AN INTEGRATED SUPPLY CHAIN MODEL FOR THE PERISHABLE ITEMS WITH FUZZY PRODUCTION RATE AND FUZZY DEMAND RATE

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1 Yugoslav Jounal of Oeations Reseach (0), Numbe, DOI: 0.98/YJOR0047S AN INEGRAED SUPPLY CHAIN MODEL FOR HE PERISHABLE IEMS WIH FUZZY PRODUCION RAE AND FUZZY DEMAND RAE Chaman SINGH Assistant Pofesso, Det. of Mathematics, A.N.D. College, Univesity of Delhi S.R. SINGH Reae, Det. of Mathematics, D.N.(P.G.) College, Meeut Receive: August 009 / Accete: Mach 0 Abstact: In the changing maket scenaio, suly chain management is getting henomenal imotance amongst eseaches. Stuies on suly chain management have emhasize the imotance of a long-tem stategic elationshi between the manufactue, istibuto an etaile. In the esent ae, a moel has been eveloe by assuming that the eman ate an ouction ate as tiangula fuzzy numbes an items eteioate at a constant ate. he exessions fo the aveage inventoy cost ae obtaine both in cis an fuzzy sense. he fuzzy moel is efuzzifie using the fuzzy extension incile, an its otimization with esect to the ecision vaiable is also caie out. Finally, an examle is given to illustate the moel an sensitivity analysis is efome to stuy the effect of aametes. Keywos: Fuzzy numbes, fuzzy eman, fuzzy ouction, integate suly chain. MSC: 90B30. INRODUCION oay, the stuy of the suly chain moel in a fuzzy envionment is gaining henomenal imotance aoun the globe. In such a scenaio, it is the nee of the hou that a eal suly chain be oeate in an uncetain envionment an the omission of any effects of uncetainty leas to infeio suly chain esigns. Inee, attention has been focuse on the anomness asect of uncetainty. Due to the incease awaeness an

2 48 C. Singh, S.R. Singh / An Integate Suly Chain Moel moe ecetiveness to innovative ieas, oganizations toay ae constantly looking fo newe an bette avenues to euce thei costs an incease evenues. his aticula stuy shows how oganizations in a suly chain can use thei esouces fo the best ossible outcome. In the cis envionment, all aametes in the total cost such as holing cost, set-u cost, uchasing ice, ate of eteioation, eman ate, ouction ate etc. ae known an have efinite value without ambiguity. Some of the business situations fit such conitions, but in most of the situations an in the ay-by-ay changing maket scenaio the aametes an vaiables ae highly uncetain o imecise. Fo any aticula oblem in the cis scenaio, the aim is to maximize o minimize the objective function une the given constaint. But in many actical situations, the ecision make may not be in the osition to secify the objective o the constaints ecisely, but athe secify them uncetainly o imecisely. Une such cicumstances, uncetainties ae teate as anomness an hanle by aealing to obability theoy. Pobability istibutions ae estimate base on histoical ata. Howeve, shote an shote ouct life cycles as well as gowing innovation ates make the aametes extemely vaiable, an the collection of statistical ata less an less eliable. In many cases, esecially fo new oucts, the obability is not known ue to lack of histoical ata an aequate infomation. In such situations, these aametes an vaiables ae teate as fuzzy aametes. he fuzzification gants authenticity to the moel in the sense that it allows vagueness in the whole setu which bings it close to eality. he efuzzification is use to etemine the equivalent cis value ealing with all uncetainty in the fuzzy value of a aamete. he fuzzy set theoy was fist intouce by Zaeh in 965. Aftewas, significant eseach wok has been one on efuzzification techniques of fuzzy numbes. In all of these techniques the aametes ae elace by thei neaest cis numbe/inteval, an the euce cis objective function is otimize. Chang et al. (004) esente a lea-time ouction moel base on continuous eview inventoy systems, whee the uncetainty of eman uing lea-time was ealt with obabilistic fuzzy set an the annual aveage eman by a fuzzy numbe only. Chang et al. (006) esente a moel in which they consiee a lea-time eman as fuzzy anom vaiable instea of a obabilistic fuzzy set. Dutta et al. (007) consiee a continuous eview inventoy system, whee the annual aveage eman was teate as a fuzzy anom vaiable. he lea-time eman was also assesse by a tiangula fuzzy numbe. Maiti an Maiti (007) eveloe multi-item inventoy moels with stock eenent eman, an two stoage facilities wee eveloe in a fuzzy envionment whee ocessing time of each unit is fuzzy an the ocessing time of a lot is coelate with its size. Bette cooination amongst the ouce, istibutos an etailes is the key to success fo evey suly chain. he integation aoach to suly chain management has been stuie fo yeas. Wee (998) eveloe a lot-fo-lot iscount icing olicy fo eteioating items with constant eman ate. Yang an Wee (000) consiee multile lot size eliveies. Yang an Wee (003) eveloe an otimal quantity-iscount icing stategy in a collaboative system fo eteioating items with instantaneous elenishment ate. Wu an Choi (005) assume sulie-sulie elationshis in the buye-sulie tia. Lee an Wu (006) eveloe a stuy on inventoy elenishment olicies in a two-echelon suly chain system. Chen an Kang (007) thought out integate veno-buye cooeative inventoy moels with vaiant emissible elay in

3 C. Singh, S.R. Singh / An Integate Suly Chain Moel 49 ayments. Singh et al. (007) iscusse otimal olicy fo ecaying items with stockeenent eman une inflation in a suly chain. Chung an Wee (007) eveloe, otimizing the economic lot, size of a thee-stage suly chain with backoeing eive without eivatives. Rau an Ouyang (008) have intouce an otimal batch size fo integate ouction-inventoy olicy in a suly chain. Kim an Pak (008) have assume eveloment of a thee-echelon SC moel to otimize cooination costs. Most of the efeences cite above have consiee single echelon o multi echelon inventoy moels with cis aametes only, an some who evelo the inventoy moel with fuzzy aamete consie only the single echelon inventoy moel. In the ast, eseaches ai no o little attention to the cooination of the ouce, the istibuto an the etailes in the fuzzy envionment. In the esent stuy, we have stive to evelo a suly chain moel fo the situations when items eteioate at a constant ate, an eman an the ouction ates ae imecise in natue. It is assume that the ouce suly n elivey to istibuto an istibuto, in tuns, sulies n eliveies to etaile in each of his elenishment. In oe to exess the fuzziness of the ouction an eman ates, these ae exesse as tiangula fuzzy numbes. Exessions fo the aveage inventoy cost ae obtaine both in cis an fuzzy sense. Late on, the fuzzy total cost is efuzzifie using the fuzzy extension incile. heeafte, it is otimize with esect to the ecision vaiables. Finally, the moel is illustate with some numeical ata.. ASSUMPIONS AND NOAIONS In this eseach, an integate suly chain moel fo the eishable items with fuzzy ouction ate an fuzzy eman ate is eveloe fom the esective of a manufactue, istibuto an etaile. We assume that the eman an the ouction ates ae imecise in natue an they have been eesente by the tiangula fuzzy numbes. Mathematical moel in this ae is eveloe une the following assumtions. Assumtions:. Moel assumes a single ouce, single istibuto an a single etaile.. he ouction ate is finite an geate than the eman ate. 3. he ouction an eman ates ae fuzzy in natue. 4. Shotages ae not allowe. 5. Deteioation ate is constant. 6. Lea time is Zeo. Notations: he following notations have been use thoughout the ae to evelo the moel:

4 50 C. Singh, S.R. Singh / An Integate Suly Chain Moel P Pouction ate P % Fuzzy ouction ate Deman ate % Fuzzy eman ate I t Single-echelon inventoy level of ouce uing eio () I () t Single-echelon inventoy level of ouce uing eio Cycle time ime eio of ouction cycle when thee is ositive inventoy ime eio of non-ouction cycle when thee is ositive inventoy θ Deteioation ate of on-han inventoy n Intege numbe of eliveies fom the ouce to the istibuto uing of inventoy cycle when thee is ositive inventoy n Intege numbe of eliveies fom the istibuto to his etaile uing each elivey he got fom the ouce I () t Single echelon inventoy level of istibuto I () t Single echelon inventoy level of etaile Q Pouce s ouction lot size Q Q Distibuto s lot size Retaile s lot size C Setu cost of the ouce e ouction cycle C Oeing cost of istibuto e oe C Oeing cost of etaile e oe C Inventoy caying cost fo the ouce e yea e unit C Inventoy caying cost fo istibuto e yea e unit C Cost of eteioate unit fo the ouce C C C C C Cost of eteioate unit fo the istibuto Cost of eteioate unit fo the etaile otal cost of the ouce otal cost of the istibuto otal cost of the etaile C he integate total annual cost C Fuzzifie integate total annual cost M % Defuzzifie integate total annual cost C

5 C. Singh, S.R. Singh / An Integate Suly Chain Moel Pouce s Inventoy Moel 3. CRISP MODEL Base on ou assumtions, the ouce stats the ouction with zeo inventoy level. Initially, the inventoy levels inceases at a finite ate (P-) units e unit time an eceases at a constant eteioation ate of (θ ), u to a time eio at which ouction is stoe. heeafte, the inventoy level eceases ue to the constant eman ate () units e unit time an at a constant eteioation ate (θ ) fo a eio of time at which the inventoy level eaches zeo level again, as shown in Figue given below. 0 ime Figue : Pouce s Inventoy Level he iffeential equations govening the single echelon ouce moel fo iffeent time uations ae as follows: I ( t ) = P θ I ( t ),0 t ()! I ( t ) = θ I ( t ),0 t ()! whee = + by solving the above equations with the bounay conitions I (0) = 0, I (0) = Q an I ( ) = 0 ouce s inventoy level I () t is given by P I ( t ) = e,0 t θ θt (3)

6 5 C. Singh, S.R. Singh / An Integate Suly Chain Moel θ ( t I ) ( t) = e,0 t θ Fom the conition I ( ) = Q = I (0), we have P θ θ e Q e θ = = θ θ [ P ( P ) e ] = ln θ Holing Cost of the Pouce is P ( ) HC = C θ θ e θ C e θ( ) θ + + θ Deteioation Cost of the Pouce is P ( ) DC = C θ θ e θ C e θ( ) θ + + θ he aveage total cost function C fo the ouce is aveage of the sum of set-u cost, caying cost an eteioation cost. C ( C + θc) ( P ) ( C + θc ) θ θ( ) C = + { e + θ } + { e θ θ (6) θ ( ) } Fo the minimization of the total cost we have (4) (5) ( C ) = 0 θ [ P + e ] his imlies that = ln, utting this value in equation (5) we θ P have, an then utting both of these values in the equation (6), we obtaine the total cost fo the ouce. 3.. Distibuto s Inventoy Moel Since the istibuto eceives a fixe quantity Q units in each of the elenishment, the istibuto s cycle stats with the inventoy levels Q units. heeafte, inventoy level eceases ue to the constant eman ate of ( ) units e n unit time an at a constant eteioation ate ( θ ), which eaches the zeo level in the time eio, as shown in Figue given below. n

7 C. Singh, S.R. Singh / An Integate Suly Chain Moel 53 0 /n /n (n -)/n n /n Figue Distibuto s Inventoy level Diffeential equations govening the istibuto s inventoy level ae as follows! I() t = θ I(),0 t t (7) n n Solving the iffeential equation with bounay conitions I ( ) = 0 gives θ ( ) () t n I t = e,0 t θ n n Maximum Inventoy of the istibuto is θ n Q = e θ n Holing cost of the istibuto in each elenishment cycle is HC C e θ n = θ n θ n n Deteioation Cost of the istibuto in each elenishment cycle is θ n DC = C e θ n θ n Distibuto s cost in each elenishment cycle is the sum of the oeing cost, caying cost an eteioation cost. Distibuto s total cost function C is the aveage of the sum of istibuto s total annual oeing cost, caying cost an eteioating cost in n elenishments. C ( C θc ) nc + θ θ n = + e θ n (8) (9) (0)

8 54 C. Singh, S.R. Singh / An Integate Suly Chain Moel 3.3. he etaile s inventoy moel Distibuto, in tuns, sulies n elenishments to the etaile in each of his elenishment cycles. In each elenishment, he sulies a fixe quantity Q to the etaile. Hence, etaile s inventoy level stats with the quantity Q an then eceases ue to the combine effect of both the constant eman an eteioation fo a time eio of at which the inventoy level eaches the zeo level, as shown in Figue 3 nn given below. 0 / n n / n n (n -)/ n n n / n n Figue 3 Retaile s Inventoy level Diffeential equations govening the etaile s inventoy level ae as follows! I() t = θ I(),0 t t () nn nn Solving the iffeential equation with bounay conitions I ( ) = 0 gives nn θ ( ) () t n n I t = e,0 t θ nn nn Maximum Inventoy of the etaile is θ nn Q = e θ nn Retaile s holing cost in each elenishment he got is HC C e θ nn = θ nn θ nn Retaile s eteioation cost in each cycle is () (3)

9 C. Singh, S.R. Singh / An Integate Suly Chain Moel 55 θ nn HC = C e θ nn θ nn Retaile s cost in each cycle is the sum of the oeing cost, holing cost an eteioation cost. Retaile s aveage total cost function C is the aveage of the sum of etaile s total annual oeing cost, caying cost an eteioation cost in nn elenishment cycles C ( C θc ) nnc + θ θ nn = + e θ nn he integate joint total cost function C fo the ouce, istibuto an etaile is the sum of C, C, an C. C = C + C + C (4) whee P θ C = C nc nnc ( C θc) ( e θ ) {( C θ θ θ( ) θ θ n { } ( ) n ( + ) θc ) e e θ C θc ( e θ ) C θc } θ nn ( e θ ) nn 3 (5) C = F ( ) + PF ( ) + F ( ) (6) C + n C + n n C = (7) ( ) F ( C + θc) θ F( ) = ( e + θ ) (8) θ ( C ) ( ) ( ) + θc θ θ C + θc θ n 3( ) { } ( F = e e θ + e θ ) + n (9) ( C ) + θc θ ( e nn θ nn ) 4. FUZZY MODEL BASED ON MODEL DEVELOPED IN SECION 3 In a eal situation an in a cometitive maket situation both the ouction ate an the eman ate ae highly uncetain in natue. o eal with such a tye of uncetainties in the sue maket, we consie these aametes to be fuzzy in natue.

10 56 C. Singh, S.R. Singh / An Integate Suly Chain Moel In oe to evelo the moel in a fuzzy envionment, we consie the ouction ate an the eman ate as the tiangula fuzzy numbes P% = ( P, P0, P) an % = (, 0, ) esectively, whee P = P Δ, P = P, P = P+Δ an 0 = Δ 3, 0 = an = +Δ 4, such that 0 < Δ < P,0 <Δ,0 <Δ 3 <,0<Δ 4 an Δ, Δ, Δ3, Δ 4 ae etemine by the ecision make base on the uncetainty of the oblem. hus, the ouction ate P an eman ate ae consiee as the fuzzy numbes Pan % % with membeshi functions P P, P P P0 P0 P P P μ % ( P) =, P0 P P (0) P P0 0, othewise, 0 0 μ ( ), % = 0 () 0 0, othewise Defuzzification of Pan % % by the centoi metho is given by M P + P + P ( 3 3 ) + + ( 3 3 ), 0 P = = P+ Δ Δ 0 M = = + Δ4 Δ3 esectively Fo fixe value of : P θ C = C nc nnc ( C C) ( e ) {( C θ + θ + + θ θ θc ) e e θ C θc ( e θ ) C θc θ( ) θ θ n { } ( ) n ( + ) } θ nn ( e θ ) nn whee C = F ( ) + PF ( ) + F ( ) 3

11 C. Singh, S.R. Singh / An Integate Suly Chain Moel 57 C + nc + nnc F ( ) = ( C + θc) θ F( ) = ( e + θ ) θ ( C ) ( ) ( ) + θc θ θ C + θc θ n 3( ) { } ( F = e e θ + e θ ) + n ( C ) + θc θ ( e nn θ nn ) Let C = y, this imlies that P = y F F 3 F y F F3 μ P% F B μ % ( ) a 3 a A 3 a Figue 4 μ ( y ) = AB C %

12 58 C. Singh, S.R. Singh / An Integate Suly Chain Moel μ % ( ) B! a 3 A! a 3 a!! Figue 5 μ ( y ) = AB C % he membeshi of the fuzzy cost function given by the extension incile is μ (y) = Su [ μ (P) μ ()] C % P% % (P,) (C) (y) () y F F3 = Su μ ( ) μ () P% F % Now Whee PF + F3 + F y, a3 a ( P P0) F y F F 3 y F F3 PF μ =, a a %P F ( P0 P) F 0 othewise y F PF y F PF y F PF a =, a = an a = 0 3 F3 F3 F3 (3) When a 0 an u, i.e. when y F+ PF + F3 y F F3 an y F+ PF 0 + 0F3, Figue exhibits the Gahs of μ P% an F μ ( ) %.

13 C. Singh, S.R. Singh / An Integate Suly Chain Moel 59 It is clea that fo evey y [ F+ PF + F3, F+ PF 0 + 0F3], μ y% ( y) = AB. he value of AB is then calculate by solving the fist equation of () an the secon equation of (3), i.e. y F F3 PF = o 0 ( P0 P) F ( y F PF )( 0 ) + ( P0 P) F = ( P P) F + ( ) F heefoe, AB = 0 y F PF F = = ( P0 P) F + ( 0 ) F3 μ ( y) When a3 an u o, i.e. when y F+ PF 0 + 0F3 y F F3 an y F+ PF + F3, Figue exhibits the gah of μ P% an F μ ( ) %. he value of A! B! is calculate by solving the secon equation of () an the fist equation of (3), i.e. PF + F3 + F y 0( P P0) F ( PF + F y)( 0) = o = ( P P ) F ( P P ) F + ( ) F heefoe, !! AB = 0 PF + F + F y 3 = = ( P P0) F + ( 0) F3 μ ( y) ( say) Membeshi function fo the fuzzy total cost is given as below: μ( y), F+ PF + F3 y F+ PF 0 + 0F3 ( ) μ ( ), C % y = μ y F+ PF 0 + 0F3 y F+ PF + F3 (6) othewisee Now let P = μ ( y ) y an R = y μ ( y ) y C % % C Defuzzification fo the fuzzy total cost, given by the centoi metho, is (5)

14 60 C. Singh, S.R. Singh / An Integate Suly Chain Moel R M ( ) C % = P = F( ) + PF( ) + F3( ) + ( Δ Δ ) F( ) + ( Δ4 Δ3) F3( ) 3 { } Whee F ( ), F ( ) an F ( ) 3 ae given by (7), (8) an (9) esectively. P + ( Δ Δ ) 3 M ( ) C nc nnc ( ) C % = C C θ θ + ( Δ4 Δ3 ) θ 3 θ( ) θ ( e + θ ) + {( C + θc) { e e θ} + θ θ θ n nn ( C + θc)( e θ ) + ( C )( ) n + θc e θ nn } (7) * o minimize the total aveage cost e unit time, otimal value of (say ) is obtaine by solving the following equation M ( ) 0 C % = which imlies that * θ ( ) ( 4 3) ( ) P+ Δ Δ + + Δ Δ e 3 3 = ln θ P + ( Δ Δ ) 3 (8) M ( ) P ( ) e C % 3 θ = + Δ Δ θ + ( ) 3 θ( ) θ ( ) + Δ Δ θ e e 4 3 an M C % ( ) > 0 * = by Hence, the cost function is minimize at = an the minimum cost is given * M ( ) C % * =

15 C. Singh, S.R. Singh / An Integate Suly Chain Moel 6 5. NUMERICAL EXAMPLE 5.. Cis Moel o illustate the oose moel, we consie that the ouce sulies five eliveies to the istibuto. Distibuto in tun sulies six eliveies to the etaile in each of the elenishments he gets fom the ouce. We assume the ouction ate is P = 0000 units e yea an the total eman is 000 units e yea while the ate of eteioation is 0.0 e yea. In this sequence, we consie that the oeing cost is $80, $400 e oe fo etaile an istibuto esectively an the ouction set-u cost is $8000 e ouction. We also assume that the caying costs e yea fo ouce, istibuto an etaile ae $0, $35 an $50 esectively. Similaly, the eteioation costs e unit fo the ouce, istibuto an etaile ae taken as $00, $50 an $00 esectively. We also consie that the time hoizon is finite, in aticula one yea. Using the above ata, the otimal values fo the ouction time with minimum total cost have been calculate an the esults ae tabulate in able. able : Results fo the cis moel: Q Q Q C C C C Fuzzy Moel In aition to the stuy on the moel in fuzzy envionment, the ouction an the eman ate ae consiee as the tiangula fuzzy numbes (7000, 0000, 5000) an (0800, 000, 4000) esectively, an all othe ata emain the same as in cis moel i.e. θ = 0.0, C = $ 8000, C = $ 400, C = $ 80, C = $ 0, C = $ 35, C = $ 50, C = $ 00, C = $ 50, C = $ 00, Δ = 3000, Δ = 5000, Δ 3 = 00, Δ 4 = 000. Using the above ata, the otimal ouction time with vaious costs has been calculate an the esults ae islaye in able Sensitivity Analysis A sensitivity analysis is efome fo the fuzzy moel with esect to vaious aametes. Results ae calculate an tabulate in the able 3.

16 6 C. Singh, S.R. Singh / An Integate Suly Chain Moel Δ able 3: Sensitivity analysis with esect to the vaious aametes fo the fuzzy moel: Paametes % Changes * Q C * * * Δ Δ 3 Δ 4 P OBSERVAIONS Base on the sensitivity analysis, it is obseve that the fuzzy execte cost is slightly highe than the cis total cost, while the otimal ouction time in the fuzzy sense is ecease. As a esult, the amount of economic ouction quantities ecease. he vaious obsevations ae shown below. he following obsevations have been mae uing the sensitivity analysis:. otal cost obtaine in the fuzzy sense is slightly highe than the cis total cost.. Otimal ouction length is slightly lowe than the cis cycle length. 3. It is obseve that the otimal manufactue quantity obtaine in the fuzzy sense is lage than the cis otimal manufactue quantity. 4. As Δ inceases total cost C * inceases an the otimal ouction quantity * Q = eceases. 5. As Δ inceases both the total cost C * * an the otimal ouction quantity Q inceases. As Δ 3 inceases, total cost C * eceases an the otimal ouction

17 C. Singh, S.R. Singh / An Integate Suly Chain Moel 63 * quantity Q inceases. As Δ 4 inceases total cost C * inceases an the otimal * ouction quantity Q eceases. As P inceases both the total cost C * an the otimal ouction quantity Q * inceases. As inceases total cost C * * inceases an the otimal ouction quantity Q eceases. he oveall obsevation fom able 3 is that in any case the total cost oes not vay much fom its oiginal value. his is the most istinguishe featue of the whole stuy. his fining is moe than sufficient to justify the whole fuzzification ocess. 7. CONCLUSIONS his stuy evelos an integate suly chain, multi-echelon eteioating inventoy moel in the fuzzy envionment. We have stive to evelo a suly chain moel fo the situations when items eteioate at a constant ate, the eman an ouction ates ae imecise in natue. It is assume that the ouce sulies n elivey to istibuto an istibuto, in tuns, sulies n eliveies to etaile in each of his elenishment. In the eveloment of inventoy moels, most of the evious eseaches have consiee the ouction ate an eman ate as constant quantity. Sometimes, a situation occus when it is not ossible to ovie exact ata, o if we consie ealistic situations, these quantities ae not exactly constant, but have little vaiations comae to the actual values. With fuzzy moels, howeve, we have the avantage that, instea of oviing the exact values fo the vaiables, we ae equie to ovie a ange with the hel of membeshi functions. his le us to eveloing a moel with fuzzy ouction ate an fuzzy eman ate. Pouction an eman ates ae taken as tiangula fuzzy numbes an the membeshi function fo the fuzzy total cost is obtaine by using extension incile. he total cost, as suggeste by the fuzzy aoach, is fa moe actical an ealistic than the cis aoach an ovies a bette chance fo attainment. he sensitivity analysis shows in able 3 that the total cost oes not vay much fom its oiginal value in any case; theefoe, the eveloe moel is vey stable an omises a bette eal to the inventoy manage. Ou analysis is the fist ste. In the next ste, we will exten ou aoach an thoughts to the suly chain moels with moe innovative ieas, such as moels with uncetain lea time oblem, the moel with shotages an atially backlogging an ice iscount with iffeent eman an eteioation ates. REFERENCES [] Chang, H.,C., Yao, J.,S., an Quyang, L.Y., Fuzzy mixtue inventoy moel with vaiable lea-time base on obabilistic fuzzy set an tiangula fuzzy numbe, Comute an Mathematical Moeling, 39 (004) [] Chang, H.,C., Yao, J.,S., an Quyang, L.,Y., Fuzzy mixtue inventoy moel involving fuzzy anom vaiable, lea-time an fuzzy total eman, Euoean Jounal of Oeational Reseach, 69 (006)

18 64 C. Singh, S.R. Singh / An Integate Suly Chain Moel [3] Chen, L.H., an Kang, F.S., Integate veno-buye cooeative inventoy moels with vaiant emissible elay in ayments, Euoean Jounal of Oeational Reseach, 83() (007) [4] Chung, C.J., an Wee, H.M., Minimizing the economic lot size of a thee-stage suly chain with backoeing eive without eivatives, Euoean Jounal of Oeational Reseach, 83() (007) [5] Dutta, P., Chakaboty, D., an Roy, A.R., Continuous eview inventoy moel in mixe fuzzy an stochastic envionment, Alie Mathematics an Comutation, 88 (007) [6] Kim, S.W., an Pak, S., Develoment of a thee-echelon SC moel to otimize cooination costs, Euoean Jounal of Oeational Reseach, 84(3) (008) [7] Lee, H.., an Wu, J.C., A stuy on inventoy elenishment olicies in a twoechelon suly chain system, Comutes an Inustial Engineeing, 5() (006) [8] Maity, M.K., an Maiti, M., wo-stoage inventoy moel with lot-size eenent fuzzy lea-time une ossibility constaints via genetic algoithm, Euoean Jounal of Oeational Reseach, 79 (007) [9] Rau, H., an Ouyang, B.C., An otimal batch size fo integate ouctioninventoy olicy in a suly chain, Euoean Jounal of Oeational Reseach, 95() (008) [0] Singh, S.R., Singh, C., an Singh,.J., Otimal olicy fo ecaying items with stock-eenent eman une inflation in a suly chain, Intenational Review of Pue an Alie Mathematics, 3() (007) [] Wee, H.M., Otimal buye-selle iscount icing an oeing olicy fo eteioating items, he Engineeing Economist Winte, 43() (998) [] Wu, M.Y., an Wee, H.M., Buye-selle joint cost fo eteioating items with multile-lot-size eliveies, Jounal of the Chinese Institute of Inustial Engineeing, 8() (00) [3] Wu, Z., an Choi,.Y., Sulie-sulie elationshis in the buye-sulie tia: Builing theoies fom eight case stuies, Jounal of Oeations Management, 4(5) (005) 7-5. [4] Yang, P.C., an Wee, H.M., Economic oe olicy of eteioate items fo veno an buye: An integal aoach, Pouction Planning an Contol Management, 6(6) (000) [5] Yang, P.C., an Wee, H.M., An Integating multi lot size ouction inventoy moel fo eteioating item, Comutes an Oeations Reseach, 30 (003)