Int. J. Appl. Math. Comput. Sci., 2005, Vol. 15, No. 2, 205–220 ANT ALGORITHM FOR FLOW ASSIGNMENT IN CONNECTION-ORIENTED NETWORKS KRZYSZTOF WALKOWIAK Chair of Systems and Computer Networks, Faculty of Electronics Wrocław University of Technology Wybrze ˙ ze Wyspia´ nskiego 27, 50–370 Wrocław, Poland e-mail: [email protected]This work introduces ANB (Ant Algorithm for Non-Bifurcated Flows), a novel approach to capacitated static optimization of flows in connection-oriented computer networks. The problem considered arises naturally from several optimization problems that have recently received significant attention. The proposed ANB is an ant algorithm motivated by recent works on the application of the ant algorithm to solving various problems related to computer networks. However, few works concern the use of ant algorithms in the assignment of static flows in connection-oriented networks. We analyze the major characteristics of the ANB and try to explain its performance. We report results of many experiments over various networks. Keywords: ant algorithms, flow assignment, computer networks 1. Introduction In recent years, we have been observing a growing role of computer networks. Due to increasing expectations for the quality of service and traffic engineering capabili- ties, new technologies like ATM (Asynchronous Transfer Mode) and MPLS (MultiProtocol Label Switching) are introduced to overcome the disadvantages of old proto- cols, e.g., the IP. Most of the new techniques apply the connection-oriented model, where routing decisions are made once, while establishing a virtual connection. Net- work resources must be provisioned effectively to facili- tate low-cost, reliable services. Furthermore, current net- works are large, and the bandwidths of links and the vol- ume of traffic grow. Therefore, methods of network op- timization, both dynamic (on-line) and static (off-line), are indispensable for the development of robust networks that can fulfil high expectations of the users. In this work we focus on capacitated static optimization of connection- oriented flows. We assume that traffic demands are known and must all be satisfied. Furthermore, we consider a ca- pacitated problem with capacity constraint, i.e., the flow of each link cannot exceed the capacity of that link. The capacitated network design problems are much more diffi- cult than the corresponding uncapacitated ones (Gendron et al., 1998). Solving static problems is crucial for ef- ficient design of new networks or updating the existing networks (Kasprzak, 2001; Grover 2004). Since various optimization problems encountered in computer networks are NP-complete and some of the ex- isting algorithms are not adequate to tackle the increas- ing complexity of such problems for large networks, a range of heuristic approaches are developed to deal with these problems. Some of the most promising approaches are algorithms taking inspiration from physics, biology or social sciences. The most significant examples are: ge- netic algorithms—simulated annealing, tabu-search, neu- ral nets, and ant systems. These heuristics apply a certain amount of repeated trials and employ one or more agends operating using a mechanism of competition-cooperation. Applications of these algorithms cover many combinato- rial optimization problems. For some examples, refer to (Colorni et al., 1996; Elbaum and Sidi, 1996). In our opin- ion the exploitation of biological concepts might lead to new approaches for many classical network optimization problems. In this work we focus on ant algorithms, a method proposed in 1991 by Dorigo et al. (1991), also called an ant system (AS) or ant colony optimization (ACO). For clarity, in the remainder of the paper we will use the term ‘ant algorithm’ to refer to a broad class of ant-based al- gorithms. The ant algorithm is a simulation of agents that cooperate to solve an optimization problem by means of communication. The inspiration comes from research on the behavior of real ants. Ants are social insects living in colonies. Their behavior is determined by the survival
This work introduces ANB (Ant Algorithm for Non-Bifurcated Flows), a novel approach to capacitated static optimizationof flows in connection-oriented computer networks. The problem considered arises naturally from several optimizationproblems that have recently received significant attention. The proposed ANB is an ant algorithm motivated by recent workson the application of the ant algorithm to solving various problems related to computer networks. However, few worksconcern the use of ant algorithms in the assignment of static flows in connection-oriented networks. We analyze the majorcharacteristics of the ANB and try to explain its performance. We report results of many experiments over various networks.
Keywords: ant algorithms, flow assignment, computer networks
1. Introduction
In recent years, we have been observing a growing roleof computer networks. Due to increasing expectationsfor the quality of service and traffic engineering capabili-ties, new technologies like ATM (Asynchronous TransferMode) and MPLS (MultiProtocol Label Switching) areintroduced to overcome the disadvantages of old proto-cols, e.g., the IP. Most of the new techniques apply theconnection-oriented model, where routing decisions aremade once, while establishing a virtual connection. Net-work resources must be provisioned effectively to facili-tate low-cost, reliable services. Furthermore, current net-works are large, and the bandwidths of links and the vol-ume of traffic grow. Therefore, methods of network op-timization, both dynamic (on-line) and static (off-line),are indispensable for the development of robust networksthat can fulfil high expectations of the users. In this workwe focus on capacitated static optimization of connection-oriented flows. We assume that traffic demands are knownand must all be satisfied. Furthermore, we consider a ca-pacitated problem with capacity constraint, i.e., the flowof each link cannot exceed the capacity of that link. Thecapacitated network design problems are much more diffi-cult than the corresponding uncapacitated ones (Gendronet al., 1998). Solving static problems is crucial for ef-ficient design of new networks or updating the existingnetworks (Kasprzak, 2001; Grover 2004).
Since various optimization problems encountered incomputer networks are NP-complete and some of the ex-isting algorithms are not adequate to tackle the increas-ing complexity of such problems for large networks, arange of heuristic approaches are developed to deal withthese problems. Some of the most promising approachesare algorithms taking inspiration from physics, biology orsocial sciences. The most significant examples are: ge-netic algorithms—simulated annealing, tabu-search, neu-ral nets, and ant systems. These heuristics apply a certainamount of repeated trials and employ one or more agendsoperating using a mechanism of competition-cooperation.Applications of these algorithms cover many combinato-rial optimization problems. For some examples, refer to(Colorni et al., 1996; Elbaum and Sidi, 1996). In our opin-ion the exploitation of biological concepts might lead tonew approaches for many classical network optimizationproblems.
In this work we focus on ant algorithms, a methodproposed in 1991 by Dorigo et al. (1991), also called anant system (AS) or ant colony optimization (ACO). Forclarity, in the remainder of the paper we will use the term‘ant algorithm’ to refer to a broad class of ant-based al-gorithms. The ant algorithm is a simulation of agents thatcooperate to solve an optimization problem by means ofcommunication. The inspiration comes from research onthe behavior of real ants. Ants are social insects livingin colonies. Their behavior is determined by the survival
K. Walkowiak206
of the whole colony while the importance of individualants is not so significant. Ants can cooperate effectivelyin a group to perform some tasks. For instance, almostblind ants are able to find the shortest paths from theircolony to feeding sources and back. It was observed thata moving ant lays some pheromone (in variable quanti-ties) on the ground, hence marking the path it follows.Next, ants moving towards the feeding area can detect thepheromone left by the previous ant, decide with a highprobability to follow it, and reinforce the selected trailwith their own pheromones. This form of indirect com-munication mediated by pheromone placing is called stig-mergy. Another principal aspect of real ants’ behavior isa coupling between the autocatalityc (positive feedback)mechanism and an implicit evaluation of solutions, i.e.,the more ants follow a trail, the more attractive that trailbecomes for being followed. A comprehensive treatmentof ants’ behavior and its impact on ant algorithms can befound in (Colorni et al., 1996; Dorigo et al., 1991; 1999).Researchers have found many applications of ant algo-rithms that cover problems such as the traveling salesmanproblem, the quadratic assignment problem, job schedul-ing, vehicle routing, graph coloring, or network routing(Dorigo et al., 1999).
Our discussion in this article focuses on the ap-plication of an ant algorithm to a non-bifurcated flowassignment (NBFA) problem. Connection-oriented net-work techniques like ATM, Frame Relay and MPLS aremodeled using the non-bifurcated multicommodity flow.These techniques have gained much attention in recentyears. Most optimization problems related to ATM andMPLS are similar to the NBFA problem. However, sinceNBFA is NP-complete, for relatively large networks onlyheuristic algorithms can be applied. In this work we dis-cuss and evaluate a new ant algorithm called ANB thatsolves the NBFA problem. Our starting point is an algo-rithm proposed in (Walkowiak, 2001b).
Network flows are usually optimized according toadditive or bottleneck weights (Szeto et al., 2002; Kar etal., 2000). The former kind of metrics assume that the costfunction is computed as a sum of weights over all links. Inthe latter approach, the objective function is given by themaximum (or minimum) value of link weights. In thiswork we concentrate on network optimization applyingadditive weights. Additive metrics arise in many settings,e.g., cost, end-to-end delay, jitter, survivability (Kasprzak,2001; Walkowiak, 2003a). The ANB algorithm minimizesthe overall network flow—the function defined as the sumof the link flows of all the links in the network. The linkflow is simply a sum of the bandwidths of the paths whichtraverse that link. However, ANB can be easily adapted touse also objective functions like network cost, delay, lostflow.
The paper is organized as follows: In Section 2 webriefly introduce non-bifurcated multicommodity flows.In Section 3 we present background and previous researchin the field of ant algorithms applied to network problems.Section 4 contains a detailed description of ANB. In Sec-tion 5 we show the results of extensive simulations andstudy three main issues that arise in applying the ANB al-gorithm: the tuning of the algorithm parameters, conver-gence and comparison with other heuristics proposed pre-viously in the literature. In the last section we draw someconclusions and outline directions for future research.
2. Multicommodity Flows
Routing algorithms used to assign network flows can beclassified as static or dynamic, and centralized or dis-tributed. Static routing assumes that the network condi-tions are time-invariant. Dynamic routing is applied innetworks where demands frequently change. Centralizedalgorithms are usually applied in legacy routing systemsand may have problems with scalability and inordinate de-mand for managing decisions requiring human attention.Distributed systems usually work locally in each networknode and use only locally available information (Kassaba-lidis et al., 2001). In this work we focus on static routingusing the centralized approach. We assume that the net-work and all demands to be sent in the network are knowna priori.
Various network or transportation problems can bemodeled as multicommodity flow problems. The multi-commodity flow problem consists in finding routes for allsuch commodities which minimize (or maximize) a per-formance function (e.g., a delay, a cost) such that a set ofconstraints (e.g., arc capacity constraints) is satisfied (As-sad, 1978; Fratta et al., 1973; Girard and Sanso, 1998;Gendron et al., 1998; Grover, 2004; Kasprzak, 2001, Ottet al., 2001; Pióro et al., 2003; Pióro and Medhi, 2004).
We are given a network ��� �� where � � �����is a directed graph with � nodes and � arcs, and � �� � �
� is a function that defines capacities of the arcs.We assume that all commodities included in a set � arenumbered from 1 to �, where � denotes the number ofall commodities. For the -th commodity, � denotesa source and �� denotes the destination of the commod-ity. Each commodity of the flow requirement �� must berouted from the node � to the node �� through a givennetwork. A multicommodity flow is a set of functions
� � �� �� � ���� � �� � � � � � (1)
for which the flow of the -th commodity in the arc ��� �� ���� �� for � �� � � � � � satisfies the following condi-
Ant algorithm for flow assignment in connection-oriented networks 207
tions:�������
���� ����
������
���� ��
�
�����
�� ��� � � ��
��� ��� � � ���
� ���� ���
(2)
���� �� � � for ��� �� � � � �� (3)
where ���� � �� � � � � and ��� �� � �� is a setof destination nodes of edges leaving the node �, and���� � �� � � � � and ��� �� � �� is a set of sourcenodes of edges entering the node �. The condition (2) iscalled the conservation of the flow at nodes. The condi-tion (3) is a non-negativity of the flow in directed edges.The definition of the multicommodity flow given by (1)–(3) is called the node-path notation.
Here ��� �� denoting the flow of the arc ��� �� isdefined as
��� �� �����
���� ��� (4)
Multicommodity flows are of two types: bifurcated andnon-bifurcated. The former is a flow in which one com-modity can be transported using many paths. Each pathcarries only a part of the commodity. For the non-bifurcated flow each commodity flows along one pathonly. An example of the bifurcated flow is the flow of theInternet using the TCP�IP protocols. Connection-orientednetworks like ATM, MPLS, and Frame Relay are exam-ples of networks applying non-bifurcated multicommod-ity flows.
Many optimization problems related to computernetworks can be modeled as multicommodity flow prob-lems. Kasprzak (2001) specifies the most important prob-lems of computer network design:
� Flow Assignment (FA),
� Capacity and Flow Assignment (CFA),
� Topology, Capacity and Flow Assignment (TCFA).
The most common method to optimally solve prob-lems of bifurcated multicommodity flow allocation is thelinear programming approach (Bienstock, 2002; Grover,2004; Ott et al., 2001; Pióro et al., 2003; Pióro andMedhi, 2004). Also, some heuristic algorithms are de-veloped for the optimization problem considered. In (Bi-enstock, 2002; Burns et al., 2003; Kasprzak, 2001; Mu-rakami and Kim, 1996), algorithms based on the Flow De-viation method proposed in (Fratta et al., 1973) are devel-oped for various problems related to bifurcated multicom-modity flows. Gendron et al. (1998) propose to apply theLagrangean relaxation to solve capacitated network prob-lems. Another popular method used for the optimization
of bifurcated multicommodity flows are soft optimizationtechniques (Corne et al., 2000; Grover, 2004). Other in-teresting algorithms for the multicommodity flow assign-ment problem are Extremal Flows (EF) (Cantor and Gerla,1974) and Gradient Projection (GP) (Schwart and Che-ung, 1976).
In this work we focus on a static flow assignment FAproblem for non-bifurcated flows. The global multicom-modity flow denoted by � � �� �� � � � � �� is definedas a vector of flows in all arcs according to constraints(1)–(4). Let �� be a set including all vectors describ-ing non-bifurcated multicommodity flows. An importantconstraint in the optimization of computer networks is thecapacity constraint defined as follows:
�� � � � � �� � (5)
The inequality (5) guarantees that in every arc the flowdoes not exceed the capacity. Let �� denote a set of allnon-bifurcated flows � �� for which the condition (5)holds. In the rest of the paper we call a flow feasible if thecapacity constraint (5) is satisfied.
The non-bifurcated multicommodity flow assign-ment (NBFA) problem can be formulated as follows:
� �
�� � �����
� � (6)
subject to � ��� (7)
The objective function (6) is the overall flow in the net-work. In the problem (6)–(7) we wish to minimizethe overall flow over all feasible non-bifurcated flows.According to (Karp, 1975), the NBFA problem is NP-complete.
In this work we apply an equivalent representationof non-bifurcated multicommodity flow assignment calledlink-path formulation (Fratta et al., 1973; Pióro et al.,2003). It is obtained by providing for each commodity� � � a set of paths �� � ���� � � �� � � � � ��� from thenode � to the node ��. For a non-bifurcated multicom-modity flow, the commodity can use only one path � �
� .Let ��� denote a ��� variable, which equals one if ���is the path for the commodity � and is equal to 0 other-wise. Another binary variable ���� indicates whether ornot the path ��� uses the arc � � �. Using this represen-tation of the multicommodity flow, the NBFA problem isas follows:
� �
�� � �����
� (8)
subject to (2), (3), and� �����
��� � �� �� � � � (9)
K. Walkowiak208
��� � ��� ��� �� � �� ��� � ��� (10)
� �����
� �����
���������� (11)
� �� � �� � �� (12)
The objective function (8) is the cost of the flow in thenetwork. To simplify the problem we assume that the costof every commodity is the same. Therefore, in (8) wesum the flows of all links in the network. Since we con-sider the non-bifurcated multicommodity flow, the con-dition (9) states that each commodity can use only oneprimary route. The constraint (10) ensures that decisionvariables ��� are binary. The condition (11) is a definitionof a link flow. Finally, (12) is a capacity constraint. If wechange the constraint (10) to
� ��� �� �� � �� ��� � ���
we will obtain the bifurcated multicommodity flow prob-lem.
The NBFA problem, like many other network designproblems, is very complex and numerically intractableeven for networks with a small number of nodes. Noticethat the NBFA problem is an integer 0–1 problem with lin-ear constraints. However, the size of the problem is verylarge even for relatively small networks. For instance, fora sample network having 10 nodes and 42 arcs the aver-age number of routes between a node pair is 237. For thelink-path formulation the number of binary variables rep-resenting the selected route is about 90���.
A popular method to solve 0–1 problems is thebranch-and-bound approach. Such algorithms were ap-plied to many problems related to NBFA (Kasprzak, 2001;Walkowiak, 2002; 2004). Nevertheless, branch-and-bound algorithms are intractable for networks of mediumand large sizes. The only way to solve the NBFA problemby an exact algorithm that can produce an optimal solu-tion is to reduce the problem size and consider only a partof all possible routes. Such an approach, called the PathGeneration technique, is discussed in (Pióro et al., 2003).Another possible method to reduce the size of the prob-lem considered is the hop-limit approach (Herzberg et al.,1995).
Some heuristic algorithms have been developed forsolving the non-bifurcated multicommodity flow problem.One of the most substantial ones is the Flow Deviation(FD) algorithm proposed in (Fratta et al., 1973). TheFD algorithm and its modifications have proven their ef-fectiveness in many network design problems (Bienstock,2002; Burns et al., 2003; Kasprzak, 2001; Murakami andKim, 1996; Walkowiak, 2002, 2003a, 2003b). Also, agenetic algorithm has been proposed for the problem con-sidered (Walkowiak, 2001a). However, it should be men-tioned that most papers on flow optimization focus on bi-
furcated multicommodity flows. Much fewer works con-sider problems of non-bifurcated flow allocation formu-lated as 0–1 integer problems. Therefore, generally, thereare not many algorithms and methods for such problems.
3. Related Work
In this section we focus on example applications of ant al-gorithms to various network problems. The related workon multicommodity flows is provided in the previous sec-tion.
First, we briefly present the main ideas of the AntColony Optimization algorithm, which is a foundationof many other ant algorithms. In ACO meta-heuristica colony of artificial ants cooperate in finding good so-lutions to discrete optimization problems. Cooperationis a major element of ACO algorithms. The choice isto allocate the computational resources to a set of rela-tively simple agents that communicate indirectly by stig-mergy. Artificial ants have some common features withreal ants—they employ pheromone trails, which are usedfor the shortest path finding in real ant colonies. How-ever, artificial ants have been enriched with some capabil-ities which are not encountered in nature. Actually, ACOis an engineering approach to the design and implemen-tation of software systems for the solution of optimiza-tion problems. Therefore, artificial ants are provided withsome capabilities that make them more efficient. The mostimportant ideas of ACO derived from real ants are theuse of a colony of cooperating individuals, a (artificial)pheromone trail for local stigmergetic communication, asequence of local moves to find the shortest paths, and astochastic decision policy using local information and nolookahead. The major differences between artificial antsused in ACO and real ants are: the discrete nature of arti-ficial ants (they live in a discrete world and have discretestates); artificial ants have internal state and memory tosave past actions; artificial ants can deposit a pheromoneaccording to the function of the quality associated with thesolution found; artificial ants can update pheromone trailsafter having generated a solution; an ACO system can usesome extra capabilities like local optimization and back-tracking (Dorigo et al., 1999).
ACO consists of a finite size colony of cooperatingartificial ants altogether seeking high quality solutions tothe optimization problem. Each individual ant constructsa solution, or an element of it, starting from an initial statechosen according to the problem considered. Each antgathers information on its own performance and on spe-cial characteristics of the problem in order to change theproblem representation, as seen by the other ants. Eachant leaves an amount of pheromone according to the qual-ity of its solution that is expressed as the shortest path
Ant algorithm for flow assignment in connection-oriented networks 209
through a graph representing the optimization problem. Asingle ant can find a feasible solution, but a much betterperformance is achieved by a set of cooperating ants. Themoves of each ant are selected according to a local opti-mization procedure that involves: ant private information,pheromone trail and problem-specific local information.Artificial ants have memory storing information about anthistory. This mechanism can be used to avoid cyclesby backtracking the ant from already visited nodes or tosave ant routes. It makes finding feasible solutions eas-ier. Two approaches are used for releasing a pheromoneby an ant: an on-line step-by-step update and a global on-line delayed update. In the former method each ant up-dates the pheromone on each visited link according to aselected formula. The latter approach assumes an updateof the pheromone trails after completing the ant’s routeand generating the whole solution. Ants make use of lo-cal available ant-decision tables including information forthe decision of each incoming ant to direct their searchtowards the most promising areas of the solution space.After generating a solution and depositing pheromone in-formation, the ant is deleted from the system. In order toencourage ants to find new routes representing solutions,the pheromone gradually evaporates. Consequently, oldsolutions, if not reinforced by new ants, successively be-come less important. Beside ants’ operations acting froma local perspective, a special daemon can be used, whichapplies a global representation of the problem in order toimprove the performance of the ant algorithm. A majorpoint in the application of the ant system to optimizationproblems is a paper problem representation. For that rea-son the first step to implement an ACO algorithm is to de-velop a representation of the problem considered proper tothe ant algorithm. Problems related to computer networkshave many features that allow the application of the ACO.The most meaningful of them are: computation distribu-tion, asynchronous evolution of the network status, andgraph representation (Dorigo et al., 1999).
The paper by Schoonderwoerd et al. (1997) is theearliest work including an application of the ant algo-rithm to a routing problem. Schoonderwoerd et al. de-veloped an ant-based control (ABC) algorithm and em-ployed it to a model of the British Telecom telephone net-work. Each node of the network has the same functional-ity as a crossbar switch with limited capacity, while net-work links have unlimited capacity. The major goal of theABC system is to find routes for new connections to ob-tain load balancing of the network and avoid the conges-tion of the network or the rejection of demands. Ants arecreated at regular temporal intervals from all the nodes to-wards randomly selected destination nodes. Ants deposita pheromone step-by-step on the links they traverse. Thepheromone is associated with routing information. Afterthat, ants moving in the opposite direction of the ant con-
sidered apply the pheromone information. An assumptionis made that the analysed network is cost-symmetric. Itsimplifies the approach and ants do not have to remem-ber their routes. The amount of the pheromone on thelink selected by an ant is reinforced according to a factorthat is a function of the ant’s age. For the normalizationof the pheromone values, which are used as probabilities,the pheromone of other links decays proportionally. Rout-ing tables for new calls are generated according to ant-decision tables. This means that new calls build routesstarting from the source node and choosing sequentiallyneighbor nodes with the highest probability value until thedestination node is reached. The node capacity is updatedaccordingly to the call setup or termination.
Di Caro and Dorigo (1998a; 1998b) developed a sys-tem called the AntNet for distributed routing in connec-tionless networks. In the AntNet real-time trips experi-enced by ants and a local-based statistical model are ap-plied to estimate the path quality. This means that antsmove through the same real network and can have delayslike real data packets. The global update of the pheromoneis used. Once a path has been found, ants lay down theon the visited nodes an amount of the pheromone propor-tional to the quality of the found path. Ants move backto source nodes using high priority queues to enable fastpropagation of pheromone information. In addition to thepheromone information, local heuristic information repre-senting the current link queues is used in the ant-decisiontable. Di Caro and Dorigo also developed an enhancedversion of the AntNet for connection-oriented networks(Dorigo et al., 1999).
White et al. (1998) developed the Routing By Ants(RBA) system for routing in connection-oriented net-works. The RBA algorithm applies a very similar ap-proach to the classical Ant System (AS) proposed in(Dorigo et al., 1991). An ant-based algorithm for dynamicpacket-switched networks is also presented in (Subrama-nian et al., 1997).
The above examples of the ant algorithm focus ondynamic routing problems. The work (Varela and Sinclair,1999) is an attempt to apply the ant algorithm to a staticrouting problem. It presents the problem of routing andwavelength-allocation (RWA) in a multi-wavelength alloptical virtual-wavelength-path routed transport network.The objective function is the network wavelength require-ment that denotes a total number of distinct wavelengthsused in the network, which is equal to the maximumnumber necessary on any link. Ants move from everysource to every destination, one link per algorithm step.When all ants reach their destination and die, a new cy-cle of the algorithm starts. Varela and Sinclair developedthree major versions of the ant algorithm according tothree update strategies: local update, global/distance andglobal/occupancy. In the first version, each ant is attracted
K. Walkowiak210
by a pheromone of its own type left by previous ants andrepelled by other ants’ pheromone. The pheromone isupdated on each algorithm step. The other two versionsapply the global update strategy. In the global/updatethe pheromone is updated inversely proportionally to thelength of the ant’s path. The global/occupancy strategyassumes that the weight of repulsion depends on the linkutilization, i.e., the number of ants of any type using thelink considered. All variants use pheromone evaporation.Simulations show that the best results are produced bythe global/occupancy algorithm. Varela et al. applied abacktracking operation, with each ant keeping a “tabu” listof previously visited nodes. Backtracking prevents dead-ends and cycles. When an ant is blocked, it analyzes its“tabu” list and tries to proceed from the previous location.This capability requires each ant’s memory to include alist of nodes visited in order.
Garlick and Barr (2003) also consider the RWA prob-lem with dynamic traffic, in which the number of wave-lengths per fiber is fixed. The objective is to mini-mize connection blocking using an ant-colony optimiza-tion (ACO) algorithm. The main goal of the algorithmis to quantify the importance of combining path-lengthand congestion information in making routing decisions.The ACO algorithm reaches lower blocking rates than anexhaustive search over all available wavelengths for theshortest path.
An overview of many ant algorithms applied to var-ious static and dynamic optimization problems can befound in (Dorigo et al., 1999). Generally, most of theliterature on the application of ant algorithms to networkproblems we have found concerns either dynamic routingor the RWA problem. To the best of our knowledge, thefirst work presenting an ant algorithm applied to the prob-lem of static flow assignment in connection-oriented net-works with capacity constraint is the work (Walkowiak,2001b). In this paper we present a much more compre-hensive study of the algorithm proposed in (Walkowiak,2001b) together with the results of extensive numericalexperiments.
4. Ant Algorithm for the Non-BifurcatedMulticommodity Flow AssignmentProblem
In this section we introduce our approach to the NBFAproblem using an ant algorithm called hereafter the ANB(Ants for Non-Bifurcated Flows) algorithm. The overallframework of the ant algorithm for connection-orientedflow problems was proposed in (Walkowiak, 2001b). TheANB algorithm presented below is a continuation of thatwork. Moreover, we also depict a second ant algorithm—ANIBS (ANB with Initial Solution)—which is a slight
modification of ANB. The only difference between ANBand ANBIS is the initialization of pheromone values. Allother procedures are the same. Therefore, if we refer toANB, also the ANBIS algorithm is kept in mind.
We use the link-path representation of the NBFAproblem presented in Section 2. For the sake of simplicity,we introduce the following function:
���� �
�� ��� � ��
� ��� � � ��
When we tackle an optimization problem with an ant al-gorithm, either the formulation of the algorithm shouldguarantee that all constraints are satisfied or constraintsof the problem must be introduced in the objective func-tion using a penalty method. To introduce the capacityconstraint (12), we modify the objective function of thenetwork flow and use the penalty method. The modifiedobjective function of the NBFA problem is as follows:
� �� � �����
� � � ��
��� � � ���
��� (13)
The �� parameter is a penalty factor. It should be notedthat to find a feasible solution of the NBFA problem, itis very important to select the value of the penalty factorin a correct way. The algorithm should be forced to findfeasible solutions. In the objective function � � we must‘promote’ feasible solutions and ‘punish’ solutions vio-lating the capacity constraint. Other constraints of NBFAconnected with the non-bifurcated nature of the multicom-modity flow are satisfied due to the formulation of theANB algorithm.
We begin the presentation of the algorithm by intro-ducing the notation. We will keep the same notation in theremainder of the paper.
Indices:
� used as a subscript, denotes the number of the antconsidered,
� used as a subscript, denotes the number of the nodeconsidered,
� used as a superscript, denotes the number of the cur-rent iteration (cycle) of the algorithm.
Algorithm parameters:
� number of ants which equals the number of com-modities,
�� bandwidth requirement for the commodity associ-ated with the ant �,
�� penalty factor used to scale the penalty function inthe objective function,
� constant used in the pheromone updating rule,
Ant algorithm for flow assignment in connection-oriented networks 211
� evaporation coefficient; at the end of each cycle � thelevel of the pheromone on all links is reduced by �(in the remainder of the paper this parameter is alsolabeled as RHO),
� parameter that allows the user to control the relativeimportance of the pheromone given by the variable���� (in the remainder of the paper this parameter isalso labeled as ALPHA),
parameter that allows the user to control the relativeimportance of the local heuristic information repre-sented by !��� (in the remainder of the paper this pa-rameter is also labeled as BETA),
NOI number of iterations of the algorithm.
Variables
��� route used by the ant � in the iteration �,
" ��� amount of the �-th ant pheromone laid on the arc tothe �-th node in the iteration �,
#�� length of the route of the �-th ant selected in the cycle�, i.e., a sum of metrics � � ������ � � ����
�� ofthe arcs belonging to the route ��� followed by theant in the cycle �,
���� weight of attraction of the neighbor node � for the�-th ant in terms of the pheromone laid on the arc �,
��� set of nodes allowed to the ant � in the time cycle �
(this set is associated with the tabu list),
!��� visibility—local heuristic information of selectingthe node � by the ant � in the iteration � (visibilityis based on strictly local information and it measuresthe attractiveness of the next node to be selected),
$��� ant decision probability of selecting the node � bythe �-th ant in the cycle � (the ant can select onlyamong all allowed nodes; since the variable $ ��� is aprobability, it is normalized).
The number of ants is equal to the number of com-modities in the network, i.e., each commodity has its ownant. Each ant deposits its own pheromone. An ant ischaracterized with the attributes of the commodity: thesource and destination nodes, and the bandwidth require-ment. Each ant possesses some memory to store the routebeing traversed. Every cycle of the algorithm starts withplacing each ant in its source node and initiating the ant’smemory. The framework of ANB and ANBIS is shown inFig. 1. The most substantial features of this algorithm areexplained in the following. It must be noted that ANB andANBIS are very similar. The only difference is included inthe InitializePheromoneValues (" ) procedure.
InitializePheromoneValues (" ): For the ANBalgorithm all pheromone values are set to the same pos-itive constant value 1 in the first iteration of the algorithm,i.e., "��� � �. For ANBIS we apply a feasible (in termsof the capacity constraint) solution in the following way:
1 InitializePheromoneValues(τ) 2 for t=1 to NOI do 3 for i=1 to p do InitializeAnt(i) 4 while(ExistAnts()==TRUE) do 5 for i=1 to p do 6 if (ExistAnt(i)==TRUE) then 7 MoveAnt(i) 8 end if 9 end for10 end while
11 PheromoneUpdate(τ)12 DaemonActions()13 end for
//i – index of analyzed ant14 procedure MoveAnt(i)15 j=SourceNodeOfAnt(i)16 do17 Ai=CreateAllowedNodesSet(i,j)18 if (Ai
t==∅) then j=019 else20 for k∈Ait do21 αikt=ComputeAttraction(i,j,k)22 ηikt=ComputeLocalVisibility(i,j,k)23 end for24 for k∈Ait do
γikt=ComputeDecisionProbability(i,j,k)25 j=SelectNextNode(γ)26 end else27 if (j==0) then28 FindShortestRoute(a)29 j=DestinationNodeOfAnt(i)30 end if31 while(j!=DestinationNodeOfAnt(i))32 end procedure
//i – index of analyzed ant; j – number ofcurrent node; k – number of //allowed nodefor the ant i which is in node j33 procedure ComputeLocalVisibility(i,j,k)34 d=DestinationNodeOfAnt(i)35 if (k==d) then36 if (ResidualCapacity(j,k)>Qi) then
ηikt=137 else ηikt=2⋅n38 else39 if (ResidualCapacity(j,k)>Qi) then
40 ηikt=LengthOfFeasibleShortestPath(i,k)41 else ηikt=2⋅n42 end else43 return ηikt44 end procedure
Fig. 1. ANB description in pseudo-code.
All pheromone values are also initially set to 1. However,after that, each ant follows the path given by the feasi-ble solution found by another algorithm. Next, we updatethe pheromone according to formulas given below. Thismeans that the pheromone values are initiated accordingto a feasible solution.
K. Walkowiak212
The main loop of the program (the lines 2–13) is re-peated for a given number of iterations. However, someother termination criteria, e.g., when the evaluation of thebest found solution gives a positive result can be applied.The variable � denoting the current cycle is a global vari-able available in all procedures.
InitializeAnt (i): Each ant is placed in its sourcenode. The memory of the ant is cleared. The loop fromthe line 4 to the line 10 is repeated until all ants reach theirdestination nodes.
MoveAnt (i): An important part of the ANB algorithm.This procedure is responsible for moving the ant throughthe network. The detailed actions of an ant are given inthe lines 14–32. The ant continues its trip across the net-work until it reaches the destination node (the lines 16–31). According to the current state and the position in thenetwork, the ant applies its decision policy and selects anext node to move to (the line 25). The ant is attracted to anode of those adjacent to its current node excluding nodescontained in the tabu list. The set ��
�, created in the line17, includes all nodes allowed to the ant �. The weight ofattraction of the node � for the �-th ant is given by
���� �" ���
����
�
" ���� (14)
Moreover, in the ant decision table we apply some localheuristic information !��� of selecting the node � by theant �. The calculation of !��� is done by the procedureComputeLocalVisibility (the line 22) discussedbelow. The values of !��� are normalized in much thesame way as ���� .
The ant decision probability of selecting the node �by the �-th ant is calculated as follows:
$��� �������
��!�����
����
�
�" ������!����
�� (15)
The parameters � and are used to find a trade-off be-tween local heuristic information and pheromone inten-sity.
Ants follow a route from a particular source nodeto a particular destination. The route must not containloops. Therefore, each ant records its route in memoryto maintain a tabu list of nodes the ant must not revisit.However, a loop may occur when the ant reaches a ‘deadend’ (Varela and Sinclair, 1999). In order to overcome thisproblem, we allow the ant to backtrack to the source nodeand find the shortest route using the shortest path algo-rithm skipping the pheromone information (the line 28).We apply the Dijkstra algorithm with the hop metric, i.e.,
each arc has a weight equal to 1. As a matter of fact, inour algorithm, we let the ant try 10 times to overcome theloop. If after 10 attempts the ant cannot reach the destina-tion node, we apply the shortest path algorithm to computethe ant’s route.
ComputeLocalVisibility (i,j,k): This proce-dure is responsible for providing some local, heuristic in-formation to the ant. While we were developing and test-ing the algorithm, we noticed that the selection of localinformation is very important for correct functioning ofthe algorithm. Therefore, we introduced some improve-ments to the initial version of the algorithm (Walkowiak,2001b). The modifications yielded much better results.The detailed actions of the procedure are shown in thelines 33–43. We use two kinds of local information: theresidual capacity of arcs and the distance from the currentnode to the destination node of he ant.
ResidualCapacity (j,k): It returns the residual ca-pacity of the arc ��� �, i.e., the difference between the arccapacity and the flow.
LengthOfFeasibleShortestPath (i,k): It re-turns the length (given in the number of arcs) of a fea-sible path from the node to the destination node of theant �. The feasibility of the route means that in order tocalculate the shortest path, we consider only those arcs inwhich the residual capacity is bigger than the bandwidthrequirement of the ant �. If such a path does not exist, theprocedure returns the value ��—it is a kind of penaltyfunction.
PheromoneUpdate (" ): Ants update pheromone.When all ants die, i.e., when ants reach their destina-tion nodes, we update the pheromone value. We use anapproach related to the ant-cycle algorithm proposed in(Dorigo et al., 1991) and the global update proposed in(Varela and Sinclair, 1999). Pheromone information isupdated after all ants have completed their routes. If weassume that in time cycle � the ant � uses the route � �� ,the variable #�� denoting the length of the ant’s route iscalculated according to the formula
#�� �����
����
� � � ��
��� � � ���
��� (16)
We use the following pheromone updating rule:
" ����� � " ��� ��
#��� (17)
where � is one of the parameters in the algorithm.
Pheromone evaporation reduces the level of thepheromone on all links by a factor � (the evaporation co-efficient) at the end of each cycle � (Dorigo et al., 1991):
" ����� � � " ��� � (18)
Ant algorithm for flow assignment in connection-oriented networks 213
DaemonActions (): When an ant reaches the destina-tion node, it dies, i.e., it is removed from the system, butthe memory of the ant’s route it transferred to the globaldaemon. Using this information, the value of the obtainedsolution can be calculated. A global daemon can computesome other information, e.g., network statistics.
The ANB algorithm has a large set of parameters thathas to be tuned in order to provide the best possible fea-sible solutions of the NBFA problem. A detailed study ofthe influence of parameter settings on the quality of solu-tions is made in the next section.
An analysis of ANB algorithms shows that manyideas are related to the algorithm presented in (Varelaand Sinclair, 1999). The most important similarities areas follows: one ant using its own type of pheromone foreach commodity, backtracking, and a global update of thepheromone. However, due to the specific capacity con-straint, which is not considered in (Varela and Sinclair,1999), we have to modify Varela and Sinclair’s algorithmand use local heuristic information in the ant decisionprobability formula. Consequently, the role of pheromoneinformation is less important in our algorithm.
It should be mentioned that many various versionsof ant algorithm were evaluated by the author, and theANB algorithm presented above is the best one in terms ofthe obtained results. Many ant algorithms developed fornetwork optimization problems could not be applied di-rectly to the NBFA problem due to the specific features ofthat problem: static optimization, the non-bifurcated flow(only one route for commodity) and capacity constraints.
5. Numerical Results
In this section, we present an analysis of the ANB andANBIS performances. Both the algorithms were imple-mented in C++. Since we had not performed a mathemat-ical analysis of the algorithm, which would have helpedto obtain an optimal parameter setting in each situation,we ran many simulations to assemble statistical data forthis purpose. The results presented in this section are ob-tained from simulations on 10 sample networks (Fig. 2).The name of each network indicates the number of nodes(two first digits) and the number of links (two last dig-its). Table 1 summarizes the parameters of all sample net-works. The heading of each column specifies the name ofthe parameter, and various tested values are listed in therespective column. In an experiment, it is assumed thatthere is a requirement to establish a connection for eachdirection of every node pair. Consequently, the total num-ber of ants in the network is equal to ������. For a givenexperiment, the value of the bandwidth requirement is thesame for each commodity. We study the performance ofthe algorithm for an increasing traffic load, examining the
evolution of the network status toward a saturation condi-tion. This means that experiments for a particular topol-ogy have different traffic demands. The last column ofTable 1 shows the number of bandwidth requirements pat-terns for each network. For instance, on the network 1034we perform 4 experiments: 10340, 10345, 1034a, 1034ewith the bandwidth requirement of each commodity 40,35, 30 and 26, respectively.
In this subsection we consider parameters that affect di-rectly or indirectly the ANB algorithm. As was mentionedabove, the number of ants has always been set as the num-ber of commodities in the network to be established. Weapply the same approach like in (Dorigo et al., 1991),i.e., for each simulation, we vary only the value of oneparameter while keeping all other parameters constant.Even though NBFA and TSP are quite different problems,we decided to use as a starting point values similar tothose in (Dorigo et al., 1991), used in the ant algorithmfor the TSP problem: �� � ��� �� �� �� ��� ���� � ���� ���� ������� � � ����� ���� ���� ���� ������� � ���� ���� �� ��� � ��� �� �� �� ��� ���. We use the follow-ing notation: Each experiment consists of 2160 simula-tions of the ANB algorithm for different values of parame-ters. An experiment differs from the others in terms of thetopology of the tested network and the traffic demand pat-tern. A simulation differs from other simulations in termsof the values of the parameters ��� �� �� � and . Dueto tested values of parameters values, the overall numberof simulations for one experiment is � � � � � �����. In all simulations, the number of iterations is set to50. A result obtained for a particular simulation is the bestresult among all 50 results obtained for each iteration ofthe algorithm.
Analyzing the obtained results, we have noticed thatparameters ��, � and � generally do not have stronginfluence on the value of the objective flow function givenby (13). Figures 3–5 show the results of running the sameset of simulations on the topology 1874 with values of theparameters � and fixed as 0.5 and 20, respectively.The other three parameters were changed according to thedefault values given above. It yields 90 simulations foreach set of the bandwidth requirement on the topology1874. The �-axis represents the aggregate flow summedover all tests. The �-axis represents the bandwidth re-quirements. For the experiments 18744, 18746, 18748and 1874a the value of the bandwidth requirement of eachcommodity is 20, 19, 18 and 17, respectively.
The general trend shown in Figs. 3–5 is that chang-ing the value of any three parameters considered does notinfluence the flow in the network. Only for the experi-ment 18744 the difference is slightly more significant thanfor other cases. For example, in the experiment 18744,the biggest gap between the values of the flow functionis 4.96%, while in experiments 18746, 18748 and 1874a,it decreases to 0.34%. Analyzing the results of all 10800simulations for various networks and different values ofparameters, we conclude that the default values of the pa-rameters ��, � and � should be fixed as 2, 100 and 0.9,respectively. However, the importance of these parame-ters was uninfluential in most of the experiments.
165000
175000
185000
195000
205000
215000
225000
18744 18746 18748 1874aexperiment
aggr
egat
e fl
ow
Pn=0 Pn=1 Pn=2 Pn=5 Pn=10 Pn=20
Fig. 3. Aggregate results for the network 1874showing the influence of the �� parameter.
340000
350000
360000
370000
380000
390000
400000
410000
420000
430000
440000
18744 18746 18748 1874aexperiment
aggr
egat
e fl
ow
R=1
R=100
R=10000
Fig. 4. Aggregate results for the network 1874showing the influence of the � parameter.
200000
210000
220000
230000
240000
250000
260000
270000
18744 18746 18748 1874aexperiment
aggr
eagt
e fl
ow
RHO=0.3
RHO=0.5
RHO=0.7
RHO=0.9
RHO=0.999
Fig. 5. Aggregate results for the network 1874showing the influence of the � parameter.
Recall that the flow function is calculated accord-ing to (13), which takes into account the penalty func-tion. If a solution is not feasible (the capacity constraintdoes not hold), the value of the flow function may be verylarge. If we calculate the aggregate flow function as a sumover many experiments, the presentation of results couldbe difficult. Moreover, for various experiments differentvalues of bandwidth requirements are used, which resultsin huge deviations of the flow function value. Therefore,to make the performance evaluation easier, we introducethe concept of the competitive ration. The competitive
Ant algorithm for flow assignment in connection-oriented networks 215
ration, which indicates how well an algorithm performsfor a given set of the parameters ��, �, �, � and ,is defined as the difference between the result obtainedfor a particular simulation and the minimum value of theflow function obtained in the experiment considered. Forinstance, if for the experiment consisting of 2160 simula-tions the minimum value of the flow function is 10000 andthe flow function of the simulation considered is 15000,the competitive ration of this simulation is calculated as������ � ������������ � ���. The competitive rationindicates the quality of the obtained result of a given sim-ulation compared with the results of other simulations fora particular experiment. A low value of the competitiveration means that the simulation result is very close to thebest results in a given experiment. For the presentationof aggregate results, we apply the aggregate competitiveration, which is the sum of competitive rations over allexperiments considered.
We now describe the influence of two parameters �and on the value of the objective function using thecompetitive ration as the performance index. To exam-ine the impact of these parameters, we fix three other pa-rameters at default values and change the values using� � ��� ���� �� �� and � ��� �� �� �� ��� ���. For clarity,we show aggregate results for all networks and for the net-work 1874 discussed above concerning the tuning of theparameters ��, � and �. Figure 6 shows the aggregatecompetitive ration for all experiments. Figure 7 depictsthe same function only for the topology 1874. In both fig-ures, the %-axis uses the logarithmical scale. The generaltrend in both figures is that the best results are obtained for� � ��� ���� �� and � ���� ���. Due to the large dif-ference in the tested topologies in terms of the number ofnodes and links, we have noticed that for networks having10 nodes the best results are obtained for � ��, whilefor larger networks the parameter should be fixed at 20.
0.0 1.0 2.0 5.0 10.0 20.0
0.0
0.51.0
5.0
0.0
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
aggr
egat
e co
mpe
titi
ve r
atio
n
BETA
ALPHA
Fig. 6. Aggregate results for all tested networks showing the in-fluence of the � and � parameters.
0.0 1.0 2.0 5.0 10.0 20.0
0.0
0.51.0
5.0
0.0
0.1
1.0
10.0
100.0
1000.0
10000.0
aggr
egat
e co
mpe
titi
ve r
atio
n
BETA
ALPHA
Fig. 7. Aggregate results for the network 1874 showing the in-fluence of the � and � parameters.
An important observation compared with previousworks in the field of ant algorithms is that good results areobtained for � � �. According to the description of thealgorithm presented in the previous section, setting � as 0means that the pheromone information is not applied andonly local heuristic data are used to determine the routes.This can be explained by the fact that local heuristic infor-mation is calculated according to a complicated formulausing a lot of data on the current state of the network.The second potential explanation of the fact is that rela-tively many commodities are established between neigh-bor nodes of networks, especially for smaller networks.For instance, in the 1042 topology 42 of 90 demands,i.e., 47% of all demands connect adjacent nodes. How-ever, for the network 1874, the ratio of such connections is24%. Since the distance between the source and destina-tion nodes is only one hop, the importance of pheromoneinformation is reduced. Consequently, local heuristic in-formation gains much more significance. To verify thishypothesis, we repeat the same simulations as presentedabove. However, we limit the number of commodities inthe network according to the minimum distance betweenthe source and destination nodes. This means that we ig-nore those demands for which end nodes are too close interms of the number of hops. Table 2 summarizes theparameters of these new simulations. As in Table 1, theheading of each column specifies the name of the parame-ter, and the various values tried are listed in the respectivecolumn. In this experiment, it is assumed that there is arequirement to establish a connection for each directionof every node pair for which the distance is at least 3 hopsor 4 hops. For a particular test, the value of the bandwidthrequirement is the same for all commodities. The last col-umn indicates the number of various traffic demand pat-terns for each network. As above, the parameters ��, �and � are fixed to default values 2, 100 and 0.9, respec-tively.
Figures 8 and 9 depict the impact of the � and parameters on the networks 1866, 1874 and 1882 forthe minimum distance between the source and destinationnodes of commodities fixed to 3 and 4 hops. The generaltrend in these figures is similar to that in previous experi-ments. However, comparing Figs. 8 and 9 against Figs. 6and 7, we see that increasing the minimum distance be-tween the end nodes of the commodities makes the resultsfor � � � relatively worse. For instance, in the experi-ment with the minimum distance fixed at 4 hops, the bestresults are obtained for � � ���, � �� and � � �, � �� (Tables 3 and 4). This proves that the good resultsobtained for � � � and presented above can be explainedpartially by the tested traffic demand pattern.
The results obtained for the ANB algorithm are notin agreement with the results of the algorithm presented in(Dorigo et al., 1991). However, since in this work we donot consider the traveling salesman problem but the non-bifurcated multicommodity flow problem, a direct com-parison is very difficult. A major difference is the capac-ity constraint of the flow problem. In our opinion, the ca-pacity constraint strongly influences the algorithm, whichresults in great importance of local heuristic informationand a lower influence of pheromone information. It must
Table 3. Aggregate competitive ration of various simulation scenarios for � � ��.
be noted that using � � � leads to an instability of thealgorithm, i.e., the algorithm cannot converge to one solu-tion.
0.0 1.0 2.0 5.0 10.0 20.0
0.0
0.51.0
5.0
0.0
0.1
1.0
10.0
100.0
1000.0
10000.0
aggr
egat
e co
mpe
titi
ve r
atio
nBETA
ALPHA
Fig. 8. Aggregate results for the topologies 1866, 1874 and1882 with the minimum hoop distance set as 3 showingthe influence of the � and � parameters.
0.0 1.0 2.0 5.0 10.0 20.0
0.0
0.51.0
5.0
1
10
100
1000
10000
aggr
egat
e co
mpe
titi
ve r
atio
n
BETA
ALPHA
Fig. 9. Aggregate results for the topologies 1866, 1874 and1882 with the minimum hoop distance set as 4 showingthe influence of the � and � parameters.
Ant algorithm for flow assignment in connection-oriented networks 217
Summarizing the process of parameter setting, wecan list the main results:
1. The influence of the parameters ��, � and � onthe performance of the ANB algorithm is relativelysmall. Only for strongly saturated networks the dif-ference between the results obtained for various val-ues of these parameters is larger than 1%.
2. The default values of the parameters ��, � and �should be fixed as 2, 100 and 0.9, respectively.
3. The influence of the parameters � and on theperformance of the ANB is strong for all tested net-works.
4. The default values of the parameters � and shouldbe � � ��� ���� �� and � ���� ���.
5. Good results obtained for � � � suggest the signifi-cance of local heuristic information.
6. The traffic demand pattern may change the param-eters setting. If we consider only commodities be-tween remote (in the number of hops) nodes, thepheromone trail becomes more significant.
5.2. ANBIS Parameter Setting
In this subsection we find the best parameter settingfor ANBIS. As in the previous subsection, we con-sider the following values of the parameters: �� ���� �� �� �� ��� ���, � � ��� ���� ������, � ������ ���� ���� ���� ������, � � ��� ���� �� ��, ���� �� �� �� ��� ���. The methodology of tests is the sameas for ANB. However, the parameter setting for ANIBSdiffers from that for ANB. The best results are obtainedfor the following values: �� � �, � � �����, � ���� �� and � ��. ANBIS is not strongly influencedby �. For instance, for the topology 1874 modifying �changes the result by no more than 1%. Recall that inANBIS we start with a feasible solution, for which the ca-pacity constraint is satisfied. Therefore, the penalty factor��, which is significant in searching for a feasible so-lution, is set to 0. The importance of � grows becausethe algorithm should stay in feasible regions of the solu-tion space. Recall that the parameter � is used in thepheromone updating rule (17). Therefore, � affects thereinforcement of the pheromone according to the route se-lected by ants in the current iteration. If � is relativelysmall, the route selected by an ant has also little influenceon the pheromone lying and it is more probable that theant will select in the next iteration another route, muchdifferent from the previous one. Consequently, in next it-erations the solution may become infeasible. On the otherhand, when � is relatively large, it is more probable thatants will use routes similar to previous ones and the ob-tained solution will stay in a feasible region.
Figure 10 shows the aggregate competitive ration forall experiments. Figure 11 depicts the same function onlyfor topology 1874. In both the figures the default values ofparameters ��, � and � are used. The general trend inboth the figures is that the best results are obtained for � ���� �� and � ��. It is consistent with the philosophyof ant algorithms—the pheromone information plays animportant role in the algorithm.
0.0
1.0
2.0
5.0
10.0
20.0
0 .0 0.51.0
5.0
0.0
0.1
1.0
10.0
100.0
aggr
egat
e co
mpe
titi
ve r
atio
n
BETAALPHA
Fig. 10. Aggregate results of ANBIS for the networks 1866,1874, 1882 showing the influence of the � and � pa-rameters.
0.0
1.0
2.0
5.0
10.0
20.0
0 .0 0.51.0
5.0
0.00
0.01
0.10
1.00
10.00
100.00
aggr
egat
e co
mpe
titi
ve r
atio
n
BETAALPHA
Fig. 11. Aggregate results of ANBIS for the network 1874showing the influence of the � and � parameters.
The comparison of ANB and ANBIS parameter tun-ing indicates that the major issue in the problem consid-ered is the feasibility of results. The ANB algorithm mustbe forced to search for the feasible solution. Therefore,local heuristic information enforced by the parameteris more significant, while � is not so important. SinceANBIS starts with a feasible solution, the major effort of
K. Walkowiak218
the algorithm is to improve the result. Thus, the parameter� becomes much more influential.
5.3. Comparison with Other Heuristics
In this subsection we compare the performance of theANB and ANBIS algorithms with that of other algo-rithms applied to the assignment of non-bifurcated flows.To assess the value of our approach applied to the non-bifurcated multicommodity flow problem, we selectedfor comparison two algorithms: a modification of theFlow Deviation (FD) algorithm presented in (Fratta et al.,1973), and a heuristic algorithm called AlgNB and devel-oped by Walkowiak (2003b). Both algorithms require aninitial feasible solution that is provided by the initial phaseof the FD algorithm (FDInit) from (Fratta et al., 1973).It must be noted that the Flow Deviation algorithm isone of the most robust methods developed for the assign-ment of multicommodity flows. The FD method and itsmodifications are widely used for optimization problemsrelated to NBFA (Bienstock, 2002; Burns et al., 2003;Kasprzak, 2001; Murakami and Kim, 1996; Walkowiak,2002; 2003a; 2003b). More details on both algorithmsand results can be found in (Walkowiak, 2003b).
ANBIS, FD and AlgNB use the output of FDInit asthe starting feasible solution. In Table 5 we report the re-sults for all tested algorithms. We show the aggregate flowobtained for all 18-nodes networks (the second row of Ta-ble 2) and detailed results for the networks 1866, 1874 and1882 (rows 3–5). We find the ANBIS algorithm to be su-perior to other algorithms. ANBIS improves the result ofFDInit from 0.09% to 1.03% for particular networks. Thesecond algorithm is AlgNB, which is only 0.10% worsethan ANBIS. The worst performance is revealed by ANB.However, the maximum difference between the best andworst results reported in Table 5 is less than 2.5%.
Table 5. Aggregate flow obtained for various algorithms.
Network ANB ANBIS FDInit FD AlgNB
18 135711 132974 133796 133106 133080
1866 32403 31770 31800 31770 31770
1874 50309 49216 49470 49176 49193
1882 52999 51988 52526 52160 52117
The results presented in Table 5 confirm that the mainproblem of ANB is the feasibility of the result. The majoreffort of ANB is to find a feasible solution—the parame-ters are set and fixed according to this issue. The tuning ofANBIS shows that if the algorithm finds a feasible solu-tion, different values of parameters are needed in order toimprove the solution. Therefore, the results obtained forANB are not satisfactory.
Another important issue is the execution time of thecompared heuristics. The comparison shows that the com-putational time of ANB and ANBIS is 15–45 times longerthan that of the other heuristics. However, our focus indeveloping the implementation was on correctness, ratherthan on program speed, and it is anticipated that this figurecould be significantly reduced. For instance, reducing thenumber of iterations could decrease the decision time ofthe ant algorithm.
6. Concluding Remarks
In this paper we have described how to apply an ant al-gorithm to a non-bifurcated multicommodity flow assign-ment problem. We have proposed and discussed com-prehensively a novel ant algorithm. Since many staticoptimization problems encountered in real connection-oriented networks can be modeled as non-bifurcated mul-ticommodity flow problems, the significance of this workis substantial. Because of the different optimization prob-lem considered, the algorithm and implementation detailstightly bound to the specific capacity constraint. There-fore, it was impossible for us to re-implement the existingant algorithms approaches proposed in previous works. Isshould be underlined that our focus in developing and test-ing ANB and ANBIS was on the correctness and evalua-tion of the algorithm, rather than issues of its effective-ness. By the correctness we mean that the algorithm canfind a feasible solution, i.e., a solution in which the flowof each link does not exceed the link capacity.
The results of many simulations run on various net-work topologies suggest that ANB can be used for capac-itated optimization of static flows. Nevertheless, there isstill some room for further improvements. We have stud-ied several issues that arise in tackling the NBFA problemby the ant algorithm. The major problem is the tuning ofthe algorithm. We observe that some parameters have aneffect on the performance of ANB and ANBIS much moresubstantially than others. We proposed the best values ofparameters and noticed that wrong selection may lead tocompletely incorrect results. A relatively strange, compar-ing with other ant algorithms, parameter setting obtainedfor ANB can be explained mainly by the specific capac-ity constraint that influences robustly the algorithm, espe-cially for congested networks in which only a very smallpart of the solution space is feasible.
We also compared ANB and ANBIS with otherheuristics. During numerical experiments we noticed sev-eral shortcomings of our approach which are listed as fol-lows:
� for highly loaded networks the ANB algorithm can-not find a feasible solution;
Ant algorithm for flow assignment in connection-oriented networks 219
� premature convergence—both algorithms stop tooearly further exploration of the search space;
� ANB performs worse than the known algorithms,and the difference in results is relatively small;
� running time of ANB and ANBIS is much longercompared with other algorithms.
Many questions remain and many problems are open:as a final remark we propose a brief catalogue of them. Infurther work, it will be possible to carry out a more de-tailed investigation of appropriate parameter settings forour algorithm according to various network topologiesand traffic demands patterns. Another issue is the role ofexperience: how to operate in parameter tuning, for bal-ancing convergence speed and good results. Adopting afeasible initial solution to pheromone initialization ratherthan setting the pheromone to the same value may provebeneficial, as ANB has sometimes trouble finding a feasi-ble solution that conforms to the capacity constraint.
In future work we plan to make improvements toANB. The first proposal is to change parameter valueswhen the algorithm runs in the following way: We startwith values providing a feasible solution. Next, we changethe parameters to values promising convergence to betterresults. The second suggestion is to modify the ANB algo-rithm according to the initial phase of the Flow Deviationalgorithm as follows: ANB finds a solution. If the capac-ity constraint is violated, the flows on all arcs are reducedproportionally, until a feasible flow is obtained. Next,ANB is run again and the flows are increased to a levelvery close to saturation. The process terminates whenone of two cases occurs: either flows have starting val-ues, or the network is saturated. In the former case ANByields a feasible solution. In the latter case the problem isclaimed to be infeasible. Furthermore, we plan to exam-ine the possibility of applying ANB to the optimizationof static flows according to other functions, e.g., the lostflow function introduced in (Walkowiak, 2002; 2003a) orfunctions using the bottleneck metric. An analytic proofand models of ANB algorithm performance are also openproblems.
Finally, we must mention that most of previous stud-ies concerning applications of ant algorithms focus on dy-namic routing problems. This paper shows that an ant al-gorithm can be applied also to static network problemswith capacity constraints. The paper (Varela and Sinclair,1999) includes an ant algorithm for a static routing prob-lem without capacity constraints. However, according to(Gendron et al., 1998) and the author’s experience, the ca-pacity constraint makes the flow allocation problem verydifficult and the algorithm by Varela and Sinclair couldnot be applied directly to the NBFA problem. Due tothe problem-dependent constraints, the original ant algo-rithm has to be modified strongly in order to guarantee
the feasibility of the results. Therefore, two developedalgorithms for ANB and ANBIS have some features ofant algorithms, but also many new elements are added.The role of local heuristics is stronger that in many otherant algorithms. However, when we tested some versionsof the algorithm without such complicated local heuris-tics, the algorithm was unable to find a feasible solution.This conclusion is consistent with the opinion presentedin (Dorigo et al., 1999) that the use of local heuristics im-proves ant algorithm performance considerably. We be-lieve that the results presented in this paper may provideadditional support for the development of ant algorithmsfor static network problems with capacity constraints.
References
Assad A. (1978): Multicommodity network flows – A survey. —Network, Vol. 8, No. 1, pp. 37–91.
Bienstock D. (2002): Potential Function Methods for Approx-imately Solving Linear Programming Problems, Theoryand Practice. — Boston: Kluwer.
Burns J., Ott T., Krzesinski A. and Muller K. (2003): Path se-lection and bandwidth allocation in MPLS networks. —Perform. Eval., Vol. 52, No. 2–3, pp. 133–152.
Cantor D. and Gerla M. (1974): Optimal routing in a packet-switched computer network. — IEEE Trans. Comm.,Vol. 23, No. 10, pp. 1062–1069.
Colorni A., Dorigo M., Maffioli F., Maniezzo V., Righini G. andTrubian M. (1996): Heuristics from nature for hard combi-natorial problems. — Int. Trans. Oper. Res., Vol. 3, No. 1,pp. 1–21.
Corne D., Oates M. and Smith D. (Eds.) (2000): Telecommuni-cations Optimization: Heuristic and Adaptive Techniques.— New York: Wiley.
Di Caro G. and Dorigo M. (1998a): Mobile agents for adaptiverouting. — Proc. 31st Hawaii Int. Conf. System, KohalaCoast, Hawaii, USA, pp. 74–83.
Di Caro G. and Dorigo M. (1998b): AntNet: Distributed stig-mergetic control for communications networks. — J. Artif.Intell. Res., Vol. 9, pp. 317–365.
Dorigo M., Maniezzo V. and Colorni A. (1991): Positive feed-back as a search strategy. — Techn. Rep. No. 91–016, Po-litechnico di Milano, Italy.
Dorigo M., Di Caro G. and Gambardella L. (1999): Ant al-gorithms for discrete optimization. — Artif. Life, Vol. 5,No. 2, pp. 137–172.
Elbaum R. and Sidi M. (1996): Topological design of local-areanetworks using genetic algorithms. — IEEE/ACM Trans.Networking, Vol. 4, No. 5, pp. 766–778.
Fratta L., Gerla M. and Kleinrock L. (1973): The flow deviationmethod: An approach to store-and-forward communica-tion network design. — Networks, Vol. 3, pp. 97–133.
K. Walkowiak220
Garlick R. and Barr R. (2003): Dynamic wavelength routing inWDM networks via ant colony optimization. — Lect. NotesComput. Sci., Vol. 2463, pp. 250–255.
Gendron B., Crainic T.G. and Frangioni A. (1998): Multicom-modity capacitated network design, In: Telecommunica-tions Network Planning (B. Sans‘o and P. Soriano, Eds.).— Norwell, MA: Kluwer, pp. 1–19.
Girard A. and Sanso B. (1998): Multicommodity flow models,failure propagation and reliable loss network design. —IEEE/ACM Trans. Networking, Vol. 6, No. 1, pp. 82–93.
Grover W. (2004): Mesh-Based Survivable Networks: Optionsand Strategies for Optical, MPLS, SONET and ATM Net-working. — Upper Saddle River, NJ: Prentice Hall.
Herzberg M., Bye S. and Utano A. (1995): The hop-limit ap-proach for spare-capacity assignment in survivable net-works. — IEEE/ACM Trans. Networking, No. 6 pp. 775–784.
Kar K., Kodialam M. and Lakshman, T.V. (2000): Minimum in-terference routing of bandwidth guaranteed tunnels withMPLS traffic engineering applications. — IEEE J. Select.Areas Commun., Vol. 18, No. 12, pp. 2566–2579.
Karp R. (1975): On the computational complexity of combina-torical problems. — Networks, Vol. 5, No. 1, pp. 45–68.
Kasprzak A. (2001): Design of Wide Area Networks. —Wrocław: Wrocław University of Technology Press, (inPolish).
Kassabalidis I., El-Sharkawi M.A., Marks II R.J., Arabshahi P.and Gray A.A. (2001): Swarm intelligence for routing incommunication networks. — Proc. IEEE Conf. Globecom,San Antonio, Texas, USA, pp. 541–546.
Murakami K. and Kim H. (1996): Virtual path routing for sur-vivable ATM networks. — IEEE/ACM Trans. Networking,Vol. 4, No. 1, pp. 22–39.
Ott T., Bogovic T., Carpenter T., Krishnan K.R. and ShallcrossD. (2001): Algorithms for flow allocation for multi pro-tocol label switching. — Telcordia Techn. MemorandumTM-26027.
Pióro M. and Medhi D. (2004): Routing, Flow, and Capacity De-sign in Communication and Computer Networks. — Ams-terdam: Morgan Kaufmann.
Pióro M., Srivastava S., Krithikaivasan B. and Medhi D. (2003):Path generation technique for robust design of wide areacommunication networks. — Proc. 10-th Polish TeletrafficSymp., Cracow, Poland, pp. 67–80.
Schoonderwoerd R., Holland O. and Bruten J. (1997): Ant-like agents for load balancing in telecommunications net-works. — Proc. Conf. Agents’97, Marina del Rey, Califor-nia, USA, pp. 209–216.
Schwartz M. and Cheung C. (1976): The gradient projectionalgorithm for multpile-routing in message switched net-works. — IEEE Trans. Comm., Vol. 24, No. 4, pp. 449–456.
Subramanian D., Druschel P. and Chen J. (1997): Ants and re-inforcement learning: A case study in routing in dynamicnetworks. — Proc. Int. Joint Conf. Artificial Intelligence,Nagoya, Aichi, Japan pp. 832–838.
Szeto W., Boutaba R. and Iraqi Y. (2002): Dynamic online rout-ing algorithm for MPLS traffic engineering. — Lect. NotesComput. Sci., Vol. 2345, pp. 936–946.
Varela N.G. and Sinclair M.C. (1999): ant colony optimisa-tion for virtual-wavelength-path routing and wavelengthallocation. — Proc. Congress Evolutionary Computation,Washington, USA, pp. 1809–1816.
Walkowiak K. (2001a): Genetic approach to virtual paths as-signment in survivable ATM networks. — Proc. 7-th Int.Conf. Soft Computing MENDEL, Brno, Czech Republic,pp. 13–18.
Walkowiak K. (2001b): Ant algorithms for design of com-puter networks. — Proc. 7-th Int. Conf. Soft ComputingMENDEL, Brno, Czech Republic, pp. 149–154.
Walkowiak K. (2002): The branch and bound algorithm forbackup virtual path assignment in survivable ATM net-works. — Appl. Math. Comput. Sci., Vol. 12, No. 2,pp. 257–267.
Walkowiak K. (2003a): A new approach to survivability of con-nection oriented networks. — Lect. Notes Comput. Sci.,Vol. 2657, pp. 501–510.
Walkowiak K. (2003b): Heuristic algorithms for assignment ofnon-bifurcated multicommodity flows. — Proc. Conf. Ad-vanced Simulation of Systems ASIS, Sv Hostyn, CzechRepublic, pp. 243–248.
Walkowiak K. (2004): A branch and bound algorithm for pri-mary routes assignment in survivable connection orientednetworks. — Comput. Optim. Applic., Vol. 27, No. 2,pp. 149–171.
White T., Pagurek B. and Oppacher F. (1998): Connection man-agement using adaptive mobile agents. — Proc. Int. Conf.Parallel and Distributed Processing Techniques and Appli-cations, Las Vegas, USA, pp. 802–809.
