Abstract
We examine mixed integer programming reformulations of the uncapacitated lot-sizing problem with backlogging. First we consider the effect of using a standard reformulation technique for fixed charge network flow problems which involves the introduction of new variables, leading to a known plant location reformulation and a shortest path reformulation. Each of these reformulations is strong in the sense that its linear programming relaxation solves the lot-sizing problem.
Secondly we attempt to treat the problem in the space of the original variables. We give an implicit description of the convex hull of solutions, and show how the problem of finding a violated cutting plane can be solved as a linear program. We also describe a family of strong valid inequalities which can be generated rapidly by a heuristic and which have proved effective in a cut generation algorithm.
The efficiency of both the shortest path formulation and the cutting plane algorithm have been tested on a series of multi-item capacitated lot-sizing problems with backlogging. Near optimal solutions have been found to problems with 8 periods and up to 100 times.
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References
I. Barany, T.J. Van Roy and L.A. Wolsey, “Strong formulations for multi-item capacitated lot sizing,”Management Science 30 (10) (1984) 1255–1261.
I. Barany, J. Edmonds and L.A. Wolsey, “Packing and covering a tree by subtrees,”Combinatorica 6, (1986) 221–233.
P.S. Dixon and E.A. Silver, “A heuristic solution procedure for the multi-item, single-level, limited-capacity, lot-sizing problem,”J. Oper. Management 2 (1981) 23–39.
G.D. Eppen and R.K. Martin, “Solving multi-item capacitated lot sizing problems using variable redefinition,” Graduate School of Business, University of Chicago, (1985).
M. Grotschel, L. Lovasz and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization,”Combinatorica 1 (2) (1981) 169–197.
N. Karmarkar, “A new polynomial time algorithm for linear programming,”Combinatorica 4 (4) (1984) 373–395.
L.G. Khachian, “A polynomial algorithm in linear programming,”Soviet Math Dokl 20 (1979) 191–194.
R.L. Rardin and V. Choe, “Tighter relaxations of fixed charge network flow problems,” Report ♯J-79-18, School of Industrial and Systems Engineering, Georgia Institute of Technology (1979).
K. Rosling, “The dynamic inventory model and the uncapacitated facility location problem,” Linköping Institute of Technology, Department of Production Economics, S-58183 Linköping, Sweden, (1983).
L.E. Schrage, “Implicit representation of variable upper bounds in linear programs,”Mathematical Programming Study 4 (1975) 118–132.
J.M. Thizy and L.N. Van Wassenhove, “Decomposition algorithm for the multi-product lot-sizing problem with capacity constraints,”IIE Transactions (1986).
T.J. Van Roy and L.A. Wolsey, “Solving mixed 0-1 programs by automatic reformulation,”Operations Research 35 (1987) 45–57.
T.J. Van Roy and L.A. Wolsey, “Valid inequalities and separation for uncapacitated fixed cost networks,”Operations Research Letters 4 (3) (1985) 105–112.
W.I. Zangwill, “A backlogging model and a multi-echelon model of a dynamic economic lot size production system—A network approach,”Management Science 15 (9) (1969) 506–527.
W.I. Zangwill, “Minimum concave cost flows in certain networks,”Management Science 14 (7) (1968) 429–450.
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Pochet, Y., Wolsey, L.A. Lot-size models with backlogging: Strong reformulations and cutting planes. Mathematical Programming 40, 317–335 (1988). https://doi.org/10.1007/BF01580738
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DOI: https://doi.org/10.1007/BF01580738