These HX convoys had been established shortly after declaration of war; and the first sailed on 16 September 1939. Ships in convoy were less vulnerable to submarine attack than ships sailing independently, but the Allies had difficulty providing an adequate number of escorting warships to establish a protective perimeter for detecting and defeating approaching submarines. British Admiralty operations research scientists evaluating convoy battles of 1941 and 1942 determined losses of ships in convoy were independent of convoy size, but varied with the number of attacking submarines and, when patrol aircraft were unavailable, with the number of escorting warships. They suggested convoy losses could be reduced by 64 percent by decreasing the frequency of convoys to increase the average number