Model a equation for following condition

[parag_patil]

I have three variables : A, B, C.

I want to write a linear equation such that

X = 1 if atleast 2 of A,B,C are ones.

X= 0 if only one of A,B,C is one.

A,B and C are binary (0,1).

Kindly suggest a linear equation for this.

Thank you.

[Manassaldi]

Hi, i think this could work. Please check it.

if atleast 2 of A,B,C are ones then X = 1
X + 1 - A + 1 - B =g= 1;
X + 1 - A + 1 - C =g= 1;
X + 1 - B + 1 - C =g= 1;

if X = 1 then 2 of A,B,C are ones
1 - X + A + B =g= 1
1 - X + A + C =g= 1
1 - X + A + B + C =g= 1
1 - X + B + C =g= 1

if only one of A,B,C is one then X=0
1 - A + B + C + 1-x =g= 1
1 - B + A + C + 1-x =g= 1
1 - C + B + A + 1-x =g= 1

if X = 0 then one of A,B,C are ones
X + A + B + C =g= 1
X + A + 1-B + 1-C =g= 1
X + 1-A + 1-B =g= 1
X + 1-A + 1-B + C =g= 1
X + 1-A + 1-B + 1-C =g= 1
X + 1-B + 1-C =g= 1
X + 1-A + 1-C =g= 1
X + 1-A + B + 1-C =g= 1
X + 1-B + 1-C =g= 1

best

[parag_patil]

Thank you for your reply.

But I only want a one set of equations which will satisfy the conditions

[Manassaldi]

If you put them together, they are a single set of equations. I don't think you will achieve your goal with a single equation.
best

Code: Select all

binary variable
X,A,B,C
;
equation
eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,
eq9,eq10,eq11,eq12,eq13,eq14,eq15,eq16
eq17,eq18,eq19,eq20
;
variable
z
;
eq1.. X + 1 - A + 1 - B =g= 1;
eq2.. X + 1 - A + 1 - C =g= 1;
eq3.. X + 1 - B + 1 - C =g= 1;
eq4.. 1 - X + A + B =g= 1     ;
eq5.. 1 - X + A + C =g= 1     ;
eq6.. 1 - X + A + B + C =g= 1 ;
eq7.. 1 - X + B + C =g= 1     ;
eq8..  1 - A + B + C + 1-x =g= 1  ;
eq9..  1 - B + A + C + 1-x =g= 1   ;
eq10.. 1 - C + B + A + 1-x =g= 1  ;
eq11.. X + A   + B   + C   =g= 1 ;
eq12.. X + A   + 1-B + 1-C =g= 1     ;
eq13.. X + 1-A + 1-B       =g= 1   ;
eq14.. X + 1-A + 1-B + C   =g= 1   ;
eq15.. X + 1-A + 1-B + 1-C =g= 1     ;
eq16.. X       + 1-B + 1-C =g= 1     ;
eq17.. X + 1-A       + 1-C =g= 1     ;
eq18.. X + 1-A + B   + 1-C =g= 1     ;
eq19.. X       + 1-B + 1-C =g= 1     ;

eq20.. z=e=1;

model test /All/
solve test using mip max z;