$2)$ A company uses three substances A$,$ B$,$ and C to produce three types of products P$1,$

P$2,$ and P$3.$ The inputs $($and unit profit$)$ for one ounce of each product are shown in

table below.

The current stock information of the three substances are as follows: at most $320$

ounces of substance A can be used, at most $150$ ounces of substance B can be used,

and at most $120$ ounces of substance C can be used to produce the three products. In

order to maximize the profit, the following LP model is developed by the company:

Let x$1=$ ounces of Product P$1$ produced; x$2=$ ounces of Product P$2$ produced; and x$3$

$=$ ounces of Product P$3$ produced.

The proposed model is solved optimally and a sensitivity analysis regarding to this LP

model is reported in the following part of the question.

a$)$ Calculate all unknowns given in the LINGO output given above $($i$.$e$.$ ounces of

product $1,$ product $2,$ and product $3,$ as well as optimal solution$).$

b$)$ If profit of product $3$ increases to $2\mathrm{.}2$$$,$ what will be new optimal solution? Will

the basis remain unchanged?

c$)$ What should profit of product $3$ be so that the model has multiple optimal

solutions?

d$)$ Which constraints are binding, and which constraints are non$-$binding? Explain

why?

e$)$ Which variables are basic, and which variables are non$-$basic? Explain why?

f$)$ What is the range of values for the profit of product P$2$ for which the current basis

remains optimal?

g$)$ What is the range of values for amount of substance A available for which the

current basis remains optimal?

h$)$ If the amount of substance B available is decreased to $100$ ounces, will the

optimal solution change? If yes, what will the new profit be$?$ If no$,$ what is the

highest RHS value of the corresponding constraint which changes current basis?

i$)$ If the amount of substance C available is decreased to $90$ ounces, what will the

new profit be$?$ If the amount of substance C available is increased to $350$ ounces,

will the current basis remain optimal? Do you think model must be reoptimized?