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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The Generated Starting Solution
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Process for Updating Current Solution by TS
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Operation on a Forbidden Move to Generate a Solution Better than $ S^{\rm best} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Experimental Design
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Performance of Algorithms
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Effects of Improvement Measure
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Effects of Parameters on Objective Function Value
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Interactive Effects of Parameters on Objective Function Value
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

$ 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
[1]

Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020034
[2]

Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134
[3]

Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021013
[4]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014
[5]

Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122
[6]

Vadim Azhmyakov, Juan P. Fernández-Gutiérrez, Erik I. Verriest, Stefan W. Pickl. A separation based optimization approach to Dynamic Maximal Covering Location Problems with switched structure. Journal of Industrial & Management Optimization, 2021, 17 (2) : 669-686. doi: 10.3934/jimo.2019128
[7]

Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109
[8]

Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic & Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047
[9]

Qiang Long, Xue Wu, Changzhi Wu. Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison. Journal of Industrial & Management Optimization, 2021, 17 (2) : 1001-1023. doi: 10.3934/jimo.2020009
[10]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[11]

Ömer Arslan, Selçuk Kürşat İşleyen. A model and two heuristic methods for The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021002
[12]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
[13]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158
[14]

Min Ji, Xinna Ye, Fangyao Qian, T.C.E. Cheng, Yiwei Jiang. Parallel-machine scheduling in shared manufacturing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020174
[15]

Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021001
[16]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[17]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278
[18]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[19]

Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130
[20]

Editorial Office. Retraction: Jinling Wei, Jinming Zhang, Meishuang Dong, Fan Zhang, Yunmo Chen, Sha Jin and Zhike Han, Applications of mathematics to maritime search. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 957-957. doi: 10.3934/dcdss.2019064

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (55)
  • HTML views (358)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences