A Forward Simplex Method for Staircase Linear Programs

Modelling planning problems that extend over many time periods as linear programs leads to a special structure called a "staircase" or "dynamic" linear program. In this special structure, the nonzero coefficients of the linear program appear in blocks along the "main diagonal" of the coefficient matrix. Such problems are commonly found in economic planning, structural design, agricultural planning, dynamic traffic assignment, production planning, and scheduling models. Forward algorithms provide an approach to solving these dynamic problems by solving successively longer finite horizon subproblems, terminating when a stopping rule can be invoked or a decision horizon found. Such algorithms are available for a large number of specific models. Here we discuss the implementation and testing of a forward algorithm for solving general dynamic staircase linear programs. Tests reported indicate that the solution time is linear in the number of periods of the staircase problem, as compared to a quadratic or cubic relationship for standard linear programming codes. Computational decision horizons are often found, and are responsible for the good performance of the algorithm.

[1]  Gerald L. Thompson,et al.  A forward algorithm and planning horizon procedure for the production smoothing problem without inventory , 1984 .

[2]  R. J. Niehaus,et al.  Managerial Tests of Conversational Manpower Planning Models. , 1975 .

[3]  James K. Ho,et al.  Computational experience with advanced implementation of decomposition algorithms for linear programming , 1983, Math. Program..

[4]  Abraham Charnes,et al.  Dynamic multiattribute models for mixed manpower systems , 1975 .

[5]  Michael A. Saunders,et al.  MINOS/AUGMENTED User's Manual. , 1980 .

[6]  Suresh Chand,et al.  Planning Horizon Procedures for Machine Replacement Models with Several Possible Replacement Alternatives , 1982 .

[7]  Deepak K. Merchant,et al.  A Model and an Algorithm for the Dynamic Traffic Assignment Problems , 1978 .

[8]  A. I. Propoi,et al.  Problems of Dynamic Linear Programming , 1976 .

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

[10]  G. Dantzig.  UPPER BOUNDS, SECONDARY CONSTRAINTS, AND BLOCK TRIANGULARITY IN LINEAR PROGRAMMING , 1955 .

[11]  André F. Perold.  Fundamentals of a continuous time simplex method , 1979 .

[12]  Richard V. Helgason,et al.  Spike Swapping In Basis Reinversion. , 1979 .

[13]  Louis W. Miller.  Using Linear Programming to Derive Planning Horizons for a Production Smoothing Problem , 1979 .

[14]  Thomas E. Morton,et al.  Universal Planning Horizons for Generalized Convex Production Scheduling , 1978, Oper. Res..

[15]  A. Propoi,et al.  The Simplex Method for Dynamic Linear Programs , 1978 .

[16]  G. Dantzig.  COMPACT BASIS TRIANGULARIZATION FOR THE SIMPLEX METHOD , 1962 .

[17]  James K. Ho,et al.  A Comparative Study of Two Methods for Staircase Linear Programs , 1980, TOMS.

[18]  James E. Kalan.  Aspects of large-scale in-core linear programming , 1971, ACM '71.

[19]  Richard D. Wollmer.  A Substitute Inverse for the Basis of a Staircase Structure Linear Program , 1977, Math. Oper. Res..

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

[21]  G. Thompson,et al.  Turnpike Horizons for Production Planning , 1980 .

[22]  Eli Hellerman,et al.  Reinversion with the preassigned pivot procedure , 1971, Math. Program..

[23]  G. Dantzig,et al.  A Basis Factorization Method for Block Triangular Linear Programs. , 1978 .

[24]  James K. Ho,et al.  A set of staircase linear programming test problems , 1981, Math. Program..

[25]  Michael A. Saunders,et al.  A FAST, STABLE IMPLEMENTATION OF THE SIMPLEX METHOD USING BARTELS-GOLUB UPDATING , 1976 .

[26]  G. Dantzig.  Programming of Interdependent Activities: II Mathematical Model , 1949 .

[27]  T. Aonuma.  A TWO-LEVEL ALGORITHM FOR TWO-STAGE LINEAR PROGRAMS , 1978 .

[28]  A. Heesterman,et al.  Special Simplex Algorithm for Linked Problems , 1965 .

[29]  Alexander Meeraus,et al.  Matrix augmentation and structure preservation in linearly constrained control problems , 1980, Math. Program..

[30]  Gene H. Golub,et al.  The simplex method of linear programming using LU decomposition , 1969, Commun. ACM.

[31]  Howard Kunreuther,et al.  Planning Horizons for Production Smoothing with Deterministic Demands: I. All Demand Met from Regular Production , 1973 .

[32]  J. Ho.  Optimal design of multi-stage structures: a nested decomposition approach , 1975 .

[33]  Wm. Orchard-Hays.  An exercise in basis inversion , 1973, SMAP.

[34]  R. Grinold.  Steepest Ascent for Large Scale Linear Programs , 1972 .

[35]  G. Dantzig.  Solving Staircase Linear Programs by a Nested Block-Angular Method , 1973 .

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

[37]  G. B. Dantzig,et al.  AT THE INTERFACE OF MODELING AND ALGORITHMS RESEARCH11Research supported by the Energy Research and Development Administration Contracts EY-76-S-03-0326 PA#18, and EY-76-S-03-0326 PA#52; the Office of Naval Research Contract N00014-75-C-0865; the National Science Foundation Grant MCS76-20019; the El , 1978 .

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

[39]  Rolf A. Lundin,et al.  Planning Horizons for the Dynamic Lot Size Model: Zabel vs. Protective Procedures and Computational Results , 1975, Oper. Res..

[40]  André F. Perold.  Extreme Points and Basic Feasible Solutions in Continuous Time Linear Programming , 1981 .

[41]  Robert Fourer,et al.  Solving staircase linear programs by the simplex method, 2: Pricing , 1983, Math. Program..