1. What if Products are Perishable?
Newsboy model:
-Too much inventory cause high cost of wastes
-Too few inventory result in high cost of shortages
Key trade-off is between over-stocking cost and under-stocking cost
Using Newsboy model, we can find the optimal service level as well as the optimal inventory level.
2. service level
Service Level is the probability that inventory is sufficient (i.e., there is no shortage/stockout).
Service level must be set between 0% and 100%.
Suppose that you decide to hold an inventory Q against customers demand D, which is unknown.
If you decide to hold a small inventory Q, the inventory Q is likely to be insufficient for the demand D: Low service level
If you decide to hold a large inventory Q, the inventory Q is likely to be sufficient for the demand D: High service level
3. Suppose that Q is inventory and D is customers’ demand.
Service Level = P(D ≤ Q)
4. Trade-Off in Newsboy Problem
Co : Over-stocking cost per unit: When inventory Q exceeds customers’ demand D (“over-stocking”), how much would be the wastage cost if you have one extra unit?
Cu : Under-stocking cost per unit: When inventory Q is below customers’ demand D (“under-stocking”), how much would be the shortage cost (or lost sales) if you have one less unit?
5. marginal analysis:
we increase one unit in inventory and see whether this one unit improves our profit or not. If it improves our profit, we should add one unit. If not, we should not. We keep adding one unit repeatedly until we find an additional unit negatively affects our profit. The inventory level we stop adding extra unit is the optimal inventory level.
•Expected cost for adding one extra unit to the current inventory Q is P(D ≤ Q) × Co
•Expected benefit for adding one extra unit to the current inventory Q is P(D > Q) × Cu
•We should add one extra unit to the current inventory Q if expected benefit exceeds expected cost: (D ≤ Q) × Co < P(D > Q) × Cu
•We should stop adding an extra unit to the current inventory Q if expected cost exceeds expected benefit: (D ≤ Q) × Co > P(D > Q) × Cu
(Note: The optimal Q* is the smallest Q that satisfies the condition (4).)
6. Finding Optimal Inventory Q*
We consider two cases: distribution of demand D is (1) discrete and given, and (2) Normal.
(1) If distribution of demand D is discrete and given
Q* = smallest quantity that the service level is at least Cu / (Cu+Co)
(2) If distribution of demand D is assumed to be Normal N (μ, σ)
z = NORMSINV(Cu / (Cu+Co))
Q* = μ + z×σ