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# Some scheduling problems with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times

• * Corresponding author: Mehmet Duran Toksari

The first author is supported by Scientific Research Fund of Erciyes University grant: FBA-2014-5397

• In recent years, significant on the past sequence dependent delivery times have been increasing for scheduling problems. An electronic component when waiting to process may be exposed to certain an electromagnetic field and is required to neutralize the effect of electromagnetism. In this case, it needs an extra time to eliminate adverse effect. In the scheduling literature, this extra time is called as past-sequence-dependent delivery times. In this paper we introduce single-machine scheduling problems with an exponential sum-of-actual-processing-time-based delivery times. By the exponential sum-of-actual-processing-time-based delivery times, we mean that the delivery times are defined by an exponential function of the sum of the actual processing times of the already processed jobs. On the other hand, the learning effect is reflected in decreasing processing times based on the job's position in schedule. In this paper, we also introduce both exponential past sequence dependent delivery times and learning effect where the job processing time is a function based on the sum of the logarithm of processing times of jobs already processed. We show that the single-machine scheduling problems to minimize makespan, total completion time, weighted total completion time and maximum tardiness with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times have polynomial time solutions.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:

• Figure 1.  The gantt chart for Example 1

Figure 2.  The gantt chart for Example 2

Figure 3.  The Gantt chart for the practical application

Table 1.  The results for Example 1

 $p_{j[r]}$ $q_{psd}$ $C_{i[r]}$ $T_{i[r]}$ $p_{2[1]}$ 6.00 $C_{2[1]}$ 6 $T_{2[1]}$ 0 $p_{2[1]}$ 4.79 $C_{1[2]}$ 10.79 $T_{1[2]}$ 0 $p_{3[3]}$ 4.08 $C_{3[3]}$ 14.87 $T_{3[3]}$ 4.87 $p_{5[4]}$ 3.76 $C_{5[4]}$ 18.63 $T_{5[4]}$ 9.63 $p_{4[5]}$ 3.92 $q_{psd}$ 14.51 $C_{4[5]}$ $C_{4[5]}$ 22.5+14.51=37.06 $T_{4[5]}$ 30.06

Table 2.  The results for Example 2

 $p_{j[r]}$ $q_{psd}$ $C_{i[r]}$ $T_{i[r]}$ $p_{2[1]}$ 6.00 $C_{2[1]}$ 6 $T_{2[1]}$ 78 $p_{2[1]}$ 4.79 $C_{1[2]}$ 10.79 $T_{1[2]}$ 97.11 $p_{3[3]}$ 4.08 $C_{3[3]}$ 14.87 $T_{3[3]}$ 104.09 $p_{5[4]}$ 3.76 $C_{5[4]}$ 18.63 $T_{5[4]}$ 93.15 $p_{4[5]}$ 3.92 $q_{psd}$ 14.51 $C_{4[5]}$ $C_{4[5]}$ 22.5+14.51=37.06 $T_{4[5]}$ 111.18

Table 3.  Laptop components of processing times and due dates

 Job Part Name Processing Time Due Date 1 CPU/GPU 24 9 2 IC Chip 6 35 3 Oscillator 9 25 4 Copper coil 12 20 5 Capacitor 70 5 6 Card slot 15 10 7 Ports 25 8 8 Cooling fan 42 6 9 Sub-PCB 35 7

Table 4.  The results for Example 2

 $p_{j[r]}$ $q_{psd}$ $C_{i[r]}$ $T_{i[r]}$ $p_{2[1]}$ 6.00 $C_{2[1]}$ 6 $T_{2[1]}$ 0 $p_{3[2]}$ 5.39 $C_{3[2]}$ 11.39 $T_{3[2]}$ 0 $p_{4[3]}$ 5.37 $C_{4[3]}$ 16.76 $T_{4[3]}$ 0 $p_{6[4]}$ 5.49 $C_{6[4]}$ 22.25 $T_{6[4]}$ 12.25 $p_{1[5]}$ 7.52 $C_{1[5]}$ 29.77 $T_{1[5]}$ 20.77 $p_{7[6]}$ 6.84 $C_{7[6]}$ 36.61 $T_{7[6]}$ 28.61 $p_{9[7]}$ 8.60 $C_{9[7]}$ 45.20 $T_{9[7]}$ 38.20 $p_{8[8]}$ 9.36 $C_{8[8]}$ 54.56 $T_{8[8]}$ 48.56 $p_{9[9]}$ 14.33 $q_{psd}$ 84.68 $C_{9[9]}$ 68.89+84.68=153.57 $T_{9[9]}$ 63.89
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