Modeling and Solving of Multi-Product Inventory Lot-Sizing with Supplier Selection under Quantity Discounts

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1 nernaonal ournal of Appled Engneerng Research SSN Volume 13, Number 10 (2018) pp Modelng and Solvng of Mul-Produc nvenory Lo-Szng wh Suppler Selecon under Quany Dscouns Naapa anchanaruangrong 1, and Chrawa Woarawcha 2 1 Deparmen of ndusral echnology, Faculy of ndusral echnology, Rajabha Rajanagarndra Unversy, 24000, haland. 2 Deparmen of ndusral Engneerng, Faculy of Engneerng, Rajamangala Unversy of echnology Lanna a, 63000, haland. Absrac hs paper proposes a mul-produc and mul-perod nvenory lo-szng problem wh suppler selecon under vehcle capacy and boh all-uns and ncremenal quany dscouns. n addon, we consder ha buyer s lmed sorage capacy. he objecves are o mnmze oal coss, where he coss nclude purchase cos of he producs, orderng cos for he supplers, ransporaon cos for he supplers and holdng cos of he remanng nvenory n each perod. s assumed ha demand of mul-produc s nown over a plannng horzon. he problem s formulaed as a mxed neger lnear programmng (MLP) model. Fnally, a numercal example s provded o llusrae he soluon procedure. he resuls, he buyer needs o deermne wha producs o order n wha quanes wh whch supplers n whch perods, n order o sasfy overall demand and mnmze he oal coss. eywords: nvenory Lo-Szng, Vehcle Capacy, Quany Dscouns, Mxed neger Lnear Programmng NRODUCON Nowadays, because of he global mare compeon, he frm can accomplsh compeveness by reducng oal logscs coss of purchasng. he mporance of he purchasng, s one of he mos sraegc acves n supply chan managemen (SCM), provdes opporunes o reduce coss and ncrease profs of companes [1]. he deermnaon of lo-szng and he suppler selecon play mporan roles n he purchasng sraegy and affec he nvenory coss [2]. n he pas several decades, nvenory lo-szng problem and suppler selecon has been suded n he leraure. nvenory lo-szng wh suppler selecon s a fas-growng offsprng of wo major problem parens n he feld of SCM [3]. Suppler selecon s consdered a very mporan ssue n nvenory lo-szng problems because of he probable dfferences beween he purchasng coss ha may arse from choosng dfferen supplers n each perod. nvenory lo-szng and suppler selecon are wo mporan decsons any purchasng of he buyer has o mae. Wh regard o nvenory lo-szng problem and suppler selecon, he buyer needs o consder mnmzng he nvenory coss [4]. n addon, he buyer ofen faces mulple supplers and quany dscoun. When quany dscoun s consdered and he order made by he buyer s large, he suppler reduces he un prce under dscoun schedule [5], [6]. Quany dscouns can be par of a prcng sraegy and can be a powerful ncenve o movae buyers o ncrease he amoun of her ordered quanes. herefore, he problems of nvenory lo-szng and suppler selecon have been verfed as havng a grave mpac on he supply chan, and araced many researchers for several years [7]. Soon aferwards, [5], [6], [8], [9] proposed a lo-szng model wh sngle produc mulple suppler and quany dscouns o mnmze oal cos over he plannng horzon. he problem was frs formulaed as a mxed neger lnear programmng (MLP) model. hs model s no able o consder he mul-produc of each suppler and vehcle capacy of each suppler. One of he mporan modfcaons we consder n hs paper s ha of nroducng he mulproduc and mul-perod nvenory lo-szng wh suppler selecon under alernave quany dscouns and vehcle capacy. hs paper exends he wor of [9], [5] [10], model. We formulae he mul-produc and mul-perod nvenory loszng wh suppler selecon consderng vehcle capacy and quany dscouns (all-uns and ncremenal dscouns). For all-uns quany dscouns, f he order sze belongs o a specfed quany level, he reduced purchase prce s appled o all-un sarng from he frs un. Dfferen wh ncremenal quany dscouns, he reduced purchase prce s appled o he uns nsde he prce brea quany [11]. he problem s formulaed as a MLP model and s solved wh opmzaon pacage LNGO 12. he objecves are o mnmze oal coss, where he coss nclude purchase cos of he producs, orderng cos for he supplers, ransporaon cos for he supplers and holdng cos of he remanng nvenory n each perod. he resuls, he buyer needs o deermne wha producs o order n wha quanes wh whch supplers n whch perods, n order o sasfy overall demand and mnmze he oal coss. 8708

2 nernaonal ournal of Appled Engneerng Research SSN Volume 13, Number 10 (2018) pp Fgure 1: he srucure of he problem under consderaon MODEL FORMULAON Model assumpons n hs secon, he problem can be brefly consdered n Fgure 1. he buyer am s o mnmze oal coss, where he coss nclude purchase cos of he producs, orderng cos for he supplers, ransporaon cos for he supplers and holdng cos of he remanng nvenory n each perod. Each suppler offers quany dscouns (all-uns and ncremenal dscouns) o movae he buyer for purchasng large quany. n addon, we consder ha he capacy each vehcle of he suppler ranspor he producs o he buyer. he proposed MLP model formulaon for he problem. Consequenly, he model assumpons, parameers and decson varables, a mahemacal formulaon are presened n followng subsecons. Model assumpons Demand of producs are nown and deermnsc. All requremens mus be fulflled n he perod n whch hey occur: shorage or bacorderng s no allowed. he supplers offer quany dscouns (all-uns and ncremenal dscouns) of each produc. Each suppler offer one quany dscouns only. Orderng cos from suppler does no depend on he varey and quany of producs nvolved. Holdng cos of produc per perod s producdependen. ransporaon cos s nown for he capacy each vehcle of he suppler. he buyer s lmed sorage capacy. nal nvenory of he frs perod and he nvenory a he end of he las perod are assumed o be zero ( 10 = 0; 20 = 0; 30 = 0). Model noaons Noaons used n he formulaon of he model are as follows: ndces: Produc ( = 1,, ). j Suppler (j = 1,, ), Where j = 1 o suppler j for all-uns quany dscouns, and suppler j = j + 1 o for ncremenal quany dscouns. Prce brea (1 =,, ). me perod ( =,, ). 8709

3 nernaonal ournal of Appled Engneerng Research SSN Volume 13, Number 10 (2018) pp Parameers: D Demand of produc n perod. P j Q j Y H Purchase cos under dscoun schedule based on he quany level of produc from suppler j wh prce brea. he upper bound quany of produc from suppler j wh prce brea. Expeced endng nvenory level of produc n perod. Expeced begnnng avalable nvenory level of produc n perod. Holdng cos of produc per perod. O j Orderng cos from suppler j. τ j ransporaon cos per vehcle from suppler j. V j Vehcle capacy from suppler j. N j Number of vehcles from suppler j n perod. s Sorage space requred for produc. M A large number. Decson varables: X j F j of he buyer. Number of produc ordered from suppler j wh prce brea n perod. A bnary varable, se equal o 1 f purchase quany from suppler j n perod, 0 oherwse. U j A bnary varable, se equal o 1 f purchase quany of produc from suppler j wh prce brea n perod, 0 oherwse. Objecve Funcon Mn = j j j' 1 OjFj τ jnj j H 2 2Y U 1 j ' 1 D P j' Q j j' Q PjXj j' 1 P jqj 1 (1) Subjec o Y Q N j j F j U j X X j, = Y D = 1 j (2) (3) (4) (5) (6) (7) (8) (9) (10) n he formulaon, he objecve funcon as shown n Equaon (1) ha consss of four pars: he oal cos (C) of orderng cos for he supplers, ransporaon cos for he supplers, purchase cos of he producs and holdng cos for remanng nvenory n each perod. n Equaon (2) he expeced endngs nvenory of produc n perod. n Equaon (3) he begnnng avalable nvenory level of produc n perod. n Equaon (4) s o se he orderng cos from suppler j n perod. n Equaon (5) se he purchase beween lower bound quany and upper bound quany prce brea of produc from suppler j n perod. n Equaon (6) s number of vehcles from suppler j n perod. n Equaon (7) he buyer sorage capacy n perod. Fnally, Equaons (8) (10) are used o force non-negave neger values and bnary varable 0 or 1 n he model. NUMERCAL EXAMPLE Example daa, j X j,,, M Fj,, j, Uj Xj QjUj, j,, 1, = Ys SXj Vj j, 0, 1, j, 0, 1,, j,, S 0, and neger, j,, he effecveness of propose MLP model s demonsraed hrough a numercal example. he daa used n hese examples 8710

4 nernaonal ournal of Appled Engneerng Research SSN Volume 13, Number 10 (2018) pp consss of he demands of hree producs over a plannng horzon of fve perods as shown n able 2. he orderng cos, ransporaon cos and vehcle capacy of each suppler s show n able 3.he quany dscouns are all-uns and ncremenal dscouns for each suppler n able 4. able 1: Demands of hree producs over a plannng horzon of fve perods Perod () ( D ) Produc , ,410 2,950 Produc ,510 2, ,850 Produc , able 2: Orderng cos, ransporaon cos, vehcle capacy of each suppler and ypes of quany dscouns Orderng cos ransporaon cos Vehcle capacy Quany dscouns Suppler 1 $200 $50 25 All-uns Suppler 2 $250 $60 30 ncremenal Suppler 3 $270 $70 35 All-uns able 3: Holdng cos and sorage space Holdng cos Sorage space Produc 1 $ Produc 2 $ Produc 3 $ oal sorage capacy of he buyer s 2,000. able 4: he quany dscouns of each suppler Prce brea Quany level () ( Q 1 j) Produc 1 Produc 2 Produc 3 Prce uns ( P 1 j) Prce brea () Quany level ( Q 2 j) Prce uns ( P 2 j) Prce brea () Quany level ( Q 3 j) Prce uns ( P 3 j) Suppler ,100 $ ,000 $ ,999 $ ,101-2,200 $ ,001-4,000 $ ,000-3,599 $ ,201-3,400 $ ,001-6,000 $ ,600-4,099 $ ,401 or more $ ,001 or more $ ,100 or more $2.45 Suppler $ $ $ ,000-2,599 $ ,000-1,999 $ ,500-2,499 $ ,600-4,099 $ ,000-2,999 $ ,500-3,499 $ ,100 or more $ ,000 or more $ ,500 or more $2.33 Suppler ,000 $ $ ,999 $ ,001-4,000 $ ,000-1,999 $ ,000-3,999 $ ,001-6,000 $ ,000-2,999 $ ,000-4,999 $ ,001 or more $ ,000 or more $ ,000 or more $2.41 Numercal resuls n hs secon, we solved a numercal example of he proposed model. o solve hs, he MLP model and he opmal soluon are found. he model s coded n opmzaon pacage le LNGO 12 and was run on compuer wh 1.80 GHz nel processor Core 3, and 4 GB RAM. he resuls of he numercal example wh wo producs, wo supplers and fve perods can be deermned. he soluon obaned by he MLP model s shown n able 5. he MLP model can provde a sasfacory soluon wh a very shor elapsed runme. he compuaonal me s 6 seconds. Fgure 2 shown resul of he numercal example solvers by LNGO 12. An example of he purchase cos of suppler 1 he buyer needs o deermne o order quanes from suppler 1, X 1132 = 2,400 uns of produc 1 s 2,400 x 2.82 = $6, he suppler 2 he buyer needs o deermne o order quanes from suppler 2, X 1211 = 230 uns of produc 1 s 230 x 3.12 = $ he suppler 3 he buyer needs o deermne o order quanes from suppler 3, X 2333 = 2,925 uns of produc 2 s 2,925 x 2.49 = $7, hus, he oal coss from suppler 1, suppler 2 and suppler 3 s $59,

5 nernaonal ournal of Appled Engneerng Research SSN Volume 13, Number 10 (2018) pp Fgure 2: he resuls bes objecve solvers by LNGO 12 able 5: he resuls of he numercal example Number of orders ( X j) Begnnng nvenory ( Y ) Endng nvenory ( ) Suppler 1 Suppler 2 Suppler 3 X 1132 = 2,400, X 1144 = 4,360, X 3114 = 1,000 Y 11 = 230, Y 12 = 2,400, Y 13 = 650, Y 14 = 4,360, Y 15 = 2, = 650, 14 = 2,950 X 1211 = 230, X 2211 = 465, X 2222 = 1,510, X 2225 = 1,850, X 3211 = 500, X 3212 = 510, X 3215 = 700 Y 21 = 465, Y 22 = 1,510, Y 23 = 2,925, Y 24 = 515, Y 25 = 1, = 515 X 2333 = 2,925, X 3313 = 475 Y 31 = 500, Y 32 = 510, Y 33 = 475, Y 34 = 1,000, Y 35 = 700 Purchase cos $46, Orderng cos $1, ransporaon cos $10, Holdng cos $1, oal cos (C) $59, CONCLUSONS n hs paper, we proposed a mul-produc and mul-perod nvenory lo-szng problem wh suppler selecon under vehcle capacy and quany dscouns. nvenory lo-szng wh suppler selecon under quany dscouns are mporan decsons any buyer has o mae. A buyer needs o deermne wha producs o order n wha quanes wh whch supplers n whch perods, n order o sasfy overall demand and mnmze he oal cos. Selecng he rgh supplers over mul-produc and mul-perod decson horzon becomes a major challenge for a buyer when supplers offer quany dscouns, and wh regard o vehcle capacy of each suppler. n addon, we consder ha buyer s lmed sorage capacy. he objecves are o mnmze oal coss, where he coss nclude purchase cos of he producs, orderng cos for he supplers, ransporaon cos for he supplers and holdng cos of he remanng nvenory n each perod. s assumed ha demand of mulple producs s nown over a plannng horzon. he problem s formulaed as a MLP model and s solved wh opmzaon pacage LNGO 12. he MLP 8712

6 nernaonal ournal of Appled Engneerng Research SSN Volume 13, Number 10 (2018) pp model can provde a sasfacory soluon wh a very shor elapsed runme s 6 seconds. Fnally, a numercal example s provded o llusrae he soluon procedure. For fuure research, a model ha aes no accoun sochasc demand or lead me, and safey soc s consdered. n addon, he budge consran s consdered for he oal coss. ACNOWLEDGMENS he auhors graefully acnowledge he ng Mongu s Unversy of echnology Norh Bango (MUNB) n sofware LNGO 12. REFERENCES [1] Mendoza, A. and Venura,. A., 2010, A seral nvenory sysem wh suppler selecon and order quany allocaon, European ournal of Operaonal Research, Vol. 207, No. 3, pp [2] Mazdeh, M.M., Emadhav, M. and Parsa,., 2015, A heursc o solve he dynamc lo szng problem wh suppler selecon and quany dscouns, Compuers & ndusral Engneerng, Vol. 85, pp [3] Rezae,., Davood, M., avasszy, L.A. and Davarynejad, M., 2016, A mul-objecve model for lo-szng wh suppler selecon for an assembly sysem, nernaonal ournal of Logscs Research and Applcaons: A Leadng ournal of Supply Chan Managemen, Vol. 19, No. 2, pp [4] Choudhary, D. and Shanar, R., 2013, on decson of procuremen lo-sze, suppler selecon, and carrer selecon, ournal of Purchasng & Supply Managemen, Vol. 19, No. 10, pp [5] Lee, A.H.., ang, H.Y., Chang, C.. and Hong, Y.W., 2013a, An negraed model for lo szng wh suppler selecon and quany dscouns, Appled Mahemacal Modelng, Vol. 37, No. 7, pp [6] Lee, A.H.., ang, H.Y. and La, C.M., 2013b, Solvng lo-szng problem wh quany dscoun and ransporaon cos, nernaonal ournal of Sysems Scence, Vol. 44, No. 4, pp [7] Mazdeh, M.M., Emadhav, M. and Parsa,., 2015, A heursc o solve he dynamc lo szng problem wh suppler selecon and quany dscouns, Compuers & ndusral Engneerng, Vol. 85, pp [8] Parsa,., hav, M.E., Mazdeh, M.M. and Mehran, S., 2013, A mul suppler lo szng sraegy usng dynamc programmng, nernaonal ournal of ndusral Engneerng Compuaons, Vol. 4, No. 1, pp [9] Ebrahm, R.M., Razm,. and Haleh, H., 2009, Scaer search algorhm for suppler selecon and order lo szng under mulple prce dscoun envronmen, Advances n Engneerng Sofware, Vol. 40, No 9, pp [10] Ayhan, B.M. and lc, H.S., 2015, A wo sage approach for suppler selecon problem n mulem/mul-suppler envronmen wh quany dscouns, Compuers & ndusral Engneerng, Vol. 85, pp [11] Woarawcha, C. and Naenna,., 2017, Mul-produc and mul-perod nvenory lo-szng wh suppler selecon under quany dscoun, nernaonal ournal of Servces and Operaons Managemen, Vol. 28, No. 2, pp