MULTI-ITEM PRODUCTION PLANNING AND MANAGEMENT SYSTEM BASED ON UNFULFILLED ORDER RATE IN SUPPLY CHAIN

Journal of the Operatons Research Socety of Japan 2007, Vol. 50, No. 3, 201-218 MULTI-ITEM PRODUCTION PLANNING AND MANAGEMENT SYSTEM BASED ON UNFULFILLED ORDER RATE IN SUPPLY CHAIN Nobuyuk Ueno Koj Okuhara Hroak Ish Hroak Shbuk Toshak Kuramoto Prefectural Unversty Osaka Unversty Mazda Motor YNS INC. of Hroshma Corporaton (Receved May 26, 2005; Revsed September 19, 2006) Abstract In the automoble ndustry, a usual busness model has a problem to realze mass customzaton, because t s dffcult to satsfy the dversfed customer needs. Ths paper proposes a mult-tem producton and nventory plannng method of the mass customzaton wth the consderaton of the restrcton of daly manufacturng capacty and so on. Ths model s formulated as a stochastc programmng problem, and then the sub problem as a lnear programmng problem. An effcent and practcal algorthm for the mult-tem model s developed. Keywords: Inventory, producton plannng, mass-customzaton, mathematcal programmng, stochastc model 1. Introducton In recent years, the automoble ndustry has pushed forward wth reducton n cost, nducton of foregn captal, competton between supplers, and so on. It becomes very mportant to remove waste of the producton actvtes and to meet the demands of an ndvdual customer and a changng market. In partcular, we must satsfy the varety of customer specfcaton n product and servce, wthout droppng the productve effcency n mass customzaton [1,5,10]. For example, n case of a certan car manufacturer, t s sad that there s possblty of 300,000,000 ways of specfcaton for a customzed product n one model of a car. On manufacturng of ths customzed product, levelng of producton load becomes very dffcult because manufacturers have to meet orders of customers based on each specfcaton. Then the productvty decreases, fnshed goods n stock ncrease, and t becomes dffcult to deal wth customer specfcaton. There have already been some manufacturng strateges for mass customzaton. However, the desgn method of producton plannng and management system for t has not yet been establshed. Under the precondton that delvery lead tme whch s expected by customers s longer than producton lead tme whch s necessary for manufacturers, Make-to-Order management system n whch manufacturng starts after recevng an order has been appled for a varety of customer specfcatons as producton plannng and management system for mass customzaton. However, t s often the case that delvery lead tme becomes shorter than producton lead tme. Therefore manufacturers have to start manufacturng before they receve an order from customers. For a varety of customer specfcatons, Make-to-Order management system and Parts Orented Producton System (POPS) type module manufacturng system are proposed [9]. 201

202 N. Ueno, K. Okuhara, H. Ish, H. Shbuk & T. Kuramoto The Materal Requrement Plannng (MRP) [11] and Advanced Plannng & Schedulng (APS) [7] are presented to plan and manage such systems. Ths MRP satsfes varous demands from customers by promotng modularzaton n producton. By makng use of modularzaton, the MRP makes producton lead tme shorter, and t manages Make-to- Order management system on precondton that delvery lead tme s longer than producton lead tme. When delvery lead tme s shorter than producton lead tme, we consder the mnmum stock as necessary beforehand. But n these management systems, the mechansm between substtutablty of unfulflled order and stocks on mass customzaton has not been dscussed quanttatvely. On precondton that delvery lead tme s shorter than producton lead tme, manufacture seat system for both prospectve stocks and order stocks has been proposed [13]. A study on the analyss of manufacture seat system coverage has also been done [6]. However, t has been ponted out that further theoretcal work about the decson method of proper manufacture seat should be done [13]. In ths paper, we take up producton plannng on the suppler sde [4,14] among manufacturer and assembly supplers n automoble ndustry. On precondton that delvery lead tme s shorter than producton lead tme, our targets are both prospectve stocks and order stocks. Our producton plannng and management system does nether dscrmnate between them nor depend on producton seat. We regard mass customzaton as fluctuaton of order quanttes from manufacturer to suppler, and we deal wth order quanttes from manufacturer, that s, the demand for suppler, stochastcally. At frst, we defne the producton plannng and management system as the stochastc programmng problem [2] to fnd producton plan mnmzng manufacture and nventory cost under both unfulflled order constrant and producton constrant. Next we show how to get practcal and effectve algorthm to the problem. Fnally, we descrbe a desgn procedure of the whole producton plannng and management system for mplementng mass customzaton. 2. Mass Customzaton Envronment The conceptual fgure of collaboraton model between manufacturer and supplers for mplementng mass customzaton for each tem (part) s shown n Fgure 1. The example n Fgure 1 shows that there are 3 types of order nformaton (forecast order) from manufacturer to suppler. The monthly forecast order gves the prospected order 3 months before, the weekly forecast order gves the predcton value 2 weeks before, and the delvery nstructon to suppler s gven as the frm order 1-3 days before delvery due date. In the mass customzaton envronment, the ncrease of customer specfcatons means the ncrease of producton specfcatons n manufacturer. Ths mples that the quanttes of frm order reflectng customer needs have large fluctuaton lke the bottom mage of Fgure 1. For suppler, the standard producton lead tme for 100-200 knds of products (or parts) usually takes about 1 week. Therefore producton lead tme becomes longer than delvery lead tme. Suppler must start producton n advance based on producton plan by usng MRP, for example, accordng to weekly forecast order. Furthermore the suppler must perform the producton plannng whle at the same tme avodng unfulflled order to the frm order whch s gven 1-3 days before delvery due date. We regard mass customzaton as fluctuaton of order from customer to manufacturer and from manufacturer to suppler. Then t can be grasped from the suppler s pont of vew as the fluctuaton of order quanttes of target part from manufacturer to suppler. Ths

Plannng Based on Unfulflled Order Rate 203 Fgure 1: Future collaboraton model between manufacturer and suppler for mplementng mass customzaton Fgure 2: Dfference between MTO and ATO means that the fluctuaton of order quanttes can be represented by standard devaton around average that forecast order gves. Fgure 2 shows the dfference between make-toorder (MTO) and assemble-to-order (ATO) n envronment under mass customzaton of producton management system. We focus on ths case, and suppler should manufacture accordng to weekly forecast order. As s shown n Fgure 3, assumng that σ 0 denotes the order fluctuaton of concerned model of car from customer to manufacturer and σ denotes the order fluctuaton about concerned producton (or parts) from manufacturer to supplers. Then the producton plannng and management system mplementng mass customzaton n supplers sde can be consdered by the problem such as how to product and to stock ndvdual parts n advance n order not to run short of supples for σ whch gves fluctuaton of frm order from manufacturer. In ths paper, we assume that σ s gven and demand s defned as a normal dstrbuton wth tme varant average and tme varant devaton of order. We here focus on the suppler s producton plannng system, so the effect of σ only are dscussed n ths model. However t s mportant to nvestgate the mpact of σ 0 on the model, whch should be ncluded n the future research.

204 N. Ueno, K. Okuhara, H. Ish, H. Shbuk & T. Kuramoto Fgure 3: Flow of order among customer, manufacturer and suppler 3. Unfulflled Order Rate for Suppler s Producton Plannng and Management System 3.1. Dervaton of unfulflled order rate In mass customzaton envronment, we formulate problem whch determnes proper producton quanttes among n perods to optmze nventory change at fnal producton stage of suppler, correspondng to fluctuaton of demand from manufacturer. In ths secton, we dscuss about one certan tem. [Notaton] : Perod ( n). d : Frm order of manufacturer at perod. x : Producton quantty at perod of suppler. S : Inventory quantty at perod of suppler. p : Manufacturng cost per module at perod of suppler. h : Inventory holdng cost per module at perod of suppler. r : Total product quantty untl perod n of suppler. R : Set of lnear producton constrants of suppler. SO : Unfulflled order rate untl perod of suppler. β : Unfulflled order rate of planned target of suppler. It s assumed that d obeys an average d and standard devaton ω where d, d j ( j) are ndependent each other, and ω = σ d, where σ s the devaton of order. Let ntal nventory be S 0. R s denoted by (x 1, x 2,, x n ) R where R denotes lnear producton constrant, and s convex. SO s probablty functon whch s n short of delvery to the frm order at least untl perod. The nventory quantty S at perod s gven by S = S 0 + x t d t (3.1) t=1 t=1 where d s random varable so that S becomes random varable. It obeys normal dstrbuton whch has the followng tme varant average and varance. S can be replaced by Average m = S 0 + x t d t, (3.2) t=1 t=1 V arance σ 2 = ωt 2. (3.3) t=1 y = S m σ. (3.4)

Plannng Based on Unfulflled Order Rate 205 The probablty functon M whch satsfes nventory quantty S derved by 0 at perod s M = 0 1 e (S m ) 2σ 2 ds = 2πσ 2 1 m σ e y 2 2π 2 dy. (3.5) Thus () M = 0.5 + () M = 0.5 These equatons can be represented by m σ 0 m σ m σ M = 0.5 + sgn(m ) 0 0 1 e y 2 2 dy (m 0), (3.6) 2π 1 e y 2 2 dy (m < 0). (3.7) 2π 1 e y 2 2 dy. (3.8) 2π We consder an unfulflled order rate SO n whch represents probablty that S 1 0, S 2 0, S 3 0,, S n 0 s not satsfed. Then, the upper bound of the unfulflled order rate SO n s gven by 1 n M t. Snce t s dffcult to obtan the true value of the unfulflled t=1 order rate, we defne the unfulflled order rate SO n by n SO n = 1 M t. (3.9) t=1 By consderng such unfulflled order rate, we can construct the producton plannng on safety sde. In addton, SO n can be rewrtten by applyng the ntegraton by parts to M t as follows; SO n = 1 A (3.10) where A = n t=1 1 2 ( ) 2 m t σ t 0.5 + sgn(m t ) e 2π k=1 m t σ t 2k+1. (3.11) 1 3 5 (2k + 1) 3.2. Property of unfulflled order rate We clarfy the property of unfulflled order rate to understand effect of some varables. Lemma 1 SO n s monotonous decrease functon of x. Lemma 2 SO n s monotonous decrease functon of S 0. Lemma 3 SO n s monotonous ncrease functon of d. Lemma 4 SO n s monotonous ncrease functon of ω n m > 0 ( n). Proof We apply partal dervatve SO n wth respect to x, S 0, d and ω. Let α be arbtrary varable. SO n α = M 1 α M 2 M n M 1 M 2 α M 3 M n M 1 M 2 M n 1 M n α. (3.12)

206 N. Ueno, K. Okuhara, H. Ish, H. Shbuk & T. Kuramoto Snce M > 0 and f(y )dy = f(g(α)) g(α) α g(α) α, (3.13) by substtutng α nto x, S 0, d and ω, we can derve the followng propertes. M α = 1 2πσ e 1 2 M α = 1 2πσ e 1 2 M α = 1 2π e 1 2 ( m σ ) 2 > 0 (α x, S 0 ), (3.14) ( m σ ) 2 < 0 (α d ), (3.15) ( ) 2 m σ m ω < 0 (α ω σ 2, m > 0). (3.16) σ Equatons (3.14) (3.16) and M > 0 ( n) lead lemmas 1 4. 4. Producton Plannng for Implementng Mass Customzaton wth Mult-tem 4.1. Problem formulaton of mass customzaton wth mult-tem In ths secton, we defne suppler s producton plannng and management system as the stochastc programmng problem to fnd producton plan mnmzng manufacturng and nventory holdng cost under both unfulflled order rate constrant and producton constrant. In order to consder the mult-tem case, prevous notatons are extended by addng nformaton about jth part of product (j m). For example, d j, x j and S j denote the frm order about jth part of manufacturer at perod wth tme-varant average d j, the producton quantty of jth part at perod and nventory quantty of jth part at perod respectvely. Also p j and h j denote manufacturng cost per module about jth part of suppler at perod and nventory holdng cost per module about jth part of suppler at perod, respectvely. Those parameters, p j and h j, wll be regarded as the same value about each part wthout losng generalty and practcalty n order to smplfy a transformaton of objectve functon. And S j 0 denotes ntal nventory quantty of jth part, β j s targeted unfulflled order rate of jth part, and Q denotes upper bound of total product quantty at perod. [Formula 1] mnmze m n n E[ { p j x j + h j S j }] (4.1) j=1 =1 =1 s.t. S j 0 + x j t d j t 0 (, j) (4.2) t=1 t=1 n x j = r j ( j) (4.3) =1 SOn j β j ( j) (4.4) m x j Q ( ) (4.5) j=1 (x j 1, x j 2,, x j n) R j ( j) (4.6) x j 0 (, j) (4.7) The evaluaton functon (Equaton (4.1)) expresses that we must fnd optmal solutons (x j 1, x j 2,, x j n), (j = 1, 2,, m) that mnmze the expected value (expectaton) of the

Plannng Based on Unfulflled Order Rate 207 sum of producton cost and nventory cost. Equaton (4.2) s non negatve condton about nventory quantty that the demand s not over the forecast order. Equaton (4.3) s constrant about target producton quantty. Equaton (4.4) gves constrant about unfulflled order rate, Equaton (4.5) denotes lmt of producton capacty at each perod and Equaton (4.6) represents lnear constrant about producton. The above problem s a manufacture/nventory problem where demand changes stochastcally [2]. Although the (s, S) strategy [3,12] s known as an effectve tool for such a problem, t s not applcable when the problem has many condtons about producton. Thus we propose practcal and effectve algorthm by solvng lnear problem as partal problems repeatedly. 4.2. Procedure of algorthm We frst substtute E[S j ] = S j 0 + t=1 x j t t=1 dj t nto total cost MC of Equaton (4.1), MC = = ( m n ) ( m n ) p j x j + h j S j 0 j=1 =1 j=1 =1 { ( m n )} { ( m n )} + h j x j t h j d j t j=1 =1 t=1 j=1 =1 t=1 ( m n ) { ( m n )} p j x j + h j x j t + MC 1 (4.8) j=1 =1 j=1 =1 t=1 where { ( m n m n MC 1 = h j S j 0 h j j=1 =1 j=1 =1 t=1 d j t )} (4.9) Assumng that h j = h j, p j = p j, = 1,, n wthout losng generalty and practcalty on smplfyng a transformaton, frst term of Equaton (4.8) can be rewrtten by m j=1 p j r j because of m j=1 p j n =1 x j. Then by usng MC s calculated by m MC 2 = h j n (n t + 1) x j t, (4.10) j=1 t=1 m MC = MC 1 + MC 2 + p j r j (4.11) j=1 where MC 1 and m j=1 p j r j are constants. Here we formulate Mult-tem P problem applyng relaxaton strategy by exceptng Equaton (4.4) n Formula 1. [Mult-tem P problem] m mnmze h j n (n t + 1)x j t (4.12) j=1 t=1 s.t. S j 0 + x j t d j t 0 (, j) (4.13) t=1 t=1

208 N. Ueno, K. Okuhara, H. Ish, H. Shbuk & T. Kuramoto Fgure 4: Basc dea of proposed algorthm for a mult-tem producton plannng model n x j = r j ( j) (4.14) =1 m x j Q ( ) (4.15) j=1 (x j 1, x j 2,, x j n) R j ( j) (4.16) x j 0 (, j) (4.17) When we obtan (x j 1, x j 2,, x j n ), (j = 1, 2,, m) as optmal soluton of mult-tem P problem, t corresponds to global optmal soluton f t satsfes Equaton (4.4). If t does not satsfy Equaton (4.4) then we consder the followng mult-tem MP problem where the unfulflled order rate holds smaller value than certan target β j about jth part at perod. [Mult-tem MP problem] mnmze m h j n (n t + 1)x j t (4.18) j=1 t=1 s.t. S j 0 + x j t d j t 0 (, j) (4.19) t=1 t=1 S j 0 + x j t d j t K j ( B j, j) (4.20) t=1 t=1 n x j = r j ( j) (4.21) =1 m x j Q ( ) (4.22) j=1 (x j 1, x j 2,, x j n) R j ( j) (4.23) x j 0 (, j) (4.24) Compared wth the Mult-tem P problem, the Mult-tem MP problem nvolves Equaton (4.20) newly. From Fgure 4, K j and B j n Equaton (4.20) can be understood as follows. We defne the target unfulflled order rate β j for each perod n plan from unfulflled

order rate of jth part β j as follows; Plannng Based on Unfulflled Order Rate 209 1 β j = n 1 β j (4.25) Let unfulflled order rate whch s calculated at the frst teraton on the mult-tem MP problem be β j(1). By usng β j(1), we derve β j(2) = β j(1) sgn(β j(1) β j ) βj(1) β j, (4.26) INC (2) j = arg max n (βj(2) ) (4.27) and B j = { (2) j }. In order to hold generally that the probablty, nventory quantty S j s less than y j at perod, s smaller than the unfulflled order rate β j n Fgure 4, we obtan y j β j = 1 2σ 2πσ j e 2 ds j, (4.28) K j = m j y j, (4.29) (Sj mj )2 B j denote the set of perod ndces, whch the unfulflled order rate of certan perod becomes smaller value than β j. Form Equaton (4.26), (4.27), (4.28) and (4.29), we fnd K j(2) satsfyng β j(2) = y j(2) 1 2πσ j e (S j mj(1) ) 2 2σ j 2 ds j, (4.30) K j(2) = m j(1) y j(2). (4.31) Thus we can reconstruct new mult-tem MP problem that the probablty, nventory quantty S j whch s less than 0 of jth part at perod (2) j, takes smaller value than β j(2) by applyng K j(2) and (2) j B j to Equaton (4.20). We wll show the reason why by solvng mult-tem MP problem repeatedly, we can fnd the approxmate soluton. Whenever Equaton (4.20) s added sequentally, we can get the mproved producton quantty x j(2) about jth part at perod that the probablty, whch nventory quantty S j(2) of the jth part at perod (2) j s less than 0, takes smaller value than β j(2) by K j(2) and (2) j β j to Equaton (4.20). Then mproved producton quantty x j(2) at one teraton makes both unfulflled order rate β j(2) of jth part at perod and the unfulflled order rate of jth part β j(2) monotonously decrease by Lemma 1 at secton 3.2. These mechansm s repeatedly by changng next step targeted unfulflled order rate β j(l) (l > 2) of jth part at perod decreasng, whch s defned lke Equaton (4.26) when l = 2. Now that, we ntroduce update parameter INC to next step unfulflled order rate n calculaton (Equaton (4.26)) of the target unfulflled order rate β j at each perod n plan. Fgure 5 shows nput factors between manufacturer and suppler. Purpose of research s to develop a suppler s producton plannng system whch s mnmzng total cost subject to unfulflled order rate and producton constrants when weekly forecast order and devaton of order σ are gven.

210 N. Ueno, K. Okuhara, H. Ish, H. Shbuk & T. Kuramoto Fgure 5: Input factors between manufacturer and suppler 4.3. Algorthm In ths secton, we descrbe algorthm for fndng the approxmate soluton of Formula 1. Step 1 Let l = 1, T j = {1, 2,, n} ( j), B j = {} ( j) and ϵ = 0.0001. T j denotes set of perod ndex. β j = 1 n 1 β j ( n, j m) (4.32) Step 2 Fndng a soluton of the mult-tem P problem, then we obtan the soluton (x j 1, x j 2,, x j n ), cost C (l) as the objectve value of Equaton (4.18), unfulflled order rate β j(l) and unfulflled order rate at each perod β j(l). Step 3 () If β j(l) β j, ( j) holds, then (x j 1, x j 2,, x j n ) s global approxmate soluton (END). () Otherwse, for certan j holdng β j(l) > β j, t s updated by then β j(l+1) = β j(l) sgn(β j(l) β j ) βj(l) β j INC, (4.33) (l+1) j = arg max n (βj(l+1) ) (4.34) {B j } = {B j } { (l+1) j }, {T j } = {T j } { (l+1) j } (4.35) and reconstruct the mult-tem MP problem by usng β j(l+1) = y j(l+1) 1 2πσ j e (S j mj(l) ) 2 2σ j 2 ds j, (4.36) K j(l+1) = m j(l) y j(l+1). (4.37) Step 4 If we get a soluton (x j 1, x j 2,, x j n ), cost C (l+1), and the unfulflled order rate β j(l+1) for the mult-tem MP problem, and the followng relaton holds then procedure s end. β j(l+1) β j(l) < ε j ( j) (4.38)

Plannng Based on Unfulflled Order Rate 211 Table 1 d, ω, σ used n numercal example 1 2 3 4 5 6 7 8 d 4 8 0 20 16 12 12 12 ω 1.4 2.8 0 7 5.6 4.2 4.2 4.2 σ 1.4 3.13 3.13 7.67 9.5 10.38 11.2 11.96 Fgure 6: Result of proposed algorthm If we can not get a soluton (x j 1, x j 2,, x j n ), cost C (l+1), and the unfulflled order rate β j(l+1) for the mult-tem MP problem, then procedure s end. Step 5 () If β j(l+1) β j, ( j) holds, then (x j 1, x j 2,, x j n ) s global approxmate soluton (END). () Otherwse, let l = l + 1, then go to Step 3 (). 5. Numercal Experments 5.1. Effect of proposed unfulflled order rate as crtera In ths secton, n order to confrm the effectveness of crtera of unfulflled order rate we propose n ths paper, we apply our algorthm to actual operaton data n auto parts suppler. Table 1 shows d, ω and σ about forecast order. It s assumed that S 0 = 24, p = 1, h = 1, r = 82, ω = d 0.35 for each and β = 0.05. We set some parameters based on the actual data such as manufacturng and nventory holdng costs to show the effcency of ths approach. Fgure 6 shows comparson of nventory change between the result of proposal method n case of m = 1, INC = 2 and the actual producton plan. We call the result of real operaton planned by producton manager n the suppler base case. Here base case s consequence of actual operaton data n suppler. From the results of Fgure 6, we can fnd that our method provde lower nventory change because of consderng nventory cost n frst half, and provde hgher nventory change for respondng to fluctuaton of demand n last half. Fgure 7 shows the transton of both unfulflled order rate and the cost at each teraton by usng proposed algorthm. The unfulflled order rate SO n decreases monotonously and soluton s obtaned after 21 teratons. The cost s plotted by relatve comparson for settng the cost of ntal state to 100%. We can fnd that the unfulflled order rate s mproved at each teraton. We can also fnd that the exstence of trade-off between the unfulflled order rate and the cost s shown n Fgure 7. Fgure 8 shows mprovement of unfulflled order rate by proposed technque when ntal nventory s changed. When ntal nventory s gven by 24, then unfulflled order rate takes

212 N. Ueno, K. Okuhara, H. Ish, H. Shbuk & T. Kuramoto Fgure 7: Result of unfulflled order rate and cost for each teraton ( : base case, : proposed algorthm, : proposed algorthm wth same cost of base case) Fgure 8: Performance of proposal algorthm 11.7% by base case, but t takes 5.8% by proposal algorthm. Partcularly, from the result about proposed method wth same cost of base case, t takes 8.5% then we can fnd our method mprovng both unfulflled order rate and cost. 5.2. Behavor of unfulflled order rate n two-tem case Here we consder the case that suppler s product conssts of two tems. Table 2 shows d j used n an example system of 2 tems, where S j 0 = 24, p j = 1, h j = 1, r 1 = 82, r 2 = 135, Q = 50, ω j = d j 0.35 and β j = 0.05. Fgure 9 shows results of nventory by proposed algorthm at 1st and 7th teratons, and fnal soluton. Compared wth obtaned result at 1st teraton, we can fnd the tendency of becomng hgher about nventory level when teraton ncreases. Fgure 9 shows typcal result. Fgure 10 shows result of producton quantty by proposed algorthm. Compared wth base case, we can fnd that proposed algorthm plans necessarly the precedence producton at 4th, 5th, 6th and 7th for tem number 1, also t plans necessarly the precedence producton at 2nd for tem number 2. Table 2 d j used n an example system of 2 tems 1 2 3 4 5 6 7 8 d 1 4 8 0 20 16 12 12 12 d 2 20 20 20 15 15 15 15 15

Plannng Based on Unfulflled Order Rate 213 < tem number 1 > < tem number 2 > Fgure 9: Results of nventory by proposed algorthm for each tem Fgure 10: Result of producton quantty by proposed algorthm

214 N. Ueno, K. Okuhara, H. Ish, H. Shbuk & T. Kuramoto Fgure 11: Result of cost and unfulflled order rate for each teraton Fgure 12: Results for each teraton by changng a step sze for update rule Fgure 11 shows result of cost and unfulflled order rate for each teraton n the case that INC =2. The unfulflled order rate decreases gradually, and soluton s obtaned after 48th teraton. Cost s represented by rate compared wth the cost obtaned at 1st teraton. From the result, we can fnd that proposed algorthm can mprove the unfulflled order rate, and we can understand exstence of alternatve between cost and the unfulflled order rate. Fgure 12 shows results for each teraton by changng a step sze for update rule INC where Q = 50. It can be found that algorthm, where INC=2, s converged faster than another. Table 3 shows results by changng Q and INC. Result when INC = 2 s qute nearly converged same as one when INC = 10 for each Q. Fgure 13 shows performance of proposed algorthm for each tem. For the tem number 1, the unfulflled order rate of base case takes 11.7%, but the unfulflled order rate obtaned by proposed method becomes 5.8%. Especally, when we compare t under the same cost of base case, proposed method realzes 8.3% about the unfulflled order rate. Smlarly, for tem number 2, the unfulflled order rate s mproved from 52.1% n base case to 14.5% n Table 3 Results by changng Q and INC SOn 1 SOn 2 cost Q INC = 2 INC = 10 INC = 2 INC = 10 INC = 2 INC = 10 60 0.0576 0.0576 0.1276 0.1276 1380.5 1380.5 50 0.0576 0.0576 0.1283 0.1283 1379.3 1379.6 40 0.0576 0.0576 0.1285 0.1286 1387.2 1385.2

Plannng Based on Unfulflled Order Rate 215 Fgure 13: Performance of proposed algorthm for each tem Table 4 d j used n an example system of 5 tems 1 2 3 4 5 6 7 8 d 1 4 8 0 20 16 12 12 12 d 2 20 20 20 15 15 15 15 15 d 3 16 16 16 20 20 8 8 8 d 4 24 24 24 12 12 12 12 12 d 5 20 20 20 16 16 16 16 16 proposed method. Furthermore, even f we compare t under the same cost of base case, proposed method realzes 36.1% about the unfulflled order rate. Proposed algorthm s better than base case from the pont of both unfulflled order rate and cost. From these results, compared wth base case, we can fnd that the proposed method can mprove both cost and unfulflled order rate under keepng manufacturng capacty restrctons and so on. 5.3. Behavor of unfulflled order rate n mult-tem case In ths secton, the behavor of unfulflled order rate n mult-tem case s nvestgated. Table 4 shows d j used n ths numercal example wth 5 tems. The other parameters are assumed to be S 1 0 = 18, S 2 0 = 24, S 3 0 = 14, S 4 0 = 24, S 5 0 = 18, p j = 1, h j = 1, r 1 = 82, r 2 = 135, r 3 = 112, r 4 = 132, r 5 = 140, Q = 100, ω j = d j 0.20 and β j = 0.05. Fgure 14 shows results when system apples INC = 2 and INC = 10 as the step sze for update rule. From ts results, we can understand that the unfulflled order rate changes decrease monotonously and converges also n case of 5 tems. 5.4. Proposal desgn method n actual case In ths secton, we show proposal desgn method wth 2 steps n actual case. As the desgn method s llustrated n Fgure 15, we frst solve the producton plannng problem by the proposed algorthm n the prevous secton, and we call ths algorthm MCPS (Masscustomzaton Producton Plannng and Management System), and by ths result we modfy orgnal due date for delvery. Almost all the case t means precedence producton. Next we execute schedulng based on modfed due date for delvery. So, we frst determne the producton plannng of fnal stage for each tem n order to obtan mnmum nventory level whle keepng target unfulflled order rate. For the due date of delvery that s gven by each order, t can be realzed by mprovng a schedule that fnshes the fnal stage. For example, fnal stage s set by the fnal process about suppler s assembly, or the delvery to manufacturer s warehouse from suppler. That s, such schedule

216 N. Ueno, K. Okuhara, H. Ish, H. Shbuk & T. Kuramoto (a) INC=2 (b) INC=10 Fgure 14: Result of unfulflled order rate and cost for each teraton ( : date that order s assgned and to be manufactured) Fgure 15: Proposal desgn method n actual case

Plannng Based on Unfulflled Order Rate 217 ams both the proper necessary precedence producton for each tem and the holdng of mnmum nventory. Next we schedule upstream process from fnal state by usng commercal software that supports system based on Make-to-Order producton system. We wll select the best software that can take nto account about the condton of suppler s producton restrcton n detal. In order to realze mass customzaton, n the case that we apply the exstng technque, for example, MRP II [11], APS [7] drectly, t s necessary to set up safety stock n advance. However, safety stock method s not realstc from the fact that forecast order changes n the planned perod usually happen n the automoble ndustry. By applyng the proposed algorthm, t s not necessary to gve the safety stock n advance. When the planned unfulflled order rate s gven n advance, then the nearly optmal producton quantty and smultaneously the mnmum nventory for planned term, nstead of safety stock, are decded. In sum, both MCPS and schedulng software solve sequentally, and realze mass customzaton. 6. Concludng Remarks In ths paper, we took up producton plannng of parts assembly suppler n auto ndustry, and defned suppler s mult-tem producton plannng and management system for mplementng mass customzaton. The problem about proper nventory quantty of mult-tem producton was formulated by stochastc programmng problem. We proposed the effectve algorthm to obtan nearly optmal nventory quantty for the producton plannng and management system mplementng mass customzaton. Further work s an applcaton of the decomposton prncple [8] to fnd the soluton of the mult-tem MP problem whch was derved, because f the number of tems becomes large then the sze of sub problem for teraton become huge. Acknowledgments The authors would lke to thank referees for ther valuable comments on the orgnal verson whch mproved ths paper very much. In addton, the authors wsh to express apprecaton to Mr. Kyozo Furuta for hs assstance. Ths research was supported by Grant-n-Ad for Scentfc Research (c) 16510118. References [1] S. Bller, E.K. Bsh, and A. Murel: Impact of manufacturng flexblty on supply chan performance n the automotve ndustry. In J.S. Song and D.D. Yao (eds.): Supply Chan Structures Coordnaton, Informaton and Optmzaton (Kluwer s Internatonal Seres, 2001). [2] J.R. Brge and F. Louveaux: Introducton to Stochastc Programmng (Sprnger-Verlag, 1997). [3] J. Bramel and D. Smch-Lev: The Logc of Logstcs (Sprnger-Verlag, 1997). [4] S. Chopra and P. Mendl: Supply Chan Management Strategy, Plannng and Operaton (Prentce-Hall, 2000). [5] J.H. Glmore and B.J. Pne: Markets of One : Creatng Customer-Unque Value through Mass Customzaton (Harvard Busness School Press, 2000). [6] Y. Kobayash and H. Tsubone: Producton seat bookng system for the combnaton of make-to-order and make-to-stock as producton envronment. Journal of Japan Industral Management Assocaton, 52 (2001), 53 59.

218 N. Ueno, K. Okuhara, H. Ish, H. Shbuk & T. Kuramoto [7] M. Kuroda: MRP to APS - nnovaton of producton management and the new role of schedulng. Proceedngs of the Schedulng Symposum 2002 (2002), 2 13 (n Japanese). [8] L.S. Lasdon: Optmzaton Theory for Large Systems (Dover Publcatons, 2002). [9] J. Nakane (ed.): BTO Producton System (Nkkan Kogyo Shmbun, 2000). [10] B.J. Pne: Mass Customzaton (Harvard Busness School Press, 1993). [11] K. Shekh: Manufacturng Resource Plannng (MRP II) (McGraw-Hll, 2003). [12] E.A. Slver, D.F. Pyke, and R. Peterson: Inventory Management and Producton Plannng and Schedulng, 3rd edton (Jon Wlley & Sons, 1998). [13] T. Tamura and S. Fujta: Producton seat system for producton management. Journal of Japan Industral Management Assocaton, 4 (1994), 5 13. [14] S. Tayur, R. Ganeshan, and M. Magazne: Quanttatve Models for Supply Chan Management (Kluwer s Internatonal Seres, 1998). Nobuyuk Ueno Faculty of Management and Informaton Systems Prefectural Unversty of Hroshma 1-1-71 Ujnahgash Mnam-ku Hroshma 734-8588, Japan E-mal: ueno@pu-hroshma.ac.jp