1 Introduction

Water allocation in a river basin is a multifaceted process aimed at balancing the competing demands of stakeholders, such as farmers, cities, and industries, with the overall health of aquatic ecosystems. Determining a fair and sustainable distribution of water resources is essential and involves navigating complex socio-economic and environmental factors. In particular, water scarcity in transboundary river basins often exacerbates the overexploitation of resources. Optimization models have emerged as valuable tools in addressing these challenges by allocating water resources among stakeholders to maximize economic benefits while minimizing socioeconomic disparities [1]. Modern water allocation planning requires a comprehensive approach that integrates socio-economic considerations with hydrology and engineering. This necessitates innovative planning techniques and new skill sets among practitioners.

Recent studies have demonstrated the application of advanced optimization models in addressing water allocation challenges. Deng et al. [2] utilized a multi-objective optimization model based on the NSGA-II algorithm to examine trade-offs between economic efficiency and equitable water distribution in the Han River Basin, China. The findings underscore the inherent tension between maximizing economic output and ensuring fairness in allocation. Similarly, Naghdi et al. [3] developed a framework combining system dynamics, optimization, and conflict resolution for managing water resources in the Najaf-Abad sub-basin, Iran. By employing NSGA-II and Nash bargaining theory, the study minimized groundwater depletion and sectoral water deficits while addressing stakeholder demands.

Qu et al. [4] explored the interplay between economic growth and water constraints in Wujiang, China, using multi-objective optimization techniques. Their study proposed strategies such as industrial restructuring, investment in water-saving technologies, and promoting less water-intensive industries for sustainable planning. Dong et al. [5] introduced a multi-objective water resource allocation model (WRAM) for the Huaihe River Basin, China, grounded in equilibrium theory. This model integrates socio-economic, water resource, and ecological systems to provide sustainable, equitable, and efficient solutions applicable to other basins. In another study, Zhang et al. [6] recommended employing the NSGA-II algorithm for multi-objective optimization of water resources in the middle and upper reaches of the Huaihe River Basin, achieving efficient, equitable, and sustainable allocation.

Further advancements include Tang et al. [7], who presented a novel multi-objective optimization model for water resource allocation that integrates fairness and water shortage risk. Using an improved NSGA-III algorithm, termed ARNSGA-III, the study applied this model to Wusu City, China, resulting in a 50.1% reduction in structural water shortage risk, a 0.2% increase in economic benefits, and a 60.5% decrease in fairness, illustrating the trade-offs among these objectives. Zhang et al. [6] formulated a multi-objective optimization framework for enhancing the sustainability of infrastructure projects. By leveraging Multi-Objective Particle Swarm Optimization (MOPSO) and NSGA-II, they identified optimal maintenance and replacement strategies for infrastructure assets, demonstrating the significant influence of annual budget constraints on decision-making. Similarly, Aalami et al. [8] applied a multi-objective optimization model to the Givi River Basin in Iran to address water scarcity. This model prioritized maximizing equality in water distribution and minimizing economic efficiency loss due to uncertainties in water availability.

Collectively, these studies highlight the importance of integrating efficiency, equity, and sustainability principles into river basin management. They underscore the need for advanced methodologies that balance economic, social, and environmental benefits, ensuring fair distribution and improved water use efficiencies across sectors [9,10,11,12,13,14,15]. While various water allocation techniques including proportional, fixed, quota-based, and priority-based methods are commonly used, each has its advantages and limitations.

The Rift Valley Lakes Basin (RVLB) in Ethiopia exemplifies the challenges of water resource management due to increasing demands from agriculture, industry, urbanization, and population growth [16]. Current practices often prioritize short-term economic gains, resulting in over-extraction, inefficient water use, social inequities, and environmental degradation. These pressures threaten the basin’s water resources, jeopardizing economic activities, ecosystem health, biodiversity, and local livelihoods.

This study introduces a multi-objective optimization framework utilizing the Non-Dominated Sorting Genetic Algorithm II (NSGA-II). This innovative approach integrates economic efficiency, equity, and environmental sustainability, enabling decision-makers to assess trade-offs and synergies among these objectives. The research provides a comprehensive solution for sustainable water management in the RVLB, addressing both human and ecological needs. Its significance lies in promoting efficient resource management, equitable water distribution, and sustainable utilization of water resources. By providing insights into trade-offs between objectives, the study supports adaptation to changing conditions, such as climate variability, population growth, and economic development. Furthermore, it informs policymakers and planners at local, regional, and national levels, fostering strategies that are both effective and equitable.

2 Methodology

2.1 Study area

The Rift Valley Lakes Basin (RVLB) is one of Ethiopia’s twelve major river basins, covering approximately 53,000 square kilometers (Fig. 1). This basin is characterized by a series of lakes of various sizes, some are terminal lakes with no surface outflow and Lake Ziway and Lake Langano are interlinked with lakeAbijata through Bulbula and Harkelo rivers. The seven lakes within the basin are Lake Ziwa, Lake Langano, Lake Abijata, Lake Shala, Lake Hawassa, Lake Abaya, and Lake Chamo, all situated to the south and southwest of Addis Ababa, Ethiopia’s capital. The RVLB spans four regional states: Oromia, Sidama, Central Ethiopia, and Southern Ethiopia.

Fig. 1
figure 1

Rift Valley Lakes Basin location map

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The Rift Valley Lakes Basin (RVLB) is primarily characterized by its graben structure, a type of block fault where the valley floor has shifted vertically relative to the valley sides. Temperature and rainfall in the RVLB vary with elevation, affecting relative humidity and potential evapotranspiration. The eastern and western edges of the Rift Valley are cooler and wetter with lower evapotranspiration rates, while the central lowlands are hotter, drier, and have higher evapotranspiration rates. Annual rainfall on the valley floor ranges from about 400mm at Chew Bahir in the south to around 700mm near the northern lakes. In contrast, areas near Geese, west of Lake Chamo, and Yirga Chefe, east of Lake Abaya, receive nearly 2,000mm of rainfall annually due to their higher elevations. The RVLB experiences two main rainfall patterns: north of Lake Abaya, the primary rains occur from July to September with a secondary peak in March or April, while south of Lake Abaya, the main rains fall between March and May. Annual potential evapotranspiration varies from approximately 1,200mm in the northeast to about 1,900mm at Chew Bahir. Across the Rift Valley floor, from Lake Ziway to Lake Chamo, the average is around 1,550mm. Average annual temperatures range from about 27°C on the valley floor near Chew Bahir to approximately 13°C at higher elevations, particularly in the northeast of the basin [17].

2.2 Conceptual framework

The model optimizes water resource allocation to enhance efficiency, equity, and sustainability within a river basin system. It addresses water supply challenges by ensuring optimal distribution to subbasins and sectors. Figure 2 presents the framework for effective water distribution. The model balances environmental sustainability, economic benefits, and equity, with policies guiding fair distribution among independent groups. Using a multi-objective programming approach, it aims to maximize fairness, ecological preservation, and economic efficiency.

Fig. 2
figure 2

Conceptual framework demand nodes of the RVLB

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2.3 Data source

Crops grown, human and livestock population data is obtained from the Central Statistics Agency of Ethiopia. Socioeconomic data are collected from district and regional authorities and agencies and used for the economic analysis of the model. The other main input of the model is hydrological data which is adopted from [18, 19]

2.4 Objective functions

2.4.1 Objective function f1(x): maximization of mean economic benefit efficiency.

Over the years, a variety of strategies for financial gains of water resources have been studied [10, 16]. The net economic benefits of agricultural water can be calculated by using a demand function in conjunction with the residual imputation approach, which calculates the economic benefits of domestic water usage. These procedures, however, are complex and include a huge number of parameters. The average yearly economic benefit per unit of water use is used to more accurately and efficiently calculate the net economic benefits of various water applications.

The first aim function is to achieve maximum economic efficiency for a calendar year with established hydrological conditions. Economic benefit is relatively straightforward to quantify and is widely used in complex water resource allocation systems. In real-world scenarios, one sector's use of water may have an impact on another. Agriculture, residential use, industry, livestock, and ecology have all been considered to help estimate water demands and allocate water resources more effectively. In this study, gross economic interest is used to measure efficiency. The first objective function represented by Eq. 1, estimated the basin’s economic benefit efficiency.

$${\varvec{maxf}}_{{{\varvec{sB}}}} = \mathop \sum \nolimits_{{{\varvec{t}} = 1}}^{{\varvec{T}}} \mathop \sum \nolimits_{{{\varvec{i}} = 1}}^{{\varvec{n}}} \mathop \sum \nolimits_{{{\varvec{j}} = 1}}^{{\varvec{m}}} \left( {{\varvec{NEB}}_{{{\varvec{i}},{\varvec{j}}}} \times {\varvec{q}}_{{{\varvec{i}},{\varvec{j}}}}^{{\varvec{t}}} } \right)$$
(1)

where:

fSB refers to system economic benefit efficiency.

NEBi,j refers to the net economic benefit per unit of water quantity of jth sector in ith catchment;

qit,j refers to the water allocated to the jth sector in the ith catchment in tth time;

T refers to the total number of months during calculation;

i refers to the total number of water intakes;

j refers to the total number of water user sectors.

2.4.2 Objective function f2(x): maximizing the water allocation equity

Equity refers to fair and just access to water and the benefits derived from its utilization. The concept of equitable water distribution amongst sectors serves as the basis for allocating water resources and is also the primary concern for achieving water sustainability. The Gini coefficient, devised by the Italian economist Gini, is extensively employed to evaluate the level of disparity in the distribution of income. Furthermore, other than assessing the equality of income distribution in the economic society, it may also be employed to inspect the equity of distribution in other fields, such as the balance of water and soil. Recently, the Gini coefficient has been employed to quantify the inequality in land and water usage. The Gini coefficient of water allocation has been applied to water allocation equity within a multi-objective decision-making system [2, 20].

In this study, the Lorenz Curve is a graphical representation of the distribution of water consumption (demand) within a population. It plots the cumulative percentage of total water consumption by the cumulative percentage of the population. If there were perfect equality, the Lorenz Curve would be a 45-degree line from the origin (this is called the line of equality). The more the Lorenz Curve deviates from this line, the higher the level of inequality. The Gini Coefficient quantifies inequality based on the Lorenz Curve. It measures the area between the line of equality and the Lorenz Curve, divided by the total area under the line of equality Eq. 2.

Mathematically, the Gini Coefficient (G) is:

$${\varvec{G}} = \frac{{\varvec{A}}}{{{\varvec{A}} + {\varvec{B}}}}$$
(2)

where:

  • A is the area between the line of equality and the Lorenz Curve.

  • B is the area under the Lorenz Curve.

The Gini Coefficient ranges from 0 to 1, where 0 represents perfect and 1 represents perfect inequality. The steps followed to Calculate the Gini Coefficient are (a) Rank the population from lowest to highest water consumption. (b) calculate the cumulative percentages of water consumption and population for each group. (c) plot the Lorenz Curve using the cumulative population on the x-axis and cumulative water consumption on the y-axis. (d) calculate the area between the line of equality and the Lorenz Curve. (e) use the formula to calculate the Gini Coefficient (Fig. 3). Equation 3 represents the average Gini coefficient between the percentage of cumulative population and the cumulative water consumption and Eq. 4 represents the average Gini coefficient between the available water resource and the cumulative percentage of water consumption.

$${\varvec{CPG}}_{{{\varvec{av}}}} = \mathop \sum \nolimits_{{{\varvec{t}} = 1}}^{{\varvec{T}}} \frac{{{\varvec{CPG}}_{{\varvec{t}}} }}{{\varvec{T}}} = 1 - \mathop \sum \nolimits_{{{\varvec{t}} = 1}}^{{\varvec{T}}} \mathop \sum \nolimits_{{{\varvec{i}} = 1}}^{{\varvec{n}}} \frac{{\left( {{\varvec{WPC}}_{{{\varvec{it}}}} + {\varvec{WPC}}_{{{\varvec{i}} - 1,{\varvec{t}}}} } \right) \times \left( {{\varvec{PPC}}_{{{\varvec{it}}}} + {\varvec{PPC}}_{{{\varvec{i}} - 1{\varvec{t}}}} } \right)}}{{\varvec{T}}}$$
(3)
$${\varvec{CWG}}_{{{\varvec{av}}}} = \mathop \sum \nolimits_{{{\varvec{t}} = 1}}^{{\varvec{T}}} \frac{{{\varvec{CWG}}_{{\varvec{t}}} }}{{\varvec{T}}} = 1 - \mathop \sum \nolimits_{{{\varvec{t}} = 1}}^{{\varvec{T}}} \mathop \sum \nolimits_{{{\varvec{i}} = 1}}^{{\varvec{n}}} \frac{{\left( {{\varvec{WPC}}_{{{\varvec{it}}}} + {\varvec{WPC}}_{{{\varvec{i}} - 1,{\varvec{t}}}} } \right) \times \left( {{\varvec{WRC}}_{{{\varvec{it}}}} + {\varvec{WRC}}_{{{\varvec{i}} - 1{\varvec{t}}}} } \right)}}{{\varvec{T}}}$$
(4)

where:

Fig. 3
figure 3

Lorenz curve for water allocation

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CPGav = the average of CPG,

CPGt = the CPG in tth time,

T = the total number of months during calculation;

n = to the total number of catchments,

WPCi,t = cumulative percentage of available water resources in ith catchment in tth time.

PPCi,t = the cumulative percentage of water consumption (population) in ith catchment in tth time.

CWGav = the average of CWG,

CWGi,t = CWG in tth time,

WRCi,t Cumulative percentage of available water resources in the ith time and it is equivalent to zero in the initial stage.

The second objective function (Eq. 5) is to minimize the comprehensive Gini coefficient to consider the impacts of various indexes on water consumption.

$${\varvec{f}}_{2} \left( {\varvec{x}} \right) = {\varvec{min}}\left( {\left( {{\varvec{\omega}}_{1} \times {\varvec{CPG}}_{{{\varvec{av}}}} } \right) + \left( {{\varvec{\omega}}_{2} \times {\varvec{CWG}}_{{{\varvec{av}}}} } \right)} \right)$$
(5)

where: f2(x) denotes the comprehensive Gini coefficient, ωi(i = 1, 2,) denotes the weighting coefficient and the sum of ωi(i = 1, 2,) is equivalent to one. The impacts of population and available water resources on water consumption can be viewed as equally important. Hence, ωi(i = 1, 2,) are equal to ½.

2.4.3 Constraints

The upper and lower limits of the constraints are based on the thresholds of the Lake levels, maximum downstream releases as presented in Eq 6, 7, 8, 9, 10.

1. water availability constraints

$$\sum {\varvec{q}}_{{{\varvec{ij}}}}^{{\varvec{t}}} \le {\varvec{AW}}_{{\varvec{i}}}^{{\varvec{t}}} \user2{ }\forall_{{{\varvec{ij}}}}$$
(6)

where: qt i,j refers to jth water use sector in ith catchment in tth time, AWit represents the available water in ith water-intake in tth time.

2. Lake storage capacity constraints

$${\varvec{V}}_{{{\varvec{k}} {\varvec{min}}}}^{{\varvec{t}}} \le {\varvec{V}}_{{\varvec{k}}}^{{\varvec{t}}} \le {\varvec{V}}_{{{\varvec{k}} {\varvec{max}}}}^{{\varvec{t}}}$$
(7)

where: Vtk, min refers to the lower bound of kth Lake in tth time, (the threshold); Vtk, max refers to the upper bound of kth reservoir in tth time, usually the maximum storage capacity below the normal storage water level in the non-flood season and the flood-control water level in the flood season, respectively.

3. Lake abstraction discharge constraints

$${\varvec{y}}_{{{\varvec{ik}}}}^{{\varvec{t}}} \le {\varvec{q}}_{{\max {\varvec{ij}}}}^{{\varvec{t}}}$$
(8)

where yti,j refers to the discharge of the kth Lake in ith catchment in tth time; qtmax i,k refers to the discharge capacity of the kth Lake in ith catchment in tth time.

4. Water demand constraints

$${\varvec{q}}_{{{\varvec{ij}}}}^{{\varvec{t}}} \le {\varvec{D}}_{{{\varvec{ij}}}}^{{\varvec{t}}}$$
(9)

where: Dti,j denotes the water demand of the jth water use sector in ith catchment in tth time

5. Non-negative constraints.

  • The amount of water allocated to a sector in a catchment cannot be negative

  • The amount of discharge from a lake cannot be negative

    $${q}_{ij}^{t}\ge 0$$
    $$y_{i} j^{t} \ge 0$$
    (10)

2.4.4 Global model

The comprehensive framework that combines the objective function and constraints to define the optimization problem. It provides a unified mathematical representation of what aim to achieve (objective function) and the limitations or conditions that must be satisfied (constraints).

$$\begin{gathered} {\text{Objective function}}\,\,\left\{ {\begin{array}{*{20}c} {max\,f_{sB} \, = \,\sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{m} {(NEB_{i,j} \times q_{i,j}^{t} )} } } } \\ {min\,f_{2} \left( x \right) = \left( {\omega_{1} \times CPG_{av} } \right) + \left( {\omega_{2} \times CWG_{av} } \right)} \\ \end{array} } \right. \hfill \\ Subjected\,\,to\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}c} {\sum {q_{i,j}^{t} \le AW_{i}^{t} } } \\ {V_{j min}^{t} \le V_{j}^{t} \le V_{j max}^{t} } \\ {y_{ij}^{t} \le q_{\max ij}^{t} } \\ \begin{gathered} q_{ij}^{t} \le D_{ij}^{t} \hfill \\ q_{ij}^{t} \ge 0 \hfill \\ y_{ij}^{t} \ge 0 \hfill \\ \end{gathered} \\ \end{array} } \right. \hfill \\ \end{gathered}$$

where: f1(x)and f2(x)are the objective functions, qti,j refer to the design variables and the remaining formulas are the constraints.

2.5 Optimization algorithm

The Non-Dominated Sorting Genetic Algorithm approach II (NSGA-II) is a widely recognized and resilient method for multi-objective optimization (MOO) that has achieved significant accomplishments in various practical scenarios. Introduced by Deb et al. [21], NSGA-II has emerged as a standard for assessing the efficiency of other algorithms for multi-objective optimization (MOP) because of its ability to effectively tackle intricate optimization problems with numerous objectives that conflict with each other. NSGA-II is characterized by its elitist and diversity-preserving techniques. The algorithm utilizes a rapid non-dominated sorting method to categorize solutions into distinct fronts according to their dominant relationships. The sorting method guarantees that solutions without any inferior aims are given higher priority than those with compromised performance. NSGA-II improves diversity preservation by integrating the calculation of crowding distance, which quantifies the concentration of solutions in the goal space. This strategy ensures a diverse range of options on the Pareto front, avoiding early convergence to a less optimal set of answers. The effectiveness of NSGA-II arises from its capacity to achieve a balance between the speed of convergence and the preservation of diversity. It effectively directs the search towards the Pareto front, while also preserving a varied collection of answers, guaranteeing that the algorithm does not become stuck in suboptimal solutions. NSGA-II is a versatile and powerful tool that can effectively address a wide range of multi-objective optimization problems in various fields such as engineering, science, and other subjects.

Water resource management has made considerable use of the evolutionary technique to develop optimal solutions to challenging multi-objective optimization situations because of the aforementioned benefits. Consequently, this study uses NSGA-II to trade off equity and efficiency. Figure 4 illustrates the NSGA-II algorithm flowchart. 500, 300, 85%, and 0.01% were allocated to the population size, number of maximal generations, crossover probability, and mutation rate, respectively. It was observed that a rigorous trial-and-error procedure is used to select the NSGA-II's parameters to produce results that will converge.

Fig. 4
figure 4

Flowchart of NSGA-II

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2.5.1 Decision-making procedures

In the Pareto front, relationships are typically trade-off-oriented. Finding a balance point between optimized goals is critical. Finding a solution that is acceptable to each target with the least amount of variation was done in this study.

2.5.2 Fast nondominated sorting

A basic method of identifying members of the first nondominated front in a population is to compare each candidate to every other solution in the group to determine if it is dominated by others.

2.5.3 Crossover

Crossover in NSGA-II is a genetic operator used to combine the genetic information of two parent solutions to generate new offspring for the next generation. It is a key mechanism that allows the algorithm to explore new areas of the solution space, potentially leading to better and more diverse solutions. In NSGA-II, the most common crossover technique is Simulated Binary Crossover (SBX). This method simulates the single-point crossover of binary strings in a real-numbered space. The SBX operator works by choosing a random crossover point and exchanging the segments of the parent solutions to create two new offspring. The probability of crossover, along with the distribution index, determines how close the offspring will be to their parents. The process of crossover in NSGA-II typically involves the selection of Parent Solutions, crossover point determination, creation of offspring, and evaluation and Selection.

2.5.4 Mutation

Mutation in NSGA-II is a crucial operator that introduces diversity into the population by randomly altering the genes of individuals. In NSGA-II, the most commonly used mutation operator is the polynomial mutation. This operator works by adding a polynomial distributed noise to the gene values, which helps in exploring the search space more effectively. Mutation in NSGA-II is a mechanism that helps maintain genetic diversity and prevents premature convergence, thereby enhancing the algorithm’s ability to find a well-distributed set of Pareto-optimal solutions.

2.5.5 Crowding distance

Crowding distance in NSGA-II is a measure used to estimate the density of solutions surrounding a particular solution in the population. It is an important concept in multi-objective optimization as it helps maintain diversity in the solution set. For each solution, the crowding distance is calculated by considering the average distance of two points on either side of this point along each of the objectives. The crowding distance essentially estimates how crowded an individual solution’s neighborhood is. A larger crowding distance implies that the solution is in a less crowded region, which is preferred during the selection process. When selecting solutions to form a new population, NSGA-II prefers solutions with a larger crowding distance. This ensures a uniform spread of solutions along the Pareto front. Equation 11 represents the crowding distance calculation formula.

$${\varvec{CD}} = \frac{{{\varvec{f}}_{{\varvec{m}}} \left( {{\varvec{x}}_{{{\varvec{i}} - 1}} } \right) - {\varvec{f}}_{{\varvec{m}}} \left( {{\varvec{x}}_{{{\varvec{i}} + 1}} } \right)}}{{{\varvec{f}}_{{\varvec{m}}} \left( {{\varvec{x}}_{{{\varvec{max}}}} } \right) - {\varvec{f}}_{{\varvec{m}}} \left( {{\varvec{x}}_{{{\varvec{min}}}} } \right)}}$$
(11)

2.5.6 Average variability

The average variability is the average value of the slopes of the lines connecting any point except endpoints with two adjacent points. Suppose the average variability of f1 and f2 is d1(i) and d2(i), respectively. Let K be the number of Pareto solution sets, which can be sequenced in order. \({f}_{1}^{(i)}\) is the value of the first objective function of the ith Pareto solution, i ∈ {1, 2,... K}; \({f}_{2}^{(i)}\) is the value of the second objective function of the ith Pareto solution, i ∈ {1, 2,... K}, then the average change rate of the above two objectives is represented in Eq. 12 and 13.

$${\varvec{d}}_{1}^{{\left( {\varvec{i}} \right)}} = \frac{1}{2}\left( {\frac{{{\varvec{f}}_{2}^{{\left( {{\varvec{i}} + 1} \right)}} - {\varvec{f}}_{2}^{{\left( {\varvec{i}} \right)}} }}{{{\varvec{f}}_{1}^{{\left( {{\varvec{i}} + 1} \right)}} - {\varvec{f}}_{1}^{{\left( {\varvec{i}} \right)}} }} + \frac{{{\varvec{f}}_{2}^{{\left( {\varvec{i}} \right)}} - {\varvec{f}}_{2}^{{\left( {{\varvec{i}} - 1} \right)}} }}{{{\varvec{f}}_{1}^{{\left( {\varvec{i}} \right)}} - {\varvec{f}}_{1}^{{\left( {{\varvec{i}} - 1} \right)}} }}} \right), {\varvec{i}} \in \left\{ {2, 3, ... {\varvec{K}} - 1\} } \right.$$
(12)
$${\varvec{d}}_{2}^{{\left( {\varvec{i}} \right)}} = \frac{1}{2}\left( {\frac{{{\varvec{f}}_{1}^{{\left( {{\varvec{i}} + 1} \right)}} - {\varvec{f}}_{1}^{{\left( {\varvec{i}} \right)}} }}{{{\varvec{f}}_{2}^{{\left( {{\mathbf{i}} + 1} \right)}} - {\varvec{f}}_{2}^{{\left( {\varvec{i}} \right)}} }} + \frac{{{\varvec{f}}_{1}^{{\left( {\varvec{i}} \right)}} - {\varvec{f}}_{1}^{{\left( {{\varvec{i}} - 1} \right)}} }}{{{\varvec{f}}_{2}^{{\left( {\varvec{i}} \right)}} - {\varvec{f}}_{2}^{{\left( {{\varvec{i}} - 1} \right)}} }}} \right), {\varvec{i}} \in \left\{ {2, 3, ...{\mathbf{K}} - 1\} } \right.$$
(13)

where: \({d}_{1}^{(i)}\) and \({d}_{2}^{(i)}\) refer to the average change rate of the first objective and the second objective, respectively. Especially, when i is equal to 1 or K, the average change rate is:

The equations provided seem to be calculating the average change rate of two objective functions, f1 and f2, across a set of Pareto solutions:

  • The first equation calculates the average change rate of the first objective function, f1, to the second objective function, f2. This is done by averaging the rate of change of f1 to f2 between the (i + 1)th and ith solutions and the ith and (i-1)th solutions (Fig. 5).

  • The second equation does the same but for the second objective function, f2, to the first objective function, f1 (Fig. 5).

For the endpoints of the Pareto front (i = 1 and K), the average variability of d1(1), d2(1), d1(K ) and d2(K ) is defined in equation 14.

$${\varvec{d}}_{1}^{1} = \frac{{{\varvec{f}}_{1}^{2} - {\varvec{f}}_{1}^{1} }}{{{\varvec{f}}_{2}^{2} - {\varvec{f}}_{2}^{1} }}, {\varvec{d}}_{2}^{1} = \frac{{{\varvec{f}}_{2}^{2} - {\varvec{f}}_{2}^{1} }}{{{\varvec{f}}_{1}^{2} - {\varvec{f}}_{1}^{1} }}, {\varvec{d}}_{1}^{{\varvec{k}}} = \frac{{{\varvec{f}}_{1}^{{\varvec{k}}} - {\varvec{f}}_{1}^{{{\varvec{k}} - 1}} }}{{{\varvec{f}}_{2}^{{\varvec{k}}} - {\varvec{f}}_{2}^{{{\varvec{k}} - 1}} }}, {\varvec{d}}_{2}^{{\varvec{k}}} = \frac{{{\varvec{f}}_{2}^{{\varvec{k}}} - {\varvec{f}}_{2}^{{{\varvec{k}} - 1}} }}{{{\varvec{f}}_{1}^{{\varvec{k}}} - {\varvec{f}}_{1}^{{{\varvec{k}} - 1}} }}$$
(14)
Fig. 5
figure 5

Average change rate of objective functions on the Pareto solutions

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2.5.7 Sensitivity ratio

The sensitivity ratio compares the average variability of a given Pareto non-inferior solution to its corresponding objective function value on the Pareto front. Specifically, the sensitivity ratio denoted as (δ1i) for Pareto non-inferior solutions is calculated as the ratio of the average variability (k1(i)) to the corresponding objective function value f1​. Similarly, δ2i is defined as the ratio of the average variability (k2(i)) to the corresponding objective function value f2​. Based on these definitions, the sensitivity ratio is computed as shown in Eqs. 15 and 16 for both objective functions.

$${\varvec{\delta}}_{1}^{{\varvec{i}}} = \frac{{{\varvec{k}}_{1}^{{\varvec{i}}} }}{{{\varvec{p}}^{{\varvec{i}}} }}, {\varvec{i}} \in \left\{ {1,2,..., {\varvec{K}}} \right\}$$
(15)
$${\varvec{\delta}}_{2}^{{\varvec{i}}} = \frac{{{\varvec{k}}_{2}^{{\varvec{i}}} }}{{{\varvec{C}}^{{\varvec{i}}} }}, {\varvec{i}} \in \left\{ {1,2,..., {\varvec{K}}} \right\}$$
(16)

where:

  • δ1.i = sensitivity ratios of f1

  • δ2.i = sensitivity ratios of f2

For the convenience of comparison, the above results need to be dimensionless. The value of the sensitivity ratio reflects the sensitivity degree of the average variability of certain objective function values as the function values change. When the functions of the MOP are performance functions, i.e., quantity, efficiency, cost, etc., the sensitivity ratio is similar to the performance-price ratio. For comparison in the next step, the sensitivity ratio must be non-dimensional.

Let ε1i and ε2i be the results of δ1i and δ2i then, the sensitivity analysis is done using Eqs. 17 and 18 for both objective functions respectively.

$${\varvec{\varepsilon}}_{2}^{{\varvec{i}}} = \frac{{{\varvec{\delta}}_{2}^{{\varvec{i}}} }}{{\mathop \sum \nolimits_{{{\varvec{i}} = 1}}^{{\varvec{k}}} {\varvec{\delta}}_{2}^{{\varvec{i}}} }},\user2{ i} \in \user2{ }\{ 1,2,...,\user2{ k}$$
(17)
$${\varvec{\varepsilon}}_{2}^{{\varvec{i}}} = \frac{{{\varvec{\delta}}_{2}^{{\varvec{i}}} }}{{\mathop \sum \nolimits_{{{\varvec{i}} = 1}}^{{\varvec{k}}} {\varvec{\delta}}_{2}^{{\varvec{i}}} }},\user2{ i} \in \user2{ }\{ 1,2,...,\user2{ k}$$
(18)

where: ε1i and ε2i refer to the dimensionless sensitivity ratios of the first objective and the second objective, respectively.

2.5.8 Bias degree based on sensitivity ratio

The bias degree is the degree of bias for different objective functions, and the value range is (0,1). Suppose K is the set of numbers of the Pareto non-inferior solutions in X. \({\omega }_{1}^{i}\) and \({\omega }_{2}^{i}\) m is the bias degrees of the Pareto non-inferior solution for objective functions f1 and f2, respectively, whose number is i in the Pareto non-inferior solutions. The bias degree is calculated using Eqs. 19 and 20 as follows:

$${\varvec{\omega}}_{1}^{{\varvec{i}}} = \frac{{{\varvec{\varepsilon}}_{1}^{{\varvec{i}}} }}{{{\varvec{\varepsilon}}_{1}^{{\varvec{i}}} + {\varvec{\varepsilon}}_{2}^{{\varvec{i}}} }},\user2{ i} \in \{ 1,2,...,\user2{ K}$$
(19)
$${\varvec{\omega}}_{2}^{{\varvec{i}}} = \frac{{{\varvec{\varepsilon}}_{2}^{{\varvec{i}}} }}{{{\varvec{\varepsilon}}_{1}^{{\varvec{i}}} + {\varvec{\varepsilon}}_{2}^{{\varvec{i}}} }},\user2{ i} \in \left\{ {1,2,...,{\varvec{K}}} \right\}$$
(20)

where: ω1i and ω2i refer to the preference degrees of ith Pareto solution for the first objective and the second objective, respectively

3 Results

3.1 Current and projected water demand

As presented in Table 1, the current overall water demand is estimated by integrating all the sectoral demands and it was found to be 1,796.34 MCM, the current water demand of different sectors is shown in Figure 6 and the projected demand ranged from 1,628.23MCM to 2,378.74 MCM in 2035 and it ranged from 2126.09MCM to 3,146.89MCM in 2050.

Table 1 Current and projected water demand of RVLB
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Fig. 6
figure 6

Water demand of the sectors in RVLB

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3.2 Non-inferior pareto front solutions

The Rift Valley Lakes Basin's monthly water allocation plan was optimized for a calendar year using the NSGA_II. The model, solved using the NSGA-II algorithm in Python, used the runoff time series and water demand data of each major catchment as input. The ideal Pareto front, or non-inferior solution set, between two objective functions is shown in Fig. 7. It suggests a favourable correlation between Economic return and the Gini coefficient. A lower Gini coefficient value denotes better water allocation equity, and a higher economic return value denotes more efficient use of available water resources. Hence, the superior efficiency objective matches the inferior equity objective, and the value of the Gini coefficient increases with the growth of economic return, translating to a worse equity condition. The trade-off between economic return and equality reveals how incompatible efficiency and equity are when it comes to allocating water resources. To effectively manage water allocation, decision-makers should analyze the Pareto front by examining trade-offs, where each point represents a viable plan. It is important to emphasize the endpoints to understand the maximum potential of each objective when the other is excluded. Engaging stakeholders is vital, involving policymakers, local communities, environmental advocates, and industry representatives to identify priorities, assess objectives, and promote social equity. The rationale behind the chosen plan should be communicated, highlighting trade-offs and stakeholder contributions. Additionally, systems for monitoring and addressing concerns during execution should be implemented. Starting with small-scale trials helps assess feasibility, and regular evaluations based on new data, feedback, or shifting priorities should guide necessary adjustments.

Fig. 7
figure 7

The Pareto fronts of the optimal water allocation

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20 Pareto set of equity and economic return objectives with even distribution is taken to analyze the correlation between different Gini coefficients, i.e., CPG and CWG, and comprehensive Gini coefficient. Similar studies have been conducted in China by Tian et al., [22].

3.3 Optimized allocation

Table 2 presents the minimum and maximum values for the two objectives on the Pareto front. The minimum Gini coefficient is 0.312, corresponding to a total economic interest of $6.549B, while the maximum Gini coefficient is 0.321, associated with a total economic interest of $6.575B. These values indicate that the best equity outcome aligns with the least efficient outcome, highlighting a conflict between the two objectives. When the study area aims for maximum economic output efficiency, i.e., optimizing for the highest monetary return, solution S1 is the ideal choice. Conversely, if the goal is to achieve the best equity, focusing on minimizing the Gini coefficient, solution S20 should be selected. However, both solutions S1 and S20 prioritize only one objective, either efficiency or equity, without balancing the two. A similar study by Deng et al., [2] in the Han River basin, China, demonstrated successful outcomes using an optimization model, and the results of this study show similarly favorable results.

Table 2 Min and max values on the Pareto front of objective functions
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As shown in Table 3, based on the dominance relationship derived from the sensitivity ratio, no solution dominates another, resulting in a new subset of non-dominated solutions. These solutions are used as inputs to calculate the preference degree of each Pareto non-inferior solution for different objective functions. The results of the preference degrees are shown in Fig. 8. These quantitative indices help decision-makers make more efficient trade-offs. For example, if the goal is to minimize f1, solution S1 can be chosen, as it is preference degree for f1 is 0.995, while for f2 it is only 0.005. Similarly, if the aim is to maximize f2, S20 is ideal, with a preference degree of 0.992 for f2 and just 0.008 for f1. For decision-makers who value both objectives equally, S10 is recommended, as it is preference degree for f1 (0.566) is closest to that for f2 (0.434) among all non-inferior solutions. The new subset of Pareto non-inferior solutions, based on the sensitivity ratio, not only narrows down the selection but also provides a quantitative evaluation, making decision-making more convenient.

Table 3 The non-inferior solution set based on sensitivity ratio with preference degree
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Fig. 8
figure 8

Distribution of the dimensionless sensitivity ratio

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Therefore, based on the optimal solution (S10), the water allocation scheme for the RVLB should be as presented in Table 4. Based on the scenarios and optimized allocation the current water allocation for irrigation should be 55.28% of the remaining 44.72%% water should be allocated to the other sectors with the livestock share being 13.14% and domestic demand taking its 7.75% share. For the planning years of 2035 and 2050 on Scenario III the irrigation demand share is declining with the use of modern irrigation technologies urbanization and industry expansion the demand share of both sectors is going to increase.

Table 4 Optimized water allocation for RVLB
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4 Conclusion

The model, implemented in Python and solved using the NSGA-II algorithm, used runoff time series data and water demand series for each major catchment as inputs. The current overall water demand in the Rift Valley Lakes Basin (RVLB) is estimated by integrating sectoral demands, resulting in a total of 1,796.34 MCM. Projections indicate that water demand will range from 1,628.23 MCM to 2,378.74 MCM in 2035 and from 2,126.09 MCM to 3,146.89 MCM in 2050. The RVLB aims to maximize economic efficiency by identifying allocation schemes yielding the highest economic returns. Solution S1 (0.995, 0.005) emerged as the most efficient allocation. Conversely, when prioritizing equity and minimizing the Gini coefficient, S20 (0.008, 0.992) was identified as the optimal scheme. However, these extreme solutions optimize only one objective, neglecting the other, and fail to integrate equity and efficiency comprehensively. Based on the optimized allocation scenarios, current water allocation should prioritize irrigation at 55.28%, with the remaining 44.72% allocated to other sectors, including 13.14% for livestock and 7.75% for domestic needs. For the planning years of 2035 and 2050 under Scenario III, irrigation demand is expected to decline due to modern irrigation technologies, while urbanization and industrial expansion will increase their respective demand shares.

In 2035 and 2050, the total water demand in the RVLB is projected to range from 2,279.49 MCM to 2,850.45 MCM and from 2,714.70 MCM to 3,867.41 MCM, respectively, reflecting a significant increase compared to the 2022 base year. Notably, agricultural and domestic water demands are expected to rise during these periods. A conflict between efficiency, equity, and sustainability in water resource allocation emerges as the Gini coefficient increases with economic benefits, resulting in reduced equity but higher monetary returns. Solutions S1 (0.995, 0.005) and S20 (0.008, 0.992) represent two extreme outcomes, each optimizing a single objective at the expense of the other. However, when both objectives, economic efficiency (f1) and equity (f2), are equally important, solution S10 (0.566, 0.434) is recommended. This balanced trade-off offers the smallest gap between preference degrees, making S10 an ideal choice for equitable and efficient decision-making. Projections for 2035 and 2050 necessitate adjusting allocation strategies to accommodate urbanization and industrial growth. This study highlights the importance of efficient water use in agriculture by promoting improved water management practices and adopting modern irrigation methods to reduce agricultural water demand, allowing more resources to be allocated to other sectors. The implementation of a multi-objective optimization model is crucial to balancing economic efficiency and equity in water distribution, ensuring sustainable and fair resource utilization.

5 Recommendations

Findings highlight critical policy implications essential for sustainable water resource management. Policymakers are advised to adopt integrated water resource management (IWRM) strategies that harmonize economic, social, and environmental goals while promoting cross-sectoral collaboration. Ensuring equity in water distribution should also be a priority through frameworks designed to provide fair access for marginalized groups and address regional disparities. Incorporating advanced optimization models such as NSGA-II into decision-making processes can help evaluate trade-offs and synergies among efficiency, equity, and sustainability. Climate resilience and sustainability should be integrated into water management policies, encouraging investments in water-saving technologies and sustainable irrigated agriculture. Preserving aquatic ecosystems and biodiversity should also be central objectives, supported by mandates for maintaining environmental flows. Transparent and participatory governance is crucial to engaging all stakeholders, it can reduce conflicts and improve decision-making. Building the capacity of water managers and policymakers is also vital for effectively implementing advanced methodologies, complemented by public awareness campaigns to promote water conservation. Economic incentives such as equitable pricing and subsidies for water-efficient technologies, paired with robust monitoring and feedback systems, are vital for adaptive and effective water management policies. These strategies collectively provide a comprehensive pathway to achieving equitable, efficient, and sustainable water resource management.