A business firm produces two lines of product, I and II, with a plant that consists of three production departments: cutting, mixing, and packaging. The equipment in each department can be used for 8 hr a day; thus we shall regard 8 hr as the daily capacity in each department. The process of production can be summarized as follows: (1) Product I is first cut, then packaged. Each ton of this product uses up hr of the cutting capacity and hr of the packaging capacity. (2) Product II is first mixed, then packaged. Each ton of this product uses up 1 hr of the mixing capacity and hr of the packaging capacity. Finally, products I and II can be sold at prices of $80 and $60 per ton, respectively, but after deducting the variable costs incurred, they yield on a net basis $40 and $30 per ton. These latter amounts may be considered either as net-revenue figures (net of variable costs) or as gross-profit figures (gross of fixed costs). For simplicity, we shall refer to them here as "profits per ton." Problem: What output combination should the firm choose in order to maximize the total (gross) profit?

Write the LP problem longhand and solve it by using the graphical solution method (both the iso-profit lines method and the corner points method).

Plot the graph with all the required lines (constraints and iso-profit lines); be precise: write all the points' coordinates on the x₁ and x₂ axes [you can plot the graph on a paper and take a photo, or you can plot it directly in Word].

Longhand representation

Feasible region

First constraint

Equation:

Coordinates of the points of intersection with the x₁-axis and the x₂-axis:

Second constraint

Equation:

Coordinates of the points of intersection with the x₁-axis and the x₂-axis:

Third constraint

Equation:

Coordinates of the points of intersection with the x₁-axis and the x₂-axis:

Iso-profit solution method

Iso-profit at an arbitrary profit level 70

Equation:

Coordinates of the points of intersection with the x₁-axis and the x₂-axis

Optimal Iso-profit (detected on the graph)

Intersection of the two equations:

Coordinates of the points of intersection of the above two equations:

Profit:

Equation:

Coordinates of the points of intersection with the x₁-axis and the x₂-axis

Corner points solution method

When all the corner points have been determined, highlight in yellow the optimal one.

First corner point

Intersection of the two equations:

Coordinates of the points of intersection of the above two equations:

Profit:

Second corner point

Intersection of the two equations:

Coordinates of the points of intersection of the above two equations:

Profit:

Third corner point

Intersection of the two equations:

Coordinates of the points of intersection of the above two equations:

Profit:

Fourth corner point

Intersection of the two equations:

Coordinates of the points of intersection of the above two equations:

Profit:

Fifth corner point

Intersection of the two equations:

Coordinates of the points of intersection of the above two equations:

Profit:

Graph

(hint: x₁ axis ranging from 0 to 28 and x₂ axis ranging from 0 to 28)