We examine a novel variation of the classic cylindrical can optimization
problem encountered in almost every first-semester calculus course: to minimize
the cost of the material used to make the can, which amounts to minimizing
the surface area given a fixed volume. The result is a height equal to
the diameter of the circular top (square front profile).
We add a distribution/storage ("ship+store") cost (based on real data) to
the material cost to get a total cost to minimize.
We show two things numerically:
• For low ship+store costs, the material cost dominates and the can has the
classic square profile.
• For ship+store costs above a critical value, the symmetry of the square
is broken and the can takes on a rectangular profile resulting from the
rectangular geometry of the distribution truck or shelf/cabinet area for
Interestingly, the can dimensions remain constant (at different values)
above and below the critical ship+store cost, exactly as with a Landau "meanfield"
phase transition in statistical mechanics. The parameter for the cost of
ship+store appears exactly as an inverse-temperature in magnetic systems,
and this Landau-type transition has the same basic phenomenology as a Landau
approximation to a magnetic system that is non-magnetic above a critical
temperature (paramagnetic) but magnetizes below a critical temperature
(ferromagnetic), much like a Curie temperature for a discontinuous phase
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