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doi: 10.3934/jimo.2020140

An efficient genetic algorithm for decentralized multi-project scheduling with resource transfers

Jingwen Zhang , , Wanjun Liu and  Wanlin Liu

School of Management, Northwestern Polytechnical University, Xián 710072, Shaanxi, China

* Corresponding author: Jingwen Zhang

Received  January 2020 Revised  July 2020 Published  September 2020

This paper investigates the decentralized resource-constrained multi-project scheduling problem with transfer times (DRCMPSPTT) in which the transfer times of the global resources among different projects are assumed to be sequence-independent, while transfers of local resources take no time within a project. First, two decision variables ($ {y_{ijg}} $ and $ {w_{ijg}} $) are adopted to express the transition state of global resources between projects. $ {y_{ijg}} $ (takes a value of 0 or 1) represents whether activity i transfers global resource g to activity j; accordingly, the transferred quantity is denoted as $ {w_{ijg}} $. Then, we construct an integer linear model with the goal of minimizing the average project delay for the DRCMPSPTT. Second, an adaptive genetic algorithm (GA) is developed to solve the DRCMPSPTT. To gain the schedules for the DRCMPSPTT, the traditional serial and parallel scheduling generation schemes (SGSs) are modified to combine with different resource transfer rules and to design multiple decoding schemes. Third, the numerical experiments are implemented to analyse the effects of eight decoding schemes, and we found that the scheme comprising the parallel SGS and maxRS rule can make the GA work the best; furthermore, the effectiveness of the GA_maxRS (GA embedded with the best scheme) is demonstrated by solving some instances with different sizes.

Keywords: Decentralized multi-project scheduling, resource transfer times, genetic algorithm, scheduling generation schemes, resource transfer rules.
Mathematics Subject Classification: Primary: 90B35, 90C11.
Citation: Jingwen Zhang, Wanjun Liu, Wanlin Liu. An efficient genetic algorithm for decentralized multi-project scheduling with resource transfers. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020140
References:
[1]

S. Adhau, M. L. Mittal and A. Mittal, A multi-agent system for decentralized multi project scheduling with resource transfers, International Journal of Production Economics, 146 (2013), 646-661.  doi: 10.1016/j.ijpe.2013.08.013.  Google Scholar

[2]

S. Adhau, M. L. Mittal and A. Mittal, A multi-agent system for distributed multi-project scheduling: An auction-based negotiation approach, Engineering Applications of Artificial Intelligence, 25 (2012), 1738-1751.  doi: 10.1016/j.engappai.2011.12.003.  Google Scholar

[3]

B. Afshar-Nadjafi and M. Majlesi, Resource constrained project scheduling problem with setup times after preemptive processes, Computers and Chemical Engineering, 69 (2014), 16-25.  doi: 10.1016/j.compchemeng.2014.06.012.  Google Scholar

[4]

C. Artigues, P. Michelon and S. Reusser, Insertion techniques for static and dynamic resource-constrained project scheduling, European Journal of Operational Research, 149 (2003), 249-267.  doi: 10.1016/S0377-2217(02)00758-0.  Google Scholar

[5]

D. Bedworth and J. Bailey, Integrated Production Control Systems Management, Analysis, Design, 2$^nd$ edition, John Wiley & Sons, Inc., New York, 1999. Google Scholar

[6]

C. Bierwirth, D. Mattfeld and H. Kopfer, On permutation representations for scheduling problems, in Parallel Problem Solving from Nature, Lecture Notes in Computer Science, 1996,310–318. doi: 10.1007/3-540-61723-X_995.  Google Scholar

[7]

P. Brucker, S. Knust, A. Schoo and O. Thiele, A branch and bound algorithm for the resource-constrained project scheduling problem, European Journal of Operational Research, 107 (1998), 272-288.  doi: 10.1016/S0377-2217(97)00335-4.  Google Scholar

[8]

Z. Cai and X. Li, A hybrid genetic algorithm for resource-constrained multi-project scheduling problem with resource transfer time, IEEE International Conference on Automation Science and Engineering, Seoul, 2012,569–574. doi: 10.1109/CoASE.2012.6386457.  Google Scholar

[9]

G. Confessore, S. Giordani and S. Rismondo, A market-based multi-agent system model for decentralized multi–project scheduling, Annals of Operations Research, 150 (2007), 115-135.  doi: 10.1007/s10479-006-0158-9.  Google Scholar

[10]

E. W. Davis, Project network summary measures constrained resource scheduling, AIIE Transactions, 7 (1975), 132-142.  doi: 10.1080/05695557508974995.  Google Scholar

[11]

E. L. Demeulemeester and W. S. Herroelen, An efficient optimal solution procedure for the preemptive resource-constrained project scheduling problem, European Journal of Operational Research, 90 (1996), 334-348.  doi: 10.1016/0377-2217(95)00358-4.  Google Scholar

[12]

L. Djerid, M. C. Portman and P. Villon, Performance analysis of permutation cross-over genetic operators, Journal of Decision Systems, 5 (1996), 131-140.  doi: 10.1080/12460125.1996.10511679.  Google Scholar

[13]

S. Hartmann and R. Kolisch, Experimental evaluation of state-of-the-art heuristics for resource-constrained project scheduling problem, European Journal of Operational Research, 127 (2000), 394-407.  doi: 10.1016/S0377-2217(99)00485-3.  Google Scholar

[14] J. B. Holland and J. H. Holland, Adaption in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Mich., 1975.   Google Scholar
[15]

J. Homberger, A multi-agent system for the decentralized resource constrained multi-project scheduling problem, International Transactions in Operational Research, 14 (2007), 565-589.  doi: 10.1111/j.1475-3995.2007.00614.x.  Google Scholar

[16]

J. Homberger, A (${\mu}$, ${\lambda}$)-coordination mechanism for agent-based multi-project scheduling, OR Spectrum, 34 (2012), 107-132.  doi: 10.1007/s00291-009-0178-3.  Google Scholar

[17]

R. Jans and Z. Degraeve, Modeling industrial lot sizing problems: A review, International Journal of Production Research, 46 (2008), 1619-1643.  doi: 10.1080/00207540600902262.  Google Scholar

[18]

R. L. Kadri and F. F. Boctor, An efficient genetic algorithm to solve the resource-constrained project scheduling problem with transfer times: The single mode case, European Journal of Operational Research, 265 (2018), 454-–462. doi: 10.1016/j.ejor.2017.07.027.  Google Scholar

[19]

J. E. Kelley, The critical-path method: Resources planning and scheduling, Industrial Scheduling, 13 (1963), 347365. Google Scholar

[20]

S. Khalilpourazari, B. Naderi and S. Khalilpourazary, Multi-objective stochastic fractal search: A powerful algorithm for solving complex multi-objective optimization problems, Soft Computing, 24 (2020), 3037–-3066. doi: 10.1007/s00500-019-04080-6.  Google Scholar

[21]

S. Khalilpourazari and S. H. R. Pasandideh, Modeling and optimization of multi-item multi-constrained EOQ model for growing items, Knowledge-Based Systems, 164 (2019), 150-162.  doi: 10.1016/j.knosys.2018.10.032.  Google Scholar

[22]

S. Khalilpourazari and S. H. R. Pasandideh, Sine-cosine crow search algorithm: Theory and applications, Neural Computing and Applications, 32 (2019), 7725–-7742. doi: 10.1007/s00521-019-04530-0.  Google Scholar

[23]

R. Kolisch, Project scheduling with setup times, in Project Scheduling Under Resource Constraints, Production and Logistics, Physica, Heidelberg, 1995,177–185. doi: 10.1007/978-3-642-50296-5_8.  Google Scholar

[24]

R. Kolisch and S. Hartmann, Heuristic algorithms for solving the resource-constrained project scheduling problem: Classification and computational analysis, in Project Scheduling, International Series in Operations Research & Management Science, Springer, Boston, 1999,147–178. doi: 10.1007/978-1-4615-5533-9_7.  Google Scholar

[25]

R. Kolisch and A. Sprecher, PSPLIB–A project scheduling library, European Journal of Operation Research, 96 (1996), 205-216.   Google Scholar

[26]

D. Kruger and A. Scholl, A heuristic solution framework for the resource constrained (multi-)project scheduling problem with sequence-dependent transfer times, European Journal of Operation Research, 197 (2009), 492-508.  doi: 10.1016/j.ejor.2008.07.036.  Google Scholar

[27]

D. Kruger and A. Scholl, Managing and modelling general resource transfers in (multi-)project scheduling, OR Spectrum, 32 (2010), 369-394.  doi: 10.1007/s00291-008-0144-5.  Google Scholar

[28]

C. J. Liao, C. W. Chao and L. C. Chen, An improved heuristic for parallel machine weighted flowtime scheduling with family set-up times, Computers and Mathematics with Applications, 63 (2012), 110-117.  doi: 10.1016/j.camwa.2011.10.077.  Google Scholar

[29]

A. Lova and P. Tormos, Analysis of scheduling schemes and heuristic rules performance in resource-constrained multiproject scheduling, Annals of Operations Research, 102 (2001), 263-286.  doi: 10.1023/A:1010966401888.  Google Scholar

[30]

A. Lova, P. Tormos, M. Cervantes and F. Barber, An efficient hybrid genetic algorithm for scheduling projects with resource constraints and multiple execution modes, International Journal of Production Economics, 117 (2009), 302-316.  doi: 10.1016/j.ijpe.2008.11.002.  Google Scholar

[31]

D. Merkle, M. Middendorf and H. Schmeck, Ant colony optimization for resource-constrained project scheduling, IEEE Transactions on Evolutionary Computation, 6 (2002), 333-346.  doi: 10.1109/TEVC.2002.802450.  Google Scholar

[32]

M. Mika, G. Waligra and J. Weglarz, Tabu search for multi-mode resource-constrained project scheduling with schedule-dependent setup times, European Journal of Operational Research, 187 (2008), 1238-1250.  doi: 10.1016/j.ejor.2006.06.069.  Google Scholar

[33]

M. L. Mittal and A. Kanda, Scheduling of multiple projects with resource transfers, International Journal of Mathematics in Operational Research, 1 (2009), 303-325.  doi: 10.1504/IJMOR.2009.024288.  Google Scholar

[34]

K. Moumene and J. A. Ferland, Activity list representation for a generalization of the resource-constrained project scheduling problem, European Journal of Operational Research, 199 (2009), 46-54.  doi: 10.1016/j.ejor.2008.10.030.  Google Scholar

[35]

M. S. Nagano, A. A. Silva and L. A. N. Lorena, A new evolutionary clustering search for a no-wait flow shop problem with set-up times, Engineering Applications of Artificial Intelligence, 25 (2012), 1114-1120.  doi: 10.1016/j.engappai.2012.05.017.  Google Scholar

[36]

K. Neumann, C. Schwindt and J. Zimmermann, Project Scheduling with Time Windows and Scarce Resources, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-540-24800-2.  Google Scholar

[37]

S. H. R. Pasandideh and S. Khalilpourazari, Sine cosine crow search algorithm: A powerful hybrid meta heuristic for global optimization, preprint, arXiv: 1801.08485. Google Scholar

[38]

J. Poppenborg and S. Knust, A flow-based tabu search algorithm for the RCPSP with transfer times, OR Spectrum, 38 (2016), 305-334.  doi: 10.1007/s00291-015-0402-2.  Google Scholar

[39]

V. Roshanaei, B. Naderi, F. Jolai and M. Khalili, A variable neighborhood search for job shop scheduling with set-up times to minimize makespan, Future Generation Computer Systems, 25 (2009), 654-661.  doi: 10.1016/j.future.2009.01.004.  Google Scholar

[40]

M. Rostami, M. Bagherpour, M. M. Mazdeh and A. Makui, Resource Pool Location for Periodic Services in Decentralized Multi-Project Scheduling Problems, Journal of Computing in Civil Engineering, 31 (2017), 04017022. doi: 10.1061/(ASCE)CP.1943-5487.0000671.  Google Scholar

[41]

V. Valls, F. Ballestin and M. S. Quintanilla, A hybrid genetic algorithm for the RCPSP, Technical report, Department of Statistics and Operations Research, University of Valencia, 2003. Google Scholar

[42]

V. Valls, F. Ballestin and M. S. Quintanilla, Justification and RCPSP: A technique that pays, European Journal of Operational Research, 165 (2005), 375-386.  doi: 10.1016/j.ejor.2004.04.008.  Google Scholar

[43]

M. Vanhoucke, Setup times and fast tracking in resource-constrained project scheduling, Computers & Industrial Engineering, 54 (2008), 1062-1070.  doi: 10.1016/j.cie.2007.11.008.  Google Scholar

[44]

K. K. Yang and C. C. Sum, A comparison of resource allocation and activity scheduling rules in a dynamic multi-project environment, Journal of Operation Management, 11 (1993), 207-218.  doi: 10.1016/0272-6963(93)90023-I.  Google Scholar

[45]

K. K. Yang and C. C. Sum, An evaluation of due date, resource allocation, project release, and activity scheduling rules in a multi project environment, European Journal of Operational Research, 103 (1997), 139-154.   Google Scholar

show all references

References:
[1]

S. Adhau, M. L. Mittal and A. Mittal, A multi-agent system for decentralized multi project scheduling with resource transfers, International Journal of Production Economics, 146 (2013), 646-661.  doi: 10.1016/j.ijpe.2013.08.013.  Google Scholar

[2]

S. Adhau, M. L. Mittal and A. Mittal, A multi-agent system for distributed multi-project scheduling: An auction-based negotiation approach, Engineering Applications of Artificial Intelligence, 25 (2012), 1738-1751.  doi: 10.1016/j.engappai.2011.12.003.  Google Scholar

[3]

B. Afshar-Nadjafi and M. Majlesi, Resource constrained project scheduling problem with setup times after preemptive processes, Computers and Chemical Engineering, 69 (2014), 16-25.  doi: 10.1016/j.compchemeng.2014.06.012.  Google Scholar

[4]

C. Artigues, P. Michelon and S. Reusser, Insertion techniques for static and dynamic resource-constrained project scheduling, European Journal of Operational Research, 149 (2003), 249-267.  doi: 10.1016/S0377-2217(02)00758-0.  Google Scholar

[5]

D. Bedworth and J. Bailey, Integrated Production Control Systems Management, Analysis, Design, 2$^nd$ edition, John Wiley & Sons, Inc., New York, 1999. Google Scholar

[6]

C. Bierwirth, D. Mattfeld and H. Kopfer, On permutation representations for scheduling problems, in Parallel Problem Solving from Nature, Lecture Notes in Computer Science, 1996,310–318. doi: 10.1007/3-540-61723-X_995.  Google Scholar

[7]

P. Brucker, S. Knust, A. Schoo and O. Thiele, A branch and bound algorithm for the resource-constrained project scheduling problem, European Journal of Operational Research, 107 (1998), 272-288.  doi: 10.1016/S0377-2217(97)00335-4.  Google Scholar

[8]

Z. Cai and X. Li, A hybrid genetic algorithm for resource-constrained multi-project scheduling problem with resource transfer time, IEEE International Conference on Automation Science and Engineering, Seoul, 2012,569–574. doi: 10.1109/CoASE.2012.6386457.  Google Scholar

[9]

G. Confessore, S. Giordani and S. Rismondo, A market-based multi-agent system model for decentralized multi–project scheduling, Annals of Operations Research, 150 (2007), 115-135.  doi: 10.1007/s10479-006-0158-9.  Google Scholar

[10]

E. W. Davis, Project network summary measures constrained resource scheduling, AIIE Transactions, 7 (1975), 132-142.  doi: 10.1080/05695557508974995.  Google Scholar

[11]

E. L. Demeulemeester and W. S. Herroelen, An efficient optimal solution procedure for the preemptive resource-constrained project scheduling problem, European Journal of Operational Research, 90 (1996), 334-348.  doi: 10.1016/0377-2217(95)00358-4.  Google Scholar

[12]

L. Djerid, M. C. Portman and P. Villon, Performance analysis of permutation cross-over genetic operators, Journal of Decision Systems, 5 (1996), 131-140.  doi: 10.1080/12460125.1996.10511679.  Google Scholar

[13]

S. Hartmann and R. Kolisch, Experimental evaluation of state-of-the-art heuristics for resource-constrained project scheduling problem, European Journal of Operational Research, 127 (2000), 394-407.  doi: 10.1016/S0377-2217(99)00485-3.  Google Scholar

[14] J. B. Holland and J. H. Holland, Adaption in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Mich., 1975.   Google Scholar
[15]

J. Homberger, A multi-agent system for the decentralized resource constrained multi-project scheduling problem, International Transactions in Operational Research, 14 (2007), 565-589.  doi: 10.1111/j.1475-3995.2007.00614.x.  Google Scholar

[16]

J. Homberger, A (${\mu}$, ${\lambda}$)-coordination mechanism for agent-based multi-project scheduling, OR Spectrum, 34 (2012), 107-132.  doi: 10.1007/s00291-009-0178-3.  Google Scholar

[17]

R. Jans and Z. Degraeve, Modeling industrial lot sizing problems: A review, International Journal of Production Research, 46 (2008), 1619-1643.  doi: 10.1080/00207540600902262.  Google Scholar

[18]

R. L. Kadri and F. F. Boctor, An efficient genetic algorithm to solve the resource-constrained project scheduling problem with transfer times: The single mode case, European Journal of Operational Research, 265 (2018), 454-–462. doi: 10.1016/j.ejor.2017.07.027.  Google Scholar

[19]

J. E. Kelley, The critical-path method: Resources planning and scheduling, Industrial Scheduling, 13 (1963), 347365. Google Scholar

[20]

S. Khalilpourazari, B. Naderi and S. Khalilpourazary, Multi-objective stochastic fractal search: A powerful algorithm for solving complex multi-objective optimization problems, Soft Computing, 24 (2020), 3037–-3066. doi: 10.1007/s00500-019-04080-6.  Google Scholar

[21]

S. Khalilpourazari and S. H. R. Pasandideh, Modeling and optimization of multi-item multi-constrained EOQ model for growing items, Knowledge-Based Systems, 164 (2019), 150-162.  doi: 10.1016/j.knosys.2018.10.032.  Google Scholar

[22]

S. Khalilpourazari and S. H. R. Pasandideh, Sine-cosine crow search algorithm: Theory and applications, Neural Computing and Applications, 32 (2019), 7725–-7742. doi: 10.1007/s00521-019-04530-0.  Google Scholar

[23]

R. Kolisch, Project scheduling with setup times, in Project Scheduling Under Resource Constraints, Production and Logistics, Physica, Heidelberg, 1995,177–185. doi: 10.1007/978-3-642-50296-5_8.  Google Scholar

[24]

R. Kolisch and S. Hartmann, Heuristic algorithms for solving the resource-constrained project scheduling problem: Classification and computational analysis, in Project Scheduling, International Series in Operations Research & Management Science, Springer, Boston, 1999,147–178. doi: 10.1007/978-1-4615-5533-9_7.  Google Scholar

[25]

R. Kolisch and A. Sprecher, PSPLIB–A project scheduling library, European Journal of Operation Research, 96 (1996), 205-216.   Google Scholar

[26]

D. Kruger and A. Scholl, A heuristic solution framework for the resource constrained (multi-)project scheduling problem with sequence-dependent transfer times, European Journal of Operation Research, 197 (2009), 492-508.  doi: 10.1016/j.ejor.2008.07.036.  Google Scholar

[27]

D. Kruger and A. Scholl, Managing and modelling general resource transfers in (multi-)project scheduling, OR Spectrum, 32 (2010), 369-394.  doi: 10.1007/s00291-008-0144-5.  Google Scholar

[28]

C. J. Liao, C. W. Chao and L. C. Chen, An improved heuristic for parallel machine weighted flowtime scheduling with family set-up times, Computers and Mathematics with Applications, 63 (2012), 110-117.  doi: 10.1016/j.camwa.2011.10.077.  Google Scholar

[29]

A. Lova and P. Tormos, Analysis of scheduling schemes and heuristic rules performance in resource-constrained multiproject scheduling, Annals of Operations Research, 102 (2001), 263-286.  doi: 10.1023/A:1010966401888.  Google Scholar

[30]

A. Lova, P. Tormos, M. Cervantes and F. Barber, An efficient hybrid genetic algorithm for scheduling projects with resource constraints and multiple execution modes, International Journal of Production Economics, 117 (2009), 302-316.  doi: 10.1016/j.ijpe.2008.11.002.  Google Scholar

[31]

D. Merkle, M. Middendorf and H. Schmeck, Ant colony optimization for resource-constrained project scheduling, IEEE Transactions on Evolutionary Computation, 6 (2002), 333-346.  doi: 10.1109/TEVC.2002.802450.  Google Scholar

[32]

M. Mika, G. Waligra and J. Weglarz, Tabu search for multi-mode resource-constrained project scheduling with schedule-dependent setup times, European Journal of Operational Research, 187 (2008), 1238-1250.  doi: 10.1016/j.ejor.2006.06.069.  Google Scholar

[33]

M. L. Mittal and A. Kanda, Scheduling of multiple projects with resource transfers, International Journal of Mathematics in Operational Research, 1 (2009), 303-325.  doi: 10.1504/IJMOR.2009.024288.  Google Scholar

[34]

K. Moumene and J. A. Ferland, Activity list representation for a generalization of the resource-constrained project scheduling problem, European Journal of Operational Research, 199 (2009), 46-54.  doi: 10.1016/j.ejor.2008.10.030.  Google Scholar

[35]

M. S. Nagano, A. A. Silva and L. A. N. Lorena, A new evolutionary clustering search for a no-wait flow shop problem with set-up times, Engineering Applications of Artificial Intelligence, 25 (2012), 1114-1120.  doi: 10.1016/j.engappai.2012.05.017.  Google Scholar

[36]

K. Neumann, C. Schwindt and J. Zimmermann, Project Scheduling with Time Windows and Scarce Resources, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-540-24800-2.  Google Scholar

[37]

S. H. R. Pasandideh and S. Khalilpourazari, Sine cosine crow search algorithm: A powerful hybrid meta heuristic for global optimization, preprint, arXiv: 1801.08485. Google Scholar

[38]

J. Poppenborg and S. Knust, A flow-based tabu search algorithm for the RCPSP with transfer times, OR Spectrum, 38 (2016), 305-334.  doi: 10.1007/s00291-015-0402-2.  Google Scholar

[39]

V. Roshanaei, B. Naderi, F. Jolai and M. Khalili, A variable neighborhood search for job shop scheduling with set-up times to minimize makespan, Future Generation Computer Systems, 25 (2009), 654-661.  doi: 10.1016/j.future.2009.01.004.  Google Scholar

[40]

M. Rostami, M. Bagherpour, M. M. Mazdeh and A. Makui, Resource Pool Location for Periodic Services in Decentralized Multi-Project Scheduling Problems, Journal of Computing in Civil Engineering, 31 (2017), 04017022. doi: 10.1061/(ASCE)CP.1943-5487.0000671.  Google Scholar

[41]

V. Valls, F. Ballestin and M. S. Quintanilla, A hybrid genetic algorithm for the RCPSP, Technical report, Department of Statistics and Operations Research, University of Valencia, 2003. Google Scholar

[42]

V. Valls, F. Ballestin and M. S. Quintanilla, Justification and RCPSP: A technique that pays, European Journal of Operational Research, 165 (2005), 375-386.  doi: 10.1016/j.ejor.2004.04.008.  Google Scholar

[43]

M. Vanhoucke, Setup times and fast tracking in resource-constrained project scheduling, Computers & Industrial Engineering, 54 (2008), 1062-1070.  doi: 10.1016/j.cie.2007.11.008.  Google Scholar

[44]

K. K. Yang and C. C. Sum, A comparison of resource allocation and activity scheduling rules in a dynamic multi-project environment, Journal of Operation Management, 11 (1993), 207-218.  doi: 10.1016/0272-6963(93)90023-I.  Google Scholar

[45]

K. K. Yang and C. C. Sum, An evaluation of due date, resource allocation, project release, and activity scheduling rules in a multi project environment, European Journal of Operational Research, 103 (1997), 139-154.   Google Scholar

Figure 1.  An example of a DRCMPSPTT
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Figure 2.  Renumbered network of the example
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Figure 3.  Individual representation for the DRCMPSPTT
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Figure 4.  A precedence feasible AL of the multi-project example
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Figure 5.  The adaptive serial SGS with transfer times
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Figure 6.  The adaptive parallel SGS with transfer times
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Figure 7.  Two-point crossover operator
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Figure 8.  Average APDs obtained by different schemes
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Figure 9.  (a) APDs obtained by the two schemes, and (b) the deviation of the APDs obtained by the two schemes varies as $ UF_{AV} $ increases
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Figure 10.  (a) The three types PREs, and (b) computation times of the solution methods
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Figure 11.  (a) mean value of the APD obtained by different methods, and (b) The three types PREs, (c) The $ PRE_{_GA} $ of the 12 problem sets, and (d) The change trend of $ PRE_{_GA} $ as $ UF_{AV} $ increases
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Table 1.  Literature review of the project scheduling with transfer 09:49:34
Authors Project Objective Transfer times Algorithm
Single Multi- Decentralized multi- Makespan Others sequence-dependent sequence-independent
Kolisch (1995) $ \surd $ $ \surd $ $ \surd $ Parallel scheme based heuristic
Neumann et al (2003) $ \surd $ $ \surd $ $ \surd $ Branch-and-bound
Vanhoucke(2008) $ \surd $ $ \surd $ $ \surd $ Branch-and-bound
Mika (2008) $ \surd $ $ \surd $ $ \surd $ Tabu search
Afshar-Nadjafi et al (2014) $ \surd $ $ \surd $ $ \surd $ GA
Poppenborg(2016) $ \surd $ $ \surd $ $ \surd $ Tabu search
Kadri and Boctor (2017) $ \surd $ $ \surd $ $ \surd $ GA
Yang and Sum(1993, 1997) $ \surd $ $ \surd $ $ \surd $ Computational experiment
Mittal and Kanda (2009) $ \surd $ $ \surd $ Cost $ \surd $ Heuristics
Kruger and Scholl(2009, 2010) $ \surd $ $ \surd $ $ \surd $ Cost $ \surd $ Priority-rule based heuristic
Cai and Li(2012) $ \surd $ APD $ \surd $ Hybrid GA
Adhau and Mittal(2013) $ \surd $ APD $ \surd $ DMAS/RIA
This research $ \surd $ APD $ \surd $ GA
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Authors Project Objective Transfer times Algorithm
Single Multi- Decentralized multi- Makespan Others sequence-dependent sequence-independent
Kolisch (1995) $ \surd $ $ \surd $ $ \surd $ Parallel scheme based heuristic
Neumann et al (2003) $ \surd $ $ \surd $ $ \surd $ Branch-and-bound
Vanhoucke(2008) $ \surd $ $ \surd $ $ \surd $ Branch-and-bound
Mika (2008) $ \surd $ $ \surd $ $ \surd $ Tabu search
Afshar-Nadjafi et al (2014) $ \surd $ $ \surd $ $ \surd $ GA
Poppenborg(2016) $ \surd $ $ \surd $ $ \surd $ Tabu search
Kadri and Boctor (2017) $ \surd $ $ \surd $ $ \surd $ GA
Yang and Sum(1993, 1997) $ \surd $ $ \surd $ $ \surd $ Computational experiment
Mittal and Kanda (2009) $ \surd $ $ \surd $ Cost $ \surd $ Heuristics
Kruger and Scholl(2009, 2010) $ \surd $ $ \surd $ $ \surd $ Cost $ \surd $ Priority-rule based heuristic
Cai and Li(2012) $ \surd $ APD $ \surd $ Hybrid GA
Adhau and Mittal(2013) $ \surd $ APD $ \surd $ DMAS/RIA
This research $ \surd $ APD $ \surd $ GA
Table 2.  Resource transfer rules
Transfer rules Priority value $ \pi_i $ Extremum
minTT $ \pi_i = \Delta_{ijg} $ min
minGAP $ \pi_i = gap_{ijg} = t - (F_i + \Delta_{ijg}) $ min
maxRS $ \pi_i = v_{ig} = u_{ig} - \sum\limits_{l \in PS} {w_{ilg}} $ max
minRS $ \pi_i = v_{ig} = u_{ig} - \sum\limits_{l \in PS} {w_{ilg}} $ min
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Transfer rules Priority value $ \pi_i $ Extremum
minTT $ \pi_i = \Delta_{ijg} $ min
minGAP $ \pi_i = gap_{ijg} = t - (F_i + \Delta_{ijg}) $ min
maxRS $ \pi_i = v_{ig} = u_{ig} - \sum\limits_{l \in PS} {w_{ilg}} $ max
minRS $ \pi_i = v_{ig} = u_{ig} - \sum\limits_{l \in PS} {w_{ilg}} $ min
Table 3.  The instances for the DRCMPSP
Problem subset NOI Characterization per instance UF$ _{AV} $
$ \left| {P} \right| $ $ \left| J_{p} \right| $ $ (\left| G \right| $; $ \left| L_{p} \right|) $
MP30_2 5 2 30 (1;3)/(2;2)/(3;1) 0.84
MP90_2 5 2 90 (1;3)/(2;2)/(3;1) 0.57
MP120_2 5 2 120 (1;3)/(2;2)/(3;1) 1.31
MP30_5 5 5 30 (1;3)/(2;2)/(3;1) 0.82
MP90_5 5 5 90 (1;3)/(2;2)/(3;1) 0.61
MP120_5 5 5 120 (1;3)/(2;2)/(3;1) 1.32
MP30_10 5 10 30 (1;3)/(2;2)/(3;1) 2.38
MP90_10 5 10 90 (1;3)/(2;2)/(3;1) 1.14
MP120_10 5 10 120 (1;3)/(2;2)/(3;1) 1.91
MP30_20 5 20 30 (1;3)/(2;2)/(3;1) 3.37
MP90_20 5 20 90 (1;3)/(2;2)/(3;1) 0.9
MP120_20 5 20 120 (1;3)/(2;2)/(3;1) 0.87
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Problem subset NOI Characterization per instance UF$ _{AV} $
$ \left| {P} \right| $ $ \left| J_{p} \right| $ $ (\left| G \right| $; $ \left| L_{p} \right|) $
MP30_2 5 2 30 (1;3)/(2;2)/(3;1) 0.84
MP90_2 5 2 90 (1;3)/(2;2)/(3;1) 0.57
MP120_2 5 2 120 (1;3)/(2;2)/(3;1) 1.31
MP30_5 5 5 30 (1;3)/(2;2)/(3;1) 0.82
MP90_5 5 5 90 (1;3)/(2;2)/(3;1) 0.61
MP120_5 5 5 120 (1;3)/(2;2)/(3;1) 1.32
MP30_10 5 10 30 (1;3)/(2;2)/(3;1) 2.38
MP90_10 5 10 90 (1;3)/(2;2)/(3;1) 1.14
MP120_10 5 10 120 (1;3)/(2;2)/(3;1) 1.91
MP30_20 5 20 30 (1;3)/(2;2)/(3;1) 3.37
MP90_20 5 20 90 (1;3)/(2;2)/(3;1) 0.9
MP120_20 5 20 120 (1;3)/(2;2)/(3;1) 0.87
Table 4.  APDs obtained by the adaptive GA with different schemes
Problem subsets Serial parallel
minTT minGAP maxRS minRS minTT minGAP maxRS minRS
MP30_2 14.8 13.7 14.8 16.1 13.9 13.3 13.7 14
MP90_2 10 8.8 10.1 11.3 6.9 6.8 7.1 7.3
MP120_2 86.3 83.6 85.1 92.3 57.2 56.7 56.6 57.2
MP30_5 34.24 34.2 36.44 39.04 23.16 23.28 22.48 23.36
MP90_5 23.96 21.76 22.92 25.44 12.4 12.4 11.76 13
MP120_5 104.04 104.44 106.56 112.64 67.72 68.32 67.56 67.8
MP30_10 131.78 126.4 134.3 145.34 85.1 85.22 83.48 87.62
MP90_10 104.14 106.72 108.96 114.3 61.34 60.4 61.12 62.1
MP120_10 252.43 254.44 256.73 262.54 164.38 166.78 166.84 171.96
MP30_20 333.61 340.69 338.15 352.6 205.06 208.3 203.74 213.16
MP90_20 65.24 65.04 66.76 72.84 44.16 43.95 44.06 44.49
MP120_20 71.28 70.49 71.48 77.56 47.48 48.28 47.24 47.52
Average 102.65 102.52 104.36 110.17 65.73 66.14 65.47 67.46
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Problem subsets Serial parallel
minTT minGAP maxRS minRS minTT minGAP maxRS minRS
MP30_2 14.8 13.7 14.8 16.1 13.9 13.3 13.7 14
MP90_2 10 8.8 10.1 11.3 6.9 6.8 7.1 7.3
MP120_2 86.3 83.6 85.1 92.3 57.2 56.7 56.6 57.2
MP30_5 34.24 34.2 36.44 39.04 23.16 23.28 22.48 23.36
MP90_5 23.96 21.76 22.92 25.44 12.4 12.4 11.76 13
MP120_5 104.04 104.44 106.56 112.64 67.72 68.32 67.56 67.8
MP30_10 131.78 126.4 134.3 145.34 85.1 85.22 83.48 87.62
MP90_10 104.14 106.72 108.96 114.3 61.34 60.4 61.12 62.1
MP120_10 252.43 254.44 256.73 262.54 164.38 166.78 166.84 171.96
MP30_20 333.61 340.69 338.15 352.6 205.06 208.3 203.74 213.16
MP90_20 65.24 65.04 66.76 72.84 44.16 43.95 44.06 44.49
MP120_20 71.28 70.49 71.48 77.56 47.48 48.28 47.24 47.52
Average 102.65 102.52 104.36 110.17 65.73 66.14 65.47 67.46
Table 5.  Parameter configurations to generate test problems
Parameter J P K $ r_j $ $ d_j $ NC RF RS
value 5, 8, 10 or 14 2 or 3 4 $ [1,8] $ $ [1,5] $ 1.5 0.25 0.2
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Parameter J P K $ r_j $ $ d_j $ NC RF RS
value 5, 8, 10 or 14 2 or 3 4 $ [1,8] $ $ [1,5] $ 1.5 0.25 0.2
Table 6.  Results of the two methods
instances CPLEX GA_maxRS (100 solutions) $ PRE_w $
(%)		 $ PRE_b $
(%)		 $ PRE_m $
(%)
opt $ ct_{ave} (s) $ worst best mean $ ct_{ave} (s) $
J5_2 1 3 0.08 3 3 3 0.11 0 0 0
2 1.5 0.03 1.5 1.5 1.5 0.14 0 0 0
3 2 0.05 2 2 2 0.1 0 0 0
4 5 0.08 5 5 5 0.15 0 0 0
5 2 0.05 2 2 2 0.13 0 0 0
J8_2 1 0.5 0.06 0.5 0.5 0.5 0.11 0 0 0
2 1 0.09 1 1 1 0.17 0 0 0
3 2.5 0.08 2.5 2.5 2.5 0.2 0 0 0
4 5 0.3 5 5 5 0.24 0 0 0
5 3 0.2 3 3 3 0.31 0 0 0
J10_2 1 2.5 2.11 2.5 2.5 2.5 0.94 0 0 0
2 9 3.72 9.5 9 9.1 0.87 6 0 1
3 10.5 2.23 12 10.5 11 1.06 14 0 5
4 7.5 3.2 7.5 7.5 7.5 0.56 0 0 0
5 7 4.83 7 7 7 1.42 0 0 0
J14_2 1 2 0.94 2.5 2 2.2 0.89 25 0 10
2 6 25.23 6 6 6 0.99 0 0 0
3 4.5 1.91 5.5 4.5 4.7 1.18 22 0 4
4 7.5 30.42 7.5 7.5 7.5 1.5 0 0 0
5 4 15.2 6 4 5.5 1.45 50 0 38
J5_3 1 4.33 0.23 4.33 4.33 4.33 0.44 0 0 0
2 2.67 0.63 3.67 3.33 3.53 0.82 37 25 32
3 4 0.86 5.33 4 4.67 0.61 33 0 17
4 3.33 0.19 4.33 3.33 3.94 0.45 30 0 18
5 3.33 0.7 3.33 3.33 3.33 0.63 0 0 0
J8_3 1 2 0.5 2 2 2 0.75 0 0 0
2 7.67 15.69 8.67 7.67 7.94 1.18 13 0 4
3 2 0.22 2 2 2 0.58 0 0 0
4 6 152.2 8.33 6 6.47 1.85 39 0 8
5 1.33 0.41 1.33 1.33 1.33 0.66 0 0 0
J10_3 1 8.67 157.94 9.33 8.76 9.01 1.86 8 1 4
2 10.67 249.08 11 10.67 10.79 1.67 3 0 1
3 4.33 320.17 5.67 4.33 4.79 1.27 31 0 11
4 8 824.63 8 8 8 1.28 0 0 0
5 6 1832.06 6 6 6 1.06 0 0 0
J14_3 1 6.33 131.98 6.33 6.33 6.33 2.14 0 0 0
2 3.33 411.58 5 3.33 3.79 2.93 50 0 14
3 5.67 3600.44 6.67 5.67 5.79 3.03 18 0 2
4 11 3600.13 13.67 11 12.1 7.33 24 0 10
5 5 450.97 7 5 5.4 3.01 40 0 8
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instances CPLEX GA_maxRS (100 solutions) $ PRE_w $
(%)		 $ PRE_b $
(%)		 $ PRE_m $
(%)
opt $ ct_{ave} (s) $ worst best mean $ ct_{ave} (s) $
J5_2 1 3 0.08 3 3 3 0.11 0 0 0
2 1.5 0.03 1.5 1.5 1.5 0.14 0 0 0
3 2 0.05 2 2 2 0.1 0 0 0
4 5 0.08 5 5 5 0.15 0 0 0
5 2 0.05 2 2 2 0.13 0 0 0
J8_2 1 0.5 0.06 0.5 0.5 0.5 0.11 0 0 0
2 1 0.09 1 1 1 0.17 0 0 0
3 2.5 0.08 2.5 2.5 2.5 0.2 0 0 0
4 5 0.3 5 5 5 0.24 0 0 0
5 3 0.2 3 3 3 0.31 0 0 0
J10_2 1 2.5 2.11 2.5 2.5 2.5 0.94 0 0 0
2 9 3.72 9.5 9 9.1 0.87 6 0 1
3 10.5 2.23 12 10.5 11 1.06 14 0 5
4 7.5 3.2 7.5 7.5 7.5 0.56 0 0 0
5 7 4.83 7 7 7 1.42 0 0 0
J14_2 1 2 0.94 2.5 2 2.2 0.89 25 0 10
2 6 25.23 6 6 6 0.99 0 0 0
3 4.5 1.91 5.5 4.5 4.7 1.18 22 0 4
4 7.5 30.42 7.5 7.5 7.5 1.5 0 0 0
5 4 15.2 6 4 5.5 1.45 50 0 38
J5_3 1 4.33 0.23 4.33 4.33 4.33 0.44 0 0 0
2 2.67 0.63 3.67 3.33 3.53 0.82 37 25 32
3 4 0.86 5.33 4 4.67 0.61 33 0 17
4 3.33 0.19 4.33 3.33 3.94 0.45 30 0 18
5 3.33 0.7 3.33 3.33 3.33 0.63 0 0 0
J8_3 1 2 0.5 2 2 2 0.75 0 0 0
2 7.67 15.69 8.67 7.67 7.94 1.18 13 0 4
3 2 0.22 2 2 2 0.58 0 0 0
4 6 152.2 8.33 6 6.47 1.85 39 0 8
5 1.33 0.41 1.33 1.33 1.33 0.66 0 0 0
J10_3 1 8.67 157.94 9.33 8.76 9.01 1.86 8 1 4
2 10.67 249.08 11 10.67 10.79 1.67 3 0 1
3 4.33 320.17 5.67 4.33 4.79 1.27 31 0 11
4 8 824.63 8 8 8 1.28 0 0 0
5 6 1832.06 6 6 6 1.06 0 0 0
J14_3 1 6.33 131.98 6.33 6.33 6.33 2.14 0 0 0
2 3.33 411.58 5 3.33 3.79 2.93 50 0 14
3 5.67 3600.44 6.67 5.67 5.79 3.03 18 0 2
4 11 3600.13 13.67 11 12.1 7.33 24 0 10
5 5 450.97 7 5 5.4 3.01 40 0 8
Table 7.  Results of GA_maxRS and DMAS/RIA
Problem subsets GA_maxRS DMAS/RIA $ PRE_{_A} $
(%)
with TT without TT $ PRE_{_GA} $
(%)		 with TT without TT $ PRE_{_D} $
(%)
MP30_2 13 10.9 19 23.3 18 29 -39
MP90_2 6.9 6.7 3 17.1 11.8 45 -43
MP120_2 56.1 55.6 1 89.6 80.3 12 -31
MP30_5 22.2 18.08 23 35.28 21.64 63 -16
MP90_5 11.68 9.36 25 22 13.32 65 -30
MP120_5 66.84 65.48 2 86.12 77.68 11 -16
MP30_10 81.58 78.16 4 92.68 83 12 -6
MP90_10 59.34 50.36 18 62.66 61.52 2 -18
MP120_10 164.28 140.3 17 162.6 144.1 13 -3
MP30_20 199.63 189.6 5 206.9 193 7 -2
MP90_20 43.38 34.92 24 42.88 43.13 -1 -19
MP120_20 46.42 39.71 17 48.09 42.11 14 -6
Average 64.28 58.26 10 74.1 65.8 13 -11
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Problem subsets GA_maxRS DMAS/RIA $ PRE_{_A} $
(%)
with TT without TT $ PRE_{_GA} $
(%)		 with TT without TT $ PRE_{_D} $
(%)
MP30_2 13 10.9 19 23.3 18 29 -39
MP90_2 6.9 6.7 3 17.1 11.8 45 -43
MP120_2 56.1 55.6 1 89.6 80.3 12 -31
MP30_5 22.2 18.08 23 35.28 21.64 63 -16
MP90_5 11.68 9.36 25 22 13.32 65 -30
MP120_5 66.84 65.48 2 86.12 77.68 11 -16
MP30_10 81.58 78.16 4 92.68 83 12 -6
MP90_10 59.34 50.36 18 62.66 61.52 2 -18
MP120_10 164.28 140.3 17 162.6 144.1 13 -3
MP30_20 199.63 189.6 5 206.9 193 7 -2
MP90_20 43.38 34.92 24 42.88 43.13 -1 -19
MP120_20 46.42 39.71 17 48.09 42.11 14 -6
Average 64.28 58.26 10 74.1 65.8 13 -11
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