A novel methodology for optimal land allocation for agricultural crops using Social Spider Algorithm

Study area

The study area considered for the current research is Coimbatore district, which is located in the western agro climatic zone. It lies between 11°01′06.00″N latitude and 76°58′2900″E longitude [TN18-COIMBATORE 31.03.2011]. The district is divided into 9 taluks, 19 blocks and 481 villages. The main rivers flowing through Coimbatore district are Bhavani, Noyyal, Amaravathi, Aliyar, Nirar, Solaar and Uppar Thirumurthi. The total area of Coimbatore district is 7,470 square kilometres, among which 3,319 square kilometres are used as cultivable areas. The major crops are millets, jowar, paddy, cowpea, bengal gram, horse gram, green gram, coconut, groundnut, sugarcane, and cotton and the main soil types are red calcareous soil, red non-calcareous soil, black soil, alluvial and colluvial soil.

Objective 1: Profit maximization

The objective function to maximize the total profit is given in Eq. (1). (1)${\displaystyle \text{Max TP}=\sum _{i=1}^n{A}_i\left[I_i-Ex{p}_i\right]}$ where TP is the total profit that could be obtained from optimal allocation of land for crops, n is the total number of crops, i is the crop which varies from 1 to n, A_i is the optimal area allocated to ith crop (square metres), I_i is the income from ith crop, Exp_i stands for expenses incurred for crop i.

I_i is calculated by using the price and production of the selected crop i, price_i and productivity_i, as given in Eq. (2). The production value of the crop, pro_i is calculated as given in Eq. (3), where the productivity values of the crops, productivity_i are collected from the website [www.coimbatore.nic.in/pdf/SHB003.pdf ]. (2)${\displaystyle I_i={\text{Price}}_i\times {\text{Productivity}}_i}$ Exp_i includes expense on seeds, expense on manure, expense on fertilize, expense on irrigation and expense on labour as given in Eq. (3). (3)${\displaystyle Ex{p}_i=C{S}_i+C{M}_i+C{F}_i+C{I}_i+CH{L}_i}$ where CS_i is the expense on seeds for ith crop, CM_i is the expense on manure for ith crop, CF_i is the expense on fertilizer for ith crop, CI_i is the expense on irrigation water for ith crop and CHL_i is the expense on labour and machinery for ith crop.

To achieve maximum profit, improved production from the crops and optimal allocation of land for the crops are highly essential. Crop production is obtained from Eq. (4). (4)${\displaystyle {\text{Max Production}}_i=A_i\times {\text{productivity}}_i.}$

Objective 2: Water requirement minimization

The secondary objective of this research is to minimize the usage of water to be used for irrigating the allocated crops due to the insufficient availability of water in the study area. (5)${\displaystyle \text{Minimize TWR}=\sum _{i=1}^n\left(CW{R}_i\times A_i\right)}$ where TWR is the total water requirement (cubic metres) and CWR_i is the crop water requirements for crop i. The constraint used is the total cropping area, which should be in the range of 2 to 4 hectares as given in Eq. (6).

Constraint 1: Total cropping area

(6)${\displaystyle max.area\ge \left(TCA=\sum _{i=1}^n{A}_i\right)\ge min.area}$ where TCA is the total cropping area, max. area is the maximum available cropping area and min. area is minimum available cropping area. Two categories of land holdings taken in this research work are 20,000 m²–40,000 m², which is called small-medium land holding category and 40,000 m²–1,00,000 m², which is called medium land holding category.

Constraint 2: Cropping area for each crop

Cropping area for each crop is constrained as given in Eqs. (7) and (8). Eq. (7) is used for small-medium land holding category and Eq. (8) is used for medium land holding category. (7)${\displaystyle A_i\ge {\text{2,000 m}}^2}$ (8)${\displaystyle A_i\ge \text{4,000}{\mathrm{m}}^2.}$

Social Spider Algorithm (SSA) for crop planning optimization

SSA is a population based algorithm proposed by Cuevas et al. (2013) to solve global optimization problems. It is based on the reproduction and cooperative behaviour of social spiders. SSA possesses certain unique characteristics, which makes it quite different and competent than many other existing bio-inspired algorithms. The algorithm employs two types of spiders as search agents, male spiders and female spiders. Each individual performs different types of operations based on their type that simulate the social behavior of the spiders within the colony. Three types of vibration operators used in SSA allow improved particle distribution in the search space, thereby enhancing the algorithm’s ability to find the global optima. Based on the vibration values, the male cooperative behavior and female cooperative behavior are determined in SSA, which in turn applies different mechanisms for exploration and exploitation during the evolution process. Another unique quality of this algorithm is the choice of new spiders for the next generation. While most of the algorithms use only the fitness of the offspring to calculate their chance to be moved to the next generation, SSA calculates and applies an influence probability in addition to the fitness, in order to assess the suitability of the individual to become a member of the new generation.

The major components of social spider colony are social members (spiders) and communal web (spider web). The communal web is considered as search Space (S). The position assigned to each spider on the web is based on its weight and fitness value (Cuevas et al., 2013). The total colony members are divided into two categories: 70% of female spiders and 30% of male spiders. The male spiders are subdivided into two classes; dominant and non-dominant males based on the fitness of the spiders. The mating operation allows the information exchange among colony members, performed by dominant male and females. The spiders generate vibrations for mating operation based on the other spider’s weight and distance between each other. The dominant male spiders mate with one or more female spiders to produce offspring (new spider). The weight of the spider is considered as the fitness of the solution. The population contains the female (f_i) and male (m_i) spiders. The number of female spiders is randomly selected within the range of 65%–90% and the number of male spiders is calculated from female spiders based on Eqs. (9) and (10). (9)${\displaystyle N_{fs}=\text{floor}\left[\left(0.9-\text{rand}\left(0,1\right)\ast 0.25\right)\ast P\right]}$ (10)${\displaystyle N_{ms}=P-N_{fs}}$ where N_fs means number of female spiders, N_ms represents the number of male spiders, P stands for entire population and floor represents the real number to an integer number. The weight (w_i) of each spider is calculated based on (11). The fitness value obtained by the spider at position s_i and the values b_s and w_s are calculated based on Eqs. (12) and (13) for maximization problems. (11)${\displaystyle W_i=\frac{J\left(s_i\right)-w_s}{b_s-w_s}.}$

The b_s is maximum fitness value and w_s is minimum fitness value obtained from the fitness values of all the spiders. (12)${\displaystyle b_s=\underset{k\in 1,2,\dots ,N}{max}J\left(s_k\right)}$ (13)${\displaystyle w_s=\underset{k\in 1,2,\dots ,N}{min}J\left(s_k\right).}$

Similarly, vibration of each spider is calculated based on the weight (w_i) and distance (d_i,j) between the individual i and member j as given in Eq. (14). (14)${\displaystyle V{b}_{i,j}=w_j.e^{-d_{i,j}^2}}$ where d_i,j is the Euclidian distance between the spiders i and j, which is calculated by using Eq. (15). (15)${\displaystyle d_{i,j}=\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}.}$

Vibrations are categorized into three types based on the relationship between the pair of individuals. These three types of vibrations are calculated to perform the cooperative behaviours of male and female spiders.

• Vibrations Vc_i—the information transmitted from individual i to member c(s_c), which is the nearest member to individual i based on Eq. (16). (16)${\displaystyle V{c}_i=w_c.e^{-d_{i,c}^2}}$

• Vibrations Vb_i—the information is transmitted between the individual i and the best member b(s_b) by using Eq. (17). (17)${\displaystyle V{b}_i=w_b.e^{-d_{i,b}^2}}$

• Vibrations Vf_i—the information is transmitted between the individual i and the nearest female s(s_f) as given in Eq. (18). (18)${\displaystyle V{f}_i=w_f.e^{-d_{i,f}^2}}$

The vibrations Vb_i and Vc_i are used to calculate the female cooperative behaviours while Vf_i is used to calculate the male cooperative behaviours. Strong vibrations are produced by the nearest spiders or big spiders. (19)${\displaystyle f_i^{k+1}=\left\{\begin{array}{c}\left\{f_i^k+\alpha .v{c}_i.\left(s_c-f_i^k\right)+\beta .\right.v{b}_i.\left(s_b-f_i^k\right)+\delta .\left(rand-\frac{1}{2}\right)\hfill \\ \text{with probability PF}\hfill \\ \left\{f_i^k-\alpha .v{c}_i.\left(s_c-f_i^k\right)-\beta .\right.v{b}_i.\left(s_b-f_i^k\right)+\delta .\left(rand-\frac{1}{2}\right)\hfill \\ \text{with probability}1-PF\hfill \end{array}\right.}$ (20)${\displaystyle m_i^{k+1}=\left\{\begin{array}{c}{m}_i^k+\alpha .v{f}_i.\left(s_f-m_i^k\right)+\delta .\left(rand-\frac{1}{2}\right)if{w}_{N_f+i}>w_{N_f+m}\hfill \\ m_i^k+\alpha .\left(\frac{\sum _{h=1}^{N_m}{m}_h^k.w_{N_f+h}}{\sum _{h=1}^{N_m}{w}_{N_f+h}}-m_i^k\right)if{w}_{N_f+i}\le w_{N_f+m}\hfill \end{array}\right.}$ where, α, β, δ represents random numbers between [0, 1], k stands for iteration number and s_c & s_b means the nearest member to i that holds the highest weight and the best individual of the entire population.

Equation (19) is used to calculate the female cooperative behaviours while Eq. (20) is utilized to calculate the male cooperative behaviours based on the threshold value. The spider movement, i.e., attraction or repulsion, is based on the random phenomena, which is represented as r_m, and it is generated within the range of [0, 1]. The r_m value is compared with the threshold value, and if the value is less than r_m, then the attraction movement will be produced, else the repulsion movement will be produced. (21)${\displaystyle r=\frac{\sum _{j=1}^n\left(P_j^{high}-P_j^{low}\right)}{2.n}}$ where r is a range to select the female spiders for mating operation, which is calculated by using Eq. (21). The dominant male spiders are arranged in set m_g andthe selected female spiders are arranged in set E^g. The sets m_g and E^g (m_gU E^g) are combined to get the reproduction. If the set E^g is empty, then there will not be any mating operation. (22)${\displaystyle P{s}_i=\frac{w_i}{\sum _{j\in T^k}{w}_j}.}$

The influence probability Psi is calculated by using the weight of each spider as given in Eq. (22). After calculating the probability, the roulette wheel method is applied to select the s_new. The weight of the new spider s_new is calculated and compared with the weight of population member. If the weight of s_new isbetter than the weight of any member in the population, then the member will be replaced by s_new. Figure 1 depicts the search space of spiders denoted by S.

The present research applies SSA for crop planning optimization. The number of crops is considered as the population size i.e., the number of spiders. Ten agricultural crops are selected for implementation in this work. Total population is split into males and females by using Eqs. (9) and (10). The value of $p_j^{\mathrm{low}}$ and $p_j^{\mathrm{high}}$ are equated for the minimum area and maximum area of cropping land considered. The social spiders are responsible for allocating optimal land for each crop and based on the land allocation, the total profit from agriculture and TWR (objective functions) are calculated. Data for the cost of seed, cost of irrigation, cost of manure, cost of fertilizer, cost of labour and machinery are obtained from the Centre for Agriculture and Rural Development Studies (CARDS), Tamilnadu Agricultural University, Coimbatore.

The weight and fitness of each spider are calculated from the objective function given in Eq. (11). The best and worst values of objective function are used for the weight calculation of each spider. Distance between each spider is calculated by utilizing Eq. (15). The values taken for the distance calculation are the weight and the total land allocation done by each spider. Similarly, vibration is calculated based on the distance between spiders. Three types of spiders chosen for the calculation of vibration are the spider with minimum distance (Vbc_i), spider with highest weight (Vbb_i) and nearest female spider (Vbf_i). Based on the vibrations, female cooperative operators and male cooperative operators are applied. The median male spider is found based on the weight to calculate the dominant and non-dominant male spiders. The spiders, whose weights are above the median male spider, are called the dominant males and those below the median are non-dominant males.

Cooperative operations are used to select the best female spiders and dominant male spiders to mating operation. The range value of dominant male spider is calculated to choose female spiders among female co-operative spiders to mating operation. Mating operation forms a new spider (brood) denoted as s_new. Each member of the new spider population suggests an optimal land allocation for multiple crops. Based on the suggestions, the objective function values are derived and the corresponding fitness and weight of the population members are calculated. The weight of the new members is compared with the weight of all the members in the population. If the new member is better than the worst member in the population, then the worst member will be replaced by the new member. This process will be continued till the optimum solution is obtained which suggests improvement in total profit and reduction in total water requirement based on the optimal land allocation done by the social spiders. Figure 2 shows the flow chart of SSA for crop planning optimization.

Conclusion and scope for further research

In the present research, an attempt is made to find the optimal land allocation for planting multiple crops in the available land area with water constraints. The optimal land allocation is performed with two objectives, profit maximization and water minimization. Social Spider Algorithm, a new and robust bio-inspired algorithm is applied for optimization and the crops cultivated in the Coimbatore region of Tamilnadu state in India are taken for the case study. Eight test cases are framed based on land availability and crop category. As the land allocation problem is considered as multi-objective problem, the results are analysed for Pareto optimal solutions which dominate both in terms of optimal profit and optimal water requirement for cropping. Rule curves are generated to depict the profit and water requirement with respect to the optimal land allocation for all the test cases analysed in this work. Promising results are obtained which proves the ability of SSA in the optimization of crop planning. Almost 98% of the available land is allocated by the algorithm for cropping, which also shows the effectiveness of the algorithm in minimizing wastage of crop land. Results of this research can be used for cropping recommendations to the farmers with varying land holdings and inadequate water availability. Further optimization techniques in crop planning such as optimization of soil, labour, transportation, climate, weed, fertilizer and pesticides are planned to be considered for future research.