Organizational Process Optimization Problem:

Maor is a jewelry store that received a special order to process $8$ lots of precious stones for faceting. In order to perform the faceting, $2$ machines were rented that perform the same cuts. The processing time for each batch of stones is shown in the following table:

$\begin{array}{cc}\text{S}\text{t}\text{o}\text{n}\text{e}& \text{D}\text{a}\text{y}\text{s}\text{r}\text{e}\text{q}\text{u}\text{i}\text{r}\text{e}\text{d}to\text{f}\text{a}\text{c}\text{e}\text{t}\text{t}\text{h}\text{e}\text{b}\text{a}\text{t}\text{c}\text{h}\\ \text{D}\text{i}\text{a}\text{m}\text{o}\text{n}\text{d}& 4\mathrm{.}2\\ \text{S}\text{a}\text{p}\text{p}\text{h}\text{i}\text{r}\text{e}& 3\mathrm{.}87\\ \text{R}\text{u}\text{b}\text{y}& 6\mathrm{.}97\\ \text{E}\text{m}\text{e}\text{r}\text{a}\text{l}\text{d}& 5\mathrm{.}31\\ \text{A}\text{m}\text{e}\text{t}\text{h}\text{y}\text{s}\text{t}& 3\mathrm{.}94\\ \text{Z}\text{i}\text{r}\text{c}\text{o}\text{n}\text{i}\text{u}\text{m}& 6\mathrm{.}89\\ \text{L}\text{a}\text{p}\text{i}\text{s}\text{L}\text{a}\text{z}\text{u}\text{l}\text{i}& 3\mathrm{.}32\\ \text{T}\text{o}\text{u}\text{r}\text{m}\text{a}\text{l}\text{i}\text{n}\text{e}& 7\mathrm{.}53\end{array}$

The machines are rented by the day $($or fraction thereof$)$ and must be returned together. They can only process one type of stone at a time, as they require specific calibration for each stone, which is a slow and variable process. For safety reasons, batches cannot be split. The first calibration always takes $5\mathrm{.}2$ hours, regardless of the type of stone.

The following table shows the number of hours required to change from one configuration to another:

$\backslash$table$[[,,$New configuration$],[,,$Diamond,Sapphire,Ruby,Emerald,Amethyst,Zircon,Lapis Lazuli,Tourmaline$],[\backslash$table$[[$Previous$],[$Configuration$]],$Diamond,,$3\mathrm{.}69,4\mathrm{.}4,2\mathrm{.}66,4\mathrm{.}12,2\mathrm{.}87,4\mathrm{.}72,4\mathrm{.}67],[$Sapphire$,3\mathrm{.}99,,2\mathrm{.}97,4\mathrm{.}98,4\mathrm{.}93,2\mathrm{.}82,2\mathrm{.}83,4\mathrm{.}76],[$Ruby$,3\mathrm{.}68,4\mathrm{.}58,,2\mathrm{.}78,2\mathrm{.}8,4\mathrm{.}46,3\mathrm{.}12,5],[$Emerald$,4\mathrm{.}75,4\mathrm{.}47,2\mathrm{.}47,,3\mathrm{.}24,2\mathrm{.}38,2\mathrm{.}74,3\mathrm{.}18],[$Amethyst$,2\mathrm{.}31,2\mathrm{.}16,2\mathrm{.}91,3\mathrm{.}92,,2\mathrm{.}4,2\mathrm{.}51,4\mathrm{.}52],[$Zircon$,3\mathrm{.}53,2\mathrm{.}01,2,2\mathrm{.}8,3\mathrm{.}75,,4\mathrm{.}3,4\mathrm{.}92],[$Lapis Lazuli,$3\mathrm{.}14,2\mathrm{.}76,4\mathrm{.}58,4\mathrm{.}28,2\mathrm{.}63,3\mathrm{.}42,,4\mathrm{.}27],[$Tourmaline$,4\mathrm{.}32,4\mathrm{.}45,4\mathrm{.}63,3\mathrm{.}35,4\mathrm{.}44,4\mathrm{.}37,2\mathrm{.}69,]]$

The following aspects must be considered:

$*$Diamond and sapphire batches require higher voltage, which can activate the circuit breaker, so they should not be the first to be processed.

$*$Lapis Lazuli and Tourmaline batches overheat the machine, so they should not be worked consecutively.

$*$The workload must be balanced between both.

$*$The objective is to assign tasks and order them so that they are completed as quickly as possible, minimizing the rental time of the machines.

Instructions: Design a model that minimizes the project costs. The model must be delivered with and without parameterization $($in the procedure, define the variables, the objective function, the restrictions, the restrictions on the nature of the variables$).$

Hint: a decision variable can have several subscripts as needed, for example, X$\_$ijk can represent the quantity of product i to be manufactured on day j to be delivered to customer k$.$