https://orcid.org/0000-0001-6074-3824
Abstract
In this paper, an integration of a fuzzy-interval credibility-constrained (FIC) model expressed by fuzzy interval sets (FIS) and interval nonlinear programming (INP), termed an FIC-INP model, was proposed and applied to two cases. After formulating the FIC-INP model, upper and lower disinfection booster costs can be obtained for two cases under various combinations of credibility levels of upper and lower limits. The results indicated that the upper and lower booster costs increased with the credibility levels of the lower limits. The upper as well as the lower booster costs increased with the booster numbers. For credibility levels of lower limit constraints ranging from 0.6 to 0.9, the upper and lower booster costs increased under enlarging/widening trapezoidal distributions, and the intervals of booster costs increased with interval uncertainty. For a lower limit constraint credibility level of 0.5, the upper and lower booster costs decreased under shrinking/narrowing trapezoidal distributions, and the booster cost intervals increased under larger interval trapezoidal distributions. The degrees of effect of the interval uncertainty are higher than the fuzzy uncertainty. The results obtained can provide more information for managers to make booster schemes under fuzzy and interval uncertainties.
HIGHLIGHTS
A fuzzy interval set was proposed and applied to two water distribution systems.
FIS based on trapezoidal fuzzy distribution was proposed.
Various trapezoidal fuzzy distributions were compared.
Four combinations of credibility levels of upper and lower limit constraints were applied to obtain upper and lower booster costs.
Effects of fuzzy and interval uncertainties of FIS on booster costs were analyzed.
INTRODUCTION
Water utilities are required to supply acceptable water quantity and water quality in water distribution system (WDSs) to consumers under stringent government regulations. To meet the requirements of consumers, tap water should be supplied without pathogenic bacteria and odor. Chlorine as a disinfectant is usually injected at the outlet of water treatment plants to reduce the risk of waterborne diseases. The injection of disinfectants may cause increasing formation of disinfection by-products (DBPs) and the residual chlorine at the far-ends of the WDS may be not enough to prevent regrowth of microorganisms due to the decay process. The water quality in WDSs can be improved by placing disinfection boosters along the WDS. The boosters can be located at intermediate locations in the WDS, which can reduce injection dose and keep chlorine residuals within proper limits (Lansey et al. 2007). In addition, boosters can control the residual chlorine distributed uniformly (Köker & Altan-Sakarya 2015; Maheshwari et al. 2018).
To optimize the locations and injection rates of disinfection boosters, large number of models were proposed (Islam et al. 2017; Fisher 2019; Xin et al. 2019). Based on the linear superposition principle, a linear optimization model was firstly applied to minimize the total injection mass with given locations (Boccelli et al. 1998). A mixed integer linear programming model was developed to optimize the locations and scheduling of booster injection stations (Tryby et al. 2002). A nonlinear optimization model was proposed with the objective function of squared difference between calculated chlorine concentrations and the minimum specified concentration and solved by a genetic algorithm (GA) (Munavalli & Kumar 2003). A multi-objective model was formulated to investigate the booster locations and injection scheduling in the WDS and solved by a nondominated sorting genetic algorithm-II (NSGA-II) (Prasad et al. 2004). A linear least-squares (LLS) model was formulated with the objective function of minimizing the sum of the squared deviations of residual chlorine concentrations from a desired target with given locations and solved using the MATLAB Optimization Toolbox (Propato & Uber 2004). Conjunctive optimal scheduling of pumping and booster chlorine injections was performed in the WDS and solved by the genetic algorithm (GA) (Ostfeld & Salomons 2006). The locations and injection rates of boosters were determined by a two-step method (Lansey et al. 2007). To optimize the booster costs, GA was integrated with EPANET-MSX to deliver water at acceptable concentrations of residual chlorine and trihalomethanes (Ohar & Ostfeld 2014).
Since in the optimization both the model nodal demands and the chlorine decay coefficients have uncertainties, a two-stage stochastic mixed integer linear programming was proposed to optimize the locations and injection rates under uncertainty (Rico-Ramireza et al. 2007). To deal with uncertain hydraulic and water quality parameters that affect the booster scheduling, a chance-constrained programming (CCP) model was proposed and applied to optimize the booster locations and injection rates (Babayan et al. 2005). However, the CCP model cannot reflect the vague information expressed by fuzzy sets (Zhao et al. 2016). As such, a fuzzy chance-constrained programming (FCCP) model was proposed to optimize the booster injections (Wang & Zhu 2021). By considering managers' attitudes of optimism and pessimism, an improved mλ-measure optimization model was proposed to obtain booster injection rates with a combination of possibility and necessity measures (Wang 2021).
However, a deterministic membership function of fuzzy sets is difficult to be obtained in many cases. The crisp values of fuzzy distribution are usually between the upper bounds and the lower bounds. As such, fuzzy interval sets (FISs) were introduced in dealing with dual uncertainties of fuzziness and intervals (Zhang et al. 2018). The FIS was integrated with FCCP to optimize the water resources to better describe the uncertainty (Zhang et al. 2018). Most FISs are based on the triangular membership function. However, it is observed that the width of the flat line in trapezoidal distribution membership functions may reflect the precision of the parameters (Guan & Aral 2005). In this paper, a trapezoidal membership function of FIS was analyzed in the optimization of booster costs of WDSs to reflect the uncertain information better.
In this paper, an integration of a fuzzy-interval credibility-constrained (FIC) model and interval nonlinear programming (INP), termed as the FIC-INP model, is proposed in Section 2. The solution of the FIC-INP model is also presented in Section 2. Next, the model is applied to two WDSs to analyze the effect of fuzzy and interval uncertainties on the booster cost, and the results and discussions are presented in Section 3. Finally, conclusions are drawn to provide more schemes to managers to obtain optimal booster costs.
METHODOLOGY
Interval nonlinear programming (INP)
and 
refer to the interval variables, r refers to the exponent, m refers to the number of decision variables, and 
and 
refer to the upper and lower bounds of interval variables. However, INP cannot deal with the fuzzy interval information (Guo et al. 2015).Fuzzy-interval credibility-constrained interval nonlinear programming (FIC-INP) model
refers to the decision variables, 
refers to the coefficients in the objective function, 
, 
, and 
are fuzzy numbers with deterministic membership functions, which are usually expressed as 
, 
, and 
based on triangular fuzzy distribution, respectively, and 
and 
refer to credibility levels for upper and lower bound constraints.
(shown in Figure 1). 
Fuzzy interval set based on trapezoidal distribution.
refers to the coefficients in the constraints, respectively, 
and 
are fuzzy-interval sets expressed as 
and 
, respectively.
is the interval objective function including operation cost (OC) ($ day−1) and construction cost (CC) ($ day−1), DRV is the return value coefficient (day −1), AI is the annual interest (%), BLD is the booster chlorination life duration (years), 
is the interval decision variables representing the injection rates at time period q at booster station p (mg min−1), 
is the time duration for period q (min), nb is the booster number, nt is the time period number, 
and 
are the interval coefficients in the cost function, and 
is the interval maximum injection rate at time period q at booster station p (mg min−1), 
is the interval response coefficients matrix of chlorine concentration at the node j at monitoring time i to the unit injection rate at time period q at booster or source location p based on the superposition principle (Boccelli et al. 2003; Lansey et al. 2007), 
and 
are the fuzzy-interval sets for acceptable upper and lower chlorine concentration limits, and 
and 
are predetermined confidence levels for constraints 
and 
, respectively.Solution method
and 
). Since the objective function is to be minimized, the lower bounds sub-model should be solved first, which is presented by Equation (6) as follows:
The general framework of this study.
Step 1: Express the parameters with intervals and FIS.
Step 2: Formulate the FIC-INP model (Equation (5)).
Step 3: Transform the FIC-INP model into two equivalent sub-models.
Step 4: Solve the sub-model corresponding to the lower and upper bounds respectively.
Case study
Layouts of water distribution systems (a) Case 1 (b) Case 2.
The fuzzy-interval sets (FIS) for upper and lower bounds are set to be 
and 
, respectively, which is termed as the initial trapezoidal distribution of S0. The nodal chlorine response coefficient matrix 
can be obtained as the 0.9 times and 1.1 times of nodal chlorine response coefficient matrix to a unit amount of chlorine at the booster/source node, which is based on the last 24-h analysis results after performing hydraulic and water quality simulation for 960 hours.
In Case 1 the WDS consisted of 12 pipes, a reservoir at a water level of 243.8 m and a cylindrical tank at a ground level of 259.1 m (shown in Figure 3(a)). The pump has a shutoff head value of 101.3 m, and a maximum flow rate of 189.3 L/s. The nodal base demands range from 6.5 L/s to 13.0 L/s with 24 h-multipliers of 1.0, 1.0, 1.2, 1.2, 1.4, 1.4, 1.6, 1.6, 1.4, 1.4, 1.2, 1.2, 1.0, 1.0, 0.8, 0.8, 0.6, 0.6, 0.4, 0.4, 0.6, 0.6, 0.6, 0.8. The pipes’ Hazen Williams roughness coefficients and chlorine decay coefficient k0 are assumed to be 100 and −1.0/day, respectively. The reservoir and nodal initial residual chlorines are set to be 1.0 mg/L and 0.5 mg/L, respectively. In Case 2, the WDS is composed of one source node with a pump station, 34 consumer nodes, one storage tank, and 40 pipes (shown in Figure 3(b)). The pump located at node 1 has a negative demand of 4,400 × 105 m3/s with a certain pump demand multiplier. Node 1 is the source node, and node 9 and node 25 are considered to be probable booster locations, which are in accordance with other studies on the same WDS (Boccelli et al. 1998; Köker & Altan-Sakarya 2015; Wang & Zhu 2021) The global bulk and wall decay coefficients are set to be kb = −0.53/day and kw = −5.1 mm/day, respectively. The FIC-INP model can be solved by ‘Solver’ add-on in Microsoft Excel.
and 
in Equation (5)) on booster costs, four combinations of credibility levels of upper and lower limit constraints were summarized in Figure 4. In Figure 4(a), the credibility level for lower limit constraints increases from 0.5 to 0.9, while the credibility level for upper limit constraints remains as 0.5. In Figure 4(b), the credibility level for lower limit constraints remains as 0.5, while the credibility level for upper limit constraints increases from 0.5 to 0.9. In Figure 4(c), the credibility level for upper and lower limit constraints are the same. In Figure 4(d), the credibility level for lower limit constraints decreases from 0.9 to 0.5, while the credibility level for upper limit constraints increases from 0.5 to 0.9. 
Four combinations of credibility levels of upper and lower limit constraints (a) Increasing lower limit constraints (b) Increasing lower limit constraints (c) Increasing lower and upper limit constraints (d) Increasing lower limit constraints and decreasing upper limit constraints.
Four combinations of credibility levels of upper and lower limit constraints (a) Increasing lower limit constraints (b) Increasing lower limit constraints (c) Increasing lower and upper limit constraints (d) Increasing lower limit constraints and decreasing upper limit constraints.
(shown in Figure 5(a)) and 
(shown in Figure 5(b)), and the upper and lower bounds of chlorine concentration limits for S2 were set to be 
(shown in Figure 5(a)) and 
(shown in Figure 5(b)). In scenario 2 the trapezoidal distributions of FIS were widened and narrowed, and termed as S3 and S4, respectively. The upper and lower bounds of chlorine concentration limits for S3 were set to be 
(shown in Figure 5(c)) and 
(shown in Figure 5(d)), and the upper and lower bounds of chlorine concentration limits for S4 were set to be 
(shown in Figure 5(c)) and 
(shown in Figure 5(d)). 
The fuzzy uncertainty ((a)-(d)) and interval uncertainty ((e)-(f)) of fuzzy-interval set (FIS) (a) Enlarged and shrank trapezoidal FIS distributions for upper bounds (b) Enlarged and shrank trapezoidal FIS distributions for lower bounds (c) Widened and narrowed trapezoidal FIS distributions for upper bounds (d) Widened and narrowed trapezoidal FIS distributions for lower bounds (e) Greater FIS intervals for upper bounds (f) Less FIS intervals for lower bounds.
The fuzzy uncertainty ((a)-(d)) and interval uncertainty ((e)-(f)) of fuzzy-interval set (FIS) (a) Enlarged and shrank trapezoidal FIS distributions for upper bounds (b) Enlarged and shrank trapezoidal FIS distributions for lower bounds (c) Widened and narrowed trapezoidal FIS distributions for upper bounds (d) Widened and narrowed trapezoidal FIS distributions for lower bounds (e) Greater FIS intervals for upper bounds (f) Less FIS intervals for lower bounds.
To compare the interval uncertainty of FIS on booster costs, the FIS intervals were made bigger and smaller, and termed as S5 and S6, respectively. For S5 the upper and lower bounds of chlorine concentration limits were set to be 
(shown in Figure 5(e)) and 
(shown in Figure 5(f)), and for S6 the upper and lower bounds of chlorine concentration limits were set to be 
(shown in Figure 5(e)) and 
(shown in Figure 5(f)).
RESULTS AND DISCUSSION
Application to Case 1
). In the combination sets of A2, B2, C2, D2, and E2 as shown in Figure 4(b), the lower optimal booster costs are the same value of $ 19.42/day, and the upper optimal booster costs are the same value of $ 41.81/day (shown in Figure 6(b)). The results indicated that the lower and upper booster costs are not affected by the credibility level for upper limit constraints (
). In the combination sets of A3, B3, C3, D3, and E3 as shown in Figure 4(c), the lower bounds and upper optimal booster costs have the same values as the optimal booster costs for the combination sets of A1, B1, C1, D1, and E1 (shown in Figure 6(c)). The results also indicated that the booster costs are affected only by the credibility level for lower limit constraints (
). In the combination sets of A4, B4, C4, D4, and E4 as shown in Figure 4(d), the lower optimal booster costs are $ 25.91/day, $ 25.45/day, $ 24.98/day, $ 24.52/day, and $ 19.42/day, respectively. The upper optimal booster costs are $ 72.59/day, $ 70.39/day, $ 68.19/day, $ 65.99/day, and $ 41.81/day, respectively (shown in Figure 6(d)). The results also indicated that the booster costs decrease with the reduction of lower limit constraints (
). The reason is that that the function's objective is to minimize the booster costs. 
Booster costs for four combination sets of scenarios of Case 1 (a) Combination scenario1 (b) Combination scenario 2 (c) Combination scenario 3 (d) Combination scenario 4.
Booster costs for four combination sets of scenarios of Case 1 (a) Combination scenario1 (b) Combination scenario 2 (c) Combination scenario 3 (d) Combination scenario 4.
). As such, the constraints of Equations (6c) and (7c) are considered. For the combination set of A1, the S1 and S2 trapezoidal distribution has no effects on the lower and upper booster costs. The reason is that for S1 and S2 under the combination set of A1 with 
, the values of 
in Equations (6c) and (7c) are the same as for S0, which suggests that S1 and S2 trapezoidal distribution has no effects on the lower and upper booster costs for the combination set of A1. In addition, the values of 
increased for S1 and decreased for S2, which leads to the increase of booster costs for S1 and the decrease of booster costs for S2. The results indicated that the fuzzy uncertainty from enlarging or shrinking the trapezoidal FIS distribution can increase or decrease the lower and upper booster costs except for the condition of 
. 
Booster costs for various fuzziness from trapezoidal distributions for Case 1 (a) Lower booster costs under S0, S1, and S2 trapezoidal distributions (b) Upper booster costs under S0, S1, and S2 trapezoidal distributions (c) Lower booster costs under S0, S3, and S4 trapezoidal distributions (d) Upper booster costs under S0, S3, and S4 trapezoidal distributions (e) Lower booster costs under S0, S5, and S6 trapezoidal distributions (f) Upper booster costs under S0, S5, and S6 trapezoidal distributions.
Booster costs for various fuzziness from trapezoidal distributions for Case 1 (a) Lower booster costs under S0, S1, and S2 trapezoidal distributions (b) Upper booster costs under S0, S1, and S2 trapezoidal distributions (c) Lower booster costs under S0, S3, and S4 trapezoidal distributions (d) Upper booster costs under S0, S3, and S4 trapezoidal distributions (e) Lower booster costs under S0, S5, and S6 trapezoidal distributions (f) Upper booster costs under S0, S5, and S6 trapezoidal distributions.
Under the combination sets of A1, B1, C1, D1, and E1 (shown in Figure 4(a)) for S3 (shown in Figure 5(c) and 5(d)), the lower booster costs are $ 18.64/day, $ 25.14/day, $ 25.45/day, $ 25.76/day, and $ 26.07/day, respectively (shown in Figure 7(c)), and the upper booster costs are $ 40.24/day, $ 67.25/day, $ 69.13/day, $ 71.02/day, and $ 72.90/day, respectively (shown in Figure 7(d)). Under the combination sets of A1, B1, C1, D1, and E1 (shown in Figure 4(a)) for S4 (shown in Figure 5(c) and 5(d)), the lower booster costs are $ 20.19/day, $ 23.90/day, $ 24.52/day, $ 25.14/day, and $ 25.76/day, respectively (shown in Figure 7(c)), and the upper booster costs are $ 43.38/day, $ 64.74/day, $67.25/day, $ 69.76/day, and $ 72.27/day, respectively (shown in Figure 7(d)). Compared with the initial trapezoidal distribution (S0), the lower and upper booster costs increased for S3 except for the combination set of A1, and decreased for S4 except for the combination set of A1. The lower and upper booster costs under the combination of A1 for S3 and S4 are not the same values as the initial trapezoidal distribution (S0). The reason is that under the combination set of A1 with 
the values of 
in Equations (6c) and (7c) decreased for S3 and increased for S4, which leads to the lower and upper booster costs decreasing for S3 trapezoidal distribution and increasing for S4 trapezoidal distribution. In addition, the values of 
increased for S3 and decreased for S4, which leads to the increase of booster costs for S3 and the decrease of booster costs for S4. The results indicated that the fuzzy uncertainty of widening or narrowing the trapezoidal FIS distribution can also increase or decrease the lower and upper booster costs except for the condition of 
.
The booster costs under various intervals of FIS were obtained by performing the FIC-INP model. Under the combination sets of A1, B1, C1, D1, and E1 (shown in Figure 4(a)) for S5 (shown in Figure 5(e) and 5(f)), the lower booster costs are $ 19.42/day, $ 22.82/day, $ 23.13/day, $ 23.44/day, and $ 23.75/day, respectively (shown in Figure 7(e)), and the upper booster costs are $ 43.38/day, $ 71.02/day, $ 73.53/day, $ 76.04/day, and $ 78.55/day, respectively (shown in Figure 7(f)). Under the combination sets of A1, B1, C1, D1, and E1 (shown in Figure 4(a)) for S6 (shown in Figure 5(e) and 5(f)), the lower booster costs are $ 19.42/day, $ 26.22/day, $ 26.84/day, $ 27.46/day, and $ 28.07/day, respectively (shown in Figure 7(e)), and the upper booster costs are $ 40.24/day, $ 60.97/day, $ 62.86/day, $ 64.74/day, and $ 66.62/day, respectively (shown in Figure 7(f)). Compared with the initial trapezoidal distribution (S0), the lower booster costs for S5 decreased except for the combination set of A1, while the upper booster costs increased for S5. However, the lower booster costs for S6 increased except for the combination set of A1, while the upper booster costs decreased for S6. The reason is that under the combination set of A1 with 
, the values of 
are the same as for S0, and the values of 
increased for S5 and decreased for S6. As such, the lower booster costs for S5 and S6 are the same as for S0, while the upper booster costs increased for S5 and decreased for S6. In addition, the values of 
and 
for S5 and S6 are the same as for S0, while the values of 
and 
increased for S5 and decreased for S6. As such, the lower booster costs decreased for S5 and increased for S6, and the upper booster costs increased for S5 and decreased for S6. The results indicated that with the increase of FIS intervals, the lower booster costs decreased while the upper booster costs increased, i.e., the booster cost interval also increased. In addition, with the decrease of FIS intervals, the lower booster costs increased while the upper booster costs decreased, i.e., the booster cost interval also decreased.
and FIS trapezoidal distributions for Case 1 are shown in Figure 8. The nodal average lower and upper chlorine concentration under S0 and A1 
range from 0.25 mg/L to 0.42 mg/L and 0.34 mg/L to 0.56 mg/L, respectively (shown in Figure 8(a)). The nodal average lower and upper chlorine concentration under S0 and E1 
range from 0.34 mg/L to 0.57 mg/L and 0.59 mg/L to 0.98 mg/L, respectively (shown in Figure 8(b)). The results indicated that with the increase of 
the lower and upper booster costs increased, which leads to the increase of nodal average chlorine concentration. To compare the nodal average chlorine concentrations under various FIS trapezoidal distributions, the scenario of C1 
was selected. The results indicated that under S1 and S3 trapezoidal distributions the lower booster cost increased, which leads to the increase of nodal lower average chlorine concentration, while under S5 trapezoidal distributions the lower booster cost decreased, which leads to the decrease of nodal lower average chlorine concentration (shown in Figure 8(c)). In addition, under S1 and S3 trapezoidal distributions the upper booster cost increased, which leads to the increase of nodal upper average chlorine concentration, while under S5 trapezoidal distributions the upper booster cost increased, which leads to the increase of nodal lower average chlorine concentration (shown in Figure 8(d)). 
Nodal average chlorine concentration under various 
and FIS trapezoidal distributions for Case 1 (a) Nodal average lower and upper chlorine concentration under scenario S0 and A1 
(b) Nodal average lower and upper chlorine concentration under scenario S0 and E1 
(c) Nodal average lower chlorine concentration under scenario S0, S1, S3 and S5 and C1 
(d) Nodal upper chlorine concentration under scenario S0, S1, S3 and S5 and C1 
.
Nodal average chlorine concentration under various and FIS trapezoidal distributions for Case 1 (a) Nodal average lower and upper chlorine concentration under scenario S0 and A1 (b) Nodal average lower and upper chlorine concentration under scenario S0 and E1 (c) Nodal average lower chlorine concentration under scenario S0, S1, S3 and S5 and C1 (d) Nodal upper chlorine concentration under scenario S0, S1, S3 and S5 and C1 .
Application to case 2
In Case 2, three nodes are considered as optimal booster locations: node 1, node 9, and node 25 (shown in Figure 3(b)). By applying FIC-INP to Case 2, the lower and upper values of booster costs are obtained (shown in Table 1). Similarly to Case 1, under the combination sets of A2, B2, C2, D2, and E2 the lower optimal booster costs are the same value of $ 5.78/day, and the upper optimal booster costs are the same value of $ 10.78/day. In addition, under the combination sets of A3, B3, C3, D3, and E3, the lower and upper optimal booster costs are the same as optimal booster costs under the combination sets of A1, B1, C1, D1, and E1.
The booster costs under four combination sets for Case 2 ($/day)
| Combination sets | A1 | B1 | C1 | D1 | E1 |
|---|---|---|---|---|---|
| Cost | [5.78, 10.78] | [7.18, 16.69] | [7.30, 17.22] | [7.43, 17.76] | [7.56, 18.29] |
| Combination sets | A2 | B2 | C2 | D2 | E2 |
| Cost | [5.78, 10.78] | [5.78, 10.78] | [5.78, 10.78] | [5.78, 10.78] | [5.78, 10.78] |
| Combination sets | A3 | B3 | C3 | D3 | E3 |
| Cost | [5.78, 10.78] | [7.18, 16.69] | [7.30, 17.22] | [7.43, 17.76] | [7.56, 18.29] |
| Combination sets | A4 | B4 | C4 | D4 | E4 |
| Cost | [7.56, 18.29] | [7.43, 17.76] | [7.30, 17.22] | [7.18, 16.69] | [5.78, 10.78] |
| Combination sets | A1 | B1 | C1 | D1 | E1 |
|---|---|---|---|---|---|
| Cost | [5.78, 10.78] | [7.18, 16.69] | [7.30, 17.22] | [7.43, 17.76] | [7.56, 18.29] |
| Combination sets | A2 | B2 | C2 | D2 | E2 |
| Cost | [5.78, 10.78] | [5.78, 10.78] | [5.78, 10.78] | [5.78, 10.78] | [5.78, 10.78] |
| Combination sets | A3 | B3 | C3 | D3 | E3 |
| Cost | [5.78, 10.78] | [7.18, 16.69] | [7.30, 17.22] | [7.43, 17.76] | [7.56, 18.29] |
| Combination sets | A4 | B4 | C4 | D4 | E4 |
| Cost | [7.56, 18.29] | [7.43, 17.76] | [7.30, 17.22] | [7.18, 16.69] | [5.78, 10.78] |
). In addition, for two booster locations the lower and upper costs for node 1 and node 25 are less than the lower and upper costs for node 1 and node 9. The results indicated that a booster located far away from the source is beneficial for making residual chlorine distribute uniformly, and can decrease the injection rates so as to decrease the upper and lower booster costs. 
The lower (a) and upper (b) booster costs for Case 2 under scenario set 1.
) except for the combination set of A1. Based on the analysis for Case 1, S1 trapezoidal distribution has no effects on the lower and upper booster costs for the combination set of A1. For S1 corresponding to an enlarged fuzzy trapezoidal distribution, the values of 
in Equation (6c) and 
in Equation (7c) are greater than the values for S0, which leads to the degree of effect of S1 trapezoidal distribution on the lower and upper booster costs increasing with the increase of lower limit constraints (
). Moreover, the degree of effect on the lower booster costs ranging from 0.00 to 2.23 are higher than the degree of effect on upper booster costs ranging from 0.00 to 1.67. The reason is that although the increase of 
and 
leads to the increase of upper and lower booster costs, the increases in the right-hand side of the constraints in Equation (6c) are greater than Equation (7c), which leads to higher increases in the lower booster costs than the upper booster costs. The results also indicated that the fuzzy uncertainty achieved by enlarging the trapezoidal FIS distribution increases the booster costs, especially the lower booster costs. 
Degree of effect of fuzzy and interval uncertainties of FIS S1, S3 and S5 on booster costs for Case 2 (a) Lower boost cost (b) Upper booster cost.
Degree of effect of fuzzy and interval uncertainties of FIS S1, S3 and S5 on booster costs for Case 2 (a) Lower boost cost (b) Upper booster cost.
Based on the analysis for Case 1, compared with S0 the lower and upper booster costs under S3 trapezoidal distribution decreased for the combination set of A1, and increased for the combination sets of B1, C1, D1, and E1. However, the degrees of increase of S3 for the combination sets of B1, C1, D1, and E1 on the lower and upper booster costs were reduced with the increase of lower limit constraints (
). For S3 corresponding to widening fuzzy trapezoidal distribution, the values of 
and 
in Equations (6c) and (7c) are greater than the values for S0, which leads to the degree of effect of S3 trapezoidal distribution on the lower and upper booster costs reducing with the increase of lower limit constraints (
). The degree of effect of S3 trapezoidal distribution on the lower booster costs ranging from 0.56 to 2.35 are also higher than the upper booster costs ranging from for 0.42 to 1.83 for the other combinations. The reason is that although the increase of 
and 
leads to the increase of upper and lower booster costs, the increases in the right-hand side of the constraints in Equation (6c) are greater than Equation (7c), which leads to higher increases in the lower booster costs than the upper booster costs. The results also indicated that the fuzzy uncertainty achieved by widening the trapezoidal FIS distribution also increased the booster costs, especially the lower booster costs. The degrees of effect of S3 on lower booster costs are relatively higher than S1 under the combination sets of B1 and C1, while the degrees of effect of S3 on upper booster costs are less than S1 under the combination sets of D1 and E1.
Under the effects of S5 trapezoidal distribution, the lower booster costs are less than S0 and the upper booster costs are higher than S0, i.e., the interval of booster costs increased. Based on the analysis for Case 1, under S5 trapezoidal distribution the lower booster costs are the same with S0 for A1 and decreased for the other combinations, while the upper booster costs increased for all the combinations. In addition, the decreasing and increasing magnitudes increase with the increase of lower limit constraints (
). For S5 corresponding to larger interval trapezoidal distribution, the values of 
in Equations (6c) and (7c) are greater than the values for S0, which leads to the degree of effect of S5 trapezoidal distribution on the lower booster costs decreasing and upper booster costs increasing with the increase of lower limit constraints (
). In addition, the degrees of effect of interval uncertainty of FIS ranging are higher than the fuzzy uncertainty of FIS.
was also selected to compare the nodal average chlorine concentrations under various FIS trapezoidal distributions. The results also indicated that under S1 and S3 trapezoidal distributions the lower booster cost increased, which leads to the increase of nodal lower average chlorine concentration, while under S5 trapezoidal distributions the lower booster cost decreased, which leads to the decrease of nodal lower average chlorine concentration (shown in Figure 11(a)). In addition, under S1 and S3 trapezoidal distributions the upper booster cost increased, which leads to the increase of nodal upper average chlorine concentration, while under S5 trapezoidal distributions the upper booster cost increased, which leads to the increase of nodal upper average chlorine concentration (shown in Figure 8(d)). The lower and upper average nodal chlorine concentrations for Case 2 are greater than Case 1. The reason is due to the fact that in Case 2 the chlorine decay coefficients are much lower than in Case 1. 
Nodal average chlorine concentration under C1 and various FIS trapezoidal distributions S0, S1, S3 and S5 for Case 2 (a) Lower chlorine concentration (b) Upper chlorine concentration.
Nodal average chlorine concentration under C1 and various FIS trapezoidal distributions S0, S1, S3 and S5 for Case 2 (a) Lower chlorine concentration (b) Upper chlorine concentration.
CONCLUSION
In this paper, an integration of fuzzy-interval credibility-constrained and interval nonlinear programming (FIC-INP) was proposed and applied to two WDSs. The results indicated that only the credibility levels of lower limits have effects on lower and upper booster costs, and with the increase of the credibility levels of the lower limits, the upper and lower booster costs increased. In addition, fuzzy uncertainty from enlarging and widening the trapezoidal FIS distribution can increase the booster costs, especially the lower booster costs, while larger interval uncertainty trapezoidal distribution can decrease the lower booster costs and increase the upper booster costs. The effects of interval uncertainty are higher than the fuzziness uncertainty. With the increase of booster number, the injection rates decreased. In addition, under the same booster numbers, the location far away from the source is more economical. The results obtained can supply more information for managers to make decisions about booster schemes under fuzzy and interval uncertainties. In addition, the results can help managers determine booster numbers under limited budgets.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict of interest.
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