4 Parameterzaton of MRP for Supply Plannng Under Lead Tme Uncertantes A. Dolgu, F. Hnaen, A. Louly 2 and H. Maran Centre for Industral Engneerng and Computer Scence 2 Kng Saud Unversty College of Engneerng - Industral Engneerng Department France 2 Kngdom of Saud Araba Open Access Database www.ntehweb.com. Introducton Effcent replenshment plannng s a very mportant problem n Supply chan management. A poor nventory control polcy leads to overstockng or stockout stuatons. In the former, the generated nventores are expensve and n the later there are shortages and penaltes due to unsatsfed customer demands. Materal Requrements Plannng (MRP) s a commonly accepted approach for replenshment plannng n major companes (Axsäter, 2006). The MRP software tools are accepted readly, the majorty of ndustral decson makers are famlar wth them through all the exstng producton control system software. MRP software has a well developed nformaton system and has been proven over tme. However, MRP s based on the supposton that the demand and lead tmes are known. Ths premse of determnstc envronment seems somewhat off base snce most producton occurs stochastcally. Component and sem-fnshed product lead tmes and fnshed product demands are rarely forecasted relably. Ths s because there are some random factors such as machne breakdowns, transport delays, customer demand varatons, etc. Therefore, n real lfe, the determnstc assumptons embedded n MRP are often too lmted. Fortunately, the MRP approach can be adapted for replenshment plannng under uncertantes by searchng optmal values for ts parameters. Ths problem s called MRP parameterzaton under uncertantes. The planned lead tmes are parameters of MRP. For the case of random lead tmes, the planned lead tmes are calculated as the sum of the forecasted and safety lead tmes. These safety tmes are obtaned as a trade-off between overstockng and stockout whle mnmzng the total cost. The search for optmal values of safety lead tmes, and, consequently, for planned lead tmes, s a crucal and challengng ssue n Supply chan management wth MRP approach. In ths chapter, we present a methodology for optmal calculaton of planned lead tmes n the MRP approach. Ths methodology was developed n our prevous works (Dolgu et al., 995; Dolgu, 200; Dolgu & Louly, 2002; Louly & Dolgu, 2002; Louly & Dolgu, 2004, Source: Supply Chan,Theory and Applcatons, Book edted by: Vedran Kordc, ISBN 978-3-90263-22-6, pp. 558, February 2008, I-Tech Educaton and Publshng, Venna, Austra www.ntechopen.com
248 Supply Chan: Theory and Applcatons Louly et al., 2007) for supply chans wth random lead tmes where holdng and backloggng costs are not neglgble. 2. Inventory control n supply chans Supply chan management s a collecton of functonal actvtes that are repeated many tmes throughout the process through whch raw materals are transformed nto fnshed products (Ballou, 999). An llustraton of a Supply chan s gven n Fg.. Demand Supplers Producton Assembly Customers Suppler lead tme Producton lead tme Assembly lead tme Fgure. Supply chan As reported n number of papers, varous sources of lead tme uncertantes may exst along ths chan. To avod these uncertantes, the companes use safety stocks (safety lead tmes), whch are rather expensve. Therefore, t s desrable to develop specal methods of supply plannng whch focus on the stochastc propertes of lead tmes (Malon & Benton, 997). Supply management n ndustral applcatons s manly based on Materal Requrements Plannng (MRP), whch provdes a framework for nventory control. In the MRP approach, an mportant dstncton s drawn between demand for the end product,.e. ndependent demand, and demand for one of ts tems,.e. dependent demand (Baker, 993). The ndependent demands are known or forecasted by the methods whch are developed n the framework of "sales forecastng". The dependent demands can be calculated from the ndependent ones by usng Bll of Materal and planned lead tmes. Under MRP logc, tme s vewed n dscrete ntervals called tme buckets. The lead tme s equal to the elapsed tme buckets from the order release date to the delvery (procurement, producton, etc.) of the correspondng tem. The lot sze s the quantty of tems to be ordered. The MRP method s based on the determnstc calculaton: all the orders of tems are released at the latest possble moment, so total cost wll automatcally be mnmal. But, f random factors exst, the meanng of "at the latest possble moment" s uncertan. In ths case, for each specfc value of MRP parameters (concretely: planned lead tmes) we can have a backlog or overstock probablty. The larger the probablty of backlog s, the bgger the average backloggng cost over tme. The same s true for overstock, the larger the probablty of overstock s, the greater the holdng cost. Therefore, a challengng problem s MRP parameterzaton, n partcular, the choce of optmal values for planned lead tmes. www.ntechopen.com
Parameterzaton of MRP for Supply Plannng Under Lead Tme Uncertantes 249 3. Related works Yeung et al. (998) propose a revew on parameters havng an mpact on the effectveness of MRP systems under stochastc envronments. Yücesan & De Groote (2000) dd a survey on supply plannng under uncertantes, but they focused on the mpact of the producton management under uncertanty on the lead tmes by observng the servce level. Process uncertantes are consdered n (Koh et al., 2002; Koh & Saad, 2003). The problem of MRP parameterzaton under lead tme uncertantes has been often studed va smulaton. For example the study of Whybark & Wllams (976) suggests that a safety lead tme s mechansm may perform better than that of a safety stock n a mult-level producton-nventory system when the producton and replenshment tmes are stochastc. Nevertheless, Grasso & Taylor (984) reached another concluson and prefer safety stocks for both quantty and lead-tme uncertantes. Weeks (98) developed a sngle-stage model wth tardness and holdng costs n whch the processng tme s stochastc and demand s determnstc. The author proves that ths s equvalent to the standard Newsboy problem. Gupta & Brennan (995) show that lead tme uncertanty has a large nfluence on the total nventory management cost. Ho & Ireland (998) llustrate that lead tme uncertanty affects stablty of a MRP system no matter what lot-szng method used or demand forecast error obtaned. The statstcs from smulatons by Bragg et al. (999) demonstrate that the lead tmes nfluence the nventores substantally. Molnder (997) study the problem of planned lead tmes (safety lead tme/safety stock) calculaton va smulaton and proposes a smulated annealng algorthm to fnd approprate safety stocks and/or safety lead tmes. The smulatons show that the overestmated planned lead tmes s conducve to excessve nventory, and underestmated planned lead tmes ntroduce shortages and delays. (Grubbström, & Tang, 999) study optmal safety stocks n sngle and mult-level MRP systems, assumng the tme nterval of end tem demand to be stochastc. For seral multlevel producton systems,.e. where the prevous level supples the next and only one suppler s at each level, Yano (987a,b) suggests an approach to determne optmal planned lead tmes for MRP. In ths study, the lead tmes are stochastc, and fnshed product demand s fxed. The author presents a general procedure for two stage systems based on a sngle-perod contnuous nventory control model. The objectve was to mnmze the sum of nventory holdng costs, reschedulng costs arsng from tardness at ntermedate stages, and backloggng cost for the fnshed product. One of the man obstacles for ths approach conssts n the dffcultes to express the objectve functon n a closed form for more than two stages. In assembly systems there are several supplers at each stage, and so, there s dependence among the dfferent component nventores at the same stage. Yano (987c) consders a partcular problem for two-level assembly systems wth only two types of components at stage 2 and one type of components at stage. The delvery tmes for the three components are stochastc contnuous varables. The problem s to fnd the planned lead tmes for MRP mnmzng the sum of holdng and tardness costs. A sngle perod model and an optmzaton algorthm were developed. Tang & Grubbström (2003) consder a two component assembly system wth stochastc lead tmes for components and fxed fnshed product demand. Ths study s smlar to (Yano, 987c). However, here, the process tme at level s also assumed to be stochastc, the due date s known and the optmal planned lead tmes are smaller than the due date. The objectve s to mnmze the total stockout and nventory holdng costs. The Laplace transform procedure s used to capture the stochastc www.ntechopen.com
250 Supply Chan: Theory and Applcatons propertes of lead tmes. The optmal safety lead tmes, whch are the dfference between planned and expected lead tmes are derved. Another nterestng sngle perod model was proposed n (Chu et al., 993) whch deals wth a punctual fxed demand for a sngle fnshed product. The model gves optmal values of the component planned lead tmes for such a one-level assembly system wth random component procurement tmes. Wlhelm & Som (998) studed a two-component assembly system usng queuenng models and showed that a renewal process can be used to descrbe the end-tem nventory level evoluton. The optmzaton of several component stocks s replaced by the optmzaton of fnshed product stock. To perform ths replacement, a smplfed supply polcy for component orderng was ntroduced. Another mult-perod model s proposed n (Gurnan et al., 996) for assembly systems wth two types of components and the lead tme probablty dstrbutons are lmted to two perods. In (Dolgu & Louly, 2002; Louly & Dolgu, 2004), a smlar one-level plannng problem wth random lead tmes and fxed demand s studed, but for a dynamc mult-perod case. The authors gve a novel mathematcal formulaton and propose a generalzed Newsboy model whch gves the optmal soluton under the assumpton that the lead tmes of the dfferent types of components follows the same dstrbuton probablty, and the unt holdng costs are dentcal. In Louly et al. (2007) the authors generalze ther studes of 2002 and 2004. They present a more unversal case, when the unt holdng costs aren t the same for all components and the component lead tmes are not..d. random varables. 4. MRP approach 4. The basc prncples of MRP systems The goal of MRP s to determne a replenshment schedule for a gven tme horzon. For example, let s consder the followng bll of materals - BOM (see Fg. 2) for a fnshed product. The needs for the fnshed product are gven by the Master Producton Schedule MPS (Fg. 3), and those for the components are deduced from BOM exploson (Dolgu et al., 2005). Let s ntroduce the followng notaton: I() nventory for the perod, N() net needs for the perod, G() gross needs for the perod, Q() released orders for the perod, Δτ planned lead tme. The avalable nventory I () for the perod s gven. For each subsequent need, the value s calculated from the net needs of the prevous perod: net needs of the perod are obtaned as follows: I ( ) = max{0, N( )}, () The released order quantty: N( ) = G( ) I( ), (2) www.ntechopen.com
Parameterzaton of MRP for Supply Plannng Under Lead Tme Uncertantes 25 Q ( ) = max{0, N( Δτ )}. (3) Fnshed good Lead-tme = 2 Component Lead-tme = 3 Quantty of components 2 Component 2 Lead-tme = 2 Component 3 Lead-tme = 2 Fgure 2. Bll of materals Level 0 Fnshed good Lead tme = 2 Level Component Lead tme = 3 Level Component 2 Lead tme = 2 Perod 2 3 4 5 6 7 8 9 0 Gross need (MPS) 0 0 0 50 0 40 20 30 50 60 Avalable nventory 20 20 20 20 0 0 0 0 0 0 Net need -20-20 -20 30 0 40 20 30 50 60 Manufacturng/order 0 30 0 40 20 30 50 60 0 0 Quantty = Perod 2 3 4 5 6 7 8 Gross need (MPS) 0 30 0 40 20 30 50 60 Avalable nventory 00 00 70 60 20 0 0 0 Net need -00-70 -60-20 0 30 50 60 Manufacturng/order 0 0 30 50 60 0 0 0 Quantty = 2 Perod 2 3 4 5 6 7 8 Gross need (MPS) 0 60 20 80 40 60 00 20 Avalable nventory 40 40 80 60 0 0 0 0 Net need -40-80 -60 20 40 60 00 20 Manufacturng/order 0 20 40 60 00 20 0 0 Quantty = Fgure 3. Master Producton Schedule (MPS) 4.2 MRP under uncertantes The man problem whch often arses wth MRP systems s derved from the uncertantes (Nahmas, 997; Vollmann et al., 997) especally demand and lead tme uncertanty (see Fg. 4). www.ntechopen.com
252 Supply Chan: Theory and Applcatons Level 0 Fnshed Good Lead tme = 2 +/- Perod 2 3 4 5 Gross need (MPS) 0 0 20 5 0 Avalable nventory 20 20 20 0 0 Net needs -20-20 0 5 0 Manufacturng/order 5 Demand uncertanty Fgure 4. Input data uncertantes Lead-tme uncertanty The demand uncertanty means that the demand sn t exactly known n advance and, so the planned quanttes for a perod may be dfferent from the actual demand. The lead tme uncertanty means that the actual lead tme may be dfferent from planned lead tme, so an order planned for a perod may not arrve at the approprate date. As aforementoned, n lterature, the majorty of publcatons are devoted to the MRP parameterzaton under customer demand uncertantes. As to random lead tmes, the number of publcatons s modest n spte of ther sgnfcant mportance. The motvaton of ths chapter s to contrbute to the development of new effcent methods for MRP parameterzaton under lead tme uncertantes (see Fg. 5). Data base: - Characterstcs of lead tmes (dstrbuton probabltes). - Penaltes (holdng costs, backloggng cost, etc.). - Objectve servce level. MRP software tool MRP parameters: planned lead tmes Input A mathematcal model for the calculaton of optmal planned lead tmes. Fgure 5. Proposed approach for MRP parameterzaton www.ntechopen.com
Parameterzaton of MRP for Supply Plannng Under Lead Tme Uncertantes 253 The core of ths approach s the calculaton of planned (safety) lead tmes. When we ncrease these parameters, the stocks ncrease also. However, stocks are expensve. In contrast, f the planned lead tmes are underestmated, the rsk of stockout and consequently the backloggng cost ncreases along wth decreasng the servce level. The goal s to fnd the planned lead tme values whch provde a trade-off between holdng and backloggng costs whle mnmzng the total cost under random actual lead tmes. In the next secton, we suggest a mathematcal model for ths optmzaton. 5. MRP parameterzaton 5. Mathematcal model descrpton In the MRP approach, replenshment order dates,.e. release dates, for each component are calculated for a seres of dscrete tme ntervals (tme buckets) based on the demand and takng nto account a fxed planned lead tme: the release date s equal to the due date mnus the planned lead tme. For the case of random actual lead tmes, n ndustry, a supply relablty coeffcent ( ) s assgned to each suppler. The planned lead tmes for MRP are calculated by multplyng the contractual lead tme by the correspondng suppler relablty coeffcent. The choce of these coeffcents (whch gve safety lead tmes) s based on past experence. However, ths approach s subjectve and can be non optmal f we need to mnmze the total cost for an MRP system. The suppler relablty coeffcents (safety lead tmes and so planned lead tmes) can be calculated more precsely takng nto account nventory holdng and backloggng costs, wth a nventory control model. Such an nventory control model must be smple (to be solvable), but representatve, ntegratng all major factors nfluencng the planned lead tme calculaton. For component planned lead tme calculaton for assembly systems wth several types of components and random component lead tmes, we have ntroduced (Dolgu & Louly, 2002) the followng model and assumptons. Ths model wll help us to solve the consdered problem of MRP parameterzaton,.e. to fnd optmal planned lead tmes for components when the actual lead tmes are random varables. Fg. 6 gves an llustraton of the suggested abstract model. Component L Component 2 L 2 h h 2 Infnte capacty Assembly b Fnshed product Component n L n h n Fgure 6. Inventory control model for component planned lead tme calculaton www.ntechopen.com
254 Supply Chan: Theory and Applcatons For ths model, we assume that the fnshed product demand per perod s known and constant as well as the assembly capacty s nfnte. Several types of components are needed to assembly one fnshed product. The unt holdng cost per perod for each type of component ( ) lead tmes ( ) h and the unt backloggng cost ( ) b for the fnshed product are known. The L for orders made at dfferent perods for the same type of component are ndependent and dentcally dstrbuted dscrete random varables. The dstrbuton of probabltes for the dfferent types of components can be not dentcal. These dstrbutons are known, and ther upper values are fnte. The fnshed products are delvered at the end of each perod and unsatsfed demands are backordered and have to be treated later (when suffcent numbers of components of each type are n stock). The supply polcy for components s Lot for Lot: one lot of each type of component s ordered at the begnnng of each perod. Because the supply polcy s the Lot for Lot and the demand s consdered as constant, the same quanttes of components are ordered at the begnnng of each perod. Thus, only planned lead tmes are unknown parameters for ths model. Hence, they are the decson varables n our optmsaton approach. The model consders random component lead tmes and also the dependence among nventores of the dfferent components sutable for assembly systems (when there s a stockout of only one component, consequently, there s no possblty to assemble the fnshed product). To smplfy the equatons, wthout lost of generalty, we assume that the fnshed product demand s equal to one unt per perod, and that one fnshed product s assembled from one unt of each type of component. Let s use the followng model notatons: functon equal to f f s true, and 0 otherwse, f n E [.] h b k L number of types of components used for the assembly of one product, mathematcal expectaton operator, unt holdng cost for the component per perod, unt backloggng cost for the fnshed product per perod, reference of a perod (perod ndex), lead tme of the components (dscrete random varable), k L lead tme of the components ordered at perod k (dscrete random varable), u upper value of the lead tme for components ( L u,2,..., k N N ( u ) u = max, =,..., n ; = n ); number of orders for the component that have not yet arrved at the end of the perod k, steady state number of orders for the component that have not yet arrved at the end of a perod, X vector of the decson varables ( x,..., x n ), + Z functon equal to the maxmum of Z and 0: max(z,0). www.ntechopen.com
Parameterzaton of MRP for Supply Plannng Under Lead Tme Uncertantes 255 Note that:. Consderng that the component ordered quanttes are the same for all perods, the planned lead tme multpled by the ordered quantty, whch s equal to the fnshed product demand, gves also the ntal nventory for the correspondng component.. The optmal planned lead tmes do not depend on the fnshed product demand (the same values of optmal planned lead tmes wll be obtaned for dfferent demand amounts, f the demand s constant and other characterstcs of the problem are fxed).. Gven the fact that the order quanttes are constant,.e. the same for all perods, the crossng of orders does not complcate the problem. v. Takng nto consderaton the objectve of ths study to calculate optmal planned lead tmes for MRP controlled assembly systems under lead tme uncertantes - the assumptons on the fxed demand and nfnte assembly capacty are necessary and natural smplfcatons. v. Takng nto account the assumptons on the constant demand and nfnte capacty of the assembly system, we are n a Just n Tme (JIT) envronment,.e. there s no stockng of fnshed products. v. Consderng that the component lead tmes cannot exceed u ( =,2,..., n), the random k varables N and N can have only the followng values: 0,, 2,, u. v. The orders are gven at the begnnng of each perod and delvered components are used at the ends of perods (so an order made at perod k can be used at the end of the same perod k, f the actual lead tme s equal to ). Let s ntroduce the followng addtonal notatons: ( x x ) F N ( j) = Pr( N j), X =, 2,..., x n are decson varables, the value x = 0 sgnfes that the component s ordered at the begnnng of the target perod (.e. when assembly must be made), H = b + n h = As shown n (Louly et al., 2007), the objectve functon and constrants for ths mult-perod model for the optmzaton of planned lead tmes can be formulated as follows:. subject to: n = ( )) C ( X, N) = h x E( N + H F ( x + j n ( )) N j 0 =, (4) 0 x u, =,2, K, n. (5) The maxmal value of component lead tme s equal to u, so only the prevous u - orders may not yet be receved. Earler orders have already arrved, therefore: 0 N u. N = u u j, =, L, n, (6) L > j j = www.ntechopen.com
256 Supply Chan: Theory and Applcatons where s L s s-th varable L (these varables are d). The man dffculty of the optmzaton problems (4) (5) s that the decson varables, x, =, L,n, are ntegers and the objectve functon s non lnear. Thus, ths s a complex optmzaton problem. 5.2 Optmzaton algorthm For the problem (4) (5), we propose the approach llustrated n Fg. 7. From practcal pont of vew, an approxmate soluton of ths problem can be suffcent. So we developed several technques to calculate lower and upper lmts for the value of decson varables. By applyng these technques we reduce the space of admssble values for these varables. Often, the obtaned space s relatvely small, so an approxmate soluton s relatvely easy to fnd. Search space reducton Heurstcs (approxmate soluton) Branch and Bound (exact soluton) Fgure 7. Optmzaton technques The proposed technques for reducton of the space of admssble values for decson varables are also useful as a pre-processng procedure for an exact optmzaton algorthm (Branch and Bound n our case). The smaller the search space for the exact procedure s, the qucker the soluton can be found. Pre-processng procedure to reduce search space. We propose a pre-processng procedure whch should be used before optmsaton. Let s present the space of all feasble solutons by [A, B], where A= ( a, a 2,, a n ), B = ( b, b 2,, b n ), where a, b are mnmal and maxmal possble values for x. We wll show some technques that reduce these ntervals: a x b, =, L, n. We proved (Louly et al., 2007) that the optmal soluton X= ( x, x 2,, followng condtons: x n ) satsfes the h b max(, ) F ( x ), (7) b H h F ( x ). (8) H Therefore, the maxmum value of x whch satsfes the condton (8) gves a upper lmt b, and the mnmum value of x whch satsfes the condton (7) gves a lower lmt a for ths decson varable x. Then, the optmal value of x must respect the followng relaton: x b. a www.ntechopen.com
Parameterzaton of MRP for Supply Plannng Under Lead Tme Uncertantes 257 The pre-processng procedure based on the verfcaton of the condtons (7) and (8) can reduce the search space before applyng any optmsaton algorthm. Domnance propertes. We use a Branch & Bound approach to fnd the optmal values of the parameters x, a, =, L, n. x b It s known that domnance propertes can mprove a Branch & Bound algorthm. In our prevous works, we have ntroduced the followng partal ncrement functons: G + (X ) = C ( x,..., x +,..., x n ) - C ( x,..., x,..., x n ), (9) G (X ) = C( x,..., x,..., x n ) - C ( x,..., x,..., x n ). (0) The followng two domnance propertes () and () were proved n (Louly & Dolgu, 2007): G + A If ( ) < 0 G B If ( ) < 0, then each soluton X of [A, B] wth x = a s domnated. (), then each soluton X of [A, B] wth x = b s domnated. (2) These domnance propertes can be used to develop effcent cut procedures. Indeed, after the dvson of a node, n a Branch and Bound algorthm, two son-nodes (descendants) are created. For each son-node, some cuts can be appled to reduce agan the correspondng search spaces before the next branchng. Lower bounds on the objectve functon. Snce, we use a Branch and Bound algorthm to solve the optmzaton problem (4) (5), we need effcent Lower bounds for each tree node and root. In ths sub-secton, we develop these bounds. In (Louly et al., 2007), we have the two followng Lower Bounds on the objectve functon n the space [A, B]: LB = (A) LB = (B) 2 n + C + b a )mn( G ( b,..., b, a,..., a ), ) = (, (3) n 0 n C + b a )mn( G ( a,..., a, b,..., b ), ) = (, (4) n 0 where, f ( amn ), f x0 < amn, LB 3 = f (x0), famn < bmax, (5) f ( bmax ), f x0 > bmax, n f (x) = h ˆ( x E( )) = N n + ( b + nhˆ) Fˆ ( x + k), k 0 = www.ntechopen.com
258 Supply Chan: Theory and Applcatons hˆ = mn h, =,..., n Fˆ ( x) = max F ( x), a mn b max =,..., n = mna, =,..., n = maxb. =,... n The parameter x 0 s the largest nteger whch verfes the followng expresson: Fˆ / n b. b + nhˆ ( x ) Fˆ ( x) The followng lower bound wll be used n the Branch and Bound: LB= max (LB, LB 2, LB 3 ) (6) Node extenson procedure. A Branch and Bound (B&B) algorthm s based on the desgn of an enumeraton tree. In our algorthm, each node of the enumeraton tree represents a set of feasble solutons. Let [A, B] be a node of ths tree. The descendants of ths node are obtaned by dvdng (parttonng) the correspondng space [A, B] nto two smaller subspaces [A, B] and [A, B] as follows: - we choose such that = arg max (b -a ), - then, the descendent [A, B] (respectvely [A, B]) s the subspace gven by the vectors A a + b and B (resp. A and B) for whom the -th component satsfes a x (resp. 2 a + b + x b ). 2 After applyng ths node extenson procedure for the node [A, B] we obtan two son-nodes [A, B] and [A, B], each wth smaller space of feasble solutons. Lower Cut and Upper Cut procedures. For each node before applyng a node extenson procedure some cuts are executed. The am s to reduce the space of feasble solutons whch s assocated wth the node to be dvded. For our algorthm, the prncple s smple, as mentoned above a node corresponds to a search space [A, B]. The cut procedure reduces the soluton space [A, B] replacng A (respectvely B) by a larger (respectvely smaller) vector. Ths s equvalent to cuttng a part of the search space [A, B]. We ntroduce two procedures: one for cuttng small values (Lower Cut procedure) and second for cuttng large values (Upper Cut procedure) of the correspondng decson varables. The reducton scheme s the same for these two procedures and they return "true" when the subset [A, B] s entrely domnated (.e. by applyng the cuts we completely elmnated the node [A, B]). www.ntechopen.com
Parameterzaton of MRP for Supply Plannng Under Lead Tme Uncertantes 259 Upper bound calculaton procedure. Ths procedure to calculate an Upper Bound s a varant of depth frst search that conssts n choosng the node [A, B] to be parttoned (dvded) that has the best soluton at one of ts two extremtes (A or B). 6. Concluson and further research We studed an MRP parameterzaton problem for assembly systems under component lead tme uncertantes. Ths problem of nventory control and producton plannng under uncertantes s crucal for ndustry. Most publcatons are devoted to customer demand uncertantes. In contrast, lead tme uncertantes seem not be suffcently studed, n partcular, for the control of component nventores for assembly systems wth random component procurement tmes (lead tmes). Nevertheless, many of ndustral applcatons exst where the component lead tmes are random. For the case of assembly systems wth random lead tmes and a random demand, f the demand and component lead tmes are ndependent random varables, the followng decomposton can be made. The fnshed product stock and the component stocks for assembly can be consdered as ndependent. So, our approach s stll vald for MRP parameterzaton: for the fnshed product a safety stock s calculated usng the standard approaches, and the component planned lead tmes are deducted from the model proposed n ths chapter. Of course, t would be better to develop a model whch takes nto account the dependence between the stock of the fnshed product and component stocks n the case of random demand and lead tmes. Ths wll be one path for our future research, perhaps usng the conjecture of (Km et al., 2006). Ths conjecture, suggested for a sngle-tem model, affrms that the behavour of an nventory/producton system where both demand and lead tmes are random can be evaluated by modellng for three partcular cases wth: () determnstc demand and lead tme, () random demand and determnstc lead tme, and () random lead tme and determnstc demand. Further research should be focused on the development of a more effectve Branch- and- Bound procedures for the general case of these sngle-level assembly systems. Another path for future work would deal wth multlevel assembly systems,.e. wth mult-level bll of materal. Logcally, dffculty wll ncrease because of dependence among levels n addton to the dependence among nventores. 7. Acknowledgements The authors thank Chrs Yukna for hs help n checkng the Englsh of ths chapter. 8. References Axsäter, S., (2006), Inventory control, 2nd ed., Sprnger Ballou, R.H, Busness Logstcs Management. Prentce-Hall, New Jersey, 999. www.ntechopen.com
260 Supply Chan: Theory and Applcatons Baker, K.R, (993). Requrements plannng, n: S.C. Graves et al. (Eds.), Handbooks n Operatons Research and Management Scence. Vol. 4, Logstcs of Producton and Inventory, North-Holland, Amsterdam, pp. 57-627. Bragg, D.J., Duplaga, E.A. and Wats, C.A. (999). The effects of partal order release and component reservaton on nventory and customer servce performance n an MRP envronment. Internatonal Journal of Producton Research, 37, pp. 523-538. Chu, C., Proth, J.M. and Xe, X., (993). Supply management n assembly systems. Naval Research Logstcs, 40, pp. 933-949. Dolgu, A., Portmann M.C., and Proth, J.M. (995). Planfcaton de systèmes d assemblage avec approvsonnement aléatore en composants. Journal of Decson Systems, 4(4), pp. 255 279. Dolgu, A. (200). On a model of jont control of reserves n automatc control systems of producton. Automaton and Remote Control, 62(2), pp. 2020 2026. Dolgu, A., and Louly, M.A., (2002). A Model for Supply Plannng under Lead Tme Uncertanty. Internatonal Journal of Producton Economcs, 78, pp. 45-52. Dolgu, A., Louly M.-A., and Prodhon, C. (2005). A survey on supply plannng under uncertantes n MRP envronments. In P. Horacek, M. Smandl, and P. Ztech (Eds.), Selected plenares, mlestones and surveys. 6th IFAC World Congress, Elsever, pp. 228 239. Grasso, E.T., and Taylor, B.W. (984). A Smulaton-Based Expermental Investgaton of Supply/Tmng Uncertanty n MRP Systems. Internatonal Journal of Producton Research, 22 (3), pp. 485-497. Grubbström, R.W., and Tang, O. (999). Further development on safety stocks n an MRP system applyng Laplace Transforms and Input-Output Analysts, Internatonal Journal of Producton Economcs, 60-6, pp. 38-387. Gupta, S.M., and Brennan, L. (995). MRP Systems Under Supply and Process Uncertanty n an Integrated Shop Floor Control Envronment. Internatonal Journal of Producton Research, 33 (), pp. 205-220. Gurnan H., Akella, R. and Lehoczky, J. (996). Optmal order polces n assembly systems wth random demand and random suppler delvery. IIE Transactons, 28, pp. 865-878. Ho, C.J., and Ireland, T.C., (998). Correlatng MRP System Nervousness wth Forecast Errors. Internatonal Journal of Producton Research, 36 (8), pp. 2285-2299. Km, J.G., Chatfeld, D., Harrson, T.P., Hayya, J.C. (2006). Quantfyng the bullwhp effect n a supply chan wth stochastc lead tme. European Journal of Operatonal Research, 73 (2), pp. 67-636. Koh, S.C.L., S.M. Saad, and M.H. Jones (2002). Uncertanty Under MRP-Planned Manufacture: Revew and Categorzaton. Internatonal Journal of Producton Research, 40 (0), pp. 2399-242. www.ntechopen.com
Parameterzaton of MRP for Supply Plannng Under Lead Tme Uncertantes 26 Koh, S.C.L., and S.M. Saad, (2003). MRP-Controlled Manufacturng Envronment Dsturbed by Uncertanty. Robotcs and Computer-Integrated Manufacturng, 9 (-2), pp. 57-7. Louly, M.-A., and Dolgu, A. (2002). Generalzed newsboy model to compute the optmal planned lead tmes n assembly systems. Internatonal Journal of Producton Research, 40(7), pp. 440 444. Louly, M.A., Dolgu, A., (2004). The MPS parameterzaton under lead tme uncertanty. Internatonal Journal Producton Economcs, 90, pp. 369-376. Louly, M.A., and Dolgu, A., (2007). Calculatng Safety Stocks for Assembly Systems wth Random Component Procurement Lead Tmes: Branch and Bound Algorthm. European Journal of Operatonal Research, (accepted, n Press). Louly, M.A., Dolgu, A., and Hnaen, F., (2007). Optmal Supply Plannng n MRP Envronments for Assembly Systems wth Random Component Procurement Tmes, Internatonal Journal of Producton Research, (accepted, n Press). Malon, M.J., Benton, W.C., (997). Supply chan partnershps: opportuntes for operatons research, European Journal of Operatonal Research,0, 49-429. Molnder, A., (997). Jont Optmzaton of Lot-Szes, Safety Stocks and Safety Lead Tmes n a MRP System. Internatonal Journal of Producton Research, 35 (4), pp. 983-994. Nahmas, S., (997). Producton and Operatons Analyss. Irwn. Tang O. and Grubbström R.W., (2003). The detaled coordnaton problem n a two-level assembly system wth stochastc lead tmes. Internatonal Journal Producton Economcs, 8-82, pp. 45-429. Vollmann, T.E., W.L. Berry, and D.C. Whybark (997). Manufacturng Plannng and Control Systems. Irwn/Mcgraw-Hll Weeks, J.K. (98). Optmzng Planned Lead Tmes and Delvery Dates, 2st annual Conference Proccedng, Amercan Producton and Inventory Control Socety, pp. 77-88. Whybark, D. C., and J.G. Wllams (976). Materal Requrements Plannng Under Uncertanty. Decson Scence, 7, 595-606. Wlhelm W.E. and Som P., (998). Analyss of a sngle-stage, sngle-product, stochastc, MRP-controlled assembly system. European Journal of Operatonal Research, 08, pp. 74-93. Yano, C.A., (987a). Settng planned leadtmes n seral producton systems wth tardness costs. Management Scence, 33(), pp. 95-06. Yano, C.A., (987 b). Planned leadtmes for seral producton systems. IIE Transactons, 9(3), pp. 300-307. Yano C.A. (987c), Stochastc leadtmes n two-level assembly systems. IIE Transactons, 9(4), pp. 95-06. Yeung J.H.Y., Wong, W.C.K. and Ma, L. (998). Parameters affectng the effectveness of MRP systems: a revew. Internatonal Journal of Producton Research, 36, pp. 33-33. www.ntechopen.com
262 Supply Chan: Theory and Applcatons Yücesan, E., and De Groote, X. (2000). Lead Tmes, Order Release Mechansms, and Customer Servce. European Journal of Operatonal Research, 20, pp. 8-30. www.ntechopen.com
Supply Chan Edted by Vedran Kordc ISBN 978-3-90263-22-6 Hard cover, 568 pages Publsher I-Tech Educaton and Publshng Publshed onlne 0, February, 2008 Publshed n prnt edton February, 2008 Tradtonally supply chan management has meant factores, assembly lnes, warehouses, transportaton vehcles, and tme sheets. Modern supply chan management s a hghly complex, multdmensonal problem set wth vrtually endless number of varables for optmzaton. An Internet enabled supply chan may have justn-tme delvery, precse nventory vsblty, and up-to-the-mnute dstrbuton-trackng capabltes. Technology advances have enabled supply chans to become strategc weapons that can help avod dsasters, lower costs, and make money. From nternal enterprse processes to external busness transactons wth supplers, transporters, channels and end-users marks the wde range of challenges researchers have to handle. The am of ths book s at revealng and llustratng ths dversty n terms of scentfc and theoretcal fundamentals, prevalng concepts as well as current practcal applcatons. How to reference In order to correctly reference ths scholarly work, feel free to copy and paste the followng: A. Dolgu, F. Hnaen, A. Louly and H. Maran (2008). Parameterzaton of MRP for Supply Plannng Under Lead Tme Uncertantes, Supply Chan, Vedran Kordc (Ed.), ISBN: 978-3-90263-22-6, InTech, Avalable from: http://www.ntechopen.com/books/supply_chan/parameterzaton_of_mrp_for_supply_plannng_under_lead_t me_uncertantes InTech Europe Unversty Campus STeP R Slavka Krautzeka 83/A 5000 Rjeka, Croata Phone: +385 (5) 770 447 Fax: +385 (5) 686 66 www.ntechopen.com InTech Chna Unt 405, Offce Block, Hotel Equatoral Shangha No.65, Yan An Road (West), Shangha, 200040, Chna Phone: +86-2-62489820 Fax: +86-2-6248982