Optimization of imperfect economic manufacturing models with a power demand rate dependent production rate

Transkript

1 Sådhanå (219)44:26 Ó Indan Academy of Scences ]( ().,-volV) Optmzaton of mperfect economc manufacturng models wth a power demand rate dependent producton rate RAZIEH KESHAVARZFARD 1, AHMAD MAKUI 2, REZA AVAKKOLI-MOGHADDAM 3 and AA ALLAH ALEIZADEH 3, * 1 School of Industral Engneerng, South ehran Branch, Islamc Azad Unversty, ehran, Iran 2 Department of Industral Engneerng, Iran Unversty of Scence and echnology, ehran, Iran 3 School of Industral Engneerng, College of Engneerng, Unversty of ehran, ehran, Iran e-mal: talezadeh@ut.ac.r MS receved 1 November 218; revsed 5 May 219; accepted 31 May 219 Abstract. he constant demand rate s the most common assumpton of the basc economc producton quantty model, whch s not very frequent n practce. In real world stuatons, demand usually vares wth tme. Wth regard to the wdespread necessty of power demand pattern, demand s supposed to follow a power law. Another unrealstc assumpton s perfect qualty of all tems. hs paper presents a producton system wth defectve tems to determne the optmal replenshment quantty, cycle length and backordered sze wth a power demand rate dependent producton rate. We assume that a manufacturer may be faced wth three dfferent cases regardng to the date that defectve tems are drawn from nventory. he set-up, backorderng, nspecton, and producton costs, as well as holdng cost of both perfect and mperfect tems are accounted n the nventory system. An algorthm s offered to optmze total nventory cost and then numercal analyses are presented to demonstrate the applcablty of the proposed models. Fnally, some senstvty analyses and manageral nsghts are provded. Keywords. Producton modelng; nventory control; power demand pattern; defectve tems; mperfect manufacturng; backordered. 1. Introducton Harrs [1] frst presented the economc order quantty (EOQ) to mnmze the total nventory cost. Snce the EOQ model conssts of some assumptons, by relaxng the assumpton that all orders are obtaned together, the economc manufacturng quantty (EMQ) model s ntroduced by aft [2]. he basc EOQ and EPQ models suppose that the demand rate s constant; however, t s not realstc n practce, and generally customer s demand vares wth tme. herefore, many researchers are nterested n studyng nventory systems when demand depends on tme. For example, Slver and Meal [3] proposed an approxmaton to fnd an optmum lot sze when demand vares wth tme. Donaldson [4] presented an nventory system, whose demand has a lnear tme-varyng trend and then proposed an approach to obtan the optmal solutons of t. Rtche [5] consdered an nventory system n whch demand ncreases lnearly. Bose et al [6] nvestgated an EOQ model wth a demand that changes wth tme postvely and lnearly, consderng shortages and deteroraton. eng [7] proposed *For correspondence a method to obtan optmal nventory polces, consderng a lnear trend n determnstc demand. here was lttle publshed work on nventory systems wth decreasng trend n demand before the paper presented by Zhao et al [8]. hey presented an analytc algorthm to solve such problems. Lo et al [9] ntroduced an optmum polcy for the problem of nventory management, where demand changes lnearly and used a model called the two-equaton model. Yang et al [1] proposed a parametrc eclectc model wth demand decreasng non-lnearly. Omar and Yeo [11] consdered a manufacturng system for a stuaton that new products are manufactured usng one knd of raw materal, used products are repared and demand s assumed to vary wth tme contnuously. Maham and Kamalabad [12] adopted a demand functon dependent on both tme and prce for an nventory model wth decayng tems. Pando et al [13] studed a model wth the holdng cost non-lnearly dependent on both quantty and tme and the stock level dependent demand rate. here are dfferent ways to take out tems durng the cycle length to supply demand. hey are referred to as demand patterns. Varous types of these patterns are dscussed by researchers. When the demand rate s constant durng the schedulng perod, ts pattern s known as

2 26 Page 2 of 19 Sådhanå (219)44:26 unform. However, t s not sutable for many practcal stuatons. In real world stuatons, there are some other ways to take out products from the stock. Demand of customers can be dependent on tme, prce, stock level, etc. Snce tme s one of the most common nputs of demand functons n practce, there are some patterns consderng t as an nput lke lnear, quadratc, exponental, non-lnear and power. Demand used n ths study s supposed to follow a power law, because of the wde-rangng uses and beng more applcable than the others. hs pattern can be used for the stuatons that a hgh percentage of demand happens at the begnnng of the cycle, because of approachng to the expry date or demandng for the fresh products (e.g., prepared food, breads, fresh meat, fruts, yogurts and vegetables), or at the end of the schedulng perod, because of becomng scarce or daly use (e.g., sugar, tea, coffee and ol). Also, t conssts of the stuaton that demand occurs unformly. Several papers have been publshed n ths category. Naddor [14] studed a power demand pattern n an orderlevel system. Lee and Wu [15] analyzed an EOQ model wth backorderng, power demand for decayng tem. hen, Dye [16] completed Lee and Wu s model by proposng a model, n whch backorderng rate changes pro rata wth tme. Sngh et al [17] proposed an nventory system consderng partal backorderng and power demand pattern for pershable tems. Abdul-Jalbar et al [18] nvestgated a onewarehouse N-retaler problem, n whch the demand pattern s power and backorders are allowed. Rajeswar and Vanjkkod [19] developed a determnstc EOQ model consderng constant deteroraton, partally backlogged shortages and power demand pattern. Scla et al [2] analyzed nventory systems, n whch determnstc demand changes wth tme and follows a power pattern. hey dscussed several scenaros: the nventory systems wth and wthout shortages, the systems wth full backloggng or entrely lost sales. Scla et al [21, 22] extended a lot-szng system wth a power demand pattern for deteroratng tems. Scla et al [23] nvestgated an EOQ system, n whch deteroraton occurs wth a constant rate and determnstc demand pattern s power. Scla et al [24] developed an EPQ system, n whch demand follows a tme-dependent power law and producton rate changes pro rata wth the demand rate, allowng for backlogged shortage. San-José et al [25] optmzed an nventory system wth power demand pattern and partal backloggng. Keshavarzfard et al [26] developed an nventory-prcng model for multple products n whch the producton rate s proportonal to power demand rate. One of the conventonal assumptons n the EPQ model s perfect qualty of all produced tems. Resultng from process deteroraton, defectve raw materals or other reasons, producng tems wth mperfect qualty s unavodable. In the last years, several studes have been done to deal wth producton of defectve tems. Shh [27] dscussed the mpact of mperfect tems on the replenshment sze and the objectve functon. Schwaller [28] ncorporated nspecton costs n the EOQ system and assumed that a known proporton of an ncomng lot s defectve. Salameh and Jaber [29] consdered that defectve products are sold as one batch after fnshng 1% nspecton process. Hayek and Salameh [3] developed a manufacturng system wth a random defectve producton rate, n whch all defectve tems can be reworked. hey derved an optmal producton polcy, n whch backordered products are allowed. Goyal et al [31] presented an easy procedure to fnd the producton polcy of a vendor and buyer system wth defectve tems. hey assumed that a specfed porton of mperfect products are produced durng the replenshment process. Chu [32] nvestgated an EPQ system for a stuaton that mperfect products are produced wth a random rate and assumed that reworkng of them starts mmedately after producton tme, however a fracton of mperfect products are scrapped. Jamal et al [33] adopted two polces to fnd the optmum lot quantty n a producton model consderng rework. Ojha et al [34] studed a manufacturng process that manufactures mperfect products wth a constant rate. hey assumed that products can be delvered just after checkng qualty of entre batch and mperfect products have to be reworked. Also three scenaros were nvestgated by them. Cárdenas- Barrón [35] presented an extenson of [33] by addng planned backorders. alezadeh et al [36] studed a producton system wth lmted capacty and allowng for backorders, n whch manufacturng mperfect tems follows ether a normal or a unform probablty dstrbuton. alezadeh et al [37] modeled an nventory system wth rework process, multple products and sngle machne, to fnd the optmal lot sze. Ouyang et al [38] dscussed a stuaton that management nvests captal to mprove qualty. Also, they consdered defectve products and nspecton polcy n ther model. Furthermore, alezadeh et al [39] analyzed a manufacturng system consderng defectve tems, rework process and multple products. alezadeh et al [4] presented a producton system wth one machne, multple products, and nterrupton n process, scrap, rework and backorderng. Jaber et al [41] extended [29] to a stuaton that replacng defectve products s mpossble due to the dstance of the suppler. hey modeled two dfferent cases to deal wth ths condton. alezadeh et al [42] and alezadeh and Noor [43] suggested an nventory system for a three-layer supply chan consderng defectve tems. hey assumed three scenaros. Frst, all mperfect products are dsposed. Second, mperfect products are reworked and sold as perfect tems. hrd, scenaro conssts of sellng mperfect products as a batch wth a lower prce than prce of perfect tems. revño-garza et al [44] obtaned the optmal value for the replenshment quantty models usng two soluton procedures. hey consdered a system of both vendor and buyer and assumed that the mperfect tems are produced as well.

3 Sådhanå (219) 44:26 Page 3 of 1926 able 1. Codes and explanatons to categorze papers. erms Code Explanatons Codes Model ype (M) EOQ/EPQ/Both of them O/P/B Consumpton (CR) Known/Random K/R Rate Demand Input (DI) me, Inventory, Prce, Nothng, I, P, No Demand Form (DF) Constant, Exponental, Power, Lnear, Non- Lnear, None C, E, Po, L, NL, N Producton (PF) Constant, Power, Other C, Po, O Form Products (P) Sngle/Multple S/Mu Imperfect (II) Yes/No Y/No Items Imperfect ype (I) Reworked, Repared Out, Scrapped, Returned, Sold, Nether Holdng Cost of Imperfect tems R, RO, S, Re, So, Ne (HI) Yes/No Y/No Deteroraton (D) Yes/No Y/No Shortage (Sh) Backlogged, Lost Sales, Partal Backlogged, Nether B, L, PB, Ne Interrupton (In) Yes/No Y/No alezadeh and Wee [45] proposed a producton system by assumng one machne, multple products, manufacturng lmtatons, defectve tems, rework, and partal backloggng. a [46] analyzed an nventory system, n whch a dfferent screenng process s consdered for each sngle qualty characterstc. Each screenng process has ndependent screenng rate and defectve percentage. alezadeh et al [47] worked on an EPQ model wth multple shpments and rework of mperfect products to fnd the number of shpments, replenshment sze and the prce. Hsu and Hsu [48] studed optmal replenshment sze models wth defectve products by assumng three scenaros accordng to the tme of sellng mperfect products. alezadeh and Moshtagh [49] worked on mperfect producton processes, qualty dependent return and lot sales n a closed loop supply chan. able 1 shows some terms and ther codes and explanatons to categorze all revewed papers. hen n table 2, a categorzaton of those papers s provded. hs work dffers from the exstng papers n some drectons. Wth regard to the lterature revew untl now no research s done on the jontly consderng nventory systems wth power demand rate dependent producton rate, backloggng and defectve tems. In the real world, producng defectve tems s part of the producton process. As regards t s not ncluded n models wth a power demand pattern. Also, such problems do not nvolve the costs (e.g., nspecton and producton) that are mpartble parts of a producton process. Actually nspecton s a process tself and so that the cost assocated wth t should be consdered n the model. As well when the cost of a producton process for each tem s not assumed, the results of modelng a system may be unreal. In practce, holdng of defectve tems has cost; however, t s rarely appled n the exstng studes. herefore, frstly we extend model presented n [24] by allowng for defectve tems. Secondly, we consder three dfferent stuatons for the proposed model regardng to the date that mperfect products are drawn from the stock. hrdly, our model conssts of producton and nspecton costs as well as holdng cost of mperfect tems. he arrangement of the rest of ths work s as follows. Problem defnton s avalable n secton 2. Moreover, three developed models and the related procedure to solve them, are presented n sectons 3 and 4, respectvely. hen n secton 5, an example s nvestgated. Secton 6 conssts of some senstvty analyses and manageral nsghts. Fnally, conclusons are provded n secton Problem defnton We consder a manufacturng factory wth producton and nspecton stages. he demand of the product has a power pattern n each nventory cycle. It s supposed that the producton rate changes pro rata wth the demand rate. Due to many reasons a fracton of the produced lot s assumed to be mperfect. Such products are dscovered n the nspecton stage. Management of the factory desres to determne the mnmum nventory costs of the system, and satsfy the customer demand smultaneously. Behavor of the nventory s studed for three cases, dependent on when defectve tems are drawn from the nventory. In case I, we nvestgate the stuaton that mperfect tems are scrapped or sold at the tme that they are dentfed. So that n ths case, the holdng cost of defectve tems s zero. In order to reduce some costs (e.g., holdng cost), t seems to be better to scrap or sell mperfect products as soon as possble; however n practce, sellng or scrappng tems day-to-day may be nfeasble. However, n some ndustres (e.g., pharmaceutcal companes) t s nescapable. In cases II and III, mperfect products are held n the stock and sold when the replenshment and schedulng perods are fnshed, respectvely. he cycle length and the reorder pont are two decson varables of the system. A mnmzng approach s appled to specfy the optmum replenshment polces of the nventory system. Fgure 1 shows the system of processng a lot sze. We apply the followng notatons n our model. : Cycle length or schedulng perod (tme). s: Reorder pont per lot (unts). Q: Producton lot sze (unts). t : Producton cycle length (tme). d: otal demand durng the schedulng perod (unts).

4 26 Page 4 of 19 Sådhanå (219)44:26 able 2. Categorzaton of revewed papers. No. Paper referred M CR DF DI PF P II I HI D Sh In 1 Harrs [1] O K C No S No No Ne 2 aft [2] P K C No C S No No Ne No 3 Slver and Meal [3] O K S No No Ne 4 Donaldson [4] O K L S No No Ne 5 Rtche [5] O K L S No No Ne 6 Bose et al [6] O K L S No Y B 7 eng [7] O K L S No No B 8 Zhao et al [8] O K L S No No Ne 9 Lo et al [9] O K L S No No Ne 1 Yang et al [1] O K NL S No No Ne 11 Omar and Yeo [11] P,O K N C S No No Ne No 12 Maham and Kamalabad [12] O K L, E P, S No Y PB 13 Pando et al [13] O K N I, S No No Ne 14 Naddor [14] O K Po S No No Ne 15 Lee and Wu [15] O K /R Po S No Y B 16 Dye [16] O K /R Po S No Y B 17 Sngh et al [17] O K Po S No Y PB 18 Abdul-Jalbar et al [18] O K Po S No No B 19 Rajeswar and Vanjkkod [19] O K Po S No Y PB 2 Scla et al [2] O K Po S No No B, LS, Ne 21 Scla et al [21] O K Po S No Y 22 Scla et al [22] P K Po C S No Y Ne No 23 Scla et al [23] O K Po S No Y B 24 Scla et al [24] P K Po Po S No No B No 25 San-José et al [25] O K Po Po S No No PB No 26 Keshavarzfard et al [26] P K L, Po P, Po Mu No No B No 27 Shh [27] O K C No S Y No No Ne 28 Schwaller [28] O K C No S Y No No B 29 Salameh and Jaber [29] O,P K C No S Y No No Ne No 3 Hayek and Salameh [3] P K C No C S Y R No No B No 31 Goyal et al [31] P K C No C S Y So No No Ne No 32 Chu [32] P K C No C S Y R/S No No B No 33 Jamal et al [33] P K C No C S Y R No No Ne No 34 Ojha et al [34] P K C No C S Y R No No Ne No 35 Cárdenas-Barrón [35] P K C No C S Y R No No B No 36 alezadeh et al [36] P K C No C Mu Y S No No B No 37 alezadeh et al [37] P K C No C Mu Y R No No Ne No 38 Ouyang et al [38] P K C No C S Y No No Ne No 39 alezadeh et al [39] P K C No C Mu Y R No No PB No 4 alezadeh et al [4] P K C No C Mu Y R/S No No B Y 41 Jaber et al [41] O K C No S Y RO/So No No Ne 42 alezadeh et al [42] P K C No C S Y S/R/So No No No 43 revño-garza et al [44] P K C No C S Y No No Ne No 44 alezadeh and Wee [45] P K C No C Mu Y R No No PB No 45 a [46] O K C No S Y Re No No B 46 alezadeh et al [47] P K C No C S Y R/S No No Ne No 47 Hsu and Hsu [48] P K C No C S Y S/So Y No B No 48 alezadeh and Moshtagh [49] B K C No C S Y R Y No LS No 49 hs Paper P K Po Po S Y S/So Y No B No r: Average demand (r = d/) (unts). C h : Carryng cost ($/unt/unt tme). C b : Cost of backorderng ($/unt/unt tme). C : Cost of nspectng ($/unt). C p : Cost of producng ($/unt). C o : Setup cost per cycle ($/replenshment). k: Defectve rate. CD(t): Demand up to tme t ( t ). D(t): Demand rate at tme t ( t ). P(t): Producton rate at tme t ( t ).

5 Sådhanå (219) 44:26 Page 5 of 1926 Raw materal Process Inspecton Non-defectve Defectve Sell (rate D(t)) Case I Cases II, III Sale/Scrap Stock Non-def Def Case II Case III Sale (end of producton cycle) Sale (end of nventory cycle) Fgure 1. Processng a lot sze. I(t): Net stock level at tme t ( t ). ID(t): Stock level of mperfect products at tme t. I h ðs; Þ: Average number of tems holdng n stock (unts). I b ðs; Þ: Average number of backordered tems (unts). I d ðþ : Average amount of defectve tems carred n nventory (unts). CHðs; Þ: Cost of carryng products ($/unt tme). CBðs; Þ: Backorderng cost ($/unt tme). COðs; Þ: Setup cost ($/unt tme). CIðs; Þ: Inspecton cost ($/unt tme). CPðs; Þ: Producton cost ($/unt tme). CDðÞ: Holdng cost of defectve tems ($/unt tme). C j ðs; Þ: otal cost for case j (j =I, II, III) ($/unt tme). We keep the man assumptons gven n [24]. (1) Infnte-horzon s assumed. (2) he amount of demand throughout the nventory cycle s consdered to be d, and average demand rate s r = d/t unts per cycle. (3) he nventory system conssts of a sngle tem. (4) he demand rate s less than the producton rate. (5) he producton rate P(t) s proportonal to demand rate at any tme t ( t t ) and s defned by P(t) = ad(t) wtha [ 1. (6) We suppose that producng defectve tems s unavodable and the fracton of defectve tems or defectve rate s denoted by k, whch s a constant value. (7) he produced tems of perfect qualty are added to nventory wth rate ð1 kþpt ðþdt ðþ, durng the producton cycle. (8) o warrant that the consumer demand s totally covered by the products of perfect qualty, t s assumed that að1 kþ1 [ or1 1 a [ k. (9) Shortages are allowed and fully backordered. (1) o warrant that there s enough replenshment capacty to meet the demand, we assume that nspecton occurs mmedately after producng an tem. (11) However, the average demand per cycle d s determnstc, the number of tems wthdrawn from stock s dependent on the tme at whch they are removed. herefore, we suppose that the cumulatve demand CDðÞup t to tme t ( t ) follows a power pattern 1=n, and s gven by CDðÞ¼d t t Where d s the demand quantty durng the nventory cycle and n s the demand pattern ndex, wth \n\1. he demand rate at tme t ( t ), follows a tmepower pattern too and s the dervatve of the functon 1n rt CDðÞ, t that s Dt ðþ¼ ð Þ=n, wth t\. he nature n ð1nþ=n of ths demand pattern s completely defned by n. If the demand pattern ndex s n ¼ 1, then demand s unform (has a constant rate) and the nventory decreases lnearly. When a great porton of demand happens manly at the begnnng of the cycle, then demand follows a pattern law wth ndex n [ 1. But f a larger percentage of demand occurs at the end of the schedulng perod, then the demand of the nventory system s defned by a power pattern ndex n\1. Also by usng ths knd of demand functon, t s supposed that the demand s dependent on both tme and the length of the schedulng perod. he length of the nventory cycle or schedulng perod s a fracton of the unt tme. For example, assume that the unt tme s a year. If ¼ 1 2 ; 1 3 ;... then a year conssts of 2, 3, nventory cycles, respectvely. he decson varables are cycle length and reorder pont s. 3. Mathematcal models hs secton extends the model proposed by [24] n some drectons. Let It ðþ be the net stock level at tme tð t Þ. Inventory cycle starts wth s unts net stock at tme. Also producton perod starts at tme t= and contnues untl t ¼ t. We suppose that the demand rate s less than the producton rate n nterval t t. Wth regard to that n real world stuatons the producton process s usually mperfect, a certan fracton k of defectve tems s supposed to be produced n each producton perod. hus, the producton rate of non-defectve

6 26 Page 6 of 19 Sådhanå (219)44:26 tems can be obtaned byð1 kþpt ðþ. herefore, nventory s accumulated durng the producton perod ½; t Š, at a rate ð1 kþpt ðþdt ðþ. Under these condtons, the followng dfferental equatons govern the system: diðþ t ¼ ð1 kþpt ðþdt ðþ¼ ¼ ðð1 kþa 1Þ t t diðþ t ¼Dt ðþ¼ 1n rt ð Þ=n n ; ð1nþ=n 1n rt ð Þ=n n 1n ðð1 kþa 1ÞDt ðþ ð1þ ð Þ=n ; t t ð2þ Wth regard to boundary condtons IðÞ¼I ð Þ ¼ s, the above dfferental equatons are solved and the solutons are as follows: It ðþ¼sðð1kþa1þr t 1=n; t t ð3þ It ðþ¼sr r t 1=n; t t ð4þ he net nventory level at t, It ð Þ, specfed by both Eqs. (3) and (4) must be equal. So that t wll be found: t ¼ ð1 kþ n a n ð5þ When k ¼, Eq. (5) reduces to t ¼ a (gven n [24]), that n s less than t ¼. It s correct because when defectve ð1kþ n a n tems are produced, system needs more producton tme to meet the demand. herefore, functon I(t) s ncreasng on ½; t Þ and decreasng on (t ; Š. AlsoI(t) s a contnuous and -perodc functon on nterval ½; 1Þ. he total demand on nterval ½; Þ s computed by: Z Z r 1 t n 1 Dt ðþ ¼ ¼ r n ð6þ A producton sze Q must be added to stock at the end of each cycle as follows: Q ¼ Z t Z t r Pt ðþ ¼ a n 1 t n 1 ¼ r 1 k ð7þ When the system does not consst of defectve tems ðk ¼ Þ, the replenshment sze s less than the stuaton wth defectve tems and t needs to produce more tems to meet the demand. Snce total demand s r by producng 1k r unts n producton tme, fracton k of ths lot sze, gven by kr 1k, are defectve tems and fracton 1 k, gven by r, are nondefectve tems. herefore, r unts of the lot sze are of good qualty and t can totally fll the demand of the cycle. When the producton quantty s entrely added to stock, the maxmum stock level s obtaned and calculated by: 1=n It ð Þ ¼ s ðð1 kþa 1Þr t ð ¼ s 1 k Þa 1 r ð1 kþa ð8þ 3.1 he average nventory level and the average shortage Wth regard to the reorder pont s and the maxmum nventory level gven n Eq. (8), three dfferent behavors of system may occur. (1) If s, there are no shortages and the system only ncludes nventores. (2) If (Iðt 1k Þ and s ) or ðð Þa1Þ ð1kþa r s, the system ncludes some nventores and some shortages too. (3) If It ð Þors ðð1kþa1þ ð1kþa r, only shortages occur. If s, then only nventores are contaned. he average quantty of nventory s as follows: I h ðs; Þ ¼ 1 Z t 1 s ðð1 kþa 1Þ r Z t r s r t1=n 1=n 1=n t1=n ð ¼ s ð1 kþn a n 1Þr ðn 1Þð1 kþ n a n ð9þ Also, there are no shortages here, I w ðs; Þ ¼. If s ðð1kþa1þ ð1kþa r, only shortages exst. he average quantty of shortage: I b ðs; Þ ¼ 1 1 Z t Z t s ðð1 kþa 1Þ r r s r t1=n 1=n 1=n t1=n ð1 1 k ¼ ð Þr ðn 1Þð1 kþ n s ð1þ an Þn a n And there are no nventores carred, I h ðs; Þ ¼. Eventually, f ðð1kþa1þ ð1kþa r s, both backorders and nventores occur. Suppose that at tmes t 1 and t 2 wthn the producton perod and the perod wthout producton, respectvely, the stock level reaches zero. Snce It ð 1 Þ ¼ It ð 2 Þ ¼, from Eqs. (3) and (4) we obtan t 1 and t 2 accordng to decson varables s and :

7 Sådhanå (219) 44:26 Page 7 of 1926 ðsþ n t 1 ¼ ðð1 kþa 1Þ n r n n ð11þ t 2 ¼ ðs rþn r n n ð12þ In ths stuaton, the average nventory level s as follows: I h ðs; Þ ¼ 1 Z t s ðð1 kþa 1Þ r t 1 1 Z t 2 r s r t t1=n 1=n 1=n t1=n ¼ ðs rþn1 ðn 1Þr n n r ðn 1Þð1 kþ n a n ðs ðn 1Þðð1 kþa 1Þ n r n n And the average shortage s: I b ðs; Þ ¼ 1 Z t 1 1 Z Þ n1 s ðð1 kþa 1Þ r t 2 r s r t1=n 1=n 1=n t1=n ¼ ðs rþn1 ðn 1Þr n n r ðn 1Þ ðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n s Also, the average number of producton runs s 1 : 3.2 Costs and optmal nventory polces ð13þ ð14þ We consder three dfferent cases of nventory systems and fnd the optmal nventory polcy for them. Case I In ths stuaton, at the tme when a defectve tem s recognzed, t may be sold wth a dscount or may be scrapped. hese defectve tems are not assumed to be n nventory (see fgure 2). Now, we model the elements of cost functon n the proposed model. Notce that the unt of tme can be for example year. he producton cost s as follows: Q CPðÞ ¼ C p ¼ C r p ð15þ 1 k he nspecton cost s: I(t ) s I(t) (1- )P(t)-D(t) t1 Producton perod 1 COðÞ ¼ C o t ¼ C o ð17þ he holdng cost s gven by CHðs; Þ ¼ C h I h ðs; Þ. Wth regard to three stuatons mentoned before, three holdng costs may occur. Frst fs, from Eq. (9) the holdng cost s as follows: CHðs; Þ ¼ C h s ð1 kþn a n ð 1Þr ðn 1Þð1 kþ n a n ð18þ If ðð1kþa1þ ð1kþa r s, from Eq. (13) we have: " ðs rþ n1 CHðs; Þ ¼ C h ðn 1Þr n n r ðn 1Þð1 kþ n a n # ð19þ ðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n And f s ðð1kþa1þ ð1kþa r, wehavechðs; Þ ¼, because there are no nventores n the system. Fnally, the shortage cost s calculated by CBðs; Þ ¼ C b I b ðs; Þ.Ifs, CBðs; Þ ¼, because there are no shortages. For ðð1kþa1þ ð1kþa r s, from Eq. (14) the shortage cost s as follows: " ðs rþ n1 CBðs; Þ ¼ C b ðn 1Þr n n r ðn 1Þ # ð2þ ðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n s Fnally, f s ðð1kþa1þ ð1kþa r, from Eq. (1), the shortage cost s gven by 1 ð1 kþ n a n ð Þr CBðs; Þ ¼ C b ðn 1Þð1 kþ n a n s ð21þ he total cost s the sum of all fve costs. hat s: 7 t2 Schedulng perod Fgure 2. Net nventory level for the EPQ model wth a power demand rate dependent producton rate and defectve tems (case I). -D(t) me Q CIðÞ ¼ C ¼ C he set-up cost s: r 1 k ð16þ C I ðs; Þ ¼ CPðÞCIðÞCOðÞCHðs; Þ CBðs; Þ ð22þ

8 26 Page 8 of 19 Sådhanå (219)44:26 herefore, the total cost n three possble stuatons can be found. Frst f s, the total cost per unt tme s gven by: r C I ðs; Þ ¼ C p 1 k C r 1 k C o C h s ð1 kþn a n ð 1Þr ðn 1Þð1 kþ n a n ð23þ If ðð1kþa1þ ð1kþa r s, the total cost s calculated by: r C I ðs; Þ ¼ C p 1 k C r 1 k C o " ðs rþ n1 C h ðn 1Þr n n r ðn 1Þð1 kþ n a n # ðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n " ðs rþ n1 C b ðn 1Þr n n r ðn 1Þ # ðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n s And for s ðð1kþa1þ ð1kþa r, the total cost s as follows: ð24þ r C I ðs; Þ ¼ C p 1 k C r 1 k C o 1 ð1 kþ n a n ð Þr C b ðn 1Þð1 kþ n a n s ð25þ o fnd the mnmum of the functon C I ðs; Þ, we consder three dfferent regons of s. As the cost C I ð; Þ s always less than the cost C I ðs; Þ, the mnmum cost cannot be n the regon s. Also, sncec I ðs; Þs always greater ðð1kþa1þ thanc I ð1kþa r;, then mnmum cost cannot be at s ðð1kþa1þ ð1kþa r. So that, the optmal cost can be found at ðð1kþa1þ ð1kþa r s. From partal dervatves of objectve functon (22) wth respect to decson varables, we have: oc I ðs; Þ os oc I ðs; Þ o ¼ ðc h C b Þ C b ðs r r n n Þ n ðsþ n ðð1 kþa 1Þ n r n n ðs rþ n ðr nsþ ¼ ðc h C b Þ ðn 1Þr n n1 # nðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n1 C h r ðn 1Þð1 kþ n a n C br ðn 1Þ C o 2 ð26þ ð27þ Equalng these dervatves to zero the optmal soluton ðs ; Þ can be calculated. hus, we have: ðs r ðc h C b Þ r n n Þ n ðsþ n ðð1 kþa 1Þ n r n n C b ¼ ð28þ ðs rþ n ðr nsþ ðc h C b Þ ðn 1Þr n n1 # nðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n1 C h r ðn 1Þð1 kþ n a n C br ðn 1Þ C o 2 ¼ ð29þ Let x be a new varable defned by x ¼ s r. hen the regon ðð1kþa1þ ð1kþa r s s equvalent to x ðð1kþa1þ ð1kþa. Also, Eqs. (28) and(29) are respectvely equvalent to: ð1 x x n Þ n ðð1 kþa 1Þ n C b ¼ C h C b ð3þ ðc h C b Þð1 xþ n ð1 nxþr nc ð h C b ðn 1Þ ðn 1Þðð1 kþa 1Þ n C h r ðn 1Þð1 kþ n a n C br ðn 1Þ C o 2 ¼ ð31þ Proposton 1 Equal ð1 xþ n x n ð1k has a unque soluton x on the nterval ; Proof Please see Appendx A. Þrx n1 ð Þa1Þ C n b ðð1kþa1þ ð1kþa C b C h ¼. Use one of the numercal methods (e.g., Newton Raphson) to fnd the soluton x (see,.e., [5]). From Eq. (3), we have: x n ðð1 kþa 1Þ n ¼ ð1 xþ n C b ð32þ C h C b Now, by replacement of Eq. (32) n Eq. (31), we have: ðc h C b Þð1 xþ n r ðn 1Þ C br ðn 1Þ C o 2 ¼ nc brx n 1 ð Þ C h r ðn 1Þð1 kþ n a n ð33þ Fnally, we can obtan the best nventory cycle length, by replacement the optmal soluton of Eq. (3) n Eq. (33). hat cycle length s calculated by: vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u C o ¼ t h ðc r h C b Þð1x Þ n ðn1þ nc bx ðn1þ C h C ðn1þð1kþ n a n b ðn1þ ð34þ

9 Sådhanå (219) 44:26 Page 9 of 1926 I(t ) s I(t) he optmal shortage level s s ¼x r. Also as Q ¼ r 1k then the economc producton quantty Q s as follows: Q ¼ 1 vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u C o r th 1 k ðc h C b Þð1x Þ n Proposton 2 convex. Proof ðn1þ nc bx ðn1þ C h C ðn1þð1kþ n a n b ðn1þ ð35þ he total cost functon C I ðs; Þ s strctly See Appendx A. Case II In ths case, the tems wth mperfect qualty are preserved n nventory and sold n each cycle, when the replenshment perod s fnshed (see fgure 3). he dfference between Case I and Case II s that n stuaton II all of the mperfect products are held n nventory untl fnshng the replenshment cycle. In addton to fve dfferent costs explaned n the prevous stuaton, Case II conssts of the holdng cost of defectve tems. In ths stuaton durng the perod ½; t Š, nventory of defectve tems would have rsen at a rate kpðþ. t So that, the dfferental equaton s gven by: didðþ t kart ð 1n Þ=n ¼ kpðþ¼kad t ðþ¼ t n ; t ð1nþ=n t ð36þ When IDðÞs t the nventory level of defectve tems and IDðÞ ¼. he soluton of the above dfferental equaton s: IDðÞ¼kar t t 1=n; t t ð37þ Suppose that I d ðþ be the average amount of defectve tems carred n nventory. It can be calculated by: I d ðþ ¼ 1 Z t (1- )P(t)-D(t) t1 Producton perod kar t1=n 1=n ¼ And, the cost of carryng defectve tems s: P(t) t t2 Schedulng perod Fgure 3. Net nventory level for the EPQ wth a power demand rate dependent producton rate and defectve tems (Case II). -D(t) knr ðn 1Þð1 kþ n1 ð38þ a n me C h knr CDðÞ ¼ C h I d ðþ ¼ ðn 1Þð1 kþ n1 ð39þ a n By addng CDðÞ to the total cost of the prevous case, the total cost of ths stuaton wll be found by: C II ðs; Þ ¼ C I ðs; ÞCDðÞ ð4þ Smlar to Case I, the mnmum nventory cost, can be found n the regon ðð1kþa1þ ð1kþa r s. So, we have: r C II ðs; Þ ¼ C p 1 k C r 1 k C o " ðs rþ n1 C h ðn 1Þr n n r ðn 1Þð1 kþ n a n # ðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n " ðs rþ n1 C b ðn 1Þr n n r ðn 1Þ # ðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n s C h knr ðn 1Þð1 kþ n1 a n ð41þ Equalng partal dervatves of the total cost functon (41)to zero, we have: oc II ðs; Þ os ¼ ðc h C b Þ C b ¼ oc II ðs; Þ o ðs r r n n Þ n ðsþ n ðð1 kþa 1Þ n r n n ðs rþ n ðr nsþ ¼ ðc h C b Þ ðn 1Þr n n1 # nðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n1 C h r ðn 1Þð1 kþ n a n C br ðn 1Þ C o 2 C h knr ðn 1Þð1 kþ n1 a ¼ n ð42þ ð43þ Defnng x ¼ s 1k r, the regon ðð Þa1Þ ð1kþa r s s equvalent to x ðð1kþa1þ ð1kþa. Also, Eqs. (42) and (43) are respectvely equvalent to: ð1 x x n Þ n ðð1 kþa 1Þ n C b ¼ C b C h ð44þ

10 26 Page 1 of 19 Sådhanå (219)44:26 ðc h C b Þð1 xþ n ð1 nxþ nc ð h C b ðn 1Þ ðn 1Þðð1 kþa 1Þ n C h r ðn 1Þð1 kþ n a n C br ðn 1Þ C o 2 C h knr ðn 1Þð1 kþ n1 a n ¼ ð45þ Þrx n1 Equaton (44) s exactly same as Eq. (3). So soluton x ðð1kþa1þ wthn ; ð1kþa s unque. Now, by replacement of Eq. (32) neq.(45), we have: ðc h C b Þrð1 x nc brx ðn 1Þ ðn 1Þ C h r ðn 1Þð1 kþ n a n C br ðn 1Þ C o 2 C h knr ðn 1Þð1 kþ n1 a n ¼ Þ n ð46þ Fnally, we can obtan the optmal nventory polcy ðs ; Þ, by replacement the optmal soluton of Eq. (44) n Eq. (46). he optmal schedulng perod s calculated by: vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u C o ¼ t h ðc r hc b Þð1x Þ n ðn1þ ncbx ðn1þ C h ðn1þð1kþ n Cb a n ðn1þ C hkn ðn1þð1kþ n1 a n ð47þ he optmal shortage level s s ¼x r. Also, Q s as follows: Q ¼ 1 vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u C o r th 1 k ðchcbþð1x Þ n ðn1þ ncbx ðn1þ Ch ðn1þð1kþ n Cb a n ðn1þ Chkn ðn1þð1kþ n1 a n ð48þ Usng the second order dervatves of C II ðs; Þ respect to s and, we have the same formula and Hessan as n case I. hus, C II ðs; Þ s strctly convex (see proposton 2). Case III he behavor of ths nventory system s smlar to Case II, the only dfference s that the mperfect products are sold when each nventory cycle s fnshed (see fgure 4). In ths stuaton, producng defectve tems starts just after t ¼ and contnues up to t ¼ t, when the nventory level of defectve tems attans a maxmum level IDðt Þ. Wth respect to that the fracton k of all Q ¼ r 1k produced tems are defectve, the total number of defectve tems s as follows: IDðt Þ ¼ kr ð49þ 1 k And after reachng that, ths value doesn t change untl. So, I d ðþ s gven by: I(t ) s I(t) Fgure 4. Net nventory level for the EPQ wth a power demand rate dependent producton rate and defectve tems (Case III). 1 I d ðþ ¼ 1 Z kar t1=n kr ð tþa 1=n 1 k knr ¼ ðn 1Þð1 kþ n1 a kr n 1 k ð1 kþ n a n kr ¼ ð1 kþ n1 ð1 kþ n a n 1 a n n 1 ð5þ And, the cost of nventory of defectve tems s: C h kr CDðÞ ¼ C h I d ðþ ¼ ð1 kþ n1 ð1 kþ n a n 1 a n n 1 ð51þ By addng CDðÞ to the total cost of the prevous case, the total cost of ths stuaton wll be found by: So that, we have: C III ðs; Þ ¼ C I ðs; ÞCDðÞ C III ðs; Þ " r ¼ C p 1 k C r 1 k C o C ðs rþ n1 h ðn 1Þr n n r ðn 1Þð1 kþ n a n # " ðsþ n1 ðs rþ n1 ðn 1Þðð1 kþa 1Þ n r n n C b ðn 1Þr n n r ðn 1Þ # ðsþ n1 ðn 1Þðð1 kþa 1Þ n r n n s C hkr ð1 kþ n1 ð1 kþ n a n 1 a n n 1 ð52þ ð53þ Equalng partal dervatves of the total cost functon (52)to zero, we have: oc III ðs; Þ os (1- )P(t)-D(t) t1 Producton perod ¼ ðc h C b Þ C b ¼ P(t) t -D(t) Schedulng perod ðs r r n n t2 Þ n ðsþ n ðð1 kþa 1Þ n r n n me ð54þ

11 Sådhanå (219) 44:26 Page 11 of 1926 oc III ðs; Þ ¼ ðc h C b Þ o " # ðs rþ n ðr nsþ nðsþ n1 ðn 1Þr n n1 ðn 1Þðð1 kþa 1Þ n r n n1 C h r ðn 1Þð1 kþ n a n C br ðn 1Þ C o 2 C h kr ð1 kþ n1 ð1 kþ n a n 1 ¼ a n n 1 ð55þ Agan defnng x ¼ s r, the regon ðð1kþa1þ ð1kþa r s s equvalent to x ðð1kþa1þ ð1kþa. Also, Eqs. (54) and (55) are respectvely equvalent to: ð1 x x n Þ n ðð1 kþa 1Þ n C b ¼ C b C h ð56þ ðc h C b Þð1 xþ n ð1 nxþ nc ð h C b ðn 1Þ ðn 1Þðð1 kþa 1Þ n C h r ðn 1Þð1 kþ n a n C br ðn 1Þ C o 2 C h kr ð1 kþ n1 ð1 kþ n a n 1 a n n 1 ¼ ð57þ Þrx n1 Now, by replacement of Eq. (32) n Eq.(57), we have: ðc h C b Þrð1 x nc brx ðn 1Þ ðn 1Þ C h r ðn 1Þð1 kþ n a n C br ðn 1Þ C o 2 C h kr ð1 kþ n1 ð1 kþ n a n 1 a n n 1 ¼ ð58þ Þ n Fnally, lke the prevous cases, we have: By takng the second order dervatves of C III ðs; Þwth respect to s and, we have the same formula and Hessan as n case I. hus, the functon C III ðs; Þ s strctly convex (see proposton 2). 4. Procedure for determnng the optmal values A bref procedure s explaned to obtan the optmum values for all the three proposed cases. Notce that optmal value of varable x s obtaned by Eq. (3), or (44) or(56), that are the same for all three cases. Step 1 Enter the values of parameters. Step 2 Obtan x of equaton ð1 xþ n x n ðð1kþa1þ n C b C b C h ¼, usng a numercal method. Step 3 Specfy, usng Eq. (34) for Case I, Eq. (47) for Case II or Eq. (59) for case III. Use s ¼x r to calculate optmal reorder pont. Step 4 Determne optmal lot sze Q gven by formula (35) for Case I, (48) for Case II or (6) for Case III. Step 5 Calculate the mnmum cost CI for Case I usng Eq. (24), CII for Case II usng Eq. (41) and C III for Case III usng Eq. (53). 5. Numercal example In order to provde nput data, we consder an nventory system, n whch the parametrc values are defned as C h = $4 per unt and year, C b = $5 per unt, C o = $1 per replenshment, C = $2 per unt, C p = $6 per unt, r=12 unts per year, a = 1.4, k =.2 and n=2. Usng Eq. (3), the followng equaton must be solved. vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u C o ¼ t h ðc r h C b Þð1x Þ n ðn1þ nc bx ðn1þ C h C ðn1þð1kþ n a n b ðn1þ C hk ð1 kþ n a n 1 ð1kþ n1 a n n1 ð59þ he optmal shortage level s s ¼x r. Also, Q s as follows: ð1 xþ 2 x2 :12 2 ¼ 5 9 ð61þ Q ¼ 1 vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u C o r th 1 k ðc h C b Þð1x Þ n ðn1þ nc bx ðn1þ C h C ðn1þð1kþ n a n b ðn1þ C hk ð1 kþ n a n 1 ð1kþ n1 a n n1 ð6þ

12 26 Page 12 of 19 Sådhanå (219)44:26 here s a unque soluton for ths equaton nsde the nterval ðð1kþa1þ ; ð1kþa ¼ :17143,thatsx ¼ : Now for three dfferent cases, we have followng optmal values: Case I Usng Eq. (34), the optmal cycle length s ¼ :8927 year ¼ 325:8355 days. From Eq. (35) the economc lot sze s Q ¼ 1339:1 unts. Usng formula s ¼x r, the optmal reorder pont s s ¼72:826 unts. From Eq. (24), the optmal cost s CI ¼ $12224 per year. Case II Usng Eq. (47), the optmal cycle length s ¼ :362 year ¼ 132:13 days. From Eq. (48) the economc lot sze s Q ¼ 542:9552 unts. Usng formula s ¼x r, the optmal reorder pont s s ¼29:2266 unts. From Eq. (41), the optmal cost s CII ¼ $12553 per year. Case III Usng Eq. (59), the optmal cycle length s ¼ :2466 year ¼ 9:9 days. From Eq. (6) the economc lot sze s Q ¼ 369:983 unts. Usng formula s ¼x r, the optmal reorder pont s s ¼19:9117 unts. From Eq. (53), the optmal cost s CIII ¼ $12654 per year. Fgures 5, 6 and 7 show the total cost as a functon of two varables and s for Cases I, II and III by usng the above example s nput parameters, respectvely. Fgure 5. otal cost for Case I. 6. Numercal analyses and nsghts Some numercal analyses are done to dscover the effect of changes n parameters on the results. Bascally, the mpacts of the producton rate a, the defectve rate k and the power demand ndex n on the varables and the total cost of the system are analyzed n ths secton. Some extra examples are reported n tables 3 and 4 usng the nput data taken from [24]. Accordng to the followng values of parameters, table 3 s provded: n=3, C o = 1, r=12, C h = 4, C b = 5, C = and C p =. Optmal polces of nventory systems consderng several combnatons of parameters a and k are shown n table 3. Also, the followng values are used n table 4: r=12, C o = 1, C h = 4, C b = 5, C = 2, C p = 6 and a ¼ 1:5. he optmal solutons for varous values of n and k are calculated n table 4. he graph of the mnmum cost and lot sze changes versus the defectve rate changes for Case I s shown n fgures 8 and 9, usng table 4. In each fgure, four dfferent values of n (.5, 1, 1.5 and 2) are consdered. Fgures 1 and 11 show the total cost and the cycle length as functons of the defectve rate for Case I, when nput parameters are used from table 3. In each fgure dfferent values of a are consdered (a ¼ 1:1; 1:3; 1:5; 1:9). Fgure 6. Fgure 7. otal cost for Case II. otal cost for Case III. Fgures 12 and 13 show changes of the total cost respect to the changes of the producton rate for Cases I and II, respectvely, usng table 3. In each fgure, four dfferent values of k are assumed (k ¼ :2; :5; :2; :5).

13 Sådhanå (219) 44:26 Page 13 of 1926 able 3. Numercal analyss of producton rate a and defectve rate k. r=12, C o = 1, C h = 4, C b = 5, C =, C p = and n=3 Case Producton rate Defectve rate x Q s C I a = 1.1 k = II III I k = II III I a = 1.3 k = II III I k = II III I k = II III I k = II III I a = 1.5 k = II III I k = II III I k = II III I k = II III I a = 1.7 k = II III I k = II III I k = II III I k = II III I a = 1.9 k = II III I k = II III I k = II III I k = II III

14 26 Page 14 of 19 Sådhanå (219)44:26 able 4. Numercal analyss of the power demand ndex n and defectve rate k. r=12, C o = 1, C h = 4, C b = 5, C = 2, C p = 6 and a ¼ 1:5 Case Index of demand Defectve rate x Q s C I n =.5 k = II III I k = II III I k = II III I k = II III I n = 1 k = II III I k = II III I k = II III I k = II III I n = 1.5 k = II III I k = II III I k = II III I k = II III I n = 2 k = II III I k = II III I k = II III I k = II III Fgures 14 and 15 show changes of total cost and reorder pont respect to the changes of the power demand ndex for Case I, usng table 3. In each fgure, four dfferent values of k are assumed (k ¼ :2; :5; :2; :5). Some senstvty analyses can be expressed as follows. From table 3, we can observe that fxed the producton rate parameter a and consderng Case I, the value x, the total amount of backorders s and the mnmum cost C decrease as the defectve rate k ncreases. However, the optmal cycle length and the

15 Sådhanå (219) 44:26 Page 15 of 1926 Fgure 8. Changes of the total cost value wth respect to the changes of the defectve rate for Case I, usng table 4. Fgure 11. Changes of the cycle length value wth respect to the changes of the defectve rate for Case I, usng table 3. Fgure 9. Changes of the lot sze value wth respect to the changes of the defectve rate for Case I, usng table 4. Fgure 12. Changes of the total cost value wth respect to the changes of the producton rate for Case I, usng table 3. Fgure 1. Changes of the total cost value wth respect to the changes of the defectve rate for Case I, usng table 3. economc lot sze Q ncrease as the defectve rate k ncreases. In the same stuaton, but consderng Case II, the mnmum cost C ncreases as the defectve rate k ncreases. he reason s that n Case I we dd not Fgure 13. Changes of the total cost value wth respect to the changes of the producton rate for Case II, usng table 3. consder the holdng cost of defectve tems; however, n Case 2 we consder t and because of that the results of Case II are more reasonable. herefore, Case II can

16 26 Page 16 of 19 Sådhanå (219)44:26 Fgure 14. Changes of the total cost value wth respect to the changes n the power demand ndex for Case I, usng table 4. Fgure 15. Changes of the reorder pont value wth respect to the changes n the power demand ndex for Case I, usng table 4. show the real world stuatons much better (n both cases, we do not have nspecton and producton costs (C = C p = )). Also, fxed the defectve rate k and consderng Case I, f the producton rate a ncreases then the value x and the mnmum cost C ncrease. However, the optmal cycle length and the economc lot sze Q decrease as the producton rate a ncreases. In the same table 3, fxed the producton parameter a and the defectve rate k, we can observe that the optmal cycle length and the economc lot sze Q decrease from case I to case III. However, C ncreases n ths stuaton. In table 4, by fxng the power demand ndex n and consderng Case I, f the defectve rate k ncreases then the cycle length, the optmum lot sze Q and the mnmum cost C ncrease. However, the total omount of backorders s and the value x decrease as the defectve rate k ncreases. Also, fxed the defectve rate k and consderng Case I, f n ncreases then the value x ncreases. However, for other nventory polces, we can not fnd a standard pattern. In the same table 4, by fxng the power demand ndex n and the defectve rate k, we can observe that the optmal cycle length and the optmum producton quantty Q and the total amount of backorders s decrease from Case I to case III. However, C ncreases n ths stuaton. Comparng tables 3 and 4, we can see that f producton and nspecton costs are nvolved n the proposed model (n table 4, C = 2 $/unt and C p = 6 $/unt) then fxed the other parameters the optmum cost C ncreases as the defectve rate k ncreases. However, f producton and nspecton costs are not consdered n the proposed model (n table 3, C = $/unt and C p = $/unt), then fxed other parameters the optmal cost C decreases as the defectve rate k ncreases. Addtonally, t shows consderng these costs can help modelng the real world stuatons much better. When producton costs or holdng cost of mperfect products are not nvolved, the obtaned results are not relable enough. hs research offers several manageral nsghts: all prevous related works have focused on the EPQ model n whch demand follows a power pattern wthout consderng mperfect tems or mperfect tems are nvolved but demand s unformly dstrbuted. None of those models are applcable enough to be used n the real world stuatons. Another mportant feature of our models s consderng the tme when those defectve tems are removed from the nventory. One of the most valuable manageral nsghts of our study s that when producton cost and nspecton cost as well as holdng cost of defectve tems are equal to zero the results can not be same as the real world stuatons, as t can be seen that n such cases when defectve rate ncreases the optmal cost decreases (nstead of ncreasng). But when one of producton and nspecton costs or holdng cost of mperfect tems are nvolved the results can model the real system accurately. Another manageral nsght s that as the defectve rate ncreases optmum cycle length, best lot quantty and total cost ncrease. Also case I can be sutable for very small factores that remove the mperfect tems when they are dscovered or the frms that produce medcal tems and have to remove defectve products as soon as possble. Many other frms can not do the same thng and are forced to keep those tems untl the end of the producton cycle to be reworked or the end of the nventory cycle to be scrapped or sold wth a lower prce. 7. Conclusons and future drectons of research In ths research, we proposed EPQ models wth a power demand rate dependent producton rate, allowng for shortages completely backordered and defectve tems. hree cases are consdered for the nventory system

17 Sådhanå (219) 44:26 Page 17 of 1926 regardng to the date that defectve tems are drawn from nventory. In the frst case, we suppose that a defectve tem s elmnated from nventory at the tme when t s recognzed. In the second and thrd stuatons, t s assumed that the tems wth mperfect qualty are kept n stock and sold n each cycle, after endng the replenshment and the nventory cycles respectvely. he optmum solutons obtaned by the mathematcal models are unque and easly calculated by the proposed algorthm. When mperfect tems, the holdng cost of defectve tems, nspecton and producton costs are not consdered, the optmal nventory polces consst of the formulate obtaned by [24]. One possble extenson to the current paper can be consdered for rework process n the nventory system. Moreover, a deteroraton rate can be consdered n the model. Another research can study nventory systems wth assumng that shortages are lost sales or partally backorderd. Fnally, one can also consder the stuaton where demand depends on the prce of the products. Also we suggest to ncorporate the concept of ths work as potentals extenson to the problems or models suggested by other researchers [51 53]. Appendx A Proposton 1 Equal ð1 xþ n x n ð1k a unque soluton x ðð1kþa1þ on ;. ð1kþa ð Þa1Þ C n b C b C h ¼ has Proof Suppose that f(x) s a real functon on [,1] defned by: fðþ¼ x ð1 xþ n ðð1 kþa 1Þ n C b ðaþ C b C h f(x) s contnuous, strctly decreasng dfferentable on the nterval (,1) because (notce that accordng to assumpton 9, ð1 kþa 1 [ ): f ðþ¼n x ð1 xþ n1 ðð1 kþa 1Þ n \ ðbþ Also, we have f ðþ ¼ C h ðð1kþa1þ C b C h [ and f ð1kþa ¼ C b C b C h \. So, usng the ntermedate value theory, a pont x ðð1kþa1þ exsts n the nterval ; ð1kþa, where yðx Þ¼. Fnally, because of that the functon s decreasng on (,1), the pont x s unque. x n nx n1 Proposton 2 he total cost functon C I ðs; Þ s strctly convex. Proof Usng the second order dervatves of C I ðs; Þ respect to decson varables, we have: o 2 C I ðs; Þ os 2 ¼ ðc h C b Þ o 2 C I ðs; Þ o 2 ¼ ðc h C b 2C o 3 " # ns ð r Þn1 nðsþ n1 r n n ðð1 kþa 1Þ n r n n " # Þ ns2 ðs rþ n1 nðsþ n1 r n n2 ðð1 kþa 1Þ n r n n2 ðcþ ðdþ o 2 C I ðs; Þ ¼ ðc h C b Þ oso " # nðsþðs rþ n1 nðsþ n r n n1 ðð1 kþa 1Þ n r n n1 And the Hessan of the functon C I ðs; Þ s gven by: Hs; ð Þ ¼ o2 C I ðs; Þ o 2 2 C I ðs; Þ os 2 o 2 o2 C I ðs; Þ oso " # nsr ¼ ðc h C b Þ ð Þn1 nðsþ n1 r n n ðð1 kþa 1Þ n r n n 2C o 3 [ ðgþ Eqs. (c) to (g) are postve, because n the regon ðð1kþa1þ 1k ð Þa r s, we always have s r [. herefore, the functon C I ðs; Þ s strctly convex. References [1] Harrs F W 1913 How many parts to make at once. he Magazne of Management. 1(2): (152) [2] aft E W 1918 he most economcal producton lot. Iron Age 11(18): [3] Slver E A and Meal H C 1973 A heurstc for selectng lot sze quanttes for the case of a determnstc tme-varyng demand rate and dscrete opportuntes for replenshment. Producton and Inventory Management 14(2): [4] Donaldson W A 1977 Inventory replenshment polcy for a lnear trend n demand An analytcal soluton. Oper. Res. Q. 28(3): [5] Rtche E 1984 he EOQ for lnear ncreasng demand: A smple optmal soluton. J. Oper. Res. Soc. 35(1): [6] Bose S, Goswam A and Chaudhur K S 1995 An EOQ model for deteroratng tems wth lnear tme-dependent ðfþ