An Inventory Model for Product Life Cycle with Defective Items and Shortages Allowed

Transcription

1 Inernaional Journal of ompuaional cience and Maemaics. IN Volume, Number, pp Inernaional Researc ublicaion House p:// An Invenory Model for roduc Life ycle wi efecive Iems and orages Allowed. Krisnamoori Assisan professor in Maemaics, KGiL Insiue of Tecnology and Researc colar, Baraiar Universiy, oimbaore srivigneswar_ooy@yaoo.co.in Absrac To manage invenory, e classical economic order quaniy EO equaion can be used o decide ow muc o order. Te EO is e order quaniy a eoreically minimizes e oal of e cos of ordering and olding invenory and i assumes a e demand is consan, does no vary over ime. Tis paper deals wi an invenory model for produc life cycle wi and wiou defecive iems and sorages allowed for a single produc manufacuring /purcasing sysem. Te defecive rae is considered as a variable of known proporions. Te objecive is o minimize e oal ne invenory cos and o find e opimal quaniy in produc life cycle. Te relevan model is buil, solved and some main resuls abou e uniqueness of is soluion wi e use of rigorous maemaical meods are obained. Tis seems o be e firs ime were suc a invenory model for produc life cycle is maemaically reaed and numerically verified. Keywords: Invenory, ycle ime, orages, efecive Iems, emand and roducion. Inroducion A produc life cycle is e life span of a produc wic e period begins wi e iniial produc specificaion and ends wi e widrawal from e marke of bo e produc and is suppor. A new produc is firs developed and en inroduced o e marke. Once e inroducion is successful, a grow period follows wi wider awareness of e produc and increasing sales. Te produc eners mauriy wen sales sop growing and demand sabilizes. Evenually, sales may decline unil e produc is finally wi drawn from e marke or redeveloped. A produc s life cycle can be divided ino several sages caracerized by e revenue generaed by e

2 6. Krisnamoori produc. Te life cycle concep may apply o a brand or o a caegory of produc. Is duraion may be as sor as a few mons for a fad iem or a cenury or more for produc caegories suc as e gasoline powered auomobile. uring e inroducory sage e firm is likely o incur addiional coss ie adverising cos associaed wi e iniial disribuion of e produc. Tese iger coss coupled wi a low sales volume usually make e inroducion sage a period of negaive profis. uring e inroducion sage, e primary goal is o esablis a marke and build primary demand for e produc class. Te grow sage is a period of rapid revenue grow. ales increase as more cusomers become aware of e produc and is benefis and addiional marke segmens are argeed. Once e produc as been proven a success and cusomers begin asking for i, sales will increase furer as more reailers become ineresed in carrying i. Te markeing eam may epand e disribuion a is poin. Te mauriy sage is e mos profiable. ile sales coninue o increase ino is sage, ey do so a a slower pace. Because brand awareness is srong, adverising ependiures will be reduced. Evenually sales begin o decline as e marke becomes sauraed, e produc becomes ecnologically obsolee or cusomer ases cange. If e produc as developed brand loyaly, e profiabiliy may be mainained longer. Uni coss may increase wi e declining producion volumes and evenually no more profi can be made. Te producion of invenory is dependen on e ype of invenory sysem a e organizaion is adoping. To faciliae solving e problem of invenory, i is required o build models, wic eplain e invenory flucuaion. In some cases suc as reailer, wolesaler/disribuor, were iems are purcased eernally, if e problem of invenory eiss, en ere are wo main quesions, wic generally arise and face any organizaion. Tese are ow many o order and wen o order. Having oo muc invenory reduces bo purcases and/or ordering coss bu i may ie up capial, wic may lead o unnecessary olding cos and possibiliy of deerioraing iems. ereas aving oo lile invenory reduces e olding cos bu i can resul in los of cusomers, wic may affec e reliabiliy of e organizaion, wic minimizes is oal invenory cos. Assumpions and Noaions a. Assumpions: Te assumpion and noaions of an invenory model for produc life cycle are as follows:. Te demand rae is known, consan and coninuous.. Iems are produced and added o e invenory.. orages are allowed. 4. Te iem is a single produc, i does no inerac wi any oer invenory iems. 5. Te producion rae is always greaer an or equal o e sum of e demand rae. b. Noaions: Tis secion defines e noaions used in e invenory models roducion rae in unis per uni ime

3 An Invenory Model for roduc Life ycle 7 emand rae in unis per uni ime on and invenory level *-Opimal size of producion run p roducion /urcase os per uni rae of defecive iems from end cusomers in unis per uni ime y Y proporion of defecive iems afer disribuion o end cusomers y is beween o proporion of defecive iems from regular producion is beween o d uni scrap cos per iem of imperfec qualiy. os of cusomer reurn cos of disposal, sipmen and penaly g -Holding cos per uni/year eup cos / ordering cos uni ime in one cycle i - uni ime in periods i i,,,.. T - Toal cos TO* - Minimum oal cos per cycle Toalos TimeInOneycle roblem Formulaion Te objecive of is researc is o develop maemaical models o minimize e epeced oal cos of invenory. Iniially, e manufacurer mus define all coss suc as e cos of producion, olding cos, seup cos producion caracerisics and all capabiliies of e producion process. Tese ave o be accurae because ese variables will direcly affec e producion quaniy and oal cos. Tis paper deals wi a finie producion invenory model for a single produc imperfec manufacuring sysem. Te defec rae is considered as a variable of known proporions. Te maemaical models for opimal producion lo size in is researc can be classed as follows:. Model : EO model for produc life cycle wi sorages.. Model : EO model for produc life cycle wi sorages and defecive iems. Model : An Economic Order uaniy Model for roduc Life ycle wi orages: Te sysem operaes as follows: I sars a ime a a demand rae. Ten producion sars were e invenory level increases a a rae - in order o saisfy e demand unil ime. A is poin, e producion ceases and e invenory level reaces is maimum. Ten e demand in e consan rae of unil ime. Tereafer, e invenory level declines coninuously a a rae and becomes zero a ime end of e cycle. Te process is repeaed. Te variaion of e underlying invenory sysem for one cycle is sown in e following figure. uring producion period, invenory is increasing a e rae of and

4 8. Krisnamoori simulaneously decreasing a e rae of. Tus invenory accumulaes a e rae of - unis. Terefore, e maimum level sall be equal o. Time needed o build up unis of iems. From e diagram, ; ; ; 4 ; 5 and roducion ycle: 5 4 erefore, Figure : On-and invenory of non- defecive iems for Model. reposiion: Te opimal soluion for e invenory policy is given by [ ] roof: Te oal cos componens consis of urcase/producion cos, seup cos and olding cos and sorages cos.

5 An Invenory Model for roduc Life ycle 9 a. urcase/roducion os b. Ordering/eup cos c. Holding os: Te olding cos is derived as follows: H - d. orages os : 5 4 e. Toal os: Te oal cos is as follows: T p Te oal cos per uni ime is given below: T p arial differeniae w.r.. o, T > o A

6 4. Krisnamoori arially differeniae wi respec o, T T > subsiue in e equaion A [ ] [ ] o [ ] Illusraive Eample, consider e following parameers 5 unis/year,, unis/year, Rs.5, per unis/year, Rs., oluion:.5 unis, 5.5; Toal cos,94.7 Model : An Economic Order uaniy Model for roduc Life ycle wi orages and efecive Iems: Te proposed invenory sysem operaes as follows: Te cycle sars a ime and e invenory accumulaes a a rae -- upo ime were producion sops. Afer a, e invenory level sars o decrease due o demand and defecive iems a a rae upo ime and. Te process is repeaed. Te variaion of e underlying invenory sysem for on cycle is sown in e following figure. Te producion rae of imperfec iems: y eernal Te producion rae of good iems is always greaer an or equal o e sum of e demand rae and e rae wic defecive iems are produced. o, we mus ave:

7 An Invenory Model for roduc Life ycle 4 Figure : On-and invenory of defecive iems for Model. uring producion period, e maimum level sall be equal o. From e above figure, Time needed o build up unis of iems. From e figure, ; ; ; 4 ; 5 ; roducion ycle: 5 4 i.e. Terefore, reposiion : Te opimal soluion for e invenory policy is given by [ ] [ ]

8 4. Krisnamoori roof: Te oal coss per cycle appropriae o is model are as follows: a. urcase/roducion os b. Ordering/eup cos c. Holding os: Te olding coss sould include a of all produced iems, defecive and non-defecive. H d. orage os e. efecive os : os per defec passed forward cusomers scrap and penaly coss d y g y Toal os: Te oal cos T would be: T urcase os Ordering cos olding cos orages cos os of defecive iems T p - d XY g XY Toal os per uni ime T p

9 An Invenory Model for roduc Life ycle 4 - d XY g XY arially differeniae w.r.. T - - > T arially differeniae w.r.. T > T [ ]

10 44. Krisnamoori [ ] [ ] [ ] Illusraive Eample, consider e following parameers, unis/year,,5 unis/year, Rs.5,.5, ; y. o., per unis/year,, d 5, y.5, g 5 Table : Variaion of, and T. Rae of efecive From able, a sudy of rae of defecive iems wi lo size, manufacuring ime, consan sales ime, decrease sales ime and cycle ime. e conclude from e above able, wen e rae of defecive iems increases en e,,, and also increases bu and decreases. Table : Variaion of Rae of efecive and invenory and Toal os. Rae of urcase Ordering Holding efecive orage Toal os efecive os os cos cos cos

11 An Invenory Model for roduc Life ycle From able, a sudy of rae of defecive iems wi invenory cos and oal invenory cos. e conclude from e above able, wen e rae of defecive iems increases en purcase cos, defecive cos and oal cos also increases bu ordering cos and olding cos decreases. onclusion In is paper, an EO model for produc life cycle is considered in wic eac of e demand, producion and as well as all cos parameers are known. orages are no allowed. Te objecive is o minimize e overall oal relevan invenory cos. An eac maemaical model and a soluion procedure is esablised. An illusraive eample also eplained. Tis seems o be e firs ime were suc a invenory model for produc life cycle is maemaically reaed and numerically verified and i is conclude a wen e rae of defecive iems increases en Opimal order uaniy,,,, purcase cos, defecive cos and oal cos also increases bu e ordering cos and olding cos decreases. References [] R..Grubbsrom, Maerial Requiremens lanning and Manufacuring Resource planning, Inernaional Encyclopedia of business and Managemen Rou ledge, London, 996. [] R.. Grubbsrom and A.Erdem, Te EO wi backlogging derived wiou derivaives, Inernaional Journal of roducion Economics, Vol.59, pp59-5, 999. [] L.E. ardenas-barron, Te Economic roducion quaniy E wi sorage derived algebraically, Inernaional Journal of producion Economics, Vol.7, pp 89-9,. [4] un-hsiung Lan and Yen-cick Yu, A noe on e improved Algebraic meod for e E Model wi ocasic Lead ime, Informaion and Managemen ciences, Vol.8, No., pp.9-96, 7. [5] Jon June and Eyler R. oaes, An Economic Order quaniy model for Time varying demand, Te Inernaional Journal of Modern Engineering, Vol., Number, pring,.

12 46. Krisnamoori [6] ing arbji, arma Jiendra, ing sivraj, rofi Maimizing probabiliics Invenory model under e Effec of ermission delay, roceedings of e Inernaional Muli onference of Engineers and ompuer cieniss Vol, III, IME, Marc 7-9,, Hongkong. [7] Zaid T. Balki and Ali. Haj Bakry, A general and dynamic producion lo size invenory model, Inernaional Journal of Maemaical Models and Meods in Applied ciences, issue, volume, 9, p [8]. ugapriya, K. Jeyaraman, An EO model for non insaneous deerioraing iems in wic olding cos varies wi ime. Elecronic Journal of Applied aisical Analysis, Vol., pp 9-7, 8.