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PENN S TATE

PENN S TATE

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PENN S TATE

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  1. Optimization Approaches for Product Family Design ME 546 - Designing Product Families - IE 546 Timothy W. Simpson Professor of Mechanical & Industrial Engineering and Engineering Design The Pennsylvania State University University Park, PA 16802 USA phone: (814) 863-7136 email: tws8@psu.edu http://www.mne.psu.edu/simpson/courses/me546 PENNSTATE © T. W. SIMPSON

  2. Optimization in Product Family Design • Optimization can be a helpful tool to support design decision-making • Optimization is frequently used in product design to help determine values of design variables, x, that minimize (or maximize) one or more objectives, f(x), with satisfying a set of constraints, {g(x), h(x)} • In product family design, optimization can be used to help balance the tradeoff between commonality and individual product performance in the family • Let’s consider a motivating example to define key terms and introduce different optimization formulations

  3. Motivating Example Objective: Design a family of ten (10) universal electric motors based on a product platform to provide a variety of power and torque outputs

  4. Universal Motor Platform Example • Universal motor is most common component in power tools • Challenge: redesign the universal motor to fit into 122 basic tools with hundreds of variations • Result: a common platform where • geometry and axial profile common • stack length varied from 0.8”-1.75” to obtain 60-650 Watts • fully automated assembly process • material, labor, and overhead costs reduced from $0.51 to $0.31 • labor reduced from $0.14 to $0.02 Electric motor field components prior to standardization 650 Watts 60 0.8” Stack length 1.75” Universal motor variants

  5. Scale-based Family: Rolls Royce Engines • Rolls Royce scales its aircraft engines to efficiently and effectively satisfy a variety of performance requirements • Incremental improvements and variations made to increase thrust and reduce fuel consumption • RTM322 is common to turboshaft, turboprop, and turbofan engines • When scaled 1.8x, RTM322 serves as the core for RB550 series

  6. The new 777 is also being designed knowing a priori that it will be stretched to carry more passengers and increase range Example Leveraging Strategies: Boeing Aircraft • Boeing 737 is divided into 3 platforms: • Initial-model (100 and 200) • Classic (300, 400, and 500) • Next generation (600, 700, 800, and 900 models)

  7. 737-600 110 passengers (8 first class) 737-300 126 passengers (8 first class) 737-700 126 passengers (8 first class) 737-400 147 passengers (10 first class) 737-800 162 passengers (12 first class) 737-500 110 passengers (8 first class) 737-900 177 passengers (12 first class) Boeing 737 Interior Layouts

  8. Flight Ranges for 737-300 Flight Ranges for 737-500 Flight Ranges for 737-700 Flight Ranges for 737-600 Flight Ranges for 737-300, -500, -600, and -700 Capacity: 126 Passengers Capacity: 110 Passengers

  9. Boeing 737-300 Boeing 737-400 Boeing 737-500 Dimensions of Boeing 737-300, -400, and -500 • All three aircraft share common height and width... …but their fuselage lengths are different:

  10. Boeing 737-800 Boeing 737-600 Boeing 737-900 Boeing 737-700 Dimensions of Boeing 737-600, -700, -800, and -900 • The same holds true for the 737-600 through 900

  11. Generic Form: Find: x Minimize: f(x) Subject to: g(x) < 0 h(x) = 0 Definitions: x = design variables f(x) = objective function g(x) = inequality constraints h(x) = equality constraints Optimization for Single Product Design • For Motor Example: • Find: r, t, AA, NA, • AF, NF, I, L • Minimize: Mass • Maximize: Efficiency, h • Subject to: MagInt, H < 5000 • Mass < 2 kg • Eff, h> 70 % • r > t • Power = 300 W • Torque = 0.5 Nm

  12. Generic Form: Find: xi Minimize: fi(xi) Subject to: gi(xi) < 0 hi(xi) = 0 Definitions: i = 1, 2, …, p p = number of products in the family Optimization for Product Family Design • For Motor Family Example: • Find: ri, ti, AA,i, NA,i, • AF,i, NF,i, Ii, Li • Minimize: Massi • Maximize: Efficiencyi • Subject to: MagInt, Hi< 5000 • Massi< 2 kg • Eff, hi> 70 % • ri > ti • Poweri = 300 W • Torquei = Ti • where: • Ti = {0.05, 0.1, 0.125, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5} Nm

  13. Challenges in Product Family Optimization • The dimensionality and size of the optimization problem increases very quickly as the number of products in the family increases • For motor example, p = 10: • Number of design variables = 8 x p = 8 x 10 = 80 • Number of objective functions = 2 x p = 2 x 10 = 20 • Number of constraints = 6 x p = 6 x 10 = 60 • Using a product platform will reduce the dimensionality of the optimization problem but not the size (i.e., the number of objectives or constraints): • Number of design variables = c + (n-c) x p where: c = number of common (platform) variables n = number of design variables for each of the p products

  14. Product Platform Concept Exploration Method Overall Design Requirements Market Segmentation Grid Step 1 Create Market Segmentation Grid The PPCEM provides a Method that facilitates the synthesis and Exploration of a commonProductPlatformConcept that can be scaled into an appropriate family of products to satisfy a variety of market niches Step 2 Classify Factors and Ranges Robust Design Principles Step 3 Simulation Analysis/Metamodels Metamodeling Techniques Step 4 Aggregate Product Platform Specifications Step 5 Develop Product Platform and Family Multiobjective Optimization Product Platform and Product Family Specifications

  15. Robust Design and Scalable Product Platforms • Robust design principles are used to minimize the sensitivity of a product platform (and resulting product family) to changes in one or more scale factors Example Scaling Variables Platform High Functional • torque = fcn(motor stack length) • thrust = fcn(# compressor stages) Scale up Mid Scale down Low Platform Segment A Segment B Segment C High-End Platform Leveraging High Conceptual/configurational • # passengers on an aircraft • size of an automobile underbody Mid Low-End Platform Leveraging Low Segment A Segment B Segment C

  16. Aspiration Space Deviation Function Feasible Design Space x2 x1 Bounds Constraints Goals Compromise Decision Support Problem Given Assumptions to model domain of interest Simulation and analyses to relate X and Y Find Xii = 1, …, n di-, di+i = 1, …, m SatisfySystem constraints (linear, nonlinear) gi(X) = 0 ; i = 1, .., p gi(X) < 0 ; i = p+1, .., p+q Systemgoals (linear, nonlinear) Ai(X) + di- + di+ = Gi ; i = 1, …, m Bounds Xjmin< Xj< Xjmin; j = 1, …, n di-, di+< 0 ; di- • di+ = 0 ; i = 1, …, m Minimize Deviation Function Z = { f1(di-, di+), ..., fk(dk-, dk+) } A hybrid of Goal Programming and Math Programming used to determine the values of design variables that satisfy a set of constraints and achieve as closely as possible a set of conflicting goals Reference: (Mistree, et al., 1993)

  17. High Cost High Performance Mid-Range Vertical Scaling Low Cost Low Performance Kitchen Appliances Power Tools Lawn & Garden Kitchen Appliances Lawn & Garden Universal Motor Platform (Common Design Variable Settings) Platform Leveraging Strategy  Design a single motor platform scaled by stack length Standardizing motor interfaces will facilitate horizontal leveraging to new segments

  18. £ > < Electric Motor Family Design Problem I • Platform parameters (common to all motors): • radius of motor, r • on armature: • wire x-sectional area, AA • number of wraps, NA • Scaling variable (1/motor): i = 1, …, 10 • stack length, Li • Constraints (6/motor) and Objectives (2/motor): • thickness of motor, t • on field: • wire x-sectional area, AF • number of wraps, NF

  19. Each line represents a different product architecture, i.e., a different combination of: [ x1, x2, x3, …., xn-1, ms, ss ] [ x1, x2, x3, …., xn-1, ms, ss ] [ x1, x2, x3, …., xn-1, ms, ss] [ x1, x2, x3, …., xn-1, ms, ss] [ x1, x2, x3, …., xn-1, ms, ss] Y Upper Limit +3Y Y -3Y Lower Limit 6S S S Two-Stage Optimization Approach in PPCEM Stage 1: Identify best platform variable settings Using robust design principles, solve one optimization problem of size n+1 to find best settings of common platform parameters, allowing one scaling variable to vary (ms, ss) Stage 2: Design individual products based on platform Fix common platform parameters and instantiate each product by solving p one-dimensional optimization problems to satisfy individual constraints while trying to meet performance targets

  20. Each line represents a different product architecture, i.e., a different combination of: [r, t, Aarmature, Narmature, Afield, Nfield] [r, t, Aarmature, Narmature, Afield, Nfield] [r, t, Aarmature, Narmature, Afield, Nfield] [r, t, Aarmature, Narmature, Afield, Nfield] [r, t, Aarmature, Narmature, Afield, Nfield] T Upper Torque Limit +3T T -3T Lower Torque Limit 6L L L Optimization Problem for Motor Family Stage 1 Using robust design principles, solve one optimization problem of size 8 to find best settings of common platform parameters, allowing one scaling variable to vary (mstack_length, sstack_length) Stage 2 Fix common platform parameters and instantiate each product by solving 10 one-dimensional optimization problems to satisfy individual constraints while trying to meet performance targets

  21. Group of individually designed motors Resulting Product Family Specifications High Mid Low Platform instantiations Universal Motor Platform Product platform obtainedusing PPCEM {Nc, Ns, Awa, Awf, r, t} 1273, 61, 0.27, 0.27, 2.67, 7.75

  22. Comparison of Results: Individual Motors Benchmark Group PPCEM (s=length) 1 10 0.9 Desired Efficiency (> 70%) 9 9 8 10 0.8 7 6 8 7 5 0.7 Mass (kg) 6 4 0.6 3 5 2 0.5 4321 Desired Performance Region (i.e., targets for mass and efficiency are achieved) Desired Mass (< 0.5 kg) 0.4 1 0.3 40% 50% 60% 70% 80% Efficiency

  23. Single-Stage Optimization Approach Single-Stage Optimization Approach Optimize product platform and product family members simultaneously by determine values of c common parameters for the product platform and s scaling variables for each product by solving one optimization problem of dimension (c + s*p) where: p = # products in the family n = # design variables per product in the family s = # scaling variables per product in the family c = # common platform variables (n = c + s) • Use multiobjective optimization to formulate the product family optimization problem and resolve the tradeoff between commonality and individual performance

  24. £ > < Universal Motor Family Design Problem II • Design variables (8/motor): i = 1, …, 10 • stack length, Li • radius of motor, ri • on armature: • wire x-sectional area, AA,i • number of wraps, NA,i • Constraints (6/motor) and Objectives (2/motor): • current, Ii • thickness of motor, ti • on field: • wire x-sectional area, AF,i • number of wraps, NF,i

  25. 10 9 8 7 6 5 4 3 2 1 Comparison of Results: Individual Motors Benchmark Group PPCEM (s=length) 1 10 PhysPro (s=length) PhysPro (s=radius) 0.9 Desired Efficiency (> 70%) 9 9 10 9 10 8 0.8 7 6 7 8 8 6 7 5 0.7 5 Mass (kg) 4 6 3 4 0.6 2 3 5 2 0.5 4321 Desired Performance Region (i.e., targets for mass and efficiency are achieved) Desired Mass (< 0.5 kg) 0.4 1 1 0.3 40% 50% 60% 70% 80% Efficiency

  26. Comparison of Approaches • Single-stage approaches: + yield performance improvements over two-stage approaches + use only a single optimization to determine best settings of common and scaling variables - increases dimensionality of optimization (many local optima) - assume best scaling variables are known a priori • Two-stage approaches: + provides flexible formulation for determining best combination of common parameters and scaling variables within a family + reduces dimensionality of optimization - increases number of optimizations that must be solved - segments optimization of platform from individual products which can lead to performance degradation within family

  27. Varying Platform Commonality • Ideally, an optimization algorithm would search all possible product platform combinations: where: the number of possible combinations of making n design variables common to platform c at a time the null platform, i.e., no commonality within the family and provide the designer with information about the: 1) design variables that should be made common 2) the values that they should take 3) the values the remaining unique variables should take

  28. Genetic Algorithms • Genetic algorithms (GAs) have shown great promise in many product design and optimization applications • GAs are well suited for product family design due to the combinatorial nature of the problem, but the associated computational costs are high • What is a Genetic Algorithm? • Optimization algorithm based on evolutionary principles (survival of the fittest) that do not require gradient information • Use strings of chromosomes to represent design variables • Each chromosome is evaluated for its “fitness” where those with higher fitness reproduce to form a new population • New populations of chromosomes are generated using selection, cross-over, and mutation

  29. GA Terminology Chromosome alleles gene 0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1 Population Individuals Selection Crossover Mutation Insertion Genetic operators Generation k Generation k+1

  30. Encoding - Decoding Genotype Phenotype coded domain decision domain expression UGCAACCGU Biology sequencing (“DNA” blocks) “blue eye” decoding Design 010010011110 H encoding (chromosome) Radius R=2.57 [m] x1 x2 xn 0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1 Radius Height Material

  31. Basic Operation of a Genetic Algorithm Initialize Population (initialization) Select individual for mating (selection) Mate individuals and produce children (crossover) next generation Mutate children (mutation) Insert children into population (insertion) Are stopping criteria satisfied? n y Finish Reference: Goldberg, D.E., 1989, Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley

  32. Genetic Operators: Selection Roulette Wheel Selection Probabilistically select individuals based on some measure of their performance. 1 2 6 Sum of individual’s selection probabilities Sum 3 3rd individual in current population mapped to interval [0,Sum] 5 4 • Selection: generate random number in [0,Sum] • Repeat process until desired # of individuals areselected • Basically: stochastic sampling with replacement

  33. Genetic Operators: Selection Tournament Selection Dominant performer placed in intermediate population of survivors 2 members of current population chosen randomly n Population Filled ? y Crossover and Mutation form new population Old Population Fitness 101010110111 8 100100001100 4 001000111110 6 Survivors Fitness 101010110111 8 001000111110 6 101010110111 8

  34. Genetic Operators: Crossover and Mutation • Crossover takes 2 solutions and creates 1 or 2 more crossover point Classical: single point crossover O1 P1 0 1 1 1 1 0 1 1 0 1 O2 P2 1 0 0 1 1 1 0 0 0 1 The children (“offspring”) The parents • Mutation randomly changes one or more alleles in the chromosome to increase diversity in the population With mutation probability Pm, O2: 1 0 0 0 1  1 0 1 0 1

  35. Genetic Operators: Insertion • Replacement scheme specifies how individuals from the parent generation k are chosen to be replaced by children from next generation k+1: • Can replace an entire population at a time (go from generation k to k+1 with no survivors) • select N/2 pairs of parents • create N children, replace all parents • polygamy is generally allowed • Can select two parents at a time • create one child • eliminate one member of population (usually the weakest) • “Elitist” strategy • small number of fittest individuals survive unchanged • “Hall-of-fame” strategy • remember best past individuals, but do not use them for progeny

  36. Stopping Criteria Typical convergence Global optimum (unknown) • There are a variety of stopping criteria: • A specific number of generations completed - typically O(100) • Mean deviation in individual performance falls below a threshold sk< e (i.e., genetic diversity has become small) • Stagnation - no or marginal improvement from one generation to the next: (Fn+1 - Fn)< e Average fitness Converged too fast (mutation rate too small?) Generation

  37. Using GAs in Product Family Design • Chromosomes typically represent a single product: • For product family design, one can use multiple chromosomes to represent the products in the family: • This requires added overhead to: • make sure all products exist in equal numbers • cluster products into families within each population • ensure that selection and cross-over operators are performed only on similar products 0 1 0 1 1 0 … 1 = one motor 0 1 0 1 0 0 … 0 = motor # 1 1 1 1 1 1 0 … 1 = motor # 8 0 1 1 0 1 0 … 0 = motor # 2 0 1 1 1 1 1 … 1 = motor # 9 = motor # 3 = motor # 10 1 1 0 1 1 0 … 0 1 1 1 1 1 1 … 1

  38. Using GAs in Product Family Design (cont.) • Alternatively, you can extend a single chromosome to represent the entire product family: • Adds overhead during the decoding process, but • fitness function will be evaluated for the entire family • genetic operators can be applied with little to no modification • Challenge is to determine how to represent a platform within the family of products • Specify common/unique variables a priori during initialization? • Or let the GA vary the levels of commonality of the platform? 0 1 0 1 0 0 … 0 1 0 0 1 0 0 … 1 … 1 1 1 1 1 0 … 1 motor # 1 motor # 2 motor # 10

  39. Varying Platform Commonality with GA • Add n commonality controlling genes to chromosome • The length, L, of each chromosome in the GA is determined by the number of design variables, n, and the number of products in the family, p: L = n + np ... ... … … 0 1 0 0 1 u11 c2 u31 u41 cn u1p c2 u3p u4p cn Commonality controlling genes Design variables for Product 1 Design variables for Product p • First n genes in the chromosome control the level of platform commonality: 0=unique, 1=common to family

  40. Product Family Penalty Function • Incorporate a Product Family Penalty Function (PFPF) as an additional objective function, which provides a surrogate for manufacturing cost savings • PFPF was introduced by Martinez, Messac, & Simpson (2000) to minimize variability of design variables within a product family to promote commonality pvarj is the percent variation of the jth design variable: where:

  41. GA-Based Method for Product Family Design Step 1: Identify design variables that could be made common Step 2: Perform DOE to check for possible reduction in design variables Step 3: Identify reduced set of design variables Step 4: Make sample runs to determine GA parameters Step 5: Use GA to generate design variable configurations Step 6: Run simulation/synthesis program for product family using GA Manufacturing feasibility analysis Cost analysis No Step 8: Compute fitness values for each design configuration Step 7: Check constraint violation and design feasibility Identify Best Design Final gen? Yes

  42. Applying the GA-based Method to GAA Example • Step 1: Identify design variables that could be made common to the platform • There are 8 design variables that define each motor: x = (r, t, Aa, Na, Af, Nf, I, L) • Step 2: Perform DOE to check for possible reduction in number of design variables • Typically used if design variables are > 8-10 • Not needed for motor example • Step 3: Identify reduced set of design variables • Not necessary for this motor example

  43. Varying Platform Commonality in GAA Example • Step 4: Setup GA for varying platform commonality • Each chromosome is 88 genes long (8 + 8*10) These genes are treated as variables that can take values of {0,1} and are subject to mutation and cross-over Commonality controlling genes (0=unique, 1=common) These genes can take on any real value within each variable’s bounds 1 1 1 1 1 1 0 0 ... 2.71 120 0.25 3.32 2.71 120 0.25 4.56 7.15 750 0.28 0.95 7.15 750 0.28 3.21 Design variables for 1st motor Design variables for 10th motor

  44. Simulate Performance of GAA Families • Step 5: Use GA to generate a population of solutions • Create product family alternatives (chromosomes) using selection, cross-over, and mutation • We use NSGA-II algorithm from: <http://www.iitk.ac.in/kangal/> • Step 6: Run simulation and/or analysis for each product in the family using GA generated design variables • Developed a set of analytical equations to evaluate performance of each motor: mass, efficiency, power, torque, etc. • Step 7: Check each chromosome for constraint violation and design feasibility • Each motor is checked against the set of constraints to ensure that is feasible

  45. Compute Fitness and PFPF • Step 8: Compute the three “fitness” values for each motor family (chromosome) in the generation • Fitness Function 1 (to minimize) = SMi • Fitness Function 2 (to maximize) = Shi • Fitness Function 3 (to minimize) = Spvarj where: • Mi and hi are summed over i = 1, …, 10 • pvarj is the % variation in the jth design variable, j = 1, …, 8

  46. Result: Multiple Platforms and Multiple Families A: e-NSGA-II families(Simpson, et al., 2005) B: NSGA-II families(Simpson, et al., 2005) C: Two-stage; radius scaled(Nayak, et al., 2002) D: Single-stage; length scaled(Messac, et al., 2002) E: Hierarchical sharing(Hernandez, et al., 2002) F: Ant colony optimization(Kumar, et al., 2004) G: Preference aggregation(Dai and Scott, 2004) H: Sensitivity/cluster analysis (Dai and Scott, 2004) New challenge: which platform and family do we choose?

  47. Generalizing Commonality and Scalability Issues • Collaborating with Dr. Jeremy Michalek and Aida Khajavirad (CMU) to create an efficient and decomposable GA-based formulation that allows for partial commonality in a family Decomposable GA formulation allows for parallel implementation toimprove scalability to large families of products Source: (Khajavirad, et al., 2006)

  48. Chromosome Representations for Problem Generalized commonality requires a 2D representation to define platform variable sharing and enforce design variable sharing among the variants Product variants are represented using regular chromosome coding Source: (Khajavirad, et al., 2006)

  49. Source: (Khajavirad, et al., 2006) Sample Results • Solutions from generalized commonality formulation dominate all of the all-or-none commonality solutions 1.0 Generalized commonality 0.9 0.8 0.7 Commonality 0.6 All-or-none commonality 0.5 0.4 0.3 0.2 Performance 0.67 0.675 0.68 0.685

  50. A Valuable Lesson from the Motor Example • Optimization can provide a useful decision support tool for product family and product platform design • In motor example, the resulting family should be scaled around radius, not stack length, to achieve specified performance • So why did B&D choose stack length? • Manufacturing considerations and production costs dictated decision: it was more economical to scale the motor along its stack length and wrap more wire around it than scale it radially • Lesson: optimization can be useful for product family planning and strategic decision making, provided the right aspects are modeled for the individual products as well as the product family as a whole