A method for simultaneous production and order planning in a cooperative supply chain relationship with flexibility contracts

Transcription

1 roeeding of the 43rd Hawaii International Conferene on Sytem Siene A method for imultaneou odution and order lanning in a ooerative uly hain relationhi with flexibility ontrat obia Rut, Daniel Brüggemann Univerity of aderborn Deartment Buine Information Sytem aderborn, Germany obia.rut@hni.ub.de Daniel.Brueggemann@hni.ub.de Abtrat Nowaday a high lanning flexibility i required due to utomer-driven market and a riing number of odut variant. In ontext of a uly hain thi flexibility i alo imortant for the ourement of inut art from external ulier. In thi work a method for imultaneou odution and order lanning in a ooerative uly hain relationhi i eented. he method i art of an overall aoah whih trie to oordinate order quantitie of an inut art uing flexibility ontrat. hereby the inut art i delivered by two ulier for whih frame ontrat with quantity and quota agreement exit. he objetive i to guarantee thee quantitie in the frame ontrat horizon at minimum ot. A fine lanning aoah i eented whih lan the odution of the manufaturer imultaneouly with the order for the onidered inut art under onideration of iory defined flexibility orridor. 1. Introdution Cutomer-driven market along with a riing number of odut variant reult in high lanning unertainty and an inreaing demand of flexibility for manufaturer. o deliver odut in time and at low ot i a hallenging tak. When eential inut art of the end odut have to be urhaed thi alo lead to the need of flexibility in buyer-ulier relationhi. In the bet-ae a relationhi i ooerative and regard both buyer and ulier irumtane to reah an overall otimum for the artiiant. In traditional odution lanning firt the odution i lanned and afterward the required inut i alulated and ordered baed on the odution lan. By diregarding the interdeendenie between the ingle lanning te a otential to ave ot i wated. A aving ould be ahieved by imultaneou lanning whih neverthele alo inlude ome oblem. he firt oblem i that the effort to reah Wilhelm Dangelmaier, Dominik iker-huhzermeyer Univerity of aderborn Deartment Buine Information Sytem aderborn, Germany Wilhelm.Dangelmaier@hni.ub.de dh@mail.ub.de good olution an beome very high due to the inreaing omlexity of imultaneou lanning. he eond oblem i the unwillingne to hare data in a buyer-ulier relationhi. In thi work a method for imultaneou odution and order lanning in a ooerative uly hain relationhi i eented. o handle the omlexity oblem the order lanning aet i limited to one inut art and the oblem i olved heuritially. he method i art of an overall aoah whih trie to oordinate quantitie with flexibility ontrat. hereby the inut art i delivered by two ulier for whih frame ontrat with quantity and quota agreement exit. he aim i to guarantee thee quantitie in the frame ontrat horizon at minimum ot. he work i trutured a follow. Chater 2 introdue the onidered lanning enario and the referene to an overall aoah where the eented work i art of. Chater 3 range the aoah in the theory regarding odution lanning and quantity oordination. Chater 4 define the lanning a a mixed integer model. he heuriti olution for the lanning oblem i exlained in hater 5 before reult are hown in hater 6. Finally hater 7 give a ummary and an outlook on further work. 2. lanning ontext 2.1. lanning enario he uly hain enario in Figure 1 build the bai for the overall aoah. he utomer i a manufaturer whih odue finihed good and fae unertain demand of end onumer. He odue variant on one odution line whih an have a bottlenek art a inut. hi art i inluded in ome of the end odut, but not in all. All other inut art are diregarded, beaue it i aumed that they an eaily be odued or oured. he bottlenek art i /10 $ IEEE 1

2 roeeding of the 43rd Hawaii International Conferene on Sytem Siene oured from two ulier for whih frame ontrat with quota agreement exit. he oblem an eaily be extended by adding ulier or bottlenek art. uliera quota A inut bottlenekart odution utomer outut demand edited order quantity oordinated delivery quantity ulierb quota B aaity Figure 1. Conidered uly hain enario A atial examle for thi enario ould be a ar manufaturer who order headlight. hee headlight annot be odued by the manufaturer and have to be oured by two ulier for whih quota agreement exit. For ure a ar manufaturer need other imortant inut art, but we think that an overall ot redution an already be ahieved by intelligent lanning of eifi inut art. 2.2 Referene to overall aoah he introdued method i art of an overall aoah whih aim to reult in lower ot for the onidered uly hain etion. For thi a framework i neeary to lan the oured quantitie at lowet oible ot for the utomer and the ulier. he framework ontain ooerative arrangement whih ontrol the behaviour of the involved artie regarding order quantitie and ie. wo of thee arrangement are deribed in [1]. Aording to the flexibility of order the utomer and it ulier have ometing objetive. he utomer fae end utomer demand whih i uually known aoximately for near future, but unonfident for farther time eriod. Beaue of that unertainty the utomer need the oibility to hange it order aording to the demand he ha to fulfil. For hi uroe thi flexibility hould be a high a oible, but an dereae when a time eriod get nearer and foreat are relaed by real order. On the other hand the ulier want to have a aurate order lan a oible to avoid overaaitie and large tok. o handle thi oblem flexibility orridor are introdued (ee Figure 2).he figure how an examle of flexibility orridor for a eifi time egment and it hanging in time. In thi examle the edited order quantity doe not hange in time, o the utomer an jut onretize the orridor around the edited quantity. If thi quantity would hange the utomer would try to onretize orridor around the edited quantitie, but inide the old orridor. today hitory Figure 2. Flexibility orridor hi onet hould inreae the ulier afety in lanning. hereby the ulier hould be able to odue at lower ot whih they an give bak artly to their utomer by offering lower ie in ome time eriod. At the end all artie ould have a benefit. he utomer hould be enouraged to ommuniate hi future order early in time at high eiion. For that he ha to ay enalty ot for leaving the lanned orridor in a next rolling-horizon lanning. hereby two objetive are ahieved: he ulier get aid for unexeted adation of quantitie he utomer i willing to lan and oordinate the flexibility orridor a good a oible More information about flexibility orridor and a method whih lan the orridor an be found in [1]. he lanning i done in a rolling way and onit of a rough lanning and a fine lanning whih onretize the rough lanning (ee Figure 3).. eriod () fine lanning time egment (t) rough lanning + fine lanning Figure 3. Rolling rough and fine lanning he alulation of flexibility orridor i art of the rough lanning and lan quantitie for time egment whih ould be for examle month. he orridor are aggregated quantitie whih are an inut to the fine lanning whih lan onrete order for eriod whih are for examle day or week. he aoah eented in thi work i laed in the fine lanning art of the overall aoah. he utomer trie to lan hi odution and order imultaneouly regarding the ior defined flexibility 2

3 roeeding of the 43rd Hawaii International Conferene on Sytem Siene orridor and other influene with the objetive of ot minimization. hee order are the inut for the lat te in the aoah, a ofit-haring te where the ulier are able to ooe alternative delivery hedule and balane ayment. 3. Claifiation in theory 3.1. rodution lanning In lai odution lanning and ontrol ytem the lanning i done ueively. In a firt te the odution i lanned under the aumtion of unlimited aaity and the availability of all inut odut. Coneutively the aaitie are heked and the odution i lanned. Aording to thi odution lan order are determined for in-houe odution or for external ulier. A good overview of odution lanning an be found for examle in [2] and [3]. Some Advaned lanning ytem (AS) ue a imultaneou aoah to regard the different deendenie. he oblem of imultaneou lanning i the high omlexity of real world oblem whih make a olution in aoiate time often imoible. One oibility to olve thi omlexity oblem i to redue or divide the overall model into artial model. A detailed derition of AS ytem an be found in [4]. In ontext of thi work an aoah i eented whih doe a imultaneou lanning for the odution and the ordering of one inut fator neeary for the odution. o redue the omlexity thi i done in a big buket model where larger time eriod are ued o that ome event an be lanned in a ingle time eriod without lanning the equene of thee event. For further information about big buket model ee for examle [5] Coordinating mid-term quantitie Frame ontrat are often ued to determine right and reonibilitie of uly hain artner. A lot of different ontrat tye deal with ie and aaity agreement, buy bak reonibilitie, quantity flexibilitie and other toi. An overview of the different ontrat tye an be found in [6] and [7]. Quantity flexibility (QF) ontrat are widely ued in axi and build an intereting onet for the utomer-ulier-relation [8]. hey give the ulier a higher lanning aurane by defining quantitie at leat within a given bandwidth. he utomer i for examle a manufaturer who fae unertain demand and ovide the art ulier with a relenihment hedule whih beome the art ulier releae hedule. QF ontrat are defined between eah node. hey are arameterized by (α, β) whih lae bound on how the manufaturer an revie the relenihment hedule. All arameter are defined over time whereby the flexibility interval hould get ogreively maller the nearer a eriod get. A rolling horizon lanning with udated etimate i reeated eah eriod. Aording to [9] both artie will get a benefit and the QF ontrat an be treated a an anwer to the well-known bullwhi effet. he ulier ha to over any requet u to the maximum of the flexibility bound giving the utomer a afety buffer. On the other ide the utomer agree on the lower bound a a minimum quantity and therefore give the ulier a higher afety in lanning. he flexibility ontrat ued in the aoah of thi work are very imilar to the QF ontrat. Some additional rule were defined to oordinate the quantitie aording to different time area. hee time area are deribed in [1] Coordinating hort-term quantitie In the mid-term lanning only aggregated quantitie are lanned whih have to be onretized in hort-term lanning. Even if the mid-term quantitie were oordinated well the exat order an be ubotimal for the ulier if their ituation are diregarded. In literature an integrated lanning and ourement i often ooed to reah a good overall olution. hi integrated lanning oordinate the odution oliy of a ulier with the ordering oliy of a utomer and an either be done imultaneouly or ueively. Simultaneou aoahe an be found for examle in [10] and [11]; ueive aoahe in [12] and [13]. A lit of more aoahe an be found in [14] and [15]. hee aoahe reah good reult, but are not well aliable for thi work. he differene of the aoah of our work to integrated lanning and ourement i that the utomer lan hi odution and the required order imultaneouly onidering the flexibility orridor. he imultaneou aet i between the utomer odution and hi ourement and not between the utomer ourement and the ulier odution. In thi work the fine lanning of the utomer i influened by different ie whih are offered by the ulier for ingle time eriod. So the ulier an affet the order behaviour of the utomer under onideration of their aaity ituation. A final ofitharing te hould give the ulier the oibility to ahieve another ot redution by offering alternative odution hedule. 3

4 roeeding of the 43rd Hawaii International Conferene on Sytem Siene Uing thi aoah the utomer i able to lan it odution and it ourement imultaneouly whih hould reult in ot redution. At the ame time the ot truture of the ulier i regarded by the ie for ingle eriod and the final te of ofit-haring. hi i done in ingle te to redue the omlexity of the overall oblem. 4. Mathematial model After introduing ome imortant aumtion the arameter and variable are lited before the model i introdued. Imortant aumtion for the model are: here are two different kind of odut: the bottlenek odut (BR) ontaining a bottlenek art (BA) and the non-bottlenek odut, whih do not ontain thi art. Every BA and BR generate the ame torage ot. Eah BR ontain one idential BA. Delivery time are zero (will be added in future work). Setu ot are equene indeendent. able 1 and able 2 lit the arameter and variable whih are onidered in the model. able 1: arameter of mathematial model arameter Meaning Number of time egment Number of eriod Number of eriod er time egment r Set of odut S Set of ulier var rie of ulier for one BA in eriod fix Fixed order ot of ulier in eriod max Maximum of deliverable BA unit for ulier in eriod max Maximum of deliverable BA unit for t, ulier in time egment t LB Lower bound of flexibility orridor of t, ulier in time egment t UB Uer bound of flexibility orridor of t, ulier in time egment t unihment ot for violating the flexibility t, orridor of ulier in time egment t er unit D b d SI C SO C Delay ot er eriod and BR unit Binary arameter indiating if odut i a BR (1) or a non-br (0) Cutomer demand for odut in eriod Storage aaity for BA Storage aaity for BR SI SO SC C CD dl Storage ot er BA unit Storage ot er BR unit Setu ot for odut in eriod rodution aaity in eriod Caaity demand for one unit of odut Binary arameter indiating if delayed delivery i ermitted (1) or not (0) able 2: Variable of the mathematial model Variable Meaning l Lot ize of odut in eriod dv Number of delivered unit of odut in eriod dl Number of delayed unit of odut in eriod Ordered BA amount at ulier in eriod o BA t Number of BA tored in eriod BR t Number of tored unit of odut in eriod l Binary variable indiating if an order at ulier i laed in eriod (1) or not (0) UB df Amount of BA exeeding the uer bound of t, the flexibility orridor of ulier in time egment t LB df Amount of BA fall below the lower bound of t, the flexibility orridor of ulier in time egment t e Binary variable indiating if a etu i required to odue odut in eriod (1) or not (0) he aim of the deribed method i the ot minimization of the utomer. he goal funtion i a follow: Minimize Z = SI BA ( t + = 1 SO BR D SC t, dl,, e, r var fix (, o, +, l, )) + S UB LB ( dft, t, + dft, t, ) t= 1 S ( + + ) + he following ot are regarded: Storage ot for BA and BR Cot for delayed delivery Setu ot Variable and fixed order ot Cot for violating the flexibility orridor (1) o get a valid olution the following ontraint have to be fulfilled: 4

5 roeeding of the 43rd Hawaii International Conferene on Sytem Siene BA 0 t BA ( t + t + dl ) = 0 (2) r + t + BR BA 0,, 0, l,, C e CD = 1,... (19) r t BA SI = 1,... (3) BR SO ( t ) = 1,... (4) r o max t = ( t 1) + 1 ( o ) max I = 1,... S (5) t = 1,... S (6) ( l CD) C = 1,... (7) r BR + t l dl d ( l ) ( d ) = 1 = 1 = 1 = 1 = 1,... r (8) r (9) ( dv ) ( d ) r (10) o l max BA + 1 r = 1,... (11) S ( o t ) ( l b ) = 1,... (12) S t ( o ) ( l b ) BA -1 + S r = t BA = 1,... (13) Contraint (2) initialize the reurively defined variable with zero and the following two ontraint limit the torage utilization to the aaity in eah eriod. Contraint (5) and (6) limit the amount of BA ordered at the artiular ulier to the maximum limit of the eriod reetively the time egment. Contraint (7) aure that the required odution aaity never exeed the available aaity in any eriod. he next two ontraint are imortant for atifying the demand. he firt one enfore that eah demand an be atified by the odut unit whih have been odued u to the artiular demand eriod, if delayed delivery i forbidden. Otherwie thi ontraint i ignored. he other ontraint ha a imilar truture but onider the whole lanning horizon. hereby it i guaranteed that every demand i atified at the end of the lat time egment if a valid olution an be found. Contraint (10) atifie the imilar funtion for the delivery variable. he next ontraint fore the value of the binary lae-variable aording to the order. Contraint (12) enure that the available volume of BA unit i high enough for the lanned odution in eah eriod. he next two ontraint are the tok balane ontraint for both torage tye. Contraint (15) and (16) are imortant for determining the variane of the order amount from the flexibility orridor. Contraint (17) and (18) are neeary for etting the value of the variable ounting the odution delay to the aurate value. Contraint (19) et the binary etu variable to one, if at leat one odut i odued and et an uer bound for the lot ize. BR BR -1, + = t l dv t t LB t, ( ) t, = ( t 1) + 1 LB o df t = ( t 1) + 1 2= 1 UB t, t, ( o ) UB df ( d l ) dl 2, 2, dl dl dv + d 1, = 1,... r (14) t = 1,... S (15) t = 1,... S (16) = 1,... (17) = 1,... r (18) 5. he method 5.1. Main idea he main objetive of the introdued method i to find a feaible and good olution onerning the reulting overall ot by following iterative te. he algorithm onit of a ontrutive art and everal imoving method wherea ome of thee method are exeuted during the ontrutive alulation and ome of them try to imove the reult after a firt olution for the given oblem ha been found. Due to age retrition only the ontrutive art i deribed. he idea of the method i to tart with an emty lan and alway add the bet lot to the odution lan until all demand are fulfilled. hereby the 5

6 roeeding of the 43rd Hawaii International Conferene on Sytem Siene determination of the lot onider all relevant information like the ulier and the uly eriod in ae of a bottlenek odut. o learly define the information whih i needed to deribe the manufaturing of one odut the term odution i introdued. A odution ha to ontain the following information: 1. Suly eriod of the BA 2. rodution eriod 3. Demand eriod 4. rodued odut 5. Sulier for the BA 6. Lot ize he oeration are alway the ame: In eah te the method trie to find a odution whih reult in minimal ot in the artiular ituation. A it ha been deribed in art four, there are everal different ot whih have to be onidered imultaneouly, like for examle torage, delay or etu ot. o afford the eented oe of hooing the loally bet odution in an iterative way, there mut be a way to etimate the ot er odut whih will exetedly our by adding the hoen tule to the reulting lan. Furthermore there mut be a data truture to generate and tore the oible odution and to determine the artiular bet one in an effiient way. hi data truture i a eriodially udated die whih ontain all available odution and the ot etimation. he next etion will deribe thi truture in detail Data truture he data truture to tore and analye the oible odution in the artiular ituation i a threedimenional truture, alled ot die (ee Figure 4). Suly eriod Demand eriod Cot die element: - Etimated ot - Sulier - rodut grou Figure 4. Cot die he dimenion are the three different temoral fator, whih influene the lanning and imlementation of the odution lan: 1. Suly eriod 2. rodution eriod 3. Demand eriod hereby the truture an tore a odution and it relevant value for eah ombination of the three different timetam. he die ontain die element, whih tore the remaining item of the odution tule. o kee the ot alulation imle the odued odut i not determined and tored during the alulation oe but during the odution hoie in the ontrutive algorithm. herefore the die element imly tore if the artiular etimated ot value doe aume the odution of a BR or a non-br. hi imlifiation to inreae the effiieny of the method i valid beaue the ot of all BR and non-br are imilar aording to the aumtion in hater 4. lanning a large amount of eriod onequently lead to a large and omlex data truture, whih might be ineffiient to handle. Beaue of thi the die i ut to a manageable ize by removing illegal or uele element Calulation and udate of ot die value o alway determine the bet odution in the artiular ituation the value of the die element mut be alway u to date. he following etion will exlain how the etimated ot of a die value are alulated and when they have to be udated Calulation of ot die value he ot value of the die element alway reeent the ot er odut whih might our if a ingle element of the artiular odution i added to the reult. Before alulating the ot the algorithm ha to erform ome hek to determine if a odution of the die element i till oible. hi i neeary beaue ome element an beome imoible, beaue other odution were lanned before whih influene for examle torage or odution aaity. If at leat one ondition i violated the etimated ot value an be et to without erforming any more alulation. Otherwie all different ot mut be examined for the artiular odution. In the following exlanation the alulation of the ot for the uly eriod lying in time egment t, the odution eriod and the demand eriod d i aumed. he different ot are onidered: 1. unihment ot for violating the flexibility orridor of the ulier he unihment ot, d, are the minimum of the unihment ot of all known ulier in the, du,, artiular time egment: = min( u S ) (20),d {, d, u } 6

7 roeeding of the 43rd Hawaii International Conferene on Sytem Siene ,d,u = { t tu,, ( u, ) tu, = ( t-1) + 1 if o < LB t t, u, (, u ) t, u = ( t-1) + 1 if o UB 0, otherwie (21) he unihment ot are a aving if an order would hel to reah the flexibility orridor. Otherwie the ot are zero if the flexibility orridor ha already been reahed or they are oitive if the uer bound of the etor ha been exeeded. 2. Storage ot he ordered BA a well a the odued BR have to be tored whih lead to the torage ot S of the odution: S SI SO = ( ) + max(0,( d )) (22) 3. Fixed order ot he fixed order ot F, d, are general ot and not alied to one ingle odut unit. herefore the fixed order ot are unitized to one odut unit by dividing them through the etimated amount AM of, d, odued odut unit. How thi amount i determined will be exlained in etion fix F, u,d = (23) AM,d 4. Variable order ot he variable order ot are the ie of the BA. Beaue of the goal of ot minimization the heaet ulier i hoen and the variable ot V, d, are onequently the minimum of all available ie: V var = min( u S ) (24),d {, u } 5. Delay ot If delayed delivery of the odut to the utomer i ermitted a odution might lead to delay ot D :, d, D D = max(0,( d )) (25),d 6. Setu ot A it ha been already mentioned the etu ot S, d, are equene indeendent and therefore eay to quantify. he only oblem i that the algorithm doe not know whih odut will be odued when thi odution i hoen o the etimated etu ot are determined by alulating the average etu ot of all odut: S S r,d = (26) r After alulating the value of the different ot the etimated ot value of the die element an be, d, determined: S F V D S = (27),d d,, d,, d,, d,, d,, Udate of ot die value after adding a odution to the reult After adding a odution of the ot die to the reult there might be the need to udate everal die element. hi udate deend on the following ondition. 1. Ued die element he ontrutive algorithm determine the oduable amount by uing everal retrition. herefore at leat one of the everal different bound ha been reahed and an udate i onequently neeary for thi element. 2. Firt order of the ulier If the added odution ha oened a new order at a artiular ulier all die element with the uly eriod have to be udated beaue the fixed order ot an be removed from the alulation. Of oure thi aume that the order limit of that ulier ha not been reahed yet. 3. All demand are atified If the added odution ha atified the omlete demand in eriod d of all odut of the ued bottlenek oerty all field with the demand eriod d have to be udated beaue the etimated ot will hange due to the hange of the examined odut et. 4. Order limit of the ulier ha been reahed If the added odution ha the onequene that the order limit of the ulier in the examined eriod or time egment ha been reahed, all die element with the ued uly eriod, or even all element with a uly eriod lying in the affeted time egment, have to be udated. hi i neeary beaue another ulier with different ot ha to be hoen. 7

8 roeeding of the 43rd Hawaii International Conferene on Sytem Siene Storage aaity ha been reahed If the added odution ha reahed the aaity limit of either the odut or the art torage the ot of the die element, whih ue thi bottlenek eriod in their odution, an be et to. 6. rodution aaity ha been reahed If the added odution ha dereaed the odution aaity in eriod to a value whih i not a high a at leat one of the odut of the examined bottlenek oerty demand, the die element of the odution eriod have to be udated. he udate will either hooe the other odut et or et the ot value to. If a odution i removed whih i the ae in an iterative te the udate oedure i imilar Contrutive algorithm Figure 5 how the general truture of the ontrutive algorithm for finding a firt feaible olution for the given oblem. [another demand to fulfill and oblem feaible to thi oint] hooe next utomer demand determine bet odution for artiular ituation determine fitting odut with greatet lot ize add odut to reult udate ot die aly iterative algorithm whih have to be exeuted during ontrution [all demand atified or oblem infeaible] Figure 5. Contrutive algorithm he following two ubetion exlain the main tak of the ontruting algorithm in detail Find the bet odution in the ot die Due to the fat that the minimal ot an be found in every legal die element the algorithm ha to examine every element of the die. An imovement of exeution eed an be ahieved by jut examining a defined amount of intelligent or randomly hoen die element, but thi of oure an lead to a dereaed quality or, in ome ae, to the reult, that the oblem i infeaible although a feaible olution ould be found. Neverthele the algorithm trie to find the odution in the die with minimal ot er unit of all element whih are taken into aount. If there are everal odution with the ame ot a rule deide about the hoie of the odution. he reult of thi oe i the odution-6-tule whih ontain all information about the next te for determining the odut and lot ize. hi oe i deribed in the next ubetion Determine the oduable odut and it lot ize After identifying the odution whih hould be added to the reult in the firt te of the iteration, the algorithm now trie to determine the odut, whih hould be odued and it lot ize. he objetive i to atify the utomer demand a far a oible o the odut with the greatet lot ize will be hoen for the odution. hereby the odut grou indiated by the die element mut be onidered in thi evaluation. he lot ize of a odution with uly eriod in time egment t, odution eriod and demand eriod d ha the following uer bound: 1. Demand Beaue tok holding hould be avoided the demand d of odut in eriod d i one limitation for the d, lot ize. 2. rodution aaity he oduible amount deending on the available odution aaity in the artiular eriod and the aaity requirement of one odut unit i another uer bound for the lot ize. he aaity bound CB for the odut in the odution eriod i defined a: C CB = (27) CD 3. Storage aaity he aaity of the BA tok S, a CA in the eriod between the uly and odution eriod a well a the aaity of the BR tok S, CA in the eriod between the odution and demand eriod may not be exeeded. So they reeent a limitation for the lot ize: S,, min min ( S a ); min ( S CB,d = CAi CAi ) (28) i [; 1] i [ ; d 1] 8

9 roeeding of the 43rd Hawaii International Conferene on Sytem Siene Maximum order limit he aaitie of the heaet ulier, whih ha been determined in the firt te may not be exeeded for the uly eriod. hi alie for the eriod O, LC u and the time egment O, LC. So thee value reeent tu, an uer bound for the lot ize: O, LC = max o (29) u, u, u O, t t, u, u, u = ( t 1) + 1 LC = max ( o ) (30) 5. Flexibility orridor bound he flexibility orridor bound have an influene on the lot ize beaue exeeding the lower or uer bound hange the etimated ot of one odut unit and thereby ut the loal otimality of the hoen odution in quetion. he order limit CS L tu, for the time egment t i defined a: o t = ( o ) (31) = ( t 1) + 1, u LBtu, o CS t,u = t, u L UB o,if o < LB tu, tu, < tu,,if LB o UB,if o UB tu, (32) he lot ize LS of a odut for the eriod, d,, determined in the firt te of the iteration i the minimum of the alulated limitation: LS,d, = S O, O, CS (33) min( d ; CB ; CB ; LC ; LC ; L ) d,, d u t, u t, u he lot ize LS of the odution and the odued, d, odut an now be identified by maximizing thi value over all odut belonging to the examined odut grou: =argmax ( LS ) (34) LS r, d,, d,,, d, = LS (35) All neeary information for the new odution ha been determined. he eond te of the iteration ha finihed and the new odution an now be added to the reult et. hi oe will be reeated until a valid olution i found or until the infeaibility of the oblem i identified. 6. Reult An imlementation of the eented algorithm and oee ha been reated to demontrate their ability to find good olution in many different enario. We ued the mathematial model eented in etion 4 and imlemented it in the otimizer MOS [16] to benhmark the method reult with the otimal reult. he imlementation in the urrent tate ha the target to demontrate the quality of the reult and i not yet otimized onerning the runtime. able 3 eent the enario whih have been ued to omare the method with the otimal value. able 3: et enario S r / Seiality =4; = =4; =2 fored unihment =5; =1 fored delay =4; = =40; = =50; =10 torage bottlenek =100; =20 Column S indiate how many ulier were regarded, olumn r the number of odut, olumn / the number of eriod and time egment and the lat olumn how if a eial demand ituation wa analyzed. hee enario odue the reult in able 4 whih eent the omutation time and overall ot of the mathematial olver and the method. able 4. et reult Solver Method ime () Cot ime () Cot 1 < < < < < < he reult demontrate the following fat: 1. he quality of the overall ot reult i very good in omarion to the otimal value. 2. he imlementation of the deribed method wa fater, even if the algorithm i not yet otimized for runtime. 9

10 roeeding of the 43rd Hawaii International Conferene on Sytem Siene Conluion In thi work an aoah for a imultaneou odution and order lanning in a ooerative uly hain relationhi wa eented. Firt a derition of the lanning enario and the referene to an overall method wa given. Afterward the work wa laified in the theory of odution lanning and quantity oordination. A mathematial model wa develoed that enable an otimal olution and an be ued a a benhmark for the heuriti method whih wa introdued in detail. Firt reult were eented whih how a very good quality of the eented aoah. Additionally to the eented method we have alo imlemented ome imovement method whih try to find a better olution after a valid lan wa found. hee method ahieve imoved reult by hanging ingle odution, but were not deribed in thi work due to age retrition. In ontext of the overall rolling lanning the deribed method will be ued both for new lanning and for hange lanning. In hange lanning odution whih need to be relanned will be identified, deleted and lanned again aording to the value of the ot die. At the moment we are working on the onjuntion of the rough lanning and the fine lanning to automatially lan the flexibility orridor, the order and the odution lanning in a rolling way. hi lanning will run for the whole frame ontat horizon and will be benhmarked with an otimal olution for whih a mathematial model wa already develoed. 8. Referene [1]. Rut, W. Dangelmaier, D. Brüggemann, J.-. ater: An aoah for flexible utomer-ulier relationhi with fixed delivery ontrat and quota agreement. In: roeeding of the 13 IFAC Symoium on Information Control roblem In Manufaturing, Moow, Ruia, Elevier Siene, 2009, [2] E. A. Silver, D. F. yke, and R. eteron, Inventory Management and rodution lanning and Sheduling, 3 rd Edition, John Wiley In., New York, [3] L. J. homa, J. O. MClain, An overview of rodution lanning, In: Handbook in Oeration Reearh and Management Siene, Volume 4, Logiti of rodution and Inventory, edited by S. C. Grave, A. H. G. Rinnoy Kan and. H. Zikin, Elevier Siene ubliher, Amterdam, 1993, [4] H. Stadtler, C. Kilger, Suly Chain Management and Advaned lanning: Conet, Model, Software, and Cae Studie, 4 th Edition, Singer-Verlag, Berlin, [5] D. Quadt, Lot-Sizing and Sheduling for Flexible Flow Line, Leture Note in Eonomi and Mathematial Sytem, Singer-Verlag, Berlin, [6] A. Gomez-adilla, J. Duvallet and D. Llerena, Contrat yology a a Reearh Method in Suly Chain Management In: Kotzab, H. (ed.), Reearh Methodologie in Suly Chain Management, hyia-verlag, Heidelberg. 2005, [7] C. Keinger, H. ieer, Managing Rik with Strutured Suly Agreement In: C. An, H. Fromm (ed.), Suly Chain Management on Demand, Singer-Verlag, Berlin, 2005, [8] A.A. ay, W.S. Lovejoy. Quantity flexibility ontrat and uly hain erformane. Manufaturing and Servie Oeration Management, 1, 2, 1999, [9] H.L. Lee, V. admanabhan and S. Whang, he bullwhi effet in uly hain. Sloan Management Review, 38, 3, [10] S.K. Goyal, An integrated inventory model for a ingle ulier ingle utomer oblem. In: International Journal of rodution Reearh, 1976, Vol. 15, No. 1, [11] A.K. Agrawal, D.A. Raju, Imoved joint eonomi lot ize model for a urhaer and a vendor. In M. Khan, C.S. Wright and R. Whalley: Advaned Manufaturing roee, Sytem and ehnologie (AMS96), London, [12] J.. Monahan, A Quantity Diount riing Model to inreae Vendor rofit. In: Management Siene, 1984, Vol. 30, No. 6, [13] H.L. Lee, M.J. Roenblatt, A Generalized Quantity Diount riing Model to Inreae Sulier rofit. In: Management Siene, 1986, Vol. 32, No. 9, [14] E. Suky, Koordination in Suly Chain: Sieltheoretihe Anätze zur Ermittlung integrierter Betellund roduktionolitiken,wiebaden, [15] E. Suky, Coordinated order and odution oliie in uly hain. OR Setrum 26, 2004, [16] U. Suhl, (2000), MOS Mathematial Otimization Sytem, OR New, 8, 2000,