The New Mathematical Models for Inventory Management under Uncertain Market

Transcription

1 Research Joural of Appled Sceces, Egeerg ad Techology 4(3): , 0 ISSN: Maxwell Scetfc Orgazato, 0 Submtted: March 03, 0 Accepted: March 4, 0 Publshed: December 0, 0 The New Mathematcal Models for Ivetory Maagemet uder Ucerta Market A. Mrzazadeh Departmet of Idustral Egeerg, Islamc Azad Uversty, Karaj Brach, Karaj, Ira Abstract: Ths paper presets the ew mathematcal model for determg the optmal orderg polcy for dustral ad commercal compaes. I the prevous research, the umerous vetory models uder flatoary codtos have bee developed. I these models, the demad rate, usually, has bee cosdered costat ad well kow, tme-varyg, stock depedet or prce-depedet. But, the demad rate, usually, s ucerta the real world. Therefore, ths study, the ew flatoary vetory models uder stochastc demad codtos have bee developed. The vetory system s the state of mult-tems wth budget costrat. The umercal examples have also bee gve to llustrate ad valdate the theoretcal results. Keywords: Budget costrat, flato, vetory systems, stochastc market demad INTRODUTION Oe of the most mportat parts of Supply ha Maagemet (SM) s vetory system maagemet whch s heretly o-determstc stuato. The may departmets of orgazato such as warehouse, marketg, sale, purchasg, facal, plag, producto, mateace ad etc. are relevace to the vetory problem. The problem of vetory systems uder flatoary codtos has receved atteto recet years. Sce 975 a seres of related papers appeared that cosdered the effects of flato o the vetory system. Before the 990s, the earler efforts have bee cosdered smple stuatos. Buzacott (975) made a Ecoomc Order Quatty (EOQ) model wth flato subject to dfferet types of prcg polces. Msra (979) developed a dscouted-cost model ad cluded teral (compay) ad exteral (geeral ecoomy) flato rates for varous costs assocated wth a vetory system. Ivetored goods ca be broadly classfed to four meta-categores. Frst, obsolescece whch refers to tems those lose ther value through tme due to rapd chages of techology or the troducto of a ew product by a compettor. Secod, deterorato tems refer to the damage, spolage, dryess, vaporzato, etc. of the products. Products such as vegetables, fsh, medce, blood, gasole ad radoactve chemcals have a fte shelf lfe, ad start to deterorate oce they are produced. Thrd, amelorato refers to tems whose value or utlty or quatty crease wth tme. Fourth, tems wth o obsolescece, o deterorato ad o amelorato. If the rate of obsolescece, deterorato or amelorato s ot suffcetly low, ts mpact o modelg of such a vetory system caot be gored. There are a few papers for obsolescg ad ameloratg tems. Moo et al. (005) cosdered the ameloratg/deteroratg tems o a vetory model wth a tme-varyg demad patter. Aother research for ameloratg tems has bee doe by Saa (00). The o obsolescg, deteroratg ad ameloratg tems have bee cosdered some researches o the flatoary vetory system. Sarker ad Pa (994) surveyed the effects of flato ad the tme value of moey o order quatty wth fte repleshmet rate. Some efforts were exteded the prevous works to cosder more complex ad realstc assumpto, such as Uthayakumar ad Geetha (009), Maty (00), Vrat ad Padmaabha (990), Datta ad Pal (99), Harga (995), Harga ad Be-Daya (996) ad hug (003). The deteroratg vetory systems have bee studed cosderably the recet years. For example, hug ad Tsa (00) preseted a vetory model for deteroratg tems wth the demad of lear tred cosderg the tme-value of moey. Wee ad Law (00) derved a deteroratg vetory model uder flatoary codtos whe the demad rate s a lear decreasg fucto of the sellg prce. he ad L (00) dscussed a vetory model for deteroratg tems wth a ormally dstrbuted shelf lfe, cotuous tme-varyg demad, ad shortages uder a flatoary ad tme dscoutg evromet. Yag (004, 006) dscussed the two-warehouse vetory problem for deteroratg tems wth a costat demad rate ad shortages. hag (004) establshed a deteroratg EOQ model whe the suppler offers a permssble delay to the purchaser f the order quatty s greater tha or equal to a predetermed quatty. Mat et al. (006) proposed a vetory model wth stock-depedet demad rate ad two storage facltes uder flato ad tme value of moey. Lo et al. (007) developed a tegrated producto-vetory model wth assumptos of varyg rate of deterorato, partal backorderg, flato, mperfect producto processes 5034

2 Res. J. Appl. Sc. Eg. Techol., 4(3): , 0 ad multple delveres. A Two storage vetory problem wth dyamc demad ad terval valued lead-tme over a fte tme horzo uder flato ad tme-value of moey cosdered by Dey et al. (008). Other efforts o flatoary vetory systems for deteroratg tems have bee made by Hseh ad Dye (00), Su et al. (996), he (998), Wee ad Law (999), Sarker et al. (000), Yag et al. (00, 00), Lao ad che (003), Balkh (004a, 004b), Hou ad L (004), Hou (006), Jagg et al. (006), her et al. (008), Sarkar ad Moo (0) ad Khara et al. (0a, b). The metoed papers have cosdered a costat ad well-kow flato rate over the tme horzo. Yet, flato eters the vetory pcture oly because t may have a mpact o the future vetory costs, ad the future rate of flato s heretly ucerta ad ustable. But, there are a few works the flatoary vetory researches uder stochastc codtos, especally wth multple stochastc parameters. Mrzazadeh ad Sarfaraz (997) preseted multple-tems vetory system wth a budget costrat ad the uform dstrbuto fucto for the exteral flato rate ad Horowtz (000) dscussed a smple EOQ model wth a Normal dstrbuto for the flato rate ad the frm s cost of captal. He showed the mportace of takg to accout the flato rate ad tme dscoutg, especally whe the former s relatvely hgh or whe there s cosderable ucertaty as to ether the flato rate or the margal cost of captal. Mrzazadeh (007) compared the average aual cost ad the dscouted cost methods the vetory system's modelg wth cosderg stochastc flato. The results show that there s a eglgble dfferece betwee two procedures for wde rage values of the parameters. Furthermore, Mrzazadeh (008) aother work, proposed a vetory model uder tme-varyg flatoary codtos for deteroratg tems. Mrzazadeh (009) developed A Partal Backloggg Mathematcal Model uder Varable Iflato ad Demad. I aother study, Mrzazadeh (0) prepared a optmal producto model for a vetory cotrol system where the tme horzo.e., perod of busess, s radom ature. I the lterature, the demad rate usually has bee cosdered costat ad well kow, tme-varyg, stock depedet or prce-depedet Furthermore, some practcal stuatos, the demad rate may be ucerta. Therefore, ths paper the o-determstc vetory model wth flato uder stochastc demad has bee developed. Also, the umercal example has bee provded to llustrate ad valdate the theoretcal results. METHODOLOGY The basc assumptos ad otatos: The followg assumptos have bee cosdered the developed model: The vetory system costs are kow at the begg of the tme horzo ad wll be creasg through the flato rates. The demad rates are stochastc. A multple tems vetory system has bee cosdered. The avalable budget s costraed ad wll be creased through the flato rate. Repleshmet s stataeous,.e., the repleshmet rate s fte. The followg otatos are used the model: : The umber of tems Q : The order quatty for -th tem D : The aual demad rate for -th tem S : The orderg cost for -th tem at the begg of the tme horzo : The purchasg cost per ut for -th tem at tme zero h k : The teral (for k = ) ad the exteral (for k = ) holdg cost per ut per ut tme for -th tem at tme zero f k : The Iteral Iflato Rate (IIR) for k = ad the Exteral Iflato Rate (EIR) for k = R k : The dscout rate et of flato: R k = r- f k where r s the dscout rate B : The maxmum avalable budget at the begg of the tme horzo E [EUA ] : The total expected aual costs for -th tem, where: =,,..., The models developmet: The may studes of the vetory maagemet systems have bee reported. Aalyss of the vetory systems the lterature s carred out usg two procedures. Frst method, determe the optmal values of the cotrol varables by mmzg the average aual cost ad the alteratve (ad theory more correct) procedure determe the optmal orderg polcy by mmzg the dscouted value of all future costs. As stated prevously, Mrzazadeh (007) showed by detaled computatos that there s a eglgble dfferece betwee two procedures for wde rage values of the parameters. The average aual cost method wll be used ths paper. The total aual vetory systems costs clude purchasg, orderg ad carryg costs. The aual purchasg cost for the -th tem s equal to: [+ (! Q /D )f /] D () also, the aual orderg cost for the -th tem s as follows: 5035

3 Res. J. Appl. Sc. Eg. Techol., 4(3): , 0 [+ (! Q /D )f /](S D / Q ) () Fally, the aual carryg cost for the -th tem after calculatg wll be: k Q / Dfk / hkrq / (3) Therefore, the total aual vetory system costs for the -th tem are: / / / Q / DfkhkrQ / k Q / D f / D EUA Q D f S D Q (4) Budget costrat: The avalable budget at tme zero s B, whch wll be creasg by EIR. Also, the ut prce,, creases by EIR. Therefore, the budget costrat at the tme t s: Qe Be ft ft By smplfyg Eq. (5) we have: (5), f D d d Id d (8) Objectve s to mmzato of the total expected aual costs: where: MZ E EUA EEUA EUA( D) dd d d f (9) (0) By substtutg Eq. (4) (0) ad the substtutg Eq. (9) ad smplfyg the terms we have: rq f h fhl( d / d ) 4( d d) Qh r f/ hr f / f MZ f / d d f S f/ d d Q () Therefore, the costraed multple tem vetory model ca be stated as follow: Q B (6) M Z Q B () The demad rates Eq. ()-(4) are stochastc. The expected value method ca be used ths drecto. Therefore, the optmal orderg quattes may be calculated usg ths model: MZ E EUA ` st..: (7) Q B The expermetal results reveal that three probablty desty fuctos (pdf) are sutable for the demad rate the vetory systems: Uform Normal Expoetal These cases wll be explaed as follows. ase (I): Stochastc demad rate wth uform dstrbuto fucto: Let demad rate has a Uform pdf as follows: The problem has bee solved wth usg the Lagraga method: LQ (, ) Z Q B (3) By takg the frst devato of the above fucto respect to Q I ad l, set them equal to zero ad smplfcato we have: S f/ d d / / Q rh h f Q B For =,..., (4) The Q, for =,,, wll be obtaed by calculatg the above equatos. ase (II): Stochastc demad rate wth ormal dstrbuto fucto: Now assume the demad rate of - th tem has a Normal dstrbuto wth the mea of m ad 5036

4 Res. J. Appl. Sc. Eg. Techol., 4(3): , 0 the stadard devato of F. The Eq. (4), may be rewrtg as follows wth cosderg rf. rf.0 : S f / EUA ( f / ) Q fs rq( h h) Qf D Therefore, the total expected aual costs are: MZ E EUA (5) (6) So, the mathematcal model ca be explaed as follows: S ( f / ) MZ ( f / ) Q fs rq( h h) Qf st..: Q B (7) Smlarly case (), the Lagrage-multpler ca be used. Therefore, the optmal orderg quattes wll be obtaed wth usg these equatos: S( f / ) Q rh h f for,..., ( / ( / ) Q B (8) ase (III): Stochastc demad rate wth expoetal dstrbuto fucto: The demad rate of -th tem has the expoetal pdf wth the mea of /q. Smlarly, the prevous cases, the optmal orderg quattes ca be calculated wth usg: S( f / ) Q rh h f for,..., ( / ( / ) Q B (9) The umercal example: The followg umercal example s provded to llustrate the theoretcal results. Let = S = $00 per order S = $00 per order = $30 per ut = $0 per ut h = $0.5 per ut per year h = $0.5 per ut per year h = $.5 per ut per year h = $0.75 per ut per year Table : The umercal example ase Parameters Q * Q * 8* I d = d = 000 d = 4000 d = 8000 II µ = F = 00 µ = 5000 F = 50 III = / = /9000 B = $0000 r = 0% per ut cost per year f = 8% f = %. The models have bee solved wth cosderg the above metoed values ad the optmal values of Q, Q ad l have bee obtaed (Table ). Summary: Sce 975 a seres of related papers appeared that cosdered the effects of flato o the vetory system. I the prevous research, the demad rate has bee cosdered ostat ad well kow, Tme-varyg, Stock depedet, Prce-depedet. Furthermore, the demad rate has a o-determstc stuato the real world. The sgfcat ad uque fdgs of ths research comparso wth the prevous work, s developg the ew flatoary vetory models wth assumg stochastc demad rate. The multple tems have bee cosdered the system ad the repleshmet s stataeous,.e., the repleshmet rate s fte. The avalable budget s costraed ad wll be creased through the flato rate. The umercal example has bee provded to llustrate the theoretcal results. REFERENES Balkh, Z.T., 004a. O the optmalty of vetory models wth deteroratg tems for demad ad ohad vetory depedet producto rate. IMA J. Maage. Math., 5: Balkh, Z.T., 004b. A optmal soluto of a geeral lot sze vetory model wth deterorated ad mperfect products, takg to accout flato ad tme value of moey. It. J. Syst. Sc., 35: Buzacott, J.A., 975. Ecoomc order quattes wth flato. Oper. Res. Quart., 6: hag,.t., 004. A EOQ model wth deteroratg tems uder flato whe suppler credts lked to order quatty. It. J. Prod. Eco., 88: he, J.M., 998. A vetory model for deteroratg tems wth tme-proportoal demad ad shortages uder flato ad tme dscoutg. It. J. Prod. Eco., 55:

5 Res. J. Appl. Sc. Eg. Techol., 4(3): , 0 he, J.M. ad.s. L, 00. A optmal repleshmet model for vetory tems wth ormally dstrbuted deterorato. Prod. Pla. otr., 3: her, M.S., H.L. Yag ad J.T. Teg, 008. Partal backloggg vetory lot-sze models for deteroratg tems wth fluctuatg demad uder flato. Eur. J. Oper. Res., 9: 7-4. hug, K.J., 003. A algorthm for a vetory model wth vetory-level-depedet demad rate. omput. Oper. Res., 30: hug, K.J. ad S.F. Tsa, 00. Ivetory systems for deteroratg tems wth shortage ad a lear tred demad-takg accout of tme value. omput. Oper. Res., 8: Datta, T.K. ad A.K. Pal, 99. Effects of flato ad tme-value of moey o a vetory model wth lear tme-depedet demad rate ad shortages. Eur. J. Oper. Res., 5: Dey, J.K., S.K. Modal ad M. Mat, 008. Two storage vetory problem wth dyamc demad ad terval valued lead-tme over fte tme horzo uder flato ad tme-value of moey. Eur. J. Oper. Res., 85: Harga, M.A., 995. Effects of flato ad tme-value of moey o a vetory model wth tme-depedet demad rate ad shortages. Eur. J. Oper. Res., 8: Harga, M.A. ad M. Be-Daya, 996. Optmal tme varyg lot-szg models uder flatoary codtos. Eur. J. Oper. Res., 89: Hseh, T.P. ad.y. Dye, 00. Prcg ad lot-szg polces for deteroratg tems wth partal backloggg uder flato. Expert Syst. Appl., I Press. Horowtz, I., 000. EOQ ad flato ucertaty. It. J. Prod. Eco., 65: 7-4. Hou, K.L., 006. A vetory model for deteroratg tems wth stock-depedet cosumpto rate ad shortages uder flato ad tme dscoutg. Eur. J. Oper. Res., 68: Hou, K.L. ad L.. L, 004. Optmal vetory model wth stock-depedet sellg rate uder maxmal total preset value of profts. Proceedg of 4th IASTED Iteratoal oferece o Modelg, Smulato ad Optmzato, pp: 7-. Jagg,.K., K.K. Aggarwal ad S.K. Goel, 006. Optmal order polcy for deteroratg tems wth flato duced demad. It. J. Prod. Eco., 03: Khara S., S.K. Ghosh ad K.S. haudhur, 0a. A EOQ model for a deteroratg tem wth tme depedet quadratc demad uder permssble delay paymet. Appl. Math. omput., 8: -9. Khara, S., S.K. Ghosh ad K.S. haudhur, 0b. Optmal prce ad lot sze determato for a pershable product uder codtos of fte producto, partal backorderg ad lost sale. Appl. Math. omput., 7: Lao, H.. ad Y.K. he, 003. Optmal paymet tme for retaler's vetory system. It. J. Syst. Sc., 34: Lo, S.T., H.M. Wee ad W.. Huag, 007. A tegrated producto-vetory model wth mperfect producto processes ad Webull dstrbuto deterorato uder flato. It. J. Prod. Eco., 06: Maty, A.K., 00. Oe mache multple-product problem wth producto-vetory system uder fuzzy equalty costrat. Appl. Soft omput., : Mat, A.K., M.K. Mat ad M. Mat, 006. Two storage vetory model wth radom plag horzo. Appl. Math. omput., 83: Mrzazadeh, A. ad A.R. Sarfaraz, 997. ostraed multple tems optmal order polcy uder stochastc flatoary codtos. Proceedg of d Aual Iteratoal oferece o Idustral Egeerg Applcato ad Practce, Sa Dego, USA, pp: Mrzazadeh, A., 008. Ecoomc order terval uder varable flatoary codtos. Hamburg Iteratoal oferece of Logstcs (HIL008), Hamburg, Germay. Mrzazadeh, A., 007. Effects of ucerta flatoary codtos o vetory models usg the average aual cost ad the dscouted cost. Eghth Iteratoal oferece o Operatos & Quattatve Maagemet (IOQM-8), Bagkok, pp: 7-0. Mrzazadeh, A., 009. A partal backloggg mathematcal model uder varable flato ad demad wth cosderg deterorato ost. World Appl. Sc. J., 7 (Specal Issue for Appled Math): Mrzazadeh, A., 0. Optmal vetory cotrol problem wth flato-depedet demad rate uder stochastc codtos. Res. J. Appl. Sc. Eg. Techol., 4(4): Msra, R.B., 979. A ote o optmal vetory maagemet uder flato. Nav. Res. Logst. Q., 6: Moo, I, B.. Gr ad B. Ko, 005. Ecoomc order quatty models for ameloratg/deteroratg tems uder flato ad tme dscoutg. Eur. J. Oper. Res., 6: Saa, S.S., 00. ERLINK" com/scece?_ob=artcleurl&_ud=b6v0v- 4YM7FGM-&_user=88467&_coverDate =07%F3%F00&_ald= &_rdoc= 3&_fmt=hgh&_org=search&_cd=5656&_sort=r &_st=4&_docachor=&_ct=5&_acct= &_verso=&_urlverso=0&_userd=88467& md5=aabc86ff37e6dd7cdb7caa4afeec"demad flueced by eterprses tatves-a mult-tem EOQ model of deteroratg ad ameloratg tems. Math. omput. Model., 5:

6 Res. J. Appl. Sc. Eg. Techol., 4(3): , 0 Sarkar, B. ad I. Moo, 0. A EPQ model wth flato a mperfect producto system. Appl. Math. omput., 7: Sarker, B.R., A.M.M. Jamal ad S. Wag, 000. Supply cha models for pershable products uder flato ad permssble delay paymets. omput. Oper. Res., 7: Sarker, B.R. ad H. Pa, 994. Effects of flato ad the tme value of moey o order quatty ad allowable shortage. It. J. Prod. Eco., 34: Su,.T., L.I. Tog ad H.. Lao, 996. A vetory model uder flato for stock depedet demad rate ad expoetal decay. Oper. Res., 33: 7-8. Uthayakumar, R. ad K.V. Geetha, 009. Repleshmet polcy for sgle tem vetory model wth moey flato. Opsearch, 46: Vrat, P. ad G. Padmaabha, 990. A vetory model uder flato for stock depedet cosumpto rate tems. Eg. ost. Prod. Eco., 9: Wee, H.M. ad S.T. Law, 999. Ecoomc producto lot sze for deteroratg tems takg accout of the tme value of moey. omput. Oper. Res., 6: Wee, H.M. ad S.T. Law, 00. Repleshmet ad prcg polcy for deteroratg tems takg to accout the tme-value of moey. It. J. Prod. Eco., 7: 3-0. Yag, H.L., 004. Two-warehouse vetory models for deteroratg tems wth shortages uder flato. Eur. J. Oper. Res., 57: Yag, H.L., 006. Two-warehouse partal backloggg vetory models for deteroratg tems uder flato. It. J. Prod. Eco., 03: Yag, H.L., J.T. Teg ad M.S. her, 00. Determstc vetory lot-sze models uder flato wth shortages ad deterorato for fluctuatg demad. Nav. Res. Log., 48: Yag, H.L., J.T. Teg ad M.S. her, 00. A vetory model uder flato for deteroratg tems wth stock-depedet cosumpto rate ad partal backloggg shortages. It. J. Prod. Eco., 3: