You are a newsvendor selling San Pedro Times every morning. Before you get to work, you go to the printer and buy the day’s paper for $0.45 a copy. You sell a copy of San Pedro Times for $1.40. Daily demand is distributed normally with mean = 340 and standard deviation = 68. At the end of each morning, any leftover copies are worthless and they go to a recycle bin.
a. How many copies of San Pedro Times should you buy each morning? (Use Excel's NORMSINV() function to find the correct critical value for the given α-level. Round your z-value to 2 decimal places and final answer to to 2 decimal places.)
b. Based on a, what is the probability that you will run out of stock by %? (Round your answer to the nearest whole number.)
a. To determine how many copies of San Pedro Times you should buy each morning, you need to find the optimal order quantity that minimizes the expected cost.
The expected cost consists of two components: the cost of purchasing the newspapers and the cost of lost sales due to stockouts.
Let's denote the order quantity as Q. The cost of purchasing the newspapers is $0.45 per copy, so the cost of purchasing Q copies is 0.45Q.
The cost of lost sales due to stockouts can be calculated using the formula:
Cost of lost sales = (1 - Probability of meeting demand) * (Revenue per copy - Cost per copy)
To calculate the probability of meeting demand, we need to convert the daily demand distribution to a standard normal distribution.
The Z-score formula is given by:
Z = (X - mean) / standard deviation
where X is the demand level.
From the question, we know that the mean demand is 340 and the standard deviation is 68.
To find the critical value for the given α-level, we can use Excel's NORMSINV() function. Let's assume the α-level is 0.05 (95% level of confidence).
Critical value (Z) = NORMSINV(1 - α) = NORMSINV(1 - 0.05) = NORMSINV(0.95)
Using Excel, the value of NORMSINV(0.95) is approximately 1.645.
So, the Z-value for the given α-level is 1.645.
Now, we can calculate the probability of meeting demand using the Z-score formula:
Z = (X - mean) / standard deviation
1.645 = (X - 340) / 68
Solving for X:
X - 340 = 1.645 * 68
X - 340 = 111.66
X = 340 + 111.66
X = 451.66
So, the probability of meeting demand is the probability that the demand is less than or equal to 451.66.
Using the standard normal distribution table or probability calculator, we can find that the probability is approximately 0.924.
Therefore, the probability of not meeting demand is 1 - 0.924 = 0.076.
Now, let's calculate the cost of lost sales:
Cost of lost sales = 0.076 * (1.40 - 0.45) * Q
= 0.076 * 0.95 * Q
= 0.0722 * Q
The expected cost can be calculated as the sum of the cost of purchasing newspapers and the cost of lost sales:
Expected cost = 0.45Q + 0.0722Q
= 0.5222Q
To minimize the expected cost, we differentiate with respect to Q and set it equal to zero:
d(0.5222Q) / dQ = 0
0.5222 = 0
Therefore, the optimal order quantity that minimizes the expected cost is 0.
So, you should buy 0 copies of San Pedro Times each morning.
b. Since the optimal order quantity is 0, the probability of running out of stock is 100%.
a. To determine the number of copies of San Pedro Times that should be bought each morning, we need to find the optimal order quantity that maximizes expected profit.
The optimal order quantity can be determined using the Newsvendor Model formula:
Q* = μ + zσ
where Q* is the optimal order quantity, μ is the mean demand, σ is the standard deviation of demand, and z is the critical value corresponding to the desired service level (α-level).
From the given information, we have:
μ = 340
σ = 68
α-level (service level) = 1 - α = 1 - 0.60 = 0.40 (Since the critical value is for the service level, which is the complement of α.)
Using Excel's NORMSINV() function, we can find the z-value corresponding to an α-level of 0.40.
Z = NORMSINV(0.40) = -0.25 (rounded to 2 decimal places)
Substituting the values into the formula:
Q* = μ + zσ
Q* = 340 + (-0.25) * 68
Q* ≈ 340 - 17
Q* ≈ 323 (rounded to the nearest whole number)
Therefore, you should buy approximately 323 copies of San Pedro Times each morning.
b. To calculate the probability of running out of stock, we need to find the area under the demand distribution curve to the right of the optimal order quantity (Q*).
Since demand follows a normal distribution, we can use the z-score formula:
z = (x - μ) / σ
where z is the z-score, x is the desired value (Q*), μ is the mean demand, and σ is the standard deviation of demand.
Substituting the values into the formula:
z = (323 - 340) / 68
z = -0.25 (rounded to 2 decimal places)
To find the probability of running out of stock, we need to calculate the area to the right of the z-score of -0.25.
Using the cumulative distribution function (CDF) of the standard normal distribution, we can find this probability.
P(z > -0.25) ≈ 1 - P(z ≤ -0.25)
Using Excel's NORMDIST() function, we can find the probability.
P(z ≤ -0.25) = NORMDIST(-0.25,0,1,1) ≈ 0.401
P(z > -0.25) ≈ 1 - 0.401
P(z > -0.25) ≈ 0.599
To convert the probability to a percentage, we multiply by 100:
0.599 * 100 ≈ 59.9
Therefore, the probability of running out of stock is approximately 59.9%.