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Column generation for a cutting stock problem

  • creating new variables and adding them to constraints and objective
  • basis in- and output
  • sorting and single knapsack algorithms
  • repeat-until statement
  • use of functions
  • hiding / unhiding constraints
  • resetting (deleting) constraints
The first version of this model (cutstk.mos) solves the knapsack subproblem heuristically, the second version (paper.mos) solves it as a second optimization problem, hiding (blending out) the constraints of the master problem. The third version defines the subproblem as a separate problem with the same model, repeatedly creating new problems (papersn.mos) or re-using/redefining the same problem (papers.mos), that is:
  • defining multiple problems within a model
  • switching between problems
A fourth version (master model: paperp[r].mos, submodel: knapsack[r].mos) shows how to implement the algorithm with two separate models, illustrating the following features of Mosel:
  • working with multiple models
  • executing a sequence of models
  • passing data via shared memory

Further explanation of this example: paper.mos, papers.mos, papersn.mos: 'Mosel User Guide', Section 11.2 Column generation, and the Xpress Whitepaper 'Embedding Optimization Algorithms', Section 'Column generation for cutting-stock' (also discusses a generalization to bin-packing problems); for the multiple model versions paperp.mos, paperms.mos see the Xpress Whitepaper 'Multiple models and parallel solving with Mosel', Section 'Column generation: solving different models in sequence'.

Source Files


   Mosel User Guide Examples

   file knapsack.mos
   Knapsack (sub)model of the for a paper cutting example,
   reading data from shared memory.

   *** Can be run standalone - or run from paperp.mos ***

   Solve the integer knapsack problem                              
     z = max{Cx : Ax<=B, x<=D, x in Z^N}  

   (c) 2008 Fair Isaac Corporation
       author: S. Heipcke, 2004, rev. Aug. 2011

model "Knapsack"
 uses "mmxprs"

  NWIDTHS=5                            ! Number of different widths
  INDATA = "knapsack.dat"              ! Input data file
  RESDATA = "knresult.dat"             ! Result data file
  WIDTHS = 1..NWIDTHS                  ! Range of widths
  A,C: array(WIDTHS) of real           ! Constraint + obj. coefficients
  B: real                              ! RHS value of knapsack constraint
  D: array(WIDTHS) of integer          ! Variables bounds (demand quantities)
  KnapCtr, KnapObj: linctr             ! Knapsack constraint+objective
  x: array(WIDTHS) of mpvar            ! Knapsack variables
  xbest: array(WIDTHS) of integer      ! Solution values
 initializations from INDATA
  A  B  C  D

! Define the knapsack problem
 KnapCtr:= sum(j in WIDTHS) A(j)*x(j) <= B
 KnapObj:= sum(j in WIDTHS) C(j)*x(j)

! Integrality condition and bounds
 forall(j in WIDTHS) x(j) is_integer
 forall(j in WIDTHS) x(j) <= D(j)    ! These bounds can be omitted

 forall(j in WIDTHS) xbest(j):=round(getsol(x(j)))

 initializations to RESDATA
  xbest  z as "zbest"


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