Respuesta :
The merchant should stock 3 items in order to maximize the expected daily profit of $0.48.
What is the expected daily profit on the items stocked?
The expected daily profit can be calculated by determining her daily profit per item sold and finding the probability distribution.
Suppose we make an assumption that:
- The merchant has two (2) items per day.
- Let the profit be represented by [tex]\mathbf{(X_2)}[/tex]
Now, the daily demand for 2 items will be:
= (1.2 - 1) × 2
= 0.4
Now for the probability of 0.1, with the daily demand of 2 items, she makes a profit of 0.4
Thus, the probability distribution for [tex]\mathbf{(X_2)}[/tex] is computed as:
x 0.4 0.4 0.4
P(x) 0.1 0.4 0.5
Thus, the expected value for [tex]\mathbf{(X_2)}[/tex] is:
[tex]\mathbf{E(X_2) = (0.4 \times 0.1 ) + (0.4 \times 0.4) + (0.4 \times 0.5)}[/tex]
[tex]\mathbf{E(X_2) =0.4}[/tex]
It implies that the expected daily profit will be 0.4 provided that the merchant stocks 2 items.
Similarly, if the merchant purchase 3 items each day, the total profit [tex]\mathbf{X_3}[/tex] is:
= ((1.2 - 1) × 2) - 1 (provided she loses $1 due to the 1 item left at day end
= -0.6
The total profit for the daily demand (3) is
= (1.2 - 1) × 3
= 0.6
Now for the probability of 0.1, with the daily demand of 2 items, she makes a profit of 0.4
Thus, the probability distribution for [tex]\mathbf{(X_3)}[/tex] is computed as:
x -0.6 0.6 0.6
P(x) 0.1 0.4 0.5
Thus, the expected value for [tex]\mathbf{(X_3)}[/tex] is:
[tex]\mathbf{E(X_3) = (-0.6 \times 0.1 ) + (0.6 \times 0.4) + (0.6 \times 0.5)}[/tex]
[tex]\mathbf{E(X_3) =0.48}[/tex]
It implies that the expected daily profit will be 0.48 provided that the merchant stocks 3 items.
Finally, for 4 items bought each day, the total profit is [tex]\mathbf{(X_4)}[/tex]
Suppose daily demand = 2:
Profit = (1.2 - 1) × 2
Profit = 0.4
Provided she loses $2 at the end of the day;
Total profit = 0.4 - 2 = -1.6
Suppose daily demand = 3:
Profit = (1.2 - 1) × 3
Profit = 0.6
Provided she loses $1 at the end of the day;
Total profit = 0.6 - 1 = -0.4
Suppose daily demand = 4:
Provided she did not lose any money at the end of the day;
Profit = (1.2 - 1) × 4
Profit = 0.8
Thus, the probability distribution for [tex]\mathbf{(X_4)}[/tex] is computed as:
x -1.6 0.6 0.8
P(x) 0.1 0.4 0.5
Thus, the expected value for [tex]\mathbf{(X_4)}[/tex] is:
[tex]\mathbf{E(X_4) = (-1.6 \times 0.1 ) -(0.4 \times 0.4) + (0.8 \times 0.5)}[/tex]
[tex]\mathbf{E(X_4) =0.08}[/tex]
It implies that the expected daily profit will be 0.08 provided that the merchant stocks 4 items.
Therefore, we can conclude that the merchant should stock 3 items in order to maximize the expected daily profit of $0.48.
Learn more about calculating expected daily profit and probability distribution here:
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