Method of formulating a ***linear*** system of equations and ***linear*** constraints to find its optimal solution. # Wyndor Glass Example A glass product company has two main products in their lineup: a glass door with aluminum framing and a wood-framed window, each with their levels of profit The company owns three plants, each with varying production capacity for each product. Product 1 requires production hours from Plants 1 and 3, while Product 2 is made in Plants 2 and 3. Therefore the two products are competing for the same allocation of production hours in Plant 3. How many of each product will yield the highest profit? -tx- | | Production Time per Batch (hours) || | | :-----------------:|:---------------------------------:| :-----: | ------------------------------------- | | | **Product** || | | **Plant** | **1** | **2** | **Production Time Available per Week (hours)** | | 1 | 1 | 0 | 4 | | 2 | 0 | 2 | 12 | | 3 | 3 | 2 | 18 | | *Profit per batch* | $3,000 | $4,000 | | [Data for the Wyndor Glass Example] This can be written as: $ \begin{alignat}{3} \text{max} \quad &&& \mathrlap{z = 3x_1+5x_2}\\ \text{s.t} \quad &&&& x_1 &\leq 4 \\ &&&& 2x_2 &\leq 12 \\ &&&& 3x_1+2x_2 &\leq18 \\ &&&& x_1, x_2 &\geq 0 \end{alignat} $ Tools such as the [[Simplex Method]] can be used to determine the optimal solution.