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**Contents:**

The Long Term. Preview this book. For instructors This book is not available as an inspection copy. Select a Purchasing Option Paperback Hardcover. ISBN: After, it creates an ant colony to minimize the number of vehicles used lines 3 to In this process, each ant builds a local solution sailing through the adjacent states of the problem. The election of the nodes is done through a pseudo-random selection rule line 6 to The local solution is compared with respect to the best global solution line 18 and the updates are done line The stop conditions of lines 17 and 21 are specified with respect to the number of iterations and time of execution respectively.

A description of equation and techniques used by ACS are given below. Pheromone Update: Global update of the pheromone trail for customers is done over the best solution. It also adds a reward using the increment rs , which is the inverse of the length of the best global solution. Local update Eq. It uses 0 , which is the inverse of the product of the length of the shortest global solution, and the number of visited nodes.

Similar equations are used for the vehicle pheromone trail. Heuristic Information for Customers: The Eq. Heuristic Information for Vehicles: The Eq. Pseudo-random selection rule: An ant k located in node r selects the next node s to move. The selection is made using the pseudo-random selection rule defined in Eq. When a balancing condition is satisfied, the ant selects the best node exploiting the heuristic information and the trails of pheromone.

Otherwise, a proportional random exploration is applied. In this equation, q 0 is a parameter of balancing between exploitation and exploration, q is a random value in [0,1], is the relative importance of the heuristic information, N k r is the set of available nodes for r. When the variability of the cost associated to each path in MST is small, all the nodes form a single conglomerate. Otherwise, it forms conglomerates through a hierarchical grouping.

The value of the heuristic information rs is modified with a factor, which is a ratio of the cardinality of the groups of r and s. A sample of three real instances and its solution is shown in Table 1. The database contains instances classified by date order, and products.

Besides, the algorithm execution takes In contrast, a human expert is dedicated on doing this activity all working day. The Vehicle Routing Problem concerns the transport of items between de-pots and customers by means of a fleet of vehicles. In the VRP, the decisions to be made define the order of the sequence of visits to the customers; they are a set of routes. A route departs from the depot and it is an ordered sequence of visits to be made by a vehicle to the customers, fulfiing their orders. A solution must be verified to be feasible, checking that it does not violate any constraint, such as the one stating that the sum of the demands of the visited vertices shall not exceed the vehicle capacity.

The VRPTW is commonly found in real world applications and is more realistic than the VRP that assumes the complete availability over time of the customers. Each customer requests a given amount of goods, which must be delivered or collected at the customer location. Time intervals during which the customer is served can be specific. These time windows can be single or multiple. A vehicle can not arrive later than a given time, but it can wait if arriving early. In such a case, the goal of the objective function is to minimize the distance travelled and the waiting time.

Salvelsberg, proved that even finding a feasible solution to the VRPTW when the number of vehicles is fid is itself a NP-complete problem. The VRPTW is represented by a set of identical vehicles denoted by K, and a directed graph G, which consist of a set of customers and a depot. The nodes 0 represents the depot. The set of n vertices denoting customers is denoted N. The arc set A denotes all possible connections between the nodes including the node denoting depot. All routes start at node 0 and end at node 0. The travel time t ij includes service time at customer i. Each customer i has a time window, [a i , b i ], where a i and b i are the respective opening time and closing times of i.

No vehicle may arrive past the closure of a given time window, b i. Although a vehicle may arrive early, it must wait until the start of service time a i is possible. It generates a waiting time w k for the route. Vehicles must also leave the depot within the depot time window [a 0 , b 0 ] and must return before or, at time b 0.

These objectives often affect one another in complex, nonlinear ways. The challenge is to find a set of values for them which yields an optimization of the overall problem at hand. Having several objective functions the aim is to find good compromises, rather than a single solution. In , Vilfredo Pareto, an Italian economist, introduced the concept of Pareto dominance Pareto, The Pareto front is composed by a set of non-dominated vectors.

Figure 9 shows a comparison between two individuals of the Pareto Front, where A is less than B in objective function f 1 , but B is less than A in objective function f 2 ; therefore both solution vectors are non-dominated. The Pareto front is particularly useful in engineering: by restricting attention to the set of solutions that are non-dominated, a designer can make tradfs within this set, rather than considering the full range of every parameter.

The three objectives of the VRPTW were presented: total distance travelled, number of vehicles, and total waiting time. Tan, , and Ombuki, , employed the travel distance and the number of vehicles to be minimized. Saadah et al. This work proposes a hybrid particle swarm MOO algorithm that incorporates perturbation operators for keeping diversity in the evolutionary search and the concept of Pareto optimality for solving the VRPTW as a MOO problem with 3 objectives: minimize total travel distance, minimize number of vehicles and minimize total waiting time. Total number of function evaluations is 1,, A well-know problem set proposed by Solomon, is used to test our model.

In these problems, the travel times are equal to the corresponding Euclidean distances. One problems is chosen for the experiment: problem C in dimensions The Pareto front obtained by MO-PSO evidenced the huge waiting time accumulated by the optimal solution reported in the literature with single objective total distance approaches. The vector reports a total distance of As we mentioned above, in this problem, the travel times are equal to the corresponding Euclidean distances. Then, the total waiting time is at least 10 times greater than the total travel time performed for all routes. The total waiting time represents the Therefore, there is no waiting times at the beginning of the route.

For example, the solution vector The solution vector Finally, the last solution vector Although this solution vector increase at least 3 times the total distance and the number of vehicles of the solution vector The empirical results show that reducing the total distance often results in an increment of the total waiting time and vice versa.

This happens when two customers that are geographically close may not necessarily be close from the temporal point of view. In real-world problems, this levels of waiting times are not conforming to standard usage. Thereby, placing more emphasis on the distance travelled will often result in unnecessary waiting time. Different problems related with Combinatorial problem son explained in this section, the first is related with Logistics in special with the Bin Packing Problem, the second with the concept of Negotiation in many societies and finally the analysis of six degree of separation in a Social Networking each one implement different strategies to improve the problem and using from different point of view Culturaal Algorithms CAs.

In the fist problem CAs relies on a communication protocol that makes possible to gain access to the belief space. In relation with the evolutionary component of the cultural algorithm, either a Genetic or an Evolutionary Programming algorithm can be indistinctively used. The cultural algorithm operates at two different levels by means of inheritance: at the belief space at a macro-evolutive manner, whereas at a micro-evolutive sense at the very own population space. As a consequence, having an impact on the development of generation of new individuals through posterior epochs.

The use of five particular kinds of knowledge is employed by the cultural algorithm to find, in the search space, a possible solution for a specific problem. This knowledge can be identified as: normative knowledge acceptable behavior , circumstantial knowledge successful and disastrous experiences , domain knowledge objects and their relation in a given domain , historical knowledge timed behavioral patterns , and topographically knowledge spatial behavioral patterns Reynolds et al. The implementation of the cultural algorithm was carried out in a standard manner as shown in figure The main contribution of the cultural algorithm to the evolutionary methods is the use of a belief space, and its cultural influence in the population for finding adequate solutions for problems that employ this knowledge space.

The prototype is a hybrid intelligent system developed in Matlab 7. We started by measuring available free space in a standard distribution vehicle. Also we measured the various presentations in cubic centimeters and their volume in liters. We also took into account the demand of the different available presentations, in order to determine the income utility. Once the measurements were made, it was necessary to create an algorithm capable of finding the right selection, in terms of the freight, to optimize income utility and reduce costs of transportation equation Then the population and the belief space were initialized.

The demand of the product is calculated based on their volume and income utility Table 3. Next, the initial population is evaluated based on the restrictions shown in the same table. Any violation of the restrictions will be penalized in such a way that only the best combinations will be obtained.

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As a result, an average result is obtained and the best individuals are able to influence the next generation based on average individuals. The resultant epoch is a proposed solution for the problem, the halt condition is reached when seven consecutive epochs reach the same result. Our initial results are based in a cultural algorithm with an initial population of one hundred individuals.

Also, a belief space was created with the same number of individuals. The system was initialized with this population and belief space, and then Variation Operators were applied. The system was iterated until the halt condition was reached producing a nearly optimal solution. The program was used to determine a proper combination of the truckload to maximize profits. The selection of the truckload was based on the volumes of the various water containers and the freight capacity of the truck.

In table 4 we observe that after the epoch fifteen the algorithm found a steady solution after eight epochs without improvement. A comparison with the Simplex Method reveals that still this method performs better at this stage in the implementation of our cultural algorithm. As a result of our research we develop a hybrid intelligent system that employs a cultural algorithm with artificial evolution for the optimization of space in the delivery of purified water.

Here we demonstrated that is feasible to use a cultural algorithm in problems such as the one described here. Our main contribution resides in a real application of the cultural algorithm, which few years ago exist only at the theoretical level. Besides doesn't have any previous heuristic to try to improve the optimization. For better understanding of the selected cultural algorithm used to solve the optimization problem, now we introduce some basic concepts and representations of the artificial culture related to this problem.

These representations are abstraction levels located between the unknown part of the agent , the domain problem dimension and the agents. In the algorithm of cultural change, the space of beliefs beliefspace by means of the best paradigm BestParadigm are set to zero, representing the fact that the culture increases the quantity of pleasure associated with such spaces, giving an incentive to the behavior associated with the best paradigm BestParadigm. The paradigm could represent the best solution for the problem denoted as the best paradigm BestParadigm.

The space of beliefs is the collective representation of the real World. In other words, this is the world as it is interpreted by one culture of the community, where the agents find the way to interact and moral values. The dimension is the real world, which never can be entirely known by the agent. This contains the experimentation cost and on which the agents are able to live when the optimization is improved.

The agents belonging to one community search inside the dimension for the most appropriated place to be developed goal. The culture could then try to lead the behavior of the new generations of agents by means of this best solution. Each agent in the community is leaded by one function that allows it to select the spaces with the lower quantity of anxiety. It can be observed that the satisfaction principle does not affect the strategy for the global resolution of the problem at collective level culture. To the contrary, this links the agent with an autonomous entity.

The culture controls the behavior to be adopted as model, creating a strategy of global action —an ideology- regarding the given problem domain. The agent selects the cell with the minimum anxiety, as the indicated for the space of beliefs Beliefspace adding to this the cultural value, as:. In this research the functions represent the agent-culture interaction and are selected according with the adopted problem. Therefore, we cannot try to establish a mathematical model of how the cultural process occurs in the real world.

Adopting a random function as was shown previously to explain how we insert, into the process, a system of multiple interactions between the agent and the culture. We try to analyze other mathematical representations in our future work. To prove and validate the theoretical concepts previously presented, we developed a cultural algorithm simulator Baharastar.

Initially our intention was only to create an environment able to carry out analysis and experiments. When cultural algorithms are used, it becomes more difficult of understanding the peculiarities proposed for each solution. Each time that the system has a precise answer, the obtained solution can hardly being duplicated exactly.

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This property of the evolutionary algorithms in general and of the cultural algorithms in particular, has been little explored or discussed in the literature. For such purpose, we select 27 societies described in Memory Alpha, and we characterize their behavior using seven base attributes agility, ability to fight, intelligence, forces, stamina, speed and emotional control , those which allowed describe as well to the society as to the individual.

The development of Baharastar is based on our desire of sharing an intuitive understanding about the treatment for a new class of systems, individuals able to possess unexpected creativity, typical characteristic of living entities. Baharastar is shown in the figure 10 , the user has the possibility to choose the starting point and the goal to reach, joined to the places where one can receive an attack by part of the enemy, and the quantity of anxiety associated to each space of the dimension where the societies reside in agents' communities.

Our prototype was developed using JBuilder X platform see Figure We describe the developed experiments using Baharastar which to contribute in the sense of making evident the importance of the creation of a new methodology to prove and to analyze the obtained results. This was not a trivial task, considering the diversity of behaviors of the provided solutions by Baharastar because it resembles more than a descriptive anthropology than a simple software test. In the first experiment, we compared the performance of 27 communities of 50 agents, and on the other hand 27 communities of agents each one.

The associated points to the beginning and goal are shown in the figure The optimal number of steps from the beginning to the goal is One of the most interesting characteristics observed in this experiment is the diversity of cultural patterns established for each community. The structured scenarios associated to the agents cannot be reproduced in general due they belong to a given instant in the time and space. They represent a unique, precise and innovative form of adaptive behavior which solves a computational problem followed by a complex change of relationships.

The generated configurations can be metaphorically related to the knowledge of the community behavior regarding to an optimization problem to make alliances, to defend from a possible invasion , or a tradition with which to emerge from the experience and with which to begin a dynamics of the process. Comparing the 50 agents of the first community regarding the agents community, this last obtained a better performance in terms of the average number of steps from the beginning to the goal They also had a greater average number of changes in the paradigm 5.

In the second experiment, we consider the same scenario for the experiment one, except that after having obtained a solution from a community of 50 agents, we place five near spaces to the goal and we begin with a new community of agents. The new community was informed of the previous cultural configurations but should take into account the new scenario.

The comparison among both solutions is not immediate, from the point of view that try to solve different problems. In this experiment, it was surprising to see initially how the community of agents uses the solution offered by the 50 agents, whenever these solutions were close the optimal grade, instead of finding entirely complete new solutions. These results make evident the conservation of a global action strategy which regulates the agents. As a consequence, the ant colonies are adaptive, complex and distributed multi-agent systems.

Now exist a lot of applications like scheduling, machine learning, data mining Ponce et al. There are several variants of AS designed to solve specific problems or to extend the characteristics of the basic algorithm. The next paragraph describes the most important variants in order of appearance. ACO use two main characteristics: rs and rs. The heuristic information rs is used to measure the predilection to travel between a pair of nodes r,s.

The trails of artificial pheromone rs is used to compute the learned reference of traveling in a determined arc r,s. It is formed by three algorithms: Ant-density, Ant-quantity and Ant-Cycle. Ant-density and Ant-quantity use the update of pheromone trails in every step of the ants, while Ant-Cycle makes updates after a complete cycle of the ant. The study establishes that the optimal number of ants is equivalent to the number of nodes of the problem. Gambardela and Dorigo Gambardela, designed the Ant-Q algorithm, which is based on the principles of Q-learning. Ant-Q uses a table of values Q to indicate how good a determined movement from node r to s is.

It applies a rule to choose the next node to be visited and uses reinforcement learning to update Q with the best tour of the ants. It uses the elements of AS. However, it modifies three aspects: updating rule, pheromone values, and the next movement. The updating rule was modified to choose the best tour in every cycle, increasing the probability of early stagnation.

Maximum and minimum limits for the pheromone trails were established. These limits avoid repeated movements: bounding the influence of the trail intensity, and leading to a higher degree of exploration. Bullnheimer et al. All solutions are ranked according to their fitness. The amount of deposited pheromone is weighted for each solution, such that the best result deposits more pheromone than bad solutions.

It presents three main differences with regard to AS: transition rule, global updating and local updating. The transition-state rule is modified to establish a balance between the exploration of new arcs and the priority exploitation of a problem. The global updating rule is applied only to the arcs of the best ant tour. A local updating of the pheromone is done while ants build a solution. Figure 2 shows a general scheme of the ACS algorithm. It is well known that ACS is one of the best ant algorithms.

ACS has the best performances and the majority of references Asmar, This algorithm was adapted to solve the logistic problem approached in a section 6. In the PSO model, every particle flies over a real valued n-dimensional space of decision variables X. The particle with the best P Best value is called the leader, and its position is called global best, G Best. The next particle's position is computed by adding a velocity term to its current position, as follows:.

The velocity term combines the local information of the particle with global information of the flock, in the following way. The equation above reflects the socially exchanged information. It resumes PSO three main features: distributed control, collective behavior, and local interaction with the environment Eberhart et al.

The second term is called the cognitive component, while the last term is called the social component. The inertia weight indicates how much of the previous motion we want to include in the new one. When the flock is split into several neighborhoods the particle's velocity is computed with respect to its neighbors. The best P Best value in the neighborhood is called the local best, L Best. Neighborhoods can be interconnected in many diferent ways, some of the most popular are shown in Fig. The star topology is, in fact, one big neighborhood where every particle is connected to each other, thus enabling the computation of a global best.

The ring topology allows neighborhoods therefore it is commonly used by the PSO with local best. PSO is a fast algorithm whose natural behavior is to converge to the best explored local optima. However, attainting flock's convergence to the global optimum with high probability implies certain adaptations Garro, The approaches range from modifications to the main PSO equation, to the incorporation of reproduction operators. PSO algorithm has been extended in several directions; in a latter section we show a multiobjective optimization approach to solve a logistic problem, the vehicle routing problem with time windows.

Cultural algorithms were developed by Robert G. Reynolds in as a complement to the evolutionary algorithms, this algorithms are bio-inspired in the Cultural Evolution of the Societies, and were focused mainly on genetic and natural selection concepts Reynolds, Cultural algorithms operate at two forms: 1 a micro-evolutionary form, which consists of the genetic material that an offspring inherits from its parents children, grandsons, greats-grandchild, etc , and 2 a macro-evolutionary level, which consists of the knowledge acquired by the individuals through generations Culture.

This knowledge is used to guide the behavior of the individuals that belong to a certain population. Figure 4 illustrates the basic framework of a cultural algorithm. A cultural algorithm operate in two spaces: Population Space and Belief Space. Cultural Algorithms uses the culture like a vehicle to store accessible relevant information to the population's members during many generations, and were developed to model the evolution of the cultural component over time and to demonstrate how this learns and acquires knowledge.

The Population Space are evaluated with a performance function obj , and an acceptance function accept can help to determine which individuals are introduce in the Belief Space. Next, the beliefs are used to influence the evolution of the population. New individuals are generated under the influence of beliefs in the time that were crates. The two feedback paths of information, one through the accept and influence functions, and the other through individual experience and the obj function create a system of dual inheritance of both population and belief.

Normative knowledge A collection of desirable value ranges for the individuals in the population eg. Domain specific knowledge Information about the problem domain where cultural algorithms are applied. Situational knowledge Specific examples of important events - eg. Temporal knowledge History of the search space - eg. The use of collective intelligence is found in different areas ranging from biology, sociology, and business to computer science. The aforementioned entities range from bacteria, animals, human to computer processes.

Furthermore, the same definition can be applied, in a broad sense, to action selection and evolutionary robotics. Evolutionary Robotics employs a quasi-optimal approach to develop autonomous controllers for different kinds of robots.

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The use of genetic algorithms and neural networks are natural candidates, as the preferred methodology, for developing single evolved neural controllers. These controllers are the result of testing populations of adapted individuals during a refinement process through series of computer-program iterations.

Next, pairs or groups of individuals can be evolved together. Following this approach a change in the evolution of one individual can be affected by the change of other related individuals in the group. The latter approach has been identified, as its biological counterpart, as co-evolution that can be cooperative or competitive. A cooperative strategy can be developed to achieve a common task e. In biology diffuse co-evolution has been referred to species evolving in response to a number of other species, which in turn are also evolving in response to a set of species.

Consequently, the identification of collective intelligence is more evident when coevolving cooperative groups. The development of collective behavior in simulation can be achieved by simple scalable control systems as a form of decentralized control. Therefore, the work of Kube, exhibited group behavior without the use of explicit communication.

Elementary communication skills were evolved alongside the abilities to reach target areas. In this way, coordination is achieved through stigmergy; nevertheless more elaborated communication can be implemented Mitri, On the whole, the individual ability to communicate may be one of the requirements for the development of collective intelligence. The evolutionary approach has been sufficient to solve problems were cooperation and communication skills are necessary for solving a particular task; in contrast communication arises as a deceptive feature within a competitive scenario.

In this scenario a common setting is that of the prey and the predator where both individuals are competing for scoring points either for escaping or capturing each other. The experiment can be expanded to add more preys and predators. The action selection problem is related, to the behavior-based approach, and particularly to decision-making when a module takes control of the available actuators until is completed or proves ineffective.

In the vertebrate brain, at specific loci, specialized centers of selection can be identified. One of them are the basal ganglia, and recent works support the idea of these nuclei playing an important role in action selection Prescott et al. The basal ganglia act as a relay station in the planning and the execution of movements behavior ; hence gathering information from the cortex and motor cortex.

The basal ganglia are able to mediate cognitive and muscular processes. Not only the basal ganglia serves as an important center of action selection; in cooperation with the cerebellum and the sensory cerebrum, all them are able to veto muscular contraction by denying the motor areas sufficient activation. In turn, these individual motor elements form more complex patterns, which can be thought as essential elements in the development of intelligence. The development of intrinsic basal ganglia circuitry with evolvable behavioral modules has already been implemented Montes et al.

Cooperative individuals not only require a society interaction, but the existence of an internal mechanism e. Nonetheless, these sensory processes need to be augmented when possible. Therefore, individuals need to build up unified internal perceptions based on their available sensory capabilities in order to produce specialized behavior. The work of Montes et al. The emergence of collective intelligence based on the behavior-based approach requires stepping out from the modeling of selfish solitary individuals to social organisms.

Therefore, we need to group our robots and expose them to numerous interactions to assure complex performances at the level of the group. Here, we have identified some common elements in collective robotics: cooperation, intelligence, communication skills, and the integration of sensory information with action selection.

All of those accomplished with the use of the evolutionary robotics approach. As a consequence, we believe that we may contribute to the development of robotic collective intelligence by way of social experiments using the artificial evolutionary method. Many applications inspired in evolving compute have a great value in Logistics. In this section, we present five in special to compare their contributions, the first is related with Ant Colony in Logistic 6. Many manufacturing companies need merchandise delivered with the minimum quantity of resources and in due time.

However, this is a high-level complex problem. This type of word problem, named recently rich problem, includes several NP-hard sub-problems with multiple interrelated variants. To contribute to this area we approach the bottled-products distribution in a Mexican company. Our proposal includes an innovative ACS solution. RoSLoP involves three tasks: routing, scheduling and loading.

Previous work approached separately at the most, three variants of BPP Chan et al. To overcome these limitations, our research tries simultaneously with eleven variants of VRP Cruz et al. In order to generalize this problem, RoSLoP is formulated as follows: Given a set of customers with a demand to be satisfied, a set of depots that are able to supply them, and a set of BPP and VRP variants that restrict them, the routes, schedules and loads for vehicles needs to be designed.

The Customer demands must be completely satisfied, so the total cost is minimized and the constraints are satisfied. The assignment of routes and schedules is solved by an ACS algorithm. Three elements complement this algorithm: an auto-adaptive constrained list; an initial search, implemented with the Nearest Neighborhood; and a local search, implemented with 3-opt and Cross-Exchange.

The loads are assigned by DiPro algorithm Cruz et al.

The ACS algorithm in Fig. After, it creates an ant colony to minimize the number of vehicles used lines 3 to In this process, each ant builds a local solution sailing through the adjacent states of the problem. The election of the nodes is done through a pseudo-random selection rule line 6 to The local solution is compared with respect to the best global solution line 18 and the updates are done line The stop conditions of lines 17 and 21 are specified with respect to the number of iterations and time of execution respectively. A description of equation and techniques used by ACS are given below.

Pheromone Update: Global update of the pheromone trail for customers is done over the best solution. It also adds a reward using the increment rs , which is the inverse of the length of the best global solution. Local update Eq. It uses 0 , which is the inverse of the product of the length of the shortest global solution, and the number of visited nodes.

Similar equations are used for the vehicle pheromone trail. Heuristic Information for Customers: The Eq. Heuristic Information for Vehicles: The Eq. Pseudo-random selection rule: An ant k located in node r selects the next node s to move. The selection is made using the pseudo-random selection rule defined in Eq.

When a balancing condition is satisfied, the ant selects the best node exploiting the heuristic information and the trails of pheromone. Otherwise, a proportional random exploration is applied. In this equation, q 0 is a parameter of balancing between exploitation and exploration, q is a random value in [0,1], is the relative importance of the heuristic information, N k r is the set of available nodes for r.

When the variability of the cost associated to each path in MST is small, all the nodes form a single conglomerate. Otherwise, it forms conglomerates through a hierarchical grouping. The value of the heuristic information rs is modified with a factor, which is a ratio of the cardinality of the groups of r and s. A sample of three real instances and its solution is shown in Table 1. The database contains instances classified by date order, and products. Besides, the algorithm execution takes In contrast, a human expert is dedicated on doing this activity all working day.

The Vehicle Routing Problem concerns the transport of items between de-pots and customers by means of a fleet of vehicles. In the VRP, the decisions to be made define the order of the sequence of visits to the customers; they are a set of routes. A route departs from the depot and it is an ordered sequence of visits to be made by a vehicle to the customers, fulfiing their orders.

A solution must be verified to be feasible, checking that it does not violate any constraint, such as the one stating that the sum of the demands of the visited vertices shall not exceed the vehicle capacity. The VRPTW is commonly found in real world applications and is more realistic than the VRP that assumes the complete availability over time of the customers. Each customer requests a given amount of goods, which must be delivered or collected at the customer location.

Time intervals during which the customer is served can be specific. These time windows can be single or multiple. A vehicle can not arrive later than a given time, but it can wait if arriving early. In such a case, the goal of the objective function is to minimize the distance travelled and the waiting time. Salvelsberg, proved that even finding a feasible solution to the VRPTW when the number of vehicles is fid is itself a NP-complete problem. The VRPTW is represented by a set of identical vehicles denoted by K, and a directed graph G, which consist of a set of customers and a depot.

The nodes 0 represents the depot.