Multiobjective Optimization for Innovation in Engineering Design


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Companies daily need to optimize their products, hence optimization plays a significant role in today's design cycle. Problems related to one or more than one objective, originate in several disciplines; typically using a single optimization technology is not sufficient to deal with real-life problems, particularly when the design concerns complex and expensive products. Therefore, engineers are frequently asked to solve problems with several conflicting objective functions. The multiobjective optimization approach provides a set of non-dominant designs (Pareto optimality) where a further improvement for one objective is at the expense of all the others: this allows designers to choose the best solution for each scenario.

Solving real-world multiobjective problems is not simple, engineers must address problems connected to the non-linearity of the functions, complexity of the physics and the computational cost that snowballs as the number of parameters increases. Moreover, the coupling between disciplines for design a product can be really challenging, involving several complicating factors, such as the limitation on the computational resources, and even a lack of communication between different departments.

This tutorial is a survey on methodologies to approach design optimization process, a set of best practices intended for rapid delivery of high-quality products, with a specific focus on the numerical algorithms and post-processing used for selecting optimal design configurations.

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Multiobjective Optimization for Innovation in Engineering Design

  1. 1. Multiobjective Optimization forInnovation in Engineering DesignSilvia Poles, M. Margonari, G. BorziEnginSoft S.p.A.mail: s.poles@enginsoft.itLION 5 - Jan 17-21, 2011, Rome, Italy
  2. 2. OUR MISSION EnginSoft is a consulting company operating in the field of Computer-Aided-Engineering (CAE). Our mission is to spread the culture of digital technologies within both production and research contexts. We pursue this challenge by offering engineering consulting services, world- class CAE software, dedicated training courses and by promoting conferences, collaborations with research institutes, and publishing activity.We propose us as key partner in Design Process Innovation
  3. 3. History and businessHISTORY: Private company, founded in 1984 on the basis of other activities/structures operating since 1973ACTIVITIES: - Leading group in Italy for CAE/iDP. - Supply of software, services, consultancy, training. - Participation in industrial research projects (EU or national funding). - MIUR – acknowledged research centre for CAE/iDP technology transfer.OFFICESIN ITALY:Trento,Bergamo,Padua,Florence,Mesagne (BR)
  4. 4. The International EnginSoft NetworkEnginSoft proposes itself as partner forthe introduction of virtual prototyping intobusinesses, also on a European andInternational level, through a network ofnew companies.GERMANY | AUSTRIA | FRANCE | SCANDINAVIAN COUNTRIES |GREAT BRITAIN | SPAIN | GREECE | TURKEY | PORTUGAL | USAOfficial network website:
  5. 5. MACRO-ACTIVITIES and FIGURES ENGINEERING ACTIVITIES SOFTWARE AND KNOWLEDGE 1700 consultancy services TRANSFER succesfully carried out with over 80 expert engineers More than 1000 software licenses in ItalyTRAINING AND METHODOLOGICAL SUPPORT RESEARCH PROJECTSAn offer of more than 100 courses per Participation in over 30 research year and a portal for on-line training projects with public co-funding EnginSoft S.p.A. Company Profile 5
  6. 6. Innovation in Engineering Design
  7. 7. Obstacles to innovationWhile it is simplistic to claim that all organizations are dealing with the sameobstacles, there are repeating themes that we have noticed during the past years:• Lack of a shared vision, purpose and/or strategy• Short-term thinking• Inadequate understanding of customers• Lack of key competencies• Costs• …
  8. 8. Steps in product innovationThere are two parallel paths involved in the process:• the idea generation, product design and detail engineering;• market research and marketing analysis. Technical Idea Commercializ NEW implementa ationScreening PRODUCT tion
  9. 9. An unsupervised text classification method implemented inScilabIDEA SCREENING
  10. 10. Strategy Canvas• The strategy canvas is both a diagnostic and an action framework for building a compelling blue ocean strategy.• It captures the current state of play in the known market space.• This allows you to understand where the competition is currently investing, the factors the industry currently competes on in products, service, and delivery, and what customers receive from the existing competitive offerings on the market.Citation: Blue Ocean Strategy.Harvard Business School Press. 2005.
  11. 11. Strategy Canvas Example US Wine Industry in the late 1990sHigh Premium Wines Budget WinesLow Price Above-the-line Vineyard prestige Wine marketing range Use of enological Aging Wine complexity terminology and distinctions quality in wine communication
  12. 12. Eliminate-Reduce-Raise-Create Grid Case Study Yellow Tail Eliminate Raise Enological Terminology Price versus budget wines Aging qualities Retail stores involvementAbove-the-line Marketing Reduce Create Wine complexity Easy drinking Wine range Ease of selection Vineyard prestige Fun and adventure
  13. 13. A New Value Curve – Strategy Canvas of Yellow TailHigh Premium Wines [yellow tail] Budget Wines CREATERAISELow REDUCE Price Above-the-line Vineyard Wine Ease of ELIMINATE marketing prestige range selection Use of enological Aging Wine Easy Fun and terminology and quality complexity drinking adventure distinctions in wine communication
  14. 14. Text Mining• Text mining is a relatively new research field whose main concern is to develop effective procedures able to extract meaningful information from a collection of text documents.• A reliable document classification strategy can help in information retrieval.• The subject is undoubtedly challenging for researchers who have to consider different and problematic aspects coming out when working with text documents and natural language.14
  15. 15. A text classification using Self Organizing Maps• Many mathematical frames have been developed for the text classification: Bayes classifiers, supervised and unsupervised neural networks, learning vector machines and clustering techniques.• We use an unsupervised self organizing map (SOM) as a tool to discover possible clusters of documents• Such maps allow a 2D representation of multivariate datasets, preserving the original topology.
  16. 16. The Problem• Our personal interest for these techniques was born reading the EnginSoft newsletters.• A typical newsletter issue usually has many contributions: case studies, interviews, corporate and software news, …• A series of questions came out: – can we have a deeper insight into our community? – Can we imagine a new categorization based on other criteria? – Can we discover categories without knowing them a-priori?
  17. 17. Stem• It easy to understand that one of the difficulties that can arise when managing text is that we could consider as “different” words which conceptually can have the same meaning. optimization, optimizing, optimized, optimizes, optimisation, optimality.• It is clear that a good preprocessing of a text document should recognize that different words can be grouped under a common root (also known as stem)
  18. 18. Collect & Managing• Scilab has been used to collect and manage all the stems• A criterion to judge the importance of a stem in a document is needed.• We decided to adopt the tf_idf coefficient (term frequency – inverse document frequency) which takes into account the relative frequency of a stem in a document and the frequency of the stem within the corpus. w is the word, tf _ idf w, d tfw,d * idf w d is the document n_ij is the number of time the word i appears in nw , d the document j tfw,d N nw , k N is the total number of document k 1 C is the entire corpus N idf w ln 1 nw , C
  19. 19. Counting stemsA matrix representation of the non-zeros tf-idf coefficients within the corpus.The matrix rows collect the text files sorted in the same order as they areprocessed, the columns collect the stems added to the dictionary in thesame order as they appear while processing the files.
  20. 20. Training the SOM• We submitted the dataset with the tf-idf coefficients and ran a SOM training.• To avoid stems with high and low values, we keep only those belonging to the range 0.1-0.8 probability. The extremes are cancelled out from the dataset, ensuring a more robust training.• The dictionary decreases from 7000 to 5000 stems which are considered to be enough to describe the corpus, keeping quite common words and preserving the peculiarities of documents.
  21. 21. D-Matrix• The white diamonds give evidence of the number of files pertaining to the neuron.• The colormap represents the mean distance between a neuron’s prototype and the prototypes of the neighbor neurons. Two groups of documents (blue portions) can be detected.
  22. 22. The stems My boss contributions to the company Newsletter My contributions to the company NewsletterFor each neuron the first two stems with the highest tf-idf arereported. This highlights the main subject discussed by articles fallingin the neurons.
  23. 23. Results of text mining• With this analysis we identify the most important “stems” and their frequency and distribution.• This is very similar to analyze web pages, blogs, … to indentify the factors the industry currently competes on in products, and what customers receive from the existing competitive offerings on the market.• For any factor we can identify the importance and fill up the strategy canvas and create our new IDEA
  25. 25. What is optimization? Selection of the best option from a range of possible choices.What makes it a complex task? The potentially huge number of options to be testedWhat qualifies as an optimization technique? The search strategy
  26. 26. Optimization ProblemMathematical formulation max f1 x1 ,  , xn , f 2 x1 ,  , xn ,  , f k x1 ,  , xn gi x 0 gj x 0 subject t o gl x 0 x S Note : When k>1 and the functions are in contrast, we speak about multi-objective optimization.
  27. 27. Math and Real worldThere is a huge difference between mathematical optimization andoptimization in the real-world applications Ideal function in the mathematical world Rugged hill in the experimental world
  28. 28. VariablesVariables:Variables are the free parameters, quantities that the designer can control Continuous variables: • point coordinates • process variables Discrete variables: • components from a catalogue • number of components
  29. 29. ObjectivesObjectives are the response parameters, i.e. the quantities that the designer wish tobe MAX or MIN MAX MIN efficiency cost performance weight ... … Note : A MAX problem can always be transformed into a MIN problem.
  30. 30. Why Multiobjective optimization• Most design or problem solving activities are multiobjective by nature• Problems usually involve multiple conflicting objectives that should be considered simultaeously
  31. 31. Pareto dominance• Pareto Dominance:• Design a dominates Design b if: – [f1(a) >= f1(b) and f2(a) >= f2(b)...and fn(a) >= fn(b)] – and [f1(a) > f1(b) or f2(a) > f2(b)...or fn(a) > fn(b)]• In the Pareto frontier none of the components can be improved without deterioration of at least one of the other component.• Pareto dominance for one objective coincides with a classical optimization approach• Pareto dominance defines a group of efficient solutions: in case of n objectives, the group of efficient solutions contains at Max ∞(n-1) points
  32. 32. Pareto dominanceA dominates B if and only if: [ f1(a) >= f1(b) and f2(a) >= f2(b)... and fn(a) >= fn(b) ] and [ f1(a) > f1(b) or f2(a) > f2(b)... or fn(a) > fn(b) ] f1Red dots are all efficientsolutions f2
  33. 33. Pareto Dominated Points • Rapidly decreasing probability of having a dominated solution in a randomly generated dataset • Rapidly increasing search effort for when the number of objective is large • Fortunately, in real-case applications the number of dimensions can collapse m ( 1) k 1 m k 1 kn 1 kr Where m is number of points and n is number of m objectives
  34. 34. Weighted FunctionWeighted Function:• n objectives can be added in a single objective using weights: – F(x) = w1*Obj1+w2*Obj2+…+wn*Objn...• Pro: • simple formulation• Cons: • weights are problem-dependent and must be empirically defined • weights are connected to objectives values and might lose significance for different objectives values
  35. 35. Why is the Weighting Method Ineffective• Although this type of scalarization is widely used in many practical problems, it has a serious drawback: it cannot provide solutions for non-convex cases• Depending on the structure of the problem, the linearly weighted sum can not necessarily provide a solution that the Decision Maker (DM) desires• The DMs tend to misunderstand that a desirable solution can be obtained by adjusting the weights but there is no positive correlation between the weights and the value of functions
  36. 36. Examplemin y1 f1 ( x), y2 f 2 ( x), y3 f 3 ( x) Suppose the DM want x to reduce more y1 and 2 even a bit y2s.t. yi 1 1 i 1, 2,3The minimum of the linearly weightedsum with all the weights equal to 1 is y1, y2 , y3 1 1/ 3,1 1/ 3,1 1/ 3given by:The DM changes the weights: y1, y2 , y3 1 10/ 105,1 2 / 105,1 1/ 105 1 , 2 , 3 10,2,1 The value of y2 is worse than before, despite the weights given by the DM
  37. 37. Why is the Weighting Method Ineffective• Someone might suspect that this is due to a missing normalization of the weights!• Normalization of the weights do not solve the problem• It is usually very difficult to adjust the weights to obtain a solution as the DM wants.
  38. 38. Maximize a Mathematical function Maximize:F1 ( x, y) [1 ( A1 B1 )2 ( A2 B2 )2 ] 2 Ai (ai , j sin( j) bi , j cos( j )) j 1 2 Bi (ai , j sin( j ) bi , j cos( j )) j 1 0.5 1.0 2.0 1.5 a b 1.0 2.0 1.5 2.0 1.0 0.5 ( x, y) [ , ]
  39. 39. Mathematical functions Maximize:F2 ( x, y) [( x 3) 2 ( y 1) 2 ] x, y [ , ]
  40. 40. Weighted SumWeighted Sum:• F= (1-k)*F1+k*F2• The parameter k is varied from 0 to 1 with a step of 0.1• The weighted sum goes progressively from F1 to F2• The red zones indicate higher values for the weighted sum
  41. 41. Pareto Frontier Two variables, two objectives, infinite efficient solutions in two regions not connected in the variables definition domain.F2 F1
  42. 42. General RemarksFacing a design problem:• Rarely is there a clearly identified and unique objective• There is a vague distinction between constraints and objectives• Even if algorithms and numerical optimization theories exists in the academic world since many years, the practical impact until today was negligible and limited to very specific applications:• It is necessary to: • extend the concept of mathematical optimization to several objectives • have “robust” tools to explore the entire design configuration space
  43. 43. Evolution & Optimization• Evolutionary algorithms are direct global search methods based the model of organic evolution.• Metaheuristics methods are a new type of methods that have been developed since 1980.• These methods have the ability to solve even difficult optimization problems in the best way possible. This is an important group of methods that has significantly contributed to the renewal of multiobjective optimization.
  44. 44. EA advantages• Evolutionary algorithms (EAs) do not need derivatives information• EAs are simple to implement• EAs are flexible, may be applied to several different problems• EAs are scalable to high-dimensional optimization problems• EAs may deal with continuous, discrete and binary variables• Always converge to a good enough solution in successive, self- contained stages• Robust against noisy objective functions• Can be easily parallelized• Shortcomings – Slow convergence (but metamodeling may help!)
  45. 45. HistoryDARWIN in the wind tunnel! The first real-case application of Evolution Strategy methods used by Prof. Rechenberg Number of possible adjustments 515 = 345 025 251
  46. 46. What companies really like of EAs• EAs find a set of solutions which lie on the trade-off (Pareto frontier) – Putting the preferences after optimization – Much better understanding of the problem – Better choices f1 Engineers may choose between solutions f2
  47. 47. Evolutionary Optimization of Yagi-Uda Antennas
  48. 48. The Problem• The Yagi antenna evolved as a special configuration of an endfire array• It is a traveling wave antenna with a surface wave that propagates along the array with a phase velocity slightly less than that of the free space• It consists of a single driven element and a number of parasitic elements made up of a reflector and a set of directors
  49. 49. Optimization• The Yagi configuration has not been amenable to theoretical analysis since it is an array of elements of different lengths with non-uniform spacing and thus cannot be treated using conventional array theory• The Yagi-Uda antennas are known to be difficult to optimize due to their sensitivity at high gain and the inclusion of numerous parasitic elements• Over the years the performance of Yagi antennas has been improved very slowly
  50. 50. Antenna Parameters• Parameters: – Lenght for each element – Spacing between elements – Diameter of the wire• With N elements, we have 2N parameters
  51. 51. Software used• The original Numerical Electromagnetics Code (NEC) has been developed at the Lawerence Livermore Laboratory.• The code has always been a "card image/batch run“• SCILAB (
  52. 52. 4nec2 project 4nec2 is a completely free Nec2 windows based tool for creating, viewing, optimizing and checking 2D and 3D style antenna geometry structures and generate, display and/or compare near/far-field radiation patterns for both the starting and experienced antenna modeler.Antenna geometry edit in 4nec2.
  53. 53. Antenna Card• An antenna described to NEC is given in two parts, a structure and a sequence of controls.• The structure is simply a numerical description of where any part of the antenna is located, and how the wires are connected up. Thanks to the old fashioned structure card style input, it is very easy to automatically change the geometry.
  54. 54. Problem Setup - SCILAB• Scilab is an interactive platform for numerical computation providing a powerful computing environment for engineering and scientific applications.• Scilab is a free software!• A set of modules are available, we will use OPTIMIZATION tools.
  55. 55. Antenna Cardwire X,Y,Z start point X,Y,Z end point radiusCM NEC 2 elementsCEGW 15 10 0.0000 -0.2500 0.0000 0.0000 0.2500 0.0000 1.e-3GW 20 42 2.0000 -2.0000 0.0000 2.0000 2.0000 0.0000 1.e-3GE 0EX 0 15 5 0 1.0000 0 Means of excitationGN -1FR 0 1 0 0 299.8 0 Frequency (MHz)RP 0 37 73 1003 -180 0 5 5 Optimization Parameters Radiation Pattern
  56. 56. Define ParametersParameters Lower Upper Step Goal Expression bound bound MaxGain Maximize(min(gain))Lengths 0.1λ 1.5 λ 0.01 MinVSWR Mimize(max(VSWR))Separations 0.05λ 0.75λ 0.01Radius 2mm 6mm 1mm The VSWR, or Standing Wave Ratio, of an antenna is aFrequency 219MHz 251MHz 16 measure of how efficiently your antenna is radiating the energy it produces when you transmit.
  57. 57. Let SCILAB find the best solutions• Several different optimization methods are available (evolutionary and gradient based)• In this example we use an evolutionary algorithm because this class of methods are able to effectively search large space• We can even approach the problem as a multiobjective optimization problem where we want to maximize the gain and minimize the Voltage Standing Wave Ratio (VSWR) without any weighting function.
  58. 58. Output Results …….. - - - POWER BUDGET - - - INPUT POWER = 4.4745E-03 WATTS RADIATED POWER= 4.4745E-03 WATTS STRUCTURE LOSS= 0.0000E+00 WATTS NETWORK LOSS = 0.0000E+00 WATTS EFFICIENCY = 100.00 PERCENT - - - RADIATION PATTERNS - - - - - ANGLES - - - POWER GAINS - - - - POLARIZATION - - - - - - E(THETA) - - - - - - E(PHI) - - -THETA PHI VERT. HOR. TOTAL AXIAL TILT SENSE MAGNITUDE PHASE MAGNITUDE PHASE DEGREES DEGREES DB DB DB RATIO DEG. VOLTS/M DEGREES VOLTS/M DEGREES-180.00 0.00 -999.99 1.78 1.78 0.00000 -90.00 LINEAR 0.00000E+00 0.00 6.35890E-01 -120.91-175.00 0.00 -999.99 2.24 2.24 0.00000 -90.00 LINEAR 0.00000E+00 0.00 6.70390E-01 -119.64-170.00 0.00 -999.99 2.61 2.61 0.00000 -90.00 LINEAR 0.00000E+00 0.00 6.99122E-01 -121.61-165.00 0.00 -999.99 2.54 2.54 0.00000 -90.00 LINEAR 0.00000E+00 0.00 6.93579E-01 -124.59 ……...
  59. 59. Results• Preparation time: 1 h• Running time on a laptop: few hours• Number of runs: 1000• Initial design: Lowest gain 2.20 dB, Highest VSWR 13.43• Some Pareto Solutions: ID Gain VSWR 681 7.66 1.39 818 7.94 1.38 820 8.57 1.86 763 8.62 2.54
  60. 60. Results Initial pointsVSWR Gain
  61. 61. VSWRFrom 13.43 to 1.38 -90% Gain From 2.20dB to 7.94dB +261%
  62. 62. Conclusions• Boost your creativity with SCILAB• Push the limits of product innovation with open source software• No limits in the number of available licenses• Option for parallel computing
  63. 63. References• Kim and Mauborgne. Blue Ocean Strategy. Harvard Business School Press. 2005.• Electromagnetic Optimization by Genetic Algorithms, Y.Rahmat-Samii and E. Michielssen, eds.,Wiley,1999• Evolutionary Optimization of Yagi-Uda Antennas, Lohn, J. D. Kraus, W. F. Linden, D. S. Colombano, S. P., LECTURE NOTES IN COMPUTER SCIENCE, 2001, ISSU 2210, pages 236-243, Springer-Verlag; 1999• Design of Yagi-Uda antennas using comprehensive learning particle swarm optimisation, Baskar, S. Alphones, A. Suganthan, P.N. Liang, J.J. Sch. of Electr. & Electron. Eng., Nanyang Technol. Univ., Singapore, in: Microwaves, Antennas and Propagation Proceedings,Oct. 2005, Volume: 152- 5, pages: 340- 346• Single and Multi-objective design of Yagi-Uda Antennas using Computational Intelligence, Neelakantam V. Venkatarayalu and Tapabrata Ray., in Proceedings of the 2003 Congress on Evolutionary Computation, Volume 2, pp. 1237--1242, IEEE Press, Canberra, Australia, December 2003 .