Abstract
We study the minimum shift design problem (MSD) that arose in a commercial shift scheduling software project: Given a collection of shifts and workforce requirements for a certain time interval, we look for a minimum cardinality subset of the shifts together with an optimal assignment of workers to this subset of shifts such that the deviation from the requirements is minimum. This problem is closely related to the minimum edge-cost flow problem (MECF), a network flow variant that has many applications beyond shift scheduling. We show that MSD reduces to a special case of MECF. We give a logarithmic hardness of approximation lower bound. In the second part of the paper, we present practical heuristics for MSD. First, we describe a local search procedure based on interleaving different neighborhood definitions. Second, we describe a new greedy heuristic that uses a min-cost max-flow (MCMF) subroutine, inspired by the relation between the MSD and MECF problems. The third heuristic consists of a serial combination of the other two. An experimental analysis shows that our new heuristics clearly outperform an existing commercial implementation.
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References
Aarts, E., Lenstra, J.K. (eds.): Local Search in Combinatorial Optimization. Wiley, Chichester (1997)
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows. Prentice Hall, Englewood Cliffs (1993)
Arata, K., Iwata, S., Makino, K., Fujishige, S.: Source location: Locating sources to meet flow demands in undirected networks. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, p. 300. Springer, Heidelberg (2000)
Bar-Ilan, J., Kortsarz, G., Peleg, D.: Generalized submodular cover problems and applications. In: The Israeli Symposium on the Theory of Computing, pp. 110–118 (1996) (also in Theoretical Computer Science, to appear)
Bartholdi, J.J., Orlin, J.B., Ratliff, H.D.: Cyclic scheduling via integer programs with circular ones. Operations Research 28, 110–118 (1980)
Bellare, M.: Interactive proofs and approximation: reduction from two provers in one round. In: The second Israeli Symposium on the Theory of Computing, pp. 266–274 (1993)
Carr, R.D., Fleischer, L.K., Leung, V.J., Phillips, C.A.: Strengthening integrality gaps for capacitated network design and covering problems. In: Proc. of the 11th ACM/SIAM Symposium on Discrete Algorithms (2000)
Equi, L., Gallo, G., Marziale, S., Weintraub, A.: A combined transportation and scheduling problem. European Journal of Operational Research 97(1), 94–104 (1997)
Even, G., Kortsarz, G., Slany, W.: On network design problems: Fixed cost flows and the covering steiner problem. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, pp. 318–329. Springer, Heidelberg (2002)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman and Co., New York (1979)
Garg, N., Yannakakis, M., Vazirani, V.V.: Approximating max-flow min- (multi)cut theorems and their applications. Siam J. on Computing 25, 235–251 (1996)
Garg, N., Yannakakis, M., Vazirani, V.V.: Primal-dual approximation algorithms for integral flow and multicuts in trees. Algorithmica 18, 3–20 (1997)
Gärtner, J., Musliu, N., Slany, W.: Rota: a research project on algorithms for workforce scheduling and shift design optimization. AI Communications: The European Journal on Artificial Intelligence 14(2), 83–92 (2001)
Goethe-Lundgren, M., Larsson, T.: A set covering reformulation of the pure fixed charge transportation problem. Discrete Appl. Math. 48(3), 245–259 (1994)
Hochbaum, D.: Optimization over consecutive 1’s and circular 1’s constraints. unpublished manuscript (2000)
Hochbaum, D.S., Segev, A.: Analysis of a flow problem with fixed charges. Networks 19(3), 291–312 (1989)
Kim, D., Pardalos, P.M.: A solution approach to the fixed charge network flow problem using a dynamic slope scaling procedure. Oper. Res. Lett. 24(4), 195–203 (1999)
Krumke, S.O., Noltemeier, H., Schwarz, S., Wirth, H.-C., Ravi, R.: Flow improvement and network flows with fixed costs. In: OR 1998, Zürich (1998)
Laporte, G.: The art and science of designing rotating schedules. Journal of the Operational Research Society 50, 1011–1017 (1999)
Lau, H.C.: Combinatorial approaches for hard problems in manpower scheduling. J. Oper. Res. Soc. Japan 39(1), 88–98 (1996)
Leiserson, C.E., Saxe, J.B.: Retiming synchronous circuitry. Algorithmica 6(1), 5–35 (1991)
Magnanti, T.L., Wong, R.T.: Network design and transportation planning: Models and algorithms. Transportation Science 18, 1–55 (1984)
Musliu, N., Gärtner, J., Slany, W.: Efficient generation of rotating workforce schedules. Discrete Applied Mathematics 118(1-2), 85–98 (2002)
Musliu, N., Schaerf, A., Slany, W.: Local search for shift design. European Journal of Operational Research (to appear), http://www.dbai.tuwien.ac.at/proj/Rota/DBAI-TR-2001-45.ps
Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs (1982)
Raz, R., Safra, S.: A sub constant error probability low degree test, and a sub constant error probability PCP characterization of NP. In: Proc. 29th ACM Symp. on Theory of Computing, pp. 475–484 (1997)
Veinott, A.F., Wagner, H.M.: Optimal capacity scheduling: Parts i and ii. Operation Research 10, 518–547 (1962)
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Di Gaspero, L., Gärtner, J., Kortsarz, G., Musliu, N., Schaerf, A., Slany, W. (2003). The Minimum Shift Design Problem: Theory and Practice. In: Di Battista, G., Zwick, U. (eds) Algorithms - ESA 2003. ESA 2003. Lecture Notes in Computer Science, vol 2832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39658-1_54
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DOI: https://doi.org/10.1007/978-3-540-39658-1_54
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