SMM Company recently developed a new instant salad machine, has 282,000 to spend on advertising. The product is to be initially test marketed in the Dallas area. The money is to be spent on TV advertising blitz during one weekend (Friday, Saturday, and Sunday) in November.

The three options available are: daytime advertising, evening news advertising, and Sunday game-time advertising. A mixture of one-minute TV spots is desired.

Estimated Audience

Ad Type Reached With Each Ad Cost per Ad

Daytime 3,000 $5,000

Evening News 4,000 $7,000

Sunday Game 75,000 $100,000

SMM wants to take out at least one ad of each type (daytime, evening-news, and game-time). Further, there are only two game-time ad spots available. There are ten daytime spots and six evening news spots available daily. SMM wants to have at least 5 ads per day, but spend no more than $50,000 on Friday and no more than $75,000 on Saturday.

Considering the above limitations, write a linear programming formulation to help SMM determine the number for each ad type on the weekend so as to maximize the total audience reached.

1. Define the decision variables

DFR = number of daytime ads on Friday; DSA = number of daytime ads on Saturday; DSU = number of daytime ads on Sunday; EFR = number of evening ads on Friday; ESA = number of evening ads on Saturday; ESU = number of evening ads on Sunday; GSU = number of game-time ads on Sunday

2. Define the Objective Function Maximize the total audience reached:

Max (audience reached per ad of each type) x (number of ads used of each type)

Max 3000DFR +3000DSA +3000DSU +4000EFR+4000ESA +4000ESU +75000GSU

3. Define the Constraints Take out at least one ad of each type: (1) DFR + DSA + DSU > 1 (2) EFR + ESA + ESU > 1 (3) GSU > 1 Ten daytime spots available: (4) DFR < 10 (5) DSA < 10 (6) DSU < 10

Six evening news spots available: (7) EFR < 6 (8) ESA < 6 (9) ESU < 6

At least 5 ads per day: (10) DFR + EFR > 5 (11) DSA + ESA > 5 (12) DSU + ESU + GSU > 5

Only two Sunday game-time ad spots available: (13) GSU < 2

Spend no more than $50,000 on Friday: (14) 5000DFR + 7000EFR < 50000

Spend no more than $75,000 on Saturday: (15) 5000DSA + 7000ESA < 75000

Spend no more than $282,000 in total: (16) 5000DFR + 5000DSA + 5000DSU + 7000EFR + 7000ESA + 7000ESU + 100000GSU7 < 282000

Non-negativity: DFR, DSA, DSU, EFR, ESA, ESU, GSU > 0