Approximation Algorithms and an FPTAS for the Single Machine Problem with Biased Tardiness Penalty

Table 5

Procedure of DP algorithm in Example 8.

	𝑣0^(𝛼)		𝑣1^(𝛼)		𝑣2^(𝛼)		𝑍^∗𝛼
	State	Sequence	State	Sequence	State	Sequence
𝛼=1	[0, 0]	(  ,  ,  )	[0, 41]	(  ,  ,  2)	[3, 41]	(3,  ,   2)	41+16=57
𝛼=2	[0, 0]	(  ,  ,  )	[5, 0]; [0, 30]	(1,  ,  ); (  ,  ,  1)	[5, 45]; [3, 30]; [0, 55]	(1,  ,  3); (3,  ,  1); (  ,  3,   1)	min45+3230+2655+17=56
𝛼=3	[0, 0]	(  ,  ,  )	[5, 0]; [0, 30]	(1,  ,  ); (  ,  ,  1)	[5, 41]	(1,  ,  2)	41+17=58