How Math Models Can Turn Your Coffee Shop into a Profit Machine

1. Customer Flow Forecast Model

Accurate foot‑traffic estimates are essential for staffing, ingredient planning, and marketing. By using historical data, holidays, weather, and week number, a simple linear regression with the previous week’s flow and a holiday indicator can predict future customer numbers.

An example with ten weeks of data demonstrates how to fit the regression and forecast week 11’s traffic (non‑holiday, previous week 490 customers).

2. Menu Optimization Model

The menu is optimized with a linear‑programming model that balances profit, preparation time, cost, and customer satisfaction.

2.1 Model Elements

Decision variables: quantity of each coffee or food item.

Constraints: cost limits, preparation‑time limits, ingredient supply.

Objective: maximize total profit or customer satisfaction.

2.2 Case Study

Four coffee types (latte, mocha, cappuccino, Americano) are considered with given profit, cost, preparation time, and satisfaction scores. The model maximizes profit within an 8‑hour (480‑minute) workday and a daily budget of 3000 yuan while keeping average satisfaction ≥ 0.8.

The optimal solution (under the simplified data) suggests producing only latte (160 cups), yielding a total profit of 2400 yuan, illustrating how model simplifications can produce unrealistic recommendations.

3. Inventory Management Model

Effective inventory control reduces waste and cost. The classic Economic Order Quantity (EOQ) model is applied to coffee‑bean ordering.

3.1 Model Elements

Inputs: sales data, supply‑chain information.

Outputs: optimal order quantity and order frequency.

3.2 Case Study

Given a fixed ordering cost of 100 yuan, holding cost of 5 yuan per kilogram per month, and monthly demand uniformly distributed between 100 kg and 300 kg, the EOQ calculation yields an optimal order size of about 89.44 kg.

4. Pricing Strategy Model

Pricing directly impacts sales and profit. A demand‑elasticity model determines the optimal price for a special latte.

4.1 Model Elements

Inputs: historical sales, cost, competitor prices.

Output: optimal price.

4.2 Case Study

Data: at 10 yuan price, expected sales are 100 cups per day; each 1 yuan price increase reduces sales by 10 cups; production cost is 5 yuan per cup. The profit function is maximized at a price of 12.5 yuan, giving a maximum daily profit of 562.5 yuan.

4.3 Competitive Pricing Game

When competitor prices are known, a Bertrand‑type competition model can be used. Assuming the rival charges 20 yuan and unit cost is 10 yuan, the firm selects its price to maximize profit given the demand response.

5. Location Selection Model

Store location is critical. A linear‑programming model incorporates foot traffic, rent, and distance from existing stores to choose the best site.

5.1 Model Elements

Inputs: pedestrian flow data, rent, competitor proximity.

Output: optimal location.

5.2 Case Study

Three candidate sites (A, B, C) are evaluated with their daily foot traffic, monthly rent, and distance to the nearest same‑brand store. Solving the model selects site B as the optimal location.

Mathematical modeling provides powerful tools for coffee‑shop owners to make data‑driven decisions, though real‑world nuances must always be considered.