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An asymptotically optimal algorithm for large-scale mixed job shop scheduling to minimize the makespan

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Abstract

This paper considers the large-scale mixed job shop scheduling problem with general number of jobs on each route. The problem includes ordinary machines, batch machines (with bounded or unbounded capacity), parallel machines, and machines with breakdowns. The objective is to find a schedule to minimize the makespan. For the problem, we define a virtual problem and a corresponding virtual schedule, based on which our algorithm TVSA is proposed. The performance analysis of the algorithm shows the gap between the obtained solution and the optimal solution is O(1), which indicates the algorithm is asymptotically optimal.

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Acknowledgments

This work is supported by National Natural Science Foundation of China(Grant Nos. 11201282, 11371137 and 61304209), Humanity and Social Science Youth Foundation of Ministry of Education of China(Grant no. 10YJCZH032), Innovation Program of Shanghai Municipal Education Commission (Grant No. 14YZ127).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University, Weihai, Shandong, 264209, China

    Manzhan Gu

  2. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

    Xiwen Lu

  3. School of Mechanical, Electrical & Information Engineering, Shandong University, Weihai, Shandong, 264209, China

    Jinwei Gu

  4. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Manzhan Gu

  5. College of Economics and Management, Shanghai University of Electric Power, Shanghai, 200090, China

    Jinwei Gu

Authors
  1. Manzhan Gu
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  2. Xiwen Lu
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  3. Jinwei Gu
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Corresponding author

Correspondence to Jinwei Gu.

Appendix

Appendix

In each matrix, one line is for a route. Each line contains the number of jobs and the number of machines, listing the machine number and the processing load for each step of the route. The machines are numbered starting with 1.

$$\begin{aligned} PT1= & {} [1,9,2,5,3,52,4,24,5,7; 5,11,4,24,3,72,2,12,1,9; 2,5,4,25,3,42,5,\\&9,1,11; 4,20,2,9,1,5,3,58,5,17; 3,44,1,9,2,9,5,9,4,21].\\ PT2= & {} [1,13,2,14,3,67,4,29,5,14; 5,18,4,15,3,43,2,7,1,5; 2,14,4,13,3,\\&46,5,14,1,6;4,28,2,8,1,9,3,55,5,17; 3,53,1,10,2,6,5,6,4,14].\\ PT3= & {} [1,6,2,14,3,61,4,18,5,11; 5,11,4,27,3,51,2,14,1,9; 2,12,4,20,3,\\&64,5,11,1,13;4,27,2,6,1,8,3,49,5,17; 3,34,1,6,2,7,5,6,4,19].\\ PT4= & {} [1,21,2,14,3,120,4,52,5,34; 5,33,4,30,3,149,2,14,1,18; 2,12,4,\\&55,3,131,5,21,1,14;4,75,2,6,1,23,3,121,5,27; 3,98,1,25,2,7,\\&5,30,4,78].\\ PT5= & {} [1,12,2,14,3,133,4,51,5,26; 5,29,4,63,3,118,2,14,1,23; 2,12,\\&4,45,3,150,5,36,1,17;4,66,2,6,1,12,3,149,5,30; 3,55,1,21,2,\\&7,5,30,4,33].\\ PT6= & {} [1,19,2,14,3,79,4,85,5,30; 5,22,4,86,3,136,2,14,1,23; 2,12,4,45,\\&3,83,5,36,1,26;4,83,2,6,1,11,3,118,5,13; 3,55,1,29,2,7,5,\\&25,4,61].\\ PT7= & {} [1,30,2,14,3,293,4,111,5,85; 5,66,4,117,3,175,2,14,1,70; 2,12,\\&4,154,3,193,5,50,1,63;4,182,2,6,1,65,3,282,5,59; 3,259,1,66,2,\\&7,5,66,4,195].\\ PT8= & {} [1,63,2,14,3,328,4,275,5,88; 5,38,4,133,3,278,2,14,1,67; 2,12,\\&4,267,3,301,5,83,1,73;4,268,2,6,1,65,3,148,5,76; 3,231,1,56,\\&2,7,5,57,4,290].\\ PT9= & {} [1,63,2,14,3,128,4,149,5,87; 5,50,4,218,3,141,2,14,1,34; 2,\\&12,4,201,3,156,5,54,1,56;4,193,2,6,1,45,3,148,5,48; 3,231,\\&1,48,2,7,5,55,4,208]. \end{aligned}$$

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Gu, M., Lu, X. & Gu, J. An asymptotically optimal algorithm for large-scale mixed job shop scheduling to minimize the makespan. J Comb Optim 33, 473–495 (2017). https://doi.org/10.1007/s10878-015-9974-7

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