How a Simple Weekly Restocking Rule Reduces Piano Stockouts

Problem and Background

A piano shop sells very few units and keeps low inventory to avoid capital lock‑up. Based on experience the average weekly demand is one piano. The restocking rule is: at the end of each week, if inventory is zero, order three pianos for the next week; otherwise, order nothing. The goal is to estimate the probability of lost sales and the average weekly sales under this policy.

Analysis

Customer arrivals are independent and weekly demand follows a Poisson distribution with mean 1. The weekly inventory level at the weekend determines whether an order of three units is placed. By modeling the weekly beginning‑of‑week inventory as a Markov chain, we can compute steady‑state probabilities and thus the long‑run stockout probability and average sales.

Model Assumptions

Weekly demand ~ Poisson(λ = 1). Restocking policy: if weekend inventory = 0, order 3 units arriving at the start of the next week; otherwise, no order. The beginning‑of‑week inventory is the state variable, and transitions are memoryless.

Model Construction

State D n : weekly demand, Poisson(1).

State I n : inventory at the beginning of week n (the Markov state).

State transition rules are derived from the demand realization and the ordering policy.

Model Solution

Estimate the probability of a stockout under the policy.

In the long run, the probability of losing a sale is about 10%.

Estimate the average weekly sales under the policy.

The steady‑state average weekly sales are approximately 0.857 pianos.

Sensitivity Analysis

When the mean demand fluctuates around 1 piano per week, the results change modestly. If the mean demand increases (or decreases) by 10%, the stockout probability changes by roughly ±12%.