LEXICOGRAPHIC GOAL PROGRAMMING APPROACH FOR THE OPTIMIZATION OF CROPPING PATTERN IN YASIN REGION GILGIT, PAKISTAN

The prosperity of the agricultural sector is crucial not only for the national economy but also for regional development. For the purpose of agricultural planning and decision making, mathematical programming models are widely used. This paper aims to formulate, apply and evaluate a Lexicographic Goal Programming (LGP) model for the best allocation of land under cultivation for the major crops of central Yasin. It is an extension of the research work [1] which was conducted in this area to maximize the profit of the three crops using Linear Programming (LP). Using the LP model the target was achieved but all the land was allocated for alfalfa being the most beneficial crop among the three. But wheat is a basic need which must be cultivated to a specific amount. Considering this problem we extended the previous work by implementing the LGP approach to satisfy multiple goals such as the allocation of land for wheat, maximization of profit and minimization of capital in the agricultural activity. The LGP model is solved using the Simplex method. The LGP model provided the most satisfactory set of allocations possible and satisfied all three goals. It gave optimum profit of Rs.170003 and capital is minimized to Rs. 73687.678. Compared with the farmer’s plan, the profit of the LGP model increased by 49.39% but compared with the LP model, the profit decreased by 13.11%. Likewise, the total crop production expenditure increased by 4.97% and 1.39% respectively in comparison with the farmer’s plan and LP model.


INTRODUCTION
Agricultural planning problems are of great importance from both social and economic points of view.The available agricultural land and resources are reducing day by day because of the growing population.Due to these limitations, optimal use of land and agricultural resources is needed to optimize.This study aims to apply an LGP model for the allocation of land for the major crops of central Yasin to get an optimal solution in the context of both objectives and constraints which give us an optimal net return by maximizing the profit and minimizing the crop production cost.
Also, the basic requirements of the wheat crop will be fulfilled.

THEORETICAL FRAMEWORK AND MODEL
LGP is one of the oldest multi-criteria decisionmaking techniques and the most frequently used method in agricultural planning that aims to minimize deviations from the desired target to achieve the goals.The LGP technique is employed for the study and the data used is secondary data.This data was used by Haq et al.
[1] to apply the Linear programming model in the agricultural sector to maximize the profit of the major crops.

Study area
The research area is Central Yasin.Yasin is a valley situated in the Ghizer district which is the northernmost part of Gilgit Baltistan, Pakistan.
Climatically it is a colder region.Most of the people in the study area depend on agriculture for their livelihood.Different vegetables like potatoes, tomatoes, onions, and carrots are grown there.But the major crops which are grown largely are wheat, maize and alfalfa.Farming is their occupation therefore there is an imminent need to formulate an efficient cropping pattern in that region to meet the overall objectives based on the availability of agricultural resources.

2.2: LGP Model
The method of formulating the LGP model is similar to that of Linear Programming.
where Z is the objective function.

2.2.1: The Goals
The goals for the LGP problem with their priority level are described in Table 1.

Fertilizer Requirement
The amount of fertilizer bags used for wheat crops is 28 bags, 25 bags for maize and 32 bags for alfalfa for one season.The fertilizer constraint is given as;

Area Constraint
The total land available for agricultural purposes is 2.338 acres.The land used for the cultivation of all crops must not exceed the total land available.The constraint can be written as; To meet the people's food requirement some area of the land should be kept for wheat cultivation that is; The net profit obtained from wheat, maize and alfalfa is Rs. 33375,Rs. 30423 and Rs. 83676 respectively.The profit constraint can be written as;

2.2.4: LGP Model for the Problem
The lexicographic goal programming model formulated for the current problem is given as; The coefficients of the above GP model are summarized in Table 2.  3 and   4. Graphical representation of tables 3 and 4 is represented in figures 1 and 2 respectively.LGP best allocates the land giving maximum profit with the lowest cost.

REFERENCES
Mathematical programming models are widely used in agricultural planning.Agricultural problems generally contain multiple and conflicting objectives or goals such as minimization of cost, minimization of labor expenditure, increasing crop yields, and maximization of profits [2].Farmers use their past experiences for agricultural planning but mathematical programming is the best tool that can be employed for this purpose.LGP appeared as the most popular method for multi-objective decision-making problems and is used widely in multi-objective decisionmaking situations [3].LGP was first introduced by Charnes and Cooper [4].The basic idea used in Goal Programming is to change multiple objectives into a single goal [5].LGP introduces the desired target of each goal and optimizes the deviation from these goals to reach the desirable solution.The unwanted deviations are measured using positive and negative deviations for each of the goals [6].The fame of LGP and its use in decision-making policies, objects to optimize farming land and other agricultural resources [7].There is a range of literature on resource allocation and decision making in agricultural sector using LGP approach.Joolaie et al. [8] used Fuzzy Goal programming (FGP) technique to determine an optimal cropping pattern in North Iran by considering economic, social and environmental goals.A FGP approach was used by Mohammadian & Heydari [9] to determine an optimal cropping pattern for the selected crops of Mazadaran province under three different scenarios.LGP approach is applied to tea industry in Barak Valley of Assam India by Sinha & Sen [10].They presented six different plans of GP approach.The model is developed to maximize the production quantity, maximize the profit, demand and to minimize the expenditure and processing time of machines considering the environmental conditions.Jafari et al. [11] developed an optimal model for the rice farm in tackled by Vivekanandan et al. [12].They formulated a GP model for the optimization of cropping patterns for the Barna command area.The objectives were maximization of net return and maximization of protein and calories.To give a better cropping plan Rezayi & Mardani [13] used the FGP approach for the agricultural planning of the Atrak Watershed Agricultural development plan.The three objectives or goals considered in this model were maximization of profit, employment maximization and minimization of erosion under 88 constraints.Hassan & Sahrin [14] developed a GP model to determine the rate of mixed fertilizers and leaf fertilizers to produce the best quality pineapple.Haq et al. [15] conducted a study in District Hunza intending to explore the most beneficial cropping pattern for the three major crops wheat, potato and alfalfa grown the region and to maximize the net profit per year.The obtained results show that profit increased by 10% as compared with former's plane.

Fig. 1 .
Fig. 1.Comparison of Land allocation Fig.2.Comparison of Profit and Cost

Table 1 :
Description of goals

Table 2 :
Description of resources

Table 3 :
Comparison of land allocation of GP model with LP and farmer's plan

Table 4 :
Comparison of profit and capital of GP model with LP and farmer's plan