Manipulation of Graphs, Algebras and Pictures. Essays Dedicated to Hans-Jörg Kreowski on the Occasion of His 60th Birthday

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1 Eletroni Communiations of the EASST Volume 26 (2010) Manipulation of Graphs, Algebras and Pitures Essays Dediated to Hans-Jörg Kreowski on the Oasion of His 60th Birthday Autonomous Units for Solving the Capaitated Vehile Routing Problem Based on Ant Colony Optimization Sabine Kuske, Melanie Luderer 23 pages Guest Editors: Frank Drewes, Annegret Habel, Berthold Hoffmann, Detlef Plump Managing Editors: Tiziana Margaria, Julia Padberg, Gabriele Taentzer ECEASST Home Page: ISSN

2 ECEASST Autonomous Units for Solving the Capaitated Vehile Routing Problem Based on Ant Colony Optimization Sabine Kuske 1, Melanie Luderer 2 1 kuske@informatik.uni-bremen.de, kuske 2 melu@informatik.uni-bremen.de, Department of Computer Siene Univery of Bremen, Germany Abstrat: Communities of autonomous units and ant olony systems have fundamental features in ommon. Both onsists of a set of autonomously ating units that transform and move around a ommon environment that is usually a graph. In ontrast to ant olony systems, the ations of autonomous units are speified by graph transformation rules whih have a preisely defined operational semantis and an be visualized in a straightforward way. In this paper, we model an ant olony system solving the apaitated vehile routing problem as a ommunity of autonomous units. The presented ase study shows that the main harateristis of ant olony systems suh as tour onstrution and pheromone updates an be aptured in a natural way by autonomous units. Keywords: Graph transformation, autonomous units, ant olony optimization 1 Introdution In omputer siene there exists a large variety of relevant problems that are too omplex to be solved by a deterministi algorithm in an aeptable time. Hene, heuristis are employed that in many ases an help to find good solutions. In this ontext, swarm intelligene plays an important role where, roughly speaking, a swarm is a large number of autonomous and selfinterested agents that at and interat in parallel. In general, a swarm as a whole an produe good solutions for omplex problems whereas a stand-alone agent is not able to do so. One wellstudied kind of swarms are ant olonies whih onsist of a set of autonomously behaving artifiial ants that move around a ommon graph and make their deisions aording to the pheromone onentration in their neighborhood. They are inspired by the way how ants find short routes between food and their formiary and have been shown to be well-suited not only for the solving of shortest path problems, but for a series of more omplex problems, typially ourring in logistis (f. [DS04]). Basially, in an ant olony system, a set of ants onstruts solutions for a given problem (mostly NP-hard) by moving along the edges of an underlying graph. Aording to the quality of the onstruted solutions the ants walk bak and put some pheromone on the traversed items, i.e., The first author would like to aknowledge that her researh is partially supported by the Collaborative Researh Centre 637 (Autonomous Cooperating Logisti Proesses: A Paradigm Shift and Its Limitations) funded by the German Researh Foundation (DFG). 1 / 23 Volume 26 (2010)

3 Autonomous Units for Ant Colony Optimization the better the solution is the more pheromone is plaed by an ant. During solution onstrution the pheromone onentration as well as some further heuristi value help the ants to deide where to go in eah step. Every ant has a memory for storing important information suh as the gth of the traversed path, et. In order to prove orretness properties of ant olony optimization algorithms, a formal modeling framework with a well defined semantis is needed. Moreover, ant olony systems an be visually represented in a straightforward way so that for simulation and verifiation purposes, it is desirable to have a graphial modeling framework whose operational semantis provides graphial representations of system states. Sine ants work on graphs, graph transformation is a suitable approah to speify the ations of the ants. Not only has graph transformation a well defined semantis and a wide theory but there exist also some graph transformation tools that ould be used to implement ACO algorithms (f. [Roz97, EEKR99, EKMR99]). Moreover, a suitable onept for modeling the autonomous behavior of ants is needed. A promising onept to ahieve this is that of ommunities of autonomous units beause on the one hand they inorporate rule-based graph transformation and on the other hand autonomous units at and interat autonomously in a ommon environment (f. [KK07, KK08, HKK09]). Essentially, every autonomous unit is omposed of a set of graph transformation rules, a ontrol ondition, and a goal. Moreover, it an ask auxiliary units for help and it an be equipped with a speifiation of initial private states where the latter may be used to represent the memory of an ant. Autonomous units transform the ommon environment and their private states simultaneously while striving for their goals, an ommuniate with eah other via the ommon environment, and may at in parallel. A ommunity is omposed of a set of autonomous units, a speifiation of initial ommon environments, a global ontrol ondition, and an overall goal. A urrent state of a ommunity onsists of a urrent ommon environment plus a private state for every autonomous unit. The semantis of a ommunity onsists of all state sequenes obtained by omposing the semantis of the autonomous units in the ommunity in suh a way that the global ontrol ondition is not violated and the start state onsists of an initial ommon environment and an initial private state for every unit. A transformation proess is suessful if it reahes the overall goal. The basi omponents of ommunities are provided by a graph transformation approah onsisting of a lass of graphs, a lass of graph lass expressions, a lass of rules with a rule appliation operator, and a lass of ontrol onditions. In the literature there exists a variety of graph transformation approahes (f. [Roz97]). They all an be used as underlying approahes for ommunities. In [KLT09], it was shown that ommunities of autonomous units are suitable to model an ant olony solving the Traveling Salesperson Problem. The present paper fouses on a more ompliated problem that an be solved in an intuitive way by ant olony optimization algorithms: the Capaitated Vehile Routing Problem (CVRP) (f., e.g., [RDH04, DS04]). Conretely, we present a ommunity of autonomous units that models an ant olony that solves the CVRP. The aim of this paper is to onsolidate the onjeture that ommunities of autonomous units are suitable as a formal framework for modeling ant olony systems. The advantages of modeling ant olony systems as ommunities of autonomous units are the following. (1) Autonomous units provide ant olony systems with a well-founded operational semantis so that verifiation tehniques for graph transformation an be applied to ant olony systems. (2) The fat that ant ations an be speified as graph transformation rules allows for a Festshrift H.-J. Kreowski 2 / 23

4 ECEASST visual modeling of ant algorithms and hene for a visual representation of ant olony behavior. (3) Existing graph transformation tools suh as GrGEN [GK08] or AGG [ERT99] an be used to implement ant algorithms. This paper is organized as follows. In Setion 2, ant olony systems for the heuristi solving of optimization problems are briefly introdued and a partiular ant olony optimization algorithm for solving the CVRP is realled. Setion 3 presents a graph transformation approah that is used throughout this paper. Setion 4 introdues autonomous units and ommunities of autonomous units. Setion 5 shows how fundamental features of ant olony systems an be modeled with autonomous units by translating an ant olony system solving the CVRP into a ommunity. The onlusion is given in Setion 6. 2 Ant Colony Optimization Ant olony optimization (ACO) systems are algorithmi frameworks for the heuristi solving of optimization problems, typially problems belonging to the omplexity lass NP-hard, sine no effiient algorithms for this kind of problems are known that always solve the problem. The idea of ACO originates in the observation of how ants find short ways between food and their formiary. An individual ant an hardly see and has a very narrow perspetive of its environment. While searhing for food, it leaves a hemial substane on the ground, alled pheromone, whih an be sensed by other ants and influene their route deision. The higher the onentration of pheromone along a way, the higher the probability that an ant will hoose this way as well, thus leaving even more pheromone. The ruial point is that pheromone evaporates with time. An ant following a short route to food will return sooner to the formiary so that the pheromone onentration on shorter routes beomes more intense than on longer routes. The higher pheromone onentration makes more ants hoose the short route whih in turn raises the pheromone onentration further. Finally, almost all ants end up hoosing one short route, although not neessarily the shortest one. Sine typial optimization problems an be niely modeled as graphs, it is the prefered data struture for ACO. In this paper we use edge-labeled undireted graphs with multiple (i.e. parallel) edges. Graphs. A graph is a tuple G = (V,E,att,m), where V is a finite set of nodes, E is a finite set of edges suh that V and E are disjoint, att : E ( V k {1,2} k) is a mapping that assigns to every edge a set of one or two soures in V, and m is a mapping that assigns a label to every edge in E. 1 A graph with no nodes and no edges is alled the empty graph whih is denoted by /0. The omponents of G are also denoted by V G, E G, att G, and m G respetively. The set of all graphs is denoted by G. A solution to an optimization problem onsists typially of a tour (e.g. an ordered sequene of nodes) within the given graph. Intuitively, the omplexity of most NP-hard optimization problems lies in the exponentially growing number of possible tours when new nodes and edges are added. The lak of an effiient searh method for the best way requires an (almost) exhaustive 1 For k N, ( V k ) denotes the set of subsets of V with k elements, i.e., ( Vk ) = {V V V = k} where V denotes the number of elements in V. 3 / 23 Volume 26 (2010)

5 Autonomous Units for Ant Colony Optimization searh of all the possible tours. To solve an optimization problem with ACO, some additional information is needed. We define optimization problems as follows. Optimization Problem. An optimization problem is a 6-tuple (CG,d,τ,η,S,g) where CG G is a onstrution graph, d is a funtion that assoiates every edge with a ost value (e.g. the distane), τ is a funtion that assoiates every edge with a pheromone value, η is a funtion that assoiates every edge with a number as an heuristi value for the quality of the edge, S V is the set of solutions, and g assigns a ost g(s) to every s S. Basially, ACO works as follows. At first, a predefined number of ants are plaed randomly at some nodes. These ants deide in parallel whih edge they follow in the next step aording to a tranion rule. Let a be an index to hoose one of n ants and U a the set of all edges that an be hosen from ant a residing at some node. The deision, whih edge e U a to take, is probability-based. The probabilities are alulated as follows. p a (e) = [τ(e)]α [η(e)] β e Ua [τ(e)] α [η(e)] β e U a (1) In words this formula states that ants prefer edges with low ost and a high onentration of pheromone. The experimental parameters α and β ontrol the influene of the pheromone resp. heuristi value in the deision. In every step this formula is applied, until all the ants have onstruted a omplete tour. The next step onerns the pheromone values. Simulating the evaporation, the values of τ are redued: τ(e) (1 ρ) τ(e) e E CG where ρ is a pheromone deay parameter in the interval (0,1]. Furthermore the release of pheromone of the ants is simulated: { n 1 τ(e) τ(e)+ τ a (e), with τ a (e) = gth(tour a ),e tour a 0 otherwise a=1 where tour a is the solution onstruted by ant a. In ontrast to nature, the release of pheromone takes plae after the ants onstruted a omplete tour, sine the amount of pheromone orresponds to the overall quality of the tour (e.g. the gth of the tour). Furthermore, in some ACO systems not every ant leaves pheromone, but just the ones having onstruted the best tours. Now the ants are plaed again at some randomly hosen nodes and the algorithm starts with the modified values of pheromone. Some variants of this basi ACO yielding better performane have been proposed in the literature. Details an be found in [DS04]. 2.1 Appliation: Capaitated Vehile Routing Problem An important appliation field of ACO onerns all kinds of tour planning with the Traveling Salesperson Problem (TSP) as the most famous one. Another problem often ourring in distribution logistis is the so alled Capaitated Vehile Routing Problem (CVRP), whih an be desribed as follows. A number of ustomers must be served with some goods that are stored at a entral. A number of vehiles with finite and equal apaity is available. The aim is to find a set of tours suh that the demands of all ustomers are met and the total ost (the sum of Festshrift H.-J. Kreowski 4 / 23

6 ECEASST the distanes of the tours) is minimized. Combinatorially, a solution an be formally desribed as a partition of the ities into m routes {R 1,...,R m }. Eah route must satisfy the ondition j Ri dem j k, where dem j desribes the demand of the j-th ustomer and k is the apaity restrition of the vehiles. Within eah partition, an assoiated permutation funtion speifies the ustomer order. Relaxing the onditions by allowing any partition (respetively setting k = ), the CVRP is transformed into an instane of the Multiple Traveling Salesperson Problem. Leaving the ondition unhanged but with a ost funtion that ounts the number of partitions CVRP beomes the well-known bin paking problem. CVRP ontains in this sense two NP-hard problems, whih in pratie makes it a lot more ompliated to solve than TSP for example and it seems a good idea to use ACO. A formulation of CVRP aording to the definition of optimization problems is quikly found. Nevertheless, there are different ways to design the funtion η : E CG R. One easy possibility onsists of the reiproal ost-value of the edge. Nevertheless, sometimes other methods are used to alulate the heuristi values; one elegant way is based on the so-alled Savings algorithm. Starting from the initial (and unfavored) solution, where every route onsists of exatly one ustomer, it is alulated, how the quality of the solution hanges (how muh one would save), putting two ustomers i and j in one route. Let d i0 denote the distane between ustomer i and the and d i j the distane between ustomer i and j. Then the saving value obtained by merging the routes R i and R j together is alulated as follows: s i j = 2 d i0 + 2 d j0 (d i0 + d i j + d j0 ) = d i0 + d j0 d i j Elaborated experiments onerning the performane of ACO and Saving Algorithm for the CVRP an be found in [RDH04]. 3 A Graph Transformation Approah Graph transformation approahes provide the main ingredients for ommunities of autonomous units. They onsist of a lass of graphs, a lass of rules, a rule appliation operator, a lass of ontrol onditions, and a lass of graph lass expressions. The graphs are used to represent the ommon environments and the private states of ommunities. The rules are needed to transform these graphs. Moreover, ontrol onditions restrit the non-determinism of rule appliation, and with graph lass expressions one an speify speifi graph sets suh as initial environments or goals to be reahed. In the literature, there exists a series of different graph transformation approahes (f. [Roz97]). In the following, we present a partiular graph transformation approah that is suitable for modeling the CVRP based on ACO. Conretely, the graph lass and the rule lass together with the rule appliation operator are a variant of the double-pushout approah [CEH + 97]. 3.1 Graphs and Rules The graph lass onsists of edge-labeled undireted graphs with multiple edges as presented in Setion 2. For the modeling of the CVRP in Setion 5 we use the following types of edge labels. 5 / 23 Volume 26 (2010)

7 Autonomous Units for Ant Colony Optimization The symbol for denoting unlabeled edges; strings in {a,...,z} to denote e names; ap,,, load, feas, sum,, ant, dem, and to denote attributes suh as the apaity of the truks, the gth of a tour, et.; labels in {x : y x {τ, dist}, y R} for pheromone quantities and distanes between loations; labels in {η : y y R { }} for the values of the funtion η; labels in N and R to denote demands, apaities, loads, gths of tours, et.; and and with j N to denote ants and memories. It is worth noting that undireted graphs an be transformed into direted graphs as used in the double-pushout approah by replaing eah undireted edge by a pair of direted edges pointing in oppoe diretions. The lass of direted graphs obtained in this way is a sublass of edgelabeled direted graphs. Subgraphs and graph morphisms are defined as follows. Subgraphs and graph morphisms. For G,G G, the graph G is a subgraph of G, denoted by G G, if V G V G, E G E G, att(e) = att (e), and m(e) = m (e) for all e E G. A graph morphism g: G G is a pair (g V,g E ) of mappings with g V : V G V G and g E : E G E G suh that labels and soures are kept, i.e., for all e E G, g V (att G (e)) = att G (g E (e)) and m G (g E (e)) = m G (e). 2 The image of G in G is the subgraph g(g) of G suh that V g(g) = g V (V G ) and E g(g) = g E (E G ). In the following, the subsripts V and E of g V and g E are often omitted, i.e., g(x) means g V (x) for x V and g E (x) for x E. Graphs are depited as usual with round or boxed nodes and lines as edges. A loop is sometimes omitted by putting its label inside the node to whih the loop is attahed. Sine a node an have several loops this is always done for at most one loop per node. A node with a label x inside will also be alled an x-node. The label * is omitted in graph drawings. Graphs an be modified by rules onsisting of a negative ontext, a left-hand side, a gluing graph, and a right-hand side. Roughly speaking, the negative ontext speifies omponents that must not our in the graph to whih the rule is applied. The left-hand side, the gluing graph, and the right-hand side are used to determine whih omponents should be deleted, kept and added, respetively. In every omputation step of a ommunity, the autonomous units transform the ommon environment and their private states simultaneously. For this purpose, every unit applies pairs of rules (r 1,r 2 ), where the first rule r 1 is applied to the ommon environment and r 2 to the private state. Rules and rule pairs. A rule r is a quadruple (N,L,K,R) of graphs with N L K R where N is the negative ontext, L is the left-hand side, K is the gluing graph, and R is the right-hand side. If all omponents of r are empty, r is the empty rule. The set of all rules is denoted by R. A rule pair is a pair of rules r = (r 1,r 2 ) where r 1 is alled the global rule and r 2 the private rule. The set of all rule pairs is denoted by R. A rule pair r = (r 1,r 2 ) where r 2 is the empty rule an be regarded as a single rule. Hene, in the following, we often do not distinguish between single rules and rule pairs with an empty private rule. 2 For a mapping f : A B and C A the set f(c) is defined as { f(x) x C}, i.e., g V (att G (e)) = {g V (v) v att G (e)}. Festshrift H.-J. Kreowski 6 / 23

8 ECEASST A rule (N,L,K,R) is depited as N R where the nodes and edges of K have the same forms, labels, and relative poions in N and R. The forbidden nodes (i.e., the nodes of N that do not belong to L) are olored gray. The forbidden edges are dashed. Figure 1 shows a rule where the left-hand side onsists of a round node, a retangle a-node and an edge onneting both. The gluing graph onsists of the round node, and the right-hand side is obtained from the gluing graph by onneting the round node with a new b-node. The gray retangle node as well as its inident edges are forbidden. a b b Figure 1: A rule A rule pair r = ((N 1,L 1,K 1,R 1 ),(N 2,L 2,K 2,R 2 )) (with non-empty private rule) is depited as L 1 L 2 R 1 R 2 where the negative ontexts and the gluing graphs are represented as in single rules. A rule (N,L,K,R) is applied to a graph as follows. (1) Choose an image g(l) of L in G. (2) Chek whether g(l) has no forbidden ontext given by N up to L. (3) Delete g(l) up to g(k) from G provided that no dangling edges are produed. (4) Glue R and the remaining graph in K. This means that the subgraph K of R is identified with its image in Z. This onstrution an be defined as follows. Gluing of graphs. Let K R and h: K Z. Then the gluing of Z and R in K with respet to h is onstruted as follows. Let V be the equivae relation generated on V Z +V R by the relation {(h V (v),v) v V K } and let E be the equivae relation on E Z E R generated by {(h E (e),e) e E K }. 3 Let (V Z +V R )/ V and (E Z + E R )/ E be the respetive quotient sets. Then the gluing of Z and R in K with respet to h yields the graph D = ((V Z +V R )/ V,(E Z + E R )/ E,att,m) where for all e (E R + E Z )/ E { [attz (e)] if e = [e] for some e E 4 Z att(e) = [att R (e)] if e = [e] for some e E R E K m(e) = { mz (e) if e = [e] for some e E Z m R (e) if e = [e] for some e E R E K The appliation of a rule to a graph is formally defined as follows. 3 + denotes the disjoint union of sets 4 For a quotient set A/, []: A A/ denotes its natural assoiated funtion. 7 / 23 Volume 26 (2010)

9 Autonomous Units for Ant Colony Optimization Rule appliation. Let r = (N,L,K,R) R, let G G. Then r is applied to G by performing the following steps. (1) Choose an injetive graph morphism g: L G suh that the following onditions are satisfied. (a) If L N, there exists no g : N G with g (x) = g(x) for all x V L E L. (b) For all e E G E g(l), att G (e) V G (V g(l) V g(k) ). (2) Construt the intermediate graph Z by deleting V g(l) V g(k) and E g(l) E g(k) from G, (3) onstrut the gluing of Z and R in K with respet to g K : K Z where g K(x) = g(x) for all x V K E K. The semanti relation of r is denoted by SEM(r) and onsists of all pairs (G,G ) suh that G an be derived from G via the appliation of r. For a set P R, we define SEM(P) = r P SEM(r). For (r 1,r 2 ) R, the semanti relation is equal to {((G 1,G 2 ),(G 1,G 2 )) (G i,g i ) SEM(r i ),i = 1,2}, i.e., SEM(r 1,r 2 ) onsists of all pairs ((G 1,G 2 ),(G 1,G 2 )) where for i = 1,2 the graph G i an be obtained by applying r i to G i. The rule in Figure 1 an be applied to a graph ontaining a node v onneted to an a-node but not onneted to a b-node. Its appliation removes the a-node plus the edge to v and adds a b-node and an edge from this b-node to v. Beause of ondition (b) of the preeding definition, the a-node is only onneted to v but not to other nodes; otherwise its deletion would produe dangling edges. The desribed kind of applying graph transformation rules is a variant of the double-pushout approah presented in e.g. [CEH + 97], where also non-injetive mathings of the left-hand side are allowed and graphs are direted and node- and edge-labeled. Replaing all undireted edges by direted ones as desribed above, the appliation of a rule as presented here is performed in the same way as in the double-pushout approah restrited to injetive mathings and edge-labeled graphs. A node with a single x-loop ould be also modeled as a node with node label x in the ase where not only edge labels but also node labels are allowed. However, in the double-pushout approah, relabeling of nodes via a graph transformation rule is often not possible beause this may violate ondition (b) in the seond step of rule appliation. For this reason we use edgelabeled graphs where this problem does not our. An approah that inludes node relabeling expliitly an be found in [HP02]. In general, the autonomous units of a ommunity apply their rules in parallel. A parallel rule appliation step involving two rules an be defined as follows. Parallel rule appliation. Let G G and for i = 1,2, let r i = (N i,l i,k i,r i ) be two rules. Let g i : L i G be two injetive graph morphisms that satisfy the onditions (a) and (b) of the definition of rule appliation and the independene ondition g 1 (L 1 ) g 2 (L 2 ) g 1 (K 1 ) g 2 (K 2 ). 5 Then r 1 and r 2 an be applied in parallel to G by (1) deleting V gi (L) V gi (K) and E gi (L) E gi (K) (for i = 1,2), and (2) onstruting the gluing of the resulting graph D and R 1 + R 2 in K 1 + K 2 with respet to g: K 1 + K 2 D, where g(x) = g i (x) if x V Ki E Ki, for i = 1,2. 6 The definition of parallel rule appliation an be extended in a straightforward way from two rules to arbitrary non-empty multisets of rules. For a multiset m of rules, SEM(m) denotes the set of all (G,G ) G G where G is derived from G via the parallel appliation of the rules 5 For G 1,G 2 G the intersetion G 1 G 2 yields the pair (V,E) where V = V G1 V G2 and E = E G1 E G2. Moreover, we have (V 1,E 1 ) (V 2,E 2 ) if V 1 V 2 and E 1 E 2. 6 The morphism g may be non-injetive. Festshrift H.-J. Kreowski 8 / 23

10 ECEASST in m. A multiset m of rules will be alled a parallel rule, and for a set P R, the set of all parallel rules over P is denoted by P. For a rule pair r = (r 1,r 2 ), SEM(r m) denotes all ((G 1,G 2 ),(G 1,G 2 )) (G G ) (G G ) where G 1 is derived from G 1 by applying the multiset obtained from adding r 1 to m, and (G 2,G 2 ) SEM(r 2). 3.2 Control Conditions It is often desirable to restrit the non-determinism of rule appliation. This an be ahieved with ontrol onditions. Conretely, we use as ontrol onditions regular expressions equipped with as long as possible. Control onditions. Let ID be a set suh that P ID for some set P of rule pairs. Then the lass C (ID) of ontrol onditions over ID is indutively defined as follows: {lambda} ID {x! x P} C (ID). For, 1, 2 C (ID), we have ( ),( 1 ; 2 ),( ) C (ID). For pratial appliations, the set ID would onsist of names referring to rule pairs (or units) but for tehnial simpliity we do not distinguish between rule pairs (units) and their names. If ID onsists only of rule pairs, a semantis of ontrol onditions an be defined in an intuitive way. Roughly speaking, the ondition lambda applies no rule. Every rule pair r is a ontrol ondition that presribes one appliation of r. The ondition stands for applying 1 or 2, 1 ; 2 means that 1 must be applied before 2, applies arbitrarily often, and r! requires that the pair r be applied as long as possible. The operator! applies only to rules beause the possibility to iterate other ontrol onditions as long as possible is not needed in the following. The semantis of ontrol onditions are sequenes of graph pairs where every pair onsists of a ommon environment and a private state of the unit the ontrol ondition is part of. Eah pair in the sequene is obtained from the previous pair by one of the following ations: (1) An appliation of a rule pair ourring in the ontrol ondition; (2) an appliation of a parallel rule to the ommon environment where the parallel rule is omposed of global rules of other autonomous units in the ommunity; (3) a parallel ompoion of (1) and (2). This means in partiular that the semantis of ontrol onditions is defined w.r.t. a set of ative rules that omprises the global rules of all other units in the ommunity. Semantis of ontrol onditions. Let AR R be a set of rules alled ative rules and let P R. Then for eah ontrol ondition in C(P) its semantis is defined as follows. 1. SEM AR (lambda) onsists of all sequenes (G 0,...,G n ) of graph pairs suh that for i = 1,...,n, (G i 1,G i ) SEM(m) for some m AR SEM AR (r) onsists of all sequenes s = (G 0,...,G n ) for whih there exist some j {1,...,n} and m 1,...,m n AR suh that for i = 1,..., j 1 and i = j + 1,...,n, the pair (G i 1,G i ) is in SEM(m i ), and (G j 1,G j ) SEM(r m j ). 3. SEM AR ( ) = SEM AR ( 1 ) SEM AR ( 2 ). 7 In this transformation, the seond omponent of every graph pair remains unhanged, beause m is a multiset of single rules. 9 / 23 Volume 26 (2010)

11 Autonomous Units for Ant Colony Optimization 4. SEM AR ( 1 ; 2 ) = SEM AR ( 1 ) SEM AR ( 2 ) SEM AR ( ) = SEM AR (). 6. SEM AR (r!) onsists of all sequenes (G 0,...,G n ) SEM AR (r ) suh that r is not appliable to G n. In Setion 4 we show how this definition an be employed for the more general ase where ID ontains units, too. 3.3 Graph Class Expressions In order to use graph transformation in a meaningful way, it should be possible to speify initial and terminal graphs of graph transformation proesses with graph lass expressions. In general, a graph lass expression an be any expression that speifies a set of graphs. In partiular, the graph lass expressions used in this paper are the following. Graph lass expressions. The lass X of graph lass expressions is reursively defined as follows: all, empty, red(p) X with P R where SEM(all) = G, SEM(empty) = /0, and SEM(red(P)) onsists of all graphs G to whih no rule of P an be applied. Moreover, for I,T X, P R, and C C (P), (I,P,C,T) X where SEM(I,P,C,T) onsists of all graphs G SEM(T) for whih there is a sequene (G 0,...,G n ) suh that G n = G, G 0 SEM(I), for i = 1,...,n (G i 1,G i ) SEM(P), and (G 0,...,G n ) SEM /0 (C). 9 One example of a graph lass expression of the last type is omplete = (empty,{nodes,edges},nodes ;edges,red({edges})), where nodes and edges are the rules in Figure 2. nodes: A id edges: Figure 2: The rules nodes and edges The left-hand side and the gluing graph of the rule nodes are empty. The negative ontext onsists of a boxed node with an inident -loop, and the right-hand side is omposed of a round node, a boxed -node, and an id-edge onneting both. The appliation of nodes to a graph G inserts a new round node onneted to a new boxed -node via an id-edge provided that there 8 For sets of sequenes S,S of graph pairs, their sequential ompoion is denoted by S S, and S is defined as i N S i with S 0 = G G and S i+1 = S i S. 9 Control onditions an be used to define sequenes of graphs (instead of sequenes of graph pairs) beause, as stated before, rules an be regarded as rule pairs with empty private omponent. Festshrift H.-J. Kreowski 10 / 23

12 ECEASST is no boxed -node in G. The rule edges onnets two existing round nodes by an unlabeled edge. Given some alphabet A, the expression omplete speifies all omplete graphs omposed of round nodes in whih eah round node is assoiated with a different element from A via an id-edge. (The id-edge an be onsidered as an attribute of round nodes whih has type A.) It is worth noting that the rule edges annot produe loops beause we only use injetive morphisms to hoose a math of the left-hand side. In addition, we tehnially distinguish between round and boxed nodes by using partiularly labeled loops that indiate the respetive node type (round or boxed). 4 Communities of Autonomous Units Every ommunity is mainly omposed of a set of autonomous units that at and interat in a ommon environment (see e.g. [HKK09] where a sequential and a parallel semantis of ommunities is introdued). 4.1 Autonomous Units Autonomous units transform a ommon graph and have an additional private graph where they an store private information. Sine the rule set of an autonomous unit an be very large, struturing onepts should be provided to keep it manageable. Autonomous units allow to import auxiliary units and provide ontrol onditions as well as graph lass expressions. Auxiliary units differ from autonomous units in the sense that they do not ontain graph lass expressions. The graph lass expressions of every autonomous unit are used to speify the initial private states as well as the goal. The latter onsists of a private goal onerning the private state and a goal onerning the ommon environment that the autonomous unit wants to reah. Autonomous units. A unit of import depth 0 is a system unit = (I,U,P,C,g) where I X is the initial private graph lass expression, U = /0, P R, C C (P U), and g X X is the goal. A unit of import depth n+1 is a system unit = (I,U,P,C,g) where U is a set of units of import depth at most n, and I, P, C, and g are defined as above. A unit (I,U,P,C,g) is an auxiliary unit if I = all, g = (all,all), and every u U is an auxiliary unit. A unit (I,U,P,C,g) is an autonomous unit if every u U is an auxiliary unit. The set of autonomous units is denoted by AUT. The omponents of unit are also denoted by I unit, U unit, P unit, C unit, and g unit, respetively. Every autonomous unit an be onverted into a flattened unit with import depth zero. The rule set and the ontrol ondition of the flattened unit an be onstruted as follows. Flattening. For unit = (I,U, P,C, g) its flattened rule set Rules(unit) and its flattened ontrol ondition flc(unit) is defined as follows. If U = /0, Rules(unit) = P and flc(unit) = C. If U /0, Rules(unit) = P u U Rules(u) and flc(unit) = C[a] where a: U C ( R) is defined as a(u) = flc(u) and C[a] is obtained by replaing every ourrene of u with a(u) (for eah u U). The parallel semantis of autonomous units onsists of all sequenes of graph pairs s = ((G 0,G 0 ),...,(G n,g n)) suh that G 0 is an initial private graph and s is allowed by the flattened 11 / 23 Volume 26 (2010)

13 Autonomous Units for Ant Colony Optimization ontrol ondition with respet to some underlying set of ative rules. Moreover, s is suessful if the last graph pair in s satisfies the goal of the unit. A ommunity onsists of a set of autonomous units, a speifiation of all possible initial environments, a global ontrol ondition, and an overall goal. In the following, global ontrol onditions are regular expressions equipped with the parallel operator. Global ontrol onditions. Let Aut AUT. Then the set of global ontrol onditions GC (Aut) is reursively defined as follows: {aut 0 aut k aut i Aut,i = 0,...,k} GC (Aut). For, 1, 2 GC (Aut), we have ( ),( 1 ; 2 ), ( ) GC (Aut). Global ontrol onditions speify sequenes of states where every state onsists of a ommon environment plus a private state for every autonomous unit in a ommunity. The global ontrol ondition aut 0 aut k presribes the parallel running of aut 0,...,aut k. The semantis of the remaining ontrol onditions are defined as expeted. In the following we define states and the semantis of global ontrol onditions. Semantis of global ontrol onditions. For Aut AUT, a state is a pair (G,map) where G G and map: Aut G is a mapping. The semantis of eah global ontrol ondition in GC (Aut) is defined as follows. 1. SEM Aut (aut 0 aut k ) onsists of all sequenes ((G 0,map 0 ),...,(G n,map n )) suh that for i = 0,...,k, ((G 0,map 0 (aut i )),...,(G n,map n (aut i ))) SEM AR(auti )(flc(aut i )), where AR(aut i ) = aut {aut 0,...,aut k } {aut i } Rules(aut), and for eah aut Aut {aut 0,...,aut k }, map 0 (aut) = = map n (aut). 2. SEM Aut ( ) = SEM Aut ( 1 ) SEM Aut ( 2 ), 3. SEM Aut ( 1 ; 2 ) = SEM Aut ( 1 ) SEM Aut ( 2 ), and 4. SEM Aut ( ) = SEM Aut (). The omponents of ommunities are given as follows. Community. A ommunity is a tuple (Init,Aut,Cond,Goal) where Init,Goal X, Aut AUT, and Cond GC (Aut). The parallel semantis of a ommunity onsists of all state sequenes that are allowed by the global ontrol ondition and start with an initial state onsisting of an initial ommon environment and an initial private state for eah autonomous unit. The state sequenes are suessful if they reah the overall goal. Parallel ommunity semantis. Let COM = (Init, Aut, Cond, Goal) be a ommunity. Then the parallel ommunity semantis of COM, denoted by PAR(COM) onsists of all state sequenes s = ((G 0,map 0 ),...,(G n,map n )) suh that G 0 SEM(Init), map 0 (aut) SEM(I aut ) (for eah aut Aut), and s SEM Aut (Cond). Moreover, s is suessful if G n SEM(Goal). Festshrift H.-J. Kreowski 12 / 23

14 ECEASST 5 An ACO Community for Solving the CVRP In this setion we present the omponents of the ACO ommunity COM CVRP for modeling the Capaitated Vehile Routing Problem (CVRP) introdued in Setion 2. The initial environment speifiation of COM CVRP speifies the onstrution graph of the problem; the set of autonomous units onsists of the autonomous units Ant 1,...,Ant k (k N), and Evap&Selet; and the global ontrol ondition Cond is equal to (Ant 1... Ant k Evap&Selet). In our first approah the overall goal is equal to all. Roughly speaking, the ommunity COM CVRP works as follows. The ant units Ant 1,..., Ant k model the ants, whih in parallel traverse the graph aording to the savings heuristis introdued in Setion 2 and the urrent pheromone trails, and searh for a solution for the CVRP. When all ants have finished their searh, the autonomous unit Evap&Selet first arries out evaporation of the urrent pheromone trails. After that it selets w ants with best solutions. Now eah seleted ant leaves a pheromone trail on its solution path aording to the quality of the solution. All the units at in parallel. To ensure the desribed order we use negative ontexts as well as ontrol onditions. 5.1 The Initial Environment The underlying struture of the onstrution graph of the ACO system modeling the CVRP is a omplete graph with some additional information suh as initial pheromone onentration, distanes, et. This onstrution graph an be defined by the graph lass expression depited in Figure 3. It uses as initial expression the graph lass expression omplete introdued in Subsetion 3.3. Its rule selets the and has to be applied exatly one. The rule ust adds a number representing the demand to every ustomer node, i.e., to every node apart from the. The rule init labels every edge e of the initial graph with a distane d and it inserts two edges between eah two nodes of the graph, one labeled with the heuristi value the other with an initial pheromone value z. The rule save omputes the heuristi value of every edge based on the savings heuristis. The ontrol ondition requires that the is seleted first. The terminal graph lass expression red({init, save, ust}) guarantees that the rules ust, init, and save are applied as long as possible. The rules ust, init, and save of Constrution graph are parameterized, i.e., their labels ontain variables. Eah of these parameterized rules represents an infinite set of rules: one for eah possible instantiation of its variables. Conretely, the variable x an be instantiated with a natural number, and d, d 1, and d 2 with non-negative real numbers. (The value z is fixed and represents the initial pheromone value.) Hene, when applying a parameterized rule, a value for eah of its variables must be hosen. More information and partiular aspets onerning parameterized rules and their appliation an be found in e.g. [EEPT06, PS04, Kus02]. The meaning of the graph lass expression Constrution graph is to formally speify the lass of initial environments onsisting of all terminal graphs that an be generated from a omplete graph by the rules suh that the ontrol ondition is satisfied. In pratie, the ommunity COM CVRP would rather start its work on already existing initial onstrution graphs instead of generating them nondeterministially. 13 / 23 Volume 26 (2010)

15 Autonomous Units for Ant Colony Optimization Constrution graph initial: omplete rules: : ust: dem x N dem x τ : z init: save: η : dist : d d R dist : d η : η : d 1 + d 2 d dist : d dist : d 1 dist : d 2 dist : d 1 dist : d 2 onds: ; (ust + init + save) terminal: red({init, save, ust}) Figure 3: The graph lass expression Constrution graph 5.2 The Ant Units In general, every ant builds a solution tour by traversing the ommon environment aording to the urrent pheromone trails. It first selets its initial poion. Afterwards, it onstruts a solution tour t. Then it puts some pheromone on t if it is seleted to do so. Every ant unit Ant j uses the auxiliary units tour j, and put phero j. The ontrol ondition is equal to initial poion j ; tour j ; put phero j where initial poion j is the rule pair depited in Figure 4. id /0 id ant ap load i 0 0 Figure 4: The rule initial poion j It puts the ant Ant j to the and generates its memory where it stores the urrent load Festshrift H.-J. Kreowski 14 / 23

16 ECEASST tour j uses: feasible neighbors j, prob j rules: move: global: ant τ : x dist : d id dem η : y sum z private: m s load feas l x α y β z τ : x dist : d id dem l η : y ant m+l load s+d return: global: ant dist : d id dist : d id ant feas private: m s load 0 load s+d i feas A ap j s stop: ant s m load onds: (feasible neighbors j ; (prob j ; move + return)) ; feasible neighbors j ; stop s Figure 5: The auxiliary unit tour j of the vehile represented by Ant j (load), the apaity of the vehile (ap), its urrent loation () and the total gth of the tours (). This information is represented by edges labeled with the respetive labels (load, ap, and ), whih are eah attahed to a node with the orresponding value inside. The unit tour j is given in Figure 5. The global and private parts of the unit s rule pairs are depited one below the other. With tour j the ant builds a solution tour depending on probabilities for the next move to a feasible neighbor alulated from the savings heuristis and the urrent pheromone trails. It ontains the auxiliary units feasible neighbors j and prob j, and the rule pairs move, return and stop. The ontrol ondition requires to apply the rule pairs move or return arbitrarily often, and afterwards, the rule pair stop is applied one. Before eah appliation of 15 / 23 Volume 26 (2010)

17 Autonomous Units for Ant Colony Optimization move the unit prob j is alled. Moreover, the unit feasible neighbors j is exeuted before eah appliation of prob j and of return as well as before the appliation of the last rule pair stop. The unit feasible neighbors j is given in Figure 6. It omputes the feasible neighbors for an ant unit Ant j and stores them in the memory of the ant. Feasible neighbors are ustomer-nodes that are not yet vied and whose demand still fits into the vehile. Every appliation of the only rule pair feas adds one feasible neighbor to the memory. Moreover, it uses the auxiliary unit delete nonfeasible that removes all neighbors from the memory that are onneted via a feas-edge to and whose demand exeeds the remaining apaity of the vehile. (We assume that the demand of eah ustomer fits into one vehile.) This is neessary beause after adding a feasible ustomer to a tour, the former feasible neighbors may not fit into the vehile anymore. For reasons of spae limitations a drawing of delete nonfeasible is omitted. feasible neighbors j uses: delete nonfeasible rules: feas: ant dist : d id dem l i ap load m m+l i ant dist : d id dem l i ap load feas m onds: delete nonfeasible ; feas! Figure 6: The auxiliary unit feasible neighbors j The unit prob j is given in Figure 7. It provides the denominator of the probability that a feasible neighbor is hosen for a next move (see Equation 1 of Setion 2). The rule begin initializes this value with 0. The rule pair sum must be applied as long as possible. For not ounting a feasible neighbor several times sum hanges eah label feas into ok. At the end the unit relabel all private j (ok,feas) is applied whih undoes this relabeling, i.e., it hanges all ok-edges into feas-edges. It is very simple and hene not depited. With the rule pair move of the unit tour j the ant moves to a feasible neighbor with the probability depited under the arrow of the rule pair move in Figure 5. Moreover, in the memory the urrent load of the vehile, the path followed so far, and the total gth of the tour are updated. With the rule pair return the ant returns to the if no feasible neighbor is left and resets its urrent load to 0. Afterwards it starts to onstrut a new subtour. Finally, when all nodes are vied, the rule pair stop is applied to delete the load and the apaity from the memory as well as the edge between the ant and the in the ommon environment beause none of them are needed for the pheromone update whih is the next and last step of one run of Ant j. Moreover, the rule pair stop ommuniates the information about the gth of the found solution via the ommon environment by inserting an edge labeled with from the ant-node to a new node labeled with the gth of the solution. The unit put phero j is depited in Figure 8. It works a little different for ants, who should leave a pheromone trail and those who should not. Both kinds of ants apply different rules, Festshrift H.-J. Kreowski 16 / 23

18 ECEASST prob j uses: relabel all private j rules: begin: /0 /0 sum 0 sum: ant τ : x η : y id feas z sum ant τ : x η : y id z+x α y β sum ok onds: begin ; sum! ; relabel all private j (ok,feas) Figure 7: The auxiliary unit prob j but the struture of rule appliations is the same. In both ases the ant traverses the solution path stored in its memory and meanwhile deletes it. (Beause the path stored in the memory is shaped like a blossom with the in the middle, first the petals (subtours) are deleted and finally the.) This behavior is represented by the rule pairs start a (resp. start b) and put (resp. delete only) and the subexpression of the ontrol ondition ((start a + start b) ; (put! + delete only!)). One appliation of start followed by appliations of put (resp. delete only) as long as possible traverses one subtour of the found tour beginning and ending at the. The rule pairs delete the traversed path from the memory (leaving the ); put additionally leaves a pheromone trail in the ommon environment with the value 1/s, where s is the gth of the solution tour. Afterwards the remaining subtours are traversed until no further subtour is left in the memory. Then the respetive stop-rules an be applied, whih deletes the ant from the ommon environment as well as its omplete memory. Please note that due to the independene ondition for parallel rule appliation the rules start a and put an only be applied in parallel by different ants to different pheromone edges so that several pheromone updates of the same edge are always exeuted sequentially. 5.3 The Unit Evap&Selet Evap&Selet is given in Figure 9. It is responsible for the evaporation of pheromone trails, for the seletion of the w best solutions provided by the ants, and for marking these w ants with a put phero-loop. With the rule hek, whih is applied only one at the beginning, the unit heks whether all ants have finished their searh. This is the ase if all ants have written the gth of the found solution into the ommon environment. With the help of the unit relabel all global evaporation takes plae by multiplying the pheromone value of every pheromone edge in the ommon en- 17 / 23 Volume 26 (2010)

19 Autonomous Units for Ant Colony Optimization put phero j rules: put phero put phero start a: τ : x id a s τ : x+1/s id a s no phero no phero start b: put phero a put phero a put: τ : x id no phero a s τ : x+1/s no phero id s delete only : stop a: stop b: put phero no phero a a a s s /0 /0 /0 /0 onds: ((start a + start b) ; (put! + delete only!)) ; (stop a + stop b) Figure 8: The auxiliary unit put phero j Festshrift H.-J. Kreowski 18 / 23

20 ECEASST Evap&Selet uses: relabel all global rules: hek: A 1 A k... l 1 l k A 1 A k... l 1 l k put phero selet: A i l i l j l j > l i A i no phero rejet: A i l i A i onds: hek ; relabel all global(τ : z,τ : (1 ρ) z) ; selet w ; rejet! Figure 9: The autonomous unit Evap&Selet vironment with (1 ρ), where ρ (0,1] is a pheromone deay parameter. After that, the rule selet is applied w times (in the ontrol ondition this is abbreviated by selet w ). It finds an ant with the best solution, marks it with a loop put phero, and deletes the information about the gth of the ant s solution from the ommon environment. Eah further appliation of selet finds a next best solution. When w best solutions are found, the rule rejet is applied as long as possible to equip the remaining ant nodes with a no phero-loop. This rank-based approah ould be extended by the elitist strategy (see e.g. [DS04]). In this strategy the best solution so far is memorized and when pheromone update takes plae, this tour gets additional pheromone. (In our modeling of the CVRP, we do not onsider this strategy beause of spae limitations.) Remark. The presented modeling an be used to prove orretness properties a few of whih are informally desribed here. 1. In every exeution of initial poion j ;tour j a solution is onstruted, i.e., a set of yles of the onstrution graph is traversed by Ant j and stored in its memory suh that the belongs to every yle, and every ustomer ours exatly one in exatly one yle. 2. The unit relabel all global models pheromone evaporation. 3. The unit put phero j models pheromone update, removes Ant j from the ommon environment, and deletes its memory. 19 / 23 Volume 26 (2010)

21 Autonomous Units for Ant Colony Optimization 4. Eah exeution of (Ant 1... Ant k Evap&Selet) models an iteration of the orresponding ACO-Algorithm, i.e., (1) solution onstrution, (2) pheromone evaporation, and (3) pheromone update. For reasons of spae limitations, proofs of the first three properties are omitted and the proof of the fourth is roughly skethed. Proof sketh of the fourth property. Let Aut CVRP = {Ant 1,...,Ant k,evap&selet}. Moreover, let Rules CVRP = aut Aut CVRP Rules(aut). Before the first iteration the urrent state is equal to (G,map) with G SEM(Constrution graph) and map 0 (aut) = /0 for all aut Aut CVRP. Assume that after n iterations the urrent state is equal to (G,map) where G is obtained from G via the pheromone evaporation and the pheromone updates of the n previous iterations. Let s SEM Aut CVRP (Ant 1 Ant k Evap&Selet) be the transformation sequene of the (n+1) th iteration. Then s = ((G 0,map 0 ),...,(G m,map m )) with G 0 = G and map 0 = map and for eah aut Aut CVRP, ((G 0,map 0 (aut)),...,(g m,map m (aut))) SEM Rules CVRP Rules(aut) (flc(c aut )). This means by the definition of the semantis of ontrol onditions that at first every ant has to exeute its rule initial poion, then its tour-unit, and finally its put phero-unit. The rules in k j=1 Rules(Ant j ) ( k j=1 Rules(put phero j )) satisfy the independene ondition so that they an be applied in parallel. (Please note that aording to the definition of the semantis of global ontrol onditions, only rules of different autonomous units an be applied in parallel whih also implies that the independene ondition of parallel rule appliation has to be heked only parallel appliations of global rules.) The rules in put phero an only be applied if the orresponding ants are equipped with a put phero- or a no phero-loop. This loops an only be generated by the autonomous unit Evap&Selet whih in turn an start working as soon as every unit has finished to exeute its tour-unit, beause its first rule hek an only be applied after eah ant has applied the rule stop whih is the last rule that is applied in tour. Aording to the first property of the remark, every ant onstruts a tour by exeuting the rule initial poion and then its tour-unit. After the appliation of hek, the unit relabel all global is applied whih aording to the seond property models pheromone evaporation. Afterwards, the autonomous unit Evap&Selet exeutes selet w followed by as many appliations as possible of rejet. It an be easily shown by indution that for eah ant either the rule selet or the rule rejet is applied exatly one for every ant. As soon as an ant has got its put phero- or its no phero-loop by the rule selet or rejet, it an start to apply the rules of put phero. (Due to the independene ondition of parallel rule appliation the rules of put phero (of different ants) an be applied in parallel if and only if they do not augment the pheromone quantity of the same edge.) By the third property the exeution of put phero models pheromone update and deletes the ants from the ommon environment as well as all private states. Hene, the environment G m is obtained from G via pheromone evaporation and pheromone update. Altogether we get that eah exeution of Ant 1 Ant k Evap&Selet models solution onstrution, pheromone evaporation and pheromone update in this order. Festshrift H.-J. Kreowski 20 / 23