Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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 advertisement goals at a minimum cost?

Step-by-Step Solution

Answer

Everest should place \(x\) advertisements in Newspaper I and \(y\) advertisements in Newspaper II to achieve their advertisement goals at a minimum cost, where \(x\) and \(y\) are the optimal values obtained from solving the linear programming problem with constraints: 1. \(70,000x + 10,000y \geq 2,000,000\) 2. \(40,000x + 20,000y \geq 1,400,000\) 3. \(20,000x + 40,000y \geq 1,000,000\) 4. \(x \geq 0\) 5. \(y \geq 0\) and minimizing the objective function: \(C = 1000x + 800y\).

Step 1: Set up constraints

For Group A: \(70,000x + 10,000y \geq 2,000,000\) For Group B: \(40,000x + 20,000y \geq 1,400,000\) For Group C: \(20,000x + 40,000y \geq 1,000,000\) Since x and y represent the number of advertisements, we also need non-negativity constraints: \(x \geq 0\) \(y \geq 0\) Our constraints will be: 1. \(70,000x + 10,000y \geq 2,000,000\) 2. \(40,000x + 20,000y \geq 1,400,000\) 3. \(20,000x + 40,000y \geq 1,000,000\) 4. \(x \geq 0\) 5. \(y \geq 0\)

Step 2: Create objective function

The cost of advertising in Newspaper I is $$\$1000$$ per advertisement, and the cost of advertising in Newspaper II is $$\$800$$ per advertisement. Our goal is to minimize the total cost, which can be represented by the following objective function: Objective function: \(C = 1000x + 800y\)

Step 3: Solve the linear programming problem

Now that we have our constraints and objective function, we need to solve the linear programming problem by finding the optimal solution that minimizes the cost. You can do this graphically by plotting the feasible region of the constraints and finding the minimum value of the objective function in that region, or you can use the simplex method or online solver.

Step 4: Interpret the solution

Once you find the optimal solution for x and y, interpret your findings in the context of the exercise. The optimal solution will give you the number of advertisements Everest should place in each newspaper to achieve their goals at a minimum cost.

Key Concepts

Linear Programming

Linear programming (LP) is a mathematical method used to find the best possible outcome in a given mathematical model whose requirements are represented by linear relationships. It’s extensively used in various fields such as economics, business, engineering, and military operations. In the context of advertising strategy, linear programming can be employed to determine the most cost-effective way to reach a target audience across different advertising platforms.

Consider a company that needs to distribute its adverts in a manner that certain customer groups are adequately reached while minimizing costs. The variables in this case would represent the number or placement of adverts in each medium. The linear aspect comes from the fact that each term in the equations and inequalities is either a constant or the product of a constant and a single variable.

Optimization

Optimization is all about making the best or most effective use of resources. In linear programming, optimization typically refers to either maximizing or minimizing some quantity, such as profit maximization or cost minimization. Optimization problems require an objective function, a clearly defined goal that needs to be achieved.

In our Everest Deluxe World Travel example, the goal is cost minimization. The company seeks to minimize the total cost of advertising to the three customer groups while meeting its reach targets. Understanding optimization allows us to establish an objective function and constraints that guide decision-making.

Constraint Formulation

Constraints in linear programming are limitations or restrictions on the possible solutions to the problem. They form the boundaries within which the optimal solution must lie. Formulating constraints requires translating the problem's restrictions into linear inequalities or equations involving the decision variables.

For instance, in our exercise, the constraints ensure that each group of potential customers sees the advertisements a certain number of times. These constraints are critical as they prevent the solution from surpassing practical limits or from failing to meet necessary conditions. The art of constraint formulation involves understanding the real-world limits and converting them into mathematical expressions.

Objective Function

The objective function in linear programming is the equation that needs to be maximized or minimized—essentially, the target of the optimization. This function directly relates to the goal of the problem and incorporates the decision variables.

In the case of Everest Deluxe World Travel, the objective function represents the total cost of advertising with Newspapers I and II. This function (\(C = 1000x + 800y\)) is crucial for identifying the most economical strategy for the company while meeting their advertising reach goals. Here, 'x' and 'y' represent the numbers of advertisements in newspapers I and II, respectively, and the coefficients represent the respective costs per advertisement.