The Biggs Department Store chain has hired an advertising firm to determine the types and amount of advertising it should invest in for its stores. The three types of advertising available are television and radio commercials and newspaper ads. The retail chain desires to know the number of each type of advertisement it should purchase in order to maximize exposure. It is estimated that each ad or commercial will reach the following potential audience and cost the following amount:

Exposure

(people/ad or commercial)

Cost (US$)

TV commercial

40,000

16,000

Radio commercial

28,000

14,000

Newspaper ad

29,000

8,000

The company must consider the following resource constraints:

The budget limit for advertising is $100,000.
The television station has time available for 6 commercials.
The radio station has time available for 10 commercials.
The newspaper has space available for 5 ads.
The advertising agency has time and staff available for producing no more than a total of 12 commercials and/or ads.
The number of Newspaper ads cannot be more than twice the number of Radio commercials.

If this problem is formulated as an integer and also a linear programming problem (separately), the maximization ("Z", the optimal solution) of the exposure would be, respectively:

Question

The Biggs Department Store chain has hired an advertising firm to determine the types and amount of advertising it should invest in for its stores. The three types of advertising available are television and radio commercials and newspaper ads. The retail chain desires to know the number of each type of advertisement it should purchase in order to maximize exposure. It is estimated that each ad or commercial will reach the following potential audience and cost the following amount:

Exposure

(people/ad or commercial)

Cost (US$)

TV commercial

40,000

16,000

Radio commercial

28,000

14,000

Newspaper ad

29,000

8,000

The company must consider the following resource constraints:

The budget limit for advertising is $100,000. 
The television station has time available for 6 commercials.
The radio station has time available for 10 commercials.
The newspaper has space available for 5 ads.
The advertising agency has time and staff available for producing no more than a total of 12 commercials and/or ads. 
The number of Newspaper ads cannot be more than twice the number of Radio commercials.

If this problem is formulated as an integer and also a linear programming problem (separately), the maximization ("Z", the optimal solution) of the exposure would be, respectively:

BlackTom AI · Accepted Answer

Exposure per dollar is the key criterion: TV = 40,000/16,000 = 2.5; Radio = 28,000/14,000 = 2.0; Newspaper = 29,000/8,000 = 3.625. Thus newspapers give the highest exposure per dollar and should be filled to their allowable maximum if feasible.

1) Relevant knowledge and constraints
- Variables: x1 = TV commercials, x2 = Radio commercials, x3 = Newspaper ads.
- Maximize Z = 40,000 x1 + 28,000 x2 + 29,000 x3
- Constraints:
  16,000 x1 + 14,000 x2 + 8,000 x3 ≤ 100,000 (budget)
  x1 ≤ 6; x2 ≤ 10; x3 ≤ 5
  x1 + x2 + x3 ≤ 12 (production limit)
  x3 ≤ 2 x2 (newspaper at most twice radio)
  x1,x2,x3 ≥ 0 (and integer for IP)

2) Linear programming (continuous variables)
- Because newspaper gives best ROI, set x3 = 5 (its maximum). Then x3 ≤ 2 x2 implies x2 ≥ 2.5.
- Set x2 to its lower necessary value 2.5 (radio is less efficient than TV).
- Budget used so far: 5·8,000 + 2.5·14,000 = 40,000 + 35,000 = 75,000. Remaining budget = 25,000.
- TV cost per unit is 16,000 so x1 = 25,000 / 16,000 = 1.5625 (fits production limit).
- So LP solution: x1 = 1.5625, x2 = 2.5, x3 = 5.
- Objective Z = 1.5625·40,000 + 2.5·28,000 + 5·29,000 = 62,500 + 70,000 + 145,000 = 277,500.

3) Integer programming (integer variables)
- Again best to choose x3 = 5. Then x3 ≤ 2 x2 forces integer x2 ≥ ceil(5/2) = 3.
- With x2 = 3 and x3 = 5, cost = 3·14,000 + 5·8,000 = 42,000 + 40,000 = 82,000. Remaining budget = 18,000, allowing at most one TV (x1 = 1).
- Production: 1+3+5 = 9 ≤ 12, all caps satisfied.
- Integer solution: x1 = 1, x2 = 3, x3 = 5.
- Objective Z = 1·40,000 + 3·28,000 + 5·29,000 = 40,000 + 84,000 + 145,000 = 269,000.

4) Why other options are incorrect
- Options proposing many TVs or radios typically violate the budget or do not respect the x3 ≤ 2 x2 constraint or the production limit. For example, x1 = 6, x2 = 4, x3 = 6 violates x3 ≤ 5 and/or budget. Options with x3 = 4 yield lower total exposure because newspapers have the best ROI and reducing x3 decreases total exposure.

Conclusion: The integer optimum is x1 = 1, x2 = 3, x3 = 5 and the linear (continuous) optimum is x1 = 1.5625, x2 = 2.5, x3 = 5, which matches the provided (correct) answer.

类似问题

PA13. If demand for each ball is limited to 6,000 balls and there are 800 hours available on the machine, how many softballs should be produced.

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