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A Fuzzy Goal Programming Approach for Optimal Product Family Design of Mobile Phones and Multiple-Platform Architecture Satish Tyagi, Kai Yang, Annu Tyagi, and Anoop Verma, Student Member, IEEE

Abstract—Competitiveness of any manufacturing industry depends on its ability to respond quickly to market niches and to produce a variety of high-utility products at relatively low costs. The promising tool to achieve aforesaid goals is the development of an efficient product family design strategy. The collection of shared components across the product family is termed as a platform that allows the saving in additional cost. Unfortunately, a single platform is advantageous only up to a certain extent; firms have sensed the requirement of multiple platforms. In this context, this paper deals with the exploration of product family design and multiple-platform architecture with a view to maximize the overall utility and to minimize the total production cost. This multiobjective problem has two conflicting and incommensurate objectives; therefore, a fuzzy goal programming model is adopted for modeling. The adoption of fuzzy goal programming model aids in combining the two objectives as well as captures the inherent uncertainty involved in decision making. The problem is formulated as a mixed integer program, and, additionally, random search optimization techniques, namely, genetic algorithm, simulated annealing, and Tabu search are being used to resolve the underlying issues. Moreover, in order to illustrate the proposed framework, a hypothetical case study—a family of mobile phones—is considered. Extensive experiments are performed on the underlying case study, and computational results are reported to validate the efficacy of multiple platforms over the single platform. Index Terms—Conjoint analysis, fuzzy goal programming, multiple-platform architecture, product family design, random search optimization techniques (RSOTs).

I. INTRODUCTION ODAY’S customer-driven marketing scenario is characterized by increasing and unpredictable preferences of consumers pertaining to the product designs. In real-world situations, consumers are encountered with a wide range of likewise

T

Manuscript received June 30, 2011; revised October 19, 2011 and January 29, 2012; accepted April 13, 2012. This paper was recommended by Associate Editor X. Guan. S. Tyagi and K. Yang are with the Department of Industrial and Systems Engineering, Wayne State University, Detroit, MI 48201 USA (e-mail: [email protected]; [email protected]). A. Tyagi is with the Lifetime Mobilities Pvt. Ltd., Mumbai 400607, India (e-mail: [email protected]). A. Verma is with the Intelligent Systems Laboratory, The University of Iowa, Iowa City, IA 52242-1527 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCC.2012.2198056

products, each comprising of many attributes at certain levels. Consumer considers various (instead of single) attributes to make a decision to buy a product. In this paper, conjoint analysis is used to simulate the consumer preferences in terms of part-worth utility value of each attribute at respective levels. Therefore, the first objective is to determine such product family design (attributes level) that maximizes the overall utility. Product family is defined as a set of product variants each having some common components or technology. It is a convention that increasing in number of product family variants generally augments the total production cost at a higher rate [1], [2]. Therefore, prevailing era of acute competition is encouraging industries to explore and utilize strategies that enable them to manufacture a variety of products at a cost comparable with that of mass-produced goods. As a result, various strategies have emerged in the recent past. This includes platform-based product development and modular design [3], [4] and delayed product differentiation [5], [6]. A platform entails components or even the production steps that are common to the product family. The advent of platform strategy has facilitated the reduction of production costs and shortening of development time, thereby leading to its adoption by various industries across the globe. Moreover, the worth is evident from implementation by a leading company, Volkswagen, to produce over four millions of passenger cars [7]. Sony has used platform strategy to produce a family of more than 250 consumer products in U.S. during 1980s [8]. Boeing has also utilized this approach to produce blended wing body aircrafts [9]. Hewlett-Packard is producing upgraded inkjet printer family by implementing platform strategy [10]. Group Technology [11] is the precursor of product platform selection problem that uses existing commonalities across many dimensions, e.g., geometric design, assembly design characteristics, etc. Traditionally, single platform is used to produce the product family which is attractive for organizations since it reduces the production cost and time. However, it is capable to diminish them up to a certain extent, since there is only one option to produce the product family. Consequently, it does not provide those ample advantages that can be gained by assigning product variants to appropriate platform. Thus, in order to alleviate the aforementioned problem, it is essential and imperative to apply multiple-platform strategy that provides an opportunity in assigning products to appropriate platform. This strategy further reduces production cost and development time which depends on optimal number of platform and their architectures. Therefore, the second aim is to resolve the aforesaid issues with

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a view to minimize the total production cost. The production cost generally involves cost of components, cost of mass assembly, and cost of adding/removing components to form a product family. Owing to the significance of these two objectives, the underlying decision problem is considered as an example of multiobjective optimization problem [12] and their incommensurate nature constrains explicit summation. Therefore, keeping in mind their nature, a fuzzy goal programming (GP) approach is presented to model the underlying problem because of its flexibility in modeling of multiobjective problems. In addition to this, the problem at hand is a combinatorial optimization problem [13] that is NP-hard in nature. Owing to the large number of possible combinations, conventional enumeration-based optimization techniques are not efficiently applicable. However, random search optimization techniques (RSOTs) have been widely used to solve such hard combinatorial optimization problems [14]. Therefore, to resolve the problem at hand, this paper uses three established algorithms: genetic algorithm (GA), simulated annealing (SA), and Tabu search (TS). In order to illustrate the behavior of aforesaid algorithms, a case study—a family of mobile phones—is considered. Obtained results are compared, and after comparison, it is found that GA outperforms other algorithms as far as resulting value and convergence to near-optimal solution are concerned. After rigorous computational experiments, it is found that the use of multiple platforms further reduces the total production cost in various scenarios. The structure of this paper is as follows: In Section II, the relevant literature related to product family design and platform architecture is reviewed. Section III describes the problem environment in which modeling of objective function is discussed. In Section IV, background of fuzzy GP and its modeling for problem at hand is discussed. In Section V, the implementation procedure of GA is described in detail. The results that are obtained by implementing aforementioned algorithms are detailed in Section VI. Finally, Section VII concludes this paper and provides direction for further search. II. LITERATURE SURVEY Products according to their usage are categorized into three major groups. The first group includes a static product that emphasizes on consumer’s adjustment for its use. The product of second group is quite flexible in use and serves many purposes simultaneously, whereas a stream of product variants for a selected group of people is classified in the third group. In [5] and [15], the authors demonstrated that cost-effective product variety can be offered by producing a product family instead of a single product by employing platform(s) where different products share components, subassemblies, or even the production steps [16]. Therefore, a platform design has become an issue of paramount importance for both academia and industry. As far as academia is concerned, Simpson [17] offered a summary of product platform theory as well as discussed its advantages and disadvantages. Jose and Tollenaere [18] provided a literature survey on the platforms method for a product family design. Kusiak et al. [19] presented a product configurations

planning method for a single product exploiting the clustering algorithms, while keeping an eye on different categories of customers. The product family design problem has been solved using multiobjective evolutionary algorithm [20] and product family evaluation graphs [21]. The active research perspectives of a platform design comprises of platform design process development [22], [23], optimization of platform architecture [24], [25], and platform portfolio optimization [26], [27]. The aim of platform design and selection is to achieve various goals, such as cost minimization [17] or life cycle design improvement [28], and maximization of profit with reducing time-to-market [29]. Ishii and Martin [30] developed a methodology by employing design for variety to decide the product platform architectures. The product platform concept exploration method has also been proposed to design the product platform which can be used for the formation of product family by scaling and stretching [31]. Generally, two types of research works are conducted in the area of product platform architecture [32]. This comprises of 1) platform development based on qualitative approach [25], [33], [34], [35], and 2) product-oriented studies based on qualitative approach [37], [38]. In qualitative approach, engineering or design aspect of products are taken into account. Moreover, the research on quantitative approach is subdivided into three groups. The first group contains research on scalable (parametric) platforms [22], [36], whereas research on module-based platform formation is classified in the second group [37], [38]. In addition, the third group contains research using a hybrid approach [39] of previously mentioned two approaches, i.e., scalable and module-based platform approaches. In general, aforementioned two research works, i.e., a product family design optimization and platforms development, are carried out at single stage and two stages. The single-stage optimization refers to concurrently optimization of a product family design and product platforms, whereas two-stage optimization involves the optimization of product design in the first stage and that of platform architecture in the second stage [40]. Owing to the combinatorial hard nature of a product platform design optimization problems, various RSOTs, such as GA [41] and SA [42], have been utilized for its effective solutions. In addition, a multiagent framework [43] has also been introduced. A similar research has been conducted in [44]–[46], where the authors considered the multiple-platform design without simultaneously considering the product design (product family design). In this paper, first an integrated mathematical model is formulated to estimate the overall utility of product family design and then to calculate total cost of producing them using multiple-platform strategy for a single-stage optimization. Further, a fuzzy GP model is adopted to capture the inherent uncertainty involved with the objectives. This adoption also helps in adding the two incommensurate objectives. In order to solve this computationally complex problem, three different RSOTs, viz., GA, SA, and TS, are utilized and compared on the basis of returned objective values at the end of C++ code. It is shown in Section VI that GA outperforms other algorithms in almost all scenarios.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TYAGI et al.: FUZZY GOAL PROGRAMMING APPROACH FOR OPTIMAL PRODUCT FAMILY DESIGN OF MOBILE PHONES

3

Full cover

PCB

Input

Radio Processor

Build-in-memory

Fig. 2.

RAM

Example of product platform.

A. Product Family Design (Overall Utility)

Fig. 1.

Configuration of considered example of mobile phones.

III. PROBLEM ENVIRONMENT A stream of product variety is produced on a single platform by accounting the commonality among them. These variants constitute a family that is termed as the product family. Thus, a set of common assets including components design among the product family can be defined as a platform [47]. This paper spawned new views to platform as the collection of common components across the variants of a product (product family) which is used as an input for their formation. Therefore, platform acts as an input on which components are added or removed to develop a product family. Apparently, design of a product platform should be flexible so that whenever components are required to be added/removed in designing product variants of a family, it can be easily carried out. Consider a product family, for example, having five variants as illustrated in Fig. 1. In this example, each product variant comprises of components from a set of available components {1, 2, . . ., 10}. A platform to produce underlying product family is shown in Fig. 2. The first product variant P1 is created after removing radio and adding main screen to the platform. However, removal of radio and addition of camera as well as main screen to the platform results in product variant P5 . This paper explores such a product family design and architecture of multiple platforms that, respectively, results in maximization of overall utility and minimization of production costs.

Part-worth utility value of each attribute level is determined with the help of products ranking provided by the responders. This ranking acts as an input for the conjoint analysis which, in turn, provides the part-worth utility [48]. The input data were collected on the basis of a survey that was conducted among 100 sophisticated users of mobile phones. They were asked to give a score to the various attributes on a scale of 10, where 10 represent the most favored product and 1 the least. Out of these various attributes, ten highly scored attributes were chosen for experimental purpose. Apparently, by taking into account all attributes at their respective levels, approximately a total number of 5 × 2 × 5 × 5 × 3 × 3 × 2 × 4 × 5 × 3 = 270 000 (see Table IV) possible combinations can be constructed for any product. It is cumbersome to handle so many combinations at a time. Therefore, to overcome this difficulty, orthogonal product profiles are constructed in the initial phase of conjoint analysis which reduced the master designs from 270 000 to 64 (using SPSS software). Estimated utilities are used to evaluate the overall utility Uover of product family by applying a simple composition rule defined as follows: Uover =

Pr M p Ai

wi · uik · Yik

(1)

p=1 i=1 k =1

where wi is the preference weight for i attribute, uik is the utility of the ith attribute at kth level, Mp is the total number of attributes in pth product variant, Ai is attribute level for Mpth attribute in product p, and Yik is the binary decision variable Yik = 1, if the ith attribute is present at the kth level in product p 0, otherwise i = 1, 2, ..., Ct , k = 1, 2, ..., K, p = 1, 2, ..., Pr m component, set m ∈ Ct = {1, 2, 3,. . ., Ct }, Ct maximum number of components;



TABLE I PAR MATRIX FOR THE CONSIDERED CASE STUDY (p = {1, . . ., 5) A A B C D E F G H I J

B

C

D

E

1

1

1

1

F

G

H

1

1

1

I

J

1

1

B. Multiple-Platform Architecture (Total Production Cost) It is assumed that considered product family can be produced by using platform, or by adding (removing) components manually. In general, removal or addition of a component to the platform costs more as compared with that of mass assembly. However, problem associated with it is that the architecture of platform must follow the part assembly sequence of product family. To tackle this problem, part assembly relationship (PAR) matrix is introduced, which is shown in Table I, for the problem at hand. An integer programming formulation of underlying problem is provided. Following notations are used for the mathematical formulation of total production cost: purchasing price for mth component at ith level; demand for pth product variant; platform in set of platforms l ∈ Pt = {1, 2, 3, . . ., Pt };

where Pt ρp ρa ρr e

maximum number of platforms; cost of creating a platform by adding component (mass assembly); cost of manually adding component to platform; cost of manually removing a component from platform.

Integrality of decision variables: All the decision variables possessing the value of 0 and 1 are as follows: 1, if platform l comprise of component m P Clm = 0, otherwise P Plp =

ADlm p

l = 1, 2,...,Pt , m= 1, 2,...,Ct 1, 0,

RElm p =

⎩

0,

if component m is removed from platform l tocreate product variant p otherwise

l = 1, 2,...,Pt , m = 1, 2,...,Ct , p = 1, 2,...,Pr .

p product variant of product family p ∈ Pr ={1, 2, 3,. . ., Pr }; Pr maximum number of product variant in product family.

φm i λp l

⎧ ⎨ 1,

if platform l is used to create product variant p otherwise

l = 1, 2,...,Pt , p = 1, 2,...,Pr ⎧ ⎨ 1, if component m is added to platform l to = create product variant p ⎩ 0, otherwise l = 1, 2,...,Pt , m = 1, 2,...,Ct , p = 1, 2,...,Pr

The formation cost of a single platform includes the cost of purchasing φm i of each component m at a particular level i and cost of mass assembly ρp of all the components. Such condition is true only if platform l comprises of component m for making a specific product variant p which in return is assigned to same platform l for production. Total number of platforms required is equal to the total demand of all products variants λp of family. Therefore, total formation cost of all platforms is expressed as follows: COSTplat =

Pt Pr Ct

(ρp + φm i ) · P Clm · θlp · λp . (2)

l=1 m =1 p=1

After the mass assembly of platforms, if component m is added manually to platform l, then additional cost of manually adding component ρa to the platform l is incurred. Cost of manual assembly includes the cost of manually adding Ct components and cost of purchasing φm i of each component m at a particular level i. Such condition is true only if component m is added to platform l to create product variant p after mass assembly, otherwise no cost is incurred. Therefore, total cost of manual assembly for making product family variants is expressed as COSTadd =

Pt Pr Ct

(ρa + φm i ) · ADlm p · θlp · λp . (3)

l=1 m =1 p=1

However, it may be required to remove the component m from platform l to produce a product variant p. In that case, additional cost of manually removing component ρr e is also charged and cost of purchasing φm i of each component m at a particular level i is saved. Such condition is true only if component m is removed from platform l to create product variant p after mass assembly, otherwise no cost is incurred.Therefore, total cost of t λp product family removing the components for making Pl=1 variants is expressed as COSTre =

Pt Pr Ct

(ρr e − φm i ) · RElm p · θlp · λp . (4)

l=1 m =1 p=1

Setup cost to build the platform is also included to determine the overall production cost: COSTsetup =

Pt

πp .

(5)

p=1

Total production cost of the product family is the summation of four types of costs: COSTtotal = COSTplat + COSTadd + COSTr e + COSTsetup

(6)


Pt Pr Ct

COSTtotal =

IV. SOLUTION METHODOLOGY

(ρp + φm i ) · P Clm · θlp · λp

l=1 m =1 p=1

+

Pt Pr Ct

(ρa + φm i ) · ADlm p · θlp · λp

l=1 m =1 p=1

+

Pt Pr Ct

(ρr e − φm i )

l=1 m =1 p=1

· RElm p · θlp · λp +

Pt

πp .

(7)

p=1

This objective function is subject to following constraints: ADlm p = (1 − P Clm ) · ηm l · θlp ∀ l ∈ Pt , m ∈ Ct , p ∈ Pr

(8)

RElm p ≤ (1 − ηm l ) · P Clm · θlp ∀ l ∈ Pt , m ∈ Ct , p ∈ Pr Pt

θlp=1

∀l ∈ Pt

(9) (10)

l=1

P Clm ≥ Rlm p · θlp · P Cl p ∀ l ∈ Pt , m ∈ Ct , p ∈ Pr

ηm l =

Rlm p =

(11)

RElm p , P Clm , ADlm p , θlp ∈ {0, 1}

(12)

Pt ∈ {1, 2, . . . , Pr }

(13)

1, 0,

if product p comprise of component m otherwise p = 1, 2, . . . , Pr

1, 0,

5

m = 1, 2, . . . , Ct

if component m is precedes l in product p otherwise

l = 1, 2, . . . , Pt , m = 1, 2, . . . , Ct , p = 1, 2, . . . , Pr . Equation (8) ensures the condition that component m is added to platform l to create product variant p (if component m is not already present on the platform p). In contrast, if the product variant does not entail the component m, then the component is removed from the platform. Constraint in (9) ensures that component m is removed from the platform l, while product variant p is assigned to it. Equation (10) ensures that one product is allocated to single platform for production. Equation (11) sets the condition that each product variant maintains the assembly sequence, i.e., it follows the PAR matrix. For example, product P2 is made on platform, and it is required to add the component main screen; then, the input and PCB must be present on the platform. Equation (12) sets the value of decision variables to binary, whereas (13) ensures the integrality of optimal number of platform(s) and restricts it not to exceed from the maximum number of product variants of the product family.

The need to provide competitive alternatives at the earliest has motivated the development and implementation of RSOTs on complex problems. Because of its ability to explore highly efficient and competitive solutions in a very short span of time, these techniques have gained much importance in the recent past. This section presents a systematic description of one of the mentioned solution methodologies for the computation of decision variables. In this paper, three RSOTs (i.e., GA, SA, and TS) are exploited to deal with the problem of optimization of combined objective function. In the next section, description about fuzzy GP is provided. A. Fuzzy Goal Programming Charms and Cooper [49] presented the GP model to obtain the satisfactory results in case of incommensurable objectives simultaneously. In this model, goals are decided for each criterion by the decision maker (DM) in the initial stage, which is followed by calculating the value of decision criteria and measuring their deviations from respective prearranged goals. Now, these deviations act as the decision criteria and are subjected to optimization. In realistic problems, it is cumbersome to define exact goals for the decision criteria because of their imprecise nature. However, fuzzy set theory, which was first introduced by Zadeh [50], has been widely accepted in modeling of some of the vague phenomenon and relationships that are stochastic in nature. In such cases, fuzzy mathematics has an advantage over classical mathematics in that it provides concepts and techniques to deal with the modes of reasoning that are approximate in nature rather than exact. Hence, it is a tool to deal with the imprecise information of the DM. This paper estimates the compromise solution by integrating two different models, namely, fuzzy and GP, to propose a fuzzy GP model. In the present approach, the imprecise objective values are determined in terms of certain scalar membership function value, and thus, vector optimization is transformed into scalar kind. Next, different weights are also assigned to each decision criteria by the DM, followed by adding them using simply additive rule. A typical formulation of a generalized multiobjective GP in fuzzy term can be expressed as follows: ⎫ Γn (G) , n = 1, . . . , δ0 ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ Γn (G) ≺ , n = δ0 + 1, . . . , δ1 ⎪ ⎪ ⎬ n ∼ , n = δ1 + 1, . . . , δ2 ⎪ . (14) Γn (G) = ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ Subject to ⎪ ⎭ yi (G) ≤ Hi , i = 1, . . . , k In (14), G is an s-dimensional vector. Here, Γn (G), yi (G) are the goal constraints and system constraints, respectively. Here, the goal constraints refer to the number of objectives (goals) which the DM wants to achieve by allowing certain level of variation from target value. The target value according to goal n is represented by Πn , and the allowable range of variation is defined by (15), (16), or (17), fitting to the characteristic of goal.



The characteristic of goal constraints can be fuzzy-maximum, fuzzy-minimum, and fuzzy-equal type, and their numbers are denoted by δ 0 , δ 1 – δ 0 , and δ 2 – δ 1 , respectively. However, the system constraints refer to the number of constraints imposed on the objective function which must be strictly met. The value of strictly required limitation of the system constraints is represented by Hi . The notation “” represents (fuzzy maximum) approximately greater than or equal to certain aspiration level and implies that the DM is satisfied even if less than up to a certain limit. The symbol “≺” represents (fuzzy minimum) less than or equal to the aspiration level and implies that the DM is satisfied even if more than a certain limit. The symbol “∼ =”refers fuzzy-equal-type Γn that should be in the neighborhood of aspiration level, i.e., the DM is satisfied even if Γn (G) is less than or greater than a certain limit. Therefore, according to aforementioned types, following three members setup functions are defined. 1) For fuzzy maximum: n (Γn (G)) ⎧ 1, if Γn (G) < ⎪ ⎪ n ⎪ ⎪ ⎪ ⎨ Γ n (G )− n , if ≤ Γn (G) ≤ +TnR = 1− R T n ⎪ n n ⎪ ⎪ ⎪ ⎪ R ⎩0, if Γn (G) > +Tn .

(15)

n

2) For fuzzy minimum: n (Γn (G)) = ⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎨ Γ n (G )−(

n

⎪ ⎪ ⎪ ⎪ ⎩ 1,

if Γn (G) > −T nL )

T nL

, if

n

n

−TnL

−TnL ≤ Γn (G) ≤

if Γn (G) >

n

(16)

⎧ δ2 ⎪ ⎪ ⎪ max wn ·M V n ⎪ ⎪ ⎪ ⎪ n =1 ⎪ ⎪ ⎨ subject to : ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

M Vn ≤ n (Γn (G)) yi (G) ≤ Hi , i = 1, . . . , k

(18)

0 ≤ M Vn ≤ 1 n = 1, 2, . . . , δ2 .

In (18), wn is the dominance factor assigned to each goal by the DM. The objective is to minimize the weighted sum of membership function value MVn corresponding to all goals. The estimated membership function value is less than or equal to the value calculated from the equations depending on the type of goal constraint like minimum, maximum, or equal and always lies in the range of 0 or 1. For this paper, the objective is to optimize the combined objective value of product family cost and total utility of product variants (δ2 = 2) and is subjected to a group of seven constraints (k = 6). The next section presents the fuzzy GP formulation of objection function. B. Fuzzy Goal Programming Formulation of the Problem The problem described earlier is formulated as the bicriterion objective, out of which one represents total production cost and other indicates overall utility of product family. It may be desirable to have production cost lower than a certain level and overall utility higher than other fixed goal. In this paper, the goals are synchronized to determine the objectives: 1) optimal number of platform and their architecture; 2) assignment of each product to apposite platform; 3) optimal design (attribute level) of product family. The first goal is to minimize the total production cost of product family:

. Minimize f1 COSTtotal ≤ g-production cost.

n

3) For fuzzy equal: n (Γn (G)) = ⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎪ Γ n (G )−( n −T nL ) ⎪ ⎪ ⎪ , ⎪ T nL ⎪ ⎪ ⎪ ⎨ 1, ⎪ ⎪ ⎪ Γ n (G )− n ⎪ ⎪ 1 − , ⎪ R Tn ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0, ⎩

ing function for maximization kind of problem:

(19)

The second goal is to maximize the total utility of product variants: if Γn (G) < if

n

n

if

−TnL

Maximize f2 Uover ≥ g-utility .

−TnL ≤ Γn (G) ≤

if Γn (G) =

≤ Γn (G) ≤

if Γn (G) >

n

Thus, the combined objective function takes the following mathematical form:

n

(17)

n

n

(20)

n

+TnR

+TnR .

These membership functions 1–3 are strictly monotone, i.e., either increasing or decreasing and continuous with respect to Γn (G). n (Γn (G)) is the membership value, and TnR and TnL are the maximum tolerance limit to target value. Furthermore, formulation that is described in (1) is reformulated into follow-

Maximize Oob j

function

= (f1 , f2 )

Maximize Oob j

function

=

2

wn × M Vn .

(21) (22)

n =1

In (22), wn dominance or priority factor assigned to goals by the DM, and MVn is the membership function of goals: total production cost and overall utility, respectively. “g-” indicates the goal set by the DM for both objectives. The fuzzification of goals for the underlying objectives is explained by (23). This


7

TABLE II CONFIGURATION OF MOBILE PRODUCT FAMILY

Comp. no. 1 2 3 4

Comp. name

P1

P2

P3

P4

P5

Ex. Cover Input Main screen Camera

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

0 0 1 1 1

0 0 1 1 1

1 1 1 1 1

1 0 1 1 1

0 1 1 1 1

Printed circuit board

6 7 8 9 10

Bluetooth Radio Build-in-memory CPU RAM

TABLE III UTILITIES VALUE OF EACH ATTRIBUTE AT DIFFERENT LEVELS OBTAINED BY THE CONJOINT ANALYSIS Comp

Importance Level1

Level2

Level3

Level4

Level5

no.

Fig. 3.

Flowchart of GA.

equation captures the vague nature of the goals in the problem at hand

⎧ 1, if Γn (G) < ⎪ ⎪ n ⎪ ⎪ ⎨ Γn (G) − n , if ≤ Γn (G) ≤ +TnR MV n = 1 − R Tn ⎪ n n ⎪ ⎪ ⎪ R ⎩ 0, if Γn (G) > +Tn n

values

1

-2.910

-3.368

6.278

2

-7.500

-3.438

1.875 -

3

3.462

-8.663

4

2.250

.938

5

0.587

-7.100

6

-3.469

3.469

7

0.825

-7.925

8.300

8

-2.111

-5.111

7.222

9

-3.656

3.656

10

-1.854

-4.313

10.100 2.3133 -7.288

0.0753 9.063

0.1294

2.213

13.088

0.1812

0.1254

0.8755

0.0356

1.525

12.275

0.1528 0.0542

5.325

.075

0.1357 0.0963 0.0571

6.1673

0.0819

(23)

where TnR is the tolerance limit for goals, and n is the index for goals. n is the value of the nth goal set by the DM, and Γn (G) is the estimated value of the function. The next section details all the three optimization methodologies.

V. IMPLEMENTATION DETAILS A. Implementation Procedure of Genetic-Algorithm-Based Methodology In this section, the aim is to define the terminologies and to discuss various design issues, such as representation, initialization, function evaluation, etc., pertaining to GA with reference to its application in solving the underlying problem. Steps involved in GA-based solution methodology are as follows. Step 1 (Encoding): Representation of search space in terms of the parameters is the initial step in implementation of RSOTs. In general, to represent the information, binary coding is used. However, in case of combinatorial optimization, real coding is preferred. Therefore, real coding has been utilized in this research to represent the attribute levels of the product family. A part of the string for the product family, having Pr product

variants and each variant having nf number of attributes at kth levels, is represented as follows. 5 1 1 5 3 0 2 4 3 0 1 2 4 5 3 2 0 5 3 Here, each continuous fragment of a chromosome, i.e., a substring of 10 bits, represents the configuration of a product variant contained in the product family. Integer coding is done so that each bit represents the level of an attribute which is present in product variants. “0” bit indicates the absence of a particular attribute in a specific variant. Since an attribute has a limited number of levels, a corresponding constraint is applied, while randomly generating the product family design. Each string corresponds to a potential solution to the problem. Arbitrarily, the architecture for a fixed number of platforms is generated. For this binary string, coding (0, 1) has been used in this paper, where “1” indicates that a particular component is present in platform and vice versa. Thereafter, the feasibility



TABLE IV COST OF EACH ATTRIBUTE AT DIFFERENT LEVELS No.

Attribute

1

Bluetooth

2

Build-inmemory

3

Camera

CPU

4

level 1 2 3 22 MB 26 MB 31 MB 40 MB 1 2 3 4 5

Cost 2.00$ 2.50$ 3.00$ 1.20$ 1.80$ 2.60$ 3.40$ 1 .0 0 $ 1.50$ 2.00$ 2.50$ 3.00$

No.

Attribute

5

Ex. Cover

6

Input

7

Main screen

8

PCB

180 MHz

1.24$

200 MHz

1.92$

220 MHz

2.40$ 9

250 MHz

2.80$

300 MHz

3.10$

10

TABLE V TOTAL COST OF PRODUCTION FOR DIFFERENT SETUP COSTS Demand

Setup costs

100

# platforms

CostTotal

1

52575

2

50800

3

50275

[250,

4

51000

250,250,250,250]

5

50475

1000

1

53475

2

52600

3

52975

4

53975

5

54975

of each platform architecture is checked. If the platform architecture does not maintain the sequence of PAR matrix, then candidate solution is termed as infeasible solution, and, to make them feasible, appropriate corrective actions are applied which can be in the form of addition or removal of components from them. Step 2 (Population Initialization): In the manner as described in Step 1, a random population having N chromosomes is randomly generated. In the considered case study, there are five variants; therefore, chromosomes have 50 bits. Step 3 (Fitness Calculation): The chromosomes are evaluated to provide the corresponding fitness value. This fitness value comprises of two factors: overall utility and optimal production cost of product family. These two factors are combined by using the fuzzy GP model. In this model, the deviation from

level Blue Black Silver Golden Keypad T. screen 1 2 3 4 5 Semi-metallic

Cost 1.30$ 1.90$ 2.10$ 2.40$ 4.00$ 5.00$ 1.30 $ 1 .6 0 $ 2. 00$ 2.40 $ 2.70 $ 2.50$

Metallic

3.20$

Pure Metallic

4.00$

Type1

3.00$

Type2

2.00$

Radio

RAM

32 MB

2.85$

64 MB 128 MB

3.25$ 3.60$

the goals acts as the design criteria. This combined value gives cost value of each chromosome. The Uover of a product family is determined on the basis of part-worth utilities. The production cost of each variant is calculated as if it is produced by using each platform. Afterward, product variant is assigned to platform having minimum production cost. The summation of production cost provides the cost value of chromosome, i.e., minimum production cost for product family. Store the minimum production cost and correspondingly the architecture of platforms, as well as assignment of product variants to them. Step 4 (Selection): Select the chromosomes for the next step on the basis of the fitness value. This selection of chromosomes is carried out by using Roulette Wheel Strategy. Step 5 (Genetic Operators): Apply the genetic operators like one-point crossover and mutation for the production of an offspring. Step 6 (Termination Criteria): Check the condition that if prearranged maximum number generations is achieved, then the optimal solution in the form a product family design and architecture of platforms is obtained. Thus, the underlying issues, those are to be resolved, have been achieved. Following the basic procedure, the underlying problem is solved iteratively as depicted by the flow chart of the procedure in Fig. 3. B. Parameter Settings In case of GA, a population size of ten chromosomes, a crossover probability of 0.7, and a mutation probability of 0.02 have been utilized. For SA, the values of initial and final temperatures are set equal to 5000 and 5, respectively. The convergence rate is controlled by selection of a suitable cooling schedule that is taken as the Boltzmann function in the present


9

TABLE VI TOTAL PRODUCTION COST FOR VARIOUS INSTANCES USING GA ρp, ρa ρre

Setup cost

Single platform

Demand

Cost ($)

1111101111

Optimal platforms

Multiple Platform

Cost ($)

1111111111[2,3,4]* 52600 1111001111[1,5] 1111001111 1111111111[1,5] 96475 2 94975 [1500, 250,250,250,250] 1111001111[2,3,4] 1111101111 1111111111[3,4] 110100 110350 2 [250, 1500,250,250,250] 1111101111[1,2,5] 1111001111[1,2,5] 2.5, 5, 3 1111111111 108475 2 106975 1000 [250, 250,1500,250 ,250] 1111111111[3,4] 1111001111 102850 1111001111[1,2,5] 101350 2 [250, 250,250,250,1500] 1111111111[3,4] [250, 250, 250, 250, 250] [--]** 42850 -[--] 42850 2,1.8, 2.5 [250, 1500, 250, 250, 250] [--] 96600 -[--] 96600 1111101111[ 1,5] 1111101111 52575 3 1111111111[2] 50275 [250, 250, 250, 250, 250] 1111001111[3,4] 1111001111[1,5] [250, 250, 1500, 250, 250] 1111111111[3,4] 104650 1111111111 107575 3 2.5, 5, 3 1111101111[2] 100 1111101111[2] 380800 [250, 250, 10000, 250, 250] 3 111100111[1,5] 11111111111[3,4] *represents that [x, y] product is assigned to that platform; ** [--] represents that no component is assigned to the platform, i.e., a null platform.

Toal procution cost

[250, 250,250,250,250]

55000

100

54000

1000

53000 52000 51000

2

to the underlying case study. In order to adjust the various parameters including goals of both the objective functions, a series of preliminary tests was executed. The coding is performed in C++, and the complied program is executed on a system specification 2.2-GHz Pentium IV processor and 768-MB RAM. A description of the dataset used in this study is provided in the next section. A. Dataset for the Case Study

50000 1 2 3 4 5 Number of platform s

Fig. 4.

53475

Variation in COSTT O TA L for different setup costs.

case. Moreover, before making a transition from T(k) to T(k + 1), 500 neighbors of the current solution were evaluated by the acceptance function. A Tabu list of ten particles is utilized for TS. The next section summarizes the computational experience of RSOTs on the aforementioned product family design and multiple-platform architecture problem. VI. RESULTS AND DISCUSSIONS In this section, first the results obtained after implementing the RSOTs on the illustrative case study for various scenarios are detailed. The RSOTs employ three criteria: the total production cost alone, over all utility alone, and a combination of aforesaid two criteria to authenticate the obtained results. Furthermore, the comparison among algorithms is made on the basis of combined objective value returned by each algorithm when applied

In this paper, a case study that deals with the family of mobile phones (see Fig. 1) is considered to demonstrate the underlying issues. Although current market is filled with abundant varieties of mobile phones, five standard examples are used that have its own purpose to be served. The configuration of each product variant is shown in Table II. Various levels of each attributes are shown in Table III. It is assumed that variables, such as demand of each product variant as well as cost of each attribute at their respective levels, are already known and are shown in Table IV. The description of each variant is provided in Table II, whereas the PAR matrix is shown in Table I. B. Obtained Results and Their Discussions The results obtained for total production cost alone, at varying setup costs and at the fixed demand of all the product variants, are listed in Table V. From here, it is clearly evident that when the setup cost is low, it is more advantageous to increase the number of platforms to obtain the minimum total production cost. In contradiction, when setup cost is high, a smaller number of platforms result in the minimum production cost. The whole scenario is portrayed through a graph between the total production cost and number of platforms (see Fig. 4). The production cost for various demand scenarios is shown in Table VI. This table represents the obtained results by GA for



TABLE VII RESULTS OBTAINED BY THREE RSBOTS Demand

1000

[250,---]

ρp, ρa, ρre 2.5,6,3

Platforms

Fuzzy goals

1

2

5

TS

COSTtotal

Uover

COSTtot

Uover

COSTtotal

Uover

COSTtotal

16-21

40,000-50000

19.736

47625

17.039

49675

16.6853

52400

19-24

37000-47600

19.806

46550

17.357

52275

16.7092

50825

18-23

37000-47600

19.123

46025

19.480

53750

17.6319

50725

16-21

40700-47600

15.915

45850

15.771

47400

15.3537

48475

37000-47600

15-20

4

SA

Uover

18-25 3

GA

40,000-50000

23.495

47025

18.917

49450

18.6153

49625

16.893

47250

17.183

54075

18.9987

51600

18-23

37000-47600

18.492

46850

17.919

55200

17.2698

53425

20-25

37000-47600

21.249

46825

20.293

52975

19.9067

51425

16-21

40,000-50000

18.074

47650

17.367

52625

16.3363

51700

19-24

37000-47600

19.069

47825

18.381

54500

19.1157

52975

TABLE VIII COMBINED AVERAGE OBJECTIVE VALUE ON GENERALIZED GOALS Fuzzy goals

COSTtotal

Uover

GA

Platforms 1 2

5000056000

3

15-25

4 5

SA

TS

Uover

COSTtotal

CV

Uover

COSTtotal

CV

Uover

COSTtotal

CV

0.4736 0.4806 0.4123 0.0915 0.8495 0.1893 0.3492 0.6249 0.3074 0.4069

0.7426 0.8480 0.8995 0.9166 0.8014 0.7794 0.8186 0.8210 0.7401 0.7230

0.6081 0.6643 0.6559 0.5040 0.8255 0.4843 0.5839 0.7230 0.5238 0.5650

0.2039 0.2350 0.4484 0.0773 0.3917 0.2183 0.2919 0.5293 0.2367 0.3381

0.5416 0.2867 0.1421 0.7647 0.5637 0.1102 0.0727 0.2181 0.2524 0.0686

0.3727 0.2612 0.2950 0.4209 0.4777 0.1642 0.1823 0.3737 0.2445 0.2033

0.1685 0.1709 0.2631 0.0353 0.3615 0.3998 0.2269 0.4906 0.1336 0.4115

0.2745 0.4289 0.4387 0.6593 0.5465 0.3529 0.1740 0.3700 0.3431 0.2181

0.2215 0.2999 0.3509 0.3473 0.4540 0.3764 0.2005 0.4303 0.2383 0.3148

TABLE IX OPTIMAL CONFIGURATION OF PRODUCTS AND OPTIMAL ARCHITECTURE OF MULTIPLE PLATFORMS

Demand

Setup cost

RSBOT

# platforms

Overall utility

Optimal production cost

GA

3

23.495 7

47025

3

18.917 9

49450

1000 Costs

ρp,ρa ρre [250,--] 2.5,6,3 SA

TS

3

18.615 3

49625

Configuration of product family 5241000452 5155300433 1125121351 5215310412 5155002153 2225000421 5145300323 5135332233 4235110431 5255001241 3224000241 5155300413 5243331411 5151320443 5235002452

Architecture of multiple platforms 1111111011[4] 1111001111[1,5] 1111111111[2,3]

1111001111[1,2,5] 1111111011[3,4] 1100101111[] 1111001111[1,5] 1111111111[2,3,4] 1100101101[]


various instances of demands and setup cost at varying number of platform. As previously mentioned, from this table, it is also concluded that first increase in number of platforms reduces the total production costs; however, further increase in number of platforms enhances the total production cost. For example, in case of same demand, i.e., 250 for each product variant, total production cost is minimum when two platforms are employed at the setup cost of $1000. Furthermore, if the setup cost is reduced from $1000 to $100, the optimal number of platform increases from 2 to 3 and minimum production cost reduces from 52 600 to 50 275. Moreover, if ρp is more than ρa , it is not favorable to use platform to produce the product family to obtain the minimum production cost. The optimal architectures of multiple platform and allocation of product variants are also provided in Table VI. The combined results obtained from all the three RSOTs for two groups of goals are shown in Table VII. This table reveals that GA performs better than the rest two algorithms in case of all the five product platforms. The optimal combined result is obtained as 47 025 for total production cost and 23.4957 for the overall utility at three platforms when the goals were assigned in the range of 37 000–47 600 and 18–25 for the total production cost and overall utility. Goal values of both the objectives corresponding to each case are also shown in the same table. Table VIII demonstrates the details of comparison among RSOTs on the basis of obtained average normalized results from computational experiments. For the purpose of comparison, both the objective values were calculated on the same range of goals for each platform which are provided in Table VIII. The optimal number of platforms and their architecture, the minimum production cost, overall utility, and the configuration of each product variant, is shown separately in Table IX. VII. CONCLUSION This paper has investigated two major issues pertaining to multiple-platform architecture and product family design problem with a view to minimize the total production cost, while maximizing the overall utility associated with the product family. In order to determine the overall utility, part-worth utilities were estimated by using the conjoint analysis. To combine these goals, the fuzzy GP model has been adopted that has a wider applicability and work with deviations from preassigned goals. RSOTs have been applied to test their efficacy over a considered case study of family of mobile phones. The results from the preliminary experiments revealed that algorithms, such as GA, SA, and TS that have been utilized on the underlying case study, have demonstrated satisfactory performance. On the basis of obtained results, this study suggests an expansion from single to multiple platforms. Therefore, the pragmatic contribution of this paper lies in the advantages offered by multiple platforms to reduce the total production cost. Additionally, in case of combined objective value, the multiple platforms perform better than the single platform. From the results, it can also be deduced that when setup cost is low, increment in the number of platforms is more advantages. Fol-

11

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Satish Tyagi received the B.S. degree from the National Institute of Foundry and Forge Technology, Ranchi, India, and the M.S. degree from the University of Louisiana at Lafayette, in 2008 and 2010, respectively. He is currently working toward the Ph.D. degree with Department of Industrial and Systems Engineering, Wayne State University, Detroit, MI. His papers appeared in journals such as Engineering Applications of Artificial Intelligence, International Journal of Production Research, Virtual and Physical Prototyping, etc. His research interests include integrated product development, lean six-sigma implementation, artificial intelligence-based optimization, supply chain, and rapid prototyping.

Kai Yang received the M.S. and Ph.D. degrees in industrial and operations engineering from the University of Michigan, Ann Arbor, in 1985 and 1990, respectively. He is currently a Professor with the Department of Industrial and Systems Engineering, Wayne State University, Detroit, MI, where he is also the Director of Center of Healthcare System Engineering Research. He is an author of more than six scholarly published books on six-sigma and quality management. His papers appeared in journals such as IEEE TRANSACTION ON RELIABILITY, IIE Transactions, Engineering Applications of Artificial Intelligence, International Journal of Production Research, Quality Engineering, etc. His research interests include quality engineering, reliability engineering, lean product development/lean knowledge management, surgery operation modeling and optimization, and reusable medical supply management.

Annu Tyagi received the B.S. degree in microbiology from Korba City College, Korba, India, in 2008, and the MBA degree in human resources (HR), retail business management, and marketing from the MIT College of Management, Pune, India, in 2010. She is currently an HR Manager with Lifetime Mobilities Pvt. Ltd., Mumbai, India. Her paper appeared in journals such as Engineering Applications of Artificial Intelligence. Her current research interests include artificial intelligence-base optimization, supply chain management, and quality/reliability engineering. Ms. Tyagi was the recipient of prestigious Gold Medal Award for gaining the best academic performance in MIT Pune (Dec. 2010).

Anoop Verma (S’10) received the B.S. degree from the National Institute of Foundry and Forge Technology, Ranchi, India, and the M.S. degree from the University of Cincinnati, Cincinnati, OH, in 2007 and 2009, respectively. He is currently working toward the Ph.D. degree with Department of Mechanical and Industrial Engineering, University of Iowa, Iowa City. His papers appeared in journals such as IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, Engineering Applications of Artificial Intelligence, International Journal of Production Research, Expert Systems With Applications, ASME Journal of Applied Mechanics, etc. He is acting as a reviewer of various international journals. His current research interests include data mining, artificial intelligence, and process optimization applied to wind energy.