Question

Demand for a six-foot trees at Home Depot is approximated by a random variable with density function given below (units in hundred trees). Suppose Home Depot buys six-foot trees for $20 and sells them for $45. Any trees not sold are shredded. Find the number of trees that should be ordered to maximize expected profit.

cx, for 5 <= x <= 10

F(x)=

0, elsewhere

Answer 1

Solution

Back-up Theory

Expected profit = [Σ{Profit (p)} x P(p)], sum over all possible values of p.

Now, to work out the solution,

Let q = number of trees ordered and x = demand for trees.

Then, using the given density function, the numerical probabilities are as follows:

X (in hundreds)	5	6	7	8	9	10	Total
Probability density	5c	6c	7c	8c	9c	10c	45c
Numerical Probability	0.1111	0.1333	0.1556	0.1778	0.2000	0.2222	1.0000

Note that the total probability must be 1. Hence, 45c = 1 or c = 1/45.

Now, profit

= (45 - 20) = 25q if it sells, i.e., x ≥ q

= 25x - (q - x)(20) = 5x – 20q if x < q

Expected profit

q			x
	5	6	7	8	9	10
5	125	125	125	125	125	125
6	105	150	150	150	150	150
7	85	130	175	175	175	175
8	65	110	155	200	200	200
9	45	90	135	180	225	225
10.00	25	70	115	160	205	250
Probability	0.1111	0.1333	0.1556	0.1778	0.2000	0.2222	Expd Profit
profit x probability - q = 5	13.889	16.667	19.444	22.225	25.000	27.778	125.003
6	11.667	20.000	23.333	26.670	30.000	33.333	145.003
7	9.444	17.333	27.222	31.115	35.000	38.889	159.004
8	7.222	14.667	24.111	35.560	40.000	44.444	166.004
9	5.000	12.000	21.000	32.004	45.000	50.000	165.004
10	2.778	9.333	17.889	28.448	41.000	55.556	155.004

Since expected profit of $16600.40 is maximum, optimum number of trees to be ordered is 8. Answer

DONE