I believe that binary variables are more efficient than Or constraints for this sort of problem, when done by linear programming. For example, the linear programming approach to Sudoku has 9 binary variables for each cell

this looks it might be useful. it's a staff scheduling problem done with LP

https://www.youtube.com/watch?v=F9UKIN16HA0

Diego, no the order is unknown. If the order was known, the NMinimize expression that Okkes Dulgerci was using would return pretty quickly.

Frank, that sounds like a good idea. I don't know exactly how to implement it. But the LinearProgramming function does allow Integers, and it has a better chance to deal with all of the possibilities with your idea. My only reluctance is that the real number algorithms are more reliable than the discrete ones. (as far as I know)

Hi Todd, Thanks again for your reply. It solves for 3 trucks and 2 port

In[2]:= NMinimize[{total, 
  t11 >= 0 && t12 >= 0 && t21 >= 0 && t22 >= 0 && t31 >= 0 && 
   t32 >= 0 &&
   (t11 + n[[1, 1]] <= t12 || 
     t12 + n[[1, 2]] <= t11) && (t21 + n[[2, 1]] <= t22 || 
     t22 + n[[2, 2]] <= t21) &&
   (t31 + n[[3, 1]] <= t32 || t32 + n[[3, 2]] <= t31) &&

   (t11 + n[[1, 1]] <= t21 || t21 + n[[2, 1]] <= t11) &&
   (t11 + n[[1, 1]] <= t31 || t31 + n[[3, 1]] <= t11) &&

   (t21 + n[[2, 1]] <= t31 || t31 + n[[3, 1]] <= t21) &&

   (t12 + n[[1, 2]] <= t22 || t22 + n[[2, 2]] <= t12) &&
   (t12 + n[[1, 2]] <= t32 || t32 + n[[3, 2]] <= t12) &&

   (t22 + n[[2, 2]] <= t32 || t32 + n[[3, 2]] <= t22) &&

   total >= t11 + n[[1, 1]] && total >= t12 + n[[1, 2]] && 
   total >= t21 + n[[2, 1]] && total >= t22 + n[[2, 2]] && 
   total >= t31 + n[[3, 1]] && total >= t32 + n[[3, 2]]}, {t11, t12, 
  t21, t22, t31, t32, total}]

Out[2]= {41., {t11 -> 0., t12 -> 22., t21 -> 19., t22 -> 0., 
  t31 -> 28., t32 -> 13., total -> 41.}}

But when it comes 5 trucks and 4 port it takes hours, an I have to quit the program..

NMinimize[{total, 
   t11 >= 0 && t12 >= 0 && t13 >= 0 && t14 >= 0 && t21 >= 0 && 
    t22 >= 0 && t23 >= 0 && t24 >= 0 && t31 >= 0 && t32 >= 0 && 
    t33 >= 0 && t34 >= 0 && t41 >= 0 && t42 >= 0 && t43 >= 0 && 
    t44 >= 0 && t51 >= 0 && t52 >= 0 && t53 >= 0 && t54 >= 0 &&

    (t11 + n[[1, 1]] <= t12 || t12 + n[[1, 2]] <= t11) &&
    (t11 + n[[1, 1]] <= t13 || t13 + n[[1, 3]] <= t11) &&
    (t11 + n[[1, 1]] <= t14 || t14 + n[[1, 4]] <= t11) &&

    (t12 + n[[1, 2]] <= t13 || t13 + n[[1, 3]] <= t12) &&
    (t12 + n[[1, 2]] <= t14 || t14 + n[[1, 4]] <= t12) &&

    (t13 + n[[1, 3]] <= t14 || t14 + n[[1, 4]] <= t13) &&

    (t21 + n[[2, 1]] <= t22 || t22 + n[[2, 2]] <= t21) &&
    (t21 + n[[2, 1]] <= t23 || t23 + n[[2, 3]] <= t21) &&
    (t21 + n[[2, 1]] <= t24 || t24 + n[[2, 4]] <= t21) &&

    (t22 + n[[2, 2]] <= t23 || t23 + n[[2, 3]] <= t22) &&
    (t22 + n[[2, 2]] <= t24 || t24 + n[[2, 4]] <= t22) &&

    (t23 + n[[2, 3]] <= t24 || t24 + n[[2, 4]] <= t23) &&

    (t31 + n[[3, 1]] <= t32 || t32 + n[[3, 2]] <= t31) &&
    (t31 + n[[3, 1]] <= t33 || t33 + n[[3, 3]] <= t31) &&
    (t31 + n[[3, 1]] <= t34 || t34 + n[[3, 4]] <= t31) &&

    (t32 + n[[3, 2]] <= t33 || t33 + n[[3, 3]] <= t32) &&
    (t32 + n[[3, 2]] <= t34 || t34 + n[[3, 4]] <= t32) &&

    (t33 + n[[3, 3]] <= t34 || t34 + n[[3, 4]] <= t33) &&

    (t41 + n[[4, 1]] <= t42 || t42 + n[[4, 2]] <= t41) &&
    (t41 + n[[4, 1]] <= t43 || t43 + n[[4, 3]] <= t41) &&
    (t41 + n[[4, 1]] <= t44 || t44 + n[[4, 4]] <= t41) &&

    (t42 + n[[4, 2]] <= t43 || t43 + n[[4, 3]] <= t42) &&
    (t42 + n[[4, 2]] <= t44 || t44 + n[[4, 4]] <= t42) &&

    (t43 + n[[4, 3]] <= t44 || t44 + n[[4, 4]] <= t43) &&

    (t51 + n[[5, 1]] <= t52 || t52 + n[[5, 2]] <= t51) &&
    (t51 + n[[5, 1]] <= t53 || t53 + n[[5, 3]] <= t51) &&
    (t51 + n[[5, 1]] <= t54 || t54 + n[[5, 4]] <= t51) &&

    (t52 + n[[5, 2]] <= t53 || t53 + n[[5, 3]] <= t52) &&
    (t52 + n[[5, 2]] <= t54 || t54 + n[[5, 4]] <= t52) &&

    (t53 + n[[5, 3]] <= t54 || t54 + n[[5, 4]] <= t53) &&


    (t11 + n[[1, 1]] <= t21 || t21 + n[[2, 1]] <= t11) &&
    (t11 + n[[1, 1]] <= t31 || t31 + n[[3, 1]] <= t11) &&
    (t11 + n[[1, 1]] <= t41 || t41 + n[[4, 1]] <= t11) &&
    (t11 + n[[1, 1]] <= t51 || t51 + n[[5, 1]] <= t11) &&

    (t21 + n[[2, 1]] <= t31 || t31 + n[[3, 1]] <= t21) &&
    (t21 + n[[2, 1]] <= t41 || t41 + n[[4, 1]] <= t21) &&
    (t21 + n[[2, 1]] <= t51 || t51 + n[[5, 1]] <= t21) &&

    (t31 + n[[3, 1]] <= t41 || t41 + n[[4, 1]] <= t31) &&
    (t31 + n[[3, 1]] <= t51 || t51 + n[[5, 1]] <= t31) &&

    (t41 + n[[4, 1]] <= t51 || t51 + n[[5, 1]] <= t41) &&

    (t12 + n[[1, 2]] <= t22 || t22 + n[[2, 2]] <= t12) &&
    (t12 + n[[1, 2]] <= t32 || t32 + n[[3, 2]] <= t12) &&
    (t12 + n[[1, 2]] <= t42 || t42 + n[[4, 2]] <= t12) &&
    (t12 + n[[1, 2]] <= t52 || t52 + n[[5, 2]] <= t12) &&

    (t22 + n[[2, 2]] <= t32 || t32 + n[[3, 2]] <= t22) &&
    (t22 + n[[2, 2]] <= t42 || t42 + n[[4, 2]] <= t22) &&
    (t22 + n[[2, 2]] <= t52 || t52 + n[[5, 2]] <= t22) &&

    (t32 + n[[3, 2]] <= t42 || t42 + n[[4, 2]] <= t32) &&
    (t32 + n[[3, 2]] <= t52 || t52 + n[[5, 2]] <= t32) &&

    (t42 + n[[4, 2]] <= t52 || t52 + n[[5, 2]] <= t42) &&

    (t13 + n[[1, 3]] <= t23 || t23 + n[[2, 3]] <= t13) &&
    (t13 + n[[1, 3]] <= t33 || t33 + n[[3, 3]] <= t13) &&
    (t13 + n[[1, 3]] <= t43 || t43 + n[[4, 3]] <= t13) &&
    (t13 + n[[1, 3]] <= t53 || t53 + n[[5, 3]] <= t13) &&

    (t23 + n[[2, 3]] <= t33 || t33 + n[[3, 3]] <= t23) &&
    (t23 + n[[2, 3]] <= t43 || t43 + n[[4, 3]] <= t23) &&
    (t23 + n[[2, 3]] <= t53 || t53 + n[[5, 3]] <= t23) &&

    (t33 + n[[3, 3]] <= t43 || t43 + n[[4, 3]] <= t33) &&
    (t33 + n[[3, 3]] <= t53 || t53 + n[[5, 3]] <= t33) &&

    (t43 + n[[4, 3]] <= t53 || t53 + n[[5, 3]] <= t43) &&

    (t14 + n[[1, 4]] <= t24 || t24 + n[[2, 4]] <= t14) &&
    (t14 + n[[1, 4]] <= t34 || t34 + n[[3, 4]] <= t14) &&
    (t14 + n[[1, 4]] <= t44 || t44 + n[[4, 4]] <= t14) &&
    (t14 + n[[1, 4]] <= t54 || t54 + n[[5, 4]] <= t14) &&

    (t24 + n[[2, 4]] <= t34 || t34 + n[[3, 4]] <= t24) &&
    (t24 + n[[2, 4]] <= t44 || t44 + n[[4, 4]] <= t24) &&
    (t24 + n[[2, 4]] <= t54 || t54 + n[[5, 4]] <= t24) &&

    (t34 + n[[3, 4]] <= t44 || t44 + n[[4, 4]] <= t34) &&
    (t34 + n[[3, 4]] <= t54 || t54 + n[[5, 4]] <= t34) &&

    (t44 + n[[4, 4]] <= t54 || t54 + n[[5, 4]] <= t44) &&

    total >= t11 + n[[1, 1]] && total >= t12 + n[[1, 2]] && 
    total >= t13 + n[[1, 3]] && total >= t14 + n[[1, 4]] && 
    total >= t21 + n[[2, 1]] && total >= t22 + n[[2, 2]] && 
    total >= t23 + n[[2, 3]] && total >= t24 + n[[2, 4]] && 
    total >= t31 + n[[3, 1]] && total >= t32 + n[[3, 2]] && 
    total >= t33 + n[[3, 3]] && total >= t34 + n[[3, 4]] && 
    total >= t41 + n[[4, 1]] && total >= t42 + n[[4, 2]] && 
    total >= t43 + n[[4, 3]] && total >= t44 + n[[4, 4]] && 
    total >= t51 + n[[5, 1]] && total >= t52 + n[[5, 2]] && 
    total >= t53 + n[[5, 3]] && total >= t54 + n[[5, 4]]}, {t11, t12, 
   t13, t14, t21, t22, t23, t24, t31, t32, t33, t34, t41, t42, t43, 
   t44, t51, t52, t53, t54, total}] // Timing

I don't know what to do.. And there should be easy way to set up equations and constraints, doing by hand is very painful and easy to make mistakes..

Regards,

Thanks for your quick reply. After I posted the problem I realized that I can use the NMinimize. But the problem is I could not set up equations and constraints.