$title Sea Distances for a World Phosphate Model (PHOSDIS,SEQ=78)

$onText
This model has been used to estimate the sea distances between phosphate
mines, fertilizer plants and final product markets. This example
demonstrates how one can expand the existing data set of distances from
basic nautical distance tables.


Meeraus, A, and Smeers, Y, Transport Cost Estimates via Linear
Programming. The World Bank, 1985.

Keywords: linear programming, micro economics, transportation problem,
          network optimization, shortest route, scenario analysis
$offText

Set
   n 'nodes'
     / odessa    , trieste   , aden      , alexandria, alger     , al-aqabah , al-bahrayn
       al-basrah , annaba    , antalya   , ashdod-yam, belawan   , belem     , bishop-rck
       bombay    , bunbury   , bur-safaja, calcutta  , cartagena , casablanca, christ-isl
       colombo   , c-g-hope  , c-leeuwin , dakar     , douala    , durban    , fastnet
       freetown  , guayaquil , inchon    , inishtrahu, istanbul  , kiel-canal, lagos
       leningrad , lombok-str, lome      , lourenco-m, masqat    , mobile    , mombasa
       montreal  , new-york  , oulu      , panama    , paramaribo, pentland-f, perth
       pnt-arenas, portland  , port-said , prt-madryn, rio-de-jan, rotterdam , sfax
       shang-hai , singapore , skagens-o , sunda-str , sydney    , s-florida , s-gibraltr
       tampa     , tampico   , tarabulus , tokyo     , torres-str, townsville, ushant
       valparaiso, vancouver , veracruz  , vladivosto, wetar-psg , wilson-prm, yucatan-ch /
   orig(n,n)  'origins'
   cnmap(*,n) 'closest country and node mapping'
              / algeria.   ( alger, annaba )
                argentina. ( prt-madryn )
                australia. ( bunbury, c-leeuwin, perth, sydney, torres-str, townsville, wilson-prm )
                bahrain.   ( al-bahrayn )
                brazil.    ( belem, rio-de-jan )
                cameroon.  ( douala )
                canada.    ( montreal, vancouver )
                chile.     ( pnt-arenas, valparaiso )
                china.     ( shang-hai )
                colombia.  ( cartagena )
                cuba.      ( yucatan-ch )
                denmark.   ( skagens-o )
                egypt.     ( alexandria, bur-safaja, port-said )
                equator.   ( guayaquil )
                france.    ( ushant )
                finland.   ( oulu )
                germany.   ( kiel-canal )
                india.     ( bombay, calcutta )
                indonesia. ( belawan, lombok-str, sunda-str, wetar-psg )
                iraq.      ( al-basrah )
                ireland.   ( fastnet, inishtrahu )
                israel.    ( ashdod-yam )
                japan.     ( tokyo )
                jordan.    ( al-aqabah )
                kenya.     ( mombasa )
                korea.     ( inchon )
                lebanon.   ( tarabulus )
                mexico.    ( tampico, veracruz )
                morocco.   ( casablanca )
                mozambique.( lourenco-m )
                netherland.( rotterdam )
                nigeria.   ( lagos )
                oman.      ( masqat )
                panama.    ( panama )
                s-africa.  ( c-g-hope, durban )
                senegal.   ( dakar )
                sier-leone.( freetown )
                sinagapore.( singapore )
                spain.     ( s-gibraltr )
                sri-lanka. ( colombo )
                surinam.   ( paramaribo )
                togo.      ( lome )
                tunesia.   ( sfax )
                turkey.    ( antalya, istanbul )
                uk.        ( bishop-rck, pentland-f )
                usa.       ( mobile, new-york, portland, s-florida, tampa )
                ussr.      ( leningrad, vladivosto )
                yemen.     ( aden )
                italy.     ( trieste )
                ussr.      ( odessa )                                        /;

Alias (n,np,npp);

Parameter dist(n,np) 'arc distances (nautical miles)'
* Only the upper triangle will be entered. The alphabetical ordering is used
* to assure triangularity. For additions the lower triangle should be entered
* at the beginning.
******************** insert additions *****************
/  odessa.istanbul          352

   trieste.port-said       1293
   trieste.s-gibraltr      1687
   trieste.alger           1254
   trieste.istanbul        1063
   trieste.tarabulus        921

   aden.al-aqabah          1251
   aden.al-basrah          1979
   aden.bombay             1678
   aden.bur-safaja         1109
   aden.colombo            2105
   aden.c-g-hope           4175
   aden.c-leeuwin          4973
   aden.lourenco-m         2973
   aden.mombasa            1619
   aden.port-said          1400
   aden.singapore          3646
   aden.sunda-str          3842
   aden.torres-str         6034

   alexandria.alger        1393
   alexandria.annaba       1169
   alexandria.antalya       349
   alexandria.ashdod-yam    266
   alexandria.istanbul      742
   alexandria.port-said     164
   alexandria.s-gibraltr   1222

   alger.annaba             246
   alger.istanbul          1393
   alger.port-said         1514
   alger.s-gibraltr         437

   al-aqabah.port-said      380

   al-bahrayn.belawan      3316
   al-bahrayn.c-g-hope     4957
   al-bahrayn.perth        5215
   al-bahrayn.port-said    3132
   al-bahrayn.singapore    3652

   al-basrah.bombay        1594
   al-basrah.colombo       2359
   al-basrah.c-g-hope      5211
   al-basrah.masqat         773
   al-basrah.port-said     3371
   al-basrah.singapore     3916

   annaba.istanbul         1168
   annaba.port-said        1289
   annaba.s-gibraltr        675

   antalya.port-said        346
   antalya.s-gibraltr      1822

   ashdod-yam.port-said     126
   ashdod-yam.s-gibraltr   2031

   belawan.bombay          2109
   belawan.calcutta        1326
   belawan.c-g-hope        5246
   belawan.lombok-str      1342
   belawan.port-said       4688
   belawan.singapore        368
   belawan.sunda-str        870
   belawan.torres-str      2839

   belem.bishop-rck        3760
   belem.c-g-hope          4368
   belem.fastnet           3741
   belem.inishtrahu        3972
   belem.montreal          3709
   belem.new-york          2975
   belem.paramaribo         747
   belem.pentland-f        4219
   belem.pnt-arenas        4363
   belem.rio-de-jan        2174
   belem.s-florida         2653
   belem.s-gibraltr        3285
   belem.ushant            3748
   belem.yucatan-ch        2739

   bishop-rck.cartagena    4214
   bishop-rck.c-g-hope     5790
   bishop-rck.dakar        2195
   bishop-rck.fastnet       149
   bishop-rck.freetown     2673
   bishop-rck.inishtrahu    395
   bishop-rck.kiel-canal    727
   bishop-rck.montreal     2989
   bishop-rck.new-york     3019
   bishop-rck.panama       4388
   bishop-rck.paramaribo   3602
   bishop-rck.pentland-f    636
   bishop-rck.pnt-arenas   7019
   bishop-rck.rio-de-jan   4846
   bishop-rck.rotterdam     454
   bishop-rck.skagens-o     851
   bishop-rck.s-florida    3876
   bishop-rck.s-gibraltr    969
   bishop-rck.yucatan-ch   4193

   bombay.calcutta         2128
   bombay.colombo           895
   bombay.c-g-hope         4679
   bombay.c-leeuwin        4059
   bombay.durban           3826
   bombay.masqat            852
   bombay.mombasa          2596
   bombay.perth            3998
   bombay.port-said        3743
   bombay.singapore        2441
   bombay.sunda-str        2664
   bombay.torres-str       4849

   bunbury.c-leeuwin         97
   bunbury.lombok-str      1510
   bunbury.port-said       6343

   bur-safaja.c-g-hope     5016
   bur-safaja.port-said     310
   bur-safaja.singapore    4729

   calcutta.colombo        1258
   calcutta.c-g-hope       5491
   calcutta.c-leeuwin      3796
   calcutta.lombok-str     2641
   calcutta.port-said      4695
   calcutta.singapore      1649
   calcutta.sunda-str      2141

   cartagena.new-york      1853
   cartagena.panama         324
   cartagena.s-gibraltr    4087
   cartagena.tampa         1295
   cartagena.tampico       1575
   cartagena.yucatan-ch     891

   casablanca.dakar        1333
   casablanca.s-gibraltr    166
   casablanca.ushant        960

   christ-isl.panama       4751
   christ-isl.sydney       3620
   christ-isl.torres-str   3500

   colombo.c-g-hope        4447
   colombo.c-leeuwin       3203
   colombo.durban          3703
   colombo.mombasa         2555
   colombo.port-said       3668
   colombo.singapore       1581
   colombo.sunda-str       1781
   colombo.torres-str      3978

   c-g-hope.c-leeuwin      4660
   c-g-hope.dakar          3610
   c-g-hope.durban          764
   c-g-hope.freetown       3190
   c-g-hope.inishtrahu     6141
   c-g-hope.lombok-str     5461
   c-g-hope.lourenco-m     1057
   c-g-hope.masqat         4464
   c-g-hope.mombasa        2472
   c-g-hope.montreal       7134
   c-g-hope.new-york       6801
   c-g-hope.pentland-f     6389
   c-g-hope.perth          4767
   c-g-hope.pnt-arenas     4618
   c-g-hope.port-said      5340
   c-g-hope.rio-de-jan     3323
   c-g-hope.singapore      5579
   c-g-hope.sunda-str      5164
   c-g-hope.s-florida      6770
   c-g-hope.s-gibraltr     5082
   c-g-hope.torres-str     6854
   c-g-hope.ushant         5746

   c-leeuwin.lombok-str    1571
   c-leeuwin.perth          193
   c-leeuwin.port-said     6389
   c-leeuwin.sunda-str     1815
   c-leeuwin.townsville    3023
   c-leeuwin.wilson-prm    1524

   dakar.inishtrahu        2546
   dakar.lagos             1550
   dakar.lome              1500
   dakar.montreal          3566
   dakar.new-york          3335
   dakar.panama            3738
   dakar.pentland-f        2794
   dakar.pnt-arenas        4928
   dakar.rio-de-jan        2763
   dakar.s-florida         3768
   dakar.s-gibraltr        1487
   dakar.ushant            2152

   douala.s-florida        5526

   durban.lourenco-m        307
   durban.mombasa          1718
   durban.port-said        4567
   durban.singapore        4872

   fastnet.montreal        2872
   fastnet.new-york        2913
   fastnet.paramaribo      3554
   fastnet.pentland-f       625
   fastnet.rio-de-jan      4873
   fastnet.s-florida       3773
   fastnet.yucatan-ch      4087

   freetown.inishtrahu     3024
   freetown.montreal       4019
   freetown.new-york       3757
   freetown.pentland-f     3272
   freetown.pnt-arenas     4731
   freetown.rio-de-jan     2613
   freetown.s-florida      4137
   freetown.s-gibraltr     1966
   freetown.ushant         2630
   freetown.yucatan-ch     4229

   guayaquil.panama        1074
   guayaquil.pnt-arenas    3057
   guayaquil.shang-hai     8872
   guayaquil.singapore    10726
   guayaquil.sydney        7268
   guayaquil.tokyo         7987
   guayaquil.torres-str    8270
   guayaquil.valparaiso    1730

   inchon.lombok-str       2885
   inchon.panama           8474
   inchon.shang-hai         503
   inchon.singapore        2560
   inchon.sunda-str        2913
   inchon.tokyo            1070
   inchon.wetar-psg        2767

   inishtrahu.montreal     2964
   inishtrahu.new-york     3032
   inishtrahu.pentland-f    272
   inishtrahu.pnt-arenas   7315
   inishtrahu.rio-de-jan   5143
   inishtrahu.s-florida    3892
   inishtrahu.s-gibraltr   1350
   inishtrahu.yucatan-ch   4207

   istanbul.port-said       804
   istanbul.s-gibraltr     1821

   kiel-canal.leningrad     779
   kiel-canal.oulu          918
   kiel-canal.pentland-f    565
   kiel-canal.rotterdam     323
   kiel-canal.ushant        716

   lagos.lome                50
   lagos.new-york          4883
   lagos.rio-de-jan        2850
   lagos.s-florida         5225
   lagos.s-gibraltr        3100
   lagos.ushant            3746

   leningrad.skagens-o      971

   lombok-str.perth        1452
   lombok-str.shang-hai    2577
   lombok-str.singapore     963
   lombok-str.sunda-str     677

   lome.rio-de-jan         2800

   lourenco-m.mombasa      1508
   lourenco-m.port-said    4348
   lourenco-m.singapore    4769

   masqat.singapore        3153

   mobile.s-florida         471
   mobile.veracruz          825
   mobile.yucatan-ch        558

   mombasa.port-said       3006
   mombasa.singapore       3987

   montreal.new-york       1520
   montreal.panama         3204
   montreal.paramaribo     3150
   montreal.pentland-f     3214
   montreal.pnt-arenas     7527
   montreal.rio-de-jan     5357
   montreal.s-florida      2531
   montreal.s-gibraltr     3283

   new-york.panama         2018
   new-york.paramaribo     2334
   new-york.pentland-f     3303
   new-york.pnt-arenas     6947
   new-york.rio-de-jan     4770
   new-york.s-florida      1237
   new-york.s-gibraltr     3208
   new-york.yucatan-ch     1366

   oulu.skagens-o           979

   panama.paramaribo       1691
   panama.pnt-arenas       3932
   panama.portland         3869
   panama.prt-madryn       6012
   panama.shang-hai        8648
   panama.singapore       10505
   panama.sydney           7674
   panama.s-gibraltr       4351
   panama.tampico          1528
   panama.tokyo            7682
   panama.torres-str       8451
   panama.valparaiso       2616
   panama.vancouver        4032
   panama.veracruz         1463
   panama.vladivosto       7833
   panama.wilson-prm       7842
   panama.yucatan-ch        855

   paramaribo.rio-de-jan   2713
   paramaribo.s-florida    2014
   paramaribo.s-gibraltr   3275
   paramaribo.ushant       3613

   pentland-f.pnt-arenas   7563
   pentland-f.rio-de-jan   5391
   pentland-f.rotterdam     495
   pentland-f.skagens-o     450
   pentland-f.s-florida    4100
   pentland-f.s-gibraltr   1598
   pentland-f.yucatan-ch   4406

   perth.port-said         6322
   perth.sunda-str         1703
   perth.torres-str        2543

   pnt-arenas.portland     6813
   pnt-arenas.prt-madryn    760
   pnt-arenas.rio-de-jan   2238
   pnt-arenas.shang-hai   10465
   pnt-arenas.sydney       5398
   pnt-arenas.s-florida    6783
   pnt-arenas.s-gibraltr   6352
   pnt-arenas.tokyo        9642
   pnt-arenas.torres-str   8348
   pnt-arenas.ushant       6986
   pnt-arenas.valparaiso   1432
   pnt-arenas.vancouver    6978
   pnt-arenas.wilson-prm   5820
   pnt-arenas.yucatan-ch   6794

   portland.singapore      7142
   portland.sydney         6737
   portland.tokyo          4338
   portland.valparaiso     5764
   portland.vancouver       371
   portland.vladivosto     4278
   portland.wilson-prm     7121

   port-said.sfax          1131
   port-said.singapore     5035
   port-said.sunda-str     5232
   port-said.s-gibraltr    1943
   port-said.tarabulus      268
   port-said.torres-str    7423

   rio-de-jan.s-florida    4612
   rio-de-jan.s-gibraltr   4180
   rio-de-jan.ushant       4815
   rio-de-jan.yucatan-ch   4622

   rotterdam.skagens-o      447
   rotterdam.ushant         444

   sfax.s-gibraltr         1043

   shang-hai.singapore     2207
   shang-hai.sunda-str     2535
   shang-hai.sydney        4642
   shang-hai.tokyo         1117
   shang-hai.valparaiso   10148
   shang-hai.vladivosto     997
   shang-hai.wetar-psg     2553

   singapore.sunda-str      532
   singapore.tokyo         2889
   singapore.vancouver     7078
   singapore.vladivosto    3004
   singapore.wetar-psg     1587

   skagens-o.ushant         833

   sunda-str.vladivosto    3357
   sunda-str.wetar-psg     1330

   sydney.tokyo            4330
   sydney.torres-str       1754
   sydney.townsville        950
   sydney.valparaiso       6294
   sydney.vladivosto       5105
   sydney.wilson-prm        440

   s-florida.s-gibraltr    4024
   s-florida.tampa          232
   s-florida.tampico        832

   s-gibraltr.tarabulus    2047
   s-gibraltr.ushant        918
   s-gibraltr.yucatan-ch   4185

   tampa.tampico            921
   tampa.yucatan-ch         404

   tampico.yucatan-ch       716

   tokyo.valparaiso        9280
   tokyo.vancouver         4272
   tokyo.vladivosto         962

   torres-str.townsville    662
   torres-str.valparaiso   7670
   torres-str.wetar-psg     881
   torres-str.wilson-prm   2183

   townsville.wilson-prm   1499

   valparaiso.vladivosto   9606
   valparaiso.wilson-prm   6031

   veracruz.yucatan-ch      654

   vladivosto.wetar-psg    3133 /;

Parameter darc(n,np) 'directed arcs (nautical miles)';
darc(n,np) = max(dist(n,np),dist(np,n));
orig(n,n)  = yes;
display darc, orig;

Variable
   x(n,np,npp) 'flow on arcs'
   cost        'total cost or length';

Positive Variable x;

Equation
   nb(n,np) 'node balance'
   cd       'cost definition';

Set
   s(n) 'source nodes - dynamic'
   d(n) 'destination nodes - dynamic';

s(n) = no;
d(n) = no;

nb(s,n)$(not orig(s,n))..
   sum(np$darc(np,n), x(s,np,n)) =g= sum(np$darc(n,np), x(s,n,np)) + 1;

cd..
   cost =e= sum((s,n,np), darc(n,np)*x(s,n,np));

Model route 'shortes route model' / all /;

* Two distance matrices will be needed: (1) producer to producer and (2)
* producer to market. Instead of solving two problems we find all needed
* origins and then remap the result in to the i and j sets. To fit
* smaller machines the problem is solved piecemeal. Also observe the use
* of the solveOpt=replace option to discard previous solutions.
Set
   j 'markets'
     / wn-america, en-america, mex+cam   , ws-america, es-america, n-europe, s-europe
       ee+ussr   ,  m-east   , n-e-africa, nw-africa , w-africa  , s-africa, e-africa
       persian-g , ws-asia   , es-asia   , asean     , oceania   , china   , japan+kor /
   i 'producers'
     / n-america , mexico   , w-europe  , peru     , brazil   , senegal   , morocco
       caribbean , oth-samer, algeria   , tunisia  , togo     , oth-africa, egypt
       jordan    , israel   , ot-mideast, syria    , ee+ussr  , india     , china
       christ-isl, japan+kor, oceania   , oth-easia, persian-g                      /
   jn(j,n) 'market port mapping'
           / wn-america.vancouver , en-america.tampa     , mex+cam.yucatan-ch
             ws-america.valparaiso, es-america.rio-de-jan, n-europe.rotterdam
             s-europe.trieste     , ee+ussr.odessa       , m-east.tarabulus
             n-e-africa.alexandria, nw-africa.s-gibraltr , w-africa.lagos
             s-africa.c-g-hope    , e-africa.mombasa     , persian-g.al-basrah
             ws-asia.bombay       , es-asia.calcutta     , asean.singapore
             oceania.sydney       , china.shang-hai      , japan+kor.tokyo     /
   in(i,n) 'producer port mapping'
           / n-america.tampa      , mexico.veracruz     , w-europe.rotterdam
             peru.guayaquil       , brazil.rio-de-jan   , senegal.dakar
             morocco.casablanca   , caribbean.yucatan-ch, oth-samer.valparaiso
             algeria.annaba       , tunisia.sfax        , togo.lome
             oth-africa.c-g-hope  , egypt.bur-safaja    , jordan.al-aqabah
             israel.ashdod-yam    , ot-mideast.istanbul , syria.tarabulus
             ee+ussr.odessa       , india.bombay        , china.shang-hai
             christ-isl.christ-isl, japan+kor.tokyo     , oceania.sydney
             oth-easia.inchon     , persian-g.al-basrah                        /

Alias (i,ip);

Set ss(n) 'all sources';

Scalar count, iter;

ss(n) = sum(i, in(i,n));
d(n)  = ss(n) + sum(j, jn(j,n));
display ss, d;

option iterLim = 8000, solveOpt = replace;

Parameter sroute(n,np) 'shortest sea distance (nautical miles)';

* get first four sources and set-up model run
for(iter = 1 to 7,
   count = 0;
   loop(ss$(count < 4),
      s(ss) = yes;
      count = count+1;
   );
   ss(s) = no;
   solve route minimizing cost using lp;
   sroute(s,d) = round(-nb.m(s,d),-1);
   s(s)  = no;
);

display sroute;

* remap from nodes into producing and demand regions
Parameter
   disii(i,ip) 'interplant distances (nautical miles)'
   disiia(i,n) 'intermediate step'
   disij(i,j)  'market distances     (nautical miles)';

disiia(i,n) = sum(np$in(i,np), sroute(np,n));
disii(i,ip) = sum(n$in(ip,n), disiia(i,n));
disij(i,j)  = sum(n$jn(j,n), disiia(i,n));

display disii, disij;