Question

2.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 36 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 11 of these barrels were full of honey, 17 were half full, and 8 were empty, this was quite a puzzle, especially because each brother objected to taking less than 2 or more than 7 barrels of the same description (full, half full, or empty). Solve this puzzle with an integer model

Answer 1

Integer model is formulated as below:

Let F1, F2, F3 be the number of full barrels to be given to the three sons respectively.

H1, H2, H3 be the number of full barrels to be given to the three sons respectively.

E1, E2, E3 be the number of full barrels to be given to the three sons respectively.

F1+F2+F3+H1+H2+H3+E1+E2+E3 = 36

s.t.

F1+F2+F3=11

H1+H2+H3=17

E1+E2+E3=8

F1+H1+E1-F2-H2-E2=0

F1+H1+E1-F3-H3-E3=0

F1+.5*H1-F2-.5H2=0

F2+.5*H2-F3-.5H3=0

F1, F2, F3, H1, H2, H3, E1, E2, E3 >= 2

F1, F2, F3, H1, H2, H3, E1, E2, E3 <= 7

Solution using Excel Solver is following:

fx SUMPRODUCT(B3:J3,SB$14:SI$14) K3 K L M ЕЗ F1 F2 F3 H1 H2 H3 E1 SSolver Parameters 3 Tota 1 36 Set Objective: Max O Min ⓔ Value Of: | 36 To: By Changing Variable Cells: SBS14:SS14 Subject to the Constraints: SKS5:SKS11 SMS5:SMS11 Add SBS14:SJS14 10 0.5 0.5 SBS14:SJS147 0.5 -0.5 Change Delete 13 14 Result: 3 Reset AlII 15 16 Make Unconstrained Variables Non-Negative 17 Select a Solving Method: Simplex LP Options 18 19 Solving Method Select the IPOPT Nonlinear engine for Solver Problems that are smooth nonlinear.Select the LP Simplex engine for linear Solver Problems. 21 23 Solve

FORMULAS:

K3 =SUMPRODUCT(B3:J3,$B$14:$J$14) copy to K5:K11

Optimal solution:

Barrels should be distributed as beow:

	Full	Half-full	Empty	Total barrels	Total honey
Son 1	3	7	2	12	6.5
Son 2	3	7	2	12	6.5
Son 3	5	3	4	12	6.5
Total	11	17	8