## Deterministic and stochastic team formation problems

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In various organizations, physical or virtual teams are formed to perform jobs that require diﬀerent skills. The success of a team depends on the technical capabilities of the team members as well as the quality of communication among the team members. We study diﬀerent variants of the team formation problem where the goal is to build the best team with respect to given criteria. First, we study a deterministic team formation problem which aims to construct a capable team that can communicate and collaborate eﬀectively. To measure the quality of communication, we assume the candidates constitute a social network and we deﬁne a cost of communication using the proximity of people in the social network. We minimize the sum of all pairwise communication costs, and we impose an upper bound on the largest communication cost. This problem is formulated as a constrained quadratic set covering problem. Our experiments show that a general-purpose solver is capable of solving small and medium-sized instances to optimality. We propose a branch-and-bound algorithm to solve larger sizes: we reformulate the problem and relax it in such a way that it decomposes into a series of linear set covering problems, and we impose the relaxed constraints through branching. Our computational experiments show that the algorithm is capable of solving large-sized instances, which are intractable for the solver. Second, we consider a two-stage stochastic team formation problem where the objective is to minimize the expected communication cost of the team. We as-sume that for a subset of pairs the communication costs are uncertain but they have a known discrete distribution. The ﬁrst stage is a trial stage where the decision-maker chooses a limited number of pairs from this subset. The actual cost values of the chosen pairs are realized before the second stage. Hence, the uncertainty in this problem is decision-dependent, also called endogenous, be-cause the ﬁrst stage decisions determine for which parameters the uncertainty will resolve. For this problem, we give two formulations, the ﬁrst one contains a set of non-anticipativity constraints similar to the models in the related lit-erature. In the second, we are able to eliminate these constraints by changing the objective function into a quadratic one, which is linearized by a set of extra binary variables. We show that the size of instances we can solve with these for-mulations using a commercial solver is limited. Therefore, we develop a Benders’ decomposition-based branch-and-cut algorithm that exploits decision-dependent nature to partition scenarios and use tight linear relaxations to obtain strong cuts. We show the eﬃciency of the algorithm presenting results of experiments conducted with randomly generated instances. Finally, we study a multi-stage team formation problem where the objective is to minimize the monetary cost including hiring and outsourcing costs. In this problem, stages correspond to projects which are carried out consecutively. Each project consists of several tasks each of which requires a human resource. We assume that due to incomplete information there is uncertainty in people’s performances and consequently the time a person needs to complete a task is random for some person-task pairs. When a person is assigned to a task, we learn how long it takes for this person to ﬁnish the task. Hence, the uncertainty is again decision-dependent. If the duration of a task exceeds the allowable time for a project then the manager must hire an external resource to speed up the process. We present an integer programming formulation to this problem and explain that the size of the formulation strongly depends on the number of random parameters and scenarios. While this deterministic equivalent formulation can be solved with a commercial solver for small-sized instances, it easily becomes intractable when the number of random parameters increases by one. For such cases where exact methods are not promising, we investigate heuristic methods to obtain tight bounds and near-optimal solutions. In the related literature, diﬀerent Lagrangian decomposition methods are developed for such stochastic problems. In this study, we show that the convergence of existing methods is very slow, and we propose an alternative method where a relaxation of the formulation is solved by a decomposition-based branch-and-bound algorithm.