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**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**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**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**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**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**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)**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**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**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:**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**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**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**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**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**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**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)**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**£**> < 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**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 +3Y Y -3Y Lower Limit 6S 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**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 +3T T -3T Lower Torque Limit 6L 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**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**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**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**£**> < 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**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**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**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**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**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**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**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**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**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**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**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**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**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**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**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**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:**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**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**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**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**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**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?**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)**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)**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**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