TomSym Choice of Expansion Projects
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This page is part of the TomSym Manual. See TomSym Manual. |
Problem description
The large company Tatayo in the north of Italy has specialized in the construction of cars for more than ten years. The company wishes to expand and has issued internally a call for proposals for expansion projects for a planning period of five years. Among the many, often cranky, propositions the management has retained five projects. Every project has an annual cost and is designed to produce a benefit after five years. The first table below gives a list of the projects with short descriptions and the expected benefit after five years. The forecast annual costs of the projects for the next five years are detailed in the second table below, together with the funds available. Which project(s) should the management choose now to maximize the total benefit after five years?
Estimated benefits of the projects (in million $)
+-------+------------------------------+----------------+ |Project|Description |Expected benefit| +-------+------------------------------+----------------+ | 1 |Expand assembly line | 10.8 | | 2 |Reorganize the main shop | 4.8 | | 3 |New painting facilities | 3.2 | | 4 |Research for a new concept car| 4.44 | | 5 |Reorganize the logistics chain| 12.25 | +-------+------------------------------+----------------+
Annual costs of projects and available funds (in million $)
+-------+------+------+------+------+------+ |Project|Year 1|Year 2|Year 3|Year 4|Year 5| +-------+------+------+------+------+------+ | 1 | 1.8 | 2.4 | 2.4 | 1.8 | 1.5 | | 2 | 1.2 | 1.8 | 2.4 | 0.6 | 0.5 | | 3 | 1.2 | 1.0 | 0.0 | 0.48 | 0.0 | | 4 | 1.4 | 1.4 | 1.2 | 1.2 | 1.2 | | 5 | 1.6 | 2.1 | 2.5 | 2.0 | 1.8 | +-------+------+------+------+------+------+ |Funds | 4.8 | 6.0 | 4.8 | 4.2 | 3.5 | +-------+------+------+------+------+------+
Variables
benefit Expected benefit budget Funds available each year costmat Cost per project and year
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
benefit = [10.8 4.8 3.2 4.44 12.25]'*1e6;
budget = [ 4.8 6.0 4.8 4.2 3.5]'*1e6;
costmat = [1.8 2.4 2.4 1.8 1.5;...
1.2 1.8 2.4 0.6 0.5;...
1.2 1.0 0.0 .48 0.0;...
1.4 1.4 1.2 1.2 1.2;...
1.6 2.1 2.5 2.0 1.8]*1e6;
n = length(benefit); %projects
choose = tom('choose',n,1,'int');
% All variables are integer.
bnds = {0 <= choose <= 1};
% Cost constraints
con = {(choose'*costmat)' <= budget};
% Objective
objective = -benefit'*choose;
constraints = {bnds, con};
options = struct;
options.solver = 'cplex';
options.name = 'Choice of Expansion Projects';
sol = ezsolve(objective,constraints,[],options);
PriLev = 1;
if PriLev > 0
names = [' Expand assembly line ' ;
' Reorganize the main shop ' ;
' New painting facilities ' ;
' Research for a new concept car' ;
' Reorganize the logistics chain'];
idx = find(sol.choose);
disp('The management should choose the following projects:')
disp(names(idx,:))
end
% MODIFICATION LOG
%
% 051201 med Created
% 060117 per Added documentation
% 090308 med Converted to tomSymProblem type appears to be: mip
Time for symbolic processing: 0.0096977 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Choice of Expansion Projects f_k -19890000.000000000000000000
f(x_0) 0.000000000000000000
Solver: CPLEX. EXIT=0. INFORM=101.
CPLEX Branch-and-Cut MIP solver
Optimal integer solution found
FuncEv 4
Elapsed time: 0.003000 sec.
The management should choose the following projects:
New painting facilities
Research for a new concept car
Reorganize the logistics chain
