Multi Product Dynamic Lot-Sizing with Supplier Selection under Quantity Discount and Budget Constraints

Transcription

1 Inernaonal Journal of Appled Engneerng Research ISSN Volume 13, Number 2 (2018) pp Research Inda Publcaons. hp:// Mul Produc Dynamc Lo-Szng wh Suppler Selecon under Quany Dscoun and Budge Consrans Sahaporn Suryan 1* and Vcha Rungreunganun 1 1 Deparmen of Indusral Engneerng, Faculy of Engneerng, Kng Mongku s Unversy of Technology Norh Bangkok, Bangkok 10800, Thaland. *ORCID: Absrac Ths sudy s am s o deermne he produc purchasng by nvenory lo-szng as a problem of suppler selecon wh quany dscouns. The obecve s o mnmze he nvenory managemen coss wh he use of mxed neger lnear programmng. Also, he selecon of supplers wh dfferen producs wll be presened, and he lead-me for supplers wll also be consdered, ncludng he condons of budge. Even so, he resuls obaned from hese ess showed ha he deermnaon of he purchasng sze and he duraon for orderng purchases were correc and were able o be followed hrough sasfacorly. Fnally, hs sudy also esed he effcency of he deermnaon of lo szes of purchases by expandng he problem o larger szes. I was hen dscovered ha he effcences n calculaons and he resuls obaned were n good and sasfacory crerons. Keywords: Lo-Szng, Suppler Selecon, Mul-Produc, Mul-Suppler, Quany dscouns INTRODUCTION Invenory Lo-Szng [1] wll be he model for purchasng producs n los. Each lo-szng wll have a suable number of producs and are suffcen of use. Ths mehod of nvenory loszng wll keep holdng coss low as well as he oal cos wh he mnmal nvenory managemen for he problem of sngle ems hrough mul-perods wh rregular demands n each perod, where hey dd no lm he exsng resources under he gven condons[2],[3]. The presenaon was done mahemacally wh lmaons n he area of sorage space, where he quanes of he nvenory combned wh he ordered quanes mus no exceed he maerals sorage space[4]. The problems for deermnng he lo producons were also suded, wh a se equaon ha mnmzes he orderng, holdng, producon, and overme coss alogeher under he condon n whch he desred capacy mus no exceed normal and overme producon. Oher han he mehods for orderng producs, he suppler selecon s also mporan, snce presenly supplers ha sell producs are n hgh demands n he marke, whch could allow varous he organzaon o have opporunes n purchasng producs from he supplers ha are mos benefcal. Each suppler wll have dfferen produc prces and orderng coss. Thus, he model n suppler selecon of nvenory-lo szng s hghly mporan, snce wll allow an awareness of he quany of producs n each lo-szng as well as he suppler ha was seleced for ha purchase. The am of hs model s o mnmze he nvenory managemen coss[5], [6]nvenory managemen s also aded by fndng economc order quany, whch sudes he problems of mul-produc manufacurng as well ha of mul supplers and nvenory lo-szng for suppler selecon n he form of varous producs and supplers wh lmed sorage space and budges. Ths was done usng lnear programmng and genec algorhms. [7]Laer on, he model was developed for orderng producs wh purchase-relaed quany dscouns of each suppler, also wh he am o mnmze oal coss. The model had been developed for he problem of suppler selecon of nvenory lo-szng n he case of produc dscouns ha consdered he purchased quanes and prces durng each dscoun level, [8] and he buyer mgh fnd dfferen supplers as well as dfferen levels of dscouns. When dscouns are praccal for general effcency wh he reducon of produc coss per un, producs wll be purchased a lower prces per un. Ths s done wh purchasng excessve quanes of produc [9]. Dynamc lo-szng. For one ype of produc wh respec o he problem of suppler selecon, 2 cases of he sudy was conduced, whch was one ha dd no consder dscouns and one ha consdered boh ncremenal and all-un dscouns[10], and a consderaon of models wh he use of mxed neger non-lnear programmng [11] as well as he use of mxed neger lnear programmng ha consdered backorderng, [12] he problems of lo-szng wh mulple supplers under quany dscouns, and nsuffcen producs wh he Slver-Meal heursc mehod [13]. The managemen of nvenory lo-szng for he problem of suppler selecon n he purchasng process consss of 3 acves alogeher, whch were he decsons for lo-szng, suppler selecon decsons, and holdng decsons [14] wh he ams of reducng he nvenory and coss wh produc nsuffcency by selecng he mos approprae me and loszng. Even so, he lead-me s anoher mporan maer o be consdered, snce wh he condon of he ndusry n he presen, mos supplers wll have a lead-me for he delvery of raw maerals for varous processes such as he preparaon, fndng, pre-delvery qualy assurance, me for he delvery of raw maerals, ec. Regardng he problem of selecon supplers wh varous producs among many supplers, he model wll consder quany dscouns and lead-me as mean values [15]. 1373

2 Inernaonal Journal of Appled Engneerng Research ISSN Volume 13, Number 2 (2018) pp Research Inda Publcaons. hp:// Suppler 1 Suppler 2 Suppler Suppler J All-Un Quany Dscoun Prce break k Prce break k Prce break k Prce break k vehcle vehcle vehcle vehcle Supplers have dfferen producs Produc 1 Produc 2 Produc Produc I Lead-me Lead-me Lead-me Lead-me Buyer Budge Fgure 1: The general dagram of he problem under consderaon Suppler selecon s anoher mporan maer, snce he selecon of purchasng producs n ha perod mean ha here are dfferen coss along wh produc lo-szng and poenally dscouns of he model, especally for agences wh demands for dfferen ypes of producs, whch resuls n he generaon of hgher coss n sorage and orderng. These aforemenoned problems o dae have no sudes regardng he selecon of supplers n he case ha each suppler s unable o fnd he same produc, wh consderaon of each of her lead-mes. Ths sudy wll presen mahemacal developmens wh mx neger lnear programmng and suppler selecon for nvenory lo-szng problem wh quany dscouns [16] wh mulproducs hrough mul-perods wh he consderaon of suppler lead-mes and under he condons of purchasng budges. Furhermore, he mahemacal model wll also consder he selecon of supplers wh dfferen producs among oher dfferences, where he ams of he sudy leaned owards mnmzng he oal coss of nvenory managemen (orderng cos, holdng cos, purchasng produc cos and ransporaon cos). The resul also alers us o he deermnaon of orderng lo-szng and he perod for purchasng wh he consderaon of lead-me as well as beng responsve o he demands as a whole. MODEL DEVELOPMENT Ths secon wll explan suppler selecon for nvenory loszng wh quany dscouns by consderng he condons of suppler lead-mes and budges, where he problems are llusraed as per fgure 1 wh he followng assumpons. The demands of each produc are ndependen and made known durng perods Produc shorage or backorderng are no allowed Each suppler wll presen all-un quany dscouns The orderng coss wll depend on each suppler and are unrelaed o produc quanes The holdng coss of each perod depend on he produc The ransporaon coss depend on he number of vehcles of each suppler Lead-me of each suppler s known over a plannng horzon Invenores are allowed o sar respondng o demands n he frs perod (Inal nvenory = 0 when he leadme = 0) Each supplers have dfferen produc and canno make dencal produc delvers MODEL PARAMETERS AND DECISION VARIABLES The noaons used n he formulaon of he model are as follows: Indces se of ndex of producs (1,, I) se of ndex of supplers (1,, J) k se of ndex of prce break (1,, K) se of ndex of me perods (1,, T) Parameers he demand of produc n perod D PC k he produc cos for one un under he all-un dscoun schedule based on he quany level of produc from suppler wh prce break k Q k he upper bound quany of produc from suppler wh prce break k I he expeced endng nvenory level of produc n perod Y he expeced begnnng avalable nvenory level of 1374

3 Inernaonal Journal of Appled Engneerng Research ISSN Volume 13, Number 2 (2018) pp Research Inda Publcaons. hp:// produc n perod he holdng cos of produc per perod H O S V II M L B he orderng cos of produc from suppler he ransporaon cos per vehcle from suppler he maxmum ransporaon (vehcle) from suppler he nal nvenory a large number lead-me he purchasng budge n perod Decson varables X k he number of produc ordered from suppler wh prce break k n perod F a bnary varable, se equal o 1 f purchase quany of L produc from suppler n perod -L, 0 f no U k a bnary varable, se equal o 1 f purchase quany of produc from suppler wh prce break k n perod, 0 f no MATHEMATICAL FORMULATION The obecve funcon o mnmze oal nvenory coss, whch conss of produc coss, orderng coss, nvenory holdng coss, and ransporaon coss. I consders producs of () ypes from supplers of () sources, and consders he enre perod of plannng horzon n () perod n equaon (1). I J T I J K T MnmzeTC O F PC X Subec o II L k k k I K J T Xk I T k H S 2Y D V 2 L D 1 (1) (2) I Y D, (3) 1 k k J K Y I X, K X k M F L,, k (4) (5) Qk 1U k Xk QkU k,, k, (6) I J K PCk X k B k (7) F L 0 or 1,, (8) Uk 0 or 1,,, (9) k Consran n equaon (2), whch s an equaon ha llusrae he number of nal nvenory s quany when he me perod s 0, snce he produc mus mach he demands n a me when orders are unable o be made. Consran n equaon (3) s an equaon ha llusrae he nvenory a he end of nsallmen. Consran n equaon (4) s a condoned equaon ha llusrae he nvenory a he begnnng of he nsallmen. Consran n equaon (5) s a condoned equaon ha llusrae produc orders, n whch each order mus no exceed he maxmum number. Consran n equaon (6) s a condoned equaon ha llusrae produc orders, n whch each order mus be done n he correc amoun wh dscouned produc levels. Consran n equaon (7) s a condoned equaon ha llusrae he budge for produc orders n each perod, where he oal produc coss mus no exceed he oal budge n each perod. Fnally, equaons (8) - (9) are equaons wh he bnary varables 0 or 1 for use n decson-makng for he model. NUMERICAL EXAMPLE Ths secon wll nroduce mahemacal problem solvng of he model by usng LINGO12, whch wll consder a suaon wh 3 producs and 5 supplers n a plannng horzon wh 5 perods, where daa was dsplayed as follows. Produc demands and coss are llusraed n able 1. Orderng coss, ransporaon coss, and he maxmum capacy of he ransporaon vehcles are llusraed n able 2. Table 3 dsplays he coss of holdng he 3 ypes of producs. Table 1: Demands of hree producs over a plannng horzon of fve perods and budge Plannng Horzon (Fve Perods) Perods() Produc Produc Produc Budge(B ) Table 2: The orderng cos (O ), ransporaon cos (S ) and maxmum ransporaon (V ) of each suppler Suppler O S V Table 3: The holdng cos (H ) of hree producs producs H

4 Inernaonal Journal of Appled Engneerng Research ISSN Volume 13, Number 2 (2018) pp Research Inda Publcaons. hp:// The quany dscouns of each suppler wll be dsplayed as allun dscouns. Ths model wll presen he selecon of supplers n he case ha each suppler has dfferen producs. Ths sudy wll use he 4 (k=4) as he dscoun level for each suppler, where able 4 dsplays he producs of each suppler, he dscoun levels, and produc prces. Table 4: The quany dscoun s all-un of hree produc and produc cos producs Produc Suppler Prce break Quany level Un produc cos Produc 1 Suppler or more 2.74 Suppler or more 2.60 Suppler or more 2.50 Produc 2 Suppler or more 2.83 Suppler or more 2.76 Suppler or more 2.75 Produc 3 Suppler or more 2.75 Suppler or more 3.00 Suppler or more

5 Invenory Level Inernaonal Journal of Appled Engneerng Research ISSN Volume 13, Number 2 (2018) pp Research Inda Publcaons. hp:// COMPUTATIONAL RESULTS Ths secon wll nroduce mahemacal problem solvng of he model wh mxed neger lnear programmng by usng LINGO12 on a compuer wh he specfcaons of Inel Core GHz and memory/ram 4 GB. The resuls of he problem solvng concerns he varables of he quany () of 3 ypes of producs, 5 supplers (), 4 quany dscoun levels (k), hrough 5 plannng perods (). Regardng he resuls of he orderng of 3 producs, hs modellng es wll label he lead-me of supplers as 1 (L=1), where he begnnng of he model s process mus frs decde he sarng quany of he produc nvenory o respond o demands n he frs me perod (II 1= 230, II 2=465 and II 3=500). Then, he sysem wll begn o consder makng orders accordng o demands n he nex me perod. Fgure 2 dsplays he level of produc 2 nvenory, whch dsplays he behavor of he nvenory levels a he sar and he fllng of produc n he begnnng of he me perods n whch here are demands. Table 5 dsplays he resuls of calculaons wh LINGO12 for he problem wh dmensons The resuls show ha he orderng process begns a perod 1 snce hs model consders suppler lead-mes. Thus, he orderng of he produc mus always be done n he F -L perod n order for produc o be flled n he nvenory n he begnnng of he me perods n whch here are demands. Ths wll llusrae he resuls of produc 1 (=1, where he resuls wll deermne ha he sarng nvenory (II 1) has 230 uns, snce he lead-me s equal o 1 perod. hus, 2029 uns of produc were bough from he suppler 4 (=4) a he 2 dscoun level (k=2) n he perod 2 (=2), along wh 371 uns from suppler 1 (=1) a he 1 dscoun level (k=1) n he perod 3 (=3), 4297 uns from suppler 4 (=4) a he 4 dscoun level (k=4) n he perod 4 (=4), and fnally 63 uns from suppler 1 (=1) a he 1 dscoun level (k=1) n he perod 5 (=5) Table 5: The resuls of he buyer order quanes of produc 1, 2 and 3 Produc Procuremen Lo-sze X 1422=2029 X 2222=1510 X 3212=700 Orderng cos 1800 Holdng cos Produc cos Transporaon cos Toal cos X 1113=371 X 2133=2470 X 3213=300 X 1444=4297 X 2214=455 X 3214=800 X 1115=63 X 2135=1850 X 3215=1000 Perods Fgure 2: The example nvenory level of produc

6 Inernaonal Journal of Appled Engneerng Research ISSN Volume 13, Number 2 (2018) pp Research Inda Publcaons. hp:// Regardng produc 2, he resuls wll deermne ha he sarng nvenory (II 2) has 465 uns, snce he lead-me s equal o 1 perod. hus, 1510 uns of produc were bough from he suppler 2 (=2) a he 2 dscoun level (k=2) n he perod 2 (=2), along wh 2470 uns from suppler 1 (=1) a he 3 dscoun level (k=3) n he perod 3 (=3), 455 uns from suppler 2 (=2) a he 1 dscoun level (k=1) n he perod 4 (=4), and fnally 1850 uns from suppler 1 (=1) a he 3 dscoun level (k=3) n he perod 5 (=5). For produc 3, he resuls wll deermne ha he sarng nvenory (II 3) has 500 uns, snce he lead-me s equal o 1 perod. Thus, 700 uns of produc were bough from he suppler 2 (=2) a he 1 dscoun level (k=1) n he perod 2 (=2), along wh 300 uns from suppler 2 (=2) a he 1 dscoun level (k=1) n he perod 4 (=4). There are also purchases of 800 and 1000 uns n he 4 and 5 me perods respecvely (=4, 5), boh form he 2 suppler (=2) a he 1 dscoun level (k=1) ANALYSIS PROBLEMS SIZE The model of nvenory lo-szng for hs maer of suppler selecon was allocaed 120 mnues of me for problem solvng calculaons [5] wh ndces,, k, and, where = produc, = suppler, k = dscoun level, and = perod. Regardng each dscoun level (k), hs sudy wll use a dscoun level of 4 (k=4) for each problem n analyzng he sensvy of problem solvng me esng. Ths analyss wll begn wh he smalles problems (3,3,4,5) and hen he larges problem (5,5,4,50), where he resuls obaned from he usage of LINGO12 n problem solvng (me lmed a 120 mnues) dscovered ha he larger problem canno be answered n he gven me. Thus, equaon (10), whch s an equaon used o sudy he error percenage from he large problem, was used o calculae he answer of he oal cos, n whch he larger problem wll oban 2 answers, whch are upper and lower lms. Percenage error of LINGO12 Upper bound - Lower bound 100 Upper bound (10) Table 6 llusrae problem szes, soluon mes, oal coss, and error percenages. I can be seen ha he larger problem wll resul n a hgher soluon mes and he answer canno be dsplayed (Lm = 120 mnues). The resul was he oal cos n he form of upper bound and lower bound. Table 6: Compuaonal resuls by LINGO12 Problem sze Soluon Tme (mnue) Toal cos % Error Problem sze Soluon Tme (mnue) Toal cos % Error 3,3,4, ,3,4, a, b ,3,4, ,3,4, a, b ,3,4, ,3,4, a, b ,4,4, ,4,4, ,4,4, ,4,4, ,4,4, a, b ,5,4, ,5,4, ,5,4, ,5,4, a, b ,5,4, ,5,4, a, b ,3,4, ,3,4, ,3,4, ,3,4, a, b ,3,4, a, b ,3,4, a, b ,4,4, ,4,4, ,4,4, ,4,4, a, b ,4,4, ,4,4, a, b

7 Soluon me(mnue) Inernaonal Journal of Appled Engneerng Research ISSN Volume 13, Number 2 (2018) pp Research Inda Publcaons. hp:// 3,5,4, ,5,4, a, b ,5,4, ,5,4, a, b ,5,4, a, b ,5,4, a, b ,3,4, ,3,4, , b ,3,4, a, b ,3,4, a, b ,3,4, a, b ,3,4, a, b ,4,4, a, b ,4,4, a, b ,4,4, a, b ,4,4, a, b ,4,4, ,4,4, a, b ,5,4, a, b ,5,4, a, b ,5,4, ,5,4, a, b ,5,4, a, b ,5,4, a, b a Upper bound, b Lower bound by LINGO Plo of he problem sze vs. soluon me and % error % Error Soluon me % error Problem sze(,, k, ) Fgure 3: Plo of he problem sze vs. soluon me and % error Thus, he researcher used equaon (10) o fnd he error percenage beween he lower and upper lms o fnd he degree of error. I was dscovered ha he hghes error percenage was 5,5,4,50 whch was equal o 1.370%, whch was he larges problem of hs sudy (5 produc, 5 supplers, 4 dscoun levels, 50 perods). Fgure 3 dsplays he relaonshp beween problem szes, soluon mes, and percenage error obaned from resuls evaluaon wh LINGO12. I was dscovered ha he smalles problem could be solved n a shor me (less han 120 mnues), whch s he problem wh 5 produc ypes, 5 supplers, and 20 perods. (=5, =5, k=4, =20) I can be seen ha he solvng of he smalles problem (3x4x4x5) ook only 0.03 mnues. The larger problems wh 20 perods onwards wll mosly have hgher soluon mes and canno fnd he approprae answer n he alloed me. Thus, her error percenages were suded. Fgure 3 shows ha afer 20 perods, he error percenage becomes hgher han ha of smaller problems, where he problem wh parameers 5x5x4x50 had he hghes error percenage. CONCLUSIONS Ths paper presened he deermnaon orderng szes for nvenory lo-szng wh problems of suppler selecon n cases wh produc dscouns by consderng he lead-mes of each suppler and each supplers have dfferen produc and 1379

8 Inernaonal Journal of Appled Engneerng Research ISSN Volume 13, Number 2 (2018) pp Research Inda Publcaons. hp:// canno make dencal produc delver. There were also condons of budges for produc orderng, cases of mulproducs, and mul-perod. An obecve funcon was used n he sudy o oban he lowes oal cos, whch conssed of he orderng cos, holdng cos, produc cos, and ransporaon coss. Resuls llusraed ha hs model was suffcen o calculae and deermne orderng lo-szng accuraely o respond o demands sasfacorly. Fnally, he problem szes were expanded n order o es ables n fndng answers, whch dscovered ha he deermnaon of orderng lo-szng were bes whn he frs 20 me perods. In he case of larger problems when esed wh percenage errors, her answers were sll found o be sasfacory and accepable. REFERENCES [1] Wagner H.M. and Whn T.M., 1958, Dynamc verson of he economc lo-sze model, Managemen Scence, Vol. 5, pp [2] Herandez W.and Suer G. A., 1999, Genec algorhms n lo szng decsons, Evoluonary Compuaon, In Proceedngs IEEE., 2002, Washngon, D.C., pp [3] Guerrez J., e al., 2002, A new characerzaon for he dynamc lo sze problem wh bounded nvenory, Compuers and Operaons Research, pp [4] Xe J.and Dung J., 2002, Heursc genec algorhms for general capacaed lo-szng problems, Inernaonal Journal Compuer and Mahemacs wh Applcaons, pp [5] Basne C.and Leung J. M. Y., 2005, Invenory loszng wh suppler selecon. Compuers and Operaons Research. pp [6] Rezae J. and Davood M., 2006, Genec Algorhm for Invenory Lo-Szng wh Suppler Selecon under Fuzzy Demand and Coss, Proceedngs n 19 h Inernaonal Conference on Indusral Engneerng and Oher Applcaons of Appled Inellgen Sysems, pp , June. [7] Woarawcha C. Kullpaaranran T. and Rungreunganun V., 2011, Invenory Lo-Szng Problem wh Suppler Selecon under Sorage Space and Budge Consrans, IJCSI, Vol. 8, Issue 2, pp [8] Choudhary D. and Shankar R., 2013, Jon decson of procuremen lo-sze, suppler selecon, and carrer selecon, Journal of Purchasng & Supply Managemen, Vol.19, pp [9] Lee A. H. I., e al, 2013, An negraed model for lo szng wh suppler selecon and quany dscouns, Appled Mahemacal Modellng, Vol.37, pp [10] Mazdeh M.M. Emadkhav M.and Parsa I., 2015, A heursc o solve he dynamc lo szng problem wh suppler selecon and quany dscouns, Compuers & Indusral Engneerng, Vol. 85, pp [11] Soo A. V., e al., 2017, Mahemacal modelng and hybrdzed evoluonary LP local search mehod for loszng wh suppler selecon, nvenory shorage, and quany dscouns, Compuers & Indusral Engneerng, Vol.109, pp [12] Ghanabad M.and Maznan A., 2017, Dynamc lo szng wh mulple supplers, backloggng and quany dscouns; Compuers & Indusral Engneerng, Vol.110, pp [13] Alfares K and Turnad R., 2016, General model for sngle-em lo-szng wh mulple supplers, quany dscouns, and backorderng, Proceda CIRP, Vol.56, pp [14] Choudhary D.and Shankar R., 2014, A goal programmng model for on decson makng of nvenory lo-sze, suppler selecon and carrer selecon; Compuers & Indusral Engneerng, Vol.71, pp [15] Oay F. C. I, 2016, A wo-sage fuzzy approach for suppler evaluaon and order Allocaon problem wh quany dscouns and lead me, Informaon Scences, Vol.339, pp [16] Woarawcha C.and Naenna T., 2017, Mul-produc and mul-perod nvenory lo-szng wh suppler selecon under quany dscoun, In. J. of Servces and Operaons Managemen, Vol.28, No