Question

If a company selects either of Project 1 or Project 2 (or both), then either Project 3 or Project 4 (or both) must also be selected. Which of the following constraints enforce this condition?

a. X1 + X2 ≤ 2(X3 + X4)
b. X1 + X2 ≤ X3 + X4
c. X1 − X3 = X2 − X4
d. X1 + X2 + X3 + X4 ≤ 2

Answer 1

The correct mathematical constraint that ensures selecting either or both of Project 1 or Project 2 requires selecting either or both of Project 3 or Project 4 is 'b. X1 + X2 ≤ X3 + X4'.

Step-by-step explanation:

The condition mentioned requires that if either Project 1 or Project 2 (or both) are selected, then it is a requirement to also select either Project 3 or Project 4 (or both). To represent this situation in a mathematical constraint, let's denote X1, X2, X3, and X4 as binary variables where 1 represents selecting the project and 0 represents not selecting it. The appropriate constraint from the ones listed that enforces this condition is:

b. X1 + X2 ≤ X3 + X4

This constraint states that the sum of projects 1 and 2 must be less than or equal to the sum of projects 3 and 4. It makes sure that if one or both of the first two projects are selected, then at least one of the last two projects is also selected. If neither project 1 nor project 2 is selected (both X1 and X2 are 0), this constraint will still hold true as it does not necessitate the selection of projects 3 or 4 in that case.

Answer 2

b. X1 + X2 ≤ X3 + X4

Step-by-step explanation:

b. X1 + X2 ≤ X3 + X4

The situation tells us that the sum of both 1 and 2 must be equal to the sum of 3 and 4 which is true, because if both projects 1& 2 are selected then the both projects 3 & 4 must also be selected.

The rest of the choices are wrong because

a. X1 + X2 ≤ 2(X3 + X4)

The sum of projects 1& 2 is not less than 2 times sum of projects 3 & 4 .

c. X1 − X3 = X2 − X4

If we leave project 3 then project 4 cannot be left out.

d. X1 + X2 + X3 + X4 ≤ 2

any two projects out of the four cannot be chosen. 1 must be chosen from 1&2 and the other from 3 &4.