Question

You are starting a Christmas tree business. You can purchase trees from local farmers at $25 per tree. You are able to sell the trees at $75 per tree. Any leftover trees can be sold to the local wood chip company at $5 per tree.

Assume that the demand follows a Normal distribution ∼N(327,31), which means a mean of 327 and a standard deviation of 31. How many trees should you stock through the holiday season?

Answer 1

The selling price per tree = $75

The cost price per tree = $25

The salvage value per tree = $5

The cost of overestimating demand or overage (C_o) = Cost price per tree - Salvage value per tree

= 25 - 5

= $20

The cost of underestimating demand or underage (C_u) = Selling price per tree - Cost price per tree

= 75 - 25

= $50

We will now calculate the critical ratio and will use the value to find the Z statistic value from a normal distribution table.

Critical ratio = C_u / (C_u + C_o)

= 50 / (50 + 20)

= 0.7142

The Z statistic for a critical ratio value of 0.7142 is 0.57 which we will use to calculate the number of trees that we should purchase so as to maximize expected profit.

The mean of demand = 327 units

The standard deviation of demand = 31 units

The number of trees that we should purchase so as to maximize expected profit = Mean + Z * Standard deviation

= 327 + 0.57 * 31

= 344.67 trees i.e. 345 trees