Hierarchical Benders Decomposition for Open-Pit Mine Block Sequencing

The open-pit mine block sequencing problem (OPBS) models a deposit of ore and surrounding material near the Earth’s surface as a three-dimensional grid of blocks. A solution in discretized time identifies a profit-maximizing extraction (mining) schedule for the blocks. Our model variant, a mixed-integer program (MIP), presumes a predetermined destination for each extracted block, namely, processing plant or waste dump. The MIP incorporates standard constructs but also adds not-so-standard lower bounds on resource consumption in each time period and allows fractional block extraction in a novel fashion while still enforcing pit-wall slope restrictions. A new extension of nested Benders decomposition, “hierarchical” Benders decomposition (HBD), solves the MIP’s linear-programming relaxation. HBD exploits time-aggregated variables and can recursively decompose a model into a master problem and two subproblems rather than the usual single subproblem. A specialized branch-and-bound heuristic then produces high...

[1]  Renaud Chicoisne,et al.  A new algorithm for the open-pit mine scheduling problem , 2009 .

[2]  G. Infanger,et al.  Planning under uncertainty solving large-scale stochastic linear programs , 1992 .

[3]  Laurence A. Wolsey,et al.  Production Planning by Mixed Integer Programming , 2010 .

[4]  D. Krige.  A statistical approach to some basic mine valuation problems on the Witwatersrand, by D.G. Krige, published in the Journal, December 1951 : introduction by the author , 1951 .

[5]  Horand I. Gassmann,et al.  Mslip: A computer code for the multistage stochastic linear programming problem , 1990, Math. Program..

[6]  Michel Gendreau,et al.  Interior point stabilization for column generation , 2007, Oper. Res. Lett..

[7]  Alexandra M. Newman,et al.  Tailored Lagrangian Relaxation for the open pit block sequencing problem , 2014, Ann. Oper. Res..

[8]  J. Karlssona,et al.  Short-term harvest planning including scheduling of harvest crews , 2003 .

[9]  Kazuhiro Kawahata.  New algorithm to solve large scale mine production scheduling problems by using the Lagrangian relaxation method, A , 2016 .

[10]  David P. Morton,et al.  An enhanced decomposition algorithm for multistage stochastic hydroelectric scheduling , 1996, Ann. Oper. Res..

[11]  Martin W. P. Savelsbergh,et al.  Branch-and-Price: Column Generation for Solving Huge Integer Programs , 1998, Oper. Res..

[12]  Alan S. Manne,et al.  Nested decomposition for dynamic models , 1974, Math. Program..

[13]  Samir Elhedhli,et al.  The integration of an interior-point cutting plane method within a branch-and-price algorithm , 2004, Math. Program..

[14]  C. R. Glassey.  Nested Decomposition and Multi-Stage Linear Programs , 1973 .

[15]  A. M. Geoffrion,et al.  Multicommodity Distribution System Design by Benders Decomposition , 1974 .

[16]  H Eivazy,et al.  A mixed integer linear programming model for short-term open pit mine production scheduling , 2012 .

[17]  George B. Dantzig,et al.  Decomposition Principle for Linear Programs , 1960 .

[18]  Evan L. Porteus,et al.  Decomposition of arborescent linear programs , 1977, Math. Program..

[19]  L. G. Stolarczyk,et al.  Definition imaging of an orebody with the radio imaging method (RIM) , 1992 .

[20]  Andrzej Ruszczynski,et al.  A regularized decomposition method for minimizing a sum of polyhedral functions , 1986, Math. Program..

[21]  Lou Caccetta,et al.  An Application of Branch and Cut to Open Pit Mine Scheduling , 2003, J. Glob. Optim..

[22]  Daniel Bienstock,et al.  Solving LP Relaxations of Large-Scale Precedence Constrained Problems , 2010, IPCO.

[23]  Graham C. Goodwin,et al.  Open-cut Mine Planning via Closed-loop Receding-horizon Optimal Control , 2007 .

[24]  Ignacio E. Grossmann,et al.  An Iterative Aggregation/Disaggregation Approach for the Solution of a Mixed-Integer Nonlinear Oilfield Infrastructure Planning Model , 2000 .

[25]  Roussos Dimitrakopoulos,et al.  Stochastic Optimisation of Long-Term Production Scheduling for Open Pit Mines with a New Integer Programming Formulation , 2018 .

[26]  Alexandra M. Newman,et al.  MineLib: a library of open pit mining problems , 2013, Ann. Oper. Res..

[27]  R. Kevin Wood,et al.  Dantzig-Wolfe Decomposition for Solving Multistage Stochastic Capacity-Planning Problems , 2009, Oper. Res..

[28]  H. Gabbay.  Multi-Stage Production Planning , 1979 .

[29]  Thys B Johnson,et al.  OPTIMUM OPEN PIT MINE PRODUCTION SCHEDULING , 1968 .

[30]  Jacques F. Benders,et al.  Partitioning procedures for solving mixed-variables programming problems , 2005, Comput. Manag. Sci..

[31]  Eduardo Moreno,et al.  A New Algorithm for the Open-Pit Mine Production Scheduling Problem , 2012, Oper. Res..

[32]  C. Roger Glassey,et al.  Dynamic Linear Programs for Production Scheduling , 1971, Oper. Res..

[33]  Ambros M. Gleixner,et al.  Solving Large-scale Open Pit Mining Production Scheduling Problems by Integer Programming , 2008 .

[34]  L. M. Giannini.  Optimum design of open pit mines , 1991, Bulletin of the Australian Mathematical Society.

[35]  Salih Ramazan,et al.  The new Fundamental Tree Algorithm for production scheduling of open pit mines , 2007, Eur. J. Oper. Res..

[36]  J. Birge,et al.  A multicut algorithm for two-stage stochastic linear programs , 1988 .

[37]  Mark Gershon.  Heuristic approaches for mine planning and production scheduling , 1987 .

[38]  R. Kevin Wood,et al.  Explicit-Constraint Branching for Solving Mixed-Integer Programs , 2000 .

[39]  Robert Entriken.  Parallel Decomposition: Results for Staircase Linear Programs , 1996, SIAM J. Optim..

[40]  I. Dumitrescu,et al.  A Multistage Stochastic Programming Approach to Open Pit Mine Production Scheduling with Uncertain Geology , 2008 .

[41]  Robert J. Wittrock.  Dual nested decomposition of staircase linear programs , 1985 .

[42]  Wilhelm Hummeltenberg.  Implementations of special ordered sets in MP software , 1984 .

[43]  Dorit S. Hochbaum,et al.  Performance Analysis and Best Implementations of Old and New Algorithms for the Open-Pit Mining Problem , 2000, Oper. Res..

[44]  J. Gondzio,et al.  Using an interior point method for the master problem in a decomposition approach , 1997 .

[45]  Mark Gershon,et al.  Optimal mine production scheduling: evaluation of large scale mathematical programming approaches , 1983 .

[46]  Alexandra M. Newman,et al.  A sliding time window heuristic for open pit mine block sequencing , 2011, Optim. Lett..

[47]  Felipe Caro,et al.  Optimizing Long-Term Production Plans in Underground and Open-Pit Copper Mines , 2010, Oper. Res..

[48]  Martin W. P. Savelsbergh,et al.  A Computational Study of Search Strategies for Mixed Integer Programming , 1999, INFORMS J. Comput..

[49]  Alexandra M. Newman,et al.  Open-Pit Block-Sequencing Formulations: A Tutorial , 2014, Interfaces.

[50]  Ioannis Minis,et al.  Efficient techniques for the multi-period vehicle routing problem with time windows within a branch and price framework , 2013, Ann. Oper. Res..

[51]  Ambros M. Gleixner,et al.  LP-based disaggregation approaches to solving the open pit mining production scheduling problem with block processing selectivity , 2009, Comput. Oper. Res..

[52]  Dorit S. Hochbaum,et al.  Solving the Convex Cost Integer Dual Network Flow Problem , 1999, Manag. Sci..

[53]  Gerald G. Brown,et al.  The Kellogg Company Optimizes Production, Inventory, and Distribution , 2001, Interfaces.

[54]  Javad Gholamnejad,et al.  A new mathematical programming model for long-term production scheduling considering geological uncertainty , 2012 .

[55]  Richard D. Wollmer,et al.  Two stage linear programming under uncertainty with 0–1 integer first stage variables , 1980, Math. Program..