Tank Loading

Contents

Problem description

Five tanker ships have arrived at a chemical factory. They are carrying loads of liquid products that must not be mixed: 1200 tonnes of Benzol, 700 tonnes of Butanol, 1000 tonnes of Propanol, 450 tonnes of Styrene, and 1200 tonnes of THF. Nine tanks of different capacities are available on site. Some of them are already partially filled with a liquid. The following table lists the characteristics of the tanks (in tonnes). Into which tanks should the ships be unloaded (question 1) to maximize the capacity of the tanks that remain unused, or (question 2) to maximize the number of tanks that remain free?

Characteristics of tanks

+---------------+---+------+---+---+---+---+---+---+---+
|Tank           |  1|    2 |  3|  4|  5|  6|  7|  8|  9|
+---------------+---+------+---+---+---+---+---+---+---+
|Capacity       |500|  400 |400|600|600|900|800|800|800|
|Current product|  -|Benzol|  -|  -|  -|  -|THF|  -|  -|
|Quantity       |  0|  100 |  0|  0|  0|  0|300|  0|  0|
+---------------+---+------+---+---+---+---+---+---+---+

Variables

capacity                   Capacity of Empty tanks
products                   Amount of incoming chemicals

Reference

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 8, 2009.

Problem setup

capacity = [500;0;400;600;600;900;0;800;800];
products = [1200-300;700;1000;450;1200-500];

l = length(products);
m = length(capacity);

load = tom('load',l,m,'int');

% Binary variables
bnds = {0 <= load <= 1};

% Load constraint, the liquids have to go
con1 = {load*capacity >= products};

% Only one product per tank
con2 = {sum(load,1) <= 1};

% Objective
objective = sum(load*capacity);

constraints = {bnds, con1, con2};
options = struct;
options.solver = 'cplex';
options.name   = 'Tank Loading';
sol = ezsolve(objective,constraints,[],options);

remaining = sum(capacity)-subs(objective,sol);

PriLev = 1;
if PriLev > 0
    disp('NB: Tank 2 and tank 7 are filled before the optimization starts.')
    temp   = sol.load';
    ships  = length(products);
    for ship = 1:ships,
        tank = find(temp(:,ship));
        disp(['Ship number ' num2str(ship) ' unloads in tank(s) ' num2str(tank')])
    end
end

% MODIFICATION LOG
%
% 051010 med   Created
% 051208 med   Added remaining the return
% 060112 per   Added documentation.
% 060124 per   Interpretation of results upgraded.
% 060125 per   Moved disp to end
% 090308 med   Converted to tomSym

Problem type appears to be: mip
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Tank Loading                   f_k    4000.000000000000000000
                                              f(x_0)      0.000000000000000000

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

FuncEv    8 
NB: Tank 2 and tank 7 are filled before the optimization starts.
Ship number 1 unloads in tank(s) 6
Ship number 2 unloads in tank(s) 9
Ship number 3 unloads in tank(s) 3  5
Ship number 4 unloads in tank(s) 1
Ship number 5 unloads in tank(s) 8