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IEEE TRANSACTIONS ON COMPONENTS, PACKAGING, AND MANUFACTURING TECHNOLOGY-PART

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A, VOL. 19, NO 4, DECEMBER 1996

Development of Modular Products Andrew Kusiak and Chun-Che Huang

A b s t r a c t - M o d u l a r p r o d ~ ~refer s to products, assemblies, and components that fulfill various functions through the combination of distinct building blocks. As companies strive to rationalize engineering design, manufacturing, and support processes and to produce a large variety of products at lower costs, modularity is becoming a focus. However, modularity has been treated in the literature at an abstract level, and it has not been satisfactorily explored in industry. Furthermore, the modular technology in electronics such as multichip modules (MCM’s) aims at decreasing the average spacing between integrated circuits rather than increasing the variety of product characteristics of other types of designs. This paper develops a methodology for determining modular products while considering cost and performance. To interpret various types of modularity such as componentswapping, component-sharing, and bus modularity, a graphical representation of the product modularity is presented, while the module components of a product set are determined by a heuristic approach. With the module components known, a rulebased fuzzy representation of the module development problem is presented, while the tradeoff between performance and cost of modules is analyzed by a fuzzy neural network approach. The approach is illustrated with the example of an MCM.

Based on the interactions within a product, three categories of modularity have been defined [5]. Component-swapping modularity occurs when two or more different basic components are paired with a module creating different product variants belonging to the same product family. Component-sharing modularity is the complementary case to component-swapping modularity. Various modules, sharing the same basic component, create different product variants belonging to different product families. Bus modularity occurs when a module can be matched with any number of basic components. Bus modularity allows for variation in the number and location of basic components in a product while component-swappingand component-sharing modularity allows only for the types of basic components to vary. Design is often defined as the creation of synthesized solutions in the form of products, processes, or systems that satisfy perceived needs through mapping between functional Index Terms- Modular products, multichip modules, fuzzy requirements (FR’s) in the functional domain and the design neural network. parameters (DP’s) of the physical domain, through the proper selection of DP’s that satisfy FR’s [6], i.e., [FR] = [A] [DP], I. INTRODUCTION where [A]is the design matrix, The conceptual design phase leads to a design object deHE manufacturing industry is undergoing a major paradigm shift that is taking it from traditional manufacturing scribed schematically with a graph of functional elements and into a world of agile manufacturing. An agile corporation their interconnections [7]. A functional element corresponds should be able to rapidly respond to all changes in the market to a subsystem (mechanism), and interconnections correspond environment. Modular products and reconfigurable processes to function flows in the function-oriented modularity repreare crucial to agile manufacturing [l]. Potential benefits of sentation. Based on the functions, six types of functional similarity are considered in the identification of modular modularity include [2]-[4]: components: geometric, temporal, force, electrical, thermal, 1) economy of scale; and photometric. In this paper, the functional similarity at the 2) increased feasibility of productkomponent change; conceptual design phase is represented with interaction graphs. 3) increased product variety; Design of modular products at the conceptual level is 4) reduced order leadtime; to determine a design matrix [A] such as the functional 5 ) decoupled risks; requirement space is mapped into the modular functional 6) easier product diagnosis, maintenance, repair, and disspace. Then, the modular functional space is mapped into posal. the module component space through the consideration of Modular products refer to products, assemblies, and compothe module performance, e.g., size, speed, and weight. The nents that fulfill various functions through the combination of mapping among three different spaces is illustrated in Fig. 1. distinct building blocks (modules) [3]. Basic components refer The elements of modular functional space are classified as to the components, subsystems, and mechanisms that interact follows (based on [31): with distinct modules resulting in different product variants. BF basic functions existent in most products, e.g., the power supply in a computer; Manuscript revised May 13, 1996. This work was supported in part by the AF auxiliary functions characteristic of variant products National Science Foundation under Grant DDM-9215259 and the Rockwell resulting from the various types of modularity, e.g., the Intemational Corporation. The authors are with the Intelligent Systems Laboratory, Department of protectiodesthetic function of a lamp cover; Industrial Engineering, the University of Iowa, Iowa City, IA 52242-1527 AdF adaptive functions which are adaptive to different modUSA. Publisher Item Identifier S 1070-9886(96)06800-X. uleshasic components, e.g., the converting function of +

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1070-9886/96$05.00 0 1996 IEEE

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Functional requirement space Modular functional Module CO space space Rg. 1. Mapping in three design spaces

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0

6 10

9 2

Interaction of force Flow of electrical signal 2. The schematic descnption of the desk lamp

a computer inference card that standardizes U 0 signals; special functions that may or may not exist, e.g., the eye protection function in a computer product; CF customer-specified function, e.g., the feedback function of vision detection of a mssile specified by the Deparment of Defense. To date, the design of modular products has not received sufficient attention in literature [3], [7], [SI. It has not been explored in industry to the same degree as, for example, design for manufacturing. Approaches to modular products have only been considered conceptually; for example, [3] summarized the development of modular products as follows. Step 1) Clarify the task: generate specifications. A module normally fulfills several main functions. Step 2) Establish functional structure: subdivide the main functions into a minimum number of similar and recurring subfunctions (BF, AF, AdF, SF, and CF) based on two constraints: i) the functional structures of the product variants considered for modularity must be logically and physically compatible; SF

Fig 3. An example of interaction graph

ii) the subfunctions determined must be interchangeable. Step 3) Determine a methodo tions. Determine s o h plementation of the v

Step 4) ules and basic com

Step 5 ) Review the constraints. Approaches need to b represent modularity, opt the impact of modularity on and management systems. may differ; the modular MCM’s is to decrease the average spacing between integrated circuits (IC’s) [9], rather than increas product characteristic of agile to form modules early in the design proc conceptual design phase. However, the info the modules might not be fully avail requirement of a module may small size and lightweight. This paper develops a methodol-

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Module 3

Module 1

Module 2

CO) Fig. 4. Modularity represented by graphs: (a) component-swapping modularity, (b) component-sharing modularity, and (c) bus modularity.

Module 3

Module 3

Module 3

Fig. 5. Products with bus modularity.

ogy for determining modular products while considering cost and performance. Section I introduces the design of modular products. Section I1 introduces a schematic graph to represent product modularity, while the module components for a set of a products are determined by a heuristic approach in Section 111. With the module components known, Section IV presents a fuzzy neural network approach for the tradeoff analysis between performance and cost in design of modules. Section V concludes the paper.

A B

48

Mouse

11. MODELING THE PRODUCT MODULARITY In this section, modular products are represented with a schematic graph. Different types of modularity are interpreted based on this graph. The key distinction between a schematic Fig. 6. Modules of a computer frame board. graph and other types of design descriptions is that the former generally do not contain the information about the phases and determining modular components when the exact geometry and material of the design. The schematic description values of design parameters are difficult to estimate, e.g., exact is useful in representing modular products at early design dimensions.

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Product 4 (C)

Fig. 7. Interaction graphs representing different types of moduianty. (a) component-swapping modularity, ( (c) bus modularity

Chips

The schematic descri Fig. 2. 1) Components 1 and4 2) Components 6, 9, 3) Components 2 trical cord, respectively. 4) Component 3 and 8 are an interna tor, respectively. 5 ) Component 5 is the 6) Component 11 Note that the interaction o tional because components at the pull and push forces.

KUSIAK AND HUANG. DEVELOPMENT OF MODULAR PRODUCTS

Fig. 9. The interaction graph of a PCB.

Fig. 10. The interaction graph of eight types of PCB’s.

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*

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m

Fig. 11. The resulting modules for eight types of PCB’s.

TABLE I CODESOF PCB COMPONENTS

Functional block Module connector Printed wiring board connector

A. Representation of Modular Products odularity is viewed in [5] as depending on two characteristics of design: 1) similarity between the physical and functional architecture of the design; 2) minimization of incidental interactions between physical components.

The characteristics above imply two t involved in the modularity concept: 1) high degree of functional interacti 2) low degree of interactions ng components of intramodules. Based on the two relationships, an interaction graph is used to represent modular products.

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Base

Mdule base Fig. 13. A module architecture.

Interaction graph ( A , E , W ) is a schematic graph with weights on the edges, where A is the node set representing components, E is the edge set representing functional interactions, and W is the weight set of frequencies of the functional interactions among components. An example of interaction graph is shown in Fig. 3, where A = {I, m, n, ...}. E = {(I, m ) ,( m , n ) , . . .}. The weight w ( l + ) = 8 indicates the function interaction from component 1 to m occurs in eight types of products. The weight density of a subgraph is defined as the ratio of the total weight of the edges included in the subgraph over the number of edges included. The weight density determines the quality of clusters. A cluster in the interaction graph is represented as a set of connected nodes N that satisfies WdN

- wzJ >

L for all

i E N and j

$ N, N CA

(1)

where: weight density of the set N of nodes (components); weight of the intra-edge from node i E N to j $ N ; L threshold index used to include a node in (or exclude from) a cluster. For example, consider cluster (module) 3 = { I C , I, m, n } in Fig. 3. The weight density of cluster 3 is w d 3 = 8. If f is set to five, no additional nodes can be included in cluster 3. If L is set to four, then nodes a, e, t , and f would be included in cluster 3. The larger the value 2, the smaller size of a cluster. The value of the module threshold index L is determined arbitrarily. The components (nodes) that do not belong to a module (cluster) are referred to as independent components (nodes). The independent components (nodes) may be the basic components (nodes) that jointly with the modular components result in different types of modular products, e.g., basic components c and d in Fig. 3. WdN

wag

B. Interpretation of DiTerent Types of Modularity The interaction graph allows to represent different types of modularity. Note that the components of modular products correspond to the nodes of an interaction graph, the interaction functions correspond to the edges, and the clusters to the modules. BF edges of an interaction graph correspond to the

basic functions (BF’s), AF edges to the auxiliary functions (AF’s), AdF edges to the adaptive functions (AdF’s), SF edges to the special functions (SF’s), and CF edges to the customer-specified functions (CF‘s). Property I . Component-Swapping Modularity: In the interaction graph representing a module (cluster) and a basic component set (basic node set) that form component-swapping modularity. 1) The nodes in the basic node set are the leaves of the cluster 2) The weight density of the cluster equals to the total weight of all AF or AdF edges between the basic node set and the cluster. Example 1: The component-swapping modularity is illustrated in Fig. 4(a), where four products use module 1 = {a,b, e}: two of them are based on module 1 and basic component c, and the remaining two products use module 1 and basic component d. The weight density of module 1, w d M 1 = 4 = Wb,c f Wb>d = 2 2. In the automotive industry, by using different audio cassette decks, windshield glass, and wheel types with the same base body of the car, different models of cars are generated. In the computer industry, the component-swapping modularity manifests itself through matching of different hard disk types, monitor types, and keyboards with the same frame board. Property 2. Component-Sharing Modularity: A cut node is defined as a basic node providing a unique connection between clusters. Removing the cut node results in disjoint clusters. In the interaction graph representing a module (cluster) set and a basic component (node) that form component-sharing modularity. 1) The node is a cut node connecting all clusters in the cluster set. 2) The total of the weight density of each cluster in the cluster set equal to the weight of the BF edge incident to the cut node. Example 2: The component-sharing modularity is illustrated in Fig. 4(b), where eight different products use independent component y and basic component k . Four of the products are based on modules 1, independent component y and basic component k , and the remaining four products use modules 2, independent component y, and basic component k . The module set = {module 1, module 2). The cut node = k . The BF edge = (y, k ) . The component-sharing modularity in automotive industry leads to the use of the same brake shoes, alternators, or spark plugs in different product families. In consumer electronics, component-sharing arises when a common power cord or a common tape transport mechanism is used in different product families. Property 3. Bus Modularity: In the interaction graph representing a module (cluster) and a basic component (basic node) set that form bus modularity. 1) All nodes in the node set are leaves of the cluster. 2) The density weight of the cluster is greater than any weight on the AF or AdF edge between the basic node in the node set and the cluster.

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ut

where eight different products use module 3 = (1, m, n}. A. Problem Formulation

2) the maximum total weight density of

sic computer frame board, where I is the input unit, M than the threshold module-swapping modularity. The operating system ROM and

o m a global bus modularity. acceptable size and c

incident to a cluster wi in the cluster. The only

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B. Heuristic Algorithm Step 0. Initialization: List all interaction functions in an interaction graph. Specify the upper bound Nu on the number of components in a module and the threshold index &. Unlabel all edges. Step 1. Labeling: Select unlabeled edges with the same largest weight and label them; i.e., remove them from the candidates for the selection set. Step 2, Clustering: Identify the connected subgraphs with all new edges labeled. Combine the nodes in a subgraph into a cluster as long as the constraints Cl), C2), C3), and C4) are satisfied. Repeat Step 1 until no more nodes can be combined. Step 3. Identification: Identify modules corresponding to the clusters in the graph. Step 4. Classification: Classify the modules generated based on the three properties from Section 11. Step 5. Termination: Stop and output the results. Example 5. Selection of MCM Module Components: Consider the set of components mounted on eight different printed wiring boards (PWB’s). Each component is a functional block composed of a number of logic chips. Some of components are in the form of MCM’s (see Fig. 8). An example of the interaction graph for a printed circuit board (PCB) is illustrated in Fig. 9. The codes of components included in PCB are illustrated in Table I. Each component is represented with a three-field code. The set of all components and its interaction graph is presented in Fig. 10. The heuristic algorithm is applied to the components of eight types of products to determine module components. The Appendix shows the steps of the heuristic algorithm. The resulting interaction graph of the components generated by the heuristic algorithm is illustrated in Fig. 11. Result: Nine modules are formed: M1 = (P2, M5, M6, M7, M8, C73, C74, C75, C76, C77, C78, C79, C70); M 2 = (P4, M14, M15, M16, M17, C31, C32, C33, C34); M3 = {Bl, PS, C90, C91, C92, C93, C94, C96, C98, C99); M4 = (Pl, M1, M2, M3, M4, C3, C4, C7, C8); M5 = (P3, M9, M10, M11, M12, C21, C22, C23, C24, C25, C26, C27, C28); M6 = (P6, C46, C47, C48, C49); M7 = (B2, P7, C80, C81, C82, C83); M8 = {Col, C02, C05, C06). Based on the three properties of Section 11, the following types of modules are determined. Module MS with single chips SC1, SC2, and empty node form the component-swapping modularity. Module M3 with module M4 and empty node form the module-swapping modularity. Module M4 with modules M8 and M9 form the moduleswapping modularity. Module M1 with module M3 and Module M7 form the module- sharing modularity.

A

Performance attributes Fig. 15. The membership function of fuzzy set {S, SM, M, ML, L}.

6) Module M2 with components C35, C36, and C37 form bus modularity. The example of a PCB is shown in Fig. 12. Iv. DEVELOPMENT OF MODULES In Section 111, components of modular products that satisfy various functions (BF’s, AF’s, and AdF’s) were determined with the heuristic algorithm. For a module to interact with other moduleshasic components and to secure the module, additional (called auxiliary) components may be needed, e.g., cover to protect the module components. In this section, a module is interpreted as the set of module and auxiiliary components combined. A generic module of an electrical product consists of the following elements (called module elements). 1) Module Components: components which are determined in Section I11 with similar functional characteristics to fulfill various functions presented in Section I, e.g., chips with similar functions. 2) Base: component(s) to package module compoinents and provide connection (called inter-connection) for them such as substrate. 3) Cover: component(s) to protect module components and maintain a suitable environment such as heal dissipation in an MCM package. 4) Z/O Channel(s): component(s) to connect with other moduleshasic components such as a conductor connection (called intra-connection) to printed wired board (PWB). 5) Module Base: components to connect all the elements included above and physically interact with other moduleshasic components such as PWB. The element fulfills the adaptive functions (AdF) of modular products. The example of an MCM presented in Fig. 13 illustrates the architecture of a module. Depending on the type of a product, some of the five elements may be combined. Cover and module base imake a combined component, and the elements of the cover and the base may not be needed for all modules. Rigid components are assembled together without covers to protect the components. The development of a module involves technology and cost factors of the module elements to match the performance attributes. The generic performance attributes of modules are as follows: 1) size and weight; 2 ) reliability;

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Module M1 formed

the electrical flow permitted in the interconnection and intra-connection such as twenty-four pins of output for the MCM; 4) delay or wear through the connections such as electrical resistance; 5) energy consumption; 6 ) waste such as heat dissipation. A module is usually generated as a result of the tradeoff between cost and performance attributes such as size, weight, or speed. The analysis of cost and performance tradeoffs is a task, specifically, the performance ath-ibutes are only in fuzzy terms early in the design process, e.g., small size, lightweight, and high speed. From the performance point of view, a large variety of design options available today to the design engineer preclude an exhaustive analysis of all viable alternatives. From the cost perspective, the treatment is usually even more cursory because of the complexity and uncertainty of cost before actual production. For example, up to 80% of cost of a product could be determined at the design phase 1. Hence, the determination of good alternatives to satisfy performance attributes with some fuzzy constraints at a reasonable cost (called the module development problem) is crucial in building modules. Note that in Section 111,the module components are selected by clustering with a heuristic algorithm. In this section, with the module components known, the module development

problem is how to de elements and package network approach is used. Neural networks mo

of these approaches, a fu approach [16] is us velopment. The FAM ap of an MCM.

A. The Module Developme Assume that module have been determined.

development problem is to det package (or assemb formance attribute space is mapped

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.

I

r

3

Fig. 17. Modules M2 and M3 formed.

elements are: module components, base, cover, U 0 channels, and module base.

B. Fuzzy Rules and Knowledge Representation The design rules for the module and technology selection at an early design phase may be essentially generated using fuzzy rules described in the following form: , C7 is A7. Premise CLis AI,(32 is A2, Conclusion 01 is BI, or 0 2 is B2, or 0, is B,, where 0, = (P2, P3, P4, P 5 , T , Cost). Notation: base; p2 cover: p3 U 0 components; P4 module base; P5 T package technology; cost cost of the module; size and weight of the module: c1 reliability of module: CZ capacity of the electrical flow permitted in c 3 the inter-connection; capacity of the electrical flow permitted in c 4 the intra-connection; delay or wear through the connections; c 5 energy consumption; c 6 e

a . . ,

waste, e.g., heat dissipation; 0 (01 . . . , Om} output alternatives, where 0, = ( P 2 , P3, P 4 , P5, T , Cost), .j = 1, , m, which is the maximum number of alternatives. ( b 2 , b3, b4, b g , t , cost), , j = 1, . . . , na; B3 base alternative E B-Seta, a set of possible b2 bases; cover alternative E B-Sets, a set of possible b3 covers; U 0 component alternative E B-Set4, a set of b4 possible U 0 components; module base alternative E B-Set5, a set of b5 possible base bases; T : package technology alternative E T-Set, a set of possible package technologies; Al, . . , A7 fuzzy members which are represented with linguistic values E {S, SM, M, ML, L}, where S: small, SM: small-medium, M: medium, ML: medium-large, L: large. The fuzzy rules describing the relations between the performance attribute space and the output alternative space of a module are represented as follows: c 7

R3 _ ( A 2 n A 3 n A g n A 5 n A s n A 7 ) ~ B ~ .

Fig 18. Module M4 formed.

Note that the rules may use only some of the seven performance attributes depending on type of a product. For example, size and weight are often specified as performance goals in many portable and aerospace products. The delay attribute of electrical signals in circuit blocks may effect the system speed. Increasing power may also increase the weight or the heat dissipation. The relation R is encoded with an encoding scheme called the correlation-minimum encoding which is often used in the fuzzy associative memory technology (161. Let UR,(u11* ' ' I u71U,) = UAI (u1) f' ug,(U,) where: U A ~ ( U , ) is the membership function of C,; UB, (vJ) is the membership function of 0,; n is the pairwise minimum operator. zy rule for an MCM is presented next: size requirement is small, the dissipated heat requirement is low, the inter- and intraconnection capacities are medium-high; HEN apply Cu or W as the conductor material in thick film or co-fired ceramic, the dielectric material for glass/ceramic, and the MCM-D technology.

that stores an arbitr where i = 1 to 7 and

=

based system without reasoning and fast dec the kth pattern p

icz, 0,)= {(CfI

C,")

= min{uc(c:), strength from neur

uo(

' . ' I

wzg

procedure of the fu

activation. This feature poor explanation prope models:

Work Construction and the Learning Procedure

The network for the generic module development problem is a two-layer feedforward heteroassociative fuzzy network

aerospace product may focus on the siz

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KUSIAK AND HUANG: DEVELOPMENT OF MODULAR PRODUCTS

(cl,

heat dissipation only. In this case, set of C2, c6, c7) is input to the network. The training process of a fuzzy neural network (learning from rules) is different from building conventional neural networks that learn directly from training data sets. In the fuzzy neural network, the premises of a training data set; e.g., performance attributes, are fuzzified through fuzzy memberships functions which can map any numeric measurement of these items into finite fuzzy variables. Fuzzification is actually the process of' compressing the widely distributed numeric information into a syntactic representation so that the training of a fuzzy nemal network is less computationally intensive than that of ai conventional neural network. The fuzzifiers (membership functions) can be generated from experience, statistics, or can be constructed by the neural networks approach. The example of membership function for the linguistic variables (S, SIM, M, ML, L) corresponding to the performance attributes is shown in Fig. 15.

D. Illustrative Example: MCM Notation: input space: perj6ormance attribute = {Cl, Cz, Cs, C4,

IF the inter-connection capacity is small-medium ( u - c ~ - 2 = l), 12 layer PCB is applied and cost is low-medium (Ucost-2 = 1). IF the inter-connection capacity is medium (u-c3-3 = 11, six layer laminate is applied and cost is medium (ucost-3:= 1). IF the inter-connection capacity is medium-large ( u - c ~ = -~ l),eight layer thin film is applied and cost is large (ucost-:,= 4 ,

1).

IF the inter-connection capacity is large (u-c3-5 = l),IC's eight layer is applied and cost is medium-large (ucOst-4:= 1). IF power consumption is low ( w C 6 - 1 = 1) and heat dissipation is high (u-c7-5 = l), FR4 (epoxy-'%' glass) (pz-1 = 1) is applied and cost is high (ucost-:,= 1) IF power consumption is medium (U-cg-3 = 1) and heat dissipation is small ( u - c ~ - 1 = l), Teflon-E glass (p2-2 = 1) is applied and cost is luw (ucOst-1 = 1) IF power consumption is small-medium (tkc6-2 = 1) and heat dissipation is small-medium ( ~ ~ 7 -=2 l), PI film (kapton H) (p2-3 = 1) is applied and cost is high (ucOst-5= 1) IF power consumption is large (U-c6-5 = 1) and heat dissipation is medium (u-c7+ = l), Gore-ply (p2-4 '= 1) is applied and cost is high (ucOst-5= 1)

c5,c6, c7).

For each (7% E {q-~ c1-2, ,

~ 1 - 3~ , 1 - 4c1-5) ,

= {S, SM, M, ML, L},

i = 1, . . . , 5 ;

output space: 0, = (P2, P4, Tcost), Cost E {Cost- 1, Cost-2, Cost-3, Cost-4, Cost-5)

= { S, SM, M, ML, L} Note that this example ignores the elements of cover and module base, 1"3 and Ps,because the base and U 0 connection, Pz and P4, are more important than P3 and Ps.

PZ E (FR4 (epoxy-"E' glass), Teflon-E glass, PI film (kapton H), Gore-ply}

= {PZ-1, p2-2 3 p 2 - 3 , p2-4) P4 E {Cu, Silver, Gold, Glass} = (P3-1) p3-2, p3-3, p3-4) T E {PWB, MCM-L, MCM-C, MCM-D}

E. Test Case

After the network in Fig. 13 has been trained, a test case is considered. Test Case: given a set of module components (included in M5 of Example 5) with the following performance requirements: small panel, high reliability, high intra-, and inter-connection capacity, low delay, small-medium power consumption, small-medium heat dissipation Input of the Test Case: (u-cl-1 = 1, u-cz-5 = 1, u-c3-5 = 1, u-cq-5 = 1, u-cS-1 = 1, ~ ~ 6 =- 1, 2 ~ ~ 7 =- 1) 2 Output of the Test Case (Two Alternatives): 01

= (Glass-fire p4-4 = 0.8, Teflon-E glass p2-2 = 0.8,

02

MCM-D T-4 = 0.8, u-cost-3 = 0.9} = {Gold p4-4 = 0.8, PI film (kapton H) p2-3 = 1, MCM-D T-4 = 0.6, u-cost-5 = 0.8)

The alternative ol, suggests applying Glass ( p ~ )Teflon, E glass ( p 2 - 2 ) , and MCM-D (T-4). The cost of alternative 01 is medium (u-cost index, u-cost-3 = 3, medium) with the U indicates the membership value, e.g., u-cl-5 = 1 indicates membership value (probability) of 0.9. The alternative 02, that the membership value of cl-5 is 1 (cl-5: the size is large suggests using Gold @4-4), PI film (kapton H) b2-3). and with probability 1). MCM-D (T-4). The cost of alternative 02 is high (u-cost index, Examples of training rules based on [18] are presented next. u-cost-5 = 5, high) with the membership value (probability) of IF the size is large (u-cl-5 = l), PWB is applied ( U T I = 1) 0.8. The final solution for the MCM example is the alternative and cost is low (ucost-l = 1). 01 because of its lower cost (3 < 5 ) with higher probability IF the size is medium-large (u-cl-4 = l),MCM-L is applied (0.9 > 0.8). (u~2 = 1) and cost is medium (ucost-3 = 1). IF the size is medium (u-cl-3 = l), MCM-C is applied V. CONCLUSION (uT3 = 1) and cost is medium-large (Ucost-4 = 1). In this paper, a methodology was developed for determining IF the size is small (u-cl-1 = l), MCM-D is applied UT^ = 1) ,and cost is high (ucOst-5= 1). modular products with the consideration of performance: and IF the inter-connection capacity is small (u-CQ-1= l),four cost. Products were represented with interaction graphs. The layer PWB is applied and cost is low (ucoSt-l= 1). module components of a product set were determined by

=(TI, T2, 7'3, 2'4)

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a heuristic approach. For the known module components, a fuzzy neural network approach was applied to analyze the between performance and cost of generic modules. M’s offer potential for increased circuit density leading to reduced size of electronic systems. However, MCM products are more expensive than the equivalent collection of a single chip components and PCB’s. Most products are often made to order. To increase the potential of using MCM’s at a reduced design and production cost, a heuristic approach etermine module components. A trade’s was presented using a fuzzy neural rules need to be defined by experts. Also, the input performance attributes need to be selected based on the characteristics of different types of products. The determination of eshold index f to bound the size of modules i s arbitr therefore further studies are needed. APPENDIX STEPSOF THE HEURISTICALGORITHM List interaction functions (correfield of each code). on the number of components in a ze consideration). The threshold index 2: 0.9

, DE

Present the interaction gra Fig. 10.) Step 1. Labeling: The unlabeled edge Fig. 16 are selected with the same maximu labeled, i.e., removed from the candidates for the selection set. Step 2. Clustering: Identify shadowed area of Fig. 16. CO odes in the subgraph into cluster MI = (M5, M6 C77, (278, C79, C70). The satisfied. Constraint C1) The cluster is not empty.

13 < the upper bound Constraint C3) The weight density on

Step 2. Clustering: Id

KUSIAK AND HUANG: DEVELOPMENT OF MODULAR PRODUCTS

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Fig. 20. Modules M8, M9 formed.

subgraphs into two clusters M2 and M3. M2 = (P4, M1 ,, M15, M16, M17, C31, C32, C33, C34). M3 = {Bl, P5, C90, C91, C92, C93, C94, C98, C99). The constraints Cl), C2), C3), C4) are !satisfied. Constraint C l ) The clusters are not empty. Constraint C2) The number of components in the clusters = 9 and 10 < the upper bound Nu = 15. Constraint C3) The weight density on the inter-edges minus the weight on the new intra-edge (5 4 = 1) is greater than the threshold index f = 0.9. Constraint C4) The interaction functions (edges) in the two clusters are identical (functions 3 and 9, respectively). Note that two clusters formed in this step rather than one cluster is due to constraints C2) and C4). The clusters are divided at the ledge (between P4 and B l ) such that two clusters have different functions (3 and 9) and the number of nodes in each cluster is; smaller than Nu = 15. Repeat from Step 1. Step 1. Labeling: The unlabeled edges (shadowed) in Fig. 18 are selected with the same maximum weight (= 4)and labeled, i.e., removed from the candidates for the selection set.

Step Clustering: - lentify the connected subgraph in the shadowed area of Fig. 18. Combine the nodes in the subgraph into cluster M4 = {Pl, M1, M2, M3, M4, C3, C4, C7, CS}. The constraints Cl), C2), C3), C4) are satisfied. Constraint C1: The cluster is not empty. Constraint C2: The number of components in a cluster = 9 < the upper bound Nu = 15. Constraint C3: The weight density of the inter-edges minus the weight of the new intra-edge (4 -2 = 2) is greater than the threshold index i = 0.9. Constraint C4: The interaction functions (edges) in the cluster are identical (function 0). Repeat from Step 1. Step 1. Labeling: The unlabeled edges (shadowed) in Fig. 19 are selected with the same maximum weight (= 3) and labeled, i.e., removed from the candidates for the selection set. Step 2. Clustering: Identify the connected subgraphs in the shadowed area of Fig. 19. Combine the nodes in the subgraphs into three clusters M5, M6, and M7. M5 = (P6, C46, C47, C48, C49). M6 = (B2, P7, C80, C81, C82, (283). M7 = (P3, M9, M10, M11, M12, C21, C22, C23, C24, C25, C26, C27, C28). The constraints Cl), C2), C3), and C4) are satisfied.

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IEEE TRANSACTIONS ON COMPONENTS, PACKAGING, AND MANUFACTURING TECHNOLOGY-PART

Constraint Cl) The clusters are not empty. Constraint C2) The number of components in the clusters = 5 , 6, and 13 < the upper bound NzL= 15. The weight density of the inter-edges minus the weight of the new intra-edge (4 - 1 = 3) is greater than the threshold index 3 = 0.9. Constraint C4) The interaction functions (edges) in the three clusters are identical (functions 2, 4, and 8, respectively). Note that three clusters formed in this step is due to constraints C2) and 64). The clusters are divided at the edge between P6 and B2 and the edge between P3 and B2 such that three clusters have different functions (2, 4, and 8) and the number of nodes in each cluster is smaller than Nu = 15 Repeat from Step 1. Step 1. Labeling: The unlabeled edges (shadowed) in Fig. 20 are selected with the same maximum weight (= 2) and labeled, i.e., removed from the candidates for the selection set. Identify the connected subgraphs in the . 20. Combine the nodes in the subgraphs shadowed area . M8 = (Col, C02, C05, COh}. . The constraints Cl), C2), C3), Constraint Cl) The clusters are not empty. Constraint C2) The number of components in the clusters = 4 and 4 < the upper bound Nu = 15. Constraint C3) The weight density of the inter-edges minus the weight of the new intra-edge (no intraconnection besides to M4) is greater than the threshold index f = 0.9. Constraint C4) The interaction functions (edges) in the two clusters are identical (functions 0 and 1, respectively) Note that three clusters formed in this step is due to straints C4). The clusters are divided at the edge between 2 and the edge between P3 and B2 such that three ave different functions (2, 4, and 8) and the number of nodes in each cluster is smaller than Nu = 15. No more nodes can be combined. Go to Step 3. Step 3. IdentGcation: Identify modules corresponding to the clusters in the graph. Modules Ml-M9 are formed. tion: Classify the modules generated

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