If a firm's safety stock is equal to one-standard-deviation, then what is the implied Cu relative to Co?

Cu Co graph-1.png

Question

If a firm's safety stock is equal to one-standard-deviation, then what is the implied Cu relative to Co?

Cu Co graph-1.png

BlackTom AI · Accepted Answer

To analyze this, we need to connect safety stock, service level, and the cost ratio in a newsvendor-like setting.
First, recall that if safety stock is set to correspond to one standard deviation of demand during lead time, the service level associated with that z-score (for a normal distribution) is about 84%. That is because z = 1 yields a cumulative probability of roughly 0.84.
In the classic newsvendor model, the critical ratio (service level) is equal to Cu / (Cu + Co). Here the service level is 0.84, so: Cu / (Cu + Co) = 0.84.
Solving for the ratio Cu/Co: Cu = 0.84(Cu + Co) => Cu = 0.84Cu + 0.84Co => 0.16Cu = 0.84Co => Cu/Co = 0.84 / 0.16 ≈ 5.25.
Thus Cu is approximately five times Co, i.e., Cu ≈ 5 Co.
Now, evaluating the options:
- Cu = 5 Co: This aligns with the derived ratio (close to 5.25), so it is plausible and consistent with a one-standard-deviation safety stock implying an 84% service level.
- Cu = 10 Co: This would imply a higher service level than 84% (closer to 0.91), not matching the one-standard-deviation assumption.
- Cu = 100 Co: This would indicate an extremely high underage cost relative to overage, inconsistent with the modest z = 1 service level.
- Cu = Co: This corresponds to a service level of 50%, which is far below the 84% implied by one standard deviation.
In summary, the option Cu = 5 Co best matches the relationship between one-standard-deviation safety stock and the implied service level via the critical ratio.

OSC 3050-007 Fall 2025 Homework: Too Much, too Little / Inventory

If a firm's safety stock is equal to one-standard-deviation, then what is the implied Cu relative to Co? Cu Co graph-1.png

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