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Through a collection of cooperative agents called ants, the near optimal solution to the multi-reservoir operation problem may be effectively achieved employing Ant Colony Optimization Algorithms (ACOAs). The problem is approached by considering a finite operating horizon, classifying the possible releases from the reservoir(s) into pre-determined intervals, and projecting the problem on a graph. By defining an optimality criterion, the combination of desirable releases from the reservoirs or operating policy is determined. To minimize the possibility of premature convergence to a local optimum, a combination of Pheromone Re-Initiation (PRI) and Partial Path Replacement (PPR) mechanisms are presented and their effects have been tested in a benchmark, nonlinear, and multimodal mathematical function. The finalized model is then applied to develop an optimum operating policy for a single reservoir and a benchmark four-reservoir operation problem. Integration of these mechanisms improves the final result, as well as initial and final rate of convergence. In the benchmark Ackley function minimization problem, after 410 iterations, PRI mechanism improved the final solution by 97 percent and the combination of PRI and PPR mechanisms reduced final result to global optimum. As expected in the single-reservoir problem, with a continuous search space, a nonlinear programming (NLP) approach performed better than ACOAs employing a discretized search space on the decision variable (reservoir release). As the complexity of the system increases, the definition of an appropriate heuristic function becomes more and more difficult this may provide wrong initial sight or vision to the ants. By assigning a minimum weight to the exploitation term in a transition rule, the best result is obtained. In a benchmark 4-reservoir problem, a very low standard deviation is achieved for 10 different runs and it is considered as an indication of low diversity of the results. In 2 out of 10 runs, the global optimal solution is obtained, where in the other 8 runs results are as close as 99.8 percent of the global solution. Results and execution time compare well with those of well developed genetic algorithms (GAs).

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This paper presents the application of an iterative penalty method for the design of water distribution pipe networks. The optimal design of pipe networks is first recasted into an unconstrained minimization problem via the use of the penalty method, which is then solved by a global mathematical optimization tool. The difficulty of using a trial and error procedure to select the proper value of the penalty parameter is overcome by an iterative use of the penalty parameter. The proposed method reduces the original problem with a priori unknown penalty parameter to a series of similar optimization problems with known and increasing value of the penalty parameters. An iterative use of the penalty parameter is then implemented and its effect on the final solution is investigated. Two different methods of fitting, namely least squares and cubic splines, are used to continuously approximate the discrete pipe cost function and are tested by numerical examples. The method is applied to some benchmark examples and the results are compared with other global optimization approaches. The proposed method is shown to be comparable to existing global optimization methods.