Question $1$: Consider the following problem.

Maximize

Z $=$ c$1$x$1+$ c$2$x$2$

subject to

a$11$x$1+$ a$12$x$2$ b$1$

a$21$x$1+$ a$22$x$2$ b$2$

x$10,$ x$20.$

The ranges of uncertainty for the parameters are shown in the table below. Parameter Range of Uncertainty

a$110\mathrm{.}91\mathrm{.}1$

a$121\mathrm{.}62\mathrm{.}4$

a$211\mathrm{.}82\mathrm{.}2$

a$220\mathrm{.}81\mathrm{.}2$

b$18\mathrm{.}59\mathrm{.}5$

b$27\mathrm{.}68\mathrm{.}4$

c$12\mathrm{.}73\mathrm{.}3$

c$23\mathrm{.}64\mathrm{.}4$

a$)$ Use robust optimization to formulate a conservative version of this model. Show your formulation explicitly.

b$)$ Solve the model you developed in part $($a$)$ using Excel Solver and generate a sensitivity report. $($Make sure to submit your Excel file$)$

c$)$ Calculate the sensitivity ranges $($i$.$e$.,$ upper and lower limits$)$ for the objective function coefficients $($c$1,$ c$2)$ and the constraint quantities on the right$-$hand side $($b$1,$ b$2)$ by referring to the sensitivity report.