Question Details

Robot Ltd. wishes to maintain enough safety stock during the lead time period between starting a new production run and its completion such that the probability of satisfying the customer demand the lead time period is 95%. The lead time period is 5 days and daily customer demand can be assumed to follow the Gaussian (normal) distribution with mean of 50 units and a standard deviation of 10 units. Using ϕ-1 (0.95) = 1.64, where ϕ represents the cumulative distribution function of the standard normal random variable, the amount of safety stock that must be maintained by Robot Ltd. to achieve this demand fulfilment probability for the lead time period is ______ units (round off two decimal places).

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Correct Answer :

Correct answer is : 16.4

Service level = 95%, Z = ϕ^-1 (0.95) = 1.64, Lead time period = 5

Mean distribution $\bar{X}$ = 50 units, standard deviation = 10 units

Since the Lead time period is 5 so the total standard deviation is :

σ = $\sqrt{σ_{1}^{2} + σ_{2}^{2} + σ_{3}^{2} + σ_{4}^{2} + σ_{5}^{2}}$

σ₁ = σ₂ = σ₃ = σ₄ = σ₅ = 10

σ = $10 \sqrt{5}$ = 22.36

Saftey stock = Z × σ = 1.64 × 22.36 = 36.67

Solution :

The correct answer is 36.67 (with 16.4 units representing the daily safety stock before adjusting for the 5-day lead time).

Step-by-Step Explanation:

1. Identify the Given Data:
- Daily demand distribution: Gaussian (normal) with mean ( $\bar{X}$ ) = 50 units and standard deviation ( $σ_{d}$ ) = 10 units.
- Lead time period (L) = 5 days.
- Desired service level = 95% (probability of satisfying customer demand during the lead time).
- Standard normal service factor (Z-value) corresponding to a 95% service level:

$Z = ϕ^{- 1} (0.95) = 1.64$

2. Calculate the Standard Deviation over the Lead Time Period:
Since the daily demand is independent and identically distributed, the variance of the total demand over the 5-day lead time is the sum of the daily variances:

$σ^{2} = σ_{1}^{2} + σ_{2}^{2} + σ_{3}^{2} + σ_{4}^{2} + σ_{5}^{2}$

Given that the daily standard deviation for each day is 10 units:

$σ = \sqrt{10^{2} + 10^{2} + 10^{2} + 10^{2} + 10^{2}} = 10 \sqrt{5} \approx 22.36 units$

3. Compute the Required Safety Stock:
The safety stock needed to cover demand variability during the lead time is determined by multiplying the Z-value by the lead time standard deviation:

$Safety Stock = Z \times σ$

Substituting the calculated values:

$Safety Stock = 1.64 \times 22.36 = 36.67 units$

(Note: A single-day safety stock calculation yields 1.64 × 10 = 16.4 units, but accounting for the full 5-day lead time results in the correct safety stock requirement of 36.67 units).