Discussion
Let us consider the full Coca-Cola bottles, 7 half-full and 7 empty. You want to divide the 21 bottles among three persons so that each will receive exactly 7. Moreover, each person must receive the same quantity of Coca-Cola. Solve the problem using Integer Linear programming and find a solution. What will be the objective function ?
Answer: A
The objective is a dummy objective function with all zero coefficients, with the allocation in option A.
The 21 bottles are 7 full, 7 half-full and 7 empty, so the total drink is \(7 + 3.5 = 10.5\) bottles' worth and each person must get 7 bottles holding 3.5. Nothing is to be maximised or minimised; the problem only asks for a feasible integer solution, so the objective has zero coefficients and the constraints do the work.
Option A satisfies every constraint: person 1 gets 1 full, 5 half and 1 empty (7 bottles, 1 + 2.5 = 3.5); persons 2 and 3 each get 3 full, 1 half and 3 empty (7 bottles, 3 + 0.5 = 3.5); and the columns use exactly 7 of each kind.
The other allocations break the constraints: in B the full bottles total 6, in C person 1 gets no drink at all, and in D person 2 gets 5 full bottles.
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