OSCS, the world's leading intercontinental ship container transportation company, wants to prepare its largest ship $($Titan $)$ for the upcoming delivery. Titan $$ has five compartments for storing cargo: bow, front, center, port and rear. These compartments have the following limits on both weight and space:

Compartment

Weight capacity $($tons$)$

Space capacity $($cubic meters$)$

Bow $11075000$

Front $15494000$

Center $225150000$

Port $16587000$

Rear $9568000$

Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the ship.

The following six cargoes are available for shipment on the next voyage:

Cargo Weight $($tons$)$ Volume $($cubic meters$/$tons$)$ Profit $($$$/$tons$)$

Cargo $11956007100$

Cargo $21427005400$

Cargo $32158007700$

Cargo $41345006200$

Cargo $51656007200$

Cargo $61816005600$

In addition, among these six cargoes, there are cargoes of two companies that are regular customers of the company OSCS. The cargo requested by these two customers is second cargo for the first customer, and sixth cargo for the second customer. Since the company wants to satisfy its regular customers, it will accept at least $50$ tons of cargo from the first customer and $60$ tons of cargo from the second customer. Apart from this regular customers demand, any proportion of these cargoes can be accepted.

$1-)$ Formulate an LP model to determine how much $($if any$)$ of each cargo should be accepted and how to distribute each among the compartments so that the total profit for the voyage is maximized. Clearly define all decision variables and constraints. Write in close format. $(50$ points$)$

$2-)$ Use OPL CPLEX language to solve your model. Copy$-$Paste your Code to your Word document. Write in close format. $(30$ points$)$

$3-)$ Answer the following questions. Explain in detail. $($Suppose that your computers are down after solving Question $2$ so that you cannot solve a new problem from scratch $)(20$ points$)$

a$)$ How much should OSCS be willing to pay for an additional ton of weight capacity in each compartment?

b$)$ How would the total profit of OSCS be affected if the minimum amount of cargo to be accepted from the first customer decreases to $40$ tons