Math CentralQuandaries & Queries


Question from Kelley, a student:

A manufacturer of skis produces two types: downhill and cross-country. Use the following table to determine how many of each kind of ski should be produced to achieve a maximum profit. What is the maximum profit? What would the maximum profit be if the time available for manufacturing is increased to 48 hours.

  Downhill Cross-country time available
manufacturing time per ski $2$ hrs $1$ hr $40$ hr
finishing time per ski $1$ hr $1$ hr $32$ hr
profit per ski $\$70$ $\$50$  

Hi Kelley,

Suppose you produce $d$ downhill skis and $c$ cross-country skis. Since the profit on each downhill ski is $\$70$ and the profit on each cross-country ski is $\$50$ the total profit is

\[P = 70 d + 50 c \mbox{ dollars.}\]

This is the objective function you are to maximize and notice that it is a linear function of $d$ and $c.$

There are some constraints in your manufacturing process. First of all $c$ and $d$ can't be negative so

\[c \ge 0 \mbox{ and } d \ge 0.\]

You are also constrained by the time available. It takes $2$ hours to manufacture a downhill ski and $1$ hour to produce a cross-country ski. You only have $40$ hours available so

\[2 d + 1 c \le 40.\]

There is a similar constraint inequality that comes from the finishing time. What is it?

Since the objective function and the four constraint inequalities are linear this is a linear programming problem.

Plot the region in the plane described by the four constraint inequalities. The theory of linear programming tells you that the maximum value of the objective function occurs at one of the vertices of the region described by the constraint inequalities. Evaluate the objective function at each of the vertices and decide which gives you the maximum profit.

Now repeat the problem with the time available for manufacturing increased to 48 hours.


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