A multi-objective mathematical model of a water management problem with environmental impacts: An application in an irrigation project.

The study proposes applying an efficient but straightforward multi-objective constrained optimization model for optimal water allocation among irrigation and environmental sectors. The model has been implemented in the Muhuri Irrigation Project (MIP), Bangladesh, where the irrigation systems lead to unjustifiable use of groundwater. This study explores how water can be optimised to increase agricultural production and sustain the local environment in the MIP. Hence, the paper has two objectives-to maximise the net return and minimise the deficit in environmental flow. The study uses a Non-Dominating Sorting Genetic Algorithm, NSGA-II, to solve the research problem. Results indicate that crops more profitable to trade should be cultivated. Furthermore, the rainfall has more impact on the net return and environmental flow deficit than water inflow. The findings of this study can help plan irrigation water and cropland resources and be a reference for further studies.

1. Introduction

The scarcity of water is one of the significant issues in the agricultural sector in Bangladesh. Although Bangladesh is low-lying, riverine and featured by heavy rainfalls, the country suffers from seasonal water scarcity, especially during winter. The agriculture sector is the highest user of water in Bangladesh. This sector uses about 88% of total available water [1]. However, irrigated agriculture has begun in the 1960s. With the introduction of plentiful varieties of crops and the irrigation systems’ modernisation, cultivation through irrigated water has become widespread [2].

Bangladesh is a low-lying country with an area of approximately 144, 170 km2. From a climatic perspective, Bangladesh has four main seasons in a year: (i) from December to February is the dry winter season, (ii) from March to May is summer, the hot and humid season, (iii) from June to September is the rainy monsoon season, and (iv) from October and November is the autumn season. Summer in Bangladesh is very humid as winds blow from the southern hemisphere, creating a lot of moisture in the atmosphere, eventually depositing heavy precipitation amounts. In contrast, winds from the northern hemisphere are arid and cold; these blow towards the warm southern oceans. Of the total rainfall in Bangladesh, about 71% occurs in the rainy season, 27% occurs in the summer and autumn, and 2% occurs in the winter [3]. Still, rainfall in summer and autumn is rare. That is why Bangladesh faces two extreme water-related events each year, namely flood and drought [4]. For cultivating required crops in periods of dry and unreliable rainfall, the country needs to increase water-use efficiency and water conservation.

Moreover, Bangladesh is a densely populated country. Its population is about 168 million. Approximately 37.2% of this population live in urban areas, and 62.8%, living in village areas. Bangladeshi villages are still agrarian. The villagers rely on agriculture and agricultural productivity to earn their livelihood and lead their life [1].

The paper engages with an agricultural project in Bangladesh, known as the Muhuri Irrigation Project (MIP), in light of the above background. This project is in Feni, a south-eastern district in Bangladesh, around the confluence of Feni, Muhuri and Kalidaskhali rivers in the coastal belt of the Bay of Bengal. The MIP consists of a closure dam and a 20-vent regulator. This project’s construction began in 1978 and was completed in 1986 at the cost of $40 million. Building this project has been to provide irrigation facilities during the winter and regulate the inflow of saline water from the Bay of Bengal into the fresh river water. The project also helps farmers grow various kinds of crops in the dry season on the banks of the Muhuri River. It also functions as a large water vessel to produce many varieties of local fish [5].

However, the MIP tends to be less productive and beneficial than was initially planned. Its water cannot be used for the cultivation of crops optimally during dry periods. The lack of a proper water supply system, poor drainage, and unplanned cropping intensities appears to be some drawbacks in making the most out of the project. Nevertheless, this project built for the betterment of the agricultural community has never drawn any academic attention. No researchers have engaged with its drawbacks or potentials either inside or outside of Bangladesh. Therefore, the present paper finds MIP an exciting research area and identifies water allocation as a research problem. In this way, the article fills a research gap and finds a solution to the water allocation problem while maintaining a balance between water and natural life within the MIP. The Lewis and Randall [6] model is adopted and improved for this research project and uses a Non-Dominating Sorting Genetic Algorithm, NSGA-II, to solve the problem. This research uses a Multi-objective Optimisation Problem (MOP) in the agriculture sector in the MIP. Thus, the study locates at the intersection of mathematics and agriculture. Its findings can contribute to the optimal distribution and allocation of water to grow agricultural production in the Feni locality.

Multi-objective Optimisation Problem (MOP) has application in water management, agriculture, industry, engineering, economics, mining and many other fields where the problem involves simultaneously optimising several conflicting objectives. For example, in agriculture, the application of multi-objective optimisation models is well-accepted.

In recent decades, researchers from various parts of the world such as Australia, India, Iran, South Africa, China, Pakistan, and Saudi Arabia have developed their models or built their agricultural water allocation research on existing multi-object optimisation models [6-16]. Lewis and Randell [6] used multi-objective evolutionary computational techniques and Pareto optimisation concepts to solve different decision problems, including environmental flow in the agricultural system of the Irrigation area at Berembed weir on the Murrumbidgee River, Australia. Wardlaw and Bhaktikul [7] developed a Genetic Algorithm (GA) for solving multi-objective water scheduling problems in irrigation in the Indira Ghandi Nahal Pariyonaja (IGNP) irrigation system located in North-West India. A rotational basis operating system is applied for optimising the water resources in the irrigation systems. Again, Xevi and Khan [8] used a multi-objective decision-making structure for solving water allocation problems with conflicting objectives in irrigation. The three conflicting objective functions of the model are minimising variable cost, maximising net return, and minimising total pumping requirements for supplementary groundwater [8]. The authors used a goal programming model with a weighted version where a single objective function is created by combining all three objective functions using different weights to solve the MOP. Ikudayisi et al. [10] presented a combined Pareto multi-objective differential evolution algorithm to optimise crop distribution and water allocation in the irrigation under inadequate water accessibility at the Vaal-Harts Irrigation Scheme (VIS) in South Africa. They used two conflicting objective functions: minimising total water allocation in the irrigation and maximise net benefit. Musa [14] applied a multi-objective model in Saudi Arabia for optimal water allocation in three sectors named domestic sector, agriculture sector, and industrial sector. A goal programming technique has been used to solve this problem. Marzban et al. [15] proposed an optimal cropping pattern of irrigation and rainfed crops by using multi-objective nonlinear programming to minimise environmental impact and maximise the revenue in Iran.

The present article builds on the Lewis and Randell model [6] to solve a multi-objective optimisation problem in water allocation in the Muhuri Irrigation Project, Bangladesh. It uses a Non-Dominating Sorting Genetic Algorithm, NSGA-II, to solve the nonlinear constraint problem to find the optimum result. This model was applied to data sourced from the literature and the Bangladesh Water Development Board (BWDB), Feni, Bangladesh. The main aims of the study are to maximise net return and minimise the deficit in environmental flow by adjusting irrigation water when seasonal water availability is limited.

The main contributions of this article are as follows.

The Lewis and Randall [6] model is adopted and improved for this research project and applied in the Muhuri Irrigation Project (MIP), Bangladesh.

Considering the scenarios of different available water resources, the results can impact the agricultural production in the MIP area.

This method is very systematic and applied to different scopes, including water resources management. However, the most important thing is that the model can be used in other irrigation projects only by modifying the parameters according to the actual situation.

The remainder of this work is organised as follows: Section 2 presents the multi-objective optimisation problem, Section 3 explores the mathematical formulation, Section 4 contains the model solution and experimental format, Section 5 illustrates the results, and finally, Section 6 presents the conclusion of the study.

2. Multi-objective optimisation problem

Optimisation refers to maximising a system’s desirable characteristics while minimising its undesirable properties [17]. Optimisation can be both single-objective and multi-objective. Still, the multi-objective optimisation model, which this research adopts, tends to be most suitable for solving real-world problems. These mainly involve several contradictory and conflicting objectives. Multi-objective Optimisation Problems (MOPs) indicate optimisation problems with more objective functions that have to be optimised systematically and simultaneously under a given feasible region. MOPs are essential for our real-life because they provide a model for the case in which we have to consider the trade-off of several conflicting objectives.

To optimise all objective functions simultaneously and find a unique solution in real-life problems is difficult. Let us consider the following MOP where f(x)≔[f1(x),…,f(x)] stands for a vector of l objective functions and x∈ℝn, where f:ℝ→ℝ, i = 1,…,l, and g: ℝ→ℝ, j = 1,…,m.

The solutions of (1) are called Pareto points [18] or efficient points [19] or nondominated solutions. A point is said to be an efficient point or Pareto point or nondominated solution for Problem (MOP) iff there is no x∈X, such that , and For some j∈{1,…,l}. The plot consisting of the images of these Pareto points in the performance (objective) region is called the Pareto front. When we cannot find any better solution in value without sacrificing some of the other objective values, the solution is called a Pareto optimal solution. From the mathematical perspective, all Pareto optimal solutions are equally acceptable as the MOP solution. Nevertheless, in the end, only one solution will be chosen out of the Pareto optimal set. The choice made to choose a desirable solution depends on a decision-maker. Someone who takes the position of the decision-maker knows the inner parts of the problem and can convey their preference relations between different solutions. However, options have to be given to the decision-maker first for them to decide.

3. List of symbols

X            Area of crop c to be planted in a hectare

Env_flow_f(m)    Environmental flow for a month (m)

TCI            Total crop income

C            Groundwater pumping and delivery cost per unit volume

P            Groundwater pumping volume in a month (m)

Vcost        Variable cost (such as seeds, fertiliser, labour,

and pesticides) per hectare excluding water cost for the crop (c)

C            Water supply costs from dam per unite volume

WREQ        Water requirement for the crop (c) in a month (m)

C            Total number of types of crops to be planted

M            Total number of months in the planning period

Ter_env_flow_f(m)    Target environmental flow for a month (m)

k        Crop coefficient for the crop (c) in a month (m)

ET         Evapotranspiration for a month (m)

Rain        Rainfall for a month (m) measured by millimetres

total_Pump        Permissible pumping for the year in the irrigated areas

T        Total cropping area available

minimum_area        Minimum plantable areas

Allocation(m)        Surface water amount which is accessible for irrigation

in a month (m)

Inflow(m)        Amount of surface (river) water available for a month (m)

4. Mathematical formulation

In this section, we present a water management model introduced by Lewis and Randall [6]. A description of the mathematical expressions used to construct the two-objective optimisation model is provided. Our goal is to formulate terms that measure the net return (NR), the shortage of irrigation water, and the environmental flow deficit (EFD).

In this article, we aim to find the planting areas per crop and corresponding optimal crop mix while maximising net return (NR) whilst minimising irrigation water and minimising deficit in environmental flow (EFD). The decision variables are X and Env_flow_f(m). The first objective of the model is to maximise net return (NR)

The first term of the objective function in Eq (2) is the total revenue and the second term is the expenditure related to the groundwater pumping and delivery cost. The third term is the expenditure, which comprises the variable cost such as fertiliser, pesticides, seeds, and other costs. Finally, the last term is related to the expenditure, including the cost of surface water supply accessible for irrigating crops in a month (m). The difference between the revenue and all expenditures gives the net return.

The second objective is to maintain enough downriver flows to sustain the environment. This objective is set to maintain a balance between water use and the life of nature in the MIP. Because if the focus is given only on irrigation but not on its environment, biodiversity will be hampered. Still, the objective focuses on how to sustain bio-diversity with minimum use of water.

The only terms in the summation of Eq (3) included are only for those months where the environmental flow is less than the target; otherwise, zero is used instead. The environmental flow, Env_flow_f(m) is the river’s flow pattern necessary to sustain the ecosystem.

Water requirement

The crop water requirements per month, WREQ, is the excess of evapotranspiration with the growth duration in months over rainfall,

Problem constraints

There are several environmental and physical constraints imposed on the model, which are shown below.

The first constraint is the pumping water constraint,

This constraint ensures that pumped groundwater does not exceed the allowable pumping for the year from the irrigation area.

The second constraint is the maximum area constraint,

This constraint limits the total crop area planted to be equal to or less than the total area available.

The third constraint is the minimum area constraint,

This constraint limits the crop planted to be of at least a minimum size or zero. This means that if a crop has a minimum plantable area, the corresponding crop area, X, must be greater than this minimum area if the crop is to be planted.

The following constraint relates to the amount of groundwater pumping. The pumped groundwater needed can be obtained from the accessible surface water and the crop water requirements for irrigation of the crop in a month (m) and is given by

The last constraint is the water allocation constraint,

After the environmental flow is released from the accessible surface river water, the remaining water can be used to irrigate the crops in a month (m).

Using Eqs (4) and (9), we write (8) as

Considering Eqs (2) and (3) which are the objectives of our model and combine with (8) and (9), subject to the constraints Eqs (5)—(7), we formulate the bi-objective Problem (P) described as follows.

Subject to

5. Model solution and experimental setup

The annual average rainfall in the MIP area is 2447 mm [20]. Here we use average rainfall data collected from the Bangladesh Water Development Board (BWDB), Feni, Bangladesh, as provided in Table 1.

Table 1

Rainfall data (in mm) in the Muhuri irrigation area.

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
0	28.27	19.5	298.25	313.5	508.75	887.25	442.75	340.75	370.75	5.75	56.25

Evapotranspiration is the sum of the water evaporated from the soil and plant and transpired through the plant. Evapotranspiration reaches the maximum level in April and May when temperature, sunshine, and wind are at or close to their maximum levels for the year. Monthly evapotranspiration data was collected from [5] and provided in Table 2.

Table 2

Evapotranspiration (in mm) data in the Muhuri irrigation area.

Area	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Feni	72	89	130	143	145	115	113	117	110	106	81	68

Major rivers within the project area are the Feni, Kalidas-Pahalia, and Muhuri rivers.

In addition, there are many Khals located in the area. Other rivers outside the project area, such as Titas, Gumti, Dakatia and Meghna, act as the prominent drainage collectors. Surface water irrigation is from the three rivers and supported by storage in the rivers, drains and reservoirs in the backwater from Feni Regulator. Table 3 contains the water inflows from the three rivers which were collected from [5].

Table 3

Total water inflow in cubic meters.

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
16.5	11.4	10.0	14.1	21.1	58.3	68.8	105.2	61.9	50.4	30.5	20.1

The crop coefficient, k, is the ratio of the reference crop evapotranspiration, ET0 and crop evapotranspiration, ET.

In this research, crop coefficient data in Table 4 has been taken from [20].

Table 4

Crop coefficient k [20].

Crops	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
T. Aus	0.2	0.2	0.2	1.05	1.2	1.2	0.9	0.2	0.2	0.2	0.2	0.2
T. Aman	0.2	0.2	0.2	0.2	0.2	0.2	1.2	1.2	0.9	0.2	0.2	0.2
Boro Rice	1.2	0.9	0.9	0.2	0.2	0.2	0.2	0.2	0.2	0.2	1.05	1.2
Wheat	1.15	1.15	0.4	0.4	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.4
Potato	0.8	0.8	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.6	1.15	1.15
Oilseeds	0.3	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.35	1.05
Pulses	1.05	0.3	0.3	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.4
Sugarcane	0.4	0.4	0.2	0.2	0.2	0.2	1.15	1.15	0.9	0.9	0.6	0.6
Winter Vegetable	0.9	0.9	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.6	1.1
Summer Vegetable	0.2	0.2	0.2	0.6	1.1	0.9	0.9	0.2	0.2	0.2	0.2	0.2

Crops production (T/ha) and crop market price (AUD) data in Table 5 were collected from the Deputy Chief Extension Officer, BWDB, Feni, Bangladesh.

Table 5

Economic data for crops in the Muhuri irrigation area (1 AUD = 60 Taka).

Crops	Production (T/Ha)	Market Price (AUD)
(1) T. Aus	3.2	331
(2) T. Aman	4.25	365
(3) Boro Rice	5.85	331
(4) Wheat	2.8	206
(5) Potato	23	248
(6) Oilseeds	1.1	537
(7) Pulses	1.56	557
(8) Sugarcane	50	4965
(9) Winter Vegetable	16.5	435
(10) Summer Vegetable	14.85	383

The number of variables set in this study is the total number of crops, X which consists of ten crops and the environmental flow, Env_flow_f(m), for twelve months. The lower bound of all the variables is zero. The upper bound of the cultivable area for each crop is 70,000 ha. The minimum area is 1000 ha. The target environmental flow, Ter_env_flow_f(m) is set to 100 GL for each month.

The Problem (P) is a multi-objective nonlinear constrained optimisation problem, requiring an excellent computational method to approximate the Pareto solutions. This article uses the NSGA-II for solving the Problem (P). Deb et al. [21] developed the NSGA-II, a multi-objective genetic algorithm for solving optimisation problems. The NSGA-II works by dominance and non-dominance relation and to determine Pareto solutions. It is an extension and improvement of NSGA, proposed earlier by Srinivas and Deb [22]. Also, it is an elite and fast sorting multi-objective genetic algorithm. The NSGA-II has three unique properties: simple crowded comparison operator, fast non-dominated sorting approach, and fast crowded distance estimation procedure [21]. The pseudocode of the NSGA-II is given next.

Step1: Randomly create an initial population P0 of size N

Step2: Calculate the values of the objective of each individual P0

Step3: By using a non-domination sorting process, assign a rank of each individual P0

Step4: Generate child population Q of size N using crossover and mutation

Step5: Calculate the objective values of each child population Q

Step6: Combine the initial and child population (P = (P0∪Q)) of size 2N

Step7: Assign rank to each individual P based on the non-domination sorting process

Step8: Calculate the crowded distance of individuals in each front

Step9: Select the best N individuals base on rank and crowded distance

Step10: Repeat Step2 to Step9 until the stopping criterion met

Step11: Terminate the algorithm

The population size is a sensitive issue in the genetic algorithm (GA); smaller populations result in lower accuracy of the solution; this means little search space is available. Therefore, it is possible to reach an unwanted local optimum. The further increase in the population size increases the accuracy of the solution, but the computational load becomes high [23]. Therefore, the size of the population must be reasonable. In each computation run, the population size of the algorithm in this study is set at 100.

The crossover rate (probability) is a genetic operator used to vary the programming of a chromosome or chromosomes from one generation to the next, i.e., the chance that two chromosomes exchange some parts if crossover probability is 100%, then all offspring are made by crossover. If it is 0%, a whole new generation is made from exact copies of chromosomes from the old population, except those that resulted from the mutation process. The crossover rate is in the range of [0, 1] [24]. The crossover rate in this study is set at 0:2.

The mutation is another vital operator which takes place after the crossover is done. The mutation rate decides how many chromosomes should be mutated in one generation. The mutation rate is in the range of [0, 1] [25]. In our study, the mutation scaling factor is set at 1.

The number of generations refers to the number of cycles before the algorithm stops. It depends on the type of optimisation problem and its complexity. In this case, the NSGA-II algorithm is iterated for 500 generations. It is to note here that setting the frequency of change based on the number of generations sometimes makes the comparison unfair. However, our experience shows that the more the population size and the number of generations, the more the results converge. Therefore we use the number of generations instead of function evaluations.

For evolutionary algorithms like GA, there are seven kinds of stopping criteria [26]. In this research, the maximum number of iterations is set for stopping criteria, and it is 300 iterations.

6. Results and discussion

In Section 4, we have demonstrated the multi-objective optimisation problem (P) for the Muhuri Irrigation Project (MIP). Our objectives have been maximising net return (NR) and minimising deficit in environmental flow (EFD) under constraints. We have adopted the NSGA-II algorithm for solving the Problem (P). Our experimental results are as follows:

Results

The test run was carried out using 300 iterations. The Pareto front obtained for 300 iterations is demonstrated in Fig 1, and we have considered this Pareto front as a base level solution. The information on the number of solutions, the computational time, and the range of objective function values obtained are in Table 6 for the NSGA-II algorithm. The Pareto front is taken from NSGA-II, representing 34 non-dominated solutions for net return in units of 10 million Australian dollars and environmental flow deficit in units of 100 GL.

Fig 1

Pareto front for 300 iterations.

Table 6

Summary for 300 iterations.

Objectives	Net return (NR)	Environmental flow deficit (EFD)
Mean	1877.12×107 AUD	10.01 GL
Maximum output	1877.48×107 AUD	35.53 GL
Minimum output	1876.68×107 AUD	0.00 GL
Number of solutions	34
Computational time	706.59 minutes

Table 6 shows that when the maximum net return is 1877.48×107 AUD, the environmental flow deficit increases to a maximum of 35.53 GL. In such a case, one needs to compromise with the environmental flow. On the other hand, we can keep EFD on it lower, in which case, the net return would be 1876.68×107 AUD, which is the lowest net return on the Pareto front.

The solution of the MOPs is a set of efficient solutions, which are also Pareto optimal solutions. There is a role of a decision-maker in choosing a solution among many options. We cannot say one solution is better than the other in this experiment. Only the decision-maker identify the best solution depends on their preference.

According to Fig 1, the analysis of all the 34 solutions, solution 1 (A in Fig 1) shows the best in terms of net return (NR) but worst in terms of environmental flow deficit (EFD). Whilst solution 34 (B in Fig 1) is the best in EFD but worst in NR.

Crop area

Cropping patterns are used for the MIP to approximate the Pareto front shown in Fig 1. The 1st solution (A in Fig 1) of the S1 Table included in the Supporting information file suggests that T. Aus, T. Aman, Boro Rice, Wheat, Potato, Oilseeds, Pulses, Sugarcane, Winter Vegetables, and Summer Vegetables should be planted in 1452.18 (ha), 1516.63 (ha), 13504.46 (ha), 2555.28 (ha), 48610.52 (ha), 6567.29 (ha), 1072.37 (ha), 69228.00 (ha), 69227.79 (ha) and 16982.25 (ha) areas of land respectively. When we inspect the solution for the crop mix, we see that the maximum areas, 69228.00 (ha) and 69227.79 (ha), are devoted to growing Sugarcane and Winter Vegetables. The reason becomes clear as both crops are highly profitable and the production is a high per hectare of 50 tonnes and 16.5 tonnes respectively. Also, Sugarcane and Winter Vegetables provide a gross return of AUD 4965 and AUD 435 per hectare.

The 34th solution (B in Fig 1) has the lowest net return of AUD 1876.68×107 with zero GL deficit in environmental flow. The cropping pattern of the 34th solution as provided in the S2 Table included in the Supporting information file, suggests that T. Aus, T. Aman, Boro Rice, Wheat, Potato, Oilseeds, Pulses, Sugarcane, Winter Vegetables, and Summer Vegetables should be planted in 1453.53 (ha), 1529.59 (ha), 13451.38, (ha), 2854.91 (ha), 48197.63 (ha), 6631.99 (ha), 1072.15 (ha), 69227.99 (ha), 69227.88 (ha) and 17026.76 (ha) areas of land respectively. A few differences between these two solutions are noticeable. The 1st solution (A) presents the planting area of Wheat, Potato, and Summer Vegetables at 2555.28 (ha), 48610.52 (ha), and 16982.25 (ha), respectively. However, in the 34th solution (B), a slightly different scenario is seen for planting these three crops. Here 2854.91 (ha), 48197.63 (ha), and 17026.76 (ha) areas of land are devoted to these crops.

Environmental flow

The environmental flow for the Pareto front of Fig 1 is provided in the Supporting Information file. The 1st solution (A in Fig 1) of the S3 Table in the Supporting Information file shows that the highest amount of water, i.e. approximately 250 GL, is required for environmental flow in November. The second and third most elevated amount of water is needed for June and October, and their amount is approximately 164 GL and 145 GL, respectively. About 129 GL water is required for the month of May. Finally, the environmental flow is almost the same (near 100 GL) in the remaining months.

We see a slight difference of environmental flow in GL of the 34th solution (B in Fig 1) is given in the S4 Table. In November, approximately 256 GL of water is needed for the environmental flow, which is the highest amount of water across all other months. About 162 GL and 157 GL are required for June and October, respectively. About 145 GL of water is needed for May. Finally, the environmental flow is almost the same for approximately 100 GL for the rest of the year.

Effect of rainfall

The results for five Pareto front curves when rainfall is varied by 10% and 20% above and below the base level using 300 simulation run is shown in Fig 2.

Fig 2

Pareto fronts for different rainfall.

Fig 2 illustrates that if rainfall is 10% and 20% below the base level, then for the 1st solution, NR will decrease 0.47% and 0.76%, respectively, whereas EFD will increase 67.95% and 71.02%, respectively.

Also, if rainfall is 10% and 20% above the base level, NR will increase 0.54% and 0.77%, respectively. On the other hand, EFD will decrease by 37.63% and 28.27%.

The crop pattern for the 1st solution in the net return for different rainfall using 300 simulation runs is provided in Fig 3A. According to Fig 3A, the land area for cultivating Sugarcane is the same for all five conditions at approximately 69228 ha. However, the most significant difference is observed for Potatoes and Summer Vegetables. In the base level rainfall, we see the highest amount of land is devoted to cultivating crop 5 (Potatoes), but the opposite scenario is seen for Summer Vegetables. When rainfall decreases or increases, the cultivation of Potatoes continuously decreases, but the opposite happens for Sugarcane. For other crops, the differences are minor but still varied.

Fig 3

Effect of different rainfall on crops and environmental flow.

(a) Crops pattern for different rainfall. (b) Environmental flow for different rainfall.

The environmental flow for the 1st solutions in the context of the net return for different rainfall using 300 simulation runs are provided in Fig 3B. As observed from Fig 3B, when rain is 20% above the base level, the highest environmental flow is required for the month 5 (May) at approximately 290 GL. On the other hand, the lowest environmental flow is needed for month 3 (March) when rainfall is 10% below the base level at about 50 GL.

In light of the above discussion, it can be argued that if it rains more, profits will increase, and the cost of irrigation and water supply for environmental flow will decrease.

Effect of water inflow

The results for five Pareto front curves when water inflow is varied by 10% and 20% above and below the base level using 300 simulation run is shown in Fig 4.

Fig 4

Pareto front for different water inflow.

Fig 4 illustrates that if water inflow is 10% and 20% below the base level, NR will decrease 0.21% and 0.37%, and EFD will decrease 11.71% and 60.66%, respectively. In addition, if water inflow is 10% and 20% above the base level, NR will increase 0.36%, and 5.32×10−9 % and EFD will decrease by 2.81% in both cases.

According to Fig 4, the highest environmental flow is required for less than 10% water inflow from the base level in the month 6 (June). The same scenario is seen for base-level water inflow in month 11 (November). For the case of 10% more water inflow, we see more than 200 GL water is required for environmental flow in months 2 (February) and 12 (December). From the above discussion, we conclude that more water inflow brings more profit.

The crops pattern for the 1st solutions in the context of the net return for different water inflows using 300 simulation runs is provided in Fig 5A. Based on Fig 5A, we see the same scenario with slight differences. For different water inflow level conditions, Sugarcane is cultivated across the same area of land. However, for Potatoes and Summer Vegetables, the opposite occurs. For all other crops, there is a slight variation.

Fig 5

Effect of different water inflow on crops and environmental flow.

(a) Crops pattern for different water inflow. (b) Environmental flow for different water inflow.

The environmental flow for the 1st solutions in the context of the net return for different water inflow using 300 simulation runs is provided in Fig 5B. According to Fig 5B, the highest environmental flow is required for less than 10% water inflow from the base level in month 6 (June). The same scenario is seen for base-level water inflow in the month 11 (November). For the case of 10% more water inflow, we see more than 200 GL of water is required for environmental flow in months 2 (February) and 12 (December).

As expected, this leads to the notion that more water inflow brings more profit.

7. Conclusion

This study sought to explore the economics of optimal water allocation for irrigation and optimal cropping patterns in the MIP of Bangladesh. Although Bangladesh is not a country with widespread, year-round water scarcity, it faces severe water shortages during the dry winter season. This article aims to maximise net return and minimise the deficit in environmental flow using optimal water management policies.

Based on the framework mentioned above, the research has several outcomes. The following is a synthesis of those outcomes:

The crop which produces the most significant profitability is recommended to be cultivated to a greater extent

During the dry season, more environmental flow is required to sustain the environment and to grow crops than in the rainy season.

The decrease and increase of net return (NR) and rainfall are directly proportional to each other. However, the relationship between rainfall and environmental flow deficit (EFD) is not proportional. The decrease of rain by 10% contributes to the increase of environmental flow deficit (EFD), but the decrease of rain by 20% does not impact the environmental flow deficit (EFD) in the same way.

When water inflows increase, net returns (NR) also increase. On the other hand, the environmental flow deficit (EFD) decreases with increased water inflow and vice versa.

Details of 1–17 Pareto solutions for the crops.

This includes results that we have found from using the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) on the Multi-objective Optimisation Problem (MOP).

(PDF)

Click here for additional data file.

Details of 18–34 Pareto solutions for the crops.

This includes results that we have found from using the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) on the Multi-objective Optimisation Problem (MOP).

(PDF)

Click here for additional data file.

Details of 1–17 Pareto solutions for the environmental flow.

This includes results that we have found from using the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) on the Multi-objective Optimisation Problem (MOP).

(PDF)

Click here for additional data file.

Details of 18–34 Pareto solutions for the environmental flow.

This includes results that we have found from using the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) on the Multi-objective Optimisation Problem (MOP).

(PDF)

Click here for additional data file.

25 Apr 2021

PONE-D-21-08160

A multi-objective mathematical model of a water management problem with environmental impacts: An application in an irrigation project

PLOS ONE

Dear Dr. Ullah,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please revise the work thoroughly according to the reviewers' comments.

Please submit your revised manuscript by Jun 09 2021 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

Shouyong Jiang, PhD

Academic Editor

PLOS ONE

Journal Requirements:

When submitting your revision, we need you to address these additional requirements.

Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at

https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and

https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

PLOS requires an ORCID iD for the corresponding author in Editorial Manager on papers submitted after December 6th, 2016. Please ensure that you have an ORCID iD and that it is validated in Editorial Manager. To do this, go to ‘Update my Information’ (in the upper left-hand corner of the main menu), and click on the Fetch/Validate link next to the ORCID field. This will take you to the ORCID site and allow you to create a new iD or authenticate a pre-existing iD in Editorial Manager. Please see the following video for instructions on linking an ORCID iD to your Editorial Manager account: https://www.youtube.com/watch?v=_xcclfuvtxQ

Additional Editor Comments :

Obviously there are a number of issues which need to be well addressed for publication.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Partly

Reviewer #2: Yes

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: I Don't Know

Reviewer #2: Yes

**********

3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: This work seems very significant to address optimum water allocation in the Muhuri Irrigation Project (MIP), Bangladesh. However, it still exists several problems.

1. Only seeing the current abstract of this paper, I don,t think that readers can appreciate it since the abstract about this work is very unclear for interested readers. So, I suggest the author improve its abstract to make it more persuasive.

2. Related work needs further discussion. Some important developments

in this area in recent years do not seem to be discussed. Additionally，half of the provided literature during the current manuscript is elder, which suggests that the provided literature of the author haven't authoritative.

3. This paper hasn't provided any parameters illustration. Some common parameters should be mentioned. Please, discuss how the parameters

of this paper were settled, and if this paper is robust to parameters

values change or not.

4. Please guaranty that you explains each symbol in all formulas of this paper. If the problem is not corrected next time, I will be obliged to reject the paper.

5. Authors should add a paragraph into the introduction section. They should write, "The main contributions of this paper are: (i) ….. (ii) ……. and (iii) ……" to highlight the key works. By this way, authors should provide a stronger motivation clearly and explain the originality of the paper.

6. Setting the frequency of change based on the number of generations sometimes makes the comparison unfair. It is preferable to be based on the number of function evaluations, or you should explain why it's still fair to use the number of generations.

Reviewer #2: After reading this paper, it is recommended to consider accepting the following major revisions:

1.Check the English sentences carefully. There are too many words repeated in the sentences in a paragraph. It is recommended to use refined sentences to write. Suggest additional content in the abstract section.

2.In the article, it is recommended to increase the algorithm pseudo code, and can make improvements on the genetic operator.

3.It is found from Table 6 that the "Computational time" time is too long. Can the algorithm be improved to reduce the CPU time?

**********

6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

27 May 2021

Dear Editor and Reviewers,

We would like to thank you for the careful review and thoughtful feedback on our manuscript “A multi-objective mathematical model of a water management problem with environmental impacts: An application in an irrigation project” (PONE-D-21-08160) and the opportunity to resubmit a revised copy. We have revised the manuscript according to the comments and believe that it is substantially improved by incorporating these edits.

Below, we provide a point-by-point reply to the reviewers’ comments. We have included a marked copy of the revised manuscript that highlights changes and a clean version. We have also ensured that our manuscript meets the style requirements of the PLOS ONE.

Thank you for your consideration of our revised manuscript.

EDITOR COMMENTS

Comment 1: Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at https://journals.plos.org/plosone/s/file?id=wjVg/ POSOne_formatting_sample_main_body.pdf

and

https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

Response: We have prepared our manuscript as per the PLOS ONE journal requirements. The file name is Manuscript.docx.

Comment 2: PLOS requires an ORCID iD for the corresponding author in Editorial Manager on papers submitted after December 6th, 2016. Please ensure that you have an ORCID iD and that it is validated in Editorial Manager. To do this, go to ‘Update my Information’ (in the upper left-hand corner of the main menu), and click on the Fetch/Validate link next to the ORCID field. This will take you to the ORCID site and allow you to create a new iD or authenticate a pre-existing iD in Editorial Manager. Please see the following video for instructions on linking an ORCID iD to your Editorial Manager account: https://www.youtube.com/watch?v=_xcclfuvtxQ

Response: ORCID iD https://orcid.org/0000-0001-5491-1289

REVIEWER # 1 COMMENTS

Comment 1. Only seeing the current abstract of this paper, I don,t think that readers can appreciate it since the abstract about this work is very unclear for interested readers. So, I suggest the author improve its abstract to make it more persuasive.

Response: We have rewritten the abstract in the following way on page 2 (lines 25 -35) in the Manuscript file:

“The study proposes applying an efficient but straightforward multi-objective constrained optimisation model for optimal water allocation among irrigation and environmental sectors. The model has been implemented in the Muhuri Irrigation Project (MIP), Bangladesh, where the irrigation systems lead to unjustifiable use of groundwater. This study explores how water can be optimised to increase agricultural production and sustain the local environment in the MIP. Hence, the paper has two objectives--to maximise the net return and minimise the deficit in environmental flow. The study uses a Non-Dominating Sorting Genetic Algorithm, NSGA-II, to solve the research problem. Results indicate that crops more profitable to trade should be cultivated. The rainfall has more impact on the net return and environmental flow deficit than water inflow. The findings of this study can help plan irrigation water and cropland resources and be a reference for further studies.”

Comment 2: Related work needs further discussion. Some important developments in this area in recent years do not seem to be discussed. Additionally, half of the provided literature during the current manuscript is elder, which suggests that the provided literature of the author haven't authoritative.

Response: Three more recent articles published in 2021 are included in the reference section on page 25, reference numbers 14 - 16. Also, six lines are included about these articles on page 4 (lines 109 - 114) in Manuscript file as follows:

“Musa [14] applied a multi-objective model in Saudi Arabia for optimal water allocation in three sectors named domestic sector, agriculture sector, and industrial sector. A goal programming technique has been used to solve this problem. Marzban et al. [15] proposed an optimal cropping pattern of irrigation and rainfed crops using multi-objective nonlinear programming to minimise environmental impact and maximise the revenue in Iran.”

Comment 3: This paper hasn't provided any parameters illustration. Some common parameters should be mentioned. Please, discuss how the parameters of this paper were settled, and if this paper is robust to parameters values change or not.

Response: More information about NSGA-II parameters is given on pages 15 and 16 (lines 318 - 343) in the Manuscript file as follows:

“The population size is a sensitive issue in the genetic algorithm (GA); the use of smaller populations results in lower accuracy of the solution, this means little search space is available, and therefore it is possible to reach an unwanted local optimum. The further increase in the population size increases the accuracy of the solution, but the computational load becomes high [23]. Therefore, the size of the population must be reasonable. In each computation run, the population size of the algorithm in this study is set at 100.

The crossover rate (probability) is a genetic operator used to vary the programming of a chromosome or chromosomes from one generation to the next, i.e., the chance that two chromosomes exchange some of their parts, If crossover probability is 100%, then all offspring is made by crossover. If it is 0%, whole new generation is made from exact copies of chromosomes from old population, except those resulted from the mutation process. The crossover rate is in the range of [0, 1] [24]. The crossover rate in this study is set at 0:2.

The mutation is another vital operator which takes place after the crossover is done. The mutation rate decides how many chromosomes should be mutated in one generation. The mutation rate is in the range of [0, 1] [25]. In our study, the mutation scaling factor is set at 1.

The number of generations refers to the number of cycles before the algorithm stops. It depends on the type of optimisation problem and its complexity. In this case, the NSGA-II algorithm is iterated for 500 generations. It is to note here that setting the frequency of change based on the number of generations sometimes makes the comparison unfair. However, our experience shows that the more the population size and the number of generations, the more the results converge. Therefore, we use the number of generations instead of function evaluations.

For evolutionary algorithms like GA, there are seven kinds of stopping criteria [26]. In this research maximum, the number of iterations is set for stopping criteria, and it is 300 iterations.”

Comment 4: Please guaranty that you explain each symbol in all formulas of this paper. If the problem is not corrected next time, I will be obliged to reject the paper.

Response: We have added a new section named “List of symbols” on pages 7 and 8 (lines 167 - 188) in the Manuscript.docx file.

Comment 5: Authors should add a paragraph into the introduction section. They should write, "The main contributions of this paper are: (i) ….. (ii) ……. and (iii) ……" to highlight the key works. By this way, authors should provide a stronger motivation clearly and explain the originality of the paper.

Response: We have added a paragraph into the introduction section on page 6 (lines 123 -131) in the Manuscript.docx file as follows:

“The main contributions of this paper can be highlighted as follows.

i. The Lewis and Randall [6] model is adopted and improved for this research project and applied in the Muhuri Irrigation Project (MIP), Bangladesh.

ii. Considering the scenarios of different available water resources, the results can have an impact on the agricultural production in the MIP area.

iii. This method is very systematic and applied to different scopes, including water resources management. The most important thing is that the model can be used in other irrigation projects only by modifying the parameters according to the actual situation.”

Comment 6: Setting the frequency of change based on the number of generations sometimes makes the comparison unfair. It is preferable to be based on the number of function evaluations, or you should explain why it's still fair to use the number of generations.

Response: More information is given on page 16 (lines 25 - 35) in the Manuscript.docx file as follows: “It is to note here that setting the frequency of change based on the number of generations sometimes makes the comparison unfair. However, our experience shows that the more the population size and the number of generations, the more the results converge. Therefore, we use the number of generations instead of function evaluations.”

REVIEWER # 2 COMMENTS

Comment 1: Check the English sentences carefully. There are too many words repeated in the sentences in a paragraph. It is recommended to use refined sentences to write. Suggest additional content in the abstract section.

Response: We have checked the English sentences carefully. Also, we have rewritten the abstract in the following way on page 2 (lines 25 -35) in the Manuscript file:

“The study proposes applying an efficient but straightforward multi-objective constrained optimisation model for optimal water allocation among irrigation and environmental sectors. The model has been implemented in the Muhuri Irrigation Project (MIP), Bangladesh, where the irrigation systems lead to unjustifiable use of groundwater. This study explores how water can be optimised to increase agricultural production and sustain the local environment in the MIP. Hence, the paper has two objectives--to maximise the net return and minimise the deficit in environmental flow. The study uses a Non-Dominating Sorting Genetic Algorithm, NSGA-II, to solve the research problem. Results indicate that crops more profitable to trade should be cultivated. The rainfall has more impact on the net return and environmental flow deficit than water inflow. The findings of this study can help plan irrigation water and cropland resources and be a reference for further studies.”

Comment 2: In the article, it is recommended to increase the algorithm pseudo code, and can make improvements on the genetic operator.

Response: The pseudocode of the NSGA-II is added on pages 15 (line number 306 - 317) in the Manuscript file.

Comment 3: It is found from Table 6 that the "Computational time" time is too long. Can the algorithm be improved to reduce the CPU time?

Response: In this research total number of variables is 22, the population size is 100, and the NSGA-II algorithm is iterated for 500 generations. The maximum number of iterations was set for stopping criteria, and it was 300. When we decreased the population size, the number of generation and the total number of iteration, “Computational time” reduced. Then, we ran this program for 600 and 1000 simulations. The computational times were 1324.17 min and 2610.44 min, respectively. A simulation run of the algorithm for 1500 iterations was attempted. Unfortunately, the algorithm failed complete processing due to insufficient memory. The simulations in this research were conducted on a Windows 10 laptop with 8 GB RAM running a 1.60 GHz Intel(R) Core (TM) i5-8250U CPU.

Submitted filename: Response to Reviewers.docx

Click here for additional data file.

30 Jun 2021

PONE-D-21-08160R1

A multi-objective mathematical model of a water management problem with environmental impacts: An application in an irrigation project

PLOS ONE

Dear Dr. Ullah,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript by Aug 14 2021 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

Shouyong Jiang, PhD

Academic Editor

PLOS ONE

Journal Requirements:

Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

Additional Editor Comments:

That paper has been largely improved after revision. However, the references need to be consistent. The authors are encouraged to use a referencing style in line with the Journal.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: Although the authors already received many comments to improve the readability and presentation of the paper, the literature still needs to be normalized.

Reviewer #2: (No Response)

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

12 Jul 2021

Dear Editor and Reviewers,

We would like to thank you for the careful review and thoughtful feedback on our manuscript “A multi-objective mathematical model of a water management problem with environmental impacts: An application in an irrigation project” (PONE-D-21-08160) and the opportunity to resubmit a revised copy. We have revised the manuscript according to the comments and believe that it is substantially improved by incorporating these edits.

Below, we provide a point-by-point reply to the reviewers’ comments. We have included a marked copy of the revised manuscript that highlights changes and a clean version.

We appreciate your consideration of our revised manuscript.

Journal Requirements:

Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

Response: We have reviewed our reference list as per the PLOS ONE journal requirements in the “Vancouver” style.

Additional Editor Comments:

That paper has been largely improved after revision. However, the references need to be consistent. The authors are encouraged to use a referencing style in line with the Journal.

Response: We have reviewed our reference list as per the PLOS ONE journal requirements. Also, we have tried to be consistent in the references.

REVIEWER COMMENT :

6. Review Comments to the Author:

Reviewer #1: Although the authors already received many comments to improve the readability and presentation of the paper, the literature still needs to be normalized.

Response: We have checked the English sentences carefully and tried to improve the literature.

Submitted filename: Response to Reviewers..docx

Click here for additional data file.

19 Jul 2021

A multi-objective mathematical model of a water management problem with environmental impacts: An application in an irrigation project

PONE-D-21-08160R2

Dear Dr. Ullah,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at http://www.editorialmanager.com/pone/, click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org.

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

Kind regards,

Shouyong Jiang, PhD

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

The authors have addressed the issue carefully.

Reviewers' comments:

23 Jul 2021

PONE-D-21-08160R2

A multi-objective mathematical model of a water management problem with environmental impacts: An application in an irrigation project

Dear Dr. Ullah:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

If we can help with anything else, please email us at plosone@plos.org.

Thank you for submitting your work to PLOS ONE and supporting open access.

Kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Dr. Shouyong Jiang

Academic Editor

PLOS ONE