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Q: In my case the determinant is the summation of the variables from U1 to Ux that have been introduced as a binary variable to GAMS, in each iteration during the optimization the GAMS is putting this variables 0 > or 1 but the structure of these variables should be considered in the optimization process for example:
determinant corresponding Value 1-0-1-1 10 0-1-0-1 100 ......
and so on this setting of variables (Ux in this example) in each iteration will be changed to reach to the optimal point but each setting of variables has a cost (corresponding Value) which is corresponding in the “look up table” so this cost should be considered in the objective function.
Please have a look at the code below. The second model requires a decent GAMS/CPLEX license:
set i / 1*4 / s / 1*5 / table data(s,*) 1 2 3 4 val 1 0 1 1 0 23 2 1 0 0 1 -5 3 1 0 1 0 16 4 0 1 0 1 4 5 1 1 1 0 111 ; parameter c(i) "cost of i when on"; c(i) = -uniformInt(1,10); display c; variable b(i) "binary selection", x(s) "indicator that s is active", z "obj"; binary variables b,x; equation defz "obj", bigm_defx(s) "identification of binary setting of b", indic_defx(s) "identification of binary setting of b" ; defz.. z =e= sum(i, c(i)*b(i)) + sum(s, data(s,'val')*x(s)); bigm_defx(s).. sum(i$(1=data(s,i)), b(i)) + sum(i$(0=data(s,i)), 1-b(i)) =g= card(i)*x(s); indic_defx(s).. sum(i$(1=data(s,i)), b(i)) + sum(i$(0=data(s,i)), 1-b(i)) =e= card(i); model bigm /defz, bigm_defx/; option mip=cplex, optcr=0; solve bigm using mip max z; display z.l, b.l, x.l; model indic /defz, indic_defx/; $echo indic indic_defx(s)$x(s) 1 > cplex.opt option mip=cplex, optcr=0; indic.optfile=1; solve indic using mip max z; display z.l, b.l, x.l;