# American Institute of Mathematical Sciences

September  2020, 16(5): 2439-2457. doi: 10.3934/jimo.2019062

## Minimizing total completion time in a two-machine no-wait flowshop with uncertain and bounded setup times

 1 Department of Mathematical Sciences, Kean University, New Jersey, USA 2 Department of Industrial and Management Systems Engineering, Kuwait University, Kuwait

1Corresponding Author: Muberra Allahverdi

Received  October 2018 Revised  February 2019 Published  May 2019

We address a two-machine no-wait flowshop scheduling problem with respect to the performance measure of total completion time. Minimizing total completion time is important when inventory cost is of concern. Setup times are treated separately from processing times. Furthermore, setup times are uncertain with unknown distributions and are within some lower and upper bounds. We develop a dominance relation and propose eight algorithms to solve the problem. The proposed algorithms, which assign different weights to the processing and setup times on both machines, convert the two-machine problem into a single-machine one for which an optimal solution is known. We conduct computational experiments to evaluate the proposed algorithms. Computational experiments reveal that one of the proposed algorithms, which assigns the same weight to setup and processing times, is superior to the rest of the algorithms. The results are statistically verified by constructing confidence intervals and test of hypothesis.

Citation: Muberra Allahverdi, Ali Allahverdi. Minimizing total completion time in a two-machine no-wait flowshop with uncertain and bounded setup times. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2439-2457. doi: 10.3934/jimo.2019062
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##### References:
Flowchart of the algorithms
The distributions used for generating $s_{i, k}$ within $Ls_{i, k}$ and $Us_{i, k}$
Overall Avg. Error of Algorithms
Avg. Std. of Algorithms
Overall Avg. Error of Algorithms with respect to $H$
Overall Avg. Error of Algorithms with respect to $H$
Overall Avg. Error of Algorithm $ALG-6$ with respect to $H$
Overall Avg. Error of Algorithms with respect to distributions
Error of Algorithms for Positive Linear Distribution
 $n$ Algorithm $H$ 100 200 300 400 500 Avg. $ALG-1$ 20 9.08 9.11 9.17 9.17 9.27 9.16 $ALG-2$ 9.38 9.37 9.28 9.34 9.3 9.33 $ALG-3$ 3.05 2.98 2.95 2.96 2.97 2.98 $ALG-4$ 4.99 4.93 4.88 4.9 4.9 4.92 $ALG-5$ 4.85 4.81 4.76 4.81 4.83 4.79 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.23 4.1 4.02 4 3.95 4.06 $ALG-8$ 2.44 2.35 2.44 2.44 2.44 2.42 $ALG-1$ 30 9.17 9.33 9.42 9.4 9.4 9.34 $ALG-2$ 9.5 9.56 9.48 9.47 9.47 9.50 $ALG-3$ 3.04 3.1 3.03 3.03 3 3.04 $ALG-4$ 5.04 5.07 4.97 4.98 4.98 5.01 $ALG-5$ 4.81 4.93 4.95 4.95 4.95 4.92 $ALG-6$ 0.08 0.01 0 0 0 0.02 $ALG-7$ 4.2 4.12 4.02 4.04 3.96 4.07 $ALG-8$ 2.48 2.45 2.46 2.49 2.49 2.47 $ALG-1$ 40 9.37 9.47 9.49 9.54 9.56 9.49 $ALG-2$ 9.65 9.69 9.64 9.63 9.67 9.66 $ALG-3$ 3.01 3.07 3.06 3.05 3.08 3.05 $ALG-4$ 5.06 5.06 5.06 5.05 5.06 5.06 $ALG-5$ 4.93 5.01 4.96 5 5.05 4.99 $ALG-6$ 0.09 0.02 0 0 0 0.02 $ALG-7$ 4.24 4.17 4.03 4.1 4.12 4.13 $ALG-8$ 2.47 2.51 2.53 2.51 2.57 2.52 Avg. 4.80 4.80 4.78 4.79 4.79
 $n$ Algorithm $H$ 100 200 300 400 500 Avg. $ALG-1$ 20 9.08 9.11 9.17 9.17 9.27 9.16 $ALG-2$ 9.38 9.37 9.28 9.34 9.3 9.33 $ALG-3$ 3.05 2.98 2.95 2.96 2.97 2.98 $ALG-4$ 4.99 4.93 4.88 4.9 4.9 4.92 $ALG-5$ 4.85 4.81 4.76 4.81 4.83 4.79 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.23 4.1 4.02 4 3.95 4.06 $ALG-8$ 2.44 2.35 2.44 2.44 2.44 2.42 $ALG-1$ 30 9.17 9.33 9.42 9.4 9.4 9.34 $ALG-2$ 9.5 9.56 9.48 9.47 9.47 9.50 $ALG-3$ 3.04 3.1 3.03 3.03 3 3.04 $ALG-4$ 5.04 5.07 4.97 4.98 4.98 5.01 $ALG-5$ 4.81 4.93 4.95 4.95 4.95 4.92 $ALG-6$ 0.08 0.01 0 0 0 0.02 $ALG-7$ 4.2 4.12 4.02 4.04 3.96 4.07 $ALG-8$ 2.48 2.45 2.46 2.49 2.49 2.47 $ALG-1$ 40 9.37 9.47 9.49 9.54 9.56 9.49 $ALG-2$ 9.65 9.69 9.64 9.63 9.67 9.66 $ALG-3$ 3.01 3.07 3.06 3.05 3.08 3.05 $ALG-4$ 5.06 5.06 5.06 5.05 5.06 5.06 $ALG-5$ 4.93 5.01 4.96 5 5.05 4.99 $ALG-6$ 0.09 0.02 0 0 0 0.02 $ALG-7$ 4.24 4.17 4.03 4.1 4.12 4.13 $ALG-8$ 2.47 2.51 2.53 2.51 2.57 2.52 Avg. 4.80 4.80 4.78 4.79 4.79
Error of Algorithms for Negative Linear Distribution
 $n$ Algorithm $H$ 100 200 300 400 500 Avg. $ALG-1$ 20 9.08 9.13 9.22 9.26 9.27 9.19 $ALG-2$ 9.38 9.26 9.33 9.35 9.27 9.32 $ALG-3$ 3.05 2.97 3.04 3.02 2.99 3.01 $ALG-4$ 4.99 4.85 4.92 4.9 4.87 4.91 $ALG-5$ 4.75 4.73 4.89 4.92 4.86 4.83 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.23 4.1 4.06 4.07 3.95 4.08 $ALG-8$ 2.44 2.41 2.44 2.5 2.43 2.44 $ALG-1$ 30 9.17 9.37 9.37 9.37 9.41 9.34 $ALG-2$ 9.5 9.51 9.53 9.5 9.51 9.51 $ALG-3$ 3.04 3.05 3.04 3.04 3.05 3.04 $ALG-4$ 5.04 5 5.03 5 4.99 5.01 $ALG-5$ 4.81 4.92 4.9 4.94 4.93 4.90 $ALG-6$ 0.08 0.01 0 0 0 0.02 $ALG-7$ 4.2 4.15 4.1 4.05 4.01 4.10 $ALG-8$ 2.48 2.54 2.46 2.55 2.53 2.51 $ALG-1$ 40 9.37 9.37 9.5 9.51 9.55 9.46 $ALG-2$ 9.65 9.54 9.59 9.64 9.62 9.61 $ALG-3$ 3.01 2.96 3.04 3.09 3.07 3.03 $ALG-4$ 5.06 4.97 5.01 5.06 5.05 5.03 $ALG-5$ 4.93 4.85 4.95 4.97 5.01 4.94 $ALG-6$ 0.09 0.01 0 0 0 0.02 $ALG-7$ 4.24 4.12 4.08 4.07 4.08 4.12 $ALG-8$ 2.47 2.44 2.58 2.54 2.57 2.52 Avg. 4.80 4.76 4.80 4.81 4.79
 $n$ Algorithm $H$ 100 200 300 400 500 Avg. $ALG-1$ 20 9.08 9.13 9.22 9.26 9.27 9.19 $ALG-2$ 9.38 9.26 9.33 9.35 9.27 9.32 $ALG-3$ 3.05 2.97 3.04 3.02 2.99 3.01 $ALG-4$ 4.99 4.85 4.92 4.9 4.87 4.91 $ALG-5$ 4.75 4.73 4.89 4.92 4.86 4.83 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.23 4.1 4.06 4.07 3.95 4.08 $ALG-8$ 2.44 2.41 2.44 2.5 2.43 2.44 $ALG-1$ 30 9.17 9.37 9.37 9.37 9.41 9.34 $ALG-2$ 9.5 9.51 9.53 9.5 9.51 9.51 $ALG-3$ 3.04 3.05 3.04 3.04 3.05 3.04 $ALG-4$ 5.04 5 5.03 5 4.99 5.01 $ALG-5$ 4.81 4.92 4.9 4.94 4.93 4.90 $ALG-6$ 0.08 0.01 0 0 0 0.02 $ALG-7$ 4.2 4.15 4.1 4.05 4.01 4.10 $ALG-8$ 2.48 2.54 2.46 2.55 2.53 2.51 $ALG-1$ 40 9.37 9.37 9.5 9.51 9.55 9.46 $ALG-2$ 9.65 9.54 9.59 9.64 9.62 9.61 $ALG-3$ 3.01 2.96 3.04 3.09 3.07 3.03 $ALG-4$ 5.06 4.97 5.01 5.06 5.05 5.03 $ALG-5$ 4.93 4.85 4.95 4.97 5.01 4.94 $ALG-6$ 0.09 0.01 0 0 0 0.02 $ALG-7$ 4.24 4.12 4.08 4.07 4.08 4.12 $ALG-8$ 2.47 2.44 2.58 2.54 2.57 2.52 Avg. 4.80 4.76 4.80 4.81 4.79
Error of Algorithm for Uniform Distribution
 $n$ Algorithm $H$ 100 200 300 400 500 Avg. $ALG-1$ 20 9.05 9.32 9.34 9.37 9.41 9.30 $ALG-2$ 9.39 9.45 9.43 9.49 9.43 9.44 $ALG-3$ 3.09 3.02 2.97 3.04 3.04 3.03 $ALG-4$ 4.98 4.96 4.92 4.98 4.96 4.96 $ALG-5$ 4.86 4.88 4.87 4.94 4.95 4.90 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.24 4.08 4.09 4.09 4.05 4.11 $ALG-8$ 2.42 4.08 4.09 4.09 4.05 4.11 $ALG-1$ 30 9.09 9.47 9.52 9.64 9.56 9.46 $ALG-2$ 9.59 9.61 9.63 9.67 9.59 9.63 $ALG-3$ 3.1 3.04 3.04 3.09 3.03 3.06 $ALG-4$ 5.13 5.03 5.02 5.06 5 5.05 $ALG-5$ 4.86 4.94 4.96 5.05 5.02 4.97 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.17 4.17 4.08 4.13 4.06 4.12 $ALG-8$ 2.44 2.49 2.53 2.59 2.56 2.52 $ALG-1$ 40 9.32 9.68 9.68 9.7 9.79 9.63 $ALG-2$ 9.67 9.89 9.85 9.87 9.89 9.83 $ALG-3$ 3.05 3.06 3.06 3.08 3.1 3.07 $ALG-4$ 5.07 5.14 5.13 5.13 5.13 5.12 $ALG-5$ 4.86 5.09 5.04 5.07 5.09 5.03 $ALG-6$ 0.09 0.01 0 0 0 0.02 $ALG-7$ 4.31 4.28 4.23 4.19 4.21 4.24 $ALG-8$ 2.43 2.61 2.64 2.68 2.68 2.61 Avg. 4.81 4.86 4.86 4.89 4.88
 $n$ Algorithm $H$ 100 200 300 400 500 Avg. $ALG-1$ 20 9.05 9.32 9.34 9.37 9.41 9.30 $ALG-2$ 9.39 9.45 9.43 9.49 9.43 9.44 $ALG-3$ 3.09 3.02 2.97 3.04 3.04 3.03 $ALG-4$ 4.98 4.96 4.92 4.98 4.96 4.96 $ALG-5$ 4.86 4.88 4.87 4.94 4.95 4.90 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.24 4.08 4.09 4.09 4.05 4.11 $ALG-8$ 2.42 4.08 4.09 4.09 4.05 4.11 $ALG-1$ 30 9.09 9.47 9.52 9.64 9.56 9.46 $ALG-2$ 9.59 9.61 9.63 9.67 9.59 9.63 $ALG-3$ 3.1 3.04 3.04 3.09 3.03 3.06 $ALG-4$ 5.13 5.03 5.02 5.06 5 5.05 $ALG-5$ 4.86 4.94 4.96 5.05 5.02 4.97 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.17 4.17 4.08 4.13 4.06 4.12 $ALG-8$ 2.44 2.49 2.53 2.59 2.56 2.52 $ALG-1$ 40 9.32 9.68 9.68 9.7 9.79 9.63 $ALG-2$ 9.67 9.89 9.85 9.87 9.89 9.83 $ALG-3$ 3.05 3.06 3.06 3.08 3.1 3.07 $ALG-4$ 5.07 5.14 5.13 5.13 5.13 5.12 $ALG-5$ 4.86 5.09 5.04 5.07 5.09 5.03 $ALG-6$ 0.09 0.01 0 0 0 0.02 $ALG-7$ 4.31 4.28 4.23 4.19 4.21 4.24 $ALG-8$ 2.43 2.61 2.64 2.68 2.68 2.61 Avg. 4.81 4.86 4.86 4.89 4.88
Error of Algorithm for Normal Distribution
 $n$ Algorithm $H$ 100 200 300 400 500 Avg. $ALG-1$ 20 9.04 9.3 9.32 9.35 9.38 9.28 $ALG-2$ 9.5 9.46 9.45 9.46 9.5 9.47 $ALG-3$ 3.03 3 3 3.03 3.06 3.02 $ALG-4$ 5.01 4.97 4.92 4.96 4.95 4.90 $ALG-5$ 4.81 4.88 4.9 4.94 4.95 4.90 $ALG-6$ 0.08 0.01 0 0 0 0.02 $ALG-7$ 4.27 4.09 4.13 4.08 4.06 4.13 $ALG-8$ 2.47 2.46 2.45 2.5 2.55 2.49 $ALG-1$ 30 9.43 9.43 9.54 9.57 9.71 9.54 $ALG-2$ 9.8 9.71 9.66 9.7 9.72 9.72 $ALG-3$ 3.11 3 3.09 3.08 3.12 3.08 $ALG-4$ 5.16 5.06 5.06 5.05 5.11 5.09 $ALG-5$ 4.99 4.92 5 4.99 5.08 5.00 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.38 4.15 4.17 4.17 4.17 4.21 $ALG-8$ 2.52 2.52 2.59 2.57 2.62 2.56 $ALG-1$ 40 9.57 9.7 9.77 9.78 9.86 9.74 $ALG-2$ 10.04 9.96 9.9 9.86 9.86 9.92 $ALG-3$ 3.12 3.03 3.05 3.07 3.05 3.06 $ALG-4$ 5.25 5.17 5.14 5.11 5.12 5.16 $ALG-5$ 5.01 5.03 5.05 5.11 5.1 5.06 $ALG-6$ 0.08 0.01 0 0 0 0.02 $ALG-7$ 4.47 4.2 4.21 4.24 4.19 4.26 $ALG-8$ 2.69 2.6 2.73 2.71 2.7 2.69 Avg. 4.91 4.86 4.88 4.89 4.91
 $n$ Algorithm $H$ 100 200 300 400 500 Avg. $ALG-1$ 20 9.04 9.3 9.32 9.35 9.38 9.28 $ALG-2$ 9.5 9.46 9.45 9.46 9.5 9.47 $ALG-3$ 3.03 3 3 3.03 3.06 3.02 $ALG-4$ 5.01 4.97 4.92 4.96 4.95 4.90 $ALG-5$ 4.81 4.88 4.9 4.94 4.95 4.90 $ALG-6$ 0.08 0.01 0 0 0 0.02 $ALG-7$ 4.27 4.09 4.13 4.08 4.06 4.13 $ALG-8$ 2.47 2.46 2.45 2.5 2.55 2.49 $ALG-1$ 30 9.43 9.43 9.54 9.57 9.71 9.54 $ALG-2$ 9.8 9.71 9.66 9.7 9.72 9.72 $ALG-3$ 3.11 3 3.09 3.08 3.12 3.08 $ALG-4$ 5.16 5.06 5.06 5.05 5.11 5.09 $ALG-5$ 4.99 4.92 5 4.99 5.08 5.00 $ALG-6$ 0.1 0.02 0 0 0 0.02 $ALG-7$ 4.38 4.15 4.17 4.17 4.17 4.21 $ALG-8$ 2.52 2.52 2.59 2.57 2.62 2.56 $ALG-1$ 40 9.57 9.7 9.77 9.78 9.86 9.74 $ALG-2$ 10.04 9.96 9.9 9.86 9.86 9.92 $ALG-3$ 3.12 3.03 3.05 3.07 3.05 3.06 $ALG-4$ 5.25 5.17 5.14 5.11 5.12 5.16 $ALG-5$ 5.01 5.03 5.05 5.11 5.1 5.06 $ALG-6$ 0.08 0.01 0 0 0 0.02 $ALG-7$ 4.47 4.2 4.21 4.24 4.19 4.26 $ALG-8$ 2.69 2.6 2.73 2.71 2.7 2.69 Avg. 4.91 4.86 4.88 4.89 4.91
Error of Algoithms with respect to $n$
 n Algorithm 100 200 300 400 500 Avg. $ALG-1$ 9.27 9.39 9.45 9.47 9.51 9.42 $ALG-2$ 9.64 9.58 9.56 9.58 9.57 9.59 $ALG-3$ 3.07 3.02 3.03 3.05 3.05 3.04 $ALG-4$ 5.09 5.02 5.01 5.02 5.01 5.03 $ALG-5$ 4.88 4.92 4.94 4.97 4.99 4.94 $ALG-6$ 0.09 0.02 0.00 0.00 0.00 0.02 $ALG-7$ 4.28 4.14 4.10 4.10 4.07 4.14 $ALG-8$ 2.49 2.49 2.53 2.55 2.56 2.52
 n Algorithm 100 200 300 400 500 Avg. $ALG-1$ 9.27 9.39 9.45 9.47 9.51 9.42 $ALG-2$ 9.64 9.58 9.56 9.58 9.57 9.59 $ALG-3$ 3.07 3.02 3.03 3.05 3.05 3.04 $ALG-4$ 5.09 5.02 5.01 5.02 5.01 5.03 $ALG-5$ 4.88 4.92 4.94 4.97 4.99 4.94 $ALG-6$ 0.09 0.02 0.00 0.00 0.00 0.02 $ALG-7$ 4.28 4.14 4.10 4.10 4.07 4.14 $ALG-8$ 2.49 2.49 2.53 2.55 2.56 2.52
Error of Algorithms with respect to $H$
 $H$ Algorithm 20 30 40 Avg. $ALG-1$ 9.24 9.42 9.59 9.42 $ALG-2$ 9.40 9.60 9.77 9.59 $ALG-3$ 3.01 3.06 3.06 3.04 $ALG-4$ 4.94 5.04 5.10 5.03 $ALG-5$ 4.86 4.94 5.01 4.94 $ALG-6$ 0.02 0.02 0.02 0.02 $ALG-7$ 4.10 4.13 4.19 4.14 $ALG-8$ 2.46 2.52 2.59 2.52
 $H$ Algorithm 20 30 40 Avg. $ALG-1$ 9.24 9.42 9.59 9.42 $ALG-2$ 9.40 9.60 9.77 9.59 $ALG-3$ 3.01 3.06 3.06 3.04 $ALG-4$ 4.94 5.04 5.10 5.03 $ALG-5$ 4.86 4.94 5.01 4.94 $ALG-6$ 0.02 0.02 0.02 0.02 $ALG-7$ 4.10 4.13 4.19 4.14 $ALG-8$ 2.46 2.52 2.59 2.52
Confidence Intervals for Average Errors
 Algorithm 95$\%$ Confidence Interval on the Avg. Error $ALG-1$ (09.33-9.51) $ALG-2$ (9.50-9.68) $ALG-3$ (2.98-3.11) $ALG-4$ (4.95-5.10) $ALG-5$ (4.86-5.01) $ALG-6$ (0.02-0.03) $ALG-7$ (4.07-4.21) $ALG-8$ (2.45-2.59)
 Algorithm 95$\%$ Confidence Interval on the Avg. Error $ALG-1$ (09.33-9.51) $ALG-2$ (9.50-9.68) $ALG-3$ (2.98-3.11) $ALG-4$ (4.95-5.10) $ALG-5$ (4.86-5.01) $ALG-6$ (0.02-0.03) $ALG-7$ (4.07-4.21) $ALG-8$ (2.45-2.59)
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