1. PENN S TATE © T. W. S IMPSON PENN S TATE Optimization Approaches for Product Family Design 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 ME 546 - Designing Product Families - IE 546 © T. W. S IMPSON

2. PENN S TATE © T. W. S IMPSON 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. PENN S TATE © T. W. S IMPSON 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. PENN S TATE © T. W. S IMPSON 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 Universal Motor Platform Example Electric motor field components prior to standardization Universal motor variants 0.8” 1.75” 60 650 Watts Stack length

5. PENN S TATE © T. W. S IMPSON 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. PENN S TATE © T. W. S IMPSON 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) The new 777 is also being designed knowing a priori that it will be stretched to carry more passengers and increase range

7. PENN S TATE © T. W. S IMPSON Boeing 737 Interior Layouts 737-300 126 passengers (8 first class) 737-400 147 passengers (10 first class) 737-500 110 passengers (8 first class) 737-600 110 passengers (8 first class) 737-700 126 passengers (8 first class) 737-800 162 passengers (12 first class) 737-900 177 passengers (12 first class)

8. PENN S TATE © T. W. S IMPSON Flight Ranges for 737-300, -500, -600, and -700 Flight Ranges for 737-700 Flight Ranges for 737-300 Capacity: 126 PassengersCapacity: 110 Passengers Flight Ranges for 737-600 Flight Ranges for 737-500

9. PENN S TATE © T. W. S IMPSON Dimensions of Boeing 737-300, -400, and -500 Boeing 737-300Boeing 737-400Boeing 737-500 All three aircraft share common height and width... …but their fuselage lengths are different:

10. PENN S TATE © T. W. S IMPSON Boeing 737-600 Dimensions of Boeing 737-600, -700, -800, and -900 The same holds true for the 737-600 through 900 Boeing 737-900 Boeing 737-700 Boeing 737-800

11. PENN S TATE © T. W. S IMPSON Optimization for Single Product Design 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 For Motor Example: Find:r, t, A A, N A, A F, N F, I, L Minimize:Mass Maximize:Efficiency,  Subject to:MagInt, H < 5000 Mass < 2 kg Eff,  > 70 % r > t Power = 300 W Torque = 0.5 Nm

12. PENN S TATE © T. W. S IMPSON Optimization for Product Family Design Generic Form: Find: x i Minimize:f i (x i ) Subject to:g i (x i ) < 0 h i (x i ) = 0 Definitions: i = 1, 2, …, p p = number of products in the family For Motor Family Example: Find:r i, t i, A A,i, N A,i, A F,i, N F,i, I i, L i Minimize:Mass i Maximize: Efficiency i Subject to:MagInt, H i < 5000 Mass i < 2 kg Eff,  i > 70 % r i > t i Power i = 300 W Torque i = T i where: T i = {0.05, 0.1, 0.125, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5} Nm

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

15. PENN S TATE © T. W. S IMPSON 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 Functional torque = fcn(motor stack length) thrust = fcn(# compressor stages) Conceptual/configurational # passengers on an aircraft size of an automobile underbody Example Scaling Variables Platform High Mid Low Segment ASegment BSegment C High Mid Low Segment ASegment CSegment B Platform Scale down Scale up Low-End Platform Leveraging High-End Platform Leveraging

16. PENN S TATE © T. W. S IMPSON Compromise Decision Support Problem 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 Given Assumptions to model domain of interest Simulation and analyses to relate X and Y Find X i i = 1, …, n d i -, d i + i = 1, …, m Satisfy System constraints (linear, nonlinear) g i (X) = 0 ; i = 1,.., p g i (X) < 0 ; i = p+1,.., p+q System goals (linear, nonlinear) A i (X) + d i - + d i + = G i ; i = 1, …, m Bounds X j min < X j < X j min ; j = 1, …, n d i -, d i + < 0 ; d i - d i + = 0 ; i = 1, …, m Minimize Deviation Function Z = { f 1 (d i -, d i + ),..., f k (d k -, d k + ) } Given Assumptions to model domain of interest Simulation and analyses to relate X and Y Find X i i = 1, …, n d i -, d i + i = 1, …, m Satisfy System constraints (linear, nonlinear) g i (X) = 0 ; i = 1,.., p g i (X) < 0 ; i = p+1,.., p+q System goals (linear, nonlinear) A i (X) + d i - + d i + = G i ; i = 1, …, m Bounds X j min < X j < X j min ; j = 1, …, n d i -, d i + < 0 ; d i - d i + = 0 ; i = 1, …, m Minimize Deviation Function Z = { f 1 (d i -, d i + ),..., f k (d k -, d k + ) } Feasible Design Space Deviation Function Aspiration Space Constraints Bounds Goals x1x1 x2x2 Reference: (Mistree, et al., 1993)

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

18. PENN S TATE © T. W. S IMPSON Electric Motor Family Design Problem I Platform parameters (common to all motors):  radius of motor, r  on armature: – wire x-sectional area, A A – number of wraps, N A Scaling variable (1/motor): i = 1, …, 10  stack length, L i Constraints (6/motor) and Objectives (2/motor):  thickness of motor, t  on field: – wire x-sectional area, A F – number of wraps, N F   

19. PENN S TATE © T. W. S IMPSON Two-Stage Optimization Approach in PPCEM 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 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 (  s,  s ) SS 6S6S YY +3  Y -3  Y Y S Lower Limit Upper Limit Each line represents a different product architecture, i.e., a different combination of: [ x 1, x 2, x 3, …., x n-1,  s,  s ]

20. PENN S TATE © T. W. S IMPSON 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 (  stack_length,  stack_length ) TT +3  T -3  T T L Lower Torque Limit Upper Torque Limit LL 6L6L Each line represents a different product architecture, i.e., a different combination of: [r, t, A armature, N armature, A field, N field ] Optimization Problem for Motor Family 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. PENN S TATE © T. W. S IMPSON Resulting Product Family Specifications Universal Motor Platform {N c, N s, A wa, A wf, r, t} 1273, 61, 0.27, 0.27, 2.67, 7.75 High Mid Low Product platform obtained using PPCEM Platform instantiations Group of individually designed motors

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

23. PENN S TATE © T. W. S IMPSON 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) Single-Stage Optimization Approach Use multiobjective optimization to formulate the product family optimization problem and resolve the tradeoff between commonality and individual performance

24. PENN S TATE © T. W. S IMPSON Universal Motor Family Design Problem II Design variables (8/motor): i = 1, …, 10  stack length, L i  radius of motor, r i  on armature: – wire x-sectional area, A A,i – number of wraps, N A,i Constraints (6/motor) and Objectives (2/motor):  current, I i  thickness of motor, t i  on field: – wire x-sectional area, A F,i – number of wraps, N F,i   

25. PENN S TATE © T. W. S IMPSON 10 Comparison of Results: Individual Motors 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 40%50%60%70%80% Efficiency Mass (kg) Desired Efficiency (> 70%) Desired Mass (< 0.5 kg) 10 8 6 5 4 3 1 9 7 2 1 2 4 5 67 8 3 9 8 7 6 5 4321 Desired Performance Region (i.e., targets for mass and efficiency are achieved) 10 9 8 7 6 5 4 3 2 1 Benchmark Group PhysPro (s=length) PPCEM (s=length) PhysPro (s=radius) 9

26. PENN S TATE © T. W. S IMPSON 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. PENN S TATE © T. W. S IMPSON 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. PENN S TATE © T. W. S IMPSON 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. PENN S TATE © T. W. S IMPSON GA Terminology Chromosome Population Generation kGeneration k+1 Selection Crossover Mutation Insertion Genetic operators Individuals gene 0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1 alleles

30. PENN S TATE © T. W. S IMPSON Encoding - Decoding PhenotypeGenotype Biology Design “blue eye” UGCAACCGU (“DNA” blocks) 010010011110 expression (chromosome) decoding encoding Radius R=2.57 [m] H sequencing coded domaindecision domain 0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1 RadiusHeightMaterial x1x1 x2x2 xnxn

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

32. PENN S TATE © T. W. S IMPSON Genetic Operators: Selection Roulette Wheel Selection 1 2 3 4 5 6 Probabilistically select individuals based on some measure of their performance. Sum Sum of individual’s selection probabilities 3rd individual in current population mapped to interval [0,Sum] Selection: generate random number in [0,Sum] Repeat process until desired # of individuals areselected Basically: stochastic sampling with replacement

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

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

35. PENN S TATE © T. W. S IMPSON 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. PENN S TATE © T. W. S IMPSON Stopping Criteria Generation Global optimum (unknown) Converged too fast (mutation rate too small?) Average fitness Typical convergence 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  k <  (i.e., genetic diversity has become small)  Stagnation - no or marginal improvement from one generation to the next: (F n+1 - F n )< 

37. PENN S TATE © T. W. S IMPSON 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 = motor # 1 = motor # 2 = motor # 3 = motor # 8 = motor # 9 = motor # 10 0 1 0 1 0 0 … 0 0 1 1 0 1 0 … 0 1 1 0 1 1 0 … 0 1 1 1 1 1 0 … 1 0 1 1 1 1 1 … 1 1 1 1 1 1 1 … 1

38. PENN S TATE © T. W. S IMPSON 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 … 01 0 0 1 0 0 … 11 1 1 1 1 0 … 1 motor # 1motor # 2motor # 10

39. PENN S TATE © T. W. S IMPSON 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... Commonality controlling genes Design variables for Product 1 Design variables for Product p 0100... 1u 11 c2c2 u 31 u 41 … cncn u 1p c2c2 u 3p u 4p … cncn First n genes in the chromosome control the level of platform commonality: 0=unique, 1=common to family

40. PENN S TATE © T. W. S IMPSON 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 pvar j is the percent variation of the j th design variable: where:

41. PENN S TATE © T. W. S IMPSON 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 7: Check constraint violation and design feasibility Step 8: Compute fitness values for each design configuration 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 Identify Best Design Final gen? Yes No GA-Based Method for Product Family Design

42. PENN S TATE © T. W. S IMPSON 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, A a, N a, A f, N f, 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. PENN S TATE © T. W. S IMPSON 2.71 Step 4: Setup GA for varying platform commonality  Each chromosome is 88 genes long (8 + 8*10) Varying Platform Commonality in GAA Example Commonality controlling genes (0=unique, 1=common) Design variables for 1st motor 1 These genes are treated as variables that can take values of {0,1} and are subject to mutation and cross-over These genes can take on any real value within each variable’s bounds 1111100 7.157500.28 1200.253.32 0.95 2.71 7.157500.28 1200.254.56 3.21 Design variables for 10th motor...

44. PENN S TATE © T. W. S IMPSON 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: 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. PENN S TATE © T. W. S IMPSON Step 8: Compute the three “fitness” values for each motor family (chromosome) in the generation  Fitness Function 1 (to minimize) =  M i  Fitness Function 2 (to maximize) =  i  Fitness Function 3 (to minimize) =  pvar j where: – M i and  i are summed over i = 1, …, 10 – pvar j is the % variation in the j th design variable, j = 1, …, 8 Compute Fitness and PFPF

46. PENN S TATE © T. W. S IMPSON Result: Multiple Platforms and Multiple Families New challenge: which platform and family do we choose? A:  -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)

47. PENN S TATE © T. W. S IMPSON 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 to improve scalability to large families of products Source: (Khajavirad, et al., 2006)

48. PENN S TATE © T. W. S IMPSON 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. PENN S TATE © T. W. S IMPSON Sample Results Solutions from generalized commonality formulation dominate all of the all-or-none commonality solutions Commonality Performance 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.670.6750.680.685 All-or-none commonality Generalized commonality Source: (Khajavirad, et al., 2006)

50. PENN S TATE © T. W. S IMPSON 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

51. PENN S TATE © T. W. S IMPSON Ongoing and Future Research Directions Classification of product family optimization problems:  Number of stages in optimization process  Platform defined a priori or a posteriori  Single or multiple objectives  Type of optimization algorithm  Number of products in the family and type of family  Module and/or scale-based product family (  configuration and/or parametric variety) Create a product family optimization testbed (on web) Incorporate multiple disciplines (e.g., manufacturing, marketing) in product family optimization problems Approaches for designing multiple platforms in a family Extend to product portfolio assignment problems involving multiple families and multiple platforms

52. PENN S TATE © T. W. S IMPSON

53. PENN S TATE © T. W. S IMPSON Physical Programming Designer formulates the optimization problem in terms of physically meaningful parameters

54. PENN S TATE © T. W. S IMPSON Implementation of Physical Programming Designer enters physically meaning preferences Numbers express desirability ranges

55. PENN S TATE © T. W. S IMPSON Showing all of these different objectives/ preferences gives a feel for what physical programming is capable of handling Number of objectives: 2 motors: 12 objs. 3 motors:18 objs. 5 motors:30 objs. 10 motors:60 objs. Physical Programming Preferences for Motor Family