# Composing Flight Crews

## Contents

## Problem description

During the Second World War the Royal Air Force (RAF) had many foreign pilots speaking different languages and more or less trained on the different types of aircraft. The RAF had to form pilot/co-pilot pairs (‘crews’) for every plane with a compatible language and a sufficiently good knowledge of the aircraft type. In our example there are eight pilots. In the following table every pilot is characterized by a mark between 0 (worst) and 20 (best) for his language skills (in English, French, Dutch, and Norwegian), and for his experience with different two-seater aircraft (reconnaissance, transport, bomber, fighterbomber, and supply planes).

Ratings of pilots

+----------+-------------------------+--+--+--+--+--+--+--+--+ | |Pilot | 1| 2| 3| 4| 5| 6| 7| 8| +----------+-------------------------+--+--+--+--+--+--+--+--+ |Language |English |20|14| 0|13| 0| 0| 8| 8| | |French |12| 0| 0|10|15|20| 8| 9| | |Dutch | 0|20|12| 0| 8|11|14|12| | |Norwegian | 0| 0| 0| 0|17| 0| 0|16| +----------+-------------------------+--+--+--+--+--+--+--+--+ |Plane type|Reconnaissance |18|12|15| 0| 0| 0| 8| 0| | |Transport |10| 0| 9|14|15| 8|12|13| | |Bomber | 0|17| 0|11|13|10| 0| 0| | |Fighter-bomber | 0| 0|14| 0| 0|12|16| 0| | |Supply plane | 0| 0| 0| 0|12|18| 0|18| +----------+-------------------------+--+--+--+--+--+--+--+--+

A valid flight crew consists of two pilots that both have each at least 10/20 for the same language and 10/20 on the same aircraft type.

Question 1: Is it possible to have all pilots fly?

Subsequently, we calculate for every valid flight crew the sum of their scores for every aircraft type for which both pilots are rated at least 10/20. This allows us to define for every crew the maximum score among these marks. For example, pilots 5 and 6 have marks 13 and 10 on bombers and 12 and 18 on supply planes. The score for this crew is therefore max(13 + 10, 12 + 18) = 30.

Question 2: Which is the set of crews with maximum total score?

Compatibility graph for pilots Nodes = pilots Arcs = possible combinations (with scores)

1 ----30--- 2 ----27--- 3

| \ / | / | | 24 28 | / | | \ / 27 26 | | | / | | 4 | / 30 25 | / | | / \ | / | | 29 21 | / | | / \ | / |

5 ---30---- 6 -----28-- 7

\ | / \ | / \ | / \ 36 / \ | / 30 | 25 \ | / \ | / \ | /

8

## Variables

scores Scores for language and piloting skills arcs_out/in For an arc i arcs_out(i) is its starting node and arcs_in(i) ts target node arcs_score Strength of an arc

## 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

scores = [ 20 14 0 13 0 0 8 8;... 12 0 0 10 15 20 8 9;... 0 20 12 0 8 11 14 12;... 0 0 0 0 17 0 0 16;... 18 12 15 0 0 0 8 0;... 10 0 9 14 15 8 12 13;... 0 17 0 11 13 10 0 0;... 0 0 14 0 0 12 16 0;... 0 0 0 0 12 18 0 18]; arcs_in = [1 1 1 2 2 2 3 3 4 4 5 5 6 6 7]'; arcs_out = [5 4 2 4 6 3 6 7 5 6 6 8 8 7 8]'; arcs_score = [25 24 30 28 27 27 26 30 29 21 30 30 ... 36 28 25]'; n = length(arcs_in); n1 = size(scores,2); fly = tom('fly',n,1,'int'); % All variables are integers. bnds = {0 <= fly <= 1}; % Orig constr. con = cell(n1,1); for i=1:n1 idx_in = find(arcs_in == i); idx_out = find(arcs_out == i); idx = unique([idx_in;idx_out]); con{i} = {sum(fly(idx)) <= 1}; end % Objective objective = -arcs_score'*fly; constraints = {bnds, con}; options = struct; options.solver = 'cplex'; options.name = 'Composing Flight Crews'; sol = ezsolve(objective,constraints,[],options); PriLev = 1; if PriLev > 0 idx = find(sol.fly); for crew = 1:idx, id = idx(crew); disp(['crew ' num2str(crew) ' has pilots ' ... num2str(arcs_in(id)) ' and ' num2str(arcs_out(id))]) end end % MODIFICATION LOG % % 051021 med Created % 060113 per Added documentation % 060125 per Moved disp to end % 060203 med Updated print level % 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: Composing Flight Crews f_k -125.000000000000000000 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=101. CPLEX Branch-and-Cut MIP solver Optimal integer solution found FuncEv 9 crew 1 has pilots 1 and 2 crew 2 has pilots 3 and 7 crew 3 has pilots 4 and 5