Cane Sugar Production

Contents

Problem description

The harvest of cane sugar in Australia is highly mechanized. The sugar cane is immediately transported to a sugar house in wagons that run on a network of small rail tracks. The sugar content of a wagon load depends on the field it has been harvested from and on the maturity of the sugar cane. Once harvested, the sugar content decreases rapidly through fermentation and the wagon load will entirely lose its value after a certain time. At this moment, eleven wagons all loaded with the same quantity have arrived at the sugar house. They have been examined to find out the hourly loss and the remaining life span (in hours) of every wagon, these data are summarized in the following table.

Table: Properties of the lots of cane sugar

Lot             1  2  3  4  5  6  7  8  9 10 11
Loss (kg/h)    43 26 37 28 13 54 62 49 19 28 30
Life span (h)   8  8  2  8  4  8  8  8  8  8  8

Every lot may be processed by any of the three, fully equivalent production lines of the sugar house. The processing of a lot takes two hours. It must be finished at the latest at the end of the life span of the wagon load. The manager of the sugar house wishes to determine a production schedule for the currently available lots that minimizes the total loss of sugar.

Variables

proclines                  the number of processing lines
proctime                   time in hours to process a wagon
loss                       loss of sugar in kg per hour
lifespan                   how long the sugar in a wagon will last

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

proclines   = 3;
proctime    = 2;
loss        = [43;26;37;28;13;54;62;49;19;28;30];
lifespan    = [8;8;2;8;4;8;8;8;8;8;8];

n1 = length(loss);
n2 = ceil(n1/proclines);

proc = tom('proc',n1,n2,'int');

% All slots are integers
intcons = {0 <= proc <= 1};

% Production constraint, all lots processed
prodcons1 = {sum(proc,2) == 1};

% Production constraint, No more than proclines concurrently running
prodcons2 = {sum(proc,1) <= proclines};

% Maximum slot number for a lot
slotcons = {sum(proc.*repmat(1:n2,n1,1),2) <= lifespan/proctime};

% Objective
objective = sum(sum(repmat(1:n2,n1,1).*repmat(loss,1,n2).*proctime.*proc));

constraints = {intcons, prodcons1, prodcons2, slotcons};
options = struct;
options.solver = 'cplex';
options.name   = 'Cane Sugar Production';
sol = ezsolve(objective,constraints,[],options);

f_k = subs(objective,sol);

PriLev = 1;
if PriLev > 0
    lots      = length(lifespan);
    timeslots = ceil(lots/proclines(1));  % even a fraction means work
    temp      = sol.proc;

    disp(['To minimize loss (' num2str(f_k) ') use this schema'])

    for time = 1:timeslots,
        disp(['at ' num2str((time-1)*proctime(1)+8) '.00:'])
        idx = find(temp(:,time));
        disp(['   process the lots ' num2str(idx') ])
    end
    disp(['at ' num2str((time)*proctime(1)+8) '.00:'])
    disp( '   harvesting completed')
end

% MODIFICATION LOG
%
% 051007 med   Created
% 051208 med   Lifespan factor wrong
% 060109 per   lifespan(9) changed to 8
% 060109 per   Added documentation.
% 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: Cane Sugar Production          f_k    1602.000000000000000000
                                              f(x_0)      0.000000000000000000

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

FuncEv   19 
CPU time: 0.015625 sec. Elapsed time: 0.015000 sec. 
To minimize loss (1602) use this schema
at 8.00:
   process the lots 3  6  7
at 10.00:
   process the lots 1  5  8
at 12.00:
   process the lots 4  10  11
at 14.00:
   process the lots 2  9
at 16.00:
   harvesting completed