agent-based optimization for product family design

Ann Oper Res(2006)143:147–156DOI10.1007/s10479-006-7378-xAgent-based optimization for product family designRahul Rai·Venkat AlladaC Springer Science+Business Media,Inc.2006Abstract This paper presents a two-step approach to determine the optimal platform level for a selected set of product families and their variants.Thefirst step employs a multi-objective optimization using an agent-based framework to determine the Pareto-design solutions for a given set of modules.The second step performs a post optimization analysis that includes application of the quality loss function(QLF)to determine the optimal platform level.The post optimization analysis yields the optimal platform level for a related set of product families and their variants.We demonstrate the working of the proposed method by using an example problem.Keywords Product family.Agent-based optimization.Quality loss function.Mass customization and intense competition is forcing many industries to shift their fo-cus from the design of single products to the design of product families.Several design and manufacturing strategies for offering product variety appear in the literature(Martin and Ishii,1997;Meyer and Lehnerd,1997;Dahmus et al.,2000).Simpson et al.(1999) proposed a model to design a product family based on the concept of scalable platforms. Platform-based design strategy is commonly used to create a product family.Two major con-tributions in the area of modular platforms are by Fujita et al.(1999)and Zugasti and Otto (2000).Fujita et al.(1999)developed a method that determines the optimum mix of existing modules(modules that have been already designed).Zugasti and Otto(2000)developed a method to accommodate both existing and new modules(modules with new functionality that have not yet been designed).Quite often,there are trade-offs to be made in regards to the performance of each individual variant of a product family to obtain the system level ben-efits of a platform-based ing the modular platform across a product family may R.RaiMechanical Engineering Department,University of Texas—Austin,USAV.Allada( )Engineering Management&Systems Engineering Department,University of Missouri-Rolla,USAe-mail:allada@compromise the performance of individual product variants,thereby,inducing inferiority in the design solutions.The inferiority induced in some of the product variants may lead the product variant’s performance to deviate from its target value.We refer to this as the“quality loss.”Hence,even though modular platforms have a number of benefits,they may instill some quality loss in the product variants.One of the critical aspects of product family design is to ensure an appropriate degree of module commonality that is shared by two or more variants of the product family.We refer to this as the‘level of platforming.’The proposed agent-based Pareto-optimization framework has been developed to solve modular product family design problem using existing modules.The agent-based system helps agents to have their own autonomous objectives and strategies to search for optimal design solutions.The methodology presented in this paper takes into account important issues such as“quality loss”to make appropriate decisions about the level of platforming.1.Proposed approachIt is assumed that we have a complete list of the existing module set that is required to produce feasible product variants.Each module type used in the product variants can have more than one instance.The proposed two-step methodology for determining a product platform comprises the following(Rai and Allada,2003):1.Pareto-optimization:This step generates the Pareto-optimal design solutions while sat-isfying all the design requirements through proper module configuration.(Refer to Figure1a).2.Post-optimal techniques:Once the set of Pareto-optimal solutions is obtained,a higher-level decision making can be used to solve various problems such as quality loss mini-mization related to modular platform design(Refer to Figure1b).For a problem having more than one objective function(say f i,i=1,2....,N),any two solu-tions x1and x2can have one of two possibilities:one dominates the other or none dominates the other.There are two types of solution sets:a non-dominated set and a Pareto-optimal set.A non-dominated set is defined with respect to a portion of the search space,whereas, a Pareto-optimal set is a non-dominated set with respect to the entire search space.The agent-based Pareto-optimization system consists offive main components,namely,Creator Agents,Interface1,Preserver Agents,Interface2,and Destroyer Agents that collaborate with each other in an iterative process.The creator agents mimic the creation process of design evolution.The number of creator agents is set equal to the number of module types available for design purposes.The creator agents work directly with the input module instance spec-ifications to produce a population of design solution(product)alternatives.Each product is represented as a vector of modules.Interface1helps to maintain an a priori number of design solutions and also checks for functional completeness of the design solutions.Interface1also eliminates any redundant copies of identical design solutions.Finally,Interface1sorts design solutions into Pareto-optimal solutions and non-Pareto solutions.The non-Pareto solutions are passed on to the preserver agents.The Pareto and the non-Pareto solution set during every iteration is determined using the definition of non-dominated and dominated set described previously.There exists a non-dominated Pareto solution set during every iteration which can change in the subsequent iterations due to emergence of new“superior”design solutions. There are two types of preserver agents:innovator agents and random agents.The innovator agents act on good design solutions and improve the design solutions by making suitable modifications along at least one of the objective functions.The random agents randomlyFig.1Two step methodology a)Agent based Pareto-optimization framework b)Post-optimal techniques.(without motive)change the design solutions allocated to them.Interface2classifies the non-Pareto solutions(that have been modified by the preserver agents)into two categories: good solutions and poor solutions.The poor solutions are eliminated from the system.Good design solutions(those that dominate on at least one objective but not on all objectives)are passed on to the Destroyer Agents.Finally,Interface2creates the Pareto-optimal surface (from the Pareto-optimal solutions provided by Interface1)for the given iteration.The de-stroyer agents act on the good design solutions provide by the Interface2.The destroyer agents fragment the design solution in order to further improve the good design solution.The number of destroyer agents is equal to the number of objectives pertaining to the problem at hand.Each destroyer agent type individually selects a design to be modified and removes the modules from the design solution that are believed to be reducing the design’s worth.Various fragmenting tactics differentiate the destroyer agent types from one another.Predominantly, these agents differ in their choice of the objectives.For example,some destroyer agents have an objective of removing expensive modules while others may remove only the heavy weigh-ing modules(termed as“faulty”modules in Figure1a.).The incomplete designs generated by the destroyer agents are sent to the creator agents to be completed into a complete design solution.The iterative process continues until the system converges or the resource and time constraints require the acceptance of the current best set of design solutions.2.Example problemLet us consider the design of a family of power screwdrivers and electric knives subject tocertain design constraints.The reason for selecting these products is that they are primarilydisparate products,but still have a few common modules that are utilized by both products. The module set from which these products can be created is presented in Table1a.Table 1a also consists of the actual components that are present in each of these modules.Some refinements were made to introduce new entries as shown in Table1a.Sometimes refinement to the module list provided by the heuristics(function-module mapping heuristics)is required as they may yield incomplete product configuration.Refinement is done to provide appropri-ate supporting modules to complete all the functionality required for a design solution.For example,the electrical supply module essentially comprises a battery and a switch whereas the actuating module consists of the battery alone.In order to generate feasible solutions that satisfy all the functionality with the actuating module,we should have a switch as an independently available module.The user-defined objectives for the design problem are as follows:minimize cost,minimize weight,maximize torque,and maximize serviceability.The assumed attributes for each of the module instances related to the four objectives mentioned above are presented in Table 1b.It is assumed that the number of instances available for each of the modules equals two;however,increasing or decreasing the number of instances associated with each of the modules does not affect the overall methodology.It was further assumed that there exist three different market segments for the power screwdriver and four different market segments for the electric knife.The market segmenta-tion can be viewed as identification of product configurations to suit different markets(which are assumed to differ from one another only due to different weight vectors of the objective functions).The weights for different market segments for both the products are presented in Table1a Module set a)module names and actual components b)module attributes.Motor torque(in/lbs):Instance10a=26,Instance10b=26,Instance3a=32Notes:r Highlighted entries indicate common modules shared by the electrical knife and the power screwdriver. r The∗indicates the modules added after the refinement process.r Serviceability rating is on a0-1scale for each individual module instance.r Module number set{2,3,4,10,11,17,18,19,20,21}is used for generating power screwdriver design solutions,while the others are used for generating electric knife design solutions.Table2a Selection of Pareto-design solution according to weight vector method.Table2a.Table2a also presents the unit sales volume(assumed)for each of these market segments.2.1.Pareto-optimization processThe number of creator agents was set at22agents with each agent corresponding to one module type.Each agent represents one unique module type(as indicated in Table1a there are22unique module types).Each agent is capable of instantiating different instances of a given module type.In order to structure the feasible design solution generation process and account for module compatibility,the AND/OR logic was used.The example of an exclusive OR collaboration can be seen between creator agent17(corresponding to module17)and creator agent21(corresponding to module21).This type of collaboration means that only one of two agents can be used in the design solution.The example of an AND collaboration can be seen between creator agent19and creator agent11.This type of collaboration means that both agents are necessary to produce a functional design solution.The AND/OR collaboration between the creator agents enable us to generate the feasible design solutions in a simple and effective way while checking the feasibility of the overall design solution produced by the agents.The number of design solutions processed for each of the iterations was limited to20. The percentage of design solutions that is altered by the innovator agents was kept at80% while the random agents alter the remaining20%of the design solutions.The number of destroyer agents for the problem equaled four corresponding to the four objective functions. Table2b presents the Pareto-design solutions for both the power screwdriver and the electric knife.There are six Pareto-design solutions each for the power screwdriver and the electric knife.2.2.Post-optimal techniqueOnce the set of Pareto-optimal solutions is obtained,usually some higher-level decision-making considerations(often based on the research question posed)are used to pick a solutionor a set of solutions.The use of the post-optimal technique is illustrated by determining the product platform for a product family.The weighted method was used to identify the Pareto-design solution that suits each of the market segments(shown in Table2a).The result of the application of the weighting method on the Pareto solutions is shown by the mapping from Table2a to Table2b.The product platform for the product family(for both the power screwdriver and the electric knife)is presented in Table3.Table3represents a case of“low”platforming.In order to increase the platforming level,one has to increase the number of shared modules between the product variants and/or product families.This can be achieved by reducing the number of modules instances of a given module type that can generate the entire family of products.Reducing the module instances to produce the entire product family not only increases the level of platforming,but also helps the design teams to effectively concentrate on fewer modules.However,reducing the number of modules instances will violate the Pareto-optimality of solutions generated and may induce a“quality loss.”So, in order to make decisions about reducing the module instances one should also take into account quality losses associated with such a decision.The issue of quality loss is discussed in Blackenfelt(2000)in which the actual loss due to“quality loss”is modeled as a fraction of the product price.In this paper,we present a detailed formulation for measuring the“quality loss”of the product family.This leads us to the following research question:How to determine the minimum number of module types and instances that can generate an entire family of products for all the market segments while minimizing the quality function loss(QLF)? 2.3.QLF formulationThe entire module set to produce the family of products for all the market segments of both the products(electrical knife and power screwdriver)is given in Table3.Only module{11a} serves as the common modular platform.Referring again to Table2b,it can be seen that the module instance{11a}was common in all the Pareto-solutions.A careful examination of Table3suggests that the number of modules instances to generate the entire family of electrical knives could be further reduced by making appropriate decisions about two possible cases presented below.1.Same module but different instances:This decision is related to the selection of a singlemodule instance from the set of instances of the same module type for all the design solutions.One such example is to choose between module instances{12a}and{12b}.2.Same function1different module combination:This decision is related to the selectionof a combination of modules or a single module from the set of instances of different module types that serves a particular function.An example of this case is the selection of one among following three module/module combinations,i.e.,{10a},{10b},{2b,3b, and4b}.Thus,the number of substitution alternatives,J,for this case is three.It is to be noted that the module combination{2b,3b,and4b}provides the same functionality as modules{10a}or{10b}.In this study,the decision to select the minimum number of module types and instances is based on a higher-level objective of minimizing the quality loss(specified in dollars)for the entire product platform.The quality loss concept is used to provide a better estimate of 1Two modules may deliver the same functions;however,they may differ in their functional performance and other characteristics.Table3Determination of product platform for power screwdriver andelectrical knife product variantsProduct platform Module set for the whole(module product familyinstances)(Platform)+(Other modules)Power(11a)(11a)screwdriver+(2b,3b,4b,5a,6b,7a,8a,9a,and10a,10b,electric knife12a,12b,13a,14a,15a,17a,18a,19b,20a,21a,22amonetary loss incurred by the manufacturer as the product performance deviates from its target value.The formulation of the quality loss function(QLF)for post-optimal solution analysis is described next.Assume that there exists a desired family consisting of N product variants corresponding to N different market segments.The product variant set is denoted by an integer set P n,where n=1to N.Let the integer set S i represent selection of one out of the j possible substitutes corresponding to i th substitution case.Corresponding to i th substitution case a decision is to be made to select one out of the J(i)possible substitutes.For example, the decision to select one module/module combination from the alternatives{{10a},{10b}, {2b,3b,and4b}}represents one such possible substitution case.For this substitution case, the number of possible substitutes,j,equals3.The value of j,varies with different i and are defined by the user.Let L n i j denotes the loss in$for the n th market segment due to j th case of the i th substitution alternative being selected.Also,we define the combination selection functionγj i such thatγj i=1If j th case of theıth substitution alternative is selected0Otherwise(1)Then,the objective function could be formulated as follows:minNn=1I,Ji=1,j=1L n i j SV nγj i(2)where SV n represents the normalized weight of the sales volume of the n th market segment.The variable L n i j is calculated from the quality loss function(QLF).Generally,the QLF is represented by a quadratic function(Fowlkes and Creveling1995):L(y)=k(y−m)2(3) where L(y)is the loss in dollars due to a deviation from the targeted performance,y is the measured response for a product,m is the target value of the product’s response,and k is a constant known as quality loss coefficient.The quality loss coefficient k is determined byfinding out the functional limits or customer tolerance for y,the measured response.Readers are referred to the text written by Phadke (1989)for further information on quality loss functions.In the case of nominal-the-best (NTB),smaller-the-better(STB),and larger-the-better(LTB),it is assumed that50%of the customers will not buy the product.Let the total loss at m± 0be equal to A0dollars,whichmay be incurred by the manufacturer as a consequence of degraded performance,deployment of more expensive modules,etc.The QLF for the NTB case can be defined as follows:L(y)=A0(y−m)2(4)L n i j is calculated by summing the quality loss associated with each design objective.If there are T objectives that are being considered,then L n i j is given by equation(8).L n i j=Tt=1L n i jt(5)where L n ijt denotes the loss in$due to t th objective in the n th market segment due to j th combination of the i th case being selected.For the example problem,the values of A0, 0,and m for the four objectives are listed in Table4.The stepwise procedure to determine the minimum set of modules is listed below: 1.Input the total number of market segments,N.In the example problem,N=7(threemarket segments of the power screwdriver and four market segments for the electric knife).2.Input the Pareto-design solutions for the whole product family for different market seg-ments(shown by the seven mapping arrows depicted in Table2(a-b)),and data related to QLFs(shown in Table4).3.Determine the set S i(set of possible substitution decisions).For the example problem, we have assumed that the set S i={({10a},{10b},{2b,3b,4b}),and({12a},{12b})}.The selection of S i set is user dependent.From Table2b,we specify thefirst substitution deci-sion as being to select one combination from the following set:{{10a},{10b},{2b,3b,4b}} since each member of the set performs the same functionality.In this case,j(1)=3.Sim-ilarly,the second substitution decision case is to select between module instances{12a}, and{12b}.So,j(2)=2for this case.Therefore,there exists L i j=j(1)+j(2)=5cases for each market segment.4.Determine the quality loss(in$)in each market segment related to different attributes(dif-ferent attributes correspond to different QLFs)for all substitution decisions and all substi-tute combinations,i.e.,all L n i j’s.For the example problem,the number of market segments are7(from step1)and the number of cases,L i j=5(from step3).Therefore,there areTable4Data for target value of product response,cost due to degradation,and functional limits for different attributes of productsObjectivefunction m A0 0 Power Cost18$200.1 screwdriver Weight325$100.3Torque32$500.4Serviceability 2.22$1000.35 Electric Cost14.65$200.1 knife Weight477$100.3Torque32$500.4Serviceability 4.05$1000.357×5=35L n i j ’s.Each L n i j is calculated by summing the quality losses (in $)associated with each attribute of the modules for the n th market segment.There are four design objec-tives that are being considered,hence T =4.For example,the value of L 212is calculated by using equation 5and is 13,838.item Determine the total quality loss across the whole prod-uct family i.e.,the value of loss function ( N n =1 I ,J i =1,j =1L n i j SV n γj i )shown by equation 2,for each combination of γj i for different i s by taking into account the sales volumes of each market segment.The number of possible combinations for the example problem are as fol-lows:{{γ11=1,γ12=1},{γ11=1,γ22=1},{γ21=1,γ12=1},{γ21=1,γ22=1},{γ31=1,γ12=1},{γ31=1,γ22=1}}.The value of the function ( N n =1 I ,J i =1,j =1L n i j SV n γj i )for the module combination ({γ21=1,γ12=1})which uses the modules {10b }and {11a }minimizes the quality loss.The quality loss for this solution is $28,390.We also found that if only one module instance {11a }was used as a product platform (which represented the optimal solution before the QLF application)then the quality loss is $562,250.The QLF is higher for the later case because three module instances,namely,{2b,3b,4b }were used instead of just one module {10b }.3.ConclusionsThis paper proposes a new two-step approach for designing modular product families.An agent-based optimization technique is used for determining pareto-design solutions from a given module set.It is to be noted that our Pareto-optimization technique will work only for convex surface frontiers and not for concave or discontinuous frontiers.Extending our approach to deal with non-convex Pareto solution frontiers is an area that requires further investigation.The post-optimal analysis is used to allocate pareto-designs to different market segments.The post-optimal technique is also used to determine the modular platform for a given product family.The QLF is used as a post-optimal tool to check if an improvement in the level of platforming also leads to decreased quality loss across the product family.The overall methodology proposed is elaborated through a design problem of designing a family of power screwdrivers and electric knives.The example demonstrates the validity and effectiveness of the proposed methodology.The AND/OR collaboration used in our approach took an average 10−3seconds for generating a single feasible design solution.While this estimate of the solution time may seem reasonable,we have not specifically concentrated on the computational performance aspects of our proposed approach.Acknowledgments This work is supported in part by the National Science Foundation grant DMI #9900226.The authors are thankful to Dr.Robert Stone and Mr.Ravi Yekula,University of Missouri-Rolla,for providing access to the design repository and laboratory facilities.ReferencesBlackenfelt,M.(2000).“Profit Maximization While Considering Uncertainty by Balancing Commonality andVariety Using Robust Design—The Redesign of a Family of Lift Tables.”Proceedings of ASME Design Engr.Technical 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