Coffee Blending. The Coffee Peddler has 1440 lb of Sumatran...

Coffee Blending. The Coffee Peddler has 1440 lb of Sumatran coffee and 700 lb of Kona coffee. $A$ batch of Hawaiian Blend requires 8 lb of Kona and 12 lb of Sumatran, and yields a profit of $\$ 90 .$ A batch of Classic Blend requires 4 lb of Kona and 16 lb of Sumatran, and yields a $\$ 55$ profit. How many batches of each kind should be made in order to maximize profit? What is the maximum profit?

Key Concepts

Linear Programming

Linear programming is a mathematical approach used to determine the best possible outcome in a given situation with constraints. This method involves defining a linear objective function that needs to be maximized or minimized, subject to a set of linear constraints, which represent the limitations on available resources or requirements.

Decision Variables

Decision variables are the unknowns in a linear programming model that represent the quantities to be determined. They are essential because they directly influence the value of the objective function and must satisfy all the constraints of the problem.

Objective Function

The objective function is a key component of linear programming that represents the quantity to be optimized, such as profit or cost. It is a linear expression composed of decision variables and is the focus of the optimization process.

Constraints

Constraints refer to the linear equations or inequalities that impose restrictions on the decision variables. They model the limitations or requirements of the resources available, ensuring that any solution adheres to these predefined restrictions.

Feasible Region and Corner Points

The feasible region is the set of all points that satisfy the constraints of a linear programming problem. Optimal solutions are often found at the corner points (vertices) of this region, because these points are where the boundaries of the constraints intersect and typically yield the maximum or minimum value of the objective function.

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