In Search Of Diverse And Connected Teams: A Computational Approach To Assemble Diverse Teams Based On Members Part 4

In this implementation, we use a variety of metrics to assess teams' diversity provided by C categorical variables, and disparity metrics to assess teams' diversity provided by U numerical variables. To measure each team's variety metrics by its members' categorical attributes C, we use the Blau index (Bt; ci ) [30]. 

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This index quantifies the probability that two team members randomly selected from the team would be in different categories. A low score means members fall into the same category, whereas a high score means members fall into different categories. 

We denote pci j as the proportion of members who fall into a particular category j in the categorical attribute ci. Given that the number of categories in ci is oci, where j ¼ 1; :::; oci, the Blau Index formula for the team t is:

Bt;ci ¼ 1 Xoci j pci 2 j ð2

To measure each team's disparity metrics by its members' numerical variables U, we use the coefficient of variation (CVt;ui ) [30], which is defined as the ratio of the standard deviation to the mean of the attribute i, ui 2 U. 

A low coefficient of variation score means that all team members t have similar levels of the attribute, whereas a high score means all team members t have different levels of the attribute. For a team t with members j = 1, 2, . . ., k, and with u�i as the team's mean value of the attribute i, the formula is the following:

CVt;ui ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xk j2t ðui ðpj Þ u�iÞ 2 =k vuut =u�i ð

These two team diversity measures are useful because they do not change when the input data is scaled linearly, and they both tend to stay around the same values. Given that the team formation problem considers C categorical variables and U numerical variables, the diversity measures can be weighted to prioritize differences within a specific variable. 

The vector of weights W has |C| + |U| elements, where W ¼ ðwu1 ; :::;wcm ;wc1 ;wc2 ; :::; wcl Þ. Based on these measures, we aggregate diversity for different attributes into a single value. We define the team diversity score V of a team t as the weighted sum of the Blau indexes for all C categorical variables and the coefficient of variation for all U numerical variables. The formula is:

Vt ¼ Xm i¼1 wui � CVt;ui þXl i¼1 wci � Bt;ci ð

Multi-objective problem statement. We formulate the problem as a multi-objective problem to find a set of r team solutions P ¼ fT1; T2; T3; :::; Trg, where each T represents a potential solution with q teams. 

The decomposition of the assessment function into both objectives-minimizing communication costs and maximizing team diversity score-allows us to find multiple solutions that the single-objective approach cannot reach. As a result, we expect to find not a unique solution T but a set of solutions P for which there is not another feasible solution better in both objective functions. 

This set of solutions P is also known as a Pareto front, where (a) there exists no other set of solutions T0 with more diverse and connected teams, and (b) each solution Ti; i 2 P is not superior to all the other solution in P concerning both diversity and communication costs objectives. Having this set of team solutions P makes it possible to assess each of them individually, so a team builder can choose the most appropriate teams possible to assemble for the given context and circumstances.

In sum, the team formation problem addressed in this paper is to find the Pareto front P of team solutions, where each solution T is composed of q teams (T = {t1, t2, t3, . . ., tq}). The dual objective is maximizing teams' diversity based on the categorical attributes C and numerical attributes U and minimizing the communication cost based on G. We can model this problem as:

Since finding teams from graph G while minimizing the sum of shortest path lengths and team allocation problems is proven to be an NP-hard problem [57, 68], this multi-objective problem is also an NP-hard problem.

NSGA-II implementation

Pareto fronts' shapes provide helpful information about the degree of trade-off between different objectives and how much compromise is needed from some criteria to improve others. 

Determining the exact Pareto front for multi-objective combinatorial optimization problems is difficult since it is necessary to compute all the possible combinations to find the true Pareto front [63]. For this reason, the goal is to find an approximation of the true Pareto front using heuristic algorithms. A critical assumption for these algorithms is that the Pareto front is sufficiently populated. 

The quality of this approximation depends upon (1) the proximity of the points on the approximated front to the points on the true Pareto front; and (2) the diversity of the solutions on the approximated front, where more diversity is typically better. Although the true Pareto front is unknown, solutions that dominate others are close to the theoretical true Pareto front. Therefore, the diversity of the solutions will provide a more extensive range and granularity of the Pareto front.

Genetic algorithms (GA) are commonly used to find approximations of Pareto fronts [69]. By imitating evolution in nature, this method optimizes a population of initial solutions into better solutions through natural selection. Each solution is characterized as a chromosome (i.e., a vector of attributes), which can be mutated and altered in each iteration. The best solutions will endure after they mutate over time. Genetic algorithms are ideal for finding solutions for optimization problems in large and highly non-linear spaces [70].

The genetic algorithm starts from a population of randomly generated solutions evolving into new solutions through an iterative process. The population created in each iteration is also known as a generation. In each generation, the algorithm evaluates each population's chromosome according to the objective function in the resolved optimization problem. 

The chromosomes with the highest scores are selected from the current generation and used to form a new generation. This process continues until a maximum number of iterations is achieved or by a threshold function defined for the solutions.

We implemented a genetic algorithm called Non-dominated Sorting Genetic Algorithm-II (NSGA-II) formulated by Deb et al. [71]. NSGA-II allows finding an approximation of the Pareto front, having different team solutions P that vary according to the communication costs and diversity score specified. The NSGA-II approach is based on sorting the populations into a hierarchy of sub-populations using Pareto dominance criteria. 

Then, chromosomes for the next iteration are selected according to the mentioned hierarchy. This elitist selection guarantees that potential good chromosomes are kept in the population, and the solution quality obtained does not decrease from one iteration to the next. The solutions are also ordered according to the similarity among their chromosomes, removing redundant ones to promote diversity in the Pareto front. 

As a result, NSGA-II can converge on a high-performing Pareto front after a few iterations. Previous work has shown that NSGA-II provides solutions with high levels of efficiency running in O(n2 ).

In this implementation, each population P contains r team solutions P ¼ fT1; T2; :::; Trg, and each chromosome represents a potential set of q teams Ti = {t1, t2, . . ., tq}. We use "chromosome" and "team solution" interchangeably throughout this paper. 

We characterize a chromosome as a vector of individuals partitioned into q parts to obtain the teams (Fig 2). As a result, each chromosome's length equals the number of people n, representing q teams of size k (q� k = n). We adapted this algorithm to our specific diverse team formation problem, and we outline these steps in Algorithm 1.

Initialization. The algorithm starts by initializing a population of chromosomes P and having teams assembled randomly. Its input parameters are the total number of chromosomes r to include in the population P, the list of people P, the number of teams q to form, and the number of iterations to perform g. 

Chromosomes are stored as two-dimensional arrays of shape (q, k), where q is the number of teams possible to assemble, and k is the number of members per team. Each chromosome is a potential solution to the diverse team formation problem, and the goal is to find a set of chromosomes with high levels of diversity and low communication costs. 

After the initial population is created, the algorithm creates the offspring and searches for the Pareto fronts iteratively until the maximum number of generations g is reached.

Crossover step. 

In each generation, the algorithm takes two random chromosomes (p1 and p2) from the existing population P and randomly selects q teams from this union. As a result, the algorithm will have a child chromosome with q teams. Since the child's teams are randomly selected from two different chromosomes, individuals may get selected twice, coming from p1 and p2. 

The algorithm replaces repeated individuals with others who were not assigned to a team. It explores each member of the child's chromosome and counts how many times an individual is part of a team. If an individual is counted more than once, this individual is randomly replaced by a missing member. At the end of this revision process, the algorithm will have the child chromosome with all the members of P assigned to one team.

These random samplings provide sufficient mutation for the algorithm to introduce diversity into the population without adding another mutation step. We outline the proposed crossover method in Algorithm 2.

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