Hi I’m a newbe of GAMS and I want to figure out optimal solution X1 and X2 (both of them must be integer)
I think it is quite simple problem but I’m struggling with it.
Please find out wrong sentence and give me help.

Your problem is a mixed integer non linear program (MINLP). I used Baron to solve it and I think it is free for for a limited number of variables. Run the code below and tell me if it works:

$TITLE Test Problem
$OFFSYMXREF
$OFFSYMLIST
integer VARIABLES X1, X2 ;
variable Z;
EQUATIONS CON1, CON2, CON3, OBJ;
CON1.. 1*(X1) =L= 48;
CON2.. X2 =L= 48;
CON3.. (1-((0.2)**(X1))) + ((0.2)**(X1))*(1-((0.3)**(X2))) =G= 0.99;
OBJ.. Z =E= 1*60*(X1) + ((0.2)**(X1))*(X2)*(20);
X1.lo = 0;
X2.lo = 0;
MODEL ENGAGEMENT / ALL /;
options MINLP = Baron
SOLVE ENGAGEMENT USING MINLP MINIMIZING Z;

Thanks petros.p I tried your code but I got same error message.
Exec Error at line 22 : Problem extracting Short path, result contains extended ASCII codes