OPTIMIZING PRODUCTION POLICIES FOR FLEXIBLE MANUFACTURING SYSTEM WITH NON-LINEAR HOLDING COST

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1 OPIMIZING PRODUCION POLICIE FOR FLEXIBLE MANUFACURING YEM WIH NON-LINEAR HOLDING CO ABRAC Leena Praher, Reearch cholar, Banahali Vidayaeeh (Raj.) Dr. hivraj Pundir, Reader, D. N. College, Meeru (UP) hi aer deal wih roducion invenory model for a deerioraing iem over infinie lanning horizon where he demand rae i a funcion of on hand invenory, roducion rae i a linear funcion of demand and hu non-linear funcion of on hand invenory. he radiional arameer of uni iem co and ordering co are ke conan and comaraive udy of linear and non- linear holding co i dicued o oberve he effec of non-lineariy of holding co in flexible manufacuring yem. Keyword: Producion olicy, Non-linear holding co, Flexible manufacuring yem INRODUCION In he reen conumeri ociey and a cu-hroa comeiion in he marke, he manufacurer are no only emloying newer mehod of diribuion bu alo newer forma of diribuion. he comanie are enering rural marke, emi-urban area and reaching ou o he unexeced egmen of oenial cuomer. In addiion o generaing a ur in demand, he comanie are uing innovaive markeing raegie and innovaive markeing acic wih varying degree of effecivene. A far a diribuion i concerned, new dearmenal ore and new hoing mall are rouing u even in he unrereened geograhical area. A a reul of all hi, he viibiliy and reach of he brand/roduc ha increaed manifold and i caue udden flucuaion in demand. A uch here i a rong need for a flexible manufacuring yem ha can ake care of he above realiie and adju ielf o he realiie of he marke. LIERAURE REVIEW In he claical economic roducion lo ize (EPL) model, he manufacuring yem i regarded o be inflexible. In recen year, flexibiliy in he roducion yem ha become an imoran aec of he invenory model a hey rereen more realiic and racical iuaion. Advancemen in he echnology and inan communicaion ha ranformed he marke mechanim and lace remendou reure in making he availabiliy of roduc inananeouly and imely. A er he emerging marke rend, organizaion end o arac cuomer by dilaying he iem in large amoun and in influenial manner a we oberve in big mall and ore. he aracive dilay of iem in large amoun and in more varieie influence and moivae he eole o buy more. he endency of buying more, reul in greaer demand and hu he demand i affeced by he ock dilayed. he changing rend in demand affec he roducion rae. o ui he recen marke rend, reearcher are being araced oward volume 114

2 Preige Inernaional Journal of Managemen & I- anchayan, Vol. (1), 013. IN: (Prin), (Online) flexible invenory yem in which roducion rae i a deciion variable and deend on variou facor. Gua and Vra (1986) were he fir o develo model for ock deenden conumion rae. Mandal and Phaujdar (1989) develoed an economic roducion quaniy model for deerioraing iem wih conan roducion and ock deenden conumion rae. Daa and Paul (1990) focued on he analyi of he invenory yem which decribe he demand rae a a ower funcion deenden on he level of on hand invenory and conan holding co. chweizer, P.J. and eidmann (1991) eablihed oimizing roceing rae for volume flexible manufacuring yem. Khouja and Meraj (1995) exended he EPL model wih variable roducion rae and imerfec roducion. Giri and Chaudhary (1997) develoed deerminiic model of erihable invenory wih ock deenden demand and non-linear holding co. Wee (1999) develoed he deerioraing invenory model wih quaniy dicoun, ricing and arial back-ordering. Mandal and Maii (1999) conidered he invenory of damageable iem wih variable relenihmen rae and ock deenden demand and ome uni in hand. kouri (000) rooed an invenory model for deerioraing iem wih ime varying demand and arial exonenial ye backlogging. kouri and Paachrio (003) enlighened he conce of oimal oing and rearing roducion ime for an economic order quaniy model wih deerioraing iem and ime deenden arial backlogging. ana and Chaudhary (004) develoed a volume flexible roducion model for deerioraing iem wih ime deenden demand, horage, and ock deenden demand. eng and Chang (005) rooed economic roducion model for deerioraing iem wih rice and ock deenden demand. ana (007) develoed volume flexible invenory model wih imerfec roducion roce. ana (010) develoed muli-iem EOQ model of deerioraing iem wih demand influenced by enerrie iniiaive. ana (011) exended an EOQ model for alemen iniiaive wih ock and rice eniive demand. arkar and Moon (011) develoed an EPQ Model wih inflaion in an imerfec roducion yem. ingh (010) develoed a uly chain model wih ochaic lead ime under imrecie arially backlogging and fuzzy ram-ye demand for exiring iem. ingh (010) conribued on an invenory model for deerioraing iem wih horage and ock-deenden demand under inflaion for wo-ho under one managemen. AUMPION AND NOAION he following Aumion and Noaion are ued o develo he Model: AUMPION 1. Producion rae i linear funcion of demand.. he ime horizon of he invenory yem i infinie. Only a yical lanning chedule of lengh i conidered, all remaining cycle are idenical. 115

3 Oimizing Producion Policie For Flexible Manufacuring yem Wih Non-Linear Holding Co 3. he demand rae i deerminiic and i a known funcion of he on hand invenory q. he funcional relaionhi beween he demand rae f(q) and he invenory level q() i given by he following exreion F(q) = Dq, D>0, 0 1, q > 0 Where denoe he hae arameer and i a meaure of reonivene of he demand o change in he level of on hand invenory and D denoe he cale arameer. 4. A conan fracion, aumed mall, of he on hand invenory ge deerioraed a er uni ime. NOAION q() : on hand invenory level a any ime. f(q) : Demand rae, f(q) = Dq, D>0, 0 1, P : Producion rae, P = l f(q) where l i a cale arameer, P > f(q), l >I K : Ordering co er order. HC : Holding co er cycle DC : Deerioraion co er cycle. AC : oal average relevan invenory co er uni ime. : ime when invenory reache i maximum. c : Producion co er iem. h : Invenory holding co er uni er uni ime MAHEMAICAL MODEL - A A he beginning of each cycle, he invenory ock i zero and roducion ar a he very beginning of he cycle. A roducion coninue invenory begin o ile u coninuouly afer meeing demand and deerioraion. Producion o a. he accumulaed invenory i ju ufficien enough o accoun for demand and deerioraion over he inerval [, ]. Each cycle coni of wo age: he inananeou ae of q() over he cycle ime i given by he following fir order non-linear differenial equaion. dq l 1 f q q 0, q 0 0 (1) d dq q f q, q Q () d olving equaion (1) & Uing iniial condiion q(0) = 0 116

4 Preige Inernaional Journal of Managemen & I- anchayan, Vol. (1), 013. IN: (Prin), (Online) log 1 q l1 D Where 1 On exanion of he righ hand ide, he fir order aroximaion of θ give q q l D l D 1 1, 0 (3) Aume be he ime when invenory reache i maximum level (ay Q) Q Q 1 l D l D q 1 1 olving equaion () & uing condiion Q D D q e e Q e Where =1-β 0< <1 Uing q( ) = 0, he ime o comlee cycle i given by = Q 1 Q D D, (4) Q Q 1 l D l D where 1 1 (5) o make a comaraive udy, boh cae: (i) linear holding co and (ii) non linear holding co, have been dicued. CAE 1 When a holding co i conidered a linear funcion of on hand invenory: Holding co in inerval [0, ] = h qd (6) Making ue of (3) in (6) and alo he boundary condiion, we have h qd = h Q Q l D l D, ] imilarly, finding holding co in inerval [ hc = h qd Uing exanion, he fir order aroximaion of θ in (4) give, (7) 117

5 Oimizing Producion Policie For Flexible Manufacuring yem Wih Non-Linear Holding Co = h qd Q h D 1 Q q Q q D D Q D 1 oal holding co in comlee cycle [0, ] uing (7) & (8) h Q Q Q Q h HC = l 1 D 1 l 1 D 1 D 1 D 1 (9) oal deerioraion co in comlee cycle [0, ] DC = c qd o = c qd qd o (10) he oal average invenory co er uni ime i, herefore given by K HC DC AC = Our roblem i o deermine he ime o o he roducion when q ake i maximum value Q, which minimize AC of he invenory yem. he neceary condiion for AC o be a minimum i d dq AC 0 d d d dq dq dq Which give HC DC K HC DC 0 ubiuing he value of, HC &DC uing (5), (9) & (10) in (11), we have Q 1 Q Q Q 1 Q h c Q D D l 1 D l 1 D D D = Q Q Q Q d K h c l 1 D 1 l 1 D 1 D 1 D 1 dq Where d dq Q D l l 1 Conidering ecial cae a θ 0 Q Q Q (8) (11) l 1 D D D l 1 D D Q Q 1 1 1

6 Preige Inernaional Journal of Managemen & I- anchayan, Vol. (1), 013. IN: (Prin), (Online) 1 KD ( 1)( l 1) Q hl & Ql ( l1) D 1 hlq k ( l 1) D( 1) AC = Conidering ecial cae a 0, 1, we have KD l 1 Q hl A l increae, roducion occur a a more raid rae. Hence for large l, he model hould aroach he inananeou delivery iuaion of he EOQ model. For large l l ; hu a l increae oward infiniy; he oimal run ize for he model aroache he EOQ. CAE II Non-linear ock deenden holding co; when holding co i conidered a a ower funcion of he on hand invenory: n HC hq d O n hq hq d (1) n d O Making ue of (3) & (4) in (1) oal holding co in comlee cycle [0, ] n n n n h Q Q Q Q h n k( l 1) D( n ) & Q nhl = l 1 D n l 1 D 1 D n D n A n 1 holding co of cae II aroache ha of cae I Q * and he correonding value of cycle ime and AC can be deermined numerically. Numerical examle 119

7 Oimizing Producion Policie For Flexible Manufacuring yem Wih Non-Linear Holding Co In hi ar we reen comuaional reul obained uing Mahemaica 7.0 which give inigh abou he behavior of oimal run ize Q *, roducion cycle ime and he oal average co AC. he arameer value in boh he model are aken a D=.0, C=$10.0 er uni, h=$0.5 er uni, K=$00erorder, and θ= 0.00 able 1 reen he effec of he hae arameer β for variou value of l on he aroximae oimal oluion and able reen he effec on oimal oluion when holding co i aken non linear. Following obervaion are made from able1: For a aricular β, generally decreae bu Q and AC increae a l increae. A β increae Q and AC alo increae bu decreae. Following obervaion are made from able1: For a aricular β and aricular n, generally decreae bu Q and AC increae a l increae. For a aricular β and aricular l, Q and decreae while AC increae a he degree of non lineariy in he holding co increae. Comaraive obervaion: AC i much higher when he nonlinear ock-deenden holding co i included in he invenory model. Q and are much le in he invenory yem wih non linear holding co a comared o invenory yem wih linear holding co. ABLE 1: Effec of =1-β and l on (Q,, AC) for cae I l /α Q AC Q AC Q AC ABLE : Effec of =1-β, l and n on (Q,, AC) for cae II l = n/α Q AC

8 Preige Inernaional Journal of Managemen & I- anchayan, Vol. (1), 013. IN: (Prin), (Online) 4 Q AC l =4 n/α Q AC Q AC MODEL B: horage are allowed. Along wih he aumion in Model A, if we add one more aumion ha horage are allowed. Each cycle now coni of four age. he iniial ock in each cycle i zero and roducion ar a he very beginning of he cycle. A roducion coninuou, invenory begin o build u coninuouly afer meeing demand and deerioraion. Producion i oed a a cerain ime. he accumulaed invenory i conumed and ulimaely become zero due o demand and deerioraion. Producion doe no rear a hi age and invenory horage coninue o accumulae for ome ime. hereafer, roducion ar and horage are gradually cleared afer meeing curren demand. he cycle end wih zero invenorie. NOAION: Along wih reviou noaion : Maximum horage level : ime when invenory reache i maximum level C : horage co er cycle : ime when horage ar m : ime when maximum horage occur : ime of comleion of cycle Mahemaical model B: Forward manufacuring yem he four age are rereened by four differenial equaion below: (1b) dq q l 1Dq, d q 0 0, 0 (b) dq q Dq, d q () = Q, (3b) dq Dq, q 0, m d (4b) dq q l 1Dq, q m, d m 11

9 Oimizing Producion Policie For Flexible Manufacuring yem Wih Non-Linear Holding Co (1b) and (b) ha already been olved in Model A. olving (3b) & (4b) and uing boundary condiion, he four oluion are a below: (1b) = 1 1 (b) q q, l D l D 0 D D q e e Q e, q D, m l 1 D l 1 D m q e m (3b) (4b) 1, he cycle coni of four age; ime for each age and he cycle ime have been calculaed in he ame manner a in model A and are a below: i he ime when roducion i oed & invenory reache i maximum. Q Q 1 l D l D 1 1 i he ime when horage ar & accumulaed invenory level i nil. m Q Q 1 D D i he ime when maximum horage occur & again roducion i ared; m = m D 1 l D 1l D 1 Which rereen comlee cycle ime. Now we calculae he variou aociaed co, holding co for he new invenory model remain ame a i wa in model A a here i no invenory during he new added eriod of horage [, ]. o find co of horage in inerval [, m ] Uing (3b) and boundary condiion, we obain horage co in [, m ]

10 Preige Inernaional Journal of Managemen & I- anchayan, Vol. (1), 013. IN: (Prin), (Online) c = m c qd = 1 D 1 c (13) Again roducion ar a = m and horage are comleely backlogged afer meeing he curren demand and deerioraion. horage co in inerval [ m, ] c c qd m 1 1 l1 D 1 l1 D 1 Adding (13) & (14) he oal horage co in he comlee cycle [0, ] are a below C c l c Dl l D Deerioraion co in [ m, ] c qd m c 1 1 l1 D 1 l1 D 1 he oal invenory co er uni ime i herefore given by K HC DC C AC Q, AC (Q,) i a funcion of Q & & our roblem i o find he ime o o he roducion when q ake maximum value Q and o find he ime o again ar he roducion when maximum horage accumulae. he neceary condiion for exreme value of AC (Q, ) are: Q 0, 0 AC AC (14) (15)

11 Oimizing Producion Policie For Flexible Manufacuring yem Wih Non-Linear Holding Co Uing CU 0 C DC K HC DC C 0 (16) Where, Q 1 Q Q 1 Q 1 l 1 D l 1 D D D D 1 l D 1 l D ubiuing all he relevan value in (16), we have Q Q Q Q K h c l 1 D 1 l 1 D 1 D 1 D l c Dl 1 1 l1 D 1 c l1 D 1 l1 D D 1 l D 1 l D 1 l D 1 l D 1 1 Q 1 Q Q 1 Q 1 l 1 D l 1 D D D D 1 l D 1 l D c c c c D l 1 D l 1 D l 1 D l 1 D Conidering he ecial cae when 0, 1 and ubiuing all he relevan value in (16), we have l hq l c l Q ( c) K l 1 D l 1 Dl 1 AC Now uing 0 Q 14

12 Preige Inernaional Journal of Managemen & I- anchayan, Vol. (1), 013. IN: (Prin), (Online) HC DC K HC DC C Q Q Q Dividing (16) & (17) and 0, 1 c hq ubiuing c = hq in (16) * KD 1 c Q 1 h l h c A l Q KDc h h c *, we ge a relaion in and Q Which i reul obained in EOQ model when horage are allowed. (17) CONCLUION In hi aricle, an invenory model wih flexible manufacuring yem, ock deenden demand rae i develoed for an infinie lanning horizon. In Model A, boh cae (i) linear holding co (ii) non-linear holding co have been dicued and oimal oluion are derived for boh he cae. In aricular, a numerical examle ha been reened o dicu he effec of non-linear holding co. I i oberved ha oal invenory co i much high when he non-linear ock deenden holding co i encounered in he invenory yem. In Model B, horage are allowed and oimal oluion i derived. he rooed aricle can be exended by including revere manufacuring, inflaion and dicouning and oher aumion. REFERENCE Boe,., Gowami, A. and Chaudhuri, K.. (1995), An EOQ model for deerioraing iem wih linear ime deenden demand rae horage under inflaion and ime dicouning. J. Oer. Re. ociey, Vol. 96, Chun-Jen Chung a, Hui-Ming Wee (011), hor life-cycle deerioraing roduc remanufacuring in a green uly chain invenory conrol yem, In. J. Producion Economic, Vol. 19, Daa,.K. and Pal, A.K. (1990), Deerminiic invenory yem for deerioraing iem wih invenory level-deenden demand rae and horage. Oearch, Vol. 7, Daa,.K. and Pal, A.K. (1991), Effec of inflaion and ime value of money on an invenory model wih linear ime deenden demand rae and horage. Eur. J. Oer. Reearch, Vol. 5, Gua, R. and Vra, P. (1986), Invenory model for ock deenden conumion rae. Oearch, Vol. 3, Himani Dem, ingh,.r. (01), A wo Warehoue Producion Model wih Qualiy conideraion, Procedia Engineering Vol. 38, Khouja, M. (1995). he economic roducion lo ize model under volume flexibiliy. Comuer and Oeraion Reearch, Vol. (5),

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