Inernaional Journal of Approximae Reasoning 50 (009) 59 540 Conens liss available a ScienceDirec Inernaional Journal of Approximae Reasoning journal homepage www.elsevier.com/locae/ijar Fuzzy economic order ime models wih random demand T. Vijayan, M. Kumaran * Deparmen of Saisics, Nehru Ars and Science College, Kanhangad, Kerala 7138, India aricle info absrac Aricle hisory Received 9 June 008 Received in revised form 8 Ocober 008 Acceped 3 November 008 Available online 4 November 008 Keywords Invenory Fuzzy se Economic order ime Lagrangian opimizaion Trapezoidal fuzzy number Invenory models in which he ime period of sales is a decision variable are considered in fuzzy environmens. Arrival of cusomers and he number of cusomers in he planning period are boh random. Several cases of he model wih one or more of he componens fuzzy are discussed. Models wih rapezoidal and riangular fuzzy numbers are considered. Opimum policy of he models are derived and numerical examples are provided in order o ascerain he sensiiveness in he decision variable wih respec o fuzziness in he componens. Ó 008 Elsevier Inc. All righs reserved. 1. Inroducion The firs scienific approach o invenory managemen problem was he Harris Wilson mehod popularly known as he economic order quaniy (EOQ) formula. In he EOQ invenory sysem, he inpu is made in equal sizes agains coninuous wihdrawal of iems a a consan rae. The coss o be considered are he seup cos, producion cos and invenory holding cos. The EOQ formula gives he order quaniy so as o mee cusomer service levels while minimizing he oal invenory cos. This formula is generally recommended in problems where demand is relaively seady. Being very simple o implemen, sock manufacurers use he EOQ formula for fixing he quaniy o be produced while he sock disribuors use i for fixing he quaniy o be purchased. A number of auhors have considered several variaions in he sandard EOQ model. An EOQ model for iems wih an exponenially decaying invenory was invesigaed by Ghare and Schrader []. Cover and Philip [5] and Tadikamalla [17] sudied he EOQ model for iems wih weibull and gamma ype deerioraion respecively. Liberaore [11] developed an EOQ model where he uncerainies in he lead ime is represened sochasically. Goyal [7] esablished a single iem EOQ model under he condiion of permissible delay in paymens. Hariga [9] described an EOQ model for deerioraing iems wih shorage and ime varying demand. An EOQ invenory model for perishable iems was developed by Padmanabhan and Vra [13] under sock dependen selling rae. Recenly, Chen [4] sudied an EOQ model under random demand. One or more componens of an invenory model ofen appear o be vague and imprecise and hence for geing realisic models all such componens are o be represened by fuzzy ses. A number of researchers have applied he fuzzy se conceps o deal wih he EOQ problems. Park [14] developed a fuzzy EOQ model where he ordering and holding coss are represened by rapezoidal fuzzy numbers. Vujoevia e al. [19] considered EOQ model under fuzzy cos componens. Roy and Maii [15] rewroe he problem of classic EOQ ino a form of nonlinear programming problem and inroduced fuzziness boh in he objecive funcion and consrains of sorage area. Lee and Yao [10] developed an EOQ model where he demand and order * Corresponding auhor. E-mail address mknas@rediffmail.com (M. Kumaran). 0888-13X/$ - see fron maer Ó 008 Elsevier Inc. All righs reserved. doi10.101/j.ijar.008.11.00
530 T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 quaniies are represened by fuzzy ses. Mandal and Maii [1] described muli-iem EOQ models wih objecive funcion, consrains and invenory coss are represened by fuzzy ses. An EOQ model wihou backorders wih fuzzy oal demand and fuzzy sorage cos was discussed by Yao and Chiang []. Chang [1] sudied an EOQ model wih imperfec qualiy iems where fuzziness is inroduced in he defecive rae and annual demand. Yadavalli e al. [1] considered a muli-iem EOQ model wih fuzzy cos. Wanga e al. [0] developed an EOQ problem wih imperfec qualiy iem characerized as a fuzzy random variable while he seup and holding coss as fuzzy variables. Shiang [1] sudied an EOQ model under fuzzy demand quaniy and fuzzy cos. An EOQ model for perishable iems wih fuzzy parial backlogging facor and fuzzy deerioraion rae was developed by Halim e al. [8]. In his paper, an invenory sysem in which he ime period of sales is he decision variable proposed by Chen [4] is reconsidered assuming he componens of he model as fuzzy ses. Arrival of cusomers and he number of cusomers in he planning period are boh random. The auhor derived he opimal lengh of he selling period so as o minimize he average invenory cos per uni ime. I was poined ou ha he number of cusomers arriving in he planning ime period in Chen s model is equivalen o he order quaniy in he radiional EOQ model. As such, Chen s model can be considered as an EOQ model wih quaniy represening he ime period, hence we name i as he economic order ime (EOT) model. Secion briefly presens he EOT model under random demand and random purchasing ime. The Lagrangian mehod of opimizaion is described in Secion 3. Secion 4 reviews he basic conceps of fuzzy se heory. The fuzzy equivalen of he EOT model wih all componens fuzzy is described in Secion 5. Secion describes he fuzzy model wih crisp ime period and model under fuzzy mean arrival rae is given in Secion 7. The defuzzified value of fuzzy cos funcion is derived by adoping he graded mean inegraion represenaion of fuzzy numbers. The las Secion presens he numerical illusraions of he developed models followed by some concluding remarks.. EOT model wih random demand The basic EOQ model deermines he economic order quaniy which minimize he oal cos based on he assumpions ha he oal demand is consan and shorage is no permied. The Wilson Harry s opimal invenory size formula is given by rffiffiffiffiffiffiffiffi Ka Q ¼ ð1þ h where K h and a are he seup cos, holding cos per uni and oal demand per uni ime, respecively. However, he radiional EOQ model seems o be effecive if he demand and cos componens are compleely known. Since he oal demand is usually uncerain, i is more realisic o replace he consan demand by he expeced value of he oal demand. Chen [4] reconsruced he EOQ model based on he random demand. The following noaions and assumpions are used. Noaions ½0 Š he selling period, being he lengh of he given period. C he purchasing cos per uni of goods. K he seup cos. h he uni holding cos per uni ime. Assumpion. (i) The purchasing ime of he cusomer is a uniform ð0 Þ random variable. (ii) The number of cusomers ha arrive in he uni ime inerval (quaniy of demand) follows a Poisson disribuion wih mean arrival rae per uni ime h. (iii) The oal number of cusomers in he inerval ½0 Š follows a Poisson disribuion wih parameer h. (iv) The purchasing imes of he cusomers are independenly and idenically disribued and hey are muually independen wih he oal number of cusomers in he selling period. The expeced oal cos in he whole period ½0 Š is given by Chen [4] ZðÞ ¼K þ Ch þ hh ðþ The expeced oal cos per uni ime is AðÞ ¼ ZðÞ K hh þ Ch þ ð3þ The objecive is o find he lengh of he selling period which minimize he average cos per uni ime. The necessary condiion for Eq. (3) o be minimum, oaðþ ¼ 0 implies ha o
a which rffiffiffiffiffiffi K ¼ ð4þ hh o AðÞ o ¼ K > 0 ð5þ 3 Hence, given by Eq. (4) minimizes he average cos in Eq. (3). I is noed ha variaions in he purchasing cos will no affec he opimum ime period. The above model can be considered as an EOQ model. Le N denoe he number of cusomers arriving in he ime inerval ð0 Þ when equals he opimum value given in Eq. (4). Since he probabiliy disribuion of N is assumed o be Poisson, expeced value of N or expeced quaniy of demand is given by rffiffiffiffiffiffiffiffiffi EðN Þ¼h ¼ Kh ðþ h I should be noed ha, Eq. () is also equivalen o he resul of radiional EOQ formula, provided ha he quaniy of oal demand a per uni ime in Eq. (1) is replaced by h, he expeced demand per uni ime. Thus EðN Þ is he expeced economic order quaniy. The fuzzy equivalen of he above model is described in Secion 5. In he following secion, he Lagrangian opimizaion echnique, needed o solve he fuzzy model is described. 3. Lagrangian opimizaion mehod T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 531 The echniques for idenifying he saionary poins of a nonlinear programming problem subjec o inequaliy consrains is based on he Lagrangian mehod. The Karush Kuhn Tucker condiions are necessary and sufficien condiions for minimizaion problem if boh he objecive funcion and soluion space are convex. In he case of minimizaion problem wih non negaive consrains, he soluion space is convex if he consrain funcion is concave and he Lagrangian mulipliers are non negaive. In such case, he Lagrangian funcion mus be convex and he resuling saionary poin yields a global consrained minimum. We adop he exended Lagrangian mehod o solve he non linear programming problem wih inequaliy consrains. This mehod is described in several sandard ex books, Taha [18] is one of he laes references. The general idea of exended Lagrangian procedure is ha if he unconsrained opimum problem does no saisfy all he consrains, he consrained opimum mus occur a he boundary poin of he soluion space. This means ha a leas one consrain mus be saisfied in equaion form. In his case, a he opimal poin, he Karush Kuhn Tucker necessary condiions indicae ha he negaive of he gradien of he objecive funcion (represen he direcion of seepes descen) mus be expressible as a posiive linear combinaion (he coefficiens are he Lagrangian mulipliers) of he gradien of he acive consrains. The bes o be hoped for using he exended Lagrangian mehod is a good feasible soluion. If he problem possess a unique consrained opimum, he procedure can be recified o locae he global opimum. For he minimizaion problem, Minimize Y ¼ f ðxþ subjec o g i ðxþ P 0 i ¼ 1... M, where he nonnegaiviy consrains X P 0, if any, are also included in he M consrains, he procedure of exension of Lagrangian mehod involves he following seps. Sep (i) Solve he unconsrained problem Minimize Y ¼ f ðxþ. If he resuling opimum saisfies all he consrains, sop he procedure. Oherwise se he number of consrains K ¼ 1 and go o sep (ii). Sep (ii) Acivae any K consrains by convering hem ino equaliies and minimize f ðxþ subjec o he K acive consrains by he Lagrangian mehod. If he resuling soluion is feasible wih respec o he remaining consrains, sop i is a local opimum. Oherwise, ake anoher se of K consrains and repea he sep. If all ses of acive consrains aken a a ime are considered wihou encounering a feasible soluion, go o sep (iii). Sep (iii) IfK ¼ M, sop no feasible soluion exiss. Oherwise se K ¼ K þ 1 and go o sep (ii). Some preliminary conceps of fuzzy se heory required in he developmen of our models are described below. 4. Fuzzy se In a universe of discourse X, a fuzzy subse A e on X is defined by he membership funcion l ea ðxþ which maps each elemen x in X o a real number in he inerval [0,1]. l ea ðxþ denoes he grade or degree of membership and i is usually denoed as l ea X!½0 1Š. A fuzzy se is said o be normal if he larges grade obained by any elemen in ha se is 1. Tha is, here mus exis a leas one x for which l ea ðxþ ¼1. The suppor of A e is defined as he crisp se ha conains all elemens of X ha have non zero membership grades. A fuzzy se A e on X is convex iff l ea ðkx 1 þð1 kþx Þ P minðl ea ðx 1 Þ l ea ðx ÞÞ for all x 1 x ex and for ke½0 1Š, where min denoes he minimum operaor.
53 T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 4.1. Fuzzy number A fuzzy number is a fuzzy subse of he real line which is boh normal and convex. In addiion, he membership funcion of a fuzzy number mus be piecewise coninuous. The membership funcion of a fuzzy number e A is usually represened as l ea ðxþ ¼lðxÞ x < m ¼ 1 m x n ¼ uðxþ x > n ð7þ where lðxþ is coninuous from he righ, sricly increasing for x < m and here exis m 1 < m such ha lðxþ ¼0 for x m 1 and uðxþ is coninuous from he lef, sricly decreasing for x > n and here exis n 1 P n such ha uðxþ ¼0 for x P n 1. lðxþ and uðxþ are called he lef and righ reference funcions, respecively. The fuzzy number e A is said o be a rapezoidal fuzzy number if i is fully deermined by ða 1 a a 3 a 4 Þ of crisp numbers such ha a 1 < a < a 3 < a 4, wih membership funcion, represening a rapezoid, of he form l ea ðxþ ¼ x a 1 a a 1 a 1 x a ¼ 1 a x a 3 ¼ x a 4 a 3 a 4 a 3 x a 4 ¼ 0 oherwise ð8þ where a 1 a a 3 and a 4 are he lower limi, lower mode, upper mode and upper limi respecively of he fuzzy number e A. When a ¼ a 3, he rapezoidal fuzzy number becomes a riangular fuzzy number. 4.. Fuzzy arihmeic operaions Some fuzzy arihmeic operaions under he funcional principle Chen [] for rapezoidal fuzzy numbers are given below. Le f A 1 ¼ða 11 a 1 a 13 a 14 Þ and f A ¼ða 1 a a 3 a 4 Þ be wo rapezoidal fuzzy numbers. Then (i) Addiion fa 1 þ f A ¼ða 11 þ a 1 a 1 þ a a 13 þ a 3 a 14 þ a 4 Þ (ii) Muliplicaion If a 11 a 1 a 1 a a 13 a 3 a 14 and a 4 are all posiive real numbers, hen fa 1 f A ¼ða 11 a 1 a 1 a a 13 a 3 a 14 a 4 Þ (iii) Subracion f A ¼ð a 4 a 3 a a 1 Þ f A1 f A ¼ða 11 a 4 a 1 a 3 a 13 a a 14 a 1 Þ (iv) Division If a 11 a 1 a 1 a a 13 a 3 a 14 and a 4 are all posiive real numbers, hen 1 ¼ A f 1 ¼ 1 1 1 1 ea a 4 a 3 a a 1 ea 1 ¼ a 11 a 1 a 13 a 14 ea a 4 a 3 a a 1 (v) Scalar muliplicaion Le k be a real number, hen for k P 0 k f A 1 ¼ðka 11 ka 1 ka 13 ka 14 Þ and k < 0 k f A 1 ¼ðka 14 ka 13 ka 1 ka 11 Þ. 4.3. Defuzzificaion In order o draw ulimae conclusions for decision making, he fuzzy resuls are o be convered ino crisp values. The mehod of exracing crisp resuls from he fuzzy models is known as defuzzificaion. In his paper, we adop he graded mean inegraion represenaion inroduced by Chen and Hseih [3] for defuzzificaion. The exension principle o find he membership funcion of fuzzy oal cos funcion is hough direc, is no simple in mos of he cases. As he membership funcion does no change under fuzzy arihmeic operaions, i is possible o evaluae he defuzzified value direcly by graded mean inegraion mehod hrough arihmeic operaions. I is more reasonable o discuss he grade of each poin of suppor se of fuzzy number for represening he fuzzy number. Chen and Hseih s mehod is effecive in he sense ha i grades as he degree of each poin of suppor se of fuzzy number and i is possible o measure he degree of similariy beween fuzzy numbers in erms of graded mean inegraion values.
For he fuzzy number A e in Eq. (7), le l 1 and u 1 denoe he inverse funcions of l and u, respecively. The graded mean c level value of A e is 1 ðcðl 1 ðcþþu 1 ðcþþ. The graded mean represenaion of A e is given by.ðaþ¼ e R 1 Z! 1 l 1 ðcþþu 1 ðcþ 1 cdc cdc ð9þ 0 0 For he rapezoidal fuzzy number e A ¼ða 1 a a 3 a 4 Þ l 1 ðcþ ¼a 1 þða a 1 Þc and u 1 ðcþ ¼a 4 ða 4 a 3 Þc. The graded mean represenaion of rapezoidal fuzzy number e A ¼ða 1 a a 3 a 4 Þ from Eq. (9) is given by.ðaþ¼ e a 1 þ a þ a 3 þ a 4 ð10þ In he case of riangular fuzzy number A e ¼ða 1 a a 3 Þ,.ðAÞ¼ e a 1 þ 4a þ a 3 ð11þ The Secions 5 7 describe differen cases of he EOT model of Secion in fuzzy environmens. The model wih all he componens fuzzy is considered in Secion 5 while he one wih crisp ime period is developed in Secion. 5. Fuzzy EOT model wih fuzzy ime period T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 533 In his secion, we consider model in Secion wih all he five parameers C K h and h as fuzzy and hey are represened by rapezoidal fuzzy numbers, as follows ec ¼ðC j 1 C j C þ j 3 C þ j 4 Þ K e ¼ðK j5 K j K þ j 7 K þ j 8 Þ ~h ¼ðh j 9 h j 10 h þ j 11 h þ j 1 Þ ~ ¼ð m 1 m þ m 3 þ m 4 Þ and ~h ¼ðh m 5 h m h þ m 7 h þ m 8 Þ j i i ¼ 1... 1 and m i i ¼ 1... 8 are arbirary posiive numbers which saisfy j 1 > j j 3 < j 4 j 5 > j j 7 < j 8 j 9 > j 10 j 11 < j 1 m 1 > m m 3 < m 4 m 5 > m and m 7 < m 8. Fuzzy expeced cos per uni ime from Eq. (3) is given by K fað~þ ¼ e ~ þ ~ hc e ~h þ ~ h~ ð1þ where e K ~ ~ hc e and h ~ ~ h~ are given by he fuzzy arihmeic operaions in Secion 4 as, he rapezoidal fuzzy numbers ek ~ ¼ K j 5 K j K þ j 7 K þ j 8 þ m 4 þ m 3 m m 1 ð13þ ~h C e ¼ððh m 5 ÞðC j 1 Þ ðh m ÞðC j Þ ðh þ m 7 ÞðC þ j 3 Þ ðh þ m 8 ÞðC þ j 4 ÞÞ ð14þ and ~h ~ h~ ¼ððh m 5 Þðh j 9 Þð m 1 Þ ðh m Þðh j 10 Þð m Þ ðh þ m 7 Þðh þ j 11 Þð þ m 3 Þ ðh þ m 8 Þðh þ j 1 Þð þ m 4 ÞÞ ð15þ Using he above Eqs. (13) (15) in Eq. (1), we have he rapezoidal fuzzy number where fað~þ ¼ðA 1 A A 3 A 4 Þ A 1 ¼ K j 5 þm 4 þðh m 5 ÞðC j 1 Þþ ðh m 5Þðh j 9 Þð m 1 Þ A ¼ K j þm 3 þðh m ÞðC j Þþ ðh m Þðh j 10 Þð m Þ A 3 ¼ Kþj 7 m þðhþm 7 ÞðC þ j 3 Þþ ðhþm 7Þðhþj 11 Þðþm 3 Þ A 4 ¼ Kþj 8 m 1 þðhþm 8 ÞðC þ j 4 Þþ ðhþm 8Þðhþj 1 Þðþm 4 Þ The graded mean inegraion value of he fuzzy number in Eq. (1) is obained from Eq. (10) as.ðfað~þþ ¼ 1 K j 5 þðh m 5 ÞðC j 1 Þþ ðh m 5Þðh j 9 Þ 1 4 þ K j þðh m ÞðC j Þþ ðh m Þðh j 10 Þ 3 þ K þ j 7 þðhþm 7 ÞðC þ j 3 Þþ ðh þ m 7Þðh þ j 11 Þ 3 þ 1 K þ j 8 þðhþm 8 ÞðC þ j 4 Þþ ðh þ m 8Þðh þ j 1 Þ 4 1 ð1þ ð17þ
534 T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 where i ¼ m i i ¼ 1 i ¼ þ m i i ¼ 3 4 C j i > 0 i ¼ 1 K j i > 0 i ¼ 5 h j i > 0 i ¼ 9 10 and h m i > 0 i ¼ 5. The defuzzified value,.ðfað~þþ, is aken as he crisp esimae of fuzzy model in Eq. (1). In order o find he parameers which minimizes.ðfað~þþ, we have o solve he following parial derivaives of.ðfað~þþ wih respec o ~ ¼ð 1 3 4 Þ each equaed o zero. o.ðfað~þþ o 1 o.ðfað~þþ o o.ðfað~þþ o 3 and ¼ ðh m 5Þðh j 9 Þ 1 ¼ ðh m Þðh j 10 Þ ¼ ðh þ m 7Þðh þ j 11 Þ o.ð fað~þþ o 4 K þ j 8 1 ¼ ðh þ m 8Þðh þ j 1 Þ 1 ðk þ j 7Þ ðk j Þ 3 K j 5 4 Solving he above, we ge sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk þ j 8 Þ ðk þ j 7 Þ ðk j Þ ðk j 5 Þ 1 ¼ ¼ 3 ¼ and 4 ¼ ð19þ ðh m 5 Þðh j 9 Þ ðh m Þðh j 10 Þ ðh þ m 7 Þðh þ j 11 Þ ðh þ m 8 Þðh þ j 1 Þ Noe ha 1 > > 3 > 4 and hence he consrain 0 < 1 < < 3 < 4 is no saisfied. Hence, we adop he Lagrangian mehod described in Secion 3. For his, we conver he inequaliy consrain 1 P 0 ino equaliy consrain 1 ¼ 0 and minimize.ðfað~þþ subjec o 1 ¼ 0. We have he Lagrangean funcion as Lð 1 3 4 Þ¼.ðfAð~ÞÞ kð 1 Þ where k is he Lagrangian muliplier. Taking he parial derivaives of Lð 1 3 4 Þ wih respec o 1 3 4 and k and equae o zero, we ge sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk þ j 8 þ ðk þ j 7 ÞÞ ðk j Þ ðk j 5 Þ 1 ¼ ¼ 3 ¼ and 4 ¼ ð1þ ðh m 5 Þðh j 9 Þþðh m Þðh j 10 Þ ðh þ m 7 Þðh þ j 11 Þ ðh þ m 8 Þðh þ j 1 Þ Since 3 > 4, he above soluion is no a local opimum. We ge he similar resul if repea he procedure by selecing any one of he oher inequaliy consrains. Hence, we conver wo of he inequaliy consrains 1 P 0 and 3 P 0as equaliy and minimize.ðfað~þþ subjec o 1 ¼ 0 and 3 ¼ 0. The Lagrangian funcion wih mulipliers k 1 and k as Lð 1 3 4 Þ¼.ðfAð~ÞÞ k 1 ð 1 Þ k ð 3 Þ The soluion obained by seing he derivaives of Lð 1 3 4 Þ in Eq. () wih respec o 1 3 4 k 1 and k are all equal o zero is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk þ j 8 þ ðk þ j 7 ÞþðK j ÞÞ 1 ¼ ¼ 3 ¼ and ðh m 5 Þðh j 9 Þþðh m Þðh j 10 Þþðhþm 7 Þðh þ j 11 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk j 5 Þ 4 ¼ ð3þ ðh þ m 8 Þðh þ m 1 Þ I may be noed from Eq. (3) ha 1 ¼ ¼ 3 > 4. Tha is, he soluion given above is no local opimum as i does no saisfy he consrain 0 < 1 3 4. We ge he similar resul if we repea by selecing any wo of he inequaliy consrains. Hence, he inequaliy consrains 1 P 0 3 P 0 and 4 3 P 0 are convering ino equaliies, 1 ¼ 0 3 ¼ 0 and 4 3 ¼ 0. The Lagrangian funcion wih k i i ¼ 1 3 4 mulipliers is Lð 1 3 4 Þ¼.ðfAð~ÞÞ k 1 ð 1 Þ k ð 3 Þ k 3 ð 4 3 Þ In order o minimize Lð 1 3 4 Þ in Eq. (4), we ake he parial derivaives of Lð 1 3 4 Þ wih respec o 1 3 4 k 1 k and k 3 and equae o zero. Thus, we have 1 ¼ ¼ 3 ¼ 4 ¼, where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðk þ j 8 þ ðk þ j 7 ÞþðK j ÞþK j 5 Þ ð5þ ðh m 5 Þðh j 9 Þþðh m Þðh j 10 Þþðhþm 7 Þðh þ j 11 Þþðhþm 8 Þðh þ j 1 Þ Because he soluion saisfies all inequaliy consrains, he procedure erminaes wih as he local opimum soluion o he problem. Since he above local opimum soluion is he only one feasible soluion, i is he opimum soluion of he model. If he lower and upper modes are equal, he rapezoidal fuzzy number reduces o he riangular fuzzy number. In such case, he rapezoidal fuzzy numbers C e K~ e h ~ and ~ h are represened by he riangular fuzzy numbers, C e ¼ðC j1 C C þ j 4 Þ K e ¼ðK j 5 K K þ j 8 Þ h ~ ¼ðh j 9 h h þ j 1 Þ~ ¼ð m 1 þ m 4 Þ, and ~ h ¼ðh m 5 h h þ m 8 Þ, where C > j 1 K > j 5 h > j 9 > m 1 and h > m 5. ð18þ ð0þ ðþ ð4þ
T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 535 Fuzzy expeced cos per uni ime is represened as a riangular fuzzy number where fað~þ ¼ðA 1 A A 3 Þ A 1 ¼ K j 5 þ m 4 A ¼ K and A 3 ¼ K þ j 8 m 1 þðh m 5 ÞðC j 1 Þþ ðh m 5Þðh j 9 Þð m 1 Þ þ hc þ hh þðhþm 8 ÞðC þ j 4 Þþ ðh þ m 8Þðh þ j 1 Þð þ m 4 Þ The defuzzified value of he fuzzy number in Eq. () is obained from Eq. (11) as.ðfað~þþ ¼ 1 K j 5 þðh m 5 ÞðC j 1 Þþ ðh m 5Þðh j 9 Þ 1 þ 4 K hh þ hc þ 4 þ 1 K þ j 8 þðhþm 8 ÞðC þ j 4 Þþ ðh þ m 8Þðh þ j 1 Þ 4 1 Proceeding as in he case Eq. (17), he opimum ime period is obained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðþ ð7þ ðk þ j 8 j 5 Þ ð8þ ðh m 5 Þðh j 9 Þþ4 h h þðhþm 8 Þðh þ j 1 Þ. Fuzzy EOT model wih crisp ime period This is a replicaion of he model described in Secion 5 wih he only difference ha he ime period is regarded as crisp consan. Following he same noaions as in Secion 5, he fuzzy average cos per uni ime wih crisp ime period is given by K faðþ ¼ e þ ~ hc e ~h þ ~ h ð9þ where ek ¼ K j 5 K j K þ j 7 K þ j 8 ð30þ ~h ~ h ¼ððh m 5 Þðh j 9 Þ ðh m Þðh j 10 Þ ðh þ m 7 Þðh þ j 11 Þ ðh þ m 8 Þðh þ j 1 ÞÞ ð31þ and, ~ h e C is he same as given by Eq. (14). The above Eqs. (30), (31) and (14) reduce he Eq. (9) ino a rapezoidal fuzzy number as faðþ ¼ðA 1 A A 3 A 4 Þ where he componens A 1 ¼ K j 5 A ¼ K j þðh m 5 ÞðC j 1 Þþ ðh m 5Þðh j 9 Þ þðh m ÞðC j Þþ ðh m Þðh j 10 Þ A 3 ¼ K þ j 7 þðhþm 7 ÞðC þ j 3 Þþ ðh þ m 7Þðh þ j 11 Þ A 4 ¼ K þ j 8 and þðhþm 8 ÞðC þ j 4 Þþ ðh þ m 8Þðh þ j 1 Þ ð33þ As before, he defuzzified value of faðþ is given by.ðfaðþþ ¼ 1 K j 5 þ K j þ K þ j 7 þ 1 þðh m 5 ÞðC j 1 Þþ ðh m 5Þðh j 9 Þ þðh m ÞðC j Þþ ðh m Þðh j 10 Þ þðhþm 7 ÞðC þ j 3 Þþ ðh þ m 7Þðh þ j 11 Þ K þ j 8 þðhþm 8 ÞðC þ j 4 Þþ ðh þ m 8Þðh þ j 1 Þ ð34þ ð3þ
53 T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 In order o find he value of which minimizes.ðfaðþþ, we equae he derivaive of Eq. (34) wih respec o o zero. Tha s o.ðfaðþþ ¼ 0 o which reduces o sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðk þ j 8 þ ðk þ j 7 ÞþðK j ÞþK j 5 Þ ð3þ ðh m 5 Þðh j 9 Þþðh m Þðh j 10 Þþðhþm 7 Þðh þ j 11 Þþðhþm 8 Þðh þ j 1 Þ wih he second-order derivaive o.ðfaðþþ ¼ ðk j 5ÞþðK j ÞþðK þ j 7 ÞþðK þ j 8 Þ > 0 ð37þ o 3 3 I should be noed ha he opimum soluion of he fuzzy model for crisp ime period given by Eq. (3) is same as he soluion of fuzzy model for fuzzy ime period in Eq. (5). In he case of he riangular fuzzy number, he fuzzy average cos per uni ime is represened by where fað~þ ¼ðA 1 A A 3 Þ A 1 ¼ K j 5 A ¼ K A 3 ¼ K þ j 8 þðh m 5 ÞðC j 1 Þþ ðh m 5Þðh j 9 Þ þ hc þ hh þðhþm 8 ÞðC þ j 4 Þþ ðh þ m 8Þðh þ j 1 Þ The defuzzified value of he fuzzy number in Eq. (38) is.ðfað~þþ ¼ 1 K j 5 þ 1 and he opimum ime period is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þðh m 5 ÞðC j 1 Þþ ðh m 5Þðh j 9 Þ K þ j 8 þðhþm 8 ÞðC þ j 4 Þþ ðh þ m 8Þðh þ j 1 Þ ð35þ ð38þ þ 4 K hh þ hc þ ð39þ ðk þ j 8 j 5 Þ ð40þ ðh m 5 Þðh j 9 Þþ4 h h þðhþm 8 Þðh þ j 1 Þ The opimum ime periods given by Eqs. (40) and (8) are he same. Nex we shall examine he model for he impac of fuzziness in he arrival rae alone. 7. EOT model wih fuzzy arrival rae We consider he model given by Eq. (3) wih he arrival rae h is represened by he rapezoidal fuzzy number ~ h in Secion 5. The fuzzy expeced cos funcion becomes faðþ ¼ K þ ~ hc þ h ~ h ¼ K þððh m 5ÞC ðh m ÞC ðh þ m 7 ÞC ðh þ m 8 ÞCÞþ Eq. (41) reduces o he rapezoidal fuzzy number, where faðþ ¼ðA 1 A A 3 A 4 Þ A 1 ¼ K þðh m 5ÞC þ ðh m 5Þh A ¼ K þðh m ÞC þ ðh m Þh A 3 ¼ K þðhþm 7ÞC þ ðh þ m 7Þh and A 4 ¼ K þðhþm 8ÞC þ ðh þ m 8Þh ðh m 5Þh ðh m Þh ðh þ m 7Þh ðh þ m 8Þh ð41þ ð4þ
T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 537 The defuzzified value of faðþ using Eq. (10) is given by.ðfaðþþ ¼ K þ 1 ðh m 5ÞC þ ðh m 5Þh þ ðh m ÞC þ ðh m Þh þ ðh þ m 7ÞC þ ðh þ m 7Þh þ 1 ðh þ m 8ÞC þ ðh þ m 8Þh ð43þ For necessary condiions of minima of.ðfaðþþ, we mus have o.ðfaðþþ ¼ 0 o which gives he value of as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1K ¼ ð45þ ðh m 5 þ ðh m Þþðh þ m 7 Þþh þ m 8 Þh The second-order derivaive o.ðfaðþþ ¼ K o 3 > 0 3 When h m ¼ h þ m 7, he rapezoidal number ~ h becomes he riangular fuzzy number ~ h ¼ðh m 5 h h þ m 8 Þ. Then he fuzzy cos funcion in Eq. (41) reduces o he riangular fuzzy number where and faðþ ¼ðA 1 A A 3 Þ A 1 ¼ K þðh m 5ÞC þ ðh m 5Þh A ¼ K hh þ hc þ A 3 ¼ K þðhþm 8ÞC þ ðh þ m 8Þh The defuzzified value of faðþ in Eq. (47) and he opimum ime period are respecively given by.ðfaðþþ ¼ K þ 1 ðh m 5ÞC þ ðh m 5Þh þ 4 hh hc þ þ 1 ðh þ m 8ÞC þ ðh þ m 8Þh and ð48þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1K ¼ ð49þ ðh m 5 þ m 8 Þh The resuls of exensive numerical sudy performed are presened in he following secion. ð44þ ð4þ ð47þ 8. Numerical sudy Consider an invenory sysem wih crisp parameer values C ¼ $300 per uni, K ¼ $50 per seup/year, h ¼ $0 per uni/ year and h ¼ 10 unis. The opimum ime period and expeced oal cos obained from Eqs. (3) and (4) are ¼ 85 monhs and AðÞ ¼$3141. We use he following ses of rapezoidal fuzzy numbers, based on arbirary choices of j i and m j i ¼ 1... 1 j ¼ 5 7 8 o represen he componens of fuzzy models. The graded mean inegraion value (defuzzified value) and he corresponding percenage difference under fuzzy case (based on he defuzzified value) from he crisp value denoed by P c for he componen c, are also shown along wih he fuzzy numbers. C.ðCÞ P C h.ðhþ P h (5,5,305,315) 180 40 (,3,1,) 1 40 (100,10,350,400) 40 0 (,7,5,30) 1 0 (10,150,350,500) 70 10 (5,10,4,35) 18 10 (180, 00, 400, 00) 330 +10 (10, 15, 5, 4) +10 (150, 50, 430, 50) 30 +0 (1, 15, 31, 40) 4 +0 (00, 80, 530, 700) 40 +40 (1, 18, 35, 50) 8 +40
538 T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 K.ðKÞ P K h.ðhþ P h (4,8,5,5) 30 40 (1,1.5,10.5,11) 40 (10,0,0,70) 40 0 (1,4,1,15) 8 0 (15,35,55,75) 45 10 (3,,11,17) 9 10 (0, 40, 70, 90) 55 +10 (4, 7, 14, 0) 11 +10 (30, 40, 75, 100) 0 +0 (, 8, 15, 0) 1 +0 (35, 45, 85, 15) 70 +40 (, 9, 18, 4) 14 +40 The above fuzzy numbers reduce o he riangular fuzzy numbers when he lower and upper modes are equal o he crisp value. The opimum ime period along wih oal cos for he fuzzy EOT model in Secion 5 are compued from Eqs. (17), (5), (7) and (8). We use Eqs. (34), (3), (39) and (40) for he fuzzy EOT model wih crisp ime period. Under fuzzy arrival rae, Eqs. (43), (45), (48) and (49) give he opimum and he oal cos. Tables 1 4 reveal ha he opimum ime for fuzzy model wih fuzzy ime period is as same as fuzzy model wih crisp ime period. Table 3 shows variaions in he opimum decision variable and he expeced oal cos due o fuzziness in all he componens (excep ) of he model. I reveals ha he opimum value and he average oal cos are highly sensiive o he level of fuzziness in he componens. The percenage changes in he opimum coss are found o increase wih he increasing percenage changes in he level of fuzziness of he componens. The posiive change in he componens due o fuzziness decrease while increases for he negaive changes. Table 5 exhibis he variaions in he opimum values of he decision variable and expeced oal cos wih respec o changes in he levels of he fuzzy mean arrival rae. The opimum decision variable considerably decreases while he opimum average oal cos increases wih respec o increase in he fuzzy mean arrival rae (in erms of defuzzified value). Noe Table 1 Opimum policy under fuzzy EOT model for rapezoidal fuzzy numbers. P C P h P K P h (Monhs) fað~þ 40 40 40 40 8.7 (31,110,3354,330) 0 0 0 0 7.9 (11,50,4390,55) 10 10 10 10 7.8 (388,973,400,7009) 00 00 00 00 8.5 (3141,3141,3141,3141) +10 +10 +10 +10 7.3 (75,1498,58,1,403) +0 +0 +0 +0 7.1 (97,103,714,13,40) +40 +40 +40 +40.5 (184,47,988,17,35) Table Opimum policy under fuzzy EOT model for riangular fuzzy numbers. j 1 j 4 j 5 j 8 j 9 j 1 m 5 m 8 (Monhs) faðþ 75,15 18, 4, 9,1 9.1 (31,3141,330) 00,100 18,10 40,0 9,5 8.78 (11,3141,55) 180,00 15,15 35,5 7,7 8.34 (388,3141,7009) 0,0 0,0 0, 0 0,0 8.50 (3141,3141,3141) 10,300 10, 30,40,10 7.9 (75,3141,1,403) 150,350 8,0 0,50 4,10 8.09 (97,3141,13,40) 100,400 18,0 15,75 4,14 7.91 (184,3141,17,35) Table 3 Opimum policy under fuzzy EOT model wih crisp ime period for rapezoidal fuzzy numbers. P C P h P K P h (Monhs) % Change in.ðfaðþþ % Change in.ðfaðþþ 40 40 40 40 8.7.35 175 43.8 0 0 0 0 7.9 7.0 98 14.1 10 10 10 10 7.8 8.4 8 10. 00 00 00 00 8.5 0.00 3141 0.00 +10 +10 +10 +10 7.3 14.1 344 10.3 +0 +0 +0 +0 7.1 1.5 5435 73.0 +40 +40 +40 +40.5 3.5 778 131.7
T. Vijayan, M. Kumaran / Inernaional Journal of Approximae Reasoning 50 (009) 59 540 539 Table 4 Opimum policy under fuzzy EOT model wih crisp ime period for riangular fuzzy numbers. j 1 j 4 j 5 j 8 j 9 j 1 m 5 m 8 (Monhs) % Change in.ðfaðþþ % Change in.ðfaðþþ 75,15 18, 4, 9,1 9.1 7.3 339 5.5 00,100 18,10 40,0 9,5 8.78 3.3 77 11.8 180,00 15,15 35,5 7,7 8.34 1.9 310 1.4 0,0 0,0 0, 0 0,0 8.50 0.0 3141 0.00 10,300 10, 30,40,10 7.9.4 37 17.0 150,350 8,0 0,50 4,10 8.09 4.8 3830 1.9 100,400 18,0 15,75 4,14 7.91.9 45 35.8 Table 5 Opimum policy under fuzzy arrival rae for rapezoidal fuzzy numbers. P h (Monhs) % Change in.ðfaðþþ % Change in.ðfaðþþ 40 10.95 8.8 1910 39. 0 9.487 11.7 57 19. 10 8.945 5.9 835 9.80 00 8.500 0.00 3141 0.00 +10 8.090 4.8 3447 9.80 +0 7.74 8.87 3755 19. +40 7.171 15. 437 39. Table Opimum policy under fuzzy arrival rae for riangular fuzzy numbers. m 5 m 8 (Monhs) % Change in.ðfaðþþ % Change in.ðfaðþþ 9 1 9.1 7.3 73 13.0 9 5 8.78 3.3 947.18 7 7 8.50 0.0 3141 0.00 0 0 8.50 0.0 3141 0.00 10 8. 3.3 334.53 4 10 8.09 4.8 3448 9.77 4 14 7.8 7.5 353 1.3 ha he percenage changes in are nearly half and hose in he average oal cos are nearly equal o he percenage changes in he mean arrival rae a various levels. When he rapezoidal fuzzy number collapses o he riangular fuzzy number (Tables 4 and ), he sensiiveness in he opimum decision variable and he average oal cos is comparaively less due o fuzziness in he componens. The case, where he lower limi m 5 increases while he upper limi m 8 decreases, he opimum expeced oal cos increases when fuzziness is allowed in he arrival rae (Table ). When m 5 ¼ m 8 (~m is symmeric), fuzziness in he arrival rae has no impac as i reduces o he crisp case. 9. Conclusion We conclude ha he soluion of he EOT model wih all componens fuzzy is he same as hose under he fuzzy model wih crisp ime period. Hence, he fuzziness in he opimum period has no much relevance. The decision variable and he average oal cos are highly sensiive due o fuzziness in he cos componens and arrival rae when considered ogeher. Under fuzzy arrival rae, he percenage change due o fuzziness cause approximaely equal percenage changes in he oal cos and he decision variable is nearly half of he changes in he componen. The case where he upper and lower modes of he rapezoidal number coincides, he sensiiviy in he opimum decision variable and average oal cos reduces. The decision maker should adop a beer rade of judgemen for accouning flexibiliy in he characerisics of he model in order o ackle he uncerainy which fis o he real siuaions. Acknowledgemen The auhors graefully acknowledge he valuable commens and suggesions of he referees which helped us o improve he qualiy of he earlier version of his aricle.
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