Optimizing Setup Time Reduction Rate in an Integrated JIT Lot-Splitting Model by Using PSO and GS Algorithms for Single and Multiple Delivery Policies

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1 Internatonal Journal of Industral Engneerng & Producton Research March 2013, Volume 24, Number 1 pp IN: ownloaded from jepr.ust.ac.r at 17:34 IRT on Tuesday eptember 4th 2018 Optmzng etup Tme Reducton Rate n an Integrated JIT Lotplttng Model by Usng PO and G Algorthms for ngle and Multple elvery Polces M.J. Taroh *, P. Motamed & F. Bagher Mohammad Jafar Taroh, epartment of Industral Engneerng, K.N.Toos Unversty of Technology, Tehran, Iran, Pegah Motamed epartment of Industral Engneerng, K.N.Toos Unversty of Technology, Tehran, Iran, motamed_p@mapnamd2.com, Fatemeh Bagher, Engneerng epartment of Golestan Unversty, Gorgan, Iran, f.bagher@gu.ac.r KEYWOR Jont economc lotszng (JEL, etup tme reducton, Partcle swarm optmzaton, Optmal aggregate total cost ABTRACT Ths artcle develops an ntegrated JIT lotsplttng model for a sngle suppler and a sngle buyer for only one product. The relatonshp between optmal lot sze and setup tme reducton s an mportant subject n such problems. In ths model we analyze the effect of setup tme reducton n the ntegrated lot splttng strategy. Two cases, ngle elvery ( case, and Multple elvery (M case are nvestgated before and after setup tme reducton. The Gradent earch (G and Partcle warm Optmzaton (PO are used n proposed model to determne the optmal order quantty (, optmal rate of setup reducton (, and the optmal number of delveres (N* just for multple delveres case. These optmum values are calculated by mnmzng the total cost for both buyer and suppler. Fnally numercal example and senstvty analyss are provded to compare the aggregate total cost for two cases and effectveness of the consdered algorthms. The results show that whch polcy for lotszng s leadng to lower total cost. Results show that the aggregate total cost n ngle delvery polcy s obtaned 1.3% lower when we used the optmzed setup tme reducton rate IUT Publcaton, IJIEPR, Vol. 24, No. 1, All Rghts Reserved. 1. Introducton Jont economc lot szng: (JEL, the problem of determnng producton and procurement quanttes s one that has to face when the suppler and the buyer has agreed to cooperate n a producton system networ. Goyal [1] has consdered an ntegrated * Correspondng author: Mohammad Jafar Taroh Emal: mjtaroh@ntu.ac.r Paper frst receved ep. 11, 2011, and n accepted form Jul. 07, nventory model for a sngle product and perhaps t s the frst contrbuton n ths feld. A semnal wor n the area of ntegrated nventory models s that of Banerjee [2] who proposed the concept of a jont economc lot sze. Also the studes of Goyal [3] are related to ths concept. He has expanded ths model, where a buyer s order quantty s delvered by the suppler n equal several shpments, as well as Km and Ha [4]. Earler wors focused on the potental savng for both partes (the vendor and the buyer smultaneously. A comprehensve lterature revew of ths wor s

2 M.J. Taroh, P. Motamed & F. Bagher Optmzng etup Tme Reducton Rate n an Integrated 38 ownloaded from jepr.ust.ac.r at 17:34 IRT on Tuesday eptember 4th 2018 presented n Goyal & Gupta [5], Abad [6], Parlar & Wang [7], Aderohunmu & others [8], Lu [9], Goyal [10], Hll [11], Vswanathan [12], Byla [13], and Goyal & Nebebe [14]. nce Goyal [1] ntroduced the ntegrated nventory model between a suppler and a buyer, many researchers have developed ths concept for varous cases, such as Banerjee [2], Goyal [3] and Hll [15]. The frst study on setup reducton s due to Porteus [16] wth an economc order quantty (EO model. pence & Porteus [17] found the optmal rate of setup reducton n a multproduct EO model and Km and others [4] the economc manufacturng quantty (EM model, respectvely. Recently, there has been an nvestgaton an EO model whch consders setup cost reducton n the varable lead tme envronment. Eyler et al. [18] extended the prevous research n two areas. Frst, the EO model wth setup cost reducton n the varable lead tme envronment. econd, nvestgaton to a more realstc stuaton where there s only a fnte number of opportuntes for setup cost reducton nvestment. enzel et al. [19] developed a dynamc lotszng model M where the values of the setup costs can be reduced by varous amounts dependng on the level of funds R commtted to ths reducton. Yang and eane researched on dependence of setup tme reducton and compettve advantage n a closed manufacturng cell [20]. Partcle swarm optmzaton (PO: s a populatonbased swarm ntellgence algorthm. It was frst ntroduced by Kennedy and Eberhart [21] as a smulaton of the socal behavor of socal organsms, such as brd flocng and fsh schoolng. PO uses the physcal movement of the ndvduals (partcles n the swarm and has a flexble and wellbalanced mechansm to enhance and adapt to global and local exploraton n contnuous space, whle some wor has been done recently n dscrete domans. Recent complete surveys for PO can be found n [22, 23, 24]. everal successful applcatons of PO to unclear problems reported n [2528] motvated us to use PO n ths wor. In [2528], the advantages of PO are demonstrated over other wellestablshed PBM (Populaton Based Metaheurstc. PO has been appled successfully to schedulng problems such as job shop schedulng, [29, 30], flow shop schedulng [31,32], assembly schedulng [33], and resourceconstrant project schedulng [34]. The wde use of PO manly durng the last few years s due to the number of advantages of ths method compared wth other optmzaton methods. ome of the ey advantages are as follows. Ths optmzaton method does not requre the calculaton of dervatves. The nowledge of good solutons s retaned by all partcles and the partcles n the swarm share nformaton among themselves. Furthermore PO s less senstve to the nature of the objectve functon, whch can be used for stochastc objectve functons and also can easly escape from local mnma. The rest of the paper s organzed as follows: ecton 2 addresses the notatons and assumptons of the proposed model. The descrpton of the setup tme reducton formulaton s gven n secton 3. ecton 4 descrbes the PO algorthm. The jont economc lotszng model, setup tme reducton, and Gradent search algorthm are descrbed for a sngle suppler and a sngle buyer wth sngle delvery and multple delveres n sectons 5 and 6, respectvely. In secton 7, numercal examples and senstvty analyss are presented. Conclusons are summarzed n secton Notatons and Assumptons Jont economc lot szng model allows the suppler and the buyer to reduce ther total costs. At the other hand, small lot szng s a way to mplementng successful JIT leadng to mnmum supply chan costs. In ths study we extend Km & Ha s model [4] by consderng setup tme reducton as a decson varable n a jont economc lotszng (JEL model wth both ngle delvery and several delveres Notatons: Followng notatons are consdered n ths paper: : buyer s demand rate per unt tme, determnstc P: suppler s producton rate per unt tme, (P> A: buyer s orderng cost per order : suppler s setup tme C: unt cost for suppler s setup tme : buyer s order quantty (producton lot sze H B : buyer s holdng cost per unt tme H : suppler s holdng cost per unt tme F: fxed transportaton cost per trp V: unt varable cost for order handlng and recevng N: number of delveres per batch cycle (nteger number q: delvery sze per trp, q N 2.2. Assumptons: 1 We consder sngle suppler and sngle buyer for only one product. 2 All necessary nformaton of the buyer and suppler are gven to both sdes. 3 Bacorders and shortages are not allowed. 4 The buyer pays transportaton and order handlng cost to facltate frequent delveres. 5 Product s manufactured wth a fnte producton rate P and P>. (f P<, we cannot satsfy buyer s demand and the problem would be nfeasble. 6 All cost parameters are nown and constant. Internatonal Journal of Industral Engneerng & Producton Research, March 2013, Vol. 24, No. 1

3 39 M.J. Taroh, P. Motamed & F. Bagher Optmzng etup Tme Reducton Rate n an Integrated ownloaded from jepr.ust.ac.r at 17:34 IRT on Tuesday eptember 4th No quantty dscount s allowed and unt prce s fxed. (demand rate and producton rate are nown, constant and determnstc. 8 H B > H, therefore t s not optmal to send any shpment when the buyer has some nventory. 9 The number and sze of transportaton vehcles has no constrants. 10 The transportaton and recevng cost of each shpment s a lnear functon of the shpped quanttes at a fxed cost. 11 There s no lead tme. In the ngle elvery case: 12 Every tme the buyer requests an order, the suppler mae the producton set up on a lot for lot bass. In the Multple elveres: 13 When the buyer places an order, the suppler splts the order quantty nto small lot szes and send them n equal shpments. 3. etup Tme Reducton At frst Porteus [16] ntroduces the relatonshp between optmal lot sze and setup tme reducton. Followng Porteus, to reduce the setup tme the cost equaton have been modfed by Kreng and Wu [35] as follows: C ( t x y ln( t for 0 t t (1 The rate of setup tme reducton s calculated by Kreng and Wu [35]: R t t 1 for 0 R 1 (2 YE NO x, y and t are postve constants. t and t are the orgnal setup tme before reducton the setup tme and after reducton, respectvely. A fxed cost s needed to reduce setup tme by a fxed percentage. Ths fxed percentage s, and the cost to reduce the ncrement of fxed percentage setup reducton s M, whch has constant value. Therefore, the Eq.(1, C, redefned as: M C( t ln(1 R ln(1 where R s consdered as the decson varable. (3 4. PO Algorthm In the mplementaton of the PO, the populaton s referred to as a swarm and each ndvdual as a partcle. It s ntalzed wth a random partcles group: Fg.1. the flowchart of general PO Algorthm and then searches the soluton space for optmal value by updatng generatons. The general PO algorthm s represented n Fgures 1 and 2. In PO, each partcle n a socal structure eeps n mnd ts best poston and uses ths as a factor for affectng ts speed. A partcle gans speed toward ts ndvdual best poston consderng how far away from that pont. It also shows the same behavor for the global best poston. In other words, whle t s scannng the surface, t s affected by the global best poston and adjusts ts own speed. If the partcle s far from the global best poston, there wll be a hgher chance n ts speed and drecton. Indvduals (partcles of a swarm show nclnaton to change ther movements by usng the nformaton below. Internatonal Journal of Industral Engneerng & Producton Research, March 2013, Vol. 24, No. 1

4 M.J. Taroh, P. Motamed & F. Bagher Optmzng etup Tme Reducton Rate n an Integrated 40 ownloaded from jepr.ust.ac.r at 17:34 IRT on Tuesday eptember 4th 2018 Poston of the th partcle n th teraton s x (=0, ter max and =1,,N. peed of the partcle n teraton s V. Best poston of the partcle (local best s pbest. Best poston of the partcle group (global best s gbest. Each ndvdual's speed changes accordng to the formula n Eq.(4; V ( wv C *( Rnd*( Pbest x 1 C *( Rnd*( Gbest x 2 (4 Where w and C are nerta functon and th nerta factor, respectvely. Rnd s a random number between 0 and 1. Inerta value of the equaton changes n each teraton. Ths changng s based on the logc of decreasng from the value determned to mnmum value accordng to nerta functon. The objectve s to converge the created speed by dmnshng on the further teratons; hence more smlar results can be obtaned [33]. Inerta functon s obtaned as follows: w w (5 max mn w wmax ( * termax w max frst nerta force w mn mnmum nerta force ter max maxmum teraton number The values of C nerta factor and w max and w mn nerta forces are nvestgated by h and Eberhart [37,38]. It s found that these values should not be changed from a problem to another. They fxed the values of these parameters as; C =2, w max =0.9 and w mn =0.4. Therefore, we use these values n our study. Postons of the partcles change by speeds as shown n Eq.(6 x x v (6 1 1 ame procedure s reterated for each dmenson. As t can be seen above, the advantages of the PO are easness to mplement and havng few parameters to adjust. However, there are some dffcultes related wth applyng PO on constrcted models. But the PO has been successfully appled n many areas, such as functon optmzaton, artfcal neural networ tranng, fuzzy system control, and other areas [39]. n our model, we also use ths algorthm for optmzng obtaned functons. Intalzaton (for =0 For =1 to N partcles Assgn partcles randomly n soluton space ( Generate ntal solutons ( x Assgn pbest = ntal solutons ( x Fg.2. Algorthmc schema for general PO x Assgn gbest = the obtaned best soluton among all Generate ntal veloctes randomly ( 5. ngle elvery ( case 5.1. Jont Economc Lot zng (JEL Model: In ths secton we frst present a lot for lot nventory polcy. In lot for lot model, the suppler produces optmal lot sze at one setup and delvers t to the buyer at one shpment. The buyer s total cost s composed of orderng cost, holdng cost, transportaton cost and order recevng cost: TC( Buyer A H B ( F V 2 (7 The suppler s total cost conssts of setup cost and holdng cost: TC( uppler C H ( 2 P (8 The total cost functon for a jont economc lot szng model conssts of all costs of both buyer and suppler. Hence, by addng Eq.(7 and Eq.(8, aggregate total cost functon wll be found. V Add veloctes to the correspondng partcles ( Intalzaton (for =0 etermne the nerta weght ( For =1 to N Update veloctes ( V w 1 Modfy the current postons ( x Update the pbest Update the gbest Fnalze the algorthm (=ter max Assgn the gbest =ubest and stop 1 x Internatonal Journal of Industral Engneerng & Producton Research, March 2013, Vol. 24, No. 1

5 41 M.J. Taroh, P. Motamed & F. Bagher Optmzng etup Tme Reducton Rate n an Integrated TC( aggregate ( A C ( H B H ( 2 P F V (9 TC(, N Buyer A H B 2N N ( F V N (14 ownloaded from jepr.ust.ac.r at 17:34 IRT on Tuesday eptember 4th 2018 By tang the frst dervatve of Eq.(9, wth respect to and set t equal to zero, optmal order quantty wll be obtaned. 2 ( A C F HB H( P ( etup Tme Reducton for Polcy We now s the suppler s setup tme and C s the unt cost for setup tme. By consderng s as the setup cost per unt tme and t as the setup tme per producton run before reducton, followng equatons are yeld: C=s t and t = t (1R Then, the aggregate total cost for sngle delvery polcy wth consderng the setup tme reducton can be redefned as follows: TC(, R ( st (1 R A aggregate s M ( H H ( F V K ln(1 R 2 B P ln(1 (11 K s the amortzaton of the setup reducton captal and a fxed reducton percentage,, can be acheved whenever the unt ncremental cost of M s made Gradent earch for Polcy In order to obtan optmal and n sngle delvery case after setup tme reducton, we should tae the partal dervatves of Eq.(11 wth respect to and R, By settng the dervatves equal to zero the optmum values for and R are obtaned as follows: The suppler s total cost conssts of setup cost and holdng cost: TC(, N H 2N uppler C (2 N N 1 P (15 Addng Eq.(14 and Eq.(15 yelds the total cost for the suppler and the buyer as follows: TC(, N aggregate ( A C (2 N H B H ( N 1 2N P N F V Fg 3. InventoryTme plot of buyer for M case (16 2 ( A st (1 R F HB H( P (12 KM 1 st.ln(1 (13 6. Multple elvery (M Case 6.1. Jont Economc Lot zng (JEL Model: In multple delveres case, the order whch s produced by quantty of, s delvered to buyer over N tmes, n small quanttes q. o we have: Nq. mall lot szng s a way to mplementng successful JIT. The buyer s total cost s: Fg 4. InventoryTme plot of suppler for M case Note that f the number of delveres, N, n Eq.(16 s equal to one, the M case becomes dentcal to Eq.(9 for polcy. Hence, n ths case, we assume that N 2. Internatonal Journal of Industral Engneerng & Producton Research, March 2013, Vol. 24, No. 1

6 M.J. Taroh, P. Motamed & F. Bagher Optmzng etup Tme Reducton Rate n an Integrated 42 ownloaded from jepr.ust.ac.r at 17:34 IRT on Tuesday eptember 4th 2018 Accordng to calculatons of Km & Ha [4], by tang the frst dervatves of Eq.(16 wth respect to and N, we obtan followng optmum values for N and : N* ( A C P( H H 2H B F( P H 2 N( A NF C H H B ((2 N N 1 P (17 (18 N* denote the optmum nteger value of N and s the optmum value of. If N* n the Eq.(17 s not an nteger number, we should choose N, whch yelds mn TC( N, TC( N n Eq.(16, where N + and N represent the nearest ntegers larger and smaller than the N* respectvely. The mnmum aggregate total cost s obtaned by substtutng N* and nto Eq.(16. The optmal delvery sze q*, whch remans the same over multple delveres polcy, s obtaned by dvdng by N* from Eq.(17 and Eq.( etup Tme Reducton for M Polcy Integratng Eq.(2 and Eq.(3 to Eq.(16, the aggregate total cost for multple delveres wth consderng setup tme reducton s obtaned as follows: TC(, N, R aggregate ( sts(1 R A (2 N N H B H ( N 1 F 2N P M V K ln(1 R ln(1 (19 The above equaton conssted of buyer s orderng cost, buyer s holdng cost, transportaton cost, order recevng cost, suppler s setup cost after setup tme reducton, suppler s holdng cost, and total setup reducton captal. The objectve s to mnmze the sum of these costs Gradent earch Algorthm for M Case We can determne the optmal order quantty,, optmal rate of setup tme reducton,, and optmal number of delveres from the aggregate total cost after setup tme reducton from Eq.(19 wth regardng that TC(,N,R aggregate s a convex functon. The optmum values are found by tang partal dervatves of Eq.(19 wth respect to R, N, and, and settng the dervatves equal to zero. N* ( A st (1 R P( H H 2H B F( P H (20 2 N( st (1 R A NF H B H ((2 N N 1 P KM 1 st ln(1 (21 (22 7. Numercal Examples In ths secton, we use an example whch orgnally comes from Banerjee [2]. It was modfed by Km & Ha [4] and we gathered addtonal values from example of Kreng & Wu [35] for analyzng our model. We consder a buyer, a suppler and a sngle product. Buyer s annual demand s 4800 unts and the order cost for each order s 25$. Fxed transportaton cost whch buyer pays for each trp s 50$ and the unt varable cost for order handlng and recevng s 1.00$/unt. Annual producton capacty of suppler s unts. The cost of suppler s setup tme s 400$ per unt. We assume that H B and H are 7$ and 8$ per unt per year. The setup tme before reducton polcy s 4 and the setup cost per unt tme s 100$, where the amortzaton of the setup reducton captal s It s assumed that a rate of 30% of fxed reducton can be acheved whenever the unt ncremental cost of 2000$ s made. In summary: A= 20 C = 400 = 4800 s= 100 t = 4 F= 60 M= 2000 V= 1 P= K= 0.35 H = 6 H B = 7 = Example for Polcy Before and After etup Tme Reducton Table 1 presents the effect of the rate of setup tme reducton on and aggregate total cost. For sngle delvery polcy, by usng PO algorthm and also Gradent earch algorthm (Eq.(11 and Eq.(19 From ths table, we can nterfere that f the setup tme s reduced from t =4 to 0 and all other parameters reman unchanged, then by usng G algorthm, the optmal soluton wll be = 0.4, t = 2.4, = , and TC*(,R= ; and by usng PO algorthm, the optmal soluton wll be = 0.4, t = 2.4, = , & TC*(,R= Example for M Polcy Before and After etup Tme Reducton Usng the gven parameters n PO and G algorthms, we calculate N* and. Then by decreasng t from 4 Internatonal Journal of Industral Engneerng & Producton Research, March 2013, Vol. 24, No. 1

7 After setup tme reducton before R=1t /t N* N Δ TC(,N mn TC(,N opt(n ΔTC Δ TC(,N mn TC(,N opt(n ΔTC After setup tme reducton 43 M.J. Taroh, P. Motamed & F. Bagher Optmzng etup Tme Reducton Rate n an Integrated ownloaded from jepr.ust.ac.r at 17:34 IRT on Tuesday eptember 4th 2018 to 0, we fnd the optmal value of R. table 2 presents ths results. From the results shown n table 2, we can nterfere that by usng the G algorthm, optmal value of R s 0.1. In other word, f the setup tme s reduced from t =4 to 0 and all other parameters reman unchanged, the optmal soluton wll be = 0.1, N*=2, t = 3.6, = , and TC*(,R,N= From the results of Partcle warm Optmzaton Algorthm presented n table 2, we can nterfere that by usng ths algorthm, optmal value of R s 0.1. In other word, f the setup tme s reduced from t =4 to 0 and all other parameters reman unchanged, the optmal soluton wll be = 0.1, N*=2, t = 3.6, = , and TC*(,R,N= Tab. 1. The effects of the rate of setup tme reducton on and aggregate total cost n sngle delvery R= t s t s t s /t s Gradent earch PO 1t /t Δ TC( ΔTC Δ TC( ΔTC before ts ts Tab. 2. The effects of R & on TC n multple delvery polcy usng G and PO algorthm Gradent earch PO ts /ts Internatonal Journal of Industral Engneerng & Producton Research, March 2013, Vol. 24, No. 1

8 Change (% TC*( TC*( P (*10 3 TC*(,N TC*(,N (*10 3 Change (% TC*( TC*( (*10 3 TC*(,N TC*(,N (*10 3 M.J. Taroh, P. Motamed & F. Bagher Optmzng etup Tme Reducton Rate n an Integrated 44 ownloaded from jepr.ust.ac.r at 17:34 IRT on Tuesday eptember 4th enstvty Analyss A senstvty analyss s performed to study the effects of changes n the parameters system on the optmal order quantty, rate of setup tme reducton, and number of delveres. Ths analyss s performed by ncreasng or decreasng the parameters by 10%, 20%, and 30% tang one at a tme, eepng the remanng parameters at ther orgnal values. The effects of changes n parameters on case and M case are nvestgated. Followng ratos are beng calculated for dfferent quantty of these parameters: TC * Gradentearch TC * PO r1 100 (23 TC * 30% 20% 10% PO G TC * TC * r2 TC * ( PO M( PO M( PO TC * TC * r3 TC * ( G M( G M( G Tab. 3. enstvty analyss over parameter M PO r (24 (25 whch r 1 shows the devaton of PO soluton from Gradent earch soluton. r 2 and r 3 determne the dfference of optmal cost functon n M and case, by usng PO and G algorthms, respectvely % % % % G G Tab. 4. enstvty analyss over parameter P PO G M PO PO r1 r2 r3 r1 r1 r 2 r 3 30% 20% 10% % % % % The followng nferences can be made from the results of table 1 and 2. and senstvty analyss based on table 2 and 3. By usng PO algorthm, whch s a metaheurstc algorthm and gves an approxmate soluton, and Gradent earch algorthm, whch gves an exact soluton, and comparng the results of these methods by consderng r 1, we can nterfere that the values computed for the aggregate total cost are approxmately smlar. Internatonal Journal of Industral Engneerng & Producton Research, March 2013, Vol. 24, No. 1

9 45 M.J. Taroh, P. Motamed & F. Bagher Optmzng etup Tme Reducton Rate n an Integrated ownloaded from jepr.ust.ac.r at 17:34 IRT on Tuesday eptember 4th 2018 To compare effectveness of dfferent delvery polces, and M, we should compare jont total costs of and M case n tables 1 and 2. nce TC* M <TC*, consequently, the polcy of frequent shpment results n less total cost than the sngle shpment polcy. TC* M = by N=2, whle TC* = From table 4 we can nterfere that by ncreasng the value of, whle other parameters reman unchanged, the optmal jont total cost of M polcy ncreases, whle the optmal jont total cost of polcy decreases. Increasng the parameter P results to more jont total cost for both sngle delvery and multple delvery polces. 8. Conclusons The effects of setup tme reducton n the ntegrated lot splttng strategy have been analyzed n ths study. The proposed model determnes optmal order quantty, optmal rate of setup reducton, and optmal number of delveres on the ntegrated total relevant cost for sngle delvery and multple delveres polces, the results were nferred by comparng the optmal values of these two polces whch obtaned by usng PO and G algorthms. The results show that whch polcy for lotszng s leadng to lower total cost. Results show that the aggregate total cost n ngle delvery polcy s obtaned 1.3% lower when we used the optmzed setup tme reducton rate. The proposed model can be extended n future studes by consderng multple products, multple buyers and supplers, or probablstc parameters. Appendx: The expresson of holdng cost s derved by Joglear (1988. From Fg.3 the holdng cost of suppler s derved as follows: q q q H ABCE AE 2... ( N 1 = Nq q q N q q H Nq ( N 1 ( ( N 1 = Nq P 2P 2 Nq H ( 2 N N 1 = Nq 2 P H ( 2 N N 1 2N P References [1] Goyal,.K., An Integrated Inventory Model for a ngle uppler ngle Customer Problem, Internatonal Journal of Producton Research. 15(1, 1976, pp [2] Banerjee, A., A Jont EconomcLotze Model for Purchaser and Vendor, ecson cences, 17, 1986, pp [3] Goyal,.K., A Jont EconomcLotze Model for Purchaser and Vendor: A Comment, ecson cences, 19 (1, 1988, pp [4] Km,., Ha,., A JIT Lotplttng Model for upply Chan Management: Enhancng Buyeruppler Lnage, Internatonal Journal of Producton Research 86, 2003, pp [5] Goyal,.K., Gupta, Y., Integrated Inventory Models: The Buyer Vendor Coordnaton. European Journal of Operatonal Research 41, 1989, pp [6] Abad, P.L., uppler Prcng When the Buyer s Annual Requrements are Fxed. Computers & OR. 21 (2, 1994, pp [7] Parlar, M., Wang,., scountng ecsons n uppler Buyer Relatonshp wth a Lnear buyer s emand. IIE Transactons 26 (2, 1994, pp [8] Aderohunmu, R., Ayodele, M., Bryson, N., Jont Vendor Buyer Polcy n JIT Manufacturng. Journal of the OR ocety 46, 1995, pp [9] Lu, L., A OneVendor MultBuyer Integrated Inventory Model. European Journal of Operatonal Research 81 (2, 1995, pp [10] Goyal,.K., A OneVendor MultBuyer Integrated IInventory Model: A Comment. European Journal of Operatonal Research 82 (1, 1995, pp [11] Hll, R.M., The nglevendor nglebuyer Integrated Producton Inventory Model wth a Generalzed Polcy. European Journal of Operatonal Research 97, 1997, pp [12] Vswanathan,., Optmal trategy for the Integrated Vendor Buyer Inventory Model. European Journal of Operatonal Research 105, 1998, pp [13] Byla,., A ynamc Model for the nglevendor, MultpleBuyer Problem. Internatonal Journal of Producton Economcs 59, 1999, pp [14] Goyal,.K., Nebebe, F., etermnaton of Economc Productonhpment Polcy for a nglevendor ngle Buyer ystem. European Journal of Operatonal Research. 121(1, 2000, pp [15] Hll, R.M., The nglevendor nglebuyer Integrated Producton Inventory Model wth a Generalzed Polcy. European Journal of Operatonal Research 97, 1997, pp [16] Porteus, E.L., Investgatng n Reduced etups n the EO Model, Management cence. 30(8, 1985, pp [17] pence, A.M., Porteus, E.L., etup Reducton and Increased Effectve Capacty, Management cence, 33(10, 1987, pp Internatonal Journal of Industral Engneerng & Producton Research, March 2013, Vol. 24, No. 1

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