Using the Econometric Models in Planning the Service of Several Machines at Random Time Intervals. Authors:

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1 Usng the Econometrc Models n Plnnng the Servce of Severl Mchnes t Rndom Tme Intervls. Authors: ) Ion Constntn Dm, Unversty Vlh of Trgovste, Romn ) Mrce Udrescu, Unversty Artfex of Buchrest, Romn

2 Interntonl Journl of Mngement Scences nd Busness Reserch Volume, Issue 0-0 ISS (6-835) Abstrct In ndustrl prctce, there re frequent cses when servng the mchnes by performer s done t rndom (concdentl) tme ntervls, whch prevents the elements of crryng out the processng procedure to be predetermned. Stutons of rndom nterventons on the mult-served mchnes occur more often n textle ndustry on the spnnng mchnes, looms, knttng mchnes, wre wevng nd nterlcng mchnes, etc. Key words: the dscplne of wtng, the verge number of mchnes, verge wtng tme of the mchnes n the system, the verge number of the mchnes n the system, Posson lw, the test X. Clssfcton: C50, D0, L3. ITRODUCTIO In order to mthemtclly shpe the mult-servng of severl mchnes by one performer t rndom tme ntervls (Askn, R., & Stndrdge, C., 993), t s requred to know the mn elements feturng tht process: - dvents (rrvls) represent the requests for servng the mchnes by performer; t s stted by the verge number of mchnes requrng servng n the tme unt. The process of servng severl mchnes by performer n ths cse s of rndom feture, the number of requests beng dfferent from ntervls to ntervls. The verge number of requests for the tme unt chosen from the relton: n fn n fn or () f n f nj () where: n - o ny ctegory of requests; fnj- frequency of requests n ctegory (the number of cses n tht ctegory - n ); f nj - the sum of ll frequences of requests representng the number () of tme ntervls when the occurrence of requests ws notced n - the number of unts requred for ctegory. fnj - the servce represents the cton by mens of whch one or severl performers: resolve the servngs nd s stted by the verge number of servces provded on the set tme unt; t s noted ; wth - servce fctor s determned from the rto between the number of unts served verge number of servces. (3) nd If, t mens tht the performer s overloded, nd the number of mchnes wtng shll contnuously ncrese. If, the performer s servng cpcty s good (Chse, R.B., Aqulno,.J., & Jcobs, F.R., 004), but due to the rndom frequency of requrements for servng the mchnes n the system, they shll hve certn wtng tme untl they shll be served; - the dscplne of wtng defned the order n whch servng the mchnes s met, nd nmely: on frst come frst served bss. For studyng the mult-servng condtons, t s necessry to lso stte the followng ndctors: the verge degree of performer s dsenggement T e ; the verge number of mchnes f from the wtng tme of servng wthn the tme ntervl tken nto ccount (L, S.Z., 995); verge wtng tme of the mchnes n the system T s the verge tme when mchne s wtng to be served; the verge number of the mchnes n the system Pge 67 f s shows the verge

3 number of mchnes tht re wthn the tme ntervl stted n the wtng strng nd t the performer, to be served. When orgnsng the ctvty of servng severl mchnes by performer, t s pursued to smultneously meet two requrements; the wtng tmes of the mchnes to be mnmum nd performer s occuptonl verge degree to be s hgh s possble.( Modrk, V., & Pndn, R.S., 0) In order to meet these requrements, t s requred the functon f m of the costs due to the tme wsted, hlts nd performer s wtng to be mnmsed, tht s: (4) fm Wh Te Ph f Tf where: W h - represents the productvty of the hourly verge work of the worker servng the mchne; P h - the vlue of the verge producton per hour cheved on the mchne unt served. The functon of optmston f m s summton of some losses, stted s vlue n le per hour. For the servngs to be performed bsed on the Posson lw, t s requred to meet the condton: 0 (5) where: - represents the mthemtcl expresson of the reltonshps exstng between the frequency of the ctegores of requests nd probbltes of ther occurrence dependng on certn number of freedom degrees nd certn requrement so the studed phenomenon would pproch theoretc Posson-type phenomenon, the theoretcl vlues for the functon hve been clculted by Person nd re represented n ttchment III. 0 - the prctcl vlue of the functon clculted bsed on the dt of the phenomenon studed re clculted by mens of the relton: fn P n 0 (6) P n where: P n - represents the theoretcl Posson probblty determned for from the prctcl dt nd for n servng ctegores.. AALYSIS OF MODELS USED Solvng the problem for the orgnston of mult-servng severl mchnes by performer t rndom tme ntervls s possble by mens of ptterns known n lterture s closed ptterns. It s ccepted tht performer serves homogeneous mchnes of sme type nd sze, whch operte ndependently.( Lee, Y.D., & Lee, T.E., 005) Untl certn moment, these mchnes operte wthout supervson or needng the worker s nterventon, but t tht moment, t s needed for the performer to ntervene. If performer ntervenes for commssonng stopped mchne from the group of the mchnes served, the durton of n nterventon beng rndom vrble, then negtve exponentl dstrbuton follows, ccordng to the prmeter. Schemtclly, the system of servng the mchnes by performer n tme s shown n fgure, wth the followng opertng mode: Advents Mchnes on the wtng strng Performer Wtng system Fgure. The dgrm of the servng system

4 Interntonl Journl of Mngement Scences nd Busness Reserch Volume, Issue 0-0 ISS (6-835) The sequence of the mchnes stoppng s lso the sequence of commssonng by the performer; t results the dscplne of the system s met: frst come frst served. In order to estblsh the equtons feturng the stte of the system, the probblty Pn must be stted, so tht for the moment there would be n mchnes n the system. (Goldberg, D.E., 989) By notng wth A the probblty tht mchne would operte wthout the performer s nterventon n tme ntervl, t s ccepted tht the need of performer s nterventon to the mchnes s ndependent, the probblty tht no mchne from the n ones opertng would not requre the performer s nterventon shll be: n A (7) The probblty tht t lest one of the n (mchnes tht re opertng) would need the performer s nterventon shll be gven by: n A n O O where: O - hs the property tht: lm 0 The probblty to complete performer s nterventon (menng to restrt the operton of the mchne served) n tme ntervl s gven by the relton: B 0 (9) The stutons tht t the moment t there would be n mchnes n the wtng system (Slck,., Chmbers, S., & Johnston, R., 00) re the followng: t the moment t there re n mchnes n the system nd n the tme ntervl t t t there re no mchnes to be served or mchnes the servng of whch hs been completed; t the moment t there re n- mchnes n the wtng system nd n the tme ntervl from t to t t s requred to serve mchne nd no mchne hs been commssoned wthn ths ntervl; t the moment t there re t mchnes n the wtng system nd n the tme ntervl from t to t t s not requred to ddtonlly serve nother mchne, but servng mchne tht s commssoned wthout supervson s determned; wthn the tme ntervl t t t, vrton of the number of mchnes greter thn one s recorded n the wtng system, whtever the number of mchnes my be n the wtng system. (Sntn, R., 003) The performer s nctvty verge shll be: n Pge 69 (8) T e pn p0 (0) 0 The verge number of mchnes opertng: p M g 0 The probblty tht mchne would be wtng to be served, tht s the performer s occuptonl probblty s: 0 n () p p () n p 0 In order to determne the verge wtng tme of the mchnes n cse of permnent opertng regme, the verge of mchnes hlts s no longer, but g. It results from here: f g, (3) nd the verge wtng tme of the mchne to be served from the relton: T p f 0 g The verge wtng tme n the system shll be: Ts Tf. (5) p0 The optml number of mchnes opertng n rndom regme, served by performer s (4)

5 determned bsed on n economc functon. 3. THE RESULT OF USIG THE MODELS A performer s servng pttern of severl mchnes t rndom tme ntervl requres gong through the followng stges: choosng the servng re tht s to be studed; performng observtons on the number of servngs on the tme unt chosen; pplyng the test (Dm, I.C., 00) n order to verfy whether the phenomenon s of Posson type; choosng the vrble dependng on the one tht s to optmse the servng of severl mchnes (the Are); optmston tself of the process of performer servng severl mchnes, by mnmsng the functon n relton to the number of mchnes served nd clculton of ddtonl ndctors of the optml vrble (Gen & Cheng, 997). 4. EXPERIMETAL RESULT In producton unt consstng n dentcl mchnes, mkng products nd homogeneous opertons, problem rses for determnng the optml number of mchnes served by performer. From the sttstcl mesures (tb. ) t ws found out tht the number of servngs dffers from one tme ntervl to nother, beng of rndom feture. It ws lso found out tht the servng tme of mchne s te =.5 mnutes/mchne served. Tble. Frequency of the servng ctegores Ctegores of servngs notced (n) Frequency of ctegores of servngs f (n) The followng dt re lso gven: W h 50 le/hour productvty of worker s work per hour; P h 400 le/hour productvty of producton cheved by mchne durng one hour of operton. Solvng ths problem requres gong through the followng stges: Stge. For the study, the re of mchnes tht re to be mult-served s chosen. Stge. Observtons re performed on the number of hourly servngs per mchne n the re of mchnes. The expermentl dt of mchne operton performed n 96 ntervls of one hour ech re gven n tble. It results from tble tht the number of servngs dffers from one ntervl to nother, hvng rndom feture. In order to smplfy the nlyss, t ws ordered by ctegores nd nmely: 3 servngs/hour were notced 8 tmes; 4 servngs/hour 3 tmes; 8 servngs per hour 6 tmes. By usng the dt n tble, the verge number of mchnes servngs per hour s clculted; the clculton s done by mens of tble. Tble. Clculton of the verge number of servngs per hour Ctegores of servng (n) Totl ctegores of servng = 6 Frequency of servngs n ctegory f(n) f n 96 umber of unts served on ctegory (n x f(n) n fn 508 The verge number of mchnes served n one hour s n fn 508 5, Stge 3. The ssumpton on the Posson dstrbuton of the occurrence of mchne servng (Dm, I.C., Mrcncn, I.., Grbr, J., Pchur, P., Kot, S., & Mn, M., 0) s verfed wth the

6 Interntonl Journl of Mngement Scences nd Busness Reserch Volume, Issue 0-0 ISS (6-835) X test. The Posson theoretcl probbltes (determned: by clculton or from the tbles of the Posson functon) for the 6 ctegores of servng re gven n tble 3. Tble 3. The Posson theoretcl probbltes for 5, 966 Ctegory of servngs (n) P(n) 3 P(3) = P(4) = P(5) = P(6) = P(7) = P(8) = The ssumpton on the Posson dstrbuton of the occurrence of mchne servng s verfed wth the X test, by mens of tble 4. The number of freedom degrees ( gl ) s determned by the relton g n. In the exmple we looked t n = 6(3, 4, 5, 5, 7, 8); therefore 6 4 freedom degrees. g n n Tble 4. Clculton of X 0 96 f n P Pn fn fn fn t X 0.78 The sserton 0, 05s lso chosen for g 4 freedom degrees, the vlue X 9, 49 s found n the tbles for the dstrbuton X. Becuse: X X0, (9,49>,78) t my be ccepted the ssumpton of dstrbutng the mchne for servng s bsed on Posson s lw wth 5, 966 servngs per hour. Stge 4. Servng s optmsed bsed on tht vrble whch s the number of mchnes served, whch shll be determned n vrous vrnts, the functon to optmse beng f m W T P f T f mnmum! e Stge 5. The ndctors of rndom servng of severl mchnes re clculted by performer n mny servng vrnts nd by mnmsng the functon f m, the optml servng vrnt s Pge 7

7 chosen. Vrnt m = 3 If = m s the number of mchnes served, then: 3 m 35,9 5,87mchnes re to be vergely served per hour. The servng tme of mchne s te =.5 mnutes on servng of one mchne, wthn n ntervl of 60 mnutes. The performer my vergely do: servngs per hour. te,5 The servng fctor or performer s occuptonl degree shll be: 3 5,87 3 0, The probblty tht ll 3 mchnes would operte t gven moment or the probblty of not be wtng s cheved from the relton: P0 P m n 3,5 03, m! n m n! 0.85 The verge number of mchnes t stndstll n the system t gven moment: g m P, 99 0 mchnes. The verge number of mchnes wtng to be served results from: P0 0, 484 f g mchnes The verge wtng tme of mchne n the strng durng one hour shll be: f T f 0,069 hours/hour. m g The verge wtng tme n the system results from: T s g m g 0,049 hours/hour. The performer s verge wtng tme n one hour s clculted thus: Te P 0 0, 85 hours/hour. The economc functon for the vrnt m=3 mchnes s clculted thus: fm3 W h T e Ph f T f 3 f m3 74, le/hour 5 5. COCLUSIOS The vlues of the servng vrnts hve been nlogclly clculted for m ( = 4, 5, 6), whch re summrsed n tble 5. Tble 5. Tble of servng nd decsonl vrnts Indctors Servng vrnts for m

8 Interntonl Journl of Mngement Scences nd Busness Reserch Volume, Issue 0-0 ISS (6-835) p e f T f T e T e f m The optml decson s m = 4 The clcultons were not performed nymore for servng 7 nd 8 mchnes, becuse the functon f m to mnmse from m = 6 represents shrp ncrese. (Dm I.C., Grbr I., Pchur P., Kot S., Modrk V., Mrcncn I.., Mn M., 0) It results from tble 5 tht the optml servng s for m = 4 mchnes. REFERECES Askn, R., & Stndrdge, C. Modelng nd nlyss of mnufcturng systems. ew York, Y: John Wley & Sons, Inc. (993) Chse, R.B., Aqulno,.J., & Jcobs, F.R. Opertons mngement for compettve dvntge (0th ed.). Boston, MA: McGrw-Hll (004). Dm, I.C. Usng multservce of ndustrl mngement. Czestochow, Polnd: Wydwnctw Wydzlu Zrzdzn Poltechnk Czestochowskej (00). Dm, I.C., Mrcncn, I.., Grbr, J., Pchur, P., Kot, S., & Mn, M. Opertonl mngement systems of the producton cheved n flexble mnufcturng cells. Presov, Slovk: Techncl Unversty of Kosce (0). Dm I.C., Grbr I., Pchur P., Kot S., Modrk V., Mrcncn I.., Mn M. Multservng opertonl mngement system of the producton cheved n flexble mnufcturng cells, WWZPCz, Czestochow (0). Goldberg, D.E. Genetc lgorthms n serch, optmzton, nd mchne lernng. Redng, MA: Addson-Wesley (989). Lee, Y.D., & Lee, T.E. Stochstc cyclc flow lnes wth blockng: Mrkovn models. OR- Spektrum, 7(4), (005). L, S.Z. Mrkov rndom feld modelng n computer vson. Sprnger-Verlg (995). Modrk, V., & Pndn, R.S. Opertons Mngement Reserch nd Cellulr Mnufcturng Systems. Hersey: IGI Globl (0). Sntn, R. A Mrkov network bsed fctorzed dstrbuton lgorthm for optmzton. Proceedng of the 4th Europen Conference on Mchne Lernng; Berln, Germny (003). Slck,., Chmbers, S., & Johnston, R. Opertons mngement. Hrlow, MA: Fnncl Tmes/Prentce Hll (00). Pge 73