Family Budget

Contents

Problem description

The mother of a family wishes to use her son’s computer. On the internet she has found optimization software and would now like to formulate a mathematical model to help plan their annual budget. She has prepared a list of the monthly expenses and receipts of money. Every month the expenses for living are $ 550. The monthly rent for the apartment is $ 630. She also budgets $ 135 for telephone every two months, $ 850 for gas and electricity bills every six months, $ 340 per month for the car and $ 100 of tax every four months. Her receipts of money are a monthly payment of $ 150 of state allowance for families with dependent children and a net salary of $ 1900 per month. She knows that she pays at least $ 165 for leisure every month (subscription to the swimming pool for the older children, football club for the youngest, gym for herself) but she would like to spend more (restaurant, cinema, holidays). How should the budget be balanced all through the year to maximize the money available for leisure?

Variables

costs                      Various costs
costfreq                   Frequency of the costs
income                     Various incomes
incomefreq                 Frequency of the income
minhobby                   Mimial cost for hobbies

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

costs      = [550 630 135 850 340 100]';
costfreq   = [  1   1   2   6   1   4]';
income     = [150 1900]';
incomefreq = [  1    1]';
minhobby   = 165;

m = 12; % Hobby, months

hobby = tom('hobby',m,1);
save  = tom('save',m,1);

% No variables are integer.
bnds = {hobby >= minhobby, save >= 0};

% Months constraints
incomefreqcount = incomefreq;
costfreqcount = costfreq;
upper = zeros(m,1);
value = 0;
for i=1:m
    for j=1:length(incomefreq)
        if incomefreq(j) == i
            value = value + income(j);
            incomefreq(j) = incomefreq(j) + incomefreqcount(j);
        end
    end
    for j=1:length(costfreq)
        if costfreq(j) == i
            value = value - costs(j);
            costfreq(j) = costfreq(j) + costfreqcount(j);
        end
    end
    upper(i,1) = value;
    value = 0;
end

con1 = {hobby(1) + save(1) <= upper(1)};
con2 = {hobby(2:end) + save(2:end) - save(1:end-1) <= upper(2:end)};

% Objective
objective = -sum(hobby);

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

PriLev = 1;
if PriLev > 0
    m       = 12 ; % number of months
    leisure = sol.hobby;
    save    = sol.save;
    disp('Money for leisure and for savings per month')
    for i = 1:m,
        disp(['month ' num2str(i) ', leisure ' num2str(leisure(i))  ', save ' num2str(save(i))])
    end
    disp(['total leisure this year: ' num2str(sum(leisure))])
    disp(['total to save this year: ' num2str(sum(save))   ])
end

% MODIFICATION LOG
%
% 051201 med   Created.
% 060116 per   Added documentation.
% 060117 per   Minor change of documentation.
% 060125 per   Moved disp to end
% 090308 med   Converted to tomSym

Problem type appears to be: lp
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Family Budget                  f_k   -3550.000000000000000000
                                              f(x_0)      0.000000000000000000

Solver: CPLEX.  EXIT=0.  INFORM=1.
CPLEX Dual Simplex LP solver
Optimal solution found


Money for leisure and for savings per month
month 1, leisure 165, save 365
month 2, leisure 165, save 595
month 3, leisure 165, save 960
month 4, leisure 165, save 1090
month 5, leisure 165, save 1455
month 6, leisure 165, save 835
month 7, leisure 165, save 1200
month 8, leisure 165, save 1330
month 9, leisure 1735, save 125
month 10, leisure 165, save 355
month 11, leisure 165, save 720
month 12, leisure 165, save 0
total leisure this year: 3550
total to save this year: 9030