Aim: To optimize the production of vegetable crops on a fixed portion of land and to maximize the revenue generation.
Mathematical principle: Development of a Linear programming problem with a specific objective function with a set of linear constraints
Introduction: Problems are generally faced by farmers like what to plant, how much to plant and where to plant. This study proposes a linear programming crop mix model. Given the limited resources l;ike budget and land acerage, the crop mix model is formulated and transformed into a multi period linear programming problem. The objective is to maximize the total returns at the end of the planning horizon.
Basic Assumptions:
Since it is not possible to incorporate all the factors affecting crop planning into the linear programming problem, to ease the formulation, the following assumptions were made:
1. The availability of resources like land, water labour etc. do not change.The level of technology do not change,
2. The land is assumed to be equally suitable for all the selected crops.
3. The crop mix depended only on the type of crops that were grown and not on the method of cultivation that was used,
4. The crop prices do no change during the planning period.
Notations and variables.
Let us consider a piece of land on which different selected combinations of a single harvest crops of different maturity age may be cultivated.
Let A= Area(in acres) of the farmland available for mix cropping scheme.
T= No of time periods in the planning period.
K= Capital or budget available.
S= Monthly administrative expenditure required to manage the farm which included wages, salaries and other type of expenses. For a planning horizon of T periods the decision variable X(m,c,t) can be assumed in general to denote the acerage allocated to crop C of maturity age m planted in the period t=1,2,3,……..T-m+1. This will generate T-m+1 decision variables for the crop C. Without the loss of generality and for other illustrative purposes crops of maturity ages of 1,2 and 3 time period were considered. It was assumed that there were three crops, Crop J with a maturity age of one month(like spinach), Crop k with a maturity age of 2 months(like lady finger, lettuce or cucumber) and crop L with a maturity age of 3 months(like French beans and long beans). Thus the following decision variables were obtained.Land allocations to crops of maturity age of 1,2 or 3 time periods for a planning horizon of T=12 time periods.
Time periods in the planning horizon.
Crop types t=1 t=2 t=3 t=4 t=5 t=6
J (1,J,1) (1,J,2) (1,J,3) (1,J,4) (1,J,5) (1,J,6)
K (2,K,1) (2,K,3) (2,K,5)
K (2,K,2) (2,K,4) (2,K,6)
L (3,L,1) (3,L,4)
L (3,L,2) (3,L,5)
L (3,L,3) (3,L,6)
Time periods in the planning horizon.
Crop types t=7 t=8 t=9 t=10 t=11 t=12
J (1,J,7) (1,J,8) (1,J,9) (1,J,10) (1,J,11) (1,J,12)
K (2,K,7) (2,K,9) (2,K,11)
K (2,K,8) (2,K,10)
L (3,L,7) (3,L,10)
L (3,L,8) (3,L,11)
L (3,L,9)
The linear programming model.
The linear programming model for the crop mix model can therefore be presented as follows.
Objective function: The objective of this model is to maximize the total contributions or revenues accumulated at the end of the whole planning period.
(Maximize) Z= ΣRj X(1,J,T) + ΣRk X(2,K,T-1)+ΣRLX(3,L,T-2)+Wt
Where ΣRj X(1,J,T) is the revenue generated by all crops of type J planted in the period T
ΣRkX(1,k,T-1) is the revenue generated by all crops of type J planted in the period T-1.
ΣRLX(1,L,T-2) is the revenue generated by all crops of type J planted in the period T-2.
Wt is the cash in hand in the period T.
System of constraints. The total acres of land under cultivation at any time period must not exceed the total available land of A acres therefore-
For t=1: ΣX(I,J,1)+ ΣX(I,k,1)+ ΣX(I,L,1)≤A
For t=2 ΣX(I,J,2)+ (ΣX(2,K,1)+ X(2,K,2))+ ΣX(3,L,1)+ ΣX(3,L,2)≤A
:
:
:
:
For t=T-1 ΣX(I,J,t-1)+ (ΣX(2,K,T-1)+ X(2,K,T-2))+ ΣX(3,L,T-2)+ ΣX(3,L,T-3)≤A
Other equations. All revenue accumulated at time t and cash in hand at the time t-1 can be utilized for crop production cost, administrative requirements and cash in hand thus:
ΣRj X(1,J,T-1) + ΣRk X(2,K,T-2)+ΣRLX(3,L,T-3)+Wt-1= ΣCj X(1,J,T-1) + ΣCk X(2,K,T-2)+ΣCLX(3,L,T-3)+Wt+S.
However for t=1 we had an initial capital K
Therefore- ΣCj X(1,J,1) + ΣCk X(2,K,1)+ΣCLX(3,L,1)+Wt = K-S
Non negativity constraints. All the decision variables are non negative.
Using the above data the crop mix model for a planning period of
12 months was solved using a Linear Programming Software called LINDO
(Linear interface discrete optimizer) and the results analyzed.
12 months was solved using a Linear Programming Software called LINDO
(Linear interface discrete optimizer) and the results analyzed.
Conclusion: In this study a vegetable crop planning scheme was formulated as an LPP and later solved with the help of a software called LINDO. The results indicate promising returns even for a relatively short period of 12 months and ifm properly implemented will enhance farm income and provide beneficial contribution to the farming societies.
The results of this study are short run in nature and subject to the assumptions and limitations. However it opens up new avenues for further explorations. A longer planning period of several years might be more appropriate if a sufficiently large capital is available initially. A farm size of a few hundred acres can shift farming into a very profitable commercial activity. Incorporating more multi harvesting crops into this model can provide promising alternatives to the farmers.
Good Project.
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