Australan Journal of Basc and Appled Scences, 5(7): 863-873, 0 ISSN 99-878 Determne the Optmal Order Quantty n Mult-tems&s EOQ Model wth Backorder Babak Khabr, Had Nasser, 3 Ehsan Ehsan and Nma Kazem Department of Mathement, Islamc Azad Unversty, Jouybar Branch, Jouybar, Iran. Department of Mathematcal Scences, Mazandaran Unversty, Babolsar, Iran. 3, Department of Industral Engneerng, Alghadr Unversty, Tabrz, Iran. Abstract: In ths paper we survey the nventory problem for tem s whch delay occurred for supply them. In ths stuaton the shortage of nventory happen and order whch encounter shortage, wll ncrease and make backorders and mpose new costs on nventory system and mpact on optmal quantty of order n EOQ model. Snce n practce and n real stuatons we don't have accurate value for much data of model, we determne the optmal order quantty n fuzzy sense. For ths objectve frst we consder an EOQ model wth fuzzy costs and then we consder EOQ model wth fuzzy backorder and for deffuzfy of these two fuzzy models we employ sgned dstance, a defuzzfcaton method for fuzzy number. At the end of ths paper for mply the applcaton of proposed model we menton a numercal example and furthermore we compare the result of fuzzy case wth those of crsp case for one tem. Key words: Inventory; Backlog Demands; Fuzzy theory; Sgned dstance; Mult tem's EOQ INTRODUCTION In the classcal Economc Order Quantty (EOQ) models, t's assumed that the systems don t encounter the shortage of nventory and ever order delver to customer on tme and wthout delay. Whle ths capablty don't always exsts n the nventory systems. In producton process problems such as; mpossblty of on tme supply of raw materal for entrance n producton flow, unforcastable defect rate of machnes and mpossblty of precse forecast of market demand wth due to the term of socety that lead to the shortage of nventory. In the nventory control, orders that can't satsfy n order to mentoned reasons called backorders. These orders mpose many costs on nventory system that called shortage costs. Shortage cost s consstng of two costs: a fxed cost of nventory shortage and shortage whch happen n the cycle of tme. Ordnary n systems whch are face to the shortage of nventory due to backlog orders, cost coeffcent, annual demand, take crsp nto account n mult tems EOQ model. In the classcal nventory models, demands, costs and etc descrbe wth crsp statstcal dstrbutons. Whle n real stuatons and n real world most of these parameters are vague and mprecse. For these knds of uncertantes t s possble to use fuzzy numbers nstead of probablstc approaches. In recent years, several researchers have appled the fuzzy sets theory and technque to develop and solve nventory problems. For example Change, (00) worked out fuzzy modfcatons of the model of Salameh and Jabber, (000) whch took the defectve rate of goods nto account Björk, (009). Yao and Lee, (996) fuzzfed the order quantty to the trangular fuzzy number, whle the backorder quantty s an ordnary varable Ln, (008). Yao and Lee, (999) and Yao and Lee, (996) fuzzfed the order quantty q as a trangular fuzzy number and trapezod fuzzy number and kept shortage quantty as a crsp real varable n the total cost of nventory wth backorder Chang, et al., (005). Other relevant contrbutons n ths feld are Roy and Mat, (997) and Vjayan and Kumaran, (008), for nstance. Fnally Björk and Carlsson, (005) solved the case, where the lead tmes are fuzzy, but the demand kept crsp. Ths paper addresses an EOQ-model wth backorders, where the demand and the lead tmes are kept fuzzy (general trangular fuzzy numbers) Björk, (009). In the past research the authors had less attenton on mult tems model whle n real world most of cases have more than one tem. The purpose of the research s to determne the optmal quantty of order n mult tem case wth backorder. The paper s organzed as follows; n secton the classcal Economc order quantty (EOQ) model n mult tem case wth backorders s ntroduced. In secton 3 some defntons and propertes about fuzzy number and sgn dstance method are ntroduced. In secton, model wth fuzzy costs s presented and solved. In the next secton a numercal example s provded to llustrate the results of proposed model accompaned by compare between fuzzy and crsp case for one tem and senstvty analyss. Correspondng Author: Had Nasser, Department of Mathematcal Scences, Mazandaran Unversty, Babolsar, Iran E-mal: nasser@umz.ac.r 863
Aust. J. Basc & Appl. Sc., 5(7): 863-873, 0. Mult Item EOQ Model wth Backorders: In stuaton that we face to shortage of nventory, arrved orders add to each other and make backlog orders. Whenever order come to depostory, before satsfy any other order, frst backlog orders must satsfy. It s trval that f backlog order doesn t have any cost then the optmal way s that the nventory n hand shouldn't be bgger than Zero. On the other hand t may ths cost be so expensve, meanwhle f backlogged orders be between these two borders the optmal way s that some backlog order must exst at the end of T cycle. The cost of each backlog request contan a fxed cost ( ˆ ) plus proportonate cost wth tme that order can't satsfy. The behavor of ths model s shown n fg.. Also parameters of ths model are: ˆ : Fxed cost of each unt shortage π : Tme Dependent cost for each unt shortage T : Tme perod that we don't have shortage T : Tme perod that we encounter shortage S ; Quantty of shortage. Q : Quantty of each order. W : Cost of rent depostory that depend on the quantty of order. O : Annual demand. T : Tme perod of cycle C o : Cost order n each order C h : Holdng cost n tme cycle per each unt. Fg. : The behavor of nventory model n length cycle Form trangular () we have: Qs Qs Tan D T T D From trangular () we have: S S Tan D T T D The mean nventory n a cycle s equvalent to the bottom surface of nventory bent dvde to T. It's mean that: QS QS ( QS). ( ) Mean nventory = D D Q S () T Q Q D The mean backlog demand n a cycle s trangular's backlog demand's surface dvde to T gven by : 86
Aust. J. Basc & Appl. Sc., 5(7): 863-873, 0. S S. S S T Mean backlog demand= S D D () T T Q Q D S. ˆ s Q ˆ. SD. S Shortage cost = constant cost + varable wth tme cost = (3) T T Q Q The total cost of a system n a cycle s: K = total cost = set up cost + Holdng nventory cost + shortage cost + rent cost + purchase cost ( ).. (, ) D Q KQS C ( ˆ O S C S D S h ) QW. CD.. Q Q Q Q Snce the goal s determne the quantty of economc order shortage s optmal quantty ore as follows: () (5) (6) (7) If system be contanng of mult tem nstead of one tem (6), (7) change as follows: (8) (9) whch shows the -th tem n a nventory system. Defntons and property about fuzzy number: Some bascs from fuzzy set theory need to be ntroduced n order to make the followng model development self Contaned. Much of these can be found n Chang, (00). Defnton.: Consder the fuzzy set A=(a,b,c) where a<b<c and defned on R, whch s called a trangular fuzzy number f the membershp A s gvent by: ( x a) a xb ( b a ) ( x) ( c x) b x c A ( c b) 0 other wse (0) 865
Aust. J. Basc & Appl. Sc., 5(7): 863-873, 0 Defnton.: Let B be a fuzzy set on R and 0 # α #. The α - cut B(a) of B s all the pont x such that ( x ),.e. B B( ) { x ( x) } B Defnton 3.: For 0 # α #, the fuzzy set functon of [ a, b ] ( ) [ a, b ] x 0 s gven by a xb other wse [ a, b ] () defned on R s called an a - level fuzzy nterval f the membershp () Decomposton Prncple: Let B be a fuzzy set on R and 0 # α #. Suppose the a-cut of B to be a closed nterval [B L (α)b U (α)], that s, B(α)=[B L (α)b U (α)]. Then we have: B B( ) 0 or ( x ) C B( )( x ) B 0 Where () α B (α) s a fuzzy set wth membershp functon B( ), x B( ) ( x) 0, other wse (3) () () C ( ) B( ) x s a characterstc functon of B(α), that s C B( ), x B( ) ( x) 0, x B( ) Remark.: From the decomposton Prncple and (), we obtan B B( ) [ B ( ), B ( )] 0 0 or ( B C ( x) ( x) L B( ) [ B ( ), ( ) ] 0 0 L BU U (5) (6) For any a,b,c,k 0R, a < b and c<d, the nterval operaton are as follows: ( ) [ a, b]( )[ c, d] [ ac, bd] ( )[ a, b]( )[ c, d] [ a d, b c] [ ka, kb], k 0, ( ) k(.)[ a, b] [ kb, ka], k 0, 866
Aust. J. Basc & Appl. Sc., 5(7): 863-873, 0 Further, for a>0 and c>0. In order to fnd non fuzzy values for model n the next secton we need to use some dstance measures as n Cheng [], we wll use sgned dstance method. Defntons 3: For any a and 0MBOL06\f"Symbol"\sR,defne the sgn dstance from a to 0 as d 0 (a,0). If a>0, the dstance from a to 0 s d 0 (a,0)=a; f a<0, the dstance from a to 0 s -a=-d 0 (a,0). Hence d 0 (a,0)=a s called the sgned dstance from a to 0. Let Ω be the famly of all fuzzy sets B defned on R for whch the a - cut are contnuous functons on α0[0,]. Then, for any 0 B B [ B ( ), B ( ) ] L, we have: From Chang [ ] t can be fnally stated how to calculate the sgned dstance. (7) Defnton.: For B, defne the sgned dstance of B to (.e, y axs) as db (,0 ) d([ B( ), ( ) ],0 0 L B ) d ( B 0 ( ) B( )) d Accordng to defnton, we obtan the followng property: (8) Property : For the trangular fuzzy number A = ( a, b, c), the a - cut A s A U (a) = c (c-b)a. The sgned dstance of A to s: d( A,0 ) ( a bc) (9) Fuzzy Mult Item EOQ Model wth Backorders: In ths secton we fuzzfy mult tem case of (). In ths model fuzzy cost s consst of : holdng cost of each unt tem (C h ), set up cost n each order for tem (C o ), fxed shortage cost for each unt tem ( ˆ ), tme dependent shortage cost for each unt tem (π), rent cost for each unt tem (w ) and purchase cost of each unt tem (C ). The mult tem of () s: D ( Q S ) S D S X ( Q, S ) ( C C wq CD) m ˆ o h Q Q Q Q The Cost coeffcent take consder as trangular fuzzy number and nter n (0). The result gven by () m D ( Q S) SD S X XQS (, ) ( ˆ C o C h QW DC ) () Q Q Q Q Where (0) () 867
Aust. J. Basc & Appl. Sc., 5(7): 863-873, 0 ˆ ˆ, ˆ, ˆ 5 6,, 7 8 W W, W, W 9 0 C C, C, C (3) () (5) (6) (7) Also and Δ j, j=,,, are determned by decson makers. Now we deffuzfy () and for ths objectve we used sgned dstance method based on property, the sgned dstance of X to 0 s gven by: D ( Q S ) S D S dx (,0 ) [ dc (,0 ) dc (,0 ) d(,0 ) d(,0 ) QdW (,0 ) DdC (,0 )] m ˆ ˆ o h Q Q Q Q (8) Where dc (,0 ), dc (,0 ), d( ˆ,0 ), d(,0 ), dw (,0 ), dc (,0 ) o h Obtan accordng to property calculated as follows: dc (,0 o ) [( Co ) Co ( Co )] Co ( ) (9) dc (,0 h ) [( Ch 3 ) Ch ( Ch )] Ch ( 3 ) (30) d( ˆ,0 ˆ ˆ ˆ ˆ ) [( 5) ( 6)] ( 6 5) (3) d(,0 ˆ ) [( 7) ( 8)] ( 8 7) dw (,0 ) ( W 9) W ( W 0) W ( 0 9) dc (,0 ) [( C ) C ( C )] C ( ) Based on (9), (30), (3), (3), (33), (3) and substtoton n (8) we Wll have the followng : D ( Q S ) (, ) (, ) [ ( ( )) ( ( )) m * X Q S d X O Co Ch 3 Q Q SD S ( ( )) ( ( )) ( ( )) ˆ 6 5 8 7 Q W 0 9 Q Q D( C ( )] (3) (33) (3) (35) 868
Aust. J. Basc & Appl. Sc., 5(7): 863-873, 0 In order to obtan we must mprove that X * ( Q, S ) s convex n pont. It mean that we must mprove the second dervatve of X * ( Q, S ) s greater than zero. Snce the cost functon has tow varable we must examne ts convexty through the hessan matrx. Therefore frst we have to compute the frst and second dervatve. (36) X ( Q, S ) D S S D [ ( ) ( )( ) ( ) * 3 6 5 C ˆ o Ch Q Q Q Q S ( ) ( )] Q 8 7 0 9 W X ( Q, S ) D S S D ( ) ( ) ( ) * 3 6 5 C ˆ 3 o C 3 h 3 Q Q Q Q S Q ( ) 8 7 3 + + - + (37) (38) (39) (0) () + + () A= + + + + + + + + + 869
Aust. J. Basc & Appl. Sc., 5(7): 863-873, 0 A= + + + () If we let A>0 have: + + ] (5) It s mean that equaton (6) must be true to X * ( Q, S ) be convex n pont. Whle decson maker want to ntend about, frst they should examne obtaned relaton. Therefore for determne. we must solve ths equaton gven by: =0 (6) + + + + (7) + + - =0 + + =0 + + + (8) 870
Aust. J. Basc & Appl. Sc., 5(7): 863-873, 0 we let (8) n (9) then we have: + + + + After smplfy, we can wrte: (9) If we have closer look on (9), the optmal order quantty n fuzzy case s close to crsp case. If dentcal to the crsp case. Hence the fuzzy case s expandng of the crsp case. then the fuzzy case s 5. Comparson and Senstvty Analyss of Model s Parameters n Fuzzy and Crsp Case: In the followng some examples provde for comparson between the result of proposed model and those of crsp model n addton to senstvty analyss. Snce the proposed model s mult tem, ths comparson can apply for all tems n an nventory system, but we perform operaton for one tem for example. Assume the tem A s a product n a company and nventory control department want to order ths product for company and obtan data are gven by: D A =056 unt/year, C OA = 500 $/cycle, C ha = $/unt/year,, π A = 3$unt/year, W A = 500$/year C A = 0.33$/unt/year. Furthermore assume that expert's vewpont about these values are respectvely set as: * * In Table the optmal order quantty ( ) and optmal total cost n each year ( TC ) are calculate n both fuzzy and crsp case and varaton between them are measured by Q A 87. The results n Table * * ndcate that there s an ncrease wth 0.% n ( Q A ) and 0.58% n ( TC ) n fuzzy case rather than crsp case. Now n order to perform senstvty analyss we take consder two parameters of model n each table. In table 5 tow parameters, annual demand D A whch s consder as crsp, and order cost C oa whch s one the fuzzy parameters of proposed model, are consder. If we decrease wth 5% n D A and C oa result show that there s a decrease respectvely 3.3% and 3.38% wth D A and C oa. Keepng ths If we ncrease wth 5% n D A and C oa result show that there s an ncrease respectvely.79% and.79% wth D A and C oa. Hence
Aust. J. Basc & Appl. Sc., 5(7): 863-873, 0 varaton n these two parameters s almost dentcal. It s mean that f we ncrease the mentoned values, the optmal order quantty wll ncrease and nversely f we decrease these values the optmal quantty wll decrease. The same result could be deducted for optmal total cost (see Table 5). Accordng to analyss of Table 5, the other tables can be analyzed. Table : Comparson between fuzzy and crsp case Q * A TC * A Crsp 338.757 33077.8905 Fuzzy 339.5 336. Increase 0. 0.58 Table : Senstvty analyss of two parameters * A Q A TC * A W A Q * A TC * A -5-3.3-3.5-5 -3.38-3.0 +5.79.96 +5.79.63 Table 3: Senstvty analyss of two parameters * A Q A TC * A W A Q * A TC * A -5 0.00008 -.3 0-7 -5 5.9-3.3 +5 0.00006 0.05 +5-0.56.66 Table : Senstvty analyss of two parameters * A Q A TC * A W A Q * A TC * A -5 0.00008 -.3 0-7 -5 5.9-3.3 +5 0.00006 0.05 +5-0.56.66 Conclusons: In many problems n nventory context, exstence vague n many classcal model causes to unrelablty to result of models, therefore use of fuzzy models sound more desrable. In ths paper we fuzzfy classcal EOQ model wth backorders n case whch costs are fuzzy and other value are crsp. In ths case we obtan optmal quantty of order for each tem. Meanwhle for deffuzfy proposed model we apply sgned dstance method. One cases of the model wth sx cost component fuzzy and all others crsp are consdered for composton between fuzzy and crsp case and senstvty study. The concluson from the comparsons was that there s an ncrease to 0.% n optmal order quantty whle ths value s 0.58% for total annual cost. Future research ncludes the task to cover the combnaton of more fuzzy and crsp parameters and also composton between dfferent methods of dffuzfcaton. ACKNOWLEDGMENT The frst authors thanks to the Research Center of Algebrac Hyperstructures and Fuzzy Mathematcs, Babolsar, Iran for ts support. REFERENCES Björk, K.M., 009. An analytcal soluton to a fuzzy economc order quantty problem, Internatonal Journal of Approxmate Reasonng, 50: 85-93. Björk, K-M., C. Carlsson, 005. The outcome of mprecse lead tmes on the dstrbutors, n: Proceedngs of the 38th Annual Hawa Internatonal Conference on System Scences (HICSS 05), Track 3, HICSS, pp: 8-90. Chang, H-C., 00. An applcaton of fuzzy sets theory to the EOQ model wth mperfect qualty tems, Computers and Operatons Research, 3: 079-09. 87
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