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TSP

Description
A flight tour starts from airport 1, visits all the other airports and then comes back to the starting point. The distances between the airports are symmetric. In which order should the airports be visited to minimize the total distance covered?

Further explanation of this example: 'Applications of optimization with Xpress-MP', Section 11.5 'Planning a flight tour'


Source Files

Data Files





tsp_graph.mos

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

   file tsp.mos
   ````````````
   TYPE:         Symmetric traveling salesman problem
   DIFFICULTY:   5
   FEATURES:     MIP problem, loop over problem solving, TSP subtour 
                 elimination algorithm; 
                 procedure for generating additional constraints, recursive 
                 subroutine calls, working with sets, `forall-do', 
                 `repeat-until', `getsize', `not', graphical representation
                 of solutions
   DESCRIPTION:  A flight tour starts from airport 1, visits all 
                 the other airports and then comes back to the starting point. 
                 The distances between the airports are symmetric. In which 
                 order should the airports be visited to minimize the total 
                 distance covered?     
   FURTHER INFO: `Applications of optimization with Xpress-MP', 
                 Section 11.5 `Planning a flight tour',
                 Section 7.5 `Paint production'
  
   (c) 2008 Fair Isaac Corporation
       author: S. Heipcke, 2002, rev. Sep. 2017
*******************************************************!)

model "Tour planning"
 uses "mmxprs", "mmsvg"

 forward procedure break_subtour
 forward procedure print_sol
 forward procedure draw_sol

 declarations
  NCITIES = 7
  CITIES = 1..NCITIES                    ! Cities
  
  DIST: array(CITIES,CITIES) of integer  ! Distance between cities
  NEXTC: array(CITIES) of integer        ! Next city after i in the solution
  X,Y: array(CITIES) of integer          ! x-y-coordinates for sol. drawing
  
  fly: array(CITIES,CITIES) of mpvar     ! 1 if flight from i to j
 end-declarations

 initializations from 'tsp.dat'
  DIST [X,Y] as 'POS'
 end-initializations

 forall(i,j in CITIES | i<j) DIST(j,i):=DIST(i,j)

! Objective: total distance
 TotalDist:= sum(i,j in CITIES | i<>j) DIST(i,j)*fly(i,j)

! Visit every city once
 forall(i in CITIES) OneVisitI(i):= sum(j in CITIES | i<>j) fly(i,j) = 1
 forall(j in CITIES) OneVisitJ(j):= sum(i in CITIES | i<>j) fly(i,j) = 1

 forall(i,j in CITIES | i<>j) fly(i,j) is_binary

! Solve the problem
 minimize(TotalDist)

! Eliminate subtours
 break_subtour

 svgwaitclose("Close browser window to terminate model execution.", 1)

!-----------------------------------------------------------------

 procedure break_subtour
  declarations
   TOUR,SMALLEST,ALLCITIES: set of integer
  end-declarations

  forall(i in CITIES) 
   NEXTC(i):= integer(round(getsol(sum(j in CITIES) j*fly(i,j) )))

! Print the current solution
  print_sol
  draw_sol
  
! Get (sub)tour containing city 1
  TOUR:={}
  first:=1
  repeat
   TOUR+={first}
   first:=NEXTC(first)
  until first=1
  size:=getsize(TOUR)
 
! Find smallest subtour
  if size < NCITIES then
   SMALLEST:=TOUR
   if size>2 then
    ALLCITIES:=TOUR 
    forall(i in CITIES) do
     if(i not in ALLCITIES) then
      TOUR:={}
      first:=i
      repeat
       TOUR+={first}
       first:=NEXTC(first)
      until first=i
      ALLCITIES+=TOUR
      if getsize(TOUR)<size then
       SMALLEST:=TOUR
       size:=getsize(SMALLEST)
      end-if
      if size=2 then
       break
      end-if 
     end-if 
    end-do        
   end-if
    
! Add a subtour breaking constraint
  sum(i in SMALLEST) fly(i,NEXTC(i)) <= getsize(SMALLEST) - 1

! Optionally, also use these additional constraints
! Also exclude the inverse subtour
  if SMALLEST.size>2 then
   sum(i in SMALLEST) fly(NEXTC(i),i) <= getsize(SMALLEST) - 1
  end-if

! A stronger subtour elimination constraint
   sum(i in SMALLEST,j in CITIES-SMALLEST) fly(i,j) >= 1
!)

! Re-solve the problem
   minimize(TotalDist)

! Closing the graphical display will interrupt the algorithm
   if not svgclosing then break_subtour; end-if

  end-if 
 end-procedure
 
!-----------------------------------------------------------------

! Print the current solution
 procedure print_sol
  declarations
   ALLCITIES: set of integer
  end-declarations
   
  writeln("Total distance: ", getobjval)
  ALLCITIES:={}
  forall(i in CITIES) do
   if(i not in ALLCITIES) then
    write(i)
    first:=i
    repeat
     ALLCITIES+={first}
     write(" - ", NEXTC(first))
     first:=NEXTC(first)
    until first=i
    writeln 
   end-if
  end-do        
 end-procedure

!-----------------------------------------------------------------

! Draw the current solution
 procedure draw_sol
  declarations
   ALLCITIES: set of integer
  end-declarations

  svgerase
  minx:=min(i in CITIES) X(i); miny:= min(i in CITIES) Y(i)
!  svgsetgraphviewbox(minx-10, miny-10,
!          max(i in CITIES) X(i)+20-minx, max(i in CITIES) Y(i)+20-miny)

  ! Draw the tour(s)
  ALLCITIES:={}; ct:=0
  forall(i in CITIES) do
   if(i not in ALLCITIES) then
    svgaddgroup("T"+ct, "Tour "+(ct+1))
    first:=i
    repeat
     ALLCITIES+={first}
     svgaddline(X(first), Y(first), X(NEXTC(first)), Y(NEXTC(first)))
     first:=NEXTC(first)
    until first=i
    ct+=1
   end-if
  end-do        

  ! Draw the cities
  svgaddgroup("C", "Cities", SVG_BLACK)
  forall(i in CITIES) svgaddtext(X(i), Y(i), text(i))
  svgaddtext(minx-5,miny-5, "Tour length: "+getobjval)

  svgsetgraphscale(5)
  svgrefresh
  ! Uncomment to pause at every iteration:
  if ct>1 then svgpause; end-if
 end-procedure

end-model

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