Optimizing under- and out-of-warranty products decisions in the finite planning horizon. Mohsen Afsahi. Tehran, Iran Tel:

Transcription

1 Optmzng under- and out-of-warranty products decsons n the fnte plannng horzon Mohsen Afsah M.AFSAHI@MODARES.AC.IR Faculty of Industral and Systems Engneerng, Tarbat Modares Unversty, Tehran, Iran Tel: Al Hussenzadeh Kashan* A.KASHAN@MODARES.AC.IR Faculty of Industral and Systems Engneerng, Tarbat Modares Unversty, Tehran, Iran *Correspondng Author: Tel: Bakhtar Ostad BOSTADI@MODARES.AC.IR Faculty of Industral and Systems Engneerng, Tarbat Modares Unversty Tehran, Iran Tel:

2 Abstract In ths paper, we consder a manufacturer that produces products n a fnte horzon tme and sells products wth non-renewng free replacement warranty polcy. The manufacturer s responsble to provde spare parts for faled products, whether the products are under or out of warranty. Prevous research on warranty optmzaton has focused on maxmzng manufacturer proft wthout consderng the spare part market for out-of-warranty products. Ths study proposes a novel nonlnear model that maxmzes manufacturer proft by optmzaton of prce, warranty length and spare part nventory for under- and out-of-warranty products n a manufacturng/remanufacturng system. Due to the model s unque structure, we propose a new two-stage approach that combnes metaheurstc and an exact method, n whch the frst stage s to determne product s prces and warranty length wth metaheurstc algorthm and n the second stage the remanng nventory related problem s transferred to a Mnmum Cost Network Flow Problem whch s solved for spare part nventory control. To llustrate the effectveness of the suggested method, the model s solved for a case study of Iranan SANAM electronc company wth two dfferent metaheurstc algorthms and a senstvty analyss s conducted to study the effect of varous parameters on the optmal soluton. Keywords: non-renewng free replacement warranty; dynamc prcng; spare part Inventory control; remanufacturng. 1 Introducton Wth tougher competton, technology advances, and shftng customer preferences, t s more crucal than ever that companes use warranty as a competton advantage n order to ncrease ther market share. Warranty sgnfes the product s qualty n the eye of customers, therefore leadng to growth n customer s satsfacton and wllngness to buy.warranty has two man functons of protecton and promoton. Regardng the former, warranty protects the manufacturer from excessve clams and protects the customer from purchase rsks. Regardng the latter, warranty s a compettve advantage to dfferentate the manufacturer from ts compettors [1]. Relatvely long warranty perod wll ncrease customer s wllngness to buy, but manufacturer can t propose long warranty because they are responsble for product's falure durng warranty perod [2]. Also they must take nto account product s relablty, because undesrable relablty may leadng to hgh cost for manufacturer [3]. Therefore, they should determne warranty length n order to optmze ther proft and customer satsfacton. In the lterature, prce and warranty length are mentoned as two key factors affectng manufacturer s proft [4-7]. Obvously, longer warranty perods and lower prce lead to ncreased sales but t also tends to decrease the manufacturer s margnal proft. As a result, smultaneous decson about these two factors n order to optmze the proft s mportant and wdely studed n the lterature. 2

3 Glckman and Berger [8] proposed a model for maxmzng manufacturer s proft by determnng prce and warranty length, assumng that customers are homogeneous and ther demand s an exponental functon of prce and warranty length. nasrollah and asgharzadeh [9] solved a mult-objectve problem wth goal programmng usng ths demand functon for estmatng warranty length. The problem n Ln and Shue [10] was determnng prce and warranty length where the demand s a functon of prce, warranty length, and cumulatve sales and the objectve s maxmzng proft. They consdered dfferent product lfe dstrbutons, whle n [6, 7], the dstrbuton of product lfetme s consdered Normal and Gamma, respectvely, and soluton approaches based on maxmum prncple were presented. Huang, Lu [11] took account of relablty n addton to prce and warranty length n modelng, and studed the problem both n stable and dynamc market scenaros. Manna [12] consdered the problem wth prce and warranty length decson varables and proposed a method to extend the model to twodmensonal warranty. The model proposed n [5] ncorporates prce, warranty length, and producton rate as decson varables wth the objectve of maxmzng manufacturer's proft. Wu, Chou [13] also developed a model wth these decson varables where product lfetme s consdered webull and the objectve s maxmzng proft. Zhou, L [4] determned the optmal prce and warranty length where the product s reparable and compared fxed-length warranty polcy wth dynamc polcy. Fang and Huang [14] proposed a Bayesan decson model by whch the ntegrated prce of products, producton quantty and term of warranty are determned under the stuaton that the manufacturer does not have suffcent hstorcal data. Shafee and Chukova [15] developed a mathematcal optmzaton model to determne prce, warranty length and upgrade strategy for second hand products. Fardmehr and Nak [16] nvestgates optmal polces for prce, warranty length and producton rate n both statc market and dynamc market. They calculate optmal polces by maxmum prncpal approach. Mahmoud and Shavand [17] propose a b-objectve model for maxmzng the manufacturer proft and mnmzng the watng tme n queue. Also, they formulate the demand functon as a fuzzy system. Tsao, Teng [18] consdered the problem of determnng retal prce and nventory level for h-tech products when the warranty length s predetermned. We, Zhao [19] proposed fve decentralzed models and determned equlbrum wholesale prces, and retal prces, and warranty perods usng game theoretcal approaches. The problem n [20] s to determne prcng polcy for returned used products, along wth ther remanufacturng level, and prcng and warranty polcy for remanufactured products. Besdes the proft obtaned through the sale of the man products, aftermarket plays an mportant role n the manufacturer s proftablty. Sellng spare parts to out-of-warranty products can lead to substantal proft, but t s very challengng to estmate the demand of spare parts due to ts greater uncertanty compared to the uncertanty n the product s demand [21]. To the best of our knowledge, there s not any research that consders sellng spare parts to out-of-warranty products, whle n many ndustres such as automoble and electronc devces, the company could ncrease ts profts up to 25 percent from sellng spare parts to out-of-warranty products [22]. 3

4 Although prcng and warranty length are two key factor that effect spare parts nventory management, only a few researchers have consdered ths nterdependency n the lterature. Km and Park [23] proposed a two-stage optmal control model to jontly determne product s prce, warranty length, and spare part nventory for under-warranty products. They dvded the plannng horzon nto two parts: product s lfe cycle and end of lfe perod. Ther study just consdered under-warranty products, but a consderable porton of proft comes from sellng spare parts for out-of-warranty products. Also, they assumed all spare parts were produced by manufacturer, but n practce, components can be refurbshed by remanufacturng wth lower amount of cost. Char, Dallo [24] developed a mathematcal optmzaton model to maxmze manufacturer s total expected proft by optmzaton of warranty length, the sale prce, the age of recondtoned components, and the proporton of recondtoned components to be used. They assumed renewng free replacement warranty and statc prcng strategy for ther model. Also, they ddn t consder the role of out-of-warranty products on manufacturer's proft. In order to present a compact revew of prevous studes and demonstrate the characterstcs of the proposed approach as compared to the lterature, Table 1 llustrates a state-of-the-art survey of Prcng and Warranty Inventory Optmzaton. Although the spare parts nventory decsons have a drect mpact on warranty length and prce decsons, (As shown n Table 1), many of the proposed approaches seek to optmze warranty length and prce wthout consderng nventory of spare parts. To the best of the authors knowledge, there s no research consderng the effect of out-of-warranty products on manufacturers revenue. The purpose of ths paper s to develop a new mathematcal model for optmzaton product prce (n dfferent stages of a product s lfe cycle), warranty length, and spare part nventory control for under-warranty and out-of-warranty products n a manufacturng/remanufacturng system wth the objectve of maxmzng manufacturer s proft. Plannng horzon conssts of three man parts: (1) product lfe cycle, (2) end of lfe (EOL) and, (3) guarantee perod for spare part avalablty. The product's demand s consdered a functon of product's prce, tme, and warranty length. The sgnfcant ssue for producer s to determne the prce n each prcng perod (pp) of the product's lfe cycle n order to gan the maxmum proft. Another challenge s to determne the warranty length, where longer warranty length perod tends to ncrease sales but at the same tme ncreases warranty relevant costs. Although falures of under- and out-of-warranty products change stochastcally n each perod, the model can calculates a good estmaton of falures n each nventory control perod (ICP) of plannng horzon n order to effectvely manage the spare part nventory. In real word, a percent of faled tem can rectfy by remanufacturng; therefore, we assume spare parts can be obtaned from two sources: (1) producton by orgnal manufacturer and (2) remanufacturng faled products. Fundamentally, ths paper ams to perform the followng: Proposed a new model that consderng out-of-warranty products as a man source of manufacturer revenue. 4

5 Coordnaton between prce, warranty length and spare part nventory decsons as an ntegrated model for under- and out-of-warranty products. Proposed a novel two stage approach wth combnaton of metahuerstc and exact method for solvng the proposed model. Identfyng how the changes of product s lfe cycle affect warranty length. Table 1 should be placed here The rest of ths paper s organzed as follows: Secton 2 presents the problem defnton. Secton 3 explans the mathematcal modelng. A soluton method s ntroduced n Secton 4. Secton 5 demonstrates the applcablty of the presented mathematcal model by a real-world numercal example taken from the Iranan SANAM electronc company along wth senstvty analyses. The paper concludes n Secton 6. 2 Problem defnton The problem of ths paper s defned for maxmzng the proft of the manufacturer, whch conssts of a set of revenue and cost elements. The manufacturer s revenue ncludes: (1) product s sale n ts lfe cycle, and (2) spare parts sale for out-of-warranty products. Moreover, the cost s composed of four man elements, whch are: (1) producton cost, (2) spare parts nventory cost, (3) spare parts remanufacturng cost, and (4) dsposal cost. The product s sale and ts producton cost are drectly related to the market demand n ts lfe cycle. The demand tself s dependent upon the tme, sales prce, and length of warranty. Sales prce and warranty length are, respectvely, nversely and drectly proportonal to the market demand. Therefore, the smultaneous decson about sales prce and length of warranty s of consderable sgnfcance n order to maxmze the proft. Addtonally, spare-part related elements affect the manufacturer s revenue and cost. Effectve spare-part nventory control decsons play an mportant role n reducng manufacturer s cost. A challenge here s to estmate the number of product s falure n each nventory control perod (ICP) n order to optmze the nventory level. Before presentng the model, the assumptons made for formulatng the problem are gven as below: 1) All ICP are equal and less than product s lfe cycle, 2) All clams durng the warranty perod are vald, 3) Warranty polcy for products s non-renewng Free Replacement Warranty (FRW), 4) The orgnal manufacturer s also responsble for the remanufacturng of ther used products, 5) The refurbshed components return to as-good-as-new state, 6) Producton capacty s unlmted n product lfe cycle, 7) The frm s a monopolst and Customers are myopc, 5

6 8) The amount of product s sale s equal to the product s demand, 9) Products have exponental falure dstrbuton. Ths assumpton was mposed by SANAM Electronc Company s product development dvson, 10) Warranty length s a postve nteger and multple of the ICP, 11) The nventory delvery s assumed to be nstantaneous (lead tme s neglgble), 12) Shortage s not allowed to avod lost sales. Assumptons 1 to 10 are common n realty of the problem (especally n electronc devce manufacturers). However, assumptons 11 and 12 were set n order to make the problem techncally more tractable. The notatons used to formulate the problem are presented n Table 2. Table 2 should be placed here Accordng to the above-mentoned assumptons, the plannng horzon of the problem s dvded nto three segments (see Fgure 1): (1) product s lfe cycle (0, L T ), (2) end of lfe (EOL) ( LT, LT w), and (3) guarantee perod for spare part avalablty ( LT w, LT g ). In each of ICPs, an estmaton of the number of products under warranty should be made. It s a challengng task because on one hand, these estmaton condtons dffer n each of the above-mentoned segments. On the other hand, n each ICP, a number of manufactured products are added to the products under warranty and some products become out of warranty. After calculatng the number of under-warranty and out-of-warranty products n each ICP, the estmated number of falures can be calculated usng product s falure behavor, whch s employed to estmate the demand for spare parts. Fgure 1 should be placed here As can be seen n Fgure 1, plannng horzon s dvded nto T+g ICP s, n whch the spare part order sze s determned based on the spare part s demand, mnus the remanng nventory from the prevous perod and the amount of remanufactured spare parts. In addton, each ICP s dvded nto l equal sub-perods n whch the amount of producton s a functon of tme, prce and warranty length. Because the product s prce s determned n each of these sub-perods, they are named as prcng perods (PPs). If warranty length s equal to w ICPs, we may have products under warranty n the market untl at most LT w. Therefore, from LT to L T w, the decson s lmted to spare part nventory control for under-warranty and out-of-warranty products. Fnally, from LT wto L T g, spare part nventory control for out-of-warranty products s the only decson to be made. Ths perod s for ensurng the customers of the avalablty of spare parts for a fxed perod of tme after the end of all product s warranty. Spare parts nventory can be provded from two sources: (1) remanufactured faled components, and (2) manufactured components. Because the plannng horzon s fnte and the component s holdng and producton costs could change wth tme, t s essental that the manufacturer optmzes the amount of component s producton n each ICP. 6

7 Fgure 2 should be placed here The relatonshp between producton, market and spare part nventory s depcted n Fgure 2. Manufacturer supples the product accordng to market demand, whch s a functon of tme, prce, and warranty length. These products nclude I crtcal components. The amount of faled products that enter the collecton center n each ICP s dependent upon the product stuaton (to be defned later) and falure rate. Based on hstorcal data, we can determne the proporton of tmes that product s falure s due to each component. Then, based on the condtons of remanufacturng process, components that can be refurbshed are sent to the refurbshng center and rreparable ones are dsposed. Due to the assumpton that the refurbshng s perfect, refurbshed components are as good as new, so they can meet a proporton of spare part s demand. Fnally, the manufacturer satsfy spare market s demand (demands from under- and outof-warranty products) by spare parts that refurbshed and produced n factory. Snce, key elements of reverse logstcs are defned n the lterature as faled tem remanufacturng, dsposton, spare part nventory management, after-sale servce, and product prcng [25, 26], we can say that the proposed framework s a part of reverse logstcs. Therefore, manufacturer can overcome reverse logstcs challenges related to under-warranty and out-of-warranty products by usng the proposed model. 3 Mathematcal model In ths secton, we descrbe the problem s mathematcal model that s constructed based on the notatons and assumptons whch were mentoned. 3.1 Objectve functon The objectve functon maxmzes manufacturer proft that gans from sellng products and spare parts. Manufacturer just can sell spare parts to out-of-warranty product. l. T I Tg j j pw max z P c. S( j, P, w) pc. D ( s) j1 1 sw1 I T g I T g I T g I T g cr. E ( s) v. V ( s) h. X ( s) co. Q ( s) 1 s1 1 s1 1 s1 1 s1 In Eq. (1), the frst term s the proft obtaned from product s sale, whch s calculated by multplyng the net product s proft and the demand n each PP. The revenue earned from spare part s sale for out-of-warranty products s calculated n the second term. The thrd to ffth terms are the remanufacturng, dsposal, and holdng costs, respectvely. The fnal term calculates the cost of spare part s producton n each ICP. (1) 7

8 Product s Market Demand or Number of Products Sold n Each Perod Eq. (2) shows the demand behavor accordng to tme, whch s ncreasng up to and decreasng afterwards. Ths s congruent wth the product s lfe cycle n whch demand s rsng untl the maturty and s fallng afterwards. Interested reader can fnd more nformaton n [27] U, 0 j duj 1 e S( j) U, j T. l du j U 1 D 0 1 e du Eq. (3) calculates the amount of man product s demand based on the assumpton that the amount of product s sale s equal to the product s demand. Therefore, the manufacturer s sale s a functon of tme ( j ), prce ( p j ), and warranty length ( w ). We assume that S( js ) potental market demand n each PP. It s obvous that the sale s nversely proportonal to the prce and drectly proportonal to the warranty length. The element k1 ( p j Pmn ) k2wdemonstrates ths type of dependence In Eq. (3), k 1 s prce coeffcent and k 2 s warranty coeffcent n the demand functon. Constrant (4) show the lower bound and upper bound on the warranty length. (2) S( j, p, w) S( j) k ( p P ) k w for k 0, k 0. W w W mn j max 1 j mn (3) (4) Because customers are consdered myopc, the product s prce s decreasng n tme. Constrants (5) take account of ths fact, whch s called markdown prcng. Constrants (6) are the lower bound and upper bound on the product prces n each PP. P j 1 P j j {1,..., T g l} (5) Pmn p j Pmax j {1,..., T g l} (6) 3.2 Product stuatons durng each nventory plannng perod Each ICP, based on ts poston n the plannng horzon, may nclude several types of stuatons for each product. Each stuaton wll demonstrate how much tme a product s under warranty or/and out of warranty n each ICP. Each product may have one or two of fve stuatons n each ICP, where stuatons one, two, and three are for calculatng the number of faled products under warranty and stuatons four and fve are for computng the number of faled out-ofwarranty products. These stuatons are as follows: Stuaton 1 (for under-warranty products) 8

9 The products that are manufactured n an ICP have the stuaton 1. For these products, the probablty of falure n that ICP s proportonal to the amount of tme t les n that ICP. It s obvous that all ICPs n product s lfe cycle have products wth stuaton 1 because there s a producton n these ICPs. Stuaton 2 (for under-warranty products) Ths stuaton covers a case where the product s under warranty throughout the ICP. Stuaton 3 (for under-warranty products) A product wth stuaton 3 falls n an ICP n whch ts warranty expres. Stuaton 4 (for out-of-warranty products) Stuaton 4 s the complement of the stuaton 3,.e. t enables us to compute the probablty of falure when the product becomes out of warranty. Stuaton 5 (for out-of-warranty product) Under ths stuaton, the warranty has been expred n one of the pror ICPs, so the probablty of falure s proportonal to the whole ICP. Fgure 3 summarzes stuatons 1 to 5 along wth ther features for products that are produced n t 1. In ths fgure we assumed that the warranty length s equal to 4 ICPs, so the amount of products produced n t 1 s S(1, P 1,4). Products that are n s=1 are faced wth stuaton 1 and the falure probablty of these products n s=1 s proportonal to L t. In s 2,3,4 products are under stuaton 2, so ther probablty of falure s proportonal to the whole length of ICP. Because ther warranty wll expre n s 5, as long as products are under warranty, they are under stuaton 3. After they become out of warranty, they get stuaton 4 untl the end of the 5 th ICP. Fnally, these products wll be n stuaton 5 n s {6,..., T g}. Therefore, t can be concluded that the products are n the stuaton 1 when they are produced, whle they pass stuaton 2 to 5 sequentally throughout the plannng horzon. Fgure 3 should be placed here 3.3 The number of falures for products under warranty In order to calculate the number of falures of under-warranty products n each ICP, we must frst compute how much tme the product s under warranty n that ICP. As defned above, underwarranty products can have at most 3 stuatons 1, 2, and

10 Accordng to assumpton 1, the probablty of falure for product wth stuaton 1 n the s th ICP that s produced n the j th PP L s 1 t j Ls s equal to 1. Ls t j Pjs P( f Ls f t j ) 1 e. On the other hand, the number of products produced n the j th PP s denoted by S( j, p j, w ), so the number of falures for products wth stuaton 1 s bnomally dstrbuted as 1 1. y ~ b S( j, p, w), P 1 e js j js Ls tj. All products that are produced n the frst ICP have the stuaton 1, therefore, usng contnuty correcton, Eq. (7) calculates the maxmum number of 1 falures ( n ) n the j th PP wth j1 w confdence whch produced n the frst ICP. Eq. (8) computes the total number of falures n the frst ICP. n 0.5 S ( j, P, w ). P P y n P z l j 1 j j1 ( j 1 ) j 1,2,, j 1 w w 1 1 S ( j, Pj, w ). Pj 1.(1 Pj 1) n ( ). S ( j, P, w ). P.(1 P ) S ( j, P, w ). P j 1 w j j 1 j 1 j j 1 1 Dw (1) n l j1 j1 (7) (8) Due to the fact that products wth stuaton 2 are under warranty for the whole of ICP, ther falure probablty s proportonal to the length of ICP, and s calculated wth. L L P P f L f L e 2 s s 1 js ( s s1) 1. Addtonally, the random varable denotng the number of faled products wth stuaton 2 follows a bnomal dstrbuton,.e L L 1. The characterstc of the 2 nd to w th ICP s that no product wll get s s y ~ b S( j, p, w), P 1 e js j js out of warranty, because t s assumed that the warranty length s equal to w ICPs. Therefore, products n these ICPs only have stuaton 1 or 2, where all products produced before s th ICP have stuaton 2 and product produced n the s th ICP have stuaton 1., Eq. (9) and Eq. (10) calculate the maxmum number of faled products wth stuaton 2 and 1, respectvely, n the s th ICP. Fnally, Eq. (11) shows the total number of faled products n the s th ICP 1 s w js w j js js j js n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, j 1,2,, s 1. l, s 2,..., w, (9) n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, js w j js js j js j s 1. l, s 1. l 1,, s. l, s 2,..., w, s1. l sl. 2 1 js D ( s) n n, s 2,..., w w j1 js1. l js (10) (11) For ICPs between w and T, product can have stuatons 1, 2, or 3. Products produced n s th ICP have stuaton 1, those produced n s w j s 1 have stuaton 2, and those produced n s w1 j s w 1 have stuaton 3. 10

11 For products wth stuaton 3, the probablty of ther falure n the s th ICP s t jwls 1 1 and the number of falures has bnomal dstrbuton,.e. P P f t w f L e 3 js j s1 3 3 t j w Ls 1 y ~ b S( j, p, w), P 1 e js j js. Eq. (12), (13), and (14) calculate the maxmum number of faled products wth stuaton 3, 2, and 1, respectvely; n the s th ICP when ICPs are between k and T. Eq. (15) calculates the total number of faled under-warranty products n the s th ICP. n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, js w j js js j js j s k 1. l, s k 1. l 1,..., s k. l, s w 1,..., T, n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, js w j js js j js j s k. l, s k. l 1,, s 1. l, s w 1,..., T, n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, js w j js js j js 1., 1. 1,,., 1,..., j s l s l s l s w T sw. l s1. l sl w js js js jsw1. l jsw. l ts1. l D ( s) n n n, s w 1,..., T. (12) (13) (14) (15) Because the producton s stopped at the end of T, all under-warranty products have stuatons 2 and 3 nt s T w, specfcally, products produced n ( s w). l j T. l have stuaton 2 and products produced n ( s w 1). l j ( s w). l have stuaton 3. Eq. (16) and (17) calculate the maxmum number of faled products wth stuaton 3 and 2, respectvely, for T s T k and Eq. (18) calculates the total number of faled under-warranty products for the s th ICP. n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, js w j js js j js j s w 1. l, s w 1. l 1,..., s w. l,s { T 1,..., T w}, n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, js w j js js j js j s w. l, s w. l 1,..., T. l,s { T 1,..., T w}, sw. l Tl. 3 2 w js js tsw1. l jsw. l D ( s) n n, s T 1,..., T w. (16) (17) (18) 3.4 The number of falure for out of warranty products The (w+1) th s the frst ICP n whch the out-of-warranty products (that are produced n the 1 st ICP) have stuaton 4. The probablty that products wth stuaton 4 fal n the s th ICP s equal to 4 1.( Ls l tj w l P P f L f t w e ) and the number of faled products s bnomally dstrbuted,.. js s j e..( Ls l tj w l y ~ (,, ), 1 ) js b S j p j w Pjs e. Eq. (19) calculates the maxmum number of faled out-of- 11

12 warranty products wth stuaton 4 that are produced n j 1,2,, l total number of faled out-of-warranty products n sw js pw j js js j js and Eq. (20) calculates the n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, j 1,2,, l,s w 1, l 4 pw( 1), js 1. j1 D w n s w In w1 s T w, out-of-warranty products have stuatons 4 and 5; so the products produced n ( s w 1). l j ( s w). l have stuaton 4, and products produced n 1 j ( s w1). l have stuaton 5. The probablty of falure of products wth stuaton 5 s equal to. L L P P f L f L e 5 s s 1 js ( s s1) 1 and the random varable of the number of faled products wth 5 5. stuaton 5 s Ls Ls 1 y js ~ bs( j, p j, w), Pjs 1 e number of falures for products wth stuaton 5 that are produced n j 1,2,, s ( w 1). l (19) (20). Usng Eq. (21), we can compute the maxmum, whle Eq. (22) calculates the maxmum number of faled out-of-warranty products wth stuaton 4 that are produced n s ( w1). l, s ( w1). l 1,, s w. l. Fnally, Eq. (23) calculates the total number of faled out-of-warranty products n s w 2,..., T w. n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, js pw j js js j js j 1,2,, s ( w 1). l, s w 2,..., T w, n ( ). S( j, P, w). P.(1 P ) S( j, P, w). P 0.5, js pw j js js j js j s ( w 1). l, s ( w 1). l 1,, s k. l, s w 2,..., T w, s( w1). l sw. l 5 4 js j1 j s( w1). l D ( s) n n, s w 2,..., T w. pw js (21) (22) (23) Snce at the end of T+w, all products have become out of warranty, so they have stuaton 5. For these products, the maxmum number of falures can be calculated wth Eq. (24), and the total number of falures n s{ T w1,..., T g} s calculated wth Eq. (25). n ( ). S ( j, P, w ). P.(1 P ) S ( j, P, w ). P 0.5, js pw j js js j js j 1,2,, T. l, s { T w 1,..., T g}, Tl. 5 pw( ), js { 1,..., }. j1 D s n s T w T g (24) (25) 12

13 3.5 Spare Part Inventory Control In ths secton, we present the framework for spare part nventory control based on the number of falures n each ICP as calculated n the prevous secton. The number of components to send to refurbshng center s a proporton of the number of faled products. As mentoned prevously, falures are assocated wth under-warranty products untl the w th ICP. From (w+1) th ICP to (T+w) th ICP, falures are from both under- and out-of-warranty products. Fnally, after (T+w+1) th ICP, falures are assocated wth just out-of-warranty products. Eq. s (26)-(28) calculate the number of component to be sent to refurbshng center n each ICP. F ( s). D ( s) s {1,..., w}, w F ( s ). D ( s ) D ( s ) s { w 1,..., T w }, w pw F ( s). D ( s) s { T w1,..., T g 1}. pw (26) (27) (28) In refurbshng center, some percentage of components are successfully refurbshed and are taken nto account as as-good-as-new component nventory, whle others are dsposed. Eq. s (29) and (30) calculate the number of component that s sent for refurbshment and dsposal, respectvely. E ( s) F ( s) s {1,2,..., T g}, 1 V ( s) 1 F s s {1,2,..., T g}. (29) (30) Fgure 4 demonstrates the flow of spare part nventory for component n each ICP of the plannng horzon. In each ICP, the on-hand nventory conssts of the remanng nventory of the prevous ICP, the amount of products manufactured and remanufactured n the current ICP, mnus the component s demand n the current ICP. Fgure 4 should be placed here Now we have completed the task of calculatng dfferent components of the proft functon besde the requred nventory balance equatons for dfferent ICP s. The optmzaton problem s summarzed as follows: l. T I Tg I max z P c. S( j, P, w) pc. D ( s) po. X ( T g) j j pw j1 1 sw1 1 I T g I T g I T g I T g o. E ( s) v. V ( s) h. X ( s) c. Q ( s) 1 s1 1 s1 1 s1 1 s1 P j 1 P j j {1,..., T g l}, (32) Pmn p j Pmax j {1,..., T g l}, (33) (31) 13

14 W w W mn max X (0) 0 {1,..., I}, X ( s) X ( s 1) D ( s) E ( s) Q ( s) {1,..., I}, s {1,..., w} w X ( s) X ( s 1) D ( s) D ( s) E ( s) Q ( s) {1,..., I }, s { w 1,..., T w }, w pw X ( s) X ( s 1) D ( s) E ( s) Q ( s) {1,..., I}, s { T w1,..., T w}, pw X ( s), Q ( s) 0 {1,..., I}, s{1,..., T w}. (34) (35) (36) (37) (38) (39) Eq. (35) states that the nventory of all components at the begnnng of the plannng horzon s zero. Eq. s (36)-(38) are nventory balance equatons for dfferent ICP s (see Fgure 4). Fnally, Eq. (39) ensures that no shortage s encountered n all ICP s. 4 Soluton Method In ths secton we propose an effectve algorthm to solve the problem already descrbed. The objectve functon to be mnmzed and the constrants are all nonlnear wth respect to prces (P j ) and warranty length (w) varables. Once we fx the value of these varables the remanng varables whch are Q () s can be found va translatng the reduced problem to a mnmum cost network flow problem. So, we can effectvely reduce the search to the prce-warranty length space to fnd good qualty solutons of the problem. Followng the above noton, gven known the prces (n each PP) and warranty length, the amount of products sold n each PP can be calculated. Then, the number of faled under- and outof-warranty products n each ICP are calculated; ths leads to determnaton of spare part demand n each ICP. The remanng s a spare part nventory control sub-problem for whch we show t can be modeled as a network problem. Fgure 5 demonstrates the network counterpart whch can be solved by a mnmum-cost network flow algorthm such as out-of-klter. Ths method was frst ntroduced by Fulkerson [28]. It works on both the prmal problem (edges of the network) and the dual problem (nodes) n successve phases to fnd a feasble soluton, and then to optmze the problem. The pseudo code of the out-of-klter s as follows: 14

15 The out-of-klter algorthm Begn π 0; Establsh a feasble flow x n the network; Defne the resdual network G(x) and compute the klter number of arcs; Whle the network contans an out-of-klter arc do Begn Select an out-of-klter arc (p,q) n G(x) Defne the length of each arc (,j) n G(x) as max{0,c π j }; Let d(.) denotes the shortest path dstances from nodes q to all other nodes n G(x)-{(q,p}) and let P denote a shortest path from node q to node p; Update π = π() d() for all N; π If c j < 0 then Begn W = P {(p, q)}; δ = mn{r j : (, j) W} ; Augment δ unts of flow along W; Update x, G(x), and the reduced costs; End; End; End; Now we can develop an algorthm for the problem whch respectvely scale the prces and warranty length wth the ad of a search based algorthm e.g., the recently proposed Optcs Inspred Optmzaton (OIO) and scale the producton amount of each component s spare parts, relevant to the gven the prces and warranty length, by solvng a mnmum cost network flow problem optmally. Ths latter step s done to compute the ftness relevant to OIO s ndvduals. To have a comparator algorthm, we also use from Improved Partcle Swarm Optmzaton (IPSO) algorthm. Our justfcaton behnd usng these algorthms comes from the pont that PSO s a classc and popular algorthm for solvng optmzaton problems. To nclude a relatvely newer and modern algorthm, we use from the recently proposed OIO algorthm. OIO has proven as an effectve algorthm and needs few parameters, so we thnk t may be useful to compare the results of an older algorthm lke PSO besde a newer one. 4.1 Soluton representaton and ftness functon To solve the problem wth the ad of Optcs Inspred Optmzaton (OIO), an ndvdual s a vector of length lt. 1, for whch the frst ltelements. are prces sorted n descendng order, and the last element s the warranty length. All prces and warranty length should generate between ther upper bound and lower bound. When computng the ftness functon relevant to a gven 15

16 ndvdual,.e., the objectve functon value, frst the decson varables relevant to the producton th amount of each component s spare parts n s ICP, are set optmally va solvng the relevant mnmum-cost network flow problem for spare part nventory control and then the ftness value s calculated. (see Fg 6). Fgure 5 should be placed here Fgure 6 should be placed here 4.2 The Optcs Inspred Optmzaton (OIO) Algorthm Optcs Inspred Optmzaton (OIO) s an optcs nspred populaton based evolutonary algorthm that was frst proposed by Hussenzadeh Kashan [29]. The algorthm assumes that a number of artfcal lght ponts (ponts n R n+1 whose mappng n R n are potental solutons to the problem) are sttng n front of an artfcal wavy mrror reflectng ther mages. OIO treats the surface of the functon to be optmzed as the reflectng mrror composed of peaks and valleys. Each peak s treated as a convex reflectve surface and each valley s treated as a concave reflectve surface. In ths way, the artfcal ray glttered from the artfcal lght pont s reflected back artfcally by the functon surface, gven that the reflectng surface s a part of a peak or a part of a valley, and the artfcal mage pont (a new pont n R n+1 whch s mapped n R n as a new soluton n the search doman) s formed uprght (toward the lght pont poston n the search space) or nverted (outward the lght pont poston n the search space). Recently several studes use ths algorthm for ther problem optmzaton [30-33]. Fgure 7 should be placed here Fgure 7 llustrates how the new soluton s generated n OIO n the one dmensonal search space. In ths fgure t s assumed that an artfcal lght pont n the jont search and objectve space (.e., R n+1 ) s n front of the functon surface (mrror) n a partcular dstance from the vertex (values on the X-axs form the search/soluton space and values on the f(x)-axs form the objectve space. The set of all ponts n the X-f(X)-coordnate system forms the jont search and objectve space). Usng the mrror equatons of Physcs, the artfcal mage s formed n the jont search and objectve space. Then, the new soluton s generated n the search space through mappng the artfcal mage poston nto the search space. The procedure of generatng new solutons s drectly depends on the reflectng part of the functon surface (convex or concave) and the poston of the artfcal lght pont n the jont search and objectve space. Fgure 7 shows four dfferent stuatons whch may occur n generatng new solutons. The above process for generaton of a new soluton can be translated n an algorthmc manner as follows. For a gven ndvdual soluton O n the populaton, a dfferent soluton F (vertex pont) s selected randomly from the populaton. If F has a worse ftness value than O, t s treated that the surface s convex and a new soluton s generated uprght somewhere toward O, on the lne connectng O and F (See Fgure 7a). If F has a better ftness value than O then t s assumed that the surface s concave and the new soluton s generated uprght toward (see Fgure 7b) or nverted outward (see Fgure 7c and 7d) O, on the lne connectng O and F n the search space. 16

17 Wth the procedure of generatng new solutons descrbed conceptually n Fure 8, OIO s able to perform both exploraton and explotaton durng the search process. The exploraton ablty s acheved by adoptng a larger jump n the soluton space (see Fgure 7b and 7c) whle the explotaton s performed by adoptng a smaller jump over the base solutons (see Fgure 7a and 7d). The detaled and ready to mplement flowchart of OIO has been shown n Fgure 8. The notatons used n Fgure 8 are descrbed as follows: O [ o o o ] t t t t j j 1 j 2 jn 1 n F [ f f f ] t t t t j j 1 j 2 jn 1 n I [ ] s t t t t j j 1 j 2 jn 1 n t j, k p q r t j, k t j, k t k the poston of artfcal lght pont j n the n dmensonal search space n teraton t (.e., the j th soluton n the populaton), a dfferent pont n the search space (.e., an ndvdual n the populaton) whch passes the artfcal prncpal axs through tself, an mage poston of the artfcal lght pont j n the search space n teraton t. The artfcal mage s formed by the artfcal mrror whose prncpal axs passes through F, t k the poston of the artfcal lght pont j (whose mage s formed by the artfcal mrror) on the functon/objectve axs (objectve space) n teraton t. The poston of artfcal lght pont j n the jont search and objectve space s thus gven by the t t t vector [ o j 1o j 2 o jn ], the dstance between the poston of artfcal lght pont j on the functon/objectve axs and the poston of artfcal mrror vertex on the functon/objectve axs n teraton t, the dstance between the mage poston of the artfcal lght pont j on the functon/objectve axs and the poston of artfcal mrror vertex on the functon/objectve axs n teraton t, the radus of curvature of the artfcal mrror whose center of curvature s on the t prncpal axs whch passes through t m the poston of the center of curvature on the functon/objectve axs (objectve k space), t HO j, k the heght of the artfcal lght pont j from artfcal prncpal axs n teraton t, t HI j, k the mage heght of the artfcal lght pont j from artfcal prncpal axs n teraton t, t j, k the value of lateral aberraton relevant to the artfcal mrror whch s reflectng the mage of the artfcal lght pont j n teraton t. F, k Fgure 8 should be placed here 4.3 The Improved Partcle Swarm Optmzaton (IPSO) Algorthm In the lterature many studes can fnd that use algorthms based on PSO to solve smlar problems [34-39]. Accordngly, we use an mproved partcle swarm optmzaton (IPSO), frst proposed by Jang, Hu [40], for comparng the results that calculatng by OIO algorthm. PSO s a populaton-based metaheurstc algorthm proposed by Kennedy and Eberhart [41]. Its concept orgnated from the socal behavor of swarms. A partcle n the swarm starts 17

18 from an ntal poston and moves n the search space accordng to the effects of two sources, namely the personal best ( p best ) and global best ( g best ). Specfcally, each partcle's velocty changes accordng to the ts dstance from the best poston (soluton) t has acheved so far (.e. p best ), and accordng to ts dstance from the best poston obtaned by the swarm ( g best ). The equatons for updatng velocty and poston of each partcle are: v 1 v C 1 ( P best x ) C 2 ( g best x ) (40) x x v (41) 1 1 The acceleraton constants C 1 and C 2 n Eq. (40) are acceleraton constants that controls the effect of P best and g best postons on the velocty. On the other hand, s the nerta factor, whch s reduced throughout the search. The calculated veloctes can be at most v max. IPSO algorthm starts wth a random populaton, whch s clustered nto some subpopulatons. Then, PSO algorthm s appled to these sub-populatons. At certan ponts n tme, the sub-populatons are merged n order to share nformaton, and reclustered agan. The steps of IPSO algorthm are as follows: Step 1: Choose p1, m 1, where, p s the number of sub-swarms and m s the number of partcles n each sub-swarm, set the Sample s=pm then calculate the objectve functon for each partcle X. Step 2: Sort the functon value of partcles n ascendng order and put them n an array E { X, f 1,..., s}. Step 3: Partton E nto p sub-swarms k k k k k j j j k p( j1) j k p( j1) 1 2 p A, A,..., A, such that: A { X, f X X, f f, j 1,..., m}, k 1,..., p. k Step 4: Evolve each A by partcle swarm optmzaton (PSO). Step 4.1: Determne the populaton sze (q) and the maxmum teraton (T). k Step 4.2: Select q partcle,..., k Y1 Y from A by ths strategy that the partcles wth better objectve k q functon have more probablty to be selected. Store them n F k { Y k, V k, u k 1,..., q}, k k k k where V s the velocty for partcle Y 1 and u s the correspondng functon value. Set G the best ndvdual of the whole swarm. k k k k k Step 4.3: Evaluate the functon value of Y and P. If Y s better, then put P Y. Evaluate the k k k k k functon value of Y and G, and f Y s better, then put G Y. Step 4.4: Update the poston and velocty of each partcle accordng to (40) and (41). Step 5: substtute 1 2 p A, A,..., A nto E. Step 6: If convergence crtera are satsfed, stop. Otherwse, go to step 4. 5 Numercal Example and Senstvty Analyss In order to see and understand the patterns of key varables optmal dynamcs, we conduct numercal analyss for LED 32" that s produced by SANAM Electronc Company. SANAM 18

19 Electronc s one of the leader company n electronc ndustry n Iran and began producng Color- Televson sets name-branded SANAM n The case company offers the followng nformaton: (1) based on the prce skmmng strategy, the prce starts from $280 (t s the maxmum prce) and prces decrease to $200 (2) the unt producton cost for LED 32" s about $150. (3) Accordng to hstorcal data and experts opnons, Man_Board and Panel are two key components. Producton cost, refurbshng cost, dsposal cost, and holdng cost for Man-Board are about $28 (per unt), $17 (per unt), $2 (per unt), and $2 (per unt/month) respectvely and these costs for Panel are $18 (per unt), $8 (per unt), $1 (per unt), and $1 (per unt/month) respectvely. (3) Sellng prces for Man_Board and Panel for out-of-warranty products are $41 and $26 respectvely. Other parameters are shown n Table 3. Table 3 should be placed here In order to solve and analyze the case study problem, parameters of the OIO algorthm and IPSO algorthm are shown n table 4 and table 5, respectvely. Table 4 should be placed here Table 5 should be placed here After solvng the problem, there are mportant dynamc relatonshps that can be derved from the numercal analyss. We frst look nto the relatonshp between T, g and optmal warranty length and company proft. We also present the prce trends, the producton amount of the LED 32 product, and the producton amount of spare parts. Fnally, we show the mpact of product s falure rate on total proft and warranty length. 5.1 The optmal proft and warranty length for a gven T and g Table 6 shows optmal proft for varous T s and g s obtaned by two algorthms, namely OIO and PSO. Accordng to hstorcal data, the product s falure rate s 0.07 and the prce and warranty coeffcents ( k 1 and k 2 ) are estmated by the company as 6 and 10. Currently, the company offers a two-year warranty for all products. We show that for each combnaton of T and g, the company must choose dfferent warranty perods. We run the proposed algorthm 10 tmes for each combnaton of T and g (720 problems n total), usng MATLAB on a Pentum 4 computer wth 8GB RAM and Core7 3.61GHz CPU, and the results were reported n Table 3. The best objectve functon obtaned, the worst, mean, standard devaton, and average tme are reported n Table 3. Table 6 shows that as product s lfe cycle and guarantee perod for spare parts avalablty ncrease, the manufacturer s proft ncreases, but ths does not necessarly mean that manufacturer can always select a longer lfe cycle or guarantee perod. Because the competton stuaton s very complex and change n factors such as technology leads to changes n customer nterest, the company must shft to new products. As Fgure 9 demonstrates, the best solutons for OIO algorthm are greater than or equal to best solutons obtaned by IPSO algorthm, whch shows the better performance of OIO n terms 19

20 of objectve functon (proft). Furthermore, t shows the mpact of g and T changes on manufacturer proft. Fgure 9 should be placed here Table 6 should be placed here Fgure 10 compared optmal warranty perod for dfferent lfe cycles and guarantee perods. It llustrates that optmal warranty perod for g=34 s greater than or equal to other cases. On the other hand, longer lfe cycles have greater warranty perods. So t can be an mportant achevement for SANAM Company, because they often consdered a fxed warranty perod for products even f the products lfe cycles dffer. Fgure 10 should be placed here Accordng to the current company s polcy, lfe cycle for LED 32" s 32 months (T=32). So, the optmal prces for ths T and dfferent g s s depcted n Fgure 11. Fgure 11 should be placed here It s worth notng that n fnal PPs, the slopes sharply ncrease. Ths s due to the fact that products produced n these perods have lower chance of falng as out-of-warranty products, so the manufacturer would better sell more products by decreasng the prces. Total demand for spare parts are depcted n Fgure. 12, where TD1 s for component 1 (Fgure 11a) and TD2 s for component 2 (Fgure 11b). Ths demand s met from two sources, namely, refurbshng (R1 and R2) and manufacturng. Q1 and Q2 represent the amount of product that should be produced. The ncreasng trend n these values are due to the cumulatve demand of all products untl a gven tme. Fgure 12 should be placed here 5.2 Senstvty analyss and product s falure rate changes In ths part, we focus on how the falure rate ( ) changes affect optmal warranty perod and total proft. In order to observe the dynamcs more clearly, we need to use a fxed product lfe cycle (T) and spare part avalablty perod (g). For the ensung analyss, we consder T=32 and g=30. Table 7 shows the optmal value of warranty length and proft for varous falure rate. Table 7 should be placed here As t can be observed n Fgure. 13, manufacturer s profts are ncreasng when falure rate ncreases or decreases. In stuatons the falure rate decreases, warranty costs decrease. So manufacturer can propose longer warranty length and sells more products. On the other hand wth larger values of falure rate, manufacturer proposes smaller warranty length. Because, shorter warranty length leads to lower warranty cost and more proft from sellng spare parts to out-ofwarranty products; It should be noted that shorter warranty length and hgher falure rate wll 20

21 cause customer dssatsfacton, whch wll weaken the company s compettveness and lessen ts market share. Fgure 13 should be placed here 6 Concluson The man purpose of the current study s to develop a new nonlnear model to ntegrate and optmze product s prce, warranty length and spare part nventory control decsons. Snce the sale of spare parts to out-of-warranty product has sensble effects on a company proft, therefore a unque ablty has been proposed n the presented model to calculate the number of out-ofwarranty spare parts demand and optmze spare part nventory decsons whch was not consdered n prevous studes. In order to solve the model, a new optmzaton approach was proposed that hybrdzes the metaheurstc algorthm wth a mnmum cost network flow optmzer. We solved the model to the real data of LED 32" by two type of algorthm. The frst one was combnaton of OIO wth MCNFP and the second one was combnaton of IPSO wth MCNFP. Expermental analyses show that f the company decdes to set a longer lfe cycle for the product, t s more proftable to propose longer warranty lengths, as compared wth the cases where the lfe cycle s shorter. Addtonally, t s recommended that the company decreases the prces more sharply n fnal perods of the lfe cycle compared to ntal perods n order to beneft from ncreased sale n fnal perods. Fnally, we found that product s falure rate s nversely proportonal to the warranty length, snce ncrease n falure rate wll lead to an ncrease n warranty costs, makng t reasonable to decrease the warranty length. As future drecton t s nterestng to conduct prcng for sellng spare part for out-ofwarranty products, snce t wll model the real-world condton more accurately. The model can also be extended to consder two-dmensonal warranty case, whch wll make the model applcable to other felds such as automoble ndustry. Other consderatons such as shortage and lost sales can also be ncorporated n the nventory control problem, whle mnmzng shortage can be consdered as another mportant objectve along wth maxmzng the proft. References 1. Murthy, D.N.P. and Blschke, W.R. "Strategc warranty management: a lfe-cycle approach". IEEE T. ENG. MANAGE., 47(1), pp (2000). 2. Nguyen, D. and Murthy, D.P. "Optmal burn-n tme to mnmze cost for products sold under warranty". IIE. Trans., 14(3), pp (1982). 3. Murthy, D.N.P. "Product warranty and relablty". Ann. Oper. Res., 143(1): p (2006). 4. Zhou, Z. L, Y. and Tang, K. "Dynamc prcng and warranty polces for products wth fxed lfetme". Eur. J. Oper. Res., 196(3), pp (2009). 21

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24 36. Yang, M.-F. and Ln, Y. "Applyng the lnear partcle swarm optmzaton to a seral multechelon nventory model". Expert. Syst. Appl., 37(3), pp (2010). 37. Mousav, S.M., et al., "A modfed partcle swarm optmzaton for solvng the ntegrated locaton and nventory control problems n a two-echelon supply chan network". J. INTELL. MANUF., 28(1), pp (2017). 38. Majumder, P. Bera, U.K. and Mat, M. "An EPQ model for two-warehouse n unremttng release pattern wth two-level trade credt perod concernng both suppler and retaler". Appl. Math. Comput., 274(Supplement C), pp (2016). 39. Bhuna, A.K. and Shakh, A.A. "An applcaton of PSO n a two-warehouse nventory model for deteroratng tem under permssble delay n payment wth dfferent nventory polces". Appl. Math. Comput., 256(Supplement C), pp (2015). 40. Jang, Y., et al., "An mproved partcle swarm optmzaton algorthm". Appl. Math. Comput., 193(1): pp (2007). 41. Kennedy, J. and Eberhart, R. "Partcle swarm optmzaton (PSO)". n Proc. IEEE Internatonal Conference on Neural Networks, Perth, Australa. (1995). 42. Huang, W. Kulkarn, V. and Swamnathan, J.M. "Coordnated nventory plannng for new and old products under warranty". Probab. Eng. Inform. SC., 21(02): pp (2007). 43. Yeo, W.M. and Yuan, X.-M. "Optmal Inventory Polcy For Products Wth Warranty Agreements. n Industral Electroncs", ISIE IEEE Internatonal Symposum on. IEEE (2007). 44. Khawam, J., Hausman, W.H. and D.W. Cheng, "Warranty Inventory Optmzaton for Htach Global Storage Technologes", Inc. Interfaces, 37(5), pp (2007). 45. Darghouth, M.N. At-kad, D. and Chelb, A. "Jont optmzaton of desgn, warranty and prce for products sold wth mantenance servce contracts". Relab. Eng. Syst. Safe., 165, pp (2017). 46. Sh, Y. and Eberhart, R.C. "Emprcal study of partcle swarm optmzaton. n Evolutonary Computaton", CEC 99. Proceedngs of the 1999 Congress on. IEEE (1999). 24

25 Al Hussenzadeh Kashan s an Assstant Professor of Industral Engneerng at Tarbat Modares Unversty. He holds degrees n Industral Engneerng from Amrkabr Unversty of Technology (PolyTechnque of Tehran), Iran. He worked as a postdoctoral research fellow at the department of Industral Engneerng and Management Systems wth the fnancal support of Iran Natonal Elte foundatons. Dr. Kashan s currently an assstant professor n the Department of Industral and Systems Engneerng, Tarbat Modares Unversty and has been actve n appled optmzaton research feld snce Hs research focuses on modelng and solvng hard combnatoral optmzaton problems n areas such as logstcs and supply networks, revenue management and prcng, resource schedulng, groupng problems, fnancal engneerng, etc. As soluton methodologes for real world engneerng desgn problems, he has ntroduced several ntellgent optmzaton procedures, whch nspre from natural phenomena, such as League Champonshp Algorthm (LCA), Optcs Inspred Optmzaton (OIO), Fnd-Fx-Fnsh-Explot- Analyze (F3EA) metaheurstc algorthm and Groupng Evoluton Strateges (GES). Dr. Kashan has publshed over 70 peer-revewed journal and conference papers, and has served as a referee for several outstandng journals such as: IEEE Transactons on Evolutonary Computatons, Omega, Computers & Operatons Research, Journal of the Operatonal Research Socety, Computers & Industral Engneerng, Internatonal Journal of Producton Research, Informaton Scences, Appled Soft Computng, Ecologcal Informatcs, Engneerng Optmzaton, Optmal Control and Applcatons etc. He has receved several awards from Iran Natonal Elte Foundaton and n 2016 he was honored by the Academy of Scences of Iran as the outstandng young scentst of Industral Engneerng. Bakhtar Ostad s an Assstant Professor of Industral Engneerng at Tarbat Modares Unversty. He receved hs PhD n Industral Engneerng from the School of Engneerng at Tarbat Modares Unversty. Also, He receved hs MSc from the Unversty of Tehran n the Industral Engneerng and hs BSc n Appled Mathematcs from K.N. Toos Unversty of Technology. Hs man areas of research nterests nclude busness process reengneerng, qualty engneerng and management, Desred organsatonal capabltes (DOCs) for mprovement projects, Created organzatonal capabltes from qualty management systems (COC-QMs), marketng, corporate fnance, Dependablty Management & RAMS and actvty-based costng. He have publshed several artcles n nternatonal academc journals and conferences ncludng Int. J. Producton Research, Computers & Industral Engneerng, Int. J. Productvty and Qualty Management, Internatonal Journal of Busness Innovaton and Research, Appled Mathematcs and Computaton, Journal of Industral Engneerng and Management, Industral Management, Journal of Industral Engneerng, Int. J. Management Practce, etc. Mohsen Afsah receved hs BS n Industral Engneerng from Kurdestan Unversty n Also, he receved MS and Ph.D. n Industral Engneerng for Tarbat modares Unversty (TMU) 25

26 n 2013 and 2017, respectvely. Currently, he s lecturer n Scence and Culture Unversty and hs research ncludes warranty nventory optmzaton, Prcng, Relablty, Mantenance and Smulaton based Optmzaton. Table captons: Table 1. Relevant prevous research works Table 2. Indces, parameters, and decson varables Table 3. Parameter values for LED 32" Table 4. OIO parameters Table 5. IPSO parameters Table 6. Optmal proft for OIO and PSO algorthm Table 7. Relatonshp between the falure rate, the warranty length and total proft Fgure captons: 26

27 Fgure 1. Demand n product lfe cycle Fgure 2. The cycle of producton, marketng and spare part nventory control Fgure 3. Products Stuatons Fgure 4. Spare part nventory system for under warranty and out-of warranty products Fgure 5. The mn cost network flow problem for spare part nventory control Fgure 6. The soluton representaton Fgure 7. The dea behnd generaton of the new solutons n OIO Fgure 8. Flowchart of OIO algorthm Fgure 9. Comparson of OIO and PSO best solutons Fgure 10. Warranty length comparson for dfferent T s and g s Fgure 11. Prce trend for T=32 Fgure 12. Spare part nventory Fgure 13. Falure Rate Analyss Table 1 Tables References prcng WLO 1 RO 2 PRO 3 Inventory Control statc dynamc Renewng Man product Spare part Sellng perod Plannng horzon EOL 4 Infnte Product s Stuaton Ln and Shue [10] Wu, Ln [7] Huang, Lu [11] Huang, Kulkarn [42] Yeo and Yuan [43] Khawam, Hausman [44] Nonrenewng Underwarranty Out-ofwarranty 27

28 References Km and Park [23] Wu, Chou [13] Ln, Wang [5] Fardmehr and Nak [16] Tsao, Teng [18] We, Zhao [19] prcng WLO 1 RO 2 PRO 3 Inventory Control statc dynamc Renewng Man product Spare part Sellng perod Plannng horzon EOL 4 Infnte Product s Stuaton Yazdan, Shahanagh [20] Char, Dallo [24] Darghouth, At-kad [45] Ths study WLO: Warranty Length Optmzaton, 2 RO: Relablty Optmzaton 3 PRO: Producton Rate Optmzaton, 4 EOL: End of Lfe Nonrenewng Underwarranty Out-ofwarranty Table 2 ndces Descrpton Counter of key components for each product, 1,2,..., I s Counter of Inventory Plannng Perods (ICP), s 1,2,..., T g j parameters Descrpton Counter of Prcng Perods (PP), j 1,..., l. T g I The number of key components n each product l The number of PP n each ICP, Ls Ls 1 l t1 t j T g L s The end of th j PP, t 1,...,. j 1 t j t 1 j l T g The number of ICP n product s lfe cycle The number of ICP that guarantee the spare part avalablty after termnate product s lfe cycle The calendar tme for end of th s ICP 28

29 Wmn Lower bound for warranty length Wmax Pmn Pmax c pc co cr v Upper bound for warranty length Lower bound for prces Upper bound for prces Per-unt procurement cost of product Unt sellng prce for component The procurement cost per unt for component The unt refurbshng cost for component The unt dsposng cost for component h The holdng cost per unt per tme unt for component, α The percent of product s falures that related to component, w The percent of component to be sent to refurbshng center The percent of Successful refurbshng for component Rate of product s falure The servce level parameter for products under warranty pw The servce level parameter for products out of warranty U D0 d Dependent varables k y js n k js S( j, Pj, w) The maxmum market demand The tme of peak demand The ntal demand A gven postve coeffcent Descrpton Random varable for number of falures for products wth stuaton k (stuatons wll be descrbed) that are produced n j th PP and faled n s th ICP Maxmum number of falures for products wth stuaton k that are produced n j th PP and faled n th s ICP Sellng functon n j th PP wth prce Pj and warranty length w D () w s Maxmum number of falures for products under warranty n D () pw s Maxmum number of falures for products out-of warranty n F ( s) E () s V () s All components that may refurbshng n th s ICP All components that have successful refurbshng n All components that may dspose n th s ICP th s ICP th s ICP th s ICP. 29

30 X () s Frst stage decson varables Pj w Second stage decson varables Q () s The number of on hand nventory at the end of Descrpton Sellng prce durng the j th PP Warranty length Descrpton Producton amount of component n th s ICP th s ICP Table 3 I l Wmn Wmax po 1 po D $11 $ w pw U Table 4 OIO Algorthm maxmum number of functon Number of lght ponts (Populaton evaluatons sze) Table 5 IPSO Algorthm maxmum number of functon evaluatons populaton sze Acceleraton constants ( c 1 and c 2 ) *the parameter used s suggested by Sh and Eberhart [46] the nerta * factor ( ) lnearly decreased from 0.9 to 0.4 the number of sub-swarms (p) 2 Table 6 OIO & out-of-klter IPSO & out-of-klter T g best sol worst sol mean std tme best sol worst sol mean std tme

31 OIO & out-of-klter IPSO & out-of-klter T g best sol worst sol mean std tme best sol worst sol mean std tme Table 7 31

32 Rate of product s falure * w Total Proft Fgures Fgure 1 32

33 S j,p,w j Manufacturer Man Product Market D D w pw Q E F Dw D pw Spare Part Warehouse Refurbshng Center Collecton Center D D w pw 1 F 1 F Spare Market Dsposal Ste Fgure 2 Fgure 3 Fgure 4 33

34 Fgure 5 p 1 p 2 3 p plt. w An OIO ndvdual Q I (1) I (2) Q (1) Q (2) Q ( T g) Q Q ( T ) The varables optmally found by a mnmum cost network flow solver Fgure 6 I g Calculate the objectve functon value as the ftness value 34