Developing a Non-gradient Based Mixed-Discrete Optimization

AIAA 2010-9174

13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference 13 - 15 September 2010, Fort Worth, Texas

13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, 13 - 15 Sep 2010, Fort Worth, Texas

Developing a Non-gradient Based Mixed-Discrete Optimization Approach for Comprehensive Product Platform Planning (CP 3) Souma Chowdhury∗, Rensselaer Polytechnic Institute, Troy, New York 12180

Achille Messac,† Syracuse University, Syracuse, NY, 13244

and Ritesh Khire‡ United Technologies Research Center (UTRC), East Hartford, Connecticut 06118 The Comprehensive Product Platform Planning (CP 3 ) framework presents a flexible mathematical model of the platform planning process, which allows (i) the formation of sub-families of products, and (ii) the simultaneous identification and quantification of platform/scaling design variables. The CP 3 model is founded on a generalized commonality matrix that represents the product platform plan, and yields a mixed binary-integer nonlinear programming problem. In this paper, we develop a methodology to reduce the high dimensional binary integer problem to a more tractable integer problem, where the commonality matrix is represented by a set of integer variables. Subsequently, we determine the feasible set of values for the integer variables in the case of families with 3 − 7 kinds of products. The cardinality of the feasible set is found to be orders of magnitude smaller than the total number of unique combinations of the commonality variables. In addition, we also present the development of a generalized approach to Mixed-Discrete Non-Linear Optimization (MDNLO) that can be implemented through standard non-gradient based optimization algorithms. This MDNLO technique is expected to provide a robust and computationally inexpensive optimization framework for the reduced CP 3 model. The generalized approach to MDNLO uses continuous optimization as the primary search strategy, however, evaluates the system model only at the feasible locations in the discrete variable space. Keywords: Binary Integer Programming, MINLP, Mixed-discrete Optimization, Platform, Product Family

I.

Introduction

A 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 could 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) ∗ Doctoral

Student, Multidisciplinary Design and Optimization Laboratory, Department of Mechanical, Aerospace and Nuclear Engineering. AIAA Student Member † Distinguished Professor and Department Chair. Department of Mechanical and Aerospace Engineering. AIAA Lifetime Fellow. Corresponding author. Email: [email protected] ‡ Senior Research Engineer. AIAA Member c 2011 by Achille Messac. Published by the American Institute of Aeronautics and Astronautics, Inc. with Copyright permission.

1 of 13 Copyright © 2010 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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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. Various optimization-based product platform planning approaches and formulations have been proposed in the literature, which includes the methods summarized by1 (methods reported until 2006). Several other approaches have been proposed in more recent years (2006 - 2010). These papers often present diverse objectives and initial assumptions that might not readily apply to a broader scenario. The Comprehensive Product Platform Planning (CP 3 ) framework, introduced by Chowdhury et al.2 seeks to coherently address different problem scenarios. The CP 3 framework presents a mathematical model of the platform planning process, which yields a Mixed Integer Non-Linear Programming (MINLP) problem. Chowdhury et al2 developed and implemented a Platform Segregating Mapping Function (PSMF) method that converts the MINLP problem into a continuous optimization problem; the approximated problem is solved using Particle Swarm Optimization (PSO). In this paper, we develop a generalized Mixed-Discrete Non-Linear Optimization (MDNLO) technique to solve the original MINLP problem presented by the CP 3 model. This MDNLO technique can be implemented using standard non-gradient based optimization algorithms, namely swarm based algorithms and evolutionary algorithms. In the case of CP 3 optimization, this MDNLO technique would allow significant reduction of the computational expense of the original MINLP problem. This paper (i) presents the reduction of the original CP 3 model into a relatively less expensive and tractable model, (ii) develops a generalized Mixed-Discrete Non-Linear Optimization (MDNLO) technique, and (ii) preliminarily explores the applicability of the proposed MDNLO technique to the reduced CP 3 model. A.

Existing Research in Product Family Design (PFD)

Depending on their designs, product families have traditionally been classified as (1) modular (module based), or (2) scalable (scale based). In a scale based product family, each individual product is comprised of the same set of physical design variables. Different products in the family are developed by scaling the non-platform features (design variables) such that each product satisfies a unique set of requirements. In a module based product family, distinct modules are added or substituted (on a common platform) to develop different products.3, 4 A popular example of a modular product family is the series of Sony Walkmans,5, 6 whereas a standard example of a scalable product family is Boeing’s 777 aircraft series.7 Major automobile manufacturers have also made efforts towards the use of scalable product families.8 Earlier scale-based product family design methodologies can be divided into two broad categories - (i) the two-step, and (ii) the exhaustive approach. A comprehensive list of different ”two-step” methods can be found in the book by Simpson.9 Both these approaches have distinct limitations that restrict their applicability to the broad scope of product family design. A handful of new methods to design scalable product families, which do not belong to the two broad categories, have also been reported in the literature;10–12 these methods address most of the limitations of the earlier methods, and also present other uniquely favorable characteristics. The design of scale based product families using physical programming13 has also been reported in the literature.14, 15 Sirisha et al.16 explored the the impact of design variable uncertainty on scaled-based product family optimization. Several well known methods exist in modular product family design, such as presented by Stone et al.,17 Dahmus et al.,18 Guo et al.,19 Fujita et al.,20 Jose et al.,21 Kalligeros et al.,22 Saron et al.,23 and Yu et al.24 B.

Existing Mixed-Discrete Optimization Approaches

A significant amount of research has been done in developing algorithms for solving Mixed-Integer NonLinear Programming (MINLP) problems. Most of these algorithms are gradient based search techniques. Three major categories of gradient based algorithms are (i) branch and bound, (ii) cutting plane, and (iii) outer approximation algorithms. A list of these algorithms, related discussion, and bibliography can be found in the websites of MINLP World25 and CMU-IBM Cyber-Infrastructure for MINLP.26 These algorithms are generally subjected to the typical advantages and limitations of gradient based optimization.27 Individual algorithms are likely to have certain unique characteristics. Binary Genetic Algorithms (GA)27, 28 have been reported to be appropriate for discrete optimization. Binary GAs convert the design variables (often, sets of design variables) into binary strings. A population of candidate solutions (each represented by a binary string) evolve over generations, through the four stages:

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(i) fitness assignment, (ii) selection, (iii) crossover, and (iv) mutation. Various fitness assignment, selection, crossover and mutation operators exist in the literature.27 One of most popular binary GA is the NSGA-II (binary), developed by Deb et al.29 Genetic algorithms have been successfully implemented on MINLP problems, representing batch plant design.30, 31 Various Ant Colony Optimization (ACO) based algorithms that deal with discrete optimization problems have also been reported in the literature.32, 33 These ACO algorithms have been applied to discrete optimization problems, such as vehicle routing, sequential ordering, and graph coloring. A discrete binary version of the Particle Swarm Optimization (PSO) algorithm was developed by Kennedy and Eberhart,34 in which a candidate solution is located at a discrete point, using a sigmoid function based probability distribution. Various other versions of discrete PSO algorithm have been reported in the literature,35, 36 as well. In the following sections, we discuss (i) the development of the generalized approach to mixed-discrete nonlinear optimization (MDNLO), (ii) the reduction of the complexity of the original Comprehensive Platform Planning (CP 3 ) model, and (iii) the exploration of discrete variable space presented by the reduced CP 3 model to the proposed MDNLO technique.

II.

Generalized Approach to Mixed-Discrete Non-linear Optimization (MDNLO)

Optimization problems that involve both continuous and discrete design variables can be termed as mixeddiscrete optimization problems. At the same time, real life engineering systems often present highly nonlinear objectives and constraint functions (also known as soft and hard criterion functions, respectively). The combination of non-linear system functions and mixed-discrete design variables lead to a Mixed-Discrete NonLinear Optimization (MDNLO) problem. Generally, this class of problems is challenging and computationally expensive to solve. Mixed-Discrete Non-Linear Optimization (MDNLO) is also closely related to integer programming and combinatorial optimization. Mixed-Integer Non-Linear Programming (MINLP) is a subclass of MDNLO. Most of the existing MINLP solvers are gradient based algorithms, which do not readily apply to complex non-linear multi-modal problems, such as presented by product family design and wind farm optimization.37 A generalized, non-gradient based, design optimization methodology can efficiently address the broad scope of MDNLO problems. In this paper, we develop a generalized heuristic technique that can use a continuous optimization (non-linear) method to deal with discrete design variables, at a reasonable computational expense. Different non-gradient based algorithms (namely, Evolutionary Optimization (EO), Particle Swarm optimization (PSO), Ant Colony Optimization (ACO), and Simulated Annealing (SA)) have uniquely helpful characteristics, suitable for distinct applications. The generalized heuristic approach to MDNLO, developed in this paper, can be implemented through a majority of these non-gradient based algorithms; hence, this approach is applicable to a broad range of MDNLO problems. In addition, this approach uses continuous optimization as the primary search strategy, and does not require any additional function evaluations at each iteration. A schematic illustration of the proposed MDNLO approach is shown in Fig.1. A majority of the non-gradient based optimization algorithms use a population of candidate solutions (designs). Over the course of iterations (generations), these candidate solutions evolve or move to better regions (in terms of objective values) in the variable space. The candidate solution movement generally occurs in a continuous domain, with the exception of binary genetic algorithms. In a mixed-discrete optimization scenario, the design space can be divided into a continuous domain and a discrete domain (corresponding to the continuous variables and the discrete variables, respectively). In the discrete domain, candidate solutions should ideally be located only at feasible discrete locations; where, feasibility pertains to the constraints imposed by the discreteness of the variable space, and not to the system constraints (hard criterion function). Hence, within a typically continuous optimization process, we can represent the location of a candidate solution (at an iteration) by the local hypercube, Hd , expressed as


 U U U xL , where xL Hd = xL m , xm 2 , x2 , . . . , 1 , x1 , (1) U L xi ≤ xi ≤ xi , ∀ i = 1, 2, . . . , m In Eq. 1, m is the number of discrete design variables, and xi ’s denote the current location of the candidate U solution in the discrete domain. The parameters xL i and xi represent two consecutive feasible values of the ith discrete variable that bound the local hypercube. The total number of vertices in the hypercube is equal to 2m . 3 of 13 American Institute of Aeronautics and Astronautics

Non-gradient based optimization

Iteration: t = t + 1 Apply continuous optimization

Evaluate system model Fi (Xci, XD-feasi)

ith candidate solution Xi

Cont. variable space location XCi

Discrete variable space location XDi

Criterion for selecting local discrete points

Approximate to nearby feasible discrete location XD-feasi

Figure 1. Process diagram of the generalized approach to MDNLO

U The values, xL i and xi , can be readily obtained from the discrete vectors that need to be specified apriori for each discrete design variable. In this paper, we develop two techniques that approximate the current discrete-domain location of the candidate solution to one of the vertices of its local hypercube, Hd (defined by Eq. 1). The values of the continuous design variables remain unchanged during this process. The primary objective of these techniques is the consistent identification of the approximating vertex (for each candidate solution, at each iteration), which would reduce the overall computational expense and increase the robustness of the algorithm. These vertex approximation techniques are discussed in the following two subsections.

A.

Nearest Vertex Approach (NVA)

The Nearest Vertex Approach (NVA) approximates the discrete-domain location to the nearest vertex of the local hypercube (Hd ), on the basis of the Euclidean distance. This approximation can be represented by ˜ = {˜ X ˜2 , . . . , x˜m } , where (x1 , x xi − xL ≤ xi − xU xL , if i i i x ˜i = xU otherwise i ,

∀ i = 1, 2, . . . , m,

(2)

˜ represents the approximated discrete-domain location (nearest hypercube vertex). An illustration In Eq. 2, X of the NVA approach for a 2-D discrete domain is shown in Fig. 2. B.

Shortest Normal Approach (SNA)

In the case of typical continuous optimization algorithms, we can determine the vector (in the discrete domain) that connects the parent/old solution(s) with the child/new solution(s) (obtained in a particular iteration). This vector is called the connecting vector, in the remainder of the proposal. Having determined the local hypercube (Hd ), the Shortest Normal Approach (SNA) approximates the discrete-domain location to the vertex with the shortest normal distance to the connecting vector. The normal distance to the

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x1 L NVA vertex

X

x2 U Local hypercube

SNA vertex

Child solution

X

x2 L x1 U

Parent solution

Shortest Euclidean Distance Shortest Normal Distance Connecting Vector Figure 2. Illustration of the NVA and the SNA approximation

ˆ j ) is given by connecting vector, h, from the j th vertex (X d T hj = U − VU T VV V , where h iT U = xˆj1 − x¯1 x , ˆjm − x ¯m ˆj2 − x¯2 · · · x h iT V = x1 − x ¯1 x2 − x¯2 · · · xm − x ¯m

j ∈ {1, 2, . . . , 2m }

(3)

In Eq. 3, U represents the vector joining the parent solution to the hypercube vertex, and V represents the connecting vector ; the parameters x ˆji ’s denote the co-ordinates of the j th vertex, and the parameters x ¯i ’s denote the co-ordinates of the parent solution. An illustration of the SNA for a 2-D discrete domain is shown in Fig. 2. For a m-dimensional discrete domain, the search for the shortest normal evaluates Eq. 3 2m number of times (for each candidate solution), which is time intensive. A bounded non-linear programming based search is likely to reduce the search time significantly. Nevertheless, the SNA preserves the direction of motion (or evolution) of the candidate solutions over generations; this property is expected to provide robustness to the overall mixed-discrete optimization strategy. C.

Modification of a Binary Integer Programming (BIP) Problem

Binary variable optimization problems are a subclass of MDNLO problems. These problems are often referred to as Binary Integer Programming (BIP) problems. Both the NVA and the SNA are favorably characterized by their ability to simultaneously (i) use the flexibility and broad scope of continuous optimization, and (ii) save computational time by evaluating the system model (criterion functions) only at the feasible discrete locations. Implementation of these approaches require explicit specification of the feasible set of discrete values for each discrete design variable, prior to optimization. From the perspective of the local vertex approximation approaches, a BIP problem, with m binary variables, would present only a single local hypercube comprised of 2m vertices. This BIP characteristic is likely to restrict the flexibility of the NVA and the SNA. Hence, we convert the mixed-BIP problem to an mixed-integer problem (mixed-IP). The binary variables can be aggregated to form one or more binary strings, as in binary genetic algorithms.27 The number of binary strings to be formed, the length of each string, and the selection of string members (i.e. which binary variable goes into which string) is likely to be problem dependent. The binary string (S), thus formed, will be converted into an integer (z), using the expression  z = s1 × 2l−1 + s2 × 2l−2 + . . . + sl × 20 ; s ∈ {0, 1} and z ∈ 0, 1, . . . , 2l − 1 (4) 5 of 13 American Institute of Aeronautics and Astronautics

where l is the length, and si is the ith element of the binary string. A binary string of length l will thereby produce a discrete variable that belongs to a feasible set of integers from 0 to 2l − 1. This “mixed-BIP to mixed-IP” conversion will be very helpful for CP 3 optimization that generally involves a large number of binary variables.

III.

Comprehensive Product Platform Planning (CP 3 )

The primary objective of the CP 3 framework2 is to address the broad scope of product family design. This framework consists of two components: (i) a comprehensive and flexible product family model that yields a Mixed Integer Non-Linear Programming (MINLP) problem, and (ii) a design optimization strategy to solve this MINLP problem. The presence of binary-integers variables can be attributed to the combinatorial nature of the platform identification process. The nonlinearity of the formulated optimization problem can be primarily attributed to the likely nonlinear nature of the system model (typical nonlinear performance functions and nonlinear constraints) for the products. The CP 3 framework, presented in this paper, has the flexibility to design most of the scale based product families that occur in practice. However, the current version addresses only the instantiation level (of the three level modular PFD approach), in the case of module based product families. A.

CP 3 Model

The principal contributions of the CP 3 model are the following: 1. This model presents a generalized and compact mathematical representation of the platform planning process, which is independent of any optimization strategy (as an evolution from most existing methods). 2. This model avoids the all-common/all-distinct restriction;11 thereby allowing the formation of subfamilies. 3. This model facilitates simultaneous (i) selection of platform/scaling design variables, and (ii) quantification of the optimal design variable values. The CP 3 model formulates a generic equality constraint (the commonality constraint ) to represent the variable based platform formation. For a product family, comprising N products and n design variables, the

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commonality constraint is expressed as X T ΛX = 0 , where  P λ1k  k6=1 1  ..   .   −λN 1 1     0     0    Λ=  0    0     0    0     0   0

···

−λ1N 1

.. . ···

.. . P Nk λ1

h

x11

x21

0

0

0

0

0 0

0 0

0 0

0 0

0 0

.. . .. .

.. .

.. .

.. .

···

−λ1N j

.. . .. .

.. .

.. .

0

0

0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0 λ1k n

0 ···

0 −λ1N n

.. . 1 −λN n

.. . ···

k6=N

0 0

0

0

P

k6=1

0 0 0 0

.. . .. .

0 0 0 0

λ1k j

.. .

1 −λN j

···

.. . 0

.. . 0

.. . 0

P

k6=N

k λN j

.. . .. . .. . 0

.. . 0

P

k6=1

0 0

k = 1, 2, ....., N ; X=

0

0 0

0 0

0 0

0 0

0 0

0 0

P

k6=N

.. . k λN n

                                  

(5)

j = 1, 2, ....., n; ···

xN 1

···

x1j

x2j

···

xN j

···

x1n

x2n

···

xN n

iT

The matrix Λ is called the commonality constraint matrix. This matrix is a symmetric block diagonal matrix, where the j th block corresponds to the j th design variable. An explicit representation of each block of the Λ matrix is given by  P 1k  λj −λ12 · · · −λ1l ··· −λ1N j j j  k6=1  P 2k    −λ21 λj · · · −λ2l ··· −λ2N  j j j   k6=2     .. .. .. .. .. ..   . . . . . .   Λj =  (6) P lk  l1 l2 lN −λj ··· −λj λj · · ·  −λj    k6=l   .. .. .. .. .. ..     . . . . . .   P Nk N1 N2 Nl λj −λj −λj · · · −λj ··· k6=N

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The commonality constraint  λ11 1  .  ..   N1  λ1    0    0   λ=  0    0    0    0    0 0

matrix is derived from the generalized commonality matrix λ, expressed as  · · · λ1N 0 0 0 0 0 0 0 0 1  .. .. . . 0 0 0 0 0 0 0 0    N · · · λN 0 0 0 0 0 0 0 0  1  .. .. .. .. ..  0 0 . . . . . 0 0 0   .. ..  1N  . 0 0 0 0 0 . λ11 · · · λ j j   .. .. .. .. .. 0 0 . . . . . 0 0 0    .. .. N1 NN 0 0 . λj · · · λj . 0 0 0    .. .. .. .. .. 0 0 . . . . . 0 0 0   (7)   0 0 0 0 0 0 0 λ11 · · · λ1N n n  ..  .. .. .  . . 0 0 0 0 0 0 0 1 N 0 0 0 0 0 0 0 λN · · · λN n n

λkl j =

(

ll l k 1 , if λkk j = λj = 1 and xj = xj 0 , otherwise

λkk j =

(

1 , if the j th variable is included in product−k 0 , if the j th variable is NOT included in product−k

)

∀ k 6= l

It can be observed from Eq. 7 that, the commonality matrix is also a symmetric block diagonal matrix. The off-diagonal elements of the commonality matrix (λkl j ) are called the commonality variables. The diagonal ) determine whether the j th variable is included in product-k. Every elements of the commonality matrix (λkk j block of the constraint commonality matrix can be expressed as a function of the corresponding commonality matrix block, which is Λj = fcon (λj ) (8) Further details of the commonality matrix formulation, and demonstration of the CP 3 model can be found in the paper by Chowdhury et al.2 B.

The Generalized MINLP Problem

For a family of N products, comprising a global set of n design variables, the generalized MINLP problem presented by the CP 3 model can be expressed as Max fperf (Y ) Min fcost (Y ) subject to X T ΛX = 0 gi (X) ≤ 0, i = 1, 2, ...., p hi (X) = 0, i = 1, 2, ...., q where Λ = fcon (λ) Y = {X, λ} h X = x11 x21 · · · xN · · · x1j 1

(9)

x2j

· · · xN j

· · · x1n

λlk j ∈ B : B = {0, 1}

x2n

· · · xN n

iT

In Eq. 9, fperf and fcost are the objective functions that represent the performance and the cost of the product family, respectively; gi and hi represent the inequality and equality constraints related to the physical design of the products; and, the matrices Λ and λ are given by Eq. 5 and Eq.7, respectively. 8 of 13 American Institute of Aeronautics and Astronautics

C.

Reduction of the Commonality Matrix

Each block of the commonality matrix (λ), corresponding to one design variable, is comprised of at least kl N (N − 1)/2 unknown commonality variables (λkl j with k 6= l). However, these λj ’s are not completely independent of each other, which leads to redundancy in the commonality matrix. The resolution of this redundancy would significantly reduce the dimensionality of the optimization problem. The interdependency of the commonality variables, pertinent to a particular design variable, xj , is illustrated in first three columns of Table 1. In this table, k, l, and i depict three different products, and the term ID means “indeterminate”.

Table 1. Interdependency of the commonality variables

If λkl j is equal to 1 1 0 0

If λki j is equal to 1 0 1 0

Then λli j is equal to 1 0 0 ID

ki li Then λkl j + λj + λj is equal to 3 1 1 0 or 1

ki li We observe from Table 1 that the feasible combinations of λkl j , λj , and λj (k 6= l 6= i) can be ensured by a constraint defined as
2 ki li λkl >0 (10) j + λj + λj − 2

Hence, a particular commonality matrix would represent a feasible combination of platform-scaling design variables for a product family, only if the corresponding commonality variables abide by the constraint defined in Eq.10. In order to determine this feasibility, the constraint in Eq.10 has to be evaluated for each unique combination of products-i, j, and k, with respect to each system design variable; therefore, the number of necessary evaluations of Eq. 10 for a candidate product family design is given by n × N C3 = n × N (N − 1) (N − 2)/6

(11)

The application of this additional constraint during the course of optimization is likely to increase the computational expense. On the contrary, a judicious use of this constraint within the scope of the MDNLO method, developed in this paper, can decrease overall computational expense significantly. To this end, we perform the “mixed-BIP to mixed-IP” conversion and further modification of the original MINLP problem presented by the CP 3 model, which is discussed in the next section.

IV.

Simplification of the CP 3 model

In this paper, we explore the simplification of the CP 3 model and the corresponding applicability of the generalized approach to MDNLO, for scale-based product families. Considering that the diagonal elements are known apriori, each block of the commonality matrix presents N (N − 1)/2 unknown binary variables. These variables are aggregated into a single binary string of length, l = N (N − 1)/2, which is then converted into an integer variable z (using Eq. 4, as discussed in Section C), such that n o z ∈ 0, 1, . . . , 2N (N −1)/2−1 (12) Thereby, a n-variable product family will produce n additional integer variables that represent individual blocks of the commonality matrix. However, based on the constraint in Eq. 10, the integer variables that correspond to infeasible combinations of commonality variables can be eliminated. This elimination produces a reduced set of feasible values for the integer variables, which should be specified prior to the application of the proposed MDNLO technique. The reduction of the CP 3 model, and subsequent implementation of the MDNLO technique is expected to significantly reduce the computational expense of product family optimization. An efficient and generic pseudocode is developed to create the appropriate reduced discrete set Z for each integer variable,

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derived from the commonality variables. This pseudocode is represented as mat = unit matrix M = N (N − 1) /2 zmax = 2M − 1 l=1 for z = 0 : zmax ∗ ∗converting z into commonality matrix block ∗ ∗ convert z into a binary string s of length M k=1 for i = 1 : N − 1 for j = i + 1 : N mat(i, j) = s(k) and mat(j, i) = s(k) k =k+1 end end ∗ ∗determining feasibility of the commonality matrix block ∗ ∗ for i = 3 : N for j = 1 : i − 2 for k = j + 1 : i − 1 if mat(j, k) + mat(j, i) + mat(k, i) = 2 discard z continue with the outermost z loop end end end Z(l) = z l =l+1 end

(13)

where mat is a commonality matrix block corresponding to a particular system design variable; and zmax represents the domain size of the integer variable, which is also equal to the total number of possible unique commonality matrix blocks. We apply this pseudocode to analyze the CP 3 model simplification for product families with 3 − 7 kinds of products. The range of an integer variable, 0 − zmax increases exponentially with the number of product kinds, where the exponent itself grows quadratically; this exponential variation is evident from Eq. 12. We found that, although the number of feasible values (due to the constraint in Eq. 10) for the integer variable also increases exponentially, the exponent grows linearly with the number of product kinds. This difference in the variation of domain size of the integer variable is illustrated in Fig. 3. The Y-axis in Fig. 3 represents the logarithm of the domain size of each integer variable (i.e. the number of unique commonality matrix blocks) Figure 3 shows that the size of the feasible set Z for the integer variables (representing feasible commonality matrix blocks) is significantly smaller than the total range for the integer variable (0 − zmax ). Determination of this feasible set and implementation of the same through the MDNLO technique is expected to (i) reduce the computational expense significantly, and (ii) provide robustness to the optimization methodology. Each combination of feasible values of the integer variables (that represent the commonality matrix blocks) represents a candidate combination of “sub-platform - platform - scaling” variables for the product family (a candidate product family structure). Interestingly, when the possibility of sub-families is accounted for, the number of such feasible candidate combinations is significantly higher than 2n . The total number of such possible combinations is rather equal to (MZ )n , where MZ represents the cardinality of the set Z, and n is the number of system design variables. We observe from Fig. 3 that, the increase in MZ with the number of product kinds, follows an exponential trend. For example, in the case of a family of 7 product kinds, the number of possible combination is 877n ; this number is substantial for medium to large scale systems (in 10 of 13 American Institute of Aeronautics and Astronautics

7

Log(number of integer variables)

6

Feasible λ blocks All unique λ blocks

5 4 3 2 1 0 1

2

3

4 5 Number of products

6

7

8

Figure 3. Domain size of the integer variable (number of unique commonality matrix blocks)

Number of feasible integer (z) values

terms of number of design variables involved). Hence, any form of exhaustive approach to product family design is likely to be practically unrealistic. Figure 4 shows the distribution of the feasible values of the integer variable (elements of the set Z), where the black circle symbols represent feasible integer values. We observe that the frequency of feasible integer

300 250 Feasible value for integer variable Number of feasible integer values

200 150 100 50 0 0

0.2

0.4

0.6

0.8 1 1.2 1.4 Integer variable, z

1.6

1.8

2

2.2 6 x 10

Figure 4. Frequency of feasible values of the integer variable

values is noticeably higher towards the lower end of the range, evident from the higher histogram bars in the range, 0 − 500, 000. In addition, the frequency of feasible integer values seem to follow a periodic trend. However, further investigation is necessary to understand the physical significance of this trend, which would require the evaluation of the feasible integer set (Z) for larger product families. Interestingly, such an effort might be computationally expensive. A preliminary analysis showed that for a family of 8 product kinds, the determination of the feasible set would take 75 hours (approximately), running on a 2.83Ghz Intel Core 2 Quad, 8Gb system (using MATLAB). The expected high computational time can be attributed to the very large domain size of the integer variable (total number of unique commonality matrix blocks) - which is zmax = 268, 435456 in the case of the 8-product family. Thus, an exploration of the scope of representing each commonality matrix block through more than one integer variable might be a helpful topic of future research.

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V.

Conclusion

The Comprehensive Product Platform Planning (CP 3 ) framework addresses a broad scope of product family design. The original CP 3 technique presents an encompassing mathematical model of the platform planning process (CP 3 model), which yields a complex Mixed Integer Non-Linear Programming (MINLP) problem. The primary component of this platform planning model is the commonality matrix (composed of binary elements) which serves as a variable in the optimization of the product family. For a “N -product n-variable”
n family, the total number of possible unique commonality matrices (size of the search space) is 2N (N −1) , which proves to be computationally expensive for optimization. In this paper, we developed a methodology to reduce the size of the commonality matrix search space by several orders of magnitude - for example, the search space size is reduced from 2097152n to 877n in the case of a 7-product family. This helpful simplification of the CP 3 model is achieved through (i) appropriate aggregation of the binary commonality variables, (ii) conversion of the binary integer problem into an integer problem, and (iii) elimination of infeasible commonality matrices (based on an additional constraint) and the corresponding values of the integer variables. The last step is achieved through an efficient pseudocode, developed in this paper. In this paper, we also developed a generalized approach to mixed-discrete non-linear optimization, which uses non-gradient based continuous optimization as the primary search strategy. In this technique, the system model is evaluated only at feasible locations in the discrete variable space; feasibility in this case is determined through prior specification of the feasible set of values for each discrete design variable, and does not refer to the system constraints. This method is expected to provide a robust optimization framework for the reduced CP 3 model. Application of the advanced CP 3 framework to design a family of universal electric motors will establish the true potential of this method.

VI.

Acknowledgements

Support from the National Foundation, from Awards CMMI-0533330, and CMII-0946765 is gratefully acknowledged.

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