A Coupled Hydrodynamic Multi-Objective Optimization Formulation for Seasonal Water-Quality Management in the Kufa River, Iraq

2.1 Kufa River section

The Euphrates River branches into the Kufa and Abbasia branches in the upper reaches of Najaf Governorate. The Kufa River flows for a distance of 75.2 km within the governorate through the districts of Kufa, Manadhira, and Mishkhab, providing drinking, industrial, and irrigation water Kareem and Alkatib [11]. The population of Al-Najaf governorate exceeds 1.9 million according to the 2024 census data issued by the Iraqi Ministry of Planning's Statistics Authority (2024). The study section extends 42.467 km from the Kufa barrage (station 0 + 00) to the Abu Sukhair regulator, including three main intakes: Najaf intake at Zarqa region with a discharge of 3.333 m³/sec (withdrawal), Kufa intake (1.111 m³/sec- withdrawal), and Manadhira intake (0.556 m³/sec- withdrawal) with seven-point discharge sources (S1-S7) involving drains, municipal water, pipes, and pumping stations. The largest discharge is the Barakiya outfall S5 (1.157 m³/sec), followed by the northern Kufa drain S2 (0.583 m³/sec). The irregular intake and discharge mean that concentrations at each intake are not governed by a single source, but rather by the cumulative mass balance of mixing, longitudinal diffusion, biochemical transformation, and runoff, making management a hydrodynamic and water quality-related distribution issue. Figure 1 shows the study area within the Al-Najaf governorate in Iraq, and Figure 2 shows the location of the stations along the Kufa River reach, after  [12]. Table 1 shows the longitudinal orders of the sources and intakes along the study reach.

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Figure 1. Layout map of Iraq and the Al-Najaf governorate

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Figure 2. Layout map for the study stations along the Kufa River reach

Table 1. Longitudinal positions of the point sources and intakes along the study reach

Note: “–” denotes boundary or intake locations without a source-station code.

2.2 Flow systems

Table 2 presents the multiple flow systems identified statistically from extreme flow frequency analysis of annual flow records for the period 2004-2022, which is the period over which the reach was characterized for the present management analysis.

Table 2. Descriptive statistics for annual flow records (2004–2022) for the Kufa River

From the results of Table 2, it is evident that the large standard deviations reflect the high annual variability and the river flow sensitivity to climatic stress. Frequency analysis with the two-parameter Weibull distribution for the annual maximum series and the three-parameter Weibull distribution for the annual minimum series provided design discharges to simulate extreme conditions: a 50-year minimum discharge of 30.495 m³/sec (for the dry scenario) and a 50-year maximum discharge of 243.159 m³/sec (for the flood scenario), surrounding the normal operation of the calibrated baseline. These 50-year quantiles are extrapolated from a 19-year record and are therefore used only as specific hydraulic stress scenarios; the water-quality optimization described below is performed on the calibrated seasonal baseline states corresponding to the two sampling rounds, not on the 50-year quantiles.

The frequency analysis for two design discharges is given in Table 3.

Seven probability distributions were fitted to each series by maximum likelihood and ranked by the Akaike information criterion, with the Bayesian criterion, the Kolmogorov-Smirnov and Anderson-Darling probabilities and confidence intervals presented in Table 3. Every distribution is accepted by the Kolmogorov–Smirnov test at the 5% level; the whole ΔAIC range is 5.29 for the maximum series and only 2.24 for the minimum series. For n = 19, such differences are not evidence of a preferred model. More importantly for the use made of these quantiles here, the design discharges adopted in this work, 243.159 and 30.495 m³/sec, lie inside the ninety-five per cent confidence interval of all seven candidates in both series, so the choice of distribution does not materially alter the flow scenarios or any result that follows from them.

Table 3. Comparison of probability distributions fitted to the annual maximum and annual minimum series of the Kufa River, 2004–2022

Note: k is the number of fitted parameters. ΔAIC is measured from the lowest value within each class. KS and AD are the Kolmogorov-Smirnov and Anderson-Darling probabilities; a value above 0.05 indicates that the fit is not rejected.

The two-parameter Weibull distribution was adopted for the annual maximum series, and the three-parameter Weibull distribution was adopted for the annual minimum series, for consistency with the antecedent hydrological analysis of the same reach [13] and because both yield stable intervals; the two-parameter form gives the narrowest interval of all seven candidates for the maxima, 204.9 to 277.3 m³/sec. Two probability distributions were examined and set aside on grounds of numerical behavior rather than fit. The generalized extreme value distribution returns an effectively unbounded upper confidence limit for the maximum series and a physically impossible negative lower limit for the minimum series, both symptomatic of an unconstrained shape parameter on a short record. The log-Pearson type III fit to the minimum series is rejected by the Kolmogorov–Smirnov test, and its Anderson-Darling probability of unity is an artefact of a negative fitted skew rather than evidence of a good fit.

The three-parameter Weibull attains the lowest AIC in both panels, but by margins of 2.94 and 0.15, respectively, and its location parameter is determined almost entirely by the smallest observation, so the fitted 50-year minimum is sensitive to a single data point. The intervals themselves are wide, covering 29% of the point estimate for the adopted maximum and 45% for the adopted minimum. Probability plots of the best-fitting distribution for the annual maximum and annual minimum series are shown in Figure 3.

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Figure 3. Fitted probability distributions for the discharge series, 2004–2022: (a) Annual maximum and (b) annual minimum

Table 4 states the correspondence between flow scenarios, simulation purpose, and water-quality data.

Table 4. Assignment of simulations and water-quality observations to flow scenarios

Note: HD: hydrodynamic module; AD: Advection–Dispersion module; ECO Lab: Ecological Laboratory module.

2.3 Water quality monitoring and governing standards

Water samples were collected at seven monitoring stations, S1–S7, of which S2–S7 represent point discharges, and S1 represents the Al-Zarqa water intake, during both the wet and dry seasons and analyzed for a series of physicochemical properties. The four components transferred to the management model, TDS, BOD, NO₃, and PO₄, were adopted because they combine importance (all limited by the Iraqi Quality Standard IQS for drinking water) with strong sensitivity to both point load and river flow. Tables 5 and 6 show the seasonal baseline concentrations at the monitoring stations. Samples were collected from 22 sites in two rounds, in March 2024 for the wet season and in August 2024 for the dry season. For each point discharge, samples were collected at the discharge location and at points 50 m upstream and 50 m downstream, so that the local effect of each source can be separated from the ambient river concentration; an additional station was located 750 m upstream of the Alboshakhair pipe (S4). Site coordinates were fixed by GPS, samples were stored in coded, pre-rinsed 1.5 L bottles and transported to the laboratory, and 22 parameters were determined and compared against the Iraqi and WHO drinking-water standards; four were carried into the management model. A single round was taken in each season; this is a limitation of the field program and is reflected in the deterministic treatment of the source terms throughout. Because an allocation problem is governed by mass rather than by concentration, the load of each discharge, L = 86.4 QC, was computed from the discharges of Table 1: in the wet season S5 carries 56% of the solids load and 83% of the PO₄ load, S2 ranks second on solids at 17% and S7 second on PO₄ at 8%, while in the dry season the totals fall to 77% and 20% of their wet-season values and the distribution flattens, the share of S5 dropping to 49% and 45% as that of S2 rises to 18% and 20% (Tables 7 and 8).

Table 5. Basic concentrations of modeled components, wet season (mg/L)

Table 6. Basic concentrations of modeled components, dry season (mg/L)

Note: TDS: Total Dissolved Solids; BOD: Biochemical Oxygen Demand.

Table 7. Wet-season pollutant loads of the six point discharges (kg/d)

Note: TDS: Total Dissolved Solids; BOD: Biochemical Oxygen Demand.

Table 8. Dry-season pollutant loads of the six point discharges (kg/d)

Note: TDS: Total Dissolved Solids; BOD: Biochemical Oxygen Demand.

Table 9. Inspection of the six-point discharges against the selection criteria

Note:  ^*The basis of the selection is administrative rather than hydraulic. TDS: Total Dissolved Solids.

The wet season carries the most severe source loading at S5 (Al-Barakiya): TDS = 1943 mg/L, and PO₄ = 9.127 mg/L, values that far exceed drinking water limits and establish S5 as the main pollution stress for the section. High PO₄ is also observed at S7 (4.04 mg/L) and S3 (2.294 mg/L). In the dry season, absolute TDS is generally lower, but chemical concentrations remain concerning, and reduced dilution increases the effect of each source, thus contributing to the seasonal irregularity. Averaged over the three controllable sources, the wet-season baselines exceed the dry-season values by factors of 1.26 for TDS, 5.09 for BOD, 1.86 for nitrate, and 4.31 for phosphate. This reversal of the usual dilution expectation is consistent with the wet-season round agreeing with the irrigation-drainage return-flow period, when the drains convey leached salts and fertilizer-derived nitrogen and phosphorus from the surrounding irrigated land.

The standard limits exerted at the intakes are the drinking water values according to the IQS417/2009: TDS ≤ 1000 mg/L, BOD ≤ 6 mg/L, and NO₃ ≤ 50 mg/L, applied equally in both seasons. Phosphates are controlled with an objective limit of 0.5 mg/L (consistent with the WHO guideline value; the strictest environmental value in the IQS is 0.4 mg/L), but are treated with a penalty rather than a strict restriction.

2.4 Selection of controllable sources and constrained intakes

Three of the six point discharges, S3, S4 and S5, are treated as controllable. Table 9 shows the position, discharge, and load rank of each source, along with the basis for the decision.

4.1 Sets, indices, and decision variables

The set of controllable sources is the discharges from S3, S4, and S5; the assessment stations S1-S7 are included, in addition to the two priority intakes (Kufa and Al-Manadhira); the component (pollutants) set is {TDS, BOD, NO₃, PO₄}; and the season set is {wet, dry}, equally weighted (γ_Wet = γ_Dry= 0.5), where Σγ_s = 1. The problem is defined by the indices as given in Table 15.

Table 15. Sets of indices in the optimization problem

The decision variables are the partial removal ratios (r):

$r(i, k, s) \in\left[0, r_{\max (i, k)}\right], s \in\{$ Wet, Dry $\}$   (2)

With one ratio for each source-component-season combination. With three sources, four components, and two seasons, the dimension of the decision vector Nd = 3 × 4 × 2 = 24. The upper bound r_max is set at 80% removal as the upper limit feasible with conventional processing, and is adjusted for each source-component pair where necessary. It is then multiplied by the investment-driven capacity factor η(i, k) ∈ (0, 1], representing the traditional processing capacity as follows:

$0 \leq r(i, k, s) \leq r_{\max .(i, k)} \times \eta(i, k)$   (3)

4.2 Objective functions

The management problem is to minimize the following multi-objective vector:

$O b j .=\left[O b j_1, O b j_2, O b j_3, O b j_4, O b j_5\right]^T$   (4)

Define the exceedance coefficients used in the entire formulation to measure the degree to which the predicted concentration is above its upper limit U_k or falls below the minimum requirement L_k:

$\Phi^{+}(j, k, s)=\max .\left\{0, C(j, k, s)-U_k\right\}$   (5)

$\Phi^{-}(j, k, s)=\max .\left\{0, L_k-C(j, k, s)\right\}$   (6)

Environmental Risk (Obj₁): A seasonally and component-weighted quadratic for all standard breaches (according to IQS) across each station captures both high excesses and deficiencies [16]:

$O b j_1=\sum_{s \in\{D r y, W e t\}} \gamma_s \sum_j \sum_k w_k\left[\left\{\Phi^{+}(j, k, s)\right\}^2+\left\{\Phi^{-}(j, k, s)\right\}^2\right]$   (7)

With component (pollutant) weights of w_k> 0, Σwₖ = 1. The quadratic formula extremely penalizes large violations, directing the search towards approximately compliant fields rather than solutions that concentrate all remaining violations at a single station.

Discharge-Weighted Removal Effort (Obj₂): The seasonally weighted operating cost of the removal plan, measured by the source discharge:

$O b j_2=\sum_{s \in\{D r y, W e t\}} \gamma_s \sum_{i, k} \beta(i, k) \times Q(i) \times r(i, k, s)$    (8)

where, β(i, k) is the unit treatment cost factor and Q(i) is the source discharge (m³/sec). Multiplying by Q(i) makes the cost proportional to the volume treated, so removals at high-flow sources such as S5 are correspondingly more expensive. The coefficient β(i, k) was set to unity for every source and parameter, a neutral choice adopted because site-specific unit costs were not available for the three controllable outfalls. Obj₂ therefore measures the discharge-weighted removal effort rather than a monetary cost, and it does not distinguish between processes whose real unit costs differ by orders of magnitude, membrane treatment for dissolved solids being far more costly than biological treatment for BOD or chemical precipitation for phosphate.

Water Quality Index at Intakes (Obj₃) is defined as the negative of the priority-weighted index and is minimized, so a more negative value corresponds to higher delivered quality [17]:

$O b j_3=-\sum_{s \in\{D r y, W e t\}} \gamma_s \sum_{j \in \text { Intakes }} \lambda(j) \times W Q I(j, s)$   (9)

where, λ(j) is the intake priority weight (higher for drinking water), and WQI(j, s) is the water quality index for each location (j) at season (s). The water quality index used in Obj₃ is the weighted arithmetic index. Three relations define it: the index itself, the sub-index of each parameter, and the weight assigned to each parameter:

$W Q I(j, s)=\frac{\sum_k w_k q_k(j, s)}{\sum_k w_k}$   (10a)

$q_k(j, s)=\frac{100\left[C(j, k, s)-C_{i e}(k)\right]}{\left[S_k-C_{i e}(k)\right]}$   (10b)

$w_k=\frac{K}{S_k}$ with $K=\frac{1}{\sum_k\left(1 / S_k\right)}$   (10c)

In Eq. (10b), the ideal concentration C_ie(k) is zero for all four parameters, so the normalization is fully specified by the standards Sₖ given in Table 16. On this scale, a larger index denotes poorer water: values below 25 are classed as excellent, 26 to 50 as good, 51 to 75 as poor, and larger values as very poor or unsuitable.

Table 16. Control limits adopted and their treatment in the optimization

Note: TDS: Total Dissolved Solids; BOD: Biochemical Oxygen Demand.

Spatial Equity (Obj₄): To prevent residual pollution from concentrating on a subset of sections, the seasonally weighted variance of the overflow load is minimized across the equity stations. With the weighted overflow load at station j,

$\begin{gathered}E s(j, s)=\sum_k w_k \times \operatorname{Max} .\left\{0, C(j, k, s)-U_k\right\} \\ \forall j \in \Omega_{e q,} s \in\{\text {Wet}, \text {Dry}\}\end{gathered}$    (11a)

The equity objective is:

$\mathrm{Obj}_4=\sum_{s \in\{\mathrm{Dry}, \text {Wet}\}} \gamma_s \cdot \operatorname{Var} \cdot\{\mathrm{Es}(j, s)\}$   (11b)

$\operatorname{Var} .\{E s(j, s)\}=\frac{1}{n} \sum[E s(j, s)-\bar{E} s]^2$   (11c)

where,

$\bar{E} s=\frac{1}{n} \sum E s(j, s)$   (11d)

n = |Ωeq.| is the number of equity monitoring stations. Minimizing this variance distributes the processing load and residual load evenly across the section.

Phosphate Penalty (Obj₅): Phosphate is not strictly constrained because dry-season mitigation is often insufficient to drive PO₄ at intakes below the target limit of 0.5 mg/L within technically feasible removals; a strict constraint would empty the feasible region. Instead, a routine quadratic penalty directs the search toward the lowest possible phosphate residues at intakes:

$\begin{gathered}O b j_5=\sum_{s \in\{D r y, W e t\}} \gamma_s \\ \sum_{j \in \Omega_{\text {Intakes}}} \sum_k\left[\operatorname{Max} .\left(0, C(j, s, k)-U_k\right)\right]^2\end{gathered}$   (12)

The minimum is set by the involved dry-season residue at the Al-Manadhira intake, which alone contributes 0.093 of the 0.113 minimum. These design strict constraints for legally feasible components, and a flexible penalty for the physically under-controllable parameters, are the methodological mark of the strict intake constraint case.

The five objectives Obj₁, Obj₄ and Obj₅ are weighted squared concentrations with the dimension (mg/L)², and range over 0.154–429.97, 0.002–35.73 and 0.113–0.144, respectively. Obj₃ is the negative of a dimensionless index on a 0–100 scale and ranges from -187.39 to -182.39. For Obj₂, site-specific unit treatment costs were not available for the three controllable outfalls, so β (i, k) = 1 for every source-parameter pair. Therefore, Eq. (7) reduces to a discharge-weighted sum of removal fractions with the dimension m³/s, ranging over 0.898–2.969. It is therefore a measure of treatment effort and not of cost. The membrane treatment for dissolved solids is far more expensive than biological treatment for BOD or chemical precipitation for PO₄, and a costed formulation would redistribute effort towards the cheaper parameters and displace the front. All five objectives are minimized.

The objectives were not normalized during the search. In gamultiobj, the crowding distance that governs diversity is accumulated over raw objective values, so an objective spanning three orders of magnitude contributes far more to the ranking than one covering 0.031. Obj₁ and Obj₄ therefore dominate the diversity measure while Obj₅ is very nearly invisible to it, with two identifiable effects: the returned front is well resolved in the risk-effort plane and lightly resolved in phosphate penalty, and the stopping rule, which observes the relative change in the range of the front, is driven almost by the two large-range objectives, so minor convergence can be reached while the small-range objective is still changing. Neither effect biases the front itself; Pareto dominance is invariant under any routine rescaling of the individual objectives, so the set of non-dominated points is unchanged, but both affect which members of that set are returned and when the search stops. The effect of this on the selected solution was tested directly: the compromise solution was identified with min–max normalization and with max-absolute scaling, and is unchanged in both cases, as reported in Section 6.1.

4.3 Constraints

The optimization problem is subject to the following constraints. Mass balancing relates the decision variables to the concentrations predicted by the response matrix [18]:

$C(j, k, s)=C^0(j, k, s)-\sum_i A_{\{i \rightarrow(j, k, s)\}} \times r(i, k, s)$   (13)

All predicted concentrations are assumed non-negative physically;

$C(j, k, s) \geq 0 \forall j, k, s($ applied as $C \leftarrow \max .\{0, C\})$    (14)

This is equivalent to the implicit removal limit:

$r(j, k, s) \leq \frac{C_0(j, k, s)}{A_{\{i \rightarrow(j, k, s)\}}}$

where, the effect factor is positive. The feasibility of the technique limits the removal ratios as in Eq. (3). The defining constraints of this case are the strict intake limits imposed on the legally feasible components (mg/L) at the intakes in both seasons:

$C(j, T D S, s) \leq 1000 ; C(j, B O D, s) \leq 6 ; C\left(j, N O_3, s\right) \leq 50$    (15)

For all j ∈{Kufa intake, Al-Manadhira intake} and s ∈{wet, dry}. Because the predicted concentration in Eq. (13) is preserved in the decision vector, these are linear inequality constraints; they are supplied to the solver in the form c(x) ≤ 0, so that any allocation violating an intake limit is infeasible and does not enter the Pareto set. Phosphate is not among them and is treated separately through the penalty objective Obj₅.

6.1 Pareto front

The NSGA-II algorithm resulted in 77 optimal Pareto front solutions. The important result of the strict intake constraints is that each of the 77 solutions satisfied all strict intake constraints for TDS, BOD, and NO₃ in both seasons, with zero exceedances in all cases. Therefore, strict compliance is not a criterion between solutions, but rather a certain property of the front; trade-offs occur between the remaining objectives. Table 18 gives the range statistics for each objective across the Pareto front.

From the results in Table 18, it is obvious that the environmental risk extends from 0.154 to 429.97, while cost varies from 0.898 to 2.969. The frontier is governed by a high risk-cost contrast where cost increases accept substantial risk reductions in the low-cost zone, but produces diminishing returns as costs rise. Second, the intake water quality index (Obj₃) takes a narrow range (-187.39 to -182.39). Because strict constraints are imposed on compliance by regulated components at intake, the resulting quality is high and relatively insensitive to the chosen solution. Thus, the real decision space is a trade-off between cost and residual risk, spatial equity, and phosphate control. Five representative solutions were obtained to characterize the front (Table 19): the lowest cost solution (No. 2), the lowest phosphate penalty solution (No. 77), the compromise solution (No. 31), the lowest risk solution (No. 4), and the best spatial justice solution (No. 3).

Table 18. Statistics for objectives across the optimal solutions on the Pareto front

The five solutions reveal a fundamental conflict. Solution 2 achieves the lowest removal effort (0.898) at the highest residual risk among the four non-compromise representatives (0.543), while Solution 4 reaches the lowest risk (0.154) at a high cost (2.90). In descending order of Obj₁ risk, the five representatives are No. 31 (32.3215), No. 2 (0.5430), No. 3 (0.4814), No. 77 (0.1719), and No. 4 (0.1542), so the compromise solution carries the highest risk of the five. Solution 77 minimizes the phosphate penalty (0.113) but again requires high costs (2.842). Solution 3 minimizes the variance of the residual exceedance load across the equity stations (Obj₄ = 0.0022); this is the equity of the residual result and does not indicate an even distribution of treatment effort across the controllable sources. Solution 31 lies in the region of the risk-cost trade-off where marginal risk reduction begins to diminish, at (risk 32.32, effort 1.436). Solution 31 is identified here as the compromise solution of the front. Four independent criteria, two based on distance to reference points [19] and two on the geometry of the front [20], were applied to the complete front of 77 solutions, and all four are reported here, whether or not they agree. The five objectives were first scaled to the unit interval by min-max normalization over the front, since they differ in dimension and range by up to three orders of magnitude; the sensitivity of the outcome to that normalization was then tested by repeating the calculation with max-absolute scaling in place of min–max normalization (see Table 20).

Table 19. Results of the trade-offs between objectives

Table 20. Selection criteria applied to the full Pareto front and the solution each one selects

Table 21. Position of Solution 31 within the front, by objective

Three of the four criteria select Solution 31, and this selection is unchanged when min–max normalization is replaced by max-absolute scaling. Solution 31 is therefore the compromise solution of the front; it is not the knee in the geometric sense, which the curvature-based criteria place at Solutions 36 and 2. The two curvature-based criteria disagree with each other and with the rest, and the reason is instructive rather than troubling: both operate on the two-dimensional Obj₁-Obj₂ projection and are therefore blind to the remaining three objectives. Maximum curvature selects Solution 36, which ranks nineteenth by distance to the ideal point and carries almost three times the environmental risk of Solution 31 for a similar removal effort. Maximum bend selects Solution 2, the extreme lowest-effort point, which ranks sixtieth of 77; this is a known failure of the chord method when one end of the front is far more densely populated than the other. Neither result recommends a different compromise solution; both show why a criterion defined across all five objectives is appropriate for a five-objective problem.

In normalized terms, it lies at 0.075 of the risk range and 0.260 of the cost range, securing most of the achievable risk reduction for about a quarter of the reached cost increase, while satisfying the TDS, BOD, and NO₃ intake limits in full. It is therefore presented as a preliminary model-based reference scenario, pending the validation described in Section 3.4, rather than as an operational direction (See Table 21).

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Figure 5. Comparison of the risk objective (Obj₁) with the cost objective (Obj₂) for the five representative solutions

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Figure 6. Pareto front for Obj₁ (risk) versus Obj₂ (cost)

The margin is narrow, with Solution 31 leading Solution 63 by 0.019 in normalized distance, or six per cent of its own distance, and no other solution lies within five per cent of the best value. The four nearest alternatives occupy the same region of the front, with risk between 16 and 39 and effort between 1.43 and 1.59, so the choice among them is a matter of preference rather than of dominance. Solution 31 is accordingly presented as a preliminary model-based reference scenario, pending the validation described in Section 3.4, identified by an explicit and reproducible rule, and not as the unique optimum. Figure 5 relates the risk and cost values for the five representative solutions. Figure 6 shows the full Pareto front for the 77 solutions, showing the inverse relationship between risk, cost, and the position of the compromise solution near (32, 1.44); the position is determined in this section.

6.2 Intake compliance

Table 22 presents the predicted intake concentrations and the phosphate exceedance for all five representative solutions in both seasons.

TDS, BOD, and NO₃ show zero exceedance at both intakes in both seasons in all five solutions, and the strict limits hold exactly as designed. The only constant exceedance is phosphate, which stays above the 0.5 mg/L target at both intakes throughout the dry season, with residues of 0.199 to 0.313 mg/L at Kufa and 0.432 mg/L at Al-Manadhira. The Al-Manadhira value is equal in all five solution points to a physical rather than an economic bound, and it alone accounts for 0.093 of the front-wide minimum of 0.113 in Obj₅. Re-computing Obj₅ independently from these residues through Eq. (12) produces the tabulated objective to four decimal places for every solution, which shows that the objective values, the intake concentrations, and the stated formulation are mutually consistent.

Table 22. Predicted intake concentrations and phosphate exceedances (mg/L) for the five representative solutions

Note: TDS: Total Dissolved Solids; BOD: Biochemical Oxygen Demand.

Several nitrate and phosphate records in Table 16 are negative because the linear form C = C₀-A_r yields a negative value whenever the attributed removal exceeds the baseline concentration, so their presence illustrates that the projection [C←max(0, C)] was applied when the constraints were evaluated but not to the reported concentration field. They are retained because they define the valid range of the response matrix, and because every one of them falls where the parameter lies far below its limit, so no compliance decision reported here depends on them.

6.3 Optimal removal tables

Table 23 lists the optimal removal ratios r*(i, k, s) at the controllable sources S3, S4, and S5 for the representative solutions.

Table 24 presents the seasonal comparison by parameter, counting the source-parameter pairs in which the dry-season ratio exceeds the wet-season ratio across all sixty pairs of Table 19.

Table 23. Optimal removal ratios at the controllable sources for the five representative solutions

Note: TDS: Total Dissolved Solids; BOD: Biochemical Oxygen Demand.

Table 24. Seasonal comparison of the optimal removal ratios by parameter

Note: TDS: Total Dissolved Solids; BOD: Biochemical Oxygen Demand.

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Figure 7. Variation of removal fraction ratios per pollutant for wet and dry seasons

Three main patterns emerge. First, removal effort is higher in the dry season for most parameters. Across the 60 paired dry–wet comparisons in Table 23 (five solutions × three sources × four parameters), the dry-season ratio exceeds the wet-season ratio in 38 cases (63.3%; Table 24), and for PO₄ in 14 of 15. The pattern follows the dilution deficit. Although the wet-season baselines at the three controllable sources exceed the dry-season values by factors of 1.26 for TDS, 5.09 for BOD, 1.86 for NO₃, and 4.31 for PO₄, the reduced dry-season discharge concentrates whatever load remains, so a larger fraction must be removed to hold the same concentration at the intakes. PO₄ shows this most severely, with its dry-season removal reaching the feasible bound at all three sources in the low-phosphate solutions. Second, the low-risk solutions (No. 4 and No. 77) distribute removal comparatively evenly across the three sources, Solution 4 spanning 0.304 to 0.800 and Solution 77 covering 0.252 to 0.800, consistent with sustained operation of three treatment points; the compromise solution (No. 31) instead shows an irregular longitudinal pattern, with high NO₃ removal at S3 and S4 but an obvious drop at S5, which explains its higher risk value. Third, PO₄ control is limited by treatment capacity rather than priority: the algorithm applies the maximum feasible removal, and the remaining residue is a physical limit of the dry-season regime. Figures 7, 8 and 9 show the behavior of the optimal removal ratios by parameter and by source.

Figure 7 shows the variation in removal ratios according to the component for the wet and dry seasons; Figure 8 shows their variation according to the source stations (S3, S4, S5), and Figure 9 compares the removal ratios by parameter between the two seasons, resolving the seasonal comparison by parameter and by source rather than attributing it to dilution alone.

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Figure 8. Variation of removal fraction ratios per source station for wet and dry seasons

Table 25. Comparison of the two constraint scenarios

Note: ^*constraints at the intakes; ^**constraints at all stations.

6.4 Sensitivity of the Pareto set to the range of the constraints

The problem was re-solved with the IQS limits enforced not at the two intakes alone but at all seven river stations in both seasons, the controllable sources, the decision vector, and the objective set being left unchanged. Because the front itself is generated from the same data, the 77 non-dominated points are common to both runs, and the scenarios differ only in which of them survive the feasibility test. Table 25 sets out the comparison.

The effect is substantial, applying the limits at every station reduces the admissible set from all 77 solutions to 13, or 16.9%, and moves the compromise solution from No. 31 to No. 35. The reference plan changes nature with it: environmental risk falls from 32.32 to 0.161 and the variance of the residual exceedance load falls from 3.354 to 0.00426, while the removal effort rises by 46%, from 1.436 to 2.099. The intake water quality index and the phosphate penalty improve only marginally, and the dry-season PO₄ residue at Al-Manadhira remains at 0.432 mg/L in both scenarios, which is further evidence that this residue is physically bound rather than a consequence of the constraint set. Scenario (2) therefore achieves a near-complete elimination of in-river risk and a nearly uniform distribution of treatment for a little under half as much effort, and the choice between the two is a scheme question rather than a computational one.

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Figure 9. Comparison between removal fraction ratios per pollutant for wet and dry seasons