A COLLABORATIVE STRATEGY FOR A THREE ECHELON SUPPLY CHAIN WITH RAMP TYPE DEMAND, DETERIORATION AND INFLATION

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1 OPERAIONS RESEARCH AND DECISIONS No. 4 DOI:.577/ord45 Narayan SINGH* Bindu VAISH* Shiv Raj SINGH* A COLLABORAIVE SRAEGY FOR A HREE ECHELON SUPPLY CHAIN WIH RAMP YPE DEMAND, DEERIORAION AND INFLAION A supply chain systm has bn invstigatd in which a singl manufacturr procurs raw matrials from a singl supplir, procsss thm to produc finishd products, and thn dlivrs th products to a singl rtailr. h customr s dmand rat is assumd to b tim-snsitiv in natur (ramp typ) that allows two-phas variation in th dmand and production rat. Our adoption of ramp typ dmand rflcts a ral markt dmand for a nwly launchd product. Shortags ar allowd with partial backlogging of dmand (only for th rtailr), i.. th rst rprsnt lost sals. h ffcts of inflation of th cost paramtrs and dtrioration ar also considrd sparatly. W show that th total cost function is convx. Using this convxity, a simpl algorithm is prsntd to dtrmin th optimal ordr quantity and optimal cycl tim for th total cost function. h rsults ar discussd with numrical xampls and particular cass of th modl discussd brifly. A snsitivity analysis of th optimal solution with rspct to th paramtrs of th systm is carrid out. Kywords: thr chlon supply chain, ramp typ dmand, dtrioration, backlogging, inflation. Introduction hr has bn a growing intrst in supply chain managmnt in rcnt yars. h supply chain which is also rfrrd to as th logistic ntwork, consists of supplir distribution cntrs and rtailr outlts, raw matrials, work in procss invntory, as wll as finishd goods that flow btwn th facilitis. Quit a lot of rsarchrs hav shown intrst in this fild of study and many companis hav also invstd a lot capital in improving thir supply chain managmnt systm. *Dpartmnt of Mathmatics, D.N. (P.G.) Collg, C.C.S. Univrsity, Mrut (U.P.), India-5, -mails: narayansingh98@gmail.com, bindu58@rdiffmail.com, shivrajpundir@yahoo.com

2 78 N. SINGH t al. Historically, th thr ky mmbrs of th supply chain: th supplir, distributor and rtailr hav bn managd indpndntly buffrd by larg invntoris. Incrasing comptitiv prssur and dcrasing marginal profitability ar forcing firms to dvlop supply chains that can quickly rspond to customr nds and furthrmor rduc th cost of holding invntory. hrough co-ordination btwn ths mmbrs, th numbr of dlivris is drivd in co-opration with ach othr to achiv th minimum ovrall intgratd cost. Clark and Scarf [] wr th first authors to considr a multi-chlon supply chain in invntory rsarch and th assumption of a constant dmand rat is usually valid in th matur stag of a product lif cycl. In th growth and nd stag of th lif cycl, th dmand rat may b approximatd by a linar function wll. Rsh t al. [] and Donaldson [8] wr th first who studid a modl with linarly tim varying dmand. In most paprs, two typs of tim varying dmand rat hav bn considrd in th supply chain of an invntory modl: (i) Linar positiv/ ngativ trnd in dmand rat (ii) Exponntially incrasing/dcrasing dmand rat. Howvr, dmand cannot incras (or dcrss) continuously ovr tim. Hill [] proposd an invntory modl with incrasing dmand followd by a constant dmand. Aftr that, svral authors discussd tim dpndnt dmand in EOQ/EPQ (conomic ordr quantity/conomic production quantity) invntory modls, as wll as in modls of a multi-chlon supply chain invntory,.g. Goyal and Gunaskaran [] considrd an intgratd production invntory markting modl to dtrmin th conomic production quantity and conomic ordr quantity for raw matrials in a multi-chlon production systm. Rsarch into dtrioration and shortag of invntory ar bcoming mor important. his is bcaus in ral lif, dcay and dtrioration occur in almost all products, such as mdicins, fruits and vgtabls. Modls of dtriorating invntory hav bn widly studid by svral authors in rcnt yars. Ghar and Schradr [] wr th first rsarchrs to considr xponntially dcaying invntory whn th dmand is constant. Covrt and Philip [4] xtndd th modl to considr dtrioration with th Wibull distribution. W [9] drivd modl that taks into account intgration btwn th vndor and buyr and th dtrioration of itms. Wu [] invstigatd an invntory modl with a ramp typ dmand rat, Wibull distributd dtrioration rat and partial backlogging. Iida [5] considrd a dynamic multi-chlon invntory modl with non-stationary dmands. Yang and W [] analyzd a singl vndor, multipl-buyrs production invntory policy for dtriorating itms with a constant production and dmand rat. Khanra and Chaudhuri [6] proposd a quadratic tim dpndnt pattrn to diminish th xtraordinarily high rat of chang in dmand for xponntial tim dpndnt dmand. Manna and Chaudhari [8] hav dvlopd a production invntory with a ramp typ, two tim priods classifid dmand pattrn, whr th finit production rat dpnds on th dmand. Zhou t al. [] addrssd a modl of co-ordination in a two chlon supply chain with on manufacturr and on rtailr, whr th dmand for th product by th rtailr is dpndnt on th on-hand

3 Collaborativ stratgy for a thr chlon supply chain 79 invntory. Skouri t al. [] dvlopd an invntory modl with a gnral ramp typ dmand rat, th Wibull dtrioration rat and partial backlogging of unsatisfid dmand. hy discussd two cass in thir modls: according to th first thr is initially no shortag and according to th scond thr is initially a shortag. Singh and Singh [] discussd a supply chain modl with a stochastic lad tim undr imprcis partial backlogging and fuzzy ramp-typ dmand for xpiring itms. H t al. [4] dvlopd a modl for a two chlon supply chain invntory of dtriorating itms whr goods ar sold to multipl markts with diffrnt slling sasons. Singh t al. [, 4] discussd tim snsitiv dmand, a Parto distribution for dtrioration and backlogging undr a trad crdit policy. Rcntly, alizadh t al. [8] invstigatd an invntory modl for a multi-product, multi-chanc constraint, multi-buyr and singlvndor systm, considring a uniformly distributd, lot siz dpndnt dmand with a lad tim and partial backlogging. Singh t al. [5] discussd shortag in an conomic production lot-siz modl with rworking and flxibility. Galanc t al. [] analyzd a quantitativ managmnt support modl of a crtain production-supply systm including boundary conditions. Sarkar [6] xtndd an EOQ modl with tim-varying dmand and dtrioration by including discounts on purchasing costs undr th nvironmnt of dlay-in-paymnts. Sinha [7] hav solvd som dtrministic invntory modls considring a finit horizon. Goyal t al. [] discussd a production policy for amlioration/dtriorating itms with ramp typ dmand. Chung and Cardnas-Barron [7] simplifid th solution procdur for a modl with dtriorating itms undr stock dpndnt dmand and two lvl trad crdits in supply chain managmnt. Morovr, th ffcts of inflation and th changing valu of mony as tim procsss ar vital in any practical nvironmnt, spcially in dvloping countris with high inflation. hrfor, th ffct of inflation and th changing valu of mony cannot b ignord in ral situations. o rlax th assumption of no inflationary ffcts on costs, Buzacott [] and Mishra [9] simultanously dvlopd an EOQ modl with a constant inflation rat for all associatd costs. Birman and homas (977) thn proposd an EOQ modl undr inflation that also incorporatd th discount rat. Mishra [] thn xtndd th EOQ modl to tak into account diffrnt inflation rats for various associatd costs. Lo t al. [7] dvlopd a thr chlon supply chain modl with an imprfct production procss and th Wibull distributd dtrioration undr inflation with partial backlogging for th rtailr. Chrn t al. [5] proposd partial backlogging invntory lot-siz modls for dtriorating itms with fluctuating dmand undr inflation. Chung t al. [7] dvlopd an invntory modl with noninstantanous rcipt and xponntially dtriorating itms for an intgratd thr layr supply chain systm with two lvls of trad crdit. h modl proposd by th author is concrnd with th intgration btwn th supplir, manufacturr and rtailr, and taks into considration diffrnt rats of dtrioration in thr stags of supply chain. W considr ramp typ dmand and production rats in a thr chlon supply chain with partial backlogging and inflation.

4 8 N. SINGH t al.. Assumptions and symbols usd h following assumptions and notations ar considrd to dvlop th modl... Assumptions A singl supplir, singl manufacturr and singl rtailr ar considrd. h production rat P(t) is dmand dpndnt and th dmand rat d(t) is a ramp typ function of tim givn by d t, f t F, t t whr f (t) is a positiv, continuous function of t, t (, ] and dfind by bt f t a and P kd t, a, b, k h dtrioration rat is constant and dtrioratd itms ar not rpaird or rplacd during a givn cycl. Partial backlogging is allowd only for th rtailr. h partial backlog is rplnishd by th nxt dlivry. h modl considrs th ffct of inflation. Multipl dlivris pr ordr ar considrd. h planning horizon is finit and cycls during th planning horizon ar continuous. Sinc on cycl is considrd, th itms includd in th first dlivry ar mad in th prvious cycl. Supply is instantanous and a singl good is considrd... Symbols B r Q w Q m Q r fraction of rtailr s dmand backordrd inflation rat quantity raw matrials ordrd pr ordr quantity of finishd goods producd by manufacturr pr production cycl quantity rcivd by th rtailr from th manufacturr pr dlivry dtrioration rat for th raw matrial dtrioration rat for finishd goods stord by th manufactur dtrioration rat for goods stord by th rtailr

5 Collaborativ stratgy for a thr chlon supply chain 8 n th numbr of dlivris pr ordr I w (t i ) invntory lvl of raw matrials at any tim t i, whr t i i I ( t ) manufacturr s invntory of finishd goods lvl at tim t i, t i i, i =, mi i I ( t ) rtailr s invntory lvl of finishd goods at tim t i t i i, i =, 4 ri C w C m C r C w C m C r C C 4 C w C m C r i MI m MI r C w C m C r C supplir s ordring cost pr ordr cycl manufacturr s ordring and st-up cost pr ordr cycl rtailr s ordring cost pr ordr cycl holding cost for a unit of raw matrial pr unit tim manufacturr s holding cost for a unit of finishd goods pr unit tim rtailr s holding cost for a unit of finishd goods pr unit tim rtailr s backlog cost for a unit of finishd goods pr unit tim rtailr s cost for lost sals of finishd goods pr unit tim cost of raw matrials pr unit cost to manufacturr of finishd goods pr unit cost to rtailr of finishd goods pr unit manufacturr s maximum invntory lvl of finishd goods rtailr s maximum invntory lvl of finishd goods prsnt valu of supplir s total cost pr unit tim prsnt valu of manufacturr s total cost pr unit tim prsnt valu of rtailr s total cost pr unit tim prsnt valu of total cost pr unit tim. Drivation of th modl h intgratd flow of matrials is shown in Fig.. Bcaus w focus on coopration btwn th supplir, manufacturr and rtailr, thr ar two stags in our modl. h first stag is th manufacturr s production systm. h manufacturr purchass raw matrials from outsid supplirs and dlivrs fixd quantitis of finishd goods with multipl dlivris to th rtailr ovr a fixd tim intrval. Fig.. Intgratd flow of matrials

6 8 N. SINGH t al... h manufacturr s raw matrial invntory A supplir procurs th raw matrial and dlivrs fixd quantitis Q w to th manufacturr s warhous at fixd tim intrvals. h manufacturr withdraws raw matrials from th warhous. During th tim priod, th invntory lvl dcrass du to both th manufacturr s dmand and dtrioration. Fig.. Manufacturr s raw matrial invntory: a), b) > h manufacturr s raw matrials invntory from Fig. a, b at any tim t can b rprsntd by th following diffrntial quation wi t P(t) I t, t () wi with th boundary condition I wi ( i ) =. hr ar two possibl rlations btwn th paramtrs and : (i) (ii) >. Each cas implis a diffrnt ordring cost, holding cost and dtrioration cost. Lt us discuss thm sparatly. In this cas, Eq. () bcoms t w Cas I ( ) ka I t t () bt w,

7 Collaborativ stratgy for a thr chlon supply chain 8 t w b ka I t, t () w with th boundary conditions, I and I I w h solutions of Eqs. () and () ar w w. b bt t ka b ka I t b b, t b w (4) b t w, ka I t t (5) h maximum invntory lvl of raw matrials is Q w, whr Q w = I w () b ka ka w b b Q b b ka b (6) hr is an initial ordring cost at th start of th cycl. h prsnt valu of th ordring cost is givn by OR w = c w (7) Invntory is hld during th tim priod. h prsnt valu of holding cost is givn by rt rt rt HDw cwiwt cw Iw t Iwt HD c ka b w w (8) h costs includ losss du to dtrioration, as wll as th cost of th itms sold. Bcaus th ordr is carrid out at t =, th prsnt valu of itm cost is givn by I c Q c ka b (9) w w w w

8 84 N. SINGH t al. h prsnt valu of th total cost during th cycl is th sum of th ordring cost (OR w ), th holding cost (HD w ) and th itm cost (I w ). Hnc, for th raw matrial, th prsnt valu of total cost pr unit tim is givn by Cw ORw HDw Iw () Cas II ( ) In this cas, Eq. () rducs to th form; t w ka I t t () bt w, with th boundary condition Iw. h solution of Eq. () is Sinc I w w ka I t t b t bt w, b Q, thn () ka b w b Q ka b () rt w w w w HD c I t c ka b (4) I c Q c ka b w w w w (5) For th raw matrial, th prsnt valu of total cost pr unit tim is Cw ORw HDw Iw (6).. Manufacturr s systm for storing finishd goods h manufactur s invntory systm dpictd in Fig. a, b can b dividd into two indpndnt tim intrvals dnotd by and (which also dnot th lngth of ths

9 Collaborativ stratgy for a thr chlon supply chain 85 intrvals). his mthod rducs th complxity of our problm, togthr with th drivation and analysis of th solution. Each phas has its own tim t i, i =,, which starts from th bginning of th phas i. During tim priod, invntory builds up and hnc dtrioration occurs. At tim t =, production stops and th invntory lvl incrass to its maximum, MI m. hr is no production during tim priod, and th invntory lvl dcrass du to dmand and dtrioration and bcoms zro at t =. Fig.. Manufacturr s invntory systm: a), b) h manufacturr s systm for storing finishd goods at any tim t can b rprsntd by th following diffrntial quation (Fig. ) mi t mi P t d t I t, t (7) t mi d t I t, t (8) mi with th boundary conditions I m () = and I m ( ) =. hr ar two possibl rlations btwn th paramtrs and ; (Fig. a), (Fig. b). Each cas implis a diffrnt ordring cost, holding cost and dtrioration cost. Lt us discuss thm sparatly blow. Cas III ( ) In this cas, Eqs. (7) and (8) bcom

10 86 N. SINGH t al. t m t m m t bt k a I t, t (9) b m k a I t, t () m b a I t, t () m with th boundary conditions I m () =, I m ( ) = I m ( + ), and I m ( ) =. h solutions of Eqs. (9) () ar k a I t, t bt t b m b b k a k a I t b t b t m, () () b t m, a I t t (4) Basd on Fig. a, th maximum invntory lvl of finishd goods is a MI I MI a b b m m, m (5) h quantity producd in a cycl is m Q P t P t P t Qm ka b b b b (6) At th start of th cycl, th cycl has an initial production st-up cost, c m. h prsnt valu of th st-up cost is

11 Collaborativ stratgy for a thr chlon supply chain 87 SE = c m (7) Invntory is hld during th tim priod and. If this systm dos not considr th rtailr, all of th holding costs blong to th manufactur. hy ar givn by th first two trms in Eq. (8) and Eq. (9). If this systm considrs th rtailrs, th holding cost for th itms that ar dlivrd to th rtailr blong to th rtailr. hy should b subtractd from th manufacturr s costs. hy ar givn by th last trm in Eq. (8) and Eq. (9). h prsnt valu of holding cost is (whn ) HD c I t c I t c I t n rt r t rt ir5 m m m m m m r i rt r t rt cm Imt Imt cm Imt c I t I t rt rt m r r From Sct.., Cas V, and Eq. (56). n i ir5 br b HDm cm k a k a a r nc a b a b m (8) Whn 5 rt r t HDm cm Imt Imt n rt rt ir5 cm Imt cm Ir t i br b HDm cm k a k a a nacm b r (9) From Sct.., Cas VI, and Eq. (7).

12 88 N. SINGH t al. h itm cost includs th loss du to dtrioration, as wll as th costs of th itms sold. Bcaus st up is don at t =, th prsnt valu of th itm cost is I c Q c ka b b m m m m () hrfor th prsnt valu of total cost during th cycl is th sum of th st-up cost (SE), th holding cost (HD m ) and th itm cost (I m ). h prsnt valu of total cost pr unit tim ovr th cycl is givn by Cm SE HDm Im,, () Cm SE HDm Im,, 5 () In this cas, Eqs. (7) and (8) bcom t m m t Cas IV ( ) bt k a I t, t ) m t m bt m, a I t t (4) b a I t, t (5) m with th boundary conditions I m () =, I m ( ) = I m ( + ), and I m () =. h solutions of Eqs. () (5) ar k a I t, t bt t b m b bt t a b a I t b b, t b m (6) (7)

13 Collaborativ stratgy for a thr chlon supply chain 89 b t m, a I t t (8) Basd on Fig. b, th maximum invntory lvl of finishd goods is MI I. m m k a MI k a b b b m (9) h quantity producd in a cycl is givn by whn. bt m (4) Q P t ka ka b HD c I t c I t c I t n rt r t rt ir5 m m m m m m r i rt r t r t cmi mt cm Imt Imt c I t I t From Sct.., Cas V, and Eq. (56). n rt rt ir5 m r r i HDm cm k a a b a r br (4) whn 5. nc a b a b m

14 9 N. SINGH t al. HD c I t c I t c I t n rt r t rt ir5 m m m m m m r i HD c I t c I t rt r t m m m m m I t c I t n r t rt ir5 m m r i (4) HDm cm k a a b r br a nac b From Sct., Cas VI, and Eq. (7) m I c Q c ka b m m m m (4) h prsnt valu of total cost pr unit tim ovr a cycl is Cm SE HDm Im,, (44) Cm SE HDm Im,, 5 (45).. Rtailr s systm for storing finishd goods (whn ) h chang in th rtailr s invntory lvl is dpictd in Fig. 4a, b. Sinc P > d, w assumd that th initial dlivry to th rtailr s invntory systm is mad at t =. Part of th stock dlivrd is usd to satisfy prvious ordr, laving a balanc of MI r units in th initial invntory. During tim priod, th invntory lvl dcrass du to dmand and dtrioration. At t =, th invntory lvl is zro. During th tim priod 4, part of th short-

15 Collaborativ stratgy for a thr chlon supply chain 9 ag is backloggd and part of it rsults in lost sals. Only th backloggd itms ar rplacd by th nxt dlivry. hr ar n dlivris in th = + tim priod. Fig. 4. Rtailr s invntory systm: a), b) 5 h rtailr s invntory systm (Fig. 4) at any tim t can b rprsntd by th following diffrntial quation; ri t () I t, t (46) ri 5 with th boundary condition I ri ( ) =. hr ar two possibl rlations btwn th paramtrs, 5 and : (Fig. 4a), 5 (Fig. 4b). Each cas implis a diffrnt ordring cost, holding cost and dtrioration cost. Lt us discuss thm sparatly blow. In this cas, Eq. (46) bcoms r t r t Cas V ( ) a I t t (47) bt r, a I t t (48) b r,

16 9 N. SINGH t al. t b r 4 with th boundary conditions I r ( ) = and I r () =. h solutions of Eqs. (47) (49) ar 4 Ba, t (49) 4 4 b bt t a b a I t b b, t b r (5) b t r, a I t t (5) b I t Ba t, t (5) r Basd on Fig. 4 (a), th rtailr s maximum invntory lvl is MI r Ir b a b a MI b b b r (5) h quantity supplid to th rtailr pr dlivry is Q MI Ba (54) b r r 4 Dlivry has an initial ordr cost (c r ) incurrd at th start of th dlivry. h prsnt valu of ordring cost is OR = c r (55) Invntory is hld during th tim priod. h prsnt valu of th holding cost is rt rt rt HDr cr Ir t cr Irt I rt b HD c a b a r r (56) Shortag occurs during th tim priod 4. h prsnt valu of backlog cost is

17 Collaborativ stratgy for a thr chlon supply chain 9 4 r t4 b r BA c I t Ba c r r (57) Lost sals occur during th tim priod 4. During this tim priod, th complt 4 d t and th partial backlogging is shortag is Bd t. h diffrnc btwn thm quals th amount of lost sals. h prsnt valu of th cost of lost sals is 4 r t4 b LS c d t Bd t B a c r r (58) h itm cost includs loss du to dtrioration, as wll as th costs of th itms sold. Bcaus th ordr is carrid out at t = and t = + 4, th prsnt valu of th itm cost is 4 I c MI c Ba b r r r r 4 r r 4 b 4 4 c a b Ba r (59) h prsnt valu of th total cost of a dlivry is th sum of th ordring cost (OR), th holding cost (HD r ), th backlog cost (BA), th lost sal cost (LS) and th itm cost (I r ). h prsnt valu of th total cost pr unit tim for a singl dlivry is! C r! Cr ORr HDr BA LS Ir (6) hr ar n dlivris pr cycl. h fixd tim intrval btwn th dlivris is 5 = /n. h prsnt valu of th total cost pr unit tim ovr th cycl at t = is C OR HD BA LS I (6) n r! ir 5 r r r r Cr r5 i Cas VI ( 5 ) In this cas, Eq. (46) bcoms r t a I t t (6) bt r,

18 94 N. SINGH t al. t bt4 r 4 4 Ba, t (6) t b r 4 4 Ba, t (64) 4 5 with th boundary conditions I r ( ), I r () =, and I r ( ) = I r ( ). h solutions of Eqs. (6) (64) ar b bt a t a r, b b I t t (65) Ba bt4 Irt4, t (66) b b I t Bab Bat, t (67) r Basd on Fig. 4b, th rtailr s maximum invntory lvl is MI r Ir MI a b r h quantity supplid to th rtailr pr dlivry is Q MI Ba b b r r 5 (68) (69) rt r r r r HD c I t ac b (7) 4 r t4 r BA c I t r t4 r t4 c Irt4 4 Irt4 4 (7) r b r c Ba Ba 5 r5 Bab r r 5 5

19 Collaborativ stratgy for a thr chlon supply chain 95 4 r t4 b LS c d t Bd t B a c b (7) 4 r 4 Ir crmi r cr Bd t cr a b Ba b r (7) h prsnt valu of th total cost incurrd by a singl dlivry is th sum of th ordring cost (OR), th holding cost (HD r ), th backlog cost (BA), lost sal cost (LS) and th itm cost (I r ). h prsnt valu of th total cost incurrd pr unit tim by a singl dlivry is!! Cr ORr HDr BA LS Ir (74) hr ar n dlivris pr cycl. h fixd tim intrval btwn th dlivris 5 is /n. hus, th prsnt valu of th total cost pr unit tim ovr a cycl at t = is C OR HD BA LS I (75) n r!! ir 5 r r r r Cr r5 i It is obvious that th rsults of Sction ar not nough to driv th total systm cost for this cas. hus th dtrmination of th total cost, systm cost rquirs th furthr xamination of th ordring rlations btwn th tim paramtrs,,, 4, 5,, and. Now w hav all th quantitis ndd to formulat th total systm cost and procd with its optimization. 4. h optimal rplnishmnt policy h rsults in th prvious sub-sction lad to th following total systm cost ovr th tim intrval [, ]; Cas A, C h C (a) C (b) C C C C, C C C C w m r w m r (76)

20 96 N. SINGH t al. Cas B, 5 C g C (a) C4 (b) (77) * and th problm is C C C C C, C C C C w m r 4 w m r min. W hav usd a scond dgr polynomial to approximat an xponntial function. h rsulting modl with a singl supplir, a singl rtailr and a singl manufacturr is dvlopd to driv th optimal production policy and lot siz. Sinc = +, 5 = /n, and it is assumd that 4 =, whr < <<, its solution rquirs sparatly studying ach of th branchs and thn combining th rsults to obtain th optimal policy. It is asy to chck that C( ) is continuous at th point. h first ordr condition for a minimum of C ( ) is dc d br b w w b b ar c kac a k c ac Suppos th drivativ C d b m w m m. ka c b kac b kac b ac ac m m m dc d = at. with.. For this w hav b br m w m m b ar c kac a k c ac which shows th convxity of th function C. Solution procdur h problm is to dtrmin th valu of n and that minimiz th C. Sinc th numbr of dlivris pr ordr, n, is a discrt variabl, th following procdur is proposd to dtrmin th optimal production policy:

21 Collaborativ stratgy for a thr chlon supply chain 97 Stp. Sinc th numbr of dlivris, n, is an intgr valu, w start by choosing an intgr valu of n. Stp. Dtrmin th first drivativs of C i (i =,,, 4) with rspct to and quat thm to zro. Stp. Find th optimal valu of for givn n, which is dnotd by * (n) or simply * whn thr is no ambiguity rgarding th valu of n. h total cost in this cas is givn by C(n, * ). Stp 4. Rpat stps for all th possibl valus of n until th minimum C is * * * * * * found such that C( n, ) C( n, ) and C( n, ) C( n, ). 5. Numrical xampls and snsitivity analysis In this sction, w provid som numrical xampls to illustrat th thortical rsults obtaind in th prvious sctions. In addition, w also carry out a snsitivity analysis for th ffct of th valus of th most important paramtrs on th optimal ordr quantity and total systm cost. Exampl h input paramtrs ar: c w = $ pr ordr, c m = $ 9 pr ordr, c r = $ 5 pr ordr, c w = $ pr unit pr wk, c m = $ 5 pr unit pr wk, c r = $ 6 pr unit pr wk, c w = $ pr unit, c m = $ 5 pr unit, c r = $ pr unit, =.5, =.6, =.9, B =.8, r =.6, c = $ 5 pr unit, c 4 = $ 5 pr unit, k =, a =, b =, = wks, = wk, =.. Using Eq. (76a), w find th optimal valus of = 5. wks, and = 4.9 wks for n =5, =., =. wks, 4 =.67 wks and th optimal valu of th total systm cost is C = $ 44.. W can s that th rsults w hav found from this analysis satisfy th condition of convxity and th conditions of Eq. (76a) such as,. Using Eq. (76b), w found th optimal valus of =.5 wks, and = 6.48 wks for n = 4, =, = 4.6 wks, 4 =.8, and th optimal valu of th total systm cost C = $,7. Hr,,. Exampl In this, xampl, 5, th input paramtrs ar th sam as in Exampl. Using Eq. (77a), w found th optimal valus of = 4. wks, and = 5.89 wks

22 98 N. SINGH t al. for n =5, =, =.4, =. wks, 4 =.67 wks and th optimal valu of th total systm cost is C = $ his mans that thr is no fasibl solution in this cas that satisfis both of th conditions 5 and. Using th data from Exampl, th snsitivity analysis is prformd to xplor th ffct of changs in som of th modl paramtrs (a, b,, ) on th optimal policy (i.. on th optimal ordr quantity and optimal total systm cost). h rsults ar prsntd in abl and som intrsting findings ar summarizd as follows: Incrass in th first dmand paramtr (a) hav no impact on th optimal production tim, whil th total systm cost also incrass. But incrass in th scond dmand paramtr (b) lad to a dcras in th production tim and an incras in th total systm cost. Incrass in th tim paramtr () lad to a dcras in and an incras in th total systm cost. Incrass in th paramtr lad to incrass in both and th total systm cost. h changs in th optimal total systm cost indicat that th modl is highly snsitiv to th changs on a, b, and. abl. Snsivity analysis Paramtr chang Prcntag C Paramtr chang Prcntag C a b Concluding rmarks A modl for a thr chlon invntory with dtriorating itms and a ramp typ dmand rat and ramp typ production rat undr inflation is studid. In this modl, th rtailr is allowd to hav shortags which ar partially backloggd. h modl assums an individual dtrioration rat for ach party. h possibl ordring rlations btwn th tim paramtrs lad to four diffrnt situations. h optimal production policy was drivd for on of thm. Convxity was also provd for on cas. An asy to us algorithm to find th optimal production policy and optimal production tim is prsntd. Som numrical xampls ar studid to illustrat th proposd modl. h snsitivity of th solution to changs in th valu of diffrnt paramtrs has also bn

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