Optimizing Production with Fixed‑Cost Problems: A Clothing Factory Case Study

1 Fixed Cost Problem

1.1 Concept

The fixed‑cost problem seeks the production plan that minimizes total cost, which consists of a fixed cost (depending on the chosen production method) and a variable cost (depending on output). It is an exclusive‑objective problem that cannot be modeled with standard linear programming.

1.2 Model

Assume there are n possible production methods. For method i, let F_i be the fixed cost, c_i the variable cost per unit, and x_i the production quantity. Introducing a binary variable y_i (1 if method i is selected, 0 otherwise) allows the total cost to be expressed as Σ(F_i y_i + c_i x_i) with appropriate constraints.

2 Case Study

2.1 Problem Statement

A garment factory can produce three products A, B, and C. Each product requires a specific equipment rental, material cost, selling price, labor hours, and equipment hours. The available labor and equipment hours are limited (A: 600/800/600, B: 800, C: 600). The goal is to determine the production quantities that maximize profit.

2.2 Analysis

Because equipment rental is incurred only when a product is produced and does not depend on the quantity, it represents a fixed cost, making this a fixed‑cost problem.

2.3 Modeling

2.3.1 Decision Variables

y_i – binary variable indicating whether product i is produced. x_i – integer quantity of product i produced.

2.3.2 Objective and Constraints

The objective is to maximize total profit: Σ(selling_price_i · x_i − c_i · x_i − F_i · y_i). Constraints include labor‑hour limits, equipment‑hour limits, and the logical relationship between y_i and x_i (x_i ≤ M·y_i for a sufficiently large M).

2.4 Solution

The model is solved with the Python PuLP library. The optimal production plan is:

Product A: 60 units

Product B: 300 units

Product C: 75 units

Total revenue equals 8,700 yuan.

3 Summary

The article demonstrates how to formulate and solve a fixed‑cost production planning problem using a clothing‑factory example.