The design of a water network involves the selection of pipe diameters that satisfy pressure and flow requirements while considering cost. A variety of design approaches can be used to optimize for hydraulic performance or reduce costs. To help designers select an appropriate approach in the context of gravity-driven water networks (GDWNs), this work assesses three cost-minimization algorithms on six moderate-scale GDWN test cases. Two algorithms, a backtracking algorithm and a genetic algorithm, use a set of discrete pipe diameters, while a new calculus-based algorithm produces a continuous-diameter solution which is mapped onto a discrete-diameter set. The backtracking algorithm finds the global optimum for all but the largest of cases tested, for which its long runtime makes it an infeasible option. The calculus-based algorithm's discrete-diameter solution produced slightly higher-cost results but was more scalable to larger network cases. Furthermore, the new calculus-based algorithm's continuous-diameter and mapped solutions provided lower and upper bounds, respectively, on the discrete-diameter global optimum cost, where the mapped solutions were typically within one diameter size of the global optimum. The genetic algorithm produced solutions even closer to the global optimum with consistently short run times, although slightly higher solution costs were seen for the larger network cases tested. The results of this study highlight the advantages and weaknesses of each GDWN design method including closeness to the global optimum, the ability to prune the solution space of infeasible and suboptimal candidates without missing the global optimum, and algorithm run time. We also extend an existing closed-form model of Jones (2011) to include minor losses and a more comprehensive two-part cost model, which realistically applies to pipe sizes that span a broad range typical of GDWNs of interest in this work, and for smooth and commercial steel roughness values.

A gravity-driven water network (GDWN) is commonly used to deliver potable water from a source at a high elevation, such as a natural spring or reservoir, to households or public tap stands (Fig. 1). When feasible, gravity-driven water networks are attractive in comparison to pumped networks because of their simplicity and lower capital, operational, and maintenance costs. In addition, in many locations where GDWN are considered, there may be little or no access to reliable grid-based electrical power for pumps. To improve reliability, networks may be designed with loops or multiple water sources, although often material cost considerations restrict attention to single-source branched networks.

Water networks are modeled as a collection of nodes, each representing a point of water demand or supply, which are connected with links representing pipes. The geometrical layout of the site is known and fixed, including water source and demand locations and elevations of all nodes. For the present work, design flow rates are determined from community survey data, which are extrapolated for future population growth. Networks in this category are referred to as “demand-driven” designs. Bhave (1978, 1983) refers to these as “Q-specified” designs. Thus, to design a network of this type, pipe diameters for each link must be chosen such that acceptable but arbitrary minimum (positive) pressure heads are maintained at each node given the design flow rate at the node. Furthermore, application of the energy equation to each link in the network demonstrates that the design problem is nonunique; i.e., choosing different pressure heads at the nodes will result in a different pipe diameter solution for the network, and thus different network costs. Minimizing network cost will produce a unique solution to the design problem, i.e., unique link diameters and nodal pressure heads.

In practice, gravity-driven water networks are commonly designed by a marching method, where diameters for each link of the network are chosen sequentially. After selecting a reasonable diameter for each link, the designer calculates the pressure head at the link outlet, and proceeds to the next link if this result is acceptable. In this way, the designer marches through the network until all pipe diameters have been selected. This method produces a feasible solution, but not a cost optimized one. As noted by Bhave (2003), cost savings of 20–30 % can result from the use of optimization techniques. In developing regions, the cost of a water network can be prohibitive, adding to the importance of optimizing network design.

Within the provided framework, the global optimum can be found through an exhaustive search of the solution space, known as complete enumeration, although this is infeasible when considering networks with many links and diameter choices (Kadu et al., 2008; González-Cebollada et al., 2011). To reduce the computational time required by enumeration, authors have proposed various partial enumeration methods which prune the search space (Kadu et al., 2008), although some of these techniques may remove the global optimum (Simpson et al., 1994). The most common types of algorithms that have been applied to optimize water network design include traditional deterministic methods, heuristic methods, metaheuristic methods, multi-objective methods, and decomposition methods (Zhao et al., 2016).

Element schematic of a GDWN.

Deterministic methods include linear programming (LP), dynamic programming, and nonlinear programming (NLP), and typically involve rigorous mathematical approaches (Zhao et al., 2016). A brief overview and comparison of these algorithms is given in Kansal et al. (1996), who use a single-part cost correlation for metric pipe diameters between 100 and 350 mm. Linear programming techniques have relatively low computational complexity and allow each link to be composed of two diameters, called a split-pipe solution, although these may not always be practical to implement (Bhave, 1983; Kessler and Shamir, 1989; Swamee and Sharma, 2000; Samani and Mottaghi, 2006). LP can also get stuck in a local optimum (Zhao et al., 2016), although combining LP with metaheuristic techniques can help with the problem's non-smoothness properties (Krapivka and Ostfeld, 2009). Dynamic programming has been used by Yang et al. (1975) and Martin (1980) to optimize networks in stages. This approach begins at the discharge nodes, proceeding to select feasible diameters and joints for upstream stages and storing these partial candidates in memory until the source node is reached. At this point, the algorithm reviews the feasible segment design options and selects a combination of stage solutions producing the lowest cost overall solution. This method, however, requires the designer to allow a relatively narrow range for the design pressure of each node, or otherwise store a large set of feasible candidate solutions in memory and also allow adjoining branches to arrive at different heads at the same node.

Nonlinear programming, a calculus-based method, deals with each link's diameter as a continuous variable. Using Lagrange multipliers and a one-part, pipe-cost model with minor-lossless flow, Swamee and Sharma (2000) developed systems of equations for both continuous and discrete pipe diameters for branch networks, assuming a constant friction factor. When solved, the solution gives diameter values that minimize distribution main cost, not network cost. In carrying out the solution, iteration is required to update the value of the friction factor. For the discrete diameter case, large computational times were noted by Swamee and Sharma because of the stiffness of the mathematical system. Cases where one or more nodal pressure heads are not acceptable need to be treated manually by the designer in various ways as discussed by the authors. For branching networks, Jones (2011) showed that by restricting the focus to smooth-turbulent (turbulent flow in a smooth pipe) minor-lossless flow, and the use of a one-part, pipe-cost model, a simple nonlinear algebraic equation for each internal node in the distribution main could be developed. The development of this algorithm, as well as solution methodology, differs from that of Bhave (1978), which assumes constancy in several terms and thus requires iteration to solve. The Jones algorithm has been extended in the present work to include minor losses and rough pipe. When solved simultaneously with the energy equation for each link, a unique solution for all link diameters and nodal pressure head values is obtained which produces minimum network cost, as opposed to the distribution main cost as in Swamee and Sharma (2000). The method of Jones also applies to serial and loop networks because of its generality.

Heuristic methods follow specific rules to incrementally build better solutions, although the rules are not strictly formulated to trend towards local or global optima. An approach by Monbaliu et al. (1990) sets all network pipes to their minimum size, where the pipe that has a maximum head loss gradient is incremented to its next-highest size until all nodal head requirements are satisfied. Similarly, an algorithm by Keedwell and Khu (2006) selects an initial solution and iteratively responds to nodal head deficits and surpluses by incrementing or decrementing pipe sizes accordingly until a feasible solution is found. Suribabu (2012) proposed a heuristic that identifies pipes to increment or decrement in size based on flow velocity and alternative metrics such as proximity to the source node, achieving acceptable cost results with computational efficiency. While these algorithms are typically computationally efficient, they do not guarantee a global optimum.

Metaheuristic optimization methods allow for a set of solutions to evolve through random processes that are guided with an objective function which rewards low network costs and penalizes hydraulic insufficiencies. Examples include evolutionary algorithms, which are most commonly genetic algorithms (Krapivka and Ostfeld, 2009; Simpson et al., 1994; Kadu et al., 2008; Prasad and Park, 2004), simulated annealing (Vasan and Simonovic, 2010; Tospornsampan et al., 2007), ant colony optimization (Maier et al., 2003), and differential evolution (Vasan and Simonovic, 2010). As reviewed by Nicklow et al. (2010), evolutionary algorithms are an emerging popular alternative to the deterministic methods, and they offer the opportunity to accommodate unique constraints and multiple design objectives. The main challenges for evolutionary algorithms are the difficulty of incorporating constraints into objective functions, the optimum selection of parameters, and a relatively large amount of computational effort. In addition to optimizing for cost, multi-objective methods, often based on evolutionary algorithms, allow the designer to choose from a Pareto optimal front of objectives, such as cost and reliability (Prasad and Park, 2004). In addition to water network design, metaheuristic algorithms have been used for a range of problems in water resources engineering, such as rainfall and runoff modeling (Taormina and Chau, 2015).

Decomposition methods involve the partitioning of networks into smaller sub-networks which are each optimized using one of many types of techniques and then combined into an overall solution. In some cases, the loops in the sub-networks are removed, producing branching trees which are then optimized individually. Techniques used to optimize the sub-networks can involve multiple methods, including linear programming (Saldarriaga et al., 2013) and differential evolution (Zheng et al., 2013), with a later stage optimizing the network as a whole using the sub-network solutions as inputs. Note that another distinct use of the term “decomposition” refers to the approach of iteratively solving “inner” and “outer” mathematical problem formulations, and has been used in the literature by Krapivka and Ostfeld (2009) who traces its use in this context back to Alperovits and Shamir (1977).

In the present study, we present three algorithms, each from one of three major categories of methods applied to the cost optimization of water distribution networks, and compare their performance on five cases adapted from real GDWNs. These algorithms include (1) the calculus-based (CB) optimization model of Jones (2011), an NLP method; (2) backtracking (BT), a partial enumeration method; and (3) a genetic algorithm (GA), a metaheuristic method. Major distinguishing features of these algorithms include their working use of continuous diameters (CB) versus discrete diameters (BT and GA), their deterministic (CB and BT) versus stochastic (GA) search process, and their relative scalability as better (CB, GA) and worse (BT) for larger networks. In terms of their ability to find a global optimum solution for the problem formulation, CB finds a global optimum for continuous diameters but cannot guarantee a discrete diameter global optimum in its mapped solution, BT can guarantee a discrete global optimum, and GA cannot guarantee an optimum. For a direct comparison of techniques, the pipe costs used for all algorithms are found by interpolating a two-part cost formula based on a curve fit of real cost data for available diameter values. The three algorithms are tested against networks adapted from field data on five actual GDWNs installed in Panama, Nicaragua, and the Philippines.

Within the broader context of water network problem formulations, this paper is concerned with finding cost-optimal single-diameter solutions to branching water distribution networks with steady-state demand flows and pre-specified pipe locations. By implication of being gravity-driven, the problem does not involve the use of pumping stations. This problem formulation is directly applicable to typical gravity-driven water networks, and is also useful for multi-objective algorithms, the consideration of sub-networks in a decomposition technique, pumped networks, and looped system optimization, which can involve reformulating the problem into a branching configuration.

The results of this study highlight the advantages and weaknesses of each GDWN design method including closeness to the global optimum, the ability to prune the solution space of infeasible and suboptimal candidates without missing the global optimum, and also computational time. We present two pre-processors which discrete-diameter search methods can use to reduce the search space without pruning the global optimum. To the authors' knowledge, this is the first implementation of “pre-processor 1” in enumeration methods and the first implementation of “pre-processor 2” in any water network design method. We also extend the Jones closed-form model to include minor losses, a more comprehensive two-part cost model, which realistically applies to pipe sizes that span a broad range typical of GDWNs of interest in this work, and for smooth and commercial steel roughness values.

Branching networks are considered (Fig. 1), where all branches connect a
distribution main node with a delivery node, shown as tap stands or houses.
For each link in a network of

For all nodes, pressure head,

The pressure upper bound is not incorporated into the optimization process. Worst-case pressure conditions occur under hydrostatic conditions, which are directly related to the maximum elevation change in the network and where no flow occurs. Therefore, before the optimization process is undertaken, the selections of appropriate pressure ratings for the pipe and, if needed, break-pressure tanks are left to the correct judgment of the designer under no-flow conditions. In addition, precautions against water hammer are left to the designer.

In this section we develop a new calculus-based algorithm for pipe diameters
that minimize overall pipe cost for the network. First appearing in Jones (2011), this algorithm is solved simultaneously with the energy
equation for each link to produce unique solutions for

We assume continuous pipe diameters in this section; values that result from the solution of the energy equation. Mapping between continuous diameters and the discrete nominal sizes, required to complete the design, will be addressed below.

Consider the three-pipe network shown in Fig. 2. Pipes 1–2, 2–3, and 2–4
meet where head

Three-pipe branch network.

To facilitate insight, we at first assume turbulent flow (which can be
verified post-calculation if necessary) in smooth pipe and that minor losses
are negligible. Two sources for the friction factor for smooth-turbulent
flow are considered, namely the classical Blasius equation (reported in
Streeter et al., 1998),

With Eq. (4) the general expression for the total cost for the pipe
material,

The mathematical basis for a unique solution for

The cost

The derivatives like

In Eq. (15) the indices

Minor losses using the equivalent-length method can be included in the above
developments by artificially extending the length of the link by

With the inclusion of the two-part cost model and minor loss term, Eq. (15)
becomes

PVC pipe cost from 2011 data.

Equation (18), and its simpler form Eq. (15) for minor-lossless flow and a
single-part pipe-cost model (it is easy to show that Eq. (18) regresses to
Eq. (15) for these conditions), is the root of the calculus-based
optimization in this work and is applied at all internal nodes to uniquely
determine

Bhave (1978) first proposed an algorithm like Eq. (15) using slightly
different notation than here. For clarity, we re-present Eq. (15) using
Bhave's notation as

Bhave (1978) index notation at an internal node,

Backtracking (BT) and genetic algorithm (GA) assess candidate solutions
composed of discrete diameters from a commercially available set. These
candidates are represented by a vector of

To increase the efficiency of BT and GA, it is advantageous to limit the
number of pipe diameters in the available set, especially those outside of
the range of the optimal solution. For the BT algorithm in particular,
larger diameters can require considerable computational effort, since they
tend not to violate static head requirements and require multiple-link
partial candidates for the algorithm to reject them once their cost exceeds
that of an already-found viable candidate. Therefore, a pre-processor is
used to provide a maximum diameter (

A second pre-processor adjusts the minimum pressure head requirement for
each internal node by considering the total head required at downstream
nodes. It can be recognized that, without the use of a pump, the total head
cannot increase at nodes downstream of a given node

After the values for

The backtracking algorithm is a partial enumeration method that employs a systematic search of candidate solutions to find a global optimum. The algorithm works recursively to incrementally build candidate solutions while checking the candidates for hydraulic and cost acceptability. The strength of the BT is that, upon discovery of an infeasible partial candidate, all extensions of that candidate can be eliminated from consideration. In this way, many solutions can be pruned from the solution tree to achieve greater computational efficiency.

Two backtracking methods in the literature are those by Gessler (1985) and González-Cebollada et al. (2011). The algorithm proposed by Gessler proposes a pipe-grouping strategy which speeds up the algorithm but risks pruning the global optimum. Additionally, pipe grouping represents its own optimization problem (Raad, 2011). The González-Cebollada algorithm does not include such pipe-grouping criteria, although it does halt its search after finding the first feasible solution, thus it does not guarantee a global optimum. The present study's BT algorithm, once run to completion, does guarantee a global optimum. It operates similarly to the method presented by González-Cebollada et al. (2011), with the major differences being that the algorithm continues searching once it has found its first feasible solution and uses pre-processors 1 and 2 to further reduce the search space. This implementation of BT, however, scales poorly with larger network sizes and would not be appropriate for use on large urban networks. Its appropriateness is shown here for many of the GDWNs encountered in practice, as evidenced by its use on real-world GDWN test cases in this paper. Moreover, it serves as a benchmark against which other algorithms can be compared.

BT uses two rejection criteria to discard candidate solutions from further consideration. The first rejection criterion is that when a candidate violates pressure head constraints, all candidates with equal or lesser diameter sizes can be discarded. This condition is leveraged even more effectively with pre-processor 2 above, which can increase pressure heads at individual nodes by anticipating the head requirements at surrounding nodes. The second rejection criterion is that once a feasible candidate has been found, all other partial candidates with a higher cost can also be discarded. The BT algorithm further extends this second criterion by considering that the links yet to be considered in a partial candidate, an “extension” to the partial candidate, will cost at a minimum that of the entire extension being composed of the smallest available diameter.

The backtracking algorithm begins its search of the solution tree by
considering the partial candidate with the smallest diameter size assigned
to the first network link. The pressure head and the partial candidate cost
at the outlet node are calculated with the

A modification to the BT algorithm was made to further improve its computational speed, although at the risk of pruning the global optimum from the search. This modified algorithm (BT-NoUp) rejects all candidates that feature a smaller diameter that is upstream of a larger diameter when an equal or smaller flow rate is present in the downstream link. Typically, a network designer would not consider such designs, and in cases where a single source feeds into a network with constant-length links, it is advantageous (or equivalent) to place larger diameters upstream of smaller diameters. However, due to the discrete nature of diameter choices and link lengths, an optimization problem may, in fact, have an optimal candidate with a larger diameter downstream from smaller ones. For this reason, the BT-NoUp algorithm, unlike the BT algorithm, may miss the global optimum at the expense of its greater computational efficiency.

Genetic algorithms are stochastic optimization techniques that mimic the process of natural selection, and numerous variations of GAs have demonstrated acceptable performance on WDN design (Nicklow et al., 2010). Given their popularity, the GA included in this study is meant to provide a point of comparison to the BT and CB algorithms when applied to GDWNs.

When implemented in water network design, each candidate solution represents
a selection of pipe diameters. The algorithm is initialized with a
population of candidates of size

To allow for a hydraulic penalty coefficient to produce similar results in
both small-scale (inexpensive) network and a large-scale (more expensive)
cases, the hydraulic penalty coefficient is made directly proportional to
the average solution cost. With each generation,

Characteristics of test cases.

In this study, the genetic algorithm parameters used were

Six cases were studied based on actual GDWN in Panama, Nicaragua, and the Philippines. Global characteristics of each network are presented in Table 1 and the details of each network are presented in Table 4a–f. Each network is a branching type without loops. The total lengths of the networks range from less than 1 to over 15 km. Two serial networks are tested to demonstrate the effect of a local high point on the algorithm solutions. Elevation plots for each case are shown in Fig. 5.

The choice of

Network elevation (

The mapping between continuous diameters and the discrete nominal pipe sizes
was accomplished in our solution in one of the following ways:

For small and moderate size networks, the designer may manually adjust the
pipe sizes (downward, normally one pipe size) starting from the first link
downstream from the source and continuing along the rest of the distribution
main to the end in a step-by-step manner. A nearby plot of the pressure
heads compared with the theoretical

Based on the theoretical

The current study evaluated three types of algorithms that optimize the design of gravity-driven water networks (GDWN). The algorithms include the calculus-based (CB) algorithm (Eq. 18), a backtracking algorithm (BT) and its modified version (BT-NoUp), and a genetic algorithm (GA). The algorithms were applied to six test cases that are based on real GDWNs. Our results show that the CB, GA, and BT-NoUp algorithms could find solutions to the GDWNs within 25 % of the BT global optimum. All cases assume minor-lossless flow and a two-part pipe-cost model. Solution costs from each algorithm are shown in Table 2 and runtime statistics are shown in Table 3. BT could run to completion in < 1 min in all but the largest case (case 6 with 59 links), which did not complete after 7 days. As such, cost comparisons to BT are not made for case 6.

Solution costs for each algorithm.

Runtime and size of solution space for each algorithm.

Diameter sizes from calculus-based (CB-Disc) solutions compared with global optimum solutions (from backtracking, BT). A global optimum for case 6, Los Mangos, is not included since BT did not complete after 7 days of runtime.

The CB algorithm based on Eq. (18), unlike the other algorithms in this
work, finds a solution with theoretical diameters that are drawn from a
continuous domain (CB-Theor). For all test cases, the costs of the CB-Theor
solutions was less when compared with the BT discrete-diameter global
optimum (5.5 to 2.6 % lower cost than BT). In fact, because of the
discrete pipe sizes needed for an actual network, the continuous model will always produce the smallest theoretical network cost. The CB algorithm then
maps this solution to a commercially-available discrete set (CB-Disc). The
mapping process used in this study simply mapped each theoretical diameter
to the nearest available diameter of a larger size, thus producing a
solution which still satisfies static head requirements but with a higher
associated material cost. This tended to oversize the diameters, although
the CB-Disc solutions were always within two diameters of the known global
optimum solutions, as shown in Fig. 6. From all the combined test cases with
known global optima, all but one (71 out of 72) of the diameter selections
were within one diameter of the global optimum. More sophisticated mapping
schemes, like independently adjusting

Case network properties, diameter (D) results (inch nominal sizes,
with CB-Theor in inches), and nodal

Continued.

Continued.

N/A – not available.

BT-NoUp, a modified version of BT which does not consider smaller diameters
upstream of large diameters, completed itself within 4 s for all
cases, and found solutions which matched or came very close to the BT global
optimum. BT-NoUp missed the global optimum in cases 2 and 3, although by a
small percentage increase in cost (2.1 and 0.4 % respectively).
BT-NoUp, however, finished its search in a shorter amount of time in
comparison to BT, a benefit that becomes relevant on problems with larger
solution spaces, such as cases 3 (1.0

GA was run on each case a total of 100 times, each run itself evolved 200 candidates for 500 generations. The lowest-cost candidate amongst the final population that did not violate the pressure head condition was chosen as the GA solution. Because GA is a stochastic search algorithm producing different results from run-to-run, the costs of the optima from all 100 runs were averaged, with this averaged value presented in Table 2. Overall, GA costs came close to the global optima (within 3 %) for cases 1–5 where the global optimum was known from BT. GA solution costs increased with larger network sizes, with its solution cost 18 % higher than CB-Theor for case 6, the largest case run. Each GA run finished consistently within 1–5 s, not including about 2 seconds of pre-processor time. We note that variations of GAs have been reported in the literature and several of these may improve upon the GA results obtained in this study. Potential improvements to the GA a self-adapting penalty function (Wu and Walski, 2005), the use of elitism to preserve the best solutions (Kadu et al., 2008), and a reduction in the search space (Kadu et al., 2008). One reported improvement, the scaling of the fitness function to magnify the rewards towards slightly fitter candidates at later generations (Dandy et al., 1996), was attempted for case 2 but did not result in a noticeable effect on performance.

Optimization results from Bhave (1978) algorithm. LHS sum and RHS sum are the left and right sides of his Eq. (19), which should be equal.

To visually compare the algorithm solutions, the hydraulic grade lines from
BT, BT-NoUp, CB-Theor, and CB-Disc are presented in Fig. 5 along with the
network elevation for each test case. For clarity, the hydraulic grade lines
of branch links are omitted from the figure. In addition, the GA solutions
are omitted since 100 solutions were obtained for each test case.
Collectively, the hydraulic grade lines reveal a close alignment of the BT
solution (the global optimum) with the CB-Theor solution which utilizes a
continuous diameter set. Furthermore, the mapping scheme used to generate a
CB-Disc solution is shown to increase pipe sizes in some cases far beyond
the limit imposed by

We compared the CB results for the Los Mangos network with those from the
Bhave (1978) optimization algorithm (see Table 5). Like Eq. (15) in the
present work, Bhave's optimality equation (his Eq. 19) equates the sum of
a weighted term for all links entering and leaving each internal node in the
distribution main. In the present work the term is proportional to the
hydraulic gradient and the weighting factor is proportional to flow rate. In
Bhave's case the term is the ratio of pipe cost to head loss, where the
weighting factor is pipe-cost exponent

Algorithms to optimize the cost of branching gravity-driven water networks are evaluated on six test cases from real networks in the Philippines, Nicaragua, and Panama. A calculus-based algorithm produced a solution composed of theoretical diameters from a continuous set (CB-Theor), which are then mapped onto discrete commercially available diameters (CB-Disc). Backtracking (BT), a recursive algorithm, systematically searches discrete candidate solutions and, when run to completion, is guaranteed to find the global optimum by following rules that prune only higher-cost or hydraulically infeasible candidates. The BT algorithm was modified (BT-NoUp) to improve computational speed by rejecting all candidates that included a small diameter directly upstream of a larger diameter but allowed for the possibility of missing the global optimum. The third type of algorithm evaluated was a genetic algorithm (GA) that used single-point crossover and tournament selection.

BT could find the global optimum in most test cases with relatively little computational effort, although its poor scaling to larger networks is evidenced by its inability to find a solution to case 6, a network with 60 nodes and 59 links. The BT-NoUp completed its search in less time than BT and could find a solution to case 6. Based on case 1–5 results, the extra pruning condition adopted in BT-NoUp sacrificed only small cost increases. Both BT and BT-NoUp, however, could become prohibitively time-consuming when dealing with networks with significantly more links, diameter choices, or an unfavorable layout. While the test cases represent the range of GDWN sizes encountered in the authors' experience, future work would be needed to verify the suitability of the BT and BT-NoUp algorithms on larger GDWNs. The calculus-based algorithm produced consistently good results for the networks tested, although a more robust mapping scheme from theoretical diameters to discrete diameters would further improve on these results as discussed above. In potential future work, the CB-Theor solutions could be used to prune the BT search space, like Kadu et al. (2008), by only including the two diameters above and below the CB-Theor diameters, producing four diameter choices per link. The calculus-based methodology provides an additional benefit to the designer by explicitly revealing the sensitivities to cost for a design. The calculus-based algorithm requires greater computational effort than backtracking for smaller networks, however, this effort scales more linearly with the number of network links, while backtracking scales exponentially. Furthermore, backtracking's computational time is sensitive to the number of available diameters. Still, when applied to the present study's GDWN test cases with a modest number of links (23), backtracking quickly found a global optimum. In addition, because it is guaranteed to find the global optimum, it can be useful for benchmarking the performance of other algorithms which scale better with more network links. While the genetic algorithm produced solutions with good proximity to the global optimum, its solution costs tended to be further from the global optimum in cases with more links.

For all test cases, the calculus-based algorithm's theoretical diameter
solutions (CB-Theor) produced a lower cost than the discrete-domain global
optimum. This result is made possible because it is not constrained to a
discrete set of diameters. As such, the CB-Theor results represent a
lower-bound on the optimum solution within the problem formulation, which
could be approached with a finer selection of pipe diameters. We also
demonstrated good agreement between the CB-based optimization algorithm
developed here and that of Bhave (1978). Though Bhave's algorithm and
Eq. (18) in the present work appear quite different due to the different
ways each was developed, both produce optimality for the networks considered
in this paper. The key distinction between the two developments is that Bhave
assumed exponent

All survey data from the network cases tested are available in Table 4.

The authors declare that they have no conflict of interest.

This work was partially supported by the Villanova Undergraduate Research Fellowship Program and the Goldwater Foundation. Edited by: Luuk Rietveld Reviewed by: three anonymous referees