APPLICATION OF IMPROVED K-MEANS ALGORITHM IN THE DELIVERY LOCATION

Transcription

1 An Open Access, Onlne Internatonal Journal Avalable at Vol. 6 (2) Aprl-June, pp /Sh Research Artcle APPLICATION OF IMPROVED K-MEANS ALGORITHM IN THE DELIVERY LOCATION *Mng Sh Department of Mathematcs and Informaton Scence, Hebe Unversty, Baodng , Chna *Author for Correspondence ABSTRACT Ths paper had proposed an mproved k-means algorthm and appled t to a lvng eample of delvery locaton. At frst, establshed an undrected weghted graph based on the actual path; Then, appled Floyd algorthm to calculate a nearest path; Fnally, utlzed a mproved k-means algorthm whch the Eucld dstance was substtuted by the nearest path to cluster and acqured a ratonal outcome. Therefore, t verfed ths algorthm could be employed n practcal stuaton. Keywords: Undrected Weghted Graph, the Shortest Path Matr, Floyd Path, K-Means Algorthm INTRODUCTION Quantty of goods delvery s drectly related to the epress company's earnngs. Therefore, whether the substaton and ther delvery locaton s reasonable becomes mportant. These stes whch the path between of them s shortest are the most mportant crtera to select these ponts. At present, there are a number of researchers had carred out research and eploraton n ths area. Accordng to the superorty-nferorty of the estng k-means, (Gu et al., 2010) desgned a new equlbrum dstrbuton regon dvson method, and verfed the practcablty and valdty of ths method; (Yan et al., 2000) obtaned a shortest path algorthm of cty road network, and verfed ts practcalty and relablty through the GIS; (Bao, 2001) adusted apcal order of the estng method, and optmzed the Dkstra algorthm, sgnfcantly mproved the computng speed and algorthmc effcency. Combned wth actual stuaton, the artcle puts forward an mproved k-means algorthm, usng the Floyd path nstead of the commonly Eucldean dstance, to determne the number k and the clusterng center grounded on the actual crcumstance. It can avod the result s blndness or rratonalty, and the result conforms to the actual need, therefore can be appled nto the actualty. Floyd Shortest Path Algorthm Almost all of the classc clusterng algorthms are based on the Eucldean dstance. The Eucldean dstance evaluates the smlarty of data s obtaned from the lnear propertes between two ponts n space. In actually, the real dstance between two pont restraned by the actual crcumstance s possble far greater than the Eucldean dstance. It only obtans a result dstorted the fact by utlzng the Eucldean dstance under the crcumstance. Therefore, t s the cru whether the adopted dstance s reasonable to realze the algorthm. Calculaton of dstance algorthm s named the shortest path algorthm, ncludng Dkstra algorthm, A* algorthm, Johnson algorthm, Floyd algorthm (Zhang and Wu, 2009) and so on. These algorthms had been systematc classfcaton and comparson by (Lu, 2001). Centre for Info Bo Technology (CIBTech) 11

2 An Open Access, Onlne Internatonal Journal Avalable at Vol. 6 (2) Aprl-June, pp /Sh Research Artcle Floyd algorthm can be eploted to fnd the shortest path between every par of vertces n a weght graph wth postve or negatve edge weghts but no aggregatve cycles. It s endowed wth concson and elegance, and pretty easy to understand. In addton, t s smple and etra effectve, especally on dense graph. Related Concepts Some of the defntons used n ths paper are derved from (Bondy and Murty, 2008). Defnton 1 A undrected weghted graph G, showed n Fgure 1, s an ordered par ( e(g), v(g), G) consstng of a set v(g) of vertces and a set e(g), dsont from v(g), of edges, together wth an ncdence functon G that assocates wth each edge of G an unordered par of (not necessarly dstnct) vertces of G. If e s an edge, u and v are vertces, such that G( e) u, v, then e s sad to on u and v, and the vertces u and v are called the ends of e. We denote the numbers of vertces and edges n G by v(g) and e(g); these two basc parameters are called the order and sze of G, respectvely. Fgure 1: Undrected Graph Here, u, v, w, and y represents vertces, and a, b, c, d, e, f, g, h represents the edge. Defnton 2 The ncdence matr M of G, s showed at the left of the table 1, s the n mmatr M : m, where m ve s the number of tmes (0, 1, or 2), and verte v and edge e are G ve ncdent. Clearly, the ncdence matr s ust another way of specfyng a graph. Defnton 3 The adacency matr A of G, s showed at the rght of the table 1, s the n n matr A : G, where a uv s the number of edges onng vertces u and v, each loop countng as two a uv edges. Error! Reference source not found.aerror! Reference source not found.error! Reference source not found.table 1: Incdence Matrces M and Adacency Matrces A of a Graph G M A b c d e f g u v w y A u v w y u v w y Defnton 4 The shortest path matr A, can be seen from the Table 2, s defned as a n n matr. If Centre for Info Bo Technology (CIBTech) 12

3 An Open Access, Onlne Internatonal Journal Avalable at Vol. 6 (2) Aprl-June, pp /Sh Research Artcle vertces v(1 n), v(1 n), there s at least one path. A s the value of the element and d s the Floyd path of the vertces v to the vertces v. If vertces v and vertces v does not est a path between, than the value of element A s denoted. That s: A d mn W P Path from v to v for P n G When the vertces v to v s not reachable Table 2: The Shortest Path Matr A A Algorthm Prncple Floyd algorthm compares all possble paths through the graph G wth vertces V between each par of vertces. The shortest path d that returns the shortest possble path from to usng vertces only from the set {1, 2,,k} as ntermedate ponts along the path. For any vertces and any vertces, the true shortest path could be ether: (1) a path that goes drectly from to, or (2) a path that goes from through a number of others vertces to. Assumng d s the shortest path to the vertces to vertces, for each vertces k, eamnes the formula dk dk d to speculate whether t stll holds. When t s, eplans the path from vertces to vertces k and then to vertces s shorter than from vertces drectly to vertces, then sets d dk dk. When 3 traverses all vertces k, d s the shortest path that we wants to know. Its tme complety s On, and the 3 space complety s On. Process Some symbols gven by (Wang et al., 2010) used n the paper are gven as follows: w represents undrected graph weghted matr; v 0 and v t represents an arbtrary source verte and target verte respectvely; d(v 0, v t) s the shortest path to the source verte v to the target verte v (1 n, 1 n) ; A s the weght matr of the shortest path; Input: The weght matr w of the undrected graph; the source verte v 0; the target verte v t. Output: The source verte v 0 to the target verte v t, the shortest path d(v 0, v t). (0) (0) Step 1, ntalze the weght matr A d (, =1,2, n; k 1) ( Mao and Sh, 2016). nn Centre for Info Bo Technology (CIBTech) 13

4 An Open Access, Onlne Internatonal Journal Avalable at Vol. 6 (2) Aprl-June, pp /Sh Research Artcle w(, ) ( n) Among, d 0 =, s not adacent or have no way to go Step 2, when k, (k from 1 to n), for all the and, checks whether the d d k or d dk, f meet, then sets k=k+1, and contnues step2; otherwse, step 3; Step 3, compare the sze of d and d k d k, replace d wth the smaller, that s d mn d, dk dk, turn step 4; Step 4, to determne whether the k s less than n, f establshed, return to Step 2; otherwse, step 5; Step 5, output source pont v 0 to the target pont v t the shortest path d(v 0, v t). K-Means Clusterng Algorthm K-means algorthm (Lu, 2011) s popular for cluster analyss n data mnng. It dvdes n observatons nto k clusters n whch each observaton belongs to the cluster wth the nearest mean, servng as a prototype of the cluster. Prncple Gven a set of observatons X m m 1,2, total K, the sample of X s denoted by d descrpton attrbute A 1, A 2, A 3, A d and these attrbutes belong to contnuous attrbute. Data sample s ( 1, 2,... ), d ( 1, 2,... d ). Among them, 1, 2,... d and 1, 2,... d denote the specfc value of d descrpton attrbutes A 1, A 2, A 3, A d respectvely corresponded sample and sample. The smlarty between sample The defnton s as follow: and sample s usually denoted by ther Eucldean dstance d(, ). d k k 2 (1) k 1 d(, ) ( ) The smlarty s calculated by the average of obects n one cluster: 1 C (2) C Where C s the number of data pont n cluster. Clusterng performance s assessed by the crteron functon E: 1 E... (3) 2 k 1 C C For a data set of sze n, the specfed number of clusters s k, D s the dmenson of the data obect, the Centre for Info Bo Technology (CIBTech) 14

5 An Open Access, Onlne Internatonal Journal Avalable at Vol. 6 (2) Aprl-June, pp /Sh Research Artcle total tme complety of the algorthm s Ondk. Process Input: the number k of clusters D and a data set contanng n obect. Output: a set of k clusters that the crteron functon s the least. Step 1. Assgns arbtrary k obects from data D as the ntal cluster centers; Step 2. Calculates the smlarty between the remanng obects to the k cluster centers, and dvdes these obects nto a cluster to that the smlarty s least; Step 3. Recalculates k cluster center based on the crteron functon of every cluster; Step 4. Contnue to cluster based on k new cluster center; Step 5. Repeat; Step 6. Untl crteron functon s no more obvous changes. Improved K-Means Algorthm K-means s a classc algorthm, t has the advantages of smple prncple, and t s easy to mplement, especally for large data sets, hgh effcency and so on (Kanungo and Mount, 2002). It has been wdely used. But the algorthm also has many problems due to lmted: (1) It need a gven class number n advance; (2) The ntal cluster centers nfluence the effect and qualty of clusterng drectly; (3) to deal wth categorcal data; (4) It s senstve to outlers and can only be found n the sphercal classes; (5) sometmes fall nto local optmal soluton, and can't get the global optmal soluton. In order to mprove these shortcomngs, an mproved k-means algorthm s proposed that the Eucldean dstance s replaced by the Floyd path. Eample In order to verfy the feasblty and effectveness of the mproved k-means clusterng algorthm, the eample are gven to verfy the effectveness of the algorthm. Fgure 2 dsplays undrected weghted graph of a cargo transport lne. In ths fgure, seven crcles of v1 v2 v3 v4 v5 v6 v7 respectvely represents the customer's locaton, the number represents actual requred tme between the two vertces, the unt s hour (h), therefore, the fgure can be vewed as a tme path dagram. In order that the courer can complete the delvery task n the shortest tme, there are two vertces n these locatons that should be chosen as a dstrbuton ste. To ths end, Floyd algorthm s appled to calculate the shortest path matr A, as shown n Table 3. Fgure 2: Undrected Weghted Graph Centre for Info Bo Technology (CIBTech) 15

6 An Open Access, Onlne Internatonal Journal Avalable at Vol. 6 (2) Aprl-June, pp /Sh Research Artcle Table 3: Shortest Path A A Then we use the mproved k-means clusterng analyss algorthm to fnd the two dstrbuton stes: In the above matr A, d( v, v ) denotes the Floyd s dstance between the vertces v and the vertces v. Set the clusterng category k equals 2, and the clusterng algorthm process s as follows: Step 1: respectvely calculates the dstance between the vertces v3, v4, v5, v6, v7 and centrod of the v1, v2. As to centrod of v1, there s d(v1, v2) 8, d(v1, v3) 3,d(v1, v4) 7,d(v1, v5) 9,d(v1, v6) 12, d(v1, v7) 9. As to centrod of v2, there s d(v2, v3) 5,d(v2, v4) 9,d(v2, v 5)=7,d(v2, v6) 4,d(v2, v7) 8 Inde of smlarty s served as the Floyd path, t s easy to know from the shortest path matr, d(v1, v2) d(v 2, v1), d(v1, v3) d(v2, v3), d(v1, v4) d(v2, v4), d(v1, v5) d(v2, v5), d(v1, v6) d(v 2, v6), d(v1, v7) d(v2, v7). Therefore, the result of the clusterng s C1={v1, v3, v4} as a cluster, and C2={v2,v5, v6, v7} as another cluster. Step 2: updates the centrod of the cluster C1 and the cluster C2. d(, ) C 3 7 Accordng to the formula, obtan Average (v1, ) 5,.where d (v1, v3) =3 C 31 and d(v1, v 4) =7, the vertces closest approach to 5 s respectvely the vertces of v3 and v4, chooses the vertces v4 at random to substtute the quondam centrod v1 here. In the same way, chooses the vertces v7 to substtute the quondam centrod v2. Step 3: repeat Step 2, untl the v6 as the centrod of the cluster v6, v4, v5, v7 and v2 as the center of the clusterv2, v1, v 3, the mean found that the two clusters of the center of mass wll not change, so the end of the cluster, cluster and cluster s the fnal result. The clusterng results are showed n Fgure 3. Fgure 3: Clusterng Results Centre for Info Bo Technology (CIBTech) 16

7 An Open Access, Onlne Internatonal Journal Avalable at Vol. 6 (2) Aprl-June, pp /Sh Research Artcle The redness and the blackness represent there are two clusters, v4, v5, v6 and v7 s one cluster, and v1, v2, v3 s another cluster. The sold rm v6 and v2 are reprehensvely the centrod of the two clusters. Concluson In ths paper, based on the dstrbuton locaton center of epress company, started at the nfluencng factors that courer requred tme s the shortest n the logstcs dstrbuton, we apply Floyd algorthm to calculate the shortest tme path of courer requred, then adopts the mproved k-means clusterng algorthm to acheve ths goal. The customer locaton acts as the obect clustered of the k-means clusterng algorthm here, and the k nput value s n accordng to the actual stuaton choce rather than the eperence, to avod the blndness of the optonal k value. ACKNOWLEDGMENTS Ths paper s granted by NSF of Chna ( ) and NSF of Hebe provnce (A ). REFERENCES Bao PM (2001). A Optmzaton Algorthm based on dkstra s algorthm n search of shortcut. Journal of Computer Research & Development 38(3) 307. Bondy JA and Murty USR (2008). Graph Theory, (San Francsco: Sprnger Press, Calforna) Gu W, Zhang Q and Hu R (2010). Research of logstcs dstrbuton regon partton method based on mproved k-means clusterng. Unversty of Scence and Technology Beng 13(24). Kanungo T and Mount D (2002). An effcent k-means clusterng algorthm: Analyss and mplementaton. IEEE Transactons on Pattern Analyss and Machne Intellgence 24(7) Lu GH (2011). A Dkstra Dstance-based Clusterng Algorthm and Applcaton n Logstcs, (Chna, Lanzhou: Lanzhou Unversty) (n Chnese). Lu F (2001). Shortest Path Algorthms: Taonnomy and Advance n Research. Acta Geodaetcal Et Cartographc Sncal 30(3) Mao H and Sh M (2016). The applcaton of artcle K-th shortest tme path algorthm. Internatonal Journal of Physcs and Mathematcal Scences 6(1) Wang HY, Huang Q, L CT and Chu BZ (2010). Graph Theory Algorthm and ts MATLAB Implementaton, (Beng: Beng Behang Unversty Press, Chna) Yan HB and Lu YC (2000). A new algorthm for fndng short cut n a cty s road net based on GIS technology Chnese. Journal of Computers (n Chnese) 23(2) Zhang DQ and Wu GL (2009). Optmzed Floyd Algorthm for Shortest Paths Problem. Journal of Xuchang Unversty 28(2) Centre for Info Bo Technology (CIBTech) 17