Can Aggregate Production Planning (APP) Be Modified to Be as Good as Master Production Scheduling (MPS)?

155 ACC JOURNAL 2017, Volume 23, Issue 2 DOI: 10.15240/ ul/004/2017-2-012 CAN AGGREGATE PRODUCTION PLANNING (APP) BE MODIFIED TO BE AS GOOD AS MASTER PRODUCTION SCHEDULING (MPS)? Tho s en Vi z hum1; Tho s en Claus2; F ank He mann3 1,2Technische Uni e si ä D esden (TU D esden) Facul y o Business and Economics, In e na ional Ins i u e Zi au Ma k 23, 02763 Zi au, Ge many 3Os baye ische Technische Hochschule Regensbu g (OTH Regensbu g) Facul y o Compu e Science and Ma hema ics Inno a ion and Compe ence Cen e o P oduc ion Logis ics and Fac o y Planning (IPF) P ü eninge S . 58, 93049 Regensbu g, Ge many e-mail: 1[email p o ec ed]; 2[email p o ec ed]; 3 ank.he[email p o ec ed] Abs ac In his pape , he signi icance o mas e p oduc ion scheduling (MPS) o hie a chical p oduc ion planning should be b ough ou . The e o e, se e al modi ica ions o agg ega e p oduc ion planning (APP) a e made and ealiza ion o he planning esul s o APP and MPS a e compa ed. The goal is o imp o e he ealiza ion o APP in such a way ha he ealiza ion is as good as he ealiza ion o MPS. Imp o emen s o indi idual planning si ua ions could be achie ed, bu no gene al solu ion could be ound. Especially he use o a sui able capaci y educ ion ac o (CRF) combined wi h he conside a ion o lead ime leads o an imp o emen o he solu ion. Howe e , inding a sui able CRF o mo e han one p oduc is di icul . This usually depends on he unde lying demand p o ile and mus be e-de e mined o each cons ella ion. Keywo ds Hie a chical p oduc ion planning; Agg ega e p oduc ion planning (APP); Mas e p oduc ion scheduling (MPS); Ma e ial equi emen planning; Resou ce p o iles. In oduc ion P oduc ion planning is a e y complex. A high numbe o pa ame e s and hei mul iple dependencies a e simul aneously o be aken in o accoun . This esul ing planning p oblem can’ be sol ed in an app op ia e ime e en wi h mos powe ul compu e s and sophis ica ed so wa e [1]. In he mos comme cial p oduc ion planning and con ol sys ems, he complex planning o p oduc ion planning is eplaced by he hie a chical p oduc ion planning app oach [6]. In his case he complex singula p oblem is decomposed in se e al manageable p oblems. These p oblems a e sol ed successi ely and he indi idual solu ions a e combined in o one o e all solu ion a he end. The hie a chical p oduc ion planning app oach is a ibu ed o Hax and Meal in he yea 1975 [5]. D exl, Fleischmann, Gün he and S adle sugges ed in 1994 o expand his concep by limi ed capaci ies o esou ces [2]. On his planning concep , he pape is based. Typical hie a chical p oduc ion planning is shown in Figu e 1. 156 Scheduling P ojec shop Job shop FMS Flow shop JIT p oduc ion ... Linked p oduc ion segmen s Capaci y-o ien ed PPC-Sys em Agg ega e Mas e Planning P oduc ion segmen (Loca ion 1) Linked p oduc ion segmen s Capaci y-o ien ed PPC-Sys em Agg ega e Mas e Planning P oduc ion segmen (Loca ion N) Loca ion-in eg a ed P oduc ion- P ocu emen - and T anspo a ion planning (Supply Ne wo k Planning, Mas e Planning, En e p ise Planning) agg ega ed de ailed De e minis ic View S ochas ic View P ocu emen Dis ibu ion Shipmen s Shipmen sSupply Ne wo k Demand o ecas ing, Demand ul illmen (A ailable- o-p omise), Wa ning-Moni o ing Suppo i e modules Bu e ing, Sa e y S ocks, Sa e y Times ... Mas e P oduc ion Planning Ma e ial Requi emen s Planning P ojec shop Job shop FMS Flow shop JIT p oduc ion ... Cus ome Supplie Sou ce: [4] Fig. 1: Typical hie a chical p oduc ion planning Agg ega ion plays an impo an ole in his planning app oach. The e a e h ee di e en ypes o agg ega ion. The agg ega ion o ime especially he pe iod size, decision a iables like g ouping o p oduc s o cons ain s like g ouping o machines o wo k cen e s o p oduc ion si es [3][7]. This planning concep consis s o he ollowing s eps: Agg ega e p oduc ion planning (APP), mas e p oduc ion scheduling (MPS), ma e ial equi emen s planning and scheduling. APP p edic s he planned independen equi emen s o one p oduc ion si e on a p oduc g oup le el based on demand o ecas s. MPS p edic s he planned independen equi emen s o inal p oduc s o a p oduc ion si e. The e o e, p oduc g oups a e disagg ega ed in o inal p oduc s. Th ough he use o esou ce p o iles p oduc ion lead ime and he capaci y equi emen s o each wo k cen e is conside ed [9]. In he pape E idence o he ele ance o mas e p oduc ion Scheduling o hie a chical P oduc ion Planning [8] he ele ance o MPS was demons a ed h ough a case s udy. MPS can a oid sho ages h ough a close examina ion o he capaci y. I was sugges ed o in oduce a capaci y es ic ion and o conside he exac o se pe iods on APP le el o possibly a oid sho ages. This will be ca ied ou . Capaci y on APP le el will be educed by a capaci y educ ion ac o (CRF). The poin o his pape is bes illus a ed by ollowing a clea example om a case s udy. A he beginning o he case s udy solu ion and ealiza ion o APP and MPS a e compa ed. In he nex s ep, a CRF o APP is in oduced and he exac o se pe iods a e conside ed. In he end, i is examined wha happens i o al demand o all p oduc s is he same, bu he demand o each p oduc is di e en . 157 1 Case S udy One p oduc ion si e wi h wo wo k cen e s is conside ed. The capaci y o each wo k cen e is 500 hou s pe pe iod. So, he p oduc ion si e has a capaci y o 1000 hou s pe pe iod. Human capaci y is equal echnical capaci y and he e is no addi ional capaci y. In o de o be able o compa e esul s which each o he , he pe iod size o agg ega e p oduc ion planning (APP) and mas e p oduc ion scheduling (MPS) is he same and i is assumed ha he p edic ed planned independen equi emen s a e iden ical o he eal cus ome equi emen s. The de i ed equi emen s o all componen s a e p oduced jus in ime. The model o a closed p oduc ion is used. Closed p oduc ion is cha ac e ized by he ac ha each planned o de mus be inished and s o ed, be o e i can be u he p ocessed o shipped [6]. Two p oduc g oups which a e p oduced in one p oduc ion si e a e conside ed. Each p oduc g oup consis s o one inal p oduc . The inal p oduc (E) needs 3 hou s o capaci y pe uni in wo k cen e (A) and he componen (V) needs 7 hou s o capaci y pe uni in wo k cen e (B). I ollows ha p oduc g oup (P) needs 10 hou s o capaci y pe uni in p oduc ion si e. The inal p oduc (F) needs 5 hou s o capaci y pe uni in wo k cen e (A) and he componen (V) needs 5 hou s o capaci y pe uni in wo k cen e (B). I ollows ha p oduc g oup (Q) needs also 10 hou s o capaci y pe uni in p oduc ion si e. The es ima ed lead ime o all p oduc s and componen s is 1 pe iod. Se -up imes and cos s a e no conside ed. The planning ho izon o he case s udy s a s in pe iod 1 and ends wi h he las cus ome demand in pe iod 10. The demands o p oduc g oup (P) espec i ely p oduc (E) and p oduc g oup (Q) espec i ely p oduc (F) a e shown in Table 1. Tab. 1: Demand o he case s udy Pe iod ( ) 1 2 3 4 5 6 7 8 9 10 ∑ Demand (𝑑𝐸,𝑡 𝐴)[uni s] 0 0 0 0 40 50 60 50 40 60 300 Demand (𝑑𝐹,𝑡 𝐴)[uni s] 0 0 0 0 40 50 60 50 40 60 300 Sou ce: Own As shown in [8] due o he agg ega ed conside a ion o capaci y equi emen o p oduc g oups, APP o e es ima es he numbe o p oduc s which can be p oduced in o one pe iod. This was shown o one p oduc g oup which is ela ed o one inal p oduc . To a oid his CRF o 0.83 is in oduced in combina ion wi h he conside a ion o he lead ime o 2 pe iods. The solu ion and he ealiza ion o his APP is as good as he one o MPS. The Solu ion o APP wi h a CRF o 0.83 and MPS is shown in Table 2. The objec i e is €872. Tab. 2: Solu ion o APP wi h capaci y educ ion ac o 0.83 and MPS Pe iod ( ) 1 2 3 4 5 6 7 8 9 10 ∑ 𝑑𝐸,𝑡 𝐴 [uni s] 0 0 0 0 80 100 120 100 80 120 600 𝑥𝐸,𝑡 𝐴 [uni s] 0 19 83 83 83 83 83 83 83 0 600 𝑦𝐸,𝑡 𝐸 [uni s] 0 0 19 102 105 88 51 34 37 0 436 Sou ce: Own The solu ion can be ealized wi hou exceeding he pe iod capaci y and no sho age occu s in he indi idual pe iods. 2 Resul s o he Resea ch In his case s udy, he planning esul s o wo p oduc g oups as desc ibed in he case s udy a e compa ed. Fi s he Solu ion o APP which is p o ided by ILOG is p esen ed. Nex , an 158 op imal solu ion o APP wi hou sho ages and delay is sea ched and p esen ed. A he end he esul s a e compa ed wi h he esul s o MPS. Tab. 3: Solu ion o APP wi h CRF 0.83 (ILOG) Pe iod ( ) 1 2 3 4 5 6 7 8 9 10 ∑ 𝑑𝑃,𝑡 𝐴 [uni s] 0 0 0 0 40 50 60 50 40 60 300 𝑥𝑃,𝑡 𝐴 [uni s] 0 19 0 21 83 27 50 77 23 0 300 𝑦𝑃,𝑡 𝐸 [uni s] 0 0 19 19 0 33 0 0 37 0 108 𝑑𝑄,𝑡 𝐴 [uni s] 0 0 0 0 40 50 60 50 40 60 300 𝑥𝑄,𝑡 𝐴 [uni s] 0 0 83 62 0 56 33 6 60 0 300 𝑦𝑄,𝑡 𝐸 [uni s] 0 0 0 83 105 55 51 34 0 0 328 Sum 𝑥𝑃,𝑡 𝐴+ 𝑥𝑄,𝑡 𝐴 0 19 83 83 83 83 83 83 83 0 600 Sou ce: Own dE,5=40 dE,6=50 dE,7=60 dE,8=50 dE,9=40 dE,10=60 dF,5=40 dF,6=50 Pe iod ( )41 2 3 5 6 107 8 9 11 12 WCB WCA V_1 E_1 W_1W_1 V_2 V_4 V_6V_5 V_7V_3V_3 W_2 W_5W_4W_3 W_6 E_2 E_3 E_5 F_1F_1 F_2 F_3 E_7E_7 F_4 E_6E_6 F_5 F_6F_6 E_4 dF,7=60 dF,8=50 dF,9=40 dF,10=60 Delay Sou ce: Own Fig. 2: Realiza ion o APP wi h CRF 0.83 (ILOG) In Figu e 2 he ealiza ion o Table 3 is illus a ed as Gan cha . The lead ime is aken in o accoun du ing he ealiza ion. Figu e 2 shows ha he demand o p oduc (E) o pe iod 6 is ul illed wi h 1 pe iod delay. The same beha io can be seen o he demand o p oduc (F) o he pe iods 7, 9 and 10. The o al delay in pe iods is 4. Besides he o al sho age o 510 uni s he in en o y holding cos s inc ease om €872 o €1266. ILOG inds an op imal solu ion. This may no be he only solu ion, he e may be o he solu ions. Ano he solu ion is shown in Table 4 and he ealiza ion is illus a ed in Figu e 3. This solu ion can be ealized wi hou sho age and delay. The in en o y holding cos s co espond o he objec i e o €872. Bo h solu ions ha e in common ha a maximum o 83 uni s o bo h p oduc s can be p oduced pe pe iod. The exac p oduc ion quan i ies can be ound in ables unde “Sum 𝑥𝑃,𝑡 𝐴+ 𝑥𝑄,𝑡 𝐴”. Fu he mo e, he esul s a e compa ed wi h he solu ion and ealiza ion o he case s udy wi h one p oduc g oup shown in he pape o Vi z hum & He mann [8]. We see ha in his case s udy we ha e he same p oduc ion quan i ies and he same in en o y holding cos s o €872. 159 Tab. 4: Solu ion o APP wi h CRF 0.83 (al e na i e) Pe iod ( ) 1 2 3 4 5 6 7 8 9 10 ∑ 𝑑𝑃,𝑡 𝐴 [uni s] 0 0 0 0 40 50 60 50 40 60 300 𝑥𝑃,𝑡 𝐴 [uni s] 0 6 42 42 42 42 42 42 42 0 300 𝑦𝑃,𝑡 𝐸 [uni s] 0 0 6 48 50 42 24 16 18 0 204 𝑑𝑄,𝑡 𝐴 [uni s] 0 0 0 0 40 50 60 50 40 60 300 𝑥𝑄,𝑡 𝐴 [uni s] 0 13 41 41 41 41 41 41 41 0 300 𝑦𝑄,𝑡 𝐸 [uni s] 0 0 13 43 55 46 27 18 19 0 232 Sum 𝑥𝑃,𝑡 𝐴+ 𝑥𝑄,𝑡 𝐴 0 19 83 83 83 83 83 83 83 0 600 Sou ce: Own dE,5=40 dE,6=50 dE,7=60 dE,8=50 dE,9=40 dE,10=60 dF,5=40 dF,6=50 dF,7=60 dF,8=50 dF,9=40 dF,10=60 dE,5=40 dE,6=50 dE,7=60 dE,8=50 dE,9=40 dE,10=60 dF,5=40 dF,6=50 dF,7=60 dF,8=50 dF,9=40 dF,10=60 Pe iod ( )41 2 3 5 6 107 8 9 11 12 WCB WCA V_1 V_2 V_3 V_4 V_6V_5 V_7 V_8 E_1 E_2 E_3 E_4 E_6E_5 E_7E_7 E_8 W_1 W_2 W_4W_3 W_5 W_6W_5 W_6 W_7 W_8 F_1 F_2 F_4F_3 F_5 F_6F_5 F_6 F_5 F_6F_5 F_6 Sou ce: Own Fig. 3: Realiza ion o APP wi h CRF 0.83 (al e na i e) Finally, he esul s o he APP a e compa ed wi h he esul s o he MPS. The esul s o he MPS a e shown in Table 5 and illus a ed in Figu e 4. Tab. 5: Solu ion o MPS Pe iod ( ) 1 2 3 4 5 6 7 8 9 10 ∑ 𝑑𝐸,𝑡 𝐴 [uni s] 0 0 0 0 40 50 60 50 40 60 300 𝑥𝐸,𝑡 𝐴 [uni s] 0 15 71 42 35 30 35 42 30 0 300 𝑦𝐸,𝑡 𝐸 [uni s] 0 0 15 86 88 73 43 28 30 0 363 𝑑𝐹,𝑡 𝐴 [uni s] 0 0 0 0 40 50 60 50 40 60 300 𝑥𝐹,𝑡 𝐴 [uni s] 0 0 0 41 51 58 51 41 58 0 300 𝑦𝐹,𝑡 𝐸 [uni s] 0 0 0 0 1 2 0 1 2 0 6 Sum 𝑥𝐸,𝑡 𝐴+ 𝑥𝐹,𝑡 𝐴 0 15 71 83 86 88 86 83 88 0 600 Sou ce: Own 160 dE,5=40 dE,6=50 dE,7=60 dE,8=50 dE,9=40 dE,10=60 dF,5=40 dF,6=50 dF,7=60 dF,8=50 dF,9=40 dF,10=60 dE,5=40 dE,6=50 dE,7=60 dE,8=50 dE,9=40 dE,10=60 dF,5=40 dF,6=50 dF,7=60 dF,8=50 dF,9=40 dF,10=60 Pe iod ( )41 2 3 5 6 107 8 9 11 12 WCB WCA V_1 V_2 V_3 V_4 V_6V_5 W_1 W_2 W_5 W_6W_4W_3 V_7 V_8 E_1 E_2 E_3 E_4 E_6E_5 E_7E_7 E_8E_1 E_2 E_3 E_4 E_6E_5 E_7 E_8 F_1 F_2 F_5 F_6F_4F_3F_1 F_2 F_5 F_6F_4F_3 Sou ce: Own Fig. 4: Realiza ion o MPS The ealiza ion can be done wi hou sho age and delay. The in en o y holding cos s a e €738. Compa ed o he APP, he s o age cos s ha e dec eased by €134. This means a sa ing o 15.37%. In o de o unde s and he di e ences, he planed independen equi emen s o APP and MPS ha e o be compa ed. APP has a maximum o 83 uni s which can be p oduced independen ly o he p oduc which is p oduced. MPS has a maximum o 88 uni s, bu i depends on he p oduc mix, which is p oduced. 3 Discussion One ad an age o MPS is he conside a ion o lead ime. Bu e en i APP akes lead ime in o accoun , we do no ge any plans ha can be ealized wi hou sho ages and delay. APP es ima es he a ailable capaci y o he wo k cen e w ongly, because o he agg ega ed planning o he p oduc ion si e. No mally APP o e es ima es he a ailable capaci y. One way o co ec his is CRF. A good chosen CRF will p o ide easible plans. Due o he gene al educ ion, a ough es ima e is made. The e is no di e en ia ion o capaci y equi emen s by p oduc s. The main ad an age o he MPS is he use o esou ce p o iles. A esou ce p o ile gi es in o ma ion abou he capaci y equi emen s o inal p oduc s in e ms o wo k cen e s and o se pe iods. Fo his eason, he capaci y o he p oduc ion p og am is aken in o accoun mo e p ecisely. The ac is bes demons a ed in compa ison o he maximum p oduc ion a e by p oduc . While APP doesn’ di e be ween p oduc (E) and p oduc (F) and has maximum p oduc ion a e o 83 uni s he MPS di e s and has a maximum p oduc ion a e o 71 uni s o p oduc (E) and a maximum p oduc ion a e o 100 o p oduc (F). Conclusion MPS is supe io o APP due o he de ailed conside a ion o he capaci y equi emen . Th ough he use o esou ce p o iles an alloca ion o he capaci y equi emen by wo k cen e and pe iod based on p oduc s is made, while APP has a ixed p oduc ion a e o all p oduc s. MPS has a a iable p oduc ion a e o he p oduc s which di e s by he p oduc mix o he p oduc ion p og am. Fo his eason MPS can use he a ailable capaci y mo e e icien ly han APP. 161 Li e a u e [1] CLAUS, T.; HERRMANN, F.; MANITZ, M. (eds.): (2015). P oduk ionsplanung und - s eue ung: Fo schungsansä ze, Me hoden und de en Anwendungen. Be lin, Heidelbe g: Sp inge Gable . [2] DREXL, A.; FLEISCHMANN, B.; GÜNTHER, H.-O.; STADTLER, H.; TEMPELMEIER, H.: (1994). Konzep ionelle G undlagen kapazi ä so ien ie e PPS- Sys eme. Zei sch i ü be iebswi scha liche Fo schung. Vol. 46, pp. 1022–1045. [3] GEBHARD, M.: (2009). Hie a chische P oduk ionsplanung bei Unsiche hei . 1. Au lage. Wiesbaden: Sp inge Gable . DOI: 10.1007/978-3-8349-8227-8 [4] GÜNTHER, H.-O.; TEMPELMEIER, H.: (2014). P oduk ion und Logis ik: Supply Chain und Ope a ions Managemen . 11. Au lage. No de s ed : Books on Demand. ISBN 978-3735721952. [5] HAX, A. C.; MEAL, H. C.: (1975). Hie a chical In eg a ion o P oduc ion Planning and Scheduling. In: M. A. Geisle (ed.) TIMS s udies in he managemen sciences. Vol. 1. Logis ics. pp. 53–69. Ams e dam u.a.: No h-Holland. [6] HERRMANN, F.: (2011). Ope a i e Planung in IT-Sys emen ü die P oduk ionsplanung und -s eue ung: Wi kung, Auswahl und Eins ellhinweise on Ve ah en und Pa ame e n. [Mi Online-Se ice] 1. Au lage. Wiesbaden: Vieweg + Teubne . [7] STADTLER, H.: (1988). Hie a chische P oduk ionsplanung bei losweise Fe igung. Physica-Sch i en zu Be iebswi scha . Vol. 23. Heidelbe g: Sp inge . [8] VITZTHUM, T.; HERRMANN, F.: (2017). E idence o he ele ance o mas e p oduc ion Scheduling o hie a chical P oduc ion Planning. In: P oceedings o he 31 h Eu opean Con e ence on Modelling and Simula ion, ECMS 2017. pp. 475–481. ISBN 978-0-9932440-4-9. DOI: 10.7148/2017-0475 [9] VOLLMANN, T. E.; BERRY, W. L.; WHYBARK, D. C.; JACOBS, F. R.: (2004). Manu ac u ing planning and con ol o supply chain managemen . 5 h ed. McG aw- Hill in e na ional edi ions. Bos on, Mass.: McG aw-Hill. ISBN 978-0072299908. Tho s en Vi z hum, M.Sc.; P o . D . habil. Tho s en Claus; P o . D .-Ing. F ank He mann 162 MŮŽE BÝT AGREGOVANÉ PLÁNOVÁNÍ PRODUKCE MODIFIKOVÁNO TAK, ABY BYLO TAK DOBRÉ JAKO PLÁNOVÁNÍ PROGRAMU HLAVNÍ PRODUKCE? V om o příspě ku je pomocí případo é s udie zdů azněn ýznam pláno ání p og amu hla ní p odukce p o hie a chické pláno ání p odukce. Je učiněn pokus o zlepšení ýsledku ag ego aného celko ého pláno ání změnou jedno li ých pa ame ů ak, aby o odpo ídalo ýsledku pláno ání p og amu hla ní p odukce. V ámci še ření se ak docílilo něk e ých zlepšení jedno li ých si uací pláno ání, a šak obecné řešení se ješ ě nepodařilo naléz . Z láš ě použi í hodného ak o u k edukci kapaci y e spojení se zohledněním p ůběhu ede ke zlepšení ýsledku. Při pláno ání íce p oduk ů je šak ěžké nají hodný ak o p o edukci kapaci y, p o ože en je od islý od pop á ky. KANN DIE AGGREGIERTE GESAMTPLANUNG SO MODIFIZIERT WERDEN, DASS DAS PLANUNGSERGEBNIS DEM DER HAUPTPRODUKTIONSPROGRAMMPLANUNG ENTSPRICHT? In diesem Bei ag wi d anhand eine Falls udie die Bedeu ung de Haup p oduk ionsp og ammplanung (HPPLAN) ü die hie a chische P oduk ionsplanung he ausgea bei e we den. Dazu wi d e such das E gebnis de agg egie en Gesam planung (AGGRPLAN) du ch Ve ände ung einzelne Pa ama e so zu e besse n, dass es dem E gebnis de HPPLAN en sp ich . Im Rahmen de Un e suchungen konn en dabei Ve besse ungen ü einzelne Planungssi ua ionen e ziel we den, eine gene elle Lösung konn e jedoch nich ge unden we den. Insbesonde e de Einsa z eines geeigne en Kapazi ä s eduk ions ak o s in Ve bindung mi de Be ücksich igung de Du chlau zei üh zu eine Ve besse ung de Lösung. Bei de Planung on meh als einem P oduk is alle dings schwie ig einen geeigne en Kapazi ä s eduk ions ak o zu inden, da diese abhängig on de Nach age is . CZY ZAGREGOWANE PLANOWANIE PRODUKCJI MOŻE ZOSTAĆ ZMODYFIKOWANE TAK, BY BYŁO TAK DOBRE JAK PLANOWANIE PROGRAMU GŁÓWNEJ PRODUKCJI? W niniejszym op acowaniu na pods awie s udium p zypadku wskazano znaczenie planowania p og amu głównej p odukcji w hie a chicznym planowaniu p odukcji. Podję o p óbę pop awienia wyniku zag egowanego ogólnego planowania pop zez zmianę poszczególnych pa ame ów, by odpowiadało o wynikowi planowania p og amu głównej p odukcji. W amach p zep owadzonych badań osiągnię o pewne udoskonalenia poszczególnych sy uacji planowania, jednak nie udało się na azie znaleźć kompleksowego ozwiązania. Wynik pop awia w szczególności zas osowanie odpowiedniego czynnika do zmniejszenia wielkości w połączeniu z uwzględnieniem p zebiegu. Podczas planowania kilku p oduk ów udno jednak znaleźć odpowiedni czynnik do zmniejszenia wielkości, ponieważ zależny jes on od popy u.