Hassan Ghaziri AUB, OSB Beirut, Lebanon Key words Competitive self-organizing maps, Meta-heuristics, Vehicle routing problem,

Transcription

1 COMPETITIVE PROBABIISTIC SEF-ORGANIZING MAPS FOR ROUTING PROBEMS Haan Ghaziri AUB, OSB Beirut, ebanon Abtract In thi paper, we have applied the concept of the elf-organizing map (SOM) algorithm, to a routing problem called the vehicle routing problem with backhaul (VRPB). Uually SOM i baed on a ingle map to repreent data. The main contribution of thi paper i to introduce a new architecture formed of multiple competing map and to deign the correponding learning algorithm uch that the Kohonen map can be applied to a family of routing problem. Thoe problem are known to be NP-Hard problem. We have applied thee concept to the vehicle routing problem (VRP) and to a variant of thi problem called the VRP with Backhaul in which cutomer are of two ort : linehaul and backhaul cutomer. In deigning the route for the variou vehicle, linehaul cutomer hould be viited firt and then the backhaul cutomer. We benchmarked the performance of our approach with the mot powerful meta-heuritic and obtained excellent reult. Key word Competitive elf-organizing map, Meta-heuritic, Vehicle routing problem, 1 Introduction The VRPB problem we are conidering in thi paper conit of deigning delivery route to erve two type of cutomer; linehaul and backahul cutomer The following information i given: The Cutomer. The poition of the cutomer i provided through their coordinate in the Euclidean pace. Along with their poition, the weight or quantity of good, of each cutomer and it type are alo provided. There are two type, the backhaul and linehaul cutomer. A certain quantity of good i delivered from the depot to the linehaul cutomer and a certain quantity of good mut be picked up from each backhaul cutomer and brought to the depot. The Vehicle. A fleet of homogeneou vehicle i given. It mean that the number of vehicle i known and the capacity of all vehicle i the ame. Each vehicle can not hip a quantity of good that exceed it capacity. The Depot. The location of the depot i known and it coordinate in the Euclidean pace are given. Each vehicle tart it route from the depot viit the linehaul cutomer aigned to it for delivery and then collect the correponding good from each backhaul cutomer and bring them back to the depot.

2 WSOM 2005, Pari 2 SOM for Routing Problem: a Brief Review Artificial Neural network were ued to olve routing problem, namely the Traveling Saleman Problem (TSP). Hopfield (1995) network wa among the firt ued to tackle thi problem. In pite of it novelty, thi approach wa not practical and not competitive with exact method and metaheuritic. The application were retricted to mall ized problem not exceeding 20 to 50 citie. The architecture that i uually ued for routing problem and that wa introduced by Fort (1988) i the ring architecture. It conit of a et of neuron placed on a deformable ring. The concept of tour i embedded in the ring architecture, ince the location of a neuron on the ring can be identified with the poition in the viit a hown in the figure 2. Thi i the mot important advantage of thi approach. The ring can be conidered a a route for an ideal problem. The interaction of the network with it environment, here the cutomer and the adaptation of it neuron will force iteratively the ring to repreent the real tour that will viit equentially the cutomer. From a practical point of view, the SOFM algorithm tart by pecifying the architecture of the network, which conit of a one ring upon which the artificial neuron are patially ditributed. The ring i embedded in the Euclidian pace. Each neuron i identified by it poition in the Euclidian pace and it poition on the ring. The Euclidian ditance will be ued to compare the poition of neuron with the poition of citie. The lateral ditance will be ued to define the ditance on the ring between two neuron. The lateral ditance between any two neuron i defined to be the mallet number of neuron eparating them plu one. In the firt tep, a cutomer i randomly picked up, hi poition i compared to all poition of neuron on the ring. The nearet neuron to the cutomer i then elected and moved toward him. The neighbor of the elected neuron move alo toward the cutomer with a decreaing intenity controlled by the lateral function Kohonen (1982). An extenive analyi of thi algorithm could be found in the work publihed by Fort (1988), Angeniol et al. (1988) and Smith (1999). 3 SOM for the VRP with Backhaul The claical SOM approach conit of a neural network with a defined architecture that i interacting with it environment to repreent it according to the following principle: Two neighboring input hould be repreented by two neighboring neuron. The neighborhood relationhip between input i defined according to the Euclidean ditance in thi paper. The concept of neighboring neuron refer to neighborhood in the map formed by the neuron. Thi relationhip i defined according to connectivity. From a routing perpective, it i clear that two neighboring cutomer hould be aigned cloe poition in the routing chedule in order to minimize the total ditance traveled by the vehicle. In thi ection, we will explain how to extend the SOM to VRPB. Thi extenion i baed on the deign of a new architecture in which the TSP ring i replaced by a certain number of ring. Each ring repreent a vehicle. For more detail refer to [1,2]. In order to repreent the concept of linehaul cutomer and backhaul cutomer, each ring will conit of two part. The firt part i a equence of neuron that will interact excluively with linehaul cutomer. They are called linehaul neuron. The econd part of the ring conit of backhaul neuron, which mean that thee neuron will interact excluively with backhaul cutomer. Experiment how that thi architecture lack flexibility preventing the network from evolving adequately. Therefore, the concept of ring ha been replaced by the concept of chain. There will be two type of chain. inehaul chain are formed of linehaul neuron and backhaul chain formed of backhaul neuron. The linehaul cutomer will interact excluively with the linehaul chain,

3 Competitive Probabilitic Self-Organizing Map for Routing Problem the backhaul cutomer will interact excluively with the backhaul chain. It i clear that thoe chain will not form tour. Therefore a procedure ha to be implemented in order to repect the following requirement: Ring hould be formed to repreent the equence according to which the cutomer are viited Each ring mut be formed of a linehaul equence followed by a backhaul equence. The linehaul equence repreent the chedule of viiting the linehaul cutomer erved by the correponding vehicle and the backhaul equence repreent the chedule of viiting the backhaul cutomer erved by the ame vehicle. Each ring mut pa by the depot. Accordingly, the procedure conit of connecting the chain uch that the requirement are repected. Conequently, four type of interaction are introduced to generate a feaible VRPB tour. et u introduce the following notation: = {l = (x 1, x i i i, x ), for i =,.., N i 1} be the et of N 1 inehaul cutomer where ( x 1i, x 2i ) are the coordinate of the inehaul cutomer l i and x 3i it weight. B = { b =(y, y 2, y 3 ), for = 1,..,N 2} 1 be the et of N 2 backhaul cutomer where ( y, 1 y 2 ) are the coordinate of the backhaul cutomer b and y 3 it weight. D = (x d, y d) be the coordinate of the depot. V = { vk, for k = 1,,N v} be the et of vehicle where Nv i the number of vehicle Q i the vehicle capacity. Q i fixed becaue we have a homogenou fleet. W m : i the current amount of good delivered to inehaul cutomer by the vehicle vm, where m=1,...,n v G n : i the current amount of good picked up at backhaul cutomer by the vehicle vn, where n=1,,n v N k = {N kl, N kb } i the et of cutomer erved by vehicle vk, where N kl i the et of linehaul cutomer and N kb i the et of backhaul cutomer, where k =1,...,N v Cm = { =( X X ) for = 1,..., Nm } be the et of N m connected neuron forming the inehaul m m m, 1 2 m m chain of neuron., where m = 1,..., N y and ( X 1, X 2 ) are the coordinate in the Euclidean pace of the neuron m. ~ n n n n 1 2 C = { B = ( Y, Y ), for = 1,..., Nn} the et of N n connected neuron forming the Backhaul chain n n of neuron., where n = 1,..., N y and ( Y1, Y 2 ) are the coordinate in the Euclidean pace of the neuron B n. (i) Interaction between the chain C m and the inehaul cutomer in. In thi interaction, the inehaul cutomer in will be preented to the chain C m one by one in a random order. In order to chooe the chain that will interact with the preented cutomer, we have to conider two factor: 1) The ditance of the nearet neuron from each chain to the cutomer 2) The current weight of each chain. Each time a cutomer i aigned to a certain chain it current weight wm will be increaed by the correponding weight. The winning chain will be elected randomly according to a probability ditribution taking into conideration

4 WSOM 2005, Pari thee two factor. Once a chain i elected, the poition of it neuron will be aduted according to the adaptation rule. (ii) Interaction between the chain C n and the backhaul cutomer in B. In thi interaction the backhaul cutomer in B will interact with C n in a imilar way to the interaction of type (i) and ue to ame adaptation rule. (iii) Interaction between the chain C m and C n. Uing the previou type of interaction, the chain will evolve independently. Nothing i forcing them to be connected in order to form a feaible route. For thi reaon, an interaction between the two chain C m and C n, i introduced. We aume that each chain ha a head and a tail. The tail and the head are repreented by the lat and the firt neuron repectively. After preenting all backhaul and inehaul cutomer, the chain C m will interact with the chain C n having the nearet neuron tail to the C m neuron head. The obective of thi interaction i to make the tail of the linehaul chain and the head of the correponding backhaul chain converge. Thi convergence will allow the formation of a ingle ring repreenting a tour viiting the inehaul and backhaul cutomer conecutively. The firt neuron of the backhaul chain i aigned a the winner neuron in thi interaction. Thi mean that the algorithm at thi level i not anymore a competitive algorithm but a upervied one in the ene that the lat neuron of the linehaul chain ha to be attracted by the firt neuron of the backhaul chain. After thi aignment, the adaptation rule ha to be applied on the neuron of C n.we apply the ame procedure to the backahul chain, by preenting the firt neuron of C n to the firt chain, aigning the lat neuron of C m a the winner neuron and updating the poition of the neuron of C m according to the ame adaptation rule. (iv) Interaction between the two type of chain and the depot. Thi type of interaction i imilar to the lat one, where the depot i preented to the linehaul chain. The firt neuron of each C m i aigned to the depot and conidered a the winner neuron. Once thi neuron i aigned, we update thi neuron and it neighboring neuron according to the uual adaptation rule. The ame procedure i applied to the lat neuron of each chain C n. The poition of the neuron in the chain will give the poition of the cutomer in the route. 3 The CP-SOM Algorithm In thi ection the CP-SOM algorithm for VRPB i ketched in peudo-code. et u introduce the following additional notation: d: i the lateral ditance. δ, β: parameter to control the probability function ή, α : parameter to control adaptation rule t : iteration number Step 1: Initialization Step 2: Select a city randomly Step 3: Select a Winner neuron If (C belong to ) Then

5 Competitive Probabilitic Self-Organizing Map for Routing Problem I. Selection of the nearet neuron for each C m, et m be the winning neuron belonging to C m, i.e. Such that m m = ( X, X ) 1m 2m ( X X ) + ( X X ) ( X X ) + ( X X ) c 1m 2c 2m 1c 1i 2c 2i i = 1,,N 1 m = 1,,N v. d (C, m ) i the Euclidean ditance between C and II. Select the aigned chain according to the probability: 2 d (C, m) exp( m+ wm+ x3 k) exp( ) 1+ δ() t β() t PCC (, ) = m 2 d (C, m) exp( m+ wm+ x3 k) exp( ) 1+ δ() t β() t m m = 1,,N v. Aume C to be the aigned chain w = w + x3k Add C to N l III. Update the coordinate of each neuron in et C for example the x- coordinate i updated by the following rule: X 1 (t+1) = X 1 (t) + ή (t) Γ(C, ) ( x 1 C X 1 (t)), = 1,..., N. where 2 (, ) 1 d Γ ( C, ) = exp σ ( t ) If (C belong to B) Then I. Selection of the nearet neuron for each C n, B n et Bn be the winning neuron belonging to C n, i.e. uch that m B n = ( Y Y ) + ( Y Y ) ( Y Y ) + ( Y Y ) c 1m 2c 2m 1c 1i 2c 2i i = 1,,N 2 n = 1,,N v. II. Select the aigned chain according to the probability: 2 n 3i d (C, B ) exp( m+ g + y ) ~ exp( ) 1+ δ ( t) β ( t) P( C, Cn ) = d (C, B ) 2 exp( m+ gn+ y3i) exp( ) 1+ δ ( t) β ( t) n n = 1,,N v. ( Y, Y ) 1n 2n

6 WSOM 2005, Pari ~ Aume C to be the aigned chain g =g + y3k Add C to N b III. Update the coordinate of each neuron in et C ~ for example the x- coordinate i updated by the following rule: Y 1 (t+1) = Y 1 (t) + ή (t) Γ (C, = 1,..., N. where B ) ( Y 1 Y 1 (t)), 2 1 d ( B, B ) Γ ( C, B ) = exp σ ( t ) Step 4. Extremitie interaction: Apply the different type of Interaction Step 5. End-Iteration Tet: If Not {all cutomer are elected at the current iteration} Then go to Step 2. Step 6. Stopping Criterion: If {all cutomer are within 10-4 of their nearet neuron in the Euclidean pace} Then Stop 4. Computational Experience Our computational experience i deigned to analyze the performance of the CP-SOM in term of olution quality and computational requirement. The computational reult are reported uing a et of 33 intance, which were propoed in Toth & Vigo [8]. Thee intance are generated from the 11 claical intance of the VRP literature. The VRPB intance range in ize between 21 and 100 cutomer. For each VRP problem intance, three VRPB intance ρ = B are generated with ratio N -the number of backhaul cutomer over the total number of backhaul and linehaul cutomer- ranging from 50, 66 and 80%. Thee intance were alo ued to report the experimental experience of many reearcher, Toth and Vigo [9] and Waan and Oman [6]. The parameter are choen experimentally and are lowly decreaed at each iteration by 1%. The Algorithm i robut in term of parameter. The propoed algorithm for the VRPB are coded in C and run on a PC Intel Pentium MMX 233 MHz. The quality of an algorithm i meaured by the relative percentage deviation (RPD) of the olution value from it optimal olution, or bet-known value publihed in the literature and by the average of RPD over all intance (ARPD). Three SOM variant, CP-SOM, CP-SOM1 and CP-SOM2, were implemented for the VRPB. The main difference between them i baed on the way the local 2-opt optimization i embedded, while keeping invariant other parameter. The CP_SOM algorithm i the baic algorithm uing SOM principle. In C_SOM1 a 2-opt procedure i ued to improve the CP- SOM generated olution, whereacp-som2 call periodically the 2-opt procedure every 50 iteration within the CP-SOM implementation rather than only at the end a in CP-SOM1. The reult comparing the performance of the 3 variant are given in table 1. We can oberve that C

7 Competitive Probabilitic Self-Organizing Map for Routing Problem the pot-optimization ha improved the reult by 5% percent for large intance and around 2% for intance of 50 point. For mall intance CP-SOM i till getting the bet known reult. Thi mean, that the performance of the neural approach i good in the allocation tage but can be improved at the cheduling tage by a local earch procedure. CPU time for potoptimization i between 2 and 3% above the CP-SOM CPU time. Thi confirm that the neural network provide the olution while the improvement are due to the local earch. Comparing the CPU time with other technique uch a the reactive tabu earch [6] i not eay becaue of the ue of different machine and programming language. However, uing ome reaonnable approximation, it can be een that the CP-SOM algorithm require more CPU time than other algorithm when the vehicle capacity i tight and when the number of linehaul and backhaul cutomer are not balanced. Thi can be een epecially for the intance correponding to 100 cutomer. However, on average CP-SOM conume 20% CPU time le than the reactive tabu earch method for a performance that i 1.66% better. 5 Concluion The SOM heuritic variant CP-SOM1 and CP-SOM2 are deigned and implemented for the VRPB. Their comparion with the bet exiting heuritic how that they are competitive with repect to olution quality, but they require more computational effort, imilar to other neural network in the literature. In particular, CP-SOM1 heuritic i the bet performing algorithm for mall-ized intance up to 32 cutomer. Therefore, it mut be recommended for uch intance. For the medium to large intance, the performance of CP-SOM1 wa enhanced by embedding a periodic-improvement trategy within it implementation leading to CP-SOM2. In general, our SOM baed neural approach i by far more powerful, flexible and imple to implement than the Hopfield-Tank neural network method. Reference [1] J.J. Hopfield, & D.W. Tank, Neural computation of deciion in optimization problem. Biological Cybernetic, Vol. 52, pp , [2] J.C. Fort, Solving a combinatorial problem via elf-organizing proce: An application to the traveling aleman problem. Biological Cybernetic, Vol. 59, pp , [3] H. Ghaziri, Superviion in the elf-organizing feature map: application to the vehicle routing problem. In: I.H. Oman and J.P. Kelly, Meta-Heuritic: Theory and Application, ( pp ). Kluwer Academic publiher, Boton, [4] H. Ghaziri, & I. Oman, A neural network algorithm for traveling ale man problem With backhaul. Computer & Indutrial Engineering. Vol. 44, pp , [5]T. Kohonen, Self-Organizing Map. Springer-Verlag: Berlin, [6] I.H. Oman, & N.A. Waan, A reactive tabu earch for the vehicle routing problem with Backhaul. Journal of Scheduling, Vol 5, pp , [7] K.A. Smith, Neural network for combinatorial optimization: A review of more than a decade of reearch. INFORMS Journal on Computing, Vol. 11, pp , [8] P. Toth, & D. Vigo, A heuritic algorithm for the vehicle routing problem with backhaul. In Advanced Method in Tranportation Analyi, Bianco, pp , [9] P. Toth, & D. Vigo, A heuritic algorithm for the ymmetric and aymmetric vehicle routing problem with backhaul. European Journal of Operational Reearch, Vol. 113, pp , 1999.

8 WSOM 2005, Pari Table 1. Comparing the performance of the 3 SOM variant with the bet known reult CP-SOM1 N RPD CPU CP-SOM RPD CPU CP-SOM RPD CPU