# Minimum concave-cost network flow problems: Applications, complexity, and algorithms

Article

- 1.1k Downloads
- 79 Citations

## Abstract

We discuss a wide range of results for minimum concave-cost network flow problems, including related applications, complexity issues, and solution techniques. Applications from production and inventory planning, and transportation and communication network design are discussed. New complexity results are proved which show that this problem is NP-hard for cases with cost functions other than fixed charge. An overview of solution techniques for this problem is presented, with some new results given regarding the implementation of a particular branch-and-bound approach.

### Keywords

Concave-cost network flow global optimization complexity theory NP-hard transportation problems## Preview

Unable to display preview. Download preview PDF.

### References

- [1]U. Akinc and B.M. Khumawala, An efficient branch and bound algorithm for the capacitated warehouse location problem, Management Sci. 23 (1977) 585–594.Google Scholar
- [2]A. Balakrishnan and S.C. Graves, A composite algorithm for a concave-cost network flow problem, Networks 19 (1989) 175–202.Google Scholar
- [3]M.L. Balinski, Fixed cost transportation problems, Naval Res. Logistics Quarterly 8 (1961) 41–54.Google Scholar
- [4]M.L. Balinski and R.E. Quandt, On an integer program for a delivery problem. Oper. Res. 12 (1964) 300–304.Google Scholar
- [5]R.S. Barr, F. Glover and D. Klingman, A new optimization method for large scale fixed charge transportation problems. Oper. Res. 29 (1981) 448–463.Google Scholar
- [6]W.J. Baumol and P. Wolfe, A warehouse-location problem Oper. Res. 6 (1958) 252–263.Google Scholar
- [7]I. Baybars and R.H. Edahl, A heuristic method for facility planning in telecommunications networks with multiple alternate routes, Naval Res. Logistics Quarterly 35 (1988) 503–528.Google Scholar
- [8]D.P. Bertsekas and J.N. Tsitsiklis,
*Parallel and Distributed Computation*(Prentice Hall, New Jersey, 1989).Google Scholar - [9]H.L. Bhatia, Indefinite quadratic solid transportation problem, J. Inform. Optimiz. Sci. 2 (1981) 297–303.Google Scholar
- [10]C.T. Bornstein and R. Rust, Minimizing a sum of staircase functions under linear constraints, Optimization 19 (1988) 181–190.Google Scholar
- [11]V.P. Bulatov, The immersion method for the global minimization of functions on the convex polyhedra,
*Int. Symp. on Engineering, Mathematics, and Applications*, Beijing, PR China (July 1988) pp. 335–338.Google Scholar - [12]A.V. Cabot and S.S. Erenguc, Some branch-and-bound procedures for fixed-cost transportation problems, Naval Res. Logistics Quarterly 31 (1984) 145–154.Google Scholar
- [13]L. Cooper and C. Drebes, An approximate solution method for the fixed-charge problem. Naval Res. Logistics Quarterly 14 (1968) 101–113.Google Scholar
- [14]G. Daeninck and Y. Smeers, Using shortest paths in some transshipment problems with concave costs, Math. Programming (1977) 18–25.Google Scholar
- [15]G.B. Dantzig,
*Linear Programming and Extensions*(Princeton University Press, Princeton, New Jersey, 1963).Google Scholar - [16]P.S. Davis and T.L. Ray, A branch-and-bound algorithm for the capacitated facilities locations problem, Naval Res. Logistics Quarterly 16 (1969) 331–344.Google Scholar
- [17]D.R. Denzler, An approximate algorithm for the fixed charge problems, Naval Res. Logistics Quarterly 16 (1969) 411–416.Google Scholar
- [18]N. Driebeek, An algorithm for the solution of mixed integer programming problems. Management Sci. 12 (1966) 576–587.Google Scholar
- [19]M.A. Efroymson and T.L. Ray, A branch-and-bound algorithm for plant location. Oper. Res. 14 (1966) 361–368.Google Scholar
- [20]H.G. Eggleston,
*Convexity, Cambridge Tracts in Mathematics and Mathematical Physics*No. 47 (Cambridge University Press, Cambridge, MA, 1963).Google Scholar - [21]R.E. Erickson, C.L. Monma and A.F. Veinott, Jr., Send-and-split method for minimum-con-cave-cost network flows, Math. Oper. Res. 12 (1987) 634–664.Google Scholar
- [22]D. Erlenkotter, Two producing areas—dynamic programming solutions, in:
*Investment for Capacity Expansion: Size, Location and Time Phasing*, ed. A.S. Manne (MIT Press, Cambridge, MA, 1967) pp. 210–227.Google Scholar - [23]J.E. Falk and R.M. Soland, An algorithm for separable nonconvex programming problems. Management Sci. 15 (1969) 550–569.Google Scholar
- [24]M. Florian, An introduction to network models used in transportation planning,
*Transportation Planning Models*(Elsevier, New York, 1984).Google Scholar - [25]M. Florian, Nonlinear cost network models in transportation analysis, Math. Programming Study 26 (1986) 167–196.Google Scholar
- [26]M. Florian and M. Klein, Deterministic production planning with concave costs and capacity constraints. Management Sci. 18 (1971) 12–20.Google Scholar
- [27]M. Florian, J.K. Lenstra and A.H.G. Rinnooy Kan. Deterministic production planning: algorithms and complexity, Management Sci. 26 (1980) 669–679.Google Scholar
- [28]M. Florian and P. Robillard, An implicit enumeration algorithm for the concave cost network flow problem, Management Sci. 18 (1971) 184–193.Google Scholar
- [29]M. Florian, M. Rossin-Arthiat, and D. de Werra, A property of minimum concave cost flows in capacitated networks, Can. J. Oper. Res. 9 (1971) 293–304.Google Scholar
- [30]C.O. Fong and M.R. Rao, Capacity expansion with two producing regions and concave costs, Management Sci. 22 (1975) 331–339.Google Scholar
- [31]L.R. Ford, jr. and D.R. Fulkerson,
*Flows in Networks*(Princeton University Press, Princeton, New Jersey, 1962).Google Scholar - [32]G. Gallo, C. Sandi and C. Sodini, An algorithm for the min concave-cost flow problem, Europ. J. Oper. Res. 4 (1980) 249–255.Google Scholar
- [33]G. Gallo and C. Sodini, Adjacent extreme flows and application to min concave-cost flow problems. Networks 9 (1979) 95–121.Google Scholar
- [34]M.R. Garey and D.S. Johnson,
*Computers and Intractability: A Guide to the Theory of NP-Completeness*(W.H. Freeman, San Francisco, CA, 1979).Google Scholar - [35]S.C. Graves and J.B. Orlin, A minimum concave-cost dynamic network flow problem with an application to lot-sizing. Networks, 15 (1985) 59–71.Google Scholar
- [36]P. Gray, Exact solution of the site selection problem by mixed integer programming, in:
*Applications of Mathematical Programming Techniques*, ed. E.M.L. Beale (The English Universities Press, London, 1970) pp. 262–270.Google Scholar - [37]P. Gray, Exact solution of the fixed-charge transportation problem. Oper. Res. 19 (1971) 1529–1538.Google Scholar
- [38]G.M. Guisewite and P.M. Pardalos, Algorithms for the uncapacitated single-source minimum concave-cost network flow problem, Technical Report, Department of Computer Science, Pennsylvania State University (1990).Google Scholar
- [39]K.B. Haley, The solid transportation problem, Oper. Res. Quarterly 10 (1962) 448–463.Google Scholar
- [40]R.W. Hall, Graphical interpretation of the transportation problem, Transportation Sci. 23 (1988) 37–45.Google Scholar
- [41]G.L. Hefley and B.O. Barr, Fixed charge transportation problem: a survey, presented to the 43rd ORSA National Meeting in Milwaukee, Wisconsin (May 1973).Google Scholar
- [42]D.S. Hochbaum and A. Segev. Analysis of a flow problem with fixed charges. Networks 19 (1989) 291–312.Google Scholar
- [43]R. Horst and H. Tuy,
*Global Optimization: Deterministic Approaches*. (Springer, 1989) to be published.Google Scholar - [44]T. Ibaraki and N. Katoh,
*Resource Allocation Problems, Algorithmic Approaches*(The MIT Press, Cambridge, MA., 1988).Google Scholar - [45]J.J. Jarvis, V.E. Unger, R.L. Rardin and R.W. Moore, Optimal design of regional wastewater systems: a fixed charge network flow model. Oper. Res. 26 (1978) 538–550.Google Scholar
- [46]R. Jogannathan and M.R. Rao, A class of deterministic production planning models, Management Sci. 19 (1973) 1295–1300.Google Scholar
- [47]L.A. Johnson and D.C. Montgomery,
*Operations Research in Production Planning, Scheduling, and Inventory Control*. (Wiley, New York, 1974) pp. 212–224.Google Scholar - [48]D. Kennedy, Some branch-and-bound techniques for nonlinear optimization, Math. Programming 42 (1988) 147–169.Google Scholar
- [49]J. Kennington, The fixed-charge transportation problem: a computational study with a branch-and-bound code. AIIE Trans. 8 (1976) 241–247.Google Scholar
- [50]J. Kennington and E. Unger, The group-theoretic structure in the fixed-charge transportation problem. Oper. Res. 21 (1973) 1142–1152.Google Scholar
- [51]J. Kennington and E. Unger, A new branch-and-bound algorithm for the fixed-charge transportation algorithm. Management Sci. 22 (1976) 1116–1126.Google Scholar
- [52]B.M. Khumawala, An efficient branch-and-bound algorithm for the warehouse location problem. Management Sci. 18 (1972) B718-B731.Google Scholar
- [53]D. Klingman, P.H. Randolph and S.W. Fuller, A cotton ginning problem, Oper. Res. 24 (1976) 700–717.Google Scholar
- [54]H. Konno, Minimum concave series production system with deterministic demands: a backlogging case. J. Oper. Res. Soc. Jpn. 16 (1973) 246–253.Google Scholar
- [55]H. Konno, Minimum concave cost production system: multi-echelon model. Math. Programming 41 (1988) 185–193.Google Scholar
- [56]H.W. Kuhn and W.J. Baumol, An approximate algorithm for the fixed-charge transportation problem, Naval Res. Logistics Quarterly 9 (1962) 1–16.Google Scholar
- [57]B. Lamar, An improved branch-and-bound algorithm for minimum concave cost network flow problems, Working Paper, Graduate School of Management and Institute of Transportation Studies, University of California, Irvine (1989).Google Scholar
- [58]L. LeBlanc, Global solutions for a non-convex, non-concave rail network model, Math. Sci. 23 (1978) 131–139.Google Scholar
- [59]S. Lee and P.H. Zipkin, A dynamic lot-size model with make-or-buy decisions. Management Sci. 35 (1989) 447–458.Google Scholar
- [60]N. Linial, Hard enumeration problems in geometry and combinatorics, SIAM J. Alg. Discr. Meth. 7 (1986) 331–335.Google Scholar
- [61]L.S. Linn and J. Allen, Minplex — a compactor that minimizes the bounding rectangle and individual rectangles in a layout,
*23rd Design Automation Conf.*(1986) pp. 123–129.Google Scholar - [62]S.A. Lippman, Optimal inventory policy with multiple setup costs. Management Sci. 16 (1969) 118–138.Google Scholar
- [63]S.F. Love, A facilities in series inventory model with nested schedules. Management Sci. 18 (1972) 327–338.Google Scholar
- [64]S.F. Love, Bounded production and inventory models with piecewise concave costs, Management Sci. 20 (1973) 313–318.Google Scholar
- [65]D.D. Lozovanu, Properties of optimal solutions of a grid transport problem with concave function of the flows on the arcs. Engineering Cybernetics 20 (1983) 34–38.Google Scholar
- [66]T.L. Magnanti and R.T. Wong. Network design and transportation planning: models and algorithms, Transportation Sci. 18 (1984) 1–55.Google Scholar
- [67]A.S. Manne and A.F. Veinott, Jr., Optimal plant size with arbitrary increasing time paths of demand, in:
*Investments for Capacity Expansion: Size, Location and Time Phasing*, ed. A.S. Manne (MIT Press, Cambridge, MA, 1967) pp. 178–190.Google Scholar - [68]P. McKeown, A vertex ranking procedure for solving the linear fixed-charge problem Oper. Res. 23 (1975) 1183–1191.Google Scholar
- [69]K.G. Murty, Solving the fixed charge problem by ranking the extreme points. Oper. Res. 16 (1968) 268–279.Google Scholar
- [70]K.G. Murty and S.N. Kabadi, Some NP-complete problems in quadratic and non-linear programming. Math. Programming 39 (1987) 117–129.Google Scholar
- [71]P.M. Pardalos, Enumerative techniques for solving some nonconvex global optimization problems. OR Spectrum 10 (1988) 29–35.Google Scholar
- [72]P.M. Pardalos, J.H. Glick and J.B. Rosen, Global minimization of indefinite quadratic problems, Computing 39 (1987) 281–291.Google Scholar
- [73]P.M. Pardalos, A.T. Phillips and J.B. Rosen,
*Parallel Computing in Mathematical Programming*, (SIAM, Philadelphia, PA, 1990) to appear.Google Scholar - [74]P.M. Pardalos and J.B. Rosen, Global minimization of large-scale constrained concave quadratic problems by separable programming, Math. Programming 34 (1986) 163–174.Google Scholar
- [75]P.M. Pardalos and J.B. Rosen. Global concave minimization: a bibliographic survey. SIAM Rev. 28 (1986) 367–379.Google Scholar
- [76]P.M. Pardalos and J.B. Rosen.
*Constrained Global Optimization: Algorithms and Applications*. Lecture Notes in Computer Science 268 (Springer, Berlin, 1987).Google Scholar - [77]P.M. Pardalos and G. Schnitger, Checking local optimality in constrained quadratic programming is NP-hard, Oper. Res. Lett. 7 (1988) 33–35.Google Scholar
- [78]A.T. Phillips, Parallel algorithms for constrained optimization, Ph.D. Dissertation, University of Minnesota (1988).Google Scholar
- [79]A.T. Phillips and J.B. Rosen, A parallel algorithm for constrained concave quadratic global minimization. Math. Programming 42 (1988) 421–448.Google Scholar
- [80]J. Plasil and P. Chlebnican, A new algorithm for the concave cost flow problem, Working Paper, Technical University of Transport and Communications. Zilina, Czechoslovakia (1989).Google Scholar
- [81]R. Polyak, On a nonconvex problem, Sov. Math. Doklady 17a (1968) 786–789.Google Scholar
- [82]P. Rech and L.G. Barton, A non-convex transportation algorithm, in:
*Applications of Mathematical Programming Techniques*, ed. E.M.L. Beale (The English Universities Press, London, 1970) pp. 250–260.Google Scholar - [83]C. Revelle, D. Marks and J.C. Liebman, An analysis of private and public sector location models, Management Sci. 16 (1970) 692–707.Google Scholar
- [84]J.B. Rosen, Global minimization of a linearly constrained concave function by partition of feasible domain. Math. Oper. Res. 8 (1983) 215–230.Google Scholar
- [85]M.J. Sobel, Smoothing start-up and shut-down costs: concave case, Management Sci. 17 (1970) 78–91.Google Scholar
- [86]R.M. Soland, Optimal facility location with concave costs. Oper. Res. 22 (1974) 373–382.Google Scholar
- [87]K. Spielberg. An algorithm for the simple plant location problem with some side conditions, Oper. Res. 17 (1969) 85–111.Google Scholar
- [88]K. Spielberg, Plant location with generalized search origin. Management Sci. 16 (1969) 165–178.Google Scholar
- [89]K. Spielberg, On solving plant location problems, in:
*Applications of Mathematical Programming Techniques*, ed. E.M.L. Beale (The English Universities Press, London, 1970) pp. 216–234.Google Scholar - [90]J.W. Stroup, Allocation of launch vehicles to space missions: a fixed-cost transportation problem. Oper. Res. 15 (1967) 1157–1163.Google Scholar
- [91]C. Swoveland, A deterministic multi-period production planning model with piecewise concave production and holding-backorder costs. Management Sci. 21 (1975) 1007–1013.Google Scholar
- [92]H.A. Taha, Concave minimization over a convex polyhedron. Naval Res. Logistics Quarterly 20 (1973) 533–548.Google Scholar
- [93]P.T. Thach, An efficient method for min concave cost flow problems under circuitless single-source uncapacitated networks, Technical Report. Technical University of Graz. Austria (1989).Google Scholar
- [94]N.V. Thoai and H. Tuy. Convergent algorithms for minimizing a concave function. Math. Oper. Res. 5 (1980) 556–566.Google Scholar
- [95]J.A. Tomlin, Minimum-cost multicommodity network flows. Oper. Res. 14 (1966) 45–51.Google Scholar
- [96]H. Tuy, Concave-programming under linear constraints. Sov. Math. Doklady (1964) 1437–1440.Google Scholar
- [97]A.F. Veinott, Jr., Minimum concave-cost solution of Leontief substitution models of multifacility inventory systems. Oper. Res. 17 (1969) 262–291.Google Scholar
- [98]H.M. Wagner, On a class of capacitated transportation problems, Management. Sci. 5 (1959) 304–318.Google Scholar
- [99]H.M. Wagner, A postscript to: Dynamic problems in the theory of the firm. Naval Res. Logistics Quarterly 7 (1960) 7–12.Google Scholar
- [100]H.M. Wagner and T.M. Whitin, Dynamic version of the economic lot size model, Management Sci. 5 (1958) 89–96.Google Scholar
- [101]W.E. Walker, A heuristic adjacent extreme point algorithm for the fixed charge problem, Management Sci. 22 (1976) 587–596.Google Scholar
- [102]B. Yaged, Jr., Minimum cost routing for static network models, Networks 1 (1971) 139–172.Google Scholar
- [103]N. Zadeh, On building minimum cost communication networks. Networks 3 (1973) 315–331.Google Scholar
- [104]N. Zadeh, On building minimum cost communication networks over time, Networks 4 (1974) 19–34.Google Scholar
- [105]W.I. Zangwill, A deterministic multi-period production scheduling model with backlogging, Management Sci. 13 (1966) 105–119.Google Scholar
- [106]W.I. Zangwill, Production smoothing of economic lot sizes with non-decreasing requirements. Management Sci. 13 (1966) 191–209.Google Scholar
- [107]W.I. Zangwill, Minimum concave-cost flows in certain networks Management Sci. 14 (1968) 429–450.Google Scholar
- [108]W.I. Zangwill, A backlogging model and a multi-echelon model of an economic lot size production system — a network approach, Management Sci. 15 (1969) 506–527.Google Scholar

## Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1990