Article Contents

2021, Volume 17, Issue 5: 2451-2474. Doi: 10.3934/jimo.2020077

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

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
^* Corresponding author: Nengmin Wang

Received: May 2019

Revised: November 2019

Early access: April 2020

Published: September 2021

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

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Abstract

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:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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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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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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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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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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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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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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