# Flight Connections at a Hub

## Contents

## Problem description

The airline SafeFlight uses the airport Roissy-Charles-de-Gaulle as a hub to minimize the number of flight connections to European destinations. Six Fokker 100 airplanes of this airline from Bordeaux, Clermont-Ferrand, Marseille, Nantes, Nice, and Toulouse are landing between 11am and 12:30pm. These aircraft leave for Berlin, Bern, Brussels, London, Rome, and Vienna between 12:30 pm and 13:30 pm. The numbers of passengers transferring from the incoming flights to one of the outgoing flights are listed in the table below.

Numbers of passengers transferring between the different flights

+----------------+---------------------------------------+ |Origins | Destinations | | +------+----+--------+------+----+------+ | |Berlin|Bern|Brussels|London|Rome|Vienna| +----------------+------+----+--------+------+----+------+ |Bordeaux | 35 | 12 | 16 | 38 | 5 | 2 | |Clermont-Ferrand| 25 | 8 | 9 | 24 | 6 | 8 | |Marseille | 12 | 8 | 11 | 27 | 3 | 2 | |Nantes | 38 | 15 | 14 | 30 | 2 | 9 | |Nice | – | 9 | 8 | 25 | 10 | 5 | |Toulouse | – | – | – | 14 | 6 | 7 | +----------------+------+----+--------+------+----+------+

For example, if the flight incoming from Bordeaux continues on to Berlin, 35 passengers and their luggage may stay on board during the stop at Paris. The flight from Nice arrives too late to be re-used on the connection to Berlin, the same is true for the flight from Toulouse that cannot be used for the destinations Berlin, Bern and Brussels (the corresponding entries in the table are marked with '–').

How should the arriving planes be re-used for the departing flights to minimize the number of passengers who have to change planes at Roissy?

## Variables

origdest People remaining in plane at transfer idximp Impossible combinations

## 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

origdest = [ 35 12 16 38 5 2;... 25 8 9 24 6 8;... 12 8 11 27 3 2;... 38 15 14 30 2 9;... 0 9 8 25 10 5;... 0 0 0 14 6 7]; idximp = find(origdest == 0); n = size(origdest,1); cont = tom('cont',n,n,'int'); % All variables are integers. bnds = {0 <= cont <= 1, cont(idximp) == 0}; % Orig constr. con1 = {sum(cont,1) == 1}; % Dest constr. con2 = {sum(cont,2) == 1}; % Objective objective = -sum(sum(origdest.*cont)); constraints = {bnds, con1, con2}; options = struct; options.solver = 'cplex'; options.name = 'Flight Connections at a Hub'; sol = ezsolve(objective,constraints,[],options); PriLev = 1; if PriLev > 0 p = 6; % the number of planes in_names = ['Bord';'Cler';'Mars';... % city names in 'Nant'; 'Nice'; 'Toul']; out_names = ['Berl';'Bern';'Brus';... % city names out 'Lond';'Rome';'Vien']; transfers = sol.cont; for i = 1:p, % in city has number i for o = 1:p, % out city has number o if transfers(i,o) == 1, disp(['plane from ' in_names(i,:) ... ' continues to ' out_names(o,:) ]) end end end end % MODIFICATION LOG % % 051021 med Created. % 060113 per Added documentation. % 060125 per Moved disp to end % 060203 med Print level updated % 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: Flight Connections at a Hub f_k -112.000000000000000000 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=101. CPLEX Branch-and-Cut MIP solver Optimal integer solution found FuncEv 15 plane from Bord continues to Lond plane from Cler continues to Bern plane from Mars continues to Brus plane from Nant continues to Berl plane from Nice continues to Rome plane from Toul continues to Vien