A Study of Inventory System with Ramp Type Demand Rate and Shortage in The Light Of Inflation I

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1 Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 A Sudy of Invenory Sysem wih Ramp ype emand Rae and Shorage in he Ligh Of Inflaion I Sangeea Gupa, R.K. Srivasava, A.K. Singh 3 Assisan Professor, Sharda Universiy, Greaer Noida Assosciae Professor, Agra College, Agra 3 Assisan Professor, FE, RBS College, Agra Absrac: In his paper, we develop an order-level invenory sysem for deerioraing iems under inflaion wih ramp ype demand funcion and parial exponenial ypebacklogging funcion of ime. hree coss are considered under inflaion as significan: deerioraion, holding, shorage. he backlogging rae is an exponenially decreasing, imedependen funcion specified by a parameer. For his model we derive resuls, which ensure he exisence of a unique opimal policy and provide he soluion procedure for he problem. he mehod is illusraed by numerical example, and sensiiviy analysis of he opimal soluion wih respec o he parameers of he sysem is carried ou. Key-words: Invenory, eerioraing iems, Inflaion, Ramp ype demand. Inroducion: Mos of he lieraure available during h cenury in he field of invenory managemen has no aken ino accoun he effec of inflaion. Perhaps his has happened mosly because of he percepion and belief ha inflaion would no influence he policy variables o any significan degree. Bu during s cenury, he moneary siuaion of mos of he counries, affluen or oherwise has changed o such an exen due o large scale inflaion and consequen sharp decline he purchasing power of money, ha i has no been possible o ignore he effecs of inflaion and so several effors have been made by researchers o reformulae he opimal invenory managemen polices aking ino accoun inflaion. he firs aemp in his direcion was done by Buzaco [ ], where he deal wih an economic order quaniy model wih inflaion subjec o differen ypes of pricing policies. Afer Buzaco [ ], several oher researchers have exended his approach o various ineresing siuaions aking ino consideraion he inflaion rae. In his connecion he works of Misra [5,6], Aggarwal [], Jeya Chandra and Bahner [ 3] ec. are worh menioning. Bu in all hese sudies, he marked demand rae has been assumed o be consan and unsaisfied demand is compleely backlogged. However, for fashionable commodiies and high-ech producs wih shor produc life cycle, he willingness for a cusomer o wai for backlogging during a shorage period is diminishing wih he lengh of he waiing ime. Hence he longer he waiing ime, smaller he backlogging rae would be. o reflec his phenomenon, Papachrisos and Skouri [7 ] esablished a parially backlogged invenory model in which he backlogging rae decreases exponenially as he waiing ime increases. ISSN: hp:// Page 96

2 Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 In a recen communicaion Mandal and pal [4] aemped o sudy on order-level invenory model for deerioraing iems, where he demand rae is a ramp-ype funcion of ime. Kun shan Wu and Liang Yuh Ouyang [9] exended heir work where invenory sars wih shorages. his ype of demand paern is generally seen in he case of any new brand of consumer goods coming o he marke. he demand rae for such iems increases wih ime up o a cerain ime period and hen ulimaely sabilizes and becomes consan. I is believed ha his ype of demand rae is quie realisic he above invesigaion led us o develop an inflaionary model for deerioraing iems wih ramp ype demand rae and parial exponenial ype backlogging. We aemp o provide he exac soluion for he problem in he ligh of numerical example followed by sensiiviy analysis of he opimal soluion wih respec o he parameers of he sysem. ASSUMPIONS AN NOAIONS: he mahemaical model of he deerminisic invenory replenishmen problem wih ramp ype demand rae is based on he following assumpions: i. he replenishmen rae is infinie, hus replenishmens are insananeous. ii. he lead-ime is zero. iii. he on hand invenory deerioraes a a consan rae θ (< θ <) per uni ime. he deerioraed iems are wihdrawn immediaely from he warehouse and here is no provision for repair or replacemen. iv. he rae of demand R () is ramp ype demand funcion of. Where H( µ) is Heaviside's funcion defined as follows: v. Unsaisfied demand is backlogged a rae e -αx, where x is he ime up o he nex replenishmen and 'α' a parameer vi. R H H he uni price is subjec o he same inflaion rae as oher relaed coss. he following noaions are used hroughou his invesigaion: he fixed lengh of each ordering cycle. S he maximum invenory level for each ordering. r he inflaion rae. C h he invenory holding cos per uni per uni of ime. ISSN: hp:// Page 97

3 Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 C s C d I() CI I SI he shorage cos per uni per uni of ime. eerioraion cos per uni of deerioraed iem. he on hand invenory a ime over [, ). he amoun of invenory carried during a cycle. he oal number of iems which deeriorae during a cycle he amoun of shorage during a cycle. MAHEMAICAL MOELS AN SOLUIONS: he objecive of he invenory problem here is o deermine he opimal order quaniy so as o keep he oal relevan cos as low as possible under inflaion has been subjeced o he invenory sars wihou shorages. he work of furher invesigaion and we shall discuss he invenory model for deerioraing iems under inflaion, where he invenory sars wihou shorages. emand Rae µ µ ime Figure. A ramp ype funcion of he demand rae [Adaped from Mandal and Pal [4]] ISSN: hp:// Page 98

4 Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 Figure. Graphical represenaion of invenory model he flucuaion of he invenory level in he sysem is given in figure. Replenishmen is made a line =, when he invenory level is a is maximum S. he invenory a = gradually reduces o zero a ime unis. he depicion of invenory level during he inerval [, ] is due o he join effec of he demand and he deerioraion of iems. A, he invenory level reaches zero, hereafer shorage are allowed o occur during he ime inerval [, ] and he demand during period [, ] is parially backlogged. he oal number of backlogged iems is replaced by he nex replenishmen. he invenory level of he sysem a ime over period [, ) can be described by he following equaions: d and d I R I d I d e R For µ<, he above equaion reduce o ISSN: hp:// Page 99

5 Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 d I d I 3 d I d I ; in view of I() S 4 and di d e µ ; in he ligh of ( ) I 5 Since (3), (4) and (5) are firs order liner differenial equaions i is fairly easy o derive heir soluions as I S e e ; 6 I e ; 7 and I e e ; 8 From equaions (6) and (7), he value of I() a =µ should coincide, which implies ha S e e 9 he amoun of invenory carried during he period [, ] is CI, where CI I d I d I d he oal number of iems which deeriorae during [, ] is I, where I S R d I d And he amoun of shorage during he period [, ) SI I d ISSN: hp:// Page

6 Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 he presen value of he invenory holding cos during he period [, ] is h h r r r C e I d C e I d e I d r S Ch e e r r r r r e r e r r r he presen value of he deerioraion cos during he period [, ] is 3 r Cd e I d r S Cd e e r r r r e r r e r r r 4 he presen value of he shorage cos during he period [, ) is r r CS e I d C S e e e d CS e e e r r r r r r r 5 Also he order quaniy during he period [, ) is given by Q S I e e e 6 Hence he oal relevan cos of he sysem during he ime inerval [,), can be pu as, r S X C C e e r r r r h d e r r r r r e C S e e e r r r r r r r 7 ISSN: hp:// Page

7 Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 hus, he average oal cos per uni ime is C 8 X o minimize he average oal cos per uni of ime, he opimal value of can be obained by solving he following equaion d C d 9 his also saisfies he condiion Equaion (9) is equivalen o d C d e r r e Ch Cd e e r r r r r r r r CS e e his is a non-linear equaion. his equaion can be easily solved using any ieraive mehod when he value of he parameers is prescribed. By using he opimal value *, he opimal value of S*, he minimum average oal cos per uni of ime and he opimal order quaniy can be obained from equaion (9), equaion (8) and equaion (6) respecively. iscussion and Conclusion:- In he ligh of numerical example adoped by Kun-Shan Wu and Liang- Yuh- Ouyang [9], he inpu parameers in our case are as follows :- Ch = $3 per uni year, Cd = $5 per uni, Cs = $ 5 per uni per year = unis, µ =. years θ =. = year α =.8 r =.5 * =.876, S* = 9.53, Q* =.699, C* = ISSN: hp:// Page

8 Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 Where * sands for opimal values. Evidenly opimal oal cos of our invenory model is less han ha of Kun-Shan Wu and Liang-Yuh-Ouyang [ 9]. Hence our model is believed o be beer one. Sensiiviy Analysis:- We will now sudy he sensiiviy of he opimal soluion o changes in he values of he differen parameers associaed wih he invenory sysem in example. he resuls are shown in able. Sensiiviy Analysis of numerical example: Parameer % change % change in S* Q* C* r +5% -.46% -.3% -.% +5% -.% -.% -.75% -5% +.3% +.3% +.% -5% +.7% +.% +.45% α +5% -.3% -.65% -.53% +5% -.65% -.3% -.8% -5% +.65% +.3% +.3% -5% +.3% +.64% +.45% C h +5% -8.68% -.% % +5% -4.59% -.59% % -5% +4.89% +.5% -.64% -5% +.8% +.96% -45.5% C d +5% -.6% -.4% +5% -.3% +.4% -5% -.33% -5% +.3% -.73% C s +5% +6.63% +.68% +6.5% +5% % +.43% +3.53% -5% -5.94% -.79% % -5% -6.33% -.55% -4.85% θ +5% +.% +.8% -.36% +5% +.% -.8% -5% -.% -.% -.64% -5% -.% -.8% -.87% µ +5% +44.4% +45.% +48.7% +5% +.56% +3.% +4.8% -5% -3.54% -3.8% -4.93% -5% -48.5% -48.4% % +5% +5.% +5% % +5% +5% +5% +4.74% -5% -5% -5% -5.8% -5% -5% -5% -5.6% ISSN: hp:// Page 3

9 Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 A careful sudy of able reveals he following poins.. I is eviden ha S* is insensiive o change in he value of he parameer θ, α, C d. I is moderaely and highly sensiive o change in he value of parameers µ and.. I is fairly easy o observe ha he opimal order quaniy Q* is insensiive o changes in he value of he parameers C h, C s, θ, r, α. I is highly sensiive o changes in he value of parameers μ and. I is no affeced by he change in he value of he parameer C d. 3. Finally i is obviously clear ha he opimum oal cos is insensiive o changes in he values of parameers C d, θ, α. I is moderaely sensiive o change in he value of parameer Cs, r and highly sensiive o change in he value of parameers C h, µ and. REFERENCES []. Aggarwal, S.C. (98), Purchase invenory decision models for inflaionary condiions. Inerfaces, 8-3. []. Buzaco, J.A. (975), Economic order quaniies wih inflaion. Operaional Research Quarerly 6(3) (i), [3]. Chandra, J.M. and Bahner, M.L. (985), he effecs of inflaion and he ime value of money on same invenory sysems. Inernaional Journal of Producion Research 3(4), [4]. Mandal, B. and Pal, A.K. (998), Order level invenory sysem wih ramp ype demand rae for deerioraing iems. Journal of Inerdisciplinary Mahemaics, ; [5]. Misra, R.B. (975), A sudy of inflaionary effecs on invenory sysems. Logisics Specrum 9(3), [6]. Misra, R.B. (979), A noe on opimal invenory managemen under inflaion. Naval Research Logisics Quarerly 6, [7]. Papachrisos, S. and Skouri, K. (), An opimal replenish-men policy for deerioraing iems wih ime-varying demand and parial exponenial ype backlogging. Operaion Research Leer 7, [8]. eng, J.., Chang, H.J., ye, C.Y. and Hung, C.H. (), An opimal replenishmen policy for deerioraing iems wih ime varying demand and parial backlogging. Operaion Research Leer 3, [9]. Wu, K.S. and Ouyang, L.Y. (), A replenishmen policy for deerioraing iems wih ramp ype demand rae. Proc. Nal. Sci. Counc. ROC (A), Vol 4(4), ISSN: hp:// Page 4