A Mississippi farmer is considering adding a special breed of bull to his farm. Let $X_{1}=1$ imply that the bull will be purchased and 0 otherwise. If it is purchased, then the pounds of organic soy purchased per day (denoted by $\mathrm{X}_{2}$ ) and the pounds of organic corn purchased per day (denoted by $\mathrm{X}_{3}$ ) should be at most 50. If the bull is not purchased, this constraint will be immaterial. The constraint that implements this would be ( $\mathrm{M}$ stands for a very big positive number and assume non-negative variables)
A) $X_{2}+X_{3} \leq 50+M^{*} X_{1}$
B) $X_{2}+X_{3} \leq 50+M *\left(1-X_{1}\right)$
C) $X_{2}+X_{3} \leq 50 * X_{1}$
D) $X_{2}+X_{3} \leq 50 *\left(1-X_{1}\right)$

Question

Quizplus · Accepted Answer

The correct constraint is $X_{2}+X_{3} \leq 50+M *\left(1-X_{1}\right)$. This is because when $X_{1}=1$ (bull is purchased), the term $M *\left(1-X_{1}\right)$ becomes 0, making the constraint $X_{2}+X_{3} \leq 50$, which is the desired condition. When $X_{1}=0$ (bull is not purchased), the constraint becomes $X_{2}+X_{3} \leq 50+M$, effectively making the constraint non-binding due to the large value of $M$.

A Mississippi Farmer Is Considering Adding a Special Breed of Bull