The Modification of EOQ Model under the Spare Parts Discrete Demand: A Case Study of Slow Moving Items

Transcription

1 Proceedigs of the World Cogress o Egieerig ad Coputer Sciece 11 Vol II WCECS 11, October 19-1, 11, Sa Fracisco, USA The Modificatio of EOQ Model uder the Spare Parts Discrete Dead: A Case Study of Slow Movig Ites Sako Wogogkolrit, ad Bordi Rassaeethes Abstract Geerally, EOQ ivetory odel is well kow as a approach usig for ivetory cotrol ad spare parts stockig policy. However, EOQ odel is basically thought o the basis of cotiuous dead. But for the discrete dead, EOQ odel ay ot perfor. This research study will be used for explaatio about the costraits of EOQ odel with discrete dead or slow ovig ites. Accordig to this study, the odificatio of EOQ odel is origially studied by based o spare parts discrete dead. This is the study of forig the extesio of EOQ odel coforig to discrete dead. I additio, the odificatio of EOQ odel will be proved i accordig to test agaist real equipet. Hopefully, this study will be used for fulfillet a little iche of EOQ odel. Ad it will be the ew directio ad useful for all aufacturers exploitig this odel as a ivetory aageet policy. Idex Ters discrete dead, spare parts, slow ovig ites, EOQ odel, ivetory cost I. INTRODUCTION The order size that iiizes the total ivetory cost is kow as the Ecooic Order Quatity (EOQ). The classical ivetory odel assues the idealized situatio show as Fig.1, where Q is a order size. Upo receipt of a order, the ivetory level is Q uits. Uits are withdraw fro ivetory at a costat dead rate, which is represeted by the egative slopig lies. Whe the ivetory reaches the reorder poit (), a ew order is placed for Q uits. After a fixed tie period, the order is received all at oce ad placed ito ivetory. The vertical lies idicate the receipt of a lot ito ivetory. The ew lot is received just as the ivetory level reaches zero, so the average is Q/ uits [1]. Iteratioal Graduate Progra Idustrial Egieerig, Faculty of Egieerig, ad Departet of Operatios Maageet, Faculty of Busiess Adiistratio, Kasetsart Uiversity, Bagkhe 19, Bagkok, Thailad. Sako Wogogkolrit s cotact eail: [email protected] ISBN: ISSN: (Prit); ISSN: (Olie) T Fig. 1. Classical ivetory odel cocept. Source: adapted fro [1]. Q T Accordig to Fig.1, this is the classical ivetory odel cocept. The optial lot size Q * is cocered i order to iiize the total ivetory cost. To obtai the iiu total cost, the lot size Q * is equal to: Q * SD (1) CI Where: S is orderig cost per order. D is aual dead i uits. C is part uit cost. I is aual holdig cost as a fractio of uit cost. Ideed, the lot sizig techiques is developed for cotiuous ad idepedet dead ites such EOQ assue that dead occurs with certaity as a costat rate, whilst discrete dead occurs at discrete itervals or poits i tie rather tha cotiuously over a tie horizo. Dead requireets are usually tie-phased i equal tie icreets over a fiite tie horizo. Ueve or lupy dead requireets occurrig over a fiite tie horizo coplicate the lot sizig decisio such as the utilizatio of spare parts cosuptios or spare parts dead [1]. II. REEVANT ITERATURES AND REATE WORKS I area of ivetory cotrol study, several works ad case studies i literatures o the ivetory aageet decisios were applied EOQ odel such as [-6]. Alost of previous studies of EOQ odel is possessed by cotiuous dead. Whilst, the studies of discrete dead are alost cofored to MRP approach such as ot-for-ot orderig [7], Wager- Withi algorith [8], east Period Cost odel [9], east Uit Cost odel [1], Silver-Meal algorith [11], etc. I geeral, EOQ odel is developed for cotiuous dead ad it is ever thought by based o etire discrete dead. However, a lot of previous studies of aterials T WCECS 11

2 Proceedigs of the World Cogress o Egieerig ad Coputer Sciece 11 Vol II WCECS 11, October 19-1, 11, Sa Fracisco, USA cotrol for aiteace actios are still applied the EOQ odel by the assuptio that spare parts dead is cotiuous patter such as [1-15]. Because of EOQ odel is siple ad easy to uderstad the it is still popular to use tha the other approaches which are the coplicated odel with difficulty usig or hard to ipleet. Actually, alost of spare parts cosuptios look quite siilar to discrete patter rather tha cotiuous patter. Accordig to this study, the odificatio for EOQ odel is origially thought by based o the etire discrete dead. A case study will be reflected to real equipet. The followigs details are the explaatio about these atters. Additioally, the idepedet dead ites will be also cosidered i order to eet it variatios. The, this is our believig that is a cogitive thikig to establish the origial cotext to be sustaied the extesio of EOQ odel. III. OCCURRENCE OF SPARE PART DEMAND Firstly, the cogitive thikig about spare parts dead should be cosidered i ter of equipet failure rate ad ea tie betwee failures. A. Failure occurreces Equipet (or part, device) failure will be happe o aytie, ad it is to be the probabilistic patter uder the exact stadard deviatio () with average tie of failure occurrece (or ea tie betwee failure: M TBF ). Chace of equipet failure per oe cycle tie t 1 t t 3 t 4 t 5 1 set ( uits) of 1 set ( uits) of keepig part(s) as spare keepig part(s) as spare High failure rate M TBF ow failure rate t t is represeted as probability of equipet failure Fig.. Occurreces of equipet failure M TBF Accordig to Fig., equipet (part) ay probably be failure o aytie. The failure patter is probabilistic. So, tie betwee failures at tie t (TBFt) is equal to average tie of failure occurreces plus failure variatio as: TBF t = M TBF (a) Where: is ultiplicatio uber of stadard deviatio. If tie betwee failures is fallig ito low failure rate period, the tie betwee failures is: TBFt = M TBF + (b) If tie betwee failures is fallig ito high failure rate period, the tie betwee failures is: TBFt = M TBF (c) If there are k idetical parts i the syste the ea tie betwee failure of the syste () is becoe to be: MTBF (3) k Ad, tie betwee failures of syste (TBF S ) is: TBF S = S (4) Where: S is failure stadard deviatio of the syste. B. Dead of spare parts Spare parts dead (D) coes fro equipet failure. It ca be easured as frequet ties of failure occurreces withi aual that is called as failure rate (), ad equal to: D = = 1/M TBF (5) If there are uits of equipet (or parts) which are siultaeously chaged withi sae tie such as chagig of batteries, or double drivig belt are siultaeously chaged. So, dead of spare parts is: D = (6a) If there are k idetical equipet istalled i the syste the (6a) ca be re-equated as: D = k = (6c) C. The discrete dead Basically, slow ovig ite (or discrete dead) ca be iplied as o-cotiuous patter. Or, parts (devices) are scarcely used. Ideed, the exact separatio betwee discrete ad cotiuous patter (of dead) is ot surely defied. But there are too ay text books, articles, case studies which were give the defiitios of discrete dead i ay differet types or several forats such as parts which are cosued less tha oe uit withi a aual [16-17], ites withdrawal less tha oe ite i a quarter [18], part has bee issued or sold at least oe withi last oe year [19], a ite where the ea tie betwee deads is uch loger tha te ties of average lead tie []. However, our poit of view is to go alog with the defiitio of []. Or, the equatio ca be show as: M TBF /k (or ) > 1. IV. INVENTORY MODE A. Geeral EOQ odel Accordig to Fig.1 ad (1), purchasig quatities (Q * ) of EOQ odel ca be rewritte i ter of failure rate as: Q * Sk (7) CI Where: is aual failure rate. is equipet with siultaeous chaged. k is aout of idetical equipet i a syste. The Re-Order Poit () is ordered at equipet (or part/device) delivery lead tie (). Ad, total ivetory cost coposes of spare part orderig cost ad holdig cost: TC EOQ = Orderig cost + Holdig cost ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 11

3 Proceedigs of the World Cogress o Egieerig ad Coputer Sciece 11 Vol II WCECS 11, October 19-1, 11, Sa Fracisco, USA * k Q TC EOQ S CI (8) * Q B. The Extesio of EOQ odel Basically, if spare parts dead is the discrete patter, the geeral EOQ odel ay ot perfor. Supposedly, there are 4 sets of circuit breaker istalled i the syste with M TBF of each breaker is years. So, aual dead (or failure rate) is. sets. Whilst, uit price of circuit breaker is 3, Baht, orderig cost is 1, Baht, ad fractio of holdig cost is 4 percet. Thus, purchasig quatity is equal to sets (if it is calculated by usig geeral EOQ odel). For this case, every te years will purchase sets of breaker (or oe set of circuit breaker have bee used for every five years). Recall to (8), total ivetory cost is: * k Q TC EOQ S CI * Q = 1,+1, =, Baht per year Or: TC EOQ = 44, Baht per years (or M TBF ) 5 1 Year DWT TC to : Thus: dtc d dtc ad set it as zero, the: d Sk CI Sk C IMTBF MTBF S (1) C I Ad: p = j* (11) Where: p is actual purchasig quatity. j is the roudig uber of, Or: j is actual purchasig lot-size (of uits) Accordig to (9), orderig cost is, Baht, holdig cost is 36, Baht, ad total ivetory cost (to have circuit breaker as spare part) is 56, Baht. C. The odificatio for the extesio of EOQ odel Practically, the exact tie of equipet failure caot be surely kow. The, ew part(s) will be replaced whe the existig part(s) is breakdow. Ad the average purchasig tie is still equal to ea tie betwee failures. Safety stock Stock-out See sec. A See sec. B Critical part No- Critical part 1 15 Fig. 3. Cosideratio for Holdig cost Year Accordig to Fig.3, each gray block is represeted by parts stockig (or holdig cost) which is equal to aterial or spare part uit cost (C) ultiplyig by holdig cost fractio (I). Thus, each block is equal to 3,*.4 = 1, Baht. There are totally 3 blocks, so total holdig cost is 36, Baht per years (M TBF ). Durig years, circuit breakers are equally purchased by ties (year, ad the ed of year 1). The, orderig cost is equal to *1, =, Baht (for years). Cosequetly, total ivetory cost is 56, Baht. This value is ot as sae as the previous. If is defied as the optial purchasig quatity. So, the extesio EOQ odel ca be show as the followigs: Total Cost = Orderig Cost + Holdig Cost TC k S k S CI i1 CI M i k 1 S CI TBF M TBF 1 MTBF (9) Sectio A uits Fig. 4. Cosideratio for Equipet failure uits Regardig to EOQ odel, spare parts ca be separated ito groups as critical part ad o-critical part. The critical part is a iportat part i syste, ad should have the safety stock for esurig that part(s) ca be i had at all. Otherwise, o-critical part is a uiportat part i syste, ad parts ca allow beig shortage (o safety stock). Accordig to Fig.4, ew order (repleishet) ca be delayed util spare part(s) is epty (for o-critical part) ad ew order will be purchased at period after spare part(s) stocked-out. Or, ew order will be purchased whe spare parts are touched safety stock (for critical part) or period after safety stock is begu to use. By this way, total holdig cost will be reduced at all. Sectio B Part(s) holdig ow failure rate Noral failure rate (Average) Part(s) shortage High failure rate ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 11

4 Proceedigs of the World Cogress o Egieerig ad Coputer Sciece 11 Vol II WCECS 11, October 19-1, 11, Sa Fracisco, USA Recall to (9), if it is cosidered by applyig Fig.(4), so this equatio ca be rewritte as: TC k 1 S C I EOQ MTBF (1) Note: The explaatios ad exaple case study for (1) ad (1) are show i appedix 1. DWT TC EOQ to ad set it as zero. dtceoq k CI The: S MTBF d Sk C I MTBF S C I Refer to (1), this equatio will be true whe spare part lead tie is very short (or assued to be zero) if copare with syste ea tie betwee failure, (Or << ). Ad, purchasig lots (or ) ust be over tha oe ( > 1). So, lot size of purchasig is sae as previous but total cost is ot sae. Particularly, ter of holdig cost is always reduced, because part ca be allowed beig shortage (for o-critical part), or safety stock is firstly used (for critical part). The purchasig quatity is referred as (1). Safety stock is issued by based o failure variatio, ad it ust cofor to discrete patter the Poisso distributio is applied. Hece, the variatio of equipet failure o lead tie is: Px( t ) x S s x! e (13) For this case, safety stock (SS) is always based o failure rate durig lead tie which is equal to parts usage durig lead tie plus its variatio that is equal to: SS = (1+ Px(t = )) (14) Where: SS is safety stock. Px is probability of occurreces. S is syste failure rate. x is probability of occurreces. is equipet delivery lead tie. Note: The exaple equipet ad uerical result is show i ext sectio. Whilst, Re-Order Poit () for this case is equal to safety stock, because this cocept is thought by based o parts borrowig fro safety stock. Ad after repleishet, parts will be refilled back to the safety stock. These two cotributios to kowledge ca be illustrated as the followigs: Max. evel Mi. evel Safety Stock Stock-out area Fig. 5. Extesio of EOQ odel for critical part Average Actual Max. j j evel Mi. = j Stock-out area Fig. 6. Extesio EOQ odel for No-Critical part p p Average Actual j Regardig to Fig.5 ad Fig.6, there are two cases for cosideratios. The first case is ivetory aageet for critical part ad the secod is ivetory aageet for o-critical part. For the critical part, safety stock is applied as it buffer, while o-critical part is ot (or parts ca be shortage). Both of the are still applied with cocept of Extesio EOQ odel (which is show i previous sectio). VI. NUMERICA RESUT The case study is spare parts for autoatic baggage sortig achie (which is called as Tilt Tray Sorter) of the Baggage Hadlig Syste i Suvarabhui Iteratioal Airport, Bagkok, Thailad. Three equipet are the exaples as Auxiliary Switch 1o/1c (for circuit breaker), Battery (for PC), ad uder voltage coil (3VAC). The autoatic baggage sortig achie coposes of 4 closed loops of Tilt Tray Sorter. Whole syste coposes of 1 pieces of auxiliary switch, 4 sets of PC (each PC eeds 8 cells of battery for backup), ad 8 sets of uder voltage coil. The equipet details with variables are show i table 1. (j-1) (j-1) V. APPICATION Ideed, the ost iportat useful of this study is How ca this cocept be applied to spare parts ivetory aageet?, ad this cocept ca be used for reducig the coplicated thikig about previous studies. Therefore, the cotributios to kowledge of this study are: 1. The Re-Order Poit ad Purchasig Quatity. The Safety Stock cocept. ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 11

5 Proceedigs of the World Cogress o Egieerig ad Coputer Sciece 11 Vol II WCECS 11, October 19-1, 11, Sa Fracisco, USA Variable TABE I Equipet details ad Variables Auxiliary cotact Battery (for PC) Uder voltage Uit price (C) Baht ,5 Aout of parts i the syste Nuber of idetical parts (k) Usage per each chagig () uit 1 eachs 3 cells 8 sets uit uit 3 8 ead tie () day 3 8 Mea tie betwee failure (MTBF) Syste ea tie betwee failure () Uit Equipet year days Orderig Cost (S) Baht 1,5 5, Fractio of % of 3% 15% 1% Holdig Cost (I) uit cost Note: Currecy is 3 Thai Baht per 1 US$ (o Jue, 11). TABE II The safety stock for each equipet Equipet Auxiliary cotact 1o+1c Battery (for PC) Uder voltage coil (3 VAC) Syste failure rate No.of Failure occureces durig lead tie Probability of occureces Cuulative or Service level Safety stock (ea) Regardig to table, safety stock is calculated by usig (14). For this case, if the required service level is supposedly equal to 1 percet the the gray stripe is represeted the safety stock for each equipet. Result Equipet Auxiliary cotact 1o+1c Battery (for PC) Uder voltage coil Optial ot Size ( ) TABE III Suary of cotrol Actual ot Size (j) Actual Purchasig (p ) Safety Stock (SS ) Re-Order Poit ( ) whe parts thouch SS whe parts thouch SS whe parts thouch SS Table 3 is represeted the ivetory aageet for each equipet. All parts are supposedly to be the critical part the safety stock is issued i order to eet failure variatio durig lead tie (see table ). APPENDIX Firstly, the thikig about the lowest total ivetory cost for geeral EOQ odel ad extesio EOQ odel will be issued i ter of the copariso betwee both odels. The exaple equipet is the uder voltage coil which ca be show as: Cost 35, 3, 5,, 15, 1, 5, Orderig Cost HC (Ge. EOQ) HC (Ext. EOQ) TC (Ge.EOQ) TC (Ext. EOQ) , No-practical ( < 1) -1, (or j) Fig.7. Plottig of Orderig cost, Holdig cost, ad Total ivetory cost (for uder voltage coil equipet) agaist lots for EOQ odel. Accordig to Fig.7, this is the plottig of uder voltage coil. The orderig cost, holdig cost, ad total cost are plotted agaist lots (or j). Full gray lie is represeted the plottig of holdig cost, ad dot gray lie is represeted the plottig of total cost for geeral EOQ odel. Whilst, full black lie is represeted the plottig of holdig cost, ad log dot black lie is represeted the plottig of total cost for extesio EOQ odel. Ad short dot black lie is represeted the plottig of orderig cost (It is the sae value for both of geeral EOQ odel ad extesio EOQ odel). The lowest total cost of geeral EOQ odel is sae poit as extesio EOQ odel. At the lowest total cost for ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 11

6 Proceedigs of the World Cogress o Egieerig ad Coputer Sciece 11 Vol II WCECS 11, October 19-1, 11, Sa Fracisco, USA extesio EOQ odel, the holdig cost is ot equal to orderig cost. ot size () Fig. 8. Siulatio of ot sizig agaist Material uit cost. Fig.8 is used for cofiratio about the aterial uit cost which is ore tha the break-eve poit, thus it will let the optial lot size () to be less tha oe. Total Cost 1, 8, 6, 4,, Material uit cost (C) Fig.9. Siulatio of Total cost agaist Material uit cost. Fig.9 is used for explaatio about the curve of total cost is turig back (or diiishig retur curve) after passed the break-eve poit aterial (spare part) uit cost is ore tha C * (for this case is equal to 8, Baht). C * No-practical area ( < 1), 4, 6, 8, 1, 1, 14, C * Material uit cost (C) ACKNOWEDGMENT This research study is supported by Iteratioal Graduate Progra Idustrial Egieerig (IGPIE), Kasetsart Uiversity, Bagkhe Capus, 19, Bagkok, Thailad. REFERENCES [1] R.J. Tersie, Priciple of ad Materials Maageet, 4th ed., PTR Pretice Hall, Eglewood Cliff, NJ, Ch.4, pp.177-3, [] A.A. Kraeburg. 6. cotrol uder syste availability costraits. A Ph.D dissertatio, Techische Uiversiteit, Eidhove, Netherlad. [3] Z. We-Yog, X. Yig, ad S. Big, Study o pare parts ivetory cotrol by quatitative aalysis i the eviroet of erp syste, Proc. of the 11 It. Cof. of Busiess Maageet ad Electroic Iforatio (BMEI), pp , 11. [4] A. K. Pal ad B. Madal, A EOQ odel for deterioratig ivetory with alteratig dead rates, Joural of Applied Matheatics ad Coputig, Vol. 4(), pp , [5] R. Begu, A EOQ Model for Deterioratig ites with Weibull distributio deterioratio, uit productio cost with quadratic dead, J. Applied Matheatical Sci., Vol.4(6), pp.71 88, 1 [6] E.Porras, ad R. Dekker, A ivetory cotrol syste for spare parts at a refiery: A epirical copariso of differet re-order poit ethods, Eur. J. Oper. Res., doi:1.116/j.ejor , 7. [7] M. Oar, ad S.S. Supadi, A ot-for-ot Model with Multiple Istalets for a Productio Syste uder Tie-Varyig Dead Process, Joural Mateatika, 3, Jilid 19, Bil., pp [8] S.J. Sadjadi, M.B.Gh. Aryaezhad, ad H.A. Sadeghi, A Iproved WAGNER-WHITIN Algorith, Iteratioal Joural of Idustrial Egieerig & Productio Research, pp , 9. [9] J.C. Ho, Y.. Chag, ad A.O. Solis, Two odificatios of the least cost per period heuristic for dyaic lot sizig, Joual of Operatio Research Society, vol. 57(8), pp , 6. [1] D.ouit, R. Pascual, D.Bajevic, ad A.K.S. Jardie, Dyaic lot sizig with product returs ad reaufacturig, It. J. of Productio Research, Vol.44(), pp (4), 6. [11] M. Oar, ad M. M. Deris, The Silver-Meal Heuristic Method for Deteriistic Tie-Varyig Dead, Joural of Mateatika, Vol.17(1), pp. 7 14, 1. [1] W. Cheagkul, K. Darograt, ad D. Muaga, Reliability based aiteace aageet. SE-Educatio PC, Bagkok, Thailad, ch.9, pp , 1. [13] S. Suppaogkol, Spare parts optiizatio. SE-Educatio Public Copay iited, Thailad, ch.1, pp , 4. [14] W. Cheagkul, ad K. Darograt, Maiteace the Profit Maker. SE-Educatio PC, Bsgko, Thailad, ch.5, pp.8-9, 3. [15] M. Sakaguchi, ad Ma. Kodaa, Sesitivity aalysis of a ecooic order quatity for dyaic ivetory odels with discrete dead, It. J. of Maufacturig Techology ad Maageet, Vol. 18(4), pp , 9. [16] S. Siasiriwattaa, W. Cheagkul, ad K. Darograt, Efficiecy of Maiteace. SE-Educatio PC,Thailad, ch.16, pp , 6. [17].Pitelo,.Gelders, ad F.V.Puyvelde. Maiteace Maageet. Acco euve/aersfoort Belgiu, ch.8, pp , [18] P. Gopalakrisha, Hadbook of Materials Maageet. Pretice- Hall of Idia Private iited,new Delhi, Ch.9, pp , 5. [19] P. Gopalakrisha, ad A.K. Baerji, Maiteac ad Spare parts aageet. Pretice-Hall of Idia Private iited, New Delhi, Ch.3, pp , 6. [] N.A.J Hastigs, Physical Asset Maageet. Spriger-Verlag odo iited. New York, Ch., pp.95-31, 9. ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 11