# How to formulate constraints in zero-one linear programming

There is a factory which produces 5 types of ice cream. If the $$i_{th}$$ ice cream is produced then $$b_i=1$$ otherwise $$b_i=0$$

How can I express the following constraints:

• The simultaneous production of all 5 types ice cream is not possible
• The type 1 and type 2 must be either both produced or none be produced ($$b1 \oplus b2$$)
• If type 4 is produced, type 5 must be produced too, but if type 4 isn't produced, type 5 can be produced or not.
• If type 4 isn't produced, type 5 must not be produced, otherwise, if type 4 is produced, type 5 can be produced or not

• $$b_1+b_2+b_3+b_4+b_5 \le 4$$
• $$b_1-b_2=0$$
• $$b_4- b_5 \le 0$$
• $$b_5- b_4 \le 0$$