# On Complexity and Exact Solution of Production Groups Formation Problem

## 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.

## Keywords

Integer programming Optimization on graphs Production groups Branch and bound algorithm## Notes

### Acknowledgement

This research is supported by RFBR projects 16-01-00740 and 17-07-00513.

## References

- 1.Afanasyeva, L.D., Kolokolov, A.A.: Design and analysis of algorithm for solving some formation of production groups problems. Omsk Sci. Bull.
**2**(110), 39–42 (2012). (in Russian)Google Scholar - 2.Afanasyeva, L.D., Kolokolov, A.A.: Study and solution of a production groups formation problem. Vestnik UGATU.
**5**, 20–25 (2013). (in Russian)Google Scholar - 3.Borisovsky, P.A., Delorme, X., Dolgui, A.: Balancing reconfigurable machining lines via a set partitioning model. Int. J. Prod. Res.
**52**(13), 4026–4036 (2013). https://doi.org/10.1080/00207543.2013.849857CrossRefGoogle Scholar - 4.Burkard, R.E., Dell’Amico, M., Martello, S.: Assignment problems. SIAM, Philadelphia (2009). https://doi.org/10.1137/1.9781611972238
- 5.Eremeev, A.V., Kel’manov, A.V., Pyatkin, A.V.: On the complexity of some Euclidean optimal summing problems. Doklady Math.
**93**(3), 286–288 (2016). https://doi.org/10.1134/S1064562416030157MathSciNetCrossRefzbMATHGoogle Scholar - 6.Eremeev, A.V., Kel’manov, A.V., Pyatkin, A.V.: On complexity of searching a subset of vectors with shortest average under a cardinality restriction. In: Ignatov, D.I., et al. (eds.) AIST 2016. CCIS, vol. 661, pp. 51–57. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-52920-2_5CrossRefGoogle Scholar
- 7.Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, San Francisco (1979)zbMATHGoogle Scholar
- 8.Il’ev, V., Il’eva, S., Kononov, A.: Short survey on graph correlation clustering with minimization criteria. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 25–36. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44914-2_3CrossRefGoogle Scholar
- 9.Kolokolov, A.A., Afanasyeva, L.D.: Research of production groups formation problem subject to logical restrictions. J. Siberian Federal Univ. Math. Phys.
**6**(2), 145–149 (2013)Google Scholar - 10.Kolokolov, A.A., Rubanova, N.A., Tsygler, I.A.: Research and solution of some small groups formation problems based on discrete optimization. OMSK Sci. Bull.
**4**(145), 139–142 (2016). (in Russian)Google Scholar - 11.Kolokolov, A.A., Rubanova, N.A., Ziegler, I.A.: Solution of personnel management problems with respect to some binary relations. In: Proceedings of 12th International School-Seminar “Optimization Problems of Complex Systems”, Novosibirsk, pp. 278–284 (2016). (in Russian)Google Scholar
- 12.Kolokolov, A.A., Rubanova, N.A., Ziegler, I.A.: Solution of small production groups formation problems using discrete optimization. In: Proceedings of the 4th International Conference “Information Technologies for Intelligent Decision Making Support”, (ITIDS 2016), vol. 1, pp. 215–218. UGATU, Ufa (2016). (in Russian)Google Scholar
- 13.Novikov, D.A.: Mathematical Models of Teams Building and Functioning. Phismathlit, Moscow (2008). (in Russian)Google Scholar
- 14.Sigal, I.C., Ivanova, A.P.: Introduction to Applied Discrete Programming. Phismathlit, Moscow (2007). (in Russian)Google Scholar
- 15.Ziegler, I.A.: On some algorithms of solving goal-oriented groups formation problems. Young Russia Adv. Technol. Ind.
**2**, 51–53 (2017). OmSTU, Omsk, (in Russian)Google Scholar