$2x+3y10,$ then $x=1$

$1$f $x=1,$ then $5x-3y6$

$x=0,$ then $2x+3y>10$

ant

$x=0$

$x=1$

$5x-3y6$

c

$2x+3y>10-M$

$5x-3y6+M$

$x=0$

Q$5)$ Which set of constraints correctly models: $"$If $2x+3y10$ then $5x-3y6"$ where $x$ and $y$ are nonnegative variables, $$ is a binary variable, and $lon$ is a small positive number?

a$)2x+3y10+M(1-)$

a$)5x-3y6+M$

b$)2x+3y10+lon-M$

d$)2x+3y10-M(1-)$

b$)5x-3y6+M(1-)$

e$)2x+3y10-lon-M$

c$)2x+3y10-M$

$5x-3y6+lon-M(1-)$

Q$6)$ Let $,,$ all be binary variables, and $M$ be a large enough number.

Which of the following correctly models the logical statement $"x+6y13$ or $4x+y10"?$

a$)x+6y13+M(1-)$ and

d$)x+6y13+M(1-)$ and

b$)x+6y13+M(1-)$ and

$4x+y10+M(1-)$

b$)4x+y10+M(1-)$ and $+=1$

e$)$ Both $($a$)$ and $($b$)$ are correct.

c$)x+6y13+M(1-)$ and

$4x+y10+M(1-)$

$4x+y10+M(1-)$

Q$7)$ Given two non$-$negative variables $x,y10$ and a constraint $2x+3y7+M(1-)$ where $$ is a binary variable, the smallest possible valid value of $M$ is:

$\backslash$table$[[\backslash$table$[[$a$)32],[$b$)50]],$d$)43],[$c$)10,$e$)21]]$

Q$8)$ Consider the following statements for an Integer Program $($IP$)$ with a minimization objective:

I: The optimal objective value of the LP relaxation is a lower bound on the optimal value of the IP$.$

II: Branch and Bound should stop if we find an integer$-$feasible solution with objective value equal to that of the lower bound.

III: Any feasible solution to the IP will give a lower bound on the optimal value of the IP$.$

a$)$ I is false, II is false and III is false

b$)$ I is false, II is false and III is true

c$)$ I is false, II is true and III is false

d$)$ I is true, II is false and III is false

e$)$ I is true, II is true and III is false

Q$9)$ Consider the following simplex tableau for the LP relaxation of a maximization integer program $($assume all the coefficients and variables are integers in the original IP formulation$)$:

$\backslash$table$[[$Basic$,x_1,x_2,x_3,x_4,x_5,x_6,x_7,$RHS$],[z,0,0,0,0,\frac{2}{3},2,\frac{20}{3},\frac{80}{3}$