A manufacturer of refrigerators must ship at least 100...

A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Each warehouse holds a maximum of 100 refrigerators. Warehouse A holds 25 refrigerators already, while warehouse $B$ has 20 on hand. It costs $\$ 12$ to ship a refrigerator to warehouse $\mathrm{A}$ and $\$ 10$ to ship one to warehouse B. How many refrigerators should be shipped to each warehouse to minimize cost? What is the minimum cost?

Key Concepts

Feasible Region

The feasible region is the set of all possible solutions that satisfy the constraints of a linear programming problem. This area, often represented graphically as a polygon, contains all the candidate solutions among which the optimal solution is located.

Corner Point Theorem

The corner point theorem states that if there is an optimal solution to a linear programming problem, it will occur at one of the vertices (corner points) of the feasible region. This theorem is fundamental to solving linear programming problems because it reduces the search for the optimal solution to these key points.

Objective Function

The objective function is the formula that defines the goal of the optimization problem; it is the function to be minimized or maximized. In many applications, including cost minimization, it encapsulates the quantities and costs or benefits associated with various decision variables.

Linear Programming

Linear programming is a mathematical method used to determine the best possible outcome, such as minimum cost or maximum profit, in a model whose requirements are represented by linear relationships. It involves optimizing a linear objective function subject to a set of linear constraints.

Constraints

Constraints are the conditions or limits placed on the decision variables in a linear programming problem. They represent real-world restrictions such as resource capacities, demand limits, or other operational boundaries that any feasible solution must satisfy.

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