## Travelling to work

The easiest **trade-offs** to consider are those where there is a simple and obvious **mathematical model **to compare with reality. A good example is “**travel to work**” (Suen and Navlakha 2019).

People live in various parts of a city and travel in to the centre every day.

The **expensive solution** is **many roads** to the centre, one for each person. The total length of road is **large** although each journey is **short**.

The** cheap solution** is a **single road **linking all the people with the middle of the city. The road is **shorter** but most journeys are **longer**, especially for the person living at the far end of the road.

The** trade-off** is a **branching road network** linking all the people with the city centre, **minimising** both the total **length** of road and the average of the **distance** travelled by each user.

For a given distribution of people, the **length of road** for the two extremes plus a number of **intermediate** **networks** can be calculated and the **distances** plotted. The resulting curve – the **Pareto Front** – represents the **best design** of the road network for **all possibilities** given the distribution of people.

A closely similar case is the **venation of a leaf**, where points around the leaf are connected to the stem. Modelling and analysis again produce a **Pareto Front** (Conn *et al*. 2017 https://pubmed.ncbi.nlm.nih.gov/28750198/).

The **trade-off** is a universal statement of a problem. In order for this to be useful, it is necessary to know what **factors** can be used to **design** or **maintain** a **trade-off**.