An Efficient Extension of Network Simplex Algorithm

Transcription

1 Journal of Industral Engneerng 2(2009)1-9 An Effcent Extenson of Network Smplex Algorthm Hassan Rashd a, Edward P. K. Tsang b a School of Computer Scence and Electronc Systems Engneerng, Unversty of Essex, Colchester CO4 3SQ, U.K., Emal: hrash@essex.ac.uk b School of Computer Scence and Electronc Systems Engneerng Unversty of Essex, Colchester CO4 3SQ, U.K., Emal: Tel: Receved 3 Nov., 2008; revsed 15 Jan., 2009; accepted 12 Feb Abstract In ths paper, an effcent extenson of network smplex algorthm s presented. In statc schedulng problem, where there s no change n stuaton, the challenge s that the large problems can be solved n a short tme. In ths paper, the Statc Schedulng problem of Automated Guded Vehcles n contaner termnal s solved by Network Smplex Algorthm (NSA) and NSA+, whch extended the standard NSA. The algorthms are based on graph model and ther performances are at least 100 tmes faster than tradtonal smplex algorthm for Lnear Programs. Many random data are generated and fed to the model for 50 vehcles. We compared results of NSA and NSA+ for the statc automated vehcle schedulng problem. The results show that NSA+ s sgnfcantly more effcent than NSA. It s found that, n practce, NSA and NSA+ take polynomal tme to solve problems n ths applcaton. Keywords: Schedulng, Contaner Termnals, Mnmum Cost Flow Problem, Network Smplex Algorthm, Optmzaton Methods. 1. Introducton The Mnmum Cost Flow (MCF) problem s the problem of flowng resources from a set of supply nodes, through the arcs of a network, to a set of demand nodes at mnmum total cost, wthout volatng the lower and upper bounds on flows through the arcs (whch represent the capactes of the arcs). Ths problem arses n a large number of ndustres, ncludng agrculture, communcatons, defence, educaton, energy, health care, manufacturng, medcne, retalng, and transportaton[1]. Ths paper has been motvated by a need to schedule Automated Guded Vehcles (AGVs) n contaner termnals. The contaner termnal components that are relevant to our problem nclude quay cranes (QC), contaner storage areas, rubber tyred gantry crane (RTGC) or yard crane, and a road network [5], [39]. A transportaton requrement n a port s descrbed by a set of jobs, each of whch beng characterzed by the source locaton of a contaner, the destnaton locaton and ts pck up or drop-off tmes on the quay sde by the quay crane. Gven a number of AGVs and ther avalablty, the Task s to schedule the AGVs to meet the transportaton requrements. Network Smplex Algorthm (NSA) s the fastest algorthm to tackle the MCF model [1]. Prcng scheme s certanly an mportant step n NSA snce the total computatonal effort to solve a problem heavly depends on ts choce. Ths step does two thngs. It checks whether the optmalty condtons for the non-basc arcs are satsfed, and f not t selects a volated arc to enter the spannng tree structure [1]. The selected arc has a potental of mprovng the current soluton. Accordng to the theory [1] the NSA termnates n a fnte number of teratons regardless of whch proftable canddate s chosen f degeneracy s treated properly. Some wellknown schemes n NSA are the steepest edge scheme (by Goldfarb and Red [21]), the Mulvey s lst (by Mulvey [21]), the block prcng scheme (by Grgorads [15]), the BBG Queue prcng scheme (by Bradley, Brown and Graves [21]), the clusterng technque (by Eppsten [11]), the multple prcng schemes (by Lobel [24]), the general prcng scheme (by Istvan [20]). In ths paper we present a new prcng scheme, whch sgnfcantly reduces the CPU-tme requred to tackle the MCF model. By usng the new prcng scheme, we obtan an effcent extenson of 1

2 Hassan Rashd et al. /An Effcent Extenson of Network Smplex Algorthm NSA, whch called Network Smplex plus Algorthm (NSA+). The structure of ths paper s as follows. Secton 2 revews the schedulng problem of Automated Guded Vehcles (AGV) n contaner termnals. Secton 3 presents two algorthms to tackle the MCF-AGV model, namely Network Smplex Algorthm (NSA) and Network Smplex plus Algorthm (NSA+). Expermental results from applyng the two algorthms to tackle the model are compared n Secton 4. Secton 5 s consdered to summary and concluson. 2. The schedulng problem of Automated Guded Vehcles (AGV) n Contaner Termnals In order to test that the new extenson of Network Smplex Algorthm s effcent, we choose the most challengng problem n contaner port. The problem s the AGV schedulng problem n the contaner termnals and t s the same as the problem presented n [37]. Here, we have an overvew on the problem. For more detal, readers can refer to [36]. The most mportant reason for choosng ths problem s that the effcency of a contaner termnal s drectly related to use the AGVs wth full effcency The Assumptons The followng assumptons are consdered to defne the AGV schedulng problem n the contaner termnals: Assumpton-1: It s assumed that the problem nvolves only one shp. For the shp, n contaners jobs must be transported from the quay-sde to the year-sde or vce versa. The source and destnaton of the contaners jobs as well as ther appontment tme on the quay-sde are gven. To load/unload the contaners from a vessel or n the yard, a QC or RTGC s used. Assumpton-2: The RTGCs or yard crane resources are always avalable,.e., the AGVs wll not suffer delays n the storage yard locaton or watng for the yard cranes. Assumpton-3: There s a predetermned crane job sequence, consstng of loadng jobs, or unloadng/dschargng jobs, or a combnaton of both for every QC. Gven a specfed job sequence, the correspondng drop-off (for loadng) or pckup (for dschargng) tmes of the jobs on the quaysde depends on the work rate of the quay cranes. After the shp docked at the quay-sde, the appontment tme of the j th job s calculated by the followng expresson : AT j = Shp-docked-tme + j W. The Shp-docked-tme s the tme at whch the shp s ready for dscharge/loadng on the quay-sde. The tme wndow W s the duraton of dschargng/loadng a contaner. Assumpton-4: We are gven a fleet of V={1,2,.., V } vehcles. Each vehcle transports only one contaner. At the start of the process, the vehcles are assumed to be empty. Assumpton-5: It s assumed the vehcles move wth an average speed so that there are no Collsons, Congeston, Lve-locks, Deadlocks 0[35] and breakdown problem. Assumpton-6: We assumed the contaner jobs are dstrbuted n the termnal so that each pckup/drop-ff) pont s vsted once only by a vehcle. In other word, a QC and RTGC are not busy n each node by dfferent contaner jobs at the same tme. Assumpton-7: In ths schedulng problem, our goal s to deploy the AGVs such that all the mposed appontment tme constrants are met wth mnmum cost. Our objectves are to mnmze (1) the total AGV watng tme on the quay sde; (2) the total AGV travelng tme n the route of port; (3) the total lateness tmes to serve the jobs The formulaton Snce the vehcles are Sngle-Load AGVs (see the Assumpton-4), the problem can be converted to a Mnmum Cost Flow (MCF) problem. For more detals on the MCF problem and the schedulng AGV problem, readers can refer to [37], [36]. The MCF s a well-known problem n the area of network optmsaton,.e. the problem s to send flow from a set of supply nodes, through the arcs of a network, to a set of demand nodes, at mnmum total cost, and wthout volatng the lower and upper bounds on flows through the arcs. The problem for two vehcles and four jobs s demonstrated n the Fgure-2. In the fgure the supply nodes are denoted by A1 and A2. Each of these nodes has a one unt supply. There s only a demand node n the MCF problem. Ths node has -2 unts demand. The drected arcs from A1 and A2 to the demand node must be added to the network model. These arcs show that an AGV can reman dle wthout servng any job. Therefore, a cost of zero s assgned to these arcs. The lower bound, upper bound and cost of each arc are noted by the trplex [Lower Bound, Upper Bound, Cost]. Solvng the MCF problem generates 2 paths (the number of vehcles), each of whch commences from a vehcle node and termnates at the demand node. Each path determnes a job sequence of every vehcle. Suppose that for some values of arc costs, the paths gven by a soluton are A and A Ths states that AGV 1 s assgned to serve jobs 1 and 4, and AGV 2 s assgned to serve jobs 2 and 3, respectvely. 2

3 Journal of Industral Engneerng 2(2009) The Algorthms In ths secton, two algorthms to tackle the problem, Network Smplex Algorthm (NSA) and Network Smplex plus Algorthm (NSA+) are presented. NSA+ s an extended NSA wth three enhanced features Network Smplex Algorthm (NSA) Every connected graph has a spannng tree [1]. The network smplex algorthm mantans a feasble spannng tree at each teraton and successfully goes toward the optmalty condtons untl t becomes optmal. At each teraton, the arcs n the graph are dvded nto three sets; the arcs belong to the spannng tree (T); the arcs wth flow at ther lower pound (L); the arcs wth flow at ther upper bound (U). A spannng tree structure (T, L, U) s optmal f the reduced cost for every arc (,j) L s greater than zero and at the same tme the reduced cost for every arc (,j) U s less than zero [3]. Wth those condtons, the current soluton s optmal. Otherwse, there are arcs n the graph that volate the optmal condtons. An arc s a volated arc f t belongs to L (U) wth negatve (postve) reduced cost. The algorthm n Fgure-2 specfes steps of the method [21]. Fg. 1. The MCF model for 2 AGVs and four contaner jobs. To create the ntal or Basc Feasble Soluton (BFS), an artfcal node 0 and artfcal arcs are appended to the graph. The node 0 wll be the root of spannng tree (T) and the artfcal arcs, wth suffcently large costs and capactes, connect the nodes to the root. The set L conssts of the man arcs n the graph, and the set U s empty [1]. Appendng the enterng arc (k, l), whch s a volated arc, to the spannng tree forms a unque cycle, W, wth the arcs of the bass. In order to elmnate ths cycle, one of ts arcs must leave the bass. The cycle s elmnated when we have augmented flow by a suffcent amount to force the flow n one or more arcs of the cycle to ther upper or lower bounds. By augmentng flow n a negatve cost augmentng cycle, the objectve value of the soluton s mproved. The frst task n determnng the leavng arc s the dentfcaton of all arcs of the cycle. The flow change s determned by the equaton θ = mn { f j for all (, j) W}. The leavng arc s selected based on cycle W. The substtuton of enterng for the leavng arc and the reconstructon of new tree s called a pvot. After pvotng to change the bass, the reduced costs for each arc (, j) T s calculated. If the reduced costs for all (, j) {L + U} satsfy the optmalty condton, then the current basc feasble soluton s optmal. Otherwse, an arc (, j) where there s a volaton should be chosen and operatons of the algorthm should be repeated. Dfferent strateges are avalable for fndng an enterng arc for the basc soluton. These strateges are called prcng rules. The performance of the algorthm s affected by these strateges. The standard textbook [1] provded a detaled account of the lterature on those. Grgorads [15] descrbes a very smple arc block prcng strategy based on dvdng the arcs nto a number of subsets of specfed sze. At each teraton, the enterng arc s selected from a block wth most negatve prce. Andrew [4] studed practcal mplementaton of mnmum cost flow algorthms and clamed that hs/her mplementatons worked very well over a wde range of problems [4]. Masakazu [25] used a prmal-dual symmetrc pvotng rule and proposed a new scheme n whch the algorthm can start from an arbtrary par of prmal and dual feasble spannng tree [25]. Eppsten [11] presented a clusterng technque for parttonng trees and forests nto smaller sub-trees or clusters [11]. Ths technque has been used to mprove the tme bounds for optmal pvot selecton n the prmal network smplex algorthm for mnmum-cost flow problem. Lobel [24] developed and mplemented the 3

4 Hassan Rashd et al. /An Effcent Extenson of Network Smplex Algorthm multple prcng rules to select an enterng arc, a mxture of several szes for the arc block [24]. A general prcng scheme for the smplex method has been proposed by Istvan [20]. Hs prcng scheme s controlled by three parameters. Wth dfferent settngs of the parameters, t creates a large flexblty n prcng and applcable to general and network smplex algorthms. Ahuja et al. [3] 1: Algorthm Network Smplex Method 2: Begn 3: Create Intal BFS; (T, L, U) 4: (k, l) enterng arc {L + U } 5: Whle (k, l) <> NULL Do 6: Fnd Cycle W {T + (k, l) } 7: θ Flow Change 8: (p, q) Leavng Arc W 9: Update Flow n W by θ 10: Update BFS; Tree T 11: Update node potentals 12: (k, l) enterng arc {L+ U} 13: End whle 14: End Algorthm developed a network smplex algorthm wth O(n) consecutve degenerate pvot [3]. He presented an antstallng pvot rule, based on concept of strong feasble spannng tree. The bass structure (T, L, U) s strongly feasble f we can send a postve amount of flow from any node to root along arcs n the spannng tree wthout volatng any of the flow bounds. Step 0: Create a Basc Feasble Soluton Step 1: Select an enterng arc Step 2: Determne the leavng arc Step 3: Exchange the enterng and leavng arc Fg. 2. The Network Smplex Algorthm (NSA) Istvan revewed a collecton of some known prcng schemes n the orgnal smplex algorthm [20]. They are Frst mprovng canddate, Dantzg rule, Partal prcng, Multple prcng and Sectonal prcng. These schemes can be appled to NSA. Frst mprovng canddate chooses the frst volate arc as the enterng arc. It s cheap but t usually leads to a very large number of teratons. In Dantzg rule all non-basc arcs are checked (full prcng) and one whch volates the optmalty condton the most s selected. Ths rule s qute expensve but overall s consderably better than the prevous method. The Partal prcng scans only a part of the non-basc arcs and the best canddate from ths part s selected. In the next step, the next part s scanned, and so on. In Multple prcng, some of the most proftable canddates (n terms of the magntude) are selected durng one scannng pass. They are updated and a sub-optmzaton s performed nvolvng the current bass and the selected canddates usng the crteron of greatest mprovement. The Sectonal prcng behaves as a knd of partal prcng, but n each teraton sectons or clusters of arc are consdered The Network Smplex plus Algorthm (NSA+) NSA+ s an effcent extenson of NSA. Compared wth the standard verson of NSA by Grgorads s blockng scheme [15] and mantanng the strongly feasble spannng tree [1], NSA+ has three new features. These features are concerned wth the startng pont/block for scannng volated arcs, the memory technque and the scannng method. The prcng scheme of NSA+ s desgned based on these features. There s a functon for the prcng scheme to fnd out an enterng arc. The pseudo-code for ths functon s llustrated n Fgure-3. The arcs n the graph of MCF model are dvded nto several blocks wth the same sze and each block s dentfed by a specfc number, known as Block-Number. For each problem, the number of blocks s calculated by dvdng the number of arcs n the graph nto the block s sze. At frst teraton, when the ntalzaton s needed and the packet s empty, the number of blocks s calculated and the frst one to be scanned for the optmalty condton s chosen (see the lnes 2-5). The functon selects the frst block randomly or by a heurstc method (based on locaton of the bggest cost, for example). Note that at frst teraton the lnes 6-9 don t perform anythng because the packet s empty (these wll be actvated from the second teraton and when the packet s not empty). Scannng of the arcs for volaton among dfferent blocks s chosen crcularly. At each scan one volatng arc (at most) from each block s put nto the packet as long as t has empty place and there s any volated arc (see the lnes 10-14). The capacty of the packet s more than the block s sze and the most volatng arcs are kept at the top of the packet. At the end of functon, f the packet s empty, the current soluton s optmal (see the lnes 15-17). Otherwse the packet wll be sorted n descendng order, based on the absolute value of the reduced costs, and the most volated arc wll be chosen as the enterng arc (see the lnes 18-19). 4

5 Journal of Industral Engneerng 2(2009)1-9 The memory technque wll be actvated from the second teraton. It uses a few elements at the top of the packet of the last teraton. The sze of ths memory may be a percentage of the block s sze. The reduced costs of the most volated arcs n the prevous teraton are recalculated (see the lne 6). If they volate the optmalty condtons agan, they are kept n the packet. Otherwse they must be removed from the packet, whch can be replaced by new volatng arcs (see the lnes 7-9). The reamng part of the functon acts as before. As we mentoned, there are two optons to choose the frst block to be scanned; Randomly and Heurstcally. Hence, NSA+ has two extensons: (a) NSA+R: The enterng arc functon chooses the frst block by Random selecton; (b) NSA+H: The enterng arc functon chooses the frst block by a Heurstc method (based on locaton of the largest cost n the graph). 1: arc Enterng_Arc_Functon 2: If Intalzaton s needed Then // the packet s empty 3: Calculate the number of blocks 4: Choose the Block-Number // Randomly or by a Heurstc method 5: End f 6: Recalculate the Reduced Costs of the most volated Arcs n the Packet 7: If the most volated elements satsfy the optmalty condtons Then 8: Remove the elements from the packet 9: End f 10: Whle the Packet has empty place AND there s any volated arc n the graph Do 11: Calculate the reduced cost of an arc from the block assocated wth the Block-Number. 12: Put the arc nto the Packet f t volates the optmalty condton. 13: Increase the Block-Number crcularly. 14: End Whle 15: If the Packet s Empty Then 16: Return Null // The Current Soluton s Optmal 17: End If 18: Sort the Packet Descendng // Based on the absolute value of the reduced costs by Quck Sort 19: Return the frst element of the Packet 20: End Functon Fg. 3. Pseudo-code of selectng an enterng arc n Network Smplex plus Algorthm 3.3 The dfferences between NSA and NSA+ The man dfference between NSA and NSA+ are n the prcng scheme and the enterng arc procedure. As we mentoned, the role of the prcng scheme s that how the enterng arc to be selected from the volated arcs n the graph. The dfferences between NSA and NSA+ are as flows: At each teraton, a packet of volated arcs from dfferent blocks s collected n NSA+ and the most volated arc s selected as the enterng arc, whereas NSA selects the most volated arc from one block. There s no memory technque n NSA whle NSA+ uses a few elements at the top of the packet for the next teraton. It benefts from the current volated arcs for the next teraton. The frst block s selected Randomly or by a Heurstc method n NSA+, whereas NSA always chooses the frst block for scannng the volated arcs. 4. Expermental Results from the mplementaton and runnng the algorthms We mplemented the standard verson of Network Smplex Algorthm (see Fgure-2). As we mentoned, the prcng rule or scheme to choose the enterng arc n Step 1 determnes the speed of algorthm. In the lterature, we revewed the prcng rules. Actually, there s the trade-off between tme spent n prcng at each teraton and the goodness of the selected arc n terms of reducng the number of teratons requred to reach the optmal soluton. The Frst mprovng canddate and Dantzg rule represent two extreme choces for the enterng arc. Other prcng schemes strke an effectve comprse between these two extremes and have proven to be more effcent n practce [1]. Kelly and Nell [21] mplemented several prcng schemes and ran ther software for dfferent classes of mnmum cost flow problems. In ther results, the block prcng scheme provded a better performance compared wth others. We therefore chose the block prcng scheme. Ths scheme s based on dvdng the arcs of the graph nto a number of subsets of specfed sze. A block sze of between 1% and 8.5% of the sze of the arcs n the graph has been recommended by Grgorads [21], for large MCF problems. We set the number to 5% by the try and error. To test the model and make a comparson between NSA and NSA+, a hypothetcal port was desgned. The parameters n Table-1 were used to defne the port. 5

6 Hassan Rashd et al. /An Effcent Extenson of Network Smplex Algorthm Table1 Value of Parameters for the smulaton Descrpton of the Parameters Values Number of Vehcles n the port 50 Number of Quay Cranes 7 Number of Blocks n the yard (Storage area nsde the port) 32 Tme Wndow of the Cranes Travellng Tme between every two ponts n the port (see Assumpton 1) 120 seconds Random between 1 and 100 seconds We mplemented our the software n Borland C++. Then, has been run to solve several random problems. The sources and destnatons of jobs were chosen randomly. The CPU-Tme requred to solve the problems by the two algorthms has been drawn n Fgure-4 and Fgure-5, accordng to the number of jobs and the number of arcs, respectvely. Also the power estmaton for those two curves has been shown on the fgures. All experments were run on a Pentum 2.2 GHz PC wth 1 GMB RAM. From the fgures, we can observe that: Observaton 1: NSA and NSA+ are run n polynomal tme to solve the MCF-AGV model, n practce. Observaton 2: NSA+ s fast and more effcent than NSA. There are two dfferent types of teraton n NSA, degenerate and non-degenerate [1]. In every nondegenerate teraton, the value of the objectve functon s decreased whereas degenerate teratons do not change the objectve functon s value. In the degenerate teratons, a flow change of zero causes cyclng. In the lterature, Grgorads experenced that cyclng s rare n practcal applcaton [15]. Observaton 1 confrms the experence. In order to confrm that NSA s run n polynomal tme to solve the MCF-AGV model (Observatons 2), we estmated complexty of the algorthm. The result shows that the CPU-Tme requred to tackle the problem, s a functon wth degree 3 of the number of jobs n the problem [36] The percentage of mprovement n CPU-Tme requred to tackle the problem In order to calculate the average CPU-Tme requred to solve the problems and to compare performance of the algorthms n ths applcaton, we ntroduce the followng terms: T NSA : The CPU-Tme used to solve the problem by NSA. T NSAH : The CPU-Tme used to solve the problem by NSA+H. 6 T NSAR : The CPU-Tme used to solve the problem by NSA+R. PIH : The Percentage of Improvement n CPU-tme used to solve the problem by NSA+H compared wth NSA. PIR : The Percentage of Improvement n CPU-tme used to solve the problem by NSA+R compared wth NSA. TPIH: The Total Percentage of Improvement n CPU- Tme used to solve the problems by NSA+H compared wth NSA. TPIR: The Total Percentage of Improvement n CPU- Tme used to solve the problems by NSA+R compared wth NSA. W: The Weght of mprovement for the problem. In ths experment we consder the number of arcs n the MCF-AGV model for the weght. Gven N jobs and M AGVs n the problem, the number of arcs s M+M N+N (N-1)+2 N. Now we calculate the percentage of mprovements n the CPU-Tme used for problem by the followng terms: PIH PIR 100 * ( T = T NSAH NSA 100 * ( T = T NSAR NSA T T NSA NSA The total percentages of mprovement n the CPU- Tme used to solve the problems by NSA+H and NSA+R, compared wth NSA, are calculated by the followng expressons: TPIH = TPIR = 32 = 1 32 = 1 W PIH = 1 = 1 W W PIR W = 35.16% = 21.28% 4.2. Statstcal test for the comparson The CPU-tme requred to solve the problems by the two algorthms, NSA and NSA+, were analysed statstcally. We tested the null hypothess that the means ) )

7 Journal of Industral Engneerng 2(2009)1-9 produced by the two algorthms were statstcally ndfferent. Snce we cared the change (the dfference between the two means) was postve or negatve, Onetal test was chosen. The result of Pared T-test along wth the crtcal values of T-dstrbuton for the partcular degree of freedom are shown n Table-2. The T-test confrms that NSA+ s sgnfcantly better than NSA wth 95% degree of confdence. CPU-Tme requred to solve the MCF-AGV Model Second Number of Jobs NSA NSA+H NSA+R Fg. 4. CPU-Tme to solve the statc problem by NSA and NSA+, based on the number of jobs Second CPU-Tme requred to solve the MCF-AGV Model Thousands Number of Arcs n the MCF-AGV NSA NSA+H NSA+R Fg. 5. CPU-Tme to solve the statc problem by NSA and NSA+, based on the number of arcs Table 2 The result of T-Test for the comparson between both algorthms, NSA and NSA+ Statstcal Parameters Number of Samples/Observatons T-Test (Pared Two Sample For Means ) NSA+H vs. NSA NSA+R vs. NSA Degree of Freedom Crtcal T-Value Complexty of NSA+ Gven N jobs and M AGVs n the problem (N>>M), the complexty of the NSA+ s calculated as follows: Assume that the maxmum flow, MF, n each of the m arcs, at maxmum cost, C, for the mnmum cost flow model. So there s an upper bound on the value of the objectve functon. Ths upper bound s gven by m C MF. There are two dfferent types of pvots n the algorthm, non-degenerate and degenerate pvots. The former s bounded by m C because the number of non-degenerate pvots n the algorthm s bounded by m C MF (MF=1 n

8 8 Hassan Rashd et al. /An Effcent Extenson of Network Smplex Algorthm the MCF-AGV model). The number of degenerate pvots s determned by the sum of nodes potental and mantanng the strongly feasble spannng tree. Gven n as the number of nodes n the graph model, the sum of nodes potental s bounded by n 2 C. It s decreased at each teraton when the spannng tree s strongly feasble [ 2]. A seres of degenerate pvots may occur between each par of non-degenerate pvots, and thus a bound on the total number of teratons s m n 2 C 2. Fnd the enterng arc s O(m) and sortng the packet s O(K LogK) operaton (K s sze of the packet, K=225 ). Fndng the cycle, amount of flow change, leavng arc and updatng the tree are O(n) operatons. Hence the complexty of each pvot s O((m + n) K LogK). Based on the complexty of the number of teratons and the complexty of each pvot, the total complexty of ths algorthm s determned by the followng equaton: 2 2 O (( m + n) mn C KLogK) Snce m=o(n 2 ) ; n=o(n), the total complexty of NSA+ to tackle the MCF-AGV model s O(N 6 ). 5. Concludng Summary In ths paper, two algorthms, NSA and NSA+, were appled to the automated guded vehcles schedulng problem n contaner termnals. Our expermental results suggested that NSA could fnd the global optmal soluton for 2,600 jobs and 7 mllons arcs n the graph model wthn 70 seconds by runnng on a 2.2 GHz Pentum PC. NSA+ has enhanced features over NSA and t s faster. The most effectve feature of NSA+ s a memory technque and scannng method, whch can be appled to Orgnal Smplex Algorthm n Operaton Research. References [1] R. K. Ahuja, T. L. Magnant, J. B. Orln, Network Flows: Theory, Algorthms and Applcatons, Prentce Hall, [2] R. K. Ahuja, J. B. Orln, M. S. Govann, P. 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