The document describes a linear programming problem to maximize the total net return for three kibbutzim (Israeli communal farms) by allocating crops across the farms under constraints of available land, water allocation, and crop quotas. Crops include sugar beets, cotton, and sorghum. The problem is formulated as a linear program to determine the optimal number of acres for each crop at each kibbutz while satisfying the restrictions.

1. OPTIMIZATION TECHNIQUES
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2. Linear Programming Problem – Regional Planning
 The Southern Confederation of Kibbutzim is a group of three
kibbutzim (communal farming communities in Israel. Overall
planning for this group is done in its Coordinating Technical
Office. This Office currently is planning agricultural production
for the coming year.
 The agricultural output of each kibbutz is limited by both the
amount of available irrigable land and the quantity of water
allocated for irrigation by the Water Commissioner . These
data are given as follows
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3. Problem Description contd…
Kibbutz Usable Land(Acres) Water Allocation(Acre
Feet)
1 400 600
2 600 800
3 300 375
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4. Problem Description contd…
 The crops suited for this region include sugar beets , cotton ,
and sorghum. These crops differ primarily in their expected
net return per acre and their consumption of water.
 In addition , the Ministry of Agriculture has set a maximum
quota for the total acreage that can be devoted to each of
these crops by the Southern Confederation of Kibbutzim , as
follows
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5. Problem Description contd…
Crop Maximum Water Net Return($
Quota(Acres) Consumption(Ac /Acre)
re Feet/Acre)
Sugar beets 600 3 1,000
Cotton 500 2 750
Sorghum 325 1 250
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6. Problem Description contd…
 Because of the limited water available for irrigation ,the Southern
Confederation of Kibbutzim will not be able to use all its irrigable land for
planning crops.
 To ensure equity between the three kibbutzim ,it has been agreed that
every kibbutz will plant the same proportion of its available irrigable land.
 Any combination of the crops may be grown at any of the kibbutzim.
 The job facing the Coordinating Technical Office is to plan how many acres
to devote to each crop at the respective kibbutzim while satisfying the
given restrictions.
 The objective is to maximize the total net return to the Southern
confederation of kibbutzim as a whole.
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7. Formulation as a Linear Programming Problem
Kibbutz
Allocation (Acres)
Crop 1 2 3
Sugar Beets X1 X2 X3
Cotton X4 X5 X6
Sorghum X7 X8 x9
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8. Formulation as a Linear Programming Problem
Maximize Z = 1000 (x1 + x2 + x3 )+ 750( x4 + x5 + x6)+ 250( x7 + x8 + x9)
Subject to following constraints :
1. Usable land for each kibbutz :
x1+x4+x7 <= 400;
x2+x5+x8 <= 600;
x3+x6+x9 <= 300;
2. Water allocation for each kibbutz :
3 x1 + 2 x4 + x7 <= 600;
3 x2 + 2 x5 + x8 <= 800;
3 x3 + 2 x6 + x9 <= 375;
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9. Formulation contd….
3. Total acreage for each crop :
x1+x2+x3 <= 600;
x4+x5+x6 <= 500;
x7+x8+x9 <= 325;
4. Equal proportion of land planted :
3 (x1+x4+x7)-2 (x2+x5+x8) = 0;
(x2+x5+x8)-2 (x3+x6+x9) = 0;
4 (x3+x6+x9)-3 (x1+x4+x7) = 0;
5. Non negativity :
X j >= 0 ,for j = 1, 2, ………,9
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10. Standard form of LPP
1. Usable land for each kibbutz :
x1+x4+x7 +s1 = 400;
x2+x5+x8 +s2= 600;
x3+x6+x9 +s3= 300;
2. Water allocation for each kibbutz :
3 x1 + 2 x4 + x7 +s4= 600;
3 x2 + 2 x5 + x8 +s5= 800;
3 x3 + 2 x6 + x9 +s6= 375;
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11. Standard form of LPP
3. Total acreage for each crop :
x1+x2+x3 +s7= 600;
x4+x5+x6 +s8= 500;
x7+x8+x9 +s9= 325;
4. Equal proportion of land planted :
3 (x1+x4+x7)-2 (x2+x5+x8) = 0;
(x2+x5+x8)-2 (x3+x6+x9) = 0;
4 (x3+x6+x9)-3 (x1+x4+x7) = 0;
5. Non negativity :
X j >= 0 ,for j = 1, 2, ………,9
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12. Thank you
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