Problem 9 - Applied Finite Mathematics

Question

A diet is to contain at least 8 units of vitamins, 9 units of minerals, and 10 calories. Three foods, Food A, Food B, and Food C are to be purchased. Each unit of Food A provides 1 unit of vitamins, 1 unit of minerals, and 2 calories. Each unit of Food \(\mathrm{B}\) provides 2 units of vitamins, 1 unit of minerals, and 1 calorie. Each unit of Food C provides 2 units of vitamins, 1 unit of minerals, and 2 calories. If Food A costs \(\$ 3\) per unit, Food B costs \(\$ 2\) per unit and Food C costs \(\$ 3\) per unit, how many units of each food should be purchased to keep costs at a minimum?

Step-by-Step Solution

Answer

The exact answer depends on the vertices of the feasible region when graphed and calculating the cost function at those points. This will provide the number of units of Food A, B, and C to be obtained for the diet valid within the constraints.

Step 1: Define the objective function and constraints

Let \(x\), \(y\), \(z\) represent the quantities of Food A, B, and C respectively required in the diet. The objective is to minimize the cost function: \(f(x, y, z) = 3x + 2y + 3z\). The constraints based on the nutritional requirements are: \(x + 2y + 2z \geq 8\) (vitamin requirement), \(x + y + z \geq 9\) (mineral requirement), \(2x + y + 2z \geq 10\) (calorie requirement), and \(x, y, z \geq 0\)(quantity of food should be non-negative).

Step 2: Graphing and Identifying feasible region

After defining the objective function and constraints, graph the constraints on a x-y-z coordinate system. Where all the constraints overlap represents the feasible region. This will not be included in the solution due to the complexity of 3D graphing but can be done easily by using any graphing software or online graphing tool.

Step 3: Find the minimum cost

The optimal solution lies at one of the vertices of feasible region or along the boundary of the feasible region. Utilize a software or manually iterate through these points to plug them into the cost function, \(f(x, y, z)\), calculated in Step 1. The ones that yield the minimum cost are the desired quantities of each food.

Step 4: Present the result

The values of \(x\), \(y\), \(z\) yielding the minimum cost are the number of units of Foods A, B, and C to be purchased for the diet.

Key Concepts

Objective Function Optimization

When it comes to making decisions, especially those involving finances, efficiency is key. In our everyday lives, we aim to get the most value out of our resources, and the same applies to linear programming in nutrition. The objective function represents what we are trying to achieve in mathematical terms—in this case, minimizing the cost of a diet given certain nutritional requirements.

Optimization of the objective function, such as the cost function provided which is \(f(x, y, z) = 3x + 2y + 3z\), involves adjusting the variables—here, the amounts of foods A, B, and C—to find the minimum possible value. The optimal solution must simultaneously satisfy all constraints, such as nutritional requirements and budget limits. Effective optimization techniques, including simplex algorithm and linear programming software, can assist in finding the point where the cost is lowest without violating any constraints.

Constraint Satisfaction

The essence of any diet plan is not only to cost less but also to fulfill dietary requirements. In linear programming, this translates into what we call constraints—the conditions that must be satisfied for the solution to be feasible. For the case at hand, we have several nutritional constraints explicitly outlined as mathematical inequalities, such as \(x + 2y + 2z \geq 8\) for vitamins, \(x + y + z \geq 9\) for minerals, and \(2x + y + 2z \geq 10\) for calories.

The key to solving this nutrition-based problem is to find the intersection of all constraints, ensuring that we purchase enough of each food to meet the minimum required nutrients, while also considering that each variable \(x, y, z\) must be non-negative, indicating you can't purchase a negative quantity of food.

Feasible Region Identification

Imagine a painter with a color palette having different shades that represent all the permissible combinations meeting the painting's requirements. Similarly, in linear programming, the feasible region is akin to the palette where each point represents a potential combination of choices that satisfy all the constraints. However, identifying the feasible region can be quite challenging, especially with more than two variables, as it involves visualizing the problem in multidimensional space.

Although the exercise doesn't graphically represent the feasible region due to the complexity of 3D graphing, we understand it to be the zone where all the constraints of our nutritional problems overlap. Computer software can create a visual representation of this region, greatly simplifying the process of finding the optimal solution.

Mathematical Modeling

Mathematical modeling is the art of translating real-world problems into mathematical formulas and equations. It turns abstract requirements, like nutrition requirements in this case, into a series of equations and inequalities. This transformation is the foundation of using linear programming to solve optimization problems. Our diet problem here involves understanding the nutritional content and cost per unit of each food option and expresses these relationships mathematically.

Applying this model, we set up the objective function and constraints, then use linear programming methods to systematically explore the possible combinations of food purchases. By doing so, we are able to pinpoint the precise quantities of foods A, B, and C that will satisfy the dietary requirements at the lowest cost, showcasing the power of mathematical modeling in making efficient and informed decisions.