New formulation of Ant Colony Optimization Algorithm for optimal design of sanitary pumped sewer network

In this paper, a heuristic single phase method with an ad-hoc engineering concept is proposed for simultaneous optimal layout and size design of sanitary pumped/gravitational sewer network using new the formulation of Ant Colony Optimization Algorithm named Arc based Ant Colony Optimization Algorithm (ABACOA). The ABACOA formulation has two significant advantages of efficient implementation of the exploration and exploitation features and also an easy and straightforward definition of the heuristic information, a useful component of the Ant Colony Optimization Algorithm (ACOA), for the ants over the usual form of the ACOA. To solve optimization problem using ACOA, the first and main step is the problem graph definition. Definition a suitable graph for the problem will allow for all feature of the ACOA to be presented. The problem graph is represented by defining decision points, options available at each decision point and the costs associated with each of these options. In the proposed formulation, upstream and downstream nodal elevations of sewer network pipes are taken as decision variables of the problem. With considering the upstream and downstream nodal elevations of the sewer network pipes as the decision variables of the problem, in this formulation, the sewer network pipes are decision points of the problem leading to an easy definition of a new form of the graph. Here, to define the problem graph, the allowable range of the decision variables is discretised into a fixed number of discrete cover depths leading to discrete nodal elevations. The options available at each decision point are defined by the aggregation of all the arcs joining the upstream and downstream nodal elevations of each sewer network pipe. By determination of upstream and downstream nodal elevations of sewer network pipes using ABACOA, a heuristic single phase method with an ad-hoc engineering concept is proposed for layout determination using known pipe slopes as follows. At first, flow direction of each sewer network pipe is defined from the node of higher elevation to the node of lower elevation. It should be noted that, for pipes with nodes of the same elevation, the flow direction is decided upon randomly. Second, by starting from an arbitrary node, each node is checked in turn for the number of leaving pipes. If more than one pipe leaves from the node, the pipe with the steepest slope is considered as the main sewer network pipe and the other pipes should be cut from the network at either end to construct the tree structure for sewer network layout. Finally, a dummy node should be used to define the cut end emphasizing on the existence of the cut. If no pipe leaves from the underlying node, the constructed layout is considered as an infeasible solution. This approach is continued until all nodes are covered leading to construct a directed tree-like layout for sewer network. In order to complete the layout construction, the location of the resulting dummy nodes and corresponding nodal cover depths should be defined. In the proposed method, the cover depths of the dummy nodes are taken equal to those of the adjacency nodes in which it requires that the pipes are cut at the end with higher nodal elevation leading to construct tree layout for sewer network. By determining the nodal cover depths, nodal elevations, pipe slopes and tree-like layout of sewer network, the sewer network pipes diameters should be determined to complete the design process. In the proposed method, the pipe diameters are calculated explicitly such that all the constraints are fully satisfied, if possible. Therefore, by starting the design processes from inlets, the smallest commercially available pipe diameter fully satisfying problem constraints, is taken as sewer pipe diameter. Proposed method is used to solve two hypothetical test examples and the results are presented and compared with those of obtained with using usual form of the ACOA for determination of the upstream and downstream nodal elevations of the sewer network pipes and a heuristic method with ad-hoc engineering concept for layout determination. The results indicate the effectiveness and efficiency of the proposed method to solve the problem of simultaneous optimal layout and size design of sanitary pumped/gravitational sewer network. In other words, while both formulations show good performance for solving the problem of simultaneous optimal layout and size design of sanitary pumped/gravitational sewer network, the proposed ABACOA formulation is shown to produce better results with the same computational effort.