A product with an annual demand of 1000 units has Co = $25.50 and Ch = $8. The demand exhibits some variability such that the lead-time demand follows a normal probability distribution with μ = 25 and σ = 5. 1. What is the recommended order quantity? 2. What are the reorder point and safety stock if the firm desires at most a 2% probability of stock-out on any given order cycle? 3. If a manager sets the reorder point at 30, what is the probability of a stock-out on any given order cycle? How many times would you expect a stock-out during the year if this reorder point were used?
sdsds
To answer these questions, we need to use the economic order quantity (EOQ), reorder point, safety stock, and probability calculations. Let's go through each question step by step:
1. Recommended Order Quantity:
The formula for EOQ is:
EOQ = √[(2 * Co * D) / Ch]
Where:
Co = Ordering cost per order
D = Annual demand
Ch = Holding cost per unit per year
Given the values:
Co = $25.50
D = 1000 units
Ch = $8
Using the formula, we can calculate the EOQ:
EOQ = √[(2 * 25.50 * 1000) / 8]
EOQ = √(51000 / 8)
EOQ = √6375
EOQ ≈ 79.92
Therefore, the recommended order quantity is approximately 80 units.
2. Reorder Point and Safety Stock:
The reorder point is the inventory level at which a new order should be placed. To calculate the reorder point, we need to consider the lead time demand, which follows a normal distribution.
Given the parameters:
μ (mean) = 25
σ (standard deviation) = 5
Desired service level = 1 - probability of stock-out
First, we need to determine the z-score corresponding to the desired service level. Since the demand follows a normal distribution, we can use a z-table or invNorm function in a statistical software to find the z-score.
In this case, the desired service level is 1 - 0.02 = 0.98. The z-score corresponding to this service level is approximately 2.05.
Reorder Point = μ + (z * σ)
Reorder Point = 25 + (2.05 * 5)
Reorder Point = 25 + 10.25
Reorder Point ≈ 35.25
Now, we can calculate the safety stock, which is the additional inventory held to mitigate stock-out risk during the lead time. It is the difference between the reorder point and the average demand during lead time.
Safety Stock = Reorder Point - (μ * Lead Time)
Safety Stock = 35.25 - (25 * 1)
Safety Stock ≈ 10.25
Therefore, the reorder point is approximately 35 units, and the safety stock is approximately 10 units.
3. Probability of Stock-Out and Expected Stock-Outs:
If the manager sets the reorder point at 30 units, we can calculate the probability of stock-out using the normal distribution.
Probability of Stock-Out = P(Lead Time Demand > Reorder Point)
Probability of Stock-Out = P(Lead Time Demand > 30)
We need to calculate the z-score using the same formula mentioned earlier with a different value for the reorder point.
z = (30 - μ) / σ
z = (30 - 25) / 5
z = 1
Now, looking up the z-score of 1 in the z-table, we find that the corresponding probability is approximately 0.8413.
Therefore, the probability of stock-out on any given order cycle would be approximately 1 - 0.8413 = 0.1587, or 15.87%.
To calculate the expected number of stock-outs during the year, we need to multiply the probability of stock-out by the number of order cycles in a year. Assuming the demand is evenly spread throughout the year, we divide the annual demand by the EOQ.
Number of Order Cycles = D / EOQ
Number of Order Cycles = 1000 / 80
Number of Order Cycles = 12.5 (approximately)
Expected Stock-Outs = Probability of Stock-Out * Number of Order Cycles
Expected Stock-Outs = 0.1587 * 12.5
Expected Stock-Outs ≈ 1.98 (approximately)
Therefore, if the reorder point is set at 30 units, we can expect approximately 2 stock-outs during the year.