Shift finished

Stars: 1 = at least 90 % of customers served. 2 = average queue wait under 3 minutes and at most 5 % spoiled. 3 = at most 2 % of sales lost to empty shelves and every price in the fair range.

The science

Everything here is a model. Numbers marked "model" are chosen for the game, not measured in real shops.

1. Reorder point and safety stock

Demand over the lead time is roughly normal, so the reorder point is s = d·L + z·σ·√L, with z = Φ-1(0.95) = 1.645. Example: d = 12, σ = 4, L = 2 gives s = 33.3 and safety stock 9.3. When the inventory position falls to s, order up to S = s + Q. In the game d and σ are estimated from the customer model (model values).

2. Checkout queue (Erlang C, M/M/c)

Customers arrive at random at rate λ, each cashier serves at rate μ. With a = λ/μ, rho = a/c < 1: P_wait = [ac/(c!(1-rho))] / [Σk<c ak/k! + ac/(c!(1-rho))] and W_q = P_wait/(cμ - λ). Little's law L = λW links queue length and wait. Example: λ = 30/h, μ = 12/h, c = 3 gives P_wait = 0.702 and W_q = 7.0 min.

3. Spoilage with FIFO

Each batch has an expiry time. Sales take the oldest batch first; expired units are removed and counted as spoilage at cost q·c. Units are simulated per batch.

4. Price and demand

Demand follows a constant-elasticity curve D(P) = D_ref·(P/P_ref)-ε (ε is a model value). Profit per day is (P - c)·D(P); for ε = 2 the best price is P* = c·ε/(ε-1) = 2c.

Simplified on purpose

• No staff schedule; cashiers never tire.
• Customers do not choose stores by queue length and never walk out (kids mode).
• Shelf fit is by facings and layers, not by weight or falling products.
• One shift is one day; shelf life is shortened to hours for a short game.

Independent checks