Skip to main content
Log in

Convex optimization problems in supply chain planning and their solution by a column generation method based on the Frank Wolfe method

  • Original Paper
  • Published:
Operational Research Aims and scope Submit manuscript

Abstract

Many problems in supply chain optimization concern the minimization of a differentiable convex objective function subject to a set of linear constraints. The aim of this work is to present a number of such problems and to propose an efficient method for their solution. The proposed method is based on improvements of the well known Frank–Wolfe algorithm. The computational results of the proposed algorithm demonstrate its effectiveness and efficiency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Allen OA (1978) Probability, statistics and queuing theory with computer science applications. Academic Press Inc, Orlando

    Google Scholar 

  • Balinski ML (1961) Fixed cost transportation problem. Nav Res Logist Q 8:41–54

    Article  Google Scholar 

  • Bazaraa MS, Shetty CM (1979) Nonlinear programming. Theory and algorithms. Wiley, New York

    Google Scholar 

  • Bertsekas DP (1982a) Constrained optimization and lagrange multiplier methods. Academic Press, New York

    Google Scholar 

  • Bertsekas DP (1982b) Projected Newton methods for optimization problems with simple constraints. SIAM J Control Optim 20:221–246

    Article  Google Scholar 

  • Bertsekas DP, Gafni EM (1983) Projected Newton methods and optimization of multicommodity flows. IEEE Trans Automat Control AC–28:1090–1096

    Article  Google Scholar 

  • Bertsekas DP, Gafni EM, Gallager RG (1984) Second derivative algorithms for minimum delay distributed routing in networks. IEEE Trans Commun COM–32:911–919

    Article  Google Scholar 

  • Bienstock B, Raskina O (2002) Asymptotic analysis of the flow deviation method for the maximum concurrent flow problem. Math Program 91:479–492

    Article  Google Scholar 

  • Brucker P (1984) An O(n) algorithm for quadratic knapsack problems. Oper Res Lett 3:163–166

    Article  Google Scholar 

  • Canon MD, Cullum CD (1968) A tight upper bound on the rate of convergence of the Frank–Wolfe algorithm. SIAM J Control 6:509–516

    Article  Google Scholar 

  • Cantor DC, Gerla M (1974) Optimal routing in a packet-switched computer network. IEEE Trans Comput C–23:1062–1069

    Article  Google Scholar 

  • Chen RR, Roundy RO, Zhang RQ, Janakiraman G (2005) Efficient auction mechanism for supply chain procurement. Manag Sci 51:467–482

    Article  Google Scholar 

  • Cooper L, LeBlanc LJ (1977) Stochastic transportation problems and other network related convex problems. Nav Res Logist Q 24:327–336

    Article  Google Scholar 

  • Damberg O, Migdalas A (1997) Distributed disaggregate simplicial decomposition—a parallel algorithm for traffic assignment. In: Hearn OD et al. (eds) Network optimization, (lecture notes in economics and mathematical systems 450. Springer, pp 172–193

  • Dunn JC (1979) Rate of convergence of conditional gradient algorithms near singular and nonsingular extremals. SIAM J Control Optim 17:187–211

    Article  Google Scholar 

  • Frank M, Wolfe P (1956) An algorithm for quadratic programming. Nav Res Logist Q 3:95–110

    Article  Google Scholar 

  • Gallego G (1990) Scheduling the production of several items with random demands in a single facility. Manag Sci 36:1579–1592

    Article  Google Scholar 

  • Garcia R, Marin A, Patriksson M (2003) A class of column generation/simplicial decomposition methods in convex differentiable optimization, I: convergence analysis. Optimization 52:171–250

    Article  Google Scholar 

  • Garcia-Ródenas R, Marin A, Patriksson M (2011) Column generation algorithms for nonlinear optimization II: numerical investigations. Comput Oper Res 38:591–604

    Article  Google Scholar 

  • Geoffrion AM, Powers RF (1995) Twenty years of strategic distribution system design: an evolutionary perspective. Interfaces 25:105–127

    Article  Google Scholar 

  • Hearn DW, Lawphongpanich S, Ventura JA (1985) Finiteness in restricted simplicial decomposition. Oper Res Lett 4:125–130

    Article  Google Scholar 

  • Hearn DW, Lawphongpanich S, Ventura JA (1987) Restricted simplicial decomposition: computation and extensions. Math Program Study 31:99–118

    Article  Google Scholar 

  • Helgason K, Kennington J, Lall H (1980) A polynomially bounded algorithm for a singly constrained quadratic program. Math Program 18:123–133

    Article  Google Scholar 

  • Holmberg K, Jornsten J (1984) Cross decomposition applied to the stochastic transportation problem. Eur J Oper Res 17:361–368

    Article  Google Scholar 

  • Holmberg K (1995) Efficient decomposition and linearization method for the stochastic transportation problem. Comput Optim Appl 4:293–316

    Article  Google Scholar 

  • Holloway CA (1974) An extension of the Frank and Wolfe method of feasible directions. Math Program 6:14–27

    Article  Google Scholar 

  • Karakitsiou A (2007) Coordination and competition in supply chain: optimization and game theoretic approaches. Ph.D. thesis, Department of Production Engineering and Management, Technical University of Crete (in Greek)

  • Karakitsiou A, Mavrommati A, Migdalas A (2005) Efficient minimization over products of simplices and its application to nonlinear multicommodity network problems. Oper Res J 4:99–118

    Article  Google Scholar 

  • Kleinrock L (1964) Communication nets: stochastic message flow and delay. McGraw-Hill, New York

    Google Scholar 

  • Larsson T, Migdalas A (1992) An algorithm for nonlinear programs over Cartesian product sets. Optimization 21:535–542

    Article  Google Scholar 

  • Larsson T, Migdalas A, Michael P (2015) A generic column generation principle: derivation and convergence analysis. Oper Res. doi:10.1007/s12351-015-0171-3

    Google Scholar 

  • Larsson T, Patriksson M (1992) Simplicial decomposition with disaggregated representation for traffic assignment problem. Transp Sci 26:4–17

    Article  Google Scholar 

  • LeBlanc SSLJ, Helgason RV, Boyce DE (1985) Improved efficiency of the Frank–Wolf algorithm for convex network programs. Transp Sci 19:445–462

    Article  Google Scholar 

  • Migdalas A (1994) A regularization of the Frank–Wolfe method and unification of certain nonlinear programming methods. Math Program 65:331–345

    Article  Google Scholar 

  • Migdalas A (2006) Nonlinear programming in telecommunication. In: Pardalos, Resende (eds) Handbook of optimization in telecommunications. Springer, New York, pp 27–66

  • Nash JF (1951) Noncooperative games. Ann Math 54:286–298

    Article  Google Scholar 

  • Pardalos PM, Kovoor N (1990) An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds. Math Program 46:321–328

    Article  Google Scholar 

  • Patriksson M (2001) Simplicial Decomposition algorithms . In: Floudas CA, Pardalos PM (eds) Encyclodepia of optimization. Springer, NY, pp 2378–2386

  • von Hohenbalken B (1977) Simplicial decomposition in nonlinear programming algorithms. Math Program 13:49–68

    Article  Google Scholar 

  • Wentzel E, Ovcharov L (1986) Applied problems in probability theory. Mir Publishers, Moscow

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. Migdalas.

Additional information

This work is dedicated to the memory of prof. K. Paparrizos.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karakitsiou, A., Migdalas, A. Convex optimization problems in supply chain planning and their solution by a column generation method based on the Frank Wolfe method. Oper Res Int J 16, 401–421 (2016). https://doi.org/10.1007/s12351-015-0205-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12351-015-0205-x

Keywords

Navigation