Question

Please turn in the solution for this problem. (No need to solve the LP)

The MSS Company, which manufacturers a new instant salad machine, has $350,000 to spend on advertising. The product is only to be test marketed initially in the Dallas area. The money is to be spent on a television advertising campaign for the Super Bowl weekend (Friday, Saturday, and Sunday.)

The company has three options of advertisement available: daytime advertising, evening news advertising, and the Super Bowl. Even though Super Bowl is a national telecast, the Dallas Cowboys will be playing in it, and hence the viewing audience will be especially large in the Dallas area. A mixture of one-minute TV spots is desired. The table below provides some relevant data:

Options of Advertisement	Cost Per Advertisement	Estimated New Audience Reached With Each Ad
Daytime	$5,000	3,000
Evening News	$7,000	4,000
During Super Bowl	$100,000	75,000

MSS has decided to place at least one ad in each of the three options. Further, there are only two spots during Super Bowl available to MSS. There are ten daytime spots and six evening news spots available on each day. Moreover, MSS wants to have at least five ads per day but intends to spend no more than $50,000 on Friday and no more than $75,000 on Saturday. Formulate a linear program to help MSS decide how the company should advertise over the Super Bowl weekend with the objective of maximizing the total new audience reached. Use the following decision variables in your formulation:

1. X1 = number of daytime ads on Friday

2. X2 = number of daytime ads on Saturday

3. X3 = number of daytime ads on Sunday

4. X4 = number of evening ads on Friday

5. X5 = number of evening ads on Saturday

6. X6 = number of evening ads on Sunday

7. X7 = number of Super Bowl ads (Sunday)

Note: For those who need clarification, Super Bowl is played on Sunday. The daytime and evening ads on Sunday do not include the ads aired during Super Bowl.

Answer 1

The decision variables are defined as,

X1 = if number of daytime ads on Friday

X2 = number of daytime ads on Saturday

X3 = number of daytime ads on Sunday

X4 = number of evening ads on Friday

X5 = number of evening ads on Saturday

X6 = number of evening ads on Sunday

X7 = number of Super Bowl ads (Sunday)

The objective of LP is to maximize the total new audience reached,

From the data values, the objective function is defined as,

The constrains of LP are,

Constraint 1: at least one ad during daytime

Constraint 2: at least one ad during evening news

Constraint 3:  at least one ad during super bowl

Constraint 4:  two spots during Super Bowl available

Constraint 5:  daytime spots available on Friday

Constraint 6: daytime spots available on Saturday

Constraint 7: daytime spots available on Sunday

Constraint 8:  Evening news spots available on Friday

Constraint 9: Evening news spots available on Saturday

Constraint 10: Evening news spots available on Sunday

Constraint 11: at least five ads on Friday

Constraint 12: at least five ads on Saturday

Constraint 13: at least five ads on Sunday

Constraint 14: no more than $50,000 on Friday

Constraint 15: no more than $75,000 on Friday

Constraint 16: Total cost

Inequality constraints:

Decision variable	x1	x2	x3	x4	x5	x6	x7
Objective, Maximize	3000	3000	3000	4000	4000	4000	75000
Constraint 1: at least one ad during daytime	1	1	1	0	0	0	0	>=	1
Constraint 2: at least one ad during evening news	0	0	0	1	1	1	0	>=	1
Constraint 3: at least one ad during super bowl	0	0	0	0	0	0	1	>=	1
Constraint 4: two spots during Super Bowl available	0	0	0	0	0	0	1	<=	2
Constraint 5: daytime spots available on Friday	1	0	0	0	0	0	0	<=	10
Constraint 6: daytime spots available on Saturday	0	1	0	0	0	0	0	<=	10
Constraint 7: daytime spots available on Sunday	0	0	1	0	0	0	0	<=	10
Constraint 8: Evening news spots available on Friday	0	0	0	1	0	0	0	<=	6
Constraint 9: Evening news spots available on Saturday	0	0	0	0	1	0	0	<=	6
Constraint 10: Evening news spots available on Sunday	0	0	0	0	0	1	0	<=	6
Constraint 11: at least five ads on Friday	1	0	0	1	0	0	0	<=	5
Constraint 12: at least five ads on Saturday	0	1	0	0	1	0	0	<=	5
Constraint 13: at least five ads on Sunday	0	0	1	0	0	1	0	<=	5
Constraint 14: no more than $50,000 on Friday	5000	0	0	5000	0	0	0	<=	50000
Constraint 15: no more than $75,000 on Friday	0	7000	0	0	7000	0	0	<=	75000
Constraint 16: Total cost	5000	5000	5000	7000	7000	7000	100000	<=	350000
Inequality	1	0	0	0	0	0	0	>=	0
Inequality	0	1	0	0	0	0	0	>=	0
Inequality	0	0	1	0	0	0	0	>=	0
Inequality	0	0	0	1	0	0	0	>=	0
Inequality	0	0	0	0	1	0	0	>=	0
Inequality	0	0	0	0	0	1	0	>=	0
Inequality	0	0	0	0	0	0	1	>=	0