Example: Fighter Aircraft Problem – Linear Assignment Method

From the original decision matrix Fighter Aircraft Problem we can obtain attributewise preferences:

Rank	f1	f2	f3	f4	f5	f6
weights	0.2	0.1	0.1	0.1	0.2	0.3
1^st	A2	A2	A3	A3	A3	A1
2^nd	A4	A3	A1 A4	A5	A1 A4	A3
3^rd	A1	A4		A1		A2 A4
4^th	A3	A1	A2	A2	A2

Three criteria f₃,f₅, and f₆ have tied attributewise rankings. These can be equalised:

Rank	f31f32	f51f52	f61f62
1^st	A3 A3	A3 A3	A1 A1
2^nd	A1 A4	A1 A4	A3 A3
3^rd	A4 A1	A4 A1	A2 A4
4^th	A2 A2	A2 A2	A4 A2

Each of these rankings gets half of weight of the tied ranking and the matrix is:

	1^st	2^nd	3^rd	4^th
A1	0.3	0.15	0.45	0.1
A2	0.3	0	0.15	0.55
A3	0.4	0.4	0	0.2
A4	0	0.45	0.40	0.15

The LP formulation is:

The optimal solution is a permutation matrix P*:

The optimal order is than: (A₃,A₄,A₁,A₂).