Multi-objective optimization model for uncertain crop production under neutrosophic fuzzy environment: A case study

: In real world uncertainty exist in almost every problem. Decision-makers are often unable to describe the situation accurately or predict the outcome of potential solutions due to uncertainty. To resolve these complicated situations, which include uncertainty, we use expert descriptive knowledge which can be expressed as fuzzy data. Pakistan, a country with a key geographic and strategic position in South Asia, relies heavily on irrigation for its economy, which involves careful consideration of the limits. A variety of factors can affect yield, including the weather and water availability. Crop productivity from reservoirs and other sources is affected by climate change. The project aims to optimize Kharif and Rabbi crop output in canal-irrigated areas. The optimization model is designed to maximize net profit and crop output during cropping seasons. Canal-connected farmed areas are variables in the crop planning model. Seasonal crop area, crop cultivated area, crop water requirement, canal capacity, reservoir evaporation, minimum and maximum storage, and overflow limits affect the two goals. The uncertainties associated with the entire production planning are incorporated by considering suitable membership functions and solved using the Multi-Objective Neutrosophic Fuzzy Linear Programming Model (MONFLP). For the validity and effectiveness of the technique, the model is tested for the wheat and


Introduction
Modeling of complex dynamic systems can pose some challenges given their main attributes, so there needs to be a requirement of the development of methods regarding qualitative analysis to handle the dynamics and behavior of such systems besides the constructing of efficient control algorithms toward efficient operation, classification, recognition, identification, optimization and simulation. Several types of uncertainty representation can be addressed such as interval, fuzzy, granular as well as combined uncertain sets. The concept of the fuzzy set given by Zadeh [1] to deal with ambiguity and vagueness is valid for situations where true grades of membership exist. However, all physical and logical models may not rely completely on the valuations of membership. The generalization of fuzzy sets was introduced as intuitionistic fuzzy set [2], being more effective in dealing with ambiguity. In contrast to the traditional the fuzzy set theory, it assigns membership and non-membership to generic elements. Fuzzy optimization is an improved method that handles the uncertainty and imprecision associated with any optimization problem by involving parameters, arithmetic operations and relations governed from fuzzy sets. Bellman and Zadeh [3] were the first individuals to combine the ideas of programming and fuzziness. Tanaka et al. [4] and Zimmerman [5] extended the work from single objective to several objective functions. Angelov [6] established intuitionistic optimization technique that depends on the remodeling of single objective minimizations problem in fuzzy environment. The single objective is converted into two objectives that are to maximize the membership degree and minimize the non-membership degree. Using intuitionistic fuzzy sets is one way to tackle uncertainty since it yields the guarantee of less violation of risks emerging from vagueness during decision-making processes. While assigning and designating the priorities in multi-objective situations, flexibility can be generated corresponding to each objective and evaluation [7]. Complex optimization problems pose multiple unknown parameters that occur due to uncontrollable and unavoidable factors. Climate, weather and water storage are some of these elements which are bound by such uncontrollable factors. Imprecise data that are reliant of different parameters can be well represented by fuzzy numbers and membership grades. More complexity occurs when there is uncertainty involved in that process regarding the membership degree with parameters being uncertain as well. Thus, the degrees of membership function are important in decision-making under uncertainty and vagueness [8]. The neutrosophic optimization technique involves degree of truth, falsity and indeterminacy memberships. Degrees of truth and falsity memberships are not complemented of each other. They are, in fact, independent of degree of indeterminacy.
Founded by Florentin Smarandache in 1998, the neutrosophic theory constitutes a further generalization of fuzzy sets, triangular dense fuzzy sets [9,10], picture fuzzy sets [11,12] and spherical fuzzy sets [13] among others. Accordingly, the study [14] developed robust neutrosophic programming model to deal with multi-objective intuitionistic fuzzy optimization problem where the approach is based on intuitionistic fuzzy numbers and ranking functions for these numbers. The model proposed in the study is effective in such a way that it includes neutral thoughts during decision-making. The study, with its practical implications, provides the different types of membership functions depicted for the marginal evaluation of each objective concurrently. Another relevant study [15] has the aim of showing how neutrosophic optimization technique can be employed for the solution of a nonlinear structural problem. The problem involves the requirement of investigating the consequence of nonlinearity of the truth, indeterminacy and falsity membership function in view of multi-objective optimization problem under consideration. The study considers a non-linear three bar truss design problem. A further study [16] involves the development of an algorithm to evaluate the multi-level and multi-objective fractional programming problems by making use of the notion of a neutrosophic fuzzy set. The authors construct a neutrosophic fuzzy goal model to minimize the group tolerance of a satisfactory degree and attain the optimal degree for truth, falsity and indeterminacy of each kind of the prearranged membership functions goals to the most possible through minimizing their corresponding deviational variables, with the ultimate goal of obtaining the optimal solutions. The study puts forward the key benefit of the neutrosophic fuzzy goal programming algorithms proposed, which can be stated that if the attained optimal solution is refused by the relevant stakeholders repeatedly, the problem can be reevaluated by defining suitable and more appropriate membership functions until the desired result is obtained.
Uncertainty and counter intuition in geological interpretations generate an often-unquantified risk for related industries and activities. The challenge related to quantifying such interpretation for uncertainty has been addressed using various methods including the empirical quantification of uncertainties as derived from comparison of interpretations of different complex data. Pakistan, in that regard, holds a significant position in South Asia in terms of its geographic and geo-political conjuncture. Moreover, the country is highly vulnerable to the climatic changes that cause unpredictable weather patterns. Droughts and floods have frequently been observed in the last two decades resulting in a huge loss in the country's GDP. Pakistan is classified as an agricultural to semiagricultural country, to put it differently, an agrarian country. Agriculture is the major supplier to food security and acts as a major contributor in Pakistan's economy [17]. More than 60% of the population living in rural areas of Pakistan is dependent on agriculture of the country [18,19]. Approximately 22 million hectors (Mha) of land are cultivated out of the total land and is a major user of water resources. Water used with the ratio of 17:5 Mha of a total 22 Mha is irrigated and rain is fed, respectively. Pakistan's irrigation system relies primarily on fresh surface water and it possesses the world's largest network of canals and reservoirs. With three major reservoirs, 19 barrages, 12 link canals, 46 main canals and thousands of hydraulic structures, Pakistan has the world's largest continuous irrigation system [20]. The most imported source of water for irrigation in Pakistan is the Indus basin which rises from Gilgit in Pakistan (see Figure 1). Flowing through the North in a southerly direction along the entire length of Pakistan, it falls into the Arabian Sea near Pakistan's port city of Karachi. It is also aided by the four other large rivers like Jehlum, Chenab and Sutluj as well as several some small rivers like Kabul, Swat, Haro, Kunhar and Chitral [21].
In terms of agricultural production, Kharif and Rabbi make up the two types of seasonal crops in Pakistan. The growing of Kharif starts from May and ends in October. Rice, sugar cane, cotton, maize, and millet belong to the class of Kharif crops, whereas Rabbi Season starts from November and ends in April. Wheat, gram, tobacco, rapeseed, barley and mustard belong to the category of Rabi crops [22]. The season of these crops is influenced by different climatic variables such as temperature and rainfall. Excess of any climatic variables has unfavorable impacts on the agricultural crops [23]. Climate change severely impacts water and land, which can cause a change in output as much as 60%. It can also stimulate the timing of agricultural seasons, water stress, the magnitude and duration of heat [24]. Several uncertainties must be considered when analyzing the effects of climate change. Most impact assessments rely on general circulation models (GCM) forecasts [25]. In an unpredictable situation, exact reservoir capacity and reservoir operation policies are among the climate change adaptation solutions. Hence, uncertainty analysis is regarded as an essential component of reservoir yield analysis [26]. The multi-objective optimization problem (MOP) is widely used in water management, agriculture, industry, engineering, economics, mining and many other fields where it is necessary to optimize several conflicting objectives at the same time. Multi-objective optimization models are extensively used in agriculture under the influence of climate change [27][28][29][30][31][32][33]. A two-stage stochastic fuzzy-interval credibility constraint programming method (ISFICP) has been developed that can be applied to the allocation of water resources under uncertain and complex situations. There may be trade-offs between system benefits and risk of violation in this strategy and the many complications of water resource management. To address the problem of farmland and ecological water allocation in irrigation regions under uncertainty, Pan et al. [34] proposed an interval multi-objective fuzzy-interval credibility-constrained nonlinear prograg (IMFICNP) model. The water demand of ecological vegetation is upgraded from a site-specific sample to a spatial decision-making unit (DMU), which offers a full range of spatial data for constraint inputs and the water requirement of ecological vegetation is divided into three categories using remote sensing (RS) and geoinformation system (GIS) tools, including forest land, grassland and shrubland. Jin et al. [35] developed an improved teaching-learning-based optimization algorithm for the purpose of capturing more robust scheduling schemes. The authors aimed to obtain promising solutions with score values of the uncertain completion times on each machine having been compared and optimized. The definition of distinct levels of fluctuations or uncertainties on processing times was performed in testing Kim's benchmark instances and the performance of computational results was analyzed in the related study. Moreover, Ren et al. [36] proposed an improved interval multi-objective programming system for dealing with numerous objectives and uncertainties in irrigation water resource allocation. The model was tested in a case study in China's Shaanxi Province's Jinghuiqu basin. Maximizing economic benefit and lowering energy use are two of the planning goals. Various planting structure optimization strategies were found in multiple typical hydrological years. They found that a lack of water resources negatively influenced the development of irrigation systems in that study. The Fuhe River basin was used to test a multi-objective water resource allocation (GWAS) model, which considered the goals of socioeconomic water consumption, power generation and river biological flow. Yue et al. [37,38] designed the models to cope with the water shortage caused by diverse natural conditions and ineffective irrigation water management Type-2 fuzzy mixed-integer bi-level and full fuzzy-interval credibility-constrained nonlinear programming approach. Sahoo et al. [39] discussed the Genetic algorithm and particle swarm optimization in combined form to solve mixed integer nonlinear reliability optimization issues in series, series-parallel, and bridge systems.
Indeterminacy allows flexibility in the decision-making process since the function and decision variable obtain optimum indeterminate outcomes. Improved and advanced methods are required as fuzzy framework has shown an ever-growing expansion. Fuzziness, ambiguity, uncertainty and vagueness connected with real-life data and the desire to get the optimal solution under these factors are what determine the formulation of fuzzy optimization. Neutrosophic set and logic appear to be a comparatively new idea introduced by Smarandache [40] and further flourished by Wang et al. [41].
The complexity of environmental and socio-economic factors has increased agricultural production uncertainty, especially under present climate change, highlighting the need for robust multiobjective optimization models. Crop production is subject to a variety of uncertainties, including crop yield, crop quality, input parameters, and output objectives. Multi-objective optimization models based on unified frameworks must be devised to address these uncertainties. A multi-objective optimization model for uncertain crop production in a neutrosophic fuzzy environment is proposed to integrate various uncertainty handling strategies to address the various crop production uncertainties. The proposed model will use neutrosophic fuzzy logic to account for uncertainty in input parameters, fuzzy logic to account for uncertainty in output objectives, and optimization techniques to optimize the multiobjective problem. The proposed model will provide an effective method for optimizing the multiobjective problem in the presence of several sources of uncertainty like climate change and active reservoirs, thereby addressing the issue of uncertain crop production. Furthermore, the proposed model can be used to assess the effectiveness of various crop management practices in a variety of situations.

Linear programming
There are many challenges involved in linear programming, including maximizing or minimizing an objective function with constraints. The constraints are linear equality/inequality. It consists of the following parts: Decision variables set ; Objective Function: a linear function ( ) to be maximized (minimized);

Multi-objective linear programming
Both the objective function and constraints can be different in multi-objective linear programming [5]. Multi-objective linear programming can be shown in mathematics as: In general, there will not exist a single point ( ) = ( , , . . . , ) which optimizes each objective function individually. Multi-objective linear programming problems (MOLP) can be used to mimic a variety of real-world issues. A multiple objective linear programme is a linear programme with numerous objective functions.

Fuzzy linear programming
Fuzzy linear programming [5] represented as: Where are decision variables in the constraints and are non-negative fuzzy number. Mathematical modelling, manufacturing, environmental management, supply chain management, and transportation management are all applications of fuzzy linear programming. Fuzziness is considered in fuzzy linear programming's objective function and constraint equations.

Neutrosophic set
Neutrosophic set [8] is a generalized concept in which each component ∈ to a set has a membership degree ( ), non membership degree ( ) as well as a degree of indeterminacy ( ), where ( ), ( ) and ( ) are real standard or nonstandard subsets of ] , [.

Multi-objective crop production model
The two related functions are constructed in this section of the study. Water shortage, being the main issue for crop production and availability of water during a specific period in Pakistan, determines the productivity of the sectors in the country concerned with agriculture. Within this framework, modern agriculture is characterized by several parameters including conflicting optimization criteria (total cost, net benefit, production, and so forth).

The related parameters
The set of parameters used in this study are presented in Table 1.

The first objective concerned with maximizing net profits
The first objective is to maximize net profit obtained by cultivating various crops in canalconnected areas Where = Kharif Crop index, = Rabbi Crop index, , =Kharif canal-connected cultivated area, ℛ , = Rabbi Crop canal-connected cultivated area. Here , is the profit obtained from Kharif and Rabbi crops, respectively. Profit is calculated by subtracting the total cost from the gross income. Calculation of the total cost from Kharif and Rabbi Crops is performed as follows: (

The second objective concerned with maximizing yield of crops
Secondly, the multi-objective optimization model maximizes the yield of various crops grown in the connected regions to the dam irrigation system during the Rabi and Kharif seasons. An area is directly related to the amount of crop productivity.
Where denotes the average yield of crop in tones/ha. Where ℛ and represent the maximum areas of the canals during the Rabbi and Kharif seasons, respectively.

Social constraint
In the dam-operating region, staple crops must be grown to satisfy social needs. Therefore, the social constraint must be used. Where , and ℛ , are the minimal area to be planted for crop with canal in both seasons.

Crop water requirement constraint
Modified Penman method is used to estimate the amount of water required for different crops in the area connected to the canal in each season. Water loss during the entire process is also considered while estimating the total irrigation requirement. The constraints for kharif and rabbi seasons are given as follows: Where ℛ , is the water discharge from the canal, ℛ , is the water required for irrigation in the time for the cultivation area , and ℛ , of the crop. The time regarding the Rabi and Kharif crops is considered, respectively. and are the application and transmission efficiencies.

Canal capacity constraint
Water discharged for irrigation from the dam in the time should be less than or equal to the maximum capacity of the canal. We have: ℛ , ≤ = 1, . . . ,12: = 1, . . . , .

Where
is the maximum capacity of the canal .

Reservoir evaporation constraint
The water loss caused by evaporation (Ѐ , ) from reservoir in the period can be calculated by counting the final and initial storage in the given period. The reservoir evaporation constraint is presented as:

Where
, and , are the regression coefficients. Ɣ , and Ɣ , are reservoir water levels during the time .

Minimum and maximum storage constraint
Physically for the safety of any reservoir, water storage level at any time should be kept under the maximum storage capacity and to compete with the demand it should be above the minimum storage level. The storage constraint is conveyed as follows: Ɣ , ≤ Ɣ , ≤ Ɣ , = 1, . . . ,5; = 1, . . . ,12 where Ɣ , and Ɣ , are precisely the extreme storage capacities of the reservoir .

Overflow constraint
Natural phenomena like excessive rains, floods and melting of glaciers turn into heavy water inflow to the reservoirs in that case the water beyond maximum storage capacity need to release immediately. The overflow constraint is stated as follows:

Multi-objective neutrosophic fuzzy linear programming model
The programming model previously noted is based on the formulation of membership and nonmembership function but there are several situations where these two components are not enough to incorporate all the necessary information about the problem. It is obvious that there will be some sorts on neutrality or indeterminacy within the information about the data under consideration. To deal with such scenario the intuitionistic fuzzy subset is upgraded in terms of the neutrosophic sets. As defined earlier in a neutrosophic set, each ∈ to a set is characterized by Computational algorithm based on multi-objective neutrosophic linear programming The computation algorithm for multi-objective neutrosophic linear programming is based on the following six steps: Step 1: Solve the objective function , , ℛ , as a single objective subject to the given set of constraints and decision variables.
Step 3: Solve the objective function , , ℛ , as a single objective subject to the given set of constraints and decision variables. Use the value of decision variables and compute the value of the first objective function. Suppose ( , * , ℛ , * ) and ( , * , ℛ , * ) be values of decision variables computed by considering and as a single objective function subject to the given set of constraints, respectively. Using these values, the pay-off matrix will be * ( , * , ℛ , * ) ( , * , ℛ , * ) ( , * , ℛ , * ) * ( , * , ℛ , * ) .
Step Step 6: At this step, it is necessary to solve the following neutrosophic linear programming problem Maximize + − such that The accuracy of multi-objective neurotrophic linear programming model is higher than the accuracy of other related models while dealing with imprecise date. These steps are presented in the Figure 2.

Case study: Multi-objective neutrosophic model for crop production
Pakistan has two distinguish cropping seasons and despite of modern irrigation technologies still a large fragment of agriculture is highly depending upon water flow from dams and canals. Climatic changes affect the water supply and ultimately the crop production in the adjacent areas. There is a need to construct a Multi-objective Neutrosophic Model for Rabbi and Kharif crop production using water reservoirs active storage under yield uncertainty. The primary data was taken from the agriculture marketing information services to calculate and maximize net profit from Rabbi and Kharif crops and crop production. For the validation of the constructed model, the two main crops play a significant role in the agriculture of Pakistan: wheat and rice from Rabbi and Kharif crops, respectively. The cultivation areas are in Figures 3 and 4.  First, the profit from Rabbi and Kharif crops using active storage water reservoirs is separately calculated since profit is the subtraction of gross income and the total cost used for crop production. Various factors are involved in cost production.

Marginal insights and interpretations
The first step of crop production is the land preparation involving deep ploughing, cultivator, planking, laser levelling, wet ploughing and wet planking are included. Rabbi's seed and sowing costs include seed treatment, ploughing, planking, bund making and tractor with the drill as depicted in Table 2. In contrast, for Kharif seeding, uprooting, transplanting and transporting, manure weedicides and seed treatment are shown in Table 3. Fertilizer cost comprises urea, DAP, transportation, fertilizer application, zinc sulphate and plant protection as presented in Table 4. Furthermore, irrigation costs are based on canal water rate, private tube-well, labor for irrigation and cleansing, as followed by Table 5. Moreover, harvesting and threshing cost is composed of harvesting and threshing cost for both. Additionally, miscellaneous costs involved land rent, agriculture income tax and management charges are presented in Table 6. Overall, the total cost and gross income for the Rice (Kharif) and Wheat (Rabbi) is calculated as given in Table 7. While using linear optimization, intuitionistic optimization and neutrosophic optimization the profit for cultivation wheat and rice and average production needed for second objective function in the case of Rice (Kharif) and Wheat (Rabbi) crop is given by Table 8. In this study, the models were solved using MATLAB software version 2018a on a device with the specification Core(TM) i3-8130U CPU and 4GB RAM. Whereas graphical representation of linear optimization, intuitionistic optimization and neutrosophic optimization the profit is given by Figure 5 and average production needed, is given by Figure 6.  Comparing the results in Table 8 and Figure 5, the profit computed by employing neutrosophic technique is higher than the profit computed by linear and intuitionistic optimization model. Thus, the proposed model is more accurate and efficient in maximizing profit and gross production.

Conclusions
It can be concluded that modelling of complex dynamic systems points some challenges considering their main attributes. Hence, the need of the development of methods regarding qualitative analysis to address the dynamics and behaviors of these systems along with the constructing of efficient control algorithms toward efficient operation, classification, recognition, identification, optimization and simulation becomes conspicuous. Several types of uncertainty representation can be noted such as interval, fuzzy, granular as well as combined uncertain sets. Furthermore, uncertainty is apparently prevalent as a challenge under the unpredictable and uncontrollable climatic conditions, so provision Neutrosophic optimization techniques have the objective of providing how neutrosophic optimization technique can be employed to solve a structural problem which requires the investigation of the effect of conflicting constraints. Accordingly, our study has aimed to investigate the profitability and marketing of Rabbi and Kharif crops in Pakistan. Firstly, we proposed a multi-objective model in which standard parameters values are taken from agriculture marketing information services. To ensure the applicability of research findings, a selection of rice and wheat crops was undertaken. The analysis in our study adopts a neutrosophic technique that considers the degrees of truth, falsity and indeterminacy. The maximum net profit was obtained by using water reservoirs active storage at 285460 PKR. Ton/ ha. Our study has put forth the benefits of neutrosophic fuzzy algorithm which has been proposed and the analyses derived can be stated to deal with yield uncertainty in the neutrosophic environments more effectively considering the different parameters in nonlinear and dynamic settings in which abrupt changes may occur any time. It is worth noting that decision-making problems manifests the challenge of being imprecise, vague, or ambiguous. Yield production is highly crucial considering the ever-changing dynamics of our current time which affects yield uncertainty. Since uncertainty is considered by a mathematical model, the neutrosophic environment is more appropriate and applicable since it deals with uncertainty more efficiently and effectively. Wh le Uncerta nty n crop product on s d ff cult to capture n the model as t s based on assumpt ons rather than emp r cal data. Cl mate change w ll ncrease the frequency and ntens ty of extreme weather events, such as droughts and floods, wh ch can lead to reduce crop y elds. Th s w ll reduce the amount of food produced, mak ng t more d ff cult for countr es to feed the r populat ons. Increased temperatures and more frequent extreme weather events can also lead to ncreased crop losses due to ncreased pests and d seases, and heat and drought stress. Cl mate change can also cause changes n the su tab l ty of d fferent crop var et es, as some may become better adapted to the chang ng cl mate wh le others may struggle. Th s could have a s gn f cant effect on crop product on, as some crops may no longer be su table for certa n reg ons. Cl mate change can also lead to water scarc ty, as ncreased temperatures can cause more water to evaporate and less prec p tat on. Th s can cause water stress for crops, lead ng to decreased y elds. Cl mate change can also lead to so l degradat on, as ncreased temperatures can cause so l to dry out, lead ng to nutr ent loss and reduced so l fert l ty. Th s can make t more d ff cult for crops to grow. The following suggestions as future directions can be outlined to improve Rabbi and Kharif crops' agriculture profitability and marketing using active water reservoirs' storage: first, the government should provide subsidies on pesticides, fertilizer, and other nutrients. There is also a need for appropriate crops farming guidance for farmers. Thus, t would be a v able opt on f the government could ass gn act ve experts and extend departments for relevant gu dance to be able to conceptual ze, mplement, ma nta n and control the appl cable strateg es and dec s on-mak ng processes n a systemat c, exped t ous and accurate way. In future, us ng the proposed methodology crop product on models can be des nged for the ra n fed areas to meet the food requ rements of the grow ng populat on.