Enormous State University's Business School is buying...

Enormous State University's Business School is buying computers. The school has two models from which to choose, the Pomegranate and the iZac. Each Pomegranate comes with $400 \mathrm{MB}$ of memory and $80 \mathrm{~GB}$ of disk space; each iZac has $300 \mathrm{MB}$ of memory and $100 \mathrm{~GB}$ of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least $48,000 \mathrm{MB}$ of memory and at least $12,800 \mathrm{~GB}$ of disk space. If the Pomegranate and the iZac cost $\$ 2,000$ each, how many of each should the school buy to keep the cost as low as possible? HINT [See Example 4.]

Key Concepts

Feasible Region

The feasible region is the set of all possible solutions that satisfy the constraints of the linear programming problem. It represents all combinations of the decision variables that meet the minimum memory and disk space requirements, and the optimal solution must lie within this region.

Constraint Formulation

This concept involves translating the problem's requirements into mathematical inequalities or equations. For the given scenario, constraints are formulated based on the minimum required total memory and disk space, creating inequalities that the decision variables (the numbers of each computer model purchased) must satisfy.

Integer Programming Considerations

Since the decision variables represent the number of computers purchased, they must be whole numbers. This introduces an integer programming aspect to the problem, where the solution not only has to be optimal in the continuous case but also have integer values to be practically implementable.

Linear Programming

Linear programming is a mathematical technique used to optimize an objective function, such as minimizing cost or maximizing profit, subject to a set of linear inequality or equality constraints. It involves decision variables whose values are determined within a feasible region that satisfies all the imposed constraints.

Objective Function

The objective function is the expression that needs to be optimized (minimized or maximized) in a linear programming problem. In this context, it represents the total cost of purchasing computers, and the goal is to minimize this cost while meeting the specified resource requirements.

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