Combining different modes of transport

Contents

Problem description

A load of 20 tonnes needs to be transported on a route passing through five cities, with a choice of three different modes of transport: rail, road, and air. In any of the three intermediate cities it is possible to change the mode of transport but the load uses a single mode of transport between two consecutive cities. The following table lists the cost of transport in $ per tonne between the pairs of cities.

Transport costs with different modes

 Pairs of cities
+----+---+---+---+---+
|    |1–2|2–3|3–4|4–5|
+----+---+---+---+---+
|Rail| 30| 25| 40| 60|
|Road| 25| 40| 45| 50|
|Air | 40| 20| 50| 45|
+----+---+---+---+---+

The next table summarizes the costs for changing the mode of transport in $ per tonne. The cost is independent of location.

Cost for changing the mode of transport

+---------+----+----+---+
|from \ to|Rail|Road|Air|
+---------+----+----+---+
|Rail     |  0 |  5 | 12|
|Road     |  8 |  0 | 10|
|Air      | 15 | 10 |  0|
+---------+----+----+---+

How should we organize the transport of the load at the least cost?

Variables

transpcost                 Transport costs
changecost                 Cost to change mode of transport
demand                     Load to transport

References

Applications of optimization... Gueret, Prins, Seveaux

% Marcus Edvall, Tomlab Optimization Inc, E-mail: tomlab@tomopt.com
% Copyright (c) 2005-2009 by Tomlab Optimization Inc., $Release: 7.2.0$
% Written Oct 7, 2005.   Last modified Apr 11, 2009.

Problem setup

transpcost = tomArray([ 30 25 40 60 ;...
    25 40 45 50 ;...
    40 20 50 45]);

changecost = tomArray([  0  5  12;...
    8  0  10;...
    15 10   0]);

demand = 30;

m = size(changecost,1); % mode
l = size(transpcost,2); % legs
n = size(changecost,2); % mode

i  = tomArrayIdx('i',1:m);    % index over all modes to change from
j  = tomArrayIdx('j',1:l);    % index over all legs
jc = tomArrayIdx('jc',1:l-1); % index over all legs that involve a change
k  = tomArrayIdx('k',1:n);    % index over all modes to chang to

change = tom('change',m*n*(l-1),1,'int');
change = tomArray(change,[m,n,l-1]);

use = tom('use', m*l, 1, 'int');
use = tomArray(use,[m,l]);

% All variables are binary.
bnds = {0 <= change <= 1, 0 <= use <= 1};

% Single mode of transport
con1 = {sum(use(i,j),i) == 1};

% Single change of mode
con2 = {sum(sum(change(i,k,jc),i),k) == 1};

% Change to mode relationship
con3 = use(i,jc) + use(k,jc+1) >= 2*change(i,k,jc);

% Objective
objective = sum(vec(transpcost(i,j).*use(i,j))) + ...
    sum(vec(changecost(i,k).*change(i,k,jc)));

constraints = {bnds, con1, con2, con3};
options = struct;
options.solver = 'cplex';
options.name   = 'Combining Diff Modes of Transp';
[sol, Result] = ezsolve(objective,constraints,[],options);

PriLev = 1;
if PriLev > 0
    modes  = reshape(sol.use,3,4);
    means  = ['rail';'road';'air '];
    disp(['the min cost (' num2str(Result.f_k) ') is found by,'])
    for m = 1:size(modes,2),
        disp(['   going from town ' num2str(m) ' to town ' num2str(m+1) ...
            ' by ' means(find(modes(:,m)),:)])
    end
end

% MODIFICATION LOG
%
% 051020 med   Created.
% 060112 per   Added documentation.
% 060125 per   Moved disp to end
% 090407 med   Converted to tomSym

Problem type appears to be: mip
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Combining Diff Modes of Transp  f_k     150.000000000000000000
                                               f(x_0)      0.000000000000000000

Solver: CPLEX.  EXIT=0.  INFORM=101.
CPLEX Branch-and-Cut MIP solver
Optimal integer solution found

FuncEv   20 
CPU time: 0.156250 sec. Elapsed time: 0.156000 sec. 
the min cost (150) is found by,
   going from town 1 to town 2 by rail
   going from town 2 to town 3 by rail
   going from town 3 to town 4 by rail
   going from town 4 to town 5 by road