Research Article Optimization and Customer Utilities under Dynamic Lead Time Quotation in an M/M Type Base Stock System

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1 Hndaw Mathematcal Problems n Engneerng Volume 2017, Artcle ID , 10 pages Research Artcle Optmzaton and Customer Utltes under Dynamc Lead Tme Quotaton n an M/M Type Base Stock System Koch Nakade and Hrok Nwa Department of Archtecture, Cvl Engneerng and Industral Management Engneerng, Nagoya Insttute of Technology, Gokso-cho, Showa-ku, Nagoya , Japan Correspondence should be addressed to Koch Nakade; nakade@ntech.ac.jp Receved 23 February 2017; Revsed 10 Aprl 2017; Accepted 2 May 2017; Publshed 25 May 2017 Academc Edtor: Huanqng Wang Copyrght 2017 Koch Nakade and Hrok Nwa. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. In a manufacturng and nventory system, nformaton on producton and order lead tme helps consumers decson whether they receve fnshed products or not by consderng ther own mpatence on watng tme. In Savaşanerl et al. (2010), the optmal dynamc lead tme quotaton polcy n a one-stage producton and nventory system wth a base stock polcy for maxmzng the system s proft and ts propertes are dscussed. In ths system, each arrvng customer decdes whether he/she enters the system based on the quoted lead tme nformed by the system. On the other hand, the customer s utlty may be small under the optmal quoted lead tme polcy because the actual lead tme may be longer than the quoted lead tme. We use a utlty functon wth respect to beneft of recevng products and watng tme and propose several knds of heurstc lead tme quotaton polces. These are compared wth optmal polces wth respect to both profts and customer s utltes. Through numercal examples some knds of heurstc polces have better expected utltes of customers than the optmal quoted lead tme polcy maxmzng system s profts. 1. Introducton In manufacturng systems the producton and nventory control must be approprate to reduce the producton cost. Informaton on producton such as advance demand, amounts of work-n-process and fnshed products, and machne s falure s mportant to the control. For customers buyng products, the nformaton of order lead tme s mportant to decde whether he/she buys a product or not. In addton, the consumer wll be unsatsfed when the actual lead tme sgreaterthanthenformedleadtme.thequotatonoflead tme causes the actual number of customers to vary because the long quoted lead tme leads to decrease of actual demand, and thus approprate nformaton makes the proft of the system ncrease. Recently, the nformaton on nventory or lead tme to customers s dscussed. For example, Duenyas and Hopp [1] develop a dynamc lead tme quotaton problem n a make-to-order system as an M/M/1 queue usng an MDP formulaton. Ata and Olsen [2] consder a make-to-order system where customers are dynamcally quoted lead tmes. They recommend quotaton polces for convex, concave, and convex-concave delay cost functons. Kapuscnsk and Tayur [3] consder two classes of customers, where the hgh prorty customers brng more proft to the system but delay for them leads to the hgher penalty on the system. Wu et al. [4] consder a newsvendor problem wth nformaton of prce and quoted lead tme and determne optmal amounts of orders, prces, and quoted lead tme. In Selçuk [5] a cost effectve dynamc lead tme quotaton procedure n a sngle-stage controlled manufacturng system s consdered and gudelnes are dscussed for settng the number of kanbans and the frequency of updatng lead tme. Slotnck [6] dscusses an optmal lead tme quotaton polcy when reputaton of frm affects whether each consumer orders products or balks and dscusses the relatonshps among order sze, reputaton, lead tme decson, and so on. In addton, Hafızoğlu et al. [7] consder prce and lead tme decsons n a make-tostock system wth contract and spot customers under Posson arrvals and exponental servce tmes. The optma polcy on prce and lead tme s characterzed. Zhao et al. [8] consder the make-to-order manufacturng system whch gves two

2 2 Mathematcal Problems n Engneerng lead tme and prce quotaton strateges, one of whch offers a sngle lead tme and prce, and the other gves the menu of pars of lead tme and prces. They dscuss the better strateges under the prce-senstve and lead tme senstve consumers. The other lead tme quotaton and decson models are found n Kesknocak et al. [9], Ata [10], Charnsrsakskul et al. [11], Chaharsoogh et al. [12], and Xao et al. [13]. In Savaşanerl et al. [14], the optmal lead tme quotaton n an M/M/1 base stock nventory queue for maxmzng the system s proft s dscussed. In ths system, f the lead tme quoted to the arrvng customer s long, the probablty that the customer enters the system and receves servce becomes small. In ther paper, the model s formulated as a Markov decson process and the property of the optmal lead tme quotaton polcy s dscussed. Lterature on optmal quotaton polces focuses on the optmal lead quotaton polces for the system. Under such optmal lead tme quotaton polces, however, the exact lead tme nformaton s not quoted to customers. One reason s that the producton process s under uncertanness because of falureorreparofthemachne,andthusactualproducton tme s stochastc. The other and mportant reason s that, under the optmal lead tme polcy, the system manager may not gve the mean of actual lead tme, even f the delay cost for quoted lead tme s consdered. Thus, when ths optmal polcy s appled, the customer s satsfacton may be small, because he/she leaves the system by quotaton of long lead tme when the actual lead tme s small and hs/her actual watng tme for tems may be longer than quoted leadtmebecauseofthesettngofshorterquotedleadtme than the actual lead tme. Thus, for the system manager, t s mportant to maxmze not only the proft of the system but also customer s satsfacton. Most of researches n lterature, however, do not dscuss ths knd of satsfacton of actual customers deeply. As one of the customer s utlty on watng tme, the effect of delay nformaton on the customer s satsfacton s formulated n Guo and Zpkn [15, 16]. In these papers, the utlty functon of each customer s defned, whch shows the degree of hs/her satsfacton on watng tme n a queue. It ncludes the stochastc parameter representng hs/her mpatence for watng tme, and f the value of ths parameter s hgh, then he/she s mpatent of watng. They defned several types of delay nformaton and ther effects are dscussed theoretcally and numercally. In Nakade and Nwa [17], an M/M/1 base stock manufacturng-nventory system s developed, but the evaluaton of the utlty functon s napproprate and thus the relatonshp between the utlty and average cost s unclear. In fact, the utlty functon tself only ncludes the watng tme cost, but the evaluaton of the utlty as performance measure ncludes the delay cost for quoted lead tme, whch s confusng. In ths paper, the effect by the lead tme quotaton n an M/M/1 base stock manufacturng-nventory queue nto the customer s utlty functon s dscussed. Posson arrvals are popular because t s well known that the arrval process from large populaton wth small probablty that each person wll arrve at a system approxmately forms a Posson process (e.g., see secton 5.2 of Pnsky and Karln [18]). The utlty functon s based on the defnton n Guo and Zpkn [15], and f hs/her utlty s negatve under the lead tme nformaton then he/she leaves the system wthout recevng an tem and otherwse enters the system and receves t after possble watng tme. The model s formulated nto a Markov decson process. Its optmal lead tme quotaton polcy s derved, and several heurstc polces such as lnear, convex, and concave lead tme quotaton polces are analyzed by brth and death processes numercally. Numercal results are modfed and extended from Nakade and Nwa [17]. In comparson between profts and utltes, the performance measure of utlty does not nclude the delay for quoted lead tme. In addton, the relatonshp among expected reward, the nventory cost, and the delay cost for quoted lead tme under optmal polcy s dscussed for each base stock and fxed delay cost. The numercal examples show that the optmal lead tme quotaton polcy for maxmzng system s average proft has low customer s utlty, and the other smple heurstc quotaton polcy leads to the greater expected values of customers utlty, although t has a bt smaller system s proft than the optmal polcy. In partcular, some lead tme quotaton polcy has both more system proft and much customer s satsfacton n comparson wth the optmal lead tmequotatonpolcywththegreaternumberofbasestocks. The organzaton of ths paper s as follows. In Secton 2, a lead tme quotaton model and a utlty functon n a manufacturng-nventory system wth base stocks are defned. In Secton 3, an average cost under a gven quotaton polcy s determned and the optmzaton problem n ths system s formulated as a Markov decson process. The expected utlty s also derved. Numercal experments for developng propertes of optmal polces and other heurstc polces are gven n Secton 4, and the concludng remarks are gven n Secton Lead Tme Quotaton Model 2.1. Model Descrpton. A manufacturng-nventory system wth a sngle process s consdered. Customers arrve n a Posson process wth rate, and the producton tme has an exponental dstrbuton wth rate μ. The processng tme s mutually ndependent among products and also ndependent of the arrval process. A base stock polcy wth base stock level s s appled n ths system. Fgure 1 llustrates the model. The state of the system s nventory poston,andf<0 then the system has ( ) productsasnventory,andf>0 then denotes the number of watng customers for products. When =0, the system has no fnshed products n nventory andnowatngcustomers. When a customer arrves at the system whose state s, the system quotes hm/her lead tme d whch s an estmatedleadtmeandsbasedonthenventorylevel. After the arrvng customer receves ths nformaton, he/she determnes whether he/she enters the system or not. The decson of each customer follows hs/her own parameter θ, whch represents mpatence on the watng. Ths parameter θ has a dstrbuton functon H(θ) on [0, 1], whose densty s h(θ). Detals on how customers decde whether they buy products or not are dscussed n Secton 2.2.

3 Mathematcal Problems n Engneerng 3 Arrval rate Lead tme quotaton 1 H( r d ) H ( r d ) Departs the system Producton authorzaton d Number of watng customers Producton Leave the system after recevng a fnshed product Queue for producton Base stock polcy Intal nventors =s Fgure 1: Base stock system wth lead tme quotaton. When he/she enters the system, one tem s ordered under a base stock polcy. If there s a fnshed product, then he/she receves t mmedately and leaves the system. It s assumed that the quoted lead tme s dscrete and n a set of 0, 0.05, 0.10,...}. The servce s based on frst come, frst served bass. When a customer enters the system, the system receves reward R. An nventory cost rate h for each fnshed product n nventory s ncurred. In addton, when the delay happens aganst quoted lead tme d,thefxeddelaycostc and the delay cost rate l,whchareproportonaltodelaytme,arealso ncurred for the system. We use the followng notatons. d :quotedleadtmewhenthelevels d 0, 0.05, 0.10,...} (1) f(d): the probablty that the customer to whom the planned lead tme d s quoted enters the system, where t s assumed that f(0) = 1 and 0 f(d) 1. d max :themnmalvalued 0, 0.05, 0.10,...} whch satsfes f(d) = 0, d mn : the maxmal value d 0, 0.05, 0.10,...} whch satsfes f(d) = 1. When c=0, the model concdes wth one n Savaşanerl et al. [14] except that they do not consder the dstrbuton of utlty functon of customers. In the next secton, the utlty functon s defned Utlty. In ths paper, we assume that the utlty functon of each customer, when the mpatence degree s θ and quoted lead tme s d,sgvenby U (θ, d ) =r θd, (2) where r s the value of product whch each customer wll receve and r s assumed postve. The utlty functon s based on Guo and Zpkn [15]. The lnear senstvty of the utlty functon may be extended to nonlnear cases, but t sassumedtoexplanthemodelmoreeaslybytreatnga lead tme quotaton polcy as a smple threshold polcy. The arrvng customer decdes whether he enters the system or notbythesgnofthsvalue.thats,fu(θ, d ) 0 then the customer enters the system, and otherwse he leaves system wthout entrance nto the system. For state whch an enterng customer fnds, f < 0, when the customer enters the system there s no watng tme, and thus hs actual utlty s r. If 0thecustomerwatsforrecevngproduct wth nformaton d.thus,asequenced ; = 0,1,2,...} s system s lead tme quotaton polcy whch controls the entrance of customers. We defne θ = r/d for a gven polcy d ; =0,1,2,...}. From (2), a customer enters the system when the state s f and only f the random parameter θ of the customer s smaller than or equal to θ under ths polcy. That s, when the lead tme d s quoted, the customer wll enter the system wth probablty f(d ) = H(r/d ).

4 4 Mathematcal Problems n Engneerng f(d s ) f(d 1 ) f(d ) s s I G;R Fgure 2: Transton dagram. We denote the maxmal θ satsfyng H(θ) = 0 by θ L, whchsassumedtobepostve.snced max (<+ ) denotes the mnmal d satsfyng f(d )=0,wehave d max = mn (d : r θ L d)= r θ L. (3) Let I= s, s+1, s+2,...,0,1,..., } denote the set of possble nventory postons. When d =d max,thecustomer wllnotenterthesystem,andthusthenventorypostonwll not become +1or more. Thus,,whchsthemaxmal value of possble states, s an nventory poston n whch the decson d =d max s taken. In ths paper, the system s assumed to be n steady state, that s, reputaton of customers aganst the manufacturer s stableunderthefxedleadtmequotatonpolcy.notethat when state s negatve, the arrvng customer can receve a fnshed product mmedately, and thus d s set as Statonary Probabltes. When the quotaton polcy d= (d s,d s+1,...,d Imax ) s decded, the process of the state of system becomes a brth and death process wth parameters f(d ) and μ, as shown n Fgure 2. Thus, the statonary probablty that the nventory poston s s gven by p ( d) = ( μ ) +s f(d 1 )f(d 2 ) f(d s )p s ( d), where p s ( d) I, (4) 1 = (/μ)+s f(d 1 )f(d 2 ) f(d s ). (5) 3. Proft and Utlty 3.1. Average Proft. Under polcy d=(d s,d s+1,...,d Imax ), the expected proft obtaned from customers per unt tme s gven by R( d) = R p ( d) f (d ). (6) The expected nventory cost per unt tme, S( d),sgvenby S( d) = h 1 ( ) p ( d). (7) The probablty that the delay happens when the lead tme d s quoted to the customer n state s denoted by C (d ). When < 0, the customer receves a product mmedately and thus C (d )=0. Snce the producton tme follows the exponental dstrbuton wth rate μ,wehave (μd ) j e μd : C (d )= j! =0,1,2,..., 0: <0. Thus, the average expected fxed delay cost for quoted lead tme s gven by D( d) = c =0 p ( d) f (d ) (μd ) j j! (8) e μd } }. (9) } In the same way, the expected delay for the customer enterng n state s gven by Snce L (d ) (x d = ) μ(μd ) e μx dx d! = 0, 1, 2,... 0 < 0. L (d )= d μ +1 x +1! e μx dx d d μ +1 x e μx dx! = +1 μ e μd +d ( j) (μd ) j (j + 1)! e μd, (10) (11)

5 Mathematcal Problems n Engneerng 5 the expected delay cost on quoted lead tme s gven by T( d) = l =0 p ( d) f (d ) ( +1 μ e μd +d ( j) (μd ) j (j + 1)! e μd ) } }. } (12) Thus,theaverageexpectedproftofthesystemoverannfnte horzon under polcy d s gven by G( d) = R ( d) S ( d) D ( d) T ( d) = R p ( d) f (d ) h 1 ( ) p ( d) c l =0 =0 p ( d) f (d ) p ( d) f (d ) (μd ) j j! e μd } } } ( +1 μ e μd +d ( j) (μd ) j (j + 1)! e μd ) } }. } (13) Here, the model s formulated as a Markov decson process toderveoptmalplannedleadtmed =d } whch maxmzes the average reward G( d). By unformzaton technque, themodelcanbeformulatedntoadscrete-tmemodelwth the optmal average reward g and relatve reward V () under optmal polcy, whch satsfes g+v () = max d h +μ + +μ f(d )(R cc (d ) ll (d )+V (+1))+ +μ (1 f (d )) V () + μ +μ V (), ( = s), h +μ + +μ f(d )(R cc (d ) ll (d )+V (+1))+ +μ (1 f (d )) V () + μ +μ V ( 1), ( > s). (14) Here = max, 0}.Thend maxmzes the rght hand sde of ths equaton, and the optmal average reward s G( d )= ( + μ)g.theoptmalpolcycanbesolvedbythewell-known polcy teraton method, because for any quotaton polcy the nventory poston can return to s wth probablty one (see Puterman [19]) Expected Utlty. When a customer enters based on the utlty functon U(θ, d ) whch s gven n (2), ts actual utlty whch the customer receves s dfferent from the value of ths utlty functon, because the delay of the actual lead tme compared wth quoted lead tme may happen. The actual expected utlty of enterng customer wth parameter θ n state, denoted by u(θ, d ),sgvenby u (θ, d ) =r θ +1 μ (15) for 0.Notethatu(θ, d )=rwhen <0.The expected utlty of each customer under quotaton lead tme polcy d s gven by 4. Numercal Experments Gven parameters, μ, R, H(θ), h, l, c, f(d), r, and θ L, we derve the optmal polcy d maxmzng the average proft of the system for each fxed s, and then the optmal amount of base stock level s s derved. Then several lnear lead tme polces are proposed, where lnear lead tme polcy means that the quoted lead tme s proportonal to the nventory sze. We dscuss these lnear polces wth optmal polces regardng both proft of the system and the utlty of customers. Convex and concave type lead tme quotaton polcesarealsontroduced. The parameters used n numercal experments are set as follows: = 0.6, μ=1.0, h = 0.5, c = 1.0, u( d) = 1 where θ = r/d. p ( d) r + =0 p ( θ d) (r φ +1 θ L μ )h(φ)dφ, (16) l = 1.0, r=1.0, R=10, θ L = 0.25, d 0.00, 0.05, 0.10,...}.

6 6 Mathematcal Problems n Engneerng 1: θ θ L +1, H (θ) = θ θ L : θ L θ θ L +1, 0: θ<θ L, r θ L +1, 1: d r f (d) = d θ L : 0: d r. θ L r θ L +1 d r θ L, (17) Wthout loss of generalty, μ s set as the recprocal of the unt tme and thus μ = 1.Thus,naunttmeoneproducts produced n average. From = 0.6, the maxmal avalablty of the process, whch s a producton rate when all orders from customers are accepted, s set as /μ = 0.6. Fromthe settng of parameters, we have d max = r/θ L =4,andby(16) the expected utlty of customers becomes u( d) = 1 p ( d) r + =0 p ( d) r(θ θ L ) ( +1 2 μ )(θ 2 θ 2 L 2 )}. (18) 4.1. Optmal Quoted Lead Tme. Optmal quoted lead tme and the expected average proft for each s are shown n Table 1 n the case that c = 0 and c = 1. The expected utltes of customers under the optmal polces are also gven. From Table 1, optmal quoted lead tme s frst ncreasng andthendecreasngns, whch agrees wth the result of Savaşanerl et al. [14], n whch no delay opportunty cost s ncurred; that s, c = 0. Under the optmal polces, the values of R( d), S( d), D( d), andt( d) are also gven n Table 1. When s s small, more customers must be accepted to ncrease system s proft and thus quoted lead tme s small, and as a result the expected delay to quoted lead tme T( d) (and delay cost D( d) for c = 1)slarge.Whens ncreases, both R( d) and S( d) ncrease. For large s, many customers can receve fnshed products mmedately at ther arrvals, and thus more customers enter the system, whereas there are more fnshed products n nventory and more nventory costs are needed. Snce = 0.6 and R=10meanthat the upper bound of R( d) s 6, for large s almost arrvng customers enter the system whereas more nventory costs are needed under optmal polces, and thus the optmal total proft of the system becomes small for large s.comparedwththecase that c=0, the optmal polcy s senstve to delay when c=1, whch leads to long quoted lead tme. As a result, T( d) and R( d) are smaller compared when c=0. The utltes are ncreasng n s, becausethedelays smallerformoreamountsofbasestocks.theoptmalbase stock level maxmzng the system s proft for c = 0 s 1 whereasts2forc=1.whencncreases, the probablty that the delay to the quoted lead tme happens s more mportant, and thus more amounts of base stocks are needed. Under optmal polces for each s, the proft decreases n c, but the expected utlty ncreases. For large c the delay probablty needs to be decreased, and as a result the hgher quoted lead tme s set under the optmal polcy, and thus customer s utlty ncreases although the fracton that customers enter the system decreases. Snce d mn = r/(θ L + 1) = 0.8,whered mn s the maxmal value satsfyng f(d) = 1,forsmallstated =d mn because for all d d mn customers enter the system wth probablty LnearLeadTmeQuotatonPolcy. As a heurstc polcy, we consder a lnear lead tme quotaton polcy whch takes quoted lead tme proportonalto +1for nonnegatve state. That s, there s a constant α whch satsfes d =α +1 μ (α >0). (19) Ths quotaton polcy s denoted by d(s, α). Notethatwhen α<d mn,forsmall such that d mn > α(( + 1)/μ) t s set as d =d mn.forlarge such that d max α(( + 1)/μ), the customer does not enter under the polcy, and d s set as d max. For example, under d(s, 0.6) d 0 =d mn = 0.8, d 1 =1.2, d 2 = 1.8, d 3 = 2.4, d 4 = 3.0, d 5 = 3.6, andd =d max = 4.0 for all 6. Table 2 gves the profts and utltes for s = 0, 1, 2, 3, 4 when α = 0.6, 0.8, 1.0, 1.2 when c=1.notethatα<1mples that the expected actual lead tme s greater than the lead tme quoted to the customer. In fact, as shown n Table 2, when α ncreases, the proft decreases whereas the utlty ncreases, because the hgher quoted lead tme mples less numbers of enterng customers and less delay to the quoted lead tme Nonlnear Polcy. Here we consder nonlnear lead tme quotaton polces and compare them wth lnear quoted lead tme polces. Consdered quoted lead tme polces are gven n Table 3(a). The word lnear s the lnear lead tme quotaton polcy d(s, 0.6). The convex polcy means the quoted lead tme s convex n state from 1 to 5, whereas the concave polcy means that t s concave n state from 1 to 5. The optmal polcy n Table 3(a) represents the optmal lead tme polcy for s=2when c=1. The profts and utltes for these polces are gven n Table 3(b) when c = 1. Compared wth lnear lead tme quotaton polcy, the convex polcy has more profts and less utltes, and the concave polcy has those vce versa. From these results, when the proft s more mportant, the convex type polcy s better and when the utlty has more weghts the concave type polcy s desrable. Note that the optmal lead tme quotaton polcy has more proft and less utlty n comparson wth three heurstc polces Comparson. For the optmal polcy and the other polces, the pars (proft, utlty) for s = 1, 2, 3, 4 are plotted n (x, y)-plane, whch s shown n Fgure 3 n the case where c=1. In ths fgure, d (s) s an optmal polcy for base stock level s, and cc(s) and cv(s) are concave and convex polces for

7 Mathematcal Problems n Engneerng 7 Table 1: Optmal polcy. (a) c=0 State State c = 0.0 s Proft Utlty R(d) S(d) D(d) T(d) (b) c=1 c = 1.0 s Proft Utlty R(d) S(d) D(d) T(d) base stock s,defnednsecton4.3,respectvely.theaverage proftandtheexpectedutltyforeachpolcyareobtaned by usng statonary probabltes (5) n Secton 2.3 and (13) and (16). For s = 1, the proft of optmal polcy s smaller than the optmal polcy for s = 2,andforanypolcywth s=1, the utlty s also smaller than the optmal polcy for s=3or 4. Thus, s=1should not be selected. For s=2,the optmal polcy maxmzes the proft among all polces wth all amounts of base stocks. Compared wth the optmal polcy when s=3,thelnear quoted lead tme polces d(2, 0.6), d(2, 0.8), and concave andconvexpolcescv(2) and cs(2) have more profts and

8 8 Mathematcal Problems n Engneerng Table 2: The lnear lead tme quotaton polcy. c = 1.0 Base stock s α Proft Utlty Proft Utlty Proft Utlty Proft Utlty Table 3: Comparson of heurstc polces. (a) Polces Lnear Convex Concave Optmal (b) Profts and utltes c=1 Base stock s Lnear Proft α = 0.6 Utlty Convex Proft Utlty Concave Proft Utlty Optmal Proft Utlty more utltes. Therefore, the optmal polcy for s = 3 s not needed to be used. Smlar propertes hold between the optmal polcy for s=4and several polces wth s=3,and thus the optmal polcy for s=2wll not be selected. The best polcy depends on the thnkng of decson maker. 5. Conclusons In ths paper, the base stock model wth quoted lead tme s consdered and as performance measure the system proft and customer s utlty are consdered. It s formulated as a Markov chan when the lead tme quotaton polcy s gven, andtsalsoformulatedasamarkovdecsonprocessto derve optmal polces for maxmzng the system s proft. Utlty functon of a customer s ntroduced and the heurstc polces for ncreasng utlty of customers are proposed. The numercal results show that quoted lead tme should be set by consderng not only proft of the system but also the utlty of customers. The optmal polcy gves the smaller utltes, and n partcular f the number of base stocks s not approprate, then the other better polcy exsts on both profts and utltes than the optmal polcy. It s noted that, throughout the numercal experments, the base stock level s the most mportant factor for maxmzng profts. That s, optmal base stock level and optmal polces are needed to be derved, but ths optmal polcy leads to poor expected utltes, and thus heurstc polces such as lnear lead tme polces should be consdered to balance the system s proft and customer s expected utlty.

9 Mathematcal Problems n Engneerng d (4, 0.8) d (4, 1.2) cc(4) d (4, 1.0) d (4, 0.6) cv(4) d (4) d (3, 1.2) d (3, 1.0) d (3, 0.8) d (2, 1.2) d (2, 1.0) d (1, 1.2) d (1, 1.0) d (1, 0.8) cc(3) d (3, 0.6) cc(1) cv(3) d (2, 0.8) cc(2) d (2, 0.6) cv(2) d (3) Utlty d (1, 0.6) cv(1) d (2) cv(0) d (1) Proft s=0 s=1 s=2 s=3 s=4 Fgure 3: Comparson on profts and utltes. Theoretcal consderaton on utltes under optmal polces and the other heurstc polces wll be needed. In addton, the cases wth general dstrbutons of processng tmes and multstage manufacturng systems are also nterestng. They are left for future research. Conflcts of Interest The authors declare that there are no conflcts of nterest regardng the publcaton of ths paper. Acknowledgments Ths paper s supported by JSPS Grant-n-Ad for Scentfc Research (C) 16K References [1] I. Duenyas and W. J. Hopp, Quotng Customer Lead Tmes, Management Scence,vol.41,no.1,pp.43 57,1995. [2] B. Ata and T. L. Olsen, Near-optmal dynamc lead-tme quotaton and schedulng under convex-concave customer delay costs, Operatons Research, vol. 57, no. 3, pp , [3]R.KapuscnskandS.Tayur, Relabledue-datesettngna capactated MTO system wth two customer classes, Operatons Research,vol.55,no.1,pp.56 74,2007. [4] Z. Wu, B. Kazaz, S. Webster, and K.-K. Yang, Orderng, prcng, and lead-tme quotaton under lead-tme and demand uncertanty, Producton and Operatons Management, vol.21, no.3,pp ,2012. [5] B. Selçuk, Adaptve lead tme quotaton n a pull producton system wth lead tme responsve demand, Manufacturng Systems,vol.32,no.1,pp ,2013. [6] S. A. Slotnck, Lead-tme quotaton when customers are senstve to reputaton, Internatonal Producton Research, vol.52,no.3,pp ,2014. [7] A. B. Hafızoğlu, E. S. Gel, and P. Kesknocak, Prce and lead tme quotaton for contract and spot customers, Operatons Research, vol. 64, no. 2, pp , [8] X. Zhao, K. E. Stecke, and A. Prasad, Lead tme and prce quotaton mode selecton: Unform or dfferentated? Producton and Operatons Management,vol.21,no.1,pp ,2012. [9] P. Kesknocak, R. Rav, and S. Tayur, Schedulng and relable lead-tme quotaton for orders wth avalablty ntervals and lead-tme senstve revenues, Management Scence, vol.47,no. 2,pp ,2001. [10] B. Ata, Dynamc control of a multclass queue wth thn arrval streams, Operatons Research, vol. 54, no. 5, pp , 2006.

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