Lot-Sizing with Stock Upper Bounds and Fixed Charges

Here we study the discrete lot-sizing problem with an initial stock variable and an associated variable upper bound constraint. This problem is of interest in its own right, and is also a natural relaxation of the constant capacity lot-sizing problem with upper bounds and fixed charges on the stock variables. We show that the convex hull of solutions of the discrete lot-sizing problem is obtained as the intersection of two simpler sets, one a pure integer set and the other a mixing set with a variable upper bound constraint. For these two sets we derive both inequality descriptions and polynomial-size extended formulations of their respective convex hulls. Finally we carry out some limited computational tests on single-item constant capacity lot-sizing problems with upper bounds and fixed charges on the stock variables in which we use the extended formulations derived above to strengthen the initial mixed-integer programming formulations.

[1]  C. d'Aspremont,et al.  Household behavior and individual autonomy , 2009 .

[2]  Christian M. Hafner,et al.  Estimating Autocorrelations in the Presence of Deterministic Trends , 2011 .

[3]  Laurence A. Wolsey,et al.  The Mixing Set with Flows , 2007, SIAM J. Discret. Math..

[4]  V. Ginsburgh,et al.  Handbook of the Economics of the Art and Culture , 2006 .

[5]  Laurence A. Wolsey,et al.  Lot-sizing with production and delivery time windows , 2005, Math. Program..

[6]  P. Picard,et al.  Firms' location under taste and demand heterogeneity , 2008 .

[7]  Mathieu Van Vyve.  The Continuous Mixing Polyhedron , 2005, Math. Oper. Res..

[8]  J. Drèze,et al.  Public goods, environmental externalities and fiscal competition , 2006 .

[9]  Laurence A. Wolsey,et al.  Polyhedra for lot-sizing with Wagner—Whitin costs , 1994, Math. Program..

[10]  Alper Atamt.  LOT SIZING WITH INVENTORY BOUNDS AND FIXED COSTS: POLYHEDRAL STUDY AND COMPUTATION , 2004 .

[11]  J. Gabszewicz.  La différenciation des produits , 2006 .

[12]  J. Rombouts,et al.  Style Rotation and Performance Persistence of Mutual Funds , 2009 .

[13]  Laurence A. Wolsey,et al.  Reformulation and Decomposition of Integer Programs , 2009, 50 Years of Integer Programming.

[14]  Laurence A. Wolsey,et al.  Network Formulations of Mixed-Integer Programs , 2006, Math. Oper. Res..

[15]  Erik Schokkaert,et al.  What good is happiness , 2009 .

[16]  Andrea Silvestrini,et al.  What do we know about comparing aggregate and disaggregate forecasts , 2009 .

[17]  Stéphane Zuber,et al.  The Pareto Principle of Optimal Inequality , 2009 .

[18]  Christian Gourieroux,et al.  Simulation-based econometric methods , 1996 .

[19]  Robert Weismantel,et al.  On the 0/1 knapsack polytope , 1997, Math. Program..

[20]  Manfred W. Padberg.  (1,k)-configurations and facets for packing problems , 1980, Math. Program..

[21]  Hylke Vandenbussche,et al.  Are Your Firm's Taxes Set in Warsaw? Spatial Tax Competition in Europe , 2008 .

[22]  Egon Balas.  Disjunctive Programming , 2010, 50 Years of Integer Programming.

[23]  Hylke Vandenbussche,et al.  Financial Constraints in China: Firm-Level Evidence , 2009 .

[24]  Oktay Günlük,et al.  Mixing mixed-integer inequalities , 2001, Math. Program..

[25]  Jacques-François Thisse,et al.  Economic Geography: The Integration of Regions and Nations , 2008 .

[26]  Patrice Pieretti,et al.  On tax competition, public goods provision and jurisdictions' size CORE DP No. 2009/12 , 2011 .

[27]  Michel Lubrano,et al.  The tradeoff between growth and redistribution: ELIE in an overlapping generations model , 2009 .

[28]  Laurence A. Wolsey,et al.  Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 4th International Conference, CPAIOR 2007, Brussels, Belgium, May 23-26, 2007, Proceedings , 2007, CPAIOR.

[29]  James Renegar,et al.  A mathematical view of interior-point methods in convex optimization , 2001, MPS-SIAM series on optimization.

[30]  Jean-François Mertens,et al.  Intergenerational Equity and the Discount Rate for Cost-Benefit Analysis , 2006 .

[31]  Jean-François Mertens,et al.  Regularity and stability of equilibria in an overlapping generations model with exogenous growth , 2009 .

[32]  Yves Smeers,et al.  Equilibrium models for the carbon leakage problem , 2008 .

[33]  Vladimir M. Veliov,et al.  Prices versus quantities in a vintage capital model , 2010 .

[34]  Nicolas Gillis,et al.  Using underapproximations for sparse nonnegative matrix factorization , 2009, Pattern Recognit..

[35]  Henry Tulkens,et al.  Dynamic Core-Theoretic Cooperation in a Two-Dimensional International Environmental Model , 2009 .

[36]  Daniel Bienstock,et al.  Potential Function Methods for Approximately Solving Linear Programming Problems: Theory and Practice , 2002 .

[37]  Mathieu Van Vyve,et al.  Lot-sizing with fixed charges on stocks: the convex hull , 2004, Discret. Optim..

[38]  Laurence A. Wolsey,et al.  Projecting an Extended Formulation for Mixed-Integer Covers on Bipartite Graphs , 2010, Math. Oper. Res..

[39]  Thierry Bréchet,et al.  CAN EDUCATION BE GOOD FOR BOTH GROWTH AND THE ENVIRONMENT? , 2009, Macroeconomic Dynamics.

[40]  W. Pohlmeier,et al.  High frequency financial econometrics : recent developments , 2007 .

[41]  R. Amir.  Supermodularity and Complementarity in Economics: An Elementary Survey , 2003 .

[42]  Jean Jaskold Gabszewicz,et al.  Public goods’ attractiveness and migrations , 2008 .

[43]  J. Dávila,et al.  The taxation of capital returns in overlapping generations economies without Financial assets , 2008 .

[44]  M. Sanin,et al.  Understanding volatility dynamics in the EU-ETS market , 2009 .