International Journal of Latest Technology in Engineering, Management & Applied Science (IJLTEMAS) Volume VI, Issue VIIIS, August 2017 ISSN

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1 Inventory odel wth Dfferent Deteroraton Rates for Imperfect Qualty Items and Inflaton consderng Prce and me Dependent Demand under Permssble Delay n Payments Shtal S. Patel Department of Statstcs, Veer Narmad South Gujarat Unversty, Surat, INDIA Abstract: One of the assumptons for an economc order quantty model s that all tems receved n an order are of perfect qualty s not always fulflled. Some of the tems are of defectve qualty n the lot receved. Another assumpton s that as soon as tems are receved, payments are made. In today s compettve the suppler allows certan fxed perod known as permssble delay for payment to the retaler for settlng the amount of tems receved. Keepng ths realty, a determnstc nventory model wth mperfect qualty s developed when deteroraton rate s dfferent durng a cycle. Here t s assumed that demand s a functon of tme and prce. Numercal example s taken to support the model. Senstvty analyss s also carred out for parameters. Key Words: Inventory model, Varyng Deteroraton, me dependent demand, Prce dependent demand, Defectve tems, Inflaton, Permssble Delay I. INRODUCION ost of the tems lose ther characterstcs overtme and ths characterstc s defned as deteroraton. Ghare and Schrader [8] consdered nventory model wth constant rate of deteroraton. Covert and Phlp [7] extended the model by consderng varable rate of deteroraton. andal and Phaujdar [4] presented an nventory model for stock dependent consumpton rate. Hapng and Wang [] studed an economc polcy model for deteroratng tems wth tme proportonal demand. Patel and Parekh [7] developed an nventory model wth stock dependent demand under shortages and varable sellng prce. Other research work related to deteroratng tems can be found n, for nstance (Raafat [0], Goyal and Gr [0], Ruxan et al. []). In realty, t happens that unts ordered are not of 00% good qualty. Rosenblat and Lee [] were the frst to focus on defectve tems. Salman and Jaber [4] developed an nventory model n whch tems receved are of defectve qualty and after 00% screenng, mperfect tems are wthdrawn from the nventory and sold at a dscounted prce. Salman and Jaber s [4] model was extended by Wee et al. [6] by allowng shortages. Chang [4] studed an nventory model to nvestgate the effects of mperfect products on the total nventory cost assocated wth an EPQ model. Patel and Patel [8] developed an EOQ model for deteroratng tems wth mperfect quantty tems. Hauck and Voros [] consdered nventory model n whch percentage of defectve tems as a random varable and defned the speed of the qualty checkng as a varable. An economc order quantty model under the condton of permssble delay n payments was developed by Goyal [9]. Aggarwal and Jagg [] extended Goyal s [9] model to consder the deteroratng tems. he related work are found n (Chung and Dye [5], Salameh et al. [3], Chung et al. [6], Chang et al. [3]). he effect of nflaton and tme value of money play mportant role n practcal stuatons. Buzacott [] and shra [5] smultaneously developed nventory model wth constant demand and sngle nflaton rate for all assocated costs. shra [6] consdered dfferent nflaton rate for dfferent costs assocated wth nventory model wth constant rate of demand. An nventory model for stock dependent consumpton and permssble delay n payment under nflatonary condtons was developed by Lao et al. [3]. An EOQ model wth lnear demand and permssble delay n payments was consdered by Sngh [5]. he effect of nflaton and tme value of money were also taken nto account. An nventory model wth nflaton and permssble delay n payments was consdered by Patel and Patel [9]. Generally the products are such that there s no deteroraton ntally. After certan tme deteroraton starts and agan after certan tme the rate of deteroraton ncreases wth tme. Here we have used such a concept and developed the deteroratng tems nventory models. In ths paper we have developed an nventory model for mperfect qualty tems wth dfferent deteroraton rates. Demand of the product s tme and prce dependent for the cycle under tme varyng holdng cost. Shortages are not allowed. o llustrate the model, numercal example s Page

2 provded and senstvty analyss of the optmal solutons for major parameters are also carred out. II. ASSUPIONS AND NOAIONS NOAIONS he followng notatons are used for the development of the model: D(t) : Demand rate s a functon of tme and prce (a+bt-ρp, a>0, 0<b<, ρ>0) c : Purchasng cost per unt p : Sellng prce per unt d : defectve tems (%) -d : good tems (%) : Screenng rate SR : Sales revenue A : Replenshment cost per order z : Screenng cost per unt p d : Prce of defectve tems per unt h(t) : Varable Holdng cost (x + yt, x>0, 0<y<) : Permssble perod of delay n settlng the accounts wth the suppler I e : Interest earned per year I p : Interest pad per year R : Rate of nflaton t : Screenng tme : Length of nventory cycle I(t) : Inventory level at any nstant of tme t, 0 t Q : Order quantty θ : Deteroraton rate durng μ t μ, 0< θ< θt : Deteroraton rate durng, μ t, 0< θ< π : otal relevant proft per unt tme. ASSUPIONS: he followng assumptons are consdered for the development of model. he demand of the product s declnng as a functon of tme and prce. Replenshment rate s nfnte and nstantaneous. Lead tme s zero. Shortages are not allowed. he screenng process and demand proceeds smultaneously but screenng rate () s greater than the demand rate.e. > (a+bt-ρp). he defectve tems are ndependent of deteroraton. Deterorated unts can nether be repared nor replaced durng the cycle tme. A sngle product s consdered. Holdng cost s tme dependent. he screenng rate () s suffcently large. In general, ths assumpton should be acceptable snce the automatc screenng machne usually takes only lttle tme to nspect all tems purchased. Durng the tme, the account s not settled; generated sales revenue s deposted n an nterest bearng account. At the end of the credt perod, the account s settled as well as the buyer pays off all unts sold and starts payng for the nterest charges on the tems n stocks. III. HE AHEAICAL ODEL AND ANALYSIS In the followng stuaton, Q tems are receved at the begnnng of the perod. Each lot havng a d % defectve tems. he nature of the nventory level s shown n the gven fgure, where screenng process s done for all the receved tems at the rate of unts per unt tme whch s greater than demand rate for the tme perod 0 to t. Durng the screenng process the demand occurs parallel to the screenng process and s fulflled from the goods whch are found to be of perfect qualty by screenng process. he defectve tems are sold mmedately after the screenng process at tme t as a sngle batch at a dscounted prce. After the screenng process at tme t the nventory level wll be I(t ) and at tme, nventory level wll become zero due to demand and partally due to deteroraton. Q Also here t= () and defectve percentage (d) s restrcted to (a+bt-ρp) d - () Let I(t) be the nventory at tme t (0 t ) as shown n fgure. Fgure he dfferental equatons whch descrbes the nstantaneous states of I(t) over the perod (0, ) s gven by di(t) = - (a + bt-ρp), dt di(t) + θi(t) = - (a + bt-ρp), dt 0t μ (3) μ t (4) Page

3 di(t) + θti(t) = - (a + bt-ρp), dt t (5) wth ntal condtons I(0) = Q, I(μ ) = S and I() = 0. Solutons of these equatons are gven by I(t) = Q - (at - ρpt + bt ), a μ - t - ρp μ - t + aθ μ - t - ρpθ μ - t 3 3 I(t) = + bμ - t + bθμ - t - aθt μ - t 3 + ρpθt μ - t - bθt μ - t + S + θ μ - t 3 3 a - t - ρp - t + b - t + aθ - t 6 (8) I(t) = - ρpθ - t + bθ - t - aθt - t ρpθt - t - bθt - t 4 (by neglectng hgher powers of θ) After screenng process, the number of defectve tems at tme t s dq. So effectve nventory level durng t t s gven by I(t) = Q(- d) - (at - ρpt + bt ). (9) From equaton (6), puttng t = μ, we have Q = S + aμ- ρpμ + bμ. (0) From equatons (7) and (8), puttng t = μ, we have I(μ ) = a μ -ρp μ + aθ μ - ρpθμ + bμ bθμ - - aθμ μ 3 + ρpθ μ - - bθμ μ + S + θ μ - a - ρp + b aθ - - ρpθ 6 6 I(μ ) = bθ -μ - aθ ρpθ - - bθ - 4 () () (6) (7) So from equatons () and (), we get S = + θμ - a - ρp + b aθ - - ρpθ + bθ -μ aθ + ρpθμ - - bθ - a μ +ρpμ aθμ + ρpθμ - - bμ bθμ + aθ μ 3 - ρpθ μ + bθ μ - (3). Puttng value of S from equaton (3) nto equaton (7), we have + θ μ - t I(t) = + θ μ - a - ρp + b aθ - - ρpθ + bθ -μ aθ + ρpθ - bθμ - - a μ +ρpμ 4 - aθμ + ρpθμ - - bμ bθμ + aθμ μ 3 - ρpθ μ + bθ μ - a μ - t - ρpμ - t + aθ μ - t ρpθμ - t + bμ - t + bθμ - t. 3 - aθt μ - t + ρpθt μ - t - bθt μ - t (4) Smlarly puttng value of S from equaton (3) n equaton (0), we have Page 3

4 Q = +θμ-μ a - ρp + b + aθ - ρpθ + bθ -μ aθ + ρpθ - - bθ - - a μ +ρpμ 4 - aθ μ + ρpθ μ - b μ bθ μ + aθμ μ 3 - ρpθ μ + bθμ μ + aμ - ρpμ + bμ. (5) Usng (5) n (6), we have I(t) = θ μ - a - ρp + b aθ - - ρpθ + bθ -μ aθμ + ρpθμ - bθ - - a μ +ρpμ aθμ + ρpθμ - bμ bθμ + aθμ μ 3 - ρpθ μ + bθ μ - + a μ - t - ρpμ - t + bμ - t. (6) Smlarly, usng (5) n (9), we have I(t) = (-d) +θ μ-μ a - ρp + b aθ - - ρpθ + bθ -μ aθμ + ρpθμ - bθ - a μ +ρpμ aθμ + ρpθμ - bμ bθμ + aθμ μ ρpθ μ + bθ μ - + (-d) aμ - ρpμ + bμ - (at - ρpt + bt ). (7) Based on the assumptons and descrptons of the model, the total annual relevant proft (μ), nclude the followng elements: () Orderng cost (OC) = A (8) () Screenng cost (SrC) = zq (9) () (v) HC = (x+yt)i(t)e 0 dt t μ = (x+yt)i(t)e dt + (x+yt)i(t)e dt 0 t + (x+yt)i(t)e dt + (x+yt)i(t)e dt μ DC = c θi(t)e dt + θti(t)e dt μ (0) () (v) SR = p (a + bt - ρp)e dt + pddq. () 0 o determne the nterest earned, there wll be two cases.e. Case I: (0 ) and Case II: (0 ). Case I: (0 ): In ths case the retaler can earn nterest on revenue generated from the sales up to. Although, he has to settle the accounts at, for that he has to arrange money at some specfed rate of nterest n order to get hs remanng stocks fnanced for the perod to. (v) Interest earned per cycle: Page 4

5 IE = pie a + bt - ρp t e dt 0 Case II: (0 ): (3) In ths case, the retaler earns nterest on the sales revenue up to the permssble delay perod. So (v) Interest earned up to the permssble delay perod s: IE = p Ie a + bt-ρp t e dt + a+b-ρp - (4) 0 o determne the nterest payable, there wll be four cases.e. Interest payable per cycle for the nventory not sold after the due perod s Case I: (0 μ ): (v) IP I(t)e dt μ μ I(t)e dt + I(t)e dt + I(t)e dt μ (5) Case II: (μ μ ): (x) IP I(t)e dt I(t)e dt + I(t)e dt Case III: (μ ): (x) IP 3 (6) I(t)e dt Case IV: (>): (7) (x) IP 4 = 0 (8) (by neglectng hgher powers of θ and R) he total proft (π ), =,,3 and 4 durng a cycle conssted of the followng: π = SR - OC - SrC - HC - DC - IP + IE (9) Substtutng values from equatons (8) to (8) n equaton (9), we get total proft per unt. Puttng µ = v, µ = v n equaton (9), and value of t and Q n equaton (9), we get proft n terms of and p for the four cases wll be as under: π = SR - OC - SrC - HC - DC - IP + IE (30) π = SR - OC - SrC - HC - DC - IP + IE π 3 = SR - OC - SrC - HC - DC - IP 3 + IE π 4 = SR - OC - SrC - HC - DC - IP 4 + IE (3) (3) (33) he optmal value of * and p* whch maxmzes π can be obtaned by solvng equaton (30), (3), (3) and (33) by dfferentatng t wth respect to and p and equate t to zero.e. π (,p) π (,p) p = 0, = 0, =,,3,4 provded t satsfes the condton π (,p) π (,p) p π (,p) p > 0, =,,3,4. π (,p) p IV. NUERICAL EXAPLE (34) (35) Case I: Consderng A= Rs.00, a = 500, b=0.05, c=rs. 5, p d = 5, d= 0.0, z = 0.40, =0000, θ=0.05, x = Rs. 5, y=0.05, v =0.30, v = 0.50, R = 0.06, Ie = 0., Ip = 0.5, =0.05 n approprate unts. he optmal value of * =0.584, p* = , Proft*= Rs and optmum order quantty Q* = Case II: Consderng A= Rs.00, a = 500, b=0.05, c=rs. 5, p d = 5, d= 0.0, z = 0.40, =0000, θ=0.05, x = Rs. 5, y=0.05, v =0.30, v = 0.50, R = 0.06, Ie = 0., Ip = 0.5, =0.0 n approprate unts. he optmal value of * =0.55, p* = , Proft*= Rs and optmum order quantty Q* = Case III: Consderng A= Rs.00, a = 500, b=0.05, c=rs. 5, p d = 5, d= 0.0, z = 0.40, =0000, θ=0.05, x = Rs. 5, y=0.05, v =0.30, v = 0.50, R = 0.06, Ie = 0., Ip = 0.5, =0.0 n approprate unts. he optmal value of * =0.436, p* = , Proft*= Rs and optmum order quantty Q* = Case IV: Consderng A= Rs.00, a = 500, b=0.05, c=rs. 5, p d = 5, d= 0.0, z = 0.40, =0000, θ=0.05, x = Rs. 5, y=0.05, v =0.30, v = 0.50, R = 0.06, Ie = 0., Ip = 0.5, =0.8 n approprate unts. he optmal value of * =0.358, p* = , Proft*= Rs and optmum order quantty Q* = he second order condtons gven n equaton (35) are also satsfed. he graphcal representaton of the concavty of the proft functon s also gven. Page 5

6 Case I and Proft Case II p and Proft Graph Case I p and Proft Graph 5 Case II, p and Proft Graph Case I, p and Proft Graph 6 Case III and Proft Graph 3 Case II and Proft Graph 7 Case III p and Proft Graph 4 Graph 8 Page 6

7 Case III, p and Proft V. SENSIIVIY ANALYSIS On the bass of the data gven n example above we have studed the senstvty analyss by changng the followng parameters one at a tme and keepng the rest fxed. Graph 9 Case IV and Proft Graph 0 Case IV p and Proft Graph Case IV, p and Proft Graph Parameter a x θ A ρ R Parameter a x θ A able Case I Senstvty Analyss % p Proft Q +0% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % able Case II Senstvty Analyss % p Proft Q +0% % % % % % % % % % % % % % Page 7

8 ρ R Parameter a x θ A ρ R Parameter a -0% % % % % % % % % % % % % % % % % % able 3 Case III Senstvty Analyss % p Proft Q +0% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % able 4 Case IV Senstvty Analyss % p Proft Q +0% % x θ A ρ R -0% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % From the table we observe that as parameter a ncreases/ decreases average total proft and optmum order quantty also ncreases/ decreases. Also, we observe that wth ncrease and decrease n the value of x and R, there s correspondng decrease/ ncrease n total proft and optmum order quantty. From the table we observe that as parameter A and ρ ncreases/ decreases average total proft decreases/ ncreases and optmum order quantty ncreases/ decreases. From the table we observe that as parameter θ ncreases/ decreases, there s correspondng decrease/ ncrease n total proft and very mnor decrease/ ncrease n optmum order quantty. From the table we observe that as parameter ncreases/ decreases average total proft ncreases/ decreases and there s very mnor change n optmum order quantty. From the table we observe that as parameter ncreases/ decreases, there s very mnor ncrease/decrease n average total proft and almost no change n optmum order quantty. VI. CONCLUSION In ths paper, we have developed an nventory model for deteroratng tems wth prce and tme dependent demand wth dfferent deteroraton rates. Senstvty wth respect to parameters have been carred out. he results show that wth Page 8

9 the ncrease/ decrease n the parameter values there s correspondng ncrease/ decrease n the value of proft. REFERENCES []. Aggarwal, S.P. and Goel, V.P. (984): Order level nventory system wth demand pattern for deteroratng tems; Eco. Comp. Econ. Cybernet, Stud. Res., Vol. 3, pp []. Buzacott, J.A. (975): Economc order quantty wth nflaton; Operatons Research Quarterly, Vol. 6, pp [3]. Chang, C.., eng, J.. and Goyal, S.K. (008): Inventory lot szng models under trade credts; Asa Pacfc J. Oper. Res., Vol. 5, pp [4]. Chang, H.C. (004): An applcaton of fuzzy set theory to the EOQ model wth mperfect qualty tems; Comput Oper. Res., Vol. 3 pp [5]. Chung, H.J. and Dye, C.Y. (00): An nventory model for deteroratng tems under the condton of permssble delay n payments; Yugoslav Journal of Operatonal Research, Vol., pp [6]. Chung, K.J., Goyal, S.K. and Huang, Y.F. (005): he optmal nventory polces under permssble delay n payments depredatng on the orderng quantty; Internatonal Journal of producton economcs, Vol. 95, pp [7]. Covert, R.P. and Phlp, G.C. (973): An EOQ model for tems wth Webull dstrbuton deteroraton; AIIE ransactons, Vol. 5, pp [8]. Ghare, P.. and Schrader, G.F. (963): A model for exponentally decayng nventores; J. Indus. Engg., Vol. 4, pp [9]. Goyal, S.K. (985): Economc order quantty under condtons of permssble delay n payments, J. O.R. Soc., Vol. 36, pp [0]. Goyal, S.K. and Gr, B. (00): Recent trends n modelng of deteroratng nventory; Euro. J. Oper. Res., Vol. 34, pp. -6. []. Hapng, U. and Wang, H. (990): An economc orderng polcy model for deteroratng tems wth tme proportonal demand; Euro. J. Oper.Res., Vol. 46, pp. -7. []. Hauck, Z. and Voros, J. (04): Lot szng n case of defectve tems wth nvestments to ncrease the speed of qualty control; Omega, do:0.06/j.omega [3]. Lao, H.C., sa, C.H. and Su,.C. (000): An nventory model wth deteroratng tems under nflaton when a delay n payment s permssble; Int. J. Prod. Eco., Vol. 63, pp [4]. andal, B.N. and Phujdar, S. (989): A note on nventory model wth stock dependent consumpton rate; Opsearch, Vol. 6, pp [5]. shra,r.b. (975): A study of nflatonary effects on nventory systems; Logstc Spectrum, Vol. 9, pp [6]. shra,r.b. (979): A note on optmal nventory management under nflaton; Naval Research Logstc Quarterly, Vol. 6, pp [7]. Patel, R. and Parekh, R. (04): Deteroratng tems nventory model wth stock dependent demand under shortages and varable sellng prce, Internatonal J. Latest echnology n Engg. gt. Appled Sc., Vol. 3, No. 9, pp [8]. Patel, S. S. and Patel, R.D. (0) : EOQ model for webull deteroratng tems wth mperfect qualty and tme varyng holdng cost under permssble delay n payments; Global J. athematcal Scence: heory and Practcal, Vol. 4, No. 3, pp [9]. Patel, S.S. and Patel, R. (03): An nventory model for webull deteroratng tems wth lnear demand, shortages under permssble delay n payments and nflaton; Internatonal Journal of athematcs and Statstcs Inventon, Vol., No., pp [0]. Raafat, F. (99): Survey of lterature on contnuous deteroratng nventory model, J. of O.R. Soc., Vol. 4, pp []. Rosenblat,.J. and Lee, H.L. (986): Economc producton cycles wth mperfect producton process; IIE rans., Vol. 8, pp []. Ruxan, L., Hongje, L. and awhnney, J.R. (00): A revew on deteroratng nventory study; J. Servce Sc. and management; Vol. 3, pp [3]. Salameh,.K., Abbound, N.E., E-Kassar, A.N. and Ghattas, R.E. (003): Contnuous revew nventory model wth delay n payment; Internatonal Journal of producton economcs, Vol. 85, pp [4]. Salameh,.K. and Jaber,.Y. (000): Economc producton quantty model for tems wth mperfect qualty; J. Producton Eco., Vol. 64, pp [5]. Sngh, S. (0) An economc order quantty model for tems havng lnear demand under nflaton and permssble delay n payments; Internatonal J. of Computer Applcatons, Vol. 33, pp [6]. Wee, H.., Yu, J., and Chen,.C. (007): Optmal nventory model for tems wth mperfect qualty and shortages backorderng; Omega, Vol. 35, pp Page 9