Platform Design for Customizable Products as a Problem of Optimal Access

PLATFORM DESIGN FOR CUSTOMIZABLE PRODUCTS AS A PROBLEM OF OPTIMAL ACCESS Christopher Williams, Janet K. Allen, Farrokh Mistree1 Systems Realization Laboratory G. W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, Georgia 30332-0405

Gabriel Hernandez Siemens Westinghouse Power Corporation 4400 Alafaya Trail, MC DV 322 Orlando, FL, 32826-2399

ABSTRACT The foremost difficulty in making the transition to mass customization is how to offer product variety affordably. The answer to this quandary lies in the successful concurrent design of the product and its manufacturing production process. While product platform design methodologies have emerged as a manner in which to approach the difficulty in providing product variety, none of the techniques address the issue of designing the product’s corresponding production process. In this chapter and in Chapter #, “Improving Manufacturing Agility through the Development of a Production Platform for the Manufacture of Customizable Products,” we offer a methodology which effectively handles the dual-faceted nature of this design problem. Specifically, we propose the Product Platform Constructal Theory Methodology (PPCTM), an approach for designing platforms in order to handle variety in both the product and its production process. In this chapter we focus on the design of product platforms for customizable products, whereas our focus shifts to production process design in Chapter #. The design of product platforms for mass customization requires a balance of performance and commonality, as well as consideration of manufacturing costs and markets of non-uniform demand. Furthermore, for problems of industrial scale, a designer must be able to synthesize multiple customization techniques in order to offer customization in multiple product specifications. Current product platform design techniques do not satisfy such requirements. In this paper, we formulate the design of platforms for customizable products as a problem of optimization of access in a geometric space as per Bejan’s constructal theory. The resulting approach, the Product Platform Constructal Theory Method, allows a designer to develop systematically hierarchic product platforms. The PPCTM enables a designer to develop product platforms for customizable products while handling issues of multiple levels of commonality, multiple product specifications, inherent tradeoffs between platform extent and performance, and markets of non-uniform demand. The approach is illustrated through the design of a product platform for a line of customizable cantilever beams.

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Professor and Corresponding Author Phone: (404) 894-8412; Fax: (404) 894-9342; Email: [email protected]

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Platform Design for Customizable Products as a Problem of Optimal Access

NOMENCLATURE b Dimension of beam width, m Average cost of selling family of cantilever beams, $ Cavg cmat Cost of beam material per volume, $/m3 Cmaterial Total cost of beam material, $ Corder Total cost of placing an order for the beam material, $ Cwaste Total cost of beam material waste, $ Ctotal Total cost, $ D Demand; number of products di-, di+ Deviation variables E Modulus of elasticity of the cantilever beam material, MPa h Dimension of beam length, m I Moment of inertia of cross-section of cantilever beam, m4 k Scaling constant for multi-attribute utility function L Length of beam, m Lo Raw length of beam material, m MN A market space with N dimensions n Total number of product variants Ni Number of ith space elements Norder Number of distinct sizes of Lo to be ordered O Objective function value P Load placed on end of beam, N A product in the market space r ri The ith attribute of a concept SF Safety factor for beam design U(X) Overall, multi-attribute utility function w Deflection of cantilever beam, m Z Deviation function Range of loads in cantilever beam market space ∆P The dimension of ith space element in P direction of market space; a decision variable for a ∆P i stage i. A derived product realization in the market space ∆r Range of lengths in the cantilever beam market space ∆L The dimension of ith space element in L direction of market space; a decision variable for a ∆Li stage i. Ultimate compressive strength of cantilever beam material, GPa σu

1. DESIGN OF PLATFORMS FOR CUSTOMIZABLE PRODUCTS As manufacturing enterprises have struggled to meet demands for customized products through traditional economies of scale strategies, mass customization has emerged as a manufacturing paradigm for enterprises to efficiently and effectively satisfy customers’ requirements for variety. Typically, mass customization is accomplished through developing product platforms – a set of common components, modules or parts from which a stream of derivative products can be created (Lehnerd, 1987). The design of product platforms for customized products enables the manufacturer to maintain the economic benefits of having common parts and process while still being able to offer variety to customers. “Platform-based product development offers a multitude of benefits including reduced development time and system complexity, reduced development and production costs, and improved ability to upgrade products” (Simpson, 2003). Clearly, a key element for the efficient production of customized goods is platform design (Anderson, 1997). Various engineering approaches have been proposed to develop product platforms throughout the literature. At their highest level of abstraction, these techniques have been classified as either bottom-up or top-down (Simpson, 1998). Bottom-up approaches are the redesign and consolidation of existing products to create more competitive product families by reducing part variety and standardizing components.

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Platform Design for Customizable Products as a Problem of Optimal Access

Examples include Kalopakjian’s Group Technology (Kalpakjian, 1997), Ericsson and Erixon’s Modular Functional Deployment (Ericsson and Erixon, 1999), and Siddique and Rosen’s Product Family Reasoning System (Siddique and Rosen, 2000). In general, the main advantage of bottom-up approaches is the existing knowledge of parts and processes. Their main disadvantage is that product rationalization is achieved after a number of products have been designed and manufactured. Top-down platform design is characterized by an up-front decision to develop simultaneously a product family based on a common core and reduce redesign cost. Examples include Simpson and coauthors’ Product Platform Concept Exploration Method (Simpson et al., 1999; Simpson et al., 2001) and Nayak and coauthors’ Variation-Based Platform Design Methodology (Nayak et al., 1999). Two limitations are perceived in existing top-down approaches. First, these approaches have been used to commonalize features or components across an entire product family without concern for significant loss of performance. Secondly, these approaches have considered variety in only one product specification. Typically, products are customized for multiple specifications (e.g., the torque and the power of a motor) using multiple approaches for managing product variety (e.g., modular design, adjustable features, dimensional customization with CAD/CAM technology, etc.). As a result of these limitations, designers using the majority of existing top-down approaches are unable to systematically design the complex product platforms that are typical of those realized in industry. An example of combining multiple approaches of commonality is Matsushita’s customized bicycles (Anderson, 1997): modularity is applied in the wheels, pedals, drive chain, seats, handlebars, and controls; adjustable customization is used to allow adjustments for the height and angle of the seat and handlebars; and dimensional customization enables customized lengths of the frame tubing. Consider a more simplified example: a manufacturer wishes to offer a line of customizable cantilever beams. The manufacturer wishes to provide customers the ability to specify a beam of any desired length and of any loading condition. As is typical of markets with customized products, the manufacturer observes that the demand amongst the beam variants is non-uniform. The manufacturer has decided to offer variety via three methods, (i) to change the beam cross-section, (ii) to change the beam material, and (iii) to cut the beams to customized lengths from standard pieces. The manufacturer wants to determine the architecture of a product family (i.e., the organization and range of application of each method of offering variety) that will satisfy two conflicting goals: to provide the lowest average cost across the volatile market, and to provide the lowest average maximum beam deflection across the family. In order to solve this problem, a designer requires a platform design method that can consider non-uniform demand, multiple customizable specifications, multiple modes of managing product variety, and the tradeoff between commonality and product performance. Can currently available product platform design approaches tackle this simple problem? Simpson provides a thorough review of 32 existing optimization-based product platform design approaches wherein their different characteristics are compared and contrasted (Simpson, 2003). The following limitations are identified in Simpson’s review: 1. Two-thirds of the techniques require specification of platform a priori to optimization; 2. Half of those techniques surveyed assume that maximizing product performance maximizes demand, maximizing commonality minimizes production costs, and that resolving the tradeoff between the two yields the most profitable product family; 3. Only half of the methods integrate manufacturing costs directly; 4. Less than one-third incorporate market demand or sales into problem - those that do assume that demand is uniform, and use single objective optimization (towards minimizing cost or maximizing profit); 5. Only two methods are capable of handling multiple methods of managing variety (modularity and product scaling). The first two limitations are typical of these techniques as it makes the design space more tractable. This is not ideal, however, as not only are the assumptions listed above not always applicable, but a method should provide a designer the opportunity to offer multiple levels of commonality. Furthermore, a good product platform technique should provide a designer the ability to offer variety in multiple design specifications. The third and fourth limitations listed are critical limitations in current platform design techniques. Without accurate market and manufacturing knowledge, a designer is unable to determine the appropriate extent of commonality across the platform. Of the techniques surveyed that incorporate market modeling in the formulation of the method, several only use traditional market-based analysis to determine the most

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Platform Design for Customizable Products as a Problem of Optimal Access

effective location for a product family in a market space, but do not relate this information to the actual product architecture (Li and Azarm, 2002); (Yu et al., 1999); (Farrell and Simpson, 2003). Those techniques that use a quantitative approach for incorporating customer demand into the formulation of the product architecture (Seepersad et al., 2002); (Jiang and Allada, 2001); (Fujita and Yoshida, 2001) (Hernandez et al., 2002), only model customer demand as uniform across the market. This is not adequate since markets of mass-customized products are characterized as niche and heterogeneous. As identified above, the fifth limitation identified prevents current design techniques from being applied to larger sets of problems. Only two approaches surveyed were capable of handling the development of product families through multiple product variety techniques (namely, product scaling and modularity): that of Fujita and coauthors (Fujita and Yoshida, 2001), and the technique presented in this chapter. Since each technique surveyed by Simpson suffers from one or more of these limitations, it is evident that a designer using the currently available product platform design methods would not be able to solve the simple cantilever beam example problem as described above. In response to these limitations, we propose the Product Platform Constructal Theory Method (PPCTM): a top-down approach for developing product platforms that facilitates the realization of a stream of customized product variants, and which accommodates the issue of multiple levels of commonality and multiple customizable specifications. The result of the use of the PPCTM is a hierarchical organization of several approaches of commonality, as well as the specification of their range of application across the product platform. The PPCTM provides a designer the opportunity to synthesize multiple methods of managing variety in order to provide customization in multiple product specifications while taking into account critical design considerations such as markets of non-uniform demand, the impact of platform commonality on product performance, multiple design objectives, and manufacturing costs. In this paper we present the theoretical constructs of the Product Platform Constructal Theory Method in Section 2. In Section 3 we present the method alongside a tutorial example, the design of a family of customizable cantilever beams in order to illustrate its strengths and contributions. Closing remarks are offered in Section 4. 2. HIERARCHIC PRODUCT PLATFORM DESIGN AS A PROBLEM OF ACCESS IN A GEOMETRIC SPACE The Product Platform Constructal Theory Method was developed in order to provide designers a methodical approach for synthesizing multiple methods of offering product variety in the development of product platforms for customized products (Hernandez, 2001). The fundamental problem addressed in the PPCTM is how to determine and organize modes for managing product variety systematically in order to create a product platform for customized products? A mode for managing product variety is any generic approach in product design or its manufacturing process for achieving a product customization; common modes include, but are not limited to, modular design, platform design and standardization, dimensional customization, and adjustable customization. As a result of the PPCTM’s theoretical foundations in both hierarchical systems theory (Simon, 1996) and constructal theory (Bejan, 1996; Bejan, 1997; Bejan, 2000), the design of platforms for customizable products is represented as a problem of access in a geometric space. 2.1 Theoretical Constructs: Hierarchic Systems Theory and Constructal Theory Our approach for organizing common components for a very large number of product variants is anchored in the thesis of Herbert Simon (Simon, 1996), who observed that complex structures adapt and evolve more efficiently when they are organized hierarchically. Considering that potential for rapid adaptation is higher in complex systems when they are organized hierarchically, we propose to determine and organize commonality of components for a large number of products in a hierarchic manner. Our approach to do so is based on the representation of platform design as a problem of optimization of access in a geometric space.

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Platform Design for Customizable Products as a Problem of Optimal Access

5

Derived Product Realization

S P1

∆r

P2

r1 r2

y

O Px,y

Space of Customization

x Figure 1: Illustration of Product Customization as an Optimal Access Problem

An optimal access problem is characterized by the need to determine the optimal “bouquet of paths” that link all points, Px,y, of an area, S, with a common destination, O (Figure 1). The arrangement of these paths is usually optimized towards some objective (i.e., most efficient, least resistance, minimal travel time). Constructal theory was initiated by Adrian Bejan as a result of studying problems of optimal access (bejan – 1996 ref). Bejan’s constructal theory rests on the premise that a single principle – the constrained optimization of global performance – is the generating mechanism of organization, complexity, and hierarchic structure in nature, engineering, and even management. Bejan’s constructal theory embodies the notion that the hierarchic organization we observe in Nature is the result of a sequential optimization of access with the objective of the maximization of access (or minimization of resistance, or losses). For example, the formation of tree networks (like blood vessels) is explained as a hierarchic process of optimization: the shape that optimizes “access” at the most elementary volume occurs first, followed by an assembly of these innermost “volumes” into a second, larger-shapes, which in turn are assembled into a third volume, and so on. Following the basic tenants of constructal theory, this optimization process should proceed in a specific time directions: from the optimization of the basic elements to the higher-order assemblies of the structure. This sequential process continues until all relevant volume is connected. A common example of constructal theory is the formation of a street network as a hierarchic process of optimization. The problem entails the layout of streets so that all points within the space (i.e., a city) are accessible from a common point (i.e., a market) in the most efficient manner possible. To accomplish this, several different modes of transportation, each with their own maximum allowed velocity, need to be hierarchically organized. Furthermore, the range of their application across the space must be also determined. Applying constructal theory to this example involves the definition of an elementary space in which the street layout is optimized to maximize access (i.e., minimize the travel time). The direction of the solution by constructal theory is to assemble these geometrically optimized areas (from smallest to largest) until the entire area is covered as shown schematically in Figure 2. S2

Figure 2: Organization in Nature as a Process of Access Optimization (adapted from (Bejan, 2000))

Unfortunately, such a sequential process yields sub-optimal results (Hernandez, 2001). Bejan's constructual theory, however, provides us with the idea of formulating the design of product platforms as a problem of access in geometric space as multi-stage optimization problems for which more effective solution algorithms can be utilized. 2.2 Abstracting Constructal Theory to Product Platform Design In an attempt to abstract constructal theory and problems of optimal access to product platform development, we introduce the concept of space of customization as the geometric space set of all feasible

Platform Design for Customizable Products as a Problem of Optimal Access

combinations of values of product specifications that a manufacturing enterprise is willing to satisfy (i.e., space S in Figure 1) (Hernandez et al., 2002). Each product specification for which variety will be offered is represented as a dimension in this space (dimensions x and y in Figure 1). The magnitude of each dimension represents the amount of variety that will be offered. Each point in the space of customization represents a product variant that the manufacturer wishes to offer. Mathematically, let N be the number of quantitative parameters that define the requirements of a product. Let ri represent these parameters, where i=1,…,N. Then the space of customization, MN, is the set:

M N ≡ {(r1 , r2 ,..., rN )}

It should be noted that a space of customization is not limited to continuous variables; it can be formed by continuous, discrete or mixed-valued requirements. Based on this mathematical definition of space of customization, a product i can be represented by a unique specification of product requirements in an N-dimensional space of customization, i.e., a vector ri(ri1,…,riN): r i = ri1eˆ1 + ri 2 eˆ2 + ... + riN eˆ N

where eˆ k is the unit vector in each direction k of the space of customization. Using this representation of a product, the derivation of a new product, rj, based on an existing product, ri, is referred in this work as a “product customization,” represented by a vector in the space of customization: N

N

k =1

k =1

∆r ji = ∑ (r jk − rik )eˆk = ∑ ∆r jik eˆk This representation of product customization is illustrated in Figure 1. We refer to generic approaches to “access” points in the space of customization, i.e., to achieve product customizations from a baseline design, as modes for managing product customization (as shown in Figure 1 as ∆r). With the introduction of these definitions, the problem of designing a platform for customizable products becomes an effort to define a baseline set of components (the product platform) from which we can access all the points of a space of customization through the systematic use of a series of modes for managing product customization, and optimizing some given objective (e.g., cost, profit, product performance, etc). Drawing an analogy to the street network example presented in the previous section, the creation of a product family involves the linking all different feasible products (points in the city) within the space of customization (the city) from a common starting point (the market). The manner in which we link each possible product (instead of using streets) is through the modes of managing product customization that we defined earlier. We provide Table 1 as a graphical example of the analogy that exists between the traditional constructal theory example, street network design, and the development of product platforms for customizable products. Table 1: The Analogy of Creating a Street Network and Designing Product Platforms via the Application of Constructal Theory

Geometric Space Crux of Problem

Means of linking spaces

Street Network City Connecting all points in city together to ensure optimal access Streets

Product Platform Space of Customization Connecting all product variants in customization space to a “baseline” to ensure minimum cost. Modes of managing product variety

The fundamental problem addressed in the application of the PPCTM to product platform design is how to organize and determine the extent of application of modes for managing product variety systematically in order to create a product platform for customized products. Through the application of the tenets of constructal theory, this optimal access problem is formulated as a multi-stage decision wherein the ranges of application of each mode for managing product variety are the decision variables. The goal of each decision is to improve the objective functions in order to provide the most efficient manner of offering product variety. More detailed information on the abstraction of constructal theory to product platform

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design can be found in (Hernandez et al., 2003a). With these fundamental theoretical constructs presented, the PPCTM is presented in detail in Section 3. 3. THE PRODUCT PLATFORM CONSTRUCTAL THEORY METHOD Based on constructal theory and our abstraction of platform design as a problem of access in a geometric space of customization, the design of platforms for customizable products is formulated and solved as a multi-stage decision problem using the six steps shown in Figure 3. Step 1: Define the Geometric Space and the Demand Scenario Step 2: Define the Objective Functions Step 3: Identify the Modes for Managing Customization Step 4: Identify the Number of Stages and Define a Multi-Attribute Utility Function for Each Stage Step 5: Formulate a Multi-Stage Utility-Based Compromise Decision Support Problem Step 6: Solve the Utility-Based Compromise Decision Support Problem

Figure 3: The Product Platform Constructal Theory Method

The first step of the PPCTM involves abstracting the development of a product platform for customizable products as a problem of access in a geometric space. After the space of customization is identified, a designer then must appropriately model the demand distribution across the market space. In the second step, the objective functions are defined. Typical objective functions include the minimization of average cost, or the improvement of a product performance metric. The modes for managing variety are identified in the third step and are hierarchically organized in Step 4. The determination of the range of application of each mode for managing variety is done through the formulation and solution of a multistage utility-based compromise Decision Support Problem. With the extent of application of each mode known, a designer is capable of fully defining the product family that offers the best compromise to the objective functions. In this section we describe each step of the PPCTM alongside a simplified example problem: the design of a line of customizable cantilever beams. This example problem is kept simple in order to maintain the focus in the paper on the methodology itself. The PPCTM has been applied successfully to several complex engineering problems such as a line of customizable pressure vessels (Williams et al., 2004), a line of customizable glass-door refrigerators (Hernandez, 2001), and a line of customizable hand exercisers (Hernandez et al., 2003b). In this chapter, a simplified example problem, the design of a product platform for customizable cantilever beams, is presented as a means to illustrate the method. 3.1 Example Problem: Product Platform Design for Customizable Cantilever Beam Returning to example problem mentioned in Section 1, consider the following scenario: a manufacturer wishes to offer a line of customizable cantilever beams. The manufacturer wishes to provide customers the ability to specify a beam that ranges in length (L) from 0.5 to 10 m, and is capable of supporting a single end-load (P) from 50 to 500 N (schematic shown in Figure 4).

P h L

b

Figure 4: Schematic of Cantilever Beam Embodiment

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The manufacturer has decided to offer variety via three methods, (i) to change the beam cross-section, (ii) to change the beam material, and (iii) to cut the beams to customized lengths from standard pieces. The manufacturer wants to determine the architecture of a product family (i.e., the organization and extent of application of the multiple modes of managing variety) that will satisfy two conflicting goals: to provide the lowest average cost across the volatile market, and to provide the lowest average maximum beam deflection across the family (i.e., to improve product quality). Furthermore, as is typical of markets with customized products, the manufacturer observes that the demand amongst the beam variants is nonuniform. In order to focus on the method, the specific details of the modeling of the cantilever beam (including design constraints and all associated cost modeling) are included in the appendix of this paper. Generally, the total cost of the manufacturing of the cantilever beam (ignoring all labor and plant costs) includes three components: material purchase cost, material waste cost, and material order cost. The sole engineering constraint of this problem is that the maximum stress of the beam, with its specified length, material, cross section, and applied load, is not greater than the material’s allowable stress (including a consideration of an appropriate safety factor). In order to maintain the simplicity of the example, it is assumed that uncertainty and risk are absent from this problem. It is also noted that some values used in the example are estimates and do not change the fundamental results of this paper. The model of the cantilever beam can easily be modified to suit specific situations; however, the authors’ focus is centered on the validation of the method itself.

3.2 Step 1: Define the Geometric Space and Demand Scenario In the first step of the PPCTM, the space of customization (defined in Section 2.1) must first be defined. A space of customization is defined by three components: • the identification of the product parameters that will be varied according to the customer demands, • the range of variety that will be offered for each parameter, and • the analysis of the demand of the market. The number of dimensions of the geometric space is directly related to the number of product parameters in which variety will be offered. The range of each varied parameter determines the bounds of each dimension of the geometric space. This range is usually bounded by economic or technological limitations (e.g., there might be no need for beams that can support more than 500 N in the market considered). For our example, there are two independent design specifications that characterize the desired product customization - the length and the applied load of the cantliever beam. The length range to be considered is 0.5 to 10 m, and the load range is 50 to 500 N. This results in the two-dimensional continuous space of customization illustrated in Figure 5. Each point in this geometric space represents a unique combination of the two customizable specifications.

500

350

P (N) 200

50 0.5

L (m)

10

Figure 5: Space of Customization for Cantilever Beam Example

A designer using the PPCTM is able to handle markets of non-uniform demand by treating demand as a property of the space of customization. For this problem, the manufacturer has observed that there is significantly more demand for the medium-ranged cantilever beams. More specifically, the manufacturer has determined that the most appropriate manner of modeling demand is to treat it as a normal distribution, as shown in Figure 6.

Platform Design for Customizable Products as a Problem of Optimal Access

9

300

250

# of Beams

200

150

100

500

50

425 350 275

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 L (m) 9 9.5 10

P (N)

200 125 50

Figure 6: Non-Uniform Cantilever Beam Demand Distribution

The choice of best model for demand of a market is context based. Although not explicitly illustrated in this paper, the reader should be reassured that this methodology is robust to any non-uniform demand model that a designer may choose, as shown in (Williams et al., 2004).

3.3 Step 2: Define the Objective Functions The objective functions represent parameters to be improved by the use of the design method. As stated in the problem description, the manufacturer of the line of customizable beams has two objectives: to minimize the average cost and the average deflection of the products in the entire market space. The cost of each product variant is estimated as the sum of the cost of the material required to produce the part, the cost required to order the material, and the cost associated with the material wasted from cutting the beams to length from standardized lengths. The average total cost of the product family is calculated by averaging the cost of each individual product variant over the entire market as seen in Eq. 1 (derived from Eq. A-7):

Cavg =

1  n  Di ( Cmaterial ,i + Corder ,i + Cwaste ,i )  ∑  Dtot  i =1 

[1]

where Di is the demand of the specific product variant i, Dtot is the total demand of the products of the market space, Corder is the material cost, Corder is the order cost, and Cwaste is the cost of product waste. The average deflection of the beams of the product family is found by summing the deflection of each product variant, and then averaging that sum over the entire market, as shown in Eq. 2 (derived from Eq. A2):

w

avg =

3 1 n − PL i i ∑ n i =1 3Ei I i

[2]

where n is the total number of product variants, and P, L, E, and I are the load, length, modulus of elasticity, and moment of inertia of each beam, respectively. Obviously it is impossible to simultaneously reduce beam deflection and reduce the cost of the beam; deflection is typically reduced via the enlargement of the cross section, or the use of a more rigid (and therefore more expensive) material.

3.4 Step 3: Identify the Modes for Managing Variety This step of the PPCTM involves the consideration of how a product will be varied in order to satisfy the customized specifications of the space of customization. These modes of managing variety (defined in Section 2.1) are the linking mechanism between the product variants that compose the product family. The determination of these modes is a strategic decision that involves decision-making between design and manufacturing. For the cantilever beam example, a designer must identify methods in which to offer product variety in beam length and beam strength. After consulting with manufacturing, and evaluating the firm’s core

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competencies, designers have chosen three modes of managing product variety - two for offering variety in strength, and one for offering variety in length. Mode P1: Standardization of Beam Cross-Section Variety in beam strength can be offered by changing the cross-section of the beam. The manufacturer has decided to keep the beams’ width (b) constant at 0.15m since the height of the beam, h, has a more direct affect on the beam’s rigidity. Therefore, for this mode, the beams’ height is varied from 0.1m to 0.25m. Mode P2: Standardization of Beam Material This mode is used to achieve changes in beam strength by choosing one of three available materials: fir, pine, and hickory. Each material differs in modulus of elasticity, ultimate stress, and cost. The properties of each of these materials are presented in Table 2. The result of the application of this mode is the determination of the range of application of each material across the product family. Table 2: Material Properties for Cantilever Beam Example

Material

E (MPa)

Hickory Fir Pine

15 13 9

Ultimate Stress (GPa) 63 50 36

Cost ($/m3)

35 25 15

Mode L1: Dimensional Customization of the Beam Length For this mode, changes in length are achieved by cutting each beam from a stock piece of raw material, while maintaining a constant cross-section. The result of this mode is the use of a common cross-section and a common stock length of material for a certain range beam lengths. These three modes (Modes P1, P2, L1) are the approaches considered for accessing all the points of the space of customization generated in Step 1 of this example. The application of each of these modes presents a tradeoff for the manufacturer. While the standardization of the cross-section and beam material will help to minimize the ordering cost for the manufacturer, the material costs associated to such “overengineering” of each individual product variant prevents the complete standardization of these properties across the entire family. The standardization of the raw length across the entire platform presents a similar tradeoff: while the application of the mode minimizes the cost of ordering, it will also add a cost associated with the resulting wasted material. The manufacturer must now properly formulate a multi-stage decision to determine the range of application of each of these modes that provides the best compromise for the two conflicting objectives of the problem statement.

3.5 Step 4: Identify the Number of Hierarchy Levels and Allocate the Modes for Managing Variety to the Levels In the fourth step, it is established how and when each mode of managing customization is used. Modes that are capable of achieving the smallest variations in the varied design parameters are typically used at the lower levels of the hierarchy (i.e., before modes that can only achieve large variations in the design parameters). Economical and technological considerations place an important role in establishing the hierarchic use of the modes for managing product variety. Following the tenets of constructal theory, each level of the hierarchy represents a geometric “subspace” of the entire space of customization. The size of each sub-space represents the extent of application of each mode for managing variety. The resultant hierarchy for the cantilever beam example problem is outlined below in Figure 7.

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∆L1 ∆P1

S1 P

∆P2

S2 L Figure 7: Geometric Representation of the Organization of the Modes for Managing Variety

The First Stage and the First Space Element For the first space element, modes L1 (dimensional customization of beam length) and P1 (standardization of beam cross section) are used in concert to achieve variety in both length and strength of the beam. These modes are the cheapest means of offering extremely small changes between product variants in both customizable specifications. The size and shape of this first space element, S1, is given by the value of the variables ∆L1 and ∆P1 as shown in Figure 10. ∆L1 and ∆P1 represent fundamentally the extent to which the raw length and the beam cross-section are commonalized across the market respectively. ∆L1 and ∆P1 are therefore decision variables for this first stage. The Second Stage and the Second Space Element The second space element, S2, is composed by a number of assemblies of the first space element, S1, in the load dimension, as shown in Figure 7. The size and shape of these space elements is given by the value of the variables ∆P2 and ∆L1. ∆P2 represents the extent to which the beam material is commonalized across the market. A graphical interpretation of the hierarchic synthesis of the modes for managing product variety is provided in Figure 8. As can be seen, the range of application of each mode (∆L1, ∆P1, ∆P2) directly correlates to the number of unique components that compose the product architecture. Pine (EA, σu,A)

Fir (EB, σu,B)

h1 (Sx)

h2 (Sx)

Raw Length 1 (Lo,1)

Raw Length 2 (Lo,2)

Hickory (EB, σu,B)

… …

hx (Sx)

Raw Length y (Lo,4)

Figure 8: Hierarchic Organization of the Modes for Managing Product Customization for the Cantilever Beam

3.6 Step 5: Formulate the Multi-Stage Utility-Based Compromise Decision Support Problem Following the tenets of constructal theory, the determination of the range of application for each mode for managing variety that compose a level of the hierarchy (or sub-space) represents one stage in a multistage decision. With the order of the use of the modes established, a designer proceeds by formulating a proper multi-stage decision problem. In our work, we prefer to formulate each decision stage with a utility-based compromise Decision Support Problem. The utility-based compromise DSP (u-cDSP) is a decision support construct that is based on utility theory and permits mathematically rigorous modeling of designer preferences such that decisions can be guided by expected utility in the context of risk or uncertainty associated with the outcome

Platform Design for Customizable Products as a Problem of Optimal Access

12

of a decision. While any appropriate decision formulation technique is serviceable, we prefer to use the ucDSP because its use “provides structure and support for including human judgment in engineering decisions involving multiple attributes, while simultaneously providing an axiomatic basis for accurately reflecting the preferences of a designer with regard to feasible tradeoffs among these attributes under conditions of uncertainty (Fernández et al., 2001).” Furthermore, the u-cDSP has proven useful in previous product platform techniques as it provides a decision construct in which a designer can model multiple, conflicting objectives (Seepersad et al., 2002). The formulation of each utility-based compromise Decision Support Problem follows the four steps presented in (Seepersad et al., 2002) (Figure 9).

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6 Utility

Utility

u-cDSP Step 1: Assess Utility Functions for Each Goal

0.5

u-cDSP Step 3: Formulate System Goals E[ui ( Ai ( X ))] + di− − d i+ = 1

0.5

0.4

0.4

0.3

0.3 2

y = -0.0 101x - 0.0069x + 1.0609 0.2

0.2

2

y = -2E-05x - 0.0002x + 1.1167 0.1

0.1

0

0 0

2

4

6

8

10

0

12

50

10 0

150

200

250

300

Cost ($)

Deflection (in .)

u-cDSP Step 2: Combine Utility Functions for Individual Goals into a MultiAttribute Utility Function n

U = ∑ ki ui ( Ai )

u-cDSP Step 4: Formulate the Deviation Function n

Z = 1 − E[U ( X )] = ∑ ki (d i− − di+ ) i =1

i =i

Figure 9: The Formulation of the Utility-Based Compromise Decision Support Problem

First a utility function for each of the objectives is formulated by qualitatively and quantitatively assessing the preferences of the designers (all designers’ preferences are modeled as risk averse in this work). These individual utility functions are then combined into a multi-attribute utility function as a weighted average of the individual utilities. Finally, goal and deviation functions are developed for each stage. The deviation function of the u-DSP is formulated to minimize deviation from the target expected utility (i.e., 1, the most preferable value), which is mathematically equivalent to maximizing expected utility. The goal and deviation functions formulated for each u-cDSP inherently consider the compromise of the tradeoffs between the each objective function. With the goal of minimizing the deviation of the expected utility from the ideal value, parameters that provide the best values for this overall objective are chosen while maintaining consistency with the designer’s preferences. With the presence of the u-cDSP, designers are given the opportunity and ability to model multiple objectives in each decision stage of the PPCTM. The baseline decision of each stage is the quantitative extent of application of each mode for managing variety across the platform, ∆r(i). Geometrically, this decision is the determination of the size and shape of each subspace of the space of customization. For a problem with N varied parameters, the decision variables for any stage i, are ∆r (i ) = [ ∆r1 (i ), ∆r2 (i ),..., ∆rN (i )] In the formulation of each u-cDSP, there exists a constraint on the selection of the range of commonality for each mode of managing so that a hierarchic design is achieved: ∆rj (i + 1) ≥ ∆rj (i ) Each u-cDSP is formulated to minimize the deviation of the expected utility from the ideal value. A typical u-cDSP formulation for a decision stage of the PPCTM is presented in Figure 10. For Each Stage i Given:

Find:

The N-dimensional market space MN = {(r1, r2, … rN)} The decision variables of the previous stages ∆r(1), …, ∆r(i-1) The modes of managing product variety to be utilized at Stage i The value of decision variable x(i)=[∆r1(i), ∆r2(i), ∆rN(i)] The deviation variables,

Satisfy:

Bounds:

d x−,i and d x+,i

∆rj ,min (i ) ≤ ∆rj (i ) ≤ ∆rj ,max

Platform Design for Customizable Products as a Problem of Optimal Access

Constraints:

13

∆rj (i ) ≥ ∆rj (i − 1) −

+

−

+

d x ,i , d x ,i ≥ 0 d x ,i id x ,i = 0 E[u ( Ax ,i ( X ))] + d x ,i − d x , i = 1 −

Goals: y

+

y

Minimize: Z i = 1 − U i = ∑ k x ( d x , i − d x ,i ) ; where U i = ∑ k x , i u ( Ax , i ( X )) −

+

x =i

x =1

Figure 10: Formulation of the Multi-Stage Utility-Based Compromise Decision Support Problem for the PPCTM

In our example problem risk and uncertainty are not modeled. Although this feature of the u-cDSP is not exercised, we employ it for its ability to quantify a designer’s preference in the presence of multiple objectives. Furthermore, the formulation of the u-cDSP is done in order to maintain connectivity to the authors’ previous work and to provide readers the confidence that the method is capable of handling a more complex set of problems. The First Stage and the First Space Element The first space element is defined by the range of application of Modes L1 and P1 (dimensional customization of beam length and standardization of beam cross-section, respectively). These ranges of application are defined by the decision variables ∆L1 and ∆P1. Thus, two design variables are commonalized in the first space element: the raw length, Lo, and the height of the beam cross-section, h. The focus in this first decision stage is the determination of ∆L1 and ∆P1 that will minimize the average cost (Equation 3) and the average deflection (Equation 4). The total average cost of the entire market space is determined by summing the cost of each first space element, S1.  1  N1 Cavg = [3]  ∑ DS1 ,i Cmaterial , S1 ,i + Corder , S1 ,i + Cwaste , S1 ,i  Dtot  S1,i  The total average deflection of is determined in a similar manner:

(

)

1 N1 w avg = ∑ wavg S 1,i N1 S1 ,i

[4]

where the properties with subscript S1,i are of a single first space element. N1 is the number of first space elements in the entire space, S, and is calculated by:

N1 =

S S1

=

∆L∆P

[5]

∆L1∆P1

Dtot is the total demand of the entire space, calculated by:

Dtot =

Pmax =5000

∑

Lmax =10

∑

i = Pmin =1000 j = Lmin = 0.5

D( L j , Pi )

[6]

In order to find the best compromise between these two objectives, utility functions need to be formulated for each objective, following the process outlined in Section 3.6. Assuming that the designer has monotonically decreasing, risk-averse preferences, the following utility values are assigned to specific variable via lottery questions: Table 3: Cantilever Beam Utility Function Assessment Utility Description (Fernandez et al., 2001) Value 0 The decision-maker’s unacceptable attribute level – beyond which he/she is unwilling to accept an alternative. 0.25 The decision-maker is indifferent between obtaining a design alternative with an ‘undesirable’ attribute value for certain and a design alternative with an 50-50 chance of yielding either a tolerable or an unacceptable attribute value.

Cost ($)

Deflection (m)

75

-0.15

50

-0.08

Platform Design for Customizable Products as a Problem of Optimal Access

0.5

0.75

1

14

The decision-maker is indifferent between obtaining a design alternative with a ‘tolerable’ attribute value for a certain and a design alternative with a 50-50 chance of yielding either an unacceptable attribute value or an ideal attribute value. The decision-maker is indifferent between obtaining a design alternative with a ‘desirable’ attribute value for certain and a design alternative with a 50-50 chance of yielding either a tolerable or an ideal attribute level. The decision-maker’s ideal attribute level – beyond which the decision-maker is indifferent to further improvements in the attribute.

25

-0.05

8

-0.008

1

0

The resulting utility functions are plotted below in Figure 11. 1.2

1 0.9 0.8 0.7 0.6 U(C) 0.5 0.4 0.3 0.2 0.1 0

1 0.8 U(w) 0.6 0.4 0.2 0

0

20

40 C ($)

60

0

80

0.05

(a)

0.1 w (m)

0.15

0.2

(b)

Figure 11: Utility Curves for (a) Cost and (b) Maximum Deflection

Polynomial curves are fitted to the plotted points; the resulting utility equations are:

u (Cavg ) = 0.0001C − 0.0203C + 0.966

[7]

u ( wavg ) = 30.68w2 + 10.79 w + 0.9284

[8]

2

It is assumed that there is no interaction between a designer’s preferences for the different attributes. As a result, the designer’s preferences can be modeled with the additive multi-attribute utility function assumptions. The resulting expected utility function is: U = k C uC + k w u w [9] The scaling constants are determined by solving the following set of equations:

kC u (C0 ) + kwu ( wo ) = kC u (Cq ) + k wu ( w1 )

[10]

kC u (C0 ) + kwu ( w0 ) = kC u (C1 ) + kwu ( wq )

where the subscript 0 represents the value of the attribute at a utility of 0, the subscript 1 represents the attribute’s ideal value, and the subscript q, represents the value of the attribute at which a designer is indifferent between two alternatives (C0, w0) and (C1, wq). After solving these equations for our example, the scaling constants are kc = 0.55 and kw = 0.45. Following the final step of the formulation of the u-cDSP, the following deviation function to be minimized is:

(

)

(

Z = 1 − ( E[U (C )] + E[U ( w)]) = kC d C− − d C+ + k w d w− − d w+

)

The resulting decision formulation for this first stage is presented in Figure 12.

[11]

Platform Design for Customizable Products as a Problem of Optimal Access

Given:

Find:

Satisfy:

15

Decision 1: ∆L1, ∆P1 The two-dimensional market space S = {(L, P)} The material of the beam Mode L1: Dimensional customization of beam length Mode P1: Standardization of the beam cross-section The value of decision variable ∆L1, ∆P1 The value of h The deviation variables, dC1-, dC1+, dw1- and dw1+ 0 ≤ ∆L1 ≤ 9.5 ;

Bounds:

σu

Constraints:

≥

SF −

+

−

+

0 ≤ ∆P1 ≤ 4000

PL 6bh

2

dC1 , dC1 ≥ 0 d w1 , d w1 ≥ 0

; −

+

−

+

;

d C 1 id C 1 = 0

;

d w1 i d w1 = 0

−

+

−

+

E[u (Cavg )] + d C 1 − d C 1 = 1

Goals:

(

E[u ( wavg )] + d w1 − d w1 = 1 −

+

)

(

−

+

Minimize: Z1 = kC d C 1 − d C 1 + k w d w1 − d w1

)

Figure 12: Formulation of the Multi-Stage Compromise Decision Support Problem for the Cantilever Beam; First Stage Element

In order to evaluate the values of cost and deflection of the first subspace, the values of raw length and the height of the beam to be commonalized across the space must be determined. In order to satisfy the property of near-decomposability of hierarchic systems, the choice of these design variables for each space element must be independent of the choice for the other space elements. The decision of appropriate values to be commonalized is based on the largest values of strength and length of the subspace, or geometrically the upper-right cornerpoint of each first space element, since its solution will be sufficient for all variants within that space. For example, the value of L0 to be commonalized that will minimize cost (it has no relationship with beam deflection) is simply the largest length in the subspace. Unfortunately, the value of height of the beam, h, is not as easy to determine. Since this variable has direct effect on both goals, a ucDSP is formulated in order to determine h for each first space element. The Second Stage and the Second Space Element The second space element is defined by the range of application of Mode P2: Commonalization of Beam Material. There are two decision variables for this second stage, the range of material commonality, ∆P2, and the selection of material for each second space element. Similar to the first stage, the focus of the second decision stage is the selection of a value of ∆P2 that minimizes the average cost and deflection of the variants in the space of customization. The cost of each second space element can be calculated as the sum of the cost of the first space elements contained within itself: N1,2

CS2 = ∑ CS1 ,i

[12]

i =1

where CS1,i is the cost of each first space element, and N1,2 represents the number of first space elements within a second space element: S ∆P ⋅ ∆L1 ∆P2 N1,2 = 2 = 2 = [13] S1 ∆P1 ⋅ ∆L1 ∆P1 The same logic applies to the total deflection of each second space element: N1,2

wS2 = ∑ wS1 ,i i =1

[14]

Platform Design for Customizable Products as a Problem of Optimal Access

16

Similar to the first space, the total cost of the beams in the second space element is simply the sum of the product of the demand and the cost of each beam within the space. The average cost of the entire customization space is then calculated as:  1  N2 Cavg = [15]  ∑ DS2 ,i CS2 ,i  Dtot  S2,i  The average deflection of the customization space is therefore: 1 N2 wavg = ∑ wS2 ,i [16] n S2 ,i where N2 is the number of second space elements, S2, in the entire space, S, and is calculated by:

N2 =

S

=

S2

∆V ∆ P

[17]

∆V2 ∆P2

The determination of each common material and the range of its commonalization, ∆P2, are based on the compromise between two inherent tradeoffs: the material cost, and the order cost. The formulation of the decision of the second stage is provided in Figure 13. Given:

Find:

Satisfy:

Decision 2: ∆P2 The two-dimensional market space S = {(V, P)} The value of ∆V1, and ∆P1 from the first stage The value of h Mode P2: Commonalization of the Radius The value of decision variable ∆P2 The material of the beam The deviation variables, d2- and d2+ Bounds:

0 ≤ ∆P2 ≤ 20

Constraints:

∆P1 ≤ ∆P2 ≤ 20

σu SF

PL

≥

6bh

−

+

−

+

2

dC 2 , dC 2 ≥ 0 dw2 , dw2 ≥ 0

; −

+

−

+

;

d C 2 id C 2 = 0

;

d w 2 id w 2 = 0 −

+

−

+

E[u (Cavg )] + d C 2 − d C 2 = 1

Goals:

(

E[u ( wavg )] + d w 2 − d w 2 = 1 −

+

)

(

−

+

Minimize: Z 2 = kC d C 2 − d C 2 + k w d w 2 − d w 2

)

Figure 13: Formulation of the Multi-Stage Compromise Decision Support Problem for the Pressure Vessel; Second Stage Element

With the decisions properly formulated we are able to move onto the final step of the PPCTM, the solution of the multi-stage decisions. 3.7 Step 6: Solve the Multi-Stage Utility-Based Compromise Decision Support Problem The final step of the PPCTM is the solution of the multistage utility-based compromise Decision Support Problem formulated in the previous step. Fundamentally, the key decision of the application of the PPCTM is the determination of the values of the ranges of each mode for managing product variety (∆r(i)) that provide the smallest deviation of the expected utility function from the ideal value of 1. As can be inferred from the discussion in the previous section, the design decisions of this problem are coupled with one another. For example, a decision about the beam height of a first space element cannot be made without knowing the beam’s material; however this cannot be determined without knowledge of the size of the first space element. While the tenets of constructal theory suggest that we solve this problem in a sequential manner (Section 2.1), we seek an appropriate solution method that will decouple these decisions. In the past we have employed such methods as dynamic programming (Herandez, et al., 2003),

Platform Design for Customizable Products as a Problem of Optimal Access

17

linear physical programming (Hernandez et al., 2002), and the construction of response surfaces (Hernandez, 2001). Since this problem is relatively simple, we will use an exhaustive search algorithm. A graphical representation of the solution method for the cantilever beam example is presented in Figure 14. This solution method involves iterating through values of the modes of managing variety (∆L1, ∆P1, and ∆P2), establishing the dimensions of the sub-spaces, commonalizing the design parameters (Lo, h, and beam material) across each sub-space, evaluating the objective functions, and comparing the resulting overall utility of each iteration.

Demand Scenario

∆L1, ∆P1, ∆P2

Geometric Space MN

Objective Functions

INPUT

Assign Bounds of Each Sub-Space Lmax , Pmax

Establish SubSpaces: S1, S2

Commonalize Parameters Across Each SubSpace

Determine Values of Parameters to be Commonalized

Each subspace: Determine material and h that minimizes cost and deflection from Pmax

Each subspace: Calculate Lo from Lmax

Evaluate Objective Functions Across Each Sub-Space Oj

Iterate

Evaluate Parameters to be Commonalized

Each variant: Calculate Cost from L, Lo, material, and h

Each variant: Assess deflection

Calculate Cavg,j and wavg,j

Evaluate Objective Functions Evaluate Objectives

Calculate Order Cost for Entire Space

Calculate Expected Utility Function E[u(Cavg)] E[u(wavg)]

Calculate Cavg and wavg of Entire Space

Evaluate Result

Calculate Deviation Function Z = 1 – E[U(X)]

OUTPUT

Figure 14: Solution Algorithm for Cantilever Beam Example

3.8 Results The result of the application of the PPCTM to this example problem is the hierarchic organization of the three modes of managing variety (Section 3.4), and the determination of their range of application across the product platform that provides the smallest average cost and average deflection. The extent of application of each mode and the resulting average cost and deflection for the platform is shown in Table 4. Table 4: Range of Commonality of the Modes of Managing Variety

∆L1 (m)

∆P1 (N)

∆P2 (N)

(customize beam length)

(standardize beam cross-section)

(standardize beam material)

1.0 250 250 The results inform the manufacturing enterprise that the best configuration of the modes of managing variety to minimize cost and deflection is to commonalize the raw length for every 1 m of beam length, commonalize the beam cross section for every 250 N of loading, and to commonalize the beam material for every 250 N. This space of customization is presented in Figure 15.

Platform Design for Customizable Products as a Problem of Optimal Access

18

500 B

350

P (N) A

200

∆P1=∆P2=250

50 ∆L = 1

0.5

L (m)

10

Figure 15: Resulting Space of Customization

From these ranges of application for each mode of managing variety, the specific values of the design variables are derived by the use of the PPCTM (Table 5). Table 5: Mapping Between Product Specifications and Design Variables L 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10

P 300 500 300 500 300 500 300 500 300 500 300 500 300 500 300 500 300 500 300 500

Material Pine Pine Pine Pine Pine Pine Pine Pine Pine Fir Pine Fir Fir Fir Fir Fir Fir Fir Fir Fir

h 0.19 0.215 0.245 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

Table 5 serves as a “roadmap” for a designer; any potential combination of beam length and beam strength for the considered space of customization is connected to specific design parameters, including dimensions of raw material. For example if a customer requests a beam of 1 m that can support a load of 200N (point A in Figure 15), the manufacturer would cut a 1.5m raw fir beam of cross section 0.15m x 0.19m to 1m. If another customer requests a 3.25m beam that can support a load of 400N (point B in Figure 15), the manufacturer would cut 0.25m off the length of a 0.15m x 0.25m x 3.5m pine beam. The hierarchical nature of the resulting product platform can be observed by expanding Figure 8 to relate to these results. This hierarchy, and its specific relationship to points A and B of Figure 15, is highlighted in Figure 16.

Platform Design for Customizable Products as a Problem of Optimal Access

Fir (EB, σu,B)

Pine (EA, σu,A)

h2 (0.215m)

h1 (0.19m)

h3 (0.245m)

Raw Length 2 (2.5m)

Raw Length 1 (1.5m)

Hickory (EB, σu,B)

Raw Length 3 (3.5m)

Fir (EB, σu,B)

Pine (EA, σu,A) h1 (0.19m)

h4 (0.25m)

…

19

Raw Length 1 (1.5m)

Raw Length 10 (10m)

h2 (0.215m)

Raw Length 2 (2.5m)

Point A

Hickory (EB, σu,B) h3 (0.245m)

Raw Length 3 (3.5m)

h4 (0.25m)

…

Raw Length 10 (10m)

Point B

Figure 16: Hierarchic Organization of the Modes for Managing Product Customization for the Cantilever Beam Example Problem

Validity of the results is established through checking the constraints of each formulated u-cDSP. Through observations of the history of the exhaustive search, it is found that: • the range of application of each mode of managing variety is smaller than those modes that are applied at higher levels in the hierarchy (i.e., ∆r ≤ ∆r ) i +1

i

•

−

+

−

+

d i , d i ≥ 0 and d i i d i = 0 .

The formulation of the expected utility function insures that this

constraint will be met due to the inability of the system to meet a designer’s “ideal” preference value (i.e., the system will never “over-achieve” the goal). Although this example problem is relatively simple, it serves as an excellent opportunity to demonstrate the strengths of the PPCTM. Through the solution of this example, it has been demonstrated that a designer using the PPCTM is capable of synthesizing multiple modes of managing product variety in order to develop a product platform for a line of products with multiple customizable specifications. While the synthesis of different types of managing variety techniques (i.e., the use of product scaling and modularity techniques) wasn’t explicitly shown in this example problem, the reader should be confident that a designer is capable of incorporating this event in the formulation of the method. Furthermore, the ability of a designer to synthesize multiple methods of managing product variety has been illustrated in (Williams et al., 2004; Hernandez et al., 2003; Hernandez et al., 2001) through other examples. The ability of the PPCTM to produce product architectures that are tailored for markets of non-uniform demand distributions has also been demonstrated through the solution of the example problem. This strength is further highlighted by comparing the changes in product architecture when the demand scenario is changed to the one shown in Table 6: Table 6. Product Platform as a Result of a Random Demand Distribution 500

450 400

# of beams

350

∆L1 (m) ∆P1 (N) ∆P2 (N)

300 250 200

0.5 100 150

150 100

500 425

50

350 275

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 L (m) 8 8.5 9 9.5 10

P (N)

200 125 50

In the scenario shown in Table 6, the demand is random across the market space. This model represents a scenario in which the product demand cannot be analytically modeled, which is typical of most markets of customized products. By comparing Table 6 and Figure 6, it is observed that more products are being demanded more uniformly across the market space. As a result, the penalties resulting from

Platform Design for Customizable Products as a Problem of Optimal Access

20

commonaliziation (i.e., providing a product variant a material, or a quantity of material, that is not needed for the given requirements) are more prevalent. We see that the results from the use PPCTM reflect this (comparing Table 4 to Table 6) – the ranges of commonality are smaller than those prescribed for the normally distributed demand initially described in our example problem (Figure 6). Through its application to this problem, the PPCTM has proven successfully to handle the tradeoff between platform commonality and performance. The tradeoffs between platform commonality, cost, and quality (i.e., beam deflection) are efficiently handled via the use of the u-cDSP in the PPCTM. The use of the u-cDSP provides a designer the opportunity to quantify his or her preferences and, as a result, alter the product architecture. To further illustrate this strength, the example problem is solved again with different values of for the scaling constants (kC and kw) in the expected utility function. In Table 7 we present the results of re-running the initial problem description (normally distributed demand), however, kw = 0.55 and kC = 0.45. Table 7. Product Platform as a Result of Different Scaling Constants (kw = 0.55, kc =0.45)

∆L1 (m)

∆P1 (N)

∆P2 (N)

(customize beam length)

(standardize beam cross-section)

(standardize beam material)

1.0

225

225

This new set of scaling constants represent a manufacturer’s preference over improving the beams resistance to deflection over the cost of such an improvement. Due to the manufacturer’s more relaxed preferences to cost, the cost of making numerous orders is no longer a large factor; thus the range of commonality has shrunk (relative to the results of the original problem statement, refer to Table 5). The use of the PPCTM provides a designer a rigorous mathematical framework in which to frame the complicated decisions that are encountered when handling such a difficult design process. Unlike other platform design techniques, the PPCTM does not rely on a designer’s subjective assumptions on platform issues. The number of product variants to be applied to the market space is not predetermined. Instead, all decisions involving issues of market demand, product commonality, and extent of application of each mode of managing variety are based on numerical results from resolving the inherent compromises that are the foundation of platform design. 4. CLOSURE In this chapter we present advances in the development of a theory and method for designing platforms for customizable products with multiple levels of commonality based on the hierarchic utilization of multiple approaches for managing product customization. Specifically, drawing from hierarchic systems theory and constructal theory (described in Section 2), we show that it is feasible and useful to formulate and solve the design of hierarchic platforms for customizable products mathematically as problems of access in a geometric space. The use of the Product Platform Constructal Theory Method enables a designer to develop a product platform for customizable products. We have shown that the use of the PPCTM resolves the limitations of other product platform design techniques through the solution of the cantilever beam example problem (Section 3). Specifically, a user of the PPCTM is able to accommodate the issues of: • multiple design objectives: As shown in Section 3.3, the development of the platform requires the ability to handle multiple, conflicting objectives (i.e., minimization of cost vs. minimization of beam deflection) • multiple modes of offering variety: As illustrated in Section 3.4, a designer must synthesize multiple modes of offering variety (standardizing beam material, cross-section, and customizing beam length) in order to provide a means of achieving all variants within the space of customization. The use of the PPCTM synthesizes multiple modes of offering variety through hierarchic organization in order to offer variety efficiently. • variety in multiple product specifications: As described in Section 3.2, the designer’s desire to offer variety in both beam length and beam strength is satisfied through the use of the PPCTM. • costs of manufacturing: As shown in Section 3.3 and the Appendix, the modeling of the design problem incorporates material and ordering costs associated with manufacturing. • non-uniform demand: As shown in Sections 3.2 and 3.8, the example problem is solved with both a normal distribution and a random distribution of demand.

Platform Design for Customizable Products as a Problem of Optimal Access

•

inherent tradeoffs between platform extent and performance: As described in Sections 3.6 and 3.7, the range of application of each mode of managing variety is determined systematically through the rigorous formulation of utility-based compromise Decision Support Problems. The PPCTM is the only method that can quantitatively model the relationship between the complex demand structure of customizable products and the design of a product platform (Section 3.8). The PPCTM is the only product platform development method that achieves all of these goals for a product platform design methodology. Currently, a designer developing product platforms with the PPCTM must assume that the market is fixed; i.e., the manufacturing enterprise will not vary the span of products that will be offered to the customer. This limitation can be alleviated through the infusion of robust design concepts. This augmentation, a direction of future research, would provide a designer the ability to effectively cope with changes in demand, changes in the market, and the uncertainty involved with design parameters and objectives. In this chapter and in Chapter #, “Improving Manufacturing Agility through the Development of a Production Platform for the Manufacture of Customizable Products,” we propose the PPCTM as an approach for designing platforms in order to handle variety in both the product and its production process. In this chapter we focus on the design of product platforms for customizable products, whereas our focus shifts to production process design in Chapter #. As such, the PPCTM is the only platform design technique which handles both aspects of the concurrent nature of the design of customizable products and manufacturing processes.

ACKNOWLEDGMENTS We gratefully acknowledge the support of NSF Grant DMI-0085136. Christopher Williams is a Georgia Tech Presidents Fellow and a NSF IGERT Research Fellow through the Georgia Tech TI:GER Program (NSF IGERT-0221600). The cost of computer time was underwritten by the Systems Realization Laboratory at the Georgia Institute of Technology. REFERENCES Anderson, D. M., 1997, Agile Product Develpment for Mass Customization, Irwin Professional Publishing, Chicago. Bejan, A., 1996, "Street Network Theory of Organization in Nature," Journal of Advanced Transportation, Vol. 255, No. 7, pp. 85 - 107. Bejan, A., 1997, Advanced Engineering Thermodynamics, John Wiley & Sons, New York. Bejan, A., 2000, Shape and Structure: From Engineering to Nature, Cambridge University Press. Ericsson, A. and G. Erixon, 1999, Controlling Design Variants: Modular Platforms, ASME Press, Dearborn, MI. Farrell, R. S. and T. W. Simpson, 2003, "Product Platform Design to Improve Commonality in Custom Products," Journal of Intelligent Manufacturing, 14, pp. 541-556. Fernández, M. G., C. C. Seepersad, D. W. Rosen, J. K. Allen and F. Mistree, 2001, "Utility-Based Decision, Support for Selection in Engineering Design," 13th ASME International Conference on Design Theory and Methodology, Pittsburgh, PA. Paper Number: DETC2001/DAC-21106. Fujita, K. and H. Yoshida, 2001, "Product Variety Optimization: Simultaneous Optimization of Module Combination and Module Attributes," 27th ASME Design Automation Conference, Pittsburgh, PA. Paper Number: DETC2001/DAC-21058. Hernandez, G., 2001, Platform Design for Customizable Products as a Problem of Access in Geometric Space, Ph.D. Dissertation, G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA. Hernandez, G., J. K. Allen and F. Mistree, 2002, "Design of Hierarchic Platforms for Customizable Products," 28th ASME Design Automation Conference, Montreal, CA. Paper Number: DETC2002/DAC-34095. Hernandez, G., J. K. Allen and F. Mistree, 2003a, "Design of Platforms for Customizable Products as a Problem of Access in a Geometric Space," Journal of Engineering Optimization, Vol. 35, Iss. 3, pp. 229-254. Hernandez, G., J. K. Allen and F. Mistree, 2003b, "A Theory and Method for Combining Multiple Approaches for Product Customization," Second Interdisciplinary Congress on Mass

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Platform Design for Customizable Products as a Problem of Optimal Access

Customization and Personalization, (Piller, F.T, Reichwald, R., Tseng, M, Eds.), Technische Universität München, Munich, Germany. Jiang, L. and V. Allada, 2001, "Design for Robustness of Modular Product Families for Current and Future Markets," 6th ASME Design for Manufacturing Conference, Pittsburgh, PA. Paper Number: DETC2001/DFM-21177. Kalpakjian, S., 1997, Manufacturing Processes for Engineering Materials, Addison Wesley, New York. Lehnerd, A. P., 1987, "Revitalizing the Manufacture and Design of Mature Global Product," Technology and Global Industry: Companies and Nations in the World Economy (Guile, B.R. and Brooks, H., eds.). Li, H. and S. Azarm, 2002, "An Approach for Product Line Design Selection Under Uncertainty and Competition," Journal of Mechanical Design, Vol. 124, 3, pp. 385-392. Nayak, R. U., W. Chen, J. K. Allen and F. Mistree, 1999, "A Variation-Based Methodology for Product Family Design," 26th ASME Design Automation Conference, Baltimore, MD. Paper Number: DETC2000/DAC-8876. Seepersad, C. C., F. Mistree and J. K. Allen, 2002, "A Quantitative Approach for Designing Multiple Product Platforms for an Evolving Portfolio of Products," 28th ASME Design Automation Conference, Montreal, CA. Paper Number: DETC2002/DAC-34096. Siddique, Z. and D. W. Rosen, 2000, "Product Family Configuration Reasoning Using Discrete Design Spaces," 12th ASME Design Theory and Methodology Conference, Baltimore, MD. Paper Number: DETC00/DTM-14666. Simon, H. A., 1996, Sciences of the Artificial, The MIT Press, Cambridge, MA. Simpson, T. W., 1998, A Concept Exploration Method for Product Family Design, Ph. D. Dissertation, G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta. Simpson, T. W., 2003, "Product Platform Design and Optimization: Status and Promise," 23rd ASME Computer and Information in Engineering Conference, Chicago, Ill. Paper Number: DETC2003/DAC-48717. Simpson, T. W., W. Chen, J. K. Allen and F. Mistree, 1999, "Use of the Robust Concept Exploration Method to Facilitate the Design of a Family of Products," Simultaneous Engineering: Methodologies and Applications, (U. Roy and J. M. Usher, eds.), Chapman-Hall, New York, pp. 247-278. Simpson, T. W., J. R. A. Maier and F. Mistree, 2001, "Product Platform Design: Method and Application," Research in Engineering Design, Vol. 13, pp. 2-22. Williams, C. B., J. K. Allen, D. W. Rosen and F. Mistree, 2004, "Designing Platforms for Customizable Products in Markets with Non-Uniform Demand," 16th ASME Design Theory Methodology Conference, Salt Lake City, UT. Paper Number: DETC2004/DTM-57469. Yu, J. S., J. P. Gonzalez-Zugasti and K. N. Otto, 1999, "Product Architecture Definition Based Upon Customer Demands," Journal of Mechanical Design, Vol. 121, 3, pp. 329-335. APPENDIX Briefly presented in Section 3.1, the details of the cantilever beam modeling efforts are described in this appendix. Given a specific material, beam length (L), beam cross-section (width, b, and height, h), and an endloading condition, the cantilever beam must satisfy the following constraint in order to support a given load: σu PL [A.1] ≥ 2 SF 6bh The ultimate compression stress of the beam, σu, (divided by a safety factor, SF, of 10) must be larger than the stress placed on the beam by the given load, P. The determination of the deflection of a cantilever beam under end-loading conditions follows the following relationship:

− PL

3

wmax =

[A.2] 3 EI where E is the modulus of elasticity, and I is the moment of inertia of the beam’s cross-section, calculated by:

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Platform Design for Customizable Products as a Problem of Optimal Access

I=

1

23

3

[A.3] 12 The analysis of the manufacturing of the cantilever beam (ignoring all labor and plant costs) includes two components: material cost and order cost. The material cost includes both the cost of the material used for each beam and the cost of the material wasted by cutting the raw beam lengths to the required dimensions. The cost of material (Cmaterial) is therefore given by: Cmaterial = cmaterial Lbh [A.4] where cmaterial is the cost per m3 of the material (see Table X for specific values), L is the desired beam length, b is the width of the beam (held constant at 0.15m), and h is the height of the beam. The cost of wasted material (Cwaste) is given by: Cwaste = cwaste ( Lo − L) [A.5] where Lo and cwaste are the length and cost per m ($0.5 per m), respectively of a raw, uncut beam. There is also a cost associated with ordering the material, Corder. This cost is based on the number of different sized beams of raw material is ordered; it is not related to the quantity of beams ordered. This cost is tabulated using Equation A.11. Corder =

bh

N order

∑ $25

[A.6]

i =1

where Norder is the number of distinct sizes of beams required for the manufacturing process (i.e., the number of distinct values of Lo). The total cost is the sum of these three costs:

Ctotal = Cmaterial + Corder + Cwaste

= (cmaterial Lbh) + (corder N order ) + cwaste ( Lo − L)

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[A.7]