The Variable-Rate Decision for Multiple Inputs with Multiple Management Zones

(10)    X^*i1 =[((P2fi)+(2diP1)/Py)-(2bidi+cifi)]/(fi²-4ciei)

(11)    X^*2i =[((P1fi)+(2ciP2)/Py)-(2cidi+bifi)]/(fi²-4ciei)

Substitute these optimal input rates into equations (7, 8, and 9), substitute the resulting optimal
yields into equation (2) to determine the net return for management zone i, do the same for each
management zone, and weight these net returns based on λ_i to get R^*_VRT . For URT, substitute
equations (7), (8), and (9) into equation (3) and set X1j= X2j = X3j = Xuj (j=1 and 2). Set the
derivative of the resulting field-average yield response function equal to P_j /P_Y and solve for
Xuj^*. Substitute these optimal uniform input application rates into equation (3) and substitute the
resulting optimal field-average yield into equation (4) to get R ^*_URT . Calculation of RVRT* is
straight forward from equation (5).

Illustrative Example

To illustrate the concepts presented above, assume hypothetical fields suited to cotton
production can be classified into three management zones and that the following quadratic
functions represent cotton yield response to fertilizer nitrogen and irrigation plus initial moisture
(W) for the management zones.

(12)  Y₁ = 233.72+23.65*W₁ -0.182*W₁² +0.439*N₁ -0.0033*N₁² +0.021*W₁ *N₁

(13)  Y₂ = -1103.6+118.35*W₂ -1.63*W₂² +2.85*N₂ -0.004*N₂² -0.046*W₂ *N₂

(14)  Y₃ =-170.93+32.45*W₃ -0.022*W₃² +3.74*N₃ -0.011*N₃² +0.022*W₃ *N₃

where Y1, Y2, and Y3 are cotton lint yields (lb/acre); W1, W2, and W3 are the amounts of water
applied plus 5 inches of available preplant moisture plus 1 inch of rainfall (acre-inches); N1, N2,
and N3 are nitrogen application rates (lb/acre); and the subscripts represent the three management

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