MODELS OF PRODUCTION RUNS FOR MULTIPLE PRODUCTS IN FLEXIBLE MANUFACTURING SYSTEM

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1 Yugoslav Journal of Oeraons Research (0), Nuer, DOI: 0.98/YJOR0307I MODELS OF PRODUCTION RUNS FOR MULTIPLE PRODUCTS IN FLEXIBLE MANUFACTURING SYSTEM Olver ILIĆ, Mlć RADOVIĆ Faculy of Organzaonal Scences, Unversy of Belgrade, Sera Receved: June 008 / Acceed: Noveer 0 Asrac: How o deerne econoc roducon runs (EPR) for ulle roducs n flexle anufacurng syses (FMS) s consdered n hs aer. Egh dfferen alhough slar, odels are develoed and resened. The frs four odels are devoed o he cases when no shorage s allowed. The oher four odels are soe knd of generalzaon of he revous ones when shorages ay exs.the nuercal exales are gven as he llusraon of he roosed odels. Keywords: Econoc roducon runs, ulroduc case, deernsc nvenory odels. MSC: 90B30.. INTRODUCTION When a nuer of roducs share he use of he sae equen on a cyclc ass, he overall cycle lengh can e esalshed n a way slar o he sngle case descred n [9]. The ore general role, however, s no o deerne he econocal lengh of a roducon run for each roduc ndvdually, u o deerne jonly he runs for he enre grou of roducs whch share he use of he sae facles. If each ar or roduc run s se ndeendenly, s hghly lkely ha soe conflc of equen needs would resul unless he oerang level s soewha elow caacy, where consderale dle equen e s avalale []. The exale resenng hs suaon are flexle

2 308 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case anufacurng syses (FMS) ha us e se u o roduce dfferen szes and yes of roduc [8], ec. Conceually, he role o deerne an econocal cycle s he sae as for he one-roduc case, ha s, o deerne he cycle lengh whch wll nze he oal of achne seu coss lus nvenory holdng coss jonly for he enre se of roducs [5], [6] and [7]. The odels resened n hs aer are he deernsc nvenory odels. In hs aer, we resen he rocedure for deernaon of he nuer of roducon runs, N, for egh slar odels. The egh odels (see Tale ) are Model I: gradual relenshen, wh deand delvery durng he roducon erod, no shorages Model II: nsananeous relenshen, wh deand delvery durng he roducon erod, no shorages Model III: gradual relenshen, no deand delvery durng he roducon erod, no shorages Model IV: nsananeous relenshen, no deand delvery durng he roducon erod, no shorages Model V: gradual relenshen, wh deand delvery durng he roducon erod, wh shorages Model VI: nsananeous relenshen, wh deand delvery durng he roducon erod, wh shorages Model VII: gradual relenshen, no deand delvery durng he roducon erod, wh shorages Model VIII: nsananeous relenshen, no deand delvery durng he roducon erod, wh shorages. The frs four odels and he sevenh one, as wll e seen laer, are all secal cases of he ffh, sxh, and he eghh. Our resenaon of he egh odels egns wh odel I, he asc econoc roducon runs (EPR) odel. Fnally, odels II, III, IV, V, VI, VII, and VIII are resened as he exensons o he asc odel.. MODELS WITHOUT SHORTAGES.. The asc econoc roducon runs odel Our frs odel (odel I) descres he case where no shorages are allowed, u he deand rae s greaer han zero durng he roducon erod, and here s a fne relenshen rae. Fgure shows how he nvenory levels for hs odel vary n e. Because he fne relenshen rae usually les a roducon rae, odel I s usually referred o as an EPR odel. Whn he conex of hs dscusson, however, he EPR odel s erely an exenson of he asc econoc roducon quany (EPQ) or econoc lo sze (ELS) odel [], [3] and [4]. The oal cos analyss for he EPR odel s exacly he sae as for he EPQ odel. Invenory coss lus seu coss yeld o oal ncreenal cos. To develo he ERP odel for several roducs, he followng noaons are used:

3 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case 309 D annual requreens for he ndvdual roducs d equvalen requreens er roducon day for he ndvdual roducs daly roducon raes for he ndvdual roducs - assung, of course, ha > d,,,..., H holdng cos er un, er year for he ndvdual roducs S seu coss er run for he ndvdual roducs nuer of roducs q roducon quany for he ndvdual roducs y eak nvenory for he ndvdual roducs c roducon erod for he ndvdual roducs consuon erod for he ndvdual roducs e eween roducon runs c oal ncreenal cos N nuer of roducon runs er year N * nuer of roducon runs er year for an oal soluon T oal e erod Invenory coss. The axu nvenory for a gven roduc s ( d), and he average nvenory s ( d) /. However, q D / N. Therefore, average nvenory can e exressed as ( d) D d D ( d ) ( ) () N N The annual nvenory cos for a gven roduc s hen he roduc of he average nvenory, gven y (), and he cos o hold a un n nvenory er year, H, or HTD d ( ) N The annual nvenory cos for he enre se of roducs s, hen, he su of exressons of he for of (), or T N d HD ( ) Seu coss. The seu coss for a gven roduc are gven y S, n dollars er run. Therefore, he oal seu cos er year for ha roduc s NS. Fnally, he oal annual seu cos s he su of NS for he enre se of roducs, or ()

4 30 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case NS and snce N s he sae for all roducs, he oal annual seu cos s, N S Toal ncreenal cos. The oal ncreenal cos assocaed wh he enre se of roducs s hen T d CN ( ) N S+ HD( ) N (3) Our ojecve s o deerne he nu of he C curve wh resec o N, he nuer of roducon runs. Therefore, followng he asc rocedure for he dervaon of he classcal roducon quany odel, he frs dervave of C wh resec o N s dc dn T d ( ) 0 S HD N solvng for N, we have N* T d HD( ) S (4) The oal cos of an oal soluon, C *. The oal cos of an oal soluon s found y susung N* for N n (3), or T d C* N* S + H D ( ) N* Susung and slfyng he exresson for N* shown n (4) leads o d C* T S H D ( ).. Model II Fgure resens nvenory levels as a funcon of e for hs odel. No shorages are allowed, so each new run arrves a he oen when he roducon level wh deand delvery durng he roducon erod reaches axu nvenory level. The oal ncreenal cos analyss for hs odel s exacly he sae as for he asc EPR odel. The axu nvenory level for a gven roduc s he sae as for he one revously defned. The cos ha changes s he annual nvenory holdng cos for

5 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case 3 he enre se of roducs ecause he araeer whch s he annual nvenory holdng e changes. Therefore, he oal ncreenal cos equaon s T d CN ( ) N S+ HD( ) N Then, he nuer of roducon runs er year for an oal soluon, N*, sasfes T d N* S HD( ) N* or N* T d HD( ) S (5) The oal ncreenal cos of an oal soluon s d C* T S H D ( ) (6).3. Model III Our hrd odel (odel III) descres he case where no shorages and no deand delvery durng he roducon erod are allowed, u now here s a fne relenshen rae. Fgure 3 shows how he nvenory levels vary n e for hs odel. Now, y q,,,...,. Therefore, he oal ncreenal cos equaon s T CN ( ) N S+ HD N and he oal nuer of roducon runs, N*, s N* T H D S

6 3 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case.4. Model IV Fgure 4 resens nvenory levels as a funcon of e for hs odel. No shorages are allowed, so each new run arrves he oen when he roducon level whou deand delvery durng he roducon erod reaches axu nvenory level. For hs odel, y q,,,...,, and he annual nvenory holdng e s he sae as n odel II. The oal ncreenal cos for hs odel s he sae as n equaon (3). Also, he oal nuer of roducon runs for hs odel s equal o he oal nuer of roducon runs for odel I. 3.. Model V 3. MODEL WITH SHORTAGES In ers of he relenshen rae and he deand rae durng he roducon erod, odel V s he sae as odel I. A gradual relenshen s assued. The dfference s ha, n odel V, shorages are allowed, and a corresondng shorage cos s rovded. In he shorage suaon n hs odel, he deand ha canno e sasfed s ackordered and s o e e afer he nex shen arrves. Ths s uch dfferen fro he case of los sales, where he cusoer does no reurn, herey reducng he deand. The nvenory levels for odel V are shown n Fgure 5. Noce ha he axu shorage for a gven roduc s and he axu nvenory for a gven roduc s y, whch eans ha he fgure s he sae as Fgure, u wh all nvenory levels reduced y he aoun. Agan, coon sense should ell us ha, ecause nvenory levels and he assocaed holdng coss wll e lower han n odel I, he run quany can e ncreased and runs can e laced less ofen. To analyze hs suaon, le us defne he cos of a ackorder er un er e (year) for a gven roduc, G. Tha s, hs cos s defned n ers of uns (dollars er e er e), whch s slar o he defnon of he nvenory holdng cos. Also, he oal ncreenal cos assocaed wh he enre se of roducs, for hs odel, s slar o he oal cos for odel I, wh he addon of coss due o shorages. Cannual seu coss + annual nvenory holdng coss + annual shorage coss There s no change n he seu coss. However, he holdng cos changes due o he dfference n calculaon of he average nvenory level for hs suaon. The average nvenory level s D d ( ) N D d ( ) N

7 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case 33 and he average ackorder oson s, slarly, D d ( ) N Consequenly, he oal cos s D d HT ( ) N GT CNB (, ) NS + + D d D d ( ) ( ) N N To oan he EPR, we dfferenae he oal cos wh resec o oh N and B and solve wo sulaneous equaons, whch yeld o N* T d G HD( ) H + G S (7) G Because H + G s ore han G, he er H G <, loadng o he + decreased N, whch was execed. The deernaon of he axu nuer of deands ousandng,, s D d H ( ) N * H + G The axu nvenory, hen, s D d y ( ) (9) N The lengh of he cycle,, s T / N, as has haened revously. The cycle,, was roken down no and c for odel I, and no nvenory and shorage e n hs odel. For hs odel, all he four e are oran. As shown n Fgure 5, where roducs + ( + ) + ( ) c 3 4 e of roducng whle here s a shorage suaon for he ndvdual (8)

8 34 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case d roducs e of roducng, whle here s nvenory on hand for he ndvdual y d 3 e of ure consuon whle here s nvenory on hand for he ndvdual roducs y d 3 / 4 e of ure consuon whle here s a shorage suaon for he ndvdual roducs d 4 / 3.. Model VI Model VI allows shorages (fne shorage cos) and has an nfne rae of relenshen wh deand delvery durng he roducon erod. The nvenory levels over e for hs odel are shown n Fgure 6. The oal cos for hs odel s D d HT N GT CNB (, ) NS + + D D N N whch yelds o he followng forulas: ( ) N* T d G HD( ) H + G S (0) d G H + G () C* T S H D ( ) and a axal ackorder oson of he equaon (8). The axu nvenory also s defned as an equaon (9). For hs odel, he e of ure consuon s he sae as n odel V.

9 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case Model VII Model VII s slar o odel III. The dfference s ha, n odel VII, shorages are allowed. The nvenory levels for hs odel are shown n Fgure 7. The oal cos for hs odel s ( ) (, ) D HT GT N CNB NS D D N N + + whch yelds o he EPR forula of * G T H D H G N S + and a axal ackorder oson of * D H N H G + () The axu nvenory level, hen, s q. The cycle,, was roken down no four es, where 3 4 c c q q q q q q or 3 4 y y d d 3.4. Model VIII Model VIII allows shorages (fne shorage cos) and has an nfne rae of relenshen and no deand delvery durng he roducon erod. The nvenory levels over e are shown n Fgure 8. The oal cos s

10 36 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case D HT ( ) GT CNB (, ) N NS + + D D ( ) ( ) N d N d whch yelds o he EPR forula of equaon (7), and a axal ackorder oson of equaon (). The axu nvenory, hen, s. The e of ure consuon for odel VIII s he sae as n odel VII. 4.. Exale 3. NUMERICAL EXAMPLES Le us work ou an exale o deernae he cycle lengh y odel II for he grou of fve roducs shown n Tale, whch shows he annual sales requreens, sales er roducon day (50 days er year), daly roducon rae, roducon days requred, annual nvenory holdng cos, and seu coss. Tale 3 shows he calculaon of he nuer of runs er year calculaed y forula 5. The nu cos nuer of cycles whch resuls n hree er year, each cycle lasng aroxaely 78 days and roducng one-hrd of he sales requreens durng each run. The oal ncreenal cos go y forula 6 s C*$ Exale Wha s he effec on N* for Exale f shorage coss are G $0.0, G $0.0, G 3 $0.05, G 4 $0.04, and G 5 $0.70 er un er year? Wha s he oal ncreenal cos of hs soluon? Tale 4 shows he calculaon of he nuer of runs er year calculaed y forula 0. The nu cos nuer of cycles whch resuls s wo er year, each cycle lasng aroxaely 7 days and roducng a half of he sales requreens durng each run. The oal ncreenal cos go y forula s C*$ CONCLUSIONS The egh slar odels resened n hs aer are he EPR odels for several roducs. Alhough hsorcally, hese odels follow n he lne of aroaches on nvenory analyss, hey have found her greaes alcaon whn he FMS envronen. Models V, VI, VII and VIII are seldo used n racce. The ajor reason s he dffculy o oan an accurae esae of he shorage cos. The odels resened here are o ehasze soe of he any assuons ha can e ul no an EPR odel and

11 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case 37 o show how hese assuons can e ncororaed no he odel. Tale 5 suarzes he forulas for he egh odels. Model III s a secal case of odels I, II, and IV, wh,,,...,. I should e recognzed ha, n odel VII, G,,,...,, leads o odel III, where no shorages are allowed. Models V, VI, and VIII are he os general of all he egh odels resened. In fac, odels I, II, III, IV, and VII are all secal cases of odels V, VI, and VIII, whch allow shorages (fne shorage cos). In shor, odels I, II, IV, and VII each resens generalzaon of one assuon fro odel III, u odels V, VI, and VIII nclude oh generalzaons sulaneously. REFERENCES [] Buffa, E. S., Models for Producon and Oeraons Manageen, John Wley & Sons, New York, 963. [] Ilć, O., Lo sze odels whou shorages for a sngle roduc n MRP syses, Proceedngs of V SyOrg Conference, Vrnjačka Banja, 996, [3] Ilć, O., Lo sze odels wh shorages for a sngle sroduc n MRP syses, Proceedngs of XXIII SYM-OP-IS Conference, Zlaor, 996, [4] Ilć, O., Econoc roducon quany odels for a sngle roduc n MRP syses, n: P. Jovanovć and D. Perovć (eds.), Coneorary Trends n he Develoen of Manageen, FON, Belgrade, 007, (n Seran) [5] Ilć, O., and Radovć, M., Models of roducon runs whou shorages for he ulroduc case n FMS, Proceedngs of I SIE Conference, Belgrade, 996, [6] Ilć, O., and Radovć, M., Models of roducon runs wh shorages for he ulroduc case n FMS, Proceedngs of II SIE Conference, Belgrade, 998, [7] Radovć, M., and Ilć, O., Producon runs for several ars or roducs, Yugoslav Journal of Engneerng, 36 (3) (986) 0-7. (n Seran) [8] Rankey, P. G., Couer Inegraed Manufacurng, Prence Hall, New Jersey, 986. [9] Wess, H. J., and Gershon, M. E., Producon and Oeraons Manageen, Allyn and Bacon, Boson, 989.

12 38 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case APPENDIX Noenclaure B( ) Vecor of axu aoun shorages C Toal (annual) ncreenal cos (dollars/e) C* Toal ncreenal cos of an oal soluon d Daly deand rae of he h roduc (uns/day) d Daly ure consuon rae of he h roduc (uns/day) D Deand rae of he h roduc (uns/e) G Shorage cos of he h roduc (dollars/un-e), where e us ach deand H Holdng cos of he h roduc (dollars/un-e), where e us ach deand Nuer of roducs N Nuer of roducon runs er e (year) N* Nuer of roducon runs er e for an oal soluon Daly roducon rae of he h roduc (uns/day) q Producon run quany of he h roduc (uns/run) S Seu or fxed cos of he h roduc (dollars/run) Lengh of he cycle Te of ure consuon of he h roduc c Te of roducng of he h roduc Te of roducng whle here s a shorage suaon of he h roduc Te of roducng, whle here s nvenory on hand of he h roduc 3 Te of ure consuon whle here s nvenory on hand of he h roduc 4 Te of ure consuon whle here s a shorage suaon of he h roduc T y Toal e erod (nuer of workng days er year) Maxu nvenory level of he h roduc Tale : Assuons and odels Shorages Deand delvery Relenshen rae durng he roducon Gradual Insananeous erod No Yes Model I Model II No Model III Model IV Yes Yes Model V Model VI No Model VII Model VIII

13 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case 39 Tale : Sales, roducon, and cos daa for fve roducs o e run on he sae equen Produc Nuer D d Producon Days H S Requred ,000 0,000 5,000 5,000 4, $ $ Toal 35 $5 Tale 3: Deernaon of he nuer of runs, jonly, for fve roducs fro forula 5 Produc d d ) ( d ) HD HD ( d ) Nuer , ,5 Toal 4,6 N* 4,6 3 cycles er year x5 Tale 4: Deernaon of he nuer of runs, jonly, for fve roducs fro forula 0 Produc G HD ( d ) G HD ( d ) Nuer H + G H + G H + G Toal,853 N*, 853 cycles er year x5

14 30 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case

15 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case 3 q Sloe - d y Sloe d 0 c Te Fgure : Invenory as a funcon of e-sawooh curve, odel I q y Sloe d 0 c Te Fgure : Sawooh curve, odel II

16 3 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case q Sloe Sloe d' 0 c Te Fgure 3: Sawooh curve, odel III q Sloe d' 0 c Te Fgure 4: Sawooh curve, odel IV

17 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case 33 q y c Te Fgure 5: Sawooh curve, odel V q y c Te Fgure 6: Sawooh curve, odel VI

18 34 O. Ilć, and M. Radovć/ Models of Producon Runs for he Mulroduc Case q q c Te Fgure 7: Sawooh curve, odel VII q q c Te Fgure 8: Sawooh curve, odel VIII