Performance Improvement in CPSs over Self-similar System Structures

Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Information Control Problems in Information Control Problems in Manufacturing Manufacturing Bergamo, Italy, June 11-13, 2018 Available online at www.sciencedirect.com Proceedings,16th IFAC Symposium on Information Control Problems in Bergamo, June 11-13, 2018 Proceedings,16th IFAC Symposium on Bergamo, Italy, Italy, June 11-13, 2018 Information Control Problems in Manufacturing Manufacturing Bergamo, June 11-13, Information Control in Manufacturing Bergamo, Italy, Italy, JuneProblems 11-13, 2018 2018 Bergamo, Italy, June 11-13, 2018

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IFAC PapersOnLine 51-11 (2018) 570–575 Performance Improvement in CPSs over Self-similar System Structures Performance Improvement in CPSs over Self-similar System Structures Performance Improvement in over Self-similar System Structures Performance Improvement in CPSs CPSs over Self-similar Structures Andrea Bonci*, Massimiliano Pirani*, Chiara Mansanta**, System Sauro Longhi* Performance Improvement in CPSs over Self-similar System Structures Andrea Bonci*, Massimiliano Pirani*, Chiara Mansanta**, Andrea Bonci*, Massimiliano Pirani*, Chiara Mansanta**, Sauro Sauro Longhi* Longhi*

Andrea Andrea Bonci*, Bonci*, Massimiliano Massimiliano Pirani*, Pirani*, Chiara Chiara Mansanta**, Mansanta**, Sauro Sauro Longhi* Longhi* Andrea Bonci*, Massimiliano Pirani*, Chiara Mansanta**, Sauro Longhi* *Department of Information Engineering (DII), **Department of Industrial Engineering and Mathematics (DIISM) *Department of Engineering (DII), **Department of Industrial and *Department of Information Information Engineering (DII), **Department [email protected], Industrial Engineering Engineering and Mathematics Mathematics (DIISM) (DIISM) Università Politecnica delle Marche, 60131, Ancona, Italy (e-mail: [email protected]) *Department of Information Engineering (DII), **Department of Industrial Engineering and Mathematics Università Politecnica delle Marche, 60131, Ancona, Italy (e-mail: [email protected], [email protected]) *Department of Information Engineering (DII), **Department of Industrial Engineering and Mathematics (DIISM) (DIISM) Università Politecnica delle Marche, 60131, Ancona, Italy (e-mail: [email protected], [email protected]) *Department of Information Engineering **Department Industrial Engineering and Mathematics (DIISM) Università Politecnica delle 60131, Ancona, Italy [email protected]) Università Politecnica delle Marche, Marche, 60131, (DII), Ancona, Italy (e-mail: (e-mail:[email protected], [email protected], [email protected]) Università Politecnica delle Marche, 60131, Ancona, Italy (e-mail: [email protected], [email protected]) Abstract: The present paper presents the further developement of a solution for the performance Abstract: Theofpresent present papercyber-physical presents the the systems further developement developement of aacomplex solution productive for the the performance performance Abstract: The paper presents further of solution for improvement distributed that approximate processes, Abstract: The present paper presents the further developement of aacomplex solution for the performance improvement of distributed cyber-physical systems that approximate complex productive processes, Abstract: The present paper presents the further developement of solution for the performance improvement of distributed cyber-physical systems that approximate productive processes, introducing self-similarities in holonic systems. The proposed methodology provides a viable solution for Abstract: The present paper presentssystems. the systems further developement of acomplex solution for the performance improvement of distributed cyber-physical that approximate productive processes, introducing self-similarities in holonic The proposed methodology provides a viable solution for improvement of distributed cyber-physical systems that approximate complex productive processes, introducing self-similarities in holonic systems. The proposed methodology provides a viable solution for aimprovement manufacturing system that integrates robotics, mechatronics and automation systems at different levels. of system distributed cyber-physical systems that approximate complex productive processes, introducing self-similarities in holonic systems. The proposed methodology provides aa viable solution for a manufacturing manufacturing that integrates robotics, mechatronics and automation systems at different levels. introducing self-similarities in holonic systems. The proposed methodology provides viable solution for aPerformance system that integrates robotics, mechatronics and automation systems at different levels. measurements and bottlenecks detection on critical production paths achieve asolution pervasive introducing self-similarities in holonic systems. The proposed methodology provides a viable for aaPerformance manufacturing system integrates robotics, mechatronics and systems at levels. measurements and bottlenecks detection on critical critical production paths achieve pervasive manufacturing system that thatof integrates robotics, mechatronics andbyautomation automation systems at different different levels. Performance measurements and bottlenecks detection on production paths achieve aa pervasive control of the effectiveness associated the system-of-systems means of a well-defined sequence of aPerformance manufacturing system that integrates robotics, mechatronics and automation systems at different levels. Performance measurements and bottlenecks detection on critical production paths achieve a pervasive control of the effectiveness of associated the system-of-systems by means of a well-defined sequence of measurements and bottlenecks detection on critical production achieve a pervasive control ofand theminimal effectiveness of associated the system-of-systems byextension means oftoapaths well-defined sequence of specific corrective actions. This work presents an n-th degree of formulas Performance measurements and bottlenecks detection on critical production paths achieve a pervasive control of the effectiveness of associated the system-of-systems by means of a well-defined sequence of specific and minimal corrective actions. This work presents an extension to n-th degree of formulas control ofand the effectiveness of associated means oftoa well-defined sequence of specific minimal corrective actions. This work presents anbyextension n-th degree of formulas previously applied to tree structures of thethe 2ndsystem-of-systems degree. control ofand the effectiveness of associated means ofto sequence of specific and minimal corrective actions. This work presents presents an anbyextension extension toa well-defined n-th degree degree of of formulas previously applied to tree tree structures of the thethe 2ndsystem-of-systems degree. specific minimal corrective actions. This work n-th formulas previously applied to structures of 2nd degree. specific and minimal corrective actions. This work presents an extension to n-th degree of formulas previously applied to tree structures of the 2nd degree. © 2018, IFAC (International Federation Control) production Hosting by Elsevier All rights reserved. Keywords: Overall throughput effectiveness; systems;Ltd. fractal factory. previously applied to tree structures of of theAutomatic 2ndcyber-physical degree. Keywords: applied Overall to throughput effectiveness; cyber-physical production systems; systems; fractal fractal factory. factory. previously tree structures of the 2ndcyber-physical degree. Keywords: Overall throughput effectiveness; production Keywords: Overall Overall throughput throughput effectiveness; effectiveness; cyber-physical cyber-physical production production systems; systems; fractal fractal factory. factory. Keywords: Keywords: Overall throughput effectiveness; cyber-physical production systems; fractal factory. as the number of descendants of a node. The limit in Bonci et 1. INTRODUCTION as number of of node. limit Bonci et as the the number of descendants descendants of aa provided node. The Theup limit indegree Bonci 2. et al. (2017b) is that a solution was to ain 1. INTRODUCTION INTRODUCTION 1. as the number of descendants of a node. The limit in Bonci et al. (2017b) is that a solution was provided up to a degree 2. as the number ofcomplete descendants of a provided node. Theup limit indegree Bonci et al. (2017b) is that a solution was to a 2. 1. INTRODUCTION This paper will and extend the work for degree n>2. With the introduction of cyber-physical systems (CPSs) in This 1. INTRODUCTION as the number of descendants of a node. The limit in Bonci et al. (2017b) is that a solution was provided up to a degree 2. paper will complete and extend the work for degree n>2. al. (2017b) is that a solution was provided upfor to degree a degree 2. This paper will complete and extend the work n>2. With the introduction of cyber-physical systems (CPSs) in 1. INTRODUCTION Still for higher degrees, the formulas here presented will With the introduction of cyber-physical systems (CPSs) in productive environment, computing, networks and al. (2017b) is that a solution was provided up to a degree 2. This paper will complete complete and extend the work work for degree n>2. n>2. Still for higher degrees, the formulas here presented will With the of systems (CPSs) in will and the degree Still forimmediate higher degrees, theextend formulas here for presented will productive environment, computing, networks and With the introduction introduction of cyber-physical cyber-physical systems (CPSs) in This admitpaper implementation in databaseproductive environment, computing, networks and intelligence shall integrate the shop floor to the whole This paper will complete and extend the embedded work for degree n>2. Still for higher degrees, the formulas here presented will With the introduction of cyber-physical systems (CPSs) in admit immediate implementation in embedded databaseproductive environment, computing, networks and Still for higher degrees, the formulas here presented will admit immediate implementation in embedded databaseintelligence shall integrate the shop floor to the whole productive environment, computing, networks and centric technology, (see Pirani et al. (2016)), for tiny intelligence shall integrate the shop floor to the whole business in compliance with the notion of industrial CPS Still for higher degrees, the formulas here presented will admit immediate implementation in embedded databaseproductive environment, computing, networks and centric technology, (see Pirani et al. (2016)), for tiny intelligence shall integrate the shop floor to the whole admit immediate implementation in embedded databasecentric technology, (see Pirani et al. (2016)), for tiny business in compliance with the notion of industrial CPS intelligence shall integrate the shop floor to the whole computing systems. Moreover, this method provides a simple business in compliance with the notion of industrial CPS (ICPS), as treated by Colombo et al. floor (2017). Production admit immediate implementation in al. embedded databasecentric technology, (see Pirani et (2016)), for tiny intelligence shall integrate the shop to the whole computing systems. Moreover, this method provides a simple business in compliance with the notion of industrial CPS centric technology, (see Pirani et al. (2016)), for tiny computing systems. Moreover, this method provides a simple (ICPS), as treated by Colombo et al. (2017). Production business in treated compliance withare thenow notion of industrial CPS interfacetechnology, tosystems. the intelligence layers that enact distributed (ICPS), as by CPSs Colombo et al. (2017). Production research problems and inseparable in the new centric (see Pirani et al. (2016)), for tiny computing Moreover, this method provides a simple business in compliance with the notion of industrial CPS interface to the intelligence layers that enact distributed (ICPS), as treated by Colombo et al. (2017). Production computing systems. Moreover, this method provides a simple interface to the intelligence layers that enact distributed research problems and CPSs are now inseparable in the new (ICPS), as treated by Colombo et al. (2017). Production strategies to and policies for the business management. To this research problems andbyCPSs now inseparable in the new computing scenarios projected the are fourth industrial revolution. systems. Moreover, this method provides a simple interface the intelligence layers that enact distributed (ICPS), as treated by Colombo et al. (2017). Production strategies and policies for the business management. To research problems and CPSs are now inseparable in the new to intelligence layers that enact distributed strategies andthe policies for the business management. To this this scenarios projected the fourth industrial revolution. research andby CPSs now inseparable in the new interface end, in Bonci et al. (2017c) the same technological ground as scenariosproblems projected byshould the are fourth industrialinstrumented revolution. Production processes be pervasively to intelligence layers that enact distributed strategies and policies for business management. To this research problems andby CPSs are now inseparable in the new interface end, in Bonci et al. (2017c) the same technological ground as scenarios projected the fourth industrial revolution. strategies andthe policies for the the business management. Totheir this end, in Bonci et al. (2017c) the same technological ground as Production processes should be pervasively instrumented scenarios projected by the fourth industrial revolution. been proposed to encompass agent’s intelligence, in Production processes should be pervasively instrumented with holonic self-* (where * stands for configuring, strategies and policies for the business management. To this end, in Bonci et al. (2017c) the same technological ground as scenarios projected by the fourth industrial revolution. been proposed to encompass agent’s intelligence, in their Production processes should be pervasively instrumented end, inproposed Bonci et to al. encompass (2017c) the the same whole technological ground as been agent’s intelligence, in their with holonic self-* (where * stands for configuring, Production processes should be pervasively instrumented holistic integration within fractal factory with holonic self-* (where * stands for configuring, adjusting, optimizing, organizing, and so on)for agents designed end, inproposed Bonci et to al. encompass (2017c) the the same whole technological ground as been agent’s intelligence, in their Production processes should be pervasively instrumented holistic integration within fractal factory with holonic self-* (where * stands configuring, been proposed to encompass agent’s intelligence, in their holistic integration within the whole fractal factory adjusting, optimizing, organizing, and so on) agents designed with holonic self-* (where * stands for configuring, components. As obtained in Bonci et al. (2017b), the adjusting, optimizing, organizing, and so on) agents designed for the unexpected and collaborating with humans, as in been proposed to encompass agent’s intelligence, in their holistic integration within the whole fractal factory withtheholonic self-* organizing, (where * and stands configuring, components. As obtained obtained inthe Bonci et al. al.of (2017b), the adjusting, optimizing, so on) agents designed withinin the whole fractal factory components. As Bonci et (2017b), the for collaborating humans, as adjusting, optimizing, organizing, and sowith on)for agents designed proposed integration technique permits enabling a pervasive for the unexpected unexpected and collaborating with humans, as in in holistic Valckenaers et al. and (2016). Centralized and hierarchical holistic integration withinin the whole fractal factory components. As obtained Bonci et al. (2017b), the adjusting, optimizing, organizing, and so on) agents designed proposed technique permits the enabling of a pervasive for the unexpected and collaborating with humans, as in components. As obtained in Bonci et al. (2017b), the proposed technique permits the enabling of a pervasive Valckenaers et al. (2016). Centralized and hierarchical for the cannot unexpected and collaborating withand humans, as in components. control of technique theAseffectiveness of the system-of-systems, by Valckenaers etbeal.able (2016). Centralized hierarchical control to deal with increasing complexity. obtained in Bonci et al. (2017b), the proposed permits the enabling of a pervasive for the unexpected and collaborating with humans, as in control of the effectiveness of the system-of-systems, by Valckenaers et al. (2016). Centralized and hierarchical proposed technique permits the enabling of a pervasive control of the effectiveness of the system-of-systems, by control cannot be able to deal with increasing complexity. Valckenaers (2016). and hierarchical means of a well-defined sequence of specific and minimal control cannotetand beal.collaborative able to dealCentralized with increasing complexity. Decentralized approaches are increasingly technique permits the the enabling of and a pervasive control of the effectiveness of system-of-systems, by Valckenaers etand al.collaborative (2016). Centralized and hierarchical proposed means of a well-defined sequence of specific minimal control cannot be able to deal with increasing complexity. control of the effectiveness of the system-of-systems, by means of a well-defined sequence of specific and minimal Decentralized approaches are increasingly control cannot be able to deal with increasing complexity. actions. The nature ofthethe actions and are minimal contextDecentralized and collaborative approaches needed. Technology choices inwith thisincreasing senseare areincreasingly an issue. corrective control of the effectiveness of system-of-systems, by means of a well-defined sequence of specific control cannot be able to deal complexity. corrective actions. The nature of the actions are contextDecentralized and collaborative approaches are increasingly means of a well-defined sequence of specific and minimal corrective actions. The nature of the actions are contextneeded. Technology choices in this sense are an issue. Decentralized and collaborative approaches are increasingly and suitable for bothofhuman-centered and fully needed. Technology choices this aretechnology an issue. dependent Karnouskos et and al. (2017) notice in that the sense hardware means of a well-defined sequence of specific and minimal corrective actions. The nature the actions are contextDecentralized collaborative approaches are increasingly dependent and suitable for both human-centered and fully needed. Technology choices in this sense are an issue. actions. The nature the actions contextdependent systems, and suitable for both human-centered and fully Karnouskos et (2017) notice that the hardware needed. Technology choices in sense are an factual issue. corrective automated depending onof grade ofare automation Karnouskos et al.still (2017) notice thatthis the barriers hardware technology and its costs areal. relevant among the totechnology the actions. The nature ofwhich the actions are contextdependent and suitable for both human-centered and fully needed. Technology choices in this sense are an factual issue. corrective automated systems, depending on which grade of automation Karnouskos et al. (2017) notice that the barriers hardware technology dependent and suitable for both human-centered and fully automated systems, depending on which grade of automation and its costs are still relevant among the to the Karnouskos et al. (2017) notice that hardware technology is present. The results of this work are ready for their and its costs are still relevant among the barriers to the factual introduction of distributed control in the industrial contexts. dependent and suitable for both human-centered and fully automated systems, depending on which grade of automation Karnouskos et al. (2017) notice that hardware technology is present. The results of this work are ready for their and its costs are still relevant among barriers to the factual automated systems, depending on which grade of automation is present. Thereal results of this work are cost ready for their introduction of distributed control in the industrial contexts. and its costs of aredistributed still relevant among barrierscontexts. to the factual adoption in the field by means of low and off-theintroduction control in industrial automated systems, depending on work which grade ofand automation is present. The results of this are ready for their and its costs are still relevant among the barriers to the factual adoption in the real field by means of low cost off-theintroduction of distributed control in industrial contexts. is present. The results of this work are ready for their adoption in the real field by means of low cost and off-theIn Pirani et al. (2016), authors firstly proposedcontexts. the adoption shelf embedded technologies. introduction of distributed control in industrial is present. The results of this work are ready for their adoption in the real field by means of low cost and off-theIn Pirani et al. (2016), authors firstly proposed the adoption introduction of distributed control in industrial contexts. shelf embedded technologies. In Pirani et al. (2016), authors firstly proposed the adoption adoption in the real field by means of low cost and off-theshelf embedded technologies. of low cost and(2016), low size embedded electronics devices as a In Pirani et al. authors firstly proposed the adoption adoption in the real field by means of low cost and off-theshelf embedded technologies. of low cost and low size embedded electronics devices as a In Pirani et of al. (2016), authors firstlyelectronics proposed the adoption The paper is organised as follows. Section 2 provides, some of low cost andreference low size embedded devices as a shelf embedded technologies. major part architectures of distributed control. In Pirani et of al. (2016), authors firstlyelectronics proposed the adoption The paper is and organised asonfollows. follows. Section 22 provides, provides, some of low cost and low embedded devices as embedded technologies. The paper is organised as Section some major part reference architectures of control. of low cost and low size size embedded electronics devices as aa shelf background recall the performance improvement major part of reference architectures ofondistributed distributed control. The proposed approach relies primarily the simplification paper is organised as follows. Section 22 provides, some of low cost of and low sizerelies embedded electronics devices as a The background and recall on the performance improvement major part of reference architectures of distributed control. The paper is organised as follows. Section provides, some background and recall on the performance improvement The proposed approach primarily on the simplification major part reference architectures of distributed control. technique. Section 3 introduces the methodology. Section 4 The proposed approach relies primarily on the simplification obtained with the assumption of self-similarity properties The paper is organised as follows. Section 2 provides, some background and recall on the performance improvement major part of reference architectures of distributed control. technique. Section 3 introduces the methodology. Section 44 The proposed approach relies primarily on the simplification background and recall on the performance improvement technique. Section 3 introduces the methodology. Section obtained with the assumption of self-similarity properties The proposed approach relies primarily on the simplification presents the results and a small example in Section 5. Section obtained with the assumption ofofself-similarity properties across the numerous components the production process. background and recall on the performance improvement technique. Section 3 introduces the methodology. Section 44 The proposed approach relies primarily on the simplification presents the results and a small example in Section 5. Section obtained with the assumption of self-similarity properties technique. 3and introduces the methodology. Section theSection results a small example in Section 5. Section across thewith numerous components ofself-similarity the extended productionproperties process. obtained the assumption is devoted to conclusions and further work. across the numerous components of the production That methodology has been of further toprocess. cover 6presents technique. Section 3 introduces the methodology. Section 4 presents the results and a small example in Section 5. Section obtained with the assumption of self-similarity properties 6 is devoted to conclusions and further work. across the numerous components of the production process. the results and a small examplework. in Section 5. Section 6 is devoted to conclusions and further That methodology has been extended to cover across the numerous components of the value production That methodology has beenthefurther further extended toprocess. cover presents holistically and dynamically whole chain beyond presents the results and a small example in Section 5. Section 6 is devoted devoted to to conclusions conclusions and and further further work. work. across the numerous components of the value production That methodology has been further extended toprocess. cover holistically and dynamically the whole chain beyond That methodology has been further extended cover 66 is holistically andactors, dynamically thefound whole chain beyond 2. devoted PERFORMANCE IMPROVEMENT BACKGROUND the shop-floor as can be in value Bonci et al.to (2017a) is to conclusions and further work. That methodology has been further extended to cover holistically and dynamically the whole value chain beyond 2. PERFORMANCE IMPROVEMENT BACKGROUND the shop-floor actors, as can be found in Bonci et al. (2017a) 2. PERFORMANCE IMPROVEMENT BACKGROUND holistically and dynamically the whole value chain beyond the shop-floor actors, as can be found in Bonci et al. (2017a) and in Stadnicka et as al. can (2017a, b). in Since then, the main 2. PERFORMANCE IMPROVEMENT BACKGROUND holistically and dynamically the whole value chain beyond the shop-floor actors, be found Bonci et al. (2017a) For a more complete reference and insight of the technique, 2. PERFORMANCE IMPROVEMENT BACKGROUND and in Stadnicka et al. (2017a, b). Since then, the main the shop-floor actors, as can be found in Bonci et al. (2017a) and in Stadnicka et al. (2017a, b). Since then, the main problem at hand was on how to drive effectively and For a more complete reference and insight of the technique, 2. PERFORMANCE IMPROVEMENT BACKGROUND For a more complete reference and insight of thePirani technique, the shop-floor actors, as can be found in Bonci et al. (2017a) and in Stadnicka et al. (2017a, b). Since then, the main we suggest to refer to Bonci et al. (2017b) and et al. problem at hand was on how to drive effectively and and in Stadnicka et al. on (2017a, Since effectively then,actions, the main problem at handthe was how b). to drive and For aa more complete reference and insight of the technique, deterministically multivariate improvement let we suggest to refer to Bonci et al. (2017b) and Pirani et For more complete reference and insight of the technique, we suggest to refer to Bonci et al. (2017b) and Pirani et al. al. and in Stadnicka et al. (2017a, b). Since then, the main problem at hand was on how to drive effectively and (2016). In the following, only some primary recalls are made deterministically the multivariate improvement actions, let problem at hand was on how to drive effectively and deterministically the multivariate improvement actions, let a more complete reference and insight of thePirani technique, we suggest to refer to Bonci et al. (2017b) and et al. alone some tentative preliminary heuristics. In Bonci etand al. For (2016). In the following, only some primary recalls are made we suggest to refer to Bonci et al. (2017b) and Pirani et al. (2016). In the following, only some primary recalls are made problem at hand was on how to drive effectively deterministically the multivariate improvement actions, let to render the treatise minimally self-consistent. alone tentative preliminary heuristics. In et al. deterministically the multivariate improvement actions, alone some some tentative preliminaryanalytical heuristics. In Bonci Bonci et let al. we suggest to refer to Bonci et al. (2017b) and Pirani et al. (2016). In the following, only some primary recalls are made (2017b) a first deterministic solution and its to render the treatise minimally self-consistent. (2016). In the following, only some primary recalls are made to render the treatise minimally self-consistent. deterministically the multivariate improvement actions, let alone some tentative preliminary heuristics. In Bonci et al. (2017b) aa first deterministic analytical solution and its alone some tentative preliminary heuristics. In fundamental Bonci (2017b) first deterministic analytical solution andet al. its (2016). In the following, only some primary recalls are made to render the treatise minimally self-consistent. implementation has been provided for the four Holarchies dynamic and self-organizing to render theare treatise minimally self-consistent. hierarchies in alone some tentative preliminaryanalytical heuristics. In fundamental Bonci (2017b) aa first deterministic solution andet al. its implementation has been for the (2017b) first deterministic analytical solution its Holarchies implementation has been provided provided forMuthiah the four four dynamic and hierarchies in to rendersystems. theare treatise minimally self-consistent. production unit structures defined by etfundamental al. and (2007). Holarchies are dynamic andbeself-organizing self-organizing hierarchies in holonic They can used to represent distributed (2017b) a first deterministic analytical solution and its implementation has been provided for the four fundamental production unit structures defined by Muthiah et al. (2007). Holarchies are dynamic and self-organizing hierarchies in implementation has been provided for the four fundamental production unit structures defined by Muthiah et al. (2007). holonic systems. They can be used to represent distributed Holarchies are dynamic and self-organizing hierarchies in By representing the structures with trees, a degree is defined holonic systems. They can be used to represent distributed autonomic computing (Mella 2009) as needed in CPSs. The implementation has been provided for thea four production unit defined Muthiah et al. (2007). Holarchies are dynamic and self-organizing hierarchies in By the with trees, is holonic systems. They can be used to represent distributed production unit structures structures defined by Muthiah etfundamental al. (2007). autonomic By representing representing the structures structures withby trees, a degree degree is defined defined computing (Mella 2009) as needed in CPSs. The holonic systems. They can be used to represent distributed autonomic computing (Mella 2009) as needed in CPSs. The production unit structures defined by Muthiah et al. (2007). By representing the structures with trees, a degree is defined holonic systems. They can be used to represent distributed autonomic computing computing (Mella (Mella 2009) 2009) as as needed needed in in CPSs. CPSs. The The By representing the structures with trees, a degree is defined autonomic Copyright © 2018, 2018 IFAC 577Hosting By representing the structures withFederation trees, a degree is defined 2405-8963 © IFAC (International of Automatic Control) by Elsevier Ltd. All rights 2009) reserved. autonomic computing (Mella as needed in CPSs. The Copyright © 2018 IFAC 577 Copyright 2018 responsibility IFAC 577Control. Peer review©under of International Federation of Automatic Copyright © 2018 2018 IFAC IFAC 577 10.1016/j.ifacol.2018.08.379 Copyright © 577 Copyright © 2018 IFAC 577

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use of recursive approaches and self-similarity is the key to the programming of systems with high granularity of autonomous components (Calabrese et al., 2010). With a treebased representation of the holarchy of the factory system, a recursive computing for the improvement of the effectiveness of productive components and of the overall system can be performed. A good performance metrics can be designed starting from the definition of the overall equipment effectiveness (OEE). This indicator is composed of three major factors, relating to the availability, the speed and the quality of a unit (Muthiah et al., 2007). The overall throughput effectiveness (OTE) performance metric can be recursively computed from the OEE of the production units. These key performance indicators pertain self-similarly to different levels and sub-systems of production units, robotics, and mechatronics systems (Pirani et al., 2016). The performance of a sub-system depends on how many artefacts or actions it can produce per unit time of observation and with which quality. The four fundamental structures that express the performance information flow are the series, parallel, assembly, and expansion (Muthiah et al., 2007). These structures can be represented as trees and the OTE of a parent node depends recursively on the following variables of the children systems (Bonci et al., 2017b): Om, the OTE of the m-th children; Qm, the quality factor of the m-th children (m=1, …,n, where n is the degree of the tree); Oa, Qa, respectively the OTE and quality of the head system in an assembly; Oe, Qe, respectively the OTE and quality of the head system in an expansion; km, that appear as weighting factors on the edges of the trees in the assembly and expansion systems. In Fig. 1 a flow chart is provided to express the performance improvement process. This process is performed bottom-up for every node of the systems’ tree. At any leaf system in the tree the OTE matches the OEE.

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The performance improvement of a structure of 2-nd degree tree has been obtained in Bonci et al. (2017b). For the structures having only 2 children elements the analytical derivation of the multivariate OTE formulas is manageable. For higher degrees the number of variables raises fast, bringing into play the “curse of dimensionality”. Fortunately, machines and production units can be associated to many equivalent forms of virtual trees and so corresponding holarchies that are abstract concepts. If we achieve to transform an n-th degree structure into an equivalent nested composition of 2-nd degree structures the problem is easily solved. To this purpose, we will search simple equivalences that transform an n-th degree structure into an equivalent recursive structure made of 2-nd order structures. This problem has not unique solution. A structure can be reshaped into many others with appropriate affinity relationships. Nevertheless, to keep some kind of homogeneity, useful in the conceptual integrity of implementation, we will propose equivalences of only self-similar structures. To represent large and deep trees of structures we will rely on the handful Newick format (Olsen 1990) with some slight extensions to distinguish assembly and expansion structures. In this format the tree is represented by a sequence of printable characters. For example we suppose to represent the tree of an expansion structure with a machine A feeding production material to three production lines B, C, and D with a proportion of 10, 30 and 60 percent respectively. It can be associated to a tree like this: A(B:0.1,C:0.3,D:0.6). Contextually, we introduce the following notations: (l , g )

(l , g )

(l , g )

(l , g )

(l , g )

S X (n) ; OX (n) ≜ O( S X (n)) ; QX (n) ≜ Q( S X (n))

(1)

S denotes a system’s structure. The number n of children in such a structure is its degree. The parameters l and g are used to respectively provide an index for the tree depth level of the system in the systems-of-systems of fundamental structures and the “grouping” index. The latter refers to an aggregation operation of some parts of a system into virtually equivalent sub-system. The X subscript is assigned to S, P, A, E to respectively denote series, parallel, assembly, and expansion structure types. O and Q are the OTE and the recursive quality operators respectively, applied to the system S. Fig. 2 shows a tree example with g=0 as no grouping is made.

Fig. 2. System-of-systems tree example. 4. EXPRESSIONS OF THE N-TH DEGREE OTE

Fig. 1. Conceptual flowchart of the performance improvement process for a system. A parent unit is the controller of a structure which components are the children.

With the notations introduced in the previous section, in this section we will present a solution for the series, parallel, assembly and expansion structures of n-th degree.

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2-nd order expressions. A similar technique is used to obtain the relationships for the remaining structures in the following.

4.1 Series structure Supposing that the 4 systems in Fig. 3 are a series of units constituting a production line, the following is a first example for a 4-th degree series system rooted at the l-th level.

4.2 Parallel structure An example of the recursive representation for parallel structure is provided in Fig. 4.

Fig. 3. 4-th degree series structure equivalences. The grouping and nesting process depicted in Fig. 3 can be expressed in Newick notation as follows: (l ,0)



(l ,0)

( l ,0)

( l ,0)

(l ,0)

 

( l ,1)

(l ,0)

   ( l +1,1)

(l +1,1) (l +1,1) 

( l ,0)

Fig. 4. 4-th degree parallel structure equivalences.



S X (4)≜ S1 (1), S 2 (1), S3 (1), S 4 (1)  ≡ S X (3), S 4 (1) ≡  S1 (1) , S 2 (1), S3 (1)  , S 4 (1) 



 

  (l +1,2)

(l +1,1) 

(l ,0)

    ( l + 2,2)   

 



( l + 2,2)  (l +1,1) 

(l ,0)



  

The same Newick expressions in (2) hold for the trees in Fig. 4 with X=P. The n-th degree basic expressions of the parallel structure are:

(2)

≡   S X (2), S3 (1)  , S 4 (1)  ≡    S1 (1) , S 2 (1)  , S3 (1)  , S 4 (1) 









(l ,0)  n (l ,0)  QP (n) ≜  ∑ Qi (1)  n    i =1 

Note the use of the X subscript in (2) as it will hold both for X=S (series) and X=P (parallel). The n-th degree expressions of the series structures are straightforward if we consider the structure formulas of the O and Q operator: (l ,0)

n (l ,0)

QS (n) ≜ ∏ Qi (1)

n  (l ,0) (l ,0)  (l ,0)    (l ,0)   OS ( n) ≜ min  min Oi (1) ∏ Q j (1)  , On (1)  i n = 1,..., − 1   j = i +1  

and

i =1

(3)

PP (n) =

For imposed structures like (2) the following are obtained: (l ,1)

(l ,0)

QS ( n) = QS ( n − 1) Qn (1)

(l + g , g +1)

;

m (l + g +1, g +1)

QS ( m) = ∏

Qi (1)

(4)

(l + g , g +1)

(5)

(l + g , g +1)

m (l + g +1, g +1)  (l + g +1, g +1)    (l + g +1, g +1)   OS ( m) ≜ min  min  Oi (1) ∏ Q j (1)  , Om (1)  1,..., 1 = − i m    j = i +1 

;

(l + g , g )

(l ,0)

Om (1) = Om (1)

(l + g +1, g + 2) (l + g +1, g +1)

PP (m − 1) + m

Pm (1)

; m = 3,..., n − 1; g = 0,..., n − m − 1

(10)

; g = n −3

(11)

(l + g +1, g + 2) (l + g +1, g +1)

P1 (1)

+ 2

P2 (1)

 ( n − 1)! (l ,0) Pm (1) g = 0,1,..., n − 3  Pm (1) =  ( m − 1)! (l ,0)  g = n−2  ( n − 1)! Pm (1)

(l + g , g )

By the comparison between (3) and (4),(5),(6) the following expressions are obtained: (l ,0)

(9)

By comparing (9)-(11) with (8) it is natural to assume the following simple relationships between the original n-th degree parallel system parameters and the 2-nd degree:

(6)

with m=1,..,n-1 and g=0,…,n-3.

Qm (1) = Qm (1)

(8)

(l ,0)

PP (n − 1) + Pn (1) n

PP (2) =

(l + g , g +1)

(l + g , g )

(l ,1)

PP (m) =

i =1 (l ,0) (l ,0) (l ,0)  (l ,0)  OS ( n) ≜ min  OS ( n − 1) Qn (1), On (1)   

(l ,0)  n (l ,0)  OP (n) ≜  ∑ Oi (1)  n    i =1 

From (8) we can see that in this case O and Q operators have the same expression. For shorthand, operator P can be an alias for both O and Q in the following formulas derived from imposing the recursive structure as in (2): (l ,0)

(l ,0)

;

(12)

(7) 4.3 Assembly structure

with m=1,..,n and g=n-max(m,2). In (7) are the mappings between the original n-th degree system and all the equivalent 2-nd degree subsystems in the recursive tree as of (2). With these kind of relationships the performance of an n-th degree system is completely determined through the recursive use of

The equivalent trees for the assembly structure are in Fig. 5.

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573

(l ,0) (l ,0)   (l ,1) (l ,0) (l ,0)  O (n − 1) O (1) O (1)  OA ( n) = Qa (1) min  A(l ,1) , (ln,0) , (la,0)   k kn Qa (1)  n −1 

(18)

 (l + g +1, g + 2) (l + g +1, g +1) (l + g +1, g +1)  O (1) Oa (1)   O ( m − 1) min  (l +Ag +1, g + 2) , (l + g m , +1, g +1) (l + g +1, g +1)   k km Qa (1)  m −1  with m = 3,..., n − 1; g = 0,..., n − m − 1

(19)

 (l + g +1, g +1) (l + g +1, g +1) (l + g +1, g +1)  O (1)  O (1)  O (1) min  (l + g +11, g +1) , (l + g +21, g +1) , (l + g +a1, g +1)   k1 k2 Qa (1)  

(20)

(l + g , g +1)

(l + g +1, g +1)

O A (m ) =

(l + g , g +1)

Qa (1)

(l + g +1, g +1)

OA (2) =

Qa (1)

with g = n − 3

In the assembly case (and in the expansion) new constraints are under consideration, due to the k variable: n (l ,0)

∑

(21)

ki = 1

i =1 (l + g , g +1) (l + g , g )

km −1 + km

(22)

= 1 ; with m = 3,..., n; g = n − m

(l + g , g ) (l + g , g )

k1 +

k2

(23)

= 1 ; with g = n − 2

Fig. 5. 4-th degree assembly structure equivalences.

By developing (18)-(20), using the min operator associativity, and comparing with (14) we obtain an equality of the form:

In Newick notation, the trees in Fig. 5 follow:

min {α1, α 2 ,..., α n , β a } = min α1' , α 2' ,..., α n' , β1' , β 2' ,..., β n' −1

(l ,0)



(l ,0)

( l ,0)

( l ,0) (l ,0) ( l ,0)

( l ,0)

{

( l ,0) (l ,0)  (l ,0)



(l ,1)



( l ,1) ( l ,0)

(l ,0)  (l ,0)

αi = α1'

≡  S A (3) : k3 , S 4 (1) : k 4  S a (1)





(13)

  ( l +1,1) ( l +1,1) (l +1,1) ( l +1,1) (l +1,1) (l +1,1)  (l +1,1) (l ,1) (l ,0) ( l ,0)  (l ,0) ≡   S1 (1) : k1 , S 2 (1) : k 2 , S3 (1) : k3  S a (1) : k3 , S 4 (1) : k 4  S a (1)   

Om (1) = (l ,1) (l ,0)

Om

(1) (l + g , g ) km

(l ,0)

∏ kn −i −1 i =0 g (l + i ,i )

(l ,0)  (l ,0)





(l ,0)

(l ,0)

n (l ,0) (l ,0)

QA ( n) ≜ Qa (1) ∑ ki Qi (1) ; i =1



g −1 (l + i ,i +1)

(l + g , g +1)

Oa (1) =

(14)

km −1

km

Qm (1)  

Qa (1)

i =0 g (l + i ,i )

g = 1,..., n − 2

(27)

∏ Qa (1)

To determine the remaining expressions of the sub-systems’ parameters we use the (15)-(17) in comparison with (14), (22), and (23). At this point, it is noted that many possible solutions are available as an underdetermined problem. Among the possible solutions, a natural one is to assume: (l + g , g )

QA ( m − 1) +

(l ,0)

∏ kn −i −1 i =1

(l ,0)

m = 1, 2,..., n − 1 g = n − max(m, 2)

Qm (1) = Qm (1) ,

(l + g +1, g +1)  (l + g +1, g + 2) (l + g +1, g + 2) (l + g +1, g +1) (l + g +1, g +1) 

Qa (1)

Oa

(1) (l + g , g )

Qa (1)

(15)

  with m = 3,..., n − 1; g = 0,..., n − m − 1

Q A (m ) =

(l ,0)

(l + g , g )

With a tree similar to (13) we can impose the following recurrent 2-nd degree expressions: (l ,0) (l ,0)  (l ,1) (l ,1) (l ,0) (l ,0)  QA ( n) = Qa (1)  kn −1 QA ( n − 1) + kn Qn (1)   

(26)

i =1

The basic n-th degree formulas in the assembly case are:   (l ,0)  (l ,0)  (l ,0) (l ,0)   O (1)  O (1)  OA ( n) ≜ Qa (1) min  min  (li,0)  , (la,0)  i =1,..., n  k  Q (1)   i  a  

m = 3,..., n − 1 g = n − max(m, 2)

,

∏ Qa (1)

km

≡    S1 (1) : k1 , S 2 (1) : k 2  S a (1) : k 2 , S3 (1) : k3  S a (1) : k3 , S 4 (1) : k 4  S a (1)  



(25)

i = 1,..., n

g −1 (l + i ,i +1)

(l ,0)

(l + g , g )

(l + 2,2) ( l + 2,2) (l + 2,2)  ( l + 2,2) (l +1,2) (l +1,1) (l +1,1)  (l ,1)

; βi = β1'

From the equalities in (25) we obtain:

  (l +1,2) (l +1,2) (l +1,1) ( l +1,1)  ( l +1,1) ( l ,1) ( l ,0) (l ,0)  ( l ,0) ≡   S A (2) : k 2 , S3 (1) : k3  S a (1) : k3 , S 4 (1) : k 4  S a (1)       ( l + 2,2)

(24)

Sufficient conditions for (24) are:

S A (4) ≜  S1 (1) : k1 , S 2 (1) : k 2 , S3 (1) : k3 , S 4 (1) : k 4  S a (1)



}

(16) (l + g , g )

km

(l ,0)

max( m,2) (l ,0)

= km (1)

∑

ki ,

i =1

(28)

m = 1, 2,..., n − 1 g = n − max( m, 2)

(29)

m = 2,3,..., n − 1 g = n − m −1

(30)

(l + g , g +1) (l + g +1, g +1)  (l + g +1, g +1) (l + g +1, g +1) (l + g +1, g +1) (l + g +1, g +1) 

QA (2) = Qa (1)

 

k1

Q1 (1)

+

k2

Q2 (1)

 

(l + g , g +1)

(17)

km

with g = n − 3

580

m (l ,0)

m +1 (l ,0)

i =1

i =1

= ∑ ki

∑

ki ,

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(l + g , g )

Qa (1) = 1,

As usual, due to the self-similarity of a tree similar to (33), the following constraints are imposed:

(31)

g = 1,..., n − 2

At this point we have all the necessary relationships between the n-th and the 2-nd degrees assembly systems. Note also that with (28)-(31), the (26), (27) are simplified as follows: (l + g , g )

(l + g , g )

(l ,0)

Om (1) = Om (1)

;

Oa (1) =

(l ,0)

Oa (1) (l ,0)

Qa

(l ,0)

(l ,1) (l ,1)

(l + g , g +1)

∑

i =1

m = 1, 2,..., n − 1 ki , g = n − max( m, 2)

QE ( m − 1) +

km −1

Qm (1)

km

(36)

with m = 3,..., n − 1; g = 0,..., n − m − 1

(32)

(l + g , g +1) (l + g +1, g +1) (l + g +1, g +1) (l + g +1, g +1) (l + g +1, g +1)

QE (2) =

Q1 (1)

k1

+

k2

(37)

Q2 (1)

with g = n − 3

4.4 Expansion structure

(l ,0) (l ,0) (l ,1)  (l ,1) (l ,1)  (l ,0) (l ,0) (l ,0) (l ,0)  OE ( n) = mink n −1QE ( n − 1) Oe (1), OE ( n − 1)  +min kn Qn (1) Oe (1), On (1)  (38)    

An example of 4-th degree tree equivalence for expansion is provided in Fig. 6, following the Newick notation: (l ,0) (l ,0)  (l ,0) (l ,0) (l ,0) (l ,0) (l ,0) (l ,0) (l ,0) (l ,0)  S E (4) ≜ Se (1)  S1(1) : k1 , S 2 (1) : k2 , S3 (1) : k3 , S4 (1) : k4      (l ,0)  (l ,1) (l ,1) (l ,0) (l ,0)  ≡ Se (1)  S E (3) : k3 , S4 (1) : k4      (l ,0)  (l +1,1) (l ,1)  (l +1,1) (l +1,1) (l +1,1) (l +1,1) (l +1,1) (l +1,1)  (l ,0) (l ,0)  ≡ Se (1)  Se (1) : k3  S1(1) : k1 , S 2 (1) : k2 , S3 (1) : k3  , S4 (1) : k4         

(35)

(l + g +1, g + 2) (l + g +1, g + 2) (l + g +1, g +1) (l + g +1, g +1)

QE ( m) = n − g (l ,0)

(l ,0) (l ,0)

QE ( n) = kn −1 QE ( n − 1) + kn Qn (1)

(l + g +1, g + 2)(l + g +1, g + 2) (l + g +1, g +1) (l + g +1, g + 2)  OE ( m) = min km −1 QE (m − 1) Oe(1) , OE ( m − 1)  +  

(l + g , g +1)

(l + g +1, g +1) (l + g +1, g +1)(l + g +1, g +1) (l + g +1, g +1)  km Qm(1) Oe (1) , Om (1)  + min   

(39)

m = 3,..., n − 1 g = 0,..., n − m − 1

(33) (l + g +1, g +1)(l + g +1, g +1) (l + g +1, g +1) (l + g +1, g +1)  OE (2) = min k1 Q1 (1) Oe(1) , O1 (1)  +  

(l + g , g +1)

(l ,0)  (l +1,1) (l ,1)  (l +1,2) (1+1,2) (l +1,1) (l +1,1)  (l ,0) (l ,0)  ≡ Se (1)  Se (1) : k3  S E (2) : k2 , S3 (1) : k3  , S 4 (1) : k4  ≡        

(l + g +1, g +1) (l + g +1, g +1)(l + g +1, g +1) (l + g +1, g +1)  + min  k2 Q2(1) Oe (1) , O2 (1)   

(l ,0)  (l +1,1) (l ,1) (l + 2,2)(l +1,2)  (l + 2,2)(l + 2,2) (l + 2,2)(l + 2,2)  (l +1,1)(l +1,1)  (l ,0) (l ,0)  Se (1)  Se (1): k3  Se (1): k2  S1 (1) : k1 , S 2 (1): k2  ,S3 (1): k3  , S4 (1): k4             

(40) g = n −3

Using the (21)-(23) into (35)-(37), natural conditions are: (l ,0)

(l + g , g )

(l ,0) (l + g , g +1) m ki Qm (1) = Qm (1) ; QE (m) =∑ max(i ,2) i =1

∑

(l ,0) (l ,0)

kj

Qi (1) ;

m = 1, 2,..., n − 1 g = n − max(m, 2)

(41)

j =1

Next step is to develop (38)-(40) using again (21)-(23), (41) and the following property of the min operator repeatedly: (42)

min{a, b} + c = min{a + c, b + c}

we obtain the remaining sufficient conditions: (l + g , g )

(l ,0) n − g (l ,0)

Oe (1) = Oe (1)

∑

ki

(43)

g = 1,..., n − 2

i =1

(l ,0) (l ,0) (l ,0) (l ,0)  Om (1) = min  km Qm (1) Oe (1), Om (1)   

(l + g , g )

m = 1,..., n − 1 g = n − max(m, 2)

(44)

5. DISCUSSION AND EXAMPLE The formulas of section 4 can be used to implement a distributed and autonomic computing system for the improvement of industrial processes. The monitoring and computing units (holons) can be spread across the process in the form of a network of tiny computing CPS actors. In particular the network is used for communications between the holons and its connections to determine the hierarchy in a holarchy, a specific organization of holons. Theoretically, any holon can be described recursively by a holarchy until the desired granularity level description is reached (Calabrese et al. 2010). The lower granularity in the presented method are the leaf units (cells) where OEE is evaluated. Nonetheless, the roles of the holons can dynamically evolve and re-assemble into a holarchy depending on the knowledge

Fig. 6. 4-th degree expansion structure equivalences. The basic expressions for the expansion structure are: (l ,0)

n (l ,0) (l ,0)

QE ( n) = ∑ ki Qi (1) i =1

;

n (l ,0) (l ,0) (l ,0) (l ,0) (l ,0)  OE ( n) = ∑ min  ki Qi (1) Oe (1), Oi (1)    i =1

(34)

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flow that, at any level, is necessary to accomplish the system goal (Calabrese et al. 2010) and in response to unexpected situations. Indeed, the applicability of this method goes beyond the classical production structures. The ontological and semantic interpretation of a cell or a whole structure depends on the interpretation of a productivity measurement with respect to a goal. The nature and the meaning of the goal is local to each holon in the holarchy. It can express at the same time process flows and machines, as treated in Pirani et al. (2016), Stadnicka et al. (2017a, b), and Bonci et al. (2017a). For example, we can map two levels of a holarchy into an assembly system’s parent that controls its children structured as in Fig. 5. With an initial (arbitrary) assignment of the values of the variables’ vector [Oa , O1, O2 , O3 , O4 , k1, k2 , k3 , k4 , Qa , Q1, Q2 , Q3 , Q4 ] =

575

will be to compare this recursive, serialized, and stepwise method of performance improvement with other centralized or distributed optimization techniques. This will give a benchmark of the technique in general contexts where computing capabilities could exceed the possibilities of the tiny devices that are usually addressed from this technique. REFERENCES Bonci, A., Pirani, M., and Longhi, S. (2017a). An Embedded Database Technology Perspective in Cyber-physical Production Systems. Procedia Manuf., 11, 830-837. Bonci, A., Pirani, M., and Longhi, S. (2017b). Robotics 4.0: Performance Improvement Made Easy. In Proceedings of the 22nd IEEE International Conference on Emerging Technologies And Factory Automation, 2017 Bonci A., Pirani, M., Dragoni, A. F., Cucchiarelli, A., Longhi, S. (2017c). The Relational Model: in Search for Lean and Mean CPS technology. In Proc. of the IEEE 15th Int. Conf. of Industrial Informatics INDIN'2017. Calabrese, M., Amato, A., Di Lecce, V., and Piuri, V.,(2010). Hierarchical-granularity holonic modelling. Journal of Ambient Intelligence and Humanized Computing, 1(3), pp.199-209. Colombo, A.W., Karnouskos, S., Kaynak, O., Shi, Y., and Yin, S. (2017). Industrial Cyberphysical Systems: A Backbone of the Fourth Industrial Revolution. IEEE Ind. Electron. Mag., 11(1), 6-16. Karnouskos, S., and Leitao, P. (2017). Key Contributing Factors to the Acceptance of Agents in Industrial Environments. IEEE Trans. Ind. Informat.,13(2),696-703 Mella P., (2009). The Holonic Revolution: Holons. Holarchies and Holonic Networks: The Ghost in the Production Machine. Pavia: Scientifica. Muthiah K. M. N., and Huang, S. H. (2007). Overall throughput effectiveness (OTE) metric for factory-level performance monitoring and bottleneck detection. Int J Prod Res, 45(20), 4753-4769. Olsen, G. (1990). “Newick's 8:45” Tree Format Standard, http://evolution.genetics.washington.edu/phylip/newick_ doc.html Pirani, M., Bonci, A., and Longhi S. (2016). A scalable production efficiency tool for the robotic cloud in the fractal factory. In Industr. Electr. Society, IECON 201642nd Annual Conf. of the IEEE, 6847–6852. Stadnicka, D., Pirani, M., Bonci, A., Ratnayake, R. C., and Longhi, S. (2017a). Self-similar Computing Structures for CPSs: A Case Study on POTS Service Process. In Working Conference on Virtual Enterprises, 157-166. Springer, Cham. Stadnicka, D., Pirani, M., Bonci, and Longhi, S. (2017b). Information Management and Decision Making Supported by an Intelligence System in Kitchen Fronts Control Process. In International Conference on Intelligent Systems in Production Engineering and Maintenance, 249-259. Springer, Cham. Valckenaers, P., Van Brussel, H. (2016). Design for the Unexpected: From Holonic Manufacturing Systems Towards A Humane Mechatronics Society. ButterworthHeinemann

(45)

[0.36, 0.09, 0.16,0.25, 0.5, 0.1, 0.2, 0.3, 0.4, 0.6, 0.3,0.4,0.5,0.7]

in Fig. 7, we show an evolution of the OTE of the assembly with respect to the variation applied to the OEEs of the 5 cells in the structure. The sequence of the variations have been obtained with a filtered selection of available actions based on the type of action (to prefer variations of OEEs rather than k and Q) and their expected OTE gain. More extended examples deserve more space and are reserved for future work.

Fig. 7. An example of evolution of OTE in a 4th degree assembly system. 6. CONCLUSION In this paper are developed the OTE formulas valid for all the trees of fundamental productive units up to the n-th degree. The expressions obtained allow to transform an n-th degree problem into recursively equivalent 2-nd degree problems, which were previously solved. The technique here exposed brought about the n-th degree extension for an already well known method that let 2-nd degree nested structures to be easily computed at low cost. The formulas here used should be checked for an even possible simpler representation. The formulas will be used in future work on case studies in industrial and engineering processes. A most interesting work 582