Mathematical Programming

, Volume 150, Issue 1, pp 79–98

# Minimum concave cost flow over a grid network

Full Length Paper Series B

## Abstract

The minimum concave cost network flow problem (MCCNFP) is NP-hard, but efficient polynomial-time algorithms exist for some special cases such as the uncapacitated lot-sizing problem and many of its variants. We study the MCCNFP over a grid network with a general nonnegative separable concave cost function. We show that this problem is polynomially solvable when all sources are in the first echelon and all sinks are in two echelons, and when there is a single source but many sinks in multiple echelons. The polynomiality argument relies on a combination of a particular dynamic programming formulation and an investigation of the extreme points of the underlying flow polyhedron. We derive an analytical formula for the inflow into any node in an extreme point solution, which generalizes a result of Zangwill (Manag Sci 14(7):429–450, 1968) for the multi-echelon lot-sizing problem.

## Keywords

Min concave cost flow Grid network Lot-sizing model

## Mathematics Subject Classification

90B05 90B10 90B30 90C11 90C35

## References

1. 1.
Aggarwal, A., Park, J.K.: Improved algorithms for economic lot size problems. Oper. Res. 41(3), 549–571 (1993)
2. 2.
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Upper Saddle River (1993)
3. 3.
Atamtürk, A., Küçükyavuz, S.: Lot sizing with inventory bounds and fixed costs: polyhedral study and computation. Oper. Res. 53(4), 711–730 (2005)
4. 4.
Atamtürk, A., Küçükyavuz, S.: An $${O}(n^2)$$ algorithm for lot sizing with inventory bounds and fixed costs. Oper. Res. Lett. 36(3), 297–299 (2008)
5. 5.
Dekker, R., Fleischmann, M., Inderfurth, K., Van Wassenhove, L.N. (eds.): Reverse Logistics: Quantitative Models for Closed-Loop Supply Chains. Springer, Berlin (2004)Google Scholar
6. 6.
Erickson, R.E., Monma, C.L., Veinott Jr, A.F.: Send-and-split method for minimum-concave-cost network flows. Math. Oper. Res. 12(4), 634–664 (1987)
7. 7.
Federgruen, A., Tzur, M.: A simple forward algorithm to solve general dynamic lot sizing models with $$n$$ periods in $${O}(n\log n)$$ or $${O}(n)$$ time. Manag. Sci. 37(8), 909–925 (1991)
8. 8.
Florian, M., Klein, M.: Deterministic production planning with concave costs and capacity constraints. Manag. Sci. 18(1), 12–20 (1971)
9. 9.
Guisewite, G.M., Pardalos, P.M.: Minimum concave-cost network flow problems: applications, complexity, and algorithms. Ann. Oper. Res. 25(1), 75–99 (1990)
10. 10.
Guisewite, G.M., Pardalos, P.M.: A polynomial time solvable concave network flow problem. Networks 23(2), 143–147 (1993)
11. 11.
Kaminsky, P., Simchi-Levi, D.: Production and distribution lot sizing in a two stage supply chain. IIE Trans. 35(11), 1065–1075 (2003)
12. 12.
Martin, R.: Generating alternative mixed-integer programming models using variable redefinition. Oper. Res. 35(6), 820–831 (1987)
13. 13.
Pochet, Y., Wolsey, L.A.: Production Planning by Mixed Integer Programming. Springer, New York (2006)
14. 14.
Tuy, H., Ghannadan, S., Migdalas, A., Värbrand, P.: The minimum concave cost network flow problem with fixed numbers of sources and nonlinear arc costs. J. Glob. Optim. 6(2), 135–151 (1995)
15. 15.
Tuy, H., Ghannadan, S., Migdalas, A., Värbrand, P.: A strongly polynomial algorithm for a concave production-transportation problem with a fixed number of nonlinear variables. Math. Program. 72(3), 229–258 (1996)
16. 16.
van den Heuvel, W., Wagelmans, A.P.: Four equivalent lot-sizing models. Oper. Res. Lett. 36(4), 465–470 (2008)
17. 17.
van Hoesel, S., Romeijn, H.E., Morales, D.R., Wagelmans, A.P.: Integrated lot sizing in serial supply chains with production capacities. Manag. Sci. 51(11), 1706–1719 (2005)
18. 18.
Wagelmans, A., van Hoesel, S., Kolen, A.: Economic lot sizing: an $${O}(n\log n)$$ algorithm that runs in linear time in the Wagner–Whitin case. Oper. Res. 40(1), 145–156 (1992)
19. 19.
Wagner, H.M., Whitin, T.M.: Dynamic version of the economic lot size model. Manag. Sci. 5(1), 89–96 (1958)
20. 20.
Zangwill, W.I.: Minimum concave cost flows in certain networks. Manag. Sci. 14(7), 429–450 (1968)
21. 21.
Zangwill, W.I.: A backlogging model and a multi-echelon model of a dynamic economic lot size production system-a network approach. Manag. Sci. 15(9), 506–527 (1969)
22. 22.
Zhang, M., Küçükyavuz, S., Yaman, H.: A polyhedral study of multi-echelon lot sizing with intermediate demands. Oper. Res. 60(4), 918–935 (2012) 