Question

Consider a capital budgeting formulation where the binary variables x₁ and x₂ are used to represent the acceptance (x_i = 1) or rejection (x_i = 0) of each alternative. Which of the following constraints shows that investing in alternative 1 is contingent upon investing in alternative 2?

a. x₁ + x₂ ≤ 1

b. x₁ + x₂ ≥ 1

c. x₁ + x₂ ≥ 1

d. x₁ ≥ x₂

Answer 1

the binary variables x₁ and x₂ are used to represent the acceptance (x_i = 1) or rejection (x_i = 0) of each alternative. In other words, if one of the variables is accepted then it takes value 1 and if it is rejected then it takes value 0.

investing in alternative 1 is contingent upon investing in alternative 2 means that if there is investment made in alternative 2, then investment will be made on alternative 1 as well. In other words, either investment occurs in both (they take value 1) or in nothing (they both take value 0)

Oprion a) indicates that either x₁ or x₂ is chosen. So, this is incorrect

Option b) and c) indicates that atleast one among x₁ or x₂ is chosen. But the sum of these two cannot be 1 becuase that would indicate that only one among the two is chosen, which cannot be the case because it is given that either both (both have value 1) or nothing is chosen (both have value 0). So, this is incorrect.

Option d) is x₁ ≥ x₂ or x₁ - x₂ ≥ 0. If both are accepted then 1-1 = 0 and if both are rejected then 0-0 = 0 which satisfies the equation. So, the correct answer is d.  x₁ ≥ x₂