Chapter 7.6, Problem 37E

To determine

To find: the optimal inventory level for the model, to find the optimal profit.

Answer to Problem 37E

The optimal profit is $8295 and the optimal inventory level is 230 units of $225 model and 45 units of $250 model.

Explanation of Solution

Given:

A merchant plans to sell two models of MP3 players at prices of $225 and $250. The $225 model yield a profit of $30 per unit and the $250 model yields a profit of $31 per unit. The merchant estimates that the total monthly demand will not exceed 275 units. The maximum investment is $63000 in inventory for these products.

Calculation:

Let $x$ be number of MP3 model pieces produced which has the cost $225 and $y$ be the number of pieces produced which is having the cost of $250. So the objective function is given by:

  $p=30x+31y$

The two constraints can be converted into linear inequalities as:

  $\begin{array}{l}x+y\le 275\\ 225x+250y\le 63000\end{array}$

Since the number of pieces produced cannot be negative, we can get two more constraints of

  $x\ge 0$ and $y\ge 0$ .

The area bounded by the constraints are shown below:

  

Now, to find the co-ordinates of the point A where the two lines intersect. The lines are:

  $\begin{array}{l}x+y=\mathrm{275.............}(1)\\ 225x+250y=\mathrm{63000.........}(2)\end{array}$

From equation (1),

  $x=275-y\mathrm{............}(3)$

From equation (2),

  $x=280-\frac{10}{9}y$ ..........(4)

At the intersection point $x$ values are same. So we can equate (3) and (4) and we get , $y$ as 25 substituting this value in equation (3), we get the value of $x$ as 230.

So the co-ordinates of the intersection point is $\left(230,45\right)$

At the four vertices of the region formed by the constraints the objective function has the following values:

At $\left(0,0\right):p=30\left(0\right)+31\left(0\right)=0$ Minimum value of $p$

At $\left(0,252\right):p=30\left(0\right)+31\left(252\right)=7812$

At $\left(230,45\right):p=30\left(230\right)+31\left(45\right)=8295$ Maximum value of $p$

At $\left(275,0\right):p=30\left(275\right)+31\left(0\right)8250$

So the optimal profit is $8295 and the optimal inventory level is 230 units of $225 model and 45 units of $250 model.