................................................... 7 2.2.4 PCB scheduling................................................................................................ 8 2.2.5 PCB component allocation ................................................................................ 8 2.2.6 Feeder assignment and placement sequencing................................................... 9 2.3 Optimization Techniques ........................................................................................ 10 2.3.1 Mathematical programming ............................................................................ 11 2.3.2 Linear / Integer programming ......................................................................... 11 2.3.3 Algorithms for Integer Linear Programming....................................................... 12 2.3.4 Branch and Bound ......................................................................................... 12 2.3.5 Cutting planes ................................................................................................ 12 2.4 Assignment Problem .............................................................................................. 13 2.4.1 Generalized Assignment Problem..................................................................... 14 viii 3. Corporate Description ................................................................................................ 16 3.1 BOSCH Group ....................................................................................................... 16 3.2 Bosch in Portugal .................................................................................................. 16 3.3 Bosch CM Braga ................................................................................................... 18 3.4 Organizational structure ......................................................................................... 19 4. Problem description and analysis of the current situation.............................................. 21 4.1 Production Process ................................................................................................ 21 4.2 Production planning and control department............................................................ 22 4.3 Problems identified ................................................................................................ 23 4.4 PCB assignment problem characterization............................................................... 24 5. Models ..................................................................................................................... 26 5.1 Cost minimization model ........................................................................................ 26 5.2 SMD working number minimization ......................................................................... 27 5.3 SMD workload balance........................................................................................... 28 5.4 Optimization software and data input ...................................................................... 29 6. Computational experiments........................................................................................ 32 6.1 Results ................................................................................................................. 32 6.2 Solution’s Visual Representation ............................................................................. 35 6.3 Capacity adjustment testing.................................................................................... 38 7. Conclusion................................................................................................................ 40 References ........................................................................................................................ 41 Appendix I – New Month Visual solution ............................................................................... 44 Appendix II – New Month SMD Capacity Usage .................................................................... 46 Appendix III – Visual Solution Input Data example ................................................................. 47 ix FIGURE INDEX Figure 1: Appropriate methods for each trade-off point. Adapted from (Bhoja, 2002) ............... 11 Figure 2: Bosch business sectors, adapted from Bosch CM, internal communication............... 16 Figure 3: Bosch Portugal data, adapted from “Bosch CM Corporate Presentation” .................. 17 Figure 4: Bosch CM main landmarks, adapted from Bosch CM, internal communication ......... 18 Figure 5: Bosch CM product range, adapted from Bosch CM, internal communication ............ 18 Figure 6: Bosch client list, adapted from Bosch presentation, internal communication ............. 19 Figure 7: Bosch CM Braga organization, adapted from Bosch CM, internal communication...... 19 Figure 8: BOSCH CM plant layout, adapted from Bosch CM, internal communication .............. 20 Figure 9: MOE1 layout and PCB transportation flux ............................................................... 30 Figure 10: Final transportation matrix cost ........................................................................... 31 Figure 11: Current situation PCB line assignment ................................................................. 35 Figure 12: Model 1 PCB line assignment ............................................................................. 36 Figure 13: Model 2 PCB line assignment ............................................................................. 37 Figure 14: Model 3 PCB line assignment ............................................................................. 37 5 two reflows can be done consecutive or within a limited time space, defined for each PCB (Ho & Ji, 2007). There are additional processes outside of the SMT scope that are dependent of the product, that are explored for Bosch CM Braga production process description, in chapter 4. 2.2 PCB production management The literature on decision-making problems in the PCB production management is extensive and diverse. To solve all of the existing problems on this area as one is complex so it is necessary to decompose them into smaller problems (Ammons et al., 1993). These smaller problems may include: • Setup strategy selection • Assignment of PCB to assembly lines • Grouping of PCB into families • PCB scheduling • Component types allocation to the assembly machines • Feeder assignment and placement sequencing All of the above-mentioned problems are reviewed, with an analysis of the past research made on these subjects. They are considered complex problems due to the high amount of possible solutions and the fact that there is no efficient way to determine which one is better. The problem addressed in this dissertation is the assignment of PCB to the assembly lines, which will be introduced in section 2.2.2, and is the main subject of this dissertation. This problem consists in optimizing the assignment of a PCB or a group PCB to the assembly lines, which have their own capacity limit. The other PCB problems are interesting to study, due to also being decision-making problems in the same industry, having common situations and useful data inputs. 2.2.1 Setup Strategy Selection In all the PCB manufacturing, there are different setup strategies, differing by the industrial environment and weight given to different objectives, whether it is to minimize changeover times, effort, component placement time per board or even making the best use of the available resources, either it is physical space or feeder slots. Setup is the physical set of different components, organized in a specific movable table, whereas changeover is the act of changing these tables. According to Leon & Peters (Leon & Peters, 1998), there are four types of setup strategies that can be identified: 6 • In the cases where there is a high variety of PCB, but the lot size number is reduced, machine operation optimization does not have the same impact as focusing on minimizing the changeover times between the different PCB on the same SMD. This last is the main objective of the Minimum setup strategy. • In contrast, in the SMD where there is an allocation of a high-volume of the same PCB type, it is preferred to minimize the component placement time per board. This objective is achieved by optimizing the SMD operations with specific machine algorithms that can, for example, assign components to the SMD feeder slot or schedule the component placement into the PCB. These methods are designated as unique setup strategies. • Partial setup strategy is characterized, as the name suggests, by partially removing components of the feeder slot, when changing PCB types. The objective of this strategy is also to minimize component placement time, besides changeover time. • In industrial environments where there is a medium variety of PCB types as well as the batch size number, it is suggested, similar to the minimum setup strategy, to minimize changeover times, being the most impactful objective. It is named as group setup strategy when, instead of the setup taking place between single PCB it is between families of PCB. This grouping should take in account the common components that each PCB has, hence having no major changeover when changing from one PCB type to another from the same family. 2.2.2 PCB line assignment The PCB line assignment is, as previous mentioned, the problem of optimizing the assignment of different PCB products to the assembly lines, while not exceeding the SMD capacity. The problem has a similar structure to the generalized assignment problem, which will be detailed in section 2.4.1. The objectives of solving the problem can vary, though traditionally, it is mainly concerned in optimizing component placement, machine workload balance, PCB assignment, scheduling or minimizing changeover times. Ho and Ji (Ho & Ji, 2005) study the PCB line assignment problem and propose a genetic algorithm (GA) approach to solve it. In their literature review, they conclude that there is no research arising in a high volume production environment. The authors propose two mathematical models to optimize the assignment of PCB to assembly lines, while minimizing total production cost. A GA is also proposed and applied in order to solve the problem efficiently. The two mathematical models are solved to optimality 7 using CPLEX solver and it is demonstrated that the GA can get good quality solutions for the problem much faster, proving to have a better performance. Balakrishnan and Vanderbeck (Balakrishnan & Vanderbeck, 1993) created an integer linear programming model to assign the PCB to each SMD in order to minimize setup costs, while balancing workload. The environment where the study was done had a high mix and low volume production. The researchers reached a near-optimal solution, by combining an optimization-based method, column generation, heuristics and lower bound procedures. Hillier and L. Brandeau (Hillier & L. Brandeau, 2001) also study the assignment of PCB to the assembly lines, though their work is more focused on the assignment of the PCB components as it will be reviewed in a next sub section. 2.2.3 PCB grouping In order to reduce the number of setups from different types of PCB in the production lines it is suggested to group PCB into families with some sort of similarities, so they can be worked as the same way. When using the group setup strategy, all of the PCB in the family are processed in only one setup, hence reducing the changeover time. One of the downsides of this strategy is that the processing time is longer, when comparing to the unique setup strategy. (McGinnis et al., 1992) Maimon (Freed, Maimon, & DAR-EL, 1989) proposes a group set-up (GSU) method, which he shows, that can significantly reduce changeover times comparing to traditional PCB production methods. The GSU method created in his research suggests dividing the products into groups, which will be produced in two stages. Primarily, the set-up of common components on the machines followed by the assembly of these components on all the PCB in the group is made. Afterwards, there is another set-up of the remaining components and further assembly on each product. In environments where there is a vast number of products sharing common components, and the production volume doesn’t justify buying more than one machine of each type, the GSU method is very advantageous not only reducing changeover time but also reducing throughput time. Dino and Marco (Luzzatto & Perona, 1993) study the application of Group Technology and Cellular Manufacturing concepts in the scheduling of PCB assembly. The objective of the study is to minimize the number of setups and maximize the workload balance between different machines. The authors develop an heuristic that is tested using a simulated model and later implemented using data from a real PCB assembly system. 8 Daskin et.al (Daskin, Maimon, Shtub, & Braha, 1997) formulate an integer linear programming problem capable of managing PCB assembly resources. The objective of the research was solving the problem of grouping PCB components in order to minimize the total components and PCB loading costs, subject to the machine’s capacity constraints. Based on these assumptions an heuristic is created to find a solution and finally, different computational results are analyzed. 2.2.4 PCB scheduling Pinedo (Pinedo, 2008) defines scheduling as the allocation of jobs to machines and their corresponding sequencing, in order to optimize an objective function, being subject to some constraints such as, for example, changeover times. In a typical PCB production system, scheduling is useful, in order to assign products to daily production and the respective release to the assembly lines. Making the best use of all the production resources to improve the overall system performance is the main goal of a scheduling system. The PCB scheduling problem consists in sequencing different lot sized batches such that an objective function is minimized or maximized, considering the job processing, setup time, and the machine cycle time. Another group of researchers (Lin, Shaw, & Locascio, 1997) study the PCB scheduling problem using a two-level scheduling approach. This approach combines product-level, in order to determine the quantity and type of the products to be daily produced and board-leveling scheduling to optimize the machine capacity allocation, hence improving line productivity. The authors then compare their two-levelscheduling scheme to other existing ones and compare results with real production data. G.Á. Farkas and P. Martinek (Farkas & Martinek, 2017) introduce a new approach for the scheduling of PCB production. Their objective is to reduce the computational time needed to optimize the model, while simultaneously, minimizing the makespan. To do so, the authors propose three different methods, distinguished by the initialization and the algorithm search tuning. The first method uses an heuristic that pre-processing the input data. The second, uses a valid starting schedule, as an input for the optimization algorithm, reducing significant time on the initialization phase, hence overall computational time. The last method consists on tuning the optimization algorithm, examining the CPLEX Optimization Studio’s Failure Directed Search algorithm. 2.2.5 PCB component allocation Considering the PCB assignment and grouping as a first management process, the component allocation is the following stage when optimizing PCB production. The electric components should be grouped and allocated to the respective machines, to achieve better cycle times. In the interest of 9 balancing the workload among different placement machines, the optimization of the allocation of components in the PCB can also be very useful. Since the placement of components usually is the operation that consumes more time among the assembly operation and has the greatest potential on reducing production cost, due to the high cost of the placement machines, there is a special interest by researchers on this subject. Hillier and Brandeau (Hillier & L. Brandeau, 2001) investigate an operation assignment problem arising from a PCB assembly process. All the process information and data were taken from a HewlettPackard (HP) factory, where a vast different number of components need to be inserted in a mix of PCB. The research on how to assign the PCB and components to the machines, so as to minimize total cost while at the same time balancing workloads across machines, is solved using a binary integer formulation. The authors develop an heuristic and an optimal solution method to solve this component assignment problem, which are then used to solve a case study with the use of computational resources to show how effective the results are. Another group of researchers (Ammons et al., 1993) addresses the problem of allocating components to different machines, in order to balance workload across them, minimizing processing and setup time. The authors develop a mathematical formulation and present two alternative solution approaches to solve the problem. One of the methods is for solving a simple version of the problem with a list-processing-based heuristic, and the second method is based on solving a linear mixed integer program with an adequate solver. Both methods are then used in an industrial case study, in which the results indicate throughput improvements over the company’s current procedure. 2.2.6 Feeder assignment and placement sequencing Both feeder assignment and placement sequencing can lower total throughput time of the PCB manufacturing, hence reducing production cost. Feeder assignment problems consists in assigning component types to locations on the feeder carriage of the assembly machines, whereas placement sequencing involves specifying the order of the components on the PCB by the assembly machine. The feeder assignment problem is usually modeled as a quadratic assignment problem, whereas the placement sequencing can be modeled as a traveling salesman problem. Both the problems mentioned are known to be NP-hard and should only be solved with efficient heuristics for larger scale problems. In order to optimize the sequence of component placement on PCB and the arrangement of component types to different machine feeders, Ho and Ji (Ho & Ji, 2004) present a hybrid genetic algorithm. In their study, the objective was to minimize the total travelling distance or time of the 10 placement head. Comparing the results of other researchers to their algorithm, they can conclude that the performance in reducing travelling distance has superior results. Other researchers (Crama, Flippo, van de Klundert, & Spieksma, 1997) performed a research on the assembly of PCB where three sub problems arise, the feeder rack assignment problem, followed by the placement sequencing problem and the component retrieval plan problem, all of them studied in a Philips manufacturing plant. Taking as much of the PCB and machine characteristics they heuristically solve the feeder rack assignment problem. Using constructive heuristics and local search methods, the remaining problems are also solved. When tested in one of the Phillips plants, the solution procedure showed notable makespan improvements. Sun et al. (Sun, Lee, & Kim, 2005) focus their study on the component allocation and feeder arrangement, which they admit are the most important in cycle time optimization. The authors structure the operational decision problems and propose a genetic algorithm capable of solving the before mentioned decisions together. In order to optimize the decisions, the number of simultaneously picked up components for a multi-head module machine is maximized, or the number of pickups is minimized and, at the same time, the workload of the machine gantries is balanced. The authors propose a greedy heuristic for the work cycle formation and pickup sequencing decisions in order to evaluate the workload at each gantry and subsequently, using real industrial data, the algorithms are tested and evaluated. 2.3 Optimization Techniques In order to optimize PCB assembly systems, there are two well-known approaches to follow. In most cases, getting an optimal solution is only guaranteed, when the problems are formulated as mathematical models and solved using exact algorithms. Another approach is solving the problems using heuristics, which can generate good solutions efficiently. The figure 1, gives an idea of the trade-off between the quality of a solution and the computational effort and the appropriate methods for each trade-off point. 11 Figure 1: Appropriate methods for each trade-off point. Adapted from (Bhoja, 2002) As computational time increases, the quality of the solution increases, ideally reaching an optimal solution, whereas in the first instance, the quality of the solution is very poor, being a random solution. 2.3.1 Mathematical programming There are innumerous programming models used to solve PCB assembly problems. These models are a representation of an industrial context, which after solved, can be used to better understand the actual situation or even help the organization in the decision-making process. In PCB assembly problems, it is necessary to optimize an objective function. Either it is maximizing or minimizing something like changeover time, component placement time, makespan, just to name a few, subject to constraints such as available capacity. In this dissertation, an integer-programming model will be formulated for solving the PCB line assignment problem. 2.3.2 Linear / Integer programming Linear programming (LP) models are characterized by having the objective function and constraints as linear functions. Additionally, the decision variables need to be continuous. Integer programming (IP) is a specific case of linear programming in which at least one decision variable must take an integer value. Due to some variables being integers instead of continuous, in real life applications, IP is a common modelling method. In this dissertation developed models, a binary integer programing model is used, in which the variables can only take the value of 0 or 1 12 Both Linear and Integer programming are well present in the literature and have been studied by several researchers since the 1940s. The number of applications of LP and IP is huge, many of them in telecommunications networks, production planning and scheduling, logistics, economics, among others. (Fourer, Gay, & Kernighan, 1990) 2.3.3 Algorithms for Integer Linear Programming Different algorithms can be applied to solve an IP problem. Nowadays, the most successful one, combines branch-and-bound and cutting plane methods, methods which now detailed. 2.3.4 Branch and Bound The branch and bound (B&B) method was first proposed by Ailsa Land and Alison Doig in 1960 and consists in partitioning the feasible solution region into more manageable subregions and, if required, further partitions of the created subregions. The smaller subregions are constantly evaluated until the best solution is found (Land & Doig, 1960). First the linear relaxation of the problem is solved using a linear programming algorithm, such as simplex or interior point methods. If the solution obtained by the relaxation is an integer feasible solution, then the method may stop if the optimal solution is found. Otherwise, the B&B algorithm sets up the lower and upper bound for the optimal solution region, tightening the bounds with the branching strategy, until an optimal solution is found, or it is no longer possible to create a new sub problem. 2.3.5 Cutting planes The cutting-plane method was proposed in the 1950s by Ralph E Gomory as another method for solving integer and mixed-integer programming problems. The method requires adding constraints that repetitively reduce the feasible region until an integer optimal solution is found. The constraint that is added to the problem is referred as cut, and the IP can then be solved until an integer solution is obtained or if no optimal solution is found the model is considered infeasible. (Gomory, 1960) In the software used to solve the PCB line assignment problem in this dissertation, the method used was a combination of the branch and bound algorithms and the cutting-planes methods to tighten the linear programming relaxation designated by branch and cut, first introduced by Manfred Padberg and Giovanni Rinaldi, in (Padberg & Rinaldi, 1991). 13 2.4 Assignment Problem The assignment problem is widely applied and extensively studied as a combinatorial optimization problem, in which mathematical techniques are applied intended to find optimal solutions within a delimited set of possible solutions, usually defined by a set of restrictions (Veldkamp, 2015). It is formulated similarly as the transportation problem and has innumerous applications in different problems such as data associating problems, load balancing, telecommunications network design and graph bandwidth problems. (Allen, Ismail, Kennington, & Olinick, 2003; Shachnai, Voloshin, & Zaks, 2018) In the assignment problem there are n agents to execute n tasks. Each agent executes exactly one task, and each task is executed exactly by one agent. The task j being executed by agent i has a cost associated 𝑐𝑖𝑗. The problem is to assign each task to each agent minimizing the cost.(Munkres, 1957) The model has the following mathematical formulation: Parameters: 𝑖 – Agent, which will perform some sort of task 𝑗 – Task to be assigned 𝑐𝑖𝑗 – cost of assigning agent i to task j Decision variables: 𝑥𝑖𝑗 = {1,𝑖𝑓 𝑎𝑔𝑒𝑛𝑡 𝑖 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡𝑎𝑠𝑘 𝑗 0,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Objective Function: 𝑀𝑖𝑛 𝑧= ∑ 𝑐𝑖𝑗𝑥𝑖𝑗 𝑖𝑗∈𝐴 Subject to: ∑ 𝑥𝑖𝑗 =1,𝑖 =1,…,𝑛 𝑗:𝑖𝑗∈𝐴 ∑ 𝑥𝑖𝑗 =1,𝑗= 1,…,𝑛 𝑖:𝑖𝑗∈𝐴 𝑥𝑖𝑗 ∈{0,1},𝑖 =1,…,𝑛;𝑗 =1,…,𝑛 In order to solve the assignment problem, Harold W. Kuhn (H W Kuhn, 1955; Harold W. Kuhn, 2010), proposed a method, which became known as the Hungarian method, a combinatorial optimization 14 algorithm, able to solve the problem in polynomial time. The Hungarian algorithm iterates on primal infeasible solutions, reaching feasibility and optimality in the last iteration. 2.4.1 Generalized Assignment Problem The generalized assignment problem (GAP) differs from the assignment problem because each agent can complete one or more tasks. Moreover, the agent has a capacity limit that cannot be surpassed. In contrast with the assignment problem, the GAP is a NP-hard combinatorial optimization problem. It has been studied considerably by several researchers for more than four decades. Most studies create effective algorithms capable of solving problems of reasonable size to optimality or good quality solutions in reduced time. The GAP was first studied and solved using the branch and bound method by Ross and Soland in 1975, and it is known for its real-life applications such as vehicle routing, assigning jobs to computer networks, logistics , to name a few. (Cattrysse & Van Wassenhove, 1992) In the GAP, in which there are n tasks to be processed by m machines, each machine has a given time capacity, and depending on which machine processes which task there is a processing time associated. The GAP is the problem of assigning each task to exactly one machine, in order to minimize the total cost of processing the tasks and each machine does not surpass its available capacity. The problem can be modeled as a linear integer as the following: Indices: 𝑖 - index of an agent, which will perform some sort of task 𝑗 - index of a task to be assigned Parameters: 𝑛 - number of agents 𝑚 - number of machines 𝑐𝑖𝑗 – Cost of assigning agent i to task j, i=1,…,n; j=,…,m 𝑤𝑖𝑗 – Capacity usage by assigning agent i the task j =1,…,n; j=,…,m 𝑏𝑖 – Capacity of agent I =1,…,n Decision variables: 𝑥𝑖𝑗 = {1,𝑖𝑓 𝑎𝑔𝑒𝑛𝑡 𝑖 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡𝑎𝑠𝑘 𝑗 0,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ,𝑖 =1…𝑛,𝑗= 1…𝑚 Model 21 4. PROBLEM DESCRIPTION AND ANALYSIS OF THE CURRENT SITUATION In this chapter, a detailed description, as well as a critical analysis of the current state of the company is made. Firstly, the production process is addressed and the MOE14 role in it is described. Then, the existing problems regarding the assignment of PCB in Bosch CM are identified and the reason behind this project existence and importance is exposed. Finally, the PCB assignment problem is characterized considering both the previous review, the existing literature and the specific characteristics of the Bosch Car Multimedia Portugal (BrgP) processes. 4.1 Production Process The project, which originated this dissertation, was made within the first stage of a large and complex process, situated in MOE1, where the automatic insertion takes place. In Bosch CM Braga, the production process has some differences between the previously reviewed PCB assembly systems, in subchapter 2.1. To start, all of the process are fully automatic, being all of the sub processes connected by a conveyor. Initially, a set of PCB are placed in a PCB loader that automatically sends, individually, each PCB to the next sub-process of the assembly. The PCB goes through a laser machine, which prints two identification codes onto the PCB. Following this, the PCB has its surface covered with paste through stencil printing, with a subsequent soldering paste inspection. Afterwards, the PCB undergoes the process of the insertion of the several components, followed by the welding sub-process that occurs in an industrial oven. Finally, the PCB goes through an automatic optical inspection process, where the soldering and welding as well as the correct placement of components is verified. If the desired quality is confirmed, the PCB is unloaded into a specific container, which after fully loaded is temporarily stored in a small buffer. The container then waits for the milk-run carriage to transport it for the next SMD, for the reflow 2 process, where the opposite side is worked. In the reflow 2 process the PCB undergoes the exact same process but with additional care, as the first reflow is now on the bottom of the PCB There are additional processes outside of the SMT scope that are dependent of the product engineering, design or requirements, such as manual insertion of components and coating application. Furthermore, additional electrical and functional tests may be required for quality purposes. Finally, several products are designed to optimize the automatic insertion and increase the SMD efficiency; to do so, the PCB are designed to have multiple patterns of the final electronic device that can go from 1 to utmost 56 smaller PCB in one PCB. When this occurs, an additional and subsequent process is required 22 to detach each smaller PCB. This process is called milling where the PCB are detached and an outer frame is removed. In total, MOE1 has 38 SMD 1 , though some are better and/or more equipped than others, distinguishing them in three categories, high, medium or low capacity. This category depends on each SMD component placement per hour (CPH), a major key performance indicator (KPI) in the assembly process. This indicator has a theoretical value, specific of the manufacturer depending of some unique and ideal conditions, such as type, format and complexity of the PCB components. In reality, the CPH of the insertion machines in MOE1 has an inferior value, due to the high variety of different components and PCB. The insertion machines have to take several different pick and place patterns, which lowers the CPH value. All the production is divided in 8 hours shifts, working 24 hours from Monday to Saturday, with 3 shifts, and 16 hours on Sundays, having 2 shifts. On Sundays, it might be required some additional working hours, depending on the production volume. The production is made possible by 4 teams of operators that rotate between the shifts. 4.2 Production planning and control department The department in charge of the production planning and control in MOE1 is designated as MOE14. A team of engineers is responsible for assigning the PCB to the SMD, in a short, medium and long term. The developed project aims to improve medium and long term planning, therefore no description of short term planning is presented. The production planning is triggered by the client order, being received by the logistics department. Resultantly, MOE14 gets from them the final products needed, where they breakdown the final products in the necessary PCB. There are several different PCB types with considerable demand. After knowing the quantity and variety of PCB to be produced, it is necessary for the planners to assign the PCB to the SMD. This is accomplished by first grouping the PCB in families that share common properties and afterwards, the assignment is manually made, based upon the SMD real CPH. 1 At the time of the study, the number of SMD was 38, although this number is constantly changing, due to layout changes. Moreover, the oldest assembly lines can be deactivated when the demand is low 23 The medium and long term planning, although are not able to predict immediate client deviations or non-planned stops, are an efficient way to prepare the required production resources. That being said, it is correct to say that when the long, medium and short have low deviations between each other, the planning is accurate. Besides having to do the production planning, MOE14 is also responsible for controlling the daily production. Whether it is stock control, or the more ordinary day-to-day occurrences, MOE14 assures that the production is running smoothly. Finally, working side by side of the planning department there is also a team of programmers, deciding which programs are needed for the insertion machines software, as well which products are assembled with the same setup. They also manage the extra production such as product samples, another reason why they need to work close to the planners. 4.3 Problems identified Following the analysis of the current state of the company, it is possible to identify the problems arising from the assignment of the PCB to the assembly lines. MOE1 occupies an area of 11000 m², within a production building acquired in 2016, which doubled the production area and space between assembly lines, hence between the 38 SMD there is a long distance to travel, being possible to identify a serious problem regarding the intermediate transports between the reflow1 and reflow2 processes. The PCB were travelling big distances to get the second side worked, leading to a big waste of time, as well as accumulated intermediate stock, and consequently its associated costs. Moreover, there was a problem with the milk run carriages traffic, where occasionally, other transportations of raw materials and tools were getting delayed due to waiting times causing production delay. This traffic also had an impact in the overall security with an excess of vehicles going through the same corridors as the rest of the workforce. Furthermore, the containers transporting the PCB can weight up to 8 kg empty and 15 kg fully loaded, causing an impact on the overall workforce health. An additional problem identified was the time and resources being wasted in manually performing the PCB assignment to the assembly lines by MOE14, which can be significantly reduced by automatizing this process. In order to optimize transport distance between reflow1 and reflow2 of the PCB, the first mathematical model is formulated, in which the objective is to minimize the intermediate transport distance. 24 As the project took progress, other problems were identified, such as in times when the production volume demand is lower and not every SMD might be necessarily used. This causes unnecessary use of resources, such as energy, maintenance work, support and workforce. Accordingly, a second mathematical model is formulated, minimizing the number of used SMD. To achieve this objective, a different objective is tackled, by minimizing the number of SMD having PCB assigned, in order to make fully usage of the machines, thus saving costs from SMD that are not necessarily needed. Moreover, another problem that was identified was the unbalanced machine workload. Some SMD were getting PCB families that were taking processing times much longer than other SMD. This causes a lack of coordination between the several SMD. Accordingly, a third mathematical model is formulated, balancing the total machine workload. This could be accomplished by assigning the necessary PCB, so that the total processing time for each SMD is identical. Balancing the workload would mean that, when there is a drop in the production demand, thus gaining more, capacity for the needed production it is possible to reduce the working hours. This reduction could mean the saving of workforce, maintenance and resource costs. 4.4 PCB assignment problem characterization In this industrial context, it is considered to have a high variety of PCB types and a low production volume. There are over 5800 distinct PCB types and a demand of a little more than 3 million each month, although these numbers are in constant change. In order to minimize the total number of setups and avoiding that similar PCB are assembled in different SMD, the PCB were grouped into families, depending on the common components, board type and reflow process. Afterwards, the PCB families are assigned to a SMD ensuring the client demand requirements are met. All PCB families can be processed in any SMD, as long as its capacity is not exceeded. The PCB families take different processing times, depending on the CPH of the automatic insertion machines, number of components each PCB has and the client demand. The demand is given for any specific month of the year and the capacity of the machines is given by the number of hours available to produce in that same month. In order to measure the solutions from the developed models, a transportation cost matrix was created, in which there is a proportion of the distance between the different SMD and the total cost of transportation, which is explained better further in this dissertation in subchapter 5.4. As mentioned, before, the first mathematical model minimizes the total cost between the two processes. 25 The second model minimizes the number of SMD with jobs assigned, thus maximizing each assembly line usage. Simultaneously, the model also makes the PCB transportation cost between the reflow processes to the minimum. Finally, the third model balances the processing time between the different SMD while, same has the first and second model, minimizes the transportation cost between the reflow processes. 26 5. MODELS In this chapter, the three mathematical models for solving the PCB assignment problems previously described are presented. At a first instance, the notation is presented, followed by the models. The three models’ parameters are the following: 𝐾 – Set of PCB families, indexed by: 𝑘= 1,…,𝑛 𝐿 – Set of SMD, indexed by 𝑖 =1,…,𝑚 or 𝑗 =1,…,𝑚 𝑏𝑖 – SMD 𝑖 capacity 𝐶𝑖𝑗 – Cost of the reflow 1 process in SMD 𝑖 and reflow 2 process in SMD 𝑗 𝑝𝑖𝑘1– Processing time of reflow 1 in SMD 𝑖 of the PCB family 𝑘, 𝑖= 1,…,𝑚 𝑝𝑖𝑘2 – Processing time of reflow 2 in SMD 𝑗 of the PCB family 𝑘, 𝑖 = 1,…,𝑚 5.1 Cost minimization model In this model, the objective is to minimize the cost of product transportation, between the reflow1 and reflow 2 processes. It is assumed the cost is proportional to the distance. The decision variables are: 𝑥𝑘𝑖𝑗={1,if reflow1 of family 𝑘 is produced in SMD 𝑖 and reflow 2 in SMD 𝑗 0,otherwise , 𝑘 =1,…,𝑛;𝑖 =1,…,𝑚;𝑗 =1,…,𝑚 and the model is 𝑀𝑖𝑛 𝑍= ∑∑∑(𝑪𝒊𝒋 𝒙𝒌𝒊𝒋 𝑗∈𝐿 ) 𝑖∈𝐿𝑘∈𝐾 Subject to: ∑∑𝑥𝑘𝑖𝑗 = 1 𝑗, 𝒊 ꓯ 𝑘∈𝐾 (1) ∑∑(𝑃𝑖𝑘 𝑥𝑖𝑗𝑘 +𝑃𝑖𝑘 𝑥𝑗𝑖𝑘)≤𝑏𝑖 𝑗, 𝒌 ꓯ 𝑘∈𝐾,ꓯ 𝑖∈ 𝐿 (2) 𝑥𝑘𝑖𝑗 ∈{𝟎,𝟏} ꓯ 𝑘∈𝐾,ꓯ 𝑖,𝑗∈ 𝐿 (3) This model’s objective function minimizes the total cost of the assignment of the PCB families. 27 The first constraints (1) force each PCB family (𝑘) to be assigned to one SMD (𝑖) for the reflow 1 process and to one SMD (𝑗) for the reflow 2 process. These constraints force all PCB families to be assembled. In the second constraints (2), the sum of the reflow 1 and reflow 2 processing times of each SMD assigned PCB families, cannot exceed the SMD capacity. Finally, the third constraints (3), define the decision variables domain as binaries. 5.2 SMD working number minimization The second model has the same parameters as the previous. In this model the objective function minimizes the number of SMD with PCB families assigned. This can be very useful, for times when, the production demand is low and not every SMD might be necessary. This model has additional decision variables (𝑦𝑖) with respect to the previous one. These variables represent if each SMD is used or not. The decision variables are: 𝑥𝑘𝑖𝑗 ={1,if reflow1 of family 𝑘 is produced in SMD 𝑖 and reflow 2 in SMD 𝑗 0,otherwise , 𝑘 =1,…,𝑛;𝑖 =1,…,𝑚;𝑗 =1,…,𝑚 𝑦𝑖={1,if SMD i is used (i.e.has at least one job assigned) 0,otherwise , 𝑖 = 1,…,𝑚 𝑀𝑖𝑛 𝑍= ∑ 𝑦𝑖 𝑖∈𝐿 Subject to: ∑∑𝑥𝑘𝑖𝑗 =1 𝑗𝒊 ꓯ 𝑘∈ 𝐾 (1) ∑∑(𝑃𝑖𝑘 𝑥𝑖𝑗𝑘 +𝑃𝑖𝑘 𝑥𝑗𝑖𝑘)≤𝑏𝑖 𝑗𝒌 ꓯ 𝑘∈ 𝐾,ꓯ 𝑖 ∈𝐿 (2) 𝑥𝑘𝑖𝑗 +𝑥𝑘𝑗𝑖 ≤𝑦𝑖 , ꓯ 𝑘∈𝐾,ꓯ 𝑖,𝑗 ∈𝐿:𝑖 ≠𝑗 (4) 𝑥𝑘𝑖𝑖 ≤𝑦𝑖 , ꓯ 𝑘 ∈ 𝐾,ꓯ 𝑖∈𝐿 (5) ∑∑∑(𝐶𝑖𝑗 𝑥𝑘𝑖𝑗 𝑗∈𝐿𝑖∈𝐿 )=0 ꓯ 𝑘∈𝐾,ꓯ 𝑖,𝑗∈ 𝐿 𝑘∈𝐾 (6) 𝑥𝑘𝑖𝑗 ∈{𝟎,𝟏} , ꓯ 𝑘∈ 𝐾,ꓯ 𝑖,𝑗 ∈𝐿 (3) 𝑦𝑖 ∈{𝟎,𝟏} , ꓯ 𝑖∈ 𝐿 (7) 28 In this model, the objective function, as the name suggests, minimizes the number of SMD with jobs assigned. The transportation cost between reflows is also minimized to 0, note that there are many solutions with the transportation cost equal to 0, but this model chooses the best solution amongst them, considering the objective function. The first three constraints (1) (2) (3), are the same as the first model, in which and only one PCB family can be assigned for a reflow 1 and reflow 2 process in a SMD (𝑖) and (𝑗), the sum of the processing times of both reflow processes cannot surpass the available capacity and the decision variable 𝑥𝑘𝑖𝑗 is defined as binary. Same as the third constraints, in (7) it is defined that the decision variable 𝑦𝑖 is binary. Constraints (4), the assignment of a PCB family to complete both reflows in different SMD implies that the SMD are used. Constraints (5) are the same as the previous (4), although the SMD chosen for the assignment must be the same for reflow 1 and 2. Finally, the last constraints (6) forces the assignment of the PCB family k to do the reflow 1 in SMD i and reflow 2 in SMD j , so as the transport cost is equal to 0. 5.3 SMD workload balance The third model has the same parameters as the previous ones. In this model the objective is to balance the SMD workload by minimizing the processing time deviation to the mean. This objective is useful for when the production volume justifies the use of all the SMD. In these circumstances, it is necessary to balance the workload, so that when it is needed to reduce the available capacity of working hours, the production is evenly distributed. Assembly lines could finish their tasks almost at the same time, for when the production needs to stop. This model has two more decision variables ( 𝑠𝑖 and 𝑡𝑖 ), which measure the deviation of the processing times, as it will be later explored. The decision variables are 𝑥𝑘𝑖𝑗={1,if family 𝑘 does reflow 1 in SMD 𝑖 and reflow 2 in SMD 𝑗 0,otherwise , 𝑘 =1,…,𝑛;𝑖 =1,…,𝑚;𝑗 =1,…,𝑚 𝑠𝑖 – Slack with respect to the average processing time of the SMD i 𝑡𝑖 –Surplus with respect to the average processing time of the SMD i 𝑖 = 1,…,𝑚 𝑀𝑖𝑛 𝑍=∑( 𝑠𝑖 + 𝑖∈𝐿 𝑡𝑖 ) Subject to: 29 ∑∑𝑥𝑘𝑖𝑗 =1 𝑗𝒊 ꓯ 𝑘∈ 𝐾 (1) ∑∑(𝑃𝑖𝑘 𝑥𝑖𝑗𝑘 +𝑃𝑖𝑘 𝑥𝑗𝑖𝑘) 𝑗≤𝑏𝑖 , 𝑘 ꓯ 𝑘∈ 𝐾,ꓯ 𝑖∈ 𝐿 (2) 𝑏𝑖−∑∑(𝑃𝑖𝑘 𝑥𝑖𝑗𝑘 +𝑃𝑖𝑘 𝑥𝑗𝑖𝑘) 𝑗𝒌 + 𝑠𝑖 − 𝑡𝑖 =𝑏𝑖−∑ ∑ (𝑃𝑖𝑘 𝑥𝑖𝑗𝑘 +𝑃𝑖𝑘 𝑥𝑗𝑖𝑘) 𝑗𝒌 𝐿,ꓯ 𝑖∈ 𝐿 (8) ∑∑∑(𝐶𝑖𝑗 𝑥𝑘𝑖𝑗 𝑗∈𝐿𝑖∈𝐿 )= 0 𝑘∈𝐾 ꓯ 𝑘∈𝐾,ꓯ 𝑖,𝑗∈ 𝐿 (6) 𝑥𝑘𝑖𝑗 ∈{𝟎,𝟏} ꓯ 𝑘∈𝐾,ꓯ 𝑖,𝑗∈ 𝐿 (3) This model’s objective function minimizes the sum of the two deviation variables for all SMD, so that, with the given constraints, the processing time for every SMD is as close as possible to the processing time mean. The first three constraints (1) (2) (3) are the same as the first two models, in which the decision variable 𝑥𝑘𝑖𝑗 is defined as binary, the sum of the processing times of both reflow processes cannot surpass the available capacity and only one PCB family can be assigned for a reflow 1 and reflow 2 process in a SMD (i) and (j). In the fourth constraints (8), the mean of available processing time across all the SMD must be equal to the available processing time of a specific SMD with the two new variables ( 𝑠𝑖 and 𝑡𝑖 ) adjusting to maintain this equality. The last constraints (6), same as the last model, makes the transport cost of the assignment of the PCB families to be equal to zero. 5.4 Optimization software and data input In order to solve the developed models, the Gurobi Optimization software and IBM ILOG CPLEX Optimization Studio software were used with AMPL as the modeling language. Moreover, a VBA Microsoft Excel tool was created to analyze the obtained solutions and measure the comparison between the current PCB assignment and the new, obtained from the developed models. To run the application, initially, it is necessary to define the number of SMDs being used. Afterwards, the PCB were grouped into families manually, with the help of an Excel macro, by combining the same board types, similar components, PCB width and the corresponding reflow process. After obtaining the PCB families, each SMD CPH is calculated and the available capacity is given by the working hours of the month in question. Moreover, the processing time of the PCB families in each SMD is calculated, also with the use of an Excel macro. This is possible by dividing the number of components each PCB has by the CPH of each SMD. Finally, the number of SMD available for use is also necessary to be defined. 30 In the figure 10, represented below, it is possible to observe a plant layout of the BrgP MOE1 automatic insertion, in which the rectangles are the different SMD and the blue circles are the position behind and in front of each SMD. This layout was drawn to help create a cost matrix between each SMD reflow. The cost between each position was determined based on the travelling distance of the PCB and are represented by the orange color arrows. There are a few restrictions to be considered in the movement flux, due to the plant layout traffic conditions, also defined by the point of the orange arrows. Figure 9: MOE1 layout and PCB transportation flux From the layout representation shown, it is possible to analyze the created cost matrix, in figure 10, which connects all the SMD of the plant layout and gives the associated transportation cost. When finishing the reflow 1 process, if the PCB family is transported to the position in front, the associated cost is equal to 0, and it is considered the best alternative, whereas all of the other combination costs are relative to the distance needed to travel. 37 Figure 13: Model 2 PCB line assignment In order to minimize the number of SMD having PCB assigned, the assembly lines with bigger CPH are chosen, having its capacity almost full, as verified in table 3. Finally, the last model visual solution is now presented in the figure 15 Figure 14: Model 3 PCB line assignment Same as the last two models, the transport between the reflow processes has zero cost, moreover the workload is balanced across the various SMD, as it is shown by the capacity usage in table 3, already represented above. It is also possible to better visualize this workload balance in graphic below presented. 38 Figure 16: Model 3 SMD capacity usage Analyzing the graphic, there is a mean of 62% capacity usage of every SMD. Given this condition, it is possible to adjust the working schedules equally and without production capacity prejudice. Since there are four shift teams and each shift correspond to 8 hours, in order to reduce, for example, one shift in this specific month, it would be necessary to free 32 hours. Doing so, there would be a capacity reduction of 5,06 % and one SMD would require additional working hours. In table 5, it is possible to verify these conditions, by freeing additional shifts. Table 4: Capacity reduction from freed shifts Shifts freed Freed hours Capacity reduction SMD in need of additional working hours 1 32 5,06% 1 2 64 10,12% 2 3 96 15,90% 2 4 128 20,25% 3 5 160 25,32% 4 6 192 30,38% 6 7 224 35,44% 8 8 256 40,51% 30 From a management point of view, it is needed to take in account some time wasted in production stops, maintenance, shift changes and unique occurrences. In order to do so, it is necessary to have a margin of free time, thus avoiding work not completed. 6.3 Capacity adjustment testing After an analysis of model 2, it was suggested changing the SMD capacity of this model, in order to reduce the assembly line usage, from near 100% to more advisable values. To do so, the capacity was limited from 632 hours to 508 hours. Doing so, would require more SMD used, although there is a trade0% 20% 40% 60% 80% 100% 120% 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435363738 Model 3 SMD capacity usage 39 off to more available capacity, for non-planned stops. The results of this capacity change of model 2 are represented in table 6 Table 5: Model 2 SMD capacity usage comparison SMD Model 2 Model 2.2 2 98,4% 80% 4 0,0% 0% 5 99,9% 80% 6 0,0% 0% 8 92,0% 79% 9 100,0% 73% 11 99,3% 79% 12 0,0% 0% 13 100,0% 79% 14 99,8% 79% 15 100,0% 77% 16 100,0% 80% 17 99,8% 80% 18 0,0% 79% 19 0,0% 74% 20 0,0% 0% 21 0,0% 0% 22 0,0% 80% 23 0,0% 0% 24 0,0% 0% 25 0,0% 0% 27 0,0% 79% 28 99,9% 80% 29 100,0% 80% 30 95,9% 80% 31 0,0% 0% 32 0,0% 0% 33 0,0% 80% 34 0,0% 80% 35 98,2% 80% 36 99,3% 79% 37 90,9% 80% 38 99,6% 80% As it is verified, although model 2.2 makes use of more SMD, 23 instead of the 17 in the initial model 2, there is capacity usage mean of 79%. Thus, in cases when the production needs to stop, there is still time for completing the production and a lot of saved resources, as mentioned before. To provide more trustworthy results, another month was analyzed, in which the exact same conclusions can be obtained. These results can be visualized in the appendix 1 and 2. 40 7. CONCLUSION In this dissertation, the printed circuit board (PCB) line assignment problem was addressed, in which the objective was to find the best assignment of PCB families to the different assembly lines of Bosch CM Portugal S.A – Braga automatic insertion production area. In order to solve this problem, three integer-programming models were formulated to answer different management problems. In the scope of this work, tools were also developed to better collect and treat the data and analyze the solutions obtained. The developed mathematical models have three distinct objectives. The first, aimed to minimize the transportation cost between the two main processes. The second model intended to minimize the number of surface mount devices (SMD) with PCB assigned, thus maximizing the workload of the chosen production lines when production demand decreases, while maintaining the transportation cost to a minimum. Finally, the third model proposed an assignment strategy, so that the SMD have their workload balanced, thus levelling each line stoppage time and managing required working hours, while also maintaining transportation cost to a minimum. These optimization models are proven effective in the decision-making process, improving both solutions and time effectiveness. Moreover, there are many opportunities for investment in these research operation tools in the electronic industry, as verified in the literature review. The proposed solutions are optimal, being more efficient as well as taking less time to be obtained, by opposition to the manual assignment currently being used. It is important to note that, by using the proposed solutions, the milk run traffic and security problems are improved. This is due to no longer being necessary assigning a milk run carriage to pick the PCB containers, between the reflow 1 and 2 processes. An operator can manually place the PCB containers that finished the reflow 1 process in the next SMD for the subsequent reflow, since the SMD are just in front of each other. For future works, the results obtained from this work could be improved by creating a mathematical model that would allow to group and create PCB families more efficiently and optimized. For this dissertation a specific manual strategy was used, if instead, a mathematical model could create the PCB families, a better assignment could be made, especially if integrated with the assignment presented in this work. Analysing the cost gains from the solutions obtained in the proposed solutions, is also a great way to measure its efficiency. Furthermore, a mathematical model of the PCB families scheduling, would be a next step to consider, since it increases a lot of value to the production planning. 41 REFERENCES Allen, D., Ismail, I., Kennington, J., & Olinick, E. (2003). An Incremental Procedure for Improving Path Assignment in a Telecommunications Network. Journal of Heuristics , 9 (5), 375–399. https://doi.org/10.1023/B:HEUR.0000004811.65073.d3 Ammons, J. C., Carlyle, W. M., DePuy, G. W., Ellis, K. P., McGinnis, L. F., Tovey, C. A., & Xu, H. (1993). Computer-Aided Process Planning in Printed Circuit Card Assembly. IEEE Transactions on Components, Hybrids, and Manufacturing Technology . https://doi.org/10.1109/33.237929 Balakrishnan, A., & Vanderbeck, F. (1993). A Tactical Planning Model for Mixed-Model Electronics Assembly Operations. Operations Research , 47 . https://doi.org/10.1287/opre.47.3.395 Bhoja, S. (2002). Optimization of the Assignment of Printed Circuit Cards to Assembly Lines in Electronics Assembly. International Journal of Production Research , 40 . https://doi.org/10.1080/00207540210135604 Cattrysse, D. G., & Van Wassenhove, L. N. (1992). A survey of algorithms for the generalized assignment problem. European Journal of Operational Research , 60 (3), 260–272. https://doi.org/10.1016/0377-2217(92)90077-M Chu, P. C., & Beasley, J. E. (1997). A genetic algorithm for the generalised assignment problem. Computers & Operations Research , 24 (1), 17–23. https://doi.org/10.1016/S03050548(96)00032-9 Crama, Y., Flippo, O. E., van de Klundert, J., & Spieksma, F. C. R. (1997). The assembly of printed circuit boards: A case with multiple machines and multiple board types. European Journal of Operational Research , 98 (3), 457–472. https://doi.org/10.1016/S0377-2217(96)00228-7 Daskin, M. S., Maimon, O., Shtub, A., & Braha, D. (1997). Grouping components in printed circuit board assembly with limited component staging capacity and single card setup: Problem characteristics and solution procedures. International Journal of Production Research , 35 (6), 1617–1638. https://doi.org/10.1080/002075497195164 Farkas, G. A., & Martinek, P. (2017). Scheduling of printed circuit board production with mathematical solvers . 106–109. https://doi.org/10.1109/SIITME.2017.8259868 Fourer, R., Gay, D. M., & Kernighan, B. W. (1990). A Modeling Language for Mathematical Programming. Management Science . https://doi.org/10.1287/mnsc.36.5.519 Freed, T., Maimon, O., & DAR-EL, E. (1989). Group Set-Up for Printed Circuit Board Assembly. THE INTERNATIONAL JOURNAL OF PRODUCTION RESEARCH , 27 , 1795–1810. 42 https://doi.org/10.1080/00207548908942655 Gomory, R. E. (1960). An Algorithm for the Mixed Integer Problem. In The RAND Corporation . Hillier, M., & L. Brandeau, M. (2001). Cost minimization and workload balancing in printed circuit board assembly. IIE Transactions , 33 . https://doi.org/10.1023/A:1010840432611 Ho, W., & Ji, P. (2004). A hybrid genetic algorithm for component sequencing and feeder arrangement. Journal of Intelligent Manufacturing , 15 , 307–315. https://doi.org/10.1023/B:JIMS.0000026569.88191.46 Ho, W., & Ji, P. (2005). PCB assembly line assignment: A genetic algorithm approach. Journal of Manufacturing Technology Management , 16 . https://doi.org/10.1108/17410380510609519 Kuhn, H W. (1955). The Hungarian method for the assignment problem. Naval Research Logistics Quarterly , 2 (1‐2), 83–97. https://doi.org/10.1002/nav.3800020109 Kuhn, Harold W. (2010). The Hungarian method for the assignment problem. In 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art . https://doi.org/10.1007/978-3-540-68279-0_2 Land, A. H., & Doig, A. G. (1960). An Automatic Method of Solving Discrete Programming Problems. Econometrica , 28 (3), 497–520. https://doi.org/10.2307/1910129 Leon, V. J., & Peters, B. A. (1998). A comparison of setup strategies for printed circuit board assembly. Computers & Industrial Engineering , 34 (1), 219–234. https://doi.org/10.1016/S03608352(97)00162-9 Lin, F.-R., Shaw, M. J., & Locascio, A. (1997). Scheduling printed circuit board production systems using the two-level scheduling approach. Journal of Manufacturing Systems , 16 (2), 129–149. https://doi.org/10.1016/S0278-6125(97)85676-2 Luzzatto, D., & Perona, M. (1993). Cell formation in PCB assembly based on production quantitative data. European Journal of Operational Research , 69 , 312–329. https://doi.org/10.1016/03772217(93)90018-I Madeira, J. (2017). No Title. MCGINNIS, L. F., AMMONS, J. C., CARLYLE, M., CRANMER, L., DEPUY, G. W., ELLIS, K. P., … XU, H. (1992). AUTOMATED PROCESS PLANNING FOR PRINTED CIRCUIT CARD ASSEMBLY. IIE Transactions , 24 (4), 18–30. https://doi.org/10.1080/07408179208964231 Munkres, J. (1957). Algorithms for the Assignment and Transportation Problems. Journal of the Society for Industrial and Applied Mathematics , 5 (1), 32–38. Retrieved from http://www.jstor.org/stable/2098689 43 O’Brien, R. (1998). An overview of the methodological approach of action Research. Optimal Production Planning for PCB Assembly. (2007). In Optimal Production Planning for PCB Assembly . https://doi.org/10.1007/978-1-84628-500-4 Padberg, M., & Rinaldi, G. (1991). A Branch-and-cut Algorithm for the Resolution of Large-scale Symmetric Traveling Salesman Problems. SIAM Rev. , 33 (1), 60–100. https://doi.org/10.1137/1033004 Pinedo, M. L. (2008). Scheduling: Theory, algorithms, and systems. In Scheduling: Theory, Algorithms, and Systems . https://doi.org/10.1007/978-0-387-78935-4 Prasad, R. P. (1997). Introduction to Surface Mount Technology. In Surface Mount Technology: Principles and Practice (pp. 3–50). https://doi.org/10.1007/978-1-4615-4084-7_1 Shachnai, H., Voloshin, A., & Zaks, S. (2018). Flexible bandwidth assignment with application to optical networks. Journal of Scheduling , 21 (3), 327–336. https://doi.org/10.1007/s10951-017-0514-4 Sun, D.-S., Lee, T.-E., & Kim, K.-H. (2005). Component allocation and feeder arrangement for a dualgantry multi-head surface mounting placement tool. International Journal of Production Economics , 95 (2), 245–264. https://doi.org/10.1016/J.IJPE.2004.01.003 Susman, G. I., & Evered, R. D. (1978). An Assessment of the Scientific Merits of Action Research. Administrative Science Quarterly . https://doi.org/10.2307/2392581 Veldkamp, B. P. (2015). Computerized Test Construction. International Encyclopedia of the Social & Behavioral Sciences , 510–514. https://doi.org/10.1016/B978-0-08-097086-8.43019-9 Walwadkar, S., Kovalchick, C., Hezeltine, W., Liang, F., & McAllister, A. (2014). In-situ surface mount process characterization using digital image correlation. Conference Proceedings of the Society for Experimental Mechanics Series . https://doi.org/10.1007/978-3-319-00780-9_3 44 APPENDIX I – NEW MONTH VISUAL SOLUTION Model 1 visual solution Model 2 visual solution 45 Model 3 visual representation 46 APPENDIX II – NEW MONTH SMD CAPACITY USAGE Capacity usage (%) SMD Model 1 Model 2 Model 3 Current situation 2 100% 100% 68% 70% 4 68% 0% 68% 69% 5 81% 100% 77% 32% 6 1% 0% 52% 240% 8 82% 91% 78% 71% 9 100% 100% 100% 6% 11 98% 99% 78% 10% 12 81% 0% 68% 61% 13 95% 99% 91% 106% 14 78% 100% 68% 10% 15 79% 100% 76% 10% 16 51% 100% 68% 8% 17 6% 100% 68% 236% 18 20% 96% 68% 143% 19 98% 0% 68% 6% 20 65% 0% 68% 64% 21 12% 0% 68% 48% 22 95% 0% 68% 15% 23 0% 0% 0% 40% 24 0% 0% 0% 5% 25 87% 0% 68% 322% 27 49% 0% 67% 18% 28 100% 100% 99% 160% 29 99% 100% 100% 23% 30 41% 41% 41% 62% 31 96% 0% 68% 38% 32 23% 0% 68% 154% 33 86% 100% 68% 23% 34 89% 100% 68% 146% 35 91% 95% 68% 113% 36 71% 100% 69% 8% 37 76% 84% 68% 3% 38 97% 100% 87% 0%