Replenishment Policy with Emergency Purchase and Partial Backorder

Transcription

1 r International Conference on Management, Behavioral Science an Economic Iue (ICMBSE'04) Feb. -, 04 Singaore Relenihment olicy with Emergency urchae an artial Backorer Hui Ming Wee, Yen Deng Huang, Simon Wu, an Yung-Lung Cheng Abtract Due to reventive maintenance or machine breakown, there i a oibility of machine unavailability. Uner uch circumtance, ome inutrie try to reuce facility unavailability by lanning regular reventive maintenance early in their lanning erio. Moreover, owing to the quality of rouction rocee, imerfect item may be rouce within the rouction cycle. Ditinct from traitional metho, the fixe oint aroach an renewal rewar theorem are utilize to otimize the rouction inventory iue. Keywor Deteriorating inventory, Economic rouction quantity eteriorating (EQD) moel, Emergency urchae, Facility unavailability, Inection cheule. M I. INTRODUCTION AINTENANCE i a roce to control the conition of equiment to maximize machine availability an minimize imerfect item. Without maintenance, efective equiment may reult in loe an hortage cot. One caue of tock-out i the reult of machine breakown an it corrective maintenance. The effect of machine breakown an corrective maintenance were firt invetigate by Groenevelt et al. []. Chung et al. [] tuie an economic rouction eteriorating moel with ranom machine unavailability to take inight into the effect of otimal rouction u-time erio on integrate inventory ytem an to erive otimal rouction u-time erio in orer to minimize the total ytem cot. In a rouction environment, efective item nee to be reworke an reaire (Chiu et al. []). Hayek an Salameh [4] who tuie an otimal relenihment olicy that minimize the total inventory cot er unit time for the finite rouction moel. Chung an Hou [5] eveloe a moel to etermine an otimal run time for a eteriorating rouction ytem with hortage. To rovie the bet ervice an to reuce comlaint, the manufacturer an retailer may offer warrantie for non-conforming item within a warranty erio. Lee an H. M. Wee i with Deartment of Inutrial an Sytem Engineering, Chung Yuan Chritian Univerity, Chung Li 0, Taiwan (correoning author to rovie hone: ; fax: ; weehm@cycu.eu.tw). Y. D. Huang i with Deartment of Inutrial an Sytem Engineering, Chung Yuan Chritian Univerity, Chung Li 0, Taiwan ( g9040@cycu.eu.tw). Simon Wu i with Deartment of Inutrial an Sytem Engineering, Chung Yuan Chritian Univerity, Chung Li 0, Taiwan ( wuimon@cycu.eu.tw). Y. L. Cheng i with Deartment of Inutrial an Sytem Engineering, Chung Yuan Chritian Univerity, Chung Li 0, Taiwan ( jkcheng54@yahoo.com.tw). Roenblatt [6] coniere the imerfect rouction roce in the traitional EQ moel an erive the otimal number of inection an rouction cycle time. An otimal rouction quantity moel for a eteriorating item wa firt introuce by Mira [7]. Figure eict the co-relation of eteriorating EQ moel with imerfect item, machine breakown, quantity lanning an warranty. Conequently, thi tuy evelo an economic rouction quantity moel with eteriorating item uner ranom machine breakown, tochatic reair time, rework, warranty an cheule inection. Different from the otimization aroache uing the algebraic metho (Yang an Wee [8]) or calculu (Yang et al. [9]), thi aer ue the concet of fixe oint aroach in otimization. A imle algorithm an numerical examle i rovie. A. Notation II. NOTATIONS AND ASSUMTIONS All of the arameter an variable are efine in Table I. B. Aumtion The five aumtion are: ) rouction rate i greater than eman rate. ) rouction an eman rate are contant. ) Deteriorating rate i contant. 4) There i no reair or relacement for a eteriorate item. 5) Machine reair time i ineenent of machine breakown. III. MODEL DEVELOMENT The behavior of the inventory moel i illutrate in Figure. When the machine oe not breakown uring the rouction erio, the rouct i manufacture uring the erio of T. When the inventory reache the maximum level I m, the rouction to an the inventory ecreae ue to eman an eterioration. The inventory level reache zero unit at time (T +T ), an the machine tart to rouce the item again. Since the machine ha a oibility of breakown, the machine may hutown within T erio. When the breakown occur, we aume that the rouction erio time i of T. When a machine break own, it will require time to reair an reet the machine. Since reair time i tochatic in (T +T ) erio, rouction may to an lot ale may occur uring the time erio T. An emergency urchae may be require to fulfill the backorer quantity from outie ulier. From Figure, the ifferential equation rereenting the behavior of inventory level can be written a follow: 70

2 r International Conference on Management, Behavioral Science an Economic Iue (ICMBSE'04) Feb. -, 04 Singaore I ( t) I( t) 0 t T t () Machine breakown (Machine unavailibility) Quality lanning (Inection cheule) Effect of imerfect item (Warranty olicy) Relenihment lanning (Backorer an emergent urchae conieration) Fig. Relationhi among relenihment lanning of EQ eteriorating moel an key factor TABLE I NOTATIONS OF ALL OF THE ARAMETERS AND VARIABLES Notation Decrition Notation Decrition rouction rate eman rate eteriorating rate C etu cot H holing cot er unit S Loe-ale cot er unit C eteriorating cot er unit I inventory level T rouction u-time erio T rouction own-time erio T hortage erio T machine breakown erio C R Rework cot er unit C w ot-ale warranty cot er unit backorering rate The maintenance rate I I C b V k R rk C M The arrival rate of machine breakown inventory level in rouction u-time erio inventory level in rouction own-time erio the elae time of imlementing inection time total reource-in-inection cot C b = V k/ + R rk reource-in-inection etu cot i inverely roortional to variable rik cot er unit i oitively roortional to the corrective cot er unit time for machine breakown, Defective rate before an after reetermine inection time reectively h, h C E A E Hazar rate of rouct rouce before an after reetermine inection time reectively Emergency urchae cot er unit from outie ulier Setu cot er unit time for emergency urchae Inventory level time T T T T T Fig. rouction Inventory level with machine breakown, inection an artial backorering of hortage erio The behavior of inventory level in rouction own-time erio i rereente by the following ifferential equation: T T T I ( t) I( t) 0 t T t () 7

3 r International Conference on Management, Behavioral Science an Economic Iue (ICMBSE'04) Feb. -, 04 Singaore From the bounary conition of I (0) = 0 an I (T ) = 0, the inventory level in rouction u-time erio an rouction own-time erio can be olve a: I( t) I( t) ex t 0 t T, ex T t 0 t T Since I (T ) = I (0), then one ha the relationhi between T an T i rewritten by uing the Taylor erie aroximation (Mira [7]). Hence, T in term of T can be aroximate a: () ex T [ T] E ( ) T T ex T ( T ) ex T (8) ex T Conequently, with mall value of, an, neglecting the term higher than T, T an T relenihment cycle time i, the value of total T ( ) T ( T )/ (4) 0.5 E[ T] T T (9) Thi aroximation i commonly ue in eteriorating rouction inventory roblem. Thi tuy ue the renewal rewar theorem which i utilize to coe with the variable relenihment cycle length T. One ha the execte total ytem cot er unit time E[TEC] of the rooe eteriorating economic rouction quantity (DEQ) moel a follow: TEC ETC ET E / (5) The execte total ytem cot, E[TC], conit of execte total etu cot C, holing cot H E[I h ], eteriorating cot C E[Det], inection cot C b (), corrective cot E[C ret ], rework cot E[RW], ot-ale warranty cot E[O], artial emergency urchae E[E c ] an it etu cot E[S E ] an lot-ale cot E[L c ]. The execte total ytem cot can be exree a: E[ TC] C E HEI h C EDet Cb( ) EC ret RW E[ O] E[ E ] E[ L ] E[ S ] Due to the unreictable machine breakown an reair time, the relenihment time i not a contant. The total relenihment time conit of the rouction u-time erio, the rouction own-time erio an the oible breakown-reair time. The total relenihment time for ifferent conition can be written a: C C E (6) T T t T ( t T ) f ( t) t, T T T (7) T T t T ( t T ) f ( t) t, T T The ranom variable t i the machine reair time with robability enity function f(t). For imlify, thi tuy coniere the machine reair time with exonentially itribute an it mean / i given. From Eq. (7), the execte value of total relenihment time with the exonential itribution of machine reair time can be exree a: The lot ale an efficiency of the machine utilization i a critical iue to the management of rouction an inventory control ue to the unreictable conition of reair. Thi aer icue the effect of reair time on rouction u-time erio an the relenihment time length. From Eq. (9), it i a quaratic equation of T. In orer to exre T in term of E[T], we olve Eq. (9) for T which lea to the following two root: () T A(,, ) () T A(,, ) A(,, ) A (,, ) E[ T] A(,, ) A (,, ) E[ T] where A(,, ) /( ) 0. an (0) When aroache to 0 an the reair time an eterioration are alo ignore, the rouction u-time erio of the above root will reuce to be a traition EQ moel. () limt T an () limt T. () Hence, we houl chooe the firt root a the olution of T. The execte total ytem cot for exonential itribution cae can be formulate by ubtituting Eq. (6) an (9) into Eq. T T (5). Auming that,, an Eq. (5) can be rearrange a k E TEC Where T kt Ck kt k4t T T aroach to zero, 7

4 r International Conference on Management, Behavioral Science an Economic Iue (ICMBSE'04) Feb. -, 04 Singaore S k k 4 H S C / M S S AE CE C h C C h C 0, R w S C R hcw C S A C 0 M 0.5 k 0, k 4 0, a 0. 5C R Cw h h, a R C C h, an E R E rk R w h Ck a a CS / w () The two following otimal conition are eential in eriving the otimal olution of the rooe ytem cot. Firt, Coniering Eq. (6) an taking econ erivative of E[TC] with reect to T, the reult i a follow: E[ TC]/ T k >0 () obtaining the relevant otimal value. Since an are very mall, by etting the firt erivative of E[TEC] with reect to T to zero an uing the fixe-oint methoology, an iterative olution of coniering T can be obtaine a ~ n V k ~ / U a n an (7) < 0. where Un h Cw h Cw T, n IV. CASE EXAMLE AND SENSITIVITY ANALYSIS A Taiwan thin film manufacturer (UNMC) rouce variou rouct of thin film that i ue in the automotive, rinting an electronic inutrie. A reetermine rouction u time i cheule. The rouct i manufacture with a efective rate of before inection an with a efective rate of after inection. The efective rate can be reuce after inection i imlemente. In the factory, there i a robability of machine break own. When the machine i unavailable after the reetermine rouction own time ue to reventive maintenance or unreicte reair time, hortage will occur. During rouction, efective item are etecte an reworke. From hitorical ata, the following arameter inut are rovie by the thin film manufacturer to illutrate the moel. The arameter inut are: Therefore, E[TC] i a convex function of T. Secon, Referring to Eq. (9) an taking econ erivative of E[T] with reect to T, one ha the reult a follow: E[ TC]/ T k ) ( 0.5)/ 0 0.5( (4) =0.0 =4000 unit =8500 unit =0.005 R rk =$4500 =0.00 µ =0.006 h =0.005 V k =$0 S =$0 C R =$ 9 C w =5 A E =0 λ=0.009 h =0.00 C S =$0 H S =$.5 C M =$0 Therefore, E[T] i a concave function of T. Obviouly, from the two otimal conition, the otimal T can be erive ince the execte total ytem cot E[TEC] = E[TC]/ E[T] i convex for mall value of λ, µ, an T uner the following conition: / (5) The uroe of the rooe moel i to erive the otimal rouction u-time erio, T an the otimal elae time, for imlementing inection lan. Taking the firt erivative of E[TEC] with reect to T an etting the reult to zero, the olution of rouction u-time T can be erive * T Ckk Ck k Ckk4 k Ckk4kk / kk 4 kk (6) Due to the comlexity in olving the ymbolic olution of the lanne rouction u-time T an the elae time of imlementing inection time, we eveloe a heuritic olution roceure with a fixe oint iterative metho for Uing the tate olution roceure, the otimal olution coniering eterioration i erive a ( T *, *, ETC*) = ( year, year, $,066.99) reectively. If eterioration aroximate to zero, the olution i erive a in Table II. The ercentage total cot ifference (TCD) to analyze the enitivity of the key arameter i ue. The ercentage of total cot ifference i efine a TCD ( TC TC*) TC *. The elae time of imlementing inection increae, the rouction u-time length ecreae, inicating that the manager houl carry out a frequent rouction relenihment olicy to minimize the execte total inventory ytem cot (Figure -6). When the arameter increae by 0% an 0% or ecreae by -0% an -0%, the relationhi among the known arameter, the eciion variable an the ercentage total cot ifference (TCD) are given in Fig. -6. * i the otimal total cot an {} i the bae column. The main concluion rawn from the enitivity analyi are a follow: (a) When the eterioration rate {} increae, a maller rouction-lot-ize olicy houl be imlemente to otimize the inventory ytem (Fig. 6). The achievement in 7

5 r International Conference on Management, Behavioral Science an Economic Iue (ICMBSE'04) Feb. -, 04 Singaore the eterioration rate reuction will benefit the economic rouction inventory ytem. (b) When the efective rate { } ecreae an { } increae, the total um of the rouction u-time erio an own-time erio ten to ecreae. That invetment on the efective rate reuction will influence the rouction an elivery olicie. In ractice, when the efective rate ecreae, a frequent elivery olicy houl be imlemente (Fig. 4). (c) When the variable rik cot ha a tenency to increae, the reetermine inection time erio {} ten to increae alo (Fig. ). () The total cot i oitively enitive to the eman rate {}, the eterioration rate {} an the unit holing cot {H } (Fig. 5). The reult are imilar to the arameter of the eterioration rate {}. When unit holing cot increae, the rouction u-time erio an own-time erio ten to ecreae. That i, a manager may ue a frequent an mall lot-ize olicy when the unit holing cot increae (Fig. 4). (e) The total cot i negatively enitive to the rouction rate {} an the arrival rate of maintenance {} (Fig. 5). When the mean time between breakown {/} increae, a frequent rouction olicy i encourage to be carrie out (Fig. 6). (f) When the unit holing cot {H } increae, the manager houl conier a maller rouction-lot-ize an horter cycle-time olicie to lower the rouction inventory level (Fig. 4 an 6). (g) The rouction lot ize, Q, i oitively enitive to rouction rate {}, arrival rate of maintenance {} an variable rik cot {V k }, an i negatively enitive to the eman rate {}, the unit holing cot {H } an the eterioration rate {}. In ractice, when the eman rate {} increae an rouction rate {} ecreae, the total um of the rouction u-time erio an own-time erio houl be horten (Fig. 6). TABLE II SENSITIVITY ANALYSIS WHEN DEFECTIVE RATE BEFORE INSECTION CHANGES { } {0.005} T Q T ETC TCD() -0.59% -0.9% 0.00% 0.9% 0.58% % 5.00% 0.00% 5.00% 0.00% -5.00% -0.00% -5.00% -0.00% S value TCD Change Rate H Fig.. Senitivity analyi for when inut arameter change change rate Fig. 5 TCD v. change rate for economic rouction moel with facility unreliability h h CS h h CS H T value 0. h h C 0.07 HS Change Rate Fig. 4. Senitivity analyi for T when inut arameter change Q h 550 h CS 50 H Change Rate Fig. 6 Senitivity analyi for Q when inut arameter change 74

6 r International Conference on Management, Behavioral Science an Economic Iue (ICMBSE'04) Feb. -, 04 Singaore V. CONCLUSION Traitional inventory moel uing ifferential metho are the founation in exiting reearch. A renewal rewar aroach i ignificant for reearcher to clarify the critical iue in the inventory control for ranom breakown or machine unavailability. Thi aer evelo a eteriorating economic rouction inventory moel coniering facility unavailability an the imerfect item. Thi tuy alo rovie a rigorou analyi uing the renewal rewar metho an obtain the otimal olution uing the fixe oint aroach. The enitivity analyi how that the eteriorating rate, unit holing cot an eman rate houl be weighte in etablihing the inection lan an the relenihment olicy coniering the facility unavailability an imerfect item. Further reearch can be extene to conier the ifferent nature of a artnerhi with uly chain member. retire on the en of Aug, 0. Hi reearch interet are in the fiel of lean rouction/uly chain, total quality management, olicy eloyment an trategic lanning. Yung-Lung Cheng i a octoral caniate in the Deartment of Inutrial an Sytem Engineering at Chung Yuan Chritian Univerity REFERENCES [] H. Groenevelt, L. intelon an A. Seiman, rouction lot izing with machine breakown, Mgt. Sci., vol. 8,. 04, 99. [] C. J. Chung, G.A. Wiyaana an H.M. Wee, Economic rouction quantity moel for eteriorating inventory with ranom machine unavailability an hortage, Int. J. ro. Re., vol. 49, no., , 0. [] S.W. Chiu, D.C. Gong an H.M. Wee, The effect of the ranom efective rate an the imerfect rework roce on the economic rouction quantity moel, Ja. J. In. A. Math., vol., no., , 004. [4].A. Hayek an M.K. Salameh, rouction lot izing with the reworking of imerfect quality item rouce, ro. lan. Contl., vol., no. 6, , 00. [5] K.J. Chung an K.L. Hou, An otimal rouction run time with imerfect rouction rocee an allowable hortage, Com. Oer. Re., vol. 0, , 00. [6] H.L. Lee an M.J. Roenblatt, Simultaneou etermination of rouction cycle an inection cheule in a rouction ytem, Mgt. Sci., vol.,. 5-6, 987. [7] R.B. Mira, Otimum rouction lot-size moel for a ytem with eteriorating inventory, Int. J. ro. Re., vol., , 975. [8].C. Yang an H.M. Wee, A ingle-venor an multile-buyer rouction-inventory olicy for a eteriorating item, Eur. J. Oer. Re., vol. 4, no., , 00. [9].C. Yang, H.M. Wee, J.Y. Shiau an Y.F. Teng, Otimal relenihment olicy for Hi-tech inutry with comonent cot an elling rice reuction, Lec. Not. Com. Sci., (incluing uberie Lecture Note in Artificial Intelligence an Lecture Note in Bioinformatic) vol. 4705, no.,. 7-7, 007. Hui-Ming Wee i a Ditinguihe rofeor in the Deartment of Inutrial an Sytem Engineering at Chung Yuan Chritian Univerity in Taiwan. He receive hi B.S. egree (honor) in electrical an electronic engineering from Strathclye Univerity (UK), M.Eng. egree in inutrial engineering an management from Aian Intitute of Technology (AIT), an h.d. egree in inutrial engineering from Clevelan State Univerity, Ohio (USA). Hi reearch interet are in the fiel of rouction/inventory control, otimization, renewable energy an uly chain management. Yen-Deng Huang i a octoral caniate in the Deartment of Inutrial an Sytem Engineering at Chung Yuan Chritian Univerity in Taiwan. Hi reearch interet are in the fiel of rouction/inventory control, quality control, eterminitic or tochatic otimization an heuritic algorithm. Simon Wu i an Aitant rofeor in the Deartment of Inutrial an Sytem Engineering at Chung Yuan Chritian Univerity. He receive hi h.d. in the Deartment of Inutrial an Sytem Engineering at Chung Yuan Chritian Univerity. He wa a manager of For Motor Comany, Taiwan, an 75