Using the random numbers given below, simulate demand for 20 weeks and answer the following questions: (a) If Higgins maintains a constant supply of 8 hot water heaters in any give... Using the random numbers given below, simulate demand for 20 weeks and answer the following questions: (a) If Higgins maintains a constant supply of 8 hot water heaters in any given week, how many times will he be out of stock during the 20 week simulation period? (b) What is the average number of heaters demanded per week over the 20 week intervals? Read more

Steps to Solve

1. Identify Sales Probability Distribution

From the sales data:

• 4 heaters sold for 6 weeks
• 5 heaters sold for 5 weeks
• 6 heaters sold for 9 weeks
• 7 heaters sold for 12 weeks
• 8 heaters sold for 8 weeks
• 9 heaters sold for 7 weeks
• 10 heaters sold for 3 weeks

Now, let's find the probabilities for each sales count. First, we create a table with the number of heaters sold and their corresponding probabilities.

The total weeks = 50. The probability for each sales number is given by:

$$ P(X=k) = \frac{\text{Number of weeks}}{\text{Total weeks}} $$

The probabilities are:

• $P(4) = \frac{6}{50} = 0.12$
• $P(5) = \frac{5}{50} = 0.10$
• $P(6) = \frac{9}{50} = 0.18$
• $P(7) = \frac{12}{50} = 0.24$
• $P(8) = \frac{8}{50} = 0.16$
• $P(9) = \frac{7}{50} = 0.14$
• $P(10) = \frac{3}{50} = 0.06$

2. Simulate Weekly Demand

Using the random numbers provided (which are not seen in the image), assign each random number to a sales count based on the probability distribution.

For example, if the random number falls within the cumulative probability for 4 heaters sold (1-12), allocate 4 heaters. If it falls between 13-22, allocate 5 heaters, etc. Perform this for 20 weeks.

3. Count Stock Outs

Starting with an inventory of 8 heaters each week:

• Track the number of heaters sold per week from the simulation.
• If the number of heaters sold exceeds 8, record it as a stock out and calculate how many additional heaters were needed.

4. Calculate Average Demand

After simulating for 20 weeks, sum the total heaters sold and then calculate the average demand per week as follows:

$$ \text{Average Demand} = \frac{\text{Total Heaters Sold}}{20} $$