Measuring flexibility of computer integrated manufacturing systems using fuzzy cash flow analysis

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1 Iformatio Scieces 168 (2004) Measurig flexibility of computer itegrated maufacturig systems usig fuzzy cash flow aalysis Cegiz Kahrama a, Ahmet Beskese b, *, Da Rua c a Departmet of Idustrial Egieerig, Istabul Techical Uiversity, Macka, Istabul, Turkey b Departmet of Idustrial Egieerig, Uiversity of Bahcesehir, 34538, Bahcesehir, Istabul, Turkey c Fuel Research Uit, Belgia Nuclear Research Cetre (SCK CEN), Boeretag 200, 2400 Mol, Belgium Received 16 Jue 2003; received i revised form 29 August 2003; accepted 10 November 2003 Abstract There exist a umber of methods proposed i the literature to quatify maufacturig flexibility i moetary terms ad to use a fiacial evaluatio model with a decisio criterio based o preset worth. However, most of these methods are uable to hadle problems with icomplete ad ucertai data. To obtai a sesible result i quatifyig the maufacturig flexibility i computer itegrated maufacturig systems, this paper proposes some fuzzy models based o fuzzy preset worth. The fuzzy models based o preset worth are basically egieerig ecoomics decisio models i which the ucertai cash flows ad discout rates are specified as triagular fuzzy umbers. To build such a model, fuzzy preset worth formulas of the maufacturig flexibility elemets are formed. Flexibility for cotiuous improvemet, flexibility for trouble cotrol, flexibility for work force cotrol, ad flexibility for work-i-process cotrol are quatified by usig fuzzy preset worth aalysis. Formulas for both iflatioary ad o-iflatioary coditios are derived. Usig these formulas, more reliable results ca be obtaied especially for such a cocept like flexibility that is described i may itagible dimesios. These models allow experts liguistic predicates about computer itegrated maufacturig systems. Ó 2003 Elsevier Ic. All rights reserved. * Correspodig author. Tel.: /1251; fax: address: beskese@bahcesehir.edu.tr (A. Beskese) /$ - see frot matter Ó 2003 Elsevier Ic. All rights reserved. doi: /j.is

2 78 C. Kahrama et al. / Iformatio Scieces 168 (2004) Keywords: Fuzzy cash flow aalysis; Computer itegrated maufacturig systems; Fuzzy umbers; Fuzzy itervals; Maufacturig flexibility 1. Itroductio Computer itegrated maufacturig (CIM) systems ca be viewed as a total system, which provides a automatic lik betwee product desig, maufacturig egieerig, ad the factory floor. Effective implemetatio of advaced maufacturig techologies through a CIM system is the corerstoe of factory moderizatio. CIM systems provide may importat beefits such as greater process flexibility, reduced ivetory, reduced floor space, faster respose to shifts i market demad, lower lead times, ad a loger useful life of equipmet over successive geeratios of products. Maufacturig flexibility may be defied as the ability to cope with chagig circumstaces or istability caused by the eviromet. Flexibility is emergig as oe of the key competitive stregths i today s maufacturig systems, sice it provides a critical measure of total maufacturig performace. Hece, measurig the flexibility of maufacturig systems is importat to operatios maagers egaged i decisio makig o strategic issues related to flexibility. Flexibility is widely recogized as a multidimesioal attribute. Most of the studies reported i the literature have focused o measurig separate dimesios idepedetly ad also have teded to be o-fiacial i ature. As a result, may of these measures have oly limited applicatio i strategic decisio makig. There exist various methods to measure the value of flexibility by usig surrogate measures to represet the itagible parts. However, maagers still prefer the methods by which all itagible parts of flexibility ca be quatified as far as it is possible i moetary terms. This preferece leads the decisiomakers ito a ucertai ecoomic decisio eviromet of the cotemporary busiess world, where a expert s kowledge about the cash flow iformatio usually cosists of a lot of vagueess istead of radomess. For example, to describe a sales profit that may be implicitly forecasted from past icomplete iformatio, liguistic descriptios like aroud oe millio are ofte used. The major cotributio of the fuzzy approach used i this paper is its capability of represetig vague kowledge i such a ecoomic eviromet. I real word applicatios, precise data cocerig flexibility factors of CIMs are ot available or very hard to be extracted. I additio, decisio-makers prefer atural laguage expressios rather tha sharp umerical values i assessig flexibility parameters. So, maufacturig flexibility is a iheretly fuzzy otio, which ca be measured by the sythesis of its costituets. Fuzzy logic offers a systematic base i dealig with situatios, which are ambiguous or ot well defied. Ideed, the ucertaity i expressios such as low flexibility or

3 C. Kahrama et al. / Iformatio Scieces 168 (2004) high utilizatio, which are frequetly ecoutered i the flexibility literature, is fuzziess. Tsourveloudis ad Phillis [29] are the oly researchers usig a fuzzy logic framework to measure maufacturig flexibility. They propose a fuzzy rulebased method that hadles imprecise data ad kowledge about a productio system. However, there is o paper published i refereed iteratioal jourals aimed at measurig flexibility i terms of fuzzy cash flows. With the rapid expasio of CIM ad flexible maufacturig systems (FMS), there are hudreds of studies o the defiitio, classificatio, ad quatificatio of maufacturig flexibility ad o decisio-makig tools for ivestmet aalysis of flexible automatio. Noetheless productio maagers of may firms, uder pressure to make automated maufacturig systems more flexible, are fidig that there is still a cosiderable gap betwee the beefits promised by the fiacial evaluatio tools ad the beefits realized i practice. As a value-added to the literature o the topic, this paper aims at providig practitioers with a fuzzy poit of view to the traditioal cash flow aalysis method for dealig quatitatively with imprecisio or ucertaity ad at obtaiig a fuzzy flexibility quatificatio from this poit of view which will close this gap cosiderably. Fuzzy itervals will be used to represet the cash flows ad the other o-moetary parameters of flexibility. Beskese et al. [1,2] provide a basis for this study. The paper is orgaized as follows: Sectio 2 gives a literature review for the quatificatio of maufacturig flexibility. Sectio 3 icludes a geeral kowledge about fuzzy preset worth aalysis, two applicatios of this aalysis for measurig maufacturig flexibility, ad a umerical example. Sectio 4 cocludes the curret research results ad future pla of the study reported i the paper. 2. Quatificatio of maufacturig flexibility The ecoomic justificatio of CIM has received icreased attetio i recet years. Arbitrarily high hurdle rates, compariso with the status quo, ad isufficiet beefit aalysis are cited as major efforts i applicatio. However, traditioal methods for the ecoomic justificatio of ew maufacturig techologies fail to iclude beefits such as better quality, greater flexibility ad reduced work-i-process, sice these beefits are difficult to measure ad therefore hard to quatify. Applicatio of traditioal capital budgetig methods, for example, does ot fully accout for the beefits arisig from icreased flexibility. Some special tools must be used to facilitate a thorough aalysis of tagible ad itagible beefits. I the literature, may researchers have tried to quatify the maufacturig flexibility ad to itegrate it with decisio-makig tools. Their methods of

4 80 C. Kahrama et al. / Iformatio Scieces 168 (2004) quatifyig the flexibility ca be classified ito three groups: (1) there are may itagible parts of flexibility, so it should be cosidered as blackbox [11,18,21]; (2) the itagible parts of flexibility that caot be quatified i moetary terms ca be measured by a surrogate value [8,27,30,32]; ad (3) all itagible parts of flexibility ca be quatified as far as it is possible i moetary terms [12,25,26,28]. Pyou ad Choi [25] distiguish the flexibility that is iheret i the maufacturig system from the flexibility that the user ca attai after implemetatio. They itroduce the cocepts of potetial flexibility ad realizable flexibility to obtai a deeper isight ito flexibility ad to give cosideratio to all possible aspects of user experiece ad capability. Potetial flexibility is the flexibility iheret i a maufacturig system from the system maufacturer s poit of view, before it is implemeted ad operated by the user. Realizable flexibility is the flexibility that will be realized by operatig the maufacturig system usig both potetial flexibility ad the users egieerig ad maagemet capability. Pyou ad Choi [25] classify potetial flexibility ito four elemets ad realizable flexibility ito 11 elemets. They propose a systematic procedure for quatifyig realizable flexibility i moetary terms ad use a fiacial evaluatio model with a decisio criterio based o preset worth. Although such efforts are highly appreciated, oe should be aware that there are some difficulties to quatify maufacturig flexibility by oly usig crisp methodologies. Some of them ca be metioed as: there is o effective method for the sythesis of the fuctioal parameters affectig each type of flexibility ad the flexibility types, which are observed o differet hierarchical levels ad are crucial i the determiatio of maufacturig flexibility, there is o correspodece betwee flexibility ad the physical characteristics of the productio system, ad certai operatioal characteristics have cotradictory effects o flexibility. To overcome such difficulties, this paper proposes a fuzzy flexibility quatificatio method usig fuzzy preset worth aalysis based o fuzzy iterval arithmetic. 3. Fuzzy preset worthaalysis i flexibility measuremet 3.1. Fuzzy preset worth aalysis Most ecoomic decisio problems ivolve the ucertaity feature of cash flow modelig. If sufficiet objective data is available, probability theory is commoly used i modelig cash flows ad performig decisio aalysis. Ufortuately, decisio-makers rarely have eough iformatio to perform the decisio aalysis, sice probabilities ca ever be kow with certaity ad the ecoomic decisio is attributable to may ucertai derivatios. I this

5 C. Kahrama et al. / Iformatio Scieces 168 (2004) µ p (x) 1.0 y f 1 (.) f 2 (.) 0.0 a b c X a+(b-a) y c+(b-c) y Fig. 1. A triagular fuzzy umber, ep ¼ða; b; cþ. situatio, most decisio-makers rely o experts kowledge i modelig cash flows. I the literature, triagular ad trapezoidal fuzzy umbers that are the special forms of LR-type fuzzy umbers are usually used to capture the vagueess of the parameters related to the topic. The arithmetic operatios of these types of fuzzy umbers ca be foud i [33]. I this paper, triagular fuzzy umbers (TFNs) will be used to cosider the fuzziess of the flexibility parameters. A TFN is desigated as ep ¼ða; b; cþ. It is graphically depicted i Fig. 1 i which f 1 ðþ is the left side, ad f 2 ðþ is the right side represetatio of the TFN. To deal quatitatively with imprecisio or ucertaity, fuzzy set theory is primarily cocered with vagueess i huma thoughts ad perceptios. As a alterative to covetioal cash flow models where cash flows are defied as either crisp umbers or risky probability distributios, Chiu ad Park [7] propose a egieerig ecoomics decisio model i which the ucertai cash flows ad discout rates are specified as triagular fuzzy umbers. They examie deviatio betwee exact preset worth (PW) ad its approximate form (PWA) ad perform the fuzzy project selectio by applyig differet domiace rules as show i Eqs. (1) ad (2) respectively. The result of the exact preset worth is also a fuzzy umber with a o-liear membership fuctio. It is i complex o-liear represetatios that require tedious computatioal effort [7]. For the reaso of simplicity, a TFN ca be used as a approximate form of the complex (exact) preset worth formula i Eq. (1). "! XN maxff lðyþ lðyþ ; 0g miff ; 0g PW ¼ where F lðyþ t t¼0 X N t¼0 Q t t 0 ¼0 t ð1 þ RrðyÞ t Þ þ Q t 0 t 0 ¼0ð1 þ RlðyÞ t Þ ; 0!# maxff rðyþ t ; 0g ð1 þ RlðyÞ t Þ þ miff rðyþ t ; 0g ð1 þ 0 RrðyÞ t Þ 0 Q t t 0 ¼0 Q t t 0 ¼0 is the left side represetatio, F rðyþ t the fuzzy cash flow ef at time t, adr lðyþ t 0 t ð1þ is the right side represetatio of is the left side represetatio, RrðyÞ is t 0

6 82 C. Kahrama et al. / Iformatio Scieces 168 (2004) the right side represetatio of the fuzzy iterest rate er at time t 0. N is a crisp umber deotig the project life. Whe the degree of membership (y) i Eq. (1) is equal to 0, F lðyþ t ¼ f t0, F rðyþ t ¼ f t2, R lðyþ t ¼ r t0, ad R rðyþ 0 t ¼ r t2. Whe the degree of membership (y) i Eq. 0 (1) is equal to 1, F lðyþ t ¼ F rðyþ t ¼ f t1,adr lðyþ t ¼ 0 RrðyÞ t ¼ r t1. Substitutig these to 0 the exact preset worth formula, the approximate form of the preset worth formula ca be derived as i Eq. (2). PWA is represeted usig its three parameters ad it is easier to implemet because they are i liear represetatios. PWA ¼ XN maxff t0 ; 0g Q t t¼0 t 0 ¼0 ð1 þ r þ miff t0; 0g Q t ; XN f t1 t 0 2Þ t 0 ¼0 ð1 þ r Q t t 0 0Þ t¼0 t 0 ¼0 ð1 þ r t 0 1Þ ; X N maxff t2 ; 0g Q t t 0 ¼0 ð1 þ r þ miff! t2; 0g Q t t 0 0Þ t 0 ¼0 ð1 þ r ð2þ t 0 2Þ t¼0 Chiu ad Park [7] compute the maximum deviatio as a measure of the fitess betwee PW ad PWA. They use very small icremets of y as the measuremet method istead of derivative method sice the latter is difficult to calculate. Usig a simulatio software, they calculate the deviatios for differet rages of cash flows ad discout rates, ad fid out that the deviatios are ot sigificat uless the cofidet width of discout rate is larger tha a absolute rage of ±4%. I the real world applicatios, whe the discout rates are usually estimated withi the width of ±4%, PWA ca be used i project aalysis. The deviatios of PW ad PWA are depicted i Fig. 2. I literature, some other approaches to calculate fuzzy PW ca be foud. For istace, Buckley [4] forms the membership fuctio for the fuzzy preset worth, P f W N, as i Eq. (3) lðxjp f h W N Þ¼ pw N1 ; f N1 ðyjp f W N Þ=pw N2 ; pw N2 =f N2 ðyjp f i W N Þ; pw N3 ð3þ where N is the crisp useful life of the project, pw Ni is the least, the most, ad the largest possible values of P f W N respectively for i ¼ 1; 2; 3, pw N1 < pw N2 < pw N3, f 1 ðyjp f W N Þ is a cotiuous mootoe icreasig fuctio of y ¼ lðxjp f W N Þ for 0 6 y 6 1 with f 1 ð0jp f W N Þ¼pw N1 ad f 1 ð1jp f W N Þ¼pw N2, ad f 2 ð0jp f W N Þ is a cotiuous mootoe decreasig fuctio of y for 0 6 y 6 1 with f 2 ð0jp f W N Þ¼pw N3 ad f 2 ð1jp f W N Þ¼pw N2. lðxjp f W N Þ is determied by f Ni ðyjp f h W N Þ¼f i ðyjef Þ 1 þ f k ðyj~rþ Ni ð4þ for i ¼ 1; 2 where k ¼ i for egative ef ad k ¼ 3 i for positive fuzzy future worth, ef [4]. I Eq. (4), the fuzzy iterest rate (~r) is assumed to be kept co-

7 C. Kahrama et al. / Iformatio Scieces 168 (2004) µ (PW) 1.0 y d l l d r PW PWA 0.0 a b c Fig. 2. Deviatio betwee PW ad PWA. PW stat whereas it is a parameter chagig from year to year i Chiu ad Park s [7] formula i Eq. (1). I other words, ~r i Eq. (4) represets the average aual fuzzy iterest rate alog N years. Assume that the project life en is fuzzy. If ef is the fial amout i the accout, the its membership fuctio is defied by [4] lðxjef Þ¼supðhÞ; CðxÞ ð5þ where h ¼ miðlðujp ew Þ; lðvj~rþ; lðwj en ÞÞ ð6þ CðxÞ ¼fðu; v; wþjuð1 þ vþ w ¼ xg ð7þ The defiitio of ef is simply the extesio priciple applied to Eq. (8) [4]. PWð1 þ rþ N ¼ F ð8þ For some relevat publicatios o fuzzy capital budgetig techiques, the reader is referred to [3,5,9,13,15 17,19,22] Applicatio of fuzzy preset worth aalysis for measurig flexibility I the literature, some publicatios ca be foud o measurig flexibility usig crisp preset worth aalysis. A sigificat cotributio to this research area by Pyou ad Choi [25], ad the fuzzificatio of this model are give i the followig sub-sectios Pyou ad Choi s quatificatio of flexibility value Pyou ad Choi [25] classify realizable flexibility ito three categories: (1) the iteral cotrol group, (2) the ivestmet policy group, ad (3) the marketig adaptatio group. The flexibility value is the sum of the values i these three categories. All of the formulas of the elemets of these three groups are based o preset worth. Therefore, we have decided to develop fuzzy preset worth formulas for the elemets of the iteral cotrol group oly, to

8 84 C. Kahrama et al. / Iformatio Scieces 168 (2004) costitute a example. Similar trasformatios ca be applied easily to the elemets of the other two groups. The flexibility formulas of the iteral cotrol group are give i Table 1. I the formulas i Table 1, t is the aual icome tax rate, r is the iterest rate ad N is the project life. Pyou ad Choi [25] use the flexibility value (FV) i their fiacial evaluatio model as show i Eq. (9). PWðrÞ ¼ XN ¼1 ( ð1 tþr ð1 þ rþ þ tb N ð1 tþs N þ ð1 þ rþ ð1 þ rþ þ FV N ) I ð1 þ rþ ð1 tþo 1 ð1 þ rþ þ td ð1 þ rþ where t is the icome tax rate, N is the project life, R reveue, I ivestmet cost, O operatig cost, D depreciatio cost, B book value, ad S salvage value. I this paper, oly the fuzzy flexibility formulas i the iteral cotrol group are obtaied sice the other modificatios are similar. The iteral cotrol group is categorized by the user s itegrated operatioal capability to cope with their iteral chages ad is calculated as the sum of four elemets, flexibility ð9þ Table 1 Flexibility formulas of iteral cotrol group Iteral cotrol group Flexibility formula Flexibility for cotiuous improvemet (CI) Flexibility for trouble cotrol (TC) Flexibility for workforce cotrol (WC) Work-i-process cotrol (IP) CI ¼ð1 tþ P N sc ¼1 ð1þrþ where s is the average umber of parts revisios, ad c is the average cost required for alteratio of toolig ad software i the th year TC ¼ð1 tþ P N Tb ¼1 ð1þrþ where TC ¼ 0ifC a ¼ 0 ad, T ¼ ed mifc a M ; Q g c 2, T is the average icreased productio reveue per breakdow i the th year, after subtractig off the reschedulig cost, b is the average umber of breakdows i the th year, e is the average reveue per item, d is the average period required for repair per breakdow, C a is the average maufacturig capacity available durig breakdow (% of M ), M is the maximum maufacturig capacity at the begiig of the th year, Q is the fuzzy average demad i the th year, ad c 2 is the reschedulig cost WC ¼ð1 tþ P N /JW ¼1 ð1þrþ where / is the average beefit per job, J is the average umber of differet jobs, W is the average umber of operators IP ¼ð1 tþ P N ucvv ¼1 ð1þrþ where u is the average reveue per item, c v is the ivetory carryig-cost rate, ad V is the average work i process i the Nth year

9 C. Kahrama et al. / Iformatio Scieces 168 (2004) for cotiuous improvemet, flexibility for trouble cotrol, flexibility for workforce cotrol, ad flexibility for work-i-process cotrol [25]. Pyou ad Choi [25] do ot cosider that some deviatios i the estimates of flexibility parameters, such as the average umber of parts revisios i the th year (s ), may occur. Our aim i this paper is to adapt their formulatio of realizable flexibility to the ucertaity of real-world coditios by applyig Fuzzy Sets Theory. So here, i this paper, the terms s, c, ad r are assumed to be fuzzy parameters because i a CIM eviromet, the decisio-makers usually have imprecise data about these three. The icome tax rate, t, is defied by the govermet ad usually is a fixed value (kept uchaged) for a very log time. So, it is assumed to be a crisp value. The project life, N, is also assumed to be a crisp umber here i the formulatio. However, the fuzziess for this parameter will be icorporated i the umerical example for the practical purposes. We may defie the parameters that Pyou ad Choi used i their paper [25] as triagular fuzzy umbers such as ~s ¼ðs 0 ; s 1 ; s 2 Þ. We may also defie the discout rate durig time such as ~r ¼ðr 0 ; r 1 ; r 2 Þ. The idices 0, 1, ad 2 i the represetatios of ~s ad ~r stad for the smallest possible value, the most possible value, ad the largest possible value, respectively. The exact form of the fuzzy preset worth of the flexibility for cotiuous improvemet (P f WoCI) uder o-iflatioary coditios ca be calculated as follows: " P f WoCI ¼ ð1 tþ XN ¼1 Q 0 ¼1 s lðyþ c lðyþ ; ð1 tþ ð1 þ RrðyÞ Þ 0 X N ¼1 Q 0 ¼1 # s rðyþ c rðyþ ð1 þ RlðyÞ Þ 0 ð10þ Uder iflatioary coditios, it is very importat to take the iflatio rate ito cosideratio. Hece, the average cost required for alteratio of toolig ad software (c ), which is the oly moetary parameter i Eq. (10) chages from year to year. I this case, the preset worth formula for geometric cash flows, which is frequetly used i Egieerig Ecoomics, ca be used [23]. Usig Buckley s otatio ad assumig that c 1 will icrease at a fuzzy costat rate (~g), ad ~s is fixed alog N years, Eq. (10) will chage to, " # f Ni ðyjp f 1 f k ðyj~rþ N f k ðyj~gþ N W N Þ¼ð1 tþf i ðyj~sþc 1 ð11þ f k ðyj~rþ f k ðyj~gþ Other moetary parameters i the formulas of the followig elemets of iteral cotrol group are T for trouble cotrol, / for workforce cotrol, ad u for work-i-process cotrol. Similar modificatios derived by iflatioary coditios are applied to Pyou ad Choi s formulas i Eqs. (12), (19) ad (21) to yield with formulas i Buckley s otatios i Eqs. (18), (20) ad (22), respectively. Please ote that, to achieve a more precise preset worth uder

10 86 C. Kahrama et al. / Iformatio Scieces 168 (2004) iflatioary coditios forcig us to deal with geometric growth rates, we substitute the arithmetic average values / with /, ad u with u for ¼ 1; 2;...; N, i Eqs. (20) ad (22), respectively. The secod elemet of the iteral cotrol group is flexibility for trouble cotrol (TC). It is the user s capability to adapt the maufacturig system to hadle a breakdow of the system. This flexibility is applicable oly whe the breakdow period is log eough to reschedule the maufacturig system ad the user also eeds to keep producig durig this period with reduced capacity. The terms T, b, r, e, d, C a, M, Q, ad c 2 are assumed to be fuzzy parameters. Oly t ad N are assumed to be crisp umbers for the reasos metioed i the fuzzificatio of CI above. Cosiderig these, flexibility for trouble cotrol P f WoTC is measured by " # P f WoTC ¼ ð1 tþ XN T lðaþ b lðaþ X N T rðaþ b rðaþ Q ; ð1 tþ Q ¼1 0 ¼1ð1 þ RrðaÞ Þ 0 ¼1 0 ¼1ð1 þ RlðaÞ Þ 0 ð12þ where et is obtaied by usig the followig equatio: et ¼ ~e d ~ mið ec a em ; eq Þ ~c 2 ð13þ I Eq. (13), it is ecessary to use a rakig method to compare the fuzzy umbers, ec a em ad eq. There are may rakig methods i the literature ad these methods may give differet rakig results. Some of them are Chag s [6], Chiu ad Park s [7], Dubois ad Prade s [10], Jai s [14], Kaufma ad Gupta s [20], Liou ad Wag s [24], ad Yager s [31] methods. For example, Liou ad Wag [24] propose the total itegral value method with a idex of optimism x 2½0; 1Š. Let ea be a fuzzy umber with left membership fuctio f L ad right membership fuctio f R. The the total itegral value is ea ea defied as E x ðeaþ ¼xE R ðeaþþð1 xþe L ðeaþ where ad E R ðeaþ ¼ E L ð e AÞ¼ Z b a Z d c xf R ea ðxþdx xf L ea ðxþdx ð14þ ð15þ ð16þ where 1 < a 6 b 6 c 6 d < þ1. For a triagular fuzzy umber, ea ¼ða; b; cþ, the total itegral value is obtaied by E x ðaþ¼ e 1 ½xða þ bþþð1 xþðb þ cþš 2 ð17þ

11 C. Kahrama et al. / Iformatio Scieces 168 (2004) Uder iflatioary coditios, assumig that T 1 will icrease at a fuzzy costat rate (~g) ad ~ b is fixed alog N years, Eq. (12) becomes " # f Ni ðyjp f W N Þ¼ð1 tþf i ðyj ~ 1 f k ðyj~rþ N f k ðyj~gþ N bþt 1 ð18þ f k ðyj~rþ f k ðyj~gþ The third elemet of the iteral cotrol group is flexibility for workforce cotrol (WC). It is the user s capability to maage the size ad the techical ad maagerial capability of the workforce required for operatio of the maufacturig system. This flexibility is measured by the size ad the techical ad maagerial capability of the required workforce ad by the average beefit that the user ca attai per job. The terms /, J, W, ad r are assumed to be fuzzy parameters. Oly t ad N are assumed to be crisp umbers for the reasos defied i the fuzzificatio of CI above. Cosiderig these, the fuzzy preset worth of flexibility for workforce cotrol (P f WoWC) ca be formulized as i Eq. (19). " P f WoWC ¼ ð1 tþ XN ¼1 / lðaþ J lðaþ Q 0 ¼1 W lðaþ ð1 þ RrðaÞ 0 Þ ; ð1 tþ X N ¼1 / rðaþ J rðaþ Q 0 ¼1 W rðaþ ð1 þ RlðaÞ 0 Þ # ð19þ Uder iflatioary coditios, assumig that / 1 will icrease at a fuzzy costat rate (~g) ad ~J ad ew are fixed alog N years, Eq. (19) becomes " # f Ni ðyjp f 1 f k ðyj~rþ W N Þ¼ð1 tþf i ðyjej Þf i ðyj ew N f k ðyj~gþ N Þ/ 1 ð20þ f k ðyj~rþ f k ðyj~gþ The fourth elemet of the iteral cotrol group is work-i-process cotrol (IP). It is the user s capability to miimize the work i process required for operatio of the maufacturig system. This flexibility is measured by the average amout of work i process required for operatio of the maufacturig system ad the ivetory carryig-cost rate. The terms u, C v, V, ad r are assumed to be fuzzy parameters. Oly t ad N are assumed to be crisp umbers for the reasos defied i the fuzzificatio of CI above. Cosiderig these, the fuzzy preset worth of flexibility for worki-process cotrol (P f WoIP) ca be formulized as i Eq. (21). " P f WoIP ¼ ð1 tþ XN ¼1 u lðaþ c lðaþ Q 0 ¼1 v V lðaþ ð1 þ RrðaÞ 0 Þ ; ð1 tþ X N ¼1 u rðaþ c rðaþ Q 0 ¼1 v V rðaþ ð1 þ RlðaÞ 0 Þ # ð21þ Uder iflatioary coditios, assumig that u will icrease at a fuzzy costat rate (~g) ad ev ad ~c are fixed alog N years, Eq. (21) becomes

12 88 C. Kahrama et al. / Iformatio Scieces 168 (2004) " # f Ni ðyjp f 1 f k ðyj~rþ W N Þ¼ð1 tþf i ðyj ev Þf i ðyj ew N f k ðyj~gþ N Þu 1 f k ðyj~rþ f k ðyj~gþ ð22þ Numerical example EGEY, a Turkish Motors Compay, produces trasmissio gearboxes for cars. The fuzzy average umbers of part revisios ð~s Þ, the fuzzy maximum maufacturig capacities ð em Þ, the fuzzy average demad ð eq Þ, ad the fuzzy average umber of breakdows ð ~ b Þ i the years are expected to be as i Table 2. However, the aalysis period is ot certai sice the compay is ot sure that it will produce these products for the ext six years. The productio will cotiue for en years where lð i j en Þ is give i Table 3. The fuzzy iterest rate is expected to be fixed as (8%, 10%, 12%) alog this period while the fuzzy average cost required for alteratio of toolig ad software is also expected to be fixed as (20, 30, 50) ( $1000). The fuzzy average maufacturig capacity available durig breakdow is (85%, 90%, 95%), the fuzzy average reveue per item is $(6.0, 6.5, 7.0) ( $10), the fuzzy average period required for repair per breakdow is (0.015, 0.032, 0.038) years, the fuzzy reschedulig cost is $(180, 210, 240) ad they are all assumed to be fixed through en years. The crisp aual icome tax rate is 35%. Usig Eq. (10), the exact fuzzy preset worth of the flexibility for cotiuous improvemet is calculated as follows: 8 2 P f < WoCI ¼ 0:65 ð20 þ 10yÞ4 : 2 0:65 ð50 20yÞ4 þ 1þy þ 2þy 2þy þ 1:12 0:02y ð1:12 0:02yÞ 2 ð1:12 0:02yÞ 3 3 3þy 3þy þ þ þ ð1:12 0:02yÞ 4 ð1:12 0:02yÞ 5 ð1:12 0:02yÞ 6 2 þ 3 4 y þ ð1:08þ0:02yþ ð1:08þ0:02yþ 2 ð1:08þ0:02yþ 3 4 y 4 5 y þ þ ð1:08þ0:02yþ 4 ð1:08þ0:02yþ 5 ð1:08þ0:02yþ 6 3 5; 39 = 5 ; Sice the discout rate is estimated withi the width of ±4%, P f WA ca be used i the problem uder cosideratio [7]. To obtai the approximate form of Table 2 Fuzzy expected values for es, em, eq, ad ~ b Year, es em eq ~ b 2003 (1, 2, 2) (3600, 3800, 4000) (3100, 3400, 3900) (3, 4, 5) 2004 (2, 3, 3) (3900, 4000, 4200) (3500, 3650, 3900) (4, 5, 6) 2005 (2, 3, 4) (4000, 4200, 4300) (3600, 3700, 4000) (4, 5, 7) 2006 (3, 3, 4) (4300, 4500, 4600) (3700, 3900, 4400) (5, 6, 7) 2007 (3, 4, 4) (4500, 4700, 4800) (4000, 4200, 4600) (6, 7, 7) 2008 (3, 4, 5) (4600, 4800, 4900) (4100, 4300, 4700) (7, 8, 8)

13 C. Kahrama et al. / Iformatio Scieces 168 (2004) Table 3 Possibility table for the project life N i lðn i j en Þ P f WoCI, for ¼ 4, 5, ad 6, y will be assiged a value of 0 ad 1 i the left side represetatio, ad the a value of 0 i the right side represetatio i the equatio above. Thus, the results i Table 4 are obtaied. After obtaiig both the exact ad approximate equatios for P f WoCI, the values of these two fuctios were calculated by usig very small icremets of o y axis. Spearma correlatio coefficiet was used to decide whether the approximate form ca be used istead of the exact form, or ot. The coefficiet was calculated as for the left sides, whereas it was for the right sides. Sice these coefficiets are close eough to 1.00, it was prove for our example that the approximate form ca be used istead of the exact form. I Fig. 3, the approximate fuzzy preset worth of the flexibility for cotiuous improvemet is show by its TFNs. The umeric results of the fuzzy flexibility for trouble cotrol are give i the followig: For the year 2003, ec a em 1 ¼ð3060; 3420; 3800Þ ad eq 1 ¼ð3100; 3400; 3900Þ. Usig Liou ad Wag s rakig method [24], for a moderately optimistic decisio-maker, x ¼ 0:5, the miimum of E 0:5 ð ec a em 1 Þ¼3425 ad E 0:5 ð e Q 1 Þ¼3450 is So we select e C a em 1 value ad use this value i Eq. (13). By calculatig all et s i the same way, we obtai et 1 ¼ð2514; 6904; 9928Þ; et 3 ¼ð3000; 7486; 10460Þ; et 5 ¼ð3203; 8588; 11950Þ; et 2 ¼ð2744; 7278; 10433Þ et 4 ¼ð3090; 8602; 11524Þ et 6 ¼ð3279; 8776; 12202Þ By usig these values, the exact fuzzy preset worth of flexibility for trouble cotrol is obtaied as Table 4 P f WoCI values for ¼ 4; 5; 6 Life, f ;i ðyjp f WoCI Þ y ¼ 0 ad i ¼ 1 y ¼ 1, i ¼ 1ori ¼ 2 y ¼ 0 ad i ¼ 2 4 $75, $167, $342, $97, $216, $431, $117, $260, $533,406.17

14 90 C. Kahrama et al. / Iformatio Scieces 168 (2004) µ 1.0 P WoCI ~ P WoCI ~ 75, , , , , , , , , Fig. 3. The approximate fuzzy preset worth of the flexibility for cotiuous improvemet show by its TFNs : >< P f WoTC ¼ 2 0: >: ð4390yþ2514þð3þyþ þ ð4534yþ2744þð4þyþ 1:12 0:02y ð1:12 0:02yÞ 2 þ ð4486yþ3000þð4þyþ ð1:12 0:02yÞ 3 þ ð5385yþ3203þð6þyþ ð1:12 0:02yÞ 5 þ ð5512yþ3090þð5þyþ ð1:12 0:02yÞ 4 þ ð5497yþ3279þð7þyþ ð1:12 0:02yÞ 6 ð yÞð5 yþ þ ð yÞð6 yþ 1:08þ0:02y ð1:08þ0:02yþ 2 þ ð yÞð7 2yÞ ð1:08þ0:02þ 3 þ ð yÞ7 ð1:08þ0:02yþ 5 þ ð yÞð7 yþ ð1:08þ0:02yþ 4 þ ð yÞ8 ð1:08þ0:02yþ ; >= >; To obtai the approximate form of P f WoTC, y will be assiged a value of 0 ad 1 i the left side represetatio, ad the a value of 0 i the right side represetatio i the equatio above. Thus, the least possible value, the most possible value, ad the largest possible value of the fuzzy preset worth of the flexibility for trouble cotrol are calculated as $33,844.19, $118,552.69, ad 217, for ¼ 4; $44,749, $155,880, ad $273,979 for ¼ 5, ad 56,377.70, 190,994.79, ad 330, for ¼ 6, respectively. I Fig. 4, the approximate fuzzy preset worth of the flexibility for trouble cotrol is show by its TFNs. The FV i Eq. (9) will be the sum of the results which are obtaied for the iteral cotrol group, the ivestmet policy group, ad the marketig adaptatio group. Here, i this umerical example, we iclude oly two of the

15 C. Kahrama et al. / Iformatio Scieces 168 (2004) µ P WoTC ~ , , , , , , , , , P WoTC ~ Fig. 4. The approximate fuzzy preset worth of the flexibility for trouble cotrol show by its TFNs. elemets of the iteral cotrol group, which are P f WoCI ad P f WoTC, just to show the way how the total FV amout is calculated. Thus, the result of the summatio of these two elemets is show i Fig. 5. Usig Fig. 5, the possibility of ay FV ca be foud easily. I case of havig icomplete iformatio, this possibility distributio provides the decisiomaker with a detailed iformatio source. For example, the most possible value i Fig. 5 is $372, If the decisio-maker wats to kow the iterval of flexibility values havig a possibility larger tha 0.90, he/she should make the calculatios below: 1. Derive the fuctios for the area above the desired possibility value (0.90 for our example) usig the bold lies i Fig. 5. x 1 ¼ 142; 504:32 þ 229; 517:4y x 2 ¼ 704; 982:61 332:960:89y 2. Substitute the desired possibility value ito these equatios ad fid the extreme poits of the iterval. x 1 ¼ 142; 504:32 þ 229; 517:4 0:90 ¼ 349; 069:98 x 2 ¼ 704; 982:61 332:960:89 0:90 ¼ 405; 317:81 Accordig to the results of the calculatios above, the iterval of flexibility values havig a possibility larger tha 0.90 is (349,069.98; 405,317.81).

16 92 C. Kahrama et al. / Iformatio Scieces 168 (2004) µ P WoFV ~ , , , , , , , , , P WoFV ~ Fig. 5. The approximate fuzzy preset worth of the flexibility value show by its TFNs. 4. Coclusio I this paper, the fuzzy preset worth formulas for optimal level of flexibility ad the flexibility elemets i the iteral cotrol group, which is categorized by the user s itegrated operatioal capability to cope with their iteral chages, are obtaied. Usig these formulas, as show i the give umeric example, more iformative results of flexibility measures ca be achieved. The flexibility modelig usig triagular fuzzy umbers allows experts liguistic predicate about computer itegrated maufacturig systems. Whe there are more tha oe alterative of CIM systems to compare i terms of flexibility, usig the fuzzy preset worth formulas of several types of flexibility ad applyig some domiace rules o triagular fuzzy umbers, the fuzzy flexibility evaluatio process is completed. This study ivolves discrete compoudig for calculatig the fuzzy preset worth of flexibility. Further research ca be aimed at icludig cotiuous compoudig for the same calculatio. Also, trapezoidal fuzzy umbers ca be used istead of TFNs whe the decisio-maker uses betwee istead of aroud to express his/her estimatios about the flexibility parameters. Refereces [1] A. Beskese, C. Kahrama, Z. Irai, Quatificatio of flexibility i advaced maufacturig systems usig fuzzy preset worth aalysis, i: Proceedigs of the 2d Iteratioal

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