ant colony optimization algoritham and application

1ANT COLONY OPTIMIZATION ALGORITHM AND APPLICATIONS

CHAPTER 1

INTRODUCTION

11 BACK GROUND AND RELATED WORK

Genetic Algorithms (GA) have been used to evolve computer programs for specific tasks and to design other computational structures The recent resurgence of interest in AP with GA has been spurred by the work on Genetic Programming (GP) GP paradigm provides a way to do program induction by searching the space of possible computer programs for an individual computer program that is highly fit in solving or approximately solving the problem at hand The genetic programming paradigm permits the evolution of computer programs which can perform alternative computations conditioned on the outcome of intermediate calculations which can perform computations on variables of many different types which can perform iterations and recursions to achieve the desired result which can define and subsequently use computed values and subprograms and whose size shape and complexity is not specified in advance GP use relatively low-level primitives which are defined separately rather than combined a priori into high-level primitives since such mechanism generate hierarchical structures that would facilitate the creation of new high-level primitives from built-in low-level primitives Unfortunately since every real life problem are dynamic problem thus their behaviors are much complex GP suffers from serious weaknesses Random systems chaos is important in part because it helps us to cope with unstable system by improving our ability to describe to understand perhaps even to forecast them

Ant Colony Optimization (ACO) is the result of research on computational intelligence approaches to combinatorial optimization originally conducted by Dr Marco Dorigo in collaboration with Alberto Colorni and Vittorio Maniezzo The fundamental approach underlying ACO is an iterative process in which a population of simple agents repeatedly construct candidate solutions this construction process is probabilistically guided by heuristic information on the given problem instance as well as by a shared memory containing experience gathered by the ants in previous iteration ACO has been applied to a broad range of hard combinatorial problems Problems are defined in terms of components and states which are sequences of components Ant Colony Optimization incrementally generates solutions paths in the space of such components adding new components to a state Memory is kept of all the observed transitions between pairs of solution components and a degree of desirability is associated to each transition depending on the quality of the solutions in which it occurred so far While a new solution is generated a component y is included in a state with a probability that is proportional to the desirability of the transition between the last component included in the state and y itself

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The main idea is to use the self-organizing principles to coordinate populations of artificial agents that collaborate to solve computational problems Self-organization is a set of dynamical mechanisms whereby structures appear at the global level of a system from interactions among its lower-level components The rules specifying the interactions among the systemrsquos constituent units are executed on the basis of purely local information without reference to the global pattern which is an emergent property of the system rather than a property imposed upon the system by an external ordering influence For example the emerging structures in the case of foraging in ants include spatiotemporally organized networks of pheromone trails

111 GENETIC PROGRMMING

Some specific advantages of genetic programming are that no analytical knowledge is needed and still could get accurate results GP approach does scale with the problem size GP does impose restrictions on how the structure of solutions should be formulated There are several variants of GP some of them are Linear Genetic Programming (LGP) Gene Expression Programming (GEP) Multi Expression Programming (MEP) Cartesian Genetic Programming (CGP) Traceless Genetic Programming (TGP) and Genetic Algorithm for Deriving Software (GADS)Cartesian Genetic Programming was originally developed by Miller and Thomson for the purpose of evolving digital circuits and represents a program as a directed graph One of the benefits of this type of representation is the implicit re-use of nodes in the directed graph Originally CGP used a program topology defined by a rectangular grid of nodes with a user defined number of rows and columns In CGP the genotype is a fixed-length representation and consists of a list of integers which encode the function and connections of each node in the directed graph The genotype is then mapped to an indexed graph that can be executed as a program In CGP there are very large numbers of genotypes that map to identical genotypes due to the presence of a large amount of redundancy Firstly there is node redundancy that is caused by genes associated with nodes that are not part of the connected graph representing the program Another form of redundancy in CGP also present in all other forms of GP is functional redundancy Simon Harding and Ltd introduce computational development using a form of Cartesian Genetic Programming that includes self-modification operationsThe interestingcharacteristic of CGP are 1 More powerful program encoding using graphs than using conventional GP tree-like representations the population of strings are of fixed length whereas their corresponding graphs are of variable length depending on the number of genes in use2 Efficient evaluation derived from the intrinsic feature of sub graph-reuse exhibited by graphs3 Less complicated graph recombination via the crossover and mutation genetic operators

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112 SWARM INTELLIGENCE

Swarm intelligence (SI) describes the collective behavior of decentralized self-organized systems natural or artificial The concept is employed in work on artificial intelligence The expression was introduced by Gerardo Beni and Jing Wang in 1989 in the context of cellular robotic systems

Swarm intelligence is the discipline that deals with natural and artificial systems composed of many individuals that coordinate using decentralized control and self-organization In particular the discipline focuses on the collective behaviours that result from the local interactions of the individuals with each other and with their environment Examples of systems studied by swarm intelligence are colonies of ants and termites schools of fish flocks of birds herds of land animals Some human artefacts also fall into the domain of swarm intelligence notably some multi-robot systems and also certain computer programs that are written to tackle optimization and data analysis problems

Emphasis is given to such topics as the modelling and analysis of collective biological systems application of biological swarm intelligence models to real-world problems and theoretical and empirical research in ant colony optimization particle swarm optimization swarm robotics and other swarm intelligence algorithms Articles often combine experimental and theoretical work

Fig 11

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The field of swarm intelligence deals with systems composed of many individuals that coordinate using decentralized control and self-organization In particular it focuses on the collective behaviours that result from the local interactions of the individuals with each other and with their environment It is a fast-growing field that encompasses the efforts of researchers in multiple disciplines ranging from ethology and social science to operations research and computer engineering Swarm Intelligence will report on advances in the understanding and utilization of swarm intelligence systems that is systems that are based on the principles of swarm intelligence The following subjects are of particular interest to the journal modeling and analysis of collective biological systems such as social insect colonies flocking vertebrates and human crowds as well as any other swarm intelligence systems application of biological swarm intelligence models to real-world problems such as distributed computing data clustering graph partitioning optimization and decision making theoretical and empirical research in ant colony optimization particle swarm optimization swarm robotics and other swarm intelligence algorithms

During the course of the last 20 years researchers have discovered the variety of interesting insect and animal behaviors in nature A flock of birds sweeps across the sky A group of ants forages for food A school of fish swims turns flees together etc We call this kind of aggregate motion Swarm behavior Recently biologists and computer scientists have studied how to model biological swarms to understand how such social animals interact achieve goals and evolve Furthermore engineers are increasingly interested in this kind of swarm behavior since the resulting swarm intelligence can be applied in optimization (eg in telecommunication systems) robotics track patterns in transportation systems and military applications A high-level view of a swarm suggests that the N agents in the swarm are cooperating to achieve some purposeful behavior and achieve some goal This apparent collective intelligence seems to emerge from what are often large groups of relatively simple agents The agents use simple local rules to govern their actions and via the interactions of the entire group the swarm achieves its objectives A type of self organization emerges from the collection of actions of the group

Swarm intelligence is the emergent collective intelligence of groups of simple autonomous agents Here an autonomous agent is a subsystem that interacts with its environment which probably consists of other agents but acts relatively independently from all other agents The autonomous agent does not follow commands from a leader or some global plan For example for a bird to participate in a flock it only adjusts its movements to coordinate with the movements of its flock mates typically its neighbors that are close to it in the flock A bird in a flock simply tries to stay close to its neighbours but avoid collisions with them Each bird does not take commands from any leader bird since there is no lead bird Any bird can fly in the front center or back of the swarm Swarm behavior helps birds take advantage of several things including protection from predators (especially for birds in the middle of the flock) and searching for food (as each bird is essentially exploiting the eyes of every other bird)

DEPARTMENT OF ECE


12 ANT COLONY

The complex social behaviours of ants have been much studied by science and computer scientists are now finding that these behaviour patterns can provide models for solving difficult combinatorial optimization problems The attempt to develop algorithms inspired by one aspect of ant behaviour the ability to find what computer scientists would call shortest paths has become the field of ant colony optimization (ACO) the most successful and widely recognized algorithmic technique based on ant behaviour This book presents an overview of this rapidly growing field from its theoretical inception to practical applications including descriptions of many available ACO algorithms and their uses The book first describes the translation of observed ant behaviour into working optimization algorithms The ant colony metaheuristic is then introduced and viewed in the general context of combinatorial optimization This is followed by a detailed description and guide to all major ACO algorithms and a report on current theoretical findings The book surveys ACO applications now in use including routing assignment scheduling subset machine learning and bioinformatics problems Ant Net an ACO algorithm designed for the network routing problem is described in detail The authors conclude by summarizing the progress in the field and outlining future research directions Each chapter ends with bibliographic material bullet points setting out important ideas covered in the chapter and exercises Ant Colony Optimization will be of interest to academic and industry researchers graduate students and practitioners who wish to learn how to implement ACO algorithms

Fig 12

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121 BEHAVIOUR OF ANT IN PRESENCE OF AN OBSTACLE

Ants are forced to decide whether they should go left or right The choice that is made is a random one Pheromone accumulation is faster on shortest path

Fig 13

In this picture we can see the movement of natural ants that is ants are moving in a line following each others when we place an obstacle in between the path some of the ants overcome the obstacle and create a new path of their own while moving they produces pheromone all other ants detects the pheromone and follow the same path

DEPARTMENT OF ECE


122 REAL BEHAVIOUR OF ANTS

Natural behaviour of ants have inspired scientists to mimic insect operational methods to solve real-life complex problems such as Travelling sales man problem Quadratic assignment problem Network model Vehicle routing By observing ant behaviour scientists have begun to understand their means of communicationAnts communicate with each other through tapping with the antennae and smell They are considered together with the bees as one of the most socialized animals They have a perfect social organization and each type of individual specializes in a specific activity within the colony They are thought by many as having a collective intelligence and each ant is considered then as an individual cell of a bigger organism Ants wander randomly amp on finding food return to their colony while laying ldquoPHEROMONE TRIALSrdquo

Fig 14

If other Ants find such paths they do not travel randomly but follow the Pheromone trail Ants secrete pheromone while travelling from the nest to food and vice versa in order to communicate with each other to find shortest path

Pheromone is a highly volatile substance which starts to evaporate more the time taken by the ant to travel to and fro more time the pheromone have to evaporate A shortest path gets marched over faster and thus the pheromone density remains high If one of the ants finds the shortest path from colony to food source other ants are more likely to follow the same path

DEPARTMENT OF ECE


13 ANTS

One of the first researchers to investigate the social behaviour of insects was the French entomologist Pierre-Paul Grasse In the forties and fifties of the 20-th century he was observing the behaviour of termites in particular the Bellicositermes natalensis and Cubitermes species He discovered that these insects are capable to react to what he called significant stimuli signals that activate a genetically encoded reaction He observed that the effects of these reactions can act as new significant stimuli for both the insect that produced them and for the other insects in the colony Grasse used the term stigmergy to describe this particular type of indirect communication in which the workers are stimulated by the performance they have achieved The two main characteristics of stigmergy that dierentiate it from other means of communication are

The physical non-symbolic nature of the information released by the communicating insects which corresponds to a modification of physical environmental states visited by the insects and

The local nature of the released information which can only be accessed by those insects that visit the place where it was released (or its immediate neighbourhood)

Examples of stigmergy can be observed in colonies of ants In many ant species ants walking to and from a food source deposit on the ground a substance called pheromone Other ants are able to smell this pheromone and its presence influences the choice of their path ie they tend to follow strong pheromone

Fig 15

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CHAPTER 2

ANT COLONY OPTIMIZATION

21 ANT SYSTEM

The importance of the original Ant System (AS) resides mainly in being the prototype of a number of ant algorithms which collectively implement the ACO paradigm The move probability distribution defines probabilities pιψk to be equal to 0 for all moves which are infeasible (ie they are in the tabu list of ant k that is a list containing all moves which are infeasible for ants k starting from state ι)After each iteration t of the algorithm ie when all ants have completed a solution trails are updated by means of formula τιψ (τ) = ρ τιψ (τ minus 1) + Δτιψwhere Δτιψ represents the sum of the contributions of all ants that used move (ιψ) to construct their solution ρ 0 le ρ le 1 is a user-defined parameter called evaporation coefficient and Δτιψ represents the sum of the contributions of all ants that used move (ιψ) to construct their solution The antsrsquo contributions are proportional to the quality of the solutions achieved ie the better solution is the higher will be the trail contributions added to the moves it used

211 ANT COLONY SYSTEM (ACS)

Ant System was the first algorithm inspired by real ants behavior AS was initially applied to the solution of the traveling salesman problem but was not able to compete against the state-of-the art algorithms in the field On the other hand he has the merit to introduce ACO algorithms and to show the potentiality of using artificial pheromone and artificial ants to drive the search of always better solutions for complex optimization problems The next researches were motivated by two

goals the first was to improve the performance of the algorithm and the second was to investigate and better explain its behavior Gambardella and Dorigo proposed in 1995 the Ant-Q algorithm an extension of AS which integrates some ideas from Q-learning and in 1996 Ant Colony System (ACS) a simplified version of Ant-Q which maintained approximately the same level of performance measured by algorithm complexity and by computational results Since ACS is the base of many algorithms defined in the following years we focus the attention on ACS other than Ant-Q ACS differs from the previous AS because of three main aspects

212 PHEROMONE In ACS once all ants have computed their tour (ie at the end of each iteration) AS updates the pheromone trail using all the solutions produced by the ant colony

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Each edge belonging to one of the computed solutions is modified by an amount of pheromone proportional to its solution value At the end of this phase the pheromone of the entire system evaporates and the process of construction and update is iterated On the contrary in ACS only the best solution computed since the beginning of the computation is used to globally update the pheromone As was the case in AS global updating is intended to increase the attractiveness of promising route but ACS mechanism is more effective since it avoids long convergence time by directly concentrate the search in a neighborhood of the best tour found up to the current iteration of the algorithm In ACS the final evaporation phase is substituted by a local updating of the pheromone applied during the construction phase Each time an ant moves from the current city to the next the pheromone associated to the edge is modified in the following way τ ij (t) = ρ τ ij (t -1)+ (1minusρ ) sdotτ 0 where 0 le ρ le 1 is a parameter (usually set at 09) and τ0 is the initial pheromone value τ0 is sdot

defined as τ0=(nmiddotLnn)-1 where Lnn is the tour length produced by the execution of one ACS iteration without the pheromone component (this is equivalent to a probabilistic nearest neighbor heuristic) The effect of local-updating is to make the desirability of edges change dynamically every time an ant uses an edge this becomes slightly less desirable and only for the edges which never belonged to a global best tour the pheromone remains τ0 An interesting property of these local and global updating mechanisms is that the pheromone τij(t) of each edge is inferior limitedby τ0 A similar approach was proposed with the Max-Min-AS (MMAS) that explicitly introduces lower and upper bounds to the value of the pheromone trials

213 STATE TRANSITION RULE

During the construction of a new solution the state transition rule is the phase where each ant decides which is the next state to move to In ACS a new state transition rule called pseudo-random-proportional is introduced The pseudorandom- proportional rule is a compromise between the pseudo-random state choice rule typically used in Q-learning and the random-proportional action choice rule typically used in Ant System With the pseudo-random rule the chosen state is the best with probability q0 (exploitation) while a random state is chosen with probability 1-q0 (exploration) Using the AS random-proportional rule the next state is chosen randomly with a probability distribution depending on ηij and τij The ACS pseudo-random-proportional state transition rule provides a direct way to balance between exploration of new states and exploitation of a priori and accumulated knowledge The best state is chosen with probability q0 (that is a parameter 0 le q0 le 1 usually fixed to 09) and with probability (1-q0) the next state is chosen randomly with a probability distribution based on ηij and τ ij weighted by α (usually equal to 1) and β (usually equal to 2)

DEPARTMENT OF ECE


214 HYBRIDIZATION AND PERFORMANCE IMPROVEMENT

ACS was applied to the solution of big symmetric and asymmetric traveling salesman problems (TSPATSP) For these purpose ACS incorporates an advanced data structure known as candidate list A candidate list is a static data structure of length cl which contains for a given city i the cl preferred cities to be visited An ant in ACS first uses candidate list with the state transition rules to choose the city to move to If none of the cities in the candidate list can be visited the ant chooses the nearest available city only using the heuristic value ηij ACS for TSPATSP has been improved by incorporating local optimization heuristic (hybridization)

The idea is that each time a solution is generated by the ant it is taken to its local minimum by the application of a local optimization heuristic based on an edge exchange strategy like 2-opt 3-opt or Lin-Kernighan The new optimized solutions are considered as the final solutions produced in the current iteration by ants and are used to globally update the pheromone trails This ACS implementation combining a new pheromone management policy a new state transition strategy and local search procedures was finally competitive with state-of-the-art algorithm for the solution of TSPATSP problems This opened a new frontier for ACO based algorithm Following the same approach that combines a constructive phase driven by the pheromone and a local searchphase that optimizes the computed solution ACO algorithms were able to break several optimization records including those for routing and scheduling problems

22 ANTS

ANTS are an extension of the AS proposed in which specifies some under defined elements of the general algorithm such as the attractiveness function to use or the initialization of the trail distribution This turns out to be a variation of thegeneral ACO framework that makes the resulting algorithm similar in structure totree search algorithms In fact the essential trait which distinguishes ANTS from atree search algorithm is the lack of a complete backtracking mechanism which is substituted by a probabilistic (Non-deterministic) choice of the state to move into and by an incomplete (Approximate) exploration of the search tree this is the rationale behind the name ANTS which is an acronym of Approximated Nondeterministic Tree Search In the following we will outline two distinctive elements of the ANTS algorithm within the ACO framework namely the attractiveness function and the trail updating mechanism

221 ATTRACTIVENESS

The attractiveness of a move can be effectively estimated by means of lower bounds (upper bounds in the case of maximization problems) on the cost of the completion of a partial solution

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In fact if a state ι corresponds to a partial problem solution it is possible to compute a lower bound on the cost of a complete solution containing ι Therefore for each feasible move ιψ it is possible to compute the lower bound on the cost of a complete solution containing ψ the lower the bound the better the move Since a large part of research in ACO is devoted to the identification of tight lower bounds for the different problems of interest good lower bounds are usually available

When the bound value becomes greater than the current upper bound it is obvious that the considered move leads to a partial solution which cannot be completed into a solution better than the current best one The move can therefore be discarded from further analysis A further advantage of lower bounds is that in many cases the values of the decision variables as appearing in the bound solution can be used as an indication of whether each variable will appear in good solutions This provides an effective way of initializing the trail values

23 COMBINATORIAL OPTIMIZATION

Combinatorial optimization problems involve finding values for discrete variables such that the optimal solution with respect to a given objective function is found Many optimization problems of practical and theoretical importance are of combinatorial nature Examples are the shortest path problems as well as many other important real world problems like finding a minimum cost plan to deliver goods to customers an optimal assignment of employees to tasks to be performed best routing scheme data packets in the internet When attacking a combinatorial optimization problem it is useful to know how difficult it is to find an optimal solution A way of measuring this difficulty is given by the notion of worst-case complexity a combinatorial optimization problem lsquopirsquo is said to have worst-case complexity O(g(n)) if the best algorithm known for solving lsquopirsquo finds an optimal solution to any instance of lsquopirsquo having size lsquonrsquo in a computation time bounded from above by constg(n) In particular we say that lsquopirsquo is solvable in polynomial time if the maximum amount of computing time necessary to solve any instance of size n of lsquopirsquo is bounded from above by a polynomial in n If k is the largest exponent of such a polynomial then the combinatorial optimization problem is said to be solvable in O (n ^ k) time Although some important combinatorial optimization problems have been shown to be solvable in polynomial time for great majority of combinatorial problems no polynomial bound on the worst-case solution time could be found so far For these problem the run time of the best algorithms known increases exponentially with the instance size and consequently so does the time required to find an optimal solution A notorious example of such a problem is TSP An important theory that characterizes the difficulty of combinatorial problem is that of NP completeness The theory classifies combinatorial problems in two main classes those that are known to be solvable in polynomial time and those that are not The first are said to be tractable the latter intractable

DEPARTMENT OF ECE


24 CONCEPT OF ANT COLONY OPTIMIZATION

Ant Colony Optimization (ACO) is a recently proposed metaheuristic approach for solving hard combinatorial optimization problems The inspiring source of ACO is the pheromone trail laying and following behavior of real ants which use pheromones as a communication medium

In analogy to the biological example ACO is based on the indirect communication of a colony of simple agents called (artificial) ants mediated by (artificial) pheromone trails The pheromone trails in ACO serve as distributed numerical information which the ants use to probabilistically construct solutions to the problem being solved and which the ants adapt during the algorithmrsquos execution to reflect their search experience The first example of such an algorithm is Ant System (AS) which was proposed using as example application the well known Traveling Salesman Problem (TSP) Despite encouraging initial results AS could not compete with state-of-the-art algorithms for the TSP Nevertheless it had the important role of stimulating further research on algorithmic variants which obtain much better computational performance as well as on applications to a large variety of different problems In fact there exists now a considerable amount of applications obtaining world class performance on problems like the quadratic assignment vehicle routing sequential ordering scheduling routing in Internet-like networks and so on Motivated by this success the ACO metaheuristic has been proposed as a common framework for the existing applications and algorithmic variants Algorithms which follow the ACO metaheuristic will be called in the following ACO algorithms Current applications of ACO algorithms fall into the two important problem classes of static and dynamic combinatorial optimization problems Static problems are those whose topology and cost do not change while the problems are being solved This is the case for example for the classic TSP in which city locations and intercity distances do not change during the algorithmrsquos run-time Differently in dynamic problems the topology and costs can change while solutions are built An example of such a problem is routing in telecommunications networks in which traffic patterns change all the time The ACO algorithms for solving these two classes of problems are very similar from a high-level perspective but they differ significantly in implementation details The ACO metaheuristic captures these differences and is general enough to comprise the ideas common to both application types The (artificial) ants in ACO implement a randomized construction heuristic which makes probabilistic decisions as a function of artificial pheromone trails and possibly available heuristic information based on the input data of the problem to besolved As such ACO can be interpreted as an extension of traditional construction heuristics which are readily available for many combinatorial optimization problems Yet an important difference with construction heuristics is the adaptation of the pheromone trails during algorithm execution to take into account the cumulated search experience Ant Colony Optimization (ACO) is a paradigm for designing metaheuristic algorithms for combinatorial optimization problems The first algorithm which can be classified within this

DEPARTMENT OF ECE


framework was presented in 1991 and since then many diverse variants of the basic principle have been reported in the literature The essential trait of ACO algorithms is the combination of a priori information about the structure of a promising solution with posterior information about the structure of previously obtained good solutions Metaheuristic algorithms are algorithms which in order to escape from local optima drive some basic heuristic either a constructive heuristic starting from a null solution and adding elements to build a good complete one or a local search heuristic starting from a complete solution and iteratively modifying some of its elements in order to achieve a better one The metaheuristic part permits the low-level heuristic to obtain solutions better than those it could have achieved alone even if iterated Usually the controlling mechanism is achieved either by constraining or by randomizing the set of local neighbour solutions to consider in local search (as is the case of simulated annealing or tabu) or by combining elements taken by different solutions (as is the case of evolution strategies and genetic or bionomic algorithms) The characteristic of ACO algorithms is their explicit use of elements of previous solutions In fact they drive a constructive low-level solution as GRASP does but including it in a population framework and randomizing the construction in a Monte Carlo way A Monte Carlo combination of different solution elements is suggested also by Genetic Algorithms but in the case of ACO the probability distribution is explicitly defined by previously obtained solution components The particular way of defining components and associated probabilities is problem- specific and can be designed in different ways facing a trade-off between the specificity of the information used for the conditioning and the number of solutions which need to be constructed before effectively biasing the probability distribution to favour the emergence of good solutions Different applications have favoured either the use of conditioning at the level of decision variables thus requiring a huge number of iterations before getting a precise distribution or the computational efficiency thus using very coarse conditioning information

DEPARTMENT OF ECE


241 FINDING SHORTEST PATH

Fig 21

The original idea comes from observing the exploitation of food resources among ants in which antsrsquo individually limited cognitive abilities have collectively been able to find the shortest path between a food source and the nest

1 The first ant finds the food source (F) via any way (a) then returns to the nest (N) leaving behind a trail pheromone (b)

2 Ants indiscriminately follow four possible ways but the strengthening of the runway makes it more attractive as the shortest route

3 Ants take the shortest route long portions of other ways lose their trail pheromones

In a series of experiments on a colony of ants with a choice between two unequal length paths leading to a source of food biologists have observed that ants tended to use the shortest route A model explaining this behaviour is as follows

a) An ant (called blitz) runs more or less at random around the colonyb) If it discovers a food source it returns more or less directly to the nest leaving in its path

a trail of pheromonec) These pheromones are attractive nearby ants will be inclined to follow more or less

directly the trackd) Returning to the colony these ants will strengthen the route

DEPARTMENT OF ECE


e) If there are two routes to reach the same food source then in a given amount of time the shorter one will be traveled by more ants than the long route

f) The short route will be increasingly enhanced and therefore become more attractiveg) The long route will eventually disappear because pheromones are volatileh) Eventually all the ants have determined and therefore chosen the shortest route

Ants use the environment as a medium of communication They exchange information indirectly by depositing pheromones all detailing the status of their work The information exchanged has a local scope only an ant located where the pheromones were left has a notion of them This system is called Stigmergy and occurs in many social animal societies (it has been studied in the case of the construction of pillars in the nests of termites) The mechanism to solve a problem too complex to be addressed by single ants is a good example of a self-organized system This system is based on positive feedback (the deposit of pheromone attracts other ants that will strengthen it themselves) and negative (dissipation of the route by evaporation prevents the system from thrashing) Theoretically if the quantity of pheromone remained the same over time on all edges no route would be chosen However because of feedback a slight variation on an edge will be amplified and thus allow the choice of an edge The algorithm will move from an unstable state in which no edge is stronger than another to a stable state where the route is composed of the strongest edges

The basic philosophy of the algorithm involves the movement of a colony of ants through the different states of the problem influenced by two local decision policies viz trails and attractiveness Thereby each such ant incrementally constructs a solution to the problem When an ant completes a solution or during the construction phase the ant evaluates the solution and modifies the trail value on the components used in its solution This pheromone information will direct the search of the future ants Furthermore the algorithm also includes two more mechanisms viz trail evaporation and daemon actions Trail evaporation reduces all trail values over time thereby avoiding any possibilities of getting stuck in local optima The daemon actions are used to bias the search process from a non-local perspective

25 ACO METAHEURISTIC

Artificial ants used in ACO are stochastic solution construction procedures that probabilistically build a solution by iteratively adding solution components to partial solutions by taking into account (i) heuristic information on the problem instance being solved if available and (ii) (artificial) pheromone trails which change dynamically at run-time to reflect the agents acquired search experience

The interpretation of ACO as an extension of construction heuristics is appealing because of several reasons A stochastic component in ACO allows the ants to build a wide variety of different solutions and hence explore a much larger number of solutions than greedy heuristics

DEPARTMENT OF ECE


At the same time the use of heuristic information which is readily available for many problems can guide the ants towards the most promising solutions More important the antsrsquo search experience can be used to influence in a way reminiscent of reinforcement learning the solution construction in future iterations of the algorithm Additionally the use of a colony of ants can give the algorithm increased robustness and in many ACO applications the collective interaction of a population of agents is needed to efficiently solve a problem The domain of application of ACO algorithms is vast In principle ACO can be applied to any discrete optimization problem for which some solution construction mechanism can be conceived In the following of this section we first define a generic problem representation which the ants in ACO exploit to construct solutions then we detail the ants behavior while constructing solutions and finally we define the ACO metaheuristic

251 PROBLEM REPRESENTATION

Let us consider the minimization problem (S f Ω) where S is the set of candidate solutions f is the objective function which assigns to each candidate solution s S isin an objective function (cost) value f(s t) and Ω is a set of constraints The goal is to find a globally optimal solution sopt Sisin that is a minimum cost solution that satisfies the constraints Ω The problem representation of a combinatorial optimization problem (S f Ω) which is exploited by the ants can be characterized as followsbull A finite set C = c1 c2 cNC of components is givenbull The states of the problem are defined in terms of sequences x =ltci cj ck gt over the elements of C The set of all possible sequences is denoted by X The length of a sequence x that is the number of components in the sequence is expressed by |x|bull The finite set of constraints Ω defines the set of feasible states X˜ with X˜ subeX

bull A set S of feasible solutions is given with S lowast lowast sube X˜ and S lowast sube S

bull A cost f(s t) is associated to each candidate solution s isin Sbull In some cases a cost or the estimate of a cost J(xi t) can be associated to states other than solutions If xj can be obtained by adding solution components to a state xi then J(xi t) le J(xj t) Note that J(s t) equiv f(s t) Given this representation artificial ants build solutions by moving on the construction graph G = (CL) where the vertices are the components C and the set L fully connects the components C (elements of L are called connections) The problem constraints Ω are implemented in the policy followed by the artificial ants The choice of implementing the constraints in the construction policy of the artificial ants allows a certain degree of flexibility In fact depending on the combinatorial optimization problem considered it may be more reasonable to implement constraints in a hard way allowing ants to build only feasible solutions or in a soft way in which

case ants can build infeasible solutions (that is candidate solutions in SSlowast) that will be penalized in dependence of their degree of infeasibility

DEPARTMENT OF ECE


252 ANTrsquoS BEHAVIOURAnts can be characterized as stochastic construction procedures which build solutions moving on the construction graph G = (CL) Ants do not move arbitrarily on G but rather follow a construction policy which is a function of the problem constraints Ω In general ants try to build

feasible solutions but if necessary they can generate infeasible solutions Components ci isin C

and connections lij isin L can have associated a pheromone trail τ (τi if associated to components τij if associated to connections) encoding a long-term memory about the whole ant search process that is updated by the ants themselves and a heuristic value η (ηi and ηij respectively) representing a priori information about the problem instance definition or run-time information provided by a source different from the ants In many cases η is the cost or an estimate of the cost of extending the current state These values are used by the ants heuristic rule to make probabilistic decisions on how to move on the graphMore precisely each ant k of the colony has the following properties

It exploits the graph G = (CL) to search for feasible solutions s of minimum cost That is solutions s such that ˆ fs = min_s f(s t)

It has a memory Mk that it uses to store information about the path it followed so far Memory can be used (i) to build feasible solutions (ie to implement constraints Ω) (ii) to evaluate the solution found and (iii) to retrace the path backward to deposit pheromone

It can be assigned a start state x^k and one or more termination conditions e^k Usually the start state is expressed either as a unit length sequence (that is a single component sequence) or an empty sequence

When in state x_r = lt x_rminus1 igt if no termination condition is satisfied it moves to a

node j in its neighbourhood N ^ k that is to a state ltx^r jgt isin XOften moves towards feasible states are favoured either via appropriately defined heuristic values η or through the use of the antsrsquo memory

It selects the move by applying a probabilistic decision rule Its probabilistic decision rule is a function of (i) locally available pheromone trails and heuristic values (ii) the antrsquos private memory storing its past history and (iii) the problem constraints

The construction procedure of ant k stops when at least one of the termination conditions e^k is satisfied

When adding a component cj to the current solution it can update the pheromone trail associated to it or to the corresponding connection This is called online step-by-step pheromone update

Once built a solution it can retrace the same path backward and update the pheromone trails of the used components or connections This is called online delayed pheromone update

It is important to note that ants move concurrently and independently and that each ant is complex enough to find a (probably poor) solution to the problem under consideration

DEPARTMENT OF ECE


Typically good quality solutions emerge as the result of the collective interaction among the ants which is obtained via indirect communication mediated by the information ants readwrite in the variables storing pheromone trail values In a way this is a distributed learning process in which the single agents the ants are not adaptive themselves but on the contrary they adaptively modify the way the problem is represented and perceived by other ants

253 THE METAHEURISTIC

Informally the behaviour of ants in an ACO algorithm can be summarized as follows A colony of ants concurrently and asynchronously moves through adjacent states of the problem by building paths on G They move by applying a stochastic local decision policy that makes use of pheromone trails and heuristic information By moving ants incrementally build solutions to the optimization problem Once an ant has built a solution or while the solution is being built the ant evaluates the (partial) solution and deposits pheromone trails on the components or connections it used This pheromone information will direct the search of the future ants Besides antsrsquo activity an ACO algorithm includes two more procedures pheromone trail evaporation and daemon actions (this last component being optional) Pheromone evaporation is the process by means of which the pheromone trail intensity on the components decreases over time From a practical point of view pheromone evaporation is needed to avoid a too rapid convergence of the algorithm towards a sub-optimal region It implements a useful form of forgetting

Procedure ACO metaheuristic Schedule Activities ManageAntsActivity() EvaporatePheromone() DaemonActions() Optional end Schedule Activities end ACO metaheuristic

Fig 22 The ACO metaheuristic in pseudo-code Comments are enclosed in braces The procedure DaemonActions() is optional and refers to centralized actions executed by a daemon possessing global knowledge Favouring the exploration of new areas of the search space Daemon actions can be used to implement centralized actions which cannot be performed by single ants

DEPARTMENT OF ECE


Examples are the activation of a local optimization procedure or the collection of global information that can be used to decide whether it is useful or not to deposit additional pheromone to bias the search process from a non-local perspective As a practical example the daemon can observe the path found by each ant in the colony and choose to deposit extra pheromone on the components used by the ant that built the best solution Pheromone updates performed by the daemon are called off-line pheromone updates In Figure 22 the ACO metaheuristic behaviour is described in pseudo-code The main procedure of the ACO metaheuristic manages via the Schedule Activities construct the scheduling of the three above discussed components of ACO algorithms (i) management of antsrsquo activity (ii) pheromone evaporation and (iii) daemon actions The Schedule Activities construct does not specify how these three activities are scheduled and synchronized In other words it does not say whether they should be executed in a completely parallel and independent way or if some kind of synchronization among them is necessary The designer is therefore free to specify the way these three procedures should interact

DEPARTMENT OF ECE


CHAPTER 3

ANT COLONY ALGORITHM

31 HISTORY The first ACO algorithm proposed was Ant System (AS) AS was applied to some rather small instances of the traveling salesman problem (TSP) with up to 75 cities It was able to reach the performance of other general-purpose heuristics like evolutionary computation Despite these initial encouraging results AS could not prove to be competitive with state-of-the-art algorithms specifically designed for the TSP when attacking large instances Therefore a substantial amount of recent research has focused on ACO algorithms which show better performance than AS when applied for example to the TSP In the following of this section we first briefly introduce the biological metaphor on which AS and ACO are inspired and then we present a brief history of the developments that took from the original AS to the most recent ACO algorithms In fact these more recent algorithms are direct extensions of AS which add advanced features to improve the algorithm performance

32 BIOLOGICAL ANALOGY

In many ant species individual ants may deposit a pheromone (a particular chemical that ants can smell) on the ground while walking [50] By depositing pheromone they create a trail that is used for example to mark the path from the nest to food sources and back In fact by sensing pheromone trails foragers can follow the path to food discovered by other ants Also they are capable of exploiting pheromone trails to choose the shortest among the available paths taking to the food Deneubourg and colleagues used a double bridge connecting a nest of ants and a food source to study pheromone trail laying and following behavior in controlled experimental conditions They ran a number of experiments in which they varied the ratio between the length of the two branches of the bridge The most interesting for our purposes of these experiments is the one in which one branch was longer than the other In this experiment at the start the ants were left free to move between the nest and the food source and the percentage of ants that chose one or the other of the two branches was observed over time The outcome was that although in the initial phase random oscillations could occur in most experiments all the ants ended up using the shorter branch This result can be explained as follows When a trial starts there is no pheromone on the two branches Hence the ants do not have a preference and they select with the same probability any of the two branches Therefore it can be expected that on average half of the ants choose the short branch and the other half the long branch although stochastic oscillations may occasionally favor one branch over the other

DEPARTMENT OF ECE


However because one branch is shorter than the other the ants choosing the short branch are the first to reach the food and to start their travel back to the nest But then when they must make a decision between the short and the long branch the higher level of pheromone on the short branch biases their decision in its favor Therefore pheromone starts to accumulate faster on the short branch which will eventually be used by the great majority of the ants

321 DOUBLE BRIDGE EXPERIMENT

Ant colonies can collectively perform tasks and make decisions that appear to require a high degree of co-ordination among the workers building a nest feeding the brood foraging for food and so on In the example presented here the ants collectively discover the shortest path to a food source In experiments with the ant Linepithaema humile a food source is separated from the nest by a bridge with two branches The longer branch is r times longer than the shorter branch(Fig3a)

The shorter branch is selected by the colony in most experiments if r is sufficiently large (r = 2 in Fig 3b) This is because the ants lay and follow pheromone trails individual ants lay a chemical substance a pheromone which attracts other ants The first ants returning to the nest from the food source are those who take the shorter path twice (from the nest to the source and back) At choice points 1 and 2 nest mates are recruited toward the shorter branch which is the first to be marked with pheromone If however the shorter branch is presented to the colony after the longer branch the shorter path will not be selected because the longer branch is already marked with pheromone

This problem can be overcome in an artificial system by introducing pheromone decay when the pheromone evaporates sufficiently quickly it is more difficult to maintain a stable pheromone trail on a longer path The shorter branch can then be selected even if presented after the longer branch This property is particularly desirable in optimization where one seeks to avoid convergence toward mediocre solutions In Linepithaema humile although pheromone concentrations do decay the lifetime of trail pheromones is too large to allow such flexibility

Fig 31a Fig31b

DEPARTMENT OF ECE


The double bridge was substituted by a graph and pheromone trails by artificial pheromone trails Also because we wanted artificial ants to solve problems more complicate than those solved by real ants we gave artificial ants some extra capacities like a memory (used to implement constraints and to allow the ants to retrace their path back to the nest without errors) and the capacity of depositing a quantity of pheromone proportional to the quality of the solution produced (a similar behavior is observed also in some real ants species in which the quantity of pheromone deposited while returning to the nest from a food source is proportional to the quality of the food source found In the next section we will see how starting from AS new algorithms have been proposed that although retaining some of the original biological inspiration are less and less biologically inspired and more and more motivated by the need of making ACO algorithms competitive with or improve over state-of-the-art algorithms Nevertheless many aspects of the original Ant System remain the need for a colony the role of autocatalysis the cooperative behavior mediated by artificial pheromone trails the probabilistic construction of solutions biased by artificial pheromone trails and local heuristic information the pheromone updating guided by solution quality and the evaporation of pheromone trail are present in all ACO algorithms It is interesting to note that there is one well known algorithm that although making use in some way of the ant foraging metaphor cannot be considered an instance of the Ant Colony Optimization metaheuristic This is HAS-QAP proposed in where pheromone trails are not used to guide the solution construction phase on the contrary they are used to guide modifications of complete solutions in a local search style This algorithm belongs nevertheless to ant algorithms a new class of algorithms inspired by a number of different behaviors of social insects Ant algorithms are receiving increasing attention in the scientific community as a promising novel approach to distributed control and optimization

33 CONSTRUCTION ALGORITHMS Construction algorithms build solutions to a problem under consideration in an incremental way starting with an empty initial solution and iteratively adding opportunely defined solution components without backtracking until a complete solution is obtained In the simplest case solution components are added in random order Often better results are obtained if a heuristic estimate of the myopic benefit of adding solution components is taken into account Greedy construction heuristics add at each step a solution component which achieves the maximal myopic benefit as measured by some heuristic information The function Greedy Component returns the solution component e with the best heuristic estimate Solutions returned by greedy algorithms are typically of better quality than randomly generated solutions Yet a disadvantage of greedy construction heuristics is that only a very limited number of solutions can be generated Additionally greedy decisions in early stages of the construction process strongly constrain the available possibilities at later stages often determining very poor moves in the final phases of the solution construction As an example consider a greedy construction heuristic for the traveling salesman problem (TSP)

DEPARTMENT OF ECE


In the TSP we are given a complete weighted graph G = (NA) with N being the set of nodes representing the cities and A the set of arcs fully connecting the nodes N Each arc is assigned a

value dij which is the length of arc (i j) isin A The TSP is the problem of finding a minimal length Hamiltonian circuit of the graph where an Hamiltonian circuit is a closed tour visiting exactly once each of the n = |N| nodes of G For symmetric TSPs the distances between the cities are independent of the direction of traversing the arcs that is dij = dji for every pair of nodes In the more general asymmetric TSP (ATSP) at least for one pair of nodes i j we have dij =dji A simple rule of thumb to build a tour is to start from some initial city and to always choose to go to the closest still unvisited city before returning to the start city This algorithm is known as the nearest neighbor tour construction heuristic Noteworthy in this example is that there are some few very long links in the tour leading to strongly suboptimal solutions In fact construction algorithms are typically the fastest approximate methods but the solutions they generate often are not of a very high quality and they are not guaranteed to be optimal with respect to small changes the results produced by constructive heuristics can often be improved by local search algorithms

331 LOCAL SEARCH Local search algorithms start from a complete initial solution and try to find a better solution in an appropriately defined neighborhood of the current solution In its most basic version known as iterative improvement the algorithm searches the neighborhood for an improving solution If such a solution is found it replaces the current solution and the local search continues These steps are repeated until no improving neighbor solution is found anymore in the neighborhood of the current solution and the algorithm ends in a local optimum The procedure Improve returns a better neighbor solution if one exists otherwise it returns the current solution in which case the algorithm stops procedure Iterative Improvement (s isin S)

srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s

end Iterative Improvement

Fig 32 Algorithmic skeleton of iterative improvement

DEPARTMENT OF ECE


The choice of an appropriate neighborhood structure is crucial for the performance of the local search algorithm and has to be done in a problem specific way The neighborhood structure defines the set of solutions that can be reached from s in one single step of a local search algorithm An example neighborhood for the TSP is the k-opt neighborhood in which neighbor solutions differ by at most k arcs The 2-opt algorithm systematically tests whether the current tour can be improved by replacing two edges To fully specify a local search algorithm is needed a neighborhood examination scheme that defines how the neighborhood is searched and which neighbor solution replaces the current one In the case of iterative improvement algorithms this rule is called pivoting rule and examples are the best-improvement rule which chooses the neighbor solution giving the largest improvement of the objective unction and the first-improvement rule which uses the first improved solution found in the neighborhood to replace the current one A common problem with local search algorithms is that they easily get trapped in local minima and that the result strongly depends on the initial solution

34 ANT ALGORITHM

We use the following notation A combinatorial optimization problem is a problem defined over a set C = c1 cn of basic components A subset S of components represents a solution of

the problem F sube 2C is the subset of feasible solutions thus a solution S is feasible if and only if

S isin F A cost function z is defined over the solution domain z 2C 1048774 R the objective being to

find a minimum cost feasible solution S ie to find S S isin F and z(S) le z(S) forallSisinF A set of computational concurrent and asynchronous agents (a colony of ants) moves through states of the problem corresponding to partial solutions of the problem to solve They move by applying a stochastic local decision policy based on two parameters called trails and attractiveness By moving each ant incrementally constructs a solution to the problem When an ant completes a solution or during the construction phase the ant evaluates the solution and modifies the trail value on the components used in its solution This pheromone information will direct the search of the future ants Furthermore an ACO algorithm includes two more mechanisms trail evaporation and optionally daemon actions Trail evaporation decreases all trail values over time in order to avoid unlimited accumulation of trails over some component Daemon actions can be used to implement centralized actions which cannot be performed by single ants such as the invocation of a local optimization procedure or the update of global information to be used to decide whether to bias the search process from a non-local perspective More specifically an ant is a simple computational agent which iteratively constructs a solution for the instance to solve Partial problem solutions are seen as states At the core of the ACO algorithm lies a loop where at each iteration each ant moves (performs a step) from a state ι to another one ψ corresponding to a more complete partial solution That is at each step σ each ant k computes a set Ak σ(ι) of feasible expansions to its current state and moves to one of these in probability

DEPARTMENT OF ECE


The probability distribution is specified as follows For ant k the probability pιψk of moving from state ι to state ψ depends on the combination of two values

The attractiveness ηιψ of the move as computed by some heuristic indicating the a priori desirability of that move

The trail level τιψ of the move indicating how proficient it has been in the past to make that particular move it represents therefore an a posterior indication of the desirability of that move

Trails are updated usually when all ants have completed their solution increasing or decreasing the level of trails corresponding to moves that were part of good or bad solutions respectively The ant system simply iterates a main loop where m ants construct in parallel their solutions thereafter updating the trail levels The performance of the algorithm depends on the correct tuning of several parameters namely α β relative importance of trail and attractiveness ρ trail persistence τij(0) initial trail level m number of ants and Q used for defining to be of high quality solutions with low cost The algorithm is the following1 InitializationInitialize τιψ and ηιψ forall(ιψ)2 ConstructionFor each ant k (currently in state ι) dorepeatchoose in probability the state to move intoappend the chosen move to the k-th ants set tabukuntil ant k has completed its solutionend for3 Trail updateFor each ant move (ιψ) docompute Δτιψupdate the trail matrixend for4 Terminating conditionIf not(end test) go to step 2

DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO

41 TRAVELLING SALES MAN

The TSP is a very important problem in the context of Ant Colony Optimization because it is the problem to which the original AS was first applied and it has later often been used as a benchmark to test a new idea and algorithmic variants

OBJECTIVE Given a set of n cities the Traveling Salesman Problem requires a salesman to find the shortest route between the given cities and return to the starting city while keeping in mind that each city can be visited only once

411 ANT COLONY OPTIMIZATION IN TSP

The meta heuristic Ant Colony Optimization (ACO) is an optimization algorithm successfully used to solve many NP hard optimization problems introduced in ACO ACO algorithms are a very interesting approach to find minimum cost paths in graphs especially when the connection costs in the graphs can change over time ie when the problems are dynamic The artificial ants have been successfully used to solve the (conventional) Traveling Salesman Problem (TSP) as well as other NP hard optimization problems including applications in quadratic assignment or vehicle routing The algorithmrsquos based on the fact that ants are always able to find the shortest path between the nest and the food sources using information of the pheromones previously laid on the ground by other ants in the colony When an ant is searching for the nearest food source and arrives at several possible trails it tends to choose the trail with the largest concentration of pheromones with a certain probability p After choosing the trail it deposits another pheromone increasing the concentration of pheromones in this trail The ants return to the nest using always the same path depositing another portion of pheromone in the way back Imagine then that two ants at the same location choose two different trails at the same time The pheromone concentration on the shortest way will increase faster than the other the ant that chooses this way will deposit more pheromone in a smaller period of time because it returns earlier If a whole colony of thousands of ants follows this behaviour soon the concentration of pheromone on the shortest path will be much higher than the concentration in other paths Then the probability of choosing any other way will be very small and only very few ants among the colony will fail to follow the shortest path

DEPARTMENT OF ECE


There is another phenomenon related with the pheromone concentration since it is a chemical substance it tends to evaporate so the concentration of pheromones vanishes along the time In this way the concentration of the less used paths will be much lower than that of the most used ones not only because the concentration increases on the other paths but also because its own concentration decreases

Fig 41

The traveling salesman problem plays a central role in ant colony optimization because it was the first problem to be attacked by these methods Among the reasons the TSP was chosen we find the following ones (i) it is relatively easy to adapt the ant colony metaphor to it (ii) it is a very difficult problem (NP-hard) (iii) it is one of the most studied problems in combinatorial optimization and (iv) it is very easy to state and explain it so that the algorithm behavior is not obscured by too many technicalities Ant System (AS) can be informally described as follows A number m of ants is positioned in parallel on m cities The antsrsquo start state that is the start city can be chosen randomly and the memory of each ant k is initialized by adding the current start city to the set of already visited cities (initially empty) Ants then enter a cycle which lasts NC iterations that is until each ant has completed a tour During each step an ant located on node i considers the feasible neighborhood reads the entries aij rsquos of the ant-routing table Ai of node i computes the transition probabilities and then applies its decision rule to choose the city to move to moves to the new city and updates its memory Once ants have completed a tour (which happens synchronously given that during each iteration of the while loop each ant adds a new city to the tour under construction) they use their memory to evaluate the built solution and to retrace the same tour backward and increase the intensity of the pheromone trails iquestij of visited connections lij This has the effect of making the visited connections become more desirable for future ants Then the ants die freeing all the allocated resources In AS all the ants deposit pheromone and no problem-specific daemon actions are performed The triggering of pheromone evaporation happens after all ants have completed their tours Of course it would be easy to add a local optimization daemon action likea 3-opt procedure this has been done in most of the ACO algorithms for the TSP that have been developed as extensions of AS

DEPARTMENT OF ECE

A

D

C

B


412 FLOW CHART FOR TSP USING ACO

Fig 42

DEPARTMENT OF ECE

Initialize

Place each ant in a randomly chosen city

Choose Next City(For Each Ant)

more cities to visit

For Each Ant

Return to the initial cities

Update pheromone level using the tour cost for each ant

Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ


413 ANT COLONY OPTIMIZATION ALGORITHM

procedure Ant colony algorithmSet for every pair (i j) Tij = TmaxPlace the g ants

For i = 1 to NBuild a complete tour

For j = 1 to mFor k = 1 to gChoose the next node using pk ij in (2)Update the tabu list T

EndEnd

Analyze solutionsFor k = 1 to g

Compute performance index fkUpdate globally Tij(t + m times g) using (3)

EndEnd

Euclidean distance between two locations dij is used as heuristic However within a city the traveling time tij between two machines is more relevant than distance due to traffic reasons Therefore the heuristic function is given by _ = (tij minus tmin)(tmax minus tmin) where tij is the estimated traveling time between location i and location j and tmin = min tij and tmax = max tij are the minimum and maximum travelling times considered In this way the heuristic matrix _ entries are always restricted to the interval [0 1] The objective function to minimize fk(t) is simply the sum of travelling time between all the visited locations

414 RULES FOR TRANSITION PROBABILITY

1 Whether or not a city has been visited Use of a memory(tabu list) set of all cities that are to be visited

2 = visibility Heuristic desirability of choosing city j when in city i

DEPARTMENT OF ECE

1d ij

N ij


3Pheromone trail This is a global type of informationTransition probability for ant k to go from city i to city j while building its route

a = 0 closest cities are selectedTrial visibility is hij = 1dij The intensity in the probabilistic transition is a The visibility of the trial segment is b The trail persistence or evaporation rate is given as r Trail intensity is given by value of tij

415 TSP APPLICATIONbull Lots of practical applicationsbull Routing such as in trucking delivery UAVs Manufacturing routing such as movement of parts along manufacturing floor or the amount of solder on circuit boardbull Network design such as determining the amount of cabling requiredbull Two main typesndash Symmetricndash Asymmetric

416 TSP HEURISTICS

bull Variety of heuristics used to solve the TSPbull The TSP is not only theoretically difficult it is also difficult in practical application since the tour breaking constraints get quite numerousbull As a result there have been a variety of methods proposed for the TSPbull Nearest Neighbour is a typical greedy approach

DEPARTMENT OF ECE

T ij( t )

pijk ( t )=iquest [ τ ij( t ) ]α [ηij ]

β

sumk isinallowed k

[τ ik ( t ) ]α [ηik ]β

if jisin allowedk iquestiquestiquestiquestiquest

iquest

iquest


417 ADVANTAGESbull Positive Feedback accounts for rapid discovery of good solutionsbull Distributed computation avoids premature convergencebull The greedy heuristic helps find acceptable solution in the early solution in the early stages of the search processbull The collective interaction of a population of agents

42 QUADRATIC ASSIGNMENT PROBLEM

Fig 43

The quadratic assignment problem (QAP) is one of fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics from the category of the facilities location problems

There are a set of n facilities and a set of n locations For each pair of locations a distance is specified and for each pair of facilities aweigh or flow is specified (eg the amount of supplies transported between the two facilities) The problem is to assign all facilities to different locations with the goal of minimizing the sum of the distances multiplied by the corresponding flows

The formal definition of the quadratic assignment problem is

Given two sets P (facilities) and L (locations) of equal size together with a weight function w P times P rarr R and a distance function d Ltimes L rarr R Find the bijection f P rarr L (assignment) such that the cost function

DEPARTMENT OF ECE

nnjid


is minimized

Usually weight and distance functions are viewed as square real-valued matrices so that

the cost function is written down as

Problem isbull Assign n activities to n locations (campus and mall layout)bull D= distance from location i to location jbull F= flow from activity h to activity kbull Assignment is permutationbull Minimizebull Itrsquos NP hard

QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B

Locations Facilities

How to assign facilities to locations

Lower costHigher cost

ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n

d ij f π (i )π ( j)

Π


421 ALGORITHMS

1) Ant System for the QAP Ant System (AS) uses ants in order to construct a solution from scratch The algorithm uses a heuristic information on the potential quality of a local assignment that is determined as follows two vectors d and f whose components are the sum of the distances resp flows from location resp facility i to all other locations resp facilities are computed This leads to a coupling matrix E = f middot dT where eij = fi middot dj Thus _ = 1eij denotes the heuristic desirability of assigning facility i to location j A solution is constructed by using both this heuristic information and information provided by previous ants usingpheromones At each step an ant k assigns the next still unassigned facility i to a location j belonging to the feasible neighbourhood of the node i ie the locations that are still free with a probability pk ij given by

After their run the ants update the pheromone information Tij

This algorithm uses an evaporation rate of (1 minus p) in order to forget previous bad choices at the

cost of loosing useful information too is the amount of pheromone ant k deposits on the edge (i j)

The algorithm parameters are the number of ants n the weight given to either heuristic or pheromone informationrsquos _ _ and the maximal amount of laid pheromone Q

DEPARTMENT OF ECE


2) The MAX minus MIN Ant System MAX minus MIN Ant System (MMAS) is an improvement over AS that allows only one ant to add pheromone The pheromone trails are initialized to the upper trail limit which cause a higher exploration at the start of algorithm Finally methods to increase the diversification of the search can be used for example by reinitialising the pheromone trails to _max if the algorithm makes no progress In MMAS the ant k assigns the facility i to the location j with a probability pk ij MMAS does not use any heuristic information but is coupled with a local search for every ant

422 SIMPLIFIED CRAFT (QAP) Simplification Assume all departments have equal size Notation distance between locations i and j travel frequency between departments k and h 1 if department k is assigned to location i 0 otherwise Example

Fig45

DEPARTMENT OF ECE

2

1 343

4 Location

Department (bdquoFacilityldquo)


1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a

Fig 46b423 ANT SYSTEM (AS-QAP)Constructive methodstep 1 choose a facility jstep 2 assign it to a location i Characteristicsndash each ant leaves trace (pheromone) on the chosen couplings (ij)ndash assignment depends on the probability (function of pheromone trail and a heuristic information)

DEPARTMENT OF ECE

Distance jid

Frequency hkf


ndash already coupled locations and facilities are inhibited (Tabu list)Heuristic information

The coupling Matrix

Ants choose the location according to the heuristic desirability ldquoPotential goodnessrdquo

424 AS-QAP CONSTRUCTING THE SOLUTION

AS-QAP Constructing the Solution facilities are ranked in decreasing order of the flow

potentials Ant k assigns the facility i to location j with the probability given by

where is the feasible Neighborhood of node i

When Ant k choose to assign facility j to location i it leave a substance called trace

ldquopheromonerdquo on the coupling (ij)

Repeated until the entire assignment is found

DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0

]rArr Di=iquest [ 6 iquest ] [10 iquest ] [ 12iquest ]iquestiquest

iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β

sumlisinN ik [ τ ij( t ) ]α [ηij ]

β if jisinN i

k

N ik


425 AS-QAP PHEROMONE UPDATE

Pheromone trail update to all couplings

is the amount of pheromone ant k puts on the coupling (ij)

Qthe amount of pheromone deposited by ant k

426 NETWORK MODEL

Routing (or routeing) is the process of selecting paths in a network along which to send network traffic Routing is performed for many kinds of networks including the telephone network electronic data networks (such as the Internet) and transportation networks This article is concerned primarily with routing in electronic data networks using packet switching technology In packet switching networks routing directs packet forwarding the transit of logically addressed packets from their source toward their ultimate destination through intermediate nodes typically hardware devices called routers bridges gateways firewalls or switches General-purpose computers with multiple network cards can also forward packets and perform routing though they are not specialized hardware and may suffer from limited performance The routing process usually directs forwarding on the basis of routing tables which maintain a record of the routes to various network destinations Thus constructing routing tables which are held in the routers memory is very important for efficient routing Most routing algorithms use only one network path at a time but multipath routing techniques enable the use of multiple alternative paths Routing task is performed by Routers Routers use ldquoRouting Tablesrdquo to direct the data

DEPARTMENT OF ECE

τ ij ( t+1 )=ρ τ ij( t )+sumk=1

m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk

if facility i is assigned to location j in the solution of ant k

0 otherwise

Jψk hellip the objective function value


Fig 47

427 PROBLEM STATEMENT

Dynamic Routing At any moment the pathway of a message must be as small as possible (Traffic conditions and the structure of the network are constantly changing)

Load balancing Distribute the changing load over the system and minimize lost calls

428 ALGORITHM

Increase the probability of the visited link by

Decrease the probability of the others by

Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8

If your destination is node 5 next node to 3

3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b

43 VEHICLE ROUTING PROBLEM WITH TIME WINDOWS (VRPTW)

The vehicle routing problem(VRP) is a combinatorial and integer programming problem seeking to service a number of customers with a fleet of vehicles Proposed by Dantzig and Ramser in 1959 VRP is an important problem in the fields of transportation distribution and logistics Often the context is that of delivering goods located at a central depot to customers who have placed orders for such goods Implicit is the goal of minimizing the cost of distributing the goods Many methods have been developed for searching for good solutions to the problem but for all but the smallest problems finding global minimum for the cost function is computationally complexObjective Functions to Minimize

bull Total travel distancebull Total travel timebull Number of vehicles

Subject to bull Vehicles ( Capacity time on road trip length)bull Depots (Numbers)bull Customers (Demands time windows)

DEPARTMENT OF ECE

2

4

3

7

1

5

8

6


Vehicle Routing Problem with Time Windows (VRPTW)

Fig 49

431 SIMPLE ALGORITHM

Place ants on depots (Depots = Vehicle ) Probabilistic choice

~ (1distance di Q) ~ amount of pheromone

If all unvisited customer lead to a unfeasible solution Select depot as your next customer

Improve by local search Only best ants update pheromone trial

DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]


432 MULTIPLE ACS FOR VRPTW

Fig 410

433 SINGLE MACHINE TOTAL WEIGHTED TARDINESS SHEDULING PROBLEM (SMTWTP)

In the SMTWTP n jobs have to be processed sequentially without interruption ona single machine Each job has a processing time pj a weight wj and a due date dj associated and all jobs are available for processing at time zero The tardiness of job j is defined as Tj = max0 Cj minus dj where Cj is its completion time in the current job sequence The goal in the SMTWTP is to find a job sequence which minimizes the sum of the weighted tardiness given by(summation from i=1 to n of (wi middotTi)) For the ACO application to the SMTWTP the set of components C is the set of all jobs As in the TSP case the states of the problem are all possible partial sequences In the SMTWTP case we do not have explicit costs associated with the connections because the objective function contribution of each job depends on the partial solution constructed so far

DEPARTMENT OF ECE

ACS-VEI(Min Vehicles number)

MAC-VRPTW Multi-objectives

ACS-TIME(Min Travel time)

Single objective

gb

TIMEACS VEIACS


The SMTWTP was attacked in using ACS (ACS-SMTWTP) In ACSSMTWTP each ant starts with an empty sequence and then iteratively appends an unscheduled job to the partial sequence constructed so far

Each ant chooses the next job using the pseudo-random-proportional action choice rule where the at each step the feasible neighborhood Ni^k of ant k is formed by the still unscheduled jobs Pheromone trails are defined as follows τij refers to the desirability of scheduling job j at position i This definition of the pheromone trails is in fact used in most ACO application to scheduling problems Concerning the heuristic information in the use of three priority rules allowed to define three different types of heuristic information for the SMTWTP The investigated priority rules were (i) the earliest due date rule (EDD) which puts the jobs in non-decreasing order of the due dates dj (ii) the modified due date rule (MDD) which puts the jobs in non-decreasing order of the modified due dates given by mddj = maxC + pj dj [2] where C is the sum of the processing times of the already sequenced jobs and (iii) the apparent urgency rule (AU) which puts the jobs in non-decreasing order of the apparent urgency given by auj = (wjpj) middot exp(minus(maxdj minus Cj 0)kp) where k is a parameter of the priority rule In each case the heuristic information was defined as ηij = 1hj where hj is either dj mddj or auj depending on the priority rule used The global and the local pheromone update is done as in the standard ACS where in the global pheromone update Tgb is the total weighted tardiness of the global best solution In ACS-SMTWTP was combined with a powerful local search algorithm The final ACS algorithm was tested on a benchmark set available from ORLIB Within the computation time limits ACS reached a very good performance and could find in each single run the optimal or best known solutions on all instances of the benchmark set

DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION

51 ADVANTAGES For TSPs (Traveling Salesman Problem) relatively efficient

for a small number of nodes TSPs can be solved by exhaustive search for a large number of nodes TSPs are very computationally difficult to solve

(NP-hard) ndash exponential time to convergence Performs better against other global optimization techniques for TSP (neural

net genetic algorithms simulated annealing) Compared to GAs (Genetic Algorithms)

Retains memory of entire colony instead of previous generation only Less affected by poor initial solutions (due to combination of random path

selection and colony memory) Can be used in dynamic applications (adapts to changes such as new distances etc)

Has been applied to a wide variety of applications As with GAs good choice for constrained discrete problems (not a gradient-

based algorithm)

52 DISADVANTAGES

Theoretical analysis is difficult Due to sequences of random decisions (not independent) Probability distribution changes by iteration Research is experimental rather than theoretical

Convergence is guaranteed but time to convergence uncertain Tradeoffs in evaluating convergence

In NP-hard problems need high-quality solutions quickly ndash focus is on quality of solutions

In dynamic network routing problems need solutions for changing conditions ndash focus is on effective evaluation of alternative paths

Coding is somewhat complicated not straightforward

DEPARTMENT OF ECE


Pheromone ldquotrailrdquo additionsdeletions global updates and local updates

Large number of different ACO algorithms to exploit different problem characteristics

53 SCOPE OF SEMINAR

In this paper we have given a formal description of the Ant Colony Optimization meta heuristic as well as of the class of problems to which it can be applied We have then describedtwo paradigmatic applications of ACO algorithms to the traveling salesman problem and to routing in packet-switched networks and concluded with a short overview on the currentlyavailable applications Ongoing work follows three main directions the study of the formal properties of a simplified version of ant system the development of Ant Net for Quality of Service applications and the development of further applications to combinatorial optimization problems Since Ant colony algorithm may produce redundant states in the graph itrsquos better to minimize such graphs to enhance the behavior of the inducted system A colony of ants moves through system states X by applying Genetic Operations By moving each ant incrementally constructs a solution to the problem When an ant complete solution or during the construction phase the ant evaluates the solution and modifies the trail value on the components used in its solution Ants deposit a certain amount of pheromone on the components that is either on thevertices or on the edges that they traverse The amount of pheromone deposited may depend on the quality of the solution found Subsequent ants use the pheromone information as a guide toward promising regions of the search space Ants adaptively modify the way the problem is represented and perceived by other ants but they are not adaptive themselves The genetic programming paradigm permits the evolution of computer programs which can perform alternative computations conditioned on the outcome of intermediate calculations which can perform computations on variables of many different types which can perform iterations and recursions to achieve the desired result which can define and subsequently use computed values and sub-programs and whose size shape and complexity is not specified in advanceACO is a recently proposed metaheuristic approach for solving hard combinatorial optimization problems Artificial ants implement a randomized construction heuristic which makes probabilistic decisions The cumulated search experience is taken into account by the adaptation of the pheromone trail ACO Shows great performance with the ldquoill structuredrdquo problems like network routing In ACO Local search is extremely important to obtain good results The field of ACO algorithms is very lively as testified for example by the successful biannual workshop (ANTS ndash From Ant Colonies to Artificial Ants A Series of International Workshops on Ant Algorithms) where researchers meet to discuss the properties of ACO and other ant algorithms both theoretically and experimentally

DEPARTMENT OF ECE


From the theory side researchers are trying either to extend the scope of existing theoretical results or to find principled ways to set parameters values From the experimental side most of the current research is in the direction of increasing the number of problems that are successfully solved by ACO algorithms including real-word industrial applications Currently the great majority of problems attacked by ACO are static and well-defined combinatorial optimization problems that is problems for which all thenecessary information is available and does not change during problem solution For this kind of problems ACO algorithms must compete with very well established algorithms often specialized for the given problem Also very often the role played by local search is extremely important to obtain good results Although rather successful on these problems we believe that ACO algorithms will really evidentiate their strength when they will be systematically applied to ldquoill-structuredrdquo problems for which it is not clear how to apply local search or to highly dynamic domains with only local information available A first step in this direction has already been done with the application to telecommunications networks routing but more research is necessary

DEPARTMENT OF ECE


BIBILIOGRAPHY

Dorigo M and G Di Caro (1999) The Ant Colony Optimization Meta-Heuristic In D Corne M Dorigo and F Glover editors New Ideas in Optimization McGraw-Hill 11-32

M Dorigo and L M Gambardella Ant colonies for the travelling salesman problem Bio Systems 4373ndash81 1997

M Dorigo and L M Gambardella Ant Colony System A cooperative learning approach to the travelling salesman problem IEEE Transactions on Evolutionary Computation 1(1)53ndash66 1997

G Di Caro and M Dorigo Mobile agents for adaptive routing In H El-Rewini editor Proceedings of the 31st International Conference on System Sciences (HICSS-31) pages 74ndash83 IEEE Computer Society Press Los Alamitos CA 1998

M Dorigo V Maniezzo and A Colorni The Ant System An autocatalytic optimizing process Technical Report 91-016 Revised Dipartimento di ElettronicaPolitecnico di Milano Italy 1991

L M Gambardella ` E D Taillard and G Agazzi MACS-VRPTW A multiple ant colony system for vehicle routing problems with time windows In D Corne M Dorigo and F Glover editors New Ideas in Optimization pages 63ndash76 McGraw Hill London UK 1999

L M Gambardella ` E D Taillard and M Dorigo Ant colonies for the quadratic assignment problem Journal of the Operational Research Society50(2)167ndash176 1999

V Maniezzo and A Colorni The Ant System applied to the quadratic assignment problem IEEE Transactions on Data and Knowledge Engineering 11(5)769ndash778 1999

Gambardella L M E Taillard and M Dorigo (1999) Ant Colonies for the Quadratic Assignment Problem Journal of the Operational Research Society 50167-176

Dorigo M Gambardella LM [1997] Ant colonies for the traveling salesman problem Bio Systems 43 73-81

DEPARTMENT OF ECE


DEPARTMENT OF ECE



The main idea is to use the self-organizing principles to coordinate populations of artificial agents that collaborate to solve computational problems Self-organization is a set of dynamical mechanisms whereby structures appear at the global level of a system from interactions among its lower-level components The rules specifying the interactions among the systemrsquos constituent units are executed on the basis of purely local information without reference to the global pattern which is an emergent property of the system rather than a property imposed upon the system by an external ordering influence For example the emerging structures in the case of foraging in ants include spatiotemporally organized networks of pheromone trails

111 GENETIC PROGRMMING

Some specific advantages of genetic programming are that no analytical knowledge is needed and still could get accurate results GP approach does scale with the problem size GP does impose restrictions on how the structure of solutions should be formulated There are several variants of GP some of them are Linear Genetic Programming (LGP) Gene Expression Programming (GEP) Multi Expression Programming (MEP) Cartesian Genetic Programming (CGP) Traceless Genetic Programming (TGP) and Genetic Algorithm for Deriving Software (GADS)Cartesian Genetic Programming was originally developed by Miller and Thomson for the purpose of evolving digital circuits and represents a program as a directed graph One of the benefits of this type of representation is the implicit re-use of nodes in the directed graph Originally CGP used a program topology defined by a rectangular grid of nodes with a user defined number of rows and columns In CGP the genotype is a fixed-length representation and consists of a list of integers which encode the function and connections of each node in the directed graph The genotype is then mapped to an indexed graph that can be executed as a program In CGP there are very large numbers of genotypes that map to identical genotypes due to the presence of a large amount of redundancy Firstly there is node redundancy that is caused by genes associated with nodes that are not part of the connected graph representing the program Another form of redundancy in CGP also present in all other forms of GP is functional redundancy Simon Harding and Ltd introduce computational development using a form of Cartesian Genetic Programming that includes self-modification operationsThe interestingcharacteristic of CGP are 1 More powerful program encoding using graphs than using conventional GP tree-like representations the population of strings are of fixed length whereas their corresponding graphs are of variable length depending on the number of genes in use2 Efficient evaluation derived from the intrinsic feature of sub graph-reuse exhibited by graphs3 Less complicated graph recombination via the crossover and mutation genetic operators

DEPARTMENT OF ECE






Fig 11

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DEPARTMENT OF ECE


12 ANT COLONY


Fig 12

DEPARTMENT OF ECE




Fig 13


DEPARTMENT OF ECE




Fig 14



DEPARTMENT OF ECE


13 ANTS





Fig 15

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CHAPTER 2


21 ANT SYSTEM






DEPARTMENT OF ECE






DEPARTMENT OF ECE





22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE







Fig 11

DEPARTMENT OF ECE





DEPARTMENT OF ECE


12 ANT COLONY


Fig 12

DEPARTMENT OF ECE




Fig 13


DEPARTMENT OF ECE




Fig 14



DEPARTMENT OF ECE


13 ANTS





Fig 15

DEPARTMENT OF ECE


CHAPTER 2


21 ANT SYSTEM






DEPARTMENT OF ECE






DEPARTMENT OF ECE





22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE






DEPARTMENT OF ECE


12 ANT COLONY


Fig 12

DEPARTMENT OF ECE




Fig 13


DEPARTMENT OF ECE




Fig 14



DEPARTMENT OF ECE


13 ANTS





Fig 15

DEPARTMENT OF ECE


CHAPTER 2


21 ANT SYSTEM






DEPARTMENT OF ECE






DEPARTMENT OF ECE





22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



12 ANT COLONY


Fig 12

DEPARTMENT OF ECE




Fig 13


DEPARTMENT OF ECE




Fig 14



DEPARTMENT OF ECE


13 ANTS





Fig 15

DEPARTMENT OF ECE


CHAPTER 2


21 ANT SYSTEM






DEPARTMENT OF ECE






DEPARTMENT OF ECE





22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE





Fig 13


DEPARTMENT OF ECE




Fig 14



DEPARTMENT OF ECE


13 ANTS





Fig 15

DEPARTMENT OF ECE


CHAPTER 2


21 ANT SYSTEM






DEPARTMENT OF ECE






DEPARTMENT OF ECE





22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE





Fig 14



DEPARTMENT OF ECE


13 ANTS





Fig 15

DEPARTMENT OF ECE


CHAPTER 2


21 ANT SYSTEM






DEPARTMENT OF ECE






DEPARTMENT OF ECE





22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



13 ANTS





Fig 15

DEPARTMENT OF ECE


CHAPTER 2


21 ANT SYSTEM






DEPARTMENT OF ECE






DEPARTMENT OF ECE





22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



CHAPTER 2


21 ANT SYSTEM






DEPARTMENT OF ECE






DEPARTMENT OF ECE





22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE







DEPARTMENT OF ECE





22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE






22 ANTS


221 ATTRACTIVENESS


DEPARTMENT OF ECE






DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE







DEPARTMENT OF ECE





DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE






DEPARTMENT OF ECE



DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




DEPARTMENT OF ECE



Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




Fig 21









DEPARTMENT OF ECE









DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE










DEPARTMENT OF ECE








DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE









DEPARTMENT OF ECE















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE
















DEPARTMENT OF ECE







DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE








DEPARTMENT OF ECE



DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




DEPARTMENT OF ECE


CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



CHAPTER 3





DEPARTMENT OF ECE







Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE








Fig 31a Fig31b

DEPARTMENT OF ECE




DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE





DEPARTMENT OF ECE





srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE






srsquo = Improve(s)

while srsquo= s do

s = srsquo

srsquo = Improve(s)

end

return s



DEPARTMENT OF ECE



34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




34 ANT ALGORITHM





DEPARTMENT OF ECE






DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE







DEPARTMENT OF ECE


CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



CHAPTER 4

APPLICATION OF ACO






DEPARTMENT OF ECE



Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




Fig 41


DEPARTMENT OF ECE

A

D

C

B



Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




Fig 42

DEPARTMENT OF ECE

Initialize




For Each Ant



Print Best tour

YES

No

Stoppingcriteria

yes

No

kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE


kiJ






EndEnd



EndEnd





DEPARTMENT OF ECE

1d ij

N ij





416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE






416 TSP HEURISTICS


DEPARTMENT OF ECE

T ij( t )


β

sumk isinallowed k



iquest

iquest




Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE





Fig 43





DEPARTMENT OF ECE

nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE


nnjid


is minimized




QAP Example Fig44

DEPARTMENT OF ECE

biggest flow A - B




ABC

A

B C A B

C

d i j

[ f h k ]n nf i j

C (π )= sumi j=1

n


Π


421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



421 ALGORITHMS






DEPARTMENT OF ECE




Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE





Fig45

DEPARTMENT OF ECE

2

1 343

4 Location



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



1 2 3 4 1 2

- 1 1 2 2

1 - 2 1 3 1 2 - 1 4 2 1 1 -

Fig 46a


DEPARTMENT OF ECE

Distance jid

Frequency hkf



The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




The coupling Matrix









DEPARTMENT OF ECE

Dij=[0 1 2 31 0 4 52 4 0 63 5 6 0


iquestiquest

S=[720 1200 1440 1680660 1100 1320 1540780 1300 1560 1820480 800 960 1120

] s11=f 1sdotd1=720

s34=f 3sdotd4=960

ζ ij=1sij

pijk ( t )= [ τ ij( t ) ]α [ ηij ]

β


β if jisinN i

k

N ik






426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE







426 NETWORK MODEL


DEPARTMENT OF ECE


m

Δτ ijk

Δτ ijk

Δijk= Q

J ψk


0 otherwise



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



Fig 47




428 ALGORITHM



Where

Fig 48a

DEPARTMENT OF ECE

Node1

Node3

Node4

Node7

Node8


3

5 2

46

1

2

ρ=ρold+Δρ

1+Δρ

ρ=ρold

1+Δρ

Δρ=f ( 1age )


Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



Fig 48b





DEPARTMENT OF ECE

2

4

3

7

1

5

8

6



Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




Fig 49






DEPARTMENT OF ECE

Depot Depots

[800-1230][815-930]

[1000-1145]

[800-900]

[1000-1115]

[1100-1130]

[830-1030]

[1015-1145]



Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




Fig 410



DEPARTMENT OF ECE




Single objective

gb

TIMEACS VEIACS




DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE





DEPARTMENT OF ECE


CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



CHAPTER 5

CONCLUSION








based algorithm)

52 DISADVANTAGES






DEPARTMENT OF ECE




53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE





53 SCOPE OF SEMINAR


DEPARTMENT OF ECE



DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE




DEPARTMENT OF ECE


BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



BIBILIOGRAPHY











DEPARTMENT OF ECE


DEPARTMENT OF ECE



DEPARTMENT OF ECE


ant colony optimization algoritham and application

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