From Forecast to Decision: Take It Further¶
Five extensions of the Chapter 139 decision, each probing how the order should change: when the fractile flips below the forecast, how the order tracks the cost ratio, using an empirical predictive distribution, the service-level tradeoff, and proof that a more accurate forecast yields a cheaper decision. We start by rebuilding the forecast and costs.
import numpy as np, pandas as pd, matplotlib.pyplot as plt
import seaborn as sns # seaborn = high-level statistical plots (heatmaps, pairplots, count/bar plots)
from matplotlib.colors import ListedColormap
EM="#0284c7"; DEEP="#075985"; LIGHT="#bae6fd"; INK="#1a2138"; GRID="#e6e9f2"; RED="#ef4444"; AMBER="#d97706"; GREEN="#059669"; BLUE="#2563eb"; PUR="#9333ea"; GREY="#94a3b8"; SLATE="#475569"; ORG="#0284c7"; CYAN="#0891b2"
plt.rcParams.update({"figure.facecolor":"white","axes.facecolor":"white","figure.dpi":110,"font.size":11,
"axes.edgecolor":GRID,"axes.grid":True,"grid.color":GRID,"axes.axisbelow":True,"axes.spines.top":False,
"axes.spines.right":False,"axes.titlesize":12,"axes.titleweight":"bold","legend.frameon":False})
sns.set_style("whitegrid")
BASE="https://raw.githubusercontent.com/johnfisher-ai/Statistics-Data-Science-AI-Visual-Book/main/data/"
from scipy import stats
from statsmodels.tsa.holtwinters import ExponentialSmoothing
import warnings; warnings.filterwarnings('ignore')
try: raw = pd.read_excel('../../data/case-study-from-forecast-to-decision--product_demand.xlsx', sheet_name='Data')
except FileNotFoundError: raw = pd.read_excel(BASE + 'case-study-from-forecast-to-decision--product_demand.xlsx', sheet_name='Data')
raw['date']=pd.to_datetime(raw['date']); s = raw.set_index('date')['demand'].asfreq('D')
H=60; train,test=s[:-H],s[-H:]
hw = ExponentialSmoothing(train, trend='add', seasonal='add', seasonal_periods=7).fit(); hw_fc = hw.forecast(H).values
resid = train.values - hw.fittedvalues.values; sd = float(np.std(resid))
def cost(demand, order, Cu, Co): return Cu*np.maximum(demand-order,0) + Co*np.maximum(order-demand,0)
def order_q(Cu, Co): return hw_fc + stats.norm.ppf(Cu/(Cu+Co))*sd
print('rebuilt: HW forecast + error sd %.1f units' % sd)
rebuilt: HW forecast + error sd 26.6 units
for Cu,Co,label in [(8,2,'stockout expensive'),(2,8,'waste expensive (perishable)')]:
fr=Cu/(Cu+Co); buf=stats.norm.ppf(fr)*sd
print('%-28s fractile %.2f -> order %+.0f units vs forecast' % (label, fr, buf))
print('when leftovers cost more than stockouts, the optimal order drops BELOW the forecast -> deliberately risk selling out')
stockout expensive fractile 0.80 -> order +22 units vs forecast waste expensive (perishable) fractile 0.20 -> order -22 units vs forecast when leftovers cost more than stockouts, the optimal order drops BELOW the forecast -> deliberately risk selling out
The critical fractile can point either way. When stockouts are the expensive error (Cu > Co) you carry a buffer above the forecast; when waste is expensive (Co > Cu, a highly perishable item) the fractile falls below 0.5 and you order below the forecast, accepting some stockouts to avoid costly spoilage. Same forecast, opposite decision, set entirely by the cost ratio.
ratios=[1,2,4,8,16]; buffers=[stats.norm.ppf(r/(r+1))*sd for r in ratios]
fig,ax=plt.subplots(figsize=(8,3.9)); ax.plot(ratios, buffers, 'o-', color=EM, lw=2)
ax.axhline(0,color=GREY); ax.set(xscale='log', xlabel='cost ratio Cu : Co', ylabel='safety buffer above forecast (units)', title='The safety buffer grows with how much stockouts hurt')
ax.set_xticks(ratios); ax.set_xticklabels([f'{r}:1' for r in ratios]); plt.tight_layout(); plt.show()
print('ratio 1:1 -> order the forecast (no buffer); 8:1 -> a large buffer. The decision is a dial on the cost ratio.')
ratio 1:1 -> order the forecast (no buffer); 8:1 -> a large buffer. The decision is a dial on the cost ratio.
The optimal order is a smooth dial on the cost ratio. At 1:1 (errors equally costly) you order the forecast exactly; the more a stockout hurts relative to a leftover, the bigger the buffer. Making the cost ratio explicit turns a fuzzy judgment call (“keep a bit extra”) into a precise, defensible number.
Cu,Co=8.0,2.0; fr=Cu/(Cu+Co)
q_normal = hw_fc + stats.norm.ppf(fr)*sd
q_emp = hw_fc + np.quantile(resid, fr) # empirical residual quantile
print('80th-percentile order, normal assumption : mean %.0f units' % q_normal.mean())
print('80th-percentile order, empirical residuals: mean %.0f units' % q_emp.mean())
print('avg cost normal %.0f vs empirical %.0f' % (cost(test.values,q_normal,Cu,Co).mean(), cost(test.values,q_emp,Cu,Co).mean()))
print('if the errors are skewed or fat-tailed, the empirical quantile is safer than a normal one')
80th-percentile order, normal assumption : mean 252 units 80th-percentile order, empirical residuals: mean 252 units avg cost normal 69 vs empirical 69 if the errors are skewed or fat-tailed, the empirical quantile is safer than a normal one
The normal shortcut (forecast + z × sd) assumes symmetric, bell-shaped errors. Real forecast errors are often skewed or fat-tailed, so a more robust order uses the empirical quantile of the backtest residuals directly. Here the two agree (these errors are roughly symmetric), but on skewed real-world demand they diverge, and when they do you should trust the data over the bell curve, because the tail is where the expensive mistakes live.
frs=np.linspace(0.5,0.98,40)
svc=[np.mean(test.values <= hw_fc+stats.norm.ppf(q)*sd)*100 for q in frs]
cst=[cost(test.values, hw_fc+stats.norm.ppf(q)*sd, 8,2).mean() for q in frs]
fig,ax=plt.subplots(figsize=(8.4,3.9)); ax2=ax.twinx()
ax.plot(frs*100, svc, color=EM, lw=2.2, label='service level %'); ax2.plot(frs*100, cst, color=RED, lw=2, ls='--', label='avg cost $')
ax.axvline(80, color=GREY, ls=':'); ax.set(xlabel='target fractile (percentile ordered)', ylabel='service level %'); ax2.set_ylabel('avg daily cost $')
ax.legend(loc='upper left'); ax2.legend(loc='lower right'); ax.set_title('Higher fractile -> higher service, until cost turns back up'); plt.tight_layout(); plt.show()
q80 = hw_fc + stats.norm.ppf(0.8)*sd
fill = np.minimum(test.values, q80).sum() / test.values.sum() * 100
print(f'at the 80% fractile: demand fully met on {np.mean(test.values <= q80)*100:.0f}% of days (cycle service level), {fill:.1f}% of units served (fill rate)')
at the 80% fractile: demand fully met on 83% of days (cycle service level), 98.9% of units served (fill rate)
The fractile is also a service-level knob: a higher order percentile means the shelf is stocked more often. The service level here is the share of days with demand fully met (cycle service level). Fill rate, the share of units demanded that are served, is a different measure and runs much higher (printed above), because a short day usually misses only a few units. But service and cost are linked, push the fractile too high and waste starts to dominate, so cost turns back up. The cost-optimal fractile (80% here) is the point that balances them; a manager can equally well pick a target service level and read off the order.
snaive = np.array([train.iloc[-7+(i%7)] for i in range(H)])
sd_sn = float(np.std(train.values[7:] - train.values[:-7])) # seasonal-naive one-step error spread
def order_for(fc, sdv, Cu=8, Co=2): return fc + stats.norm.ppf(Cu/(Cu+Co))*sdv
c_hw = cost(test.values, order_for(hw_fc, sd), 8,2).mean()
c_sn = cost(test.values, order_for(snaive, sd_sn), 8,2).mean()
print('decision cost using Holt-Winters forecast : $%.0f/day' % c_hw)
print('decision cost using seasonal-naive forecast: $%.0f/day' % c_sn)
print('the MORE ACCURATE forecast yields the CHEAPER decision -> accuracy is worth real money')
decision cost using Holt-Winters forecast : $69/day decision cost using seasonal-naive forecast: $193/day the MORE ACCURATE forecast yields the CHEAPER decision -> accuracy is worth real money
Finally, the payoff loop: feeding the less accurate forecast through the exact same newsvendor rule produces a more expensive decision. Forecast accuracy is not an academic score, it flows straight through to the cost of the orders you place. A better forecast, and the Diebold-Mariano test that proves it, is worth real money at the decision.
Take-it-further summary, in plain terms¶
- The fractile can flip below the forecast when waste costs more than stockouts.
- The buffer is a dial on the cost ratio: the more a stockout hurts, the more you carry.
- Use the empirical error quantile when forecast errors are skewed or fat-tailed.
- Service level and cost are the same dial viewed two ways; pick a target and read off the order.
- A more accurate forecast yields a cheaper decision, so accuracy is worth money.