Assume that you own a small factory. A critical piece of machinery in your factory will need to be replacedin 180 days. If the machinery does not show up on time, you will need to shut down until it arrives. This might cause you to permanately lose customers.When you order the part you wil need to pay the $500,000 in advance. That is a lot of money for your small business. If you keep the money in the bank, it will earn interest each month. If you spend the money now, it will leave you with very litte money on hand, and you might have to borrow money to make payroll. You know from past experience that the delivery times are normally distributed with a mean of 45 days and a standard deviation of 15 days. When should you order the part? It is your company, but you must write up an explanation for your actions that convinces your investors that your actions are best. (Unfortunately, the investors cannot afford $500,000 at this time.) Explain why 135 days from now is the date to order it if you want to be 50% sure of getting the delivery on time.
To determine the best date to order the critical machinery part, we need to consider the probability of receiving the delivery on time. In this scenario, we want to be 50% sure of getting the delivery on time.
Given that the delivery times are normally distributed with a mean of 45 days and a standard deviation of 15 days, we can use the properties of a normal distribution to find the desired order date.
To calculate the desired order date, we need to find the delivery time that corresponds to the probability of 50% in a normal distribution. This can be done using a Z-score table or a statistical software, but for this explanation, we will use the concept of Z-scores.
A Z-score tells us how many standard deviations a particular data point (in this case, the delivery time) is away from the mean. We want to find the Z-score that corresponds to a 50% probability. The Z-score corresponding to a 50% probability is 0.
The formula for calculating the Z-score is:
Z = (X - μ) / σ
Where:
Z is the Z-score,
X is the data point (delivery time),
μ is the mean (45 days),
σ is the standard deviation (15 days).
Rearranging the formula, we can solve for X:
X = Z * σ + μ
Since we want to be 50% sure of getting the delivery on time, and the corresponding Z-score is 0, we can calculate the desired delivery time:
X = 0 * 15 + 45 = 45 days
Therefore, if we order the part 45 days before the critical deadline, we will have a 50% chance of receiving the delivery on time.
However, to account for the time it takes for the bank transfer to process and the payment to reach the supplier, we need to order the part earlier. Assuming it takes around 5 working days for the payment process, we should order the part 45 days - 5 days = 40 days before the critical deadline.
Therefore, to have a 50% chance of getting the delivery on time, we should order the critical machinery part 40 days before the 180-day deadline, which corresponds to 180 - 40 = 140 days from now.
In conclusion, ordering the part 135 days from now (140 days from the deadline) will provide a 50% chance of receiving the delivery on time, allowing us to mitigate the risk of a delayed delivery and potential loss of customers.