Linear programming

Nonlinear & Inequality

#Specify the function to be maximized. By default constrOptim performs minimization but 
#the problem can be maximized by setting the fnscale to -1
LP_sol=function(X){
  Y_R=5625-2897*(0.966^X[1])-2728   #Rice N response function
  Y_M=3159-1191*(0.976^X[2])-1968   #Maize N response function
  Y_B=1415-715*(0.950^X[3])-700     #Bean N response function
  Y_FM=2100-923*(0.944^X[4])-1177    #Finger millet N response function
  Y_R1=5665-828*(0.871^X[5])-4837     #Rice P response function
  Y_M1=4474-770*(0.898^X[6])-3704     #Maize P response function
  Y_B1=1138-263*(0.848^X[7])-875      #Bean P response function
  Y_FM1=2101-537*(0.798^X[8])-1564    #Finger millet P response function
  Y_PP=2538-487*(0.758^X[9])-487     #Pigeon Pea P response function
  Y_M2=2502-251*(0.94^X[10])-2251     #Maize K response function
  Y_PP1=2535-127*(0.666^X[11])-2408   #Pigeon Pea K response function
  
  P_R=700;P_M=700;P_B=1000;P_FM=700;P_PP=1500 #Expected commodity prices
  
  P_N=1100; P_P=1200;P_K=1200; #Nutrient cost computed from fertilizer prices per Kg
  
  
  P=(Y_R*P_R-X[1]*P_N)+(Y_M*P_M-X[2]*P_N)+(Y_B*P_B-X[3]*P_N)+(Y_FM*P_FM-X[4]*P_N)+
    (Y_R1*P_R-X[5]*P_P)+(Y_M1*P_M-X[6]*P_P)+(Y_B1*P_B-X[7]*P_P)+(Y_FM1*P_FM-X[8]*P_P)+(Y_PP*P_PP-X[9]*P_P)+
    (Y_M2*P_M-X[10]*P_K)+(Y_PP1*P_P-X[11]*P_K)   #Net return to fertilizer use function
  return(P)
}
# Assume a budget of TSH 500000,the cost of using 1 kg urea, TSP and KCl to be 1000, 1200, 1200 respectively and 
#expected commodity prices per Kg for Rice, beans, finger millet, pigeon pea and maize to be 700, 1000, 700, 1500 and 700 respectively
#Inequality constraints can be stated as:
#1100*X1+1100*X2+1100*X3+1100*X4+1200*X5+1200*X6+1200*X7+1200*X8+1200*X9+1200*X10+1200*X11<=500000
#X1>=0;X2>=0;X3>=0;X4>=0;X5>=0;X6>=0;X7>=0;X8>=0;X9>=0;X10>=0;X11>=0;X12>=0
#All X should be within 0 to 150
#Constraints need to be specified in the form Ax-b>=0
A=matrix(c(-1100,-1100,-1100,-1100,-1200,-1200,-1200,-1200,-1200,-1200,-1200,
           1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,
           0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,
           0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,
           0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,
           0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,
           0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,
           0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,
           0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,
           0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,
           0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,
           0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1),ncol=11,byrow=T)
B=c(-100000,0,-150,0,-150,0,-150,0,-150,0,-150,0,-150,0,-150,0,-150,0,-150,0,-150,0,-150)
#Initial guesses of the X values to be maximized
xinit=c(1,1,1,1,1,1,1,1,1,1,1)
# Nelder-Mead method is used to perform the maximization, other options exist e.g. BFGS
xan=constrOptim(theta=xinit, f=LP_sol, grad=NULL, ui=A, ci=B, control=list(fnscale=-1),method = "Nelder-Mead")
#The optimized solution is accessed as below
xan$par