Fuzzy bi-criteria scheduling on parallel machines involving weighted flow time and maximum tardiness

Figures & data

Figure 1. Triangular fuzzy number.

Figure 2. Adjacent jobs on parallel machines.

Table 1. Jobs with fuzzy processing time

Jobs	1	2	3	4	5
Processing time (in fuzzy environment)	(9, 10, 11)	(16, 17, 18)	(9, 10, 11)	(6, 7, 8)	(11, 12, 13)
Due date (di)	13	19	18	11	15
Weightage (wi)	3	4	2	5	1

Table 2. Jobs with AHR of processing time

Jobs	1	2	3	4	5
AHR of processing time	32/3	53/3	32/3	23/3	38/3
Due date (di)	13	19	18	11	15
Weightage (wi)	3	4	2	5	1

Table 3. In–out of jobs following WSPT order

Jobs	4	2	1	3	5
M1	0–23/3			23/3–55/3
M2		0–53/3
M3			0–32/3		32/3–70/3
Due date (di)	11	19	13	18	15
Weightage (wi)	5	4	3	2	1
Tardiness (Ti)				1/3	25/3

Table 4. Jobs with fuzzy processing time

Jobs	1	2	3	4	5	6	7
Processing time (in fuzzy environment)	(7, 8, 9)	(6, 7, 8)	(10, 11, 12)	(8, 9, 10)	(6, 7, 8)	(11, 12, 13)	(8, 9, 10)
Due date (di)	12	18	19	16	10	13	15
Weightage (wi)	5	3	3	2	1	4	2

Table 5. Jobs with AHR of processing time

Jobs	1	2	3	4	5	6	7
AHR of processing time	26/3	23/3	35/3	29/3	23/3	38/3	29/3
Due date (di)	12	18	19	16	10	13	15
Weightage (wi)	5	3	3	2	1	4	2

Table 6. In–out of jobs in WSPT order

Jobs	1	6	2	3	4	7	5
M1	0–26/3				26/3–55/3		55/3–78/3
M2		0–38/3				38/3–67/3
M3			0–23/3	23/3–58/3
Due date (di)	12	18	19	16	10	13	15
Weightage (wi)	5	4	3	3	2	2	1
Tardiness (Ti)				10/3	25/3	28/3	11

Table 7. In–out of jobs following EDD order

Jobs	1	6	3	2	4	7	5
M1	0–26/3			26/3–49/3			49/3–72/3
M2		0–38/3				38/3–67/3
M3			0–35/3		35/3–64/3
Weightage (wi)	5	4	3	3	2	2	1
Due date (di)	12	18	16	19	10	13	15
Tardiness (Ti)	–	–	–	–	34/3	28/3	27/3