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Cut generation for an economic lot-sizing (ELS) problem

Description
This model implements various forms of cut-and-branch and branch-and-cut algorithms. In its simplest form (looping over LP solving) it illustrates the following features:
  • adding new constraints and resolving the LP-problem (cut-and-branch)
  • basis in- and output
  • if statement
  • repeat-until statement
  • procedure
The model els.mos also implements a configurable cutting plane algorithm:
  • defining the cut manager node callback function,
  • defining and adding cuts during the MIP search (branch-and-cut), and
  • using run-time parameters to configure the solution algorithm.
The version elsglobal.mos shows how to implement global cuts. And the model version elscb.mos defines additional callbacks for extended logging and user stopping criteria based on the MIP gap.

Another implementation (master model: runels.mos, submodel: elsp.mos) parallelizes the execution of several model instances, showing the following features:
  • parallel execution of submodels
  • communication between different models (for bound updates on the objective function)
  • sending and receiving events
  • stopping submodels
The fourth implementation (master model: runelsd.mos, submodel: elsd.mos) is an extension of the parallel version in which the solve of each submodels are distributed to various computing nodes.

Further explanation of this example: elscb.mos, elsglobal.mos, runels.mos: Xpress Whitepaper 'Multiple models and parallel solving with Mosel', Section 'Solving several model instances in parallel'.


Source Files

Data Files





elsc.mos

(!*******************************************************
   Mosel Example Problems
   ======================

   file elsc.mos
   `````````````
   Economic lot sizing, ELS, problem
   (Cut generation algorithm adding (l,S)-inequalities 
    in one or several rounds at the root node or in 
    tree nodes)

   Standard version (configurable cutting plane algorithm).

   Economic lot sizing (ELS) considers production planning 
   over a given planning horizon. In every period, there is 
   a given demand for every product that must be satisfied by 
   the production in this period and by inventory carried over 
   from previous periods.
   A set-up cost is associated with production in a period, 
   and the total production capacity per period is limited. 
   The unit production cost per product and time period is 
   given. There is no inventory or stock-holding cost.

   *** This model cannot be run with a Community Licence 
       for the provided data instance ***

   (c) 2008 Fair Isaac Corporation
       author: S. Heipcke, 2001, rev. June 2010
  *******************************************************!)

model "ELS"
 uses "mmxprs","mmsystem"

 parameters
  ALG = 6                             ! Algorithm choice [default settings: 0]
  CUTDEPTH = 10                       ! Maximum tree depth for cut generation
  EPS = 1e-6                          ! Zero tolerance
 end-parameters 

 forward public function cb_node:boolean
 forward procedure tree_cut_gen

 declarations
  TIMES = 1..15                              ! Range of time
  PRODUCTS = 1..4                            ! Set of products

  DEMAND: array(PRODUCTS,TIMES) of integer   ! Demand per period
  SETUPCOST: array(TIMES) of integer         ! Setup cost per period
  PRODCOST: array(PRODUCTS,TIMES) of integer ! Production cost per period
  CAP: array(TIMES) of integer               ! Production capacity per period
  D: array(PRODUCTS,TIMES,TIMES) of integer  ! Total demand in periods t1 - t2

  produce: array(PRODUCTS,TIMES) of mpvar    ! Production in period t
  setup: array(PRODUCTS,TIMES) of mpvar      ! Setup in period t

  solprod: array(PRODUCTS,TIMES) of real     ! Sol. values for var.s produce
  solsetup: array(PRODUCTS,TIMES) of real    ! Sol. values for var.s setup
  starttime: real
 end-declarations

 initializations from "els.dat"
  DEMAND SETUPCOST PRODCOST CAP
 end-initializations

 forall(p in PRODUCTS,s,t in TIMES) D(p,s,t):= sum(k in s..t) DEMAND(p,k)

! Objective: minimize total cost
 MinCost:= sum(t in TIMES) (SETUPCOST(t) * sum(p in PRODUCTS) setup(p,t) + 
                            sum(p in PRODUCTS) PRODCOST(p,t) * produce(p,t) )

! Satisfy the total demand
 forall(p in PRODUCTS,t in TIMES) 
   Dem(p,t):= sum(s in 1..t) produce(p,s) >= sum (s in 1..t) DEMAND(p,s)

! If there is production during t then there is a setup in t
 forall(p in PRODUCTS, t in TIMES) 
  ProdSetup(p,t):= produce(p,t) <= D(p,t,getlast(TIMES)) * setup(p,t)

! Capacity limits
 forall(t in TIMES) Capacity(t):= sum(p in PRODUCTS) produce(p,t) <= CAP(t)

! Variables setup are 0/1
 forall(p in PRODUCTS, t in TIMES) setup(p,t) is_binary 

! Uncomment to get detailed MIP output
 setparam("XPRS_VERBOSE", true)
 
! All cost data are integer, we therefore only need to search for integer
! solutions
 setparam("XPRS_MIPADDCUTOFF", -0.999)

! Set Mosel comparison tolerance to a sufficiently small value
 setparam("ZEROTOL", EPS/100)
 
 writeln("**************ALG=",ALG,"***************")

 starttime:=gettime
 SEVERALROUNDS:=false; TOPONLY:=false

 case ALG of
  1: setparam("XPRS_CUTSTRATEGY", 0)  ! No cuts
  2: setparam("XPRS_PRESOLVE", 0)     ! No presolve
  3: tree_cut_gen                     ! User branch-and-cut + automatic cuts
  4: do                               ! User branch-and-cut (several rounds),
      tree_cut_gen                    ! no automatic cuts
      setparam("XPRS_CUTSTRATEGY", 0)
      SEVERALROUNDS:=true
     end-do
  5: do                               ! User cut-and-branch (several rounds)
      tree_cut_gen                    ! + automatic cuts
      SEVERALROUNDS:=true
      TOPONLY:=true
     end-do
  6: do                               ! User branch-and-cut (several rounds)
      tree_cut_gen                    ! + automatic cuts
      SEVERALROUNDS:=true
     end-do
 end-case

 minimize(MinCost)                    ! Solve the problem
                                       
 writeln("Time: ", gettime-starttime, "sec,  Nodes: ", getparam("XPRS_NODES"),
         ",  Solution: ", getobjval) 
 write("Period  setup    ")
 forall(p in PRODUCTS) write(strfmt(p,-7))
 forall(t in TIMES) do
  write("\n ", strfmt(t,2), strfmt(getsol(sum(p in PRODUCTS) setup(p,t)),8), "     ")
  forall(p in PRODUCTS) write(getsol(produce(p,t)), " (",DEMAND(p,t),")  ")
 end-do
 writeln

!*************************************************************************
!  Cut generation loop:
!    get the solution values
!    identify and set up violated constraints
!    load the modified problem and load the saved basis
!*************************************************************************

 public function cb_node:boolean
  declarations
   ncut:integer                        ! Counter for cuts
   cut: array(range) of linctr         ! Cuts
   cutid: array(range) of integer      ! Cut type identification
   type: array(range) of integer       ! Cut constraint type
   objval,ds: real
  end-declarations

  depth:=getparam("XPRS_NODEDEPTH")

  if((TOPONLY and depth<1) or (not TOPONLY and depth<=CUTDEPTH)) then
   ncut:=0 

 ! Get the solution values
   forall(t in TIMES, p in PRODUCTS) do
     solprod(p,t):=getsol(produce(p,t))
     solsetup(p,t):=getsol(setup(p,t))
   end-do
  
 ! Search for violated constraints
   forall(p in PRODUCTS,l in TIMES) do
    ds:=0 
    forall(t in 1..l)
      if(solprod(p,t) < D(p,t,l)*solsetup(p,t) + EPS) then ds += solprod(p,t)
      else  ds += D(p,t,l)*solsetup(p,t)
      end-if
  
   ! Generate the violated inequality
    if(ds < D(p,1,l) - EPS) then
      cut(ncut):= sum(t in 1..l) 
       if(solprod(p,t)<(D(p,t,l)*solsetup(p,t))+EPS, produce(p,t), 
          D(p,t,l)*setup(p,t)) - D(p,1,l)
      cutid(ncut):= 1
      type(ncut):= CT_GEQ
      ncut+=1
    end-if   
  end-do

  returned:=false                     ! Call this function once per node
   
 ! Add cuts to the problem
   if(ncut>0) then 
    addcuts(cutid, type, cut);  
    if(getparam("XPRS_VERBOSE")=true) then
     writeln("Cuts added : ", ncut, " (depth ", depth, ", node ", 
            getparam("XPRS_NODES"), ", obj. ", getparam("XPRS_LPOBJVAL"), ")")
    end-if
    if SEVERALROUNDS then
     returned:=true                   ! Repeat until no new cuts generated
    end-if 
   end-if
  end-if
 end-function

! ****Optimizer settings for using the cut manager****

 procedure tree_cut_gen
  setparam("XPRS_PRESOLVE", 0)        ! Switch presolve off
  setparam("XPRS_EXTRAROWS", 5000)    ! Reserve extra rows in matrix
  setcallback(XPRS_CB_CUTMGR, "cb_node")  ! Set the cut-manager callback
 end-procedure

end-model

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