Abstract
A triangular intuitionistic fuzzy linear programming (TIFLP) model is formulated for the planning of sustainable fruit production system for hyperarid regions while assuming the availability of resources and existing knowledge. A remarkable advancement is achieved through the composition of intuitionistic fuzzy concept with the linear programming by considering all parameters and variables in the form of triangular intuitionistic fuzzy numbers, which provides a planning or strategic tool for handling uncertain situations with more control and in a realistic way. This fuzzy optimization model is redesigning the feasible region obtained by linear programming which is presented in graphical form. Moreover, the practical application and implementation of this fruit production system for planning in real-life scenarios are accomplished considering the case study of fruit orchards of Baluchistan, Pakistan.
1. Introduction
Have you ever imagined experiencing the world without agriculture? In that case, most of the world’s population could not outlive hunger, and the remaining ones would be hunting for food. In fact, you would no longer be here to read this paper because the path of modern civilization would be lost forever with the absence of agriculture. Agriculture is art, science, and business of all types of crop production which flourished into seven major branches named as agronomy, horticulture, forestry, animal husbandry, agricultural engineering, fishery, and home science [1]. The beginning of human civilization started with agricultural development referred to as first agricultural revolution. Later on, agriculture and farming spread into different regions around the world and broadened with livestock, industrial agriculture, agronomy, and much more. The history of human civilization is reflected by the inventions, methods, and techniques used to enhance the agriculture and its different branches in a productive manner. Throughout modification in agricultural field, it has been improved and transformed into much more ultramodern form known as “sustainable agriculture” which equally impacts the environment, society, and economy [2].
The ultimate motive of sustainable agriculture is the satisfaction of all human needs and necessities with the major contribution to economy in healthy environmental conditions. The improvement of our food security system is the mostly targeted goal for the betterment of present and future generation. The sustainable development goal is the eradication of hunger by accomplishing food security and improving the nutrition intake by 2030 [3]. A thorough analysis was carried out about the achievement of “zero hunger” goal by studying all the existing scientific literature to assess their contribution to the achievement of the sustainable development goal [4]. According to a latest study, the fourth agricultural revolution demands the balance between the agricultural production and world’s population together with the environment [5]. To eradicate the undernourishment of the world, fruit consumption rate of the world per capita should be according to diverging health conditions. The low intake of fruit and vegetable increases the worldwide burden of disease, which can be controlled through the ample amount of fruit consumption and production [6]. Analytical study reveals that approximately 22% of difference exists between the demand and supply of fruit production, whereas this percentage increases to 58% for the underdeveloped countries, which is increasing with the passage of time [7].
Pakistan, being a middle-income developing country, produces five major crops, wheat, rice, sugarcane, maize, and cotton, along with the most importantly fruits and vegetables with pulses and oilseeds [8]. The production of fruits and vegetables is approximately 12 million tons per year. More precisely, fruits contribute to agricultural gross domestic product of Pakistan, producing apples, mangoes, grapes, dates, citrus, peaches, cherries, plums, loquat, pears, and guava. According to a rough analysis, Pakistan earned million by exporting 1.165 million tons of fruits and vegetables in a year [9]. The study of Pakistan recommends investing in research and development to find innovative strategies to enhance production and quality and reduce postharvest losses in order to boost fruit and vegetable export competitiveness [10]. The global horticultural products trade for the past two decades was maximized by four times by making earnings of USD 51 billion in 2001 to USD 200 billion in 2018 [11]. The international trade competitiveness of Pakistan is evaluated through the analysis of competitive and comparative demand and supply of vegetables and fruits [12]. The overwhelming pressure on the demand of food security caused by population increase and global development results in the destruction of natural resources and food crises [13]. Additionally, COVID-19 and intense climate changes severely escalate the demand of food by decreasing the average agricultural production [14].
Real-life situations can be assessed mathematically. For modeling and management of certain scenarios, mathematical analysis of real-life occurrences utilized quantitative and qualitative methodologies. Linear programming is a generalized and renowned technique presented by Kantorovich [15] to optimize agricultural aims and objectives by allocation and restriction of certain demand and availability constraints [16]. In light of our current agricultural requirements, our objective is not only food supply but also the ample amount and quality of food provision around the world. Therefore, agricultural planning is carried out for this goal using operational mathematical approaches in the most efficient way in order to eliminate food security issues [17, 18]. It is used as a single objective as well as multiple objectives to minimize and maximize the cost and profit by the utilization and management of natural resources, labor, techniques, research, capital regarding land allocation, cropping patterns, optimization of water resources, raising livestock, and production maximization with cost minimization [19].
Food production system must be thoroughly modified and armed with resilience and adaptivity and have high diversity against different situations and factors (climate change, pest attacks and diseases, governmental policies at national and international level, social and cultural stability factors) [20]. For perfection in the precision of goals regarding planning, this area still needs much more modifications in terms of changing environmental, ecological, and social factors [21]. Globally, agricultural output continuously confronts drastic fluctuations due to which sustainable agriculture is constantly evolving with the passage of time and demand of the world is changing continuously regarding various aspects. These factors generate uncertainly and vagueness in environment, which is assessed by using the concept of fuzzy sets introduced by Zadeh [22]. Indeed, fuzzy set and its generalizations such as intuitionistic fuzzy sets [23] are utilized to present data that is fuzzy in nature. Eventually, fuzzy optimization theory was initiated by Zimmermann for effective decision making in fuzzy environment [24].
Fuzzy linear programming approach was further investigated through meticulous application to decision making and management problems considered in uncertain environment, and it obtained much more precise and feasible output [25]. Under unpredictable circumstances in energy-water nexus, an integrated fuzzy optimization approach was proposed for agricultural water and land resource management [26]. Multiobjective fuzzy methodology having three goals was considered as maximization of net benefits, agricultural output, and labor employment for Pune city of Maharashtra State, India [27]. Another study was conducted by applying intuitionistic fuzzy optimization technique in agricultural production planning, with a focus on smallholder farmers in north Bihar, India [28].
Specifically, fruit production planning by using linear programming is done, which is generalized for production maximization in hyperarid regions with available resources, labor, capital, etc. Further, in order to evaluate a targeted objective function that stays valid and optimal under the influence of climatic, social, and economic conditions, triangular intuitionistic fuzzy linear programming has been constructed more accurately and meticulously. The article is divided into five sections, where all the basic and essential information is provided in Preliminaries section. The objective function and constraints for optimal fruit production in crisp and intuitionistic fuzzy environment are defined in Methodology section. The model is then applied to a real-life example by considering fruit production data from Baluchistan province of Pakistan. The superiority of the proposed methodology is supported by comparative and postoptimal analysis.
2. Preliminaries
2.1. Fuzzy Set
Let X be the universal set. A fuzzy set [22] consists of a pair defined as , in which the first element of belongs to classical set and the second element defined as refers to the membership degree of in , called the membership function of .
2.2. Fuzzy Intuitionistic Sets
Let X be denoted as a universal set. An intuitionistic fuzzy set (IFS) [23] is defined as set of ordered triplets , in which the functions and represent membership and nonmembership degree of x in , respectively, for each element satisfying .
2.3. Triangular Intuitionistic Fuzzy Number
A triangular intuitionistic fuzzy number (TIFN) [29] is an especial IFN with the membership function and nonmembership function defined as follows:where , denoted by or TIFN. Membership and nonmembership functions of TIFN are presented in Figure 1.
2.4. Accuracy Function
The accuracy function [30] for triangular intuitionistic fuzzy numbers is defined as
3. Operations on Triangular Intuitionistic Fuzzy Number
A triangular intuitionistic fuzzy number is said to be nonnegative if and only if .
The arithmetic operations of triangular intuitionistic fuzzy number [29], i.e., addition, subtraction, multiplications, and division, are defined by considering two nonnegative triangular intuitionistic fuzzy numbers and . Two triangular intuitionistic fuzzy numbers are equal, , if and only if , , , , and .
3.1. Addition
3.2. Subtraction
3.3. Symmetric Property
3.4. Scalar Multiplication
Let be any scalar; then,
3.5. Multiplication
Remark 1. If and are not nonnegative triangular fuzzy numbers, then their multiplication will be performed aswhere
4. Linear Programming Model
General linear programming [16] is defined as
subject to the following constraints:
Condition of nonnegativity is as follows:where , and are the decision variables, coefficients of quantity which we have to maximize or minimize, constraints coefficients, and constants, respectively. This represents the crisp modeling of the problem, but for the most beneficial implementation of this model in our daily life problems, we used its modified form “triangular intuitionistic fuzzy linear programming” which is endowed with the generalized techniques for the absorbtion of fuzziness due to unpredictable and unfortunate scenario.
5. Triangular Intuitionistic Fuzzy Linear Programming Model
Triangular intuitionistic fuzzy linear programming enhances the targeted requirements by evaluating the problem specifications meticulously using the generalization of fuzzy logics intuitionistic fuzzy sets. A triangular intuitionistic fuzzy linear programming [25] can be formulated as follows:subject to the following constraints:
Condition of nonnegativity is as follows:where the model contains all coefficients, variables, and constants in the form of triangular intuitionistic fuzzy numbers; for example, , , and are triangular intuitionistic fuzzy cost coefficients, triangular intuitionistic fuzzy constraints coefficients, and constants, respectively, with being triangular intuitionistic fuzzy decision variables. Ultimately, is the maximum triangular intuitionistic fuzzy objective value.
6. Methodology
The linear programming for fruit production maximization is developed assubject to the following constraints:
Condition of nonnegativity is as follows:where is maximized fruit production; refers to activities (cutting, pruning, harvesting, thinning, leveling, sales, etc.); indicates objective coefficients (market prices of variables, product profit, etc.); denotes constraints coefficients (utilized resources and capital per unit of fruit production); and is the total available amount/units/volume of supplies per hector.
Generally defined constraints for major fruit production are further written as follows: where is the total number of fruit crops, is the total cultivated land, is the total available area for each fruit, is the total area for grapes, is the total area for apples, is the total area for cherry, is the total area for almond, is the total area for plum, is the total available hours or man-days for labor, is the area for each fruit crop, is the required working hours or man-days for each ith crop, represents the required amount of fertilizer per hector, represents the required amount of pesticide per hector, is the total cost per hector, is the amount of yields in kg per hector, and is the market selling price of yield per kg.
Then, we need much more precision regarding data and situation analysis because of changing factors and circumstances in our universe. The world we are living in is not like before; it is constantly changing, which makes it more challenging for us to change ourselves and our methods according to that change. The simple linear programming is not enough for our environment changes like climate changes, economic downfall, fluctuation of prices and demand, unsuitability of resources, pest and diseases, governmental policies, international trade agreements, topography, and political and social factors. We made a conscious effort regarding this issue especially for the hyperarid zones of Pakistan to improve our food security and GDP. Here, a triangular fuzzy linear programming is formulated according to the present situation analysis of fruit production of Pakistan for improvement.
The triangular intuitionistic fuzzy linear programming for fruit production maximization is developed assubject to the following constraints:
Condition of nonnegativity is as follows:where is the triangular intuitionistic fuzzy maximized fruit production; refers to the triangular intuitionistic fuzzy activities (cutting, pruning, harvesting, thinning, leveling, sales, etc.); indicates the objective triangular intuitionistic fuzzy coefficients (market prices of variables, product profit, etc.); represents the triangular intuitionistic fuzzy constraints coefficients (utilized resources and capital per unit of fruit production); and is the total available triangular intuitionistic fuzzy amount/units/volume of supplies per hector.
The objective function and constraints equations will be written as
By using the operations of triangular fuzzy numbers,
Further simplification was carried out using accuracy function on the triangular intuitionistic fuzzy objective function.
Ultimately, triangular intuitionistic fuzzy objective function is transmuted into linear objective function by accuracy function, and regarding that reference, the constraints are thoroughly modified into
Using the equality condition of triangular intuitionistic fuzzy number, we have
Now, the model is converted into simple linear problem which can be easily solved through LP algorithm or Excel Solver. Then, we get the values of unknowns (decision variables) that are substituted into the triangular intuitionistic fuzzy objective function to get the maximized result in the form of triangular intuitionistic fuzzy number.
7. Application
The provinces of Punjab and Baluchistan produce abundant amount of fruit where Baluchistan lies in the arid regions of Pakistan. Baluchistan is the largest province on the basis of area occupying square kilometres and located in southwest direction. The climatic conditions of Baluchistan region are characterized by very cold winter and very hot summer with maximum of to [31]. Moreover, strong windstorms and temperature make the area very hot arid zone, which is referred to as hyperarid zone. Baluchistan contributes nearly to GDP which is far less than other provinces. Recently, water availability for the expansion of sustainable agricultural land is achieved by making Mirani Dam on the Dasht River which irrigates of area [32]. For practical application of our formulated models, data for fruit production is collected from Baluchistan and is arranged in tabular form for easy further use.
8. Mathematical Model Formulation
The practical formulation of the model is carried out through the application of the above statistics that are specifically gathered from the Baluchistan province based on the data given in Tables 1–3.
Objective function is as follows: subject to the following constraints:
In this model, we used fertilizers, all types of cost, available labor hours, and average fruit yield as constraints to find the optimal fruit production. After the above developments, we used Excel Solver for the maximum yield which gives objective value kg. Afterwards, fuzzy modification of model is carried out to figure out more optimal way of modeling the existing methodology. The triangular fuzzy intuitionistic linear programming is given as follows.
Intuitionistic fuzzy objective function is as follows:subject to the following intuitionistic fuzzy constraints:
This is the mathematical formulation of triangular intuitionistic fuzzy linear programming in which all the decision variables and the regarding coefficients are triangular intuitionistic fuzzy numbers. As stated above, we cannot directly solve this model. Ultimately, we convert this model into crisp linear programming by using the accuracy function and arithmetic operations of triangular intuitionistic fuzzy numbers accordingly.
Along with the linear constraints simplification, which is carried out using the arithmetic operations of multiplication and equality of triangular intuitionistic fuzzy numbers according to the methodology, we have the crisp LP model which is simply solved through Excel Solver to find the values of decision variables. The values of decision variables obtained are
The triangular intuitionistic fuzzy objective value is obtained by putting the values of decision variables , into the triangular intuitionistic fuzzy objective function as follows:with membership and nonmembership degree as follows:
8.1. Interpretation and Comparison of Results
For comparison, the results obtained by optimization model considered in fuzzy environment should be compared with the linear programming in crisp environment. The general linear programming specifically designed for fruit production gives the output of 858880500 kg which is maximum fruit yield by consuming the available resources and inputs. The modified triangular intuitionistic fuzzy linear programming yields the result ofwhich is clearly maximum fruit production output in the form of triangular intuitionistic fuzzy number. These results are further explained and demonstrated trough detailed analysis in the form of graphical representation in Figure 2 which shows the output of both techniques. The level of satisfaction increases with the production increase from 588 714 344 to 859 026 000, reaches the maximum over 859 026 000 with membership degree 1, and then decreases afterwards to 1042 552 710. It is obvious that degree of nonmembership decreases with the increase in membership degree simultaneously. The vertical line in the graph at 858 880 500 represents the results of linear programming. In comparison, the graph already shows that 145 500 kg of yield increased by triangular intuitionistic fuzzy linear programming and the optimal region obtained from this technique is much more acceptable due to the feasibility levels at certain situations.
8.2. Postoptimality (Sensitivity) Analysis
Sensitivity analysis (postoptimality analysis) is the process of determining how changes in the optimal solution influence it, within certain limits. The sensitivity analysis is carried out by changing the coefficients of objective function and the right-hand side (RHS) values of constraints. Here, postoptimality (sensitivity) analysis of triangular intuitionistic fuzzy linear programming is assessed using the Tables 4 and 5. The solution remains optimal and feasible within the specified limits of variables and parameters. Range of optimality is dependent on the coefficients of objective function, which means that change in the coefficients of objective function affects the optimality of solution, which is represented by Table 4. This table contains the limits for the coefficients of each variable in the form of allowable increase and decrease. For example, the limit of coefficient of having original value 0 is between 10082.65797 and 333727.8781, and the solution remains optimal for this range. The cell containing value in the form of allowable increase or decrees means that there is no limit for the increase or decrease of that specific variable.
In Table 5, the range of each constraint is presented with the shadow increase in objective value, which is only valid for given ranges. A change in the right-hand side of a constraint directly changes the feasible region which perhaps influences the optimal solution. From Table 5, it is clear that our feasibility region remains feasible and the same if the constraints change within the allowable range. As observed from Table 5, the total land constraint 1 has a range between 7.275641026 and 30.41666667 in which feasibility region of the model remains unchanged. Moreover, shadow price is also given per unit increase in the right-hand side of the constraint providing improvement in the value of the optimal solution. The above analysis indicated that this technique is providing flexible optimal solution with the original data.
9. Conclusion
The comparison of methodologies, postoptimality (sensitivity) analysis, and compiled statistics stated that the triangular intuitionistic fuzzy linear programming is providing best results for management of real-life problems. The feasible region for optimal production in fuzzy environment remains feasible and optimal within sufficient range. In future, we can consider this model in different fuzzy environments to optimize production and observe the optimality and feasibility levels more accurately. To maintain the level of food security nationally or internationally, we can design a multilevel model in fuzzy environment for the achievement of best optimal agricultural production with least cost by consuming available resources.
Data Availability
Fruit production data were collected from local farmers to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
All authors contributed equally to the preparation of this manuscript.