A HYBRID GENETIC ALGORITH FOR A DYNAIC INBOUND ORDERING AND SHIPPING AND OUTBOUND DISPATCHING PROBLE WITH HETEROGENEOUS VEHICLE TYPES AND DELIVERY TIE WINDOWS by Byung Soo Km, Woon-Seek Lee, and Young-Seok Ock Deparmen of Sysems anagemen & Engneerng, Pukyong Naonal Unversy, Busan 608-737, Souh Korea Emal: ewslee@ pknu.ac.kr ABSTRACT Ths paper consders a sngle-produc problem for nbound orderng and shppng, and oubound dspachng a a hrd-pary warehouse, where he demand s dynamc over he dscree me horzon. Each demand mus be delvered no he correspondng delvery me wndow whch s he me nerval characerzed by he earles and laes delvery daes of he demand. Ordered producs are shpped by heerogeneous vehcles. Each vehcle has ype-dependen carryng capacy and he un fregh cos dependng on vehcle ype. The oal fregh cos s proporonal o he number of each vehcle ype employed. Also s assumed ha relaed cos funcons are concave and backloggng s no allowed. The objecve of hs sudy s o smulaneously deermne he opmal nbound orderng and shppng, and oubound dspachng plans ha mnmze he oal sysem cos o sasfy he dynamc demands over he fne me horzon. The problem s dffcul o solve he opmal soluon as he problem sze becomes large. Therefore, we propose a hybrd genec algorhm (H-GA) ncludng he margnal fregh cos heursc o deermne he number of each vehcle ype employed. We presen compuaonal resuls o evaluae he performance of H-GA. KEYWORDS Hybrd, Genec Algorhm, Shppng, Dspachng, Vehcle, Tme Wndows INTRODUCTION For he couple of decades, he reducon of ransporaon cos and warehousng cos have been wo mporan ssues o enhance logsc effcency and demand vsbly n a supply chan. The logsc allances and specalzed Thrd- Pary-Logsc (TPL) provders have been growng o reduce he coss n ndusry. In a dynamc plannng me perod, he ssue of ransporaon schedulng for nbound orderng and shppng of producs o TPL warehouse by proper ransporaon modes a rgh me and he ssue of lo sze dspachng of sored producs o he cusomers have become sgnfcanly mporan for procuremen and dsrbuon managemen. In hs paper, a warehouse purchases a sngle produc o sasfy demands of he produc over he dscree me horzon. Ordered produc s shpped by varous vehcle ypes no warehouse and sored producs are delvered from he warehouse o realers. Therefore, may lead o he manageral decson problems ncludng lo-szes and dspachng sze for each demand, conaner ypes used, loadng polcy n conaners, and he number of conaners used. Furhermore, each demand mus be delvered no he correspondng delvery me wndow whch s he me nerval characerzed by he earles and laes delvery daes of he demand wh no penaly cos. Ths provdes us wh a movaon o smulaneously nvesgae he opmal lo-szng, nbound shpmen schedulng, and oubound dspachng problem. The dynamc lo-szng model (DLS) has semmed from he work of Wagner and Whn (958). The majory of DLSs have no consdered any producon-nvenory problem ncorporang ransporaon acves. Several arcles have aemped o exend he classcal DLS ncorporang producon-nvenory and ransporaon funcons ogeher. Lee (989) consdered DLS allowng mulple se-up coss conssng of a fxed charge cos and a fregh cos, n whch a fxed sngle conaner ype wh lmed carryng capacy s consdered and he fregh cos s proporonal o he number of conaners used. Lee e al. (003) exended he work of Lee (989) by consderng heerogeneous vehcle ypes o mmedaely ranspor he fnshed produc n he same perod s produced. I s also assumed ha each vehcle has a ype-dependen carryng capacy and he un fregh cos for each vehcle ype s dependen on he carryng
capacy. Lee e al. (003) consdered a dynamc model for nvenory lo-szng and oubound shpmen schedulng n he Thrd-Pary Warehousng doman. They presened a polynomal me algorhm for compung he opmal soluon. Lee e al. (00) was he frs paper ha suded he DLS wh me wndows. They developed wo polynomal me algorhms for he case of a backloggng allowed and he case of a backloggng beng no allowed. Jaruphongsa e al. (004) suded a sngle em, wo-echelon DLS wh delvery me wndow. Wolsey (006) proposed polynomal me algorhm consderng boh producon and delvery me wndows. Hwang (007) developed an mproved algorhm by he model of Lee e al. (00). Hwang and Jaruphongsa (008) suded a lo-szng model for major and mnor demands n whch major demands are specfed by me wndows whle mnor demands are gven by perods. Ths paper analyzes a dynamc nbound orderng and shppng schedulng, and oubound dspachng problem for a sngle produc ha s ranspored from a suppler o TPL warehouse by varous fregh vehcles and delvered from TPL warehouse o realers n a supply chan. Demand s dynamc over he dscree me horzon. Each demand mus be delvered no he correspondng delvery me wndow whch s he me nerval characerzed by he earles and laes delvery daes of he demand. Each vehcle has ype-dependen carryng capacy and he un fregh cos dependng on vehcle ype. The oal fregh cos s proporonal o he number of each vehcle ype employed. Also s assumed ha relaed cos funcons are concave and backloggng s no allowed. The man objecve of hs sudy s o smulaneously deermne he opmal nbound lo-szes and he shpmen schedule, and oubound dspachng plans ha mnmze he oal cos whch consss of orderng, nvenory holdng, and fregh coss. ODEL FORULTION The followng noaons are defned o formulae he problem: T = lengh of he me horzon, = perod ndex (,,, T ), = number of demands, = demand ndex (,,, ), N = number of vehcle ypes, j = vehcle ype ndex ( j,,, N ) W = carryng capacy of a conaner, S = orderng cos n perod, h = un nvenory holdng cos of demand from perod o perod, F = un fregh cos of conaner n perod, x = amoun of orderng and shppng by vehcle ype j j a perod, = number of vehcle ype j employed n perod y j I = amoun of nvenory a he end of perod, d = amoun of dspachng amoun n perod, and d = amoun of delvery o demand, We assume ha all producs have he same wegh and volume specfcaons. The objecve of he problem s o deermne x, y, d ) for,,, T and,,,, j,,, N so ha all demands are sasfed a he ( j j mnmum oal cos. We can represen a mxed neger lnear programmng (ILP) model. fj yj h I () T N N N (P) n S xj p xj s.. j j j N x I j j j j j d I,,, T () x W y, j,, N;,, T (3) L E d d,,,, (4) d 0,,, ; E,, L, (5)
d 0,,, ;,, E, (6) d 0,,, ; L,, T, (7) I T I 0 0. (8) x 0, y 0, I 0,,, T,,, (9) j j The consrans ()-(9) n he model P defnes a closed and bounded convex se, and he objecve funcon () s concave, so ha aans s mnmum a an exreme pon of he convex se. Alhough, he above mxed neger programmng model s que solvable for problems of small szes, s also obvous ha a heursc alernave wll be of grea neres for many applcaon as he sze of problem ncreases. Hence, we focus on genec algorhms o oban near-opmal soluons for he problems n a reasonable CPU me. Represenaon and Inalzaon X x GENETIC ALGORITHS The proper represenaon of a soluon plays a key role n he developmen of a genec algorhm. Le and we have a se of hree decson varables x, D, y. We frs nalze dspachng amoun D and hen X s represened by 0 and. Once he purchasng amoun used y and s quany x are deermned for he each purchasng amoun X we nalze he purchasng amoun X purchasng amoun based on he dspachng amoun usng he zero-one encodng, n whch he X s deermned, he vehcle ypes usng a local search heursc. A) Inalzaon D : Snce he dspachng amouns of demand, we adop wo-dmensonal real represenaon o nalze he dspachng amoun and chalewcz (99). by d have wo dmenson such as plannng horzon and D as he work of Vgaux B) Inalzaon X : Once he dspachng amoun D s nalzed, demand of each perod can be calculaed d d k k. Based on he demand n each perod, we can nalze he purchasng amoun X o avod nonnegave nvenores I for each perod. We represen a chromosome for purchasng amoun usng he zero-one encodng. In hs represenaon, we use he srng of T dgs (where T s he number of perods). Each dg may ake 0 or. 0 ndcaes no order n correspondng perod and a ndcaes an order n he correspondng perod for s demand and he demand of all subsequen perods wh a 0 code. 0 ndcaes no order n correspondng perod and a ndcaes an order n he correspondng perod for s demand and he demand of all subsequen perods wh a 0 code. Fgure descrbes a decodng process of he zero-one encodng of a chromosome 000 for he demand of 4, 3, 6, 7, 0, and from perod o 6, respecvely. Snce he frs gene of he chromosome s and wo consecuve genes are 0, he purchasng amoun of he frs perod x s 3. Smlarly, x and 4 x are 7 and. 6 FIGURE DECODING PROCESS OF THE ZERO-ONE CHROOSOE Chromosome: (, 0, 0,, 0, ) X = 3 0 0 7 0 I = 9 6 0 0 0 0 3 4 5 6 d = 4 3 6 7 0
C) Reparng X : To manan soluon feasbly, he genes of any chromosome are always force o zero before he frs posve demand and he gene wh he frs posve demand forces o one. Whenever nfeasble soluons are found durng he crossover and muaon operaons of D and X, he reparng process s execued. D) Selecon of vehcle ypes for shppng he purchasng amoun y : Once he purchasng amoun perod s consruced from decodng of 0- chromosome, we hen have o deermne he vehcle ypes used quany x for he correspondng purchasng amoun X by he concep of he margnal cos heursc per un-produc. To deermne he vehcle ypes used y X of each y and s n each perod. The selecon of vehcle ypes s deermned and s quany x, we propose a smple heursc usng he margnal fregh cos per un-produc for each vehcle. Snce he GA ncludes a local search heursc o deermne he vehcle ype y, we call our algorhm a hybrd genec algorhm (H-GA). The dealed heursc procedure s descrbed as follows: Sep. For each seleced perod, calculae he un-fregh cos, lowes. j Sep. Oban he number of he vehcle ype p, p x w x x x Sep 3. Selec he vehcle ype p chosen, where Objecve and Fness Funcon k N wh he lowes k f, x w k F. k fk x w, oherwse f w and selec he vehcle ype p wh he j j n. Se y n and x n w. Then updae p p p p p F and se k x w y and xk x, oher vehcles s no Once he a se of decson varables x, D, y and dependng varable I s deermned based on he X, D, we can oban he objecve funcon value defned by he model P. Snce all of he problems under consderaon are mnmzng problems, however, we have o conver he objecve funcon values o fness funcon values of maxmzaon form. We use he smples procedure o ge he fness value for a chromosome (,e, F ) from he objecve funcon value of he chromosome (.e., as follows: Reproducon, Crossover and uaon begn A) The crossover operaor Z ) and he maxmum objecve funcon value among he populaon (.e., F Z max (0) Z The crossover n he dspachng amoun D s descrbed as follows: Selec wo chromosomes D d k and D d k Sep. Creae wo emporary marces P p k and q k p d d / k k k and q d d mod. k k k end. The crossover s performed n hree seps: Q as follows: Sep. Dvdes marx Q no wo marces Q and Q such ha Q Q Q, where, T T q k q k, for k. Sep3. Then we produce wo offsprng of D * D Q * and D D Q * D and The crossover n he purchasng amoun X crossover mehod. * D as follows: Z ) max used one-cu-exchange whch s he eases and he mos classcal
B) The muaon operaor uaon produces sponaneous random changes n varous chromosomes. For he muaon of he dspachng D, he enre amoun n a seleced perod whn he delvery me wndow moves o a perod n he common me wndow, n whch s co-perods n he me wndows of mulple demands. By locang dspachng amouns n dfferen demands o he common me wndow, he probably ha larger amoun can be ordered and delvered a he same perod ncreases. Then, nvenory and purchasng coss can be reduced by he muaon. The dealed muaon procedure n he dspachng D for boh splng demand and non-splng demand are he same and descrbed as follows: begn repea selec a random real value r from 0,. f r P, Sep. selec a random neger number b from,. Sep. fnd E, L b b a row b n D and change E E and L L and choose a movng poson b b s from a common me wndow E, T. Sep 3. consruc a common wndow whle ( n N ) Sep 3. selec a random p from, Sep 3. fnd E, L p p. Sep 3.3 f E, L and E, L p p has a common perod, updae E maxe, E p and L mnl, L p. end f Sep3.4 ncrease n by. end whle Sep 4. selec a random perod e a row b from E,T Sep5. remove d from perod s o e a row b add no d. bs be end f unl ( he repeang number equals T ) end The muaon n he purchasng amoun X uses a random change beween 0 and. C) The reproducon operaon Adopng he els sraegy, wo bes chromosomes are excluded from he crossover and muaon procedure and hey are drecly coped o he nex generaon. Oher chromosomes are seleced for nex generaon usng roulee wheel selecon. COPUTATIONAL RESULTS To evaluae he performance of he H-GA, he followng expermenal condons were desgned: () Se T 0,,4,8, 4 and 5%,50%,75% of each T, respecvely and, se N 3, 4, 5 () Se he sze of me wndow, TW 30%, 40%, 50% of each T, respecvely, (3) Demands were generaed from a normal dsrbuon N dsrbuon U 300,900 and k was equally lkely seleced from,, where k k k was generaed from an unform and / 5, (4) h, p 7 and S TS / was assumed whou loss of generaly and TS,3, 6 where TS denoes EOQ me supply, (5) Se W 00, 00, 300 and W j f W 0. j. W j and respecve un fregh cos was seleced from H-GA heursc was coded usng C++ and run on a lapop compuer wh an Inel(R) Penum 4 CPU.66GHz wh 48 RA. To fnd he opmal soluon, CPLEX 6.0. package was used. Crossover probably and muaon probably of a chromosome se X, D are P C 0. 5 and P 0. 05. Four replcaons were performed for each combnaon of npu parameers. Toal 60 nsances were esed n boh CPLEX and H-GA heursc. j j
Due o he lmaon of a compuer performance, he opmal soluon was no obaned whn hours for many large-szed es problems havng more han or equals o 4 perods. Whereas, for many small-szed es problems havng less han or equals o perods, he opmal soluon was found. So, CPLEX package was modeled and run so as o fnd he bes soluon whn,000,000 node lms. eanwhle, H-GA found a bes soluon no more han 0 seconds n every nsances. To evaluae he performance of he heursc, he average percen gap beween he bes soluon and he heursc soluon was compued as follows: Z Z 00 Z, () H B B where Z objecve value of he bes soluon and Z objecve value of he heursc soluon. B H Tables,, and 3 presen he average percen gap beween he bes soluon and he heursc soluon n me wndow sze of 30%, 40%, and 50% of he lengh of plannng perod, repecvely.. The negave value of he average percen gap mples ha he heursc soluon s beer han he bes soluon on he average. The dfferences of he average percen gaps beween he bes soluon n CPLEX and heursc soluon n H-GA vary from -0.6 o.5. The resuls n H-GA works more effcen as T and ncreases because CPLEX gves poor soluons due o ermnon by node-lms. The ables show ha he average gap becomes large as he number of vehcle ypes and conaner sze ncreases. The ables show ha he average gap becomes smaller as he fregh cos under same conaner wegh ncreases, whch means ha he heursc become less effecve when fregh cos per wegh has less crcal effec on he oal cos. In hs case, may no necessary o conan producs fully o save fregh cos. Snce we use he zero-one-wo encodng for he purchasng amoun, here s more possbly ha he producs are fully conaned o mee conaner capacy. Therefore, H-GA usng he zero-one encodng resuls n a poor performance when fregh cos per wegh s no sgnfcanly effec on he oal cos. The average gaps provde he mnmum a he wndow sze 40% of plannng horzon by showng 0.7%. Ths reason s ha s easy o oban he opmal soluon when he wndow sze s small. eanwhle, he searchng space n he heursc becomes larger whn he common me wndow and s dffcul o oban a good soluon, when he wndow sze s oo large. In an average sence, he heursc offers good soluons whn 0.9%, 0.7%, and 0.37% n comparson wh he bes soluon. TABLE AVERAGE GAP FRO THE BEST SOLUTION FOR 30% TIE WINDOW V W gap(%) T = 0 T = T = 4 T = 8 T = 4 5% 3 00 0.00 0.00 0.00 0. 0.04 00 0.00 0.34 0.03.4 0.09 300 0.00 0.00 0.88 0.06 0.4 4 00 0.00 0. 0.04 0.00 0.4 00 0.00 0.00 0.55 0.00 0. 300 0.00 0.00 0.00 0.0 0.0 5 00 0.00 0.06 0.34 0.08 0.03 00 0.00 0.00 0.7 0.69 0.89 300 0.00 0.00 0. 0.50 0.39 50% 3 00 0.7 0.03.48.0 0.6 00 0.4 0.00 0.4 0.6 0.74 300 0.33 0.30 0. 0.64 0.30 4 00 0.0 0.0.5 0.07 0.0 00 0.00 0.93 0.09 0.58 0.4 300 0.5 0. 0.03 0.9.4 5 00 0.03 0.08 0.08 0.4 0.59 00 0.08 0.0 0.0 0.7 0.36 300 0.00 0.4 0.4 0.40.04
75% 3 00 0.9 0.36 0.9 0.06 0.3 00 0. 0.63-0.6 0.7-0.04 300 0.5 0.9 0.30 0.00 0.5 4 00 0.8 0.05 0.5 0.0 0.94 00 0.05 0.34 0.00 0.9 0. 300 0.9 0.0 0.33 0.50 0.54 5 00 0.3 0.9 0.06 0.35-0.07 00 0.04 0.00 0.7. 0.49 300 0.63 0.07 0.9.0 0.56 Average 0.9 TABLE AVERAGE GAP FRO THE BEST SOLUTION FOR 40% TIE WINDOW V W gap(%) T = 0 T = T = 4 T = 8 T = 4 5% 3 00 0.00 0.00 0.00 0.00 0.00 00 0.00 0. 0.00.9 0.3 300 0.00 0.00 0.00 0.00 0.74 4 00 0.00 0.00 0.00 0.00 0.80 00 0.30 0.00 0.00 0.00 0.7 300 0.00 0.00 0.6 0.00 0.00 5 00 0.00 0.00 0.07 0.00 0.9 00 0.47 0.07 0.6 0.00 0.5 300 0.00 0.00 0.00 0.8 0.7 50% 3 00 0.03 0.00 0.00 0.00 0.00 00 0.00 0.00 0.00 0.00 0.0 300 0.00 0.00 0.5 0.00 0.68 4 00 0.00 0.00 0.8.06 0.83 00 0.4 0.00 0.00 0.00 0.3 300 0.00 0.6 0.83 0.66 0.35 5 00 0.00 0.00 0.5 0.50 0.53 00 0.00 0.03 0.88 0.00 0.7 300 0.00 0.0 0.00 0.00 0. 75% 3 00 0.39 0.00 0.33 0.08 0.6 00 0.00 0.00 0.00 0.00 0.06 300 0.00 0. 0.00 0.55 0.00 4 00 0.00 0.00 0.00 0.00-0.06 00 0.00 0.8 0.00 0.08 0.09 300 0.4 0.00 0.00 0.45 0.4 5 00 0.46 0.7 0.89 0.08 0.46 00 0.07 0.05 0.00 0.9 0.7 300 0.87 0.00 0. 0.36 0.6 Average 0.7
TABLE 3 AVERAGE GAP FRO THE BEST SOLUTION FOR 50% TIE WINDOW V W gap(%) T = 0 T = T = 4 T = 8 T = 4 5% 3 00 0.00 0.00 0.00 0.00 0.00 00 0.00 0.00 0.00 0.00 0.00 300 0.00 0.00 0.00 0.5 0.00 4 00 0.00 0.00 0.00 0.00. 00 0.00 0.00 0.35 0.00 0.00 300 0.00 0.00 0.00 0.00.39 5 00 0.00 0.00 0.00 0.00 0.00 00 0.00 0.00 0.00 0.00 0.7 300 0.00 0.00 0.00 0.43.3 50% 3 00 0.85 0.00 0.00 0.00.06 00 0.00 0.00 0.4 0.00.03 300 0.00 0.00 0.63 0.49 0.34 4 00 0.00 0.8 0.00 0.7 0.5 00 0.84 0.00 0.8 0.73 0.39 300 0.00 0.00.4 0.97.7 5 00 0.00 0.00 0.00 0.88 0.97 00 0.45 0.00 0.00 0.46 0.47 300 0.56.5 0.33.00.80 75% 3 00 0. 0.00 0. 0.00.0 00 0.56 0.00 0.33 0.00 0.85 300 0.00 0.7 0.64 0.00 0.85 4 00 0.43 0.50 0.44 0.00 0.95 00.7 0.00 0.4 0.00 0.33 300 0.00 0.5 0.4 0.00.5 5 00 0.4 0.00 0.33 0.00 0.87 00 0.00 0.37 0.38 0.00.06 300 0.84 0.4 0.79 0.00.50 Average 0.37 CONCLUSION Ths paper analyzed a dynamc nbound orderng and shppng schedulng problem for a sngle produc ha s ranspored from a suppler o TPL warehouse by varous fregh vehcle modes and dspached o realers n a supply chan. Snce he large-szed problem canno fnd an opmal soluon usng he convenonal opmzaon solvers, we proposed a hybrd genec algorhm (H-GA). To evaluae he performance of he heursc, we presened he compuaonal resuls from a se of smulaon expermen. In an average sence, he heursc offered good soluons whn 0.9%, 0.7%, and 0.37% n comparson wh he bes soluon for gven es problems. Furher research wll be consder a dynamc nbound orderng, and shppng schedulng, and dspachng problem for mul-produc case. ACKNOWLEDGEENTS Ths work was suppored by he Korea Research Foundaon Gran funded by he Korean Governmen (OEHRD, Basc Research Promoon Fund) (KRF-008-33-D0).
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