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A method called linear programing gradient (LPG) is presented, by which the optimal design of a water distribution system can be obtained. The system is a pipeline network, which delivers known demands from sources to consumers and may contain pumps, valves, and reservoirs. Operation of the system under each of a set of demand loadings is considered explicitly in the optimization. The decision variables thus include design parameters, i.e., pipe diameters, pump capacities and reservoir elevations, and operational parameters, i.e., the pumps to be operated and the valve settings for each of the loading conditions. The objective function, to be minimized, reflects the overall cost capital plus present value of operating costs. The constraints are that demands are to be met and pressures at selected nodes in the network are to be within specified limits. The solution is obtained via a hierarchial decomposition of the optimization problem. The primary variables are the flows in the network. For each flow distribution the other decision variables are optimized by linear programing. Postoptimality analysis of the linear program provides the information necessary to compute the gradient of the total cost with respect to changes in the flow distribution. The gradient is used to change the flows so that a (local) optimum is approached. The method was implemented in a computer program. Solved examples are presented.
Alperovits et al. (Thu,) studied this question.