A critical study of the linear programming gradient method for optimal design of water supply networks

The linear programming gradient (LPG) method originally proposed by Alperovits and Shamir (1977) and the subsequent publications based on it are briefly reviewed. The optimal solutions obtained by different investigators for a simple two-loop network, originally taken by Alperovits and Shamir as an illustrative example, are compared and it is shown that they are not even the local optimum solutions. The latest modification, introduced by Kessler and Shamir (1989), which incorporates the projected gradient of the objective function, is also briefly reviewed and is shown to lead to a truly local optimum solution as claimed by Kessler and Shamir. It is shown that it is preferable initially to suppress the minimum link flow constraint, necessary for reliability purposes, and introduce it later after obtaining the optimal solution for a branching configuration. The philosophy of the LPG method is analyzed. It is shown that the LPG method is inefficient, at least for optimization of the illustrative two-loop network, as compared to a heuristic method that initially identifies logically good branching configurations for obtaining the optimal solution.