B.5:5.4 - A relaxed problem supports a different use

A planner must choose whole jobs for one worker’s four-hour window. At most one A-job is available; it takes three hours and has stipulated value 5. At most two B-jobs are available; each takes two hours and has stipulated value 3. For this constructed problem the values and durations add, and the question is which choice satisfying the stated availability and time limits has greatest value. The reader needs elementary algebra and can enumerate the few integer choices.

Let x and y count A- and B-jobs. The intended constraints are x in {0,1}, y in {0,1,2}, and 3x + 2y <= 4; maximize V = 5x + 3y. An AI-assisted calculation instead allows real x and y with 0 <= x <= 1 and 0 <= y <= 2. It returns x = 1, y = 0.5, V = 6.5.

Recover what that calculation establishes. From the relaxed time constraint, y <= (4 - 3x)/2, so V <= 6 + 0.5x <= 6.5. The returned real-valued pair attains this bound. This is an optimum of the relaxation. The proposed half B-job is excluded by the intended whole-job condition.

For the whole-job question, x = 0 permits at most y = 2 and value 6. With x = 1, the remaining hour permits y = 0 and value 5. Two B-jobs therefore attain the integer optimum 6. The human or tool doing this reasoning can check both feasibility and the comparison directly.

The relaxation still answers a useful question: can any permitted whole-job choice reach value 7? Every integer choice is also feasible for the relaxation, whose proved upper bound is 6.5, so the answer is no. Choosing a realizable assignment needs the integer result; ruling out value 7 needs only the upper bound. With five hours instead, the same argument gives V <= 7.5 + 0.5x <= 8. The choice x = 1, y = 1 attains value 8 in both formulations. The planner can use that relaxed optimum as the whole-job answer because this returned choice satisfies the integer conditions.

Here the human–AI division follows the contribution being sought. A planner can use assisted optimization while being able to state what counts as a whole job, recover the constraints actually solved, and distinguish an attainable choice from an upper bound. If the required work is to develop the optimization Method, its construction and proof become additional capability targets. An exercise can change the time window or divisibility condition and ask the learner to choose the formulation and explain which result answers the question.