American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jimo.2020077

Tabu search and simulated annealing for resource-constrained multi-project scheduling to minimize maximal cash flow gap

Yukang He 1, , Zhengwen He 2, and  Nengmin Wang 2,,

1. 

School of Engineering and Applied Science, Aston University, Birmingham, B4 7ET, United Kingdom
2. 

School of management, Xi'an Jiaotong University, Xi'an 710049, China

* Corresponding author: Nengmin Wang

Received  May 2019 Revised  November 2019 Published  April 2020

Fund Project: The second author is supported by the National Natural Science Foundation of China grant 71871176. The third author is supported by the National Natural Science Foundation of China grants 71732006 and 71572138

In reality, a contractor may implement multiple projects simultaneously and in such an environment, how to achieve a positive balance between cash outflow and inflow by scheduling is an important problem for the contractor has to tackle. For this fact, this paper investigates a resource-constrained multi-project scheduling problem with the objective of minimizing the contractor's maximal cash flow gap under the constraint of a project deadline and renewable resource. In the paper, we construct a non-linear integer programming optimization model for the studied problem at first. Then, for the NP-hardness of the problem, we design three metaheuristic algorithms to solve the model: tabu search (TS), simulated annealing (SA), and an algorithm comprising both TS and SA (SA-TS). Finally, we conduct a computational experiment on a data set coming from existing literature to evaluate the performance of the developed algorithms and analyze the effects of key parameters on the objective function. Based on the computational results, the following conclusions are drawn: Among the designed algorithms, the SA-TS with an improvement measure is the most promising for solving the problem under study. Some parameters may exert an important effect on the contractor's maximal cash flow gap.

Keywords: Multi-project scheduling, optimization model, maximal cash flow gap, tabu search, simulated annealing.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Yukang He, Zhengwen He, Nengmin Wang. Tabu search and simulated annealing for resource-constrained multi-project scheduling to minimize maximal cash flow gap. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020077
References:
[1]

A. Alghazi, A. Elazouni and S. Selim, Improved genetic algorithm for finance-based scheduling, J. Comput. Civil Engineering, 27 (2013), 379-394.  doi: 10.1061/(ASCE)CP.1943-5487.0000227.  Google Scholar

[2]

M. M. Ali and A. Elazouni, Finance-based CPM/LOB scheduling of projects with repetitive non-serial activities, Construction Management Economics, 27 (2009), 839-856.  doi: 10.1080/01446190903191764.  Google Scholar

[3]

M. Abido and A. Elazouni, Multiobjective evolutionary finance-based scheduling: Entire projects' portfolio, J. Comput. Civil Engineering, 25 (2011), 85-97.  doi: 10.1061/(ASCE)CP.1943-5487.0000070.  Google Scholar

[4]

S. T. Al-Shihabi and M. M. AlDurgam, A max-min ant system for the finance-based scheduling problem, Comput. Industrial Engineering, 110 (2017), 264-276.  doi: 10.1016/j.cie.2017.06.016.  Google Scholar

[5]

J. Blazewicz, J. K. Lenstra and K. A. H. G. Rinnooy, Scheduling subject to resource constraints: Classification and complexity, Discrete Appl. Math., 5 (1983), 11-24.  doi: 10.1016/0166-218X(83)90012-4.  Google Scholar

[6]

T. R. Browning and A. A. Yassine, A random generator of resource-constrained multi-project network problems, J. Scheduling, 13 (2010), 143-161.  doi: 10.1007/s10951-009-0131-y.  Google Scholar

[7]

T. R. Browning and A. A. Yassine, Resource-constrained multi-project scheduling: Priority rule performance revised, Internat. J. Production Economics, 126 (2010), 212-228.  doi: 10.1016/j.ijpe.2010.03.009.  Google Scholar

[8]

R. H. Doersch and J. H. Patterson, Scheduling a project to maximize its present value: A zero-one programming approach, Management Science, 23 (1977), 882-889.  doi: 10.1287/mnsc.23.8.882.  Google Scholar

[9]

M. Engwall and A. Jerbrant, The resource allocation syndrome: The prime challenge of multi-project management?, Internat. J. Project Management, 21 (2003), 403-409.  doi: 10.1016/S0263-7863(02)00113-8.  Google Scholar

[10]

A. M. Elazouni and A. A. Gab-Allah, Finance-based scheduling of construction projects using integer programming, J. Construction Engineering Management, 130 (2004), 15-24.  doi: 10.1061/(ASCE)0733-9364(2004)130:1(15).  Google Scholar

[11]

A. Elazouni, A. Alghazi and S. Selim, Finance-based scheduling using meta-heuristics: Discrete versus continuous optimization problems, J. Finance Management Property Construction, 20 (2015), 85-104.  doi: 10.1108/JFMPC-07-2014-0013.  Google Scholar

[12]

A. Elazouni, Heuristic method for multi-project finance-based scheduling, Construction Management Economics, 27 (2009), 199-211.  doi: 10.1080/01446190802673110.  Google Scholar

[13]

A. Elazouni and M. Abido, Multiobjective evolutionary finance-based scheduling: Individual projects within a portfolio, Automat. Construction, 20 (2011), 755-766.  doi: 10.1016/j.autcon.2011.03.010.  Google Scholar

[14]

M. S. El-Abbasy, A. Elazouni and T. Z. F. ASCE, Generic scheduling optimization model for multiple construction projects, J. Comput. Civil Engineering, 31 (2017). doi: 10.1061/(ASCE)CP.1943-5487.0000659.  Google Scholar

[15]

H. Fathi and A. Afshar, GA-based multi-objective optimization of finance-based construction project scheduling, KSCE J. Civil Engineering, 14 (2010), 627-638.  doi: 10.1007/s12205-010-0849-2.  Google Scholar

[16]

F. Glover, Future path for integer programming and links to artificial intelligence, Comput. Oper. Res., 13 (1986), 533-549.  doi: 10.1016/0305-0548(86)90048-1.  Google Scholar

[17]

W. S. Herroelen, P. Dommelen and E. L. Demeulemeester, Project network models with discounted cash flows: A guided tour through recent developments, European J. Oper. Res., 100 (1997), 97-121.  doi: 10.1016/S0377-2217(96)00112-9.  Google Scholar

[18]

Z. He, R. Liu and T. Jia, Metaheuristics for multi-mode capital-constrained project payment scheduling, European J. Oper. Res., 223 (2012), 605-613.  doi: 10.1016/j.ejor.2012.07.014.  Google Scholar

[19]

Z. He, H. He, R. Liu and N. Wang, Variable neighbourhood search and tabu search for a discrete time/cost trade-off problem to minimize the maximal cash flow gap, Comput. Oper. Res., 78 (2017), 564-577.  doi: 10.1016/j.cor.2016.07.013.  Google Scholar

[20]

A. Jiang, R. R. A. Issa and M. Malek, Construction project cash flow planning using the Pareto optimality efficiency network model, J. Construction Engineering Management, 17 (2011), 510-519.  doi: 10.3846/13923730.2011.604537.  Google Scholar

[21]

P. Leyman and M. Vanhoucke, A new scheduling technique for the resource–constrained project scheduling problem with discounted cash flows, Internat. J. Prod. Res., 53 (2015), 2771-2786.  doi: 10.1080/00207543.2014.980463.  Google Scholar

[22]

P. Leyman and M. Vanhoucke, Payment models and net present value optimization for resource-constrained project scheduling, Comput. Industrial Engineering, 91 (2016), 139-153.  doi: 10.1016/j.cie.2015.11.008.  Google Scholar

[23]

P. Leyman and M. Vanhoucke, Capital- and resource-constrained project scheduling with net present value optimization, European J. Oper. Res., 256 (2017), 757-776.  doi: 10.1016/j.ejor.2016.07.019.  Google Scholar

[24]

S. S. Liu and C. J. Wang, Profit optimization for multiproject scheduling problems considering cash flow, J. Construction Engineering Management, 136 (2010), 1268-1278.  doi: 10.1061/(ASCE)CO.1943-7862.0000235.  Google Scholar

[25]

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, Equation of state calculations by fast computing machines, J. Chemical Physics, 21 (1953), 1087-1092.  doi: 10.2172/4390578.  Google Scholar

[26]

M. Ning, Z. He, T. Jia and N. Wang, Metaheuristics for multi-mode cash flow balanced project scheduling with stochastic duration of activities, Automat. Construction, 81 (2017), 224-233.  doi: 10.1016/j.autcon.2017.06.011.  Google Scholar

[27]

M. Ning, Z. He, N. Wang and R. Liu, Metaheuristic algorithms for proactive and reactive project scheduling to minimize contractor's cash flow gap under random activity duration, IEEE Access, 6 (2018), 30547-30558.  doi: 10.1109/ACCESS.2018.2828037.  Google Scholar

[28]

L. Özdamar and H. Dündar, A flexible heuristic for a multi-mode capital constrained project scheduling problem with probabilistic cash inflows, Comput. Opera. Res., 24 (1997), 1187-1200.  doi: 10.1016/S0305-0548(96)00058-5.  Google Scholar

[29]

L. Özdamar, On scheduling project activities with variable expenditure rates, IIE Transactions, 30 (1998), 695-704.  doi: 10.1023/A:1007598405238.  Google Scholar

[30]

C. Schwindt and J. Zimmermann, Handbook of Project Management and Scheduling, Springer International Publishing AG, Berlin, 2014. doi: 10.1007/978-3-319-05443-8.  Google Scholar

[31]

D. E. Smith-Daniels and V. L. Smith-Daniels, Maximizing the net present value of a project subject to materials and capital constraints, J. Oper. Management, 7 (1987), 33-45.  doi: 10.1016/0272-6963(87)90005-2.  Google Scholar

[32]

D. E. Smith-Daniels, R. Padman and V. L. Smith-Daniels, Heuristic scheduling of capital constrained projects, J. Oper. Management, 14 (1996), 241-254.  doi: 10.1016/0272-6963(96)00004-6.  Google Scholar

show all references

References:
[1]

A. Alghazi, A. Elazouni and S. Selim, Improved genetic algorithm for finance-based scheduling, J. Comput. Civil Engineering, 27 (2013), 379-394.  doi: 10.1061/(ASCE)CP.1943-5487.0000227.  Google Scholar

[2]

M. M. Ali and A. Elazouni, Finance-based CPM/LOB scheduling of projects with repetitive non-serial activities, Construction Management Economics, 27 (2009), 839-856.  doi: 10.1080/01446190903191764.  Google Scholar

[3]

M. Abido and A. Elazouni, Multiobjective evolutionary finance-based scheduling: Entire projects' portfolio, J. Comput. Civil Engineering, 25 (2011), 85-97.  doi: 10.1061/(ASCE)CP.1943-5487.0000070.  Google Scholar

[4]

S. T. Al-Shihabi and M. M. AlDurgam, A max-min ant system for the finance-based scheduling problem, Comput. Industrial Engineering, 110 (2017), 264-276.  doi: 10.1016/j.cie.2017.06.016.  Google Scholar

[5]

J. Blazewicz, J. K. Lenstra and K. A. H. G. Rinnooy, Scheduling subject to resource constraints: Classification and complexity, Discrete Appl. Math., 5 (1983), 11-24.  doi: 10.1016/0166-218X(83)90012-4.  Google Scholar

[6]

T. R. Browning and A. A. Yassine, A random generator of resource-constrained multi-project network problems, J. Scheduling, 13 (2010), 143-161.  doi: 10.1007/s10951-009-0131-y.  Google Scholar

[7]

T. R. Browning and A. A. Yassine, Resource-constrained multi-project scheduling: Priority rule performance revised, Internat. J. Production Economics, 126 (2010), 212-228.  doi: 10.1016/j.ijpe.2010.03.009.  Google Scholar

[8]

R. H. Doersch and J. H. Patterson, Scheduling a project to maximize its present value: A zero-one programming approach, Management Science, 23 (1977), 882-889.  doi: 10.1287/mnsc.23.8.882.  Google Scholar

[9]

M. Engwall and A. Jerbrant, The resource allocation syndrome: The prime challenge of multi-project management?, Internat. J. Project Management, 21 (2003), 403-409.  doi: 10.1016/S0263-7863(02)00113-8.  Google Scholar

[10]

A. M. Elazouni and A. A. Gab-Allah, Finance-based scheduling of construction projects using integer programming, J. Construction Engineering Management, 130 (2004), 15-24.  doi: 10.1061/(ASCE)0733-9364(2004)130:1(15).  Google Scholar

[11]

A. Elazouni, A. Alghazi and S. Selim, Finance-based scheduling using meta-heuristics: Discrete versus continuous optimization problems, J. Finance Management Property Construction, 20 (2015), 85-104.  doi: 10.1108/JFMPC-07-2014-0013.  Google Scholar

[12]

A. Elazouni, Heuristic method for multi-project finance-based scheduling, Construction Management Economics, 27 (2009), 199-211.  doi: 10.1080/01446190802673110.  Google Scholar

[13]

A. Elazouni and M. Abido, Multiobjective evolutionary finance-based scheduling: Individual projects within a portfolio, Automat. Construction, 20 (2011), 755-766.  doi: 10.1016/j.autcon.2011.03.010.  Google Scholar

[14]

M. S. El-Abbasy, A. Elazouni and T. Z. F. ASCE, Generic scheduling optimization model for multiple construction projects, J. Comput. Civil Engineering, 31 (2017). doi: 10.1061/(ASCE)CP.1943-5487.0000659.  Google Scholar

[15]

H. Fathi and A. Afshar, GA-based multi-objective optimization of finance-based construction project scheduling, KSCE J. Civil Engineering, 14 (2010), 627-638.  doi: 10.1007/s12205-010-0849-2.  Google Scholar

[16]

F. Glover, Future path for integer programming and links to artificial intelligence, Comput. Oper. Res., 13 (1986), 533-549.  doi: 10.1016/0305-0548(86)90048-1.  Google Scholar

[17]

W. S. Herroelen, P. Dommelen and E. L. Demeulemeester, Project network models with discounted cash flows: A guided tour through recent developments, European J. Oper. Res., 100 (1997), 97-121.  doi: 10.1016/S0377-2217(96)00112-9.  Google Scholar

[18]

Z. He, R. Liu and T. Jia, Metaheuristics for multi-mode capital-constrained project payment scheduling, European J. Oper. Res., 223 (2012), 605-613.  doi: 10.1016/j.ejor.2012.07.014.  Google Scholar

[19]

Z. He, H. He, R. Liu and N. Wang, Variable neighbourhood search and tabu search for a discrete time/cost trade-off problem to minimize the maximal cash flow gap, Comput. Oper. Res., 78 (2017), 564-577.  doi: 10.1016/j.cor.2016.07.013.  Google Scholar

[20]

A. Jiang, R. R. A. Issa and M. Malek, Construction project cash flow planning using the Pareto optimality efficiency network model, J. Construction Engineering Management, 17 (2011), 510-519.  doi: 10.3846/13923730.2011.604537.  Google Scholar

[21]

P. Leyman and M. Vanhoucke, A new scheduling technique for the resource–constrained project scheduling problem with discounted cash flows, Internat. J. Prod. Res., 53 (2015), 2771-2786.  doi: 10.1080/00207543.2014.980463.  Google Scholar

[22]

P. Leyman and M. Vanhoucke, Payment models and net present value optimization for resource-constrained project scheduling, Comput. Industrial Engineering, 91 (2016), 139-153.  doi: 10.1016/j.cie.2015.11.008.  Google Scholar

[23]

P. Leyman and M. Vanhoucke, Capital- and resource-constrained project scheduling with net present value optimization, European J. Oper. Res., 256 (2017), 757-776.  doi: 10.1016/j.ejor.2016.07.019.  Google Scholar

[24]

S. S. Liu and C. J. Wang, Profit optimization for multiproject scheduling problems considering cash flow, J. Construction Engineering Management, 136 (2010), 1268-1278.  doi: 10.1061/(ASCE)CO.1943-7862.0000235.  Google Scholar

[25]

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, Equation of state calculations by fast computing machines, J. Chemical Physics, 21 (1953), 1087-1092.  doi: 10.2172/4390578.  Google Scholar

[26]

M. Ning, Z. He, T. Jia and N. Wang, Metaheuristics for multi-mode cash flow balanced project scheduling with stochastic duration of activities, Automat. Construction, 81 (2017), 224-233.  doi: 10.1016/j.autcon.2017.06.011.  Google Scholar

[27]

M. Ning, Z. He, N. Wang and R. Liu, Metaheuristic algorithms for proactive and reactive project scheduling to minimize contractor's cash flow gap under random activity duration, IEEE Access, 6 (2018), 30547-30558.  doi: 10.1109/ACCESS.2018.2828037.  Google Scholar

[28]

L. Özdamar and H. Dündar, A flexible heuristic for a multi-mode capital constrained project scheduling problem with probabilistic cash inflows, Comput. Opera. Res., 24 (1997), 1187-1200.  doi: 10.1016/S0305-0548(96)00058-5.  Google Scholar

[29]

L. Özdamar, On scheduling project activities with variable expenditure rates, IIE Transactions, 30 (1998), 695-704.  doi: 10.1023/A:1007598405238.  Google Scholar

[30]

C. Schwindt and J. Zimmermann, Handbook of Project Management and Scheduling, Springer International Publishing AG, Berlin, 2014. doi: 10.1007/978-3-319-05443-8.  Google Scholar

[31]

D. E. Smith-Daniels and V. L. Smith-Daniels, Maximizing the net present value of a project subject to materials and capital constraints, J. Oper. Management, 7 (1987), 33-45.  doi: 10.1016/0272-6963(87)90005-2.  Google Scholar

[32]

D. E. Smith-Daniels, R. Padman and V. L. Smith-Daniels, Heuristic scheduling of capital constrained projects, J. Oper. Management, 14 (1996), 241-254.  doi: 10.1016/0272-6963(96)00004-6.  Google Scholar

Figure 1.  A Numerical Example
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Figure 2.  The Generated Starting Solution
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Figure 3.  Process for Updating Current Solution by TS
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Figure 4.  Operation on a Forbidden Move to Generate a Solution Better than $ S^{\rm best} $
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Figure 5.  Experimental Design
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Figure 6.  Performance of Algorithms
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Figure 7.  Effects of Improvement Measure
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Figure 8.  Effects of Parameters on Objective Function Value
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Figure 9.  Interactive Effects of Parameters on Objective Function Value
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Table 1.  Summary of Reviewed Literature
The positive cash flow balance is taken as a constraint The positive cash flow balance is taken as an objective
The objective is to maximize project profit The objective is to minimize project duration The objective is the optimal trade-off among multiple objectives The activity durations are constants The activity durations are stochastic variables
A contractor needs to implement a single project Doersch and Patterson ([8]); Smith-Daniels and Smith-Daniels ([31]); Smith-Daniels et al. ([32]); Özdamar and Dündar ([28]); Özdamar ([29]); He et al. ([18]); Leyman and Vanhoucke ([21]); Leyman and Vanhoucke ([22]); Leyman and Vanhoucke ([23]) Elazouni and Gab-Allah ([10]); Alghazi et al. ([1]); Ali and Elazouni ([2]); Elazouni et al. ([11]); Al-Shihabi and AlDurgam ([4]) Fathi and Afshar ([15]) He et al. ([19]) Ning et al. ([26]); Ning et al. ([27])
A contractor needs to implement multiple projects concurrently Liu and Wang ([24]) Elazouni ([12]) Elazouni and Abido ([13]); Abido and Elazouni ([3]); El-Abbasy et al. ([14]) This paper
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The positive cash flow balance is taken as a constraint The positive cash flow balance is taken as an objective
The objective is to maximize project profit The objective is to minimize project duration The objective is the optimal trade-off among multiple objectives The activity durations are constants The activity durations are stochastic variables
A contractor needs to implement a single project Doersch and Patterson ([8]); Smith-Daniels and Smith-Daniels ([31]); Smith-Daniels et al. ([32]); Özdamar and Dündar ([28]); Özdamar ([29]); He et al. ([18]); Leyman and Vanhoucke ([21]); Leyman and Vanhoucke ([22]); Leyman and Vanhoucke ([23]) Elazouni and Gab-Allah ([10]); Alghazi et al. ([1]); Ali and Elazouni ([2]); Elazouni et al. ([11]); Al-Shihabi and AlDurgam ([4]) Fathi and Afshar ([15]) He et al. ([19]) Ning et al. ([26]); Ning et al. ([27])
A contractor needs to implement multiple projects concurrently Liu and Wang ([24]) Elazouni ([12]) Elazouni and Abido ([13]); Abido and Elazouni ([3]); El-Abbasy et al. ([14]) This paper
Table 2.  Cash Flows under the $ S^{\rm star} $ and $ S^{\rm earl} $
Project 1 Project 2
$ t $ Cash outflow Cash inflow Cash outflow Cash inflow $ ACO_t $ $ ACI_t $ $ G_t $
Cash flows under the $ S^{\rm star} $ ($ G_{\rm max} $ = 8)
0 4 / / / 4 0 4
2 $ / $ $ / $ 4 $ / $ 8 0 8
3 2 5.4 $ / $ $ / $ 10 5.4 4.6
4 2 $ / $ 5 5.1 17 10.5 6.5
5 3 5.4 $ / $ $ / $ 20 15.9 4.1
7 $ / $ $ / $ $ / $ 2.55 20 18.45 1.55
9 $ / $ 6.2 $ / $ $ / $ 20 24.65 –4.65
10 $ / $ $ / $ $ / $ 6.35 20 31 –11
Cash flows under the $ S^{\rm earl} $ ($ G_{\rm max} $ = 13)
0 6 $ / $ $ / $ $ / $ 6 0 6
2 $ / $ $ / $ 7 $ / $ 13 0 13
3 2 8.1 $ / $ $ / $ 15 8.1 6.9
4 $ / $ $ / $ 2 5.1 17 13.2 3.8
5 3 2.7 $ / $ $ / $ 20 15.9 4.1
7 $ / $ $ / $ $ / $ 2.55 20 18.45 1.55
8 $ / $ $ / $ $ / $ 6.35 20 24.8 –4.8
9 $ / $ 6.2 $ / $ $ / $ 20 31 –11
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Project 1 Project 2
$ t $ Cash outflow Cash inflow Cash outflow Cash inflow $ ACO_t $ $ ACI_t $ $ G_t $
Cash flows under the $ S^{\rm star} $ ($ G_{\rm max} $ = 8)
0 4 / / / 4 0 4
2 $ / $ $ / $ 4 $ / $ 8 0 8
3 2 5.4 $ / $ $ / $ 10 5.4 4.6
4 2 $ / $ 5 5.1 17 10.5 6.5
5 3 5.4 $ / $ $ / $ 20 15.9 4.1
7 $ / $ $ / $ $ / $ 2.55 20 18.45 1.55
9 $ / $ 6.2 $ / $ $ / $ 20 24.65 –4.65
10 $ / $ $ / $ $ / $ 6.35 20 31 –11
Cash flows under the $ S^{\rm earl} $ ($ G_{\rm max} $ = 13)
0 6 $ / $ $ / $ $ / $ 6 0 6
2 $ / $ $ / $ 7 $ / $ 13 0 13
3 2 8.1 $ / $ $ / $ 15 8.1 6.9
4 $ / $ $ / $ 2 5.1 17 13.2 3.8
5 3 2.7 $ / $ $ / $ 20 15.9 4.1
7 $ / $ $ / $ $ / $ 2.55 20 18.45 1.55
8 $ / $ $ / $ $ / $ 6.35 20 24.8 –4.8
9 $ / $ 6.2 $ / $ $ / $ 20 31 –11
Table 3.  Parameter Settings
Parameter Setting
Number of projects, $ H $ 3
Number of non-dummy activities in projects, $ n^h $–2 20
Network complexity of multiple projects, $ C $ LLL, HLL, HHL, HHH, where "L" and "H" represent the network complexity of an individual project. "L" means that the network complexity of the project equals 0.14 while "H" implies it is 0.69
Number of resource types, $ K $ 4
Normalized average resource loading factor, $ NARLF $ –2, 0, 2
Modified average utilization factor, $ MAUF $ 0.8, 1.0, 1.2
Variance in $ MAUF $s of different resource, $ \sigma^2_{MAUF} $ 0, 0.25
Cost of activities, $ c_i $ Randomly selected from U[1, 9]
Earned value of activities, $ v_i $ $ \rho_v\cdot c_i $, where $ \rho_v $, which is a special parameter defined for generating $ v_i $, is randomly selected from U[1.3, 1.5]
Number of milestone activities, $ M^h $ 4, 5, 6, where the dummy end activity must be a milestone activity while other milestone activities are randomly selected from all the non-dummy activities
Compensation proportion of projects, $ \theta^h $ 0.7, 0.8, 0.9
Earliest start time of projects, $ EST^h $ Randomly selected from U[1, 5]
Deadline of projects, $ D^h $ 1.1$ \cdot CPL $, 1.3$ \cdot CPL $, 1.5$ \cdot CPL $, where $ CPL $ is the critical path length of the project network without the consideration of renewable resource constraints
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Parameter Setting
Number of projects, $ H $ 3
Number of non-dummy activities in projects, $ n^h $–2 20
Network complexity of multiple projects, $ C $ LLL, HLL, HHL, HHH, where "L" and "H" represent the network complexity of an individual project. "L" means that the network complexity of the project equals 0.14 while "H" implies it is 0.69
Number of resource types, $ K $ 4
Normalized average resource loading factor, $ NARLF $ –2, 0, 2
Modified average utilization factor, $ MAUF $ 0.8, 1.0, 1.2
Variance in $ MAUF $s of different resource, $ \sigma^2_{MAUF} $ 0, 0.25
Cost of activities, $ c_i $ Randomly selected from U[1, 9]
Earned value of activities, $ v_i $ $ \rho_v\cdot c_i $, where $ \rho_v $, which is a special parameter defined for generating $ v_i $, is randomly selected from U[1.3, 1.5]
Number of milestone activities, $ M^h $ 4, 5, 6, where the dummy end activity must be a milestone activity while other milestone activities are randomly selected from all the non-dummy activities
Compensation proportion of projects, $ \theta^h $ 0.7, 0.8, 0.9
Earliest start time of projects, $ EST^h $ Randomly selected from U[1, 5]
Deadline of projects, $ D^h $ 1.1$ \cdot CPL $, 1.3$ \cdot CPL $, 1.5$ \cdot CPL $, where $ CPL $ is the critical path length of the project network without the consideration of renewable resource constraints
Table 4.  $ ARP $(%) of Algorithms under Different Values of Parameters
Parameter Value $ \rm TS^{NIM} $ $ \rm TS^{IM} $ $ \rm SA^{NIM} $ $ \rm SA^{IM} $ SA-$ \rm TS^{NIM} $ SA-$ \rm TS^{IM} $
$ C $ LLL 8.26 3.30 7.36 2.18 6.23 1.46
HLL 7.57 2.63 7.15 2.04 6.10 1.35
HHL 7.13 2.47 6.84 1.91 5.41 1.40
HHH 6.28 1.77 5.97 1.67 5.13 1.23
$ NARLF $ –2 7.06 2.36 6.62 1.80 5.55 1.24
0 7.38 2.61 6.70 1.90 5.75 1.39
2 7.49 2.65 7.18 2.16 5.86 1.45
$ MAUF $ 0.8 8.20 3.00 7.57 2.46 6.23 1.52
1.0 7.33 2.70 6.74 1.88 5.67 1.35
1.2 6.41 1.93 6.18 1.51 5.26 1.20
$ \sigma^2_{MAUF} $ 0 7.13 2.42 6.71 1.89 5.54 1.19
0.25 7.49 2.67 6.94 2.01 5.90 1.53
$ D^h $ 1.1$ \cdot CPL $ 6.05 1.84 5.85 1.54 5.07 1.13
1.3$ \cdot CPL $ 7.25 2.45 6.76 1.96 5.61 1.35
1.5$ \cdot CPL $ 8.63 3.33 7.89 2.35 6.47 1.60
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Parameter Value $ \rm TS^{NIM} $ $ \rm TS^{IM} $ $ \rm SA^{NIM} $ $ \rm SA^{IM} $ SA-$ \rm TS^{NIM} $ SA-$ \rm TS^{IM} $
$ C $ LLL 8.26 3.30 7.36 2.18 6.23 1.46
HLL 7.57 2.63 7.15 2.04 6.10 1.35
HHL 7.13 2.47 6.84 1.91 5.41 1.40
HHH 6.28 1.77 5.97 1.67 5.13 1.23
$ NARLF $ –2 7.06 2.36 6.62 1.80 5.55 1.24
0 7.38 2.61 6.70 1.90 5.75 1.39
2 7.49 2.65 7.18 2.16 5.86 1.45
$ MAUF $ 0.8 8.20 3.00 7.57 2.46 6.23 1.52
1.0 7.33 2.70 6.74 1.88 5.67 1.35
1.2 6.41 1.93 6.18 1.51 5.26 1.20
$ \sigma^2_{MAUF} $ 0 7.13 2.42 6.71 1.89 5.54 1.19
0.25 7.49 2.67 6.94 2.01 5.90 1.53
$ D^h $ 1.1$ \cdot CPL $ 6.05 1.84 5.85 1.54 5.07 1.13
1.3$ \cdot CPL $ 7.25 2.45 6.76 1.96 5.61 1.35
1.5$ \cdot CPL $ 8.63 3.33 7.89 2.35 6.47 1.60
Table 5.  $ G_{\rm max} $ under Different Values of Parameters
Parameter Value $ G_{\rm max} $ Parameter Value $ G_{\rm max} $
$ C $ LLL 66.06 $ \sigma^2_{MAUF} $ 0 70.37
HLL 67.34 0.25 66.86
HHL 69.66 $ M^h $ 4 81.48
HHH 71.43 5 67.12
$ NARLF $ –2 70.63 6 57.26
0 68.51 $ \theta^h $ 0.7 83.76
2 66.73 0.8 68.73
$ MAUF $ 0.8 65.57 0.9 53.36
1.0 68.44 $ D^h $ 1.1$ \cdot CPL $ 72.88
1.2 71.86 1.3$ \cdot CPL $ 67.66
1.5$ \cdot CPL $ 65.33
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Parameter Value $ G_{\rm max} $ Parameter Value $ G_{\rm max} $
$ C $ LLL 66.06 $ \sigma^2_{MAUF} $ 0 70.37
HLL 67.34 0.25 66.86
HHL 69.66 $ M^h $ 4 81.48
HHH 71.43 5 67.12
$ NARLF $ –2 70.63 6 57.26
0 68.51 $ \theta^h $ 0.7 83.76
2 66.73 0.8 68.73
$ MAUF $ 0.8 65.57 0.9 53.36
1.0 68.44 $ D^h $ 1.1$ \cdot CPL $ 72.88
1.2 71.86 1.3$ \cdot CPL $ 67.66
1.5$ \cdot CPL $ 65.33
Table 6.  $ G_{\rm max} $ under Combinations of Different Values of Parameters
$ MAUF $ $ C $ $ G_{\rm max} $ $ MAUF $ $ NARLF $ $ G_{\rm max} $ $ MAUF $ $ \sigma^2_{MAUF} $ $ G_{\rm max} $ $ MAUF $ $ D^h $ $ G_{\rm max} $
0.8 LLL 61.8 0.8 –2 66.96 0.8 0 66.67 0.8 1.1$ \cdot CPL $ 71.33
HLL 64.29 0 65.46 0.25 64.46 1.3$ \cdot CPL $ 64.61
HHL 66.6 2 64.28 1.0 0 69.99 1.5$ \cdot CPL $ 60.78
HHH 69.58 1.0 –2 70.3 0.25 66.88 1.0 1.1$ \cdot CPL $ 72.2
1.0 LLL 66.08 0 68.33 1.2 0 74.46 1.3$ \cdot CPL $ 67.46
HLL 67.16 2 66.7 0.25 69.25 1.5$ \cdot CPL $ 65.65
HHL 69.48 1.2 –2 74.62 1.2 1.1$ \cdot CPL $ 75.12
HHH 71.05 0 71.75 1.3$ \cdot CPL $ 70.9
1.2 LLL 70.3 2 69.21 1.5$ \cdot CPL $ 69.57
HLL 70.58
HHL 72.9
HHH 73.67
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$ MAUF $ $ C $ $ G_{\rm max} $ $ MAUF $ $ NARLF $ $ G_{\rm max} $ $ MAUF $ $ \sigma^2_{MAUF} $ $ G_{\rm max} $ $ MAUF $ $ D^h $ $ G_{\rm max} $
0.8 LLL 61.8 0.8 –2 66.96 0.8 0 66.67 0.8 1.1$ \cdot CPL $ 71.33
HLL 64.29 0 65.46 0.25 64.46 1.3$ \cdot CPL $ 64.61
HHL 66.6 2 64.28 1.0 0 69.99 1.5$ \cdot CPL $ 60.78
HHH 69.58 1.0 –2 70.3 0.25 66.88 1.0 1.1$ \cdot CPL $ 72.2
1.0 LLL 66.08 0 68.33 1.2 0 74.46 1.3$ \cdot CPL $ 67.46
HLL 67.16 2 66.7 0.25 69.25 1.5$ \cdot CPL $ 65.65
HHL 69.48 1.2 –2 74.62 1.2 1.1$ \cdot CPL $ 75.12
HHH 71.05 0 71.75 1.3$ \cdot CPL $ 70.9
1.2 LLL 70.3 2 69.21 1.5$ \cdot CPL $ 69.57
HLL 70.58
HHL 72.9
HHH 73.67
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