Agility evaluation using fuzzy logic

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Int. J. Production Economics 101 (2006) 353–368

Agility evaluation using fuzzy logic

Ching-Torng Lin

a,

, Hero Chiu

b

, Yi-Hong Tseng

a

Department of Information Management, Da-Yeh University, 112 Shan-Jiau Rd., Da-Tsuen, Changhua 51505, Taiwan

b_{Department of Management Science, National Chiao Tung University, HsinChu, Taiwan}

Received 12 May 2003; accepted 29 January 2005 Available online 19 March 2005

Abstract

‘‘Change’’ seems to be one of enterprises’ major characteristics in this new competitive era. Agile enterprise whereby an organization can change and adapt quickly to changing circumstances is increasingly viewed as a winning strategy. However, in embracing agile enterprise, there are important questions to be asked: what precisely is agility and how can it be measured? How can one assist in achieving and enhancing agility effectively? Answers to such questions are critical to the practitioners and to the theory of agile enterprise design. The foundation of agile enterprise lies in the integration of information system/technologies, people, business processes and facilities. Due to the ill-deﬁned and vague indicators which exist within agility assessment, most measures are described subjectively by linguistic terms which are characterized by ambiguity and multi-possibility, and the conventional assessment approaches cannot suitably nor effectively handle such measurement. However, fuzzy logic provides a useful tool for dealing with decisions in which the phenomena are imprecise and vague. Thus, the novelty in the paper is development of the absolute agility index, a unique and unprecedented attempt in agility measurement, using fuzzy logic to address the ambiguity in agility evaluation. Details of the approach and a framework of a fuzzy agility evaluation will be presented. An example is also used to illustrate the approach developed.

Keywords: Agile enterprise; Agility index; Agility measuring; Fuzzy logic

1. Introduction

Nowadays, many companies are facing constant-ly increasing competition stimulated by techno-logical innovations, changing market environments

and changing customer demands. This critical situation has led to a major revision in business priorities, strategic vision, and in the viability of conventional and even relatively

contempo-rary models (Shariﬁ and Zhang, 1999). In an

increasingly competitive market, there is a need to develop and improve organizational ﬂexibility and responsiveness. In the past decade, most compa-nies adopted restructuring and re-engineering in www.elsevier.com/locate/ijpe

Corresponding author. Tel.: +886 4 851 1888x3133; fax: +886 4 851 1500.

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response to challenges and demands; however, these were not always successfully (Sutclife, 1999). Agile enterprise addresses new ways of running companies to react quickly and effectively to changing markets, driven by customized products and services. The foundation of agile enterprise is the integration of information system/technolo-gies, people, business processes and facilities into a harmonious and ﬂexible organization so as to respond quickly to changing circumstances. In-deed agile enterprise not only encompasses the whole spectrum of activities in the company but is also associated with the supply chain as well (Ren et al., 2001).

Agile enterprise in general can provide lower manufacturing costs, increase market share, satisfy customer requirements, facilitate the rapid intro-duction of new products, eliminate non-value added activities and increase company’s competi-tiveness. Thus, agile enterprise has been advocated as the 21st century’s enterprise paradigm, and is seen as the winning strategy to become national and international leaders in an ever increasing competitive market of fast changing customer requirements (Yusuf et al., 1999). However, the ability to build agile enterprise has not developed as rapidly as anticipated, because the development of technology to manage agile enterprise is still under way (Sharp et al., 1999). Thus, in embracing agile enterprise many important questions con-cerning agility need to be asked, such as: what precisely is agility and how can it be measured? How will companies know when they have it, as there are no simple metrics or indexes available? How and to what degree does the company’s attributes affect companies’ business performance? How to compare agility with competitiveness? If a company wants to improve agility, how can the company identify the principal obstacles to im-provement? How to assist in achieving agility effectively (Sharp et al., 1999;James-Moore, 1997;

Yusuf et al., 2001)? Answers to such questions are critical to the practitioners and to the theory of agile enterprise design. Therefore, the purpose of this research is to solve some of these problems, with particular focus upon agility measuring and identifying the principal obstacles to agility improvement.

To assist managers in better achieving an agile enterprise, there have been numerous studies dedicated to measure the agility of an enterprise. Some authors (Yusuf et al., 2001; Youssuf, 1993;

Van Hoek et al., 2001) proposed integration agility index methods. They deﬁned the agility index as a combination of measuring intensity levels of agility capabilities. Thus, the absolute agility level of company i is measured as

ðAGILITYindexÞi¼

XN j¼1

Aij,

where Aij is the agility level of capability j of

company i: Furthermore, the relative agility level can be obtained by standardizing all absolute agility levels of competitive companies.

Other measuring methods (Ren et al., 2000;

Meade and Rogers, 1997) were developed on the basis of analytic hierarchical process (AHP) logical concept. Using these methods, the evaluators apply a pairwise comparison technique to evaluate the agility capabilities. Finally, this evaluation is synthesized across the entire capabilities to derive overall agility indexes for each company; distin-guishing the superiority of different ﬁrms or different manufacturing operations can be done by comparing the agility of different ﬁrms or different manufacturing operations.

Furthermore, based on the characteristics of mass customization (MC) of product ing and on the requirements of agile

manufactur-ing, an MC product manufacturing agility

evaluation index system was established by Yang

and Li (2002). If Ri and Wi denote the agility

index and the weight of each agility capability, respectively, they deﬁne the agility index as ðAGILITYindexÞ ¼ XN i¼1 RiWi, where X N i¼1 Wi¼1.

The above methods are easy to implement and focus attention on the most important issues. However, due to the imprecise and vague deﬁni-tion of agility indicators, and when a situadeﬁni-tion is characterized by either lack of evidence or the

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inability of the experts to make a signiﬁcant assessment of an event, linguistic expressions are

used to estimate ambiguous events (Karwowski

and Mital, 1986). Most agility measurements are described subjectively by linguistic terms, which are characterized by ambiguity and multi-possibi-lity. Thus, the scoring of the above techniques can always be criticized, because the scale used to score the agility capabilities has two limitations: (1) such techniques do not take into account the ambiguity and multi-possibility associated with the mapping of one’s judgment to a number, and (2) the subjective judgment and the selection and pre-ference of evaluators have a signiﬁcant inﬂuence on those methods.

However, fuzzy logic provides a useful tool to deal with problems in which the phenomena are imprecise and vague (see Appendix A.1). Using fuzzy concepts, evaluators can use linguistic terms to assess the indicators in a natural language expression and each linguistic term can be associated with a membership function. Fuzzy logic by making no global assumption about the independence, exhaustiveness, or exclusiveness of facilitated evidence, tolerates a blurred boundary in deﬁnitions (Machacha and Bhattacharya, 2000). This brings hope of incorporating qualitative indicators into decision-making since it is often

vaguely deﬁned or has unclear boundaries.

Furthermore, fuzzy logic has found large

applica-tion in management decisions (Machacha and

Bhattacharya, 2000; Lin and Chen, 2004; Basim and Imad, 2003;Bu¨yu¨ko¨zkan and Feyzioglu, 2004;

Beskese et al., 2004).

From this review, to assist managers in better achieving an agile enterprise, a model on the basis of fuzzy logic is purposed to provide a means of both measuring how agile an enterprise is and identifying the principal obstacles to improve the agility level. In this approach, the performance ratings and importance weights of different agility capabilities assessed by experts are expressed in linguistic terms. Then appropriate fuzzy numbers are used to present the linguistic values, and a simple fuzzy arithmetical operation is employed to synthesize these fuzzy numbers into one fuzzy number, which is called the fuzzy-agility-index (FAI). Also, the FAI is matched with appropriate

linguistics, thereby enabling the agility level to be expressed in linguistic terms. After that the fuzzy performance-importance index (FPII) of each agility capability is devised to help managers identify the main adverse factors and calls for managers to institute an appropriate action plan to improve the agility level. As an illustration, the agility measuring of an MC product manufactur-ing is used to illustrate the approach developed, followed by a discussion on its effectiveness.

2. The conceptual model of agility enterprise The purpose of agile enterprise is to enrich/ satisfy customers and employees. An enterprise essentially possesses a set of capabilities so as to make appropriate responses to changes taking place in its business environment. However, the business conditions in which many companies ﬁnd themselves are characterized by volatile and unpredictable demand; thus, the increasing ur-gency of pursuing agility. The agility might, therefore, be deﬁned as the ability of an enterprise to rapidly respond to change in market and

customers’ demands (Sharp et al., 1999). To be

truly agile, an enterprise should possess a number of distinguishing agile enablers. From a review of the normative literature, the authors have devel-oped a conceptual model of agile enterprise, as

shown in Fig. 1, culminating in many research

propositions.

The main driving force behind agility is change. Even through change is nothing new, today’s change is taking place at a much faster speed than ever before. Turbulence and uncertainty in the business environment have become the main causes of failure in the manufacturing industry. The number of changes and their type, specifica-tion or characteristic cannot be easily determined and are probably indefinite. Different enterprises with different characteristics and in different circumstances experience different changes that are specific and perhaps unique to themselves. But there are common characteristics in changes that occur, which can bring about a general conse-quence for every enterprise. Summarizing previous studies (Sharifi and Zhang, 1999;Ren et al., 2001;

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Yusuf et al., 1999), the general areas of change in business environment are categorized as follows: (1) market volatility caused by growth of the niche market, increasing new product introduction and product lifetime shrinkage; (2) intense competition caused by a rapidly changing market, increasing cost pressure, international competitiveness and short development of new products; (3) customer requirements’ changes caused by demand for customization, quality expectation increase and quicker delivery time; (4) accelerating technologi-cal change caused by the introduction of new and efﬁcient production facilities, and system integra-tion (hardware and software); and (5) change in social factors caused by environmental protec-tion, workforce/workplace expectations and legal pressures.

Agile enterprises are concerned about change, uncertainty and unpredictability within their busi-ness environment and make appropriate re-sponses. Therefore, agile enterprises require a number of distinguishing capabilities or ‘‘ﬁtness’’

to deal with the change, uncertainty and unpre-dictability within their business environment. These capabilities consist of four principle

ele-ments (Ren et al., 2001; Yusuf et al., 1999;

Giachetti et al., 2003): (1) responsiveness which is the ability to identify changes and respond quickly to them, reactively or proactively, and recover from them; (2) competency which is the ability to efﬁciently and effectively reach enterprises’ aims and goals; (3) ﬂexibility/adaptability which is the ability to process different processes and achieve different goals with the same facilities; and (4) quickness/speed which is the ability to carry out activity in the shortest possible time. Furthermore, underpinning these fours principles is a methodol-ogy to integrate them into a coordinated, inter-dependent system, and to translate them into strategic competitive capabilities (Sharp et al.,

1999). These must be taken into account if an

organization is to carry out agile enterprise. Achieving agile enterprise requires responsive-ness in strategies, technologies, people, busiresponsive-ness

Agility capabilities: Responsiveness Competency

Agile enterprise Enrich and satisfy customers

Cost Time

Function Robustnes Changing competition in business environments (Agility drivers)

Technology Social factors Customer requirement Competition criteria Market Flexibility Quickness Agility enablers/pillars Collaborative relationships (Strategy) Master change and

uncertainty (Control) Leverage people and

information technology (Foundation)

·

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processes and facilities. Thus all areas of the company need to have some agility providers to effectively respond to changing market require-ments. In the past, to assist managers in achieving an agile enterprise, there have been numerous studies dedicated to identify agility providers from which organization leaders could select those items appropriate to their own strategies, organization business processes and information systems. For

example, Goldman et al. (1991) suggested that

agility has four underlying components of agility including delivering value to the customers, being ready for change, valuing human knowledge and skills, and forming virtual partnerships. The project ‘‘Next Generation Manufacturing’’

con-ducted byNGM Project Ofﬁce (1997)indicates six

attributes for agility: they are the customer, physical plant and equipment, human resources, global markets, core competency, and practices

and cultures. Moreover, Yusuf et al. (1999)

suggested a set of 32 agility providers as listed in

Table 1, which include four dimensions: core management competency, virtual enterprise, cap-ability for reconﬁguration, and knowledge driven enterprise. These enablers are supposed to be the aspects of agility and to determine the entire behavior of the enterprise.

From this review we can see that different researchers provide some insights into different aspects of agility. There is a high probability that there is no single set of enablers, which reﬂect all aspects. However, the most important point is to understand the relationships of the enablers, to deploy and integrate them, and ﬁnally to trans-form them into strategic competitive capabilities. Even aspects of agility enablers listed in Table 1

are by no means exhaustive and therefore new factors may be added depending on the product, industry and market characteristics.

3. Fuzzy agility evaluation approach

The fuzzy agility evaluation (FAE) framework,

shown inFig. 2, is composed of two major parts.

The ﬁrst part is the business operation environ-ments’ evaluation and agility capabilities’ identiﬁ-cation. The purpose of the business environment survey is to collect and analyze the agility drivers which are the changes in the business environment that drive a company to reconsider the company’s position, strategy and process, and in sequence maybe used to reset new strategies when running their business and building agility capabilities. The

Table 1 Agility enablers

Dimensions Related attributes Dimensions Related attributes Integration Concurrent execution of activities Change Culture of change

Enterprise integration Continuous improvement

Information accessible to employees Partnership Trust-based relationship with customers/suppliers Competence Business practice & structure are difﬁcult to

replicate

Rapid partnership formation Multi-venturing capabilities Strategic relationship with customers Team

building

Decentralized decision-making Close relationship with suppliers

Empowered individuals working in teams Market Response to changing market requirements Cross-functional team New product introduction

Teams across company borders Customer-driven innovations Technology Technology awareness Customer satisfaction

Leadership in the use of current technology Education Continuous training and development Skill and knowledge enhancing technologies Learning organization

Flexible production technology Multi-skilled and ﬂexible people Quality Quality over product life Workforce skill upgrade

Products with substantial added value Welfare Employee satisfaction First-time right design

Short development cycle times

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company’s agility capabilities are the vital abilities that would provide the required strength to make appropriate responses to changes taking place in its business, so that agility capabilities will provide for agility measuring of a company.

The second part of the framework is to evaluate agility capabilities and synthesize the ratings and the weights to obtain an FAI of an agile enterprise and to match the FAI with an appropriate agility level and to make an improvement analysis. A stepwise description is given as follows:

1. Select criteria for evaluation.

2. Determine the appropriate linguistic scale to assess the performance ratings and importance weights of the agility capabilities.

3. Measure the performance and importance of agility capabilities using linguistic terms.

4. Approximate the linguistic terms by fuzzy numbers.

5. Aggregate fuzzy ratings with fuzzy weights to obtain an FAI of an enterprise.

6. Match the FAI with an appropriate level. 7. Analyze and identify the principal obstacles to

improvement.

4. An illustrative example

In this section the FAE approach was employed to study and measure the agility of the MC product manufacturing company, Xi Dian Casting

Limited Company (XDCLC) (Yang and Li, 2002).

Step 1: Select criteria for evaluation. MC is a kind of production model, which combines the advantages of mass-production with those of customized products to satisfy the customers’ demand so that it supplies to the individual customer in random quantity, or to multi-variety small batch markets, based on mass-production with high efﬁciency. In a business environment that continues to vary very quickly, MC enterprise is required to enhance its ability to master change and uncertainty by improving its product manu-facturing agility. Thus, MC’s product manufactur-ing is aimed at enrichmanufactur-ing customers via agile response to customer demand, market change and market opportunities. MC enterprise uses a series of advanced information technology, mod-ern management technology and advanced manu-facturing technology. This makes components and technology universal and enables the rapid launch of products. Accordingly, MC enterprises are generally required to possess at least three general agility capabilities: organization management, product design and product manufacturing. Ac-cording toYang and Li (2002), the three grades of the three general agility capabilities (AC) for

XDCLC are listed inTable 2.

Step 2: Determine the appropriate linguistic scale to assess the performance ratings and importance weights of the agility capabilities. In many cases it is virtually impractical for experts to directly determine the score of a vague indicator, such as the perfect degree of information systems, the way information demand was obtained,

Agility drivers: Change in marketplace Change in competition Change in customer desire Change in technology Change in social factors Agility capabilities:

Responsiveness Competency Flexibility Quickness

Identify the needs of agility capability and determine the

assessment terms

Linguistic assessment and linguistic translation

Fuzzy numbers Management

threshold

Fuzzy ratings and crisp weights aggregation

Linguistic label bank

Match fuzzy agility index with linguistic

term Rank fuzzy performance-importance index Obstacles to improvement Agility level FPII FAI

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Table 2 Agility capab ilities for agility ind ex eva luating in MC prod uct manu factu ring 1-Gr ade index 2-Gr ade ind ex 3-Grade index MC enterprise org anizatio n man agement agilit y (AC 1 ) Info rmation manag ement agility (AC 11 ) Perf ect de gree of enterp rise information syst em (A C111 ), network connect ion exte nsiven ess (A C112 ), information and netw ork utiliz ation rate (A C113 ) Inte r-organ ization co operat ive exte nt (AC 12 ) T h e degre e o f co operat ion with other enterprises (A C121), the applic ation de gree of the VE (A C122 ) Prod uce organi zing agility (A C13 ) T h e space org anizatio nal form of the pro duction pro cess (AC 131 ), the time organi zationa l form of the prod ucti on process (AC 132 ) The agility of instit utiona l fram ework (AC 14 ) T h e form of institut ional fram ework (AC 141 ), the spe ed of the team building (AC 142 ) MC enterprise prod ucts design agility (AC 2 ) Custo mer de mand information agile to get (AC 21 ) Th e way of demand info rmation got (AC 211 ), the pro portion of information proce ssing time in pro ducts period (AC 212 ) The speed of prod ucts de sign (AC 22 ) T h e pe riod prod ucts design (AC 221 ), the pro portion of design period in prod ucts pe riod (A C222 ) Prod ucts de sign fle xibility (AC 23 ) T h e seria tion degre e o f pro ducts (AC 231 ), the similar de gree of prod ucts st ructure (AC 232 ), the un iversalization degree of the part (AC 233 ) MC enterprise proce ssing manu factu re agility (AC 3 ) Re-c onfigura ble (A C31 ) P a ckaging integrate d unit mo dular (AC 311 ), sup plement too l displac ement (AC 312 ), disp lacement co mpatib ility (AC 313 ) The speed of manu facture (AC 32 ) T h e pro portion of produc tion and technolo gy pre paring time in pro ducts period (AC 321 ), the pe riod of manu factu re (AC 322 ), the proportio n o f man ufacture period in prod ucts period (A C323 ) Man ufact ure flexib ility (A C33 ) T h e un iversalization de gree of equip ment (A C331 ), the scal able de gree of equipm ent (AC 332 )

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displacement compatibility, etc. Therefore, in this approach, linguistic terms are used to assess the performance rating and importance weights of the agility capabilities.

The ad hoc usage of linguistic terms and corresponding membership functions is always criticized by fuzzy logic. Notably, many popular linguistic terms and corresponding membership functions have been proposed for linguistic

assess-ment (Karwowski and Mital, 1986; Chen and

Hwang, 1992). For the sake of convenience, and instead of elicitation from the experts, linguistic terms and corresponding membership functions can be obtained directly from past data or used as a basis and modiﬁed to incorporate indivi-dual situations and the requirements of different users. Furthermore, it is in general suggested that linguistic levels not exceed nine levels, which represent the limits of human absolute discrimination.

On the basis of the original data of the study

conducted byYang and Li (2002)and considering

the human way of perceiving differences, the linguistic variables {Excellent [E], Very Good [VG], Good [G], Fair [F], Poor [P], Very Poor [VP], Worst [W]}, are selected to assess the performance rating of the agility capabilities. Furthermore, the linguistic variables {Very High [VH], High [H], Fairly High [FH], Medium [M], Fairly Low [FL], Low [L], Very Low [VL]}, are selected to assess the importance weights of the agility capabilities.

Step 3: Measure the performance and impor-tance of agility capabilities using linguistic terms. Once the linguistic variables for evaluating the performance ratings and the importance weights of the agility capabilities are deﬁned, according to the company policy and strategy, company proﬁle, company characteristics, business changes and practices, marketing competition information, and the experts’ experience and knowledge, the experts can directly use the linguistic terms above to assess the rating which characterizes the degree of the performance of various agility capabilities. Concurrently, the experts can evaluate the relative importance of each agility capability by compar-ison, on the basis of the company’s strategies and policies, marketing competition trend, technology

development trend and experts’ experience and knowledge.

This case study was adopted to sustain Yang

and Li’s (2002) data. Thus on the basis of their original assessment, ﬁve experts used the linguistic terms above to directly measure the performance rating and importance weight of the agility capabilities. Furthermore, median operation was used to aggregate the ﬁve experts’ assessments (since median operation is more robust in a small sample). The results, integrated performance rat-ings and integrated importance weights of agility capabilities measured by linguistic variables, are shown inTable 3.

Step 4: Approximate the linguistic terms by fuzzy numbers. Applying the approximate reason-ing of fuzzy sets theory (see the Appendix A.1), the linguistic value can be approximated by a fuzzy

number. Using previous studies (Lin and Chen,

Table 3

Aggregated performance rating and aggregated importance weight of agility capabilities

ACi ACij ACijk Wi Wij Wijk Rijk

AC1 AC11 AC111 H VH VH G AC112 VH G AC113 H G AC12 AC121 H VH G AC122 H P AC13 AC131 VH VH G AC132 VH E AC14 AC141 H VH G AC142 H F AC2 AC21 AC211 H H H VG AC212 VH VG AC22 AC221 VH VH VG AC222 VH VG AC23 AC231 H H VG AC232 VH VG AC233 FH G AC3 AC31 AC311 VH FH FH G AC312 FH F AC313 H VG AC32 AC321 VH H G AC322 VH VG AC323 VH VG AC33 AC331 VH VH VG AC332 H G

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2004; Chen and Hwang, 1992) as a basis and modifying to incorporate the MC product manu-facturing XDCLCs situations, a set of fuzzy numbers for approximating linguistic variable values was developed as listed inTable 4. (Table 4

merely presents what we assessed to be most suitable for this case study. However, the compe-titive situations and requirements vary from company to company; hence, companies must establish their unique membership function appro-priate to their speciﬁc environment and

considera-tions.) Then, applying the relation between

linguistic terms and fuzzy numbers, the linguistic terms shown inTable 3are transferred into fuzzy

numbers as shown inTable 5.

Step 5: Aggregate fuzzy ratings with fuzzy weights to obtain a FAI of the MC product manufacturing XDCLC. FAI is an information fusion, which consolidates the fuzzy ratings and fuzzy weights of all of the factors that inﬂuence agility. FAI represents overall enterprise agility. Enterprise agility increases with increasing FAI. Thus, the membership function of FAI is used to determine the agility level. According to fuzzy-weighted average deﬁnition, the fuzzy index of the agility 2-grade-capability ACijcan be calculated as

ACij¼ Xn k¼1 ðWijkACijkÞ , Xn k¼1 Wijk, (1)

where ACijk and Wijk; respectively, represent the

fuzzy performance rating and fuzzy importance weight of the agility element capability.

Then, by using the formulas in Eq. (1), the fuzzy

index of the agility 2-grade-capability ACij is

obtained. For example, the fuzzy index of the agility 2-grade capability, information manage-ment agility AC11, is calculated as

AC11¼ ½ð5; 6:5; 8Þ ð0:85; 0:95; 1:0Þ ð5; 6:5; 8Þ

ð0:85; 0:95; 1:0Þ ð5; 6:5; 8Þ ð0:7; 0:8; 0:9Þ=½ð0:85; 0:95; 1:0Þ ð0:85; 0:95; 1:0Þ ð0:7; 0:8; 0:9Þ ¼ ð5; 6:5; 8Þ.

Applying the same equation, other fuzzy indexes of agility 2-grade-capabilities ACij and the agility

1-grade-capabilities ACi are obtained as listed in

Table 6.

Finally, applying Eq. (1) again, the FAI of MC product manufacturing XDCLC is calculated as FAIXDCLC¼ ½ð4:85; 6:44; 7:91Þ ð0:7; 0:8; 0:9Þ ð6:75; 7:87; 8:94Þ ð0:7; 0:8; 0:9Þ ð5:79; 7:22; 8:54Þ ð0:85; 0:95; 1:0Þ=½ð0:7; 0:8; 0:9Þ ð0:7; 0:8; 0:9Þ ð0:85; 0:95; 1:0Þ ¼ ð5:72; 7:18; 8:52Þ:

Step 6: Match the FAI with an appropriate agility level. Once the FAI has been obtained, to identify the level of agility, the FAI can be further matched with the linguistic label whose membership function is the same as (or closest to)

Table 4

Fuzzy numbers for approximating linguistic variable values

Performance-rating Importance-weighting

Linguistic variable Fuzzy number Linguistic variable Fuzzy number

Worst (W) (0, 0.5, 1.5) Very Low (VL) (0, 0.05, 0.15)

Very Poor (VP) (1, 2, 3) Low (L) (0.1, 0.2, 0.3)

Poor (P) (2, 3.5, 5) Fairly Low (FL) (0.2, 0.35, 0.5)

Fair (F) (3, 5, 7) Medium (M) (0.3, 0.5, 0.7)

Good (G) (5, 6.5, 8) Fairly High (FH) (0.5, 0.65, 0.8)

Very Good (VG) (7, 8, 9) High (H) (0.7, 0.8, 0.9)

Excellent (E) (8.5, 9.5, 10) Very High (VH) (0.85, 0.95, 1.0)

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the membership function of the FAI from the natural-language expression set of agility label (AL).

Several methods for matching the membership function with linguistics terms have been proposed

(Eshragh and Mandani, 1979; Schmucker, 1985). There are basically three techniques: (1) Euclidean distance method, (2) successive approximation, and (3) piecewise decomposition. It is recom-mended that the Euclidean distance method (see in Appendix A.2) be utilized because it is the most intuitive form of human perception of proximity (Guesgen and Albrecht, 2000).

In this case the natural-language expression set AL ¼ {Extremely Agile [EA], Very Agile [VA], Agile [A], Fairly [F], Slowly [S]} is selected for labeling, and the linguistics and corresponding

membership functions are shown inFig. 3. Then,

by using the Euclidean distance method, the Euclidean distance D from the FAI to each member in set AL is calculated:

DðFAI; EAÞ ¼ 16984; DðFAI; VAÞ ¼ 0:0645,

DðFAI; AÞ ¼ 1:9519; DðFAI; FÞ ¼ 1:9841,

DðFAI; SÞ ¼ 1:9841.

Table 5

Linguistic terms approximated by fuzzy numbers

ACi ACij ACijk Wi Wij Wijk Rijk

AC1 AC11 AC111 (0.7, 0.8, 0.9) (0.85, 0.95, 1.0) (0.85, 0.95, 1.0) (5, 6.5, 8) AC112 (0.85, 0.95, 1.0) (5, 6.5, 8) AC113 (0.7, 0.8, 0.9) (5, 6.5, 8) AC12 AC121 (0.7, 0.8, 0.9) (0.85, 0.95, 1.0) (5, 6.5, 8) AC122 (0.7, 0.8, 0.9) (2, 3.5, 5) AC13 AC131 (0.85, 0.95, 1.0) (0.85, 0.95, 1.0) (5, 6.5, 8) AC132 (0.85, 0.95, 1.0) (8.5, 9.5, 10) AC14 AC141 (0.7, 0.8, 0.9) (0.85, 0.95, 1.0) (5, 6.5, 8) AC142 (0.7, 0.8, 0.9) (3, 5, 7) AC2 AC21 AC211 (0.7, 0.8, 0.9) (0.7, 0.8, 0.9) (0.7, 0.8, 0.9) (7, 8, 9) AC212 (0.85, 0.95, 1.0) (7, 8, 9) AC22 AC221 (0.85, 0.95, 1.0) (0.85, 0.95, 1.0) (7, 8, 9) AC222 (0.85, 0.95, 1.0) (7, 8, 9) AC23 AC231 (0.7, 0.8, 0.9) (0.7, 0.8, 0.9) (7, 8, 9) AC232 (0.85, 0.95, 1.0) (7, 8, 9) AC233 (0.5, 0.65, 0.8) (5, 6.5, 8) AC3 AC31 AC311 (0.85, 0.95, 1.0) (0.5, 0.65, 0.8) (0.5, 0.65, 0.8) (5, 6.5, 8) AC312 (0.5, 0.65, 0.8) (3, 5, 7) AC313 (0.7, 0.8, 0.9) (7, 8, 9) AC32 AC321 (0.85, 0.95, 1.0) (0.7, 0.8, 0.9) (5, 6.5, 8) AC322 (0.85, 0.95, 1.0) (7, 8, 9) AC323 (0.85, 0.95, 1.0) (7, 8, 9) AC33 AC331 (0.85, 0.95, 1.0) (0.85, 0.95, 1.0) (7, 8, 9) AC332 (0.7, 0.8, 0.9) (5, 6.5, 8) Table 6

Fuzzy index of each grade of agility capabilities

ACi ACij Ri Rij AC1 AC11 (4.85, 6.44, 7.91) (5, 6.5, 8) AC12 (3.65, 5.13, 6.58) AC13 (6.75, 8, 9) AC14 (4.1, 5.81, 7.53) AC2 AC21 (6.75, 7.87, 8.94) (7, 8, 9) AC22 (7, 8, 9) AC23 (6.32, 7.59, 8.79) AC3 AC31 (5.79, 7.22, 8.54) (4.9, 6.61, 8.21) AC32 (6.31, 7.56, 8.74) AC33 (6.1, 7.31, 8.53)

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Thus, by matching a linguistic label with the minimum D, the agility index level of the MC product manufacturing XDCLC can be identiﬁed as ‘‘very agile’’, as shown inFig. 3.

Step 7: Analyze and identify the principal obstacles to improvement. An agility evaluation not only measures how agilze an enterprise is but also, most importantly, helps managers assess distinctive competencies and identify the principal obstacles for implementing appropriate improve-ment measures. Although the agility index level of MC product manufacturing XDCLC is ‘‘very agile’’ (according to the evaluation), there were obstacles within the organization which could have impacted the agility of the MC product manufacturing XDCLC.

In order to identify the principal obstacles for improving agility level, a fuzzy performance-importance index (FPII) of agility element cap-ability, which combines the performance rating and importance weight of each agility element capability, represents an effect which will con-tribute to the agility level of an organization. The lower the FPII of a factor is, the lower the degree of contribution for this factor. Thus, the score of the FPII of a factor is used for identifying the principal obstacles.

If used directly to calculate the FPII, the importance weights Wijk will neutralize the

per-formance ratings in calculating FPII; in this case, it will become impossible to identify the actual main obstacles (low performance rating and

high importance). If Wijk is high, then the

transformation ½ð1; 1; 1ÞÞ Wijk is low.

Conse-quently, for each agility element capability ijk, the

fuzzy performance-importance index FPIIijk, is

deﬁned as

FPIIijk¼W0ijkACijk, (2)

where W0

ijk¼ ð1; 1; 1Þ W0ijk; Wijk is the fuzzy

importance weight of the agility element capability ijk:

Then, by using the formulas in Eq. (2), the FPIIs of each agility element capability are obtained as

listed in Table 7. For example, the FPII of the

perfect degree enterprise information system

AC111, is calculated as

FPII111¼ ½ð1; 1; 1Þ ð0:85; 0:95; 1:0Þ ð5; 6:5; 8Þ

¼ ð0; 0:325; 1:2Þ.

Since fuzzy numbers do not always yield a totally ordered set as real numbers do, all the FPIIs must be ranked. Many methods

have been developed to rank fuzzy numbers (Chen

and Hwang, 1992; Lee-Kwang and Lee, 1999). Here, the ranking of the fuzzy number is based on Chen and Hwang’s left-and-right fuzzy-ranking method (see in Appendix A.3), since it not only preserves the ranking order but also considers the absolute location of each fuzzy number (Chen and Hwang, 1992). By using Chen and Hwang’s left-and-right fuzzy-ranking method, the total scoring values of the 24 FPIIs of the agility element capabilities are obtained as shown in Table 7.

As mentioned in the Pareto principle, resources should be used in the improvement of critical obstacles. To identify the few critical obstacles, scale 0.8 was set as the management threshold to distinguish which critical obstacles need to be improved. Subsequently, as shown inTable 7, four capabilities have a lower performance than the threshold, namely: (1) perfect degree of enterprise information system, (2) network connection ex-tensiveness, (3) the degree of cooperation with other enterprises, and (4) the space organizational form of the production process. These capabilities represent the most signiﬁcant contributions to enhance the MC product manufacturing XDCLC and to achieve agility. Furthermore, according to this identiﬁcation, managers can select appropriate agility providers fromTable 1to implement better agility level. FAI 1.0 F( x) 0 1 2 3 4 5 6 7 8 9 10 x S F A VA EA

Fig. 3. Linguistic levels to match fuzzy-agility-index. [(S (0, 1.5, 3); F (1.5, 3, 4.5); A (3.5, 5, 6.5); VA (5.5, 7, 8.5); EA (7, 8.5, 10)].

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5. Discussion and conclusions

This paper has highlighted the question: how far toward becoming agile is an organization? How can an organization improve its agility effectively? Also, limited by the nature of agility deﬁnition, agility measurement is associated with vagueness and complexity and the conventional (crisp) assessment approaches are unsuitable and ineffec-tive for handling such evaluation. To compensate for these limitations, an FAI which concentrates on the application of linguistic approximating and fuzzy arithmetic has been developed to address agility measuring, stressing the multi-possibility and ambiguity of agility capability measurement. Three aspects, including organization manage-ment, product design and product manufacturing, form the agility index of MC product manufactur-ing. The evaluation procedures include: identifying agility capabilities, selecting linguistic variables for assessing and interpreting the values of the

linguistic variables, fuzzy rating and fuzzy weights integrating, fuzzy index labeling, and defuzzifying FPII in order to identify the main adverse factors which can inﬂuence agility achievement.

This model was developed from the concept of MCDM (Multi Criteria Decision Making) and fuzzy logic, and adopted the MC product

manu-facturing XDCLC (Yang and Li, 2002) as an

initial case study to validate the model and approach. With regard to the efﬁciency of the method to measure agility index, the result generated by both approaches seemingly leads to similar conclusions as shown inTable 8. However, the FAI generated by the fuzzy logic approach is expressed in terms of ranges of value. This rating can provide an overall picture of the pertinent possibility and ensure that the decision made in the

subsequent selection process is not biased.

Furthermore, it gives the decision-makers a high degree of ﬂexibility in decision-making. As an example in this study, the agility index had a fuzzy

Table 7

Fuzzy performance—importance indexes of 24 agility capabilities Agility

capability

Aggregated fuzzy performance rating

(1, 1, 1)_{z W}0

ijk Fuzzy performance— importance index Ranking score AC111 (5, 6.5, 8) (0, 0.05, 0.15) (0, 0.325, 1.2) 0.709 AC112 (5, 6.5, 8) (0, 0.05, 0.15) (0, 0.325, 1.2) 0.709 AC113 (5, 6.5, 8) (0.1, 0.2, 0.3) (0.5, 1.3, 2.4) 1.703 AC121 (5, 6.5, 8) (0, 0.05, 0.15) (0, 0.325, 1.2) 0.709 AC122 (2, 3.5, 5) (0.1, 0.2, 0.3) (0.2, 0.7, 1.5) 1.028 AC131 (5, 6.5, 8) (0, 0.05, 0.15) (0, 0.325, 1.2) 0.709 AC132 (8.5, 9.5, 10) (0, 0.05, 0.15) (0, 0.475, 1.5) 0.907 AC141 (5, 6.5, 8) (0, 0.05, 0.15) (0, 0.325, 1.2) 0.709 AC142 (3, 5, 7) (0.1, 0.2, 0.3) (0.3, 1.0, 2.1) 1.413 AC211 (7, 8, 9) (0.1, 0.2, 0.3) (0.7, 1.6, 2.7) 1.95 AC212 (7, 8, 9) (0, 0.05, 0.15) (0, 0.4, 1.35) 0.809 AC221 (7, 8, 9) (0, 0.05, 0.15) (0, 0.4, 1.35) 0.809 AC222 (7, 8, 9) (0, 0.05, 0.15) (0, 0.4, 1.35) 0.809 AC231 (7, 8, 9) (0.1, 0.2, 0.3) (0.7, 1.6, 2.7) 1.95 AC232 (7, 8, 9) (0, 0.05, 0.15) (0, 0.4, 1.35) 0.809 AC233 (5, 6.5, 8) (0.2, 0.35, 0.5) (1.0, 2.275, 4.0) 2.715 AC311 (5, 6.5, 8) (0.2, 0.35, 0.5) (1.0, 2.275, 4.0) 2.715 AC312 (3, 5, 7) (0.2, 0.35, 0.5) (0.6, 1.75, 3.5) 2.274 AC313 (7, 8, 9) (0.1, 0.2, 0.3) (0.7, 1.6, 2.7) 1.95 AC321 (5, 6.5, 8) (0.1, 0.2, 0.3) (0.5, 1.3, 2.4) 1.703 AC322 (7, 8, 9) (0, 0.05, 0.15) (0, 0.4, 1.35) 0.809 AC323 (7, 8, 9) (0, 0.05, 0.15) (0, 0.4, 1.35) 0.809 AC331 (7, 8, 9) (0, 0.05, 0.15) (0, 0.4, 1.35) 0.809 AC332 (5, 6.5, 8) (0.1, 0.2, 0.3) (0.5, 1.3, 2.4) 1.703

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value (5.72, 7.18, 8.52). Qualitatively, this suggests that the agility level of MC product manufacturing XDCLC approaches ‘‘very agile’’ while being far from ‘‘extremely agile’’.

In comparison with the original study, seven linguistic levels and a range value are designated to represent the linguistic ambiguity range. Actually, on the basis of the needs of cognitive perspectives and available data characteristics, the number of linguistic levels and its membership functions can be adjusted correspondingly. In general, it is suggested that one not exceed the human dis-crimination capacity consisting of nine levels.

Finally, there are some limitations to this fuzzy logic approach. The membership functions of linguistic variables depend on the managerial perception of the decision-maker. Thus, the decision-maker must be at a strategic level in the company in order to realize the importance, possibility and trends of all aspects, such as strategy, marketing and technology. Furthermore, competitive situations and requirements vary from company to company; hence, companies have to establish their unique membership function by ﬁtting in with their speciﬁc environment and considerations. In addition, the computation of a fuzzy-weighted average is still complicated and not easily appreciated by managers. Fortunately, this calculation has been computerized to increase accuracy while reducing both time and possibility of errors.

Furthermore, the contribution of this work has provided potential value to practitioners by offer-ing a rational structure to reﬂect the imprecise phenomena in agility evaluation and has taken into account the ambiguity of each agility cap-ability to ensure relatively realistic and informative information, and to researchers by demonstrating an unprecedented application of fuzzy logic. In

addition, from the example, this approach has several advantages when compared to previous methods:

(1) This method can give the analyst relatively realistic and informative information. The FAI is expressed in a range of values. This provides an overall picture about the possible agility of an organization and ensures that the decision made in selection will not be biased. As an example of this study, the agility index has a fuzzy value (5.72, 7.18, 8.52).

(2) This method can systematically identify the weak factors within an organization and provide the means for a manager to formulate a comprehensive plan for improvement. There-fore, the method can be further used in self-assessment.

(3) It provides an appropriate function structure to reﬂect the imprecise phenomena in many business environments and takes into account the uncertainty effect of each factor to ensure a more convincing and reliable evaluation. Moreover, algorithm of the proposed method can be computerized. Thus, by the decision-makers’ providing linguistic assessments through a menu-driven interface design, the agility level of an enterprise and its agility obstacles can be obtained easily.

Appendix A

A.1. Basic concept of fuzzy set theory

For the purpose of application, the basic properties of fuzzy set theory needed in this study are introduced. Additional discussion can be

found in books byKlir and Yuan (1995).

A.1.1. Fuzzy numbers

Let X be a collection of objects, called the universe, whose elements are denoted by x: A fuzzy subset A in X is characterized by a

membership function f_AðX Þ which is associated

with each element x in X and a real number in the interval [0,1]. The function value fAðX Þ represents

the grade of membership of x in A:

Table 8

Comparison of fuzzy logic approach and crisp approach Approach Agility

index

Range Linguistic labeling Fuzzy logic (5.72, 7.18, 8.52) 2.8 Very agile

Crisp approach 7.26 Very agile

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A fuzzy subset A is called a fuzzy (real) number if A is convex and there exists exactly one real number a with f_AðaÞ ¼ 1: There are many forms of fuzzy numbers to represent imprecise information. As used here, triangular fuzzy numbers are applied.

Let x; a; b; c 2 R (real line); hence, a triangular fuzzy number is a fuzzy number A in R; if its membership function f_A:R-[0,1] is f_AðxÞ ¼ ðx aÞ=ðb aÞ; apxpb; ðx cÞ=ðc bÞ; bpxpc; 0; otherwise: 8 > < > :

The triangular fuzzy number is parameterized by the triplet A ¼ ða; b; cÞ: The parameter ‘‘b’’ gives the maximal grade of f_AðxÞ; i.e., f_AðbÞ ¼ 1; which is the most probable value of the evaluation data. The parameters ‘‘a’’ and ‘‘c’’ are the lower and upper bounds of the available area for the evaluation data. For example, the triangular fuzzy number to represent ‘‘close to 5’’ can be para-meterized by A ¼ ð3; 5; 7Þ: Triangular fuzzy num-bers are mostly used because they are easily speciﬁed by experts. Furthermore, under some weak assumptions, such use immediately complies with the relevant optimization criteria.

A.1.2. Linguistic variables

The concept of a linguistic variable is very useful in dealing with situations which are too complex or too ill-deﬁned to be reasonably described in conventional quantitative expressions. A linguistic variable is a variable whose values are words or sentences in natural or artiﬁcial language. For example, ‘‘low’’ is a linguistic variable if its value is linguistic rather than numerical. Furthermore, by the approximate reasoning of fuzzy sets theory, the linguistic value can be represented by a fuzzy number. For example, for the linguistic variables, {Excellent [E], Very Good [VG], Good [G], Fair [F], Poor [P], Very Poor [VP], Worst [W]}, the fuzzy numbers approximate to these linguistic values are shown inTable 4.

A.1.3. Fuzzy number arithmetic operations

Let A1and A2 be two triangular fuzzy numbers,

where A1¼ ða1; b1; c1Þ and A2¼ ða2; b2; c2Þ:

Ac-cording to the extension principle, the triangular fuzzy-number addition, subtraction and multi-plication operations of A1 and A2 are deﬁned as

follows: Fuzzy-number addition : A1A2¼ ða1; b1; c1Þ ða2; b2; c2Þ ¼ ða1þa2; b1þb2; c1þc2Þ. Fuzzy-number subtraction : A1A2¼ ða1; b1; c1Þ ða2; b2; c2Þ ¼ ða1c2; b1b2; c1a2Þ. Fuzzy-number multiplication : A1A2¼ ða1; b1; c1Þ ða2; b2; c2Þ ¼ ða1a2; b1b2; c1c2Þ.

A.2. Euclidean distance method

The Euclidean distance method consists of calculating the Euclidean distance from the given fuzzy number to each of the fuzzy numbers representing the natural-language expressions set. Suppose the natural-language expression set is agility level (AL). Then the distance between the fuzzy number fuzzy-agility-index (FAI) and each

fuzzy number member ALiAAL can be calculated

as below: dðFAI; ALiÞ ¼ X x2p ðf_FAIðxÞ f_AL_iðxÞÞ2 )1=2 8 < : , where p ¼ fx0; x1; . . . ; xmg ½0; 10 so that 0 ¼ x0ox1o oxm¼10: To simplify, let p ¼ f0; 0:5; 1; 1:5; 2; 2:5; 3; 3:5; 4; 4:5; 5; 5:5; 6; 6:5; 7; 7:5; 8; 8:5; 9; 9:5; 10g: Then, the distance from the FAI to each of the members in the set AL can be calculated.

A.3. Chen and Hwang’s left-and-right fuzzy-ranking method

In the Chen and Hwang’s left-and-right fuzzy-ranking method, for defuzzifying a fuzzy number, the fuzzy maximizing and minimizing set are,

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respectively, deﬁned as: fmaxðxÞ ¼ x; 0pxp10; 0; otherwise; ( fminðxÞ ¼ 10 x; 0pxp10; 0; otherwise; (

When given a triangular fuzzy number FPII

deﬁned as fFPII: R-[0, 10], with a triangular

membership function, the right-and-left scores of FPII can be obtained, respectively, as

URðFPIIÞ ¼ sup x ½f_FPIIðxÞ ^ f_maxðxÞ, ULðFPIIÞ ¼ sup x ½f_FPIIðxÞ ^ f_minðxÞ.

Finally, the total score of FPII can be obtained by combining the left-and-right-scores. The total score of FPII is deﬁned as

UTðFPIIÞ ¼ ½URðFPIIÞ þ 10 ULðFPIIÞ=2.

Using the total score, the fuzzy numbers can be ranked. For example, the total scoring value of a fuzzy number FPII111¼(0, 0.325, 1.2) is

calcu-lated as

URðFPII111Þ ¼sup fFPII111ðxÞ ^ fmaxðxÞ

x ¼1:103, ULðFPII111Þ ¼sup x fFPII111ðxÞ ^ fminðxÞ ¼9:685,

UTðFPII111Þ ¼ ½URðFPII111Þ þ10 ULðFPII111Þ=2

¼0:709.

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