John White is the program scheduling manager for the television channel CCFO. John would like to plan the schedule of television shows for next Wednesday evening.

The table below lists nine shows under consideration. John must select exactly five of these shows for the period from $8$:$00$ p$.$m$.$ to $10$:$30$ p$.$m$.$ next Wednesday evening. For each television show, the estimated advertising revenue $($in $ millions$)$ is provided. Furthermore, each show has been categorized into one or more of the categories "Public Interest," "Violent," "Comedy," and "Drama." In the following table, a $1$ indicates that the show is in the corresponding category and a $0$ indicates it is not.

Number Show Revenue $($$ millions$)$ Public Interest Violent Comedy Drama

$1$ Sam's Place $$80011$

$2$ Texas Oil $$110101$

$3$ Cincinnati Law $$121001$

$4$ Jarred $$40101$

$5$ Bob & Mary $$70010$

$6$ Chainsaw $$60100$

$7$ Loving Life $$101001$

$8$ Islanders $$80010$

$9$ Urban Sprawl $$91000$

John would like to determine a revenue$-$maximizing schedule of television shows for next Wednesday evening. However, he must be mindful of the following considerations:

Consideration $1$: Exactly $5$ television shows must be selected.

Consideration $2$: The schedule must include at least as many shows that are categorized as public interest as shows that are categorized as violent.

Consideration $3$: If John schedules "Loving Life," then he must also schedule either "Jarred" or "Cincinnati Law" $($or both$).$

Consideration $4$: John cannot schedule both "Loving Life" and "Urban Sprawl."

Consideration $5$: If John schedules more than one show in the "Violent" category, he will lose an estimated $$4$ million in advertising revenues from family$-$oriented sponsors.

$($a$)$

Formulate a binary integer program that models the decisions John faces.

Let xi $=$

$1$ if show i is selected

$0$ otherwise

for

i $=1,2,3,4,5,6,7,8,9.$

Let

y $=1$

if two or more violent movies are chosen and $0$ otherwise. Give your objective function in millions of dollars. For Consideration $5,$ use the smallest $($in magnitude$)$ possible integer coefficient for

y$.)$

Min

Correct: Your answer is correct.

s$.$t$.$

Consideration $1$

Correct: Your answer is correct.

Consideration $2$

Correct: Your answer is correct.

Consideration $3$

Correct: Your answer is correct.

Consideration $4$

Correct: Your answer is correct.

Consideration $5$

Correct: Your answer is correct.

$($b$)$

Solve the model formulated in part $($a$).$

$($x$1,$ x$2,$ x$3,$ x$4,$ x$5,$ x$6,$ x$7,$ x$8,$ x$9,$ y$)=$

Incorrect: Your answer is incorrect.

What is the optimal revenue $($in millions of dollars$)?$

$$50$

Correct: Your answer is correct.

million