Applied Mathematics Problems Solved: Linear Programming, Queuing Theory, Fourier Analysis, and Insurance Risk Models

This is the final installment in a series on applied mathematics problems across industries. Below are five more cases where mathematical tools provide rigorous, practical solutions — from factory floors to audio compression to insurance actuarial tables.

Applied Mathematics Problems in Resource Allocation: Linear Programming

Problem 16. Optimal resource distribution (manufacturing, logistics)

Given linear constraints — raw materials, capacity, time — and a linear objective such as profit or output volume, the task is to find a production or logistics plan that maximizes the objective while satisfying all constraints.

Solution: Linear programming provides a precise formal framework and algorithms that are guaranteed to find the optimal solution. The method works by examining the vertices of the feasible region, where the optimal value is always located. This approach is the backbone of supply chain planning, production scheduling, and transportation optimization worldwide.

Queuing Theory: How Many Servers Does Your System Actually Need

Problem 17. How many cashiers, operators, or servers are needed to keep queues tolerable?

Given statistics on customer arrival rates and service times, the goal is to choose the number of service units so that wait times and the risk of long queues remain acceptable — while keeping infrastructure costs minimal.

Solution: Queuing theory builds probabilistic models and derives closed-form formulas for average queue length and expected waiting time. These formulas let engineers compare configurations and find the right balance between service quality and cost. The Poisson distribution is frequently central to describing arrival patterns in such models.

Computational Mesh Generation for Surface Modeling

Problem 18. Building quality computational meshes for surface simulation

Given a complex surface — an aircraft wing, a vehicle body, a 3D character model — the task is to cover it with a mesh of triangles or quadrilaterals that contains no narrow, poorly-shaped elements, so that numerical simulations remain stable and accurate.

Solution: Theorems on circle packings and related algorithms make it possible to represent a surface through mutually tangent circles and construct a well-structured mesh from them. The resulting elements have good angles and proportions, which makes numerical methods more precise and stable — a critical requirement in aerospace engineering and finite element analysis.

Fourier Analysis: Solving the Signal Compression Problem

Problem 19. Decomposing a signal into “waves” for compression and filtering

The task is to represent a sound or image as a sum of simple components, so that insignificant parts can be discarded, noise can be filtered out, and the overall data size can be reduced.

Solution: Basis decompositions — most famously the Fourier transform — concentrate the “meaning” of a signal into a relatively small number of coefficients. By truncating the smallest coefficients, the quality loss is minimal while the data size drops dramatically. This is the mathematical principle behind many compression formats and digital filters used in audio processing, image storage, and telecommunications.

Insurance Risk Models: Ruin Probability and Premium Calculation

Problem 20. Ruin probability and insurance tariff calculation

Given the frequency of insured events and the distribution of claim sizes, the goal is to set premiums and reserves such that the probability of insolvency stays below a defined threshold.

Solution: The company’s capital is modeled as a stochastic process with continuous inflows and occasional large payouts. The analysis computes the probability that this process ever falls below zero — the ruin probability. These models yield formulas and numerical methods for estimating risk, allowing actuaries to calibrate tariffs and reserves to meet regulatory requirements.

What Applied Mathematics Problems Remain Unsolved?

As the comments under previous posts in this series confirmed, this list is far from exhaustive. But having covered what mathematics has solved, the natural next question is: what applied problems has mathematics so far failed to crack?