A Multi Item Integrated Inventory Model with Reparability and Manufacturing of Fresh Products

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1 Moder Appled Scece; Vol. 1, No. 7; 216 ISSN E-ISSN Publshed by Caada Ceter of Scece ad Educato A Mult Item Itegrated Ivetory Model wth Reparablty ad Maufacturg of Fresh Products Pky Saxea 1, S. R. Sgh 2 & Isha Sagal 3 1 Departmet of Mathematcs, Krsha Isttute of Egeerg ad Techology, Ghazabad, Ida 2 Departmet of Mathematcs, C.C.S Uversty, Meerut, Ida 3 Ceter for Mathematcal Sceces, Baasthal Uversty, Rajastha, Ida Correspodece: S. R Sgh, Departmet of Mathematcs, C.C.S Uversty, Meerut, 251, Ida. Tel: E-mal: shvrajpudr@gmal.com Receved: October 18, 215 Accepted: March 5, 216 Ole Publshed: Aprl 28, 216 do:1.5539/mas.v17p74 URL: Abstract I ths paper a mult tem tegrated vetory model s preseted wth reparablty of retured tems. It s assumed here that oly a certa rato of retured tems ca be repared ad the remag stock of retured tems s salvaged. By usg these retured tems, the waste ca be reduced, whch pollute the evromet. Ths s a gree supply cha where the demad for the products s sellg prce depedet ad producto rate s take as a fucto of demad rate. The shortages are allowed here. A umercal example ad sestvty aalyss are also preseted to llustrate the model. Keywords: tegrated vetory, reparablty, mult tems, deterorato, shortages 1. Itroducto I so may dustral systems, there s more tha oe maufacturg plat ad they produce dfferet type of tems. Most of the classcal vetory system tres to calculate the ecoomc producto quatty by whch the total cost of the system ca be optmzed. But these models are developed oly for sgle tem. I ths model we have preseted a tegrated vetory model for mult tems. Cosderg mult tems vetory model, (Be-Daya & Raouf, 1993 have developed a approach for more realstc ad geeral cocept of budgetary ad floor costrats, where the demad of tems follows a uform probablty dstrbuto fucto. A mult tem vetory model for deteroratg tems s developed by (Bhattacharya, 25. (Roy et al., 1995 developed mult deteroratg tems wth costrat space ad vestmet ad obtaed some terestg results.( Balkh,29 has troduced a geeral model for mult tem producto vetory system whch the cost parameters are treated as a arbtrary fucto of tme. (Yadav et al., 21 developed a fuzzy mult tem producto vetory model wth relablty ad flexblty uder lmted storage capacty wth deterorato va geometrc programmg. (Sghal & Sgh, 213 troduced a volume flexble mult tems vetory system wth mprecse evromet. (Tayal et al., 214 preseted a mult tem vetory model for deteroratg tems wth exprato date ad allowable shortages. However, most of the classcal producto vetory models eve cocered wth mult tems the atteto s gve oly for the optmalty of separate member of the tegrated system. For ay successful supply cha coordato betwee the vedor ad the buyer s requred. Ths close relatoshp betwee vedor ad buyer s a key of success for ay busess orgazato. The, a ew approach of tegrato of all the fuctos a supply cha was detfed.( Sgh & Dksha,29 developed a tegrated cooperatve vetory model for vedor ad buyer uder progressve cre perod whch demad s assumed to be a multvarate fucto. (Hadd et al., 21 developed a tegrated producto vetory model for schedulg ad perfect mateace. Ths work tegrates, smultaeously, the decsos of prevetve mateace ad job order sequecg for a sgle mache.( So & Patel,214 preseted a optmal decso polcy for tegrated vedor-buyer vetory system cocerg defectve tems wth varable lead tme ad servce level costrat. (Tayal et al, 216 troduced a tegrated producto vetory model for pershable products wth trade cre perod ad vestmet preservato techology. Cocerg the deterorato ad reparablty a vetory model s more realstc ad geeral cocept. (Park, 74

2 Moder Appled Scece Vol. 1, No. 7; studed a producto vetory system for a sgle product wth deteroratg raw materals. (Ya & Cheg, 1998 troduced a optmal producto stoppg ad restartg tmes for a EOQ model wth deteroratg tems. (Maty & Mat, 29 developed optmal vetory polces for deteroratg complemetary ad substtute tems. (Sgh et al., 21 preseted a EOQ model wth Pareto dstrbuto for deterorato, Trapezodal type demad ad backloggg uder trade cre polcy. Sgh et al. (213 troduced a EOQ model wth volume aglty, varable demad rate, Webull deterorato rate ad flato. (Tayal et al.,214 developed a two echelo supply cha model for deteroratg tems wth effectve vestmet preservato techology after that (Tayal et al.,214 studed a vetory model for deteroratg tems wth seasoal products ad a opto of a alteratve market. (Sgh et al., 215 preseted a EPQ vetory model for o-stataeous deteroratg tem wth tme depedet holdg cost ad expoetal demad rate. (Sgh et al., 216 developed a ecoomc order quatty model for deteroratg products havg stock depedet demad wth trade cre perod ad preservato techology. Oe of the frst authors was (Schrady, 1967 who developed a smple heurstc procedure for determg the lot szes of remaufacturg ad maufacturg lots. He proposed a smple EOQ-techque that optmzes the sum of fxed ad holdg costs per tme ut. (Teuter.2 geeralzed the results of Schrady a way that he examed dfferet structures of a remaufacturg cycle. Hs aalyss cocludes that t s ot effcet f more tha oe remaufacturg lot ad more tha oe maufacturg lot are establshed the same repar cycle. (Saxea et.al. 213 preseted two-warehouse producto vetory model wth varable demad ad permssble delay paymet uder flato. (Saxea et.al.214 geeralzed producto model uder fuzzy evromet. They compared both the results obtaed cosderg crsp data as well as fuzzy data. (Sgh & Sgh, 213 preseted gree supply cha model wth product remaufacturg uder volume flexble evromet. (Sgh & Prasher, 214 troduced a producto vetory model wth flexble maufacturg, radom mache breakdow ad stochastc repar tme. I ths paper we have preseted a tegrated vetory model for mult tems a closed loop supply cha wth reparablty of retured ad collected tems. The shortages are allowed for retaler ad assumed to be completely lost. The repared tems are assumed to be equvalet to fresh products. 2. Assumptos ad Notatos 2.1 Assumptos 1. Ths s a mult tem vetory model, preseted for the tegrated producto of ew products ad reparablty of collected tems. 2. The demad for the products s a fucto of sellg prce. 3. The products after reparablty are assumed to be equvalet to ew products. 4. The producto rate s assumed to be a fucto of demad rate. 5. The used tems are collected at a rate of b( α βs. 6. A certa rato γ of collected tems, whose qualty level s acceptable for reparablty, s used for producto ad rest of the materal s salvaged. 7. The tems are assumed to be deteroratg ature. 8. The shortages are allowed for retaler ad occurrg shortages are assumed to be completely lost. 2.2 Notatos deterorato rate α, β demad parameters for th tem s 1, s 2 sellg prce per ut for the producer ad the retaler for th tem a producto parameter for th tem, a 1 b collecto parameter for th tem, b<1 t 1 tme for remaufacturg for th tem t 2 tme at whch vetory level for remaufactured th tem becomes zero t 3 tme up to whch producto of fresh th tems occur T legth of the complete cycle for th tem I r (t vetory level for remaufactured th tem at ay tme t 75

3 Moder Appled Scece Vol. 1, No. 7; 216 I m (t vetory level for fresh produced th tem at ay tme t I R (t vetory level of collected th tem at ay tme t I s (t retaler s vetory level for th tem at ay tme t v the tme at whch the vetory level of th tem becomes zero for retaler umber of repleshmet cycles for the retaler c m procuremet cost per ut for th tem c R acqusto cost per ut for th tem P m producto cost per ut for th tem P r remaufacturg cost per ut for th tem h r, h m, h R, h s holdg cost per ut for th tem for remaufactured tems, produced tems, collectve tems ad for the retaler O orderg cost per order c 1 purchasg cost per ut for th tem for the retaler c 2 lost sale cost cost per ut for th tem for the retaler K 1, K 2, K 3 set up cost for remaufacturg, fresh producto ad for the collecto I 1 al vetory level for the retaler for th tem Q 2 shortage amout for th tem 3. Mathematcal Model I ths the used tems are collected from the market ad a certa rato of these tems s repared. I the above metoed fgure the remaufacturg graph s show. Durg [, t 1 ] remaufacturg occurs ad the vetory level of collected tems decreases ad becomes zero at tme t 1. Durg [t 1, t 2 ] the vetory level of repared tems depletes due to combed effect of demad ad deterorato. After t = t 1 the vetory level of collected tems aga pled up. At t = t 2 the maufacturg of fresh products start ad occurs up to t = t 3. The below metoed fgure ( shows the vetory level of ths system. Fgure 1. Ivetory tme behavor for repared tems, fresh produced tems ad retured tems 76

4 Moder Appled Scece Vol. 1, No. 7; 216 These are the dfferetal equatos showg the vetory wth the varato tme for repared tems, fresh producto ad collected tems. dir + Ir = ( a ( α βs t t1 ( dir + Ir = ( α βs t1 t t2 (2 dim + Im = ( a ( α βs t2 t t3 (3 dim + Im = ( α βs t3 t T (4 di R + IR = ( b ( α βs t t1 (5 di R + IR = b( α βs t1 t T (6 wth boudary coos: I ( =, I ( t =, I ( t =, I ( T =, I ( t =, I ( = I ( T (7 r r 2 m 2 m R 1 R R The soluto of these above metoed equatos are gve as follow: ( a I ( ( (1 r t = α βs e t t1 (8 ( α βs ( t2 t I ( ( r t e = t1 t t2 (9 ( a ( 2 ( ( (1 t Im t α βs e = t2 t t3 (1 ( α βs ( ( ( t Im t e = t3 t T (1 ( α βs ( 1 ( ( (1 t I R t b a e = t t1 (12 b ( 1 ( ( (1 t IR t s e = α β t1 t T (13 Fgure 2. Retaler s vetory 77

5 Moder Appled Scece Vol. 1, No. 7; 216 If I s (t deotes the retaler s vetory level for th tem at ay tme t, the the dfferetal equatos showg the vetory level at ay tme t are gve as follow: dis + Is = ( α βs 2 t v (14 dis T = ( α βs2 v t (15 wth boudary coo Is ( v = (16 The soluto of these above metoed equatos are gve as follow: ( α βs2 ( ( ( v t Is t e = t v (17 T Is = ( α βs 2 ( v t v t (18 The total average cost for the th tem for the maufacturer s gve by: 1 TAC.. m = [Procuremet cost + Acqusto cost + Producto cost + T Remaufacturg cost + Holdg cost + Salvage cost + Set up cost] (19 The total average cost for the th tem for the retaler s gve by: TAC.. s = [Purchasg cost + Holdg cost + orderg cost + lost sale cost] (2 T Dfferet assocated costs for maufacturer: t3 m 1 t2 Procuremet cost = c a( α β s Procuremet cost = cma( α βs ( t3 t2 (2 Acqusto cost = c b( α β s T R Acqusto cost = crbt( α βs (22 Producto cost = t3 P a ( α β s m 1 t2 Producto cost = Pma( t3 t2 ( α βs (23 Remaufacturg cost = t1 Pa ( α β s r 1 Remaufacturg cost = Pat r 1 ( α βs (24 Set up cost = K 1 + K 2 + K 3 (25 Salvage cost = sv{(1 γ b( α βs } T (26 Holdg cost = t1 t2 t3 T t1 T h I + h I + h I + h I + h I + h I r r r r m m m m R R R R t1 t2 t3 t1 78

6 Moder Appled Scece Vol. 1, No. 7; 216 Holdg cost = 1 ( 2 1 ( t 1 ( t a e α t βs e 1 hr{ ( α βs ( t1 + + ( + ( t1 t2 } ( 2 3 ( 3 ( t t 1 ( T a e α t βs e 1 + hm{ ( α βs {( t3 t2 + } + {( ( T t3}} 1 ( 1 ( 1 t ( t b a e α T βs e 1 + hr{ ( α βs( t1 + b {( + ( T t1 }} (27 Dfferet assocated costs for retaler: Purchasg cost = ( I1 ( + Q2 c1 where I ( α β s 2 v 1 ( ( = e T Q = ( α β s η 2 2 v ( α β s 2 P.C. = { v ( T e + ( α 2 ( βs η v} c1 (28 Holdg cost = v h I s s v ( α 2 βs e 1 H. Cs = hs ( v (29 Orderg cost = O (3 T Lost sale cost = c2 ( α βs2 T LSC.. = c2( α βs2( v (3 Hece the total cost per ut tme of the gve vetory model as a fucto of t 1, t 2, t 3, v ad T say T.A.C.(t 1, t 2, t 3, v, T s gve by v 1 T. AC.. = { cma( α βs( t3 t2 + crbt ( α βs + Pma( t3 t2 ( α βs + T Pat ( α β s + K + K + K + s {(1 γ b( α β s } T+ r v 1 1 ( 2 1 ( 2 3 ( t 1 ( t t 1 ( t a e α t βs e a e 1 hr{ ( α βs( t1 + + ( + ( t1 t2} + hm{ ( α βs {( t3 t2 + } + ( T t3 t1 ( α βs e 1 ( b a 1 e {( ( T t3}} + hr{ ( α βs ( t1 + ( α β s e 1 T ( α β s T b + T t + e + s v c ( t1 T 1 2 v {( ( }} [{ ( ( α β 2 η( } 1 v ( α 2 βs e 1 T hs ( v + O + c2 ( α βs 2( v ] (32 Equato (32 deotes the cost fucto of the system terms of t 1, t 2, t 3, v ad T. To fd out the optmal soluto of ths system we have to fd out the optmal values of t 1, t 2, t 3, v ad T. We have some relatos 79

7 Moder Appled Scece Vol. 1, No. 7; 216 betwee these varables. t1 t2 t3 T v T t 1 ( t 2 t 1 (33 (34 ( a (1 e = ( e (35 ( 2 3 ( 3 ( (1 t t a e ( e T t = (36 1 ( 1 ( (1 t (1 t b a e b e T = (37 Equatos (33 ad (34 are the essetal coos for the exstece of ths model. Equatos (35 ad (36 show the vetory level I r (t ad I m (t at t=t 1 ad t=t 3. Equato (37 demostrates that the vetory level of collected ad retured tems wll be the same at t= ad t=t. Usg the equatos (35 - (37 the values of t 1, t 2 ad t 3 ca be fd the form of T, It ca be sad that t 1 = f 1 (T, t 2 = f 2 (T, t 3 = f 3 (T Therefore the total average cost wll be the fucto of varables T ad v. 4. Numercal Example A umercal example s carred out to llustrate the model. Correspodg to the below metoed parametrc values the optmal value of tme v, T ad T.A.C are obtaed for dfferet three products. Table 1. Parameters Product 1 Product 2 Product 3 α γ s s a b β h r h m h R c m c R P m P r S v K K K c h s O v T T.A.C

8 Moder Appled Scece Vol. 1, No. 7; Fgure 3. Covexvty of the T.A.C 1 fucto 5. Sestvty Aalyss Correspodg to dfferet assocated parameters, a sestvty aalyss s carred out to check the stablty of the model. The aalyss has bee doe wth the parameters, b, β 1, γ 1, α 1, s 1 ad a 1 takg oe parameter at a tme ad other varables uchaged ad s show below metoed tables. Table 2. Sestvty aalyss wth respect to deterorato parameter (: % varato v 1 T 1 T.A.C 1-2% % % % % % % % % Fgure 4. Varato T.A.C 1 wth the varato 81

9 Moder Appled Scece Vol. 1, No. 7; 216 Table 3. Sestvty aalyss wth respect to parameter b: % varato b b v 1 T 1 T.A.C 1-2% % % % % % % % % b Fgure 5. Varato T.A.C 1 wth the varato b Table 4. Sestvty aalyss wth respect to demad parameter (β 1 : % varato β 1 β 1 v 1 T 1 T.A.C 1-2% % % % % % % % % β Fgure 6. Varato T.A.C 1 wth the varato β 1 82

10 Moder Appled Scece Vol. 1, No. 7; 216 Table 5. Sestvty aalyss wth respect to parameter γ 1 : % varato γ 1 γ 1 v 1 T 1 T.A.C 1-2% % % % % % % % % Fgure 7. Varato wth the varato γ1 γ1 Table 6. Sestvty aalyss wth respect to demad parameter (α: % varato α 1 α 1 v 1 T 1 T.A.C 1-2% % % % % % % % % α Fgure 8. Varato wth the varato α1 83

11 Moder Appled Scece Vol. 1, No. 7; 216 Table 7. Sestvty aalyss wth respect to sellg prce (s1 % varato s 11 s 11 v 1 T 1 T.A.C 1-2% % % % % % % % % s Fgure 9. Varato T.A.C 1 wth the varato s 1 Table 8. Sestvty aalyss wth respect to producto parameter (a: % varato a 1 a 1 v 1 T 1 T.A.C 1-2% % % % % % % % % a Fgure 1. Varato wth the varato a1 5.1 Observatos Table 2 shows the varato T.A.C. wth the varato deterorato parameter. From ths table t s 84

12 Moder Appled Scece Vol. 1, No. 7; 216 observed that wth the cremet deterorato rate the T.A.C. of the system creases. 1. From table 3 t s observed that a cremet parameter b results a decrease T.A.C. of the system. 2. Table 4 lsts the varato demad parameter (β 1. It s observed from ths table that as the value of β 1 creases, the T.A.C. of the system shows the reverse effect. 3. Table 5 ad table 6 show the varato parameter γ ad demad parameter α ad from these t s observed that a cremet both the parameters result a cremet T.A.C. 4. Table 7ad table 8 show the varato T.A.C. wth the chages sellg prce s 1 ad producto parameter a respectvely. It s observed that a cremet sellg prce ad producto parameter result a decle T.A.C. 6. Cocluso I ths paper we have preseted a tegrated producto vetory model for mult tems. Ths s a closed loop supply cha, troduced wth the producto of ew tems ad remaufacturg of collected ad retured tems wth deterorato. The demad rate s take as a fucto of sellg prce whch shows a very realstc pheomeo. A umercal example s show to llustrate the model. The model s optmzed ad the covexty of the model s show. A sestvty aalyss s also performed to check the stablty of the model. For future scope the model ca be exteded for stochastc demad rate ad wth learg ad forgettg effects for producto ad maufacturg. Refereces Balkh, Z. T. (29. Mult- tem producto vetory systems wth budget costrats. Proceedgs of the1stiteratoal Coferece o maufacturg Egeerg, Qualty ad Producto Systems (MEQAPS 9, Be-Daya, M., & Raouf, A. (1993. O the costraed mult-tem sgle-perod vetory Problem. Iteratoal joural of Producto Maagemet, 13, Bhattacharya, D. K. (25. Producto, maufacturg ad logstcs o mult-tem vetory. Europea Joural of Operatoal Researc, 162, Hadd, L. A., Turk, U. M. A., & Rahm, M. A. (21. A tegrated cost model for producto schedulg ad perfect mateace. Iteratoal Joural of Mathematcs Operatoal Research, 3(4, Leard, J. D., & Roy, B. (1995. Mult-tem vetory cotrol: A mult-crtera vew. Europea Joural of Operatoal Research, 87, Maty, K., & Mat, M. (29. Optmal vetory polces for deteroratg complemetary ad substtute tems, Iteratoal Joural of Systems Scece, 4, Park, K. S. (1983. A tegrato producto vetory model for decayg raw materals. Iteratoal Joural of Systems Scece, 14, Saxea, P., & Sgh, S. R. (213. A two warehouse producto vetory model wth varable demad ad permssble delay paymet uder flato. Iteratoal Joural of soft computg ad Egeerg, 3(5, Saxea, P., & Sgh, S. R. (214. Fuzzy warehouse vetory model for tems wth mperfect qualty uder trade cre polcy ad flatoary coos. Iovatve Applcatos of Computatoal Itellgece o Power, Eergy ad Cotrols wth ther mpact o Humaty (CIPECH, Ghazabad, Schrady, D. A. (1967. A determstc vetory model for reparable tems Naval Research Logstcs Quarterly, 14, Sgh, N., Vash, B., & Sgh, S. R. (21. A EOQ model wth Pareto dstrbuto for deterorato, Trapezodal type demad ad backloggg uder trade cre polcy. The IUP Joural of Computatoal Mathematcs, 3(4, Sgh, S. R., & Prasher, L. (214. A producto vetory model wth flexble maufacturg, radom mache breakdow ad stochastc repar tme. Iteratoal Joural of Idustral Egeerg Computatos, 5 (4, Sgh, S. R, Sharma, R., & Chauha, A. (214. Two echelo supply cha model for deteroratg tems wth effectve vestmet preservato techology. Iteratoal Joural Mathematcs Operatoal Research, 6(,

13 Moder Appled Scece Vol. 1, No. 7; 216 Sgh, S. R., Sharma, R., & Sgh, A. P. (215. A EPQ model for o-stataeous deteroratg tem wth tme depedet holdg cost ad expoetal demad rate. Iteratoal Joural of Operatoal Research, 23(2, Sgh, S. R., Gupta, V., & Basal, P. (213. EOQ model wth volume aglty, varable demad rate, Webull deterorato rate ad flato.iteratoal Joural of Computer Applcatos, 72(23, 1-6. Sgh, S. R., Chauha, A., & Sharma, R. (214. A deteroratg producto vetory problem wth space restrcto. Joural of Iformato & Optmzato Sceces, 35(3, Sgh, S. R., Khuraa, D., & Tayal, S. (216. A ecoomc order quatty model for deteroratg products havg stock depedet demad wth trade cre perod ad preservato techology. Ucerta Supply Cha Maagemet, 4(, Sgh, S. R., & Sgh, N. (213. Gree supply cha model wth product remaufacturg uder volume flexble evromet. Proceda Techology, 1, Sgh, S. R., & Dksha. (29. Itegrated vedor-buyer cooperatve model wth multvarate demad ad progressve cre perod. Joural of Operatos Maagemet, 8(2, Sghal, S., & Sgh, S. (213. Volume flexble mult tems vetory system wth mprecse evromet. Iteratoal Joural of Idustral Egeerg Computatos, 4(4, So, H. N., & Patel, K. A. (214. Optmal polces for vedor-buyer vetory system volvg defectve tems wth varable lead tme ad servce level costrat. Iteratoal Joural of Mathematcs Operatoal Research, 6(3, Tayal, S, Sgh S. R., & Sharma, R. (214. A vetory model for deteroratg tems wth seasoal products ad a opto of a alteratve market. Ucerta Supply Cha Maagemet, 3, Tayal, S., Sgh S. R., & Sharma, R. (214. A mult tem vetory model for deteroratg tems wth exprato date ad allowable shortages. Ida Joural of Scece ad Techology, 7(4, Tayal, S., Sgh, S. R., & Sharma, R. (216. A tegrated producto vetory model for pershable products wth trade cre perod ad vestmet preservato techology. Iteratoal Joural of Mathematcs Operatoal Research, 8(2, Teuter, R. H. (2. Ecoomc orderg quattes for recoverable tem vetory system. Naval Research Logstcs, 48, Yadav, D., Pudr, S., & Kumar, R. (21. A fuzzy mult tem producto model wth relablty ad flexblty uder lmted storage capacty wth deterorato va geometrc programmg. Iteratoal Joural of Mathematcs Operatoal Research, 3(, Ya, H., & Cheg, T. C. E. (1998. Optmal producto stoppg ad restartg tmes for a EOQ model wth deteroratg tems. Joural of the Operatoal Research Socety, 49, Copyrghts Copyrght for ths artcle s retaed by the author(s, wth frst publcato rghts grated to the joural. Ths s a ope-access artcle dstrbuted uder the terms ad coos of the Creatve Commos Attrbuto lcese ( 86

Multi Objective Fuzzy Inventory Model with. Demand Dependent Unit Cost and Lead Time. Constraints A Karush Kuhn Tucker Conditions.

Multi Objective Fuzzy Inventory Model with. Demand Dependent Unit Cost and Lead Time. Constraints A Karush Kuhn Tucker Conditions. It. Joural of Math. Aalyss, Vol. 8, 204, o. 4, 87-93 HIKARI Ltd, www.m-hkar.com http://dx.do.org/0.2988/jma.204.30252 Mult Objectve Fuzzy Ivetory Model wth Demad Depedet Ut Cost ad Lead Tme Costrats A