Choice of Expansion Projects

Contents

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 tomSym

Problem type appears to be: mip
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
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 
The management should choose the following projects:
  New painting facilities       
  Research for a new concept car
  Reorganize the logistics chain