Critical Ratio Explorer

The math

You choose one order quantity $Q$ before demand $D$ is known. Each unit you are short costs $c_u$; each unit left over costs $c_o$. The expected cost is

Raise $Q$ by one unit and two things change. With probability $F(Q)$ demand would not have reached that unit, so it becomes leftover and costs $c_o$. With probability $1-F(Q)$ it is sold and saves a shortage worth $c_u$. Keep adding units while the expected saving exceeds the expected cost, and stop where they balance:

That fraction is the critical ratio. The optimal order is the critical-ratio quantile of the demand distribution, $Q^* = F^{-1}\!\left(\tfrac{c_u}{c_u+c_o}\right)$. For normal demand that is $Q^* = \mu + z\,\sigma$ with $z = \Phi^{-1}(\text{CR})$. For discrete demand such as Poisson, $Q^*$ is the smallest integer with $F(Q^*) \ge \text{CR}$.

How to read it

• The ratio, not the costs, decides the quantile. Costs of 12 and 4 give the same answer as 3 and 1. Only how much worse one mistake is than the other matters.
• A ratio above one half orders above the mean. When running short costs more than running over, you buy insurance. Below one half, you order less than you expect to sell, because the leftover risk is not worth carrying.
• Uncertainty is priced, not padded. More variability pushes $Q^*$ further from the mean in whichever direction the ratio points. Low-ratio items can end up with less stock as demand gets noisier.
• The in-stock probability equals the ratio, the fill rate is higher. A 75% ratio means a 75% chance of not running out, but because shortfalls are usually small, the share of demand served is well above 75%.

Assumptions

• One decision, one period: leftover stock does not carry over to a next order. For repeated orders, the same logic applies per review period with holding cost in $c_o$; see the explainer on order economics and planning horizons.
• Linear costs: no fixed ordering cost, no minimum order, no quantity discount.
• No starting inventory, and the full order arrives before demand.
• The demand distribution is known. In practice it comes from a probabilistic forecast, and its error flows straight into $Q^*$.
• Normal demand is taken on the whole real line, as in the textbook model. When it puts noticeable probability below zero, the explorer says so; lognormal is the safer choice for skewed, non-negative demand. Orders are clipped at zero.
• When $c_o = 0$ the ratio is 1 and no finite order is optimal: excess is free, so more is never worse.

The same calculation in Python

from scipy.stats import norm, lognorm, poisson
import numpy as np

cu, co = 12.0, 4.0                     # underage, overage cost per unit
cr = cu / (cu + co)                    # critical ratio = 0.75

# Normal demand
mu, sigma = 500, 80
q_normal = max(0.0, norm.ppf(cr, loc=mu, scale=sigma))   # 553.96

# Lognormal demand with the same mean and standard deviation
s2 = np.log(1 + sigma**2 / mu**2)
q_lognormal = lognorm.ppf(cr, s=np.sqrt(s2), scale=np.exp(np.log(mu) - s2 / 2))

# Poisson demand: smallest integer with F(Q) >= CR
q_poisson = int(poisson.ppf(cr, mu))

The explorer computes everything in your browser with tested numerical routines: the normal CDF and its inverse agree with Python's reference implementation to better than $10^{-9}$, and the optimal quantity is checked against a brute-force minimization of expected cost.