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Fiveleaper (MIP model and graphics)

Description
A fiveleaper travels over the chessboard by advancing in a single move either 5 squares in a straight line or three squares in a particular direction and 4 in an orthogonal direction. That is, with every move it travels a Euclidean distance of five squares. The problem of finding fiveleaper's tours on a chessboard is an instance of the Traveling Salesman Problem (TSP) and we adapt the subtour elimination algorithm described in the book 'Applications of optimization with Xpress-MP' (problem f5tour) for solving this problem.

You will find a description of this problem and a few other model versions in the following publication: M.J. Chlond, B. Daniel, S. Heipcke, Fiveleapers a-leaping, INFORMS Transactions on Education, Vol. 4, No. 1, 2003. http://ite.pubs.informs.org/Vol4No1/ChlondDanielHeipcke

The model version fiveleap_graph.mos displays a user graph within a webbrowser. The model fiveleap.mos simply prints out the results and will work on any platform for which Mosel is available.


Source Files





fiveleap.mos

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

   file fiveleap.mos 
   `````````````````
   Fiveleaper problem.

   Closed TSP form of 5leaper for a rectangular chessboard

   Run with parameters NROW and NCOL (numbers of squares on the edges)
   e.g.   mosel exe fiveleap NROW=9 NCOL=6
   NROW=NCOL=8 are the smallest possible values for square boards.
 
   This is a variant of the problem in "f5tour.mos" from the book
   "Applications of optimization with Xpress-MP" 

   You will find a description of this problem and a few other 
   model versions at the following address:
   http://ite.pubs.informs.org/Vol4No1/ChlondDanielHeipcke

   A square (i,j) on the rectangle is labeled
     sq := j + NCOL*(i-1)
   Given a square label
     i := ceil(sq/NCOL)
     j := sq - NCOL*(i-1)
     
   (c) 2008 Fair Isaac Corporation
       authors: Bob Daniel, S. Heipcke, March 2003
*******************************************************!)

model 'FIVELEAP'
  uses 'mmxprs', 'mmsystem'

  parameters
    NCOL = 8                            ! Number of colums
    NROW = 8                            ! Number of rows
  end-parameters

  forward function break_subtour: integer
  forward procedure print_sol
  forward procedure find_subtour(start: integer, TOUR: set of integer)

  declarations
    NSQ = NCOL*NROW                     ! Number of squares
    RSQ = 1..NSQ                        ! Range of squares
    
    nmoves, ntours, niterations, ntourstotal: integer
    nmovesfrom: array(RSQ) of integer   ! Counts possible moves leaving

    x: dynamic array(RSQ,RSQ) of mpvar  ! 1 if we leap from sq1 to sq2
    NEXTX: array(RSQ) of integer        ! Next square after sq in the solution

    square1, square2, ii, jj: integer
  end-declarations

! For each pair of squares with a valid move...
  nmoves := 0
  forall(i1,i2 in 1..NCOL, j1,j2 in 1..NROW | (i1-i2)^2+(j1-j2)^2 = 25) do
    square1 := j1+NCOL*(i1-1)
    square2 := j2+NCOL*(i2-1)
    create(x(square1,square2))          ! Create the 'x' binary decision variable
    x(square1,square2) is_binary
    nmovesfrom(square1) += 1            ! Note a possible move from square
    nmoves += 1
  end-do
  writeln("There are ", nmoves, " possible moves")

! Check that each square is reachable
  forall(sq in RSQ | nmovesfrom(sq)=0) do
    ii := ceil(sq/NCOL)
    jj := sq-NCOL*(ii-1) 
    writeln("**** Cannot go anywhere from square (", ii, ",", jj, ")")
    exit(0)
  end-do

! Objective (arbitrary)
  AnyObj := sum(sq1,sq2 in RSQ | exists(x(sq1,sq2)) ) x(sq1,sq2)

! Each square precedes one other
  forall(sq1 in RSQ) sum(sq2 in RSQ | exists(x(sq1,sq2)) ) x(sq1,sq2) = 1

! Each cell is preceded by one other
  forall(sq2 in RSQ) sum(sq1 in RSQ | exists(x(sq1,sq2)) ) x(sq1,sq2) = 1

! Optimizer settings to speed up the iterations
  setparam("XPRS_CUTSTRATEGY", 0) 
  setparam("XPRS_PRESOLVE", 0) 

! Solve the problem

  starttime := gettime
  niterations := 0

  while (TRUE) do
    minimize(AnyObj)
  
    if (getprobstat <> XPRS_OPT) then
      writeln("Problem is infeasible.")
      exit(0)
    end-if

    forall(sq in RSQ) 
      NEXTX(sq):= integer(round(getsol(sum(sq2 in RSQ) sq2*x(sq,sq2) )))

    ntours := break_subtour

    if (ntours=1) then break; end-if

    ! Print a log every 10 iterations
    niterations += 1
    ntourstotal += ntours
    if (niterations mod 10 = 0) then
      writeln(niterations, " (", gettime-starttime, " sec) tours eliminated: ",
              ntourstotal)
    end-if

  end-do
  
  print_sol

  writeln("Total time: ", gettime-starttime, " sec")
 
 
!-----------------------------------------------------------------
! Eliminate all subtours, and return number of (sub)tours
! - Find news subtours starting at each node in turn
! - If found complete tour, return 1
! - Otherwise add subtour elimination cut and continue
!
! Global variables required
!   RSQ: range of squares
!   NSQ: number of squares
!   NEXTX(i): square following i in current solution

  function break_subtour: integer
    declarations
      TOUR, ALLTOURS: set of integer
      ntours: integer
    end-declarations
 
    ALLTOURS := {}
    ntours := 0

    forall(startsquare in RSQ | startsquare not in ALLTOURS) do
      ! Find (sub)tour containing {startsquare}
      find_subtour(startsquare, TOUR)
      ALLTOURS += TOUR
      ntours += 1
      ! If found complete tour, return
      if (getsize(TOUR)>=NSQ) then break; end-if
      ! Add a subtour breaking constraint
      sum(sq in TOUR) x(sq,NEXTX(sq)) <= getsize(TOUR) - 1
    end-do
    
    returned := ntours
  end-function

!-----------------------------------------------------------------
! Find a subtour starting at square 'start'; result returned in TOUR

  procedure find_subtour(start: integer, TOUR: set of integer)
    declarations
      square: integer
    end-declarations

    TOUR := {}
    square := start
    repeat
      TOUR += {square}
      square := NEXTX(square)
    until (square=start)
  end-procedure


!-----------------------------------------------------------------
! Print the current solution
!
! Global variables required
!   RSQ: range of squares
!   NEXTX(i): square following i in current solution

  procedure print_sol
    declarations
      SEENSOFAR: set of integer
      square: integer
    end-declarations
   
    SEENSOFAR:={}
    forall(startsquare in RSQ | startsquare not in SEENSOFAR) do
      write(startsquare)
      square := startsquare
      repeat
        SEENSOFAR += {square}
        square := NEXTX(square)
        write(" - ", square)
      until (square=startsquare)
      writeln(' ')
    end-do        
  end-procedure

end-model



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