[American Institute of Aeronautics and Astronautics 51st AIAA/ASME/ASCE/AHS/ASC Structures,...

American Institute of Aeronautics and Astronautics

1

Comprehensive Product Platform Planning (CP3)

Framework: Presenting a Generalized Product Family

Model

Souma Chowdhury1, Achille Messac

2

Rensselaer Polytechnic Institute, Troy, NY, 12180

and

Ritesh Khire3

United Technologies Research Center, East Hartford, CT, 06118

Development of a family of products that satisfies different sectors of the market

introduces significant challenges to today’s manufacturing industries – from development

time to aftermarket services. A product family with a common platform paradigm offers a

powerful solution to these daunting challenges. The Comprehensive Product Platform

Planning (CP3) framework formulates a flexible product family model that (i) seeks to

eliminate traditional boundaries between modular and scalable families, (ii) allows the

formation of sub-families of products, and (iii) yield the optimal depth and number of

platforms. In this paper, the CP3 framework introduces a solution strategy that obviates

common assumptions; namely (i) the identification of platform/non-platform design

variables and the determination of variable values are separate processes, and (ii) the cost

reduction of creating product platforms is independent of the total number of each product

manufactured. A new Cost Decay Function (CDF) is developed to approximate the reduction

in cost with increasing commonalities among products, for a specified capacity of

production. The Mixed Integer Non-Liner Programming (MINLP) problem, presented by

the CP3 model, is solved using a novel Platform Segregating Mapping Function (PSMF). The

proposed CP3 framework is implemented on a family of universal electric motors.

I. Introduction

product family consists of a set of products that share certain common features that are embodied in what is

called a platform. Different products within the family are produced by customizing specific additional features

on the platform. By doing so, a group of related products can be derived from a common product platform to satisfy

a variety of market niches. Also, sharing of a common platform by different products is expected to result in: (i)

reduced overhead, (ii) lower per product cost, and (iii) increased profit. The key to a successful product family is the

effectiveness of the product platform around which the product family is derived. By sharing components and

production processes across a platform of products, companies can develop different products efficiently. This

approach also increases the flexibility and responsiveness of their manufacturing processes and takes away market

share from competitors that develop one product at a time. In addition, the domain of a product platform planning

can be extended to multiple sets/series of products; thereby providing the flexibility of creating sub-families as well.

For example, the automobile company, General Motors (GM), produces an extensive family of cars (individual

products) under several brand names (sub-families), such as Buick, Cadillac, Chevrolet and GMC. The identification

of commonalities across entire family of products can prove to be more beneficial (economically) than restricting

the product platform planning to each individual set of products (individual sub-families).

1 Doctoral Student, Department of Mechanical Aerospace and Nuclear Engineering, Student Member of AIAA. 2 Professor, Department of Mechanical Aerospace and Nuclear Engineering, 110 8th St, JEC 2049, Lifetime Fellow of AIAA. 3 Sr. Research Engineer, United Technologies Research Center, 411 Silver Lane, Senior AIAA Member.

A

51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<BR> 18th12 - 15 April 2010, Orlando, Florida

AIAA 2010-2837

Copyright © 2010 by Achille Messac. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


2

A. Product Family Design (PFD)

In the field of management, product platform research is often addressed in terms of qualitative product planning

problems1, 2

. While these product planning tools do support managerial and strategic decision making, they do not

offer practical support to system-level designers who must develop a product architecture to deliver the different

products while sharing parts and production steps across the products3. In particular, designers must translate these

leveraging strategies into useful customer requirements to guide platform-based product development. These issues

are addressed in the engineering design domain; and employ quantitative methods. A characteristic distinction

between management and engineering approaches is that the former typically rely on conceptual and/or qualitative

methods, while the latter typically rely on quantitative formal/optimization methodologies.

Over forty optimization-based Product Platform Planning approaches and formulations have been proposed in

the literature, which includes the methods summarized by Jiao et al.4 (until 2006) and other approaches proposed in

the recent years (2006 – 2010). These papers often present diverse objectives and initial assumptions that might not

readily apply to a broader scenario. Little work has been done to develop a formal framework to coherently address

different problem scenarios. In this paper, we introduce the new Product Platform Planning (CP3) framework to

address these issues from a broad perspective. In other words, the proposed CP3 framework simultaneously presents

an encompassing product family model and a generic design optimization methodology, which addresses the

different classes of problems.

Depending on their design, product families have traditionally been classified as (1) modular, or (2) scalable. In

a scalable product family, different products in the family are developed by scaling the non-platform features such

that each product satisfies a unique set of requirements. In a modular product family, distinct modules are added or

substituted (on a common platform) to develop different products5, 6

. A popular example of a modular product

family is the series of Sony Walkmans7, whereas a successful example of a scalable product family is Boeing's 777

aircraft series8. Recently, major automobile manufacturers have made efforts towards the use scalable product

families9.

B. Existing Research in PFD: Important Aspects and Limitations

1. Scale Based Product Families

Under prevailing approaches to the design optimization of product families, two critical decisions typically made

are: (i) the selection of platform and scaling design variables, and (ii) the determination of the values of these design

variables. The selection of platform and scaling design variables is combinatorial in nature; while determining the

values is continuous in nature. The combination of combinatorial and discrete aspects makes this design

optimization of product families a challenging one10

. In the literature, researchers have proposed two major classes

of methodologies to address these challenges: (i) the two-step approach, and (ii) the exhaustive approach.

A comprehensive list of different “two-step” methods can be found in Ref. [11]. In all of these methods, the

selection of platform and scaling design variables is performed separately from the optimization of the product

family, which can potentially introduce a significant source of sub-optimality. In the second class of methods,

namely the exhaustive search technique, multiple product families (each containing a unique combination of

platform design variables) is individually optimized and mutually compared. In such methods, the number of

possible product families (and the number of optimization problems) increases with the number of design variables:

(a) Two-Step Approach (b) Exhaustive Approach

Figure 1. Existing methods to design scale-based product families

Combinatorial

in nature

Continuous/Discrete

in nature

Select platform and

scaling design

variables

Determine optimal

values of platform and

scaling design variables

Step 2 Step 1

Platform/Scaling

Combination #1

(optimized)

Platform/Scaling

Combination #N

(optimized)

Compare

all N

optimal

designs and

select

overall

optimal


3

n design variables lead to 2n possible combinations of platform and non-platform design variables

12. Hence, these

methods may become computationally prohibitive for systems with a large number of design variables. The

processes of a typical two-step approach and a typical exhaustive approach are depicted in Fig. 1a and 1b,

respectively.

2. Modular Product Families

The design process of module-based product family is conceptually divided into the following three levels (i)

Architectural level: to establish a system structure and its variations (i.e., modules architecture), (ii) Configuration

level: to establish standard configuration(s), and it's variations of products and modules, and (iii) Instantiation level:

to develop a practical product family through variable quantification and combinatorial selection of the modules.

The selection of architecture and configuration has great influences on product variety optimality. However, it is

generally difficult to formulate the former tasks into a mathematical form; and descriptive or prescriptive methods

are used for this purpose. Stone, et al.13

present a heuristic method to identify modules for these product

architectures; which was later extended14

to identify functional and variational modules within a product family.

This latter work is foundational to some methods for developing modular product architectures. Comparisons of

methods for modularizing product architectures can be found in Ref. [15]. The instantiation task level is composed

of the following two phases (i) Variable quantification: To develop modules across models by quantifying variables

against acceptable ranges of specifications, and (ii) Combinatorial selection: To develop models by selecting

practical combination from feasible ones. The product family determination includes the two aspects. Based on these

aspects, module-based product family optimization problems can be categorized into the following three classes: (i)

Optimization of module attributes under fixed module combination, (ii) Optimization of module combinations using

predefined module candidates, and (iii) simultaneous optimization of module attribute and module combination.

The vast majority of the approaches for solving these optimization problems require specifying the platform

(fixed module combination, i.e., Class 1) a priori to the optimization in order to reduce the design space and make

the problem more tractable. Most other optimization approaches are geared toward Class 2 optimization problems,

e.g. Ref [16]. The assumptions involved in these two classes may lead to sub-optimal module based product

families, since designers might prefer to use optimization to simultaneously explore various module combinations

and module attributes. Very few optimization approaches exist to solve Class 3 type optimization problems, e,g. Ref.

[17].

3. Recent Generic Product Family Design (PFD) Approaches

Two PFD approaches, developed in recent years, that can be applied to a wider array of product family problems

(both scalable and modular) are: (i) the Selection Integrated Optimization (SIO) approach, and (ii) the Genetic

Algorithm (GA) approach. The SIO approach introduced by Khire et al.18

addresses the sub-optimality of two-step

methods by integrating the (i) platform identification process, and the (ii) product family optimization process. A

new Variable Segregating Mapping Function (VSMF) converts the discrete combinatorial process (of platform

identification) into a continuous process. This presents a robust and computationally inexpensive (compared to other

single stage methods) product family design optimization framework. Nevertheless, the scope of application of this

method is restricted by the assumption – a platform is formed only when the value of a design variable can be

maintained fixed across all products in the family.

The other class of recent PFD methods uses Multi-Objective Genetic Algorithms (MOGA) to design a product

family. A single stage approach, (without apriori platform identification) based on a decomposition solution strategy

that uses the binary Non-Sorting Genetic Algorithm-II (NSGA-II), is presented by Khajavirad et al.19

. This method

demonstrates flexibility in allowing the formation of a platform, whenever a design variable (value) is shared by

more than one product, and not necessarily all products in the family. This eliminates the all-common or all distinct

restriction pertaining to selection of platform /non-platform design variables. The significant computational expense

of the binary GA approach (especially in the case of large scale problems) is dealt with using a parallelized sub-GA

solution strategy. The flexibility in platform creation is also demonstrated by Chen et al.20

, a PFD method that uses a

2-Level Chromosome Genetic algorithm (2-LCGA). Chen et al.20

presents an information theoretical approach that

incorporates fuzzy clustering and Shannon’s entropy to identify platform design variables. The platform creation is

followed by performance optimization of the product family. Consequently, Chen et al.’s20

approach demonstrates

the attributes of “two-step” methods and is likely to yield sub-optimal solutions in certain cases.

In addition to the above limitations, most existing methods (including recent methods) assume that each product

is comprised of all the design variables involved in the family. This restricts the application of the method to

modular product families where different products in the family might be comprised of different modules;

subsequently, different products will comprise a different set of design variables, e.g. in a family of cars, a sports


4

coupe might have an Anti-Braking System (ABS) whereas a midsize sedan might not. Most of the existing

approaches use a penalty function or a commonality metric to account for the reduction in manufacturing costs,

resulting from platform planning. This cost penalty is estimated either using the differences in the magnitude of each

design variable across different products or using the aggregate of the number of commonalities among products.

Such an approach is limited by the following inherent assumptions (rarely stated explicitly):

• The cost reduction resulting from creating product platforms is independent of the total number of each product

manufactured.

• The cost reduction (attributed to the selection of platform design variables) is equally sensitive to each design

variable comprising the product.

At the same time, existing methods generally do not account for the influence (might be positive, negative or

neutral) of product platform formation (increasing product commonalities) on the direct cost of manufacturing. The

direct cost of manufacturing generally includes the cost of material, and the cost of manufacturing labor. As a result

of these assumptions, a product family design is likely to further deviate from commercial scenarios.

C. Comprehensive Product Platform Planning Framework (CP3)

Through the CP3 framework, our objective is to address all the pertinent attributes of product family design. This

framework consists of two components: (i) a comprehensive and flexible product family model that presents a

Mixed Integer Non-Linear Programming (MINLP) problem, and (ii) a design optimization strategy to solve this

MINLP problem. In the remainder of the paper, these components will be referred to as the “CP3 model”, and the

“CP3 optimization”, respectively. The CP

3 model does not distinguish between scalable and modular product

families. CP3 does not restrict the selection of platform/non-platform design variables to the all-common/all-distinct

scenario19

that allows platform formation only when a variable is shared by all products in the family. This

flexibility allows for the formation of sub-families within the product family, which follows from the classification

of design variables as platform, sub-platform and non-platform variables. A precise definition of each class of

variables will be provided in Section II-B.

Most importantly, the CP3 model presents a generalized MINLP problem, the solution of which would

simultaneously yield (i) the optimal identification and number of platforms, and (ii) the appropriate magnitudes of

the design variables. The presence of a combination of integers and continuous variables can be attributed to the

process of platform identification. The non-linearity of the problem can be primarily attributed to the non-linear

nature of the cost objective for a product family; typically, design of products involves non-linear performance

functions and non-linear constraints as well. To the best of authors’ knowledge, such a comprehensive and

universally applicable product family is unique in the literature.

The CP3 optimization strategy introduces the Platform Segregating Mapping Function (PSMF) to solve the

MINLP problem. In the case of a generic product family comprising N products and a total of n design variables

(that might be discrete or continuous) for each product, the resulting MINLP problem contains at least

( )2 1 2Nn C n N N× = × − integer variables. Typical MINLP solvers are usually not effective at solving such a high

dimensional non-linear optimization problem. The PSMF approach is inspired by the concept of converting the

mixed integer problem into a continuous variable problem, developed in the Selection Integrated Optimization (SIO)

method18

. However, SIO solves an approximate product family model that is restricted by the all-common/all-

distinct assumption19

. PSMF uses a Gaussian function to perform a continuous approximation of the integer

variables (that control the selection of platform/non-platform design variables).

The dependence of the cost of a product on the number of similar products (with respect to design variable

values) is expressed using a Cost Decay Function (CDF). The use of this generic CDF avoids the traditional

assumptions: the cost reduction resulting from product commonalities (i) is independent of the capacity of

production and (ii) is not sensitive to the specific design variable. For example, in an automobile family, the cost

reduction in the case of three different cars sharing a common transmission platform would be the same as the cost

reduction in the case of three different cars using a common windshield wiper, which is not the case in practice.

The CP3 optimization strategy implements the Particle Swarm Optimization (PSO) algorithm

21 to solve the

approximated MINLP problem. A robust constraint handling technique introduced by Deb et al.22

, and later adopted

by Chowdhury et al.23

, is employed to deal with the constraints involved in the optimization problem. The

approximated MINLP problem is observed to be a multimodal problem. PSO, being a stochastic search algorithm,

deals with multimodal problems significantly better than gradient based algorithms. Moreover, PSO is easy to

implement and involves fewer user defined parameters that need to be adjusted when compared to some of the

standard evolutionary optimization algorithms.

The following are discussed in the subsequent sections:


5

1. The Comprehensive Product Platform Planning (CP3) model

2. Cost analysis of the CP3 model

3. CP3 optimization strategy: Brief description of the Platform Segregating Mapping Function (PSMF), the Cost

Decay function (CDF), and the Particle Swarm Optimization algorithm

4. Brief Description of the universal electric motor and the optimization of a family of motors: Results and

Discussion

II. CP3 Model

A. General Formulation

The generalized CP3 model is built on the concept of the following general Mixed Integer Non-Linear

Programming (MINLP). Table 1 shows a representative family of two products comprised of three physical design

variables, each as shown in Table 1.

Table 1. A family of two products Physical Design Variable Product-1 Product-2 Integer Variables

1st variable 1

1x

2

1x

12

1λ

2nd variable 1

2x

2

2x

12

2λ

3rd variable 1

3x

2

3x

12

3λ

[For a design variable, k

jx , the superscript (k) and the subscript (j) represent the product number and the variable number,

respectively]

In Table 1, the λ variables are integer variables that are defined as

1 2

121 , if

0 , otherwise

j j

j

x xλ

== (1)

The general MINLP problem formulated to represent the design optimization of the product family shown in

Table 1 is give by

( )( )

( ) ( ) ( )( )( ){ }

2 2 212 1 2 12 1 2 12 1 2

1 1 1 2 2 2 3 3 3

1 1 1 2 2 2

1 2 3 1 2 3 1 2 3

Max

Min

s.t. 0

0, 1, 2,....,

0, 1, 2,....,

, , , , , , , ,

p

s

i

i

f Y

f Y

x x x x x x

g X i p

h X i q

Y x x x x x x

λ λ λ

λ λ λ

− + − + − =

≤ =

= =

=

{ }( ) { }

1 1 1 2 2 2

1 2 3 1 2 3

12 12 12

1 2 3

, , , , ,

, , : 0, 1

X x x x x x x

B Bλ λ λ

=

∈ =

(2)

where fp and fs are objective functions that represent the performance of the product family and the cost of the

product family, respectively. In Eq. (2) gi and hi represent the inequality and equality constraints contributed by the

physical design of the product, respectively. The first equality constraint in Eq. (2) that involves λjs can be termed

the commonality constraint. This formulation approach can be extended to a general product family comprising N

products and n design variables. In that case, the commonality constraint can be represented in a concise and

encompassing matrix format as


6

1 1

1 1

1

1

1 1

1 1

1

1

1 1

1

1

0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

= 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

T

k N

k

N Nk

k N

k N

j j

k

N Nk

j j

k N

k N

n n

k

N Nk

n n

k N

X X

λ λ

λ λ

λ λ

λ λ

λ λ

λ λ

≠

≠

≠

≠

≠

≠

Λ =

−

−

−

Λ

−

−

−

∑

∑

∑

∑

∑

∑

⋯

⋮ ⋮ ⋮

⋯

⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋯ ⋮

⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋯ ⋮

⋮ ⋮ ⋮ ⋮ ⋮

⋯

⋮ ⋮ ⋮

⋯

1 2 1 2 1 2

1 1 1

1, 2, .....,

TN N N

j j j n n n

k N

X x x x x x x x x x

=

= ⋯ ⋯ ⋯ ⋯ ⋯ (3)

The matrix Λ that can be called the commonality constraint matrix, is primarily a symmetric block diagonal matrix,

where the jth

block corresponds to the jth

design variable. This matrix is a function of the commonality matrix λ. A

more explicit representation of each block is given by

( )

1 12 1 1

1

21 2 2 2

2

1 2

1 2

=

k l N

j j j j

k

k l N

j j j j

k

j M jl l lk lN

j j j j

k l

N N Nl Nk

j j j j

k N

C

λ λ λ λ

λ λ λ λ

λλ λ λ λ

λ λ λ λ

≠

≠

≠

≠

− − − − − − Λ = − − − − − −

∑

∑

∑

∑

⋯ ⋯

⋯ ⋯

⋮ ⋮ ⋮ ⋮ ⋮ ⋮

⋯ ⋯

⋮ ⋮ ⋮ ⋮ ⋮ ⋮

⋯ ⋯

(4)

where CM( ) represents the commonality constraint matrix (Λ) as a function of the commonality matrix (λ). The

generalized commonality matrix (λ) is given by


7

11 1

1 1

1

1 1

11 1

1

11 1

1

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

= 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

N

N NN

N

j j

N NN

j j

N

n n

N NN

n n

kl

j

k

λ λ

λ λ

λ λλ

λ λ

λ λ

λ λ

λ

⋯

⋮ ⋮ ⋮

⋯

⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋯ ⋮

⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋯ ⋮

⋮ ⋮ ⋮ ⋮ ⋮

⋯

⋮ ⋮ ⋮

⋯

�

1 , if =1 and

0 , otherwise

1 , if variable is included in product-

0 , if variable is NOT included in product-

kk ll l k

j j j j

l

th

kk

j th

x x

j k

j k

λ λ

λ

≠

= ==

=

(5)

It can be observed from Eq. (5) that the commonality matrix is also a symmetric block diagonal matrix. The

parameters, kk

jλ , determine whether the j

th variable is included in product-k. In modular product families, different

products can have different types and different number of modules. Consequently, different products can be

comprised of physically different design variables. Certain design variables might be required in all product.

Consequently, the corresponding kk

jλ s would be known apriori to be equal to 1. Similarly, certain design variables

might not be relevant for a product; and hence, the corresponding kk

jλ s would be known apriori to be equal to 0.

However, if a design variable might or might not be included in a product (jth

variable for product-k) , the

corresponding kk

jλ is not known apriori, and ideally should be allowed to be determined during the course of the

product family design (PFD) optimization. The commonality matrix representation allows this PFD flexibility,

thereby avoiding traditional distinctions between scalable and modular product families.

B. Demonstration of a 4-product/5-variable CP3 Model

The proposed CP3 model is illustrated, using the example of a product family comprising 4 products. It is helpful

at this point to provide a precise definition of a product platform – “A product platform is said to be created when

more than one product in a family have the same magnitude of a particular design variable.” Table 2 shows the

sample product family. Each shade in Table 2 represents a platform. Hence, blocks in Table 1 displaying similar

shading imply that the corresponding products are members of a particular platform (share a common design

variable); and blocks displaying no shading (white) represent non-platform design variables. The platform shades

are defined in Table 3.

Table 2. Product Platform Classification Design Variable Product-1 Product-2 Product-3 Product-4

x1 � � � �

x2 � � � �

x3 � � � �

x4 � � �

x5 � � �

�: Variable is included in that product

Table 3. Platform Colors Platform Color Platform Color

P1 P3

P2 P4


8

This platform planning classifies design variables (in the entire family) into the following categories.

1. Platform design variable: A design variable that is shared by all the products in the family; e.g. variable x1 in

Table 2.

2. Sub-platform design variable: A design variable that is shared by a particular set of products (sub-family) in the

family; e.g. variable x5 in Table 2. It is also possible that a sub-platform leads to multiple sub-families; e.g.

variable x3 in Table 2.

3. Non-platform design variable: A design variable that is not shared between (among) any two or more products in

the family; e.g. variables x2 and x4 in Table 2.

The diagonal blocks of the commonality matrix, corresponding to each design variable for the product family

illustrated in Table 2, is given by

1 2 3 4 5

1 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0 1

1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0, , , ,

1 1 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0

1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1

λ λ λ λ λ

= = = = =

(6)

In principle, we can remove any row and column that have a common diagonal element of zero, such as the 2nd

row/column of matrix, λ4. The resulting commonality matrix would still remain a symmetric matrix. Equation (4)

yields the five diagonal blocks of the constraint commonality matrix (Λ) to be used in the MINLP problem, which

are

1 2 3

4 5

3 1 1 1 0 0 0 0 1 1 0 0

1 3 1 1 0 0 0 0 1 1 0 0, , ,

1 1 3 1 0 0 0 0 0 0 1 1

1 1 1 3 0 0 0 0 0 0 1 1

0 0 0 0 1 0 0 1

0 0 0 0 0 0 0 0 ,

0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 1

− − − − − − − − Λ = Λ = Λ = − − − − − − − −

− Λ = Λ =

−

(7)

C. Generalized CP3 MINLP Problem

The generalized MINLP problem for a family of N products comprising a global set of n design variables can be

stated as

( )( )

( )( )

( ){ }

1 2 1 2 1 2

1 1 1

Max

Min

s.t. 0

0, 1, 2,....,

0, 1, 2,....,

,

p

s

T

i

i

M

TN N N

j j j n n n

f Y

f Y

X X

g X i p

h X i q

C

Y X

X x x x x x x x x x

λ

λ

Λ =

≤ =

= =

Λ =

=

= ⋯ ⋯ ⋯ ⋯ ⋯

(8)


9

where the matrices Λ and λ are given by Eq. (3) and Eq. (5), respectively. It is noted that, although matrix λ is a

variable for the MINLP problem, some of the diagonal elements kk

jλ might be known apriori.

The cost of manufacturing (fc) a family of products generally depends on (i) the extent of product commonalities,

(ii) the values of the design variables, and (iii) the required tolerance on each design variable. However, the second

and the third factors are difficult to account for in a generic formulation, since they are specific to the product family

being studied/planned. Although the exact variation (trend) of the cost of production with different factors is product

dependent, the nature of variation nominally complies with the following generic criteria:

• The cost of manufacturing the whole family of products is a decreasing function of product commonalities.

• The cost of manufacturing per product decreases with the capacity of production (total number of products

manufactured)

An extensive cost analysis is presented in the next section that accounts for these criteria and describes the other

distinct aspects of the cost of manufacturing a product family.

III. CP3 Cost Analysis

A. Cost of a family of products

The cost of the family of products is expressed as

F FD FOC C C= +

(9)

CF : Cost of the family of products

CFD : Direct Cost of manufacturing (the product family), which includes (i) material cost and (ii) production

labor cost

CFO : Auxiliary Costs (for the product family), which includes (i) manufacturing overhead, (ii) non-

manufacturing costs (e.g. administrative overhead, publicity expenses, insurance, taxes and transportation

expenses)

B. Direct Cost of manufacturing

The direct cost of manufacturing strongly depends on the product design. This cost generally increases

monotonically with the capacity of production, and is given by

1

Nk k

FD PD

k

C m C=

=∑ (10)

CPDk : Direct Cost of manufacturing product-k

m k : Number of product-k to be manufactured

N : Number of types of products in the family

Typically, products are comprised of multiple components. A representative product family is illustrated in Fig. 2.


10

k k k

PD C AC C C= + (11)

CC k : Total Cost of fabricating all components in product-k

CA k : Cost of assembling the components in product-k

The total cost of fabricating all the components of product-k is given by

{ }{ }1

1 , if component- is included in product-

0 , if component- is NOT included in product-

pk i k i k

C C

i

i k

C C

i k

i k

δ

δ

=

=

=

∑

(12)

iCC

k : Cost of fabricating component-i belonging to product-k

p : Total number of components

( )

1 2

,

... ...

i k i k k k

C C

Tk k k k k

j n

C f X m

X x x x x

=

= (13)

x kj : j

th variable in product-k

n : Number of independent design variables in the product family ifC

k ( ) : A function specific to the product/component studied (to be determined).

( ),k k k k

A AC f x m= (14)

fAk ( ) : A function specific to the product studied (to be determined)

Therefore, the direct cost of manufacturing product-k is estimated by substituting Eq. (12), (13) and (14) in Eq. (11),

which yields

{ } ( ){ } ( )1

, ,p

k i k i k k k k k k

PD C A

i

C f x m f x mδ=

= +∑ (15)

Figure 2. A representative product family structure


11

It is observed from Eq. (15) that the direct cost of manufacturing per product is dependent on the number of

products manufactured, which is likely in commercial scenarios. Therefore, the direct cost of manufacturing all

products is given by (substituting Eq. (15) into Eq. (10))

{ } { } ( ){ } ( ) ( )

( )

1 1

1 2

11 22

ˆ, ,

... ...

ˆ ... ...

pNk i k i k k k k

FD C A FD

k i

Tk N

Tkk NN

C m f x f x f X m

m m m m m

diag

δ λ

λ λ λ λ λ λ

= =

= + =

=

= =

∑ ∑

(16)

m : Capacity vector. The capacity vector is an input to the product family design, which is generally

determined by balancing marketing and manufacturing objectives24

, and demand modeling25

.

fFD ( ) : A function specific to the product family studied.

The design vector X and the commonality matrix diagonal, λ̂ , can be derived from Eq. (3) and Eq. (5), respectively.

The function fFD ( ), in Eq. (16), might be a single continuous function or a piecewise continuous function that will

have the following generic characteristics with respect to m.

( ) ( )2

20 & 0 1, 2, ...,

( )FD FDk k

f f k Nm m

∂ ∂> ≤ ∀ =

∂ ∂ (17)

C. Auxiliary Cost

The auxiliary cost of manufacturing includes costs that involve a weak or no correlation with the design of

products (e.g. administrative overheads, publicity expenses). These costs also need not scale up with the capacity of

production. The auxiliary cost of manufacturing also includes manufacturing overheads that are significantly

reduced with increasing commonalities among products. Hence the latter demands special attention in planning a

commercial product family. The auxiliary cost for a family of products is given by

( )1 2 1 2 1 2

1 1 1

,

FO FO

TN N N

j j j n n n

C f X M

M M M M M M M M M M

M mλ

=

= =

⋯ ⋯ ⋯ ⋯ ⋯

(18)

M : Product of the commonality matrix and the capacity. Each element k

jM represents how many products in

the family will share design variable k

jx .

fDO ( ) : A function specific to the product family studied. This function will have the following generic

characteristics.

( ) ( )2

20 & 0 1, 2, ...,

1, 2, ..., & 1, 2, ...,

DO DO

k

jk

j k

f f k NM M

MM k N j n

m

∂ ∂≥ < ∀ =

∂ ∂

= ∀ = = (19)

Equations (16), (17), (18) and (19) provide substantial evidence supporting the hypothesis – “the capacity of

production (number of each product manufactured) should be an integral part of platform planning and not a post

process consideration.” Existing methods in product family design (PFD) do not generally take this factor into

consideration, and hence are likely to yield sub-optimal product family structures.


12

D. Cost Objective Function

The total cost of the product family is calculated from the direct cost and the auxiliary cost (substituting Eq. (16)

and (18) into Eq. (9)), and is given by

( ) ( )ˆ, , ,F FD DOC f X m f X Mλ= + (20)

In Eq. (20), the capacity vector m is an input to the PFD, vector M is calculated using m and the commonality matrix

λ, and λ̂ represents the diagonal of the matrix λ. Therefore, the cost objective can be represented as

( ) ( ) ( )ˆMin , , , ,s FD DOf X f X m f X Mλ λ= + (22)

IV. CP3 Optimization Strategy

A. Cost Decay function (CDF)

In the absence of specific information regarding the cost product family, it is difficult to obtain a reliable

expression for determining the function fs shown in Eq. (22). Therefore, in order to derive a practical solution

approach, the cost objective function can be simplified as

( ) ( ) ( )1 2,s s sf X f X fλ λ= × (23)

where functions fs1 and fs2 are independent functions of the design vector X and the commonality matrix λ,

respectively. Since it is difficult to account for the function fs1 without prior commercial information regarding the

product, we will assume the function to be a constant. An explicit expression for fs2 is also not possible for similar

reasons. However, considering the likely nature of the variation of the cost of manufacturing as shown by Eq. (17)

and (19), we propose a new decay function in this paper. Increase in (i) the specified capacity of production (m)

and/or (ii) commonalities (λ) in the product family tend to reduce the cost of manufacturing per product. The vector

M (Eq. 18) simultaneously accounts for both of these factors; hence the Cost Decay Function (CDF) that represents

the variation of the cost of manufacturing per product is defined as

( )( )11

1

2

3 2

3

1

1

cck k

j jc

cCDF M c c

c

M mλ

− −−

−= − +

− = (24)

c1 : coefficient that controls the rate of decrease of this cost contribution, per unit of a product

c2 : coefficient that provides the lower limit of this cost contribution

c3 : coefficient that provides the approximate number of products beyond which the cost contribution remains

practically constant.

The general nature of this decay function is apparent from the illustration shown in Fig. 3. Such a generic

function depicting the trend of cost decay is uniquely helpful. It can be observed from Fig. 3 that different

commercial scenarios can be reasonably approximated using this CDF, by appropriate specification of the

coefficients c1, c2 and c3. Nevertheless, if sufficient cost data is available while designing a particular commercial

family of products, a robust response surface model can be used to express the cost variation more accurately,

instead of the generic CDF. Using the CDFs, the total cost of manufacturing the whole product family is estimated

as

2

1 1

CDFN n

k k

s j

k j

f m= =

=

∑ ∑

(25)


13

B. Platform Segregating Mapping Function (PSMF)

The combination of discrete (binary numbers – 0 & 1) and continuous design variables in the CP3 model present

a classical mixed integer problem. In this paper, we propose a new Platform Segregating Mapping Function (PSMF)

to reduce/simplify the mixed integer problem into a continuous variable problem. Prior to the investigation of this

new solution methodology, it is necessary to reformulate the commonality constraint (from Eq. (8)) as

TX X εΛ ≤ (26)

where ε is the aggregate tolerance set to allow for platform creation. A close analysis of the commonality constraint

(Eq. (26)) illustrates how the commonality matrix variables ( kl

jλ ) present an “inverse proportionality relation” to the

squares of corresponding design variable difference ( ( ) ( )kl k l

j j jx x x∆ = − ). Hence the commonality matrix variables

can be approximated as an inverse square function of the kl

jx∆ s, which is however undefined at 0kl

jx∆ = . At the

same time, the design variable differences of any pair of products are not independent of each other. These issues

have been addressed by using a collection of Gaussian distribution functions (called PSMF) to represent the “inverse

proportionality relation.”

A Gaussian probability density function is given by

( ) ( )2

2exp

2

x bp x a

σ

− = − (27)

where the coefficients a, b and σ represent the amplitude, the mean and the standard deviation of the distribution,

respectively. The PSMF derived from the general Gaussian function is given by

( )2

2exp

2

k l

j jkl

j

j

x xaλ

σ

− = − (28)

Figure 2. Cost Decay Function

1 10 100 1000 100000.5

0.6

0.7

0.8

0.9

1.0

Number of products that share design variable xj

k (Mj

k)

Cost Decay Function for variable xjk (C

DFjk )

c1 = 0.1

c1 = 0.3

c1 = 0.5

c1 = 0.7

c1 = 0.9

c1 = 1.0

c2 = 0.5

c3 = 104


14

where coefficient a is equal to unity. Therefore, the commonality matrix can be represented by a function of the

design vector X, which can be expressed as

( )PSMF Xλ = (29)

The design variable values ( k

jx ) serve as the mean of the Gaussian kernels. The parameter σj modulates the width of

each kernel, and is itself controlled by the full width at one tenth maximum (∆x10), given by

( )

10

10

2 2ln10

110

x

p b x

σ∆

=

± ∆ = (30)

A representative plot of the PSMF with respect to a particular design variable (jth

variable, normalized using the

known variable limits) for a sample family of five products is shown in Fig. 4(a). In the CP3 optimization, initially,

an optimal design that maximizes performance is obtained for each product (in the family) separately. The design

variable values so determined are used to set a modified range (∆xj) for the application of the PSMF on each design

variable (jth

design variable), similar to the approach in the SIO technique18

. This modified range is used to calculate

the full width at one tenth maximum ((∆x10)j) for the jth

design variable, using

( ) 1010 jjx x x∆ = ∆ ×∆

(31)

where 10x∆ is the normalized (to a scale of 0 to 1) full width at one tenth maximum explicitly specified during the

execution of the algorithm.

The CP3 model is solved using a sequence of Nstage Particle Swarm Optimizations (PSOs), with decreasing

values of the parameter 10x∆ . This multistage optimization results in sharper Gaussian kernels with increasing

stages, rendering progressively rigorous application of the commonality constraint (illustrated in Fig. 4(b)).

Optimization is performed on the approximated MINLP problem

( ) ( ) ( )( )

( )( )

( )( )

1 1

1 2 1 2 1 2

1 1 1

Max 1

s.t. 0

0, 1,2,....,

0, 1,2,....,

PSMF

p s

T

i

i

M

TN N N

j j j n n n

w f X w f X

X X

g X i p

h X i q

C

X

X x x x x x x x x x

λ

λ

+ − −

Λ =

≤ =

= =

Λ =

=

= ⋯ ⋯ ⋯ ⋯ ⋯

(32)

where w1 is equal to 0.5, and PSMF(X) is given by Eq. 29. The process of application of the PSMF technique using

PSO can be represented by the pseudocode:


15

max

10 10

1. Optimize each product using PSO (maximizing performance)

2. Determine the range for implementing PSO on each

3. Initiate a random population of size

4. Set & 1

5. Simultaneo

jx

Npop

x x istage∆ = ∆ =

1min 1

1 1010 10 10 10 max

10

usly optimize products using PSO (solve Eq. 30)

6. Set , where

7. Choose the optimal configuration as one of the starting point

Nstageistage istage frac

frac

N

xx x x x

x

−+ ∆

∆ = ∆ ×∆ ∆ = ∆

s

8. Initiate a random population of size -1, & set 1

9. If go to step 5, else terminate solution

Npop istage istage

istage Nstage

= +

<

(33)

C. Constrained Particle Swarm Optimization (PSO) Algorithm

PSO is one of the most well known stochastic optimization algorithms21

, initially coined by an Electrical

Engineer (Russel Eberhart) and a Social Psychologist (James Kennedy) in 1995. Later, several improved variations

of the algorithm have appeared in literature, as well as used in popular commercial optimization packages. The PSO

algorithm used in this project has been derived from the unconstrained version presented by Colaco et al.29

. The

basic steps of the algorithm are summarized as

( ) ( )

1 1

1

1 2

t t t

i i i

t t t t

i i l i i g g i

x x v

v v r p x r p xα β β

+ +

+

= +

= + − + − (34)

xit : i

th member of the population (swarm) at the t

th iteration

r1 & r2 : random numbers between 0 and 1

pi : the best candidate solution found for the ith

member,

pg : the best candidate solution for the entire population and

�, �l, �g: user defined constants in the range [0, 1].

The technique used to deal with constraints is based on the principle of constrained non-domination, introduced

by Deb et al.22

. In this technique, solution-i is said to dominate solution-j if,

• solution-i is feasible and solution-j is infeasible or,

• both solutions are infeasible and solution-i has a smaller constraint violation than solution-j or,

• both solutions are feasible and solution-i weakly dominates solution-j.

If none of the above conditions apply (possible only in the case of a multi-objective problem), then both of the

solutions are considered non-dominated with respect to each other.

V. Results and Discussion

A. Test Problem Description: Universal Electric Motor

Universal motors are capable of delivering more torque than any other single phase motors, and can operate

using both direct current (DC) and alternating current (AC)26

. As a result of such high performance characteristics,

universal motors have been frequently used in a variety of applications, e.g. electric drills and saws, blenders,

vacuum cleaners, and sewing machines27

. Extensive analysis and detailed equations related to the design of the

universal electric motor can be found in Simpson et al.28

. In this example, the objective is to develop a scale-based

product family of five universal electric motors that are required to satisfy different torque requirements (Trq), as

specified in Table 4. Each motor is also subjected to other design constraints, regarding (i) the output power (Pout),

(ii) the total mass (Mtotal), (iii) the efficiency (η), (iv) the magnetization intensity (H), and (v) the ratio of the outer

radius (ro) to the thickness (t) of the stator.


16

Table 4. Torque requirements (TTTTrq) Motor 1 2 3 4 5

Torque (N/m) 0.1 0.2 0.3 0.4 0.5

The design optimization of the family of universal electric motors (in this paper) involves simultaneous (i)

maximization of the efficiency of the motors and (ii) minimization of the cost of the family of motors, chiefly

attributed to optimal platform planning. The cost minimization demands selection of platform/sub-platform design

variables and subsequent identification of sub-families within the product family. However, this is implicit to the

MINLP problem formulated in CP3 (CP

3 model). The design of each motor involves 8 design variables; the

corresponding variable limits are given in Table 5. The variable current (I) is not used in platform planning since (i)

current can adjust itself to satisfy design constraints (maximum power output), and (ii) current does not explicitly

contribute to the manufacturing expenses (for motors).

(a)

(b)

Figure 4. PSMF for the jth design variable (a) for five products in a particular stage (b) for one product

in a sequence of stages

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

Magnitude of jth design variable, xj

Commonality variable (λkl j)

product 1

product 2

product 3

product 4

product 5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

Magnitude of jth design variable, xj

Commonality variable (λkl j)

delx = 10.0

delx = 8.0

delx = 6.0

delx = 4.0

delx = 2.0

delx = 1.0

delx = 0.5

delx = 0.1


17

Table 5. Design variable limits

Design Variable Lower Limit Upper Limit

Number of turns on the armature (Nc) 100 1500

Number of turns on each field pole (Ns) 1 500

Cross-sectional area of the armature wire (Awa) 0.01 mm2 1.00 mm2

Cross-sectional area of the field pole wire (Awf) 0.01 mm2 1.00 mm2

Radius of the motor (ro) 10.00 mm 100.00 mm

Thickness of the stator (t) 0.50 mm 10.00 mm

Stack length of the motor (L) 1.00 mm 100.00 mm

Current drawn by the motor (I) 0.1 Amp 6.0 Amp

B. Test Problem Statement

The performance of the universal motor family (of 5 motors) is determined from the efficiency of each motor as

5

1

1

5p k

k

f η=

= ∑ (35)

where ηk is the efficiency of the kth

motor. The cost function component, fs1, is assumed to be a constant equal to

one. Three different cases, classified according to the number of each motor to be manufactured, are analyzed. The

capacity vectors for the three cases are

[ ][ ][ ]

Case 1: 10 10 10 10 10

Case 2: 100 100 100 100 100

Case 3: 10000 10000 10000 10000 10000

T

T

T

m

m

m

=

=

= (36)

The cost function for the universal motor family can be calculated as

5 7

1 1

100 CDFk

s j

k j

f= =

= ×

∑ ∑

(37)

The PFD optimization problem for universal electric motors can be summarized as

( ) ( ) ( )( )1 1Max 1

1, 2, ...,

300 N/m 1, 2, ...,

2 kg 1, 2,

s.t.

p s

k k

rq

k

out

k

total

w f X w f X

T T k N

P k N

M k

+ − −

= ∀ =

= ∀ =

≤ ∀ = ...,

Physi 5000 Amp.turns/m 1, 2, ...,

0.15 1, 2, ...,

1 1, 2, ...,

k

k

k

o

k

N

H k N

k N

rk N

t

η

≤ ∀ =

≥ ∀ = ≥ ∀ =

( )( )

cal design constraints

0

Commonality constraint

PSMF

T

M

T

C s wa wf o

X X

C

X

X N N A A r t L I

λ

λ

Λ =

Λ = =

=

(38)


18

Where w1 is equal to 0.5, and Tk is the torque generated by motor-k.

C. CP3 Optimization Results

Initially, optimization is performed on each motor separately (using PSO) in order to maximize performance

(function fp), subjected only to the physical design constraints specified in Eq. (38). The user-defined constants in

PSO (for step 1) are given in Table 6. A new set of variable limits is determined from the highest and lowest values

of the corresponding design variables, resulting from the optimal designs of the five products. The new design

variable limits are used to execute steps 4 to 9 of the pseudocode (shown in Eq. (33)). The user-defined parameters

for the PSMF and the CDF are specified in Table 7, and the PSO user-defined constants (for simultaneously

optimizing the product family – step 5) are given in Table 7.

Table 6. User-defined constants in PSO

Constant Value for step 1 Value for step 5

α 0.5 0.5

βg 1.4 1.4

βl 1.4 1.4

Population size (Npop) 160 400

Maximum function calls 25,000 40,000

Table 7. User-defined parameters in PSMF and CDF

Parameter Value

� 10-6

c1 0.2

c2 0.1

c3 10,000 max

10x∆ 1.0

min

10x∆ 0.01

Nstage 10

1. Case 1 Results

In this case, feasibility, with respect to the commonality constraint was achieved in stage 8 (when istage = 8).

The optimal platform planning yielded by CP3 optimization is illustrated in Table 8. Blocks in Table 8, displaying

identical colors, imply members of the same platform.

Table 8. Optimal product platform for Case 1 (mk = 10)

Design Variable Motor-1 Motor-2 Motor -3 Motor -4 Motor -5

Nc NP1 NP2 NP3 NP4 NP5

Ns NP6 P1 P1 NP7 NP8

Awa P2 P3 P2 P3 P2

Awf NP9 P4 P4 P4 P4

ro NP10 NP11 P5 NP12 P5

T P6 P6 P7 P6 P7

L NP13 NP14 NP15 NP16 NP17

It is observed from Table 10 that seven platforms have been formed. Platform P4 has four member products each,

platforms P2 and P6 have three member products each, and the other platforms have two member products each. It

is also observed that the number of turns in the armature (Nc) and the stack length of the motor (L) are non-platform

variables, whereas all the other design variables are sub-platform variables.

2. Case 2 Results


The optimal platform planning yielded by CP3 optimization is illustrated in Table 9.


19



Nc NP1 NP2 P1 NP3 P1

Ns NP4 P2 NP5 P2 NP6

Awa P3 NP7 P3 NP8 P3

Awf NP9 P4 P4 NP10 NP11

ro NP12 NP13 NP14 NP15 NP16

t P5 P6 P6 P6 P5

L NP17 NP18 P7 NP19 P7

Pk – kth platform; NPk – kth adaptive variable

It is observed from Table 11 that seven platforms have been formed. Platforms P3 and P6 have three member

products each, and the other platforms have two member products each. It is also observed that the outer radius of

the motor (ro) is a non-platform variable, whereas all the other design variables are sub-platform variables.

3. Case 3 Results


The optimal platform planning yielded by CP3 optimization is illustrated in Table 10.



Nc NP1 NP2 NP3 NP4 NP5

Ns NP6 NP7 NP8 NP9 NP10

Awa P1 NP11 P1 P1 NP12

Awf NP13 NP14 NP15 NP16 NP17

ro NP18 NP19 P2 P2 NP20

t P3 P3 NP21 NP22 NP23

L NP24 NP25 NP26 NP27 NP28

Pk – kth platform; NPk – kth non-platform variable

It is observed from Table 12 that only three platforms have been formed, which have two member products each.

Table 12 shows that design variables Nc, Ns, Awf, and L are non-platform variables, whereas design variables Awa, ro

and t are sub-platform variables.

The CP3 results, for the three cases, illustrate that platform planning is sensitive to the number of each different

products manufactured (specified capacity of production, m). The tendency of platform formation is also observed to

decrease with increasing capacity of production, which shows that the relative cost benefit of platform planning

decreases, as the number of products (of each type) manufactured (mk) increases. Figure 5(a) shows that the number

of variables that do not belong to any platform (termed adaptive variables) increase with the specified capacity of

manufacturing.

Another measure of the commonality in a product family is given by the Extent of Commonality (EC) which is

calculated as

( ) 1

Number of off-diagonal 'ones' in λ, divided by total number of off-diagonal elements

1 , 1, 2, ..., & 1, 2, ...,

-1

nkl

j

j k l

EC

k N l NnN N

λ= ≠

=

= = =∑∑ (39)

The extent of commonality (EC) is similar to the commonality objective used in the product family optimization

technique presented by Khajavirad et al.19

. Figure 5(b) shows that the extent of commonality decreases (as expected)

with increasing capacity of production. Nevertheless, these results are based on the nature of cost variation estimated

by the specified CDF coefficients (c1, c2 and c3). In the case of a commercial product, the actual variation of cost

with the number of products manufactured (if known) might follow a different trend.


20

VI. Conclusion

The Comprehensive Product Platform Planning (CP3) technique introduced in this paper lays the foundation of a

unified approach that captures the full potential of the product family paradigm. This approach is intended to avoid

the distinction between scalable and modular product families. The CP3 technique introduces an encompassing and

flexible product family model (CP3 model) that yields a Mixed Integer Non-linear Programming (MINLP) problem.

A commonality matrix is defined to classify (i) variables that are shared by certain products and (ii) variables that

are not shared by more than any one product. The robust formulation of the commonality matrix also allows

variables to be completely excluded from a product, thereby accounting for modular product families. In addition to

the above beneficial features, the commonality matrix seeks avoid the restrictions of an all-common/all-distinct

product platform system.

The commonality constraint, constructed from this matrix, allows the simultaneous (i) selection of platform

variables and (ii) optimization of the design variable values. To best of author’s knowledge, such a generalized

mathematical formulation of this selection process, which is independent of the solution strategy (to be applied), has

favorably unique features. The CP3 optimization technique converts the MINLP problem into a continuous problem

using a set Gaussian distribution functions (Platform Segregating Mapping Function, PSMF). A Cost Decay

Function (CDF) is also introduced to represent the variation of the cost (of the product family) with “the number of

each different product manufactured”. This CP3 framework is used to design a family of five universal electric

motors (with different torque requirements). It is found that the set of product platforms obtained in the case of

different “specified number of products manufactured” are distinct from each other.

For future work, the exact MINLP problem (instead of a continuous approximation) will be solved. A multi-

objective scenario will be investigated, to explore the trade-offs between product performances and net cost benefit

of platform planning. To this end an appropriate combination of deterministic and heuristic algorithms (hybrid

optimization algorithms) will be useful. Further exploration of module-based product family applications will

establish the true potential of this new method.

Acknowledgments

Support from the National Science Foundation (Awards CMMI-053330 and CMII-0946765) is gratefully

acknowledged.

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York, 1997. 2Meyer, M. H., and Lopez, L., “Technology strategy in a software products company,” Journal of Product Innovation

Management, Vol. 12, No. 4, 1995, pp. 294-306(13).

(a) (b)

Figure 5. Variation of (a) the number of adaptive variables and (b) the Extent of Commonality (EX) with the

capacity of production

100

101

102

103

104

15

17

19

21

23

25

27

29

Capacity of production (mk)

Number of adaptive variables

100

101

102

103

104

0.05

0.1

0.15

0.2

0.25

Capacity of production (mk)

Extent of commonality (EC)


21

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