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The muffin problem asks us to divide $m$ muffins into pieces and assign eachof those pieces to one of $s$ students so that the sizes of the pieces assignedto each student total $m/s$, with the objective being to maximize the size ofthe smallest piece in the solution. Muffin problems are a special type ofvariant of extended bottleneck transportation problem in which thetransportation time is simply the quantity transported between any source andsink and the objective is to maximize the minimum transportation time. Ofparticular interest are Three Matrix Division and Assignment Problems (3M-DAP),for which all sources have the same supply and sinks are divided into twosubsets having the same demand within each subset. Muffin problems are 3M-DAPin which all sinks have the same demand. We present a recursive algorithm forsolving any 3M-DAP, and hence any muffin problem, and demonstrate that italways produces an optimal solution. The nature of the recursive algorithmallows us to identify interesting relationships between families of suchproblems.

An Optimal Solution for the Muffin Problem