Dmitry Ivanov
email: divanov@hwr-berlin.de

Alexandre Dolgui
email: alexandre.dolgui@imt-atlantique.fr

Viability of Intertwined Supply Networks: Extending the Supply Chain Resilience Angles towards Survivability. A Position Paper Motivated by COVID-19 Outbreak

Keywords: supply chain resilience, supply chain risk management, supply chain dynamics, intertwined supply network, viability, survival, coronavirus (COVID-19)

An intertwined supply network (ISN) is an entirety of interconnected supply chains (SC) which, in their integrity secure the provision of society and markets with goods and services. The ISNs are open systems with structural dynamics since the firms may exhibit multiple behaviors by changing the buyer-supplier roles in interconnected or even competing SCs. From the positions of resilience, the ISNs as a whole provide services to society (e.g., food service, mobility service or communication service) which are required to ensure a long-term survival. The analysis of survivability at the level of ISN requires a consideration at a large scale as resilience of individual SCs. The recent example of coronavirus COVID-19 outbreak clearly shows the necessity of this new perspective. Our study introduces a new angle in SC resilience research when a resistance to extraordinary disruptions needs to be considered at the scale of viability. We elaborate on the integrity of the ISN and viability. The contribution of our position study lies in a conceptualization of a novel decision-making environment of ISN viability. We illustrate the viability formation through a dynamic game-theoretic modelling of a biological system that resembles the ISN. We discuss some future research areas.

Introduction

Coronavirus COVID-19 outbreak is an unprecedented and extra-ordinary situation that clearly shows a need for progressing the supply chain (SC) resilience research and practices. The coronavirus outbreak affects the global and local economies at a larger scale. Supply availability in global SCs has been drastically reduced and misbalanced with the demands. According to [START_REF] Araz | Data Analytics for Operational Risk Management[END_REF], the COVID-19 dispersal "is breaking many global SCs". Early in March 2020, the number of COVID-19 cases has grown exponentially all over the world resulting in border closures, quarantines, and entirely full shut downs of many crucial facilities, markets and activities in the SCs. On March 11, 2020, the World Health Organization (WHO) announced the global pandemic.

Being lean and globalized in structures, the SCs of many companies became specifically prone to coronavirus outbreaks [START_REF] Ivanov | Predicting the impact of epidemic outbreaks on the global supply chains: A simulation-based analysis on the example of coronavirus (COVID-19 / SARS-CoV-2) case[END_REF]. 94% of the Fortune 1000 companies have been affected by coronavirus-driven SC disruptions (Fortune 2020). [START_REF] Linton | Coronavirus Is Proving We Need More Resilient Supply Chains[END_REF] show on the example of data from the Resilinc system that the world's largest 1,000 SCs own more than 12,000 facilities (i.e., factories, warehouses and other operations) in COVID-19's quarantine areas. For some SCs, demand has drastically increased and the supply was not able to cope with that situation (e.g., facial masks, hand sanitizer, disinfection spray). As such, the question of market and society survivability was raised. For other SCs, the demand and supply have drastically dropped resulting in the production stops (e.g., automotive industry), the danger of bankruptcies and necessities of governmental supports [START_REF] Harbour | The coronavirus' impact on the global automotive supply chains[END_REF]. Here, the questions of SC survivability arose. Obviously, both these questions go beyond the existing state-of-the-art in SC resilience since they cannot be resolved within a narrow SC perspective but rather require an analysis at a larger scale.

One interesting research avenue to bring the discussion at the level of survivability is to consider the concept of viability. While SC resilience has gained a considerable attention [START_REF] Dolgui | Does the ripple effect influence the bullwhip effect? An integrated analysis of structural and operational dynamics in the supply chain[END_REF][START_REF] Hosseini | Review of quantitative methods for supply chain resilience analysis[END_REF], the understanding of viability remains an open research question. In this paper, we define viability as the system ability to meet the demands of surviving in a changing environment following the Viable System Model by [START_REF] Beer | Brain of the firm[END_REF] and ecology modeling perspectives [START_REF] Aubin | Viability Theory[END_REF]. The example of coronavirus COVID-19 outbreak shows that in case of extraordinary events, SC resistance to disruptions needs to be considered at the scale of survivability or viability to avoid SC and market collapses and secure the provision with goods and services.

Analysis of the viability concepts in ecology, biological systems [START_REF] Aubin | Viability Theory[END_REF] and cybernetics [START_REF] Beer | Brain of the firm[END_REF] shows that viability is mostly considered in the settings of complex systems which span and connect some individual sub-systems. In SC domain, several examples of such systems can be observed in literature and practice. [START_REF] Niu | Strategic analysis of dual sourcing and dual channel with an unreliable alternative supplier[END_REF] show that Apple and Samsung play the role of both suppliers and competing focal firms in two different but intersecting SCs. [START_REF] Wang | Toward a Resilient Holistic Supply Chain NetworkSystem: Concept, Review and Future Direction[END_REF] and [START_REF] Panetto | Challenges for the cyber-physical manufacturing enterprises of the future[END_REF] develop the concept of holistic SC networks that are composed of a group of intersecting SCs which are intertwined. The SC intertwining can also be encountered in industrial symbiosis as well as in circular and sharing economies [START_REF] Fracassia | Rethinking Resilience in Industrial Symbiosis: Conceptualization and Measurements[END_REF], Choi et al. 2020). Another example is a symbiosis of commercial and humanitarian logistics when several business and humanitarian SCs are sharing the warehouse facilities [START_REF] Altay | OR/MS research in disaster operations management[END_REF][START_REF] Dubey | Drivers of coordination in humanitarian relief supply chains[END_REF], Dubey et al. 2019b). Considering the practical environments, [START_REF] Kelly | Supply chains and ISNs. Business ecosystems come of age[END_REF] point to "complex, dynamic, interconnected supply chains" highlighting to the role of information technology in synchronizing the SCs within the value webs. As noted by Vincenzo Boccia, the president of Confindustria in Italy on March 23, 2020 [START_REF] Agi | L'economia di guerra ci costa 100 miliardi al mese, dice Boccia[END_REF], it is very difficult to overcome the epidemic crisis and determine the most essential SCs to ensure survivability since "suppliers in the automotive sector are at the same time producers of valves for respirators".

Despite the existence and an increasing practical utilization of the above concepts, the SC management literature has not framed this new kind of network integrities in a specific concept so far. In this study, we introduce for the first time the term "Intertwined Supply Network" (ISN) that encapsulates entireties of interconnected SCs which, in their integrity secure the provision of society and markets with goods and services. While the resilience is related to the level of individual SCs or supply networks, viability is considered at the ISN level. SCs rarely represent single, isolated networks but are rather open systems that are characterized by structural dynamics (Ivanov et al. 2010, Kelly and[START_REF] Kelly | Supply chains and ISNs. Business ecosystems come of age[END_REF]. In contrast to linearly directed SCs or supply networks (cf. Fig. 1) with static structures, the firms in the ISNs may exhibit multiple behaviors in buyer-supplier relations (i.e., behavioral dynamics) being buyers and suppliers in interconnected or even competing SC simultaneously. The assumption that the firms' roles in the SCs can vary might change the kind of problem settings and results to be expected from the analysis of system properties closed to resilience. Differently but supplementing to the studies on SC resilience, we consider in this paper the ISN viability. From the positions of resilience, the ISNs as a whole provide services to society (e.g., food service, mobility service or communication service) which are required to ensure a longterm survival. The analysis of survivability at the level of ISN requires a consideration at a large scale as resilience of individual SCs. The recent example of coronavirus COVID-19 outbreak clearly shows the necessity of this new perspective.

For example, a traditional understanding of automotive SCs is the car production as the final output goal. Differently, the ultimate goal of an automotive ISN is to provide a mobility service to society. In electronics industry, a traditional SC understanding yields production of some electronic devices as a desired output performance while the performance of the electronics ISN is rather related to providing communication service to society. As such automotive and electronics ISNs are responsible to provide to important services to society, i.e., mobility and communication at a global scale. Obviously, the analysis of disruption impacts at such a level is concerned with long-term securing the mobility and communication in the society, i.e., ensuring the viability, rather than with performance impact of disruptions in individual SCs in terms of revenue or annual sales, as traditional SC resilience analysis usually does.

Therefore, the ISN viability appears a timely and crucial topic which opens doors for a variety of new problem settings and solution techniques. The global pandemic and SC collapses moved the SC survivability issues through collective behavioral changes in the forefront of risk management discussions [START_REF] Keogh | COVID-19. To ensure the supply of food and consumer goods, we must 'Change the Rules of the Game[END_REF][START_REF] Ivanov | Predicting the impact of epidemic outbreaks on the global supply chains: A simulation-based analysis on the example of coronavirus (COVID-19 / SARS-CoV-2) case[END_REF]. The SC survivability in the context of such extra-ordinary events goes beyond a narrow understanding of SC performance as some profits or revenues and brings the discussion to the next level, i.e., SC performance in terms of securing the provision of the goods and services in society and long-term survival of the whole industry sectors.

The contribution of this position paper lies in conceptualization of a novel decision-making environment for SC resilience that considers ISN and viability as an integrity. Our study introduces a new angle in SC resilience research when SC resistance to extraordinary disruptions needs to be considered at the scale of survivability, or viability. We also illustrate the viability concept idea through a dynamic game-theoretic modelling inspired from a biological system that resembles the ISN. We discuss some future research areas. Conceptually, in Section 2 we show how viability is different than the resilience, and why viability appears a necessary quality to be added to the SC resilience analysis in the ISN context. Technically, building upon the resemblance of ISN to ecological systems, we illustrate in Section 3 the viability formation through dynamic game-theoretic modelling of a biological system. Section 4 is devoted to the discussion of theoretical and practical implications as well as future research avenues. We summarize in Section 5 the main ideas of the paper.

Literature review

In this section, we discuss recent literature on SC resilience in the aspects concerned with viability. Since there is no specific literature on SC viability, we consider the state-of-the-art SC resilience angles as the most appropriate methodical basement for a development of the viability concept.

Viability vs. stability, robustness, and resilience of SCs

To recapitulate, Viability is a system ability to meet the demands of surviving in a changing environment. SC literature has produced a large body of knowledge for analysis of network behaviors and their adaptations in the presence of changing environments which are related to the categories of stability, robustness, and resilience. Therefore, we present these concepts here and compare with each other. Our analysis remains at a generalized level according to the objective of this position paper, and we refer the interested reader to the survey papers by [START_REF] Ho | Supply chain risk management: a literature review[END_REF], [START_REF] Ivanov | Literature review on disruption recovery in the supply chain[END_REF], [START_REF] Dolgui | Does the ripple effect influence the bullwhip effect? An integrated analysis of structural and operational dynamics in the supply chain[END_REF], [START_REF] Bier | Methods for mitigating disruptions in complex supply chain structures: A systematic literature review[END_REF][START_REF] Duhadway | Understanding risk management for intentional supply chain disruptions: Risk detection, risk mitigation, and risk recovery[END_REF], [START_REF] Hosseini | Review of quantitative methods for supply chain resilience analysis[END_REF], for more detailed considerations.

Put simply, SC reaction to disturbances can be analysed as follows (Table 1):  Stability -ability to return to a pre-disturbance state and ensure a continuity (Ivanov andSokolov 2013, Demirel et al. 2019)  Robustness -ability to withstand a disruption (or a series of disruptions) to maintain the planned performance (Nair andVidal 2011, Simchi-Levi et al. 2018)  Resilience -ability to withstand a disruption (or a series of disruptions) and recover the performance [START_REF] Spiegler | A control engineering approach to the assessment of supply chain resilience[END_REF][START_REF] Hosseini | Review of quantitative methods for supply chain resilience analysis[END_REF]. [START_REF] Demirel | Identifying dynamical instabilities in supply networks using generalized modeling[END_REF] point to stability as a "basic desirable property of a supply network without an explicit consideration of performance" while the robustness and resilience explicitly include the performance in the analysis of disruption impacts. [START_REF] Ivanov | Control and system-theoretic identification of the supply chain dynamics domain for planning, analysis, and adaptation of performance under uncertainty[END_REF] show that robustness allows to analyse the system ability to withstand a disruption (or a series of disruptions) without any structural and parametrical changes/adaptations, while resilience analysis explicitly allows the system to employ some recovery/adaptation in order to restore the disrupted operations and performance [START_REF] Craighead | The severity of supply chain disruptions: design characteristics and mitigation capabilities[END_REF][START_REF] Zhao | Modelling supply chain adaptation for disruptions: An empirically grounded complex adaptive systems approach[END_REF].

Research in SC reaction to disturbances is related to the semantic network analysis level with a focus on structural properties, complexity roles, and node/arc criticality (Ivanov and Dolgui 2019). The studies [START_REF] Basole | Supply Network Structure, Visibility, and Risk Diffusion: A Computational Approach[END_REF]Belami 2014, Ivanov et al. 2014a,b, Kim et [START_REF] Macdonald | Supply chain risk and resilience: theory building through structured experiments and simulation[END_REF] show that SC robustness and resilience should not merely be based on a straightforward disruption magnitude analysis, but rather seek trajectories of how different disruption scenarios influence the severity in network degradation and recovery (Pavlov et al. 2019a).

The term "viability" has been widely used in ecological modelling [START_REF] Aubin | Viability Theory[END_REF][START_REF] Bene | A viability analysis for a bio-economic model[END_REF]) as a system ability to maintain itself and recover in the presence of disturbances over a long-term horizon. In Table 2, we summarize the major differences between the resilience and viability. To this end, resilience is considered a disruption-driven SC property (single, discrete, unique events). It relates to singular disruption-reaction analyses within a closed system setting. These analyses are performance-oriented for some fixed time-windows and mostly linear, single-flow directed SC systems. Viability is a behavior-driven property (continuous system change) of a system with structural dynamics. It considers system evolution (i.e., open system context) through disruption-reaction balancing in the open system context. The viability analysis is survival-oriented without fixed time windows at a long-term scale.

Intertwined supply networks (ISN)

In this study, we consider ISNs, i.e. complex supply networks with dynamically changing structures, roles and behaviors of the firms involved. For example, [START_REF] Zhao | Modelling supply chain adaptation for disruptions: An empirically grounded complex adaptive systems approach[END_REF] show that a competitor of a focal firm can also serve as the focal firm's upstream supplier echoed by [START_REF] Niu | Strategic analysis of dual sourcing and dual channel with an unreliable alternative supplier[END_REF] who consider a co-opetition SC setting on the example of Apple and Samsung with a competitive supplier (i.e., frenemy) that plays the role of both supplier and competing focal firm in two different but intersecting SCs simultaneously causing dependence asymmetry [START_REF] Dong | Opportunism in distribution networks: The role of network embeddedness and dependence[END_REF].

In early 2000s, the ideas of dynamic SC formations have found first developments in the area of virtual enterprises and collaborative networks [START_REF] Ivanov | Intelligent Supply Chain Planning in Virtual Enterprises[END_REF], 2005[START_REF] Camarinha-Matos | Collaborative networks: a new scientific discipline[END_REF][START_REF] Dekkers | Distributed Manufacturing as Co-Evolutionary System[END_REF][START_REF] Ivanov | Structure dynamics control-based framework for adaptive reconfiguration of collaborative enterprise networks[END_REF][START_REF] Sarkis | A strategic model for agile virtual enterprise partner selection[END_REF][START_REF] Dolgui | Supply chain engineering: Useful methods and techniques[END_REF][START_REF] Ivanov | The Inter-Disciplinary Modelling of Supply Chains in the Context of Collaborative Multi-Structural Cyber-Physical Networks[END_REF][START_REF] Chibani | Dynamic optimisation for highly agile supply chains in e-procurement context[END_REF]. Other relevant research streams can be found in the theories of complex adaptive systems [START_REF] Choi | Supply networks and complex adaptive systems: control versus emergence[END_REF][START_REF] Surana | Supply-chain networks: a complex adaptive systems perspective[END_REF] and SC structural dynamics [START_REF] Ivanov | A multi-structural framework for adaptive supply chain planning and operations with structure dynamics considerations[END_REF].

Most Moreover, analyses from the value point of view are often applied. A possible new way for the research from viability perspectives deals with the value web approaches developed, for example by Amazon. A value web is based on the use of highly synchronized information technologies to coordinate value chains of different firms of ISN [START_REF] Kelly | Supply chains and ISNs. Business ecosystems come of age[END_REF].

ISN analysis inspired from ecology modeling

ISNs in this context can be studied with approaches similar to ecological modelling. Ecological modelling is a research area concerned with the analysis of ecosystems in dynamics [START_REF] Gross | Enrichment and foodchain stability: the impact of different forms of predator-prey interaction[END_REF][START_REF] Gross | Generalized models reveal stabilizing factors in food webs[END_REF]. Recent literature point to resemblance of the SCs to ecosystems [START_REF] Byrne | Biologicalisation: Biological transformation in manufacturing[END_REF][START_REF] Gross | Introduction to dynamics of manufacturing supply networks[END_REF][START_REF] Demirel | Identifying dynamical instabilities in supply networks using generalized modeling[END_REF][START_REF] Nair | Revisiting the complex adaptive systems paradigm: Leading perspectives for researching operations and supply chain management issues[END_REF]. The applications of ecological modelling to SC uncertainty have been mostly focused on the stability analysis and multi-echelon SC synchronization in terms of balancing the demand and inventory levels [START_REF] Anne | Bifurcation analysis and synchronisation issues in a three-echelon supply chain[END_REF][START_REF] Demirel | Identifying dynamical instabilities in supply networks using generalized modeling[END_REF][START_REF] Mondal | A new supply chain model and its synchronization behavior[END_REF].

Ecological perspective of the resilience has been developed by Holling (2006) which is based on the ability of system to react to stressors, to absorb and withstand shocks, with an emphasis on persistence. We consider ecological perspective as the closest one to SC viability since it focuses on the ecosystem services provided to society [START_REF] Linkov | Fundamental concepts of cyber resilience: Introduction and overview[END_REF].

Major principles of ecology modeling are survival-orientation, absence of explicit time windows in analysis, and ecosystem focus. These principles appear to be very close to the ISN viability (cf. Sect. 2.1). We note that viability does not replace resilience but rather adds a new quality to analysis of SC performance and behaviors under uncertainty.

Trophic chain-based model of the ISN viability

In this section, we illustrate the viability formation in networks by an original interpretation of a three-level trophic chain model developed in the area of ecological modelling (Aubin 1991, Bonneuil and[START_REF] Bonneuil | Population viability in three trophic-level food chains[END_REF]. We note that this model is presented in our study to illustrate the concept of viability rather than to be used as an optimization tool for a particular decisionmaking setting. Our interpretation comprehensively combines major determinants of the ISNs, i.e., competition between suppliers, market demands, and behavioral dynamics.

Trophic chain

Trophic chains are a part of ecological modelling. Conceptually, trophic chains are used for modeling and analysis of prey-predator systems. Technically, the trophic chain modelling is rooted into the Lotka-Volterra first-order nonlinear differential equations, also known as the predator-prey equations [START_REF] Getz | Control analysis of trophic chains[END_REF][START_REF] Baudrot | Effects of contaminants and trophic cascade regulation on food chain stability: Application to cadmium soil pollution on small mammals -Raptor systems[END_REF].

The system of differential equations describes the trophic chain dynamics subject to some functional response (i.e., the consumption rate of the predator depending on the density rate of the preys) and interactions between the prey and predator. The Jacobian matrix of the predatorprey model can be used to analyse the stability of the ecological system and the impact of oscillations [START_REF] Gross | Enrichment and foodchain stability: the impact of different forms of predator-prey interaction[END_REF]). Moreover, the bifurcation point analysis makes it possible to observe the disturbance propagation through the trophic chain, i.e., the cascading effect [START_REF] Baudrot | Effects of contaminants and trophic cascade regulation on food chain stability: Application to cadmium soil pollution on small mammals -Raptor systems[END_REF]) and to identify the timing when the ecosystem will turn into a chaotic state under the given dynamics [START_REF] Dilao | A general approach to the modelling of trophic chains[END_REF][START_REF] Gross | Generalized models reveal stabilizing factors in food webs[END_REF], Castellanos and Chan-Lopez 2017).

It can be observed that the trophic chain exhibits some characteristics close to the SC. In literature, SCs are considered multi-stage systems which dynamics is guided through vendor-buyer operations. Generally, a firm that plays the buyer role in an upstream echelon of the supply chain becomes a supplier in the next downstream stage. We take up this observation of commonalities between the trophic chains and the SCs and bring the discussion to the next level by using the trophic chain concept for viability analysis of the ISNs.

Model

We illustrate the viability formation in an ISN through dynamic game-theoretic modelling of an ecosystem. Game-theoretical models have been considered suitable to address the SC disruption risks analysis [START_REF] Gupta | Dual sourcing under supply disruption with risk-averse suppliers in the sharing economy[END_REF]. We follow the model of [START_REF] Bonneuil | Population viability in three trophic-level food chains[END_REF] rooted in population viability analysis as a method of risk assessment frequently used in conservation biology. We describe the viability of an ISN where firms exhibit independent, time-varying survival strategies by a specific set, the viability kernel, gathering all states from which there exists at least one trajectory safeguarding each firm over a given survivability threshold (e.g., minimal levels of some financial or operational indicators which allow maintaining the firm's operations and avoiding bankruptcy) .

Consider an ISN structure (Fig. 2).

Fig 2. Intertwined supply network modelling

In an ISN ecosystem, three major roles can be distinguished, i.e., suppliers, focal firms, and market demand, i.e., the consumers. Notably, the firms can play different roles simultaneously.

As such, if we talk about a supplier, we do not mean a concrete firm but rather a role in a particular SC that can be taken by different firms situationally.

Denote inventories at suppliers, focal firms, and markets (customers) as vector functions x1(t), x2(t), x3(t), respectively. Eqs. ( 1)-( 3) represent the dynamics of firm interactions and can be considered as a dynamical game [START_REF] Bonneuil | Population viability in three trophic-level food chains[END_REF]

The dynamical game (1)-( 3) consists in having the system change so that the firms in an ISN can keep the possibility to survive jointly by selecting such strategies that allow each player to continue playing the game, i.e., to survive. In the setting of coronavirus COVID-19 pandemic, the strategies 𝑢 𝑡 ∈ 𝑢 , 𝑢 𝑈, v(t) ∈ 𝑣 , 𝑣 𝑉, and 𝑤 𝑡 ∈ 𝑤 , 𝑤 𝑊 would mean some operating policies (e.g., inventory control) at firms of different SC echelons. u(t) describes the strategies at the suppliers. v(t) and w(t) describe the buyer and customer strategies, respectively. Technically, the game consists in safeguarding a least one solution of the system (1)-(3) that can be written into the differential inclusion (4)

x t ∈ F x t (4) under constraint (5) 𝑥 t ∈ K (5)
remaining in a fixed closed set K, where

F x t ≔ 𝑓 𝑥, 𝑢, 𝑣, 𝑤 |𝑢 𝑡 ∈ 𝑈, 𝑣 𝑡 ∈ 𝑉, 𝑤 𝑡 ∈ 𝑊 (6)
is a point-to-set map, also called correspondence.

The set K describes the strategies for co-existence of the entities in the ISN. The set K is viable if suppliers, focal firms and markets are maintain themselves above certain thresholds and are able to co-exist under these conditions. In other words, there exists at least one solution

x(•) = (x1(•), x2(•), x3(•))
of the dynamical game (1)-(3) starting from x 0 and satisfying the threshold conditions. The viability kernel of the set 𝐾 is the largest closed viable set in K under the dynamic F (Eq. 6) that represents the integrity of survival strategies for the firms in the ISN.

According to [START_REF] Aubin | Viability Theory[END_REF] and [START_REF] Bonneuil | Population viability in three trophic-level food chains[END_REF], the strategies permitting the property ( 7):

for all 𝑥 t ∈ 𝐾 there exists 𝑢 𝑡 ∈ 𝑈, 𝑣 𝑡 ∈ 𝑉, 𝑤 𝑡 ∈ 𝑊 such that 𝑓 𝑥, 𝑢, 𝑣, 𝑤 , 𝑓 𝑥, 𝑢, 𝑣, 𝑤 , 𝑓 𝑥, 𝑢, 𝑣, 𝑤 ∈ 𝑇 𝑥

can be called viable strategies of the ISN, where 𝑇 𝑥 is the contingent cone at state x. All the solutions outside the viability kernel lead to loss of either internal competition between suppliers, or behavioral dynamics, or market demand fulfilment. Moreover, the viability approach make it possible to highlight the timing when the strategy change is needed for an ISN to perpetuate itself.

Discussion and implications for future research

Stability, robustness, and resilience analysis have covered a large area of SC disruption risk analysis. Though, none of them deals with ISNs mostly assuming directed network graphs and fixed time-window for analysis. While the resilience analysis of individual SCs is useful for many cases, the firms are participating in different SCs in different roles. This makes it an important research task to model the ISNs and not the individual SCs to correctly understand the impacts of disruptions and the ripple effects on the viability in the presence of extraordinary events. As such, a new terminology is obviously required.

We suggested using the term "viability" differently than the resilience in regard to the ISNs, in analogy to ecological modelling [START_REF] Aubin | Viability Theory[END_REF][START_REF] Bene | A viability analysis for a bio-economic model[END_REF]) that considers viability as a system ability to maintain itself and recover in the presence of disturbances over a long-term horizon.

In ) outbreak to describe some future research angles. The global pandemic and collapses of many SCs and the markets depict the importance of the SC viability research. While the firms have increasingly dealt with the resilience of their global SCs triggered by some severe natural and man-made disasters and established a set of useful methods such as risk mitigation inventories, subcontracting capacities, backup supply and transportation infrastructures, and data-driven, real-time monitoring and visibility systems, it is unclear how these methods can be applied to SC survivability analysis.

One can expect that some of the SC resilience actions established to cope with natural disruptions can be useful for the viability analysis, too. Though, one can also expect that the specific features of viability would require an examination of new concepts or a modification of the existing ones. For example, the cases of epidemic disruptions, proactive measures such as inventory can help only at the beginning of the crisis due to very long disruption times. Backup suppliers and subcontracting facilities can also be supposed to be less efficient because of quarantining the whole regions and even continents. As such, our sentiment is that the focus of SC viability management would rather be shifted towards the situational reactions to real-time changes rather than building some proactive redundancies. At the same time, the importance of proactive management does not disappear. Here the focus would be shifted towards creating a flexible redundancy which would make the SC networks less sensitive to external uncertainties.

One research are in this direction is the LCN (low-certainty-need) SC framework (Ivanov and Dolgui 2019). In addition, the viability goes beyond a narrow understanding of SC performance as some profits or revenues and brings the discussion to the next level, i.e., SC performance in terms of securing the life on the earth. As such, the issues of collaborative, collective survival in the presence of extra-ordinary conditions are very important, and new research areas. Some other open questions to address are, e.g.: How to analyse disruption impacts in ISNs? How to analyse disruption propagation, i.e., the ripple effect in ISNs? We note that even if viability is important, the questions of resilience, robustness and stability in ISNs are of equal importance as in the SC settings.

The concept and techniques for analysis and modeling of complex ecological networks as postulated by viability and trophic chains can find several applications to production and SC management and can stimulate some new ideas and research in SCs. The ecological modeling in general and trophic chains in particular have a potential to be applied to different areas of SC decision-making, such as:

 Risk propagation analysis in the SCs (e.g., bullwhip effect (propagation upstream) and ripple effect (propagation downstream))  Quality management control (i.e., how a quality error propagates downstream the SC)  Circular economy (i.e., cycles control)  Competition modeling (survival competition, agent modeling, learning agents, learning through evolution).

At the same time, some potential limitations need to be addressed. A direct usage of mathematics for trophic chain analysis (i.e., non-linear dynamic differential system, bifurcation points) requires a specific technical competence which is not a standard equipment of SC and operations management researchers. Besides, the analysis of the ecological system frequently results in bifurcation points derivation of which can be difficult for large-scale system. Methodically, the bifurcation point analysis does not bring new insights into the recovery optimization. This analysis ends with the insight if we loss resilience or not, and we can also understand the conditions surrounding the resilience dynamics. Such an analysis of alternative paths the system takes to save resilience can be useful for deciphering the contingency plans. However, we do not go beyond the bifurcation point -this analysis is not much helpful for recovery and is therefore restricted to the pre-disruption stage and subject to very generalized flows. Such stylized models can certainly be of methodical interest; real system modelling will require simulations of complex adaptive systems and SC structural dynamics.

Conclusion

SCs evolve towards ISNs that are characterized by structural dynamics. Differently than linearly directed SCs with static structures, the firms in ISNs may exhibit multiple behaviors in buyer-supplier relations (i.e., behavioral dynamics) in interconnected or even competing SCs simultaneously. These new dynamic, co-evolving structures require re-thinking of some traditional analysis concepts.

An intertwined supply network (ISN) is an entirety of interconnected supply chains (SC) which, in their integrity secure the provision of society and markets with goods and services. Unlike the resilience of individual SCs, the viability of ISNs has not received much attention in literature so far. The recent example of coronavirus COVID-19 outbreak shows that in the case of extraordinary events, SC resistance to disruptions needs to be considered at the scale of survivability or viability to avoid SC and market collapses and secure the provision with goods and services. In this paper, we elaborated on the integrity of these two novel angles, i.e., ISNs and viability. The contribution of this study lies in conceptualization of a novel decision-making environment that considers ISNs and viability as an integrity to ensure the survivability at a large scale. We showed how the viability is different than the resilience, and why a necessary quality to be added to the SC resilience analysis in the context of ISNs ecosystems. Building upon the analogy of ISNs to ecological systems, we illustrated the viability formation through dynamic game-theoretic modelling of an ecosystem. Some limitations exist in our approach, as with any study. We took a much generalized, "birdeye" perspective on the viability and its modelling. For concrete applications, the concept of viability and the trophic chain modelling should be detailed and extended given the context of decision-making situations. In addition, the ISNs themselves need to be thoroughly investigated in terms of methodology and practice of their formations and control. Finally, the role of ISNs survivability in the securing the provision with goods and services in the case of extra-ordinary events, such as epidemic outbreaks and global pandemics (e.g., coronavirus COVID-19). These areas can be considered promising future research avenues.

Another interesting research topic is examination of disruption outbreaks in the downstream SC parts and the resulting combined effects of the forward and backward propagations of the ripple effect. One promising research area in research on ISNs and their viability is the utilization of digital, data-driven technologies to uncover their potentials to support the decisionmaking in cases of long-term, sever disruptions such as epidemic outbreaks. In particular, digital SC twins (Ivanov and Dolgui 2020) -i.e., the computerized SC models that represent the network state for any given moment in real time -can be further investigated in this direction.

Fig. 1 .

 1 Fig. 1. Linear supply chains, supply networks and intertwined supply networks

  recently,[START_REF] Fracassia | Rethinking Resilience in Industrial Symbiosis: Conceptualization and Measurements[END_REF] point to the multiple, intersecting SCs in the industrial symbiosis which are characterized by using the waste of some SC processes as the inputs into the other SCs[START_REF] Pathak | Toward a Structural View of Co-opetition in Supply Networks[END_REF].[START_REF] Olson | Complex product manufacturing in the intelligence-connected era[END_REF] elaborate on the intelligence-connected manufacturing. Moreover, Industry 4.0 and cyber-physical manufacturing have significantly transformed the SCs and increased their intertwining[START_REF] Liao | Past, present and future of Industry 4.0 -a systematic literature review and research agenda proposal[END_REF], Ivanov et al. 2019[START_REF] Panetto | Challenges for the cyber-physical manufacturing enterprises of the future[END_REF][START_REF] Tang | The strategic role of logistics in the industry 4.0 era[END_REF].[START_REF] Choi | Game theory applications in production research in the sharing and circular economy era[END_REF] show different forms of SC interconnections in the sharing and circular economies. As such, we can conclude that many SCs evolve into ISNs based on the principles of co-creation and co-evolution. Such mechanisms are principally different from the classical SC understanding and therefore it becomes a timely and crucial research task to develop a new thinking of resilience towards viability.

  future, different topics of the ISNs in connection with viability can frame an interesting research avenue. Potential topics include but are not limited to:  Framing the ISN viability and SC survivability concepts  Network structures of ISNs and their viability  Roles and dynamics of ISNs at the times of epidemic outbreaks  Contributions of ISNs to recover the SCs after an epidemic outbreak  How long can an ISN sustain a disruption so what is the critical disruption time?  What are the most critical scenarios of epidemic propagation for the global SCs and their ISNs?  Impact of digitalization, Big Data analytics, and additive manufacturing on the ISN viability  Viability in the context of Value Web approaches  Ripple effect in the ISNs  Game-theoretic modelling of the ISN viability  Complex adaptive systems with applications to ISNs  Collaboration of humanitarian and business logistics for survivability  Ecological modelling approaches to ISN viability  Dynamic analysis of the ISN viability (simulation, control theory) We use the example of coronavirus (COVID-19

Table 1 :

 1 Major analysis concepts for SC performance under uncertainty

	Concept	Operational	Disruption in	Output per-	Recovery	Survivability
		disturbance	SC structures	formance
	Stability	+
	Robustness	+	+	+
	Resilience		+	+	+
	Viability		+	+	+	+

Table 2

 2 Differences Resilience vs. Viability

	Criterion	Resilience	Viability
	System	Close	Open
	Structure	Static	Dynamic
	Scope of analysis	Disruption-driven (single, dis-	Behavior-driven (continuous
		crete, unique events)	change)


Olga Battaïa

Alexandre Dolgui

Nikolai Guschinsky

Optimal Cost Design of Flow Lines with Reconfigurable Machines for Batch Production

Keywords: Reconfigurable manufacturing systems, Reconfigurable machine-tools, Manufacturing lines, Batch processing, Line design and balancing, Cost minimization, Optimization

Modular reconfigurable machines offer the possibility to efficiently produce a family of different parts.

This paper formalizes a cost optimization problem for flow lines equipped with reconfigurable machines which carry turrets, machining modules and single spindles. The proposed models take into account constraints related to: i) design of machining modules, turrets, and machines, ii) part locations, and iii) precedence relations among operations. The goal is to minimize equipment cost while reaching a given output and satisfying all the constraints. A Mixed Integer Program (MIP) is developed for the considered optimization problem. The approach is validated through an industrial case study and extensive numerical experiments.

Introduction

Batch production is often used in low-volume diversified manufacturing context which is gaining more and more importance due to the major trend of product customization [START_REF] Liao | Past, present and future of Industry 4.0 -a systematic literature review and research agenda proposal[END_REF][START_REF] Buer | The link between Industry 4.0 and lean manufacturing: mapping current research and establishing a research agenda[END_REF][START_REF] Xu | Industry 4.0: state of the art and future trends[END_REF][START_REF] Yin | The evolution of production systems from Industry 2.0 through Industry 4.0[END_REF], Dolgui et al., 2019). A batch is a set of different parts that are manufactured together, it means that all the parts of the same batch follow the same manufacturing path, but each part receives its own operations. In reconfigurable manufacturing systems, operating modules of machines can be adapted for each batch. However, it is preferably to not use equipment reconfigurations for the parts of the same batch. Therefore, the batches are constituted in the way that the processing of the parts within the same batch will not require any equipment reconfiguration. The reconfiguration is only realized if needed between two different batches. In this paper, we consider a design problem for a reconfigurable manufacturing system for batch production. The design objective is to decide which modules will be used for each batch of products at which machine as well as which will be the general layout of the manufacturing system. The considered manufacturing system may consist of several reconfigurable machines composed of different machining modules (spindle heads). A subset of machining modules is used for manufacturing each batch of parts. The decision on the modules to be used in the system and the number of machines has to be made taking into account the parts to be produced and their quantity (size of each batch). Machining modules of a machine can be activated simultaneously or sequentially when applied to the same part.

Simultaneous activation is realized for modules machined different accessible sides of the part. Sequential application is realized by the use of turrets which can carry several machining modules. In such a configuration, each part can be machined by at most 4 parallel modules corresponding to accessible sides of the machined part (two examples of such machines are shown in Fig. 1 and Fig. 2). Fig. 1. A machine with one vertical turret and 2 horizontal modules Fig. 2. A machine with two horizontal turrets and one horizontal module

The design problem considered in this paper concerns the choice of number of reconfigurable machines as well as the equipment to be installed on in terms of turrets and machining modules in order to machine a given number of batches. Since the location and machining axe and direction has to be decided for each machining module, the design solution defines also the orientations of each part on each machine, as well as cutting modes for each machining module.

The rest of the paper is organized as follows. Section 2 analyses the state of the art in the field. Section 3 provides a formal problem statement and Section 4 presents a mathematical model for it. A case study is described in Section 5. Finally, Section 6 concludes the study and suggests research perspectives.

Literature review

The research on reconfigurable manufacturing systems (RMS) introduced by Koren et al. 20 years ago [START_REF] Koren | Reconfigurable manufacturing systems[END_REF]) actually attracts a growing number of academic and industrial studies [START_REF] Bortolini | Reconfigurable manufacturing systems: Literature review and research trend[END_REF]. This is due to the fact that RMS offer a novel effective manufacturing paradigm for nowadays production challenges characterized by an increasing variety of customized, high-quality products. A large number of publications dedicated to the research on RMS can be found in the literature. They provide multiple insights on the research perspectives in the field: on theoretical and practical challenges in the design of RMS [START_REF] Rosio | Reconfigurable production system design-theoretical and practical challenges[END_REF], on measures of reconfigurability (Farid, 2014), on the use of artificial intelligence for the design of RMS [START_REF] Renzi | A review on artificial intelligence applications to the optimal design of dedicated and reconfigurable manufacturing systems[END_REF], on the adaptation of RMS for industrial assembly [START_REF] Huettemann | Adaptation of reconfigurable manufacturing systems for industrial assembly-review of flexibility paradigms, concepts and outlook[END_REF], on reconfigurable manufacturing on multiple levels [START_REF] Andresen | Reconfigurable manufacturing on multiple levels: literature review and research directions[END_REF], on prerequisites and barriers for the development of RMS (Andresen et al., 2016), on the current design methods for RMS [START_REF] Andersen | Towards a generic design method for reconfigurable manufacturing systems: analysis and synthesis of current design methods and evaluation of supportive tools[END_REF]. In particular, three recent surveys give a comprehensive view of challenges and research trends [START_REF] Singh | Reconfigurable manufacturing systems: journey and the road ahead[END_REF][START_REF] Bortolini | Reconfigurable manufacturing systems: Literature review and research trend[END_REF][START_REF] Brahimi | A literature review of optimization problems for reconfigurable manufacturing systems[END_REF] where the interested reader will find a historical perspective of RMS development, detailed discussions on the topics attracting the major streams of research and future trends. Due to the availability of these complete bibliographical studies in the literature, we focus our literature analysis on the work that is the closest to our research question.

As for traditional manufacturing systems, the research on RMS concerns both the design and operation levels. Since in this study, a design problem is considered, the literature review is focused on the design level where a configuration of the system has to be defined and an equipment selection has to be made. These decisions however are usually closely related to the decisions on operation assignments to manufacturing modules and machines [START_REF] Koren | Design of reconfigurable manufacturing systems[END_REF].

One of the first studies on the design of RMS with the objective to optimize the investment cost can be found in [START_REF] Youssef | Modelling and optimization of multiple-aspect RMS configurations[END_REF]. This study has been consequently extended by considering a reconfiguration smoothness metric [START_REF] Youssef | Optimal configuration selection for reconfigurable manufacturing systems[END_REF] and machine availability [START_REF] Youssef | Availability consideration in the optimal selection of multiple-aspect RMS configurations[END_REF]. [START_REF] Dou | Configuration selection of reconfigurable manufacturing system based on hybrid analytical hierarchy process[END_REF][START_REF] Dou | Optimisation for multi-part flow-line configuration of reconfigurable manufacturing systems using GA[END_REF] studied reconfigurable flow lines with the objective to optimize the total investment cost for single-and multi-part flow lines, respectively. This research was further extended to a configuration generation problem where both the total cost and the total tardiness were minimized [START_REF] Dou | Bi-objective optimization of integrating configuration generation and scheduling for reconfigurable flow lines using NSGA-II[END_REF]. [START_REF] Ashraf | Configuration selection for a reconfigurable manufacturing flow line involving part production with operation constraints[END_REF] considered the problem of optimal configuration selection of reconfigurable machine tools (RMT) for a reconfigurable serial flow line.

These studies are close to our work, however, our design problem needs a more sophisticated mathematical model since the configuration of each machining module and the corresponding cutting tools have to be defined according to the results of the optimization and not taken from a catalogue. As a consequence, more refined mathematical models are developed in the present study. In our previous work [START_REF] Battaïa | Decision support for design of reconfigurable rotary machining systems for family part production[END_REF], we studied a close problem for rotary machining systems and here we extend the obtained mathematical model to a more general machining system layout.

The estimation of the total cost of RMS over planning horizon has been investigated in several studies.

The optimization of the cost for scalability of RMS systems, in other words, their ability to be adapted to the scale of production volume was investigated by [START_REF] Son | An approach to scalability and line balancing for reconfigurable manufacturing systems[END_REF], [START_REF] Spicer | Scalable reconfigurable equipment design principles[END_REF], [START_REF] Wang | Scalability planning for reconfigurable manufacturing systems[END_REF], and Koren et al., (2016). [START_REF] Saxena | A model and optimisation approach for reconfigurable manufacturing system configuration design[END_REF] explored the structure of the cost of RMS including equipment investment, reconfiguration cost, operating cost, maintenance cost and residual marginal value over time. Xiaobo et al. (2000Xiaobo et al. ( a,b, 2001 a,b) a,b) used stochastic optimization to maximize the average expected profit for a designed RMS. Production planning of RMS with stochastic demands was investigated by Abbasi andHoushmand (2009, 2011). The problem of RMS configuration design is also addressed in [START_REF] Moghaddam | Configuration design in scalable reconfigurable manufacturing systems (RMS) a case of single-product flow line (SPFL)[END_REF], where the demand of a single product varies throughout its production life cycle, and the system configuration can change to satisfy the required demand with a minimum cost. The problem of selecting the optimal machining parameters for all operations of a batch processing line that minimize the total batch production cost while ensuring the required line throughput was studied by [START_REF] Dolgui | Optimisation of the aggregation and execution rates for intersecting operation sets: an example of machining process design[END_REF].

In terms of multi-objective optimization, the cost is often regarded jointly with the total completion time, e.g. [START_REF] Bensmaine | A non-dominated sorting genetic algorithm based approach for optimal machines selection in reconfigurable manufacturing environment[END_REF] and [START_REF] Haddou-Benderbal | Flexibility-based multi-objective approach for machine selection in reconfigurable manufacturing system (RMS) design under unavailability constraints[END_REF]. [START_REF] Goyal | Optimal configuration selection for reconfigurable manufacturing system using NSGA II and TOPSIS[END_REF][START_REF] Goyal | A novel methodology to measure the responsiveness of RMTs in reconfigurable manufacturing system[END_REF] introduced a multi-objective model for the optimal machine assignment for a single part flow line with the objective to find a trade-off between the cost and the responsiveness of the RMS. The bi-objective model by [START_REF] Dou | Bi-objective optimization of integrating configuration generation and scheduling for reconfigurable flow lines using NSGA-II[END_REF] was developed both for design and scheduling levels in order to minimize the investment and reconfiguration costs as well as total tardiness. The choice of single or multi-objective optimization depends on the criteria used in particular decision context. In our study, the industrial designers are principally interested by the criterion of the total investment cost while the production time is considered as a hard constraint.

The description of the optimization problem addressed in this paper is given in the next section. The aim of the study is not on the technology level but to facilitate the decision making process related to the preliminary design (combinatorial design) of RMS. In our study, we consider the stage of the design of RMS for a planning horizon with a known set of parts, therefore, the designed system includes all equipment required to process all known parts, it means that the reconfiguration "cost" is included in the investment cost, if the same equipment can be used for different parts it will be preferred in the design solution. The problem of reconfiguration for "unknown" parts is not considered in the present study. This optimization problem for batch production design is novel and not been addressed in the literature yet.

Moreover, the model we develop does not only help to solve this specific problem, but also offers new ideas for modelling and solving other similar optimization issues.

Problem statement

The considered design problem consists in defining the number of reconfigurable machines m and their configurations in order to produce d0 types of parts. The parts are grouped in  batches with required output O  (where  is the number of batch , =1,2,…,). Batches and parts in each batch are processed sequentially. Parts of  th batch are loaded in sequence   =( 1 ,  2 , …,   ) where  j {1,2,…,d0}, j=1,2,…,  where   is the number of parts processed in a sequence in batch  = O  Using sequences   we can define, in one-to-one manner, function

  (i,k), i=1,…,O    +m-1, of part number on k th machine after i movements of the conveyer,   (i,k){0,1,2,…,d0} (  (i,k)=0 means that no part is processed on k th machine).
Figures 3 to 5 in Section 5 show an example of batch of three different parts.

Let us recall the introduced indexes for further presentation of parameters and decision variables:

-Index for reconfigurable machines k = 1,...,m, where m is the number of machines.

-Index for parts: dD={1,2,…,d0}, where d0 is the number of types of parts.

-Index for batches : =1,2,…,, where  is the number of batches to be produced.

-Index for the positions of parts in sequence of each batch:

  =( 1 ,  2 , …,   ) where  j {1,2,…,d0}, j=1,2,…,  and   is the number of parts processed in a sequence in batch  = O  .
Let N d be the set of machining operations needed for machining of elements of d th part, dD={1,2,…,d0} located on its nd sides and d s N , s=1, 2,…,nd, is a subset of operations for machining of elements of s th side of the part d. The part d can be located at a machine in different orientations H(d) but elements of no more than one side can be processed by a machining module. Machining modules can be of 4 types: type j=1 (from above), j=2 (from the left), j=3 (from behind), j=4 (from the right) and no more than 3 types of machining modules can be installed at each machine. H(d) can be represented by a matrix of dimensions rd x nd where hrs(d) is equal j, j=0,1,2,3,4, if the operations of d s N can be processed by machining modules of type j (j=0 means that processing is not possible).

Let N be the set of all operations required for completing all considered parts, N=  ..,m, j=1,...,4, l=1,...,bkj; -the feed rate Гdkjl[max{γ1(u)|uNdkjl},min{γ2(u)|uNdkjl}] for execution of operations from Ndkjl,dD,k=1,...,m,j=1,...,4,l=1,...,bkj. We assume that if the turret of type j is installed at machine k then the execution time of operations from

Let Pdkj = (Pdkjl = (Ndkjl,Гdkjl)| l=1,...,bkj), Pdk = (Pdkj| j=1,…,4), Pd = ((Pdk, Hdk)| k = 1, ...,m), P = (Pd | dD). P is a design decision.
Ndkjl is equal to t h (Pdkj) =  g bkj +   kj b l 1 t b (Pdkjl), j=1,…,4
, where  g is an additional time for one rotation of a turret. If the spindle head is installed, then t h (Pdkj) = t b (Pdkjl), j=1,…,4. If all Ndkjl are empty then

t h (Pdkj) = 0. If bkj =1, then t h (Pdkj) = t b (Pdkj1).
The execution time t p (Pdk) is defined as

t p (Pdk) =  r + max{t h (Pdkj)| j=1,…,4}
, where  r is an additional time for part transportation and its location at a machine. Then the time T  (P) of execution of all corresponding operations for batch  is defined as

T  (P)=        1 1 m O i max{ ) ( ) , ( k k i p P t  
|k=1,,m} and the time T(P) for machining all the batches is equal to

    1 T  (P).
We assume that the given productivity is provided, if the total time T(P) does not exceed the available time T0.

Let Csm, Ctb, Cmm, Csb, and Cst be the relative costs for one machine, one turret, one machining module of a turret, one spindle box, and one workhead with a single spindle, respectively. Then the cost C(bkj) for performing set of operations Nkj by associated bkj machining modules can be assessed as follows:

C (bkj) =               . 1 if , 1 | | and 1 if , 1 | | and 1 if , 0 if 0 1 1 kj kj mm tb kj kj sb kj kj st kj b b C C N b C N b C b
The machine cost Q(P) is calculated as the total cost of all equipment used:

             1 1 1 1 1 4 1 0 ) , ( ) ( ) ( m k d d dk dk m k j kj sm H H C b C m C P Q ,
where C(Hdk,Hdk+1) is the cost incurred by changing orientation of part d between stations k and k+1 and C(Hdk,Hdk+1)=0 if Hdk = Hdk+1 and C(Hdk,Hdk+1)= Cro, otherwise (Cro is the relative cost for reorientation).

The design decision P should satisfy the following constraints:

precedence constraints which define time dependencies among operations;

inclusion constraints which require to execute some pairs of operations from N at the same machine, by the same turret, by the same machining module or even by the same spindle;

exclusion constraints which require to not assign some pairs of operations from N to the same machine, to the same turret, or to the same machining module;

constraints on the maximal number m0 of machines and on the maximal number b0 of machining modules in a turret;

constraints on feasible orientations of the part for processing each operation;

the impossibility to perform operations from d s N at one machine by machining modules of different types;

productivity constraints to provide the required output (a time limit for total completion time).

Precedence constraints are modelled by a directed graph G OR =(N,D OR ): if an arc (p,q)D OR then operation p has to be executed before operation q. It should be noted that if such operations p and q belong to different sides of the part, then they cannot be executed at the same machine without violating the precedence constraint.

Inclusion constraints are represented by undirected graphs

G SP =(N,E SP ), G ST =(N,E ST ), G SM =(N,E SM ) and
G SS =(N,E SS ): if there is an edge (p,q)E SS (or (p,q)E SM , or (p,q)E ST , or (p,q)E SP ), then operations p and q must be executed by the same spindle (the same machining module, turret, at the same machine, respectively).

Exclusion constraints are represented by undirected graphs G DM =(N,E DM ), G DT =(N,E DT ), and

G DP =(N,E DP ).
If there is an edge (p,q)E DM (or (p,q)E DT ), or (p,q)E DP )), then operations p and q cannot be executed by the same machining module (turret, at the same machine, respectively).

MIP approach

In order to solve the considered problem, we developed a Mixed Integer Programming (MIP) model

where the following notations are used:

Xpkjl
  =(O    - m0+1)/    
additional number of part sequences when all machines are occupied   =mod(O    -m0+1,  ) tpq minimal time necessary for the execution of operations p and q by the same machining module, tpq = max((p), (q))/min(2(p),2(q))+ a M an upper bound on the maximal number of operations to be potentially assigned to one

machining module } , , 1 | max{ 0 1 d d s d d n s N M     
The objective function can be expressed as follows:

Minimize            ) ) 2 ( ( 2 1 4 1 1 0 0 kjs st kj mm tb m k j kjb sb m k k sm Y C Y C C Y C Z C            1 1 1 1 4 1 3 0 0 0 m k D d dkk ro m k j b l kjl mm C Y C  ( 1 )
The following equations determine decision variables Xpkjl as well as auxiliary variables ds kj Y , Ykjl, and Zk.

1 0 0 1 4 1 1        m k j n l pkjl X ; pN ( 2 ) X pkjl=0; j=1,…,4; jJ(p) (3) qkjl l l pkjl k k j b l l j pk X X X             1 1 ' ' 1 1 ' 4 1 1 ' ' ' 0 ; (p,q)D OR ; k=1,…,m0; j=1,…,4; l=1,…,b0 ( 4 )
XpkjlXqkjl; {p,q}E SM E SS ; k=1,…,m0; j=1,…,4; l=1,…,b0

( 5 )

  0 1 b l pkjl X =   0 1 b l qkjl X ; {p,q}E ST ; k=1,…,m0; j=1,…,4 (6)     4 1 1 0 j b l pkjl X =     4 1 1 0 j b l qkjl X ; {p,q}E SP ; k=1,…,m0 ( 7 )
Xpkjl+Xqkjl1; {p,q}E DM ; k=1,…,m0; j=1,…,4; l=1,…,b0 ( 8 )

     0 0 1 1 b l qkjl b l pkjl X X +Ykj2  2; {p,q}E DT ; k=1,…,m0; j=1,…,4 (9)          0 0 1 4 1 1 4 1 b l qkjl j b l pkjl j X X  1; {p,q}E DP ; k=1,…,m0 (10)     d s N p pkjl b l X 0 1 | d s N | ds kj Y ; ds kj Y      d s N p
  4 1 1 j kj Y 3;   4 1 1 j kj Y 3Zk; Zk    4 1 1 j kj Y ; k=1,…,m0 (16)
, 1

2 1       M MY Y X kjs kj p pkj N k=1,…,m0; j=1,…,4 (17)
,

1 2 kj kjs kj kjb Y Y Y Y    k=1,…,m0; j=1,…,4 (18) Z kZk-1; k=2,…,m0 (19)
Constraints ( 2) -( 3) guarantee an assignment of each operation to exactly one machining module of compatible type. Precedence constraints, inclusion constraints for spindles, machining modules, turrets and machines as well as exclusion constraints for machining modules, turrets and machines are provided by ( 4) -( 10) respectively. Constraints ( 11) -( 12) prevent from an assignment of operations located at the same side of each part to machining modules of different types of the same machine. Constraints ( 13

1 1    d r r rdk h ; dD; k=1,…,m0 (21) 1 1 1 1          rdk r r rdk r r dkk d d d rh rh r ; rdk r r rdk r r dkk d d d rh rh r          1 1 1 1 ; dD; k=1,…,m0-1 (22)
Constraints ( 20) -( 21) defines the orientation of each part at each machine. Variables dkk+1 are equal to 0 if hrdk = hrdk+1 for all r=1,…,rd due to ( 22) and minimization of (1).

F dk   0 1 b l dkjl F + g (2Ykj2+   0 3 b l kjl Y )+b0 g ( d kj Y -1); dD; k=1,…,m0; j=1,…,4 (25) i F   ) , 1 ( 0 k i m k F     + r ; =1,…,; i=1,…,  ; k=1,…,m0 (26) i s F   ) , ( k i k F   + r ; =1,…,; i=1,…,m0-1; k=1,…,i (27) i f F   ) , ( 0 k i m O k F        + r ; =1,…,; i=2,…,m0; k= i,,m0 (28)
) ( 0 0 2 1 1 1 1 1                           m i i f m i i s i i i i F F F F  T0 (29)
Constraints ( 21) -( 22 Variables domains are defined as follows:

X pkjl pN; k=1,…,m0; j=1,…,4; l=1,…,b0 (30) ds kj Y  k=1,…,m0; dD; s=1,…,nd; j=1,…,4 (31) …,m0; j=1,…,4; l=1,…,b0; dD (38)

d kjl F  r d k t    k=1,
d k F  r d k t    k=1,…,m0; dD (39) i F   i t   i t   i=1,…,  (40) i s F   } ,..., 1 | max{ ) , ( i k t k i     } ,..., 1 | max{ ) , ( i k t k i k     i=1,…,m0-1 (41) i f F   } ,..., | max{ 0 ) , ( m i k t k i O         } ,..., | max{ 0 ) , ( m i k t k i O k         i=2,…,m0 (42) where d t =min{(p)/2(p)+ a + r | pN d }, } ,..., 1 | max{ 0 ) , ( m k t t k i i      ,                         ' 1 ' ' , 1 ' ' ' ' ' , 1 ' ' 0 / ) ( i i i i i i i t O t O T t and } ) , ( , 1 ,..., 1 , ,..., 1 | max{ 0 d k i m O i t t i d k              
.

A case study

In order to validate the developed model, it was used to design a flow line for machining 3 different parts (Fig. 3 to Fig. 5) in 2 batches with O1=34 and O2=68. The parameters of machining operations are given in Table 1. The operations to be executed for the first part are located on 4 different sides and those for the second and third parts are located on only 3 sides. The loading sequences of parts are {1,2} and {3}.

Other parameters are:  a =  g =  r = 0.1 min, Csm=20, Csh=4, Ctb=5, Cmm=2, Cro=0.5. The available time T0 is 384 min. The possible orientations of the parts are defined as follows: Precedence constraints, exclusion constraints for machining modules, turrets and machines are presented in Tables 2,3, 4, and 5, respectively. Inclusion constraints for machines, turrets and machining modules are given in Tables 6, 7 and 8. Operations to be executed by the same spindle are presented in Table 9. In all obtained solutions the flow line consists of 2 machines and 2 turrets should be installed at each machine but the assignment of operations to machining modules is quite different. The obtained optimal solution for m0=5 and its characteristics are presented in Tables 11 and12. The total cost Q(P) is equal to 2Csm+0Csh+4Ctb+12Cmm+3Cro5=85.5. In this case p1=32, p2=64, 1 =2 =0.

H(1)=                           1 1 2 1 1 4 1 1 1 1 1 4 1 1 1 4 1 4 1 1 , H(2)=              1 2 1 1 1 4 4 1 1 , H(3)=                     1 1 4 1 1 2 4 1 1 2 1 1 .
H 1k H 2k H 3k 1 1.03 0.56 3.02 -1,-1,4,1 -1,1,4 4,-1,-1 2 2.46 2.46 1.96 1,2,-1,-1 1,2,-1 -1,1,2 3 0 0 0 - - - 4 0 0 0 - - - 5 0 0 0 - - -
The proposed approach was also tested on 20 industrial examples with ||=2. The proprieties of these instances are given in Table 9 where OS OR is the order strength of the precedence constraints (the ratio The experiments were carried out for two variants of each problem that differed in matrices H(d) and sets J(p). The results are presented in Tables 10 and11 where NO is the number of rows of a matrix H of possible orientations of all parts, TC is the calculation time in seconds, NV and NC are the numbers of variables and constraints in MIP formulations, respectively. H is built on the base of matrices H(d) and takes also into account constraints D SS and D SM . The obtained results show that the tested industrial problems could be solved to optimality in reasonable time. However, the complexity of the problem and the time required to solve it strongly depends on the density of inclusion and exclusion constraints which in fine define the number of possible solutions.

Conclusions

In this paper, we developed a new mathematical model for the cost optimization problem in the design of machining flow lines composed of reconfigurable machines and used for batch production. This new model includes the definition of precedence, exclusion and inclusion constraints among operations and different types of equipment, such as machining modules, turrets, spindle boxes and workheads with single spindles. The configuration of the line is decided by optimizing the assignment of operations to the pieces of equipment while satisfying technical and technological constraints as well as the required final completion time.

The developed model is general and can be applied to different reconfigurable manufacturing systems (RMS) used in machining industry. The proposed model has been successfully validated on industrial case studies. The obtained results show that it can be used in practice for designing RMS for variable batches of parts required in low volume. In particular, the solution time for finding optimal solutions for industrial problems is acceptable. Therefore, the use of the developed model may help designers to evaluate quickly different design configurations and the investment costs required. Thus, the total cost and design time can be substantially reduced. The future research will include the integration of environmental factors into the considered problem. Despite of the fact that the investment cost is still important for decision makers in industry, solutions can be also compared on the basis of their environmental indicators such as energy consumption, waste generation, options available for the end of life treatment. However, an important effort will be needed to properly model these environmental indicators.

  pN is characterized by the following parameters: the length (p) of the working stroke for operation p, the range [γ1(p),γ2(p)] of feasible values of feed rate defining cutting speed, and the set J(p) of types of possible machining modules for execution of operation p.The considered optimization problem is to determine:-the number m of machines; -the orientation Hdk of part dD at machine k, k = 1,...,m; -the number bkj of machining modules of type j, j=1,…,4, that are installed at machine k, k=1,...,m;-the set of machining operations Ndkjl from N d , assigned to machining module l of type j at machine k, dD, k = 1,.

  Pdkjl) = L(Ndkjl)/Гdkjl + a , where L(Ndkjl) = max{(p)| pNdkjl}, and  a is an additional time for advance and disengagement of tools.

  ) calculate the execution time of the l-th machining module of type j at the k-th machine for processing part d. If a turret of type j with bkj machining modules is installed at the k-th machine then: Fdk  25) if at least one operation from N d is executed by the turret and Fdk =0, otherwise. If a spindle head of type j is installed at the k-th machine then Fdk  d kj F 1 .

Fig. 3 .

 3 Fig.3. The first part to be machined

  between the number of arcs in the closure of G OR and the number of arcs in the complete digraph with the same number of vertices. Densities of other constraints (D DM , D DT , D DP , D SS , D SM , D ST , and D SP ) are defined in the same way.

  decision variable which is equal to 1 if operation p is assigned to the lth machining module of spindle head or turret of type j at station k, pN, k=1,…,m0, j=1,…,4, l=1,,b0 hrdk decision variable which is equal to 1 if orientation r is chosen for part d at station k, r=1,…,rd,

				dD, k=1,…,m0
	ds kj
				d s N , dD, s=1,…,nd, k=1,…,m0, j=1,…,4
	d kj
				machine
	Fdk		the execution time for part d at k-th machine
	F i		total remaining processing time when processing of part  i of -th batch is finished
	i F  s	total time of first i machines after the i-th turn of the conveyer for processing -th batch
	F	f	i

Y

auxiliary variable which is equal to 1 if at least one machining module of type j is installed at station k for execution of operations from Y auxiliary variable which is equal to 1 if at least one machining module of type j is installed at station k for execution of operations from N d , dD, k=1,…,m0, j=1,…,4 Ykjl auxiliary variable which is equal to 1 if the lth machining module of type j is installed at station k, k=1,…,m0, j=1,…,4, l=1,,b0 Ykjs auxiliary variable which is equal to 1 if the workhead with a single spindle of type j is installed at station k, k=1,…,m0, j=1,…,4 Ykjb auxiliary variable which is equal to 1 if the spindle box of type j is installed at station k, k=1,…,m0, j=1,…,4 Zk auxiliary variable which is equal to 1 if station k is used, k=1,…,m0 dkk+1 auxiliary variable which is equal to 1 if part d is reoriented between stations k and k+1, dD, k=1,…,m0-1 Fdkjl execution time for part d by l-th machining module of spindle head or turret type j at the k-th  total time of last i machines after the O    +m0-i-th turn of the conveyer for processing -th batch   number of cyclically repeating part sequences when all machines are occupied

Table 1 .

 1 Operations and their parameters

	p	Hole Part Side (p) mm	γ 1 (p), mm/ min	γ 2 (p), mm/ min	J(p)	mm p Hole Part Side (p),	min mm/ γ 1 (p),	min mm/ γ 2 (p),	J(p)
	1	H3 1	1	34	37.7 63.4 {1,4}	36 H6	2	1	75	29.7 105.7 {1,4}
	2	H3 1	1	22	27.8 249.5 {1,4}	37 H7	2	2	24	24.6 83.6 {1,2}
	3	H4 1	1	34	37.7 63.4 {1,4}	38 H7	2	2	9	28.3 106.3 {1,2}
	4	H4 1	1	22	27.8 249.5 {1,4}	39 H8	2	2	24	24.6 83.6 {1,2}
	5	H5 1	1	79	22.8 81.3 {1,4}	40 H8	2	2	9	28.3 106.3 {1,2}
	6	H5 1	1	75	29.7 105.7 {1,4}	41 H9	2	2	24	24.6 83.6 {1,2}
	7	H6 1	1	79	22.8 81.3 {1,4}	42 H9	2	2	9	28.3 106.3 {1,2}
	8	H6 1	1	75	29.7 105.7 {1,4}	43 H10	2	2	24	24.6 83.6 {1,2}
	9	H7 1	2	24	24.6 83.6 {1,2}	44 H10	2	2	9	28.3 106.3 {1,2}
	10	H7 1	2	9	28.3 106.3 {1,2}	45 H15	2	3	2	18.8 62.7 {4}
	11	H8 1	2	24	24.6 83.6 {1,2}	46 H16	2	3	2	18.8 62.7 {4}
	12	H8 1	2	9	28.3 106.3 {1,2}	47 H17	2	3	2	18.8 62.7 {4}
	13	H9 1	2	24	24.6 83.6 {1,2}	48 H18	2	3	2	18.8 62.7 {4}
	14	H9 1	2	9	28.3 106.3 {1,2}	49 H4	3	1	53	39.2 62.9 {2,4}
	15	H10 1	2	24	24.6 83.6 {1,2}	50 H4	3	1	34	27.2 248	{2,4}
	16	H10 1	2	9	28.3 106.3 {1,2}	51 H5	3	1	53	39.2 62.9 {2,4}
	17	H11 1	2	25	22	82.2 {1,2}	52 H5	3	1	34	27.2 248	{2,4}
	18	H12 1	2	25	22	82.2 {1,2}	53 H6	3	1	100 22.8 81.3 {2,4}
	19	H13 1	2	25	22	82.2 {1,2}	54 H6	3	1	98	29.7 105.7 {2,4}
	20	H14 1	2	25	22	82.2 {1,2}	55 H7	3	1	100 22.8 81.3 {2,4}
	21	H15 1	3	2	18.8 62.7 {4}	56 H7	3	1	98	29.7 105.7 {2,4}
	22	H16 1	3	2	18.8 62.7 {4}	57 H8	3	1	45	22.8 81.3 {2,4}
	23	H17 1	3	2	18.8 62.7 {4}	58 H8	3	1	43	29.7 105.7 {2,4}
	24	H18 1	3	2	18.8 62.7 {4}	59 H9	3	1	100 22.8 81.3 {2,4}
	25	H19 1	4	2	18.8 62.7 {1}	60 H9	3	1	98	29.7 105.7 {2,4}
	26	H20 1	4	2	18.8 62.7 {1}	61 H10	3	2	3	15.5 51.6 {1}
	27	H21 1	4	2	18.8 62.7 {1}	62 H11	3	2	3	15.5 51.6 {1}
	28	H22 1	4	2	18.8 62.7 {1}	63 H12	3	2	3	15.5 51.6 {1}
	29	H3 2	1	34	37.7 63.4 {1,4}	64 H13	3	2	3	15.5 51.6 {1}
	30	H3 2	1	22	27.8 249.5 {1,4}	65 H14	3	2	3	15.5 51.6 {1}
	31	H4 2	1	34	37.7 63.4 {1,4}	66 H15	3	2	3	15.5 51.6 {1}
	32	H4 2	1	22	27.8 249.5 {1,4}	67 H16	3	2	30	43.7 74.1 {2,4}
	33	H5 2	1	79	22.8 81.3 {1,4}	68 H16	3	5	24	31.9 197.1 {2,4}
	34	H5 2	1	75	29.7 105.7 {1,4}	69 H16	3	5	24	26.9 161.6 {2,4}
	35	H6 2	1	79	22.8 81.3 {1,4}	70 H16	3	5	18	26.7 160.2 {2,4}

Table 2 .

 2 Precedence constraints

	Operation Predecessors	Operation Predecessors
	2	1 3 29 31	40	9 11 13 15 37 39 41 43
	4	1 3 29 31	42	9 11 13 15 37 39 41 43
	6	5 7 33 35	44	9 11 13 15 37 39 41 43
	8	5 7 33 35	50	49 51
	10	9 11 13 15 37 39 41 43	52	49 51
	12	9 11 13 15 37 39 41 43	54	53 55 59
	14	9 11 13 15 37 39 41 43	56	53 55 59
	16	9 11 13 15 37 39 41 43	58	57
	30	1 3 29 31	60	53 55 59
	32	1 3 29 31	68	67
	34	5 7 33 35	69	68
	36	5 7 33 35	70	69
	38	9 11 13 15 37 39 41 43

Table 3 .

 3 Incompatibility of operations in machining modules

	Operation	Incompatible operations
	9/10/11/12/13/14/15/16 1 2 3 4 5 6 7 8
	17	1 2 3 4 5 6 7 8 13 14
	18	1 2 3 4 5 6 7 8 17
	19	1 2 3 4 5 6 7 8 18
	20	1 2 3 4 5 6 7 8 15 16 17 19
	21/22	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
	23	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
	24	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22
	25	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
	26	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
	27	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
	28	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
	37/38/39/40/41/42/43/44 29 30 31 32 33 34 35 36
	45/46	29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
	47	29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
	48	29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 46
	61/62/63/64/66	49 50 51 52 53 54 55 56 57 58 59 60
	67/68/69/70	49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
	Table 4. Incompatibility of operations in turrets
	Operation	Incompatible operations
	9/10/11/12/13/14/15/16/17/18/19/20 1 2 3 4 5 6 7 8
	21/22/23/24	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
	25/26/27/28	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
	37/38/39/40/41/42/43/44	29 30 31 32 33 34 35 36
	45/46/47/48	29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
	61/62/63/64/65/66	49 50 51 52 53 54 55 56 57 58 59 60
	67/68/69/70	49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Table 5 .

 5 Incompatibility of operations in machines

	Operation	Incompatible operations
	1/2/3/4/5/6/7/8	21 22 23 24
	17/18/19/20	25 26 27 28
	49/50/51/52/53/54/55/56/57/58/59/60 61 62 63 64 65 66

Table 6 .

 6 Operations to be assigned to the same machine

	Operation Operations to be executed at the same machine	Operation	Operations to be executed at the same machine
	49	51 53 55 57 59	50	52 54 56 58 60

Table 7 .

 7 Operations to be assigned to the same turret

	Operation Operations to be executed by the same turret	Operation	Operations to be executed by the same turret
	25	26 27 28	67	68 69 70

Table 8 .

 8 Operations to be assigned to the same machining module

	Operation Operations to be executed by the same machining module	Operation	Operations to be executed by the same machining module
	1	3	33	35
	5	7	34	36
	6	8	37	39 41 43
	9	11 13 15	49	51
	17	19	53	55 59
	18	20	54	56 60
	29	31

Table 9 .

 9 Operations to be executed by the same spindle

	Operation Operations to be executed by the same tool	Operation	Operations to be executed by the same tool
	1	29		11	39
	2	30		12	40
	3	31		13	41
	4	32		14	42
	5	33		15	43
	6	34		16	44
	7	35		21	45
	8	36		22	46
	9	37		23	47
	10	38		24	48
	Solver CPLEX 12.2 was used to solve the corresponding problems (1) -(42) for b0=4 and m0=2, 3, 4, 5
	on PC Intel Pentium (2.40 Ghz, 4 Gb RAM). The obtained results are presented in Table 10.
	Table 10. Results of optimization
	m 0	Number of variables Number of constraints	Solution time, s
	2	1639	49700	8.18
	3	2458	74553	20.13
	4	3277	99406	40.51
	5	4096	124259	56.82

Table 12 .

 12 Characteristics of the solution

	The total time T(P) is equal to

Table 9 .

 9 The proprieties of the industrial examples

	Problem |N| OS OR D DB	D DT	D DP	D SS	D SB	D ST	D SP	|D|	    1	
	1	66	0.050 0.502 0.228 0.030 0.002 0.025 0.003 0.026 2	4
	2	58	0.015 0.356 0.000 0.039 0.000 0.033 0.004 0.031 2	4
	3	58	0.060 0.521 0.257 0.015 0.002 0.030 0.000 0.016 2	4
	4	50	0.020 0.358 0.000 0.020 0.000 0.041 0.000 0.017 2	4
	5	92	0.037 0.712 0.205 0.017 0.013 0.022 0.000 0.039 3	4
	6	82	0.013 0.640 0.000 0.022 0.014 0.028 0.000 0.048 3	4
	7	100 0.034 0.515 0.224 0.018 0.006 0.018 0.001 0.022 3	4
	8	88	0.010 0.376 0.000 0.023 0.004 0.023 0.002 0.027 3	4
	9	92	0.038 0.550 0.242 0.011 0.007 0.020 0.000 0.019 3	4
	10	80	0.013 0.408 0.000 0.015 0.005 0.026 0.000 0.023 3	4
	11	90	0.037 0.517 0.247 0.040 0.006 0.018 0.001 0.015 3	4
	12	78	0.011 0.360 0.000 0.053 0.004 0.024 0.002 0.018 3	4
	13	80	0.041 0.524 0.233 0.043 0.006 0.022 0.002 0.010 3	4
	14	70	0.012 0.380 0.000 0.056 0.005 0.028 0.002 0.014 3	4
	15	124 0.027 0.493 0.239 0.024 0.007 0.014 0.001 0.015 4	4
	16	108 0.008 0.335 0.000 0.032 0.005 0.018 0.001 0.019 4	4
	17	114 0.029 0.482 0.229 0.029 0.007 0.016 0.001 0.018 4	4
	18	100 0.009 0.329 0.000 0.037 0.006 0.020 0.001 0.022 4	4
	19	148 0.023 0.493 0.226 0.019 0.008 0.012 0.001 0.019 5	6
	20	130 0.007 0.346 0.000 0.025 0.007 0.016 0.001 0.024 5	6

Table 10 .

 10 The results for the first variant

	Problem NO	TC	Q(P) m	NV	NC
	1	20	10.9 158.5 4	3513 122608
	2	16	16.4 119.5 3	3188 72585
	3	10	14.4 143.5 4	3183 100419
	4	8	13.6 103	3	2858 56796
	5	125	10.6 136	4	3112 293550
	6	48	14.2 96.5	3	2772 206452
	7	100	16.9 159.5 4	4307 266213
	8	64	69.2 120.5 3	3977 153665
	9	50	24.8 144.5 4	3977 240794
	10	32	24.3 104	3	3647 137846
	11	40	20.2 159	4	4212 220392
	12	32	44.2 119.5 3	3887 119849
	13	40	40.1 148.5 4	3732 178124
	14	24	8.3	91	2	3402 103066
	15	200	29.2 160	4	5006 385227
	16	128	64.6 120.5 3	4676 200959
	17	200	25.2 160	4	4606 322529
	18	96	43.1 120	3	4271 173516
	19	1000 47.8 161	4	5422 536299
	20	384	76.6 121	3	5082 288161

Table 11 .

 11 The results for the second variant

	Problem NO	TC	Q(P) m	NV	NC
	1	45	48.8	147.5 4	3538 122093
	2	45	71.7	87	2	3218 72155
	3	27	95	138	4	3208 99944
	4	27	110.8 79.5 2	2888 56406
	5	405 128.6 133.5 4	3152 293090
	6	405 45.8	75.5 2	2832 206092
	7	405 95.6	148	4	4352 265598
	8	405 350.2 87.5 2	4032 153140
	9	243 202.3 138.5 4	4022 240219
	10	243 534	80	2	3702 137361
	11	135 78.6	148	4	4242 219782
	12	135 75.9	87	2	3922 119324
	13	75	157.7 148	4	3742 177714
	14	75	33.1	91	2	3422 102746
	15	1215 167.1 148.5 4	5056 384517
	16	1215 398	87.5 2	4736 200339
	17	675 62.6	152.5 4	4636 321939
	18	675 104.5 91.5 2	4316 173021
	19	6075 645.4 153	4	5472 535609
	20	6075 1152.2 92	2	5152 287571

Acknowledgements: This work was partially supported by the region Pays de la Loire (France).


Levy Batista

Mathieu Milhem

Thierry Bastogne

Fabien Clanché

Gabin Personeni

Jean-Philippe Jehl

Gérôme Gauchard

A data-driven classification solution for the timed-up and go test in risk falling assessment

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The present study addresses the issue of automated classification methods of healthy and risky subjects based on the Timed-Up and Go test. This test was introduced in 1991 by Podsiadlo and Richardson [START_REF] Podsiadlo | The timed "Up & Go": a test of basic functional mobility for frail elderly persons[END_REF] and since many other studies have proposed to improve its diagnostic performances. However, Barry et for each motion phase, and finally motion classification. The latter step was based on a logistic regression model using 12 motion features as discrimination variables. A total of 24 runs were carried out using a full factorial design of experiments.

III. RESULTS

For four different positions of the sensor, the logistic model was able to discriminate positive and negative runs with 100% of sensitivity and specificity, except for the use on the wrist. That position was not able to provide informative signals for the TUG test. The logistic model was only composed of two motion features: duration of the turn round and duration of the return phase. This pilot study has shown the proof of concept of an improved TUG test using inertial wearable sensors associated with a sensor fusion method and a logistic regression model. An automatic TUG motion segmentation was successfully tested and two discriminant motion features were identified. New experiments are ongoing on real patients to confirm the performances obtained with the age simulation suit.

  al. have shown in 2014 the poor performances of the current test[START_REF] Barry | Is the timed up and go test a useful predictor of risk of falls in community dwelling older adults: a systematic review and meta-analysis[END_REF] with a sensitivity of only 32%. In this study, our goal is to keep the simplicity of the test while improving its sensitivity performances by analyzing signals recorded during the tests by inertial wearable sensors.II. METHODSAn age simulation suit was used by four different subjects who performed the TUG test in triplicate with and without the simulation suit. Each subject was equipped with five wearable inertial sensors at five specific positions: neck, belt, right wrist, left and right foot. The inertial sensors used for the test were MetaMotionR sensors developed by Mbientlab Inc. This technology is composed of a BMI160 6-axis Accelerometer + Gyroscope, a BMM150 3-axis Magnetometer, a BOSCH 9-axis Sensor Fusion, a 8MB Flash Memory and a lithium-ion rechargeable battery. 9 signals (accelerometry, gyrometry and magnetometry) were measured during each run.The implemented method is decomposed as follows: time synchronization, axis recovery, pitch and roll estimation by a Kalman filter, followed by the yaw angle reconstruction, the probabilistic estimation of two statistics: direction and motion, the motion decomposition into four phases, feature extraction (duration, vertical accelerometry magnitude and frequency) *Research supported by MAIF Foundation. L Batista, M Milhem, and T. Bastogne are with CYBERnano, 193 avenue Paul Muller, 54180, Villers-lès-Nancy, FRANCE. T. Bastogne is with the Université de Lorraine, CNRS, CRAN, and the INRIA BIGS, F-54000 Nancy, FRANCE. J-P Jehl is with the Université de Lorraine, CNRS, IJL, F-54000 Nancy, FRANCE F Clanché, G. Personeni, and G Gauchard are with the Université de Lorraine, DevAH, F-54000 Nancy, France.

Figure 1 .

 1 Figure 1. Automatic TUG Motion Segmentation. 4 phases of the TUG test during a pilot trial. Black: vertical accelerometry. Red: yaw angle

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A data-driven classification solution for the Timed-Up and Go test in Risk Falling Assessment

Levy Batista, Mathieu Milhem, Thierry Bastogne, Fablien Clanché, Gabin Personeni, Jean-Philippe Jehl and Gérôme Gauchard