Problem 43 - Finite Mathematics and Applied Calculus

Question

\(\nabla\) Transportation Scheduling Your publishing company is about to start a promotional blitz for its new book, Physics for the Liberal Arts. You have 20 salespeople stationed in Chicago and 10 in Denver. You would like to fly at least 10 into Los Angeles and at least 15 into New York. A round-trip plane flight from Chicago to LA costs \(\$ 195 ;{ }^{.37}\) from Chicago to \(\mathrm{NY}\) costs \(\$ 182 ;\) from Denver to LA costs \(\$ 395\); and from Denver to NY costs \(\$ 166\). How many salespeople should you fly from each of Chicago and Denver to each of LA and NY to spend the least amount on plane flights?

Step-by-Step Solution

Answer

The optimal solution is to fly 10 salespeople from Chicago to Los Angeles and 15 salespeople from Denver to New York, with a minimum total cost of $4440.

Step 1: Write down the objective function and constraints

We want to minimize the cost, given by the objective function: Minimize: C(x₁, x₂, x₃, x₄) = 195x₁ + 182x₂ + 395x₃ + 166x₄ Subject to constraints: 1. x₁ + x₃ ≥ 10 2. x₂ + x₄ ≥ 15 3. x₁ + x₂ ≤ 20 4. x₃ + x₄ ≤ 10 5. x₁, x₂, x₃, x₄ ≥ 0

Step 2: Find the feasible region for the given constraints

We need to find the region that satisfies all the constraints. By plotting the constraints, we get the following inequalities: 1. x₃ ≥ 10 - x₁ 2. x₄ ≥ 15 - x₂ 3. x₂ ≤ 20 - x₁ 4. x₄ ≤ 10 - x₃ Since all x variables are non-negative, the feasible region can be found by intersecting these inequalities.

Step 3: Identify corner points of the feasible region

To find the corner points of the feasible region, we can solve the constraints as equalities: 1. x₃ = 10 - x₁ 2. x₄ = 15 - x₂ 3. x₂ = 20 - x₁ 4. x₄ = 10 - x₃ We get the following corner points: A (0, 5, 10, 10) B (10, 0, 0, 15) C (0, 20, 10, 0)

Step 4: Calculate the cost for each corner point

Using the cost function, we can calculate the cost for each corner point: C(A) = 195(0) + 182(5) + 395(10) + 166(10) = 0 + 910 + 3950 + 1660 = 6520 C(B) = 195(10) + 182(0) + 395(0) + 166(15) = 1950 + 0 + 0 + 2490 = 4440 C(C) = 195(0) + 182(20) + 395(10) + 166(0) = 0 + 3640 + 3950 + 0 = 7590

Step 5: Select the corner point with minimum cost

Comparing the cost for each corner point, we find that B (10, 0, 0, 15) has the minimum cost of $4440. So, the optimal solution is to fly 10 salespeople from Chicago to LA and 15 salespeople from Denver to NY. The total cost of plane flights will be $4440.

Key Concepts

Objective Function

In linear programming, an objective function is the heart of the problem you're trying to solve. It's a mathematical expression you want to either maximize or minimize. In the context of the exercise, the objective function represents the total cost of flying salespeople from Chicago and Denver to Los Angeles and New York. The goal is to minimize this total cost.
The objective function can be expressed as a linear equation:\[C(x_1, x_2, x_3, x_4) = 195x_1 + 182x_2 + 395x_3 + 166x_4\] Each term (e.g., \(195x_1\)) in the equation represents the cost per person for one of the flight routes multiplied by the number of people (denoted by \(x_1, x_2, x_3,\) or \(x_4\)) taking that route.
Understanding the objective function is critical because it defines what you want to achieve (minimum cost) and lays the foundation for how you direct your problem-solving efforts. Without it, you wouldn't have a clear goal to work towards.

Constraints

Constraints are the boundaries within which your solution must operate. They reflect the limitations or requirements of the problem, telling you what is possible and what is not. In this task, constraints account for the number of salespeople you have and the need to meet minimum travel requirements to LA and NY.
The constraints in this exercise are:

• \(x_1 + x_3 \geq 10\) - At least 10 salespeople must fly to LA.
• \(x_2 + x_4 \geq 15\) - At least 15 salespeople need to fly to NY.
• \(x_1 + x_2 \leq 20\) - Total salespeople flying from Chicago cannot exceed 20.
• \(x_3 + x_4 \leq 10\) - Total from Denver cannot exceed 10.
• \(x_1, x_2, x_3, x_4 \geq 0\) - You can't fly a negative number of people.

These constraints help ensure that solutions are realistic and feasible concerning the resources and logistical needs. By understanding and correctly setting up these constraints, you can ensure that your solution will be practical and applicable to the real-world scenario you are modeling.

Feasible Region

The feasible region is the set of all potential solutions that satisfy the problem's constraints. In a graphical representation, this region is usually depicted as a shape on a plane where all conditions imposed by constraints overlap.
To find the feasible region, you combine all inequalities—the constraints from the problem. By plotting these, you visualize the area where every condition is met. For this specific problem, you look at how each constraint intersects or influences the possible values for \(x_1, x_2, x_3,\) and \(x_4\).
The feasible region is important because it contains all the possible responses to the problem, and within it, you will find the optimal solution by evaluating the objective function. Only points within this region can be considered viable solutions, meaning any solution outside doesn’t meet all required constraints.

Corner Points

Corner points are the vertices or edges of the feasible region where constraints intersect. In linear programming, these points are crucial because, thanks to the convex nature of the feasible region, the optimal solution often lies at one of these corner points.
For the exercise, you identify the corner points by solving the system of equations formed by the constraints when treated as equalities. This transformation helps pinpoint where constraints meet precisely:

• A (0, 5, 10, 10)
• B (10, 0, 0, 15)
• C (0, 20, 10, 0)

Next, you calculate the cost at each corner using the objective function to determine which one offers the lowest cost, thereby providing the optimal solution. Thus, understanding and assessing these corner points effectively helps you find the solution to the linear programming problem.