$(30$ points$)$ Way's Widgets $($WW$)$ is a small manufacturer of $($what else$)$ widgets as well as a few other items. They have a small product line of five items: widgets $(2$ models$),$ wadgets, filbert flanges, and grapple grommets. There are four resources used in the production of these items: Lathe, Drilling, Heat$-$Treat, and Grinding. The resource requirements for one of each item, the availability of each resource, and the per$-$unit profitability are given in the table below. Typically, WW can sell whatever it produces.

$\backslash$table$[[,$Widgets $1,$Widgets $2,$Wadgets,$\backslash$table$[[$Filbert$],[$Flanges$]],\backslash$table$[[$Grapple$],[$Grommets$]],\backslash$table$[[$Resource$],[$Availability$]]],[$Lathe$,0\mathrm{.}2,0\mathrm{.}25,0\mathrm{.}6,0\mathrm{.}8,0\mathrm{.}25,800],[$Drilling$,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}8,0\mathrm{.}3,0\mathrm{.}1,1000],[$Heat Treat,$0\mathrm{.}4,0\mathrm{.}4,0\mathrm{.}5,0\mathrm{.}25,0\mathrm{.}2,160],[$Grinding$,0\mathrm{.}35,0\mathrm{.}4,0\mathrm{.}7,0\mathrm{.}4,0\mathrm{.}3,600],[$Finishing$,0\mathrm{.}5,0\mathrm{.}3,0\mathrm{.}2,0\mathrm{.}8,0\mathrm{.}4,1000],[$Profit$,20,22,50,40,15,]]$

Below is a formulation of the problem:

Let Wid$1=$ # of units of widgets model $1$ to manufacture. Similarly, let Wid$2,$ Wad, FF$,$ and GG $=$ # of units of widgets model $2,$ wadgets, filbert flanges, and grapple grommets, respectively, to manufacture.

MaxZ$=20$Wid$1+22$Wid$2+50$Wad$+40FF+15GG$

Subject to

$0\mathrm{.}2$Wid$1+0\mathrm{.}25$Wid$2+0\mathrm{.}6$Wad$+0\mathrm{.}8FF+0\mathrm{.}25GG800(lathe)$

$0\mathrm{.}5$Wid$1+0\mathrm{.}6$Wid$2+0\mathrm{.}8$Wad$+0\mathrm{.}3FF+0\mathrm{.}1GG1,000(drill)$

$0\mathrm{.}4$Wid$1+0\mathrm{.}4$Wid$2+0\mathrm{.}5$Wad$+0\mathrm{.}25FF+0\mathrm{.}2GG160(heat$ treat$)$

$0\mathrm{.}35$Wid$1+0\mathrm{.}4$Wid$2+0\mathrm{.}7$Wad$+0\mathrm{.}4FF+0\mathrm{.}3GG600(grind)$

$0\mathrm{.}5$Wid$1+0\mathrm{.}3$Wid$2+0\mathrm{.}2$Wad$+0\mathrm{.}8FF+0\mathrm{.}4GG1,000($finish$)$

All variables non$-$negative.

Make alterations$/$additions to the LP formulation based on each of the following scenarios. Eac situation is independent of the others. Make sure to define any new variables. There is no need to rewrite the entire LP$.$ Just indicate the changes you would make.Q$4(30$ points$)$ Refer to the question on the separate package.

a$)$ WW has the option to purchase an additional $200$ hours of grinding time at the price of $$500.$ They must purchase all or none of the additional hours.

b$)$ The process of drilling for wadgets requires a setup step. Making any wadget will cost $$25$ and $5$ hours of drilling time for setup. If they make no wadgets, they do not incur the cost or lost time.

c$)$ WW sells their widgets to a reseller. This reseller purchases product from other manufacturers and so they naturally fear having too broad a product line. For that reason, they will only buy one model style of widget from WW$.$ However, they are indifferent as to which model, and so WW is free to choose which style to produce for them. For that reason, only one model of widget should be produced.

d$)$ The heat$-$treat process is basically a large furnace used to "soften" metal that has become brittle from previous operations $($e$.$g$.$ lathe$).$ Products undergoing heat treat are placed in the furnace for several hours at several hundred degrees. All the products WW makes require a $600$ degree cycle $4-$hour cycle $($placed in the oven at $600$ degrees for $4$ hours$)$ except grapple grommets which require a $700$ degree $6-$hour cycle. For that reason, all the other products can go in the oven with each other, but grapple grommets must go in only with other grapple grommets. WW does not want to run the heat treat unless it has a full load of product. Since the heat$-$treat can fit $30$ grapple grommets in at a time, WW has a policy of always producing them in lots of $30($e$.$g$.0,30,60,$ and so on are allowable, but $35$ is not$).$