Abstract
The success of a modern enterprize is substantially determined by the effectiveness of staff selection and formation of various kinds of functional groups. Creation of such groups requires consideration of different factors depending on the activity of the groups. The problem of production groups formation, considered in this paper, asks for an assignment of workers to jobs taking into account the implicational constraints. The first result of the paper states the NP-hardness of the problem under consideration. The second result is a branch and bound method, which uses supplementary assignment problems for computing bounds. A software implementation of the algorithm is made, and a computational experiment is carried out, comparing the proposed algorithm with the CPLEX solver on randomly generated input data.
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This research is supported by RFBR projects 16-01-00740 and 17-07-00513.
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Appendix
Appendix
This appendix contains a detailed description of the algorithm CUT from [15]. The number of constraints defined by (4) and (5) can be too large for a straightforward application of a standard ILP algorithm. However, the inequalities may be added gradually, as it is done in cutting plane algorithms. This idea was implemented in the CUT procedure based on the CPLEX package.
Step 0. Solve the root assignment problem (1)–(3), (6) and go to Step 1.
Step 1. If a solution exists then go to Step 2. Otherwise, go to Step 5.
Step 2. If the obtained solution does not satisfy some of the constraints (4), (5) then we add one violated constraint to the current problem and go to Step 3. (The tie-breaking rule is described below.) Otherwise, go to Step 4.
Step 3. Solve the current problem and go to Step 1.
Step 4. The obtained solution is optimal. Stop.
Step 5. Problem (1)–(6) is infeasible. Stop
If several violated constraints (4), (5) are identified in Step 2, we need to choose one constraint to add to the current problem. To this end, for each tuple \(\langle (i,j),(i',j')\rangle \), associated to a violated constraint, we count how many times the index i occurs in other violated constraints (4), (5). Denote this number by \(N_i\). Analogous values \(N_j\), \(N_{i'}\), and \(N_{j'}\) are calculated for \(j, i'\), and \(j'\). Then we assign the weight to each violated inequality, defined as \(N_i+N_j+N_{i'}+N_{j'}\). Finally, we choose the inequality with the maximum weight and add it to the problem.
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Eremeev, A., Kononov, A., Ziegler, I. (2019). On Complexity and Exact Solution of Production Groups Formation Problem. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol 974. Springer, Cham. https://doi.org/10.1007/978-3-030-10934-9_8
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