Newsvendor Discrete Critical Fractile

The newsvendor model minimizes the expected mismatch cost between an inventory level and uncertain demand. It evaluates the expected shortage cost when demand exceeds stock and expected leftover cost when stock exceeds demand.

Worked Example

Consider a Uniform demand scenario where probabilities are 1/5 for each demand level from 0 to 4. Let Shortage cost (C_u) = 2 and Leftover cost (C_o) = 1. We test Stock Q = 3.

First, calculate expected shortage when D > 3. The only value is D=4 with p=1/5.

E[(D - 3)⁺] = (1/5) * (4 - 3) = 0.2

Next, expected leftover when D < 3 (i.e. D=0, 1, 2).

E[(3 - D)⁺] = (1/5)*(3 - 0) + (1/5)*(3 - 1) + (1/5)*(3 - 2) = (3 + 2 + 1)/5 = 6/5 = 1.2

The total Expected cost is:

2 * 0.2 + 1 * 1.2 = 0.4 + 1.2 = 1.6

The expected cost is 1.6.

Experiment

You can test various distributions using the interactive Newsvendor Discrete Critical Fractile module.

Set Demand to "Uniform".
Set Shortage cost to 2.
Set Leftover cost to 1.
Set Stock to 3. Notice the expected cost is 1.6, and the critical fractile is 0.6666666666666666.
Change Stock to 4 and observe the expected cost rise to 2.0.

Limitations

This model assumes all leftover inventory has a zero salvage value (or net salvage is baked into the leftover cost) and demand distribution is strictly discrete and finite (0 to 4). If the CDF stays equal to the critical fractile across multiple discrete support points, there can be multiple tying optimal integer stocks. Real purchasing might also consider holding costs across time instead of a single period.

References

For more details on the structural estimation of the newsvendor model, refer to this paper:

Structural Estimation of Newsvendor Models. Discusses theoretical and empirical applications of the newsvendor model.

Newsvendor Discrete Critical Fractile