Fleet Planning for Vans

Contents

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
con1 = cell(m,1);
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

for i=1:m
    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
    months = length(demand);
    c      = length(contractlength);
    temp   = sol.rent;
    disp(['minimal cost of ' num2str(f_k) ' by '])
    for m = 1:months,
        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
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
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


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