This problem illustrates the issue where change$-$over or setup is dependent on the sequence itself

Part A

Assume that a process needs to produce $A=50,B=25,C=50$ and $D=10$ units of product. The Product dependent Set$-$ up and Changeover times in hours for these four products are as follows.

$\backslash$table$[[,$A$,$B$,$C$,$D$],[$A$,0,1\mathrm{.}67,0\mathrm{.}87,1\mathrm{.}26],[$B$,1\mathrm{.}23,0,0\mathrm{.}76,1\mathrm{.}23],[$C$,1\mathrm{.}00,0\mathrm{.}89,0,2\mathrm{.}21],[$D$,1\mathrm{.}55,1\mathrm{.}22,1\mathrm{.}10,0]]$

$(1)$ List all of the product changeover sequences that are possible and calculate the total time consumed by each. $(2)$ What is the optimal product mix which can be used? $(3)$ How many different subset optimal solutions can be identified?