Andrew B. Templeman

THE COMPUTATIONAL COMPLEXITY OF THE PROBLEM OF DETERMINING LEAST CAPITAL COST DESIGNS FOR WATER SUPPLY NETWORKS

Pipes for water supply are manufactured in a set of discrete-sized diameters. This situation introduces significant difficulties to the problem of devising an algorithm for selecting pipe diameters to constitute a water supply network of least capital cost. In this paper, it is shown that, even for the very simplest type of branching network, the problem is one of a mathematical class known as NP-hard; a result which, by implication, applies to the more complex type of network containing loops. This result suggests that research aimed at devising such an algorithm is likely to be unsuccessful, and would be better directed towards developing good approximate solution methods.

A QUANTIFIED ASSESSMENT OF THE RELATIONSHIP BETWEEN THE RELIABILITY AND ENTROPY OF WATER DISTRIBUTION SYSTEMS

Several researchers have suggested that it might be possible to use entropy as a general performance indicator for water distribution systems. It has several advantages over other performance and reliability indices, for example, it is extremely rapid and far easier to calculate than other measures, has minimal data requirements and lends itself to direct incorporation into design optimization frameworks. This paper summarises the first proper attempt to investigate the apparent relationship between the entropy and reliability of water distribution systems. A maximum entropy-constrained approach was used to generate designs for a sample water distribution system which, along with traditional minimum-cost designs, formed the basis of this study. By varying the layout, number of loops and links and reversing the direction of flow in some pipes, it is shown statistically that the correlation between entropy and reliability is strong. Based on the results, a new method for sizing the pipes of water distribution systems is proposed. It is quick, easy to implement, finds optimal pipe sizes, does not require non-linear programming and always guarantees a high level of reliability.

OPTIMUM DESIGN OF FLEXIBLE WATER DISTRIBUTION NETWORKS

Abstract A method for designing flexible water distribution networks is presented. Flexibility is the extent to and ease with which a distribution network can cope with eventualities for which it was not specifically designed. This paper shows that some flexibility can be achieved by maximizing the entropy of the flows. A sample network is considered and designs for various levels of entropy are examined. Several indices including energy and head loss are used to compare the designs. The results suggest that an entropy constraint can reduce the tendency towards implicitly branched configurations characteristic of cost minimization models. A striking feature of the proposed methodology is its apparent ability to produce resilient designs without a substantial increase in cost. The results further highlight some implications for connectivity-based reliability measures and core tree approaches to layout optimization.

Modelling errors, entropy and the hydraulic reliability of water distribution systems

This paper reports on an investigation of the possible influence of modelling errors on the relationship between the entropy and hydraulic reliability of water distribution systems. The errors are due to minor differences between the design optimisation and subsequent simulation models, which lead to small discrepancies between the capacity of the network and the required supply. Pressure-dependent analysis was used for the hydraulic simulations. It is shown that any correlation between the redundancy or undercapacity due to the modelling errors and the hydraulic reliability is insignificant. The results, therefore, provide yet more evidence that the entropy-reliability relationship is strong.

A MAXIMUM ENTROPY APPROACH TO CONSTRAINED NON-LINEAR PROGRAMMING

The paper explores the use of the Shannon (informational) entropy measure and Jayness maximum entropy formalism in the solution of constrained non-linear programming problems. Through a surrogate constraint approach an entropy based update formula for the surrogate multipliers is derived. A numerical example of the method is presented. Some information-theoretic interpretations of mathematical programming are explored. Finally, through the use or surrogate duals the method is extended into an entropy augmented Lagrangean formulation.

Maximum entropy flows for single-source networks

This paper was prompted by growing evidence that Shannons measure of uncertainty can be used as a surrogate reliability measure for water distribution networks. This applies to both reliability assessment and reliability-governed design. Shannons measure, however, is a non-linear function of the network flows. Therefore, the calculation of maximum entropy flows requires non-linear programming. Hence, a simpler, more accessible method would be most useful. This paper presents an alternative and rigorous method for calculating maximum entropy flows for single-source networks. The proposed method does not involve linear or non-linear programming. Also, it is not iterative. Consequently, the method is very efficient. In this paper, the methodology is described, several examples are presented and an algorithm is suggested.

CALCULATING MAXIMUM ENTROPY FLOWS IN MULTI-SOURCE, MULTI-DEMAND NETWORKS

Abstract Previous work has shown how maximum entropy flows in a water distribution network can be calculated by maximizing a nodal entropy function. This requires the use of numerical nonlinear optimization. In the case of single-source networks a much simpler path entropy formulation exists which permits solutions to be obtained quickly by manual calculations not requiring numerical optimization. This paper extends the path entropy formulation to general multi-source, multi-demand networks and develops a simple manual calculation method for maximum entropy flows in these general networks. The method is quick, non-iterative and does not directly involve the use of numerical optimization. Examples are presented and discussed.

Calculating the reliability of single-source networks by the source head method

Abstract This paper proposes two new reliability measures that, for small networks in particular, are easy to understand and interpret and straightforward to calculate. The measures fully recognise the pressure dependency of demand and are fully capable of quantifying partial failure. The methodology is based on a recent parameter referred to as the notional net source head to fully satisfy all nodal demands. An important feature of the present formulation consists of a subtle reversal of the roles of normal and subnormal service, wherein lies the key to the proper quantification of partial failure. The final component of the methodology is a probabilistic analysis which addresses the random nature of component failures or unavailability. The final result is a sufficiently meaningful and accurate reliability measure. Damage tolerance is also characterised and quantified. The efficacy of the reliability and damage tolerance measures are highlighted using a numerical demonstration. Finally, despite the overall simplicity of the proposed approach, it is computationally efficient.

Discrete optimum structural design

Abstract This paper is concerned with some of the difficulties involved in the optimum design of engineering structures using only components which are available in discrete sizes. As an example the optimum design of trusses using rolled steel sections is used to critically examine several different methods and point out the snags in using them in a computer-aided design context. The combinatorial nature of the problem is described and heuristic methods for finding non-rigorous but very close discrete optimum designs are discussed. An investigation of the practical needs of structural designers rather than the capabilities of numerical algorithms demonstrates that the nature of the discrete design problem is in practice very different from that perceived by many researchers. The paper then examines possible ways of solving the practical optimum design problem rather than the problems most often tackled in the literature.

ENTROPY-BASED SYNTHESIS OF PRETENSIONED CABLE NET STRUCTURES

Abstract Cable nets structures exhibit highly non-linear behaviour under applied static loads. Non-linearities are caused by the changes in configuration necessary to equilibrate applied loads and by the slackening and/or yielding of the cables. In order to stiffen such cable net structures and reduce displacements under applied loads the cables are often pretensioned. Such pretensioning, however, requires more substantial cables, stronger connections and stronger supporting structures. It is therefore desirable to be able to design a cable net structure which satisfies all displacement and stress performance criteria with as small a level of pretensioning as possible This paper describes a method which sets the above design problem in a multicriteria optimization context with goals of minimum prestressing force, displacement and stress. A minimax solution is found by means of an entropy-based optimization algorithm. Illustrative examples are solved showing the applicability of the method