A large department store needs at least 3600 labor hours...

A large department store needs at least 3600 labor hours covered per week. It employs full-time staff $40 \mathrm{hr} / \mathrm{wk}$ and part-time staff $25 \mathrm{hr} / \mathrm{wk}$. The cost to employ a full-time staff member is more because the company pays benefits such as health care and life insurance. $$ \begin{array}{|l|c|c|} \hline & \text { Hours per Week } & \text { Cost per Hour } \\ \hline \text { Full time } & 40 \mathrm{hr} & \$ 20 \\ \hline \text { Part time } & 25 \mathrm{hr} & \$ 12 \\ \hline \end{array} $$ The store manager also knows that to make the store run efficiently, the number of full-time employees must be at least 1.25 times the number of part-time employees. a. Determine the number of full-time employees and the number of part-time employees that should be used to minimize the weekly labor cost. b. What is the minimum weekly cost to staff the store under these constraints?

A large department store needs at least 3600 labor hours covered per week. It employs full-time staff $40 \mathrm{hr} / \mathrm{wk}$ and part-time staff $25 \mathrm{hr} / \mathrm{wk}$. The cost to employ a full-time staff member is more because the company pays benefits such as health care and life insurance.
$$
\begin{array}{|l|c|c|}
\hline & \text { Hours per Week } & \text { Cost per Hour } \\
\hline \text { Full time } & 40 \mathrm{hr} & \$ 20 \\
\hline \text { Part time } & 25 \mathrm{hr} & \$ 12 \\
\hline
\end{array}
$$
The store manager also knows that to make the store run efficiently, the number of full-time employees must be at least 1.25 times the number of part-time employees.
a. Determine the number of full-time employees and the number of part-time employees that should be used to minimize the weekly labor cost.
b. What is the minimum weekly cost to staff the store under these constraints?

Key Concepts

Corner Point Method

The corner point method is an approach used in linear programming where the optimal solution is found by evaluating the objective function at the vertices, or corner points, of the feasible region. Due to the linear nature of the problem, the optimal solution will always lie at one of these vertex points.

Feasible Region

The feasible region is the set of all points (combinations of decision variables) that satisfy the constraints of a linear programming problem. It is typically defined by the intersection of multiple linear inequalities and represents all viable solutions among which the optimal solution is sought.

Cost Minimization

Cost minimization is a common objective in operations research where resources are allocated in a way that minimizes total expenses while satisfying all required constraints. This concept is crucial for businesses aiming to reduce operational costs while meeting service demands.

Objective Function

The objective function represents the quantity that needs to be optimized, which, in this context, is the total labor cost. It is a linear function of the decision variables, encapsulating the costs associated with different types of employees, and serves as the basis for determining the optimal allocation that minimizes expenses.

Constraints

Constraints are the conditions or limitations under which the optimization must occur. They translate real-world requirements, such as minimum labor hours and a required ratio of full-time to part-time employees, into linear inequalities. These restrictions ensure that the solution is practical and meets operational standards.

Linear Programming

Linear programming is an optimization technique used to achieve the best outcome—such as maximum profit or lowest cost—in a mathematical model whose requirements are represented by linear relationships. It involves formulating the problem through an objective function and a set of inequalities or equations as constraints, and is widely applied in operations research and management sciences.

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