Signed Minkowski decomposition is an expression of a polytope as a Minkowski sum and difference of smaller polytopes. Signed Minkowski decompositions of a polytope can be interpreted as factorizations of a max-plus (tropical) function. We review two relations about Minkowski decompositions, and we prove that any 2-dimensional integral polytopes (polygons) have a signed Minkowski decomposition which consists of integral points, integral line segments of length 1, and integral triangles of area 1/2. From this result, we also obtain that any max-plus functions with two variables and integer coefficients can be expressed as a set of specified form of simpler max-plus functions.