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.

Following a companion paper on analytical methods, this paper presents simulation as a complementary method for analyzing the reliability of water distribution networks. For this simulation, the distribution system is modeled as a network whose pipes and pumps are subject to failure. Nodes are targeted to receive a given supply at a given head. If this head is not attainable, supply at the node is reduced. Pumps and pipes fail randomly, according to probability distributions with userspecified parameters. Several reliability measures are estimated with this simulation. Confidence intervals are also supplied for some of these reliability measures. Simulation results are presented for a small network (ten nodes) and a larger network (sixteen nodes). Simulation enables computation of a much broader class of reliability measures than do analytical methods, but it requires considerably more computer time and its results are less easy to generalize. It is therefore recommended that analytical and simulation methods be used together when assessing the reliability of a system arid considering improvements.

A procedure is described that uses the history of main breaks to forecast how the number of breaks would change with time if the pipe were not replaced; a separate analysis predicts the failure rate of newly installed pipes. These forecasts are combined with cost data and a discount rate that accounts for inflation to determine the optimal replacement date. 248 MANAGEMENT AND OPERATIONS

Probabilistic reliability measures for the performance of water distribution networks are developed and analytical methods for their computation explained. The paper begins with a review of reliability considerations and measures for water supply systems, making use of similar notions in other fields. It classifies reliability analyses according to the level of detail with which the water system is modeled, and then concentrates on methods relevant to networks. Two probabilistic measures, reachability (connection of a specific demand node to at least one source) and connectivity, are explored for use in water distribution systems. Two algorithms for their computation are presented, one for series-parallel networks and one for general networks. These measures are computed for two systems, each with ten nodes. Additionally, the probability that a given point receives sufficient supply is proposed for use as a reliability measure. For the calculation of this measure, an algorithm is provided that combines a capacitated network algorithm with a method to efficiently search through network configurations involving multiple link failures. This measure is calculated for the two sample systems.

Optimal design of a water distribution network is formulated as a two‐stage decomposition model. The master (outer) problem is nonsmooth and nonconvex, while the inner problem is linear. A semi‐infinite linear dual problem is presented, and an equivalent finite linear problem is developed. The overall design problem is solved globally by a branch and bound algorithm, using nonsmooth optimization and duality theory. The algorithm stops with a solution and a global bound, such that the difference between this bound and the true global optimum is within a prescribed tolerance. The algorithm has been programmed and applied to a number of examples from the literature. The results demonstrate its superiority over previous methods.

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A theoretical analysis of the linear programming (LP) gradient method for optimal design of water distribution networks is presented. The method was first proposed by A. Alperovits and U. Shamir (1977) and has received much attention in the last 10 years. It consists of two stages that are solved in alteration: (1) a LP problem is solved for a given feasible flow distribution and (2) a search is conducted in the space of flow variables, based on the gradient of the objective function (GOF). In this paper a matrix formulation is given for both stages using well-known graph theory matrices. It is proven that the mathematical expression of the GOF is independent of the choice of the sets of loops and paths along which the head constraints are formulated. This is contrary to the claim made by I. C. Goulter et al. (1986). The original GOF expression is shown to have been an approximation of the steepest direction, but still gives good results. Finally, the search procedure is improved by using the projected gradient method.

Optimal annual operation of a coastal aquifer is determined by using a multiple objective linear programing model based on a multicell model ot the aquifer and a network representation of the hydraulic distribution system. The decision variables are pumping and/or recharge quantities in each cell. Four objective functions are based on (1) a desired groundwater surface map, (2) a desired location of the sea water-fresh water interface toe in each coastal cell, (3) a desired concentration map of a selected conservative contaminant, and (4) minimization of the energy for pumping and recharge. An approximate linearized expression of the location of the interface has been developed to enable the use of linear programing as the optimization method. A trade-off procedure is employed for identifying the most desirable solution. The model is applied to a segment of the coastal aquifer in Israel (a 44-km strip along the coast with a width of 7 to 15 km) and results are discussed. ally 290 x 10 6 m3/yr. Artificial recharge, which adds another 80 X 10 6 m3/yr consists of imported water, flood waters, and some reclaimed sewage, infiltrated through spreading basins and through wells. The water imported to the region is from the Sea of Galilee via the National Water Carrier and from another main aquifer, a Turonian limestone formation located some distance to the east. Return flow to the saturated zone of the coastal aquifer from water applied at the ground surface, primarily for irrigation, adds some 80 x 10 6 m3/yr, while about 100 x 10 6 m3/yr of fresh water flows to the sea. Present pumping from the aquifer is about 410 x 106 m3/yr, resulting in an average annual deficit of 60 x 10 6 m3/yr in the aquifer's water balance.Presently, the sea water-fresh water interface toe reaches inland a distance which varies along the coast and ranges between 250 m and 1900 m. Unless equilibrium has been reached, the interface continues its advance. The location and motion of the interface are of considerable importance in the management of the aquifer because of the strong relationship between the fresh water discharge to the sea and the length of sea water intrusion. 435