Self-adaptive multi-population-based Jaya algorithm to optimize the cropping pattern under a constraint environment

Water resources have been very restricted due to the growing population and demand from the agricultural sector. It has become essential to wisely and effectively allocate water resources, particularly in areas facing an acute water shortage. In order to maximize the agricultural yield, adequate water supply is essential for the growth of the crop. Many irrigation projects have been constructed over the past few centuries. With the growing demand, however, owing to high investment and land acquisition issues, it is not practical to build more and more water resources projects (Adeyemo & Otieno 2010). Optimizing the existing irrigation project is an alternative solution (Kumar & Yadav 2018). The conventional methods of optimization are linear programming (LP), non-linear programming (NLP) and dynamic programming (DP). LP has been the best method of optimization commonly used to obtain the optimum cropping pattern to increase the net benefits (Paul et al. 2000; Singh et al. 2001; Moradi-Jalal et al. 2007). Ghahraman & Sepaskhah (2002) used the system based on NLP and DP to find the optimum water allocation for various crop models. The NLP-based model was used by Georgiou & Papamichail (2008) to optimize water allocation and crop planning. Mainuddin et al. (1997) and Sethi et al. (2002) used deterministic linear programming (DLP) to achieve the optimum cropping pattern. Azaiez et al. (2005) applied change constrained linear programming (CCLP) to optimize the stochastic nature of the cropping pattern. Sethi et al. (2006) used DLP and CCLP to study the optimum cropping pattern. Jothiprakash et al. (2011) used CCLP to solve different dependable inflows in order to obtain an optimal cropping pattern based on the monthly and annual probability distribution models.

Although all of these methods have been widely used, they have certain limitations. LP can solve only linear equations and cannot cope with the uncertainty problem (Kumar & Yadav 2018). On the other side, NLP has not been commonly used due to the high computational time and laborious mathematics involved in its development (Singh 2012), whereas DP has a dimensionality curse (Hossain & El-shafie 2013). In addition, many times these techniques cannot provide an optimal global solution. Heuristic and metaheuristic optimization techniques were created to overcome these problems. Initially, heuristic and metaheuristic methods were created to address issues, such as nonlinearity, discontinuity discrete, multi-objective and uncertainty (Fayaed et al. 2013). Researchers have used these methods to solve optimal cropping pattern models, for example: the genetic algorithm (GA) (Yang et al. 2009; Fallah-Mehdipour et al. 2013), goal programming (Fasakhodi et al. 2010), non-dominated sorting GA (NSGAII) (Lalehzari et al. 2016), fuzzy programming (Srivastava & Singh 2015), particle swarm optimization (PSO) (Noory et al. 2012; Habibi Davijani et al. 2016), fuzzy stochastic GA (Dutta et al. 2016), differential evolution (DE) (Reddy & Kumar 2008) and fuzzy LP (Raju & Kumar 2000; Sahoo et al. 2006; Zeng et al. 2010; Regulwar & Gurav 2011).

There are some limitations to existing optimization algorithms. The main limitation is algorithm-specific parameters other than the common control parameters, i.e. the number of iterations and population size. GA and NSGAII are the most widely used methods of optimization, but these require proper tuning of the probability of mutation and crossover, as well as reproduction. PSO needs the inertia weights, social and cognitive parameters to be tuned. DE requires tuning of crossover rate and scaling factor. The artificial bee colony algorithm needs the number of bees, such as onlooker, scout and employed bees. The biogeography-based optimization algorithm requires the emigration rate and immigration rate. Similarly, ant colony optimization, the shark algorithm and improved bat algorithm involve tuning of their own parameters. These parameters are called the algorithm-specific parameters and need to be tuned. If proper tuning is not achieved, this will affect the algorithm's overall performance and will increase the computational cost and the tendency to trap in a local optimal solution (Rao & Patel 2013). To overcome these problems, the teaching–learning-based optimization (TLBO) and Jaya algorithm (JA) have been newly developed free of the algorithm-specific parameters (Venkata Rao 2016), thus reducing the user's burden. Rao et al. (2011) proposed the TLBO algorithm, which is one of the most newly developed, efficient and simple to understand. It is based on the idea of teaching in the classroom, where the teacher teaches and the student learns. JA, conversely, has been developed very recently and works on the principle that the solution should move toward the best solutions and avoid the worst solution (Venkata Rao 2016).

A multi-population-based JA has been used in the present study which was proposed by Venkata Rao & Saroj (2017). Multi-population helps to divide the entire population into different sub-population groups in order to improve the diversity of search space. The idea is to split the search space into separate fields by allocating distinct sub-population groups. Each population in the sub-population will either intensify or diversify the search space (Venkata Rao & Saroj 2017). The algorithm's performance depends largely on the sub-population number choice. To solve this issue, a self-adaptive multi-population-JA (SAMP-JA) has been used (Venkata Rao & Saroj 2017). The self-adaptive helps to monitor the problem effectively and to change the number of sub-populations based on the landscape changes in the problem, i.e. the increased or decreased number of sub-populations.

The objective of the study is to introduce the SAMP-JA, which adjusts the number of sub-populations based on the landscape changes in the problem. An optimal cropping pattern problem was chosen to explore the efficiency of the algorithm. This research seeks to create an optimum cropping pattern and area to maximize net benefits using 75 per cent dependable inflow for the Karjan irrigation scheme. The SAMP-JA performance is compared to particle swarm optimization (PSO), differential evolution (DE), Invasive weed optimization (IWO) and the firefly algorithm (FA). In addition, to check accuracy, it is compared with TLBO, JA, the elite version of JA and TLBO, i.e. elitist-JA (EJA) and elitist teaching–learning-based optimization (ETLBO) algorithms, and the actual cropping pattern from the literature (Kumar & Yadav 2019). The following section describes the methods and materials used.

JA works on the principle to get closer to a better solution by avoiding failure. The following are the steps to run the SAMP-JA:

• Step 1: The first step is to decide the population size and the iteration size.

• Step 2: Generate the initial solutions between the upper and lower bound of the variables.

• Step 3: The population is grouped into m (i.e. the number of sub-populations) groups based on the quality of the solution (initially considered as ⁠). m is updated at each iteration (bigger or smaller depending on the objective function value).

• Step 4: The next step is to identify the best and worst solution from the population list. If is the jth variable (i.e. ⁠, D is the number of design variables) for the  kth candidate (i.e. population size, n is the number of candidate solution) during the ith iteration.

• Step 5: Each sub-population modifies the solution using the following equation:

  

  (1)

  where and are the best and worst solutions, and is the newly updated solution. The term helps the algorithm to avoid the worst solution and the term helps the algorithm to move toward the best solution. and are the two random numbers between [0,1].

• Step 6: The objective function obtained by ⁠, i.e. O(best-before) and ⁠, i.e. O(best-after) are compared. The modified solution is only accepted if it is better than the old solution, then m is increased by 1, i.e. ⁠, it helps to increase the search space exploration. If m is decreased by 1, i.e. ⁠, it helps to exploit the search space, here, ⁠.

• Step 7: This completes an iteration. The cycle stops when the maximum number of generation is reached; otherwise, it repeats itself. To maintain diversity, duplicate solutions are replaced by the newly obtained solution. The SAMP-JA flowchart is presented in Figure 1.

Storn & Price (1996) proposed DE as an efficient heuristic technique used for solving optimization problems. The algorithm works as follows:

• Step 1: Decide the internal parameters and generate the initial population vector solution, i.e. ⁠.

  

  (2)

  where is the initial population at the kth component for the jth vector at the generation, n is the population size, D is the dimensional parameter vector, and are the lower and upper limit and is the random number.

• Step 2: The second step mutation operator is used to produce the mutation vector, i.e. for each target vector ⁠.

  

  (3)

  where F is the scaling factor parameter that ranges between [0, 2], are the randomly generated integers between [1, ].

• Step 3: The next crossover operation is implemented in which each target vector and its respective mutation vector produce a trail vector ⁠. The basic binomial crossover is defined as follows:

  

  (4)

  where is the randomly generated integers between [1, ] and is the crossover rate parameter varies between [0, 1].

• Step 4: Selection operation helps to select the best member for the next generation based on the vector's fitness value and its trial vector.

  

  (5)

• Step 5: Increase the generation to ⁠. The algorithm reiterates until the maximum iterations are accomplished.

PSO was proposed by Eberhart & Kennedy (1995). It is a method of optimization based on metaheuristic swarm intelligence and works similar to natural groupings like a flock of birds and a fish school. It provides the optimum solution based on individual and social behavior. The algorithm works as follows:

• Step 1: Decide the internal parameters and generate the initial group of random particles; for each particle, generate position and velocity ⁠.

• Step 2: Evaluate the fitness function of each particle and obtain the best position using Equation (6):

  

  (6)

  where F represents the fitness function, represents the individual's best position and I represents the iteration, i.e. ⁠, for the kth individual, i.e. population size ⁠.

• Step 3: Next, find the global best position from the swarm, i.e.

  

  (7)
• Step 4: Update the velocity using Equation (8) if > = ⁠; if < = ⁠:

  

  (8)

  where and are the velocity and position of the kth individual, respectively. The internal parameters are internal weight, cognitive parameters and social parameters.

• Step 5: The next step is to update the velocity using Equation (9).

  

  (9)

  This completes one cycle; the algorithm reiterates until the maximum iterations are completed.

IWO was proposed by Mehrabian & Lucas (2006) inspired by the concept of colonizing weeds. The algorithm works as follows:

• Step 1: Define the N-dimensions solution space for each variable.

• Step 2: Initialize a population, a finite number of seeds are dispersed over the search space with an initial solution.

• Step 3: Next, evaluate the fitness function and rank the population based on fitness values.

• Step 4: Based on the colony ranking, reproduce new seeds from their minimum to maximum possible seeds production, i.e. and ⁠. The number of seeds is reproduced using Equation (10):

  

  (10)

  where ⁠, and are the allowable reproduced, maximum and minimum numbers of seeds, respectively. and are the worst and best fitness function values and is the fitness function of the seed considered.

• Step 5: Disperse the new seed over the search space by varying standard deviations (SDs) using Equation (11):

  

  (11)

  where and are the initial and final SD, is the maximum number of iteration and a is the non-linear modulation index that controls IWO performance and convergence (Mehrabian & Lucas 2006).

• Step 6: Evaluate the fitness function and rank the new and the parent's seeds. The plants with low rank are eliminated from the colony.

• Step 7: The process is repeated for the next iteration from step 3 and runs until the maximum iteration is achieved or the fitness criterion is obtained.

FA is a swarm-based metaheuristic algorithm introduced by Yang (2008), inspired by the fireflies. The insects have the ability to emit light through the biochemical process called bioluminescence. Nonetheless, its purpose is still uncertain. The two fundamental functions are attracting prospective preys and mating partners. The flashing light also protects the fireflies from other predators. Three basic assumptions or rules were idealized in the development of the FA (Yang 2008). First of all, all the fireflies are unisex. Second, the flash's brightness is directly proportional to the fly's attractiveness. Third, the brightness decreases as the distance increases. The algorithm works as follows:

• Step 1: Decide the internal parameters and generate the initial firefly population n within the N-dimensions solution space for each variable.

• Step 2: Evaluate the fitness function of each firefly.

• Step 3: The firefly's attractiveness can be expressed using the following equation:

  

  (12)

  where is the firefly attractiveness, is the attractiveness at distance and is the light absorption coefficient.

• Step 4: The distance between any two fireflies at and can be represented as follows:

  

  (13)

  where and are the dimension of the and firefly's position coordinates, respectively.
• Step 5: The i firefly movement is attracted to another more attractive (brighter) firefly j is expressed as follows:

  

  (14)

  where is a randomization coefficient generated between [0, 1] and is a random number vector.

• Step 6: Evaluate the new fitness function, rank the fireflies and find the current best. This completes one cycle; the algorithm reiterates until the maximum iterations are completed.

The Karjan River originates from the Satpura hill near Mangrol Taluka's Kalvipura village in Gujarat State's Surat district, India. It has an overall length of 96.55 ⁠. The Karjan Dam is situated at a latitude of and a longitude of ⁠. The main purpose of the dam is irrigation, but later micro-hydropower generation was added. The total length and height of the dam are 911.11 and 119.70 ⁠, with a catchment area of 1,403.78 ⁠. The dam is equipped with nine radial gates. Table 1 shows Karjan Dam's salient features. Figure 2 depicts the index map of the Karjan irrigation project with its cultivable command area. The irrigation project has two canal systems, i.e. the left bank main canal (LBMC) and the right bank main canal (RBMC). The RBMC and LBMC systems are allocated in 4 and 12 zones, respectively. The total length of the RBMC is 12.6 and LBMC is 51 ⁠. The irrigation project provides water for an area of 11,412 ha under the RBMC and an area of 39,585 ha under the LBMC.

The following data were collected from the state water data center, Gandhinagar, for the Karjan River; monthly reservoir storage capacity for 18 years, monthly reservoir inflow data for 18 years, monthly evaporation losses from the reservoir's free surface for 18 years and monthly overflow data for 18 years. The various crops produced in the command area are groundnut, wheat, sugarcane, banana, millet grains/corn, pulses, vegetables, cotton and castor. The reservoir's evaporation loss is 32 ⁠. An average inflow to the reservoir is 989 ⁠. The minimum and maximum reservoir storage are 49 and 630 ⁠. Available water for irrigation is 520 ⁠. The actual cropping pattern of the RBMC and LBMC has been collected for 12 years from 1999 to 2011 and the canal carrying capacity. In addition to these net irrigation, water demands and surface water charges are also collected for each crop. The subsequent section describes the mathematical model.

The mathematical formulation of the objective function and the constraints are given in the sequential section.

With respect to the areas, the cultivation region for the biannual and annual crops is the length for both the seasons. The cropping areas under biannual and annual crops are equal in both the seasons of the years. is the cost coefficient of the crop, assuming full irrigation proposed on deducting the fertilizer and labor cost of the total benefits. The net benefit is calculated given that the job is performed by the farmer's family members.

The target function is subjected to a set of constraints. These constraints are given under:

The monthly linear models were prepared from the actual evaporation and average storage. Table 2 (Kumar & Yadav 2019) shows the observed regression coefficients with the statistical parameters r for each month, where is the reservoir storage at the beginning of period t in ⁠, is the evaporation losses from the reservoir in ⁠, is the reservoir storage at the end of period in ⁠, and and are the regression coefficients during time period t.

The dependable annual inflows were calculated from the annual probability distribution. Nine different rates of inflow were computed using the Weibull formula based on available historical monthly reservoir inflow data, namely 50%, 55%, 60%, 65%, 75%, 80%, 85% and 90%. Figure 3 depicts the annual inflow for various rates of inflow (Kumar & Yadav 2019). It was observed that 95% of rainfall occurs from June to October. Therefore, in the operation of the reservoir, the probability of June to October is more crucial than the annual probability. Of these, 75% dependable inflow was used for the study (Vasan & Raju 2009; Jothiprakash & Arunkumar 2013). Based on the actual condition of the Karjan irrigation scheme, two distinct models were created, namely (1) maximum cropping pattern, i.e. model-1 and (2) average cropping pattern, i.e. model-2. Model-1 has been prepared at 75% dependable inflow with the maximum crop area sown over the past 12 years. Model-2 has been prepared at 75% dependable inflow with an average crop area sown over the past 12 years. Here, the aim behind the maximum and average sown crop area is to check the maximum and average benefits obtained. These two model scenarios are formulated in such a manner that the farmer can choose either model depending on the cultivation area and its corresponding net benefits. To optimize both models, a SAMP-JA was used. SAMP-JA's performance has been compared to PSO, DE, IWO and FA. In addition, JA, TLBO, EJA and ETLBO algorithms and the actual cropping pattern from the literature (Kumar & Yadav 2019) have been compared to verify the accuracy. All algorithms were coded with MATLAB R2014b software, and Origin-Pro 8.5 software was utilized for plots and statistical parameters. Via a 64-bit system type, Intel^® core^{TM i}7-7700, @3.60 GHz system configuration with an 8.00 GB ram.

Various combinations of common controlling parameters, such as the number of iterations (i.e. 10, 20, 35, 50, 80 and 100) and population size (i.e. 10, 20, 30, 50, 75 and 100), were used to check algorithms’ performance over 30 independent runs. It was observed that for a population size of 100, all algorithms performed better. The internal parameters in PSO, i.e. the cognitive parameter and social parameter ⁠, were taken as 1.3 and 2.0, respectively and the inertia weight as 1. The internal parameters in DE, i.e. the lower and upper scaling factors, were taken as 0.2 and 0.8, respectively, and the crossover probability was taken as 0.2. IWO's internal parameters were the minimum and maximum number of seeds, i.e. and were taken as 0 and 3, the non-linear modulation index was taken as 2, and initial SD and final SD were taken as 0.6 and 0.001. The internal parameters in FA were the base coefficient of attraction taken as 2, the light absorption coefficient was taken as 1 and the randomized parameter was taken as 0.2. The following section describes the results and the discussion.

Two models were prepared using the objective function and its constraints as stated earlier. The algorithms were named as SAMP-JA1, PSO1, DE1, IWO1 and FA1 for model-1. Similarly, the algorithms were named as SAMP-JA2, PSO2, DE2, IWO2 and FA2 for model-2. The comparison was done between (SAMP-JA1, PSO1, DE1, IWO1 and FA1) for model-1 and (SAMP-JA2, PSO2, DE2, IWO2 and FA2) for model-2. Table 3 shows model-1 comparative results for the various number of iterations for different algorithms over 30 independent runs and past literature studies (Kumar & Yadav 2019). It can be noted that for all the comparative iterations, SAMP-JA1 performed better and was able to attend very fast convergence. Similarly, Table 4 demonstrates model-2 comparison outcomes for the various number of iterations for different algorithms over 30 independent runs and previous literature studies (Kumar & Yadav 2019). Similar findings were also noted here, where SAMP-JA2 performed better for all the comparative iterations and was able to undergo very rapid convergence. The net benefits were in Indian rupees (1 US$ = 63.96 Rs).

The convergence rates of the various algorithms for model-1 and 2 for 100 iterations are shown in Figures 4 and 5. It was noted that for both models, SAMP-JA converges quickly to an optimum solution. PSO and DE converge slower than SAMP-JA and FA, and IWOs performed poorly. Table 5 shows the comparison of net benefits derived from different algorithms and real income for both the models. In model-1, SAMP-JA1 was 15.93%, 0.59%, 0.05%, 0.06%, 0.05%, 0.04% and 0.003% better than IWO1, TLBO1, FA1, PSO1, DE1, ETLBO1 and JA1. The results of SAMP-JA1 and EJA1 were nearly the same, but as shown in Figure 6, SAMP-JA1 had a faster convergence rate. In model-2, SAMP-JA2 was 14.73%, 0.20%, 0.18%, 0.15%, 0.08%, 0.08%, 0.06% and 0.05% better than IWO2, TLBO2, FA2, ETLBO2, DE2, PSO2, JA2 and EJA2. Since the unit was in Rupees, even a small change in outcomes has resulted in a large change in net benefits. The average time to run the models were from 1 to 2 minutes for all algorithms.

Compared to the real income produced during 2010 and 2011, SAMP-JA1 was found to be 67.38% and 65.72% better, and SAMP-JA2 was 11.04% and 6.52% better than the real income generated in 2010 and 2011, respectively. Figures 7 and 8 demonstrate the convergence plot for model-1 and model-2 for all algorithms running up to 5,000 iterations. For model-1, it was observed that at 189 iterations SAMP-JA1 obtained the global optimum solution as 200.4017, and at 303 iterations PSO1 obtained the global optimum solution as 200.4017. At 5,000 iterations, DE1, FA1 and IWO1 obtained an optimal solutions as 200.4014, 200.307 and 197.0938, respectively. For model-2, it was observed that at 412 iterations SAMP-JA2 obtained the global optimum solution as 73.4872, and at 3,381 iterations PSO2 obtained the global optimum solution as 73.4872. At 5,000 iterations, DE2, FA2 and IWO2 obtained the optimum solution as 73.4869, 73.3535 and 72.9611, respectively. Model-1 can provide maximum net benefits, but more irrigation releases are needed compared to model-2. During less inflow, the dam officials are restrained to releases from the dam, and time model-2 was more suitable.

The maximum cultivated areas proposed by SAMP-JA1, DE1, PSO1, FA1 and IWO1 for model-1 were 20,598, 20,579, 20,527, 20,527 and 14,685 ha, respectively. While the maximum cultivated areas proposed by SAMP-JA2, DE2, PSO2, FA2 and IWO2 for model-2 were 8,754, 8,722, 8,722, 8,712 and 7,546 ha, respectively. The optimized crop area obtained for model-1 using SAMP-JA1, DE1, PSO1, FA1 and IWO1 for both canals is shown in Table 6. The crop area acquired for groundnut by SAMP-JA1 in the RBMC was 1.12%, 5.54%, and 1.12% higher than PSO1, IWO1 and FA1, respectively, and 78.17% higher than IWO1 in the LBMC. The optimal cropping area for other crops are bolded values in Table 6. SAMP-JA1 clearly performed better than other algorithms.

Table 7 shows the optimized cropping area for model-2 for both canals using SAMP-JA2, DE2, PSO2, FA2 and IWO2. The cultivation areas obtained for groundnut by SAMP-JA2 in the RBMC were 1.64% and 3.77% more than PSO2 and IWO2, and in the LBMC 14.07% and 62.09% more than PSO2 and IWO2, respectively. The optimal cropping area for other crops are bolded values in Table 7. The cropping area has directly affected the net benefits for both the models resulting in higher net revenues for SAMP-JA compared to PSO, DE, FA and IWO.

The cropping pattern acquired from SAMP-JA1, DE1, PSO1, FA1, IWO1, SAMP-JA2, DE2, PSO2, FA2 and IWO2 was compared with the actual cropping patterns for both RBMC and LBMC in 2010 and 2011. Tables 6 and 7 also show the comparison of optimal cropping patterns for model-1 and 2 with the actual cropping pattern for 2010 and 2011. It was found in the RBMC that SAMP-JA1 and DE1 had increased groundnut, sugarcane and vegetable areas by 4.21%, 35.13% and 88.20%, respectively. Cotton and castor areas were increased by 78.41% and 71.61% using SAMP-JA1, PSO1, DE1 and FA1. Millet grains and wheat areas were increased by 100% and 54.94% using SAMP-JA1. The percentage increase of SAMP-JA1 with the actual cropping pattern of 2010 and 2011 in the RBMC is presented in Table 8(a).

In the LBMC, wheat, banana, cotton and castor areas were found to have increased by 55.36%, 75.07%, 89.85% and 37.19% using SAMP-JA1, PSO1, DE1 and FA1. Areas of cultivation of sugarcane and vegetables had increased by 30.05% and 73.56% using SAMP-JA1 and DE1. Table 8(b) presents the percentage increase of SAMP-JA1 with the actual cropping pattern in LBMC. Since the crop type in both the canals was the same, the results were the same except for the cultivated area. The main difference between SAMP-JA1, PSO1, DE1 and FA1 from IWO1 was that the cropping areas for SAMP-JA1, PSO1, DE1 and FA1 were more for groundnut, wheat, sugarcane, banana, millet grains and castor. As a result, the net benefits obtained by SAMP-JA1, PSO1, DE1 and FA1 were more compared with IWO1.

Most heuristic and evolutionary algorithms require good tuning of algorithm-specific parameters other than common control parameters. A slight change in the tuning parameters affects the overall performance of the algorithms. This paper suggested a multi-population-based JA that does not require algorithm-specific parameters. The multi-population enables us to divide the population into sub-population groups and helps to improve the search space diversity. The algorithm's performance is highly dependent on the selected numbers of the sub-population. To solve this problem, a self-adaptive model is used, i.e. SAMP-JA. In this research, SAMP-JA is used to optimize the optimum cropping pattern and area for the Karjan reservoir project in order to maximize the net benefit. To verify the precision, SAMP-JA's performance was compared with PSO, DE, IWO, FA, JA, TLBO, EJA and ETLBO algorithms. Following are the conclusions drawn from this study:

• (i)

  In model-1, SAMP-JA1 had better net benefits than IWO1, LP1, TLBO1, FA1, PSO1, DE1, ETLBO1 and JA1. The results by SAMP-JA1 and EJA1 were nearly the same, but there was a faster convergence rate for SAMP-JA1. In model-2, SAMP-JA2 had better net benefits than IWO2, LP2, TLBO2, FA2, ETLBO2, DE2, PSO2, JA2 and EJA2. Since the unit was in ⁠, even a small change in results led to a large change in net benefits. In addition, due to the internal parameters, the time it took to tune IWO, FA, PSO and DE was very high to get the optimum solution.

• (ii)

  While comparing the optimized cropping area, it was clear that SAMP-JA1 performed better than other model-1 algorithms, and that SAMP-JA2 performed better for model-2.

• (iii)

  The cropping area has a direct effect on net benefits. Thus, it leads to a higher net benefit for SAMP-JA1, PSO1, DE1 and FA1 as compared to IWO1.

• (iv)

  The optimal cropping pattern has resulted in a substantial rise in groundnut, wheat, sugarcane, banana, millet grains and castor crops production by SAMP-JA1, PSO1, DE1 and FA1, which in turn generated more net benefits compared to IWO1, SAMP-JA2, DE2, PSO2, FA2, IWO2 and actual condition.

Based on the results of both the models, it is concluded that SAMP-JA converses faster and performs better than PSO, DE, IWO, FA, JA, TLBO, LP, EJA and ETLBO. In addition, to operate the models as a multi-objective reservoir, future studies are needed to consider the present and potential scenarios of power generation capability.

The authors acknowledge the helpful comments and suggestions made by anonymous reviewers to enhance the manuscript. They would like to thank the authorities of the Sardar Vallabhbhai National Institute of Technology (SVNIT) and the PG section of water resources engineering for their valuable support.