Double summation nminimize

Question

Please help me...

S = 100;
hv[1] = 3.5; hv[2] = 4;
c[1] = 10; c[2] = 50;
Cv[1] = 2.4; Cv[2] = 3;
Cf[1] = 3; Cf[2] = 1.2;
s[1] = 0.3; s[2] = 0.1;
th[1] = 0.01; th[2] = 0.03;
A[1, 1] = 3; A[2, 1] = 2; A[1, 2] = 1; A[2, 2] = 1.1;
d[1, 1] = 1850; d[2, 1] = 2000; d[1, 2] = 2100; d[2, 2] = 1300;
k[1, 1] = d[1, 1]*0.05 ; k[1, 2] = d[1, 2]*0.05 ; 
k[2, 1] = d[2, 1]*0.05 ; k[2, 2] = d[2, 2]*0.05 ;
so[1, 1] = 3.5; so[2, 1] = 1.5; so[1, 2] = 0.3; so[2, 2] = 0.6;
t[1, 1] = 1; t[2, 1] = 1; t[1, 2] = 1; t[2, 2] = 1;
e[1, 1] = 2; e[2, 1] = 2; e[1, 2] = 2; e[2, 2] = 2;
Y[1, 1] = 3; Y[2, 1] = 3; Y[1, 2] = 3; Y[2, 2] = 3;
hb[1, 1] = 2.4; hb[2, 1] = 1.3; hb[1, 2] = 4.5; hb[2, 2] = 3.6;
W[1] = 500; W[2] = 1000;
w[1] = 12; w[2] = 2; w[3] = 3;
P[1] = 8000; P[2] = 10000; P[3] = 6000;

NMinimize[{Sum[(S*de[i])/(m[i]*Q[i])+(hv[i]*Q[i])/2*((m[i]-1)*(1-de[i]/P[i])
+de[i]/P[i])+de[i]/(m[i]*Q[i])*(m[i]*c[i]+m[i]*Q[i]*Cv[i]+Cf[i])
+(m[i]*s[i]*de[i]*Q[i]*th[i])/2, {i, 1, 2}]
+Sum[(A[i, j]*d[i, j])/q[i, j]+(hb[i, j]*(1 - k[i, j])^2*q[i, j])/2
+(k[i, j]^2*so[i, j]*q[i, j])/2+(m[i]*t[i, j]*d[i, j])/q[i, j]
+(m[i]*e[i, j]*d[i, j])/q[i, j] + Y[i, j]*d[i, j], {i,1, 2}, {j, 1, 2}],
Table[{Table[1000 > q[i, j] > 0, {j, 1, 2}],Element[m[i], Integers], 1000 > m[i] >= 1,
Q[i] == Sum[q[i, j], {j, 2}] > 0, de[i] == Sum[d[i, j], {j, 2}],
Sum[(q[i, 1]*w[i]/m[i]), {i, 2}] <= W[1],Sum[(q[i, 2]*w[i]/m[i]), {i, 2}] <= W[2]}, {i, 1, 2}]}, 
Join[Flatten@{Table[{m[i], Table[q[i, j], {j, 1, 2}]}, {i, 1,2}]}, {Q[1], Q[2], de[1], de[2]}], MaxIterations -> 100]

I made this code for solving the problem. When i run, i got a message.

NMinimize::lvar: Variables {{{m[1],{q[1,1],q[1,2]}},{m[2],{q[2,1],q[2,2]}}}} should be a list of variables, with each element being a variable, or a list containing a variable and lower and upper bounds for the starting region for that variable. >>

I followed the tutorial of Nminimize, however, it happened. What am i wrong with this code?

Answer 1

If you see the Mathematica message, it is clear that the variable list is incomplete. It should also include Q[1],Q[2], de[1],de[2], P[1] and P[2]. The syntax therefore becomes:

Timing[NMinimize[{Sum[(S*de[i])/(m[i]*Q[i]) + (hv[i]*Q[i])/
        2*((m[i] - 1)*(1 - de[i]/P[i]) + de[i]/P[i]) + 
      de[i]/(m[i]*Q[i])*(m[i]*c[i] + m[i]*Q[i]*Cv[i] + Cf[i]) + (m[i]*
         s[i]*de[i]*Q[i]*th[i])/2, {i, 1, 2}] + 
    Sum[(A[i, j]*d[i, j])/
       q[i, j] + (hb[i, j]*(1 - k[i, j])^2*q[i, j])/
       2 + (k[i, j]^2*so[i, j]*q[i, j])/2 + (m[i]*t[i, j]*d[i, j])/
       q[i, j] + (m[i]*e[i, j]*d[i, j])/q[i, j] + Y[i, j]*d[i, j], {i,
       1, 2}, {j, 1, 2}], 
   Table[{Table[q[i, j] > 0, {j, 1, 2}], Element[m[i], Integers], 
     m[i] > 0, Q[i] == Sum[q[i, j], {j, 2}], 
     de[i] == Sum[d[i, j], {j, 2}]}, {i, 1, 2}]}, 
  Join[Flatten@{Table[{m[i], Table[q[i, j], {j, 1, 2}]}, {i, 1, 
       2}]}, {Q[1], Q[2], de[1], de[2], P[1], P[2]}]]]

Now we get an answer from NMinimize, but with a warning that at a specific point it encountered 1/0. Indeed, if you substitute the values given out by the message into the function we get 1/0. You have to therefore check if this is a valid answer, or if you need to add more constraints to your problem to skip this singularity.

Hope this helps. (I do not consider this to be complete answer to this question, but comments does not allow for proper code formatting. So I put this in as an answer)