**Introduction**

The application of scientific, mathematical, and logical ideas to the solution of military problems gave rise to operations research approaches. The origins of these techniques can be traced back to F. W. Lancaster’s work on the application of mathematical analysis to army tactics during the First World War. Lancaster investigated the relationship between a force’s victory and its advantage in firepower and numbers. Nobel Laureate Blackett (1948), provided two notes outlining some of the foundations of operations research and analysis methods. First in the United Kingdom, then in the United States, Canada, and Australia, operations research organizations were formed. Because operations research proved to be a useful tool to the allies in bringing them victory, US army facilities continued to fund and support programs to expand operations research long after the battle was done.

**What is Operation Research? **

Activities research is a scientific method of providing executive departments with a quantitative foundation for making choices about the operations under their supervision. One of the definition’s flaws is that it fails to distinguish operations research from a number of other business-related fields. Even if users combine Operations Research with Quality Control or Cost Accounting, the concept is true. According to Yates, (1949) operational research consists of the application of technical exploration to the problems arising in organization and arrangement by methods of scientific research, mean that arrangement of observation, trial, and reasoning which researchers are in the habit of using in their systematic investigations form an integral part of operation research operational research, according to Yates (1949), consists of the application of technical exploration to problems arising in organization and arrangement by scientific research methods, which means that the arrangement of observation, trial, and reasoning that researchers are accustomed to using in their systematic investigations are an integral part of operation research. Ackoff (1961) defines an operation as a sequence of actions necessary to achieve a specific goal. Communication, content, control, and structure are the four components he lists for an organization.

The following are the essential elements of operations research:

- Identification and description of the problem;
- Definition of the objective function for optimization;
- Construction of a mathematical model that satisfies the constraints based on the variables’ values.
- Obtaining empirical parameter estimations
- Solving the model and determining the best line of action for maximizing the objective function.

**Operation Research in Applied Research **

Operational research in agricultural research helps researchers, particularly agricultural economists, make better decisions. It is also necessary to have knowledge of the relationship between economics and the professional approach of operations research. To do so, you must first comprehend some of the principles of both disciplines. The operations researchers come from a variety of academic and professional backgrounds. In other words, operations research as a field has attempted to maintain its multi-professional, collaborative approach to wide objectives. As a result, operations research has become increasingly crucial in decision-making across the board, including transportation, manufacturing, purchasing, and selling. Many companies have their own operations research branch. Government and social entities have employed operations research methodologies for a variety of goals. In the subject of business management research, operations research has established itself as one of the most important sciences. It not only aids in the recognition of various natural situations and strategies but also in the listing of other courses of action available to the decision-maker and the outcomes associated with them, thereby suggesting which strategy to choose and act under a given set of circumstances for the best management.

**Operation research, like scientific research, follows a scientific approach that includes the steps below.**

**Formulating the Problem:**Operation Research is a study of the operation of a man, machine, or organization, and it must take into account the operation’s economics.

The following important components must be considered when creating a problem for an Operation Research study:

- The environment
- The objectives,
- The decision-maker,
- The many alternatives and constraints.

The environment is the most extensive of the four components since it serves as a backdrop for the other three. The operation researcher will attend conferences, pay visits, send observation reports, and conduct research in order to gather enough data to construct the problems.

**Creating a Model to Represent the Underlying System:**After the project has been accepted by the management, the following stage is to create a model for the system under investigation. The model can now be built by the operation researcher to illustrate the relationships and interrelationships between a cause and effect or an action and a reaction.

Now, the goal of an operations researcher is to create a model that allows him to predict the impact of key components in the solution of a given problem. The proposed model can be tested and changed to work within the limits of the environment. If the management is dissatisfied with the model’s performance, it can be changed.

**Extraction of a Solution from a Model:**

A solution can be retrieved from a model by performing experiments on it, such as simulation, or by using mathematical analysis. If the data isn’t correct, no model will work well. The findings of experiments or hunches based on experience may provide this knowledge.

The data collection has a considerable impact on the model’s output. When an operation researcher defines his objective and model, he should not assume that he has accomplished his aim of solving the problem. If data collecting errors are to be minimized, the required data gathering takes time to prepare.

**Validating the Model and the Resultant Solution:**

As previously said, a model is never a perfect depiction of reality. It may, however, be beneficial in providing/predicting the effect of changes in control variables on overall system efficacy if appropriately defined and adjusted.

A model’s usefulness or utility is determined by how effectively it anticipates the effects of these changes. Sensitivity analysis is the term for this type of analysis. The solution’s utility or validity can be validated by comparing the outcomes obtained without using the solution to the results obtained after using it.

**Establishing Controls over the Solution:**

The operation researcher’s next step is to present his findings to management. It should be noted that he should define the circumstances in which the solution can be used.

He should also explain up any flaws if any, so that management is aware of the risks involved in using the model to generate results. As a result, he should additionally describe the parameters within which the model’s results are valid. He should also specify the circumstances in which the model will fail.

**Implementation of Solution:**

The final step in the operation research technique is to put the solutions developed in the previous steps into action. Though decision-making in operation research is scientific, its execution is fraught with behavioral difficulties. As a result, the implementing authority is responsible for resolving the behavioral difficulties. He must persuade not only workers but also supervisors, of the value of O.R.

The distance between the O.R. scientist and management can lead to huddles, thus the gap between the one who delivers a solution and the one who wants to use it must be bridged. To do so, both management and the O.R. scientist must play a positive role. A correctly executed solution obtained through the use of O.R. approaches improves working conditions while also gaining management support.

**Operation Research in Agriculture **

- W. Thornthwaite (1954) on Seabrook farm was one of the first, intriguing, and challenging implementations of operations research in agriculture. Yates (1949) produced a paper in the field of agriculture in the United Kingdom about the use of operational research. During WWII, the United Kingdom had to not only raise its food supply through greater food production but also cut costs on imports, which included items needed to boost agricultural production. In economics and agricultural economics, operations researchers have made substantial contributions. Samuelson (1952) recognized the value of modeling approaches and established a link between mathematical programming and economic equilibrium models. To put it another way, mathematical programming has become a useful tool for economic analysis (Samuelson 1952). Operations researchers developed the GAMS modeling approach to explain general equilibrium models for evaluating national improvement programs and to present the first-grain storage models to guard against shortages (Gustafson 1958 and Brooke
*et al.*1993). The microeconomic policy analysis models were developed by Hogan, W. W. (1975), and his research team called Project Independence Evaluation System (PIES).

**Economics and Operations Research Perspectives**

Economics and operations research are two distinct professions because economists and operations researchers have opposing viewpoints and interests. Economists are primarily engaged in qualitative economic analysis, while operations researchers are more concerned with supporting decision-making through strong computational and professional coordination. Economists are concerned in quantitative numbers and attempting to quantify the impact of individual assessment-based decisions, whereas econometricians are interested in computational challenges and estimate methods. Scarf (1973) developed an economic equilibrium model based on algorithmic calculations, which is completed by studying the sensitivity of the parameters Scarf and examining various circumstances, scrutinizing alternative tactics, and examining the sensitivity of the parameters Scarf. Inventory systems have been implemented by operations researchers and computer professionals, whereas economic academics have focused on the effect of inventories in the trade sequence rather than inventory difficulties. The operations research tool for facts and envelopment analysis, as well as estimating production functions using a separate technique with distinct assumptions and goals (Charnes*et al.* 1978). Wardrop (1952) developed an equilibrium model for economic longitudinal equilibrium on the basis of times and conditions that are directly related to equilibrium situations. The majority of the material is available in transportation journals using an operations research perspective, as (Beckmann *et al.*1956) highlighted and established the relationship with economic equilibrium. Nagurney (1993) described how to describe many types of equilibrium models for various situations.

**Mathematical Programming in Agriculture **

Farming differs from other sectors in that it is heavily influenced by natural variables such as soil, climate, and so on. Plants and animals cannot be treated in the same way that other industries’ products are. The fundamental characteristic of agriculture that distinguishes it from other businesses is the time lag; one must wait years for a return on his investment. The issues of marketing and storing agricultural products differ from those of other businesses due to their bulky and perishable nature. The differences in the nature of decision difficulties confronted by a farmer and a company executive are due to all of these and other agricultural features. The purpose of this research is to look into the suitability and applicability of operations research approaches to farming decision-making. This research is identified in the following topics:

**Mathematical Programming – Linear fractional functioning and Programming****Game Theory**

The desired solution can be computed using ‘Mathematical Programming’ if the objective function can be represented in terms of the mathematical model. As the name implies, mathematical programming, like other mathematical tools, is a mathematical approach with no economic content. We will simply look at the following features of linear programming:

- a) Characteristics of linear programming problems and linear programming’s general results (theorems).
- b) A basic overview of Linear Fractional Functional Programming.
- c) Network Analysis

The structure of a system and its objective function can be characterized using mathematical models, and the intended solution can be computed using techniques grouped under the scope of mathematical programming. Linear programming (integer and non-integer, price and resource variables, perturbation approaches, and so on) and non-linear programming (e.g., quadratic programming, concave and convex programming), as well as dynamic programming, are included.

**Concept of formulation of Linear Programming **

The objective function and constraints are the foundations of linear programming. The fundamental goal of deriving linear programming is to identify the most efficient set of extreme points of a set defined by the objective function. Every extreme point in a linear programming issue is a basic possible solution under the constraints. Similarly, every extreme point of the set of feasible solutions is a basic feasible solution.

It has been observed that the simplex approach takes fewer iterations than alternative techniques for particular LP problems. We used the LP model to allocate land to the four primary cereal crops in agriculture in this study. The Simplex Algorithm is used to find the solutions.

**Selection of Optimum Value:** The optimum value (Maximization) of Ct as varies over j will be one or more of the extreme points of n.

Maximize Z = + + ………… +

Subject to:+ +……………………+ ≤

+ +……………………+ ≤

. . …………………… . . . …………………… .

++ ≤ ≥ 0 andj= 1, 2… n

Can be written as

Max. Z =

Subject to

AX ≤ b and

X ≥ 0

Where, X represents the vector of the variable, while C and b are vectors of a known matrix of Coefficient. The expression to be maximized is called the objective function in this case, the equation AX≤ b is the constraint that specifies a convex polyhedral set over which the Objective function is to be optimized. The (,, ………, )are the unit returns for the coming from each production process ( , , ………., ).

In terms of Matrix

Max. Z = ∑

i = 1 to n

∑ ≤

j = 1 to n

= [ ] m x n

= [] m x n

= [] m x n

∑

i= 1 to n

**Application of Linear Programming in Agriculture **

Since its conception, linear programming has been employed in agriculture. In agriculture, linear programming has been utilized virtually from the beginning of time. Waugh applied this strategy to the subject of minimizing feed costs for dairy difficulties in 1951. Koopmans (1951) derived a production and resource use activity analysis. On the Choice of a Crop Rotation Plan was a paper written by Hildreth and Reiter in 1951. Heady and Candler (1958) employed linear programming methods and focused solely on agricultural applications. The article “Linear Programming and Farm Management Analysis” by Boles (1955) was published. The most common use of linear programming in agriculture has been in the field of feed-mixing, with the aim of decreasing feed costs. The use of linear programming for individual farmers is known as program planning, and it is widely utilized in Europe and Japan, as well as to a lesser extent in the United States. Barker (1964) conducted a study on the use of linear programming in farm management decision-making and concluded that linear programming can be useful in farmer decision-making by providing quantitative estimates of returns for specified alternatives and levels of resource use and that the larger the farm, the more alternatives an individual has and the more probable it is that the benefits of linear programming will exceed the cost. In addition to their implementation at the micro-level, i.e., cost minimization and profit maximization on a single farm, linear programming techniques have been used to solve problems in agricultural marketing and geographical analysis at the macro level. The spatial linear programming technique has been used to study interregional production and make modifications for important crops. In applied agricultural research, transportation models, assignment models, and simple linear programming models are also used.

**Linear Fractional Functional Programming **

Linear Fractional Functional Programming is applicable when the objective function is a ratio. There are many situations in agriculture, the ratio needs to be optimized like returns per hour of man labor, maximizing profits per dollar of investment, etc. Given below is a simple example of an application of linear fractional functional programming in agriculture. A farmer proposes to raise corn and wheat on 100 acres of land. Given below is the basic data for these two crops on his farm. We assume that all costs (except for labor requirements) are identical for both crops. We further assume that whether the farmer raises any crop or not, he has to devote four hours during the crop season to the maintenance and upkeep of the equipment such as tractor, combine, etc. The farmer has 300 hours of labor available for the crop season and he is interested in maximizing returns per hour of labor. Let be the acre under corn, Let be the acre under wheat. The problem is to find out as to how much area he should have under corn and how much under wheat. In other words, the decision variables are and we have to determine their desired values. After solving the equations, we get to the conclusion that farmers should raise corn on all 100 acres of land and not grow any wheat. The linear fractional functional programming technique holds great in its application to agriculture when the objective is to maximize returns per hour of family labor, hired labor, or both, or returns per dollar of capital invested.

**Application of Network Analysis**

Network analysis has been widely used in industries for planning, scheduling, and evaluation of projects but very applications have been made in the field of agriculture. There are many areas in agriculture where it can be used for advantage, such as if a farmer has one tractor are it can be used for plowing, bringing fertilizers, fertilizing, sowing, cultivating, irrigating, and even transporting the produce. But he cannot use it for buying fertilizers from the market and plowing at the same time. There can be only one path connecting all those activities that are to be performed by the tractor. Network analysis will help us in drawing schedules for the most efficient use of the tractor.

**Game Theory**

John Von Neumann and Oscar Morgenstern are considered to be the originator of game theory. They mentioned it in the book ‘Theory of Games and Economic Behaviour’. A game is a situation in which two or more participants take part in the pursuit of certain conflicting objectives.

In this case, some players may win by getting positive gain while others may lose. In the same way in a competitive market, two or more parties make decisions with conflicting interests,s and the action of one depends on the opponent’s task. Each and every opponent ‘acts in a rational way for resolving the conflict in their own favor. Game theory resolves this conflicting situation of business and military operations. This important technique of operation research provides a basis for determining under specified conditions, the particular strategy that will result in a maximum gain or minimum loss. Thus game theory may be defined as a body of knowledge that deals with the decision-making of two or more rational opponents in the condition of conflict and competition.

Of the several models for decision-making under uncertainty, only the model for maximizing expected utilities, the game-theoretic models, the naive or econometric models, and various precautions for uncertainty provide an objective rule for obtaining an implicit or an explicit goal. However, this study has shown that they suggest plans which farmers in various problem settings may wish to follow. Research and extension personnel may want to use the model to derive farmer recommendations. Uncertainty is the usual environment for agricultural decisions. Uncertainty is introduced by technical and technological changes, price variation, and unpredictable human action. Game-theoretic techniques may have considerable use in the whole farm planning. Usually, input-output coefficients used in linear programming or budgeting are averages. They are subject to variations. This variation may affect the profitableness of the whole farm planning. Some farmers want to be sure that income levels will not fall below some minimum or feasible level. Thus, a whole farm plan based on average input-output coefficients is not acceptable. However, farmers might accept a plan which assures maximum-minimum levels each year. Such a plan may be based on the input-output coefficient derived from a game against nature by application of the Wald criterion. The solution for a crop enterprise problem would suggest a plan which assures a minimum return. The minimum return may be regarded as the output coefficient. The input coefficient is given by the combination of variety, fertilizers, and cultural practices required for the crop plan which is selected. The particular crop is an activity to be included in linear programming analysis designed to plan the whole farm operation. Operations Research (OR) can be described as the discipline of applying advanced analytical methods to help to make better decisions and has been around in the agricultural management sector since the 50s, approaching decision problems that range from more strategic level planning to farm operation issues and integrated supply chain. The presence of OR in Agriculture Management applications is already extensive but the potential for development is huge in times where resources are becoming increasingly scarce and more has to be done with less, in a sustainable way.

**APPLICATION OF OPERATIONS RESEARCH IN THE OPTIMIZATION OF AGRICULTURAL PRODUCTION**

In recent years, people in agricultural planning have come to realize the importance of increased regulation in developing agricultural countries and since computers have become more available, increased agricultural production regulations have become possible. As a result, the question to be solved with the help of the research paper was if various resources and technological equipment are fixed can the production profit increase through scientific regulation and planning? There was a) Qualitative and b) Quantitative Analysis to find the answer which consisted of Modeling:

**Variables:**Let denote the portion of the arable land to be used to cultivate the ith crop combination And Let denote the amount of livestock to be raised on the land.**Objective function:**The net income as the objective function. The coefficient of variable in the objective function is equal to the net income per year. Similarly the coefficient of equal to the net income from livestock per year.**Constraints:**Constraints come from social demand, state purchase quotas, peasants’ capacity of storing grain, and the quantity for the market, the ability to process factories, productive capacity, and inherent law in the agricultural activities mainly.**Sensitivity analysis:**To understand the influence of market prices on the structure of fanning and stock raising, the application of sensitivity analysis to the coefficients of the objective function in the model established for the Changing County. After the analysis, we found that the fluctuation of prices is sensitive to some crop combinations and almost unresponsive to others. This result is important when revising the production structure according to the change of market prices.

**LIMITING FACTORS FOR APPLICATION OF OR IN AGRICULTURE**

- Natural Calamities: OR models cannot take into account the damage done to crops by natural calamities like drought, floods, etc. Hence the application of such techniques can only be done under certain assumptions which limit us to finding the perfect solution.
- Changes in Weather Conditions: Changes in weather conditions have a direct impact upon the production of crops and it is a factor that cannot be controlled by anyone which makes it a limitation as to any effect of weather on production cannot be taken under consideration while applying OR techniques.
- Hikes in Fuel Prices: In OR models where the technique is used to find the perfect transportation or to minimize the cost of transportation the changes in fuel prices which in turn leads to an increase in the cost of transportation become a limiting factor.
- Demand for Products: Agriculture is a highly volatile market that is led by the demand for crops and manipulation of such market is an easy task that results in hiked prices or very low prices which then discourages the production of crops.
- Hence, is a limiting factor while optimizing the production and allocation of resources.
- Changes in Government Subsidies and Policies: Subsidies offered to farmers and policies of certain states may change affecting the profit of the farmers and production of crops, another factor limiting the application of OR as it is dependent on the human behavior and decision-making process.
- Strikes and Riots: As discussed the human behavior is almost unpredictable and highly volatile. Strikes and riots are the outcome of human behavior and affect the transportation of crops and production as well as being an important factor that limits the application of OR.

**Conclusion**

In the agricultural industry, operations research is extremely important, especially when it comes to decision-making. Agriculture, as we indicated in our literature review, plays a vital part in the economies of many countries, but it is now under threat from issues such as climate change, resource scarcity, soil composition changes, and so on. Many of the tests undertaken in various locations have yielded favorable results, and these techniques, if implemented on a wide scale, can result in huge improvements in the sector of agriculture. As a result, resources will be better utilized and waste will be reduced, enhancing process efficiency and profitability. If we continue to destroy our resources at this rate, there will be little left for future generations. The key to a successful future is sustainable development, and Operations Research can help us get there. In order to boost growth, use fewer inputs, and improve the quality of the food, new inventions, better procedures, and improved facilities can be developed and used in farming. We urge individuals to implement these behaviors as soon as possible in order to ensure a bright future for ourselves and future generations.

**Reference:**

Ackoff, R. L. (1961). Progress in. Operations Research. New York: John Wiley & Sons, 1, 321-31.

Barna, T. (1946). Theory of games and economic behavior.

Bajuri, N. H., &Jadoon, I. A. (2011). IzazUllah Khan. International Journal of Business and Social Science, 2(23).

Beckmann, M., McGuire, C. B., &Winsten, C. B. (1956). Studies in the Economics of Transportation, Yale University Press. New Haven, Connecticut, USA.

Blackett, P. M. S. (1948). Operational research. Advancement of science, *5*(17), 26-38.

Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European journal of operational research, 2(6), 429-444.

Nagurney, A. (1993). Network Economics: A Variational Inequality Problem.

Yates, F. (1950). Agriculture, sampling and operational research. Agriculture, sampling and operational research.

Gass, S. I., & Assad, A. A. (2005). An annotated timeline of operations research: An informal history (Vol. 75). Springer Science & Business Media.

Gustafson, R. L. (1958). Implications of recent research on optimal storage rules. Journal of Farm Economics, 40(2), 290-300.

Samuelson, P. A. (1952). Spatial price equilibrium and linear programming. The American economic review, 42(3), 283-303.

Scarf, H. (1973). The computation of economic equilibria (in collaboration with T. Hansen; Yale University Press, New Haven, CT).

Retrieved from https://www.scribd.com/document/285582570/Applications-of-Operations-Research-Techniques-in-Agriculture.

Hogan, W. W. (1975). Energy policy models for project independence. Computers & Operations Research, 2(3-4), 251-271.

Wardrop, J. G., & Whitehead, J. I. (1952). Correspondence. some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, 1(5), 767-768.

**Written by Mahamudul Hasan Millat**

**Research Scholar **

**Statistics Discipline**

**Science, Engineering & Technology School**

**Secretary, Rotaract Club of Khulna University **

**Khulna University, Khulna-9208, Bangladesh **

**Email: [email protected] **

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