Two Measures of Production System Flexibility and their Application to Identify Optimal Capacity Investment

Transcription

1 Two Measures of Producton System Flexblty and ther Applcaton to Identfy Optmal Capacty Investment Klaus Altendorfer 1, Cornna Engelhardt-Nowtzk, Herbert Jodlbauer Upper Austran Unversty of Appled Scences, Steyr, Austra Wehrgrabengasse 1-3, 44 Steyr, Austra, Tel.: +43 () , Fax.: Abstract We motvate the use of a company s capablty to handle the requrements concernng mean customer requred lead tme and customer demand varaton as measures for producton system flexblty wthn a Value Chan. Thus, the customer drven producton plannng concept s appled whch lnks together customer demand varaton, mean customer requred lead tme and servce level. Thereby nventory holdng and capacty costs for certan levels of requred customer demand varaton and mean customer requred lead tme are derved. For a servce level constrant model, the cost optmum for capacty and nventory holdng costs s shown to mply the ablty to handle a certan mean customer requred lead tme flexblty when capacty demand varaton s predefned. Further, the optmal costs are shown to ncrease n capacty demand varance. Applyng the developed method a manager can ether dentfy the mnmum mean customer requred lead tme for a predefned capacty demand varaton or he/she can dentfy the optmal values for the two flexblty measures dscussed to mnmze costs. Keywords: manufacturng, flexblty management, customer drven producton plannng, nventory, capacty nvestment analyss, Value Chan demand varaton 1. Introducton Flexblty can be defned as the ablty to react to changes n specfcaton, amount and due date of customer orders, gven a certan system specfcaton. All these measures can be transformed nto changes n the customer requred lead tme 1 Correspondng author. E-mal address: Klaus.Altendorfer@fh-steyr.at

2 and to the customer demand varaton. Therefore, the two flexblty measures for a customer suppler relatonshp wthn a Value Chan can be dentfed as: the ablty to handle short mean customer requred lead tmes and the ablty to handle a certan level of customer demand varaton. Though both measures are lnked to local producton facltes, they are of major mportance for the overall Value Chan performance n two respects: Frst, any producton ste across the whole value creaton process may be affected heavly by respectve demand varance and thus negatvely nfluence the entre Value Chan performance due to lmted performance and flexblty. Second, suchlke demand fluctuatons can exponentally amplfy across several Value Chan stages (also known as bull-whp effect, cp. Forrester (1961), Lee et al. (1997)). Ths effect can be unfavorably enforced through an nsuffcent local handlng of the balance between customer requred lead tme and assocated key ndcators. The customer requred lead tme s defned as the tme between when an order s stated (or restated f changes occur) and the requested due date (see Jodlbauer and Altendorfer (1) or Altendorfer (1)). Even though there already exsts a large set of lterature on flexblty measures as well as on ther nfluence on the company performance, a huge proporton of ths lterature s of a qualtatve nature. These papers dscuss flexblty as a rather generc and merely measurable construct that needs to be further operatonalzed (frequently cted defntons are provded e.g. n Ackoff (1971), Seth and Seth (199), Upton (1994) and, more recently, Zhang et al. (3) or Rechhart and Holweg (7)). Most of the quanttatve approaches avalable, lmt ther scope of applcaton to a very specfc context or research queston (see e.g. Das and Abdel-Marek (3), Ahlert et al. (9), Francas et al. (11) or Palomnos et al. (9)). Currently avalable flexblty lterature s extended n ths paper by developng and applyng a set of equatons concernng the nherent relatonshp between the two aforementoned general flexblty measures. The modelng frame appled s a sngle-stage producton system whereby ths sngle stage can ether be a sngle machne, a set of machnes or a whole producton plant. The man fndng s that for any customer demand varaton a certan lower bound for the mean customer requred lead tme s already mpled when mnmzng capacty and nventory costs wth respect to a servce level constrant. The practcal mplcaton s that whenever the downstream company n a Value Chan requres a certan demand flexblty from ts upstream suppler due to ts order varaton, t also mplctly defnes the mean customer requred lead tme flexblty that can be provded by ths suppler (f the suppler does not change ts capacty). Studyng the relatonshp between these two flexblty measures contrbutes to a better understandng of Value Chan coherences occurrng due to flexblty requrements, ncludng performance and cost ssues. The Paper s structured as follows: Secton revews the relevant lterature on flexblty measures and producton logstcs related key fgures. In Secton 3 the relatonshp between mean customer requred lead tme flexblty and customer

3 3 demand varaton flexblty s developed. A numercal study s provded n Secton 4 and we conclude n Secton 5.. Lterature revew The lterature lnked to the research topc of ths paper can be splt up nto flexblty-related ssues and contrbutons on producton logstcal key fgures. Especally lterature usng the two flexblty measures appled n ths paper s revewed n the frst subsecton. The lnk between these two flexblty measures from a producton logstcal pont of vew s revewed n the second subsecton..1 Defntons and applcaton of flexblty The term flexblty s dscussed heterogeneously n the relevant Supply Chan lterature (see Stevenson and Sprng (7) for an overvew). A hgh amount of contrbutons provde rather generc defntons: For nstance Zelenovc (198) descrbes the flexblty of a producton system as the capacty to adapt to changng envronmental condtons and process requrements. Upton (1994) defnes flexblty as the ablty to change or react wth lttle penalty and Rechhart and Holweg (7) nvestgate several prevously proposed flexblty defntons and summarze that flexblty s the ablty of a system to adapt to nternal or external nfluences, thereby actng or respondng to acheve a desred outcome. A more tangble, though, stll qualtatve perspectve s appled n other publcatons that prmarly apply a manufacturng-related pont of vew. These contrbutons are mostly compatble wth the aforementoned generc defntons, although, accordng to Palomnos et al. (9), the advantage of hgh compatblty s accompaned by a lack of operatonal applcablty, at least wthn a manufacturng context. For nstance Olhager et al. (1) dscuss the applcaton of chase strateges to acheve flexblty and adaptablty n make-to-order (MTO) settngs. They lnk manufacturng ssues wth demand-related aspects. Seth and Seth (199) are further specfyng flexblty as the ablty to make dfferent parts wthout major setups. In addton to the heterogeneous flexblty dscusson, other terms lke responsveness (see e.g. Rechhart and Holweg (7)) and aglty (see e.g. Bernardes and Hanna (9) and Narasmhan et al. (6)) are appled to descrbe constructs that are smlar (or correspondng) to flexblty.

4 4 Tab. 1. Relevant publcatons on flexblty. Source focus lead tme demand varaton Tang and Tomln (8) flexblty as a means for Supply Chan rsk mtgaton n dsruptve envronments (quanttatve approach) Salvador et al. mx and volume flexblty n MTO settngs mplctly (7) (emprcal nvestgaton) mpled Özbayrak et al. nvestgate MTO control polces wth (6) fluctuatng demand (smulaton model) Merzfonluoğlu determnaton of optmum demand, producton and and nventory levels (quantta- Geunes (6) tve approach) Moattar Hussen quanttatve analyss of the relaton be- et al. (6) tween resource level and output flexbl- Bsh et al. (5) ty n a Just-n-Tme envronment hedgng flexble capacty aganst forecast errors n MTO settngs whle keepng swngs n producton low Zhang et al. mpacts of logstcal flexblty on customer (5) satsfacton (emprcal evdence) Wjngaard nfluence of advance demand nformaton (4) on producton and nventory control mplctly mpled mplctly mpled mplctly mpled mplctly mpled (quanttatve approach) Zhang et al. manufacturng flexblty as crtcal ssue mplctly (3) n Value Chans (emprcal evdence) mpled Garavell (3) product flow optmzaton n Supply Chans wthout dramatcally ncreasng flexblty costs (smulaton model) Oke (3a) drvers of volume flexblty n manufacturng plants, especally customer nfluence on lead tme (emprcal evdence) Oke (3b) nventory and volume flexblty wthn order fulfllment (emprcal evdence) Barad and Sapr flexblty benefts n logstc systems (3) (quanttatve approach) Takahash and reactve Just-n-Tme order systems to Nakamura acheve agle control n the face of unstable () demand characterstcs (smulaton Koste et al. (1999) study) framework for manufacturng flexblty, usng 1 flexblty dmensons mplctly mpled mplctly mpled mplctly mpled mplctly mpled From the quanttatve pont of vew there s also a lot of lterature avalable overcomng the lmtaton of qualtatve approaches whch are often not able to operatonally specfy manageral mplcatons and measure ther outcomes. In order to motvate the two flexblty measures proposed above (the ablty to handle a short customer requred lead tme and a certan level of customer demand varaton) Tab. 1 provdes an overvew of relevant lterature addressng flexblty ssues

5 5 wthn a quanttatve context whereby column three and four, respectvely, ndcate whether a contrbuton apples/ncludes these two measures. The classfcaton reports a drect applcaton of the respectve measure n a research paper whereas mplctly mpled means that arguments wth a dstnct relaton to ths measure are dscussed, but t s not drectly mentoned. Tab. 1 shows that the capablty of beng able to cope wth demand varatons s mentoned as flexblty means n most of the quanttatve contrbutons revewed, whereas the mean customer requred lead tme flexblty s more often addressed ndrectly. In ths regard typcal arguments ndrectly addressng ths topc have ncorporated related measures such as a low utlzaton, low nventory levels or the capacty to quckly react n the face of changes all of them mplyng the capablty to operate wth short lead tmes. A man concluson s therefore, that the two ntally proposed abltes to handle short mean customer requred lead tme and a certan level of customer demand varaton are found to be reasonable measures for flexblty n an ndustral Value Chan context, at the same tme beng compatble to the generc flexblty defntons dscussed above.. Producton logstcal relatonshps concernng the defned flexblty measures The relatonshp between WIP, utlzaton and producton lead tme has been dscussed n a lot of publcatons based on queung theory, see Lttle (1961), Medh (1991), Chen and Yao (1), Tjms (3) and Hopp and Spearman (8). Some other analytcal models dscussng ths relatonshp are e.g. developed by Mssbauer et al. (1998) and Jodlbauer (8a). Addtonally, there are a lot of emprcal studes on ths relatonshp avalable ether applyng smulaton, see Jones (1973), Hopp and Roof-Sturgs (), Jodlbauer and Huber (8), Altendorfer et al. (7a), and Altendorfer et al. (7b), or analyzng real company data, see Nyhus and Wendahl (), Wendahl and Brethaupt (1999) and Wendahl et al. (5). In all these models the lnk to servce level or tardness s mssng. Ths lnk to servce level and/or tardness has recently been examned by Jodlbauer (8b), Altendorfer and Jodlbauer (11) and Jodlbauer and Altendorfer (1) applyng ether queung theory or other analytcal methods. Based on the results of Jodlbauer (8b) and Jodlbauer and Altendorfer (1) the capacty needed to reach a capacty-related servce level can be calculated based on the capacty demand fluctuaton and the customer requred lead tme dstrbuton. Ths approach, called customer drven producton plannng, s extended n the current paper to dscuss the relatonshp between the two proposed flexblty measures.

6 6 3. Model development 3.1 Assumptons and Notaton We study a mult-tem sngle-stage MTO producton system wth the structure shown n Fg. 1: Fg. 1. Producton system structure. In ths system, each tem 1,..., N has a random customer demand n peces A n for an arbtrary tme perod n, whch s ndependent of other tem demands and whch s ndependent of the prevous tme perod demands. Ths A n follows a Gauss Process (see Ross (1) or Bechelt (6) for detals on Gauss Processes) wth EAn n and Var A n n whch states that the demand for each tem n any arbtrary tme perod n s normally dstrbuted (see also Altendorfer (1) for ths formulaton of the settng dscussed n Jodlbauer and Altendorfer (1)). and denote the mean demand rate and the demand varance of tem respectvely. Followng Altendorfer (1) wth defnng N q and q 1 N 1 (whereby q s the tem dependent determnstc capacty needed to produce one pece) the mean and varance of capacty demand can be dentfed as ED n n Var D n n f and F Dn and. Dn denote the probablty densty functon (pdf) and cumulated dstrbuton functon (cdf) of the random varable Dn respectvely. Based on the order date and the random due date the customer requred lead tme L can be defned as the random customer requred lead tme of one capacty unt (one tme unt needed from the machne). For a dervaton of L dependng on the tem dependent cus-

7 7 tomer requred lead tme values L see Altendorfer (1). fl and FL denote the pdf and cdf of the random varable L respectvely. The processng stage processes orders at a rate of. Tab.. Lst of varables. Varable A n E, Descrpton random customer demand n peces for tem Var expected value and varance for a random varable respectvely,, Dn fr q L, mean demand rate and the demand varance of tem respectvely mean and varance of capacty demand rate respectvely random customer demand n capacty for tme perod n FR probablty densty functon (pdf) and cumulated dstrbuton functon (cdf) of the random varable R respectvely determnstc capacty needed to produce one pece of tem random tem dependent customer requred lead tme L random customer requred lead tme of one capacty unt processng rate of the sngle-stage P mnmum requred producton lead tme of processng stage Dn coeffcent of varaton of D(n) s servce level sp percentage of capacty whch cannot be delvered on tme (based on P) n length of capacty smoothng perod E L mn mnmum mean customer requred lead tme whch can be provded to reach a certan servce level s X work ahead wndow (parameter for work release polcy) Y random nventory n capacty unts c nventory holdng cost factor h c C capacty cost factor overall costs for capacty and nventory All orders are released to the producton system accordng to a work ahead wndow strategy (see Jodlbauer and Altendorfer (1) or Altendorfer and Jodlbauer (11)) and are processed accordng to the earlest due date (EDD) dspatchng rule. In front of the processng stage, the released materals are stored n a buffer and all orders fnshed pror to the due date, are stored n the fnshed-goodsnventory. A capacty servce level s, statng what percentage of capacty s delvered on tme, can be calculated. Jodlbauer and Altendorfer (1) argue that n such a system the capacty servce level can be appled to approxmate the often -servce level and s equal to ths -servce level whenever processng tme

8 8 s equal for each tem. The producton system s modeled as havng a mnmum requred producton lead tme P. Ths value s needed to nclude techncal constrants as well as transportaton tmes or organzatonal tmes before producton. Tab. provdes a lst of varables. In ths settng three propostons concernng the mnmum mean customer requred lead tme and the optmal costs wth respect to a predefned capacty demand varaton are stated. Frstly, for a system wth predefned capacty t s shown what the lower bound of mean customer requred lead tme s. Secondly, the problem s extended to a settng where the capacty nvestment s optmzed. In ths second settng agan a lower bound for mean customer requred lead tme and the cost mnmum are provded. Mean customer requred lead tme and optmal costs are both lnked to the capacty demand varaton showng ts nfluence wthn a Value Chan settng wth a downstream customer and ts upstream suppler. 3. Relatonshp between the two flexblty measures Followng the concept of Jodlbauer (8b), Jodlbauer and Altendorfer (1) and Altendorfer (1) t s obvous that the random capacty demand Dn wll, for some tme perods n, be greater than the avalable capacty n n ths tme perod. Any tme perod n can be nterpreted as an averagng perod for smoothng capacty demand snce the coeffcent of varaton of Dn decreases wth ncreasng n: Var Dn d 3 (1) Dn 1 Dn n EDn n dn Accordng to Altendorfer (1), there s always a certan percentage of capacty whch cannot be delvered on tme for orders wth customer requred lead tme shorter than P (see also servce level defnton n Secton 3.1): sp FL P () Based on equaton (1) and (), the followng servce level equaton shows that for a predefned capacty, the servce level reached ncreases n the capacty smoothng perod appled: F n Dn sp s (3) Followng Jodlbauer and Altendorfer (1) and Altendorfer (1) shows that for normally dstrbuted demand, the followng smoothng perod s needed n order to reach a predefned servce level s: 1 F N(,1) s s (4) P n

9 9 1 whereby F N (,1) denotes the quantle (nverse) of the standard normal dstrbuton functon. From Altendorfer (1), the followng relatonshp concernng the customer requred lead tme dstrbuton has to hold, to ensure that a certan servce level can be reached n the MTO settng: P (5) EL 1 F ( ) d n, s L For low P values ( P E L ) the followng approxmaton s stated n Altendorfer (1): P (6) 1 F ( ) d P L Furthermore, for low P values ( P E L ), the servce level loss s P s very low and s therefore neglected: s F P wth F P 1 s (7) P L L P Proposton 1: The mnmum mean customer requred lead tme E Lmn whch s stll able to handle a predefned capacty demand varaton under normally dstrbuted demand at a sngle-stage producton system wth respect to a predefned servce level s s defned as (applyng approxmaton (6) and (7)): 1 F N (,1) s (8) ELmn P Proof: Proposton 1 follows drectly from equaton (4) to (7) wth: P E L 1 F ( ) d n, s mn L E Lmn n, s P E L (9) Proposton 1 states the relatonshp between customer requred lead tme flexblty and capacty demand varance flexblty wth respect to the capacty avalable. Note that Proposton 1 s smlar to the condton stated as MTO ablty of a producton system n Jodlbauer (8b), however the relatonshp s stated there as beng an nequalty. The mnmum mean customer requred lead tme E Lmn

10 1 has a lnear ncrease n the capacty demand varance and a convex decrease n the capacty avalable, see: 1 1 de Lmn FN (,1) s de Lmn FN (,1) s (1) and 3 d d F s 1 mn N (,1) 4 de L 6 d Ths holds snce s assumed for a stable system wth utlzaton below 1%. Fg. vsualzes ths relatonshp: a) E[Lmn] Predefned b) E[Lmn] ,5,5 3 Predefned µ P 1; s.95; 1; 1.5; 1 Case a) Case b) Fg.. mn wth respect to E L and. An nterestng fndng from Proposton 1 s that the varance of customer requred lead tme has no nfluence on the mnmum mean customer requred lead tme whch can be mantaned when applyng approxmatons (6) and (7). However, when the mean customer requred lead tme s hgher than the mnmum value, the work ahead wndow X for plannng s mplctly defned as (see Altendorfer (1)): X (11) 1 F ( ) d n P L whch shows that n ths stuaton the dstrbuton of customer requred lead tme s also needed. Ths holds snce the work ahead wndow s when calculatng E L mn.

11 Optmal capacty nvestment Before optmzng the capacty nvestment, the nventory calculaton from Jodlbauer and Altendorfer (1) and Altendorfer (1) s ntroduced brefly. Based on the mnmum requred producton lead tme P and on the work ahead wndow work release polcy wth parameter X, see equaton (11), the nventory follows as (for a dervaton see Altendorfer (1)): EY P n, s (1) Term one of (1) corresponds to the nventory needed to mantan the mnmum requred lead tme of the system and term two s defned as surplus nventory and corresponds to the nventory from pre-producng on stock. When mnmzng the sum of nventory holdng cost and capacty nvestment cost under a servce level constrant s (problem (13)), Altendorfer (1) proves the followng optmal capacty nvestment and nventory (see (14)) under the condton (15) (applyng approxmatons (6) and (7)): C, X E Y c c mn (13) h, w.r.t. s s F (,1) * N s c h c 1 3 * FN (,1) s c P c h E Y * n, s EL Pn 1 3 N (,1) F s c ch X (14) (15) Proposton : The mnmum mean customer requred lead tme whch s stll able to handle a predefned capacty demand varaton under normally dstrbuted demand at a sngle-stage producton system when optmzng capacty and nventory costs wth respect to a servce level constrant s s defned as: (16) 1 F 3 N (,1) s c EL mn P c h

12 1 has a concave ncrease wth respect to the capacty demand va- and ELmn rance. Proof: Equaton (16) follows drectly from equaton (8) and (14); and 1 3 mn 1 (,1) 3 N 3 c h de L F s c d mn (,1) FN s c 3 9 c 3 h de L d shows the concave ncrease. (17) Note that ths relatonshp only states what the mnmum mean customer requred lead tme has to be when capacty nvestment can be optmzed and capacty demand varance s predefned. Comparng equaton (8) wth equaton (16) shows that n the cost optmal case, the ncrease of ELmn wth respect to s no longer ndependent of. Takng equaton (14) nto account shows that an ncrease n also leads to an n- crease n optmal capacty to nvest. Proposton 3: The optmal cost for capacty and nventory for a sngle-stage producton under normally dstrbuted demand wth respect to a servce level constrant follows as: * 1 (18) 3 3 C FN(,1) s c ch Pch c whch s a concave ncreasng functon n capacty demand varance. Proof : Equaton (18) follows from equaton (13) and (14): 1 (19) F 3 (,1) c (,1) * N s c h FN s C P c h c c h c N h h N h 1 1 (,1) (,1) F s c c c F s c Pc c

13 13 The concave ncrease follows from the frst two dervatves: * 1 dc FN(,1) s c ch d 3 dc * FN(,1) s c ch d 9 () a) 14 b), ,5 C * 6 4 µ* 1, Predefned Predefned c) d) 5 E[Y*] 15 1 E[Lmn] Predefned Predefned P1; s.95; 1; c 5; c h 1 Fg. 3. Optmal cost and flexblty wth capacty and nventory costs. For practcal applcaton, equaton (16) shows the mnmum mean customer requred lead tme whch can be handled for a predefned capacty demand varance and a predefned servce level constrant. Furthermore, the nfluence of reducng the capacty demand varance on mnmum mean customer requred lead tme whch can be handled and on optmal costs s shown. From equaton () t becomes clear that even f the optmal costs can be reached wth the occurrng mean customer requred lead tme, a hgher capacty demand varance leads to hgher costs. Fg. 3 shows the optmal costs, the mnmum customer requred lead tme to reach them, the optmal capacty nvestment and the optmal nventory wth respect to capacty demand varance.

14 Addtonal costs for lower mean customer requred lead tme In ths subsecton the addtonal costs for a shorter mean customer requred lead tme than calculated n equaton (16) are evaluated. Based on the optmal costs from equaton (18) whch only hold f E L E L mn, the addtonal costs for E L E Lmn are calculated. In ths case, constrant (15) of the optmzaton problem s bndng and therefore n s defned as: n EL P (1) Wth ths new n, the capacty to nvest follows from equaton (4) and (7) as: 1 1 FN(,1) s F N(,1) s () E L P E L P and the expected nventory can also be calculated based on the new n as (see equaton (1)): EY E L PP E L (3) whch leads to costs of: 1 F (,1) N s (4) C, E L EYch c E L c h c EL P Ths cost from equaton (4) can be compared to the optmal cost from (18) and provdes a manager wth the nformaton how expensve t s to mantan the lower mean customer requred lead tme. The followng Fg. 4 shows ths addtonal cost wth respect to E Lmn E L. a) Cost ncrease n [%] 18% 16% 14% 1% 1% 8% 6% 4% % % 5 1 b) Cost Increase n [%] 18% 16% 14% 1% 1% 8% 6% 4% % % 5 1 E[Lmn] E[L] E[L] P s EL 1;.95; 1; 1; mn 1.9 Fg. 4. Cost ncrease when E L mn. E L

15 15 From a Value Chan perspectve ths addtonal cost shows the potental cost beneft of mprovng nformaton flow, whch leads to an ncreased mean customer requred lead tme E L snce orders are stated earler. The cost reducton potental of decreasng E Lmn E L can, for example, be balanced aganst the addtonal cost of nformaton technology mprovng the nformaton flow. 4 Numercal study Ths secton dscusses the nfluence of the predefned servce level constrant, the mnmum requred producton lead tme and the capacty and nventory costs on the relatonshp between E Lmn and. In subsecton 4.1 the curves shown n Fg. 3 are compared for dfferent P, s, c and c h values. In subsecton 4. a senstvty analyss showng E Lmn wth respect to P, s, c and c h for a predefned s provded. In ths secton the cost optmal E Lmn from equaton (16) s dscussed. 4.1 Parameter comparson examples To show the nfluence of dfferent parameter sets on the relatonshp between E L mn and, the followng examples are compared for the settng n whch capacty and nventory costs are mnmzed: Example 1 basc scenaro: P1; s.95; 1; c 5; c h 1 Example producton lead tme: P ; s.95; 1; c 5; c h 1 Example 3 servce level: P1; s.9; 1; c 5; c h 1 Example 4 capacty costs: P1; s.95; 1; c 6; c h 1 Example 5 nventory holdng costs: P1; s.95; 1; c 5; c h Fg. 5 shows the results of ths study. Comparng the results of example 1 wth example shows that an ncrease n mnmum producton lead tme has only a low nfluence on the optmal capacty to nvest and t leads to a slght cost ncrease as well as to a slght ncrease n E Lmn whch means to a slght flexblty loss. Lookng at example 3 provdes the nsght that a decrease of the servce level constrant decreases costs, optmal capacty, optmal nventory and E Lmn. There-

16 16 fore, acceptng a lower servce level from the upstream suppler n a Value Chan can mprove ths suppler s flexblty concernng E Lmn and. The comparson of examples 4 and 5 wth example 1 shows an nterestng dfference n the effect of cost ncreases. Whle a cost ncrease n capacty costs leads to a worse stuaton for all parameters (costs are hgher and flexblty decreases), the cost ncrease n nventory holdng costs shows that n ths case the optmal costs ncrease, however, the flexblty also ncreases (va a decreasng E Lmn ). a) 13 b) C * µ * 1,8 1,6 1,4 1, Predefned Predefned Ex 1 Ex Ex 3 Ex 4 Ex 5 Ex 1 Ex Ex 3 Ex 4 Ex 5 c) 5 d) 5 E[Y * ] Predefned Ex 1 Ex Ex 3 Ex 4 Ex 5 E[Lmn] Predefned Ex 1 Ex Ex 3 Ex 4 Ex 5 Fg. 5. Numercal study exemplary comparson. 4. Numercal senstvty analyss To study the effect of the parameters P, s, c and c h more n detal, n ths subsecton a numercal senstvty analyss s provded. The coeffcent of varaton of capacty demand s therefore predefned wth 1 and the parameters are vared n certan ranges. All parameters, except the one vared, are set to the values of example 1 from subsecton 4.1.

17 17 a) 5 b) 5 E[Lmn] 15 1 E[Lmn] Predefned P Predefned s c) 5 5 d) E[Lmn] 15 1 E[Lmn] Predefned c µ Predefned c h Fg. 6. Numercal study senstvty analyss. Fg. 6 shows that customer requred lead tme flexblty decreases when mnmum producton lead tme, capacty costs and servce level ncrease. However, wth hgher nventory cost, there s a rse of optmal customer requred lead tme flexblty. One manageral nsght from ths analyss can be derved from the curve shape of n Fg. 6b) whch shows that especally servce level constrants above 95% lead to a mayor decrease n flexblty. Another manageral nsght can be found n Fg. 6d) whch shows that n the cost optmal settng, hgher valued products (wth hgher nventory holdng costs) already mply that a hgher customer requred lead tme flexblty s optmal. Referrng to the ntally mentoned matter of Value Chan mpacts resultng from local producton decsons and performance states (as well as the possble amplfcaton effects across the entre value creaton process), t s mportant for responsble producton managers to capture a better understandng of local senstvtes regardng ther mpact on overall Value Chan behavor.

18 18 5. Concluson In ths paper the capablty to handle a certan capacty demand varance and a certan mean customer requred lead tme s assumed to be a possble measure for flexblty n the relatonshp between a downstream company and ts upstream supplers wthn a Value Chan. For a predefned capacty and servce level t s found that capacty demand varance and mean customer requred lead tme are not drect substtutes. Wth respect to a servce level constrant, predefnng a certan capacty varance demanded by the customer(s) only leads to a lower bound of mean customer requred lead tme whch stll can be provded. Optmzng the capacty nvestment and nventory costs at the suppler ste wth respect to a servce level constrant and a predefned capacty varance agan leads to a certan lower bound for the mean customer requred lead tme whch can stll be mantaned. Further, the addtonal costs arsng from stuatons wth a customer requred lead tme below ths value are dscussed. For practcal applcaton ths paper provdes the manageral nsght that the lower bound of the mean customer requred lead tme flexblty s a functon of the customer demand varance and the (optmal) capacty avalable. Applyng the fndngs of ths study to a Value Chan shows that cost benefts can be acheved by measures for smoothng the capacty demand varance and by measures for mprovng the nformaton flow (whch leads to an ncreased mean customer requred lead tme) wthn the Value Chan. We recommend future research on the extenson of the presented method, especally regardng optmzaton of capacty demand varance and customer requred lead tme as decson varable between customers and supplers wthn a Value Chan. References Ackoff, R.L. (1971) Towards a system of systems concepts. Management Scence. 17(11), Ahlert, K.-H., Corsten, H. and Gössnger, R. (9) Capacty management n order-drven producton networks A flexblty-orented approach to determne the sze of a network capacty pool. Internatonal Journal of Producton Economcs, 118(), Altendorfer, K. (1) Capacty and Inventory Plannng for Make-to-Order Producton Systems - The Impact of a Customer Requred Lead Tme Dstrbuton, Dssertaton at Unversty of Venna, Steyr. Altendorfer, K. and Jodlbauer, H. (11) An analytcal model for servce level and tardness n a sngle machne MTO producton system. Forthcomng: Internatonal Journal of Producton Research. Altendorfer, K., Jodlbauer, H., and Huber, A. (7a) Behavour of MRP, Kanban, CONWIP and DBR under dynamc envronmental varablty. Management komplexer Materalflüsse mttels Smulaton State-of-the-Art nnovatve Konzepte,

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