A MODEL FOR OPTIMIZING ENTERPRISE S INVENTORY COSTS. A FUZZY APPROACH

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1 OPERATIONS RESEARCH AND DECISIONS No DOI: /ord Wtold KOSIŃSKI Rafał MUNIAK Wtold Konrad KOSIŃSKI A MODEL FOR OPTIMIZING ENTERPRISE S INVENTORY COSTS. A FUZZY APPROACH Applcablty of a fuzzy approach to a problem orgnatng from admnstratve accountng, namely to determne an economc order quantty (EO) n a varable compettve envronment wth mprecse and vague data, has been presented. For ths purpose, the model of ordered fuzzy numbers developed by the frst author and hs two co-workers s used. The present approach generalzes the one developed wthn the framework of convex fuzzy numbers and stays outsde the probablstc one. Keywords: economc order quantty, optmzaton problem, ordered fuzzy numbers 1. Introducton We encounter nventory management ssues n commercal and ndustral enterprses. Accordng to the accountng act, nventores nclude: a) fnshed products produced or processed by the entty, b) products n the process of producton, c) semfnshed products products that have undergone stages of closed-tech manufacturng processes, d) goods purchased for resale wthout further processng. Cost-effectve nventory management can ndrectly affect the value of the company measured as the value of expected cashflows. The level of cash flow results from the volumes of the company s ncomes and expendture. The volume of sales (revenue) depends on the avalablty of fnshed products and changes n workng captal, as growth n these s Department of Computer Scence Polsh-Japanese Insttute of Informaton Technology, ul. Koszykowa 86, Warszawa, Poland, Insttute of Mechancs and Appled Computer Scence, Kazmerz-Welk Unversty, ul. Chodkewcza 30, Bydgoszcz, Poland. Department of Computer Scence Polsh-Japanese Insttue of Informaton Technology ul. Koszykowa 86, Warszawa, e-mal addresses: rafalm@pjwstk.edu.pl, wkosn@pjwstk.edu.pl

2 40 W. KOSIŃSKI et al. assocated wth a drop n sales, whle a reducton s assocated wth an ncrease n sales. Mantanng large nventores also ncreases fxed and varable costs. The am of ths paper s to show the applcablty of a fuzzy approach to a problem orgnatng from admnstratve accountng, namely to determne an economc order quantty (EO) n a varable compettve envronment wth mprecse and vague data. For ths purpose, the model of ordered fuzzy numbers developed by the frst author [4] and hs two co-workers was used. The present approach generalzes the one developed wthn the framework of convex fuzzy numbers and stays outsde the probablstc one. For many years the only tool representng mprecse and vague notons was probablty theory. Hence, any suggeston of substtutng ths tool by an approach related to fuzzy logc and fuzzy sets leads to the queston: s t worth dong ths, and f yes, then why? In ths paper, we focus our fuzzy approach on applcatons to problems n economcs, for whch modellng the nfluence of mprecse quanttes and preferences on a decson maker s opnons s mportant. Wth the help of a fuzzy number t s possble to express ncomplete knowledge about a quantty by gvng the range n whch ts realzaton can appear, and wrtng t n the form of a (subjectve) functon of the nformaton, representng the capablty degree of any realzaton. In ths case, only one condton appears, namely capablty degrees may take values from the nterval [0, 1], where complete mpossblty s expressesed by 0, whle 1 expresses full capablty. There are no more restrctons on the form of such a functon. On the other hand, one may use a random varable but n order to descrbe and to model ths stuaton wth mprecse quanttes, the random varable has to supply nformaton about a probablty, rather than the capablty degrees of the possble realzatons (values of the quantty). Ths means that n ths probablstc case, one s forced here to gve the probablty dstrbuton of the quantty (even t s subjectve). Then we have to fulfl some constrant whch follows from the defnton of a probablty dstrbuton. In the case of a fuzzy number approach, on the other hand, the decson maker s completely free as far as the form of the capablty degree functon of the mprecse quantty s concerned. The noton of an ordered fuzzy number (OFN) arrved a decade ago, proposed by the frst author, together wth two co-workers: P. Prokopowcz and D. Ślęzak, to elmnate several drawbacks of classcal convex fuzzy numbers (CFN), such as the unbounded growth n fuzzness whch results from a large number of arthmetc operatons on fuzzy numbers and the lack of solutons to smple lnear equatons usng fuzzy numbers [4]. Ths new model s not based on the classcal noton of the membershp functon ntroduced by Zadeh. A less restrctve concept s used, namely a membershp relaton. Each convex fuzzy number possesses two representatves n the form of ordered fuzzy numbers whch dffer by ther orentaton. Ths means that each element of an OFN has an addtonal attrbute, namely ts orentaton, whch s

3 A model for optmzng enterprse s nventory costs. A fuzzy approach 41 not present n a CFN [1, 3, 8]. In ths way, we are able to reflect nformaton about the trend of changes n such mprecse quanttes n the modelng. Ths nformaton s not present whenever the CFN approach or the probablstc one are appled. In ths paper, wel propose a soluton to a problem orgnatng from admnstratve accountng,.e. nventory management usng the model of an OFN, namely to determne an economc order quantty (EO) n a varable compettve envronment wth mprecse and vague data. If the decson maker chooses hs tool for makng crsp numbers from fuzzy ones,.e. a defuzzfcaton functonal, the method proposed here gves a tool for determnng the optmal order quantty that mnmzes the total nventory cost. In ths way, a decson support tool s constructed for when data are fuzzy. The fnal result of ths paper manfests the applcablty of ordered fuzzy numbers as a tool based on whch nvestors can make nvestment decsons. The organzaton of the paper s as follows. In Secton 2 we present the problem of nventory management. We quote the basc ndcators of the qualty of such management. Then we consderthreats usngrsk analyss. Here, we specfy rsk categores n tabular form, together wth a descrpton of ther sources and levels of mpact on costs. Ths secton ends wth a lst of 3 optons that can be used to select an optmal strategy for nventory management. In Secton 3, we formulate the model for determnng the economc order quantty (EO) n a determnstc setup. Then we pass on to ts fuzzy formulaton. Secton 4 brngs some conclusons. In the Appendx, the man concepts and defntons nvolved n the model of an ordered fuzzy number are brefly presented. 2. The ssue of nventory management Effectve nventory management mproves the lqudty of a company, lowers the cost of storage and results n an unnterrupted producton process. Mantanng ether excessve or too low nventory levels leads to a reducton n the effcency of the company. Too much nventory generates costs assocated wth the mantenance of warehouse space, nventory storage costs and costs of frozen captal. Too low nventory levels can nterfere wth the producton process and sales through a lack of materals or fnshed products. Here you can fnd many sources of emergng rsks. One of the ndcators summarzng the level of nventores n a company s the nventory turnover rato W rot = Z p P + Z 2 k (1) where P revenue from the sale of goods, materals, fnshed products, Z p nventory at the begnnng of the analyzed perod, Z k nventory at end of the analyzed perod.

4 42 W. KOSIŃSKI et al. Another ndcator evaluatng the correctness of nventory management s the nventory turnover perod, gven by O rot 365 = W rot (2) The turnover perod ndcates how long on average current supples wll last to conduct operatons. An ncrease n the turnover rato means that stocks wll last for a shorter perod of sales or producton. The nventory turnover perod ndcates the number of days between renewals of the company s nventory. A long nventory turnover perod means a slow turnover of stocks, a short perod rapd turnover. A too long perod for holdng stocks results n an ncrease n the costs of producton and worsens the company s fnancal stuaton. The sze of stocks should result from the current demand for materals and goods. At the same tme, when analyzng the effcency of warehouse management, one must take nto account the adverse effects of ncreasng nventory turnover ratos such as: orderng costs ncrease due to more frequent purchases, the rsk of nventory shortages, dsruptng of the rhythm of producton, loss of customers. Costs assocated wth nventory management can be dvded nto three groups: 1. The costs of mantanng an nventory: frozen captal costs, storage costs, costs of nventory mantenance, rsk assocated wth the loss of physcal nventores and random events. 2. The costs of creatng stocks: procurement costs, transportaton costs, costs of recevng nventory. 3. The costs of nventory shortages: costs of producton downtme, costs of lost sales, costs of lost reputaton Rsk analyss In a company, an ncrease n operatonal rsk assocated wth nventory management results n a decrease n the value of the company measured as a sum of cash flows. Too low value of nventory ncreases the lkelhood of nterference to the rhythm of producton. Reducng operatonal rsk by ncreasng the value of stock

5 A model for optmzng enterprse s nventory costs. A fuzzy approach 43 ncreases fxed and varable costs whch, n turn, leads to a reducton n the value of cash flows and therefore the value of the company. Table 1 descrbes categores of operatonal rsk. Table 1. Categores of operatonal rsk Rsk category Descrpton of sources of rsk Degree of mpact on the cost Increase n stock prces Provdng steady producton Uncertanty n supply and delays Degradaton and agng of nventory Uncertanty about the prce of the materals used n the company caused by: the geopoltcal stuaton, nflaton, changes n exchange rates. These rsks are mnmzed by mantanng larger stocks. Lack of materals n stock results n ncreased costs assocated wth downtme such as loss of customers, contractual penaltes. Dsapponted customers can swtch to the competton to satsfy ther needs. Dffcultes n ensurng steady producton whch, n turn, may lead to loss of customers and proft from sales. Some stocks requre the provson of specfc condtons such as: approprate humdty, optmal temperature. Not provdng such condtons results n loss of ther utlty value. The ncrease n costs as a result of prce ncreases for example: prces of materals used n producton. The ncrease n costs ncurred by the company and the loss of revenue from sales. The ncrease n costs assocated wth mantanng optmal condtons for storng stocks and the costs of nventory lqudaton. To determne optmal management strateges, an entrepreneur should calculate whch opton generates the hghest cash flow. We dscuss three such optons. Opton 1. Mantan hgh values of stock whle beneftng from the dscounts and lower transport costs resultng from less frequent delveres of large batches of materals. Opton 2. Mnmze the value of stock by: more frequent delveres, shortenng the producton cycle, shortenng the duraton of storage of fnshed goods, whch leads to lower costs of: frozen captal n the nventory and forsaken opportuntes to nvest avalable cash, storage and management of stocks, such as rental of warehouse space, related to the ageng of stocks. Opton 3. Mantan a hgh value of stock, n order to reduce operatonal rsk whch has the effect of ncreasng the cost of: producton downtme, lost sales due to a lack of goods or fnshed products, lost reputaton due to nadequate customer servce.

6 44 W. KOSIŃSKI et al. 3. Inventory optmzaton Every frm faces the challenge of matchng ts supply volume to customer demand. How well a frm manages ths challenge has a major mpact on ts proftablty. Also, the amount of nventory held has a major mpact on the amount of cash avalable. Wth workng captal at a premum, t s mportant for companes to keep nventory levels as low as possble and to use or sell nventory as quckly as possble. Inventory s one of the most key factors that analysts take nto account when makng earnngs forecasts or recommendatons to buy and sell. The challenge of managng nventory s becomng ever more complex. Models of nventory optmzaton can be determnstc wth each set of varable states beng unquely determned by the parameters n the model, or stochastc wth varable states descrbed by probablty dstrbutons or fuzzy numbers. In ths paper, we propose a fuzzy number approach Determnstc model Inventory management wthn an enterprse s an ntegral part of ts operatonal actvtes as t affects the lqudty of ts fnancal performance and compettve advantage. The purpose of nventory management s to have stock at a hgh enough level to operate smoothly, whle ncurrng the lowest possble operatng costs. The present formulaton s wthn the general framework of the model of the economc order quantty (EO). We consder an abstract nventory tem. To estmate the cost of nventory management, we formulate the man assumptons n the EO model: the abstract nventory tem s splt nto unts, we refer to some tme unt, say one year, demand s constant n tme, sales are unform n tme and known, the next delvery arrves just as the stock level falls to zero. Let us start wth a determnstc formulaton n whch the followng objects appear: D annual nventory demand, measured n number of unts, order quantty, measured n number of unts, c unt purchase cost, r() dscount functon, K t transportaton cost of a sngle delvery, K s unt nventory cost, unt cost of loss (agng or extraordnary loss), K u

7 A model for optmzng enterprse s nventory costs. A fuzzy approach 45 θ fracton of possble unexpected nventory loss, R bankng nterest rate, used to calculate the cost of frozen captal. The assumpton about the arrval of each supply makes t possble to smplfy the problem and to state that the average level of nventory s /2. However, the statement that the annual frequency of delvery D/ s obvous. Now we can wrte the general expresson for the total cost K(), as the sum of the purchase cost K z and the storage cost K m,.e. K ( ) = K z + K m We assume that the dscount functon s a step functon of the followng form (n lne wth Optons 1 and 2 descrbed above): r0, f 0 < 1 r ( )= r1, f 1 < 2 r2, f 2 (3) where 0, 1 and 2 are fxed quanttes. Here three steps are assumed. However, more steps can also be consdered. The purchase cost functon K p () depends on the quantty n a sngle delvery, the frequency of delveres, the dscount r() and the unt prce c, and s gven by K ( p )= c( 1 r ( ) ) D = c( 1 r ( ) ) D. From the form of the dscount functon r(), we can see that ths s a pecewse constant functon. It s obvous that the cost of frozen captal depends on: the number of delveres D/R, the money spent on a sngle delvery, the bankng nterest rate R, the annual frequency of delveres. The form of the purchase cost K p leads to the followng cost K f of frozen captal: whch reduces to D R K f = ( 1 r( ) ) c D

8 46 W. KOSIŃSKI et al. ( ) K = 1 r( ) cr (4) f where the functon r() s gven by Eq. (3). We can see that the expresson K f represents a step functon, whch s pecewse lnear. Hence, the cost K z (), related to the total costs of frozen captal, together wth the purchase, and transportaton (delvery) to a warehouse of D/ delveres per year, s gven by the expresson D Kz( )= c( 1 r( ) ) D+ ( 1 r( ) ) cr+ Kt (5) The condtons of Opton 2 lead to the followng total storage cost K m K = K + K θ m s u 2 2 storage cost unexpected loss (6) Hence, the functon descrbng the total cost can be expressed as K ( )= K( ) + K ( ) z m D K ( )= c( 1 r ( ))( D+ R) + ( K + Kθ ) + K purchase cost + captal cost s u t 2 storage cost + unexpected loss transportaton cost (7) Optmzaton problem. Hence, the optmzaton problem of nventory management requres us to fnd the mnmum of the cost functon K() n Eq. (7). The argument whch gves the mnmum s the optmal value of the order quantty. Notce that n Eq. (5) the frst component depends on n a pecewse way, and the search for the optmal value should be performed n a pecewse way,.e. consderng each subnterval L0 :=(0, 1), L1:=[ 1, 2) and L2 :=[ 2, D ] Thus the global optmum s the quantty whch gves the mnmal cost over the three optmal values calculated from each subnterval. Notce that n each of these subntervals L, = 0, 1, 2 the local optmum s attaned at 1/2 KD t Ks + Kuθ q =, M = c(1 r) R+, = 0,1, 2 M 2 (8)

9 A model for optmzng enterprse s nventory costs. A fuzzy approach 47 provded a gven q belongs to the correspondng subnterval L, = 0, 1, 2. Otherwse, q s assumed to be the end pont of the correspondng nterval whch gves the mnmum cost. From these three canddates, the optmal value s calculated accordng to { ( K( q) q L = ) ( K( j) j = ) } arg mn :, 0, 1, 2,, 0, 1, 2 More complex problems can also be formulated, n whch the assumpton concernng the arrval of the next delvery wll be omtted and the so-called securty level of stock appears, and then some dynamcs wll enter our optmzaton problem. Ths more general case wll be consdered n a future paper. (9) 3.2. Fuzzy optmzaton problem Ths formulaton s wthn the framework of the model of the economc order quantty (EO) and smlar to the one proposed n the set of CFN by [9] and consdered by [7]. In terms of the fuzzy optmzaton problem, our am s to gve a general soluton wth the cost functon gven by (7) when D, K t, K s, K u and θ are fuzzy and represented by ordered fuzzy numbers (OFN). It wll be easy to see that the arthmetc of OFN manfests ts superorty over the arthmetc of convex fuzzy numbers (CFN), and the complex calculatons performed by the authors of [9] and [7] can be avoded. The only thng we need to do s to choose the defuzzfcaton functonal whch suts the decson maker the most. Let φ() be the defuzzfcaton functonal chosen by the decson maker. Then the problem of mnmzng the fuzzy cost K() gves us the economc order quantty. Wrttng ths problem explctly { φ K R } fnd arg mn ( ( )) : the followng queston arses: how can we fnd the mnmum of ths functonal? The answer s rather obvous and comes from physcs, and s formulated accordng to the statonary acton prncple: the mnmum of the functonal appears at the argument where ts frst varaton (the Gâteaux dervatve) vanshes. Calculatng the frst varaton of φ(k()) wth respect to under gven D, K t, K s, K u and θ, we get δφ( K ( )) = φ( K) K ( ) δ (10) K

10 48 W. KOSIŃSKI et al. Here Kφ( K) and K ( ) denote functonal dervatves. Due to the arbtrarness ofδ, the condton δφ ( K ( )) = 0 mples that φ( K) K( )=0 K and the argument, at whch the product of these dervatves vanshes, gves us the soluton to our optmzaton problem. To llustrate ths, let us consder a class of lnear functonals gven by Eq. (20). Let us denote the branches of the fuzzy number K() by (f K, g K ), and for the remanng quanttes we adapt the prevous notaton by usng the approprate subscrpts,.e. ( ) ( D D) t ( t t) = f, g, D= f, g, K = f, g ( ) θ ( ) K = f, g, = f, g u u u θ θ (11) Then the lnear functonal descrbng the fuzzy cost K() has the form where, see Eq. (7), ( ) ( ) 1 1 (12) φ K ( ) = φ f, g = f ( sdh ) ( s) + g ( sdh ) ( s) K K K 1 K f ()= s f ()(1 s r)( f () s + R f ()) s K c D ft() s fd() s fs() s + fufθ () s + + f ( s ), = 0, 1, 2 f () s 2 (13) and the form of g () s s analogous. Now we dfferentate Eq. (10) where the functonal s gven by Eq. (12), to get K 1 fd () s δϕ( K ( ))= fm( s) ft( s) δ f ( ) 2 sdh1 ( s) 0 ( f ( s)) 1 gd () s + gm() s gt() s δ g () 2 s dh2 () s 0 ( g ( s)) (14)

11 where ( M M ) A model for optmzng enterprse s nventory costs. A fuzzy approach 49 f ( s ), g ( s ), = 0, 1, 2 represent 3 fuzzy numbers M as OFN,.e. fs() s + fu fθ () s fm ()= s + fc()(1 s r) R (15) 2 and gm ( s ) has an analogous form. We can consder two cases: Case A. The functons h 1 and h 2 are absolutely contnuous, and Case B. The functons h 1 and h 2 are sngular,.e. h 1 () s and h () s 2 are equal to zero almost everywhere. In [2] we dscussed a less complex case and proved that n both cases the forms of h 1 and h 2 n Eq. (12) have no effect on the optmal value of. In the present case, however, we formulate the followng theorem: Theorem. If the total nventory cost K() arsng from the fuzzy unt costs of purchase c, of nventory K s, of transportaton K t, of loss K u and ts fracton θ, together wth the annual demand D, the dscount functon r(), and bankng nterest rate R, are gven by Eq. (7) and the decson maker chooses the defuzzfcaton functonal φ gven n Eq. (12), then n the case A the economc order quantty s gven by a two phase optmzaton procedure: Frst phase: the optmal values on each subnterval L 0, L 1, L 2 are found. We have ( ) ( ) q = φ K( ), where = f, g f q belongs to correspondng nterval, where f () s and g () s are gven by where 1/2 1/2 ft() s fd() s gt() s gd() s f ()= s, g ()= s, s [0,1] f M g M fs() s + fu() s fθ () s fm = fc( s) ( 1 r) R+, = 0, 1, 2, s [0,1] 2 (16) and the expresson for g M s analogous. When q as defned above does not belong to the correspondng nterval, then t s defned to be equal to the end pont of one of the ntervals L0, L1, L 2 whch gves the smallest value of the defuzzfcaton functonal.

12 50 W. KOSIŃSKI et al. Second phase: from these 6 expressons, the optmal value s calculated accordng to { ( φ f g K K ) ( φ f g j K ) } ( ( j ) K ( j ) arg mn (, ) : = 0,1, 2, (, ) : = 0, 1, 2 usng the notaton from Eqs. (9), (11), (13), and where the expressons for fk and g K n Eq. (16) are used to calculate the par ( f, g ). Remark. If the rebate functon s just a constant,.e. 0 = 1 = 2, then the economc order quantty s gven by wth ( ) ( ) q = φ K( ), where = f, g (17) where 1/2 1/2 ft() s fd() s gt() s gd() s f()= s, g()= s, s [0,1] fm gm fs() s + fu() s fθ () s fm()= s fc()1 s ( r2 ) R+, s [0,1] 2 and the expresson for g M s analogous. In the partcular case s gven by = ( K( f (1)) K( (1)))/2, g φ = φ MOM, the mnmal cost q + wth the functon K gven by Eq. (7). 4. Conclusons Here we have solved a problem orgnatng from nventory management, usng the model of ordered fuzzy numbers, and we have demonstrated ts applcablty n modellng the nfluence of mprecse quanttes and preferences of a decson maker. Thanks to the well-defned arthmetc of OFN, one can construct effcent decson support tools when data are mprecse. In our next paper, we wll ntroduce some dynamcs nto nventory management and show that OFN can be successfully appled to the presentaton of stock prces gvng a transparent mage of the stock exchange.

13 A model for optmzng enterprse s nventory costs. A fuzzy approach 51 References [1] GOETSCHEL R. Jr., VOXMAN W., Elementary fuzzy calculus, Fuzzy Sets and Systems, 1986, 18 (1), [2] CHWASTYK A., KOSIŃSKI W., Fuzzy calculus wth applcatons, Mathematca Applcanda, 2013, 41 (1), [3] NGUYEN H.T., A note on the extenson prncple for fuzzy sets, Journal of Mathematcal Analyss and Applcatons, 1978, 64, [4] KOSIŃSKI W., PROKOPOWICZ P., ŚLĘZAK D., Fuzzy numbers wth algebrac operatons: algorthmc approach, [n:] Proc. Intellgent Informaton Systems 2002, M. Klopotek, S.T. Werzchoń, M. Mchalewcz (Eds.), Sopot, Poland, June 3 6, 2002, Physca Verlag, Hedelberg 2002, [5] KOSIŃSKI W., PROKOPOWICZ P., ŚLĘZAK D., Ordered fuzzy numbers, Bulletn of the Polsh Academy of Scences, Ser. Sc. Math., 2003, 51 (3), [6] KOSIŃSKI W., On fuzzy number calculus, Internatonal Journal of Appled Mathematcs and Computer Scence, 2006, 16 (1), [7] KUCHTA D., Soft mathematcs n management. The use of nterval and fuzzy numbers n management accountng, Ofcyna Wydawncza Poltechnk Wrocławskej, Wrocław 2001 (n Polsh). [8] BUCKLEY J.J., Solvng fuzzy equatons n economcs and fnance, Fuzzy Sets and Systems, 1992, 48, [9] VUJEŠEVIĆ M., PETROVIĆ D., PETROVIĆ R., EO formula when nventory cost s fuzzy, Internatonal Journal of Producton Economcs, 1996, 45, Appendx The frst author and hs two coworkers Prokopowcz and Ślęzak [4 6] recently proposed an extended model of convex fuzzy numbers [3] (CFN), called ordered fuzzy numbers (OFN), whch does not requre any membershp functons. Usng ths model, we obtan an extenson of the CFN model, when one takes a parametrc representaton of fuzzy numbers, known snce 1986 [1]. Defnton 1. By an ordered fuzzy number we understand a par of functons (f, g) defned on the unt nterval [0,1] whch are contnuous functons (or of bounded varaton) [4 6]. Four algebrac operatons have been proposed for OFN, denoted by R (or R BV ), wth the arguments beng fuzzy numbers and crsp (real) numbers, n whch componentwse operatons are present. In partcular, f ( y)= f ( y) f ( y), g ( y)= g ( y) g ( y), (18) C A B C A B where can be replaced by +,, or, and where A B s defned f the functons and g B are strctly bounded from below by 0. Hence, any fuzzy algebrac equaton

14 52 W. KOSIŃSKI et al. A + X = C, where A and C are OFN, possesses a soluton. A convex fuzzy number corresponds to two OFNs, whch dffer by ther orentaton. A relaton of partal orderng n the space of all OFN can be ntroduced by defnng the subset of postve ordered fuzzy numbers: a number A = ( f, g) s not less than zero, and by wrtng A 0 ff f 0, g 0 (19) In ths way, the set R (or R BV ) becomes a partally ordered rng, R usng the relatons correspondng to operatons on reals. Defnton 2. A map φ from the space R (or R BV ) of all OFN s to reals s called a defuzzfcaton functonal f t satsfes: 1) φ ( c )= c, 2) φ( A + c )= φ( A) + c, 3) φ(ca) = cφ(a), 4) φ(a) 0, f A 0, for any c R and A R, where c ( s) = ( c, c), s [0,1] represents a crsp (real) number c R. The lnear functonals, MOM (mddle of maxmum), FOM (frst of maxmum), LOM (last of maxmum) are gven by specfyng h 1 and h 2 n the followng expressons: ( ) 1 1 φ f, g = f () s dh() s + g () s dh () s A A A 1 A where h 1, h 2 are non-negatve and of bounded varaton and 1 1 dh () s + dh () s = Example In [7], the author consdered the problem of mnmzng the value of the fuzzy cost K() of a frm n whch D K( ) = Dc+ Kt + K by neglectng the cost of frozen captal, dscount, as well as unexpected loss. Ths corresponds to the case consdered n the optmzaton problem from Sec. 3.1 and formula (8) for EO, but wth M = K s /2. The author of [7] frst consdered the crsp (determnstc) case wth the followng data: D = 1000, c = 10, K t = 8 and K s = 7. Accordng to her calculatons, the economc s 2

15 A model for optmzng enterprse s nventory costs. A fuzzy approach 53 order quantty k s 46 and the total cost K ( k ) correspondng to ths order s However, usng our model, we get k = 47.8 and the correspondng cost s K ( k ) = These values are dfferent from those of Kuchta n [7]. She also consdered the fuzzy case wth the same crsp values of D and c but wth the fuzzy transportaton cost K t represented by the trangular membershp functon (7, 8, 9) and the fuzzy storage cost K s represented by the trangular membershp functon (1.5, 7, 15). Determnaton of the economc order quantty n ths case s not unque, and s based on some estmaton to be done by the decson maker f he/she s suppled wth a set of fuzzy cost values determned be a formula n whch the fuzzy values K t and K s appear, together wth 2M + 1 crsp values of from the neghborhood of k, where M s a natural number determned by the decson maker (n Kuchta s paper t was 50). The decson maker then has to choose the most sutable for hm/her from those 2M + 1 fuzzy cost values. On the other hand, f we apply our method and the lnear defuzzfcaton functonal (20), then from the theorem for the case A (absolutely contnuous h 1 and h 2 ) n Eq. (12), we get an explct expresson for the fuzzy EO. To ths end, let us choose the representaton of two convex trangular fuzzy numbers K t and K s as ordered fuzzy numbers. From the Appendx, we know that each CFN corresponds to two OFNs, whch dffer n orentaton. Hence, for K t we have (7 + s, 9 s) and (9 s, 7 + s), wth s [0, 1]. On the other hand, for K s we have ( s, 15 8s) and (15 8s, s). For K t, f we take the frst OFN, whch has the so-called postve orentaton, then ths means that our estmate of the future transportaton cost s rather pessmstc: the cost s at least around 8. On the other hand, f we take the second OFN, namely (9 s, 7 + s), then we are rather optmstc: the transportaton cost s at most around 8. For further calculaton, we assume the pessmstc vewpont and defne f t (s) = 7 + s and g t (s) = 9 s, whle f s (s) = s and g s (s) = 15 8s. Notce that there are three other cases and, consequently, three other solutons for the fuzzy EO could follow. Assumng a determnstc demand of , applyng the formula for the EO wth f D (s) = 1000, f M (s) = f s (s)/2, and g D (s) = 1000, g M (s) = g s (s)/2, we obtan the fuzzy EO as the followng ordered fuzzy number ( s) ( s) 1/2 1/ ( ) ( ) [ ] f s =, g s =, s 0, s 15 8s 1 Notce that n our example D s crsp and s represented by the par of constant functons (1000, 1000 ).

16 54 W. KOSIŃSKI et al. From ths expresson we could easly calculate the fuzzy mnmal nventory cost K ( ). Notce that nether nor K ( ) can be represented n the form of a CFN wth a trangular membershp functon. We could draw fgures for them by substtutng values of s from the nterval [0, 1]. By applyng a partcular defuzzfcaton functonal, we could calculate the crsp values correspondng to and K ( ). Fnally, the characterstc values of are 1/2 1/ ( ) () () f 0 = = 96.6, f 1 = g 1 = 47.8, g ( ) 1/ = = Notce that by applyng the defuzzfcaton functonal φ = φmom to, we obtan the crsp EO φ = φmom ( ) = f (1) = 47.8, whch s equal to k from the determnstc case. Correspondng to these values, the characterstc values of the cost are: K t s f ( ) ( 0) 1000 f f ( 0) = f ( 0) + f ( 0) = = K K t s f () ( 1) 1000 f f () 1 = g () 1 = f () 1 + f () 1 = g g ( 0) = g ( 0) + g ( 0) = = K t s g ( ) Consderng the data regardng the values of the fuzzy cost n [7], we can see (Table 7.1, p. 112) that the domans of the trangular membershp functons of these values s the nterval from to Moreover, these fuzzy values of cost are related to the range of order quanttes from 91 to 36. In our calculatons, ths range s from 96.6 to Receved 28 Aprl 2013 Accepted 21 December 2013