Optimal crop distribution in Vojvodina

Instructor: Dr. Lužanin Zorana

Aleksić Tatjana Dénes Attila Pap Zoltan Račić Sanja Radovanović Dragica Tomašević Jelena Vla Katarina

Introduction

• Find optimal crop distribution subject to:

– Maximization of the total gross margin– Minimization of the total risk

Constraints

• Land constraint

• Crop rotation

• Budget limit

Start model

• Gross margin maximization

– Subject to land constraint – Rotational constraints ?!

Discrete model

• Land is divided into m parcels

• There is only 1 crop on each parcel

• Includes some uncertainty for yield

• Rotational constraint

• Budget constraint

Input

Wheat Maize Sunflower Soybean Sugar beet

Yield (average)

y1 y2 y3 y4 y5

Selling price c1 c2 c3 c4 c5Costs t1 t2 t3 t4 t5Parity p1 p2 p3 p4 p5

Main idea• Data for crop distribution for last 4 years

in terms of 0-1 3D matrices

)( ijka) (parcels m1,...,k

) (years 1,...,5j) (crops ,51,i

, 0, 1

ijka if in j th year i th crop was planted on the k th parcel

otherwise

To generate ai,j,k for j = 5 (for year 2005) we have to respect:

• Rotational constraints

If ai,j,k = 1 than ai,j+1,k = 0

If a3,j-1,k=1 or a3,j-2,k=1 or a3,j-3,k=1 than a4,j,k=0

If a4,j-1,k=1 or a4,j-2,k=1 or a4,j-3,k=1 than a3,j,k=0

If a5,j-1,k=1 or a5,j-2,k=1 or a5,j-3,k=1 or a5,j-4,k=1 than a5,j,k=0,

i=1,…,5 j=1,…,5 k=1,…,m

• Overlapping constraints

If ai0,j,k=1 than ai,j,k=0 for i<>i0 i=1,…,5 j=1,…,5 k=1,…,m

Algorithm - idea

• Program eliminates scenarios which don’t satisfy constraints

• Calculates objective function for every feasible solution

• Output is optimal solution

Output

• Optimal distribution of crops

• Profit

• Graphical presentation of crop distribution

Mathematica\MatrixUpFill.nb

Mathematica\Matrix2_paritet.nb

Modification of Algorithm

• Algorithm which calculates optimal distribution for 2 years

Mathematica\Matrix2.nb

Mathematica\Model for two years.nb

Conclusion

• Model gives optimal crop distribution, s.t. rotational limits (overlapping)

• Some tries for including uncertainty (price, yield) without stochastic

Stochastic model

• Includes risk

• Yield and price are stochastic

• Optimal solution respect to risk and profit

• Use utility function

Model

• µπi – expected profit

• σπi – standard deviation of profit

• σij – covariance of profit

• R – measure of risk

• U – utility function

5,,10 ixi

5,,1 irotNx ii

5

1

..i

i Nxts

ijijjiiiii

xxxx

Rx

i

22

2max

Model

Open questions

• Extend constraints for agricultural policy

• When to buy mechanization?

• Involving more stochastic

• Price and yield distribution?

• Measuring of risk

• Utility function?

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