Question

Consider a capital budgeting formulation where the binary variables x₁, x₂, and x₃ are used to represent the acceptance (x_i = 1) or rejection (x_i = 0) of each alternative. The requirements that at least one, but not more than two, out of the three alternatives can be accepted can be represented by which of the following constraints?

a. x1 + x2 ≤ 1 and x1 + x3 ≤ 1

b. x1 + x2 + x3 ≥ 2 and x1 + x2 + x3 ≥ 1

c. x1 + x2 ≥ 1 and x1 + x3 ≥ 1

d. x1 + x2 + x3 ≥ 1 and x1 + x2 + x3 ≤ 2

Answer 1

When a variable is accepted, it takes a value of 1 and when it is rejected, it takes a value of 0.

So, out of the three if one is accepted and the other two are rejected, then the sum of these three would be 1 because the accepted one would have value 1 and the rejected ones would have value 0.

Out of the three if two are accepted, then they both take value one and the other one which is rejected takes the value 1. So, the sum of these three would be 2.

Ateast one acceptance would mean that the sum is greater than one. No more than two acceptance means the sum has to be less than 2.

option a) indicates either  x₁ or x₂ is accepted (x₁ + x₂ ≤ 1) and either x₁ or x₃ is taken (x₁ + x₃ ≤ 1). So, this option is incorrect.

Option b) indicates that atleast two are accepted (x₁ + x₂ + x₃ ≥ 2) and atleast one is accepted (x₁ + x₂ + x₃ ≥ 1). So, this option is incorrect.

option c) indicates that indicates that atleast one among x₁ and x₂ is accepted (x₁ + x₂ ≥ 1) and either alteast one among x₁ or x₃ is taken (x₁ + x₃ ≥ 1). So, this option is incorrect.

option d) indicates that at least one, but not more than two, out of the three alternatives can be accepted. Therefore, option d) x₁ + x₂ + x₃ ≥ 1 and x₁ + x₂ + x₃ ≤ 2 is the correct answer.