Mathematical programming approaches for modeling a sustainable cropping pattern under uncertainty: a case study in Southern Iran

In recent years, the excessive and unreasonable use of chemicals, the occasional use of water, and the use of improper irrigation methods have created a worrying and unstable situation in developing countries’ agricultural activities. In the present study, the robust multi-objective fractional linear programming model (RMOLFP) was introduced to determine the sustainable optimal cropping pattern. This model was presented in the Gotvand irrigation and drainage network located in Khuzestan province, southern Iran, under two scenarios with and without considering the uncertainty to evaluate the ability of the model. The results showed that in the first scenario, the consumption of critical disruptive inputs of sustainable agriculture such as fertilizers and chemical pesticides decreased by 5.9% and 8.19%, respectively. On the other hand, the model’s uncertainty condition was applied in the second scenario in which the increase in gross margin was reduced. There is a trade-off between protecting the optimization model against system uncertainty and gross margin. Finally, the ability of the proposed model to apply uncertainty conditions was verified by the Monte Carlo simulation method. The results of this simulation confirmed the use of the RMOLFP method in determining the sustainable optimal cropping pattern for the study area.


Introduction
Over the last five decades, agricultural development policies have increased agricultural production; however, the use of disruptive inputs such as pesticides, fertilizers, and agricultural machinery has gradually replaced natural resources available inside farms (Li et al., 2020). Sustainable agriculture has been emphasized by numerous international organizations and has led to positive environmental, economic, and social impacts (Tseng et al., 2013). From an economic point of view, the emphasis on sustainable production has led to increased productivity of agricultural inputs as well as improved market opportunities. (Li et al., 2020).
One of the approaches to achieve the optimum use of off-farm inputs and disrupt the sustainable agriculture process is to determine the optimum utilization of inputs by establishing a sustainable optimal cropping pattern . Figure 1 shows the main purposes of studies that focus on the sustainable production of agricultural products and implicitly consider other objectives of crop pattern studies. The explanation for this claim lies in the definition of sustainability provided in Guideline 21 at the United Nations Conference of Environment and Development (UNCED) in 1992. In this definition, sustainability was presented in ecological, economic, and social dimensions .
Ecological sustainability means maintaining the ability of the system to fully adapt to the environment and its changes. Therefore, studies focusing primarily on environmental considerations emphasize this aspect of sustainability (Mardani Najafabadi et al., 2019). Sustainability in the economic dimension is a crucial concept in sustainable development literature. One of the most important objectives to be achieved in this dimension is to increase farmers' gross profit. In most developing countries, the agricultural sector tends to increase agricultural output while reducing unemployment (Mardani et al., 2018). This is due to ensuring the country's food security and the reduction of the adverse social effects of unemployment (Mardani Najafabadi et al., 2019). Thus, the two mentioned goals (food security and employment) can be considered in the social dimension of sustainability.
Considerable studies have been carried out to optimize the allocation of arable land using mathematical programming techniques worldwide. Most of the studies have used Fig. 1 Three dimensions of sustainable optimal cropping pattern linear programming models (Wineman & Crawford, 2017), goal programming , and multi-objective programming (Ren et al., 2019).
On many practical issues such as quantitative sustainability indicators, optimizing the ratio of criteria provides a better vision than each criterion alone . Fractional programming is the most common mathematical programming with relative objectives. On the other hand, the existence of different criteria in relational objectives has also led to the development of multi-objective linear fractional programming models. This approach has been used in various studies in agricultural management. For example, in the study of Zhang et al. (2017), the interval nonlinear multi-objective programming model with fuzzy-interval credibility constraint (FIC-INMP) method has been used to optimally allocate irrigation water in the Heihe basin in northwest China. CONNISE algorithm has been used in their study to find optimal solutions that integrate the constraint (CON) and non-inferior set estimation (NISE) methods. This method is defined based on the estimation of weak efficient boundaries in the objectives space, including some points of feasible set (Gorissen, 2015). The distance between these points should not exceed the value previously provided. This method is beneficial for finding efficient solutions, but in problems with more than two objectives, the optimal solutions lose their convergence, and the final solution is challenging to find. This problem can be solved by controlling the distance considered for a predetermined error using the controlled estimation method (CEM) proposed by Caballero and Hernández (2004).
The agricultural uncertainties are traditionally classified in areas such as production management, marketing, and investment . The application of fractional programming under uncertainty is widespread in agricultural management. Meanwhile, the use of fuzzy, interval, gray, stochastic, or a combination of these methods is more common (Tan & Zhang, 2018;Wang et al., 2019).
In the mid-1990 s, the issue of creating conservatism in mathematical programming models was introduced by limiting uncertain parameters (Ben-Tal & Nemirovski, 2000). The benefits of using robust optimization are reporting point-based optimal solution (confronting the optimal interval solutions in interval programming) (Sabouhi & Mardani, 2017) and lack of awareness of the data distribution (the challenge of coping with the need to be aware of the data distribution in the stochastic programming) (Mardani Najafabadi & Taki, 2020).
Recently, a wide range of robust optimization applications in the management of water resources and agricultural land have been implemented. For the first time, this method was used in fractional programming by Gorissen (2015). Tan & Zhang (2018) optimized the allocation of water resources and agricultural land using robust fractional programming to increase water-use efficiency in the arid northwest region of China. It should be noted that in both studies, single-objective fractional programming is considered, and the problem of drastically losing the convergence of the optimal solutions in multiobjective fractional programming in their methods remains. Using controlled estimation method (CEM) can be an appropriate approach to address this defect and enable the application of robust multi-objective fractional programming to provide a sustainable optimal cropping pattern.
Therefore, the purpose of the present study was to propose a mathematical programming model to determine the sustainable optimal cropping pattern for irrigation and drainage network covered land. For this purpose, a robust multi-objective fractional linear programming model (RMOLFP) was proposed. Thus, the innovation and importance of this study can be examined in two respects; first, a new mathematical programming model for optimal cropping pattern with emphasis on sustainability has been proposed, and second, a detailed plan has been prepared for an area that desperately needs a revision of the cropping pattern.

Study area
Gotvand irrigation and drainage network is located in southwest of Iran in Khuzestan province with geographical coordinates of 48° 81′ E longitude and 32° 24′ N latitude. According to official statistics from government agencies, fertilizer and pesticides used in the lands covered by this network are 3.6 times the average in Iran (Anonymous, 2018). The surplus irrigation water in the network that is returned to the rivers by drainage, contaminates water with a surplus of fertilizers and pesticides downstream of the network . Due to the mentioned issue, selecting this region to determine the cropping pattern that leads to optimal use of disturbing inputs to sustainable agriculture, seems appropriate. Besides, the irrigation water consumption is better managed in this network.
Gotvand irrigation and drainage network is designed for irrigation of land in three areas of Gotvand, Aghili, and Dimcheh, confined between two rivers, Karoun & Lor (Fig. 2). The purpose of this network was to manage water resources better and distribute the water equitably among the farmers in the area. The land covered by this network is 43,930 hectares, of which 34,144 hectares with over 4300 plots are used. This irrigation network directs irrigation water to 1076 plots of land in the Gotvand region, using 56 km of canals and 121 active valves with an average discharge of 68 m 3 /s. Aghili area is smaller so that by using 44 km of canals and 165 active valves, 11.23 m 3 /s of irrigation water can irrigate 1118 plots of land. Both Gotvand and Aghili regions receive their required irrigation water from the Gotvand Dam constructed on the Karoun River. The most considerable amount of infrastructure is in the Dimecheh area. The length of Fig. 2 Schematic of the Gotvand irrigation and drainage network in Khuzestan Province, Iran the canals in this area is 179 km and uses 241 valves to deliver 43 cubic meters of water to 2006 arable lands. In addition to the Gotvand Dam, the Dimcheh area feeds directly on several alternative dams from the Lor River, a tributary of the Dez River (Anonymous, 2017).

Multi-objective fractional linear programming
Fractional linear programming is a type of fractional programming, and the numerator and denominator of the objective function is affine Function (AF), and the possible set is a convex polyhedron. A multi-objective linear fractional programming is presented as Eq. (1): where c and d are the coefficients of the objective function, α and β are the constant components, A is the technical coefficient matrix, x is the decision variable, and b is rightsided values. The CONNISE method, which combines the constraint method (CON) and the Non-inferior Set method (NISE), can be used to find optimal solutions to multi-objective optimization problems. This method is also used for multi-objective fractional linear programming.
Assuming that in multi-objective programming, E is efficient set while E w is weak efficient set. The CONNISE method is defined based on the estimation of weak efficient boundaries including points of the set of possible zones in the objective space ( (E w ) ). The distance between possible points should not exceed a certain value.
Although the CONNISE method was very useful for finding efficient solution in problems with more than two aims, the solutions lose their convergence so finding the final solution is very difficult. This problem can be solved using the Controlled estimation method (CEM). In this method, the intended interval for the predetermined error is controlled. The CEM algorithm which is consist of 7 steps, proposed by Caballero & Hernandez (2004).
In the first stage, the ideal and non-ideal points (the pay-off matrix) are obtained. The boundaries of objective space are formed between ideal ( D * i ) and non-ideal ( D i− ) values. In the second stage, = max in set A . This stage is repeated for all x j . If A is null, this process is over, otherwise, the sixth step is started and m that is obtained from third stage, considered as m + 1. If A = x 1 , … , x m } , for each x j belongs to A , x k point, which is closer to x j considered instead (in stage four). This stage is repeated for all points in set A . Ultimately, in stage seventh, set A = ∅ is obtained and it goes to step three.

Robust optimization
In the present study, robust optimization applies the uncertain conditions to the proposed parameters. The linear form of the robust optimization model can be written as Eq.
(2) (Ben-Tal & Nemirovski, 2000): where J i is a subset of the parameters associated with the uncertain parameter ã ij which is specified for each constraint i. Assume that ã ij are independent, symmetric and their boundaries are in the range of [− 1, 1].
Equation (2) can be rewritten as robust optimization model that improves the reliability of systems under uncertainty (Bertsimas et al., 2010): where where for each j, ith constraint in terms of certainty. This model is based on a nonlinear form of robust optimization model. The reason for the nonlinearity of this model is additional maximization expression. This maximization statement by the controlling parameter of the amount of conservatism Γ i guarantees the model's reliability against uncertainty. Calculating, the linear form of Model 5 is used to avoid the complexity of calculations in the maximization expression as follows (Bertsimas et al., 2010): And finally, the form of robust optimization is model (5): (2)

Maximum cx
Subject to (3)

Maximum cx
where z, f, and p are non-negative additional variables and ε is a given uncertainty level in the model. In model 7, which is a linear form of optimization, n + k + 1variables and m + k + n constraints are existed. The degree of confidence in the model against uncertainty depends on the parameters Γ i . If Γ i = 0 , the maximization statement is eliminated and the constraint condition is changed from uncertainty to certainty. If Γ i = | | j i | | the degree of system protection against uncertainty is maximized and fully implemented. There are different values for Γ i and it depends on the probability level of the deviation of ith constraint ( p i ) and the number of uncertain parameters (n). Equation (7) shows this dependence (Bertsimas et al., 2010).
where Φ is cumulative distribution of the standard Gaussian variable and n is uncertainty resources in each constraint.

Objective function
In order to optimize the cropping pattern and align this model with the relative objectives, the objective functions in the present study are defined as Eq. (8): The objective function 1 to 3 are related to the sustainable use of nitrogen (N), phosphate (P) and potash (K) fertilizers. In these functions, A csr represents the cultivation area crop c in season s for region r, and Fer tcs represents the required amount of fertilizer by type t (N, P or K) to cultivate each hectare of crop c in season s for region r.
Functions 4 to 6 are related to sustainability of herbicides, fungicides, and insecticides.
In these equations, Pes zcs represents the required amount of pesticide by the type z per hectare of product c in season s for the region r.
Water resource productivity is another ecological objective of the optimal cropping pattern model that has been addressed in the objective function 7 .
NetW csr is the net water requirement for crop c in season s and region r (m 3 /ha), and Eff r is irrigation efficiency in the region r.
The ecological dimension of sustainability of crop production in the proposed model is provided by objective functions 1 to 7 .
The objective function 8 ensures the economic dimension of the sustainability of agricultural production (Eq. 11).
In this equation, Mp cr , Sp cr and Cp cr are the main crops value, the sub-crop value and the production cost for cultivating one hectare of crop c in region r, respectively.
It should be noted that the result of Mp cr + Sp cr − Cp cr is equal to the gross margin of crop c in region r ($/ha) and from now on, it will be shown with the symbol GM cr .
The social dimension of a sustainable cropping pattern model can be considered in workforce recruitment for the agricultural labor force. In this regard, the objective function 9 (Eq. 12) tries to increase the workforce recruitment in the optimal cropping pattern.
where Lab csr represents the amount of required labor to cultivate each hectare of crop c in season s and region r (man-day/ha).

Constraint sets
The scarcity of production inputs limits crop cultivation. Eqs 13,14,15,16,17,18,19 clearly define these limitations. The set of irrigation water constraint is related to the amount of available water. Equation (13) determines the allowable limit for the amount of water available for each area.
TW sr is the total amount of available water in season s and region r (m 3 ).
NetW csr Eff r A csr ≤ TW sr ∀s, r The set of constraints (14-17) are related to the allowable limit for the amount of labor, agricultural machinery, fertilizer, and pesticide, respectively.
where TLab sr , TMec sr , TFer tsr, and TPes zsr, represents the total amount of available labor (man-day), available agricultural machinery (hour), Total available fertilizer of type t (kg), and amount of available pesticide of type z (kg or liter), respectively.
Equation (18) relates to land and in this set, the cultivation area of crops is limited by the amount of available land in each region.
The TA r represents the amount of available land in the region r (ha) for all crops and Sch csr indicates the land occupancy coefficient. If the crops are cultivated each month (from planting to harvesting), the land occupancy coefficient of that crop in those months is one, and otherwise zero.
In order to rely on the optimal solutions and to accept the model presented in this study, the current profit of crop cultivation in each region should be provided (Eq. 19).
In this equation, CurProfit r is the current profit in the region r. The set of constraints 20 and 21 specifies the maximum and minimum production of agricultural products, respectively.
where Yield cr is the yield of crop c in the region r and MaxProd cr and MinProd cr are the maximum and minimum allowable amount of production of agricultural products, Lab csr A csr ≤ TLab sr ∀s, r Fer tcsr A csr ≤ TFer tsr ∀t, s, r Pes zcsr A csr ≤ TPes zsr ∀z, s, r Yield cr A csr ≥ MinProd cr ∀c, r respectively. The amount of maximum production in Eq. (20) is determined based on the potential of each region and macroeconomic considerations. The amount of minimum production in Eq. (21) is calculated based on self-consumption needs and food security policies set by the government for each region.

Uncertain data
There are many uncertain parameters in an optimization model for cropping pattern. In this study, the total amount of available water ( TW sr ) and the gross margin GM cr are considered as uncertain parameters. Therefore, by defining the parameter Γ TW sr , variables z TW sr and p TW sr and using model 6, the constraint of the total amount of available water (Eq. 13) is converted into two Eqs. 22 and 23.
where TW sr represents the nominal value of the total amount of available water.
Considering GM cr as an uncertain parameter, the two Eqs. 11 and 19 change. These equations, like Eqs. (22) and (23), are converted to robust form by defining new parameters and variables. Equation (24) and (25) are formulated to robustification the objective function 8 (maximizing gross margin) and Eq. (26) and (27) are formulated to robustification the Eq. 19.

Model evaluation
Monte Carlo simulation method is used to evaluate the proposed model (Fig. 3). For this purpose, after solving the robust multi-objective linear fractional programming, 10,000 random numbers with specified distribution for the uncertain parameters (in this study, the amount of available water and gross margin) are generated. After solving the proposed mathematical programming model, the values of the optimal cultivation area ( A * csr ) and generated numbers are replaced in Eqs. 13 and 19. If the equations were not correct after

NetW csr
Eff r A csr + z TW sr ΓTW sr + pTW sr ≤ TW sr ∀s, r (23) zTW sr + pTW sr ≥ TW sr ∀s, r cr ≥ GM cr ∀c, r these replacements, they are recorded as infeasible solution indicating model's inability to apply uncertain data. This process is performed at different levels of probability deviation of constraint (p) and given uncertainty level (ε).

Software and model solving algorithm
The proposed mathematical algorithms are modeled by a robust multi-objective fractional linear programming method and are fully coded in GAMS optimization software, solved for different levels of p and ε and generate different scenarios to apply conservatism against uncertainty. The optimization method chosen for solving this model was CONOPT4, which is an optimizer for solving large-scale nonlinear programming problems. CONOPT4 is based on the Generalized Reduced Gradient (GRG) method and is developed in GAMS software by the Danish Consulting and development firm ARKI (GAMS/CONOPT4, 2015).

Primary data processing
In this study, all the required data were obtained from various governmental agencies such as the Jihad-e-Keshavarzi organization (Anonymous, 2018) and the Great Karoun Irrigation Network Operating Company (Anonymous, 2017), which is under the supervision of Khuzestan Water and Power Organization. These types of data, which are In this irrigation network, there are generally winter and summer agricultural products in which 12 types of crops are cultivated. Some of these crops are perennial (such as alfalfa). Table 1 listed these crops along with the current cultivation area. Wheat is one of the most popular crops among farmers in this irrigation and drainage network due to its guaranteed price and its role in elimination of income risk, with about 17,000 hectares cultivation area. Although the availability of water and canal length in the Aghili area is less than in the Dimech area, the presence of more modern irrigation equipment in the Aghili area has made its current cultivation area (about 2800 hectares) the same as the area of Dimcheh. Table 2 summarizes the important parameters used in the proposed model. Table 1 and Table 2 provide findings on the relationship between the cultivation area and input consumption. Although the Corn cultivation area is lower than wheat and rice, the gross margin for corn (5864 $/ha) is more than the mentioned crops ( Table 1). The government has led a restriction on corn cultivation in the study area since the consumption of fertilizer and pesticide for its cultivation is more than the other two crops. However, this coercion was accompanied by fierce opposition from farmers (due to poverty) and led to severe tensions between government bodies and farmers. It is also evident that despite the low consumption of these inputs in barley, the cultivation area of this crop is just about 183 hectares. The simplest reason is the low gross profit of this product (192 $/ha) compared to other products, so the low gross margin has led to farmers' unwillingness to cultivate this crop.
Tables 1 and 2 show that the study area desperately needs a revision of the current cropping pattern. This change should be made in light of the problems already mentioned. Firstly, the farmers tend to participate in this change, depending on its effect on their income, (more or equal to their current income) and secondly, due to the critical conditions of fertilizers and chemical pesticides consumption in the area, their   consumption is drastically reduced. In other words, it is necessary to make a compromise between the different objectives in the area.

Cultivation area
Applying this model, a massive difference in the current and proposed patterns was observed. Referring to Table 3, the details of these changes are identified, and the crop cultivation level is presented at p = 1 and ε = 0 (scenario 1), which in fact, models the certainty conditions. Also, the model results at p = 0.1 and ε = 0.1 (scenario 2) are presented for applying the uncertainty conditions, which creates an excellent opportunity for comparing these two scenarios.
It should be noted that many uncertainty scenarios can be selected due to the nature of the parameters Γ and ε, and consequently there are many outputs for the model. Due to the reduction of model outputs and the focus on analyzing the effect of uncertainty on different aspects of the sustainable cropping pattern, two scenarios of optimism (Scenario 1) and pessimism (Scenario 2) were selected. This type of scenario analysis is also present in different studies (Mardani Najafabadi et al., 2019). Table 3 detailed the area of each crop for cultivation under two scenarios. It is observed that onion in both scenarios was excluded from the model. This can be traced to the large consumption of inputs and the low gross margin for this product. For example, the total pesticide use (herbicides, insecticides and fungicides) for this product was 4.78 l/ha, which is more than the other crops. However, the share of this product is insignificant, and its overall elimination due to regional conditions seems reasonable. Due to the significant share of wheat and rice in the current cropping pattern (about 63%), the cultivation area variation of both crops in two scenarios is analyzed. Table 3 shows that there is an increase (in scenario 1) in the cultivation area of wheat and rice, which increases the cultivation area to 2635 ha in the entire area. With the increase in the protection of the model against uncertainty, the proposed cultivation area has decreased significantly.
The cultivation area of some crops such as rice and corn has been increased in the proposed models to offset the reduction in profits in both scenarios. So, the policy that is currently being implemented by the government to limit corn cultivation area is not appropriate. This policy increases the dissatisfaction of farmers and, on the other hand, loses the alternatives. For example, the government could authorize the cultivation of corn along with barley, which has an increased cultivation area in both scenarios, which will increase farmers' interest in cultivating barley (Table 2). However, this policy should be done with caution if the whole proposed pattern is implemented.
It can be observed that by increasing the model's protection against uncertainty (decreasing the probability level p), the total cultivation area decreases; at p = 1, the cultivation area increased by 9.34%, and at p = 0.1 the mentioned area decreased by 3.63%. Table 4 presents the results of the changes in fertilizer and pesticide in the study area. However, the absolute reduction in the number of consumed inputs in optimal models does not necessarily indicate an improvement in the use of these production resources, and it just Table 3 Cultivation area of optimal cropping pattern of Gotvand irrigation and drainage network at different levels of p  happened by reducing the cultivation area. Therefore, the average use of fertilizer per hectare can be a good criterion for examining the effect of cropping pattern change on the optimal use of these resources. The results show that the fertilizers and pesticides used have decreased by approximately 6% and 8%, respectively, in both scenarios. Due to the current challenging economic situation in Iran (Poor economic growth, sanctions, and unemployment), reducing this amount of fertilizer is crucial for the agriculture sector.

System profit (economic dimension)
The vital point in examining the proposed model is that with increasing system protection against uncertainty (decreasing level of p from 1 to 0.1), the Total profit has increased. Figure 4 is presented to investigate this issue better and analyze its sensitivity at different levels of p ranging from 0.1 to 1 with steps 0.1 for each region. It is observed that as the level of p increases, the amount of total profit decreased in all three regions. In other words, there is a trade-off correlation between the degree of system protection against uncertainty (robustness) and the amount of total profit. Table 5 reports the utilization of other inputs generated by implementing the two scenarios of optimal cropping patterns and the current situation. In both scenarios and all three regions, irrigation water uses per hectare decreased. In the Gotvand region, the reduction of water use per hectare is increased with increasing system protection against uncertainty, and in other regions, it remains unchanged. In other words, there is no particular trend (such as total profit) for water consumption by increasing the system protection against uncertainty. In Gotvand and Dimcheh regions, as the level of p decreases, the utilization of labor and machinery does not have a significant change, while in the Agheli region, by decreasing the level of p, the increase in the labor is about 10% lower than the first scenario.

Evaluation of the capability of the proposed model against uncertainty
Monte Carlo simulation method was used to validate the proposed model, as shown in Fig. 3. In this regard, 10,000 random data were generated for the available water parameter with uniform (at a given distance) and normal (with 95% convergence) distributions for uncertain parameters GM jr and TW sr . Then, the feasibility of randomly generated numbers was investigated by using the simulation method. Figures 5 and 6 show the results of this simulation method for different levels of p and ε for random numbers generated using uniform and normal distributions, respectively. It is observed that in both distributions, with the decrease in the level of p, the percentage of infeasibility is reduced; So that at the 10% given uncertainty level (ε = 0.1) with a uniform distribution, with the decrease level of p from 100 to 10%, the infeasibility percentage is decreased from 56 to 17%. It is also found that by increasing the given uncertainty level in both distributions, the average infeasibility is decreased. The lower infeasibility at high levels of conservatism (lower levels of p) indicates the considerable robustness of the proposed optimal cropping pattern model to overcome the uncertainty.

Discussion
In this section, the results of the present study and previous studies are compared based on three ecological, economic, and social dimensions of the optimal sustainable cropping pattern. A summary of this discussion is provided as follows:

Ecological dimension
Increasing the cultivation area in the Gotvand irrigation and drainage network led to a decrease in the total area under the proposed model, improperly. Reducing the cultivation area in optimal cropping pattern models is not limited to this study and has been recommended in most similar studies for different regions of Iran (Sabouhi & Mardani, 2013). For example , Mardani Najafabadi et al. (2019), used the multi-objective robust optimization method in their cropping pattern programming model and found that about 16.5% of the total cultivation area in Isfahan province has been reduced. According to the results, although in the certain optimal conditions (Scenario 1) the cultivation area of wheat and rice has increased, with increasing conservatism and the application of uncertain conditions (Scenario 2) the cultivation area of these two crops will decrease. The main reason for this reduction compared to scenario 1 is the extensive water use of these two products. This result seems reasonable due to the increased protection leading to a decrease in the amount of available water.
According to statistics published by the Jihad-e-Keshavarzi organization of Khuzestan province, the use of fertilizers and pesticides in the Gotvand irrigation and drainage network has an upward trend and the consumption of these two destructive inputs will almost double. It is crucial to prevent the increase in the use of fertilizers and pesticides in this region, however, the reduction in their use in the proposed model is less than 10% (Anonymous, 2017).
In contrast, Neamatollahi et al. (2017) applied different scenarios by using a fuzzy multi-objective planning model, they found that the reduction rate of fertilizer and chemical pesticide consumption is 38% and 35%, respectively. These percentages are realized only when the demand for the industrial and livestock sectors is not taken into account. If this demand is calculated, the consumption of fertilizers and pesticides will increase sharply by 52 and 11 percent, respectively. The important point in the present study is that the need for self-consumption and food safety considerations are considered in Eq. 21 and this has led to more input consumption of fertilizers and pesticides. However, contrary to the study of Neamatollahi et al. (2017), the consumption of these two destructive inputs has decreased.
In many studies that have tried to optimize the cropping pattern, the ecological dimension has been considered only for the optimal use of irrigation water . Tan & Zhang (2018) optimized irrigation water use efficiency using robust fraction programming. The result of their study is that with increasing conservatism (reducing the level of probability of deviation), the efficiency of water consumption decreases.

Economic dimension
Sensitivity analysis of gross profit in the present study showed that with increasing system protection against uncertainty, system profit decreases. According to a review of research, this sensitivity analysis has been used in studies such as Tan & Zhang (2018) and Sabouhi & Mardani (2013) that used a robust optimization approach. Depending on the number of uncertain parameters in the models, this decrease in profit or increase in cost is distinctive.
Increasing the protection of the system against uncertainty is not limited to the robust optimization method. For example, in the study of Zhang et al. (2017), the probability of occurrence for different inflow is considered as different levels of conservatism. The profit of the system has increased by increasing the conservatism which means the protection of the system against uncertainty is reduced. Wang et al. (2019) used a bi-level multi-objective linear fractional programming model with the theory of fuzzy sets and different levels of alpha (Uncertainty condition) to redistribute water to the agricultural, industrial, urban, and ecological sectors of the Heihe River basin in China. They found that increasing the level of conservatism (alpha levels) reduced the total gross profit, especially in the agricultural sector. Filippi et al., (2017) used the mixed-integer linear programming model in similar conditions. The objective function of their model was different, so they maximized the net return function with the approach of Conditional Value-at-Risk as a safety measure. Increasing the risk-aversion parameter in this model, which can be considered as increasing the protection of the system against uncertainty, led to a decrease in profit.

Social dimension
In the present study increasing employment and food security have been considered as two aspects of social dimensions. According to the results, by implementing the cultivation pattern introduced in this study, the total employment rate will increase which shows the priority of ecological dimension in Gotvand region (especially reducing the use of fertilizers and pesticides). Although the aspect of food security in the present study is not considered as the main objective, however, in Eq. 21, the need for self-consumption as well as government policies on food security when estimating the right-hand side of this constraint (the minimum production rate of each product) is calculated.
Mardani Najafabadi et al. (2019) considered Maximization of the use of labor (employment aspect) and minimization of the net import of energy from agricultural products (food security aspect). As the result, the employment rate in the whole region increased by 9%; But its relative rate (labor per hectare) has decreased by 11 percent. In this study, the food security index did not improve due to a 2% increase in net energy imports. Considering the increase in population and the consequent increase in the need for calories, this result seems reasonable. However, Wineman and Crawford (2017) studied that due to the importance of the aspect of food security in a country, its only main objective is to maximize calorie production through the cultivation of agricultural products.

Summery
Due to many aspects of the optimal cropping pattern, Table 6 was designed to provide a summary of the status of the studies previously discussed. Although, it should be noticed that these studies have different objectives depending on the areas under study and the different priorities of each of them in solving the challenges, so considering different challenges is not the reason for the superiority of the studies.
As expressed in the conceptual model of optimal sustainable cropping pattern (Fig. 1), one of the important components is the optimal use of agricultural production resources. In this regard, Article 17 of the Performance on Determining the Duties of the Jihad-e-Keshavarzi organizations emphasizes the issues of sustainability of agricultural production resources, especially irrigation water. This is addressed in objectives 1-7 (Eqs. 8,9,10), which support minimizing the use of fertilizer, pesticide, and water, respectively. The installation of smart counter to draw irrigation water from the river and subsidy policies for fertilizers and pesticides in line with the implementation of the proposed cropping pattern can help encourage farmers to accept the pattern.
In Article 6 of the Agricultural Productivity Enhancement Performance, economic issues are of particular importance. The existence of the objective 8th (Eq. 11) is because of economic issues importance. Although, farmers' profits increase with the implementation of the proposed optimal cropping pattern but the increase in Scenario 2, where the issue of uncertainty is raised, is less than Scenario 1. To reduce the price risk for crops such as wheat, the government can buy them by determining the appropriate guaranteed prices, thus compensate for this reduction in profits.

Limitation and future research
According to Table 6, the proposed model in the present study does not apply only the supply-demand balance aspect since the transportation equations in this model are not provided. In the study of Mardani Najafabadi et al. (2019), agricultural products could be transferred from one county to another and also outside of Isfahan province. For this reason, the lack of supply of products from outside the province or other county was compensated. However, in this study, adding this feature required redesigning the model, which was not necessary due to regional priorities. As can be seen, this aspect has not been considered in most studies.
In the present study, it was not possible to create an optimal pattern at the farm level due to the lack of statistical information about farmers covered by Gotvand irrigation and drainage network. It seems necessary to create an extensive database in which accurate hydrological, soil, economic and social information about this region which helps to achieve an Table 6 Comparison of the results of the present study with others optimal sustainable cropping pattern at the farm and operator level by making changes in the structure of the proposed mathematical programming model.

Conclusion
Over the past few decades, environmentalists have consistently emphasized the need to create a sustainable agricultural system because of imbalances between water supply and demand, falling aquifers, increasing fertilizer and pesticides use, and depleting soil resources to conserve valuable environmental resources. Therefore, assessing the sustainability of farming systems to prevent the degradation of water and soil resources and reducing economic and social damages has become one of the top priorities of agricultural policymakers. Hence, this study aims to achieve an optimal cropping pattern for balancing the consumption of agricultural inputs (especially fertilizers and pesticides) by using a robust multi-objective fractional programming approach based on two scenarios (with and without uncertainty). The study area was located under the irrigation and drainage network of Gotvand in Khuzestan province, Iran, while it is dissimilar in terms of the average consumption of fertilizers and pesticides due to other provinces. The model was solved at different levels of deviation probability of each constraint (p) in two scenarios. The results showed that using the optimal cropping pattern in both scenarios made it possible to achieve the appropriate and optimal use of agricultural inputs (especially fertilizers and pesticides), which is the main objective of the study. However, the gross profit of the proposed model shows a slight increase in increasing system protection against uncertainty. Since, in the real world, the probability of uncertainty in the utilized data is higher, it is, therefore, advisable to use scenario (2) results to apply changes in the cropping pattern. Different and sometimes conflicting objectives in the structural programming scheme of the cropping pattern have been considered. It is possible to compromise between the considered objectives in the RMOLFP model, which is recommended to decision-makers. On the one hand, farmers are encouraged to implement the proposed model (due to the increase in gross profit) and on the other hand, the rate of water and chemical inputs (fertilizer and pesticides) consumption, which are in line with the implementation of the sustainable optimal cropping pattern, will be reduced.