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Everest Deluxe World Travel has decided to advertise in the Sunday editions of two major newspapers in town. These advertisements are directed at three groups of potential customers. Each advertisement in newspaper I is seen by 70,000 group-A customers, 40,000 group-B customers, and 20,000 group-C customers. Each advertisement in newspaper II is seen by 10,000 group-A, 20,000 group-B, and 40,000 group-C customers. Each advertisement in newspaper I costs $$\$ 1000$$, and each advertisement in newspaper II costs $$\$ 800$$. Everest would like their advertisements to be read by at least 2 million people from group A, \(1.4\) million people from group \(\mathrm{B}\), and 1 million people from group C. How many advertisements should Everest place in each newspaper to achieve its advertising goals at a minimum cost? What is the minimum cost?

Step by step solution

01

Declare variables and constraints

Let \(\displaystyle x_{1}\) represent the number of advertisements in Newspaper I and \(\displaystyle x_{2}\) represent the number of advertisements in Newspaper II. We have the following constraints based on the number of readers for each group: 1. Group A: \(\displaystyle 70{,}000x_{1} +10{,}000x_{2} \ge 2{,}000{,}000\) 2. Group B: \(\displaystyle 40{,}000x_{1} +20{,}000x_{2} \ge 1{,}400{,}000\) 3. Group C: \(\displaystyle 20{,}000x_{1} +40{,}000x_{2} \ge 1{,}000{,}000\) Additionally, the costs for each advertisement are: - Newspaper I: $$1000\, x_{1}$$ - Newspaper II: $$800\, x_{2}$$ Our objective is to minimize the total cost, represented by the objective function \(\displaystyle C(x_{1} ,x_{2}) =1000x_{1} +800x_{2}\).

02

Simplify constraints and express as linear inequalities in terms of x1 and x2

We can simplify the constraints by dividing each constraint by a common factor to reduce the coefficients: 1. \(\displaystyle 7x_{1} +x_{2} \ge 200 \) 2. \(\displaystyle 2x_{1} +x_{2} \ge 70 \) 3. \(\displaystyle x_{1} +2x_{2} \ge 50 \)

03

Solve the system of linear inequalities to find feasible region

The feasible region can be found by graphing the system of linear inequalities and identifying the region where all three constraints are satisfied simultaneously.

04

Use simplex method or graphical method to solve the linear programming problem

Here, we can use the graphical method to solve the linear programming problem. Graph the constraints on the coordinate plane, and look for the vertices of the feasible region. Then, evaluate the objective function at each vertex and determine the vertex that results in the minimum cost.

05

Determine the optimal number of advertisements and minimum cost

Upon solving the linear programming problem using the graphical method or another method (e.g., simplex method), we can determine the optimal number of advertisements in each newspaper and find the minimum cost. Let's say the solution is \(\displaystyle (x_{1}^{\ast} ,x_{2}^{\ast})\), then the minimum cost can be calculated as \(\displaystyle C(x_{1}^{\ast} ,x_{2}^{\ast})\). In this problem, Everest should place \(\displaystyle x_{1}^{\ast}\) advertisements in Newspaper I and \(\displaystyle x_{2}^{\ast}\) advertisements in Newspaper II to achieve its advertising goals at a minimum cost of \(\displaystyle C(x_{1}^{\ast} ,x_{2}^{\ast})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constraint Optimization

Constraint optimization is a vital component in finding the best solution to a problem while adhering to specific requirements or limitations. In the context of linear programming, constraint optimization involves creating equations or inequalities that represent the limits or demands the problem imposes. Here, Everest Deluxe World Travel has advertising goals they want to achieve without overshooting their budget.

The constraints are formulated based on groups A, B, and C, which are identified by the number of potential customers they need to reach. The constraints are as follows:

• Group A: The advertisements must be seen by at least 2 million people. The inequality is: \(70{,}000x_{1} +10{,}000x_{2} \ge 2{,}000{,}000\).
• Group B: The aim is to reach at least 1.4 million people, leading to: \(40{,}000x_{1} +20{,}000x_{2} \ge 1{,}400{,}000\).
• Group C: They need coverage for at least 1 million people, resulting in: \(20{,}000x_{1} +40{,}000x_{2} \ge 1{,}000{,}000\).

These constraints need to be satisfied to ensure effective advertising while minimizing costs.

Objective Function

The objective function is the formula in a linear programming problem that we seek to maximize or minimize. In the given problem, the objective function represents the total advertising cost that Everest aims to minimize.

This function is calculated as:

• For Newspaper I: The cost is \(1000x_{1}\), based on how many advertisements are placed.
• For Newspaper II: The cost becomes \(800x_{2}\), again correlating with advertisement counts.

Thus, the objective function is \C(x_{1}, x_{2}) = 1000x_{1} + 800x_{2}\. This expression gives the total cost of the advertisement campaign. The goal is to find values for \(x_{1}\) and \(x_{2}\) that minimize this total cost, while still meeting all the necessary viewer numbers for each customer group.

Feasible Region

The feasible region represents all possible solutions that satisfy the constraints in a linear programming problem. It is identified by graphing the constraints as linear inequalities onto a coordinate plane.

In this scenario, plotting the simplified constraints:

• \(7x_{1} + x_{2} \ge 200\)
• \(2x_{1} + x_{2} \ge 70\)
• \(x_{1} + 2x_{2} \ge 50\)

The feasible region is the overlapping area where all these inequalities are valid. This region visually represents the set of all possible \(x_{1}\) and \(x_{2}\) combinations that meet Everest's advertising goals. Finding the vertices of this region is crucial since they potentially contain the optimal solution.

Simplex Method

The Simplex Method is a systematic procedure for testing the vertices of the feasible region to find the optimal solution for linear programming problems. In the Everest Deluxe World Travel scenario, this method can be used to minimize costs effectively.

While in simpler problems a graphical approach suffices, the Simplex Method is often favored for higher-dimensional problems or more complex cases. This method evaluates each vertex of the feasible region iteratively to determine which one offers the minimal cost according to the objective function.

• Start by selecting a potential vertex (corner point) within the feasible region.
• Calculate the value of the objective function at this vertex.
• Move to adjacent vertices by following the edge of the region that improves the objective function.
• Continue this process until no better adjacent vertices are available.

By employing the Simplex Method, Everest ensures that they achieve the lowest possible cost while meeting their advertising constraints.