International Journal of Computer Science Trends and Technology (IJCST) Volume 3 Issue 6, Nov-Dec 2015

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1 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 RESEARCH ARTICLE OPEN ACCESS An EPQ Model for Two-Parameer Weibully Deerioraed Iems wih Exponenial Demand Rae and Compleely Backlogged Shorages Devyani Chaerji [], U B Gohi [] Assisan Professor [] Deparmen of Saisics S M Pael Insiue of Commerce GLS Universiy, Ahmedabad Associae Professor [] Deparmen of Saisics S Xavier s College (Auonomous), Ahmedabad Gujara - India ABSTRACT In formulaing invenory models, wo facors of he problem have been of growing ineres o he researchers, one being he variaion in he demand rae and he oher being he deerioraion of iems In he presen paper, an EPQ model wih demand following exponenial paern and wo-parameer Weibull deerioraion rae is considered The ime-varying invenory holding cos is a linear funcion of ime Shorages are allowed and hey are compleely backlogged Numerical example is used o illusrae he developed model Sensiiviy analysis and graphical analysis of he opimal soluion of various parameers are carried ou Keywords:- Exponenial demand rae, Two-parameer Weibull deerioraion rae, Compleely backlogged shorages, Time-varying invenory holding cos I INTRODUCTION One of he assumpions of radiional invenory models was ha he iems preserve heir original characerisics hroughou when hey are kep in he invenory This assumpion is rue for mos of he iems, bu no for all Mos of he physical goods like vegeables, grains, pharmaceuicals, fashion goods, radioacive subsances ec deeriorae over ime They eiher ge decayed, damaged, vaporized or affeced by some or oher facors and do no remain in a perfec condiion o saisfy he demand While formulaing invenory models, wo facors of he problem have been of growing ineres o he researchers, one being he variaion in he demand rae and he oher being he deerioraion of iems Demand is he major facor in he invenory managemen Hence, decisions of invenory problems are o be made by considering boh presen and fuure demands As demand plays a key role in deerioraing invenory models, researchers have concenraed in he sudy of variaions in demand Demand may be consan, ime-varying, sock-dependen, pricedependen ec The consan demand is valid only when he phase of he produc life cycle is maured and also for finie periods of ime Parmar and Gohi [] developed a deerminisic invenory model for deerioraing iems where ime o deerioraion has Exponenial disribuion and imedependen quadraic demand Gohi and Parmar [7] exended he above deerminisic invenory model by aking wo parameer Weibull disribuion o represen he ime o deerioraion, where shorages are allowed and parially backlogged Jani, Jaiswal and Shah [8] developed (S, qp) sysem invenory model for deerioraing iems Bhojak and Gohi [] developed an EOQ model wih ime-dependen demand and Weibully disribued deerioraion Parmar, Aggarwal and Gohi [0] developed an Order level invenory model for deerioraing iems under varying demand condiion Venkaeswarlu and Mohan [] developed an EOQ model wih parameers Weibull deerioraion, ime-dependen quadraic demand and salvage value Furher, Mohan and Venkaeswarlu [9] proposed an invenory model for ime dependen quadraic demand wih salvage considering deerioraion rae is ime-dependen Amuha and Chandrashekharan [] developed an EOQ model for deerioraing iems and quadraic demand and ime-dependen holding cos Gohi and Chaerji [6] developed an EPQ model for imperfec qualiy iems under consan demand rae and varying holding cos Parmar and Gohi [] developed an EPQ model of deerioraing iems using hree parameer Weibull disribuion wih consan producion rae and ime-varying holding cos Furher, Parmar and Gohi [] have developed an ISSN: wwwijcsjournalorg Page 8

2 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 EPQ model for deerioraing iems under hree parameer Weibull disribuion and ime dependen IHC wih shorages Chaerji and Gohi [] developed an EOQ model for deerioraing iems under wo and hree parameer Weibull disribuion and consan invenory holding cos wih parially backlogged shorages Recenly, Chaerji and Gohi [5] have developed an EOQ model for deerioraing iems under wo and hree parameer Weibull disribuion and consan IHC wih parially backlogged shorages Verma and Verma [5] developed an invenory model wih exponenially decreasing demand and linearly increasing deerioraion Yang and Wee [6] developed an inegraed muli-lo-size producion invenory model for deerioraing iems wih consan producion and demand raes Avikar, Tuli and Sharma [] developed an Opimal invenory managemen for exponenially increasing demand wih finie rae of producion and deerioraion They assumed shorages are allowed and backlogged II NOTATIONS The following noaions are used o develop he model: Q() : Insananeous rae of he invenory level a p any ime (0 ) : Producion rae per uni ime R() : Demand rae varying over ime θ() : Deerioraion rae 5 : Iniial rae of demand 6 A : Ordering cos per order during he cycle period 7 Cd : Deerioraion cos per uni per uni ime 8 Ch : Invenory holding cos per uni per uni ime 9 Cs : Shorage cos per uni per uni ime 0 p c : Producion cos per uni per uni ime S : Maximum invenory level a ime a S : Maximum invenory level during he shorage period a : Time a which shorages sar, 0 : Lengh of he replenishmen cycle 5 TC : The average oal cos for he ime period [0, ] III ASSUMPTIONS The following assumpions are considered o develop he model: A single iem is considered over he prescribed period of ime Replenishmen rae is infinie and lead ime is zero The demand rae of he produc is of exponenial paern Once a uni of he produc is produced, i is available o mee he demand 5 No repair or replenishmen of he deerioraed iems akes place during a given cycle 6 Holding cos is a linear funcion of ime and i is 7 C h r h () ( hr, 0) is he wo parameer Weibull deerioraion rae in he ime inerval [, ] where [0, ] and is scale parameer (0 ) and is shape parameer( 0) 8 Shorages are allowed and hey are fully backlogged 9 Toal invenory cos is a real and coninuous funcion which is convex o he origin IV MATHEMATICAL MODEL AND ANALYSIS In he mahemaical model iniially 0 invenory is zero A ime he producion sars and simulaneously demand is also saisfied The producion sops a ime where he maximum invenory level is aained In he inerval 0, S he invenory is accumulaed a a rae p as ae demand is following exponenial disribuion and here is a deerioraion rae of wo parameer Weibull disribuion The invenory level reaches zero level a ime Thereafer, shorages occur during he ime inerval and here becomes a backlog of, S unis, which is fully saisfied in he following ime inerval, ISSN: wwwijcsjournalorg Page 9

3 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 The differenial equaions which governs he insananeous sae of Q() over he ime inervals and,, are given by, 0,,, dq() Q( ) p ae 0 d () dq() Q( ) ae d () dq() ae d () dq() p ae d () Using boundary condiions Q Q Q 0 0 (5) Q S, Q S (6) ISSN: wwwijcsjournalorg Page 0

4 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 The soluions of equaions (), (), () and () are given by Q( ) p a a 0 a a Q( ) a (8) a Q( ) (9) a Q( ) p a (0) (7) Puing Q( ) S in equaion (7), we ge a S p a () Puing Q() = S in equaion (8), we ge a a () S a Comparing () and () we ge an expression which expresses decision variable p p a in erms of, and hence is no aken as he () Q( ) Puing S in equaion (9), we ge a S () Q( ) S Puing in equaion (0), we ge a S p a Comparing () and (5) we ge anoher expression which expresses in erms of and, and hence is no aken as he decision variable a a a a p (6) p Cos Componens: The oal cos per replenishmen cycle consiss of he following cos componens: Operaing Cos The operaing cos OC over he period 0, is OC A (7) (5) ISSN: wwwijcsjournalorg Page

5 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 Deerioraion Cos The deerioraion cos DC over he period 0, is DC C d Q ( ) d Q ( ) d 0 a DC Cd p a a (8) Holding Cos The holding cos for carrying invenory over he period IHC h rq()d h rq()d 0 a p a p a a a 8 h p a p a r 6 a r h IHC 6 ah r 8 Shorage Cos The shorage cos SC over he period, SC Cs Q( ) d Q( ) d a a p a 6 SC Cs a a p is 0, is (9) (0) 5 Producion Cos The producion cos PC during he period 0, and, PC p () c is ISSN: wwwijcsjournalorg Page

6 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 Hence, he average oal cos for he ime period 0, is given by TC OC DC IHC SC PC a p a a A C d a a h p a p a 6 a p a p a 8 r TC a r h 6 ah r 8 a a p a 6 Cs a a p pc () ISSN: wwwijcsjournalorg Page

7 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 Using he expressions () and (6), and are eliminaed from equaion () of oal cos TC Hence, TC remains a funcion of and only, which are he decision variables * and * are he opimum values of and respecively, which minimize he cos funcion TC and hey are he soluions of he equaions TC TC 0 & 0 such ha TC TC TC 0, = () TC 0, = V NUMERICAL EXAMPLE Le us consider he following example o illusrae he above developed model Taking A = 00, α = 0000, β =, h = 8, r =, p =, a =, θ = 0000, Cs = 8, pc = 0 and Cd= (wih appropriae unis) The opimal values of and are * = 9567, * = unis and he opimal oal cos per uni ime TC = unis VI SENSITIVITY ANALYSIS AND GRAPHICAL ANALYSIS Sensiiviy analysis depics he exen o which he opimal soluion of he model is affeced by he changes in is inpu parameer values Here, we sudy he sensiiviy for he cycle lengh and oal cos per ime uni TC wih respec o he changes in he values of he parameers A, α, β,h, r, p, a, θ, Cs, pc and Cd The sensiiviy analysis is performed by considering variaion in each one of he above parameers keeping all oher remaining parameers as fixed The las column of he Table gives he % changes in TC as compared o he original soluion for he relevan coss Table : Parial Sensiiviy Analysis Parameer Values TC A α β h ISSN: wwwijcsjournalorg Page

8 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 Parameer Values TC r p a θ Cs pc Cd VII GRAPHICAL PRESENTATION Figure Figure ISSN: wwwijcsjournalorg Page 5

9 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 VIII CONCLUSION From parial sensiiviy analysis and graphical analysis we can conclude ha as scale parameer, shape parameer, deerioraion rae, deerioraion cos C d, linear consans of invenory holding cos r and h, producion cos p c, operaing A cos, iniial rae of demand and shorage cos C increase, oal cos TC increases Bu we have s also observed he following poins: a Figure From Figure i is observed ha oal cos TC is highly sensiive o change in operaing cos, iniial rae of demand and A shorage cos C s From Figure i is observed ha oal cos TC is moderaely sensiive o he change in he linear consans r and h of invenory holding cos and producion cos p c a Figure 5 From Figure i is observed ha here is a mild change in he oal cos TC owards he change in scale parameer, shape parameer, deerioraion rae and deerioraion cos C d Again from he parial sensiiviy analysis and graphical analysis, from Figure 5 i can be said ha as he rae of producion increases, oal cos TC decreases Also from he -D ploing in Figure 6 of, and TC, i can be concluded ha as and increases increase, oal cos TC p always REFERENCES Figure 6 [] Amuha, R and Chandrashekharan, E, An EOQ model for deerioraing iems wih quadraic demand and ime-dependen holding cos Inernaional Journal of Emerging Science and Engineering (IJESE), Volume-, Issue-5, pp 9 678, March 0 ISSN: wwwijcsjournalorg Page 6

10 Inernaional Journal of Compuer Science Trends and Technology (IJCST) Volume Issue 6, Nov-Dec 05 [] Avikar, Tuli, RK and Sharma, A, Opimal Invenory Managemen for exponenially increasing demand wih finie rae of producion and deerioraion Inernaional Journal of Compuer, Science, Trends and Technology, Vol, Issue, Sepember 0 [] Bhojak, A and Gohi, U B, An EOQ model wih ime dependen demand and Weibull disribued deerioraion Inernaional Journal of Engineering Research and Technology (IJERT), Vol Issue 09, Sepember 05 [] Chaerji, DA & Gohi UB, Opimal EPQ model wih Weibully disribued deerioraion rae and ime varying IHC Inernaional Journal of Mahemaics Trends and Technology, Volume 5, Number Sepember 05 [5] Chaerji, DA & Gohi UB, EOQ model for deerioraing iems under wo and hree parameer Weibull disribuion and consan IHC wih parially backlogged shorages Inernaional Journal of science, Engineering and Technology Research, Vol, Issue 0, Ocober 05, pp [6] Gohi, UB & Chaerji, D, EPQ model for imperfec qualiy iems under consan demand rae and varying IHC Sankhya Vignan, NSV, No, pp 7-9, June 05 [7] Gohi, UB & Parmar, K C, Order level lo Size invenory model for deerioraing iems under quadraic demand wih ime dependen IHC and parial backlogging Research Hub Inernaional Mulidisciplinary Research Journal (RHIMRJ), Vol, Issue, 05 [8] Jani, B B, Jaiswal, MC and Shah, YK, (S, qp) sysem invenory model for deerioraing iems Inernaional Journal of Producion Research 6, pp -9, 978 [9] Mohan R and Venkaeswarlu R, Invenory Model for Time Dependen Deerioraion, Time Dependen Quadraic Demand and Salvage Value Journal of Inernaional Mahemaics Sociey, (In press) (0b) [0] Parmar, K C, Aggarwal, I & Gohi, UB (05) Order level lo Size invenory model for deerioraing iems under varying demand condiion Sankhya Vignan, (NSV ), 0-0 [] Parmar, K C & Gohi, U B, Order level invenory model for deerioraing iems under quadraic demand wih ime dependen IHC Sankhaya Vignan, NSV 0, No, pp, 0 [] Parmar, KC and Gohi U B, An EPQ model of deerioraing iems using hree parameer Weibull disribuion wih consan producion rae and ime varying holding cos Inernaional Journal of Science, Engineering and Technology Research, Vol, Issue, 05, pp [] Parmar, KC and Gohi, UB, EPQ model for deerioraing iems under hree parameer Weibull disribuion and ime dependen IHC wih shorages American Journal of Engineering Research, Vol, Issue 7, pp 6-55, 05 [] Venkaeswarlu R and Mohan R, An Invenory Model wih Weibull Deerioraion, Time Dependen Quadraic Demand and Salvage Value AIMS -0, Proceedings, Bangalore, (0a) [5] Verma, VK and Verma, BB, An invenory model wih exponenially decreasing demand and linearly increasing deerioraion Inernaional Journal of Physical, Chemical and Mahemaical Sciences, Vol, No, Jan-June 0 [6] Yang, PC and Wee, HM, An inegraed muli-lo-size producion invenory model for deerioraing iems wih consan producion and demand raes Compuers and Operaions Research, 0, 00, pp ISSN: wwwijcsjournalorg Page 7