Distance Decay Functions

The Question

“How quickly does customer interest drop off with distance?”

Three Functions

Power Function (most common)

f(d) = d^−β

Slow decay. For destination restaurants. β = 1.5-2.5 for food retail.

Exponential Function

f(d) = e^−βd

Fast decay. For convenience food (cha chaan teng, fast food). β = 0.5-1.0.

Gaussian Function

f(d) = e^−d²/2σ²

Bell curve. Flat nearby, then drops sharply. For mixed-use areas.

Source: de Smith, M., Goodchild, M. & Longley, P. — Geospatial Analysis (6th ed.)

Which One? Depends on Your Business

This is where Company Input directly changes the math:

Restaurant Type	Best Function	β	Why
Cha chaan teng / fast food	Exponential (β=0.8)	0.8	Convenience: won’t walk past a closer option
Mid-range unique cuisine	Power (β=2.0)	2.0	Destination: people walk 10+ min for something specific
High-end / special occasion	Power (β=1.2)	1.2	Quality dominates over distance
Takeaway / delivery	Flat to 2km, then exponential	—	Delivery radius IS the catchment

Caution

Choosing the wrong decay function can make a location look 3x better or worse than reality. A delivery-first concept has a completely different catchment than a dine-in-only cha chaan teng.

Key Insight

At β=2.0 (power), a restaurant 300m away is 9x less attractive than at 100m. At β=0.8 (exponential), the same restaurant is only 1.3x less attractive. The business model determines which world you live in.