Uncertain Resource-Constrained Project Scheduling Problem with Net Present Value Criterion
- Chenkai Zhao^{1},
- Hua Ke^{1}Email author and
- Zhiyi Chen^{2}
https://doi.org/10.1186/s40467-016-0054-z
© The Author(s) 2016
Received: 10 November 2016
Accepted: 28 November 2016
Published: 20 December 2016
Abstract
On the basis of uncertainty theory, plenty of researches have been done on uncertain resource-constrained project scheduling problems. Instead of minimizing the makespan, in this paper, we address the maximization of net present value of a project’s cash flows when activity durations are assumed to be uncertain. In addition to precedence constraint and resource constraint involved in resource-constrained project scheduling problem, a deadline constraint is taken into account. Thus, our aim is to maximize net present value and to satisfy the deadline constraint as well. Accordingly, we introduce three models and utilize a revised estimation of distribution algorithm to solve this problem. This work may provide net present value criterion for financial officers on project scheduling.
Keywords
Project scheduling Uncertainty theory Net present value Estimation of distribution algorithmIntroduction
Project scheduling is to assign activity starting times based on scheduling objectives [1]. Researches in this area have primarily emphasized on modeling and algorithmic developments for specific classes of project scheduling problems, such as net present value (NPV) maximization, quality maximization, and makespan minimization [2]. Most literature on project scheduling focused on arranging activity starting times in such a way that the project makespan is minimized. This objective may be unsuitable for capital-intensive IT and construction projects, where large amounts of money are invested over a long period of time. In such an environment, the wise coordination of cash flows crucially affects the profitability of a project. This suggests that in such situations, financial aspects should be the center of decision makers’ attention. The financial benefit of a project is measured by its NPV, which is determined by discounting all arising cash flows to the start time of the project. In other words, NPV can be regarded as the cash equivalent of undertaking the project. In reality, contractors sign contracts with clients to define ways of receiving payments linked with progress of related activities. Thus, different contracts lead to different time points of payments and cash flows vary correspondingly. In this paper, we assume that contractors only receive payments when related activities are completed. For instance, when a construction company undertakes a project consisting of lots of activities funded by its client, it seems unreasonable to make payments happen either at the beginning of the project or at the end of the project. Therefore, to avoid being cheated by each other and to keep the project progress on schedule, an agreement is signed to set payments on the completion time of each activity. This criterion can be applied to resource-constrained project scheduling problem (RCPSP), discrete time-cost trade-off problem (TCTP), and other classical subproblems of project scheduling problems. Scheduling projects to maximize NPV in a deterministic setting has been studied under a broad range of contractual arrangements and planning constraints [3], but in practice, there are frequently significant indeterminacies. In this paper, we focus on an uncertain resource-constrained project scheduling problem with discounted cash flows (URCPSPDC) additionally considering a due date constraint.
Then, we propose three uncertain models. The chance-constrained model is proposed to maximize NPV with a certain belief degree, which can be applied to decision makers who may be risk-averse and desire to realize the objective value with a pretty high belief degree. Besides, the expected value model aims at maximizing the expected NPV of cash flow received on the completion time of each activity. The chance maximization model is based on dependent-chance programming. Moreover, a revised estimation of distribution algorithm (EDA) is utilized in this paper. EDAs have been developed for years and successfully applied in RCPSP. For more details about EDA for RCPSP, readers may refer to [4, 5].
The remainder of this paper is as follows. In Section Literature Review, we summarize related literature. Section Preliminaries introduces some basic concepts in uncertainty theory. Section NPV Models describes URCPSPDC in detail and proposes three corresponding NPV models to satisfy the demand of financial officers. To solve these three models, revised EDAs are designed in Section Revised EDA. Section Numerical Examples conducts some numerical examples. Finally, conclusions are drawn in Section Conclusion and Future Work.
Literature Review
Maximizing the NPV of a project was first suggested by Russell [6]. Besides, Buss and Rosenblatt [7] as well as Sobel et al. [8] extended Russell’s model by including indeterminacies assuming independent and exponentially distributed activity durations. Neuman and Zimmermann [9] presented different heuristic and exact procedures for solving RCPSP with NPV criterion. Leyman and Vanhoucke [10] introduced a new schedule construction technique to improve the project NPV. Researches on RCPSP in early phase were done with the assumptions of complete information and deterministic environment. For a given project, a baseline schedule can be obtained by solving deterministic RCPSP. The deterministic RCPSP has been extensively studied and numerous exact and heuristic methods have been proposed to solve it [11, 12]. However, the baseline schedule is vulnerable when being executed in indeterminate environment. In reality, there are considerable indeterminacies (accident, resource breakdown, unreliable deliveries, etc.), which may result in an infeasible baseline schedule. Hence, it is necessary to consider indeterminate factors when solving a project scheduling problem.
Recently, Fathallahi and Najafi [13] studied fuzzy RCPSP with NPV criterion. In the fuzzy set theory, the decision can be estimated by experts based on their experiences and professional judgments. However, fuzzy set theory may lead to counterintuitive results [14]. Besides, the stochastic resource-constrained project scheduling problem (SRCPSP), the main context of stochastic project scheduling, is characterized by random activity durations and scheduling policies. Activity durations are assumed to be random variables and a scheduling policy is used to decide which activities to be started at decision points (the starting time of project and the finishing times of activities) [15]. Generally, the SRCPSP aims at minimizing the expected makespan by making a limited set of decisions during project execution. Tsai and Gemmill [16] proposed tabu search for both deterministic and stochastic RCPSP. For more details about SRCPSP, readers may refer to [12, 17, 18].
In SRCPSP, activity durations are represented by random variables. The assumption is reasonable when there are enough historical data to estimate variables’ probability distributions precisely. However, in a project, it is difficult to get enough historical data for activities seldom or never executed. This situation is common in consideration of the uniqueness of a project. Therefore, new ways instead of probability distribution are needed to describe such variables. In this case, uncertainty theory, initiated by Liu [14], was founded to rationally deal with belief degrees, which inspired a new method of describing indeterministic phenomena. For now, the new theory has been successfully applied to varieties of fields, such as, option pricing problem [19–21], stock problem [22], production control problem [23], shortest path problem [24], etc. Ke et al. [25] researched project scheduling problem in the environment with uncertainty and randomness. For more details about uncertain project scheduling problem, readers may refer to [26, 27].
In this paper, we build three NPV models for URCPSPDC additionally considering a due date constraint.
Preliminaries
Uncertainty theory, as a branch of axiomatic mathematics, has been well developed in many aspects in reality. In this section, some concepts and theorems of uncertainty theory are introduced to lay the foundation for URCPSPDC modeling.
Let Γ be a nonempty set, \(\mathcal {L}\) a σ-algebra over Γ, and each element Λ in \(\mathcal {L}\) is named as an event. Uncertain measure \(\mathcal {M}\), initiated by Liu [14] and perfected by Liu [28], is a function from \(\mathcal {L}\) to [ 10,1]. It is defined over the following four axioms.
Axiom 1
(Normality Axiom) \(\mathcal {M}\{\Gamma \}=1\).
Axiom 2
(Duality Axiom) \(\mathcal {M}\{\Lambda \}+\mathcal {M}\{\Lambda ^{c}\}=1\) for any event Λ.
Axiom 3
Axiom 4
where Λ _{ k } are arbitrarily chosen events from \(\mathcal {L}_{k}\) for k=1,2,⋯, respectively.
Definition 1
is an event.
The uncertainty distribution is indispensable to establish practical uncertain optimization models.
Definition 2
for any real number x.
An uncertainty distribution Φ is confirmed to be regular if its inverse function Φ ^{−1}(α) exists uniquely for each α∈[0,1].
Definition 3
provided that at least one of the above two integrals is finite.
Lemma 1
Lemma 2
Lemma 3
NPV Models
Problem Description
A project containing n activities can be described by an activity-on-the-node network G(N,A). The set of nodes N={1,2,⋯,n+2} represents activities, and the set of arcs A denotes finish-start, zero-lag precedence relations between activities. Activities 1 and n+2 do not consume time and resources, and only signify project starting point and finishing point, respectively. Specially, durations of all activities in URCPSPDC are represented by an uncertain vector \(\boldsymbol {d}=\{0,\tilde {d_{2}},\cdots, \tilde {d}_{n+1},0\}\) and the start times of all activities are denoted as a vector s={0,s _{2},⋯,s _{ n+1},s _{ n+2}}. Therefore, the completion time of ith activity f _{ i } can be calculated as \(s_{i}+\tilde {d_{i}}\). Moreover, there are totally K types of renewable resources and each of them has a constant availability a _{ k }, k=1,2,⋯,K. Besides, when activity i is completed, a cash flow \({C^{F}_{i}}\) will be received accordingly and we discount it to the project staring time with a discount rate r∈(0,1]. In this paper, we additionally take a due date δ into consideration.
The URCPSPDC aims at maximizing the NPV of a whole project with uncertain activity durations meanwhile satisfying the deadline constraint. Solving URCPSPDC is a dynamic decision process. The decision maker decides to start which feasible activity at each decision point, including project starting time and activity finishing times. In the decision process, the decision maker can only utilize partial information which appears before his decision point.
Therefore, s _{ i }(π,d) is strictly increasing with respect to d, then \(s_{i}(\pi, \boldsymbol {d}) + \tilde {d}_{i}\) is also strictly increasing with respect to d, and \(Npv(\pi, \boldsymbol {d})= \sum _{i=2}^{\,n+1}{C^{F}_{i}}r^{f_{i}}\) must be strictly decreasing with respect to d and s. Thus, we can have
Theorem 1
Theorem 2
Chance-Constrained Model
In the above model, the objective function is to maximize the NPV with belief degree α, enforced in the first constraint. The second constraint ensures the makespan cannot exceed the deadline with belief degree β.
Expected Value Model
Chance Maximization Model
Revised EDA
Since deterministic RCPSP is NP-hard in the strong sense, URCPSPDC, an extension of RCPSP, needs to be solved by heuristic or meta-heuristic algorithm. In this section, a revised intelligent heuristic algorithm is designed by applying NPV criterion to uncertain serial schedule generation scheme (US-SGS) in EDA. For more details about EDA for RCPSP, readers may refer to [4,5].
Revised Uncertain Serial Schedule Generation Scheme
As discussed by Kolish and Hartmann [37], there are several types of feasible solution representations for project scheduling. A former work by Ma et al. [30] designed an uncertain serial SGS (US-SGS) for uncertain RCPSP. For URCPSPDC in this paper, we choose the solution representation activity list π, which represents the executing order of activities. The US-SGS can be described as follows.
Chance-Constrained Model
It is worth mentioning that a realized makespan with confidence level β must be subject to the due date constraint and if not, the corresponding activity list has to be abandoned.
Expected Value Model
Compared with the chance-constrained model, the expected value model generates \({d_{1}^{m}},{d_{2}^{m}},\ldots,d_{n+2}^{m}\) according to the uncertainty distributions of activities’ durations Φ _{1},Φ _{2},…,Φ _{ n+2}. Denote \(\boldsymbol {d^{m}}=(0,\Phi ^{-1}_{2}(m/100),\Phi ^{-1}_{3}(m/100),\cdots,\Phi ^{-1}_{n+1} (m/100),0), m=1,2,\ldots,99\), respectively. Next, by using the decoding scheme concerned above, \(s_{n+2}^{m}\) and N p v ^{ m } can be obtained. Finally we can get \(E[\!{Npv}]=\sum _{m=1}^{99}Npv^{m}/99\). Note that an expected makespan \(E[{\!s_{n+2}}]=\sum _{m=1}^{99}s_{n+2}^{m}/99\) must be subject to the due date constraint and if not, the corresponding activity list has to be abandoned.
Chance Maximization Model
Compared with the above two models, for each activity list π, this model first obtains a realized makespan with confidence level β and a series of N p v ^{ m },m=1,2,…,99. In the condition of \(Npv^{m}\geq \overline {Npv}\), we choose the largest m among the feasible solution set. Note that the realized makespan with confidence level β must be subject to the due date constraint and if not, the corresponding activity list has to be abandoned.
Revised EDA
In this paper, to revise the EDA, here are the steps. First, NP solutions are generated according to the initial probability matrix as the initial population and we update probability matrix according to the initial population. Each solution is an activity list, where one activity can only be assigned if all of its predecessors have been finished. Second, the US-SGS is utilized to generate schedules according to activity lists, to filter infeasible solutions when realized makespan exceeds the deadline and to evaluate each solution. After evaluating the population, P<N P best individuals are selected from the population to form the elite set. And, the elite set is chosen to update probability matrix. Then, the new probability matrix is employed to sample population of the next generation. After a certain number of generations, the solution with best fitness value is reported as quasi-optimal solution.
Numerical Examples
Project information
Activity | Duration | R _{1} | R _{2} | R _{3} | R _{4} | Successors | Cash flows |
---|---|---|---|---|---|---|---|
1 | 0 | 0 | 0 | 0 | 0 | 2, 3, 4 | 0 |
2 | \(\mathcal {Z}(5,7,8)\) | 4 | 0 | 0 | 0 | 8, 10, 13 | 105 |
3 | \(\mathcal {Z}(7,9,10)\) | 10 | 0 | 0 | 0 | 5, 9, 19 | 54 |
4 | \(\mathcal {L}(1,3)\) | 0 | 0 | 0 | 3 | 6, 16, 17 | 149 |
5 | \(\mathcal {Z}(1,3,4)\) | 3 | 0 | 0 | 0 | 10, 18, 31 | 112 |
6 | \(\mathcal {L}(8,10)\) | 0 | 0 | 0 | 8 | 7, 22 | 128 |
7 | \(\mathcal {Z}(7,8,10)\) | 4 | 0 | 0 | 0 | 28 | 87 |
8 | \(\mathcal {Z}(1,3,4)\) | 0 | 1 | 0 | 0 | 11, 12 | 85 |
9 | \(\mathcal {L}(1,3)\) | 6 | 0 | 0 | 0 | 14, 27 | 115 |
10 | \(\mathcal {Z}(8,10,11)\) | 0 | 0 | 0 | 1 | 30 | 125 |
11 | \(\mathcal {L}(7,10)\) | 0 | 5 | 0 | 0 | 24 | 143 |
12 | \(\mathcal {Z}(8,10,11)\) | 0 | 7 | 0 | 0 | 15, 21 | 52 |
13 | \(\mathcal {Z}(1,3,4)\) | 4 | 0 | 0 | 0 | 17 | 80 |
14 | \(\mathcal {L}(1,3)\) | 0 | 8 | 0 | 0 | 20 | 76 |
15 | \(\mathcal {Z}(3,5,6)\) | 3 | 0 | 0 | 0 | 30 | 86 |
16 | \(\mathcal {L}(2,4)\) | 0 | 0 | 0 | 5 | 25 | 126 |
17 | \(\mathcal {L}(7,11)\) | 0 | 0 | 0 | 8 | 21 | 146 |
18 | \(\mathcal {Z}(6,8,9)\) | 0 | 0 | 0 | 7 | 29 | 121 |
19 | \(\mathcal {Z}(2,4,5)\) | 0 | 1 | 0 | 0 | 20, 23, 24 | 93 |
20 | \(\mathcal {Z}(7,10,11)\) | 0 | 10 | 0 | 0 | 21 | 140 |
21 | \(\mathcal {L}(4,6)\) | 0 | 0 | 0 | 6 | 28 | 68 |
22 | \(\mathcal {Z}(2,4,5)\) | 2 | 0 | 0 | 0 | 26 | 69 |
23 | \(\mathcal {L}(3,5)\) | 3 | 0 | 0 | 0 | 26 | 52 |
24 | \(\mathcal {Z}(3,5,6)\) | 0 | 9 | 0 | 0 | 25, 29 | 123 |
25 | \(\mathcal {L}(6,8)\) | 4 | 0 | 0 | 0 | 30 | 78 |
26 | \(\mathcal {Z}(4,5,7)\) | 0 | 0 | 4 | 0 | 28 | 129 |
27 | \(\mathcal {L}(1,3)\) | 0 | 0 | 0 | 7 | 31 | 137 |
28 | \(\mathcal {Z}(2,3,5)\) | 0 | 8 | 0 | 0 | 31 | 77 |
29 | \(\mathcal {Z}(1,2,4)\) | 0 | 7 | 0 | 0 | 32 | 125 |
30 | \(\mathcal {Z}(5,6,8)\) | 0 | 7 | 0 | 0 | 32 | 146 |
31 | \(\mathcal {Z}(4,6,7)\) | 0 | 0 | 2 | 0 | 32 | 58 |
32 | 0 | 0 | 0 | 0 | 0 | 0 |
Chance-Constrained Model
The quasi-optimal solutions for chance-constrained model
(α,β) | Quasi-optimal schedule | NPV | Makespan |
---|---|---|---|
(0.05,0.05) | 1,2,8,4,3,6,12,9,5,11,13,22,16,18,7,15,19,23, | 3421.4 | 47.75 |
24,10,14,27,29,25,20,17,26,30,21,28,31,32 | |||
(0.15,0.15) | 1,4,2,8,6,3,11,12,13,19,16,5,22,18,24,9,23,27, | 3342.4 | 52.8 |
14,15,17,10,20,25,30,26,29,7,21,28,31,32 | |||
(0.25,0.25) | 1,4,2,8,3,6,11,12,9,13,14,5,7,27,16,19,23,24, | 3311.1 | 54 |
18,15,10,20,17,22,25,26,30,29,21,28,31,32 | |||
(0.35,0.35) | 1,3,4,19,9,6,5,2,14,20,13,10,23,8,27,22,7,26, | 3255.8 | 60.2 |
18,11,12,16,24,17,21,25,15,28,31,29,30,32 | |||
(0.45,0.45) | 1,4,2,8,6,3,11,12,19,16,9,5,22,18,10,14,13,20, | 3206.6 | 65.3 |
23,7,17,26,27,24,21,15,25,28,30,31,29,32 | |||
(0.55,0.55) | 1,2,4,8,3,6,12,11,7,9,16,5,14,10,19,18,22,23, | 3199.5 | 61.4 |
13,24,27,25,26,20,15,17,30,21,28,31,29,32 | |||
(0.65,0.65) | 1,4,2,8,6,3,11,12,19,5,9,16,22,10,18,24,14, | 3163.5 | 67.8 |
13,20,23,7,27,17,15,25,21,30,26,29,28,31,32 | |||
(0.75,0.75) | 1,4,3,2,6,9,19,23,5,13,14,16,8,27,20,22,26, | 3093.5 | 69 |
11,12,10,7,17,15,24,18,21,25,28,29,31,30,32 | |||
(0.85,0.85) | 1,3,4,19,6,2,9,14,13,23,16,5,22,27,8,18,7,12, | 3086.5 | 74.4 |
26,11,15,24,10,25,17,20,21,30,28,31,29,32 | |||
(0.95,0.95) | 1,2,4,16,8,3,6,5,7,13,12,18,11,10,19,22,9,14, | 3065.4 | 77.6 |
23,24,27,25,29,15,20,26,17,30,21,28,31,32 |
The result may help risk-averse decision makers from the following two aspects: First, a financial officer can arrange the capital operation plan with NPV based on a project manager’s makespan prediction according to their own belief degrees;. Second, a given belief degree corresponds with an optimal schedule.
To conclude, given a fixed discount rate, a higher belief degree corresponds with a lower realized NPV and the optimal solution varies according to different belief degrees. Also, an optimal NPV solution is not necessary to reach a best makespan. Therefore, for risk-averse decision makers, it is considerable to choose applicable belief degrees to solve this problem.
Expected Value Model
Suppose that the discount rate is 0.99. Then the expected value model can be rewritten as:
The quasi-optimal solutions for expected value model
Generation and popsize | Quasi-optimal schedule | E[NPV] | E[makespan] |
---|---|---|---|
150 and 30 | 1,3,19,4,2,9,5,14,6,20,13,8,10,22,27,23,18, | 3172 | 65 |
12,11,17,15,16,7,21,24,29,26,28,31,25,30,32 | |||
500 and 30 | 1,4,3,19,16,2,5,9,8,13,14,27,23,6,12,11,22,7, | 3182.4 | 65.068 |
18,10,15,26,24,20,25,17,30,21,28,31,29,32 | |||
500 and 30 | 1,2,8,3,4,6,12,9,19,13,15,22,11,27,16,5,14,23, | 3144.7 | 71.263 |
17,7,26,20,21,28,18,31,24,10,29,25,30,32 | |||
500 and 30 | 1,3,4,19,2,9,23,6,5,14,13,10,20,8,16,22,7,11, | 3147.3 | 69.505 |
27,12,18,17,24,15,21,29,26,25,28,30,31,32 | |||
150 and 50 | 1,4,2,16,8,3,6,22,13,7,19,17,23,12,5,11,26,10, | 3148.4 | 63.763 |
9,14,20,27,24,25,15,21,28,31,18,30,29,32 | |||
150 and 50 | 1,2,4,8,3,6,7,22,11,12,19,5,13,23,16,26,17,9, | 3180.2 | 60.505 |
24,10,14,20,18,25,15,29,21,28,27,31,30,32 | |||
150 and 50 | 1,3,2,9,27,14,5,4,19,8,13,23,20,16,6,10,22,12, | 3151.5 | 68.753 |
17,11,7,26,21,28,15,24,18,25,31,30,29,32 | |||
150 and 50 | 1,2,4,16,8,3,6,11,12,5,9,19,13,24,17,7,27,25, | 3156.8 | 62.131 |
14,10,20,22,18,15,23,26,21,29,30,28,31,32 | |||
150 and 50 | 1,2,4,8,3,6,11,12,9,19,5,10,15,23,24,22,27,14, | 3232 | 68.258 |
20,7,13,26,16,17,25,21,28,30,18,29,31,32 | |||
150 and 50 | 1,2,4,8,3,6,12,11,13,19,16,7,22,24,5,18,9,25, | 3162.6 | 73.288 |
14,15,29,27,17,23,20,10,21,30,26,28,31,32 |
The effectiveness of the EDA
Size | The best | The worst | The average | Error |
---|---|---|---|---|
30 | 3232 | 3077.8 | 3142.30 | 4.77% |
Chance Maximization Model
With the crisp-form of the chance maximization model, 99-method can be applied as follows:
The quasi-optimal solutions for chance maximization model
\(\overline {Npv}\) | Quasi-optimal schedule | Uncertainty | NPV |
---|---|---|---|
3000 | 1,4,2,16,8,3,6,22,13,9,19,11,5,7,27,23,14,17, | 0.99 | 3021.7 |
10,12,20,15,18,26,24,25,29,21,28,31,30,32 | |||
3100 | 1,2,4,8,3,6,12,22,5,19,9,11,10,23,14,18,24,26, | 0.69 | 3104.6 |
13,16,20,7,15,17,21,28,25,27,30,31,29,32 | |||
3200 | 1,4,2,8,6,3,11,12,5,16,22,9,19,14,23,24,18,15, | 0.46 | 3203.7 |
25,27,26,7,13,17,20,21,29,10,28,31,30,32 | |||
3300 | 1,3,4,19,6,9,5,2,14,20,13,22,27,23,10,8,12,11, | 0.27 | 3301.6 |
18,26,7,16,24,15,17,25,21,30,28,31,29,32 | |||
3400 | 1,2,4,16,8,3,6,11,12,19,5,15,9,10,24,23,14,18, | 0.07 | 3403.8 |
13,22,29,7,25,20,17,21,26,27,28,31,30,32 |
According to Table 5, we can compare the results with the chance-constrained model and the two models validate each other. The result may help financial officers from the following two aspects. First, a financial officer can determine an uncertainty at which a given NPV value can be reached based on a project manager’s makespan prediction according to a belief degree. Second, a given predetermined NPV corresponds with an optimal schedule.
To conclude, for financial officers, it is considerable to set applicable NPV goals and constraint belief degrees to solve this problem.
Conclusion and Future Work
In the real project, the environment for project execution is full of indeterminacies. Considering the uniqueness of a project, it is common that activities are seldom or never executed before. As a result, it is difficult to describe activity durations by probability distributions for lack of historical data. Also, fuzzy set theory may lead to counterintuitive results. This paper considers NPV of RCPSP with uncertain durations and a deadline constraint. To satisfy different demand of financial officers, three uncertain models are built. We utilize a special SGS for our problem called US-SGS and added it into EDA. Furthermore, some numerical examples are solved with our models and algorithms. We hope our work may provide financial criterion for financial officers. For future work, we believe that it is worthwhile to take other factors as uncertain variables such as discount rate and to apply NPV criterion to other uncertain types of project scheduling problems.
Declarations
Acknowledgements
This work was supported by National Natural Science Foundation of China (No.71371141) and the Fundamental Research Funds for the Central Universities.
Authors’ contributions
CZ and HK carried out the study in the paper and drafted the first version of the manuscript. They designed the framework and with ZC performed the algorithm design and numerical analysis together. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Herroelen, W, Demeulemeester, E, De Reyck, B: A classification scheme for project scheduling. In: Project Scheduling: Recent Models, Algorithms and Applications, pp. 1–26. Kluwer, Amsterdam (1999).Google Scholar
- Kolisch, R, Padman, R: An integrated survey of deterministic project scheduling. Omega. 29(3), 249–272 (2001).View ArticleGoogle Scholar
- Herroelen, WS, Van Dommelen, P, Demeulemeester, EL: Project network models with discounted cash flows: a guided tour through recent developments. Eur. J. Oper. Res. 100(1), 97–121 (1997).View ArticleMATHGoogle Scholar
- Wang, L, Fang, C: An effective estimation of distribution algorithm for the multi-mode resource-constrained project scheduling problem. Comput. Oper. Res. 39(2), 449–460 (2012).View ArticleGoogle Scholar
- Fang, C, Kolisch, R, Wang, L, Mu, C: An estimation of distribution algorithm and new computational results for the stochastic resource-constrained project scheduling problem. Flex. Serv. Manuf. J. 27(4), 585–605 (2015).View ArticleGoogle Scholar
- Russell, A: Cash flows in networks. Manag. Sci. 16(5), 357–373 (1970).View ArticleMATHGoogle Scholar
- Buss, AH, Rosenblatt, MJ: Activity delay in stochastic project networks. Oper. Res. 45(1), 126–139 (1997).View ArticleMATHGoogle Scholar
- Sobel, MJ, Szmerekovsky, JG, Tilson, V: Scheduling projects with stochastic activity duration to maximize expected net present value. Eur. J. Oper. Res. 198(3), 697–705 (2009).MathSciNetView ArticleMATHGoogle Scholar
- Neumann, K, Zimmermann, J: Procedures for resource leveling and net present value problems in project scheduling with general temporal and resource constraints. Eur. J. Oper. Res. 127(2), 425–443 (2000).MathSciNetView ArticleMATHGoogle Scholar
- Leyman, P, Vanhoucke, M: A new scheduling technique for the resource-constrained project scheduling problem with discounted cash flows. Int. J. Prod. Res. 53(9), 2771–2786 (2015).View ArticleGoogle Scholar
- Demeulemeester, E, Herroelen, W: Project scheduling—a research handbook. Vol. 49 of International Series in Operations Research & Management Science. Kluwer Academic Publishers, Boston (2002).MATHGoogle Scholar
- Ballestín, F: When it is worthwhile to work with the stochastic RCPSP. J. Sched. 10(3), 153–166 (2007).MathSciNetView ArticleMATHGoogle Scholar
- Fathallahi, F, Najafi, A: A hybrid genetic algorithm to maximize net present value of project cash flows in resource-constrained project scheduling problem with fuzzy parameters. Scientia. Iranica. 23(4), 1893–1903 (2016).Google Scholar
- Liu, B: Uncertainty Theory. 2nd edn. Springer, Berlin (2007).View ArticleMATHGoogle Scholar
- Igelmund, G, Radermacher, FJ: Preselective strategies for the optimization of stochastic project networks under resource constraints. Networks. 13(1), 1–28 (1983).MathSciNetView ArticleMATHGoogle Scholar
- Tsai, YW, Gemmill, DD: Using tabu search to schedule activities of stochastic resource-constrained projects. Eur. J. Oper. Res. 111(1), 129–141 (1998).View ArticleMATHGoogle Scholar
- Ballestin, F, Leus, R: Resource-constrained project scheduling for timely project completion with stochastic activity durations. Prod. Oper. Manag. 18(4), 459–474 (2009).View ArticleGoogle Scholar
- Ashtiani, B, Leus, R, Aryanezhad, M-B: New competitive results for the stochastic resource-constrained project scheduling problem: exploring the benefits of pre-processing. J. Sched. 14(2), 157–171 (2011).MathSciNetView ArticleMATHGoogle Scholar
- Chen, X: American option pricing formula for uncertain financial market. Int. J. Oper. Res. 8(2), 32–37 (2011).MathSciNetGoogle Scholar
- Peng, J, Yao, K: A new option pricing model for stocks in uncertainty markets. Int. J. Oper. Res. 8(2), 18–26 (2011).MathSciNetGoogle Scholar
- Gao, Z, Wang, X, Ha, M: Multi-asset option pricing in an uncertain financial market with jump risk. J. Uncertain. Anal. Appl. 4(1), 1 (2016).View ArticleGoogle Scholar
- Bhattacharyya, R, Chatterjee, A, Kar, S: Uncertainty theory based multiple objective mean-entropy-skewness stock portfolio selection model with transaction costs. J. Uncertain. Anal. Appl. 1(1), 16 (2013).MathSciNetView ArticleGoogle Scholar
- Liu, B, Yao, K: Uncertain multilevel programming: algorithm and applications. Comput. Ind. Eng. 89, 235–240 (2015).View ArticleGoogle Scholar
- Zhang, Y, Liu, P, Yang, L, Gao, Y: A bi-objective model for uncertain multi-modal shortest path problems. J. Uncertain. Anal. Appl. 3(1), 8 (2015).View ArticleGoogle Scholar
- Ke, H, Liu, H, Tian, G: An uncertain random programming model for project scheduling problem. Int. J. Intell. Syst. 30(1), 66–79 (2015).View ArticleGoogle Scholar
- Zhang, X, Chen, X: A new uncertain programming model for project scheduling problem. Inf. Int. Interdiscip. J. 15(10), 3901–3910 (2012).MathSciNetMATHGoogle Scholar
- Ke, H: Uncertain random time-cost trade-off problem. J. Uncertain. Anal. Appl. 2(1), 23 (2014).View ArticleGoogle Scholar
- Liu, B: Uncertainty Theory. 4th edn. Springer, Berlin (2010).View ArticleGoogle Scholar
- Artigues, C, Roubellat, F: A polynomial activity insertion algorithm in a multi-resource schedule with cumulative constraints and multiple modes. Eur. J. Oper. Res. 127(2), 297–316 (2000).View ArticleMATHGoogle Scholar
- Ma, W, Che, Y, Huang, H, Ke, H: Resource-constrained project scheduling problem with uncertain durations and renewable resources. Int. J. Mach. Learn. Cybernet. 7(4), 613–621 (2015).View ArticleGoogle Scholar
- Charnes, A, Cooper, WW: Chance-constrained programming. Manag. Sci. 6(1), 73–79 (1959).MathSciNetView ArticleMATHGoogle Scholar
- Liu, B: Dependent-chance programming: a class of stochastic optimization. Comput. Math. Appl. 34(12), 89–104 (1997).MathSciNetView ArticleMATHGoogle Scholar
- Liu, B, Iwamura, K: Modelling stochastic decision systems using dependent-chance programming. Eur. J. Oper. Res. 101(1), 193–203 (1997).View ArticleMATHGoogle Scholar
- Liu, B: Uncertain programming. John Wiley & Sons, Inc., New York (1999).Google Scholar
- Liu, B: Uncertain programming: a unifying optimization theory in various uncertain environments. Appl. Math. Comput. 120(1), 227–234 (2001).MathSciNetMATHGoogle Scholar
- Liu, B: Theory and practice of uncertain programming. Springer, Heidelberg (2002).View ArticleMATHGoogle Scholar
- Kolisch, R, Hartmann, S: Heuristic algorithms for the resource-constrained project scheduling problem: classification and computational analysis. In: Project Scheduling: Recent Models, Algorithms and Applications, pp. 147–178. Kluwer Academic Publishers, Netherlands (1999).Google Scholar