An economic production model for deteriorating items and time dependent demand with rework and multiple production setups

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1 J Ind Eng Int (17) 13: DOI 1.17/s ORIGINAL RESEARCH An economic production model for deteriorting items nd time dependent demnd with rework nd multiple production setups R. Uthykumr 1 S. Thrni 1 Received: Februry 17 / Accepted: 1 April 17 / Published online: 5 April 17 Ó The Author(s) 17. This rticle is n open ccess publiction Abstrct Recently, much emphsis hs given to study the control nd mintennce of production inventories of the deteriorting items. Rework is one of the min issues in reverse logistic nd green supply chin, since it cn reduce production cost nd the environmentl problem. Mny reserchers hve focused on developing rework model, but few of them hve developed model for deteriorting items. Due to this fct, we tke up productivity nd rework with deteriortion s the mjor concern in this pper. In this pper, production-inventory model with deteriortive items in which one cycle hs n production setups nd one rework setup (n, 1) policy is considered for deteriorting items with stock-dependent demnd in cse 1 nd exponentil demnd in cse. An effective itertive solution procedure is developed to chieve optiml time, so tht the totl cost of the system is minimized. Numericl nd sensitivity nlyses re discussed to exmine the outcome of the proposed solution procedure presented in this reserch. Keywords Inventory Rework Production Time dependent demnd Introduction Mngement Scientists nd Industril Engineers hve given more importnce to three dimensions of inventory mngement. The first dimension, utiliztion, mesures the efficiency through which firms use their inventories. It is & S. Thrni thrnimths@gmil.com 1 Deprtment of Mthemtics, The Gndhigrm Rurl Institute-Deemed University, Gndhigrm, Tmilndu, Indi opertionlized by inventory specultion, mesure of the percentge of units mnufctured s inventory. The higher proportion of units, the greter the specultion, nd the lower efficiency with which the mnufcturing orgniztions fulfill customers demnds, nd built the inventory. The second performnce dimension, effectiveness, cptures the qulity of the process output. This dimension is interpreted brodly to include durbility, service level, relibility of supply, quntity per unit pckge, nd dvertising to estblish brnd imge. As customers judge the goods on the bsis of both price nd qulity, the choice of qulity is often n importnt fctor of n industry. The lower qulity results in lower effectiveness with which firm/industry meets customers demnds. The third dimension, productivity, is mesure of trnsformtion efficiency nd is reported s the inventory turnover rtio. The higher turnover cuses higher productivity with which firm/industry uses its inventories. Production-inventory model plys dominnt role in production scheduling nd plnning. The EPQ model is commonly used by prctitioners in the fields of production nd inventory mngement to ssist them in mking decision on optimum production nd totl cost. For the determintion of optiml downtime, uptime of production, nd production quntity, it is required to minimize the expected totl cost. The totl cost of production is dependent on production rte, demnd rte, nd rte of deteriorting items. Reverse logistics is for ll opertions relted to the reuse of products nd mterils. It is the process of moving goods from their typicl finl destintion for the purpose of cpturing vlue, or proper disposl. Remnufcturing nd refurbishing ctivities lso my be included in the definition of reverse logistics. Growing green concerns nd dvncement of green supply chin mngement concepts 13

2 5 J Ind Eng Int (17) 13: nd prctices mke it ll the more relevnt. The number of publictions on the topic of reverse logistics hs incresed significntly over the pst two decdes. Deteriortion, in generl, my be considered becuse of vrious effects on the stock, some of which re dmge, spoilge, obsolescence, decy, decresing usefulness, nd mny more. For exmple, in mnufcturing industries like drugs, phrmceuticls, food products, rdioctive substnces, the item deteriortes over. Although qulity hs received significnt ttention in the mnufcturing industries nd its economic benefits re beyond ny doubt, lot of questions rise such s how much to invest nd order, when to replenish nd deliver, nd in wht bsis the industries mintin sustinble competitive dvntge. A model is progressed here to guide firm/ industry is ddressing these questions. The firm/industry produces single product nd opertes in n oligopolistic competition. Demnd for the product in n industry depends on price, time, nd performnce qulity with time. In generl, incresing demnd functions over time re exponentil, stock-dependent, nd etc. In the cse of some products (e.g., new electronic chips, sesonl goods, new spre prts of mchinery systems, etc.), the demnd rte is likely to increse very fst, lmost exponentilly, with time. In ddition, in the supermrket, we often see tht lrge piles of goods displyed give the customer wider selection of goods nd increse the probbility of mking sle. The effect of this dependence is tht the retilers hve incentive to keep higher levels of inventory in spite of higher holding costs s long s the item is profitble nd the demnd is n incresing function of the inventory level. In the competitive mrketing system, price is fixed throughout fixed. The concept of qulity hs lwys emerged s focl point in business nd its populrity hs significntly incresed, since it becomes powerful competitive tool in the mrket of stndrd products. The problem of optiml plnning work nd rework processes belongs to the brod field of production-inventory model which regrds ll kinds of reuse processes in supply chins. These processes im to recover defective or used product items in such wy tht they meet the qulity level of good item. The benefits re regining the mteril nd vlue dded nd improving the environmentl protection. Mngement science literture minly recommends bout qulity s conformnce qulity. According to these, the im of qulity plnning is to flourish process cpble of meeting qulity gols under certin operting conditions. As result, in n imperfect production process, the defective items should be reworked or rejected. The defective items of some products (e.g., textile, toys, electronic goods, etc.) cn be reworked or re mnufctured t cost. The economic nd socil costs of disposl increse s lndfill gets filled up nd environmentl protection groups protest ginst dumping in third world countries. Incresing production knowledge decreses unit production cost, wheres lower vlues of product relibility fctor increse development cost. Therefore, productivity nd qulity knowledge cn be developed through induced nd utonomous lerning to strengthen compny position. The findings of our reserch extend to those of prior studies, which mostly hve concentrted only on the issue of n EPQ inventory system for defective products with the considertion of imperfect production processes, rework, nd constnt demnd. Our study indictes tht the demnd is vrible nd to develop mthemticl model nd design n itertive solution procedure to effectively increse productivity nd to reduce the expected totl cost of n EPQ inventory system involving defective products with rework. Literture review One competitive dvntge in globl competition mrket is producing high-qulity products. To produce high-qulity products, defective products should be eliminted through 1% screening. For n economic reson nd environmentl concerns, defective products re reworked to become serviceble items. Rework process is lso one importnt issue in reverse logistics where used products re reworked to reduce wste nd environment problems. The reserch on vrious replenishment policies is typiclly ddressed by developing proper mthemticl models in inventory mngement system tht consider fctors such s demnd rte, deteriortion of inventory items, permissible dely in pyments, effects of infltion nd time vlue of money, shortges, nd finite plnning horizon, to mention only few. Some of the prominent ppers re discussed below. The erliest reserch tht focused on rework nd remnufcturing process ws estblished by Schrdy (1967). Since then, studies on rework hve ttrcted mny reserchers. Khouj () delt with direct rework for economic lot sizing nd delivery scheduling problem (ELDSP). A model for two-stge mnufcturing system with production nd rework processes ws progressed by Buscher nd Lindner (7). Srkr et l. (14) flourished n economic production quntity model for defective products with bckorders nd rework process in single-stge production system. Hyek nd Slmeh (1) ssumed tht ll defective items produced re repirble nd chieved n optiml operting policy for the EMQ model under the effect of reworking ll defective items. Mishr nd Singh (11) developed 13

3 J Ind Eng Int (17) 13: production-inventory model for time dependent deteriorting item with production distribution nd gives nlyticl solution to determine the optiml production time during norml nd disrupted production s. Crdens-Brron (9) developed n EPQ type inventory model with plnned bckorders for single deteriorting product, which is mnufctured in singlestge mnufcturing system tht genertes imperfect qulity products, nd ll these defective products re reworked in the sme cycle. Crdens-Brron (9) corrected some mthemticl expressions in the work of Srker et l. (8). Wee et l. (13) revisited the work by Crdens-Brron (9). Tleizdeh et l. (1) promoted production quntity model by considering rndom defective items, repir filure, nd service-level constrints. Lter, Tleizdeh et l. (11) developed production-inventory models of two joint systems with nd without rework. Chung nd Wee (11) explined with short life-cycle deteriorting product remnufcturing in green supply chin inventory control system. Yssine et l. (1) nlyzed the shipment of imperfect qulity items during single production runs nd over multiple production runs. Some reserch on rework lso focuses on production policy to minimize production nd inventory costs. Dobos nd Richter (4) flourished production nd recycling inventory model with n number of recycling lots nd m number of production lots. Teunter (4) developed EPQ models with rework with respect to two policies. In the first policy, m number production lots re lternted with one recovery lot, (m, 1) policy; in the other policy, one production lot is lternted with n recovery lots, (1, n) policy. Lter, Widydn nd Wee (1) introduced n lgebric pproch to solve Teunter (4) models efficiently nd effectively. Mthemticl models for the optiml EPQ, optiml production, rework frequency, nd their sequence re described by Liu et l. (9). They found tht the (m, 1) policy hs bigger chnce to rech n optiml solution compre with (1, 1), (1, n), nd (m, n) policy. Srker et l. (8) compred direct rework process nd (m, 1) rework policy in multi-stge production system. Feng nd Viswnthn (11) proposed mthemticl models for generl multi mnufcturing nd remnufcturing setup policies. Hsueh (11) investigted inventory control policies for mnufcturing/remnufcturing during by considering different product life-cycle phses. Sn (11) hs integrted production-inventory model of imperfect qulity products in three-lyer supply chin. Recently, Pl et l. (1) hve developed threelyer supply chin model with production-inventory model for reworkble items. Rework is common in semiconductor, phrmceuticl, chemicl, nd food industries. The products re considered s deteriorting items, becuse their utility is lost with time of storge due to the price reduction, product useful life expirtion, decy, nd spoilge. Considertion of deteriorting items in rework process ws creted by Flpper nd Teunter (4). They developed logistic plnning model with deteriorting recoverble product. Inderfuth et l. (5) delt with n EPQ model with rework nd deteriorting recoverble products. Since the recoverble products deteriorte, it will increse rework time nd rework cost per unit. Tleizdeh et l. (1) developed n Economic Production Quntity (EPQ) model with limited production cpcity for the common cycle length of ll products with some defective productions. Tleizdeh et l. (13) described n economic production quntity (EPQ) model with rndom defective items nd filure in repir under limited production cpcity nd shortges. Tleizdeh et l. (14) considered multi-product single mchine economic production quntity (EPQ) model with rndom defective items nd filure in repir under limited production cpcity nd shortges under bckordering. Tleizdeh et l. (13) developed n imperfect, multi-product production system with rework. Tleizdeh et l. (13b) studied n inventory control model to determine the optiml order nd shortge quntities of perishble item when the supplier offers specil sle. Tleizdeh et l. (15) hve determined the optiml price, replenishment lot size, nd number of shipments for n EPQ model with rework nd multiple shipments. Tleizdeh nd Noori-dryn (16) discussed n economic production quntity model in three levels supply chin including multiple non-competing suppliers, single mnufcturer, nd multiple non-competing retilers for multiple products with rework process under integrted nd non-integrted structures. Tleizdeh et l. (16) delt n economic production quntity (EPQ) inventory model with reworkble defective items nd multi-shipment policy. A production plnning of new nd recovery defective items were evolved by Inderfuth et l. (6). They ssumed tht defective items would deteriorte while witing for rework. When the witing time of the defective items exceeds the deteriortion time limit, they cnnot be recovered nd should be disposed. Similr reserch with multiple products ws summrized by Inderfuth et l. (7). An EPQ model for deteriorting items with multiple production setups nd rework ws etched by Widydn nd Wee (1). In our recommended study, we codify rework models for serviceble deteriorted items. In our lot sizing model for deteriorted items with rework, both serviceble nd recoverble items re deteriorting with time. The rework production system is shown in Fig. 1. In this system, items re inspected fter production. Good qulity items re stocked nd sold to customers immeditely. Defective 13

4 5 J Ind Eng Int (17) 13: Fig. 1 Production process with rework items re scheduled for rework. We ssume tht ll recoverble items fter rework re recognized s new. Rework process is not done immeditely fter the production process, but it wits until determined number of production setups is completed. Nottions nd ssumptions We need the following nottions nd ssumptions to develop the mthemticl model of the proposed problem. Additionl nottions nd ssumptions will be dded up when required. Nottions Prmeters Percentge of good qulity items D T Totl deteriorting unit (unit) P Production rte (unit/yer) P r Rework process rte (unit/yer) D Demnd rte (unit/yer) n Number of production setup in one cycle h Deteriorting rte A p Production setup cost ($/setup) A r Rework setup cost ($/setup) H s Serviceble items holding cost ($/unit/yer) H r Recoverble items holding cost ($/unit/yer) D c Deteriorting cost ($/unit) Vribles I s1 Serviceble inventory level in production I s Serviceble inventory level in non-production I s3 Serviceble inventory level in rework I s4 Serviceble inventory level in non-rework I r1 Recoverble inventory level in production I r Recoverble inventory level in non- production I r3 Recoverble inventory level in rework I ts1 Totl serviceble inventory level in production I ts Totl serviceble inventory level in non-production I ts3 Totl serviceble inventory level in rework I ts4 Totl serviceble inventory level in non-rework I tr1 Totl recoverble inventory level in production I tr Totl recoverble inventory level in non- production I tr3 Totl recoverble inventory level in rework I v1 Totl recoverble inventory level in n non-production I mrp Mximum inventory level of recoverble items in production setup I mrr Mximum inventory level of recoverble items when rework process TSI Totl serviceble inventory TRI Totl recoverble inventory T 1 Production T Non-production T 3 Rework process T 4 Non-rework process Totl cost per unit time Assumptions 1. Inventory is depleted not only by the demnd but lso by deteriortion. 13

5 J Ind Eng Int (17) 13: Fig. Serviceble inventory level of five production setups nd one rework I(t) I Mx T 1 T T 3 T 4 T. Shortges re not llowed. The rte of producing good qulity items nd rework must be greter thn the demnd rte. 3. The demnd rte is dependent on the current stock level which is of the form DðtÞ ¼ þ biðtþ, [, b 1 where I(t) is the inventory level t time t in cse 1 nd is dependent on time which is of the form D ¼ e bt in cse. 4. No mchine brekdown occurs in the production run nd rework. 5. Production nd rework rtes re constnt. 6. Deteriortion rte is constnt. 7. There is replcement for deteriorted item. 8. Defective items re generted only during production. Rework process results in only good qulity items. 9. The rte of producing good qulity items should be greter thn the sum of the demnd rte nd the deteriorting rte. regulr production. The recovered inventories re pushed to survive the customer s demnd. The behvior of inventory level through time of proposed mnufcturing problem is shown in Fig. 1. The inventory level of serviceble items in five production setups is illustrted in Fig.. Production is performed during T 1 time. When production is estblished, there re ð1 ÞP products defect per unit time. The rework process strts fter predetermined production up time nd production setups. The rework process is done in T 3 time which is illustrted in Fig. 3. Since production processes of mteril nd product defect re different, rework rte is not the sme s the production rte. A prcticl usge of the proposed model is illustrted with rel-life sitution. While mnufcturing wfers, there will be some wfers with excess of photoresist. In this sitution, the rework process is crried out, using the thinner compositions which re pplied for removing excess photoresist coted on the edge side or bck side of wfers. Problem description Most trditionl pproches to the problem of determining the economic ordering quntities hve lwys ssumed implicitly tht items produced re of perfect qulity. Product qulity, however, is not lwys perfect. A prcticl sitution hs considered here in which mnufcturing system produces perfect items s well s defective items. The inventory level is zero in the initil stge. There re five production setups nd one rework setup in the proposed model. The production strts t the very beginning of the cycle. As the production continues, the finished inventories re ccumulted to scrutinize whether they re serviceble or repirble. After scrutinizing, the serviceble inventories re crried out to meet the customer s demnd nd the repirble inventories re stored nd dmitted to rework. Furthermore, it is ssumed tht ll defective items re reworkble t the end of the Mthemticl formultion Serviceble inventory level The inventory level of serviceble items is prded in the Fig.. The inventory level grows during the intervl ½; T 1 Š. Thus, the inventory level in production from the serviceble items is governed by the differentil equtions: di s1 ðt 1 Þ þ hi s1 ðt 1 Þ¼P D; t 1 T 1 ð1þ dt 1 with the initil condition I s1 ðþ ¼. The inventory level depletes long the intervl ½; T Š. The inventory level in non-production is represented by the following differentil eqution: di s ðt Þ þ hi s ðt Þ¼ D; t T ðþ dt 13

6 54 J Ind Eng Int (17) 13: Fig. 3 Recoverble inventory level of five production setups nd one rework I(t) I Mrr I Mrp T 1 T T 3 T 4 T x T with the boundry condition I s ðt Þ¼. The inventory level in rework production is represented by the following differentil eqution: di s3 ðt 3 Þ þ hi s3 ðt 3 Þ¼P r D; t 3 T 3 ð3þ dt 3 with the initil condition I s3 ðþ ¼. The inventory level in rework non-production is represented by the following differentil eqution: di s4 ðt 4 Þ þ hi s4 ðt 4 Þ¼ D; t 4 T 4 ð4þ dt 4 with the boundry condition I s4 ðt 4 Þ¼. The demnd is tken to be n incresing function nd we consider the cse of both linerly time vrying nd exponentil demnds. Cse 1: D ¼ þ biðtþ By solving the Eq. (1) nd using the initil condition, we obtin the inventory level in production t ny time t 1 during the intervl ½; T 1 Š s P I s1 ðt 1 Þ¼ 1 e ðhþbþt 1 : ð5þ By integrting the inventory level in production t ny time t 1 during the intervl ½; T 1 Š, we cn get the totl inventory in production up time s I ts1 ðt 1 Þ¼ Z T 1 ¼ P 1 e ðhþbþt 1 P T1 ; dt 1 ð6þ with Tylor series pproximtion. By solving the Eq. () nd using the boundry condition, we get the inventory in non-production s I s ðt Þ¼ e ðhþbþðt t Þ 1: ð7þ By integrting the inventory level in non-production t ny time t 1 during the intervl ½; T 1 Š, we cn get the totl inventory in non-production up time s Z T I ts ðt Þ¼ e ðhþbþðt t Þ 1 dt ¼ T ; ð8þ using Tylor series pproximtion. Since I s1 ¼ I s when t 1 ¼ T 1 nd t ¼, then we hve P 1 e ðhþbþt 1 ¼ e ðhþbþt 1 : Using the Tylor series pproximtion, we extrct n expression for T in terms of T 1 s P T ¼ T 1 T1 ; * ðþ T \\1! : ð9þ By solving the Eqs. 3 nd 4, the inventory level in rework production nd rework non-production re s follows: I s3 ðt 3 Þ¼ P r 1 e ðhþbþt 3 ð1þ nd I s4 ðt 4 Þ¼ e ðhþbþðt 4 t 4 Þ 1 ð11þ respectively. By integrting the inventory level in rework production nd rework non-production, the totl inventory during the intervls ½; T 3 Š nd ½; T 4 Š is 13

7 J Ind Eng Int (17) 13: I st3 ðt 3 Þ¼ nd Z T 3 ¼ P r 1 e ðhþbþt 3 P r T3 dt 3 Z T I st4 ðt Þ¼ e ðhþbþðt t Þ 1 dt ¼ T ; ð1þ ð13þ respectively, with the help of the Tylor series pproximtion. Since I s3 ¼ I s4 when t 3 ¼ T 3 nd t 4 ¼, then we hve P r 1 e ðhþbþt 3 ¼ e ðhþbþt 4 1 Using the Tylor series pproximtion, we extrct n expression for T 4 in terms of T 3 s T 4 ¼ P r T 3! T3 ; * ðþ T4 \\1 : ð14þ Cse : D ¼ e bt By solving the Eq. (1) nd using the initil condition, we ttin the inventory level in production t ny time t 1 during the intervl ½; T 1 Š s I s1 ðt 1 Þ¼ P h 1 e ht 1 ebt 1 e ht 1 : ð15þ By integrting the inventory level in production t ny time t 1 during the intervl ½; T 1 Š, we cn ern the totl inventory in production up time s Z T 1 P I ts1 ðt 1 Þ¼ h 1 e ht 1 ebt 1 e ht 1 dt1 P ¼ T1 ; ð16þ with Tylor series pproximtion. By solving the Eq. () nd using the boundry condition, we chieve the inventory in non-production s I s ðt Þ¼ e ðhþbþðt t Þ 1 : ð17þ By integrting the inventory level in non-production t ny time t 1 during the intervl ½; T 1 Š, we hve reched the totl inventory in non-production up time s Z T I ts ðt Þ¼ e ðhþbþðt t Þ 1 dt ¼ T ; ð18þ using Tylor series pproximtion. Since I s1 ¼ I s when t 1 ¼ T 1 nd t ¼, then we hve P 1 e ht1 h ebt1 e ht1 ¼ e ðhþbþt 1 : Using the Tylor series pproximtion, we extrct n expression for T in terms of T 1 s T ¼ 1 P T 1 h T 1 T 1 h b T1 ;! * ðþ T \\1 : ð19þ By solving the Eqs. 3 nd 4, the inventory level in rework production nd rework non-production re s follows: I s3 ðt 3 Þ¼ P r h 1 e ht 3 ebt 3 e ht 3 ðþ nd I s4 ðt 4 Þ¼ e ðhþbþðt 4 t 4 Þ 1 ð1þ respectively. By integrting the inventory level in rework production nd rework non-production, the totl inventory during the intervls ½; T 3 Š nd ½; T 4 Š is I st3 ðt 3 Þ¼ nd Z T 3 P r 1 e ðhþbþt 3 ¼ P r T 3 dt 3 Z T I st4 ðt Þ¼ e ðhþbþðt t Þ 1 dt ¼ T ; ðþ ð3þ respectively, using Tylor series pproximtion. Since I s3 ¼ I s4 when t 3 ¼ T 3 nd t 4 ¼, then we hve P r 1 e ðhþbþt 3 ¼ e ðhþbþt 4 1 : Using the Tylor series pproximtion, we extrct n expression for T 4 in terms of T 3 s 13

8 56 J Ind Eng Int (17) 13: T 4 ¼ P r T 3 T 3 Recoverble inventory level! ; * ðþ T4 \\1 : ð4þ The inventory level of recoverble items is prded in the Fig. 3. The inventory level of recoverble items in production cn be formulted s di r1 ðt r1 Þ þ hi r1 ðt r1 Þ¼ð1 ÞP; t r1 T 1 dt r1 ð5þ with the condition I r1 ðþ ¼: By the given condition, we cn chieve the inventory level of recoverble items in production s ð1 Þ I r1 ðt r1 Þ¼ P 1 e ht r1 : ð6þ h Using Tylor series pproximtion, the totl recoverble items in production up time in one setup is I tr1 ðt r1 Þ¼ ð1 ÞPT 1 : ð7þ Since there re n production setups on one cycle, the totl inventory for recoverble items in one cycle is mð1 ÞPT 1. The initil recoverble inventory level in ech production setup is equl to I Mrp nd it cn be engrved just s ð1 Þ I Mrp ¼ P 1 e ht 1 : ð8þ h Deling with Tylor series pproximtion, Eq. (8) cn be reformulted s I Mrp ¼ð1 ÞP T 1 ht 1 ð9þ The inventory level of recoverble items in non-production is di r ðt r Þ dt r þ hi r ðt r Þ¼; t r ðn 1ÞT 1 þ nt with the condition I r ðþ ¼I Mrp : ð3þ With the bove condition, the inventory level of recoverble items in production setup cn be compiled s I r ðt r Þ¼I Mrp e ht r : ð31þ The totl inventory of recoverble items in non-production cn be generted s follows: I tr ðt r Þ¼ ðn 1ÞT Z 1 þnt t r ¼ I Mrp e ht r dt r : ð3þ The totl inventory of recoverble items in n non-production s cn be ttined s I v1 ¼ Xn k¼1 ðk 1ÞT Z 1 þkt t r ¼ I Mrp e ht r dt r : ð33þ With the help of Tylor series pproximtion, we nnexed! I v1 ¼ Xn I Mrp ððk 1ÞT 1 þ kt Þ hððk 1ÞT 1 þ kt Þ : k¼1 ð34þ At the end of production cycle, the inventory level of recoverble items is equl to the mximum inventory level of recoverble items in production setup reduced by deteriorting rte during production up time nd down time. By this sttement, the inventory level cn be crved s I Mrr ¼ Xn k¼1 I Mrp e hððk 1ÞT 1þkT Þ : ð35þ With the help of Tylor series pproximtion, we reformulte Eq. (35) s! I Mrr ¼ Xn I Mrp 1 hððk 1ÞT 1 þ kt Þþ h ððk 1ÞT 1 þ kt Þ : k¼1 ð36þ The inventory level of recoverble items in rework cn be dictted s di r3 ðt r3 Þ dt r3 þ hi r3 ðt r3 Þ¼ P r ; t r3 T 3 with the condition I r3 ðt 3 Þ¼: The solution of the bove eqution is ð37þ I r3 ðt r3 Þ¼ P r h ðehðt 3 tr3þ 1Þ: ð38þ The totl inventory of recoverble items in rework cn be ccessed s I tr3 ¼ P rt3 ð39þ using Tylor series pproximtion. When t r3 ¼, the number of recoverble inventory is equl to I Mrr. We cn modify Eq. (38) s 13

9 J Ind Eng Int (17) 13: P r h ðeht 3 1Þ ¼I Mrr : ð4þ Since ht 3 \\1, Tylor series pproximtion results in T 3 ¼ I Mrr : ð41þ P r The totl serviceble nd totl recoverble inventory cn be computed s TSI ¼ ni ts1 þ ni ts þ I ts3 þ I ts4 nd TRI ¼ ni tr1 þ I v1 þ I tr3 : ð4þ ð43þ The number of deteriorting item is equl to the number of totl items produced minus the number of totl demnds. The totl deteriorting units cn be modeled s D T ¼ðnPT 1 þ P r T 3 Þ ðnðþbi ts1 ÞT 1 þ nð þ bi ts ÞT þðþbi ts3 ÞT 3 þðþbi ts4 ÞT 4 Þ: ð44þ The totl inventory cost ccumultes vrious cost such s the production setup cost, reworksetup cost, serviceble inventory cost t different time intervls, recoverble cost, nd deteriorting cost. The totl inventory cost per unit time cn be written s follows: ¼ :94, D c ¼ $3 per unit nd h ¼ :3. By iterting the vlues of n, we found the optiml solution s T 1 ¼ :1 nd ¼ 634:179 in cse 1 & T 1 ¼ :1 nd ¼ 631:135 in cse t n ¼ 4 by solving Eq. (45) for solution with the help of MATLAB (Figs. 4, 5, 6, 7, 8, 9, 1, 11, 1, 13, 14, 15, 16, 17). Algorithm for both the cses Step 1: Step : Step 3: For n ¼ 1, clculte the vlue of T 1 with the condition o ot 1 ¼. By substituting T 1 in Eq. (45), evlute the vlue of. Increse the vlue of n by n þ 1 nd clculte the vlue of T 1 with the condition o ot 1 ¼. ¼ na S þ A r þ H s TSI þ H r TRI þ hd T : ð45þ nðt 1 þ T ÞþT 3 þ T 4 The optiml solution must stisfy the following condition tht: oðn; T 1 Þ ¼ : ð46þ ot 1 In ddition, the optiml solution of n, denoted s n, must hold the following condition tht: ðn 1; T 1 Þðn ; T 1 Þðn þ 1; T 1 Þ: ð47þ Since the cost function Eq. (45) is nonliner eqution nd the second derivtive of Eq. (45) withrespecttot 1 is extremely complicted, closed form solution cnnot be derived. This mens tht the optiml solution cnnot be gurnteed. However, by mens of empiricl experiments, one cn indicte tht Eq. (45) is convex for smll vlue of T 1. The optiml T 1 vlue cn be obtined using itertive method. MATLAB 13 softwre is used to vlidte the empiricl experiment results. Numericl illustrtion Here, we incorported the more prcticl numericl exmple for the support of our model verifiction. Consider n inventory system with prmeters s A p ¼ $3 per unit, A r ¼ $5 per unit, ¼ 55, b ¼ :5, P ¼ $5 per setup, P r ¼ $3 per setup, H s ¼ $15 per unit, H r ¼ $ per unit, T.1.95 Fig. 4 Grphicl representtion of the Tble T Fig. 5 Grphicl representtion of the Tble 5 n 5 n

10 58 J Ind Eng Int (17) 13: Fig. 6 Sensitivity on production setup cost in cse Fig. 9 Sensitivity serviceble holding cost in cse Fig. 7 Sensitivity on rework setup cost in cse Fig. 1 Sensitivity on recoverble holding cost in cse 1 Step 4: By substituting the new T 1 in Eq. (45) evlute the vlue of. Step 5: Repet the steps 3 nd 4 until there is n increse in the successive vlues of. Bsed on the lgorithm, we hve nnexed tble for ech cse to obtin the optiml vlue of (Tbles 1, ) Fig. 8 Sensitivity on rework process rte in cse 1 Sensitivity nlysis The sensitivity nlysis is crried out by chnging ech of the prmeters by, 1,?1, nd þ%. At n instnt of time, one prmeter cn be chnged nd ll the others should be kept unchnged. The results re summrized in Tbles 3, 4, 5, nd 6. 13

11 J Ind Eng Int (17) 13: Fig. 11 Sensitivity on deteriorting cost in cse Fig. 14 Sensitivity on rework process rte in cse Fig. 1 Sensitivity on production setup cost in cse Fig. 15 Sensitivity serviceble holding cost in cse Fig. 13 Sensitivity on rework setup cost in cse Fig. 16 Sensitivity on recoverble holding cost in cse 13

12 51 J Ind Eng Int (17) 13: Tble 3 Sensitivity of in cse 1 Prmeters % 1% þ1% þ% A p A r b P P r H s H r h D c Fig. 17 Sensitivity on deteriorting cost in cse Bsed on our numericl results, we chieve the following mngeril phenomen: 1. In the clcultion of optiml solution, the vlue of time T 1 decreses s the vlue of n increses in both the cses.. From Tbles 3nd 4, we observe the results of sensitivity nlysis for production setup cost A p nd serviceble items holding cost H s through which we cn find tht the mnufcturer s totl cost per unit time increses significntly with the increse of production setup cost A p nd serviceble items holding cost H s in both the cses. 3. We cn find tht the demnd D(t) hs significnt impct on the mnufcturer s optiml totl cost per unit time since the term in the demnd D(t) hs greter impct thn the term b in the demnd in both the cses. 4. For both the cses, the mnufcturer s optiml totl cost per unit time is little sensitive for the increse in the prmeters such s production rte P, recoverble items holding cost H r, nd rework setup cost A r. Tble 4 Sensitivity of in cse Prmeter % 1% þ1% þ% A p A r P r H s H r D c Tble 5 Sensitivity of in cse of prmeter nd h Prmeter þ1% þ% þ3% þ4% h Especilly the sensitivity is quite significnt for rework process rte P r, deteriorting rte h, nd deteriorting cost D c in both the cses. 6. From Tble 5, we find tht the prmeters nd h re chnged in the rnge þ1 to þ4% insted of the rnge to þ%. This is due to the fct tht the Tble 1 Clcultion of Optiml solution for cse 1 n T Tble Clcultion of optiml solution for cse n T

13 J Ind Eng Int (17) 13: Tble 6 Sensitivity of in cse of prmeter b nd P Prmeter 4% 3% % 1% b P totl deteriorting cost D T becomes negtive in the rnge to 1%. It my be negtive throughout the negtive rnge. 7. From Tble 6, we observe tht the prmeters b nd P re chnged in the rnge 4 to 1% insted of the rnge to þ%. This is becuse the totl deteriorting cost D T becomes negtive in the rnge þ1 to þ%. It my be negtive throughout the positive rnge. 8. Throughout the sensitivity nlysis, we could ble to inspect tht n increse in the vlues of prmeters gives n incresing effect on the mnufcture s totl cost per unit time. Conclusion Severl mnufcturers hve to tke their products bck fter use nd remnufcture them to stisfy the demnds with new ones in recent yers. This type of remnufcturing process my prevent disposl cost nd reduce environment dilemms. To overcome this problem, n economic production quntity model hs been portryed for deteriorting items with rework nd multiple production setups. Here, the demnd is considered to be s stock dependent s well s exponentil. The optiml production time is the sme for both the cses nd the number of production setup is n ¼ 4 for ech cse. The optiml totl cost is sensitive to the chnges in production setup cost, the term in demnd nd the serviceble inventory cost, but it is not sensitive to the deteriorting rte nd deteriorting cost. This model cn further be extended by considering more relistic production scheme in ech cycle nd stochstic demnd. Open Access This rticle is distributed under the terms of the Cretive Commons Attribution 4. Interntionl License ( commons.org/licenses/by/4./), which permits unrestricted use, distribution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthor(s) nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde. References Buscher U, Lindner G (7) Optimizing production system with rework nd equl sized btch shipments. Comput Oper Res 34: Crdens-Brron LE (9) Economic production quntity with rework process t single-stge mnufcturing system with plnned bckorders. Comput Ind Eng J 57(3): Crdens-Brron LE (9) On optiml btch sizing in multi-stge production system with rework considertion. Eur J Oper Res 196(3): Chung CJ, Wee HM (11) Short life-cycle deteriorting product remnufcturing in green supply chin inventory control system. Int J Prod Econ 19:195 3 Dobos I, Richter K (4) An extended production/recycling model with sttionry demnd nd return rtes. Int J Prod Econ 9: Feng Y, Viswnthn S (11) A new lot-sizing heuristic for mnufcturing systems with product recovery. Int J Prod Econ 133: Flpper SDP, Teunter RH (4) Logistic plnning of rework with deteriorting work-in-process. Int J Prod Econ 51:51 59 Hyek PA, Slmeh MK (1) Production lot sizing with the reworking of imperfect qulity items produced. Prod Pln Control 1: Hsueh CF (11) An inventory control model with considertion of remnufcturing nd product life cycle. Int J Prod Res 133: Inderfuth K, Lindner G, Rchniotis NP (5) Lot sizing in production system with rework nd product deteriortion. Int J Prod Res 43: Inderfuth K, Jnik A, Kovlyov MY, Werner F (6) Btching work nd rework processes with limited deteriortion of recoverble. Comput Oper Res 33: Inderfuth K, Kovlyov MY, Ng NT, Werner F (7) Cost minimizing scheduling of work nd rework processes on single fcility under deteriortion of reworkbles. Int J Prod Econ 15: Khouj M () The economic lot nd delivery scheduling problem: common cycle, rework, nd vrible production rte. IIE Trns 3: Liu N, Kim Y, Hwng H (9) An optiml operting policy for the production system with rework. Comput Ind Eng 56: Mishr VK, Singh LS (11) Production inventory model for time dependent deteriorting items with production disruptions. Int J Mng Sci Eng Mng 6(4):56 59 Pl B, Sn SS, Chudhuri K (1) Three-lyer supply chin production-inventory model for reworkble items. Appl Mth Comput 19: Sn S (11) A production-inventory model of imperfect qulity products in three-lyer supply chin. Decis Support Syst 5: Srkr B, Cárdens-Brrn LE, Srkr M, Singgih ML (14) An economic production quntity model with rndom defective rte, rework process nd bckorders for single stge production system. J Mnuf Syst 33: Srker BR, Jml AMM, Mondl S (8) Optiml btch sizing in multi-stge production system with rework considertion. Eur J Oper Res 184: Schrdy DA (1967) A deterministic inventory model for repirble items. Nvl Res Logist 48: Tleizdeh AA, Wee HM, Sdjdi SJ (1) Multi-product production quntity model with repir filure nd prtil bckordering. Comput Ind Eng 59:45 54 Tleizdeh A, Njfi AA, Niki SA (1) Economic production quntity model with scrpped items nd limited production cpcity. Sci Irn Trns E Ind Eng 17(1):58 Tleizdeh AA, Sdjdi SJ, Niki STA (11) Multiproduct EPQ model with single mchine, bckordering, nd immedite rework process. Eur J Ind Eng 5:

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