Specific crossover and mutation operators for a grouping problem based on interaction data in a Regional Science context

Francisco Flórez RevueltaJosé M. Casado Díaz

Lucas Martínez Bernabeu

IEEE Congress on Evolutionary Computation,

Singapore, September 25-28, 2007

Specific crossover and mutation operators for a grouping problem based on interaction data in a Regional Science context

1. Introduction

2. Problem formulation

3. Evolutionary proposal

4. Operators

5. Experimentation

6. Behaviour of the operators

7. Conclusions

1. Introduction

Administrative areas are usually defined by boundaries derived from historical reasons

It is not assured a meaningful insight of the territorial functional reality

Alternative: use of local labour market areas (LLMAs) Aggregation of basic territorial units (counties, wards,

municipalities,…) of similar features, or Our choice: aggregations are based on the interaction

between units in terms of travel-to-work flows

1. Introduction

This delineation of the territory is very useful in a policy making context in different fields: labour, housing market, transport,…

Most developed countries have delineated official LLMAs in the last decades. Nowadays, most of them are revising these maps

1. Introduction

LLMA: area where the majority of the interaction between workers seeking jobs and employers recruiting labour occurs

Two components: the boundary of the area is rarely crossed in daily journeys to

work high degree of intra-market movement (so that the defined

market is internally active and so as unified as possible) Administrative use of LLMAs add several

requirements: contiguity absence of overlappings minimal size detail

1. Introduction

Delineation of LLMAs:Current methods have a very diverse nature:

• Inductive vs. deductive

• Hierarchical vs. multi-step

One of the most successfully applied is that of Coombes et al. (1986)

2. Problem formulation

Delineation problem is presented as the maximization of markets’ internal cohesion in terms of travel-to-work flows subject to a number of restrictions to assure self-containment and minimum size, with the aim of identifying as many independent markets as possible

2. Problem formulation

2. Problem formulation

We can define different fitness functions based on the interaction index:

Every region must fulfil several constraints of self-containment and minimum size:

3. Evolutionary proposal

1. Produce an initial population of size n. One of the individual is taken as the whole territory in one are. This ensures one valid individual to begin with

2. Evaluate fitness of all individuals and sort them accordingly

3. Generate nr new individuals by recombination

4. Generate nm new individuals by mutation

5. Evaluate fitness of all new individuals

3. Evolutionary proposal

6. Sort the whole population

7. Generate a new population choosing the n best individuals

8. Stop condition: if the best individual remains without changes for g generations, finish. Otherwise, return to step 3

3. Evolutionary proposal

Individual representation:

Selection: Ranking method

4a. Recombination Operators

We have developed 5 different operators:

4a. Recombination Operators

We have developed 5 different operators:

4a. Recombination Operators

We have developed 5 different operators:

4a. Recombination Operators

We have developed 5 different operators:We have added 2 new operators to cope with

the fact that areas characterized by lower identifiers are also assigned to the regions with lower identifiers

They are variations of Recombination2 and Recombination3

A random recoding of regions is performed before applying the recombinations

4b. Mutation Operators

We have designed an extensive set of operators, some of them specifically intended for the delineation of LLMAs

Four main functions:division of regions fusion of regions reassignment of single areas reassignment of group of areas

4b. Mutation Operators

Division1: Divides a region into two a region Ri is randomly selected

an area from Ri is randomly chosen and assigned to a new region R’i

another area from Ri is randomly chosen and assigned to a new region R’’i

The rest of the areas belonging to R are taken at random, being assigned to the region (R’i or R’’i) to which each of them has a stronger interaction index

4b. Mutation Operators

Division2: Creates a new region removing from another one enough areas to form a valid market

Division3: Divides a region to form two regions with a similar number of areas

4b. Mutation Operators

Fusion1: Two randomly selected regions are merged

Fusion2: A region is randomly chosen. Each of its constituting areas is merged with its optimal region

4b. Mutation Operators

Reassignment1: One area is randomly selected and assigned to a random region

Reassingment2: Similar to the previous one, but the destination region is the optimal one

4b. Mutation Operators

Reassignment1: One area is randomly selected and assigned to a random region

Reassignment2: Similar to the previous one, but the destination region is the optimal one

Reassignment3: An exchange of areas between regions is performed

4b. Mutation Operators

GlobalReassignment1: Removes from a region the areas that score lower in the interaction index, being assigned to their optimal regions

GlobalReassignment2: A whole group of areas is assigned to its optimal region

GlobalReassignment3-4: k regions are disintegrated, being reassigned in k new regions

4b. Mutation Operators

GlobalReassignment1: Removes from a region the areas that score lower in the interaction index, being assigned to their optimal regions

GlobalReassignment2: A whole group of areas is assigned to its optimal region

GlobalReassignment3-4: k regions are disintegrated, being reassigned in k new regions

5. Experimentation

Delineation of LLMAs in the Region of Valencia, Spain (541 municipalities)

Travel-to-work data from the 2001 Spanish Census of Population

Fitness function:

Parameters:

5. Experimentation

5. Experimentation

Traditional method Our evolutionary proposal

5. Experimentation

Traditional method Our evolutionary proposal

5. Experimentation

Our proposal reaches similar results to that obtained with the traditional method in few iterations:

6. Behaviour of the operators

Contribution of each operator is different in the reaching of the best solution:

6. Behaviour of the operators

Improvement in the value of the fitness function differs between the operators:

6. Behaviour of the operators

Temporal cost differs between operators:

6. Behaviour of the operators

Operators that generate good individuals vary along time:

7. Conclusions

Official methods were designed some decades ago

We have modelled the problem as one of optimization solved by a evolutionary algorithm

Evolution is only possible with ad-hoc operators

Our results are similar to the official ones but improving detail (number of markets)

They also satisfy official requirements

7. Conclusions

Two research areas:Regional Science: study and definition of new

fitness functions and evaluation of the resulting delineations

Evolutionary Computation: improvement in the algorithm (avoiding local maxima) and in the temporal cost:

• Self-adaptive proposal

• Other representations and grouping evolutionary approaches

• Parallel implementation

Specific crossover and mutation operators for a grouping problem based on interaction data in a Regional Science context

Francisco Flórez RevueltaJosé M. Casado Díaz

Lucas Martínez Bernabeu

IEEE Congress on Evolutionary Computation,

Singapore, September 25-28, 2007