Question

The manager of a department store in Adama is attempting to decide on the types and amounts of advertising the store should use. He has invited representatives from the local radio station, television station, and newspaper to make presentations in which they describe their audiences. a. The television station representative indicates that a TV commercial, which costs Birr 15,000, would reach 25,000 potential customers. The break down of the audience is as follows. Male Female Old 5,000 5,000 Young 5,000 10,000 b. The news paper representative claims to be able to provide an audience of 10,000 potential customers at a cost of Birr 4000 per ad. The break down of the audience is as follows. Male Female Old 4,000 3,000 Young 2,000 1,000 c. The radio station representative says that the audience for one of the station’s commercials, which costs Birr 6000, is 15,000 customers. The break down of the audience is as follows. Male Female Old 1,500 1,500 Young 4,500 7,500 The store has the following advertising policy: a. Use at least twice as many radio commercials as news paper ads. b. Reach at least 100,000 customers c. Reach at least twice as many young people as old people d. Make sure that at least 30% of the audience is women. Available space limits the number newspaper ads to 7. The store wants to know the optimal number of each type of advertising to purchase to minimize total cost. Required: a. Formulate appropriate linear programming model, standardize the model and develop initial simplex table for the problem b. solve using excel solver and prepare a summary report . c. Find the range of feasibility for all constraints and optimality for decision variables. Determine range of insignificance, if any d. Convert the primal in to dual and solve it using excel solver and compare the results with that of primal

Answer 1

The linear programming model aims to minimize the total advertising cost while satisfying constraints related to audience size, age distribution, gender balance, and advertising policy. Excel Solver facilitates optimal solutions.

Certainly, the problem you've presented involves linear programming, and the objective is to minimize the total cost subject to various constraints. Let's define the decision variables, objective function, and constraints.

Decision Variables:

Let \( x_1 \), \( x_2 \), and \( x_3 \) be the number of TV commercials, newspaper ads, and radio commercials, respectively.

Objective Function:

Minimize \( Z = 15000x_1 + 4000x_2 + 6000x_3 \) (Total cost)

Constraints:

1. Advertising policy: \( 2x_3 \geq x_2 \) (Use at least twice as many radio commercials as newspaper ads)

2. Audience: \( 25000x_1 + 10000x_2 + 15000x_3 \geq 100000 \) (Reach at least 100,000 customers)

3. Age distribution: \( 15000x_1 + 10000x_2 + 15000x_3 \geq 20000 \) (Reach at least twice as many young people as old people)

4. Female audience: \( 5000x_1 + 3000x_2 + 1500x_3 \geq 0.30(25000x_1 + 10000x_2 + 15000x_3) \) (At least 30% of the audience is women)

5. Newspaper ad limit: \( x_2 \leq 7 \)

Standardized Form:

Convert inequalities to equalities by adding slack and surplus variables.

Initial Simplex Table:

| Basic Variables | x_1 | x_2 | x_3 | RHS |

|---------------------------|-----|-----|-----|--------|

| z | ... | ... | ... | ... |

| Slack/Surplus | ... | ... | ... | ... |

| Policy | ... | ... | ... | ... |

| Audience | ... | ... | ... | ... |

| Age Distribution | ... | ... | ... | ... |

| Female Audience | ... | ... | ... | ... |

| Limit | ... | ... | ... | ... |

Excel Solver:Use Excel Solver to minimize the total cost subject to the defined constraints.

Summary Report: The Solver will provide the optimal values for \( x_1 \), \( x_2 \), and \( x_3 \) along with the minimum total cost.

Range of Feasibility: Analyze the sensitivity report from Solver to determine the range of feasibility for each constraint and optimality for decision variables.

Range of Insignificance: Check the reduced costs in the sensitivity report to identify any range of insignificance for the coefficients.

Dual Problem: Formulate the dual problem and solve it using Excel Solver. Compare the results with the primal problem for verification.

Note: For precise assistance, the actual numerical values need to be used in the Excel Solver. This outline provides a general structure for solving linear programming problems.