The water distribution system serves as a basic necessity for society. Due to its large size and involvement of various components, it is one of the most expensive civil infrastructures and thus demands optimization. Much work has been done to reduce the distribution system cost. However, with only one objective, the obtained solutions may not be practical to implement. Thus, improving cost along with the efficiency of the network is the demand of the hour. The present work introduces a unique parameter-less methodology for generating Pareto fronts without involving the concept of non-dominance. The methodology incorporates the Jaya optimization model for a bi-objective problem, one being the reduction in network cost and the other improving the reliability index of the network. The efficiency of the proposed work is analyzed for three different benchmark problems. The Jaya technique is found to be very efficient and fast when compared with the other evolutionary technique applied for the same networks. The parameter-less nature of the Jaya technique smooths the process to a very large extent as no synchronization of algorithm parameters is required.

  • Jaya algorithm is applied for the very first time to a multi-objective optimization problem in the distribution network.

  • A priori approach is proposed to generate a set of non-dominated solutions.

  • The suitability of the Normalized objective function defined in previous research is validated.

  • Jaya converges much faster compared to another evolutionary algorithm.

  • Present work results in highly reliable solutions.

Graphical Abstract

Graphical Abstract
Graphical Abstract

As humankind is evolving, there is always a demand for a much better solution. The solution that may fit only one objective is not seen as a satisfactory option as multiple goals need to be fulfilled. Meeting multiple goals from a single solution in technical terms is known as multi-objective optimization. A priori and posteriori are two approaches that can be used to solve any multi-objective optimization problem. In the a priori approach, the decision-maker must decide the importance of all objectives i.e. weightage or priority before the start of the analysis, and corresponding solutions are obtained then. A different set of solutions can be obtained by varying the weights. However, a non-dominated set of Pareto Front is directly obtained in the a posteriori approach and a decision-maker is allowed to choose a solution from this set. The a priori approach uses the scalarization method to create the objective function which further utilizes an evolutionary algorithm (EA) to reach the optimal points. A multi-objective evolutionary algorithm (MOEA) is used to find the optimal results for the a posteriori approach. MOEA's working principle is similar to an EA except these are based on the concept of non-dominance and crowding distance to find the optimal solution, which is considered complex when compared with the scalarization method (Jahanpour et al. 2018).

A water distribution system (WDS) comprises various parts such as reservoirs, pumps, pipes, valves, and so on, that are required to supply the water at the consumer end with adequate pressure, of sufficient quality and quantity. Since WDS plays an important role for any developed society, it is obvious to state that the optimal design of a water distribution network is one of the most researched fields. The main objective in the design of any distribution network is to reduce network costs. Hence, optimization of WDN is often seen as a single objective problem with the main aim of reducing the network cost. However, the optimal solution (optimal solution corresponds to the minimum network cost while satisfying all constraints of the problem) is not practical if implemented since it comprises diameters that are designed to just satisfy the minimum pressure head at the demand node. A slight change in this pressure may occur due to pipe breakout or sudden increase in demand node that may lead to network failure. Since WDN are a costly infrastructure, it is not possible to redesign them. In addition to this, the expansion of the network is not possible in all places due to the constraint of the space. Thus, at the design stage, if only the efficiency of WDN is considered, it may save a lot of effort and cost at the later stage. Hence, minimizing the network cost should not be the only objective to satisfy a longer-running network. Problems that are associated with satisfying more than one objective from a solution are known as multi-objective optimization problems. It was not until the authors Gessler & Walski (1985) that the first multi-objective optimization model was suggested for the distribution system. In most cases, authors chose reduction of network cost as the first objective. The second objective can be a reduction in emission of greenhouse gases from any WDN, reduction of pressure deficit, improving the reliability of the network, and so on (Jetmarova et al. 2018). Gheisi et al. (2016) presented detailed literature on different types of reliability and various measures to calculate them. Liu et al. (2017) conducted a comparative analysis of four different reliability indices in addition to introducing two new reliability indices, namely API (available power index), PHRI (pipe hydraulic resilience index). In the present work, scalarization is adopted to optimize the network using different weights for a two-objective optimization problem. The scalarization method combines all the objectives of the problem into one by assigning different weights to different objectives (Gunantara 2018), thus converting a multi-objective problem into a single objective problem and hence easy to analyze. The a priori method of optimization has proved its efficiency for generating a set of non-dominated solutions that may be used as a pivot point for different techniques (Torres et al. 2012).

Suribabu (2017) developed a normalized function combining the cost and reliability of the network as two objectives into one. The author found the function to be efficient using Differential Evolution (DE) as an optimization technique and under different weights of cost and reliability. The present work validates this normalized function to obtain a set of non-dominated solutions for three different benchmark problems using the Jaya optimization technique as the algorithm emerged to be an efficient performer when tested on various optimization problems. Dede (2018) incorporated the single objective optimization of steel structures using Jaya and found it to be a better performer. Hao et al. (2021) adopted the Jaya technique for the calcination process of cement. The detailed reviews on the various applications of Jaya as the optimization technique can be found in Alsajri et al. (2018).

There is not a well-defined definition for the term “reliability of distribution network” (Xu & Goulter 1999). However, WDS is designed considering that total energy supplied by the pumps or elevated reservoir must be able to deliver the water at sufficient pressure despite all the losses occurring in the pipe. However, due to sudden breakout in the network, which may occur due to a sudden increase in the demand capacity such as fire demand or pipe failure, losses occurring in the pipes may increase dramatically and may affect the users downstream of the failure junction. Nonetheless, if the network can absorb this head loss and continue its supply, the failure may not lead to any drastic effects and may be fixed during odd hours of the day. These networks that absorb such failure losses are said to be more resilient (Goulter & Morgan 1984) and thus have immense hydraulic power. Todini (2000) took advantage of this surplus power in any distribution network and introduce a technique to determine the reliability of a WDN and therefore becomes the first to introduce the surrogate measure of calculating the reliability of the WDN. Thereafter many surrogate measures are developed and have been implemented for the optimal design of WDN. Prasad & Park (2004) introduce the network resilience index with the addition of diameter uniformity in Todini's index of reliability. Raad et al. (2010) carried out the study of comparing various reliability indices and found Todini's RI to be more reliable.

The present work incorporates the RI for the optimal design of the three benchmark networks.

Resilience Index (RI) – Todini's RI is still the most popular surrogate reliability measure among all the indices developed (Raad et al. 2010). An RI is defined as the ratio of surplus hydraulic power to the available hydraulic power. The RI can range anywhere between 0 and 1 for the feasible solution, 0 being the least resilient and 1 being the most resilient. However, for a non-feasible solution where havl < hmin RI can even take a negative value. Such a situation arises more frequently while designing the distribution network by any evolutionary algorithm as it begins with random numbers. However, after certain iterations, the algorithms tend to give a positive value of RI.
formula
(1)
where qi is the demand at node i, havl and hmin are the available and minimum pressure head respectively at any j, Qr is the flow from the reservoir to the network, hres is the head available at the reservoir, Pb is pump capacity if any are present in the network, v is the specific weight of the liquid, which is generally water, and N is the number of demand nodes in the network.
The main objective of the problem is to minimize the total cost (TC) of the network per unit of reliability (R). Thus, the objective function is formulated as given in Equation (2)
formula
(2)
TC of the network depends on the per-unit cost of the commercial diameters obtained in optimization for the pipes and their corresponding length. TC can be calculated using Equation (3).
formula
(3)
where Li = length of pipe i (m); Ci (Di) = per meter cost of a pipe for a given diameter; Di = diameter of chosen pipe (m); np = total number of pipes in the network. Since the optimal design of WDN is a constrained problem, the following constraints are to be satisfied for each solution in the population.

Continuity at nodes

For any node, continuity of flow must be satisfied and can be written as
formula
(4)
where Qin and Qout = inflow and outflow at any node k (m3/s), qk=demand at node k (m3/s); nn = number of nodes.

Energy conservation in loops

Total head loss for a closed-loop is always equal to zero. However, for serial pipes between two reservoirs with fixed heads, the head loss is calculated as the difference between those fixed heads.
formula
(5)
where nl = number of loops in the network; hfi = head loss because of friction in the pipe and fittings i (m). Present work incorporates the Hazen-Williams formula for determining the pressure loss in pipes that occurs due to friction.

Minimum pressure at nodes

For the flow to occur, pressure at any node must be greater than the minimum pressure head required at that node.
formula
(6)
where = simulated pressure head at node k, = prescribed minimum pressure head at node k.

Pipe size availability

Since the optimal design of distribution networks belongs to the category of discrete optimization, the diameters must always be selected from the set of commercially available diameters. A larger set leads to an increase in search space.
formula
(7)
where = number of commercial pipe diameters, np = number of pipes in the network.

The Scalarization method converts a multi-objective problem into a single objective by assigning weights to each objective and combining them. Selecting a higher weight for any objective prioritizes it over the other objectives. Tchebycheff scalarization using interactive weights was proposed by Steuer & Choo (1983). Thereafter Tchebycheff has been used extensively for scalarization. Many new scalarization approaches that are used to generate Pareto solutions for non-convex problems can be found in Xia et al. (2021). The author also developed a combined scalarization technique. Kasimbeyli et al. (2019) compared six different scalarization techniques, namely – weighted sum, epsilon constraint, Chebyshev method, Pascoletti-Serafini, conic and Benson method. The present work uses weighted sum by assigning higher weights to the cost and then reducing it subsequently. The decrease in weight of cost is equal to the increase in weight of reliability, such that the summation of all weights is always equal to one.

However, to maintain a level of fairness among the objectives, it is important to normalize the objective function. Suribabu (2017) presents a normalized function incorporating the cost and resilience of the network.
formula
(8)
where
formula
(9)

C represents the cost of the network obtained, Cmin is optimal network cost determined by substituting the optimal diameters that are obtained by considering minimizing the network cost as the only objective. Cmax is the maximum network cost, obtained by substituting the maximum available commercial diameter of the network to all the pipes and thus calculating its cost, RMmax is a resilience measure of the network that corresponds to Cmax, RMmin is the resilience measure of the network that corresponds to Cmin. RM is the resilience measure of the network at cost C. w1 and w2 are weights assigned to the cost and resilience measure of the network, respectively. The summation of w1 and w2 is always equal to 1. Since EA starts with random numbers, the probability that the RM value of a solution is less than RMmin is very high. For such situations, RM – RMmin will become negative, which reduces the value of Zmin, and such a solution will be selected as the best solution. To avoid this during any time in the optimal design, if RM<RMmin is found, RM is substituted as RMmin+0.000001 so that RM – RMmin will always be equal to 0.000001. A similar concept may help in dealing with solutions that have zero resilience value. Cmin, Cmax, Rmin, Rmax values for the Two Loop, Hanoi, and Go-Yang networks are given in Table 1.

Table 1

Normalized function parameters for Two Loop, Hanoi and Go-Yang Networks

NetworkCminCmaxRminRmax
Two Loop 4,19,000 44,00,000 0.2104 0.9038 
Hanoi 60,81,087 1,09,70,586 0.1920 0.3537 
Go-Yang 1,77,010 3,29,725.64 0.4944 0.9941 
NetworkCminCmaxRminRmax
Two Loop 4,19,000 44,00,000 0.2104 0.9038 
Hanoi 60,81,087 1,09,70,586 0.1920 0.3537 
Go-Yang 1,77,010 3,29,725.64 0.4944 0.9941 
Almost all the evolutionary techniques developed involve various constants that need to be tuned for the proper functioning of the algorithm. The algorithm cannot function without these constants and may lead to incorrect results if tuned improperly. The tuning is a tedious process and involves huge computational efforts. In addition to this, these constants are to be newly tuned for each optimization problem. This led to the development of techniques that require minimum interaction of algorithm constants such as parameterless or self-adaptive techniques. Taking the above into consideration, Rao (2016) developed the Jaya technique, which is not only parameterless; that is, no parameters/constants are involved thus it totally escapes the process of synchronization, but is extremely easy to incorporate into any optimization model. Like any other meta-heuristic technique, Jaya is also a population-based optimization algorithm that utilizes the best and worst solutions from the population set and uses them to generate the new solutions. For a minimization problem, the solution with the minimum objective value is selected as the best and the maximum value is selected as the worst. Once the best and worst solutions are generated, a new generation of the population is obtained using Equation (10)
formula
(10)

r1 and r2 in Equation (10) are random numbers that are generated between 0 and 1. The best and worst solutions are represented by indexes b and w respectively. Index i represents iteration number, j represents the position of the variable in any solution, k is the index of the position of solution in a population set. As the algorithm advances in the iterations, the best and worst values are improvised and an optimal solution is obtained, thus the algorithm becomes victorious as the termination criterion is reached and hence named Jaya (A Sanskrit word that means – Victory over the evil). The pseudo-code for the methodology is given below.

Initialize trials, iterations, population size, number of variables, w1 and w2

Trial t=1

while t is less than t-maximum

  Generate initial population matrix at random

  Iteration i=1

  while i is less than i-maximum

    Initialize EPANET and determine cost matrix along with Resilience index (using Equations(3) and (1))

    Determine objective function using Equations (2) and (8)

    Identify best, worst and their corresponding indices

    Update the population using Equation (10)

    Compare updated solution with previous solution

    if updated solution is better than previous solution

      update the population and move to next iteration

    else

      retain the population and move to next iteration

    i++

  end while

  n++

end while

  • 1.

    The problem is initialized by the number of variables, number of iterations, weighing factor w1 and w2, lower and upper limit of the decision variable.

  • 2.
    Initial solutions are generated randomly using Equation (11). The solutions are generated randomly to avoid any biasedness.
    formula
    (11)

xij is generated random value of the decision variable, is the lower and are the upper limits of the decision variables. is a random number generated between 0 and 1 and distributed uniformly.

  • 3.
    WDN belongs to the category of non-combinatorial discrete optimization, but the solutions that are generated using Equation (11) are continuous and are thus converted to discrete values using Equation (12).
    formula
    (12)
is a set of commercially available discrete diameters that are arranged in increasing order.
  • 4.

    EPANET 2 (Rossman 2000) is called and simulated pressure heads, demand are determined at each node for every solution in the population set.

  • 5.
    If solutions are found to violate the minimum pressure requirement constraint, it is penalized. The penalty for such a solution is determined using Equation (13).
    formula
    (13)
    where Pmin is the minimum pressure value among all the nodes for a solution; that is, simulated using EPANET 2, Preq is the required minimum pressure for any node, λ is a penalty constant.
  • 6.
    Total Cost; that is, the fitness function, is calculated as a sum of penalty value and the cost of the network.
    formula
    (14)
  • 7.

    Reliability is calculated using the resilience index defined in Equation (1).

  • 8.

    Cost by Reliability is further calculated, which becomes the new fitness value for the Jaya algorithm. For normalized objective function Zmin is calculated.

  • 9.

    The solution giving the minimum and maximum cost by reliability or Zmin value (normalized function) is taken as the best and worst solutions respectively.

  • 10.

    Once the best and the worst solutions are identified, a new population is generated using Equation (10).

  • 11.

    Steps 4–8 are followed for the new set of solutions. The new population is compared with the old population and the better-performing (with minimum fitness value) solutions are selected and passed to the next generation.

  • 12.

    The process for a pair of weights is continued until a stopping criterion is reached.

Two loop network

Two loop network (Alperovits & Shamir 1977) is a hypothetical network with a reservoir fixed at an elevation of 210 m. The network has 14 commercially available pipe diameters and 8 pipes each 1,000 m long with a Hazen-William coefficient of 130, arranged in two loops, which leads to a total search space of 1.48 * 109. A minimum pressure head of 30 m is required for the network at each node. The other hydraulic data of the network such as demand and node elevation are given in Figure 1(a). The optimal cost of the network is 419,000 (Cunha & Sousa 1999) with the optimal diameters as given in Table 2. Different solutions obtained by the Jaya technique by varying the weights are given in Table 3. As can be seen from Table 3 for w1 = 0.95 & w2 = 0.05, the optimized cost increase by 32.21% over the optimal cost, while reliability increases to 91.17%, which is very appealing. It is also to be noted that for w1 = 0.80, 0.75, and 0.70 similar cost and reliability are achieved. A similar trend is seen for w1 = 0.45, 0.40, 0.35. No improvement in diameters is obtained for such solutions by varying the weights. Thus, considering the designer's preference for the importance of the two objectives, any solution can be picked. Reliability as high as 0.8998 is obtained when w2 becomes 0.95, which is very near to the Rmax; however, the cost at this reliability is 25,52,000, which is less when compared with the Cmax; that is, 44,00,000. Thus, maximum reliability can be attained with much less cost. Optimal diameters obtained for a certain set of weights are given in Table 4. No comparison is available in the literature for this normalized function and hence it has not been compared here. For every pair of weights, 10 trial runs are performed with a population size of 50 and 200 iterations as the termination criteria. Convergence is obtained in just 30–50 iterations for every trial run, which highlights the searching capability of Jaya for a non-linear discrete optimization problem. The Pareto Front obtained for the network is shown in Figure 1(b).

Table 2

Comparison of diameters for single objective and multi-objective including reliability

Pipe No.Optimal diameters (in mm)
Diameters for minimizing total cost by reliability (in mm)
Two Loop NetworkHanoi NetworkGo-Yang NetworkTwo Loop NetworkHanoi NetworkGo-Yang Network
1. 457.2 1016.0 200 508.0 1016 300 
2. 254.0 1016.0 125 406.4 1016 250 
3. 406.4 1016.0 125 406.4 1016 200 
4. 101.6 1016.0 100 25.4 1016 200 
5. 406.4 1016.0 80 355.6 1016 200 
6. 254.0 1016.0 80 25.4 1016 150 
7. 254.0 1016.0 80 406.4 762 150 
8. 25.4 1016.0 80 304.8 762 200 
9.  1016.0 80  609.6 80 
10.  762.0 80  762 80 
11.  609.6 80  609.6 80 
12.  609.6 80  508 80 
13.  508.0 80  609.6 100 
14.  406.4 80  762 80 
15.  304.8 80  762 100 
16.  304.8 80  1016 80 
17.  406.4 80  1016 80 
18.  609.6 80  1016 80 
19.  508.0 80  1016 100 
20.  1016.0 80  1016 80 
21.  508.0 80  609.6 80 
22.  304.8 80  406.4 80 
23.  1016.0 80  1016 80 
24.  762.0 80  762 80 
25.  762.0 80  609.6 80 
26.  508.0 80  304.8 80 
27.  304.8 80  609.6 80 
28.  304.8 80  609.6 80 
29.  406.4 80  406.4 80 
30.  304.8 80  304.8 80 
31.  304.8   304.8  
32.  406.4   406.4  
33.  406.4   406.4  
34.  609.6   609.6  
Cost (units and $) 4,19,000 60,81,087 1,77,010 5,54,000 70,67,326 1,87,378.93 
Reliability Index 0.2104 0.1920 0.4944 0.6091 0.3142 0.9594 
Pipe No.Optimal diameters (in mm)
Diameters for minimizing total cost by reliability (in mm)
Two Loop NetworkHanoi NetworkGo-Yang NetworkTwo Loop NetworkHanoi NetworkGo-Yang Network
1. 457.2 1016.0 200 508.0 1016 300 
2. 254.0 1016.0 125 406.4 1016 250 
3. 406.4 1016.0 125 406.4 1016 200 
4. 101.6 1016.0 100 25.4 1016 200 
5. 406.4 1016.0 80 355.6 1016 200 
6. 254.0 1016.0 80 25.4 1016 150 
7. 254.0 1016.0 80 406.4 762 150 
8. 25.4 1016.0 80 304.8 762 200 
9.  1016.0 80  609.6 80 
10.  762.0 80  762 80 
11.  609.6 80  609.6 80 
12.  609.6 80  508 80 
13.  508.0 80  609.6 100 
14.  406.4 80  762 80 
15.  304.8 80  762 100 
16.  304.8 80  1016 80 
17.  406.4 80  1016 80 
18.  609.6 80  1016 80 
19.  508.0 80  1016 100 
20.  1016.0 80  1016 80 
21.  508.0 80  609.6 80 
22.  304.8 80  406.4 80 
23.  1016.0 80  1016 80 
24.  762.0 80  762 80 
25.  762.0 80  609.6 80 
26.  508.0 80  304.8 80 
27.  304.8 80  609.6 80 
28.  304.8 80  609.6 80 
29.  406.4 80  406.4 80 
30.  304.8 80  304.8 80 
31.  304.8   304.8  
32.  406.4   406.4  
33.  406.4   406.4  
34.  609.6   609.6  
Cost (units and $) 4,19,000 60,81,087 1,77,010 5,54,000 70,67,326 1,87,378.93 
Reliability Index 0.2104 0.1920 0.4944 0.6091 0.3142 0.9594 
Table 3

Zmin values for different pairs of weights for Two Loop network

S.No.W1W2Cost of network (in units)Resilience IndexZmin
1. 0.95 0.05 5,54,000 0.6091 0.1192 
2. 0.90 0.10 6,42,000 0.6800 0.1981 
3. 0.85 0.15 6,72,000 0.6952 0.2686 
4. 0.80 0.20 8,32,000 0.7771 0.32277 
5. 0.75 0.25 8,32,000 0.7771 0.3837 
6. 0.70 0.30 8,32,000 0.7771 0.4397 
7. 0.65 0.35 9,02,000 0.7957 0.4935 
8. 0.60 0.40 9,54,000 0.8063 0.5461 
9. 0.55 0.45 9,94,000 0.8133 0.5969 
10. 0.50 0.50 12,84,000 0.8590 0.6432 
11. 0.45 0.55 13,24,000 0.8642 0.6857 
12. 0.40 0.60 13,24,000 0.8642 0.7274 
13. 0.35 0.65 13,24,000 0.8642 0.7690 
14. 0.30 0.70 15,12,000 0.8780 0.8094 
15. 0.25 0.75 15,12,000 0.8780 0.8476 
16. 0.20 0.80 15,82,000 0.8809 0.8858 
17. 0.15 0.85 16,82,000 0.8846 0.9218 
18. 0.10 0.90 20,42,000 0.8934 0.9546 
19. 0.05 0.95 25,52,000 0.8998 0.9823 
S.No.W1W2Cost of network (in units)Resilience IndexZmin
1. 0.95 0.05 5,54,000 0.6091 0.1192 
2. 0.90 0.10 6,42,000 0.6800 0.1981 
3. 0.85 0.15 6,72,000 0.6952 0.2686 
4. 0.80 0.20 8,32,000 0.7771 0.32277 
5. 0.75 0.25 8,32,000 0.7771 0.3837 
6. 0.70 0.30 8,32,000 0.7771 0.4397 
7. 0.65 0.35 9,02,000 0.7957 0.4935 
8. 0.60 0.40 9,54,000 0.8063 0.5461 
9. 0.55 0.45 9,94,000 0.8133 0.5969 
10. 0.50 0.50 12,84,000 0.8590 0.6432 
11. 0.45 0.55 13,24,000 0.8642 0.6857 
12. 0.40 0.60 13,24,000 0.8642 0.7274 
13. 0.35 0.65 13,24,000 0.8642 0.7690 
14. 0.30 0.70 15,12,000 0.8780 0.8094 
15. 0.25 0.75 15,12,000 0.8780 0.8476 
16. 0.20 0.80 15,82,000 0.8809 0.8858 
17. 0.15 0.85 16,82,000 0.8846 0.9218 
18. 0.10 0.90 20,42,000 0.8934 0.9546 
19. 0.05 0.95 25,52,000 0.8998 0.9823 
Table 4

Optimal diameters (in mm) obtained for different pairs of weights for the Two Loop network

Pipe No.W1 = 0.8, W2 = 0.2W1 = 0.6, W2 = 0.4W1 = 0.4, W2 = 0.6W1 = 0.2, W2 = 0.8
1. 558.8 558.8 609.6 609.6 
2. 406.4 508 508 508 
3. 508 508 508 558.8 
4. 355.6 254 25.4 406.4 
5. 406.4 406.4 508 508 
6. 25.4 25.4 254 25.4 
7. 355.6 457.2 508 508 
8. 355.6 355.6 355.6 457.2 
Cost (units) 8,32,000 9,54,000 13,24,000 15,82,000 
RI 0.7771 0.8063 0.8642 0.8809 
Pipe No.W1 = 0.8, W2 = 0.2W1 = 0.6, W2 = 0.4W1 = 0.4, W2 = 0.6W1 = 0.2, W2 = 0.8
1. 558.8 558.8 609.6 609.6 
2. 406.4 508 508 508 
3. 508 508 508 558.8 
4. 355.6 254 25.4 406.4 
5. 406.4 406.4 508 508 
6. 25.4 25.4 254 25.4 
7. 355.6 457.2 508 508 
8. 355.6 355.6 355.6 457.2 
Cost (units) 8,32,000 9,54,000 13,24,000 15,82,000 
RI 0.7771 0.8063 0.8642 0.8809 
Figure 1

Two Loop Network and its Pareto Front.

Figure 1

Two Loop Network and its Pareto Front.

Hanoi network

The Hanoi network in the present work is taken from Fujiwara & Khang (1990). The network consists of 34 pipes and a reservoir at an elevation of 100 m (Figure 2(a)). There are six commercial sizes available for the Hanoi network, as shown in Table 5, which leads to a search space of 2.865 * 1026, which is large and computationally expensive and hence is a challenging problem for any computational technique. The hydraulic data of the network is given in the supplementary material. The optimal cost of the network is obtained as 6.081 M$ (Zheng et al. 2013) when only minimizing the network cost is considered. The diameters corresponding to this cost are shown in Table 2. The optimal diameters obtained when Equation (2) is optimized using Jaya are also shown in Table 2. The cost and resilience index for different weights to obtain a Pareto Front is shown in Table 6. As the weightage of the resilience index is increased, higher-cost solutions are obtained, since larger size diameters are needed to meet the resilience of the network. There is an increase of 83.85% in reliability for resilience as 0.3530 and cost of $9798084.93 i.e 61.21% more than optimized cost using cost and resilience index weights as 0.05 and 0.95 respectively. The designer can choose among any of the solutions given in Table 6 based on the availability of cost and desired level of reliability to be achieved. The results obtained by Jaya are similar to those obtained by Suribabu (2017). The slight change in diameter values from Suribabu (2017) is marked in bold in Table 7. DE applied by Suribabu (2017) took as many as 2,000 iterations to reach these results; Jaya converges to the same results in less than 100 iterations for all the successful trial runs, which again highlights the computational efficiency of Jaya. The comparison of diameters obtained by Suribabu (2017) to Jaya is shown in Table 7. For every pair of weights, 10 trial runs are performed with a population size of 75 and 200 iterations as the termination criteria. The Pareto Front obtained for the network is shown in Figure 2(b).

Table 5

Commercially available pipe diameters along with the unit cost for Two Loop, Hanoi, and Go-Yang networks

Two Loop Network
Hanoi Network
Go-Yang Network
Diameter (mm)Cost (units)Diameter (mm)Cost ($/m)Diameter (mm)Cost ($/m)
25.4 304.8 45.73 80 37.890 
50.8 406.4 70.40 100 38.933 
76.2 508.0 98.38 125 40.563 
101.6 11 609.6 129.33 150 42.554 
152.4 16 762.0 180.80 200 47.624 
203.2 23 1016.0 278.30 250 54.125 
254.0 32   300 62.109 
304.8 50   350 71.524 
355.6 60     
406.4 90     
457.2 130     
508.0 170     
558.8 300     
609.6 550     
Two Loop Network
Hanoi Network
Go-Yang Network
Diameter (mm)Cost (units)Diameter (mm)Cost ($/m)Diameter (mm)Cost ($/m)
25.4 304.8 45.73 80 37.890 
50.8 406.4 70.40 100 38.933 
76.2 508.0 98.38 125 40.563 
101.6 11 609.6 129.33 150 42.554 
152.4 16 762.0 180.80 200 47.624 
203.2 23 1016.0 278.30 250 54.125 
254.0 32   300 62.109 
304.8 50   350 71.524 
355.6 60     
406.4 90     
457.2 130     
508.0 170     
558.8 300     
609.6 550     
Table 6

Zmin values for different pairs of weights for the Hanoi network

S.No.W1W2Cost of network ($)Resilience IndexZmin
1. 0.95 0.05 64,39,485.15 0.2630 0.1837 
2. 0.90 0.10 66,56,523.05 0.2908 0.2698 
3. 0.85 0.15 67,63,503.90 0.3000 0.3435 
4. 0.80 0.20 67,77,493.90 0.3010 0.4110 
5. 0.75 0.25 67,94,518.05 0.3019 0.4776 
6. 0.70 0.30 71,15,456.99 0.3166 0.5379 
7. 0.65 0.35 72,27,908.24 0.3210 0.5917 
8. 0.60 0.40 74,17,895.00 0.3281 0.6399 
9. 0.55 0.45 75,61,088.00 0.3329 0.6838 
10. 0.50 0.50 76,04,991.15 0.3341 0.7255 
11. 0.45 0.55 77,35,780.40 0.3372 0.7657 
12. 0.40 0.60 77,98,595.60 0.3384 0.8042 
13. 0.35 0.65 79,65,846.80 0.3413 0.8402 
14. 0.30 0.70 81,12,091.70 0.3432 0.8741 
15. 0.25 0.75 84,67,134.20 0.3470 0.9058 
16. 0.20 0.80 86,82,326.70 0.3488 0.9329 
17. 0.15 0.85 89,67,423.89 0.3507 0.9560 
18. 0.10 0.90 91,12,978.43 0.3514 0.9765 
19. 0.05 0.95 97,98,084.93 0.3530 0.9934 
S.No.W1W2Cost of network ($)Resilience IndexZmin
1. 0.95 0.05 64,39,485.15 0.2630 0.1837 
2. 0.90 0.10 66,56,523.05 0.2908 0.2698 
3. 0.85 0.15 67,63,503.90 0.3000 0.3435 
4. 0.80 0.20 67,77,493.90 0.3010 0.4110 
5. 0.75 0.25 67,94,518.05 0.3019 0.4776 
6. 0.70 0.30 71,15,456.99 0.3166 0.5379 
7. 0.65 0.35 72,27,908.24 0.3210 0.5917 
8. 0.60 0.40 74,17,895.00 0.3281 0.6399 
9. 0.55 0.45 75,61,088.00 0.3329 0.6838 
10. 0.50 0.50 76,04,991.15 0.3341 0.7255 
11. 0.45 0.55 77,35,780.40 0.3372 0.7657 
12. 0.40 0.60 77,98,595.60 0.3384 0.8042 
13. 0.35 0.65 79,65,846.80 0.3413 0.8402 
14. 0.30 0.70 81,12,091.70 0.3432 0.8741 
15. 0.25 0.75 84,67,134.20 0.3470 0.9058 
16. 0.20 0.80 86,82,326.70 0.3488 0.9329 
17. 0.15 0.85 89,67,423.89 0.3507 0.9560 
18. 0.10 0.90 91,12,978.43 0.3514 0.9765 
19. 0.05 0.95 97,98,084.93 0.3530 0.9934 
Table 7

Comparison of diameters for different pairs of weights for the Hanoi network

Pipe No.W1 = 0.8, W2 = 0.2
W1 = 0.6, W2 = 0.4
W1 = 0.4, W2 = 0.6
W1 = 0.2, W2 = 0.8
Present WorkSuribabu (2017) Present WorkSuribabu (2017) Present WorkSuribabu (2017) Present WorkSuribabu (2017) 
1. 1016 1016 1016 1016 1016 1016 1016 1016 
2. 1016 1016 1016 1016 1016 1016 1016 1016 
3. 1016 1016 1016 1016 1016 1016 1016 1016 
4. 1016 1016 1016 1016 1016 1016 1016 1016 
5. 1016 1016 1016 1016 1016 1016 1016 1016 
6. 1016 1016 1016 1016 1016 1016 1016 1016 
7. 1016 1016 1016 1016 1016 1016 1016 1016 
8. 1016 1016 762 762 1016 1016 1016 1016 
9. 762 762 762 762 762 762 1016 1016 
10. 762 762 762 762 762 762 1016 1016 
11. 609.6 609.6 762 762 762 762 1016 1016 
12. 508 508 609.6 609.6 609.6 609.6 762 762 
13. 304.8 304.8 508 508 609.6 609.6 609.6 609.6 
14. 508 508 609.6 609.6 762 762 762 762 
15. 508 609.6 762 762 762 762 1016 1016 
16. 1016 1016 1016 1016 1016 1016 1016 1016 
17. 1016 1016 1016 1016 1016 1016 1016 1016 
18. 1016 1016 1016 1016 1016 1016 1016 1016 
19. 1016 1016 1016 1016 1016 1016 1016 1016 
20. 1016 1016 1016 1016 1016 1016 1016 1016 
21. 609.6 609.6 609.6 609.6 762 762 762 762 
22. 406.4 406.4 406.4 406.4 508 508 508 508 
23. 762 762 1016 1016 1016 1016 1016 1016 
24. 508 508 762 762 762 762 1016 1016 
25. 304.8 304.8 508 508 609.6 609.6 762 762 
26. 609.6 609.6 406.4 406.4 508 508 508 508 
27. 762 762 609.6 609.6 762 762 762 762 
28. 762 762 762 762 762 762 1016 1016 
29. 406.4 406.4 508 508 508 508 609.6 609.6 
30. 304.8 304.8 406.4 406.4 406.4 406.4 508 304.8 
31. 304.8 304.8 304.8 304.8 304.8 304.8 304.8 406.4 
32. 406.4 406.4 304.8 304.8 406.4 406.4 406.4 406.4 
33. 406.4 406.4 406.4 406.4 508 508 508 508 
34. 609.6 609.6 609.6 609.6 762 762 762 762 
Cost ($) 67,77,493 67,10,999 74,17,985 74,17,236 77,98,595 77,97,775 86,82,326 86,81,431 
RI 0.3010 0.2980 0.3281 0.3281 0.3384 0.3384 0.3488 0.3487 
Pipe No.W1 = 0.8, W2 = 0.2
W1 = 0.6, W2 = 0.4
W1 = 0.4, W2 = 0.6
W1 = 0.2, W2 = 0.8
Present WorkSuribabu (2017) Present WorkSuribabu (2017) Present WorkSuribabu (2017) Present WorkSuribabu (2017) 
1. 1016 1016 1016 1016 1016 1016 1016 1016 
2. 1016 1016 1016 1016 1016 1016 1016 1016 
3. 1016 1016 1016 1016 1016 1016 1016 1016 
4. 1016 1016 1016 1016 1016 1016 1016 1016 
5. 1016 1016 1016 1016 1016 1016 1016 1016 
6. 1016 1016 1016 1016 1016 1016 1016 1016 
7. 1016 1016 1016 1016 1016 1016 1016 1016 
8. 1016 1016 762 762 1016 1016 1016 1016 
9. 762 762 762 762 762 762 1016 1016 
10. 762 762 762 762 762 762 1016 1016 
11. 609.6 609.6 762 762 762 762 1016 1016 
12. 508 508 609.6 609.6 609.6 609.6 762 762 
13. 304.8 304.8 508 508 609.6 609.6 609.6 609.6 
14. 508 508 609.6 609.6 762 762 762 762 
15. 508 609.6 762 762 762 762 1016 1016 
16. 1016 1016 1016 1016 1016 1016 1016 1016 
17. 1016 1016 1016 1016 1016 1016 1016 1016 
18. 1016 1016 1016 1016 1016 1016 1016 1016 
19. 1016 1016 1016 1016 1016 1016 1016 1016 
20. 1016 1016 1016 1016 1016 1016 1016 1016 
21. 609.6 609.6 609.6 609.6 762 762 762 762 
22. 406.4 406.4 406.4 406.4 508 508 508 508 
23. 762 762 1016 1016 1016 1016 1016 1016 
24. 508 508 762 762 762 762 1016 1016 
25. 304.8 304.8 508 508 609.6 609.6 762 762 
26. 609.6 609.6 406.4 406.4 508 508 508 508 
27. 762 762 609.6 609.6 762 762 762 762 
28. 762 762 762 762 762 762 1016 1016 
29. 406.4 406.4 508 508 508 508 609.6 609.6 
30. 304.8 304.8 406.4 406.4 406.4 406.4 508 304.8 
31. 304.8 304.8 304.8 304.8 304.8 304.8 304.8 406.4 
32. 406.4 406.4 304.8 304.8 406.4 406.4 406.4 406.4 
33. 406.4 406.4 406.4 406.4 508 508 508 508 
34. 609.6 609.6 609.6 609.6 762 762 762 762 
Cost ($) 67,77,493 67,10,999 74,17,985 74,17,236 77,98,595 77,97,775 86,82,326 86,81,431 
RI 0.3010 0.2980 0.3281 0.3281 0.3384 0.3384 0.3488 0.3487 
Figure 2

Hanoi Network and its Pareto Front.

Figure 2

Hanoi Network and its Pareto Front.

Go-Yang network

The Go-Yang network (Figure 3(a)) consists of 30 pipes and 22 nodes arrange in 9 loops with a ground reservoir fixed at an elevation of 71 m. A pump of 4.52 kW capacity is provided to supply the water to the network. A 15 m minimum pressure head is required above the ground elevation at all the nodes in the network. The Hazen-Williams coefficient for all the pipes in the network is taken as 100. The hydraulic data of the network such as ground elevation, nodal demand, and so on, are given in the supplementary material. The optimal cost of the network (when only minimizing the network cost is considered) is reported as $177,010 (Zheng et al. 2012) with the optimal diameters as given in Table 2. Equation (2) is optimized using Jaya for the Go-Yang network and with just 5.85% increase in cost from the optimal value, 94.05% increase in resilience index is obtained, with the diameters given in Table 2, which is captivating and thus highlights the importance of considering reliability in the optimization model and to not rely on just minimizing the network cost. Table 8 shows the value of the cost and resilience index obtained for different pairs of weights. The maximum cost of the network is $329725.64 with an RI of 0.9941; however, for the weights w1 = 0.95 and w2 = 0.05 a cost as high as $211604.5 is reported with an RI of 0.9922, which highlights that by giving just 5% weightage to the reliability, maximum reliability can be achieved with a much lesser cost. With nearly the same resilience index, $118,121.14 is saved, which is huge and is more than the optimal network cost. The diameters obtained for a certain set of weights are shown in Table 9. For each set of weights, 10 trial runs are performed with a population size of 75 and 200 iterations as the termination criteria. For all the trials, the results are obtained in less than 100 iterations. The Pareto Front for the obtained results is shown in Figure 3(b) for the Go-Yang network.

Table 8

Zmin values for different pairs of weights for the Go-Yang network

S.No.W1W2Cost of network ($)Resilience IndexZmin
1. 0.95 0.05 1,81,046.2 0.8731 0.0911 
2. 0.90 0.10 1,81,851.9 0.8927 0.1539 
3. 0.85 0.15 1,84,618.5 0.9347 0.2125 
4. 0.80 0.20 1,85,131.4 0.9407 0.2663 
5. 0.75 0.25 1,86,339.7 0.9524 0.3184 
6. 0.70 0.30 1,86,699.6 0.9552 0.3696 
7. 0.65 0.35 1,87,378.9 0.9594 0.4201 
8. 0.60 0.40 1,88,932.4 0.9665 0.4701 
9. 0.55 0.45 1,90,179.2 0.9717 0.5183 
10. 0.50 0.50 1,90,946.3 0.9742 0.5662 
11. 0.45 0.55 1,93,016.0 0.9794 0.6136 
12. 0.40 0.60 1,94,532.6 0.9827 0.6597 
13. 0.35 0.65 1,94,692.3 0.9829 0.7052 
14. 0.30 0.70 1,96,250.4 0.9852 0.7502 
15. 0.25 0.75 1,96,370.3 0.9854 0.7947 
16. 0.20 0.80 1,96,466.3 0.9855 0.8393 
17. 0.15 0.85 2,01,970.3 0.9893 0.8824 
18. 0.10 0.90 2,06,007.5 0.9910 0.9243 
19. 0.05 0.95 2,11,604.5 0.9922 0.9646 
S.No.W1W2Cost of network ($)Resilience IndexZmin
1. 0.95 0.05 1,81,046.2 0.8731 0.0911 
2. 0.90 0.10 1,81,851.9 0.8927 0.1539 
3. 0.85 0.15 1,84,618.5 0.9347 0.2125 
4. 0.80 0.20 1,85,131.4 0.9407 0.2663 
5. 0.75 0.25 1,86,339.7 0.9524 0.3184 
6. 0.70 0.30 1,86,699.6 0.9552 0.3696 
7. 0.65 0.35 1,87,378.9 0.9594 0.4201 
8. 0.60 0.40 1,88,932.4 0.9665 0.4701 
9. 0.55 0.45 1,90,179.2 0.9717 0.5183 
10. 0.50 0.50 1,90,946.3 0.9742 0.5662 
11. 0.45 0.55 1,93,016.0 0.9794 0.6136 
12. 0.40 0.60 1,94,532.6 0.9827 0.6597 
13. 0.35 0.65 1,94,692.3 0.9829 0.7052 
14. 0.30 0.70 1,96,250.4 0.9852 0.7502 
15. 0.25 0.75 1,96,370.3 0.9854 0.7947 
16. 0.20 0.80 1,96,466.3 0.9855 0.8393 
17. 0.15 0.85 2,01,970.3 0.9893 0.8824 
18. 0.10 0.90 2,06,007.5 0.9910 0.9243 
19. 0.05 0.95 2,11,604.5 0.9922 0.9646 
Table 9

Optimal diameters (in mm) obtained for different pairs of weights for the Go-Yang network

Pipe No.W1 = 0.8, W2 = 0.2W1 = 0.6, W2 = 0.4W1 = 0.4, W2 = 0.6W1 = 0.2, W2 = 0.8
1. 300 350 350 350 
2. 200 250 300 300 
3. 200 200 250 250 
4. 200 200 250 250 
5. 150 200 200 250 
6. 150 150 200 200 
7. 125 150 200 200 
8. 200 200 250 250 
9. 80 80 80 80 
10. 80 80 80 80 
11. 80 80 80 80 
12. 80 80 80 100 
13. 80 100 125 150 
14. 80 80 80 80 
15. 100 100 125 125 
16. 80 80 80 80 
17. 80 80 80 80 
18. 80 80 80 80 
19. 100 100 100 125 
20. 80 80 80 80 
21. 80 80 80 80 
22. 80 80 80 80 
23. 80 80 80 80 
24. 80 80 80 80 
25. 80 80 80 100 
26. 80 80 80 80 
27. 80 80 80 80 
28. 80 80 80 80 
29. 80 80 80 80 
30. 80 80 80 80 
Cost ($) 1,85,131.4 1,88,932.4 1,94,532.6 1,96,466.3 
RI 0.9407 0.9665 0.9827 0.9855 
Pipe No.W1 = 0.8, W2 = 0.2W1 = 0.6, W2 = 0.4W1 = 0.4, W2 = 0.6W1 = 0.2, W2 = 0.8
1. 300 350 350 350 
2. 200 250 300 300 
3. 200 200 250 250 
4. 200 200 250 250 
5. 150 200 200 250 
6. 150 150 200 200 
7. 125 150 200 200 
8. 200 200 250 250 
9. 80 80 80 80 
10. 80 80 80 80 
11. 80 80 80 80 
12. 80 80 80 100 
13. 80 100 125 150 
14. 80 80 80 80 
15. 100 100 125 125 
16. 80 80 80 80 
17. 80 80 80 80 
18. 80 80 80 80 
19. 100 100 100 125 
20. 80 80 80 80 
21. 80 80 80 80 
22. 80 80 80 80 
23. 80 80 80 80 
24. 80 80 80 80 
25. 80 80 80 100 
26. 80 80 80 80 
27. 80 80 80 80 
28. 80 80 80 80 
29. 80 80 80 80 
30. 80 80 80 80 
Cost ($) 1,85,131.4 1,88,932.4 1,94,532.6 1,96,466.3 
RI 0.9407 0.9665 0.9827 0.9855 
Figure 3

Go-Yang network and its Pareto Front.

Figure 3

Go-Yang network and its Pareto Front.

In this paper, a new methodology is proposed that eases the process of generating a set of non-dominated solutions for a multi-objective optimization problem. The computational methodology presented is known to converge to the results previously reported in the literature in many fewer iterations. A normalized function is applied for multi-objective optimization in three different benchmark problems using the Jaya algorithm, a newly developed parameter-less technique. The proposed methodology can generate a set of non-dominated solutions by deciding the priority of the objectives between them. The technique is found to be more convenient when compared with the a posteriori approach. No knowledge of the non-dominance is required, and different sets of Pareto Fronts are obtained that may be used as a pivot solution for further analysis. In addition to this, the involvement of more objectives in a problem may make it very hard to solve using the concept of non-dominance. However, for the method described in the present work, the designer just needs to decide the weights of every objective to obtain the desired results. It is worth mentioning that the a posteriori approach can lead to the non-dominance set in a single run while to generate the same Pareto Front in the a priori approach, different runs are made by varying the weights of the objective that makes the a priori approach more convenient and simpler. The present work indicates that by including the resilience index in the optimization model, better solutions are obtained that are less expensive but highly reliable. Jaya belongs to a parameterless algorithm and hence involves no constants, which further eases the analysis as no efforts are spent on tuning the algorithm. Similar results are obtained in 40–50 iterations using Jaya when compared with the 2,000 iterations taken by Differential Evolution for the Hanoi network; this further reaffirms the trust in Jaya as an evolutionary algorithm capable of solving multi-objective optimization.

All relevant data are included in the paper or its Supplementary Information.

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