TomSym Fleet Planning for Vans

This page is part of the TomSym Manual. See TomSym Manual.

Problem description

A chain of department stores uses a fleet of vans rented from different rental agencies. For the next six months period it has forecast the following needs for vans (table below):

Requirements for vans for six months

+---+---+---+---+---+---+
|Jan|Feb|Mar|Apr|May|Jun|
+---+---+---+---+---+---+
|430|410|440|390|425|450|
+---+---+---+---+---+---+

At the 1st January, the chain has 200 vans, for which the rental period terminates at the end of February.

To satisfy its needs, the chain has a choice among three types of contracts that may start the first day of every month: 3-months contracts for a total cost of $1700 per van, 4-months contracts at $2200 per van, and 5-months contracts at $2600 per van. How many contracts of the different types need to be started every month in order to satisfy the company�s needs at the least cost and to have no remaining vans rented after the end of June?

Running contracts in month 5 (May)   .........
                                     :       :
                             +-------:-------:-------+
                             | Rent34:       :       |
                     +-------+-------:-------:-------+
                     |     Rent33    :       :
                     +---------------:-------:-------+
                     | Rent43        :       :       |
             +-------+---------------:-------:-------+
             |    Rent42             :       :
             +-----------------------:-------:-------+
             | Rent52                :       :       |
     +-------+-----------------------:-------:-------+
     |    Rent51                     :       :
     +-------------------------------:-------:
             .       .       .       :       :       .
Month   Jan  :  Feb  :  Mar  :  Apr  :  May  :  Jun  :
     :.......:.......:.......:.......:       :.......:
                                     :.......:

Variables

demand                     Demand of vans per month
initialsupply              Vans per month
contractlength             Contracts available
contractcost               Cost of contracts

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

demand         = [ 430; 410; 440; 390; 425; 450];
initialsupply  = [ 200; 200;   0;   0;   0;   0];
contractlength = [ 3; 4; 5];
contractcost   = [ 1700; 2200; 2600];

c = length(contractlength); % contracts
m = length(demand);         % months

rent = tom('rent',c,m,'int');

% All variables are integers.
bnds = {0 <= rent};

% Demand constraint
for i=1:m
    for j=1:c
        idx1(i,j) = max(1,i-contractlength(j)+1);
        idx2(i,j) = min(i,m-contractlength(j)+1);
    end
end

con1 = cell(m,1);
for i=1:m
    con1{i} = 0;
    for j=1:c
        con1{i} = con1{i} + sum(rent(j,idx1(i,j):idx2(i,j)));
    end
    con1{i} = {con1{i} >= demand(i) - initialsupply(i)};
end

% Objective
objective = sum(contractcost'*rent);

constraints = {bnds, con1};
options = struct;
options.solver = 'cplex';
options.name   = 'Fleet Planning for Vans';
sol = ezsolve(objective,constraints,[],options);

f_k = subs(objective,sol);

PriLev = 1;
if PriLev > 0
    Nmonths = length(demand);
    c      = length(contractlength);
    temp   = sol.rent;
    disp(['minimal cost of ' num2str(f_k) ' by '])
    for m = 1:Nmonths,
        if sum(temp(:,m)) > 0,
            rent = find(temp(:,m));
            disp(['   renting ' num2str(temp(rent,m)) ' vans for ' ...
                num2str(contractlength(rent)) ' months in month ' num2str(m)])
        end
    end
end

% MODIFICATION LOG
%
% 051021 med   Created.
% 060112 per   Added documentation.
% 060125 per   Moved disp to end
% 090308 med   Converted to tomSym

Problem type appears to be: mip
Time for symbolic processing: 0.052209 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Fleet Planning for Vans        f_k 1261000.000000000000000000
                                              f(x_0)      0.000000000000000000

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


Elapsed time: 0.002000 sec. 
minimal cost of 1261000 by 
   renting 230 vans for 3 months in month 1
   renting 210 vans for 4 months in month 3
   renting 240 vans for 3 months in month 4