A Fuzzy Approach to Co-Ordinated and Non Co-Ordinated Two Stage Supply Chain

Transcription

1 Internationa Journa of Pure and Appied Mathematica Sciences. ISS Voume 0, umber (07), pp. -3 Research India Pubications A Fuzzy Approach to Co-Ordinated and on Co-Ordinated Two Stage Suppy Chain Tahseen Jahan A. and *, Maragatham, M. Department of Mathematics, Justice Basheer Ahmed Sayeed Coege for Women, Chennai, India. Department of Mathematics, Periyar EVR Coege, Tiruchy, India. Abstract This research paper focuses on the fuzziness factor of demand in a two-stage suppy chain network. The genera fuzzy number technique is appied to exhibit customer demand. This paper expores the optimization of the verticay integrated two stage suppy chain under rationa co-ordination between the whoesaer and retaier of any business firm. Even the noncoordination case is aso discussed. A numerica exampe is iustrated to exhibit the optimum profit performance of both cases. Keywords: Suppy chain, Fuzzy umber, Whoesaer, Retaier, Co-ordination Fuzzy mode.. ITRODUCTIO To tacke uncertainty of different integrities of suppy chain design and panning, fuzzy approach is appied to traditiona suppy chain modeing giving emphasis on the randomness factor of uncertainty. Pethora of probabiistic modeing techniques has been evoved to tacke uncertainty but fuzzy modeing has been effectivey appied in suppy chain design and panning to yied best optimum resuts. In actua suppy chain design and panning, one of the most compex issues encountered by the decision makers is to forecast the market demand. In the recent past, extensive research endeavours have been focused to surmount this chaenge.

2 Tahseen Jahan A. and Maragatham, M. The avaiabe research in the iterature mosty expains the modes from a probabiistic aspect, in which a premise is that requisite data is not avaiabe to make a reasonabe concusion to be made to the demand distribution and aso past data are not aways accessabe or pertinent due to market voatiity or technoogica growth. Diminished product ife cyces aong with escaating innovation rather make the demand extremey fuctuating and needed statistica data is not forthcoming is today s competitive word. At this juncture standard probabiistic modeing that uniquey appies a repeated approach may not be the right choice. Thus an aternate presentation of uncertainty is required. One method is appying subjective probabiities that represent the degree of beief of the decision maker yet another approach that is appied here is fuzzy set theory, where the information avaiabe for vaues of market demand is interva- vaued. In the area of suppy chain modeing with fuzzy demand there have been few research studies. Wang and Shu [] deveoped fuzzy suppy chain mode by using six-point fuzzy numbers [3] to represent the fuctuating customer demand, uncertain processing time and unreiabe suppy deivery. They appied the genetic agorithm approach to determine the optima stock order-up-to-eves, and the simuation approach to vaidate the deveoped concept. Recenty, Aiev et a. [] deveoped a fuzzy integrated muti-period and muti-product production and distribution mode in suppy chain, which is formuated in terms of fuzzy programming and the soution is provided by genetic agorithm. In a these papers isted above approximate agorithms were adopted to derive the soution of their modes such as the simuation techniques used in [4, 9, 0] and the genetic agorithm in [, ]. In our proposed mode we use genera fuzzy number, namey, Left-Right (LR) -type fuzzy number, to describe the estimate for the externa demand, and deveop the decision modes to study the coordination probem for two stage suppy chain. The cosed form soution for the optima profit of the verticay integrated two stage suppy chain is obtained. The reationships of a whoesaer and retaier under both coordination and non-coordination scenarios, are considered and we prove that the maximum suppy chain profit in the coordination situation is greater than that in the non-coordination situation. In order to share the obtained profit, we give a simpe scheme such that both payers can benefit from the coordination. Finay, numerica resuts are presented to demonstrate the performance of our techniques. PRELIMIARIES Basicay most suppy chain probems do not have enough historica data to construct the probabiity distribution hence in this paper we deveop the decision modes for singe period two stage suppy chain which consists of the whoesaer and retaierand

3 A Fuzzy Approach to Co-Ordinated and on Co-Ordinated Two Stage Suppy Chain 3 we aso enumerate the method to maximise their suppy chain profit in the case of coordination and non-coordination of their business deas to overcome the aspect of demand uncertainty. We use genera fuzzy number namey LR- type fuzzy number to estimate the externa demand. Figure : LR type fuzzy number DEFIITIO An LR-type fuzzy number D can be described with the foowing membership function[8]: L(x) if d x<m if x [m,m r] D R(x) if m r<x dr 0 otherwise () where[ m, m r] is the range of most ikey vaues of D ; m and m r are the ower and upper moda vaues ; Lx ( ) and R(x) are the increasing and decreasing continuous functions respectivey. The pictoria representation of the above membership function is fig. The cosure of the support of D is exacty [ d, d ] and the eve set of D can be denoted as D [L ( ),R ( )], [0,], which is the cosed interva on rea number set R. In order to measure the mean vaue of a fuzzy number D,we use the fuzzy mean introduced by Dubois and Prade[0]:

4 4 Tahseen Jahan A. and Maragatham, M. Where * * E ( D) E ( D) d ( ) d ( ) E( D) d () * [ *( ),E ( )] 0 E D D is the interva-vaued expectation, and [ d( ), d( )] is the eve set of D. OTATIOS AD ASSUMPTIOS In this proposed paper a singe period two stage suppy chain, namey the whoesaerand retaier suppy chain setting is considered.demand is fuzzy.to start with, the whoesaer fixes his unit price and then the retaier his unit price and then makes his own repacement poicy to maximise his own profit.the whoesaer s profit wi depend on the retaier s ordering quantity and in turn the retaier s profit depends on the consumer demand.foowing are the notations used in our mode: p - Whoesaer s unit price c - Whoesaer s unit production cost w - Retaier s unit price h - Retaier s unit hoding cost s - Retaier s unit shortage cost q - Retaier s order quantity D - Fuzzy demand FORMULATO OF THE PROBLEM Let the retaier s fore-casted demand be D ( d, m, m, d ) r r LR then for order quantity q units the saes voume, hoding and shortage quantity is denoted as min{ q D} max{ q D,0} and max{ D q,0} respectivey, by Zadeh sextension principe. Hence the retaier s tota profit Rq ( ) becomes R( q) wmin{q, D} h max{q D,0} s max{d q,0} pq (3) As D is LR - fuzzy number, so is Rq ( ) we have to sove the optimisation probem to obtain the optima order quantity q so as to get the maximum profit. Three cases arise for the optimisation probem max E( R( q)) which are as foows:- q0

5 A Fuzzy Approach to Co-Ordinated and on Co-Ordinated Two Stage Suppy Chain 5 Case : d q m In this case the - eve sets of fuzzy saes voume,hoding and shortage quantity can be respectivey expressed as - [L (λ),q] for 0< L(q) (min{q,d}) [q,q] for L(q)< - [0,q-[L (λ)] for 0< L(q) (max{ qd,0}) [0,0] for L(q)< (max{ Dq,0}) - [0,R (λ)-q] for 0< L(q) - - [L (λ)-q,r (λ)-q] for L(q)< Simiary the - eve sets of R( q ) are cacuated as foows: If 0 Lq ( ),then R( q ) = wmin(q, D) h max(q D,0) s max(d q,0) pq = w[l ( ), q] h[0, q L ( )] s[0, R ( ) q] [ pq, pq] = [( w h)l ( ) sr ( ) ( h s p) q.( w p) q] If Lq ( ),then R s q R q pq pq (q) w[q,q] h[0,0] [L ( ), ( ) ] [, ] = [(w s p)q s R ( ),(w s p)q sl ( )] It foows that the fuzzy mean of R (q) by (), is L(q) ( (q)) [(w h)l ( ) s R ( ) (h s p)q (w p)q]d E R 0 [(w s p)q s R ( ) (w s p)q sl ( )]d L (q) L(q) [(w h)l ( ) s R ( )]d 0 R q w h s ql q L(q) (s ( ) L ( ))d (w s p) ( ) ( ) Find the first and second derivatives of ER ( (q)) with respect to q as foows:

6 6 Tahseen Jahan A. and Maragatham, M. de( R(q)) ( w s p) (w h s) L( q) (4) dq d E R Using first order conditions, we have Ceary ( w s p ) (w hs) ( (q)) (w h s ) L ( q ) (5) dq ( w s p) Lq ( ) (w h s) 0 Since w p So, if ( w s p ) (w hs), namey w s (6) h p Then we have q ( w s p) L (w hs) (7) d E( R(q)) On the other hand, if dq 0 since ( ) Lq is increasing in[ d, m ]. Then the function maximum vaue is ER ( (q)) reaches its maximum at q if w s h p, and the ( wsp) (w hs) E( R(q)) ( w h) L ( )d s R ( )d L ( )d 0 0 ( wsp) (w hs) Case: m q mr For q ying in the range of (8) m and mr we have (min{ q, D}) [L ( ), q] (max{,0}) [0, L ( )] q D q [0, ) q] (max{d q,0}) R ( It foows that the eve set of retaier s fuzzy profit: R(q) wmin{q, D} h max{ q D,0} s max{ D q,0} pq = w q h q s R [L ( ), ] [0, L ( )] [0, ( ) q] pq = [(w h)l ( ) s R ( ) (s h p)q,(w p)q]

7 A Fuzzy Approach to Co-Ordinated and on Co-Ordinated Two Stage Suppy Chain 7 ow, the fuzzy mean of retaier s profit is given by And the first derivative ( (q)) [( )L ( ) ( )]d ( ) ] E R w h sr w s h p q 0 de( R(q)) (w s h p ) (9) dq If w s h p 0, then ER ( (q)) gets its maximum at q mr ;if w s h p 0, then ER ( (q)) reaches its maximum at q m ; and if w s h p 0, then ER ( (q)) reaches its maximum for any q [ m, m ] Case 3: mr q dr r Hence again, the - eve sets of fuzzy saes voume, hoding and shortage quantity can be respectivey expressed as - [ L ( ), q] for 0< R(q) min{q,d}) = [L ( ), R ( )] for R(q)< - [0,q L ( )] for 0< R(q) (max{q D, 0}) = [ q R ( ),q L ( )] for R(q)< - [0, R ( ) q] for 0< R(q) (max{d q,0}) = [0, 0] for R(q)< Simiary the - eve sets of R(q) are cacuated as foows: - [(w h)l ( ) sr ( ) (s h p)q,(w p)q] for 0< R(q) R (q) = (0) [(w h)l ( ) (h p) q,( w h) R ( ) (h p)q] for R(q)< ow the fuzzy mean of R(q) is cacuated as foows: ER ( (q)) R(q) [(w h) L ( ) s R ( ) (w s h p) q]d 0 (w h)(l ( ) R ( )) (h p) q] d Rq ( ) R( q) ( w h)l ( )d sr ( ) d (w h) R ( ) d 0 0 R( q)

8 8 Tahseen Jahan A. and Maragatham, M. ( h p) q ( w h s) qr( q) () The first and second order derivatives of ER ( (q)) with respect to q are as foows: de( R(q)) (w h s ) R ( q ) ( h p ) () dq d E R ( (q)) (w h s ) R ( q ) 0 (3) dq If de( R(q)) 0 then we get dq (h p) R(q r ) (4) w h s (h p) If, namey, w s h p, then the Retaier optima order quantity wi w h s be as foows: q r (h p) R (w hs) And the retaier s optima profit is (5) (h p) whs E( R( qr )) ( w h) L ( ) d R ( ) d s R ( ) d 0 (h p) 0 whs Combining a the 3 cases we have the foowing inferences for optima order quantity and its corresponding optima profit of the Retaier which is as foows: q* - (w+s-p) L if w+s h p w+h+s [m,m r ] if w+s = h p - h+p R if w+s > h p w+h+s (6)

9 A Fuzzy Approach to Co-Ordinated and on Co-Ordinated Two Stage Suppy Chain 9 ( wsp) whs ( ) ( )-s ( ) ( )d if w+s w h L R d L h p 0 0 ( wsp) whs [(w+h)l - ( )-sr - E( R( q*)) ( )]d if w+s = h p 0 h p whs ( ) ( )d ( )d w h L R s R ( ) d if w+s > h p 0 h p 0 whs (7) Case of non-coordinated suppy chain Both the whoesaer and the retaier are independent in their decisions of how much to pace the orders and fix their costs or profits accordingy once they take independent decisions the whoesaer has to suppy the order quantity required by the retaier and in that case first the whoesaer s profit is cacuated as foows: E (W( q*)) ( p c) q* - (w+s-p) (p-c)l if w+s h p w h s (p-c)[m,m r ] if w+s = h p - h+p (p-c)r if w+s > h p w h s (9) The corresponding whoe suppy chain profit is T ( q*) E( R( q*)) E (W( q*)) (0) Case of coordinated suppy chain In this case the whoesaer and retaier coordinate and jointy discuss their demands and requirements so as to maximise their combined profit and hence in this scenario the externa demand, whoesaer unit production cost, retaier s hoding and penaty cost is same for both of them hence the whoe suppy chain profit is now cacuated as foows: qt - (w+s-c) L if w+s hc w h s [m,m r ] if w+s = h c - h+c R if w+s > h c w h s ()

10 0 Tahseen Jahan A. and Maragatham, M. Corresponding optima profit is ( wsc) whs ( ) ( )d -s ( ) ( )d if w+s w h L R d L h c 0 0 ( wsc) whs [(w+h)l - ( )-sr - Ec(W( qt)) ( )]d if w+s = h c 0 hc whs ( ) ( )d ( )d w h L R s R ( ) d if w+s > h c 0 hc 0 whs In continuation to the above two cases discussed, we can aso enumerate a sharing scheme between the whoesaer and the retaier to get the maximum tota profit as foows: Let E( R( q*)) A T (q*), E (W( q*)) B where 0<A,B< and A+B=. T (q*) Ceary we have A( Ec( W( qt)) E(R( q*)) and B( Ec( W( qt)) E ( W( q*)) Hence in our inferences we wi be concuding that in a verticay integrated two stage suppy chain the totaprofit forecasted in coordinated case is greater than that in the non-coordinated case which we wi prove in the foowing numerica exampe umerica Iustration Let us assume a eectronics whoesae deaer forecasts the demand for its rechargeabe battery components which he represents as trapezoida fuzzy number as D (45000,50000,60000,65000) and the a the costs per unit are given by w 00, h 0,s 80,c 30 In the non- coordinated case, suppose the whoesaer sets the its price as p 60then the retaier s optima order quantity is ( h p) 4 w h s 5 q* R R 6000 And the optima fuzzy mean profit () ( h p) whs s E( R( q*)) ( w h) L ( ) d R ( ) d R ( ) d,560,000 (3) 0 ( h p) 0 whs

11 A Fuzzy Approach to Co-Ordinated and on Co-Ordinated Two Stage Suppy Chain Since w s h p. Correspondingy, the whoesaer s profit is E ( W( q*)) ( p c) q*,830,000 (4) Therefore, the whoe suppy chain profit under non-coordination situation is T ( q*) E( R( q*)) E ( W( q*)) 3,390,000 (5) On the other hand, in the coordination situation, the optima order quantity and corresponding optima suppy chain profit can be computed and (q T ) 65,000 E ( W( q )) 3,4,500 c T According to the sharing scheme, the retaier and the whoesaer can obtain their respective profit as foows: and E( R( q*)) AEc ( W ( qt )) Ec ( W ( q T )),570,400 T ( q*) (6) E( W ( q*)) BEc ( W ( qt )) Ec ( W ( q T )),84,00 T ( q*) (7) Tabe Optima Soutions. Order quantity Retaier s profit Whoesaer s profit Tota profit on-coordination Coordination Tabe The Effects of Demand Fuzziness. Order quantity Retaier s profit Whoesaer s profit Tota profit on-coordination Coordination

12 Tahseen Jahan A. and Maragatham, M. It is obvious that the channe s profit is increased by coordination and both payers get more profit in the coordination situation than in the non-coordination situation. The resuts are summarized and reported in Tabe ow we decrease the fuzziness of demand and observe its effects. Suppose that the fuzzy demand is D (45000,50000,60000,6000) and other parameters remain as before. The resuts obtained by using our fuzzy modes are given in Tabe Comparing the resuts presented in Tabe and, we can see that a retaier s profit wi increase when the demand fuzziness decreases. The reason is that the ess demand uncertainty, the ess hoding and shortage cost the retaier has to pay, and the more profit he obtains. We aso note that when demand fuzziness decreases the whoesaer s profit drops sighty in the non-coordination situation. This is because the whoesaer cannot adjust the whoesae price quicky in response to the change of the demand uncertainty. However the whoesaer sti obtains more profit once he shares more market information and cooperates with the retaier. COCLUSIO This study takes into account the subjective estimation of both the whoesaer and retaier s decisions and judgements in which their demand is fuzzy rather than stochastic. Genera fuzzy number is appied to exhibit the estimate for the externa demand and evove for the decision modes to determine the optima profit for both retaier and integrated suppy chain. The cosed form soutions for both modes are got and can be specified to meet non- fuzzy situation when the fuzzy demand in our modes is crisp rea number. Based on the cosed form soutions, both noncoordination and co-ordination situations are evauated and it is exhibited that coordination enhances the anticipated profit. The merit of the cosed form soutions is that they remove the need for enumeration over aternative vaues and carify the reations among the mode parameters. In comparison to that of traditiona probabiistic approach, the proposed mode requires imited data to predict the uncertain demand and can make use of the subjective assessment on the basis of decision maker s perception, experience and judgement. In view of this the proposed methodoogy provides an aternative for suppy chain panning under uncertain situations. It is apt when the scenario is voatie and there is absence of past records. REFERECES [] Arnod, B.F., Hauenschid,., and Stahecker, P., Monopoistic price setting under fuzzy information Eur.J.Oper.Res.54(004)

13 A Fuzzy Approach to Co-Ordinated and on Co-Ordinated Two Stage Suppy Chain 3 [] Aiev, R.A., Fazoahi, B., Guirimov, B.G., and Aiev, R.R., Fuzzy-genetic approach to aggregate production-distribution panning in suppy chain management, Inform. Sci. 77 (007) [3] Fortemps, P., Jobshop scheduing with imprecise durations: A fuzzy approach, IEEE Trans. Fuzzy Syst. 5(997) [4] Giannoccaro, I., Pontrandofo, P., and Scozzi, B., A fuzzy echeon approach for inventory management in suppy chain. Eur. J. Oper. Res.49(003) [5] Harrison, T.P., Principes for the strategic design of suppy chains in: T.P Harrison, Where Theory andappication Converge, Kuwer Academic Pubishing,003 [6] Hassan Shavand, Fuzzy set theory and its appication in industria engineering and management, i, the pubication [7] Liou, T. and Wang, M.J., [99b], Ranking fuzzy number with integra vaue: fuzzy Sets and Systems, 50(3), pp [8] Min, H., Zhou, G., suppy chains modeing: past, present and future computing, Eng,43(00) [9] Petrovic, D. Roy, R., and Petrovic, R., Suppy chain Modeing using fuzzy sets.int. J. Prod. Econ. 59(999) [0] Petrovic, D., Roy, R., and Petrovic, R., Modeing and simuation of a suppy chain in an uncertain environment, Eur. J. IOper.Res. 09(998) [] Wang, J., and Shu, Y., Fuzzy decision modeing for a suppy chain management, fuzzy sets syst.50(005) 07-7 [] Yano, C.A., Gibert, S.M. coordinated pricing and production /procurement decisions: a review in: A.K. Chakravarty.J. Eiashberg (Eds) Managing Business Interfaces, Kuwer Academic Pubishing 004. [3] Zadeh, L.A., Fuzzy sets, inform.contr.8(965) [4] Zadeh, L.A., Fuzzy sets, Information and Contro, 8, ,968. [5] Zimmermann, Fuzzy set theory and its appication, Boston: Kuwer Academic Pubisher, 99.

14 4 Tahseen Jahan A. and Maragatham, M.