PROCEEDINGS OF THE 108 EUROPEAN STUDY GROUP WITH INDUSTRY ... · PROCEEDINGS OF THE 108 EUROPEAN STUDY GROUP WITH INDUSTRY (108 ESGI) The 108 ESGI was organized in Andalucía and

PROCEEDINGS OF THE

108 EUROPEAN STUDY GROUP WITHINDUSTRY (108 ESGI)

Sevilla, 17th-20th February 2015

Editors

Tomás Chacón RebolloUniversidad de Sevilla

[email protected]

Laureano Escuedero BuenoUniversidad Rey Juan Carlos

[email protected]

Carlos Parés MadroñalUniversidad de Málaga

[email protected]

Justo Puerto AlbandozUniversidad de Sevilla

[email protected]

PROCEEDINGS OF THE

108 EUROPEAN STUDY GROUP WITHINDUSTRY (108 ESGI)

The 108 ESGI was organized in Andalucía and it was held in collaborationbetween the Universities of Sevilla, Málaga and Almería and the SpanishNetwork for Mathematics and Industry (math-in). It was also fundedby the Mathematics Institute of the University of Sevilla (IMUS), theInternational Campus of Excellence Andalucía TECH and the ThematicNetwork RTmath-in, granted by the Spanish Ministry of Economy and Financewithin the dynamic actions "Networks of Excellence" at 2014 call.

Index

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Optimization in the design of the "Antesclusa" in the port ofSeville to minimize sedimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Ángel Pulido Hernández, Director of the Autoridad Portuaria de SevillaEnrique D. Fernández Nieto, University of Sevilla

Implementation of the seismic capacity of buildings . . . . . . . . . . . . . . . . . . 9

Diego Fernández, HABITECMaría Luisa Rapún Banzo, Technical University of Madrid

Production scheduling optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Tamara Borreguero, Airbus Defence and SpaceVíctor Blanco, University of Granada

Geolocation and positioning in sports training . . . . . . . . . . . . . . . . . . . . . . . . . 33

Carlos Padilla, RealTrack SystemsAndrei Martínez Finkelshtein and Fernando Reche Lorite, University ofAlmería

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Introduction

The 108 European Study Group with Industry (108 ESGI), held in Sevillafrom 17 -20th February 2015, was co-organized by the Mathematics Instituteof the University of Sevilla (IMUS) and the Spanish Network for Mathematicsand Industry (math-in) in collaboration with the Universities of Sevilla,Málaga and Almería.

Initiated in Oxford in 1968, Study Groups with Industry provide a forum forindustrial scientists to work alongside academic mathematicians on problemsof direct industrial relevance.

The success of the ESGI lies in its unique format which has been copiedaround the world, and which allows Mathematicians to work on reduced groupsto study problems presented by industry. These problems arise from anyeconomic sector thanks to the versatility of Mathematics.

The objective is to present the capabilities of Mathematics and itsapplicability in a large part of the challenges and needs that industry presents.It aims to bring small, medium and large companies a technology with greatpotential, used by highly qualified researchers and which does not require largeinvestments to use.

Therefore, collaboration between industry experts and researchers is keyto address technological innovation issues by using successful mathematicaltechniques. ESGI contributes to the promotion of mathematics and helpscompanies to use Mathematics to improve their processes.

The objectives set to achieve at the ESGI are:

• find solutions and insights into existing industrial problems;• establish lasting and productive working links between applied

mathematicians researchers and industry;• propose new lines of research based on business challenges;• reinforce the importance of mathematics in industry and the

incorporation of mathematics to companies; and• stimulate greater awareness in the wider community of the power of

mathematics in providing solution paths to real-world problems.

Finally, it should be pointed out that 56 researchers, students, professorsand company technicians contributed to a successful 108 ESGI.

Seville on 20th February, 2015

Members of the Scientific Committee:- Tomás Chacón Rebollo. Department of Differential Equations and

Numerical Analysis, Universidad de Sevilla

- Laureano Escudero Bueno. Universidad Rey Juan Carlos and memberof the Management Board of the Spanish Network for Mathematics andIndustry (math-in)

- Carlos Parés Madroñal. Department of Mathematical Analysis(Universidad de Málaga) and Vice-president of the Spanish Network forMathematics and Industry (math-in)

- Justo Puerto Albandoz. Department of Statistics and OperationsResearch, Universidad de Sevilla

Optimization in the design of the "Antesclusa" inthe port of Seville to minimize sedimentation

Academic Coordinator Enrique D. Fernández NietoUniversity University of Sevilla

Business Coordinator Ángel Pulido HernándezCompany Director of the Autoridad Portuaria de Sevilla

Specialist María Carmen Molina González, consultant in river studies.Team: Hortensia Almaguer (Universidad Juárez Autónoma deTabasco),Patricio Bohórquez (Universidad de Jaén),Luca Bonaventura(Politécnico di Milano), Manuel J. Castro (Universidad de Málaga), TomásChacón (Universidad de Sevilla), Enrique Delgado (Universidad de Sevilla),Daniel Franco (Universidad de Sevilla), Macarena Gómez (Universidad deSevilla), Marino González (Universidad de Sevilla), Francisco Guillén(Universidad de Sevilla), Juan Roberto Hernández (Universidad JuárezAutónoma de Tabasco), Tomás Morales (Universidad de Córdoba), GladysNarbona (Universidad de Sevilla), Carlos Parés (Universidad de Málaga), M.Ángeles Rodríguez (Universidad de Sevilla), Samuele Rubino (Universidad deSevilla).

Problem Description: Access to the port of Seville is via the route E.60.02Eurovia Navigable Guadalquivir, registered in the riverbed, getting to theport facilities through a lock. Very high levels of sedimentation appears in thearea antesclusa because of, among other reasons, to the low velocity of waterin this area, due to the characteristics of geometry and connection to themain channel, and the high load suspended sediment transported by the river.This implies the need for major dredging. The goal was to propose solutionsto reduce levels of sedimentation in the antesclusa the port of Seville.(Paper not available)

Implementation of the seismic capacity of buildings

Academic Coordinator María Luisa Rapún BanzoUniversity Universidad Politécnica de Madrid

Business CoordinatorDiego FernándezCompany Project manager of HABITECTeam Victor Compán (Universidad de Sevilla), Bosco García–Archilla(Universidad de Sevilla), David González (Universidad de Zaragoza), CarlosParés (Universidad de Málaga), María–Luisa Rapún (Universidad Politécnicade Madrid), Enrique Vázquez (Universidad de Sevilla)

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Implementation of the seismic capacity ofbuildings

Victor Compán∗, Bosco García–Archilla†, David González‡,Carlos Parés§, María–Luisa Rapún¶, Enrique Vázquez∥

Abstract

In this work we deal with the selection of a suitable (fast and accurate)numerical method for the computation of the maximum displacementof a building when a seismic acceleration at its base is considered.The problem was proposed by Habitec, a Spanish non profit privatefoundation, during the ESGI–108.

Keywords | Seismic Analysis of Buildings; Seismic Resistance Code; SeismicQualification of buildings.

AMS classification | 86A15, 86A17

1. Introduction

Seismic codes in building design are codes to protect the life, health andproperties of people in case of earthquakes. The current seismic code in Spainis the “Norma de Construcción Sismorresistente” (NCSE–02 in short), that waspublished in 2002, see Ref. [9].

The NCSE–02 establishes the modal response spectrum analysis as thereference method for the seismic analysis of structures. It also provides asimplified method that can be used in buildings that satisfy some requirements(geometrical regularity, mechanical regularity, ...). The challenges proposed byHabitec deal with the cases when the application of the simplified method isnot possible.

∗[email protected]†[email protected]‡[email protected]§[email protected]¶[email protected]∥[email protected]

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108 Study Group with Industry (108 ESGI)

The paper is organized as follows. We briefly describe the responsespectrum method and its simplification in Sections 2 and 3, respectively.Section 4 is devoted to the description of the challenges proposed by Habitec.Our recommendations are summarized in Section 4. The paper ends with ourconclusions and possible ideas for future research.

2. Response spectrum method

The response of a structure can be approximated as a linear combinationof vibration modes. The response spectrum method for the prediction ofdisplacements in seismic analysis involves the calculation of only the maximumvalues of the displacements in each mode of vibration using smooth designspectra that are the average of several earthquake motions.

The recommended method in the NCSE–02 code is as follows (see Refs. [1,9]). Consider a discrete model with n degrees of freedom where the equationof motion is discretized as

[M ]{u(t)}+ [C]{u(t)}+ [K]{u(t)} = {f(t)}. (1)

Here {u(t)} ∈ Rn is the displacement vector at the time instant t; [M ], [C], [K]are n×n matrices: the mass, damping and stiffness matrices, respectively; andthe vector {f(t)} models the resultant of the external forces. Then, the modes{ϕi}, i = 1, . . . , n, and their natural frequencies ωi, i = 1, . . . , n, are thesolutions of the generalized eigenvalue problem

ω2[M ]{ϕ} = [K]{ϕ}. (2)

For the numerical solution of problem (2) we refer to Chapters 7 and 8 ofRef. [4].

It can be proved that the set of modes {{ϕi}, i = 1, . . . , n} is a basisof Rn, and therefore, the displacement vector can be decomposed as a linearcombination of such modes:

{u(t)} =

n∑i=1

ζi(t){ϕi}. (3)

Furthermore, modes associated to the lowest frequencies contain more elasticdeformation energy (thus conditioning the response of the system to a largeextent) while modes associated to the highest frequencies are prone to addnumerical errors. For these reasons, only the first r modes are usually kept inthe expansion (3). The NCSE–02 code lays down that the number r can betaken as the smallest number such that "the sum of the effective mass of thefirst r modes exceeds the 90% of the mass that is moved by the earthquake

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motion". Mathematically, this condition can be written asr∑

i=1

τ2i {ϕi}⊤[M ]{ϕi} ≥ 0.90{J}⊤[M ]{J},

where {J} is the influence vector (its components are the rigid bodydisplacements of the degrees of freedom of the structure when its baseexperiences a unit displacement in the direction of the earthquake motion),and τi = {ϕi}⊤[M ]{J}/{ϕi}⊤[M ]{ϕi} is the participation coefficient of thei–th mode.

Once the number r is selected, the maximum displacement is estimated as

{umax} =r∑

i=1

α(Ti)τiω2i

{ϕi},

where α(Ti) is the value of the response spectrum α at the vibration periodTi associated to the i–th mode, that is, Ti = 2π/ωi. In Fig. 1 we include anormalized elastic response spectrum α extracted from the NCSE–02 code.

Figure 1: Elastic response spectrum, see Ref. [9].

3. Simplified model

The NCSE–02 code contains a simplified method that can be applied when thefollowing requirements are satisfied (see Section 3.5.1 of Ref. [9]):

1. The number of floors is less than 20.2. The height of the building is less than 60m.3. The floor layout and the elevation are regular.4. Supports are continuous, uniformly distributed and do not have abrupt

stiffness changes.5. The distribution of mass, stiffness and resistance presents mechanical

regularity, in such a way that the centers of gravity and torsion in all thefloors are approximately in the same vertical line.

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6. The eccentricity between the centers of mass and torsion is less than a10% of the dimension of the floor layout.

This simplification avoids the computation of the vibration modes and ofthe associated frequencies. That is, one does not have to solve the eigenvalueproblem (2).

The vibration period Ti (Ti = 2π/ωi) associated to the i–th mode, isestimated as

Ti = TF /(2i− 1),

where TF is the fundamental period of the structure, and is computed usingvery simple formulae involving the height of the building (H), the numberof floors (n),... For instance, for a reinforced concrete framed buildingwithout stiffening walls, TF = 0.09n, while in presence of stiffening walls,TF = 0.07n

√H/(B +H), where B is the dimension of the stiffening wall. For

other structures we refer to Section 3.7.2.2 of Ref. [9].The number r of selected modes is also estimated in terms of TF : r = 1 if

TF ≤ 0.75, r = 2 if 0.75 < TF ≤ 1.25, and r = 3 if TF > 1.25. For example, fora reinforced concrete framed building without stiffening walls, if the numberof floors is up to eight, then r = 1.

Finally, the horizontal displacement is computed as a product of severalterms that are either coefficients (that are provided in the NCSE–02 code,like the ductility coefficient), or simple expressions involving those coefficientsand the parameters determining the structure (that are straightforward toevaluate). For instance, one has to compute a linear combination of functionsof the form

sin((2i− 1)πhk/(2H)), (4)

where hk is the height of the k–th floor and H is the total height of thestructure. In particular, the use of the functions (4) avoids the computationof the vibration modes.

In few words, the simplified model provides very simple formulae that canbe evaluated with a standard calculator in one or two minutes, or in one secondusing a computer program.

4. Challenges description

Habitec is interested in estimating the maximum displacement of a buildingfrom the knowledge of the seismic acceleration at its base in view to implementa software of seismic qualification. Habitec would like to implement a very fastmethod (in terms of computational time) able to:

(i) Appraise existing buildings that were built before the study of theseismic risk were compulsory in Spain, that is, to evaluate the seismicvulnerability and potential seismic risk level of existing buildings thatwere not constructed under the consideration of any seismic code.

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(ii) Guarantee that projects of new buildings verify the NCSE–02 code.In case the (existing or designed) building verifies the requirements to apply

the simplified method, this method provides a very fast estimation of themaximum displacement. In principle, this is the method that Habitec hasselected to be implemented. The challenges proposed by Habitec deal with thecase of non applicability of the method.

4.1. First challenge

Habitec would like to generalize the simplified model to the case of buildingsthat do not satisfy the third requirement: "the floor layout and the elevationare regular", and especially when the the sixth requirement "the eccentricitybetween the centers of mass and torsion is less than a 10% of the dimension ofthe floor layout" fails. Fig. 2 gives examples of these lacks of regularity.

Figure 2: Two situations that do not satisfy the requirements to apply thesimplified model.

4.2. Second challenge

In case of dealing with highly irregular buildings, where the simplified model isby no means applicable, Habitec would like to be recommended the preferablemethod in terms of precision and time speed to solve problem (1).

5. Recommendations

In this section we summarize our recommendations for the two proposedproblems.

5.1. First challenge

When dealing with buildings where the eccentricity e between the centers ofmass and torsion is higher than a 10% of the dimension of the floor layout, it

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could be possible to transform the problem in an equivalent one, considering anew structure satisfying the requirements of the NCSE–02 code to apply thesimplified method. In that case, the maximum displacement of the originalbuilding could be calculated from the maximum displacement of the newstructure, corrected by the value of e. However, we think that it is not worth totry to generalize the simplified model and we recommend to apply always theresponse spectrum method (even when the NCSE–02 code allows to use thesimplified model). Our main reasons to discard the idea of such generalizationare the following:

1. The new method cannot be at the same time universal and very simple.Therefore, it would be restricted to a certain type of buildings, with acertain rage of eccentricity (for example, with an eccentricity between a10% and a 15% of the floor layout).

2. The design of the generalized method involves a detailed study of thebehavior of such class of structures, and several coefficients have to beexperimentally adjusted.

3. Our main reservation about the design of a generalized method is thatwe strongly believe that any generalization is going to be inaccurate,because the original simplified method is inaccurate.

4. Nowadays computing the maximum displacement of a conventionalbuilding is computationally inexpensive (a few seconds–one minute ofcomputational time) using the response spectrum method. Besides, thisis the recommended method in the NCSE–02 code.

Figure 3: Structure of the considered buildings.

To illustrate the inaccuracy of the simplified model, we have considered twofour-floor reinforced concrete framed buildings without stiffening walls locatedin Seville (Spain). Their structure is represented in Fig. 3. The coefficientsand parameters that characterize both structures are given in Fig. 4. Theonly difference between the buildings is that for the first one the pillars are ofsize 50cm × 50cm and the beams of size 30cm × 50cm, while for the secondone, the size of both the pillars and the beams is 30cm × 30cm. Whenapplying the response spectrum method, we find that in both buildings the

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Figure 4: Seismic loads. The definition of the parameters is given in theNCSE-02 code.

number of required modes to estimate the maximum displacement is 5. Themaximum displacement of the first building is 1.607cm, while for the secondone is 10.444cm.

The application of the simplified model to the two buildings involves theuse of just one mode, providing in both cases a maximum displacement of1.7cm. Notice that the simplified model does not take into account some ofthe characteristics of the buildings. In particular, it does not take into accountthe size of the pillars and beams. In comparison with the response spectrummethod, we observe that for the first building, the estimate is reasonable,while for the second one, the simplified method completely underestimates themaximum displacement.

Remark. We want to emphasize that the NCSE–02 code was not createdfor appraising existing buildings. Furthermore, the simplified methodwas conceived to estimate the contribution of a possible earthquake inthe equivalent resultant of external forces, not to estimate the maximumdisplacement of a building. In addition, the response spectrum method isthe recommended method in the NCSE–02 code, even when it allows the useof the simplified model.

5.2. Second challenge

In the NCSE–02 code, the study of the seismic risk level involves the useof the elastic response spectrum (see Sect. 1.1 and Fig. 1). For highlyirregular buildings, the simplified method does not apply. Therefore, to testif an (existing or prospected) irregular building satisfies the NCSE–02 codeit is compulsory to compute the modes and the natural frequencies of thesystem. Such computation is the most time consuming part of the processand it is unavoidable. Once the modes and the frequencies are computed, the

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application of the response spectrum method is straightforward (in terms ofcomputational complexity) and immediate (in terms of computational time).In consequence, we recommend the application of the response spectrummethod for highly irregular buildings.

6. Conclusions

With the current computational resources, the use of the response spectrummethod for conventional buildings only involves seconds or at most a fewminutes in a standard PC. Furthermore, the method is the preferred in theNCSE–02 code. Therefore, we think that it makes no sense to try to find asimplified version of it in case of dealing with slightly irregular structures, orto apply a different method in case of dealing with highly irregular structures.

Figure 5: An application for tablets.

In order to apply this method for assessment of existing buildings, anapplication for tablets or smartphones could be of interest. We think thatthe architect would benefit from having an application where he only has tosketch the building design and to introduce the relevant data of it (number offloors, height of the floors, size of pillars,...) to obtain the maximum expecteddisplacement when applying the response spectrum method, see Fig. 5 foran illustration. We think that current tablets would be able to provide theresult in a few minutes when dealing with standard buildings (more than the99% residential buildings in Spain). Furthermore, in order to avoid onlinecomputations, that is, to solve the problem once the data are introduced, itis also possible to generate previously (offline) a multiparametric database(a response surface). This means that the designers of the application canselect the most common cases to be considered (buildings with up to 10floors, typical sizes of beams and pillars, etc.), and solve the correspondingproblems in a cluster to save all the results in a multiparametric database.

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The computational time required to generate such database could be huge(probably months). But once the database is created, from the introduceddata one can recover the maximum displacement in a few seconds, since theproblem has been previously solved, and the response is obtained from thedatabase.

We think that the design of the application for tablets that solve theproblem online is feasible and easy. The design of the more sophisticatedapplication that searches the solution in a previously generated databaserequires a more careful study. In this direction, we point out that for anoptimal selection of parameters in terms of obtaining a robust but not hugedatabase it would be of interest to consider Taguchi’s methods, see Refs. [8, 14].To optimize the memory requirements to save the database, the high ordersingular value decomposition is highly recommended, see Refs. [4, 6].

Finally, to end this section we want to give some ideas dealing with thesolution of the problem

[M ]{u(t)}+ [C]{u(t)}+ [K]{u(t)} = {f(t)}, (5)

for a given accelerogram. Notice that the use of the elastic response spectrumis defined from the average of several earthquake motions, and in the end,one only solves the stationary problem (2). In case one wants to solve thetime–dependent problem (5) for a given accelerogram in a time interval t1 ≤t ≤ t2, then it is not necessary to use the response spectrum method. Forconventional buildings we still recommend the use of the mode decompositiondescribed in the response spectrum method. Modes form a basis of Rn thatis M–orthogonal, K–orthogonal and C–orthogonal, see Ref. [1]. Then, we canrepresent u as a linear combination of the first r modes

{u(t)} =

r∑i=1

ζi(t){ϕi},

introduce this expression in (5), and multiply by the vector {ϕi}⊤ to find asystem of r uncoupled equations for the coordinates ζi(t):

miζi(t) + ciζi(t) + kiζi(t) = fi(t), i = 1, . . . , r,

where

mi := {ϕi}⊤[M ]{ϕi}, ci := {ϕi}⊤[C]{ϕi},ki := {ϕi}⊤[K]{ϕi}, fi(t) := {ϕi}⊤{f(t)}.

For non–conventional buildings, where a huge number of degrees offreedom is involved, the previous method could very time–demanding. Somealternatives can be explored, specially, the use of reduced order models forthe acceleration of the time integration of the equations. To this end, instead

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of using vibration modes, some other modes could be of interest, based onProper Generalized Decompositions (see Refs. [2, 3, 5]), Dynamical ModeDecompositions (see Ref. [13]), adaptive methods based on Proper OrthogonalDecomposition (see Refs. [11, 12])... Furthermore, domain decompositiontechniques to parallelize computations would also decrease the computationaltime (see Refs. [7, 10]).

References

[1] J.M. Canet, A.H. Barbat, Estructuras sometidas a acciones sísmicas.Ed. Centro Internacional de Métodos Numéricos en Ingeniería, Barcelona(1994).

[2] F. Chinesta, P. Ladeveze, E. Cueto. A short review on model orderreduction based on proper generalized decomposition. Arch. Comput.Methods Eng. 18 (2011), 395-404.

[3] F. Chinesta, E. Cueto. PGD–based modeling of materials, structures andprocesses. Springer International Publishing, Switzerland (2014)

[4] G.H. Golub, C.F. Van Loan, Matrix Computations. Fourth edition. JohnsHopkins Studies in the Mathematical Sciences. Johns Hopkins UniversitiyPress, Baltimore, M.D. (2013).

[5] D. Gonzalez, E. Cueto, F. Chinesta. Real–time direct integration of reducedsolid dynamics equations. International Journal for Numerical Methods inEngineering 99 (2014) 633–653.

[6] L. de Lathauwer, B. de Moor, J. Vanderwalle, A multilinear singular valuedecomposition. SIAM J. Matrix Anal. Appl. 21 (2000), 1253–1278.

[7] P. le Tallec. Domain decomposition methods in computational mechanics.Computational Mechanics Advances 1 (1994), 121–220.

[8] V.N. Nair, B. Abraham et al, Taguchi parameter design – a paneldiscussion, Technometrics 34 (1992), 127–161.

[9] Norma de Construcción Sismorresistente: Parte general y edificación(NCSE-02). http://www.fomento.gob.es/MFOM.CP.Web/handlers/pdfhandler.ashx?idpub=BN0222

[10] A. Quarteroni, A. Valli. Domain decomposition methods for partialdifferential equations. OUP Oxford, (1999)

[11] M.L. Rapún, F. Terragni, J.M. Vega. Adaptive POD–based lowdimensional modeling supported by residual estimates. Int. J. Numer.Math. Engng. 104 (2015), 844–868.

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[12] M.L. Rapún, J.M. Vega. Reduced order models based on local PODplus Galerkin projection. Journal of Computational Physics 229 (2010),3046–3063.

[13] P.J. Schmidt, Dynamic mode decomposition of numerical and experimentaldata. J. Fluid Mech. 656 (2010), 5–28.

[14] K.L. Tsui, An overview of Taguchi method and newly developed statisticalmethods for robust design, IIE Transactions 24 (1992), 44–57.

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Production scheduling optimization

Academic Coordinator Víctor Blanco IzquierdoUniversity Universidad de Granada

Business Coordinator Tamara BorregueroCompany Airbus Defence and SpaceTeam Manuel Arana (Universidad de Cádiz), Eduardo Conde (Universidadde Sevilla), Alfredo García (Universidad de Sevilla), Rocío González(Universidad de Sevilla), Yolanda Hinojosa (Universidad de Sevilla), MarianoLuque (Universidad de Málaga), Juan Miguel Mendoza Moreno (Universidadde Sevilla), Manuel Muñoz (Universidad de Cádiz), Miguel Ángel Olivero(Universidad de Sevilla), Diego Ponce (Universidad de Sevilla), Justo Puerto(Universidad de Sevilla), Nicolás Robayo (Universidad de Sevilla), JuliánSierra (Universidad de Sevilla).

Figure 6: Part of the P3 Team during the event.

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24 |


Víctor Blanco ∗

Abstract

Airbus manufactures several aircraft aerostructures in different plants.The production process of each of those articles assumes a specificdemand (with associated production rate), assembling line sharing,a maximum number of workers, a fixed deadline, etc. Hence, it is acomplicated task that may need automatization and some optimalityanalysis by means of minimizing production costs, manufacturing timeand stockage control. In this report we reflect the work of the groupon this problem during the 108th ESGI held in Seville in February2015.

Keywords | scheduling, mathematical programming, production planning.

AMS classification 2010 | 90B35, 90C90, 90B70

1. Introduction

Airbus Defence and Space is a division of Airbus Group formed by combiningthe business activities of Cassidian, Astrium and Airbus Military. Thenew division is Europe’s number one defence and space enterprise, thesecond largest space business worldwide and among the top ten globaldefence enterprises. It employs some 40,000 employees generating revenuesof approximately 14 billion euros per year.

Airbus manufactures some of the aircraft aerostructures of its airplanesin different production plants through the Spanish geography. In particular,this report concerns the Tablada (Sevilla) production plant. The high volumeof demand of the products, the high cost of the workers, the length of theproduction processes and the assumption of minimum stockage in the plantmakes necessary to carefully planning the production of the articles in theplant.

∗[email protected]

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In the plant, four different articles are produced (two different articles, butfor each of the two sides of the plane- left and right). Since each pair of articles(left and right side) are actually both necessary for the plane (and then, to theclient), a maximum and minimum delay is also fixed between the completions ofthe two sides of each of the two articles. To complete the production of a singlearticle, several (known) tasks must be performed at different platforms of theplant and with different maximum available number of workers. Furthermore,some of the products share platforms/workers for some of the tasks in theproductions and some tasks for some of the products are preformed in exclusiveplatforms. Hence, since the average times of completion of the tasks aresomehow reliable and almost fixed, a delay queue is created increasing the totalproduction times of the articles. Although one may think that if the uniquegoal is to shorten the overall production time one may increase as necessarythe workers in the platform, a maximum capacity of workers is allowed ateach platform. Moreover, the number of workers wants also to be minimizedat each of the working times (two different working periods: 7am-3pm and3pm-11pm) although some of the tasks may be done during the night withno workers (resting tasks). Hence, the planning of such a production processis not trivial. At this moment, the company is making the planning basedon the previous experience by filling a Gantt chart [1] that allows to correctinterceptions between tasks/platforms/product at each of the working periods.Such a by-hand planning is performed trying to parallelize the production ofthe two sides of the same article to assure the minimum delay between thecompletion of the two products, when possible, although it does not assurethe best planning in terms of completion time or number of workers. Thisproblem was submitted to the the 108th European Study Group in Industryheld in Sevilla in February 2015. This report reflects the work of the groupduring those days on this problem.

This problem has many common elements with the so-called Job ShopScheduling Problem (JSSP). Given a set of independent jobs, each having itsown processing order through a set of machines. Each job has an orderedset of operations, each of which must be processed on a predefined machine.JSSP consists of sequencing operations on the machines so that the maximumcompletion time over all jobs is minimized. Considerable research has beendevoted to this problem in the literature (see [2]). However, JSSP does notconsider many of the peculiarities of the Airbus planning problem, so severalmodification must be done to cast the actual scheduling problem.

This report is organized in four sections. The first section is theIntroduction. In Section 2 we detail the problem. A first mathematicalprogramming problem is presented in Section 3, both to try to solve theproblem but also to understand its nature and difficulties. Finally, someconclusions are drawn in Section 4.

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2. Challenge description

Two different articles are produced (1 and 2) in the plant and for each of thema right and a left side must be produced. Hence, initially, four products areconsidered:• Product 1 – Left Side (L1)• Product 1 – Right Side (R1)• Product 2 – Left Side (L2)• Product 2 – Right Side (R2)Actually, not only four articles are manufactured but several copies of each

of them through the time horizon, so the articles to be produced are in theform:

L1, . . . , L1, R1, . . . , R1, L2, . . . , L2, R2, . . . , R2.

for a given number of copies for each of them.For each of the articles, ten tasks may be performed to complete its

production. This sequence is fixed and known:

[->,>=stealth’,shorten >=0.1pt,auto,node distance=1.1cm, thick, scale=0.5]every state=[color=black, text=blue, minimum size=0.6cm, inner sep=0pt][state] (A) 1; [state] (B) [right of=A] 2; [state] (C) [right of=B] 3; [state] (D)[right of=C] 4; [state] (E) [right of=D] 5; [state] (F) [right of=E] 6; [state] (G)[right of=F] 7; [state] (H) [right of=G] 8; [state] (I) [right of=H] 9; [state] (J)

[right of=I] 10;A)edgeB); B)edgeC) ; C)edgeD); D)edgeE); E)edgeF); F)edgeG); G)edgeH);

H)edgeI); I)edgeJ);

Each of the 10 above tasks are known to be performed in a platform ora set of possible platforms as detailed in Figure 1 provided by Airbus. Froma first observation, one may consider that the platforms that allow differentsimultaneous tasks can be duplicated in order to allow only one task perplatform and in the mathematical programming model one can impose whichplatforms are available to perform each of the tasks.

The capacity of each platform for each of the products is known as well asthe maximum number of workers. The processing time for worker at each ofthe stages is also known for each article.

Hence, the goal is to find a production planning (for a given time horizon)by:• Mínimizing the number of workers by time period.• Minimizing the overall completion time.• Minimizing the stockage of articles.

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Figure 1: Platforms and tasks in the Tablada plant.

The main differences between the problem and JSSP are: not onlycompletion times must be taken into account but also the number of workers,the platforms have different behavior with respect to different products sincesome of the platforms allow simultaneous processing, and other not, some ofthe tasks do not need workers but the processing times must be taken intoaccount to compute the overall completion time (in some cases is convenientto perform those tasks in the night but, in general, not allways).

3. A mathematical programming model

We propose a mathematical programming model to deal with this problem. Westate the optimization problem for any number of articles, tasks, time horizon,etc, in order to be easily adaptable to different scenarios or situations. Theparameters of the problem are:• Aircraft aerostructures to be produced: I0 = {1, . . . , n}. A given number

of items is produced of each type of article. Hence, the goal is to producethe articles I = {1, . . . ,m1} ∪ {m1 + 1, . . . ,m1 + m2} ∪ {m1 + m2 +1, . . . ,m1 +m2 +m3} . . .. mi is the number of copies for product i ∈ I0.• J set of tasks that must be processed for the completion of an article.

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• K set of machines/platforms where the tasks are processed. In practice,we assume that each machine is operated by a single worker. Thosemachine that allows more workers will be duplicated accordingly.• Oi: operations needed for the completion of product i ∈ I.• Mj : possible platforms for task j ∈ J .• tijk: processing times of task j ∈ J in machine k ∈ K for product i ∈ I.• Minimum and maximum delay between the completion times of

consecutive productions of left and right sides of article i: d−i , d+i , for

i ∈ I.• Working periods (workers): [u1r, u2r] = [24r, 24r + 16], r = 0, . . ..• Night periods: [v1r, v2r] = [24r + 16, 24r + 24], r = 0, . . ., when only

repose tasks are allowed.The variables of our problem are:• Continuous Variables:

– Sijk: Starting time of task j ∈ Oi in machine k ∈ Mj for articlei ∈ I.

– Cijk: Finalization time of task j ∈ Oi in machine k ∈Mj for articlei ∈ I.

• Binary Variables:– Xijk = 1 if task j ∈ Oi for product i ∈ I is processed in machinek ∈Mj ; 0, otherwise.

– Yiji′j′k = 1 if in machine k ∈ Mj ∩Mj′ the task j ∈ Oi of articlei ∈ I is processed before the task j0 ∈ Oi′ of product i′ ∈ I.

– δijkl = 1 if of task j ∈ Oi in machine k ∈ Mj for article i ∈ I isprocessed in the working day l (for tasks that need workers).

– γijkl = 1 if of task j ∈ Oi in machine k ∈ Mj for article i ∈ I isprocessed the lth night (for tasks that do nor need workers).

First Goal: Minimize the maximum number of tasks (∼ workers) in all theplatforms:

min maxl

∑ijk

δijkl

The variables above are related using the constraints that may be imposedto the problems which are:• Each operation is processed in an unique platform.• Starting times are zero if the machine is not chosen to perform the task.• The completion time of each tasks is the starting time plus the processing

time.• Each pair of tasks processed in the same machine must be sorted (not

processed in parallel).• Task j precedes task j + 1, for j ∈ J .• The completion time of each product coincides with the completion time

of the last task of such an item.

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• The difference of completion times of a left side and consecutive rightsice must be in the given interval.• Some task must be processed during the day and some others are allowed

to be processed during the nights.• Adequate definition of variables: nonnegativity of times, binary variables,

etc.These “requirements” are allowed to be written as a mixed integer linear

programming (MINLP) problem solvable in any commercial software for smallto medium sizes (in terms of articles, tasks and machines). As a first attempt,it was implemented in XPRESS-FICO.

4. Conclusions

During the 108th European Study Group, the participants analyzed aninteresting optimization problem that consist of finding an optimal productionplanning for the production of a given set of articles through a fixed timehorizon with respect to the maximum number of workers, the minimum overallcompletion time and the minimum stockage. The problem has been initiallymodeled a mixed integer linear programming problem. The size of the proposedmodel for a toy simulated problem with only two products (product 1 withside right and left) and 3 copies for each of them and that need the 10 tasksto be completed and using 6 platforms (15 machines) require 6355 variablesand 14154 constraints. For 5 copies of the 4 actual products with the sameoperations but 21 machines require 48101 variables and 126672 constraints.Since the number of variables and constraint are large and increase considerablywith the number of product/machines/operations, although solvable in anycommercial software, to quickly solve large size problem a further analysisis needed. In particular, reducing the variables of the problem seems to bepossible by re-modelling the problem by discretizing the time as only 8-hourworking times per worker are allowed each day. Secondly, we may analyzeapproximative approaches that allows to solve the problem at the price ofpossible loss quality (optimality) of the solutions. One of the possible optionsis the use of metaheuristic algorithms. For the sake of implementing this typeof approaches some items must be analyzed carefully: the adequate codificationof the solutions, define mutations of solutions to provide improvement betweenthem, control the feasibility of the considered solutions and define the correctobjective function to be optimized with the process.

Moreover, this problem belongs by definition to the class of optimizationproblems known as multiobjective optimization problems due to the need tomaximize/minimize several objectives at the same time. Thus, tools frommultiobjective optimization may be taken into account to deal with usefulsolutions to this problem.

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5. Acknowledgements

All contributors would like to thank Tamara Borreguero from Airbus Space& Defence for introducing the problem, answering questions and participateactively during the event.

References

[1] Gantt, H.L., A graphical daily balance in manufacture. ASMETransactions 24 (1903), 1322–1336

[2] Jain, S.S., Meeran, S., Deterministic job-shop scheduling: past, presentand future, Eur. J. Oper. Res. 113 (1999), 390–434.

[3] C. Özgüven, L. Özbakir, Y. Yavuz, Mathematical models for job-shopscheduling problems with routing and process plan flexibility, AppliedMathematical Modelling 34 (6) (2010), 1539–1548.

[4] PAN, C.H., A study of integer programming formulations for schedulingproblems, International Journal of Systems Science, 28:1 (1997), 33–41.

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Geolocation and positioning in sports training

Academic Coordinator Andrei Martínez Finkelshtein and Fernando RecheLoriteUniversity Universidad de Almería

Business Coordinator Carlos PadillaCompany RealTrack Systems

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Geolocation and positioning in sportstraining

Andrei Martínez Finkelshtein ∗ Fernando Reche Lorite †

1. Introduction

Our team worked on some problems put forward by RealTrack Systems, atechnological company from Almería, a mid-size town in the South-East ofSpain.

This company has been developing a small wireless device called WIMU(Figure 1) which is able to supply a large amount of real-time data aboutphysical activity.

Figure 1: WIMU device

Although WIMU is very versatile, the main current target of this companyare football (soccer) clubs, athletes and medical teams. WIMU is a source ofa valuable information about training sessions, vital signs, etcetera, which canimprove their assessment of the training sessions and sports activities. In somesituations, this information was only available so far by expensive devices or inspecialized laboratories. Hence, WIMU is an affordable solution for a real-timeinformation.

WIMU contains in its interior several sensors supplying data onsome kinematic variables (velocity, acceleration, distance) as well as somephysiological variables (heart rate), with different frequencies. This wealth ofinformation on each variable is conveniently managed using QÜIKO, a softwarealso developed by RealTrack Systems, which is able to show dynamically theresults in a graphical interface.

Among the sensors included in WIMU are:∗[email protected]†[email protected]

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Figure 2: QÜIKO output

• GPS/Galileo, providing data at 5 Hz.• Accelerometer 3D, up to 2G, providing data at 1000 Hz.• Accelerometer 3D, up to 8G, providing data at 1000 Hz.• Gyroscope 3D, with precision of 2000/s and 440/s, providing data at 140

Hz.• Magnetometer 3D, up to 4 Gauss, providing data at 50 Hz.• Barometer, up to 120 kPa, providing data at 9 Hz.

2. Problems

Two main questions have been raised by RealTrack Systems during theintroductory meeting: the first one considered WIMU as an isolated device,and the second one deals with the situation when several WIMU ’s are usedwith a group of athletes. For the sake of brevity, we will refer to them as theIndividual WIMU problem and the Group WIMU problem, respectively.• Problem 1: Individual WIMU

– Merge data supplied by the inertial sensors of WIMU with dataprovided by the built-in GPS. In particular, correct the sensors’error using the information supplied by the GPS.

• Problem 2: Group WIMU– Define measures of “harmony” in sports activity in order to study

possible patterns in training sessions. This information couldimprove the efficiency and suggest variations in activities.

– Discover collective attitudes and cooperation patterns in orderto control automatically the teamwork during training sessions.Although nowadays the use of WIMU and analogous devices duringfootball matches is forbidden, the situation might change in a near

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future, when we will be able to use this device to supply relevantinformation during competition.

3. Possible solutions and lines of research

3.1. Problem 1: Individual WIMU

The team worked in several lines of research studying this problem. Wehave tried to focus on mixing information concepts to merge information fromdifferent sources and types of sensors [5].

The first option was filtering signals by means of the Kalman filters [2].After that we will make a weighted mix of information supplied by GPS andWIMU device depending on the “credibility” in each moment. For example, ifwe are in a sport hall, GPS signal in less reliable than WIMU sensors. In thiscase the weight of WIMU data is bigger than GPS data.

The second option explored during the sessions was the use of fuzzy neuralnetwork methodology [1, 6, 7]. Fuzziness is a concept suitable to modelimprecise information. With fuzzy neural networks we try to make a dynamiclearning process setting a threshold depending on the GPS precision and using“fuzzification” and “defuzzification” procedures.

Finally, we proposed other methodologies like Bayesian networks [3] asclassifiers: Näive Bayes, Tree Augmented Näive Bayes with continuous models[4]. Bayesian networks supply us a powerful tool which can be applied in severalframeworks. In this problem the can be used as classification tool setting themost probable situation based on data.

3.2. Problem 2: Group WIMU

In order to tackle this problem we needed first to understand the concept of“harmony”. This task was not easy because it is a very personal concept, butwe focused on football and we tried to explore “harmony” in this popular sport,with the purpose to extend it later to other settings.

With this goal in mind we had an intense meeting with a football coachand an engineer from RealTrack Systems, trying to find out how the trainingsessions are organized and what kind of behavior they consider as “harmonious”,seeking for some common patterns. Clearly, a general definition of “harmony”was beyond the reach of this meeting, we were able to define some parametersor markers of harmonic collective behavior in football. We hope that thesemeasures could be used in training sessions or in competitions.

The direct collaboration and participation of the RealTrack Systems wascrucial in this stage, since the makers were implemented and tested almost inreal time.

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Some of the parameters suggested during our work sessions, computableby means of the sensors available in WIMU, were:• Convex hull (see Figure 3):

– Compactness index:P 2

A

where P is the perimeter and A the area of the convex hull of theplayers.

– Alignment index (normalized):

0 ≤ 4πA

P 2≤ 1.

Figure 3: Convex hull in a football team trainig session

• Centroid (center of mass):The information supplied by the centroid can be used to define someindexes for detecting collective behaviour in some group, subgroups orindividuals with respect to the global group.– Global group:∗ Location of the centroid:

We can see in Figure 4 how the centroid is positioned duringtraining session. Our group of football players moves mainlythrow three places of the field in this training session.∗ Group spreading with respect to the centroid:

Dt =

∑|xti − xt|n

– Subgroups:∗ Spreading with respect to the centroid Ds

t .– Individuals:

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Figure 4: Path built by the centroid

∗ Leave-one-out methodology, which allows us to detect theindividuals responsible for breaking the “harmony”.

• Delaunay triangulation (dual of Voronoi diagram).– Sum of lengths of the triangulation edges.– Formation index: if At = At

n , where A is the total area and n isthe number of triangles in the triangulation, then we can define theformation index as

It =

∑|Ati − At|n

This index allows us to measure the group consistency throughtriangle areas dispersion with respect to the average of the areas.

4. Future work

As possible future lines we plan to explore methods for pattern detection ofgroups, subgroups or individuals inside the group. Classifiers in time seriescould be used to check time lag in activities. Clustering methodologies couldbe useful to attack this problem.

References

[1] Rafik Aziz Aliev, Babek Ghalib Guirimov. Type-2 fuzzy neural networksand their applications. (2014) Springer International Publishing.

[2] Francois Caron, Emmanuel Duflos, Denis Pomorski, Philippe Vanheeghe.GPS/IMU data fusion using multisensor Kalman filtering: introduction ofcontextual aspects.. (2006) Information Fusion, Vol. 7, Issue 2, pp. 221-230.

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[3] Finn V. Jensen. Bayesian Networks and Decision Graphs (second edition.(2007) Springer Verlag.

[4] Helge Langseth, Thomas D. Nielsen, Rafael Rumí, Antonio Salmerón.Parameter estimation and model selection for mixtures of truncatedexponentials. (2010) International Journal of Approximate Reasoning 51,485-498.

[5] H.B. Mitchel. Multi-Sensor Data Fusion. (2007) Springer Verlag.

[6] David Retana Pz, Aldo Enrique Vargas Moreno. (2010) Estabilizaciun helicro a escala mediante sistemas neuro-difusos., PhD Thesis,Universidad auta de Mco. (http://vargasmoreno.com/aldo/Tesis/Tesis.pdf)

[7] Puyin Liu, Hongxing Li. Fuzzy neural network and application. (2004)Series in machine perception and artificial intelligence, 59. WorldScientific.

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Acknowledgements

The Scientific Committee wishes to thank to the company speakers, theacademic coordinators and the researchers of each working team for theirinvaluable contributions to the scientific success of the 108 European StudyGroup with Industry.

We also want to express our gratitude to the Mathematics Institute of theUniversity of Sevilla (IMUS) which held the 108 European Study Group withIndustry.

Finally, we would also like to express our gratitude to Gladys Narbona,coordinator of the Organizing Committee and Guadalupe Parente, Technologytranslator of math-in, whose meticulous work and dedication have contributedto the success of this 108 European Study Group with Industry.

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