Uncertain random time-cost trade-off problem
- Hua Ke^{1}Email author
DOI: 10.1186/s40467-014-0023-3
© Ke; licensee Springer. 2014
Received: 22 September 2014
Accepted: 8 November 2014
Published: 25 November 2014
Abstract
Since for most projects, ‘new’ parts (with little or no historical data) and ‘old’ parts (with sufficient historical data) always coexist in the same project, uncertainty and randomness should be considered simultaneously in time-cost trade-off problem. In this paper, combined with uncertainty theory and dependent-chance programming, an uncertain random time-cost trade-off model is built. A crisp equivalent model is also given for some special case. Besides, uncertain random simulation and genetic algorithm are integrated for solving the proposed model.
Keywords
Project scheduling Time-cost trade-off Uncertain variable Uncertain random programming Dependent-chance programmingIntroduction
Time-cost trade-off problem (TCTP), a specific type of project scheduling problem, desires to balance two important goals in real-life projects, project completion times, and project total costs to achieve decision-making requirements. For real-life projects, project managers always consider the trade-offs among the performance goals (e.g., project cost, project completion time, project quality, etc.) for project scheduling and control. The time-cost trade-off problem takes into account the project time-cost trade-off by crashing or prolonging project activity durations. In 1961, Kelly [1] first did research on the TCTP. In the following years, many researches have been done on the TCTP with assumption of deterministic activity durations [2-5].
In real-life projects, activity durations may be variational for many external factors, such as technological advance, weather change, labor force shortage, etc. For some projects, since many similar projects have ever been fulfilled before, there are enough historical data for deducing probability distributions for describing imprecise quantities in projects. Hence, probability theory was introduced into the TCTP for estimating nondeterministic factors. In 1985, Wollmer [6] discussed a stochastic version of the deterministic linear TCTP. Following that, many authors discussed the TCTP with stochastic environmental factors [7-9].
In project management, one of the most important characteristics of the concept ‘project’ is that there is always something new in a project. That is, there are always some parts which are never performed before or only similarly performed. For such parts in projects, historical data for estimating probability distributions of some imprecise quantities are always not enough when facing external indeterminacy, which means that probability theory is no longer valid for describing imprecise quantities (e.g., activity durations) in such project parts. For this case, project activity durations were originally described by fuzzy variables, and many researchers studied the TCTP with fuzzy parameters [10-13].
In the past years, subjective probability theory and fuzzy set theory were applied to estimate imprecise quantities (or indeterminacy) with no or little statistical data in decision systems, including project scheduling and control. However, actually both these two theories were proved not suitable for the above case, illustrated by some examples [14]. Since such a type of indeterminacy cannot be depicted via randomness or fuzziness, a new theory is needed. On this occasion, Liu founded uncertainty theory in 2007 [15] and refined it in 2010 [16], which is a branch of axiomatic mathematics for modeling human uncertainty. Since its foundation, uncertainty theory has been widely applied in many decision systems, such as portfolio selection [17], inventory problem [18], facility location [19], supply chain integrated production-inventory problem [20], investment problem [21], differential games [22], and coalitional game [23]. Especially, Ke [24] applied uncertain measure for estimating activity durations in time-cost trade-off problem.
For most projects, though they always include some ‘new’ parts, meanwhile they may contain some common parts with many other finished projects. With enough statistical data from other projects, probability theory may be introduced to estimate imprecise quantities in these common parts. Then for many projects, integrated by some ‘new’ parts and some other ‘old’ (common) parts, randomness and uncertainty may coexist. For such cases, the concepts of uncertain random variable and chance measure and the philosophy of uncertain random programming were presented by Liu [25] and Liu [26], respectively. Uncertain random programming has been further studied in some aspects, e.g., uncertain random graph and uncertain random network [27], uncertain random risk analysis [28], uncertain random multi-objective optimization [29], and uncertain random multilevel programming [30]. In this paper, we introduce chance measure and uncertain random programming to model the TCTP with coexisted randomness and uncertainty. As Huang and Ding [31] illustrated that the standard path algorithms (e.g., the well-known Dijkstra method) were not able to arrive at solutions for searching critical path of this problem, genetic algorithm (GA) is applied in this paper.
The remainder of this paper is organized as follows: in Section ‘Preliminaries of uncertainty theory’, some concepts and useful theorems of uncertainty theory are presented. Section ‘Description of time-cost trade-off problem’ briefly describes the problem with uncertain random parameters. In Section ‘Uncertain random time-cost trade-off model with dependent-chance programming’, with dependent-chance programming, an uncertain random time-cost trade-off model is built and a crisp TCTP model is also given for some special case of uncertain random variable. Section ‘Algorithm description with numerical experiment’ conducts a numerical experiment to illustrate the proposed model. Finally, Section ‘Conclusions’ draws some concluding statements.
Preliminaries of uncertainty theory
In this section, we introduce some basic concepts and theorems for building uncertain random TCTP model. Let Γ be a nonempty set, $\mathcal{L}$ a σ-algebra over Γ, and each element Ʌ in $\mathcal{L}$ is called an event.
Definition1.
(Liu [15]) The set function is called an uncertain measure if it satisfies:
Axiom 1 (Normality Axiom). $\mathcal{M}\{\Gamma \}=1$ for the universal set Γ.
Axiom 2 (Duality Axiom). $\mathcal{M}\left\{\u0245\right\}+\mathcal{M}\left\{{\u0245}^{c}\right\}=1$ for any event Ʌ.
Besides, the product uncertain measure on the product σ-algebra $\mathcal{L}$ was defined by Liu [32] as follows:
where Ʌ_{ k } are arbitrarily chosen events from ${\mathcal{L}}_{k}$ for k=1,2,⋯, respectively.
Definition2.
is an event.
Definition3.
for any Borel sets B_{1}, B_{2},⋯, B_{ n }.
Sometimes, to describe real-world optimization problems with uncertain parameters, it is sufficient to know the uncertainty distribution rather than the uncertain variable itself.
Definition4.
for any real number x.
Definition5.
Liu [16] also defined the inverse function Φ^{-1} as the inverse uncertainty distribution of uncertain variable ξ.
Theorem1.
Based on the definitions of uncertain variable and random variable, the concept of uncertain random variable is given as follows:
Definition6.
(Liu [25]) An uncertain random variable is a function ξ from a chance space $(\Gamma ,\mathcal{L},\mathcal{M})\times (\times ,\mathcal{A},Pr)$ to the set of real numbers such that {ξϵB} is an event in $\mathcal{L}\times \mathcal{A}$ for any Borel set B.
Example 1 A random variable is a special uncertain random variable since any real value is a special uncertain variable.
Example 2 The sum of a random variable and an uncertain variable is an uncertain random variable.
Definition7.
Liu [25] proved that the chance measure is self-dual.
Definition8.
for any xϵR.
For some special case, some operational laws are presented as follows:
Theorem2.
where F(x;y_{1},⋯, y_{ m }) is the uncertainty distribution of the uncertain variable f(y_{1},⋯, y_{ m }, τ_{1},⋯, τ_{ n }) for any real numbers y_{1},⋯, y_{ m }.
Theorem 3.
Description of time-cost trade-off problem
Whether a project is successful or not depends on the opinion of the responsible managers. Some of them may focus more on whether the project is on time or before. Some others may pay more attentions to the project budget. For some projects, e.g., China’s manned space program, the highest priority is technical performance. Generally, the success of a project is measured by three dimensions, i.e., time, cost, and performance (quality or agreed-upon specifications). For many projects, since project managers only need to meet some quality standards, the dimension of quality is not so important for project success. In this case, project managers are required to make trade-off between project cost and project completion time with the project progress. Furthermore, project cost and project completion time are always in some relation. Sometimes, project managers may make decisions to finish projects sooner with project cost augment by accelerating project schedule, which is also named as project crashing in project management. In other cases, motivated by reducing project costs, managers may be conscripted to sacrifice with prolonging project completion times. To sum up, it is natural for project managers to find some schedules to complete projects with the balance of project costs and project completion times.
Uncertain random time-cost trade-off model with dependent-chance programming
where α_{0} is the confidence level given in advance, T^{0} is the due date of the project, C^{0} is the budget, l_{ ij } and u_{ ij } are integers given in advance, and T(x , ξ ) and C(x , ξ ) are defined by Equations 1 and 2, respectively.
Algorithm description with numerical experiment
Since Huang and Ding [31] demonstrated that the standard path algorithms were not able to approach the critical path for project scheduling problem with random activity durations, it is more difficult to solve uncertain random TCTP. Hence, uncertain random simulation and GA are integrated for solving the proposed chance maximization model in this section. An uncertain random simulation algorithm for chance measure (e.g., Ch{C(x , ξ )≤C^{0}}) can be obtained as follows: Uncertain random simulation for chance measure: Step 1. Set e=0. Step 2. Randomly generate ω from the probability space according to the probability distribution. Step 3.$e\leftarrow e+\mathcal{M}\left\{C\right(\mathit{x},\mathit{\xi}(\u2022,\Omega ))\le {C}^{0}\}$. Step 4. Repeat the second and third steps N times, where N is a sufficiently large number. Step 5. Return e/N as the chance measure.
The above designed uncertain random simulation approach can be embedded in GA simply for solving the proposed model.
Activity durations and costs of project
Activity | Normal duration | Normal cost | Additional cost |
---|---|---|---|
(i, j) | ξ _{ ij } | c _{ ij } | d _{ ij } |
(1,2) | $\mathcal{L}(7,10)$ | 170 | 200 |
(1,3) | $\mathcal{U}(9,12)$ | 300 | 280 |
(1,4) | $\mathcal{L}(8,10)$ | 65 | 70 |
(2,5) | $\mathcal{U}(8,12)$ | 270 | 300 |
(3,5) | $\mathcal{L}(10,15)$ | 135 | 150 |
(3,6) | $\mathcal{L}(9,13)$ | 75 | 90 |
(3,7) | $\mathcal{U}(11,15)$ | 150 | 100 |
(4,7) | $\mathcal{L}(10,13)$ | 600 | 400 |
(5,8) | $\mathcal{U}(10,14)$ | 85 | 100 |
(6,8) | $\mathcal{L}(12,14)$ | 300 | 400 |
(7,8) | $\mathcal{U}(9,14)$ | 95 | 90 |
Uncertain random simulation and GA are employed to search for the optimal solution. For the above project, the parameters in GA are set as the population size 70, the mutation probability 0.8, and the crossover probability 0.5. After a run of 1,000 generations, the optimal solution and value are obtained: x^{*}=(1,-1,0,3,0,-3,1,3,-1,3,-1) and Ch{C(x , ξ )≤24100}=0.938, respectively.
Conclusions
In project management, one of the characteristics of the concept ‘project’ is that there is always something new in a project, i.e., a project always has some ‘new’ parts with little or no historical data. Meanwhile, most projects may contain some common or ‘old’ parts (with enough historical data) ever performed by many other finished projects. Hence, uncertainty and randomness should be considered simultaneously in project scheduling problem. In this paper, an uncertain random time-cost trade-off model with dependent-chance programming was built with a crisp equivalent model for the case that uncertain random parameters in the problem are partly random variables and partly uncertain variables. Furthermore, uncertain random simulation and GA were integrated for solving the proposed model.
The TCTP with uncertain random parameters can be regarded as the extension of the fuzzy TCTP, random TCTP, uncertain TCTP, and fuzzy random TCTP. Besides, since in this paper continuous time-cost trade-off relationship was assumed, in future research, discrete or mixed time-cost trade-off relationships can be introduced for modeling other project optimization problems.
Declarations
Acknowledgements
The work was partly supported by the National Natural Science Foundation of China (71371141, 71001080).
Authors’ Affiliations
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