Question

the manufacturer of a particular bicycle model has the following costs associated with the management of this products inventory. in particular, the company currently maintains an inventory of 1000 units of this bicycle model at the beginning of each year. if x units are demanded each year and x is less than 1000, the excess supply, 1000 - x units, must be stored until next year at a cost of $50 per unit. if x is greater than 1000 units, the excess demand, x - 1000 units, must be produced separately at an extra cost of $80 per unit. assume that the annual demand (x) for this bicycle model is normally distributed with mean 1000 and standard deviation 75. a. find the expected annual cost associated with managing potential shortages or surpluses of this product. use simulation to approximate the answer. run the simulation 10,000 times. an exact solution using probability arguments is beyond the level of this book. (hint: ) b. find two annual total cost levels, equidistant from the expected value found in part a, such that 95% of all costs associated with managing potential shortages or surpluses of this product are between these values. round your answers to the nearest dollar, if necessary. (continue to use simulation.) 2.5th percentile $11790 $11870 $23660 97.5th percentile $119 $48401 c. comment on this manufacturers annual production policy for this bicycle model in light of your findings in part b. the total cost can vary greatly, but its mean is quite stable

Explanation:

Step1: Define cost - function

Let the cost function $C(X)$ be calculated as follows. If $X\geq1000$, the cost of surplus is $80(X - 1000)$; if $X<1000$, the cost of shortage is $50(1000 - X)$. The annual demand $X$ is normally distributed with mean $\mu = 1000$ and standard - deviation $\sigma=75$.

Step2: Set up simulation

Use a software or programming language (e.g., Python with NumPy and Sci - Py) to generate 10000 random numbers from a normal distribution with mean 1000 and standard deviation 75 to represent the annual demand $X$. For each simulated value of $X$, calculate the cost $C(X)$ according to the above - defined cost function.

Step3: Calculate expected cost

Sum up all the calculated costs from the 10000 simulations and divide by 10000 to get the expected annual cost.
Let's assume we use Python code:

import numpy as np
n_simulations = 10000
mu = 1000
sigma = 75
total_cost = 0
for _ in range(n_simulations):
    X = np.random.normal(mu, sigma)
    if X >= 1000:
        cost = 80*(X - 1000)
    else:
        cost = 50*(1000 - X)
    total_cost += cost
expected_cost = total_cost/n_simulations

Step4: For part b

Since the distribution is normal, we know that for a normal distribution $N(\mu,\sigma)$, the $z$ - scores corresponding to the 2.5th and 97.5th percentiles are $z_{1}=- 1.96$ and $z_{2}=1.96$ respectively.
We use the formula $X=\mu+z\sigma$.
For the 2.5th percentile: $X_1=\mu + z_1\sigma=1000-1.96\times75 = 1000 - 147=853$.
For the 97.5th percentile: $X_2=\mu + z_2\sigma=1000 + 1.96\times75=1000 + 147 = 1147$.
The two equidistant values from the expected value (mean = 1000) are 853 and 1147.

Step5: Comment on production policy

The large variation in total cost (from the simulation results and the wide range of percentiles) indicates that the manufacturer's annual production policy for this bicycle model is risky. The mean cost is stable, but the spread of possible costs is significant, which means there is a high degree of uncertainty in the cost associated with managing shortages and surpluses.

Answer:

a. The expected annual cost can be found using the simulation as described above. (The actual numerical value depends on the simulation results. In the Python code example, the variable expected_cost holds the result).
b. The two annual total cost levels equidistant from the expected value are associated with demand levels of 853 and 1147 units.
c. The manufacturer's annual production policy is risky due to high cost - variability despite a stable mean cost.