Question

Please solve in excel

WA 339.94)              With a little thought you should be able to “eyeball” the solution.   However the question asks to develop a linear integer model to solve the problem. This can be set up in less than 10 minutes. INCLUDE ALL CONSTRAINTS.

An aged merchant of Baghdad was much respected by all who knew him. He had three sons, and it was a rule of his life to treat them equally. Whenever one son received a present, the other two each received a present of equal value. One day this worthy man fell sick and died, bequeathing all of his possessions to his three sons in equal shares. The only difficulty that arose was over the stock of honey. There were exactly 21 barrels. The old man left instructions that each son should not only receive an equal quantity of honey, but each son should receive exactly the same number of barrels, and no honey should be transferred from barrel to barrel on account of the waste involved. Now, as 7 of these barrels were full of honey, 7 were half full, and 7 were empty, this was quite a puzzle, especially because each brother objected to taking more than 4 barrels of the same description (full, half full, or empty). Develop an integer model to solve this puzzle.

Answer 1

We need to formulate the problem first. For this, we need to define the decision variables, and the objective function. Firstly we can denote the decision variable between two parameters

X_ij where i={full, half, empty} and j={son1, son2, son3}

X_ij denotes the number of barrels taken by each i and j values.

Our objective is to make sure that the honey is equally distributed and nothing is wasted. This means,

?X_ij = 21

This is our objective function.

The constraints are

?X_ison1 = ?X_ison2 = ?X_ison3

This means the number of barrels by each of them should be equal.

X_fullson1 + 0.5 * X_halfson1 + 0 * X_emptyson1 = X_fullson2 + 0.5 * X_halfson2 + 0 * X_emptyson2 = X_fullson3 + 0.5 * X_halfson3 + 0 * X_emptyson3

This means that each of them receive same amount of honey

?X_ison1,?X_ison1,?X_ison1 <= 4 where i={full, half, empty}

This means that each of them will not take more than 4 barrels of the same type

X_ij >= 0

This means the number of barrels cannot be in negative

Now we can use excel and plot this model. The setup and the answer is shown in the image below.

Hope this helps. Cheers!