9.2 Mathematical Structure of the Problem
221
SURPLUS(10, 0, 0, 20)φ 1,10 (x 1 )S(x 2 )S(x 3 )R(x 4 )
+SURPLUS(10, 20, 0, 20)φ 1,10 (x 1 )R(x 2 )S(x 3 )R(x 4 ) +
SURPLUS(10, 0, 20, 20)φ 1,10 (x 1 )S(x 2 )R(x 3 )R(x 4 )
+SURPLUS(10, 20, 20, 20)φ 1,10 (x 1 )R(x 2 )R(x 3 )R(x 4 ) ;
(9.9)
the third:
SURPLUS(0, 10, 0, 0)S(x 1 )φ 1,10 (x 2 )S(x 3 )S(x 4 )
+SURPLUS(20, 10, 0, 0)R(x 1 )φ 1,10 (x 2 )S(x 3 )S(x 4 ) +
SURPLUS(0, 10, 20, 0)S(x 1 )φ 1,10 (x 2 )R(x 3 )S(x 4 )
+SURPLUS(20, 10, 20, 0)R(x 1 )φ 1,10 (x 2 )R(x 3 )S(x 4 ) +
SURPLUS(0, 10, 0, 20)S(x 1 )φ 1,10 (x 2 )S(x 3 )R(x 4 )
+SURPLUS(20, 10, 0, 20)R(x 1 )φ 1,10 (x 2 )S(x 3 )R(x 4 ) +
SURPLUS(0, 10, 20, 20)S(x 1 )φ 1,10 (x 2 )R(x 3 )R(x 4 )
+SURPLUS(20, 10, 20, 20)R(x 1 )φ 1,10 (x 2 )R(x 3 )R(x 4 ) ;
(9.10)
the fourth:
SURPLUS(0, 0, 10, 0)S(x 1 )S(x 2 )φ 1,10 (x 3 )S(x 4 )
+SURPLUS(20, 0, 10, 0)R(x 1 )S(x 2 )φ 1,10 (x 3 )S(x 4 ) +
SURPLUS(0, 20, 10, 0)S(x 1 )R(x 2 )φ 1,10 (x 3 )S(x 4 )
+SURPLUS(20, 20, 10, 0)R(x 1 )R(x 2 )φ 1,10 (x 3 )S(x 4 ) +
SURPLUS(0, 0, 10, 20)S(x 1 )S(x 2 )φ 1,10 (x 3 )R(x 4 )
+SURPLUS(20, 0, 10, 20)R(x 1 )S(x 2 )φ 1,10 (x 3 )R(x 4 ) +
SURPLUS(0, 20, 10, 20)S(x 1 )R(x 2 )φ 1,10 (x 3 )R(x 4 )
+SURPLUS(20, 20, 10, 20)R(x 1 )R(x 2 )φ 1,10 (x 3 )R(x 4 ) ;
(9.11)
and the fifth:
SURPLUS(0, 0, 0, 10)S(x 1 )S(x 2 )S(x 3 )φ 1,10 (x 4 )
+SURPLUS(20, 0, 0, 10)R(x 1 )S(x 2 )S(x 3 )φ 1,10 (x 4 ) +