Two empirical uncertain models for project scheduling problem
© 2014 Operational Research Society Ltd. All rights reserved. 0160-5682/14
Journal of the Operational Research Society (2014), 1–10
www.palgrave-journals.com/jors/
Two empirical uncertain models for project scheduling problem Chunxiao Ding and Yuanguo Zhu* Nanjing University of Science and Technology, Nanjing, China The project scheduling problem with uncertain activity durations is considered, and two types of models for uncertain project scheduling problems are established according to different management requirements. These models are transformed to their crisp forms, which may be solved by classical optimization methods. For the models that could not be transformed to their crisp forms, an uncertain simulation is employed to approximate uncertain functions. Finally, two numerical examples are given to illustrate the usefulness of proposed models. Journal of the Operational Research Society advance online publication, 26 November 2014; doi:10.1057/jors.2014.115 Keywords: project scheduling; uncertain theory; uncertain programming; genetic algorithm; uncertain simulation
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Project scheduling problem is to determine the schedule of allocating resources so as to balance the total cost and the completion time. This question has been studied by many researchers. In 1961, Kelley presented a function relationship between the total costs and activity durations, which established the mathematical foundation for project scheduling problem. Following that, Charnes and Cooper (Charnes and Cooper, 1957; Charnes et al, 1964) studied project scheduling problem via chance constrained programming. Kelley (1963) formulated an approach to model project scheduling problem with the objective of minimizing the total cost. All studies mentioned above worked in crisp environments. It is generally known that non-deterministic factors exist in project scheduling problem. The non-determinacy in activity durations was regarded as randomness by Freeman (1960). From then on, the stochastic project scheduling problem was studied widely using probability theory. Ke and Liu (2005) built three types of models for project scheduling problem with stochastic activity durations. Although probability theory has been successfully applied into dealing with project scheduling problem and got lots of attentions, it was unreasonable to treat activity durations as random variables in some cases. For instance, the statistical data may be insufficient, or some probability distributions of activity durations may be unknown and so on. The activity durations in project scheduling problem were assumed to fuzzy variables by Prade (1979). Then Ke and Liu (2007, 2010) presented three types of fuzzy models for
project scheduling problem and provided a hybrid intelligent algorithm to solve them. In order to deal with human non-determinacy, uncertainty theory, a branch of axiomatic mathematics was founded by Liu in 2007 (Liu, 2007) and refined in 2010 (Liu, 2010). Nowadays, uncertainty theory has been applied to uncertain programming (Liu, 2009a; Gao, 2011; Rong, 2011; Gao, 2012; Sheng and Yao, 2012; Gao et al, 2013; Yang and Zhou, 2014), uncertain logic (Chen and Ralescu, 2011), uncertain entropy (Yao, 2014), uncertain process (Liu, 2008; Yao and Li, 2012), uncertain calculus (Liu, 2009b), uncertain differential equations (Chen et al, 2012; Liu, 2008; Peng and Yao, 2011), uncertain optimal control (Zhu, 2010), and so on. Uncertain project scheduling problem was proposed by Liu (2010) in uncertainty theory. In 2012, Zhang and Chen (2012) proposed a new expected model of uncertain project scheduling problem. On the basis of previous studies, this paper presents two new empirical uncertain models for uncertain project scheduling problem. Influenced by human behaviour, it is more reasonable to let activity durations be uncertain variables, whose uncertainty distributions are analysed by expert’s experimental data. In addition to inverse uncertainty distribution, an uncertain simulation (Zhu, 2012) will be employed to deal with uncertain programming models. The rest frame is organized as follows. In next section, some concepts and properties in uncertainty theory are reviewed. In Section 3, we state the uncertain project scheduling problem, and obtain the empirical uncertainty distributions of activity durations. In Section 4, two new programming models, uncertain multi-objective pessimistic value model and uncertain time range measure optimization model, are presented. In Section 5, some theorems are proposed to convert these uncertain optimization models into their equivalent crisp forms. In Section 6, an uncertain simulation is introduced to approximate uncertain
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1. Introduction
*Correspondence: Yuanguo Zhu, Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China. E-mail: [email protected]
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measure of the completion time constrained in a given time range. In Section 7, two numerical examples are suggested to illustrate the usefulness of the models and the efficiency of the algorithm. At last a brief summary is given.
Let ξ be an uncertain variable. Assume that we have obtained a set of expert’s experimental data ðx1 ; α1 ; x2 ; α2 ; ¼ ; xn ; αn Þ
2. Preliminary
that meet the following consistence condition (perhaps after a rearrangement) x1 < x2 < <xn ; 0 > < i + 1 - xi Þ ; if αi ⩽ α ⩽ αi + 1 ; 1 ⩽ i > > : if αn > > < ΦðxÞ ¼ αi + > > > : 1;
k¼1
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Axiom 4: (Product measure axiom) Let Γk be non-empty sets on which Mk are uncertain measures, k = 1, 2, … , respectively. Then the product uncertain measure M is an uncertain measure on the product σ-algebra L1 ´ L2 ´ ¼ , satisfying ( ) 1 Y 1 Λk ¼ ^ Mk fΛk g M k¼1
That is, for each event Λ ∈ L, we have
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8 > > > >
> > > :
sup
for all α ∈ [0, 1]. Theorem 2 (Liu, 2010) Let ξ1, ξ2, …, ξn be independent uncertain variables with uncertainty distributions Φ1, Φ2, …, Φn, respectively. If f(x1, x2, …, xn) is strictly increasing with respect to x1, x2, …, xm, and strictly decreasing with respect to xm + 1, xm + 2, …, xn, then ξ = f(ξ1, ξ2, …, ξn) has
min Mk fΛk g
if
Λ1 ´ Λ2 ´ Λ 1 ⩽ k0:5
Λ1 ´ Λ2 ´ Λ 1 ⩽ k > > > :
inf c
MfAi g;
1 P
if
MfAi g; if
B ∪ Ai i¼1
0:5;
1 P
inf
B ∪ Ai i¼1
inf c
Definition 6 (Zhu, 2012) An uncertain vector ξ = (ξ1, ξ2, … , ξn) is common if every uncertain variable ξi is common for i =1, 2, …, n.
MfAi g < 0:5
1 P
B ∪ Ai i¼1
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Each element Ai emerging in the sequel is in ϱ. The uncertainty distribution of f(ξ) for a common uncertain variable ξ or a common uncertain vector is as follows.
MfAi g < 0:5
otherwise
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(4)
Theorem 3 (Zhu, 2012)
1 P
MfAi g;
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inf
ff ðξÞ ⩽ xg ∪ Ai i¼1
1> > > > > > :
1 P
inf
ff ðξÞ>xg ∪ Ai i¼1
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ΨðxÞ ¼ Mf f ðξÞ ⩽ xg ¼
8 > > > > > >
xg ∪ Ai i¼1
MfAi g < 0:5
MfAi g < 0:5
(5)
otherwise
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0:5;
if
inf
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(ii) Let f : 0:5 > > 1⩽k⩽n > Λ1 ´ Λ2 ´ ´ Λn Λ 1 ⩽ k ⩽ n > < Λ1 ´ Λ2 ´ ´ Λn Λ ¼ 1min Mk fΛk g if sup min Mk fΛk g > 0:5 sup > Λ1 ´ Λ2 ´ ´ Λn Λc 1 ⩽ k ⩽ n Λ1 ´ Λ2 ´ ´ Λn Λc 1 ⩽ k ⩽ n > > > : 0:5; otherwise
where Λ = f − 1(− ∞, x), and each Mk fΛk g is derived from 8 > > > > > > < MfΛk g ¼
inf
1 P
Λk ∪ Ai i¼1
1> > > > > > :
inf c
MfAi g;
1 P
Λk ∪ Ai i¼1
0:5;
MfAi g;
if if
inf
1 P
Λk ∪ Ai i¼1
inf c
1 P
Λk ∪ Ai i¼1
otherwise
MfAi g < 0:5 MfAi g < 0:5
(7)
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of activity ðnode 1; node jÞ 2 A is its loan time, denoted by
Generally, a project scheduling problem could be depicted by a directed acyclic network graph G ¼ ðV; AÞ; in which nodes set V ¼ fnode 1; node 2; ¼ ; node ðn + 1Þg corresponds to milestones and arcs set A ¼ fðnode i; node jÞ j ðnode i; node jÞ 2 Ag stands for activities (see Figure 1). In order to model project scheduling problem with uncertain activity durations, we introduce the following indices and parameters:
The starting time of the whole project is the minimal value of T1j (x, ξ). The starting time Tij (x, ξ) of activity ðnode i; node jÞ 2 A is Tij ðx; ξÞ ¼ xij _
max
ðTki ðx; ξÞ + ξki Þ
ðnode k;node iÞ2A
By a recursive process, we can obtain the completion time T(x, ξ) of total project is T ðx; ξÞ ¼ max Tk;n + 1 ðx; ξÞ + ξk;n + 1 (8) ðnode k; node n + 1Þ2A
3.2. Uncertain total cost The total cost of project could be presented as follows: X
Cðx; ξÞ ¼
cij ð1 + rÞdT ðx; ξÞ - xij e
(9)
ðnode i; node jÞ2A
Where dae represents the minimal integer greater than or equal to a, and T(x, ξ) is the completion time of project mentioned in (8).
3.3. Expert’s experimental data for uncertain activity durations The expert’s experimental data are collected by questionnaire survey and uncertain statistics. We invite m domain experts to provide experimental data of uncertain activity durations.
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uncertain duration of activity ðnode i; node jÞ 2 A; ξij : Φij (t): uncertainty distribution of ξij; inverse uncertainty distribution of ξij; Φij−1(α): ξ ¼ fξij ; ðnode i; node jÞ 2 Ag: vector of uncertain durations; xij: decision variable representing loan time of activity ðnode i; node jÞ 2 A; x ¼ fxij ; ðnode i; node jÞ 2 Ag: vector of decision variables; uncertain starting time of activity ðnode i; Tij(x, ξ): node jÞ 2 A; Ψij− 1(x,α): inverse uncertainty distribution of Tij(x, ξ); T(x, ξ): uncertain completion time of total project; Ψ − 1(x,α): inverse uncertainty distribution of T(x, ξ); cij: a given loan amount of activity ðnode i; node jÞ 2 A; c ¼ fcij ; ðnode i; node jÞ 2 Ag: the vector of cost variables; r: interest rate; C(x, ξ): uncertain total cost of project; ϒ - 1 ðx; αÞ: inverse uncertainty distribution of C(x, ξ).
T1j ðx; ξÞ ¼ x1j ; ðnode 1; node jÞ 2 A
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3. Uncertain project scheduling problem
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In mathematics, the lexicographic or lexicographical order (also known as lexical order, dictionary order, alphabetical order, or lexicographical product) is a generalization of the way that the alphabetical order of words is based on the alphabetical order of their component letters (http://en.wikipedia.org/wiki/ Lexicographical_order). Assume that the unit of loan time is month. So xij is a non-negative integer. We rank ξij in ξ, xij in x, and cij in c with lexicographical (alphabetical) order of ij for all (node i, node j) in A, respectively.
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y1ij ; α11 ij
6 6 6 1 21 6 yij ; αij 6 6 6 .. 6 . 6 4 1 yij ; αm1 ij
y2ij ; α12 ij y2ij ; α22 ij
.. . y2ij ; αm2 ij
ynij ; α1n ij
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7 7 7 7 ynij ; α2n ij 7 7 7 .. .. 7 . . 7 5 n mn yij ; αij
3.1. Uncertain completion time
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In this section we formulate the completion time and the total cost of uncertain project scheduling problem. The starting time
where y1ij, y2ij … ynij mean predetermined possible data of uncertain activity duration ξij, and αijkl is the kth expert’s belief degree for that uncertain activity duration ξij is less than or equal to a predetermine duration yijl, k =1, 2, … , m, l =1, 2, … , n, ðnode i; node jÞ 2 A. The expected value of m experts’ experimental data is considered to be the experimental data of uncertain duration time ξij, written as
Figure 1 A project.
y1ij ; α1ij ; y2ij ; α2ij ; ¼ ; ynij ; αnij
P kl where αijl ¼ m k¼1 αij =m; l ¼ 1; 2; ¼ ; n. On the basis of the experts’ experimental data and Definition 4, the empirical uncertainty distribution of uncertain activity duration ξij could
Chunxiao Ding and Yuanguo Zhu—Two empirical uncertain models for project scheduling problem
be written as 8 0; if t < y1ij > > > > > > l+1 l l > > α α t y > ij ij ij < l αij + ; Φij ðtÞ ¼ ylij+ 1 - yijl > > > > > if yijl ⩽ t ⩽ ylij+ 1 ; l ¼ 1; 2; ; n - 1 > > > > : 1; if t > ynij
(10)
and its inverse uncertainty distribution is 8 1 yij ; if α < α1ij > > > > > > > ðα - αijl Þðylij+ 1 - yijl Þ > < yl + ; ij αijl + 1 - αijl (11) Φij- 1 ðαÞ ¼ > > > l l+1 > if αij ⩽ α ⩽ αij ; l ¼ 1; 2; ; n - 1 > > > > : n yij ; if α > αnij
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where α and β are two given confidence levels, T(x, ξ) is the uncertain completion time defined by (8), and C(x, ξ) is the uncertain total cost defined by (9). The most common approach to multi-objective optimization is the weighted sum method, in which all objective functions are combined to form a single function. Also weighted sum criterion is a type of utility function in which parameters are used to describe the preferences and keep balance between two objectives. Thus the uncertain multi-objective pessimistic value model (12) could be transformed into the following model, 8 min min w1 T + w2 C > > > x > > > > > < subject to (13) M T ðx; ξÞ ⩽ T ⩾ α > > > > > M Cðx; ξÞ ⩽ C ⩾ β > > > : x is an integer vector where w1 and w2 are the weights of total cost and completion time.
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4.2.Uncertain time range measure optimization model In some practical projects, a project may be required to be completed in a predetermined time range. Decision makers tend to maximize the uncertain measure of the event that completion time belongs to a given time range, under the condition that the uncertain measure of total cost controlled in a given budget is not less than a predetermined confidence level. In this case, the uncertain time range measure optimization model is formulated as follows. 8 max MfT1 ⩽ T ðx; ξÞ ⩽ T2 g > > x > > > < subject to (14) > > M C ð x; ξ Þ ⩽ C ⩾ α f g 0 > > > : x is an integer vector
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In many projects, especially large-scale ones, the costs are obtained by loans. So how to design the schedule of allocating loans is always a significant problem to decision makers. Because the models with uncertain variables cannot be calculated as crisp ones, uncertain measure is employed to formulate the uncertain models in this paper. In this section, two kinds of methods are presented to design the optimal schedule of allocating loans according to different management goals.
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4. Uncertain programming models
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4.1. Uncertain multi-objective pessimistic value model
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In order to constrain uncertain measure in a given confidence level, uncertain pessimistic value is defined as follows.
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Definition 7 (Liu, 2007) The pessimistic value of an uncertain variable ξ is defined as minfrjMfξ ⩽ rg ⩾ αg
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where α is a predetermined confidence level.
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In practice, decision makers may want to minimize the total cost, as well as to finish the project as soon as possible. We optimize the schedule of allocating loans in order to minimize the hybrid uncertain pessimistic value of total cost and completion time under uncertain measure constraints. The multiobjective pessimistic value model can be written as 8 minfðmin T; min CÞg > > > x > > > > > < subject to (12) M T ðx; ξÞ ⩽ T ⩾ α > > > > > > M C ðx; ξÞ ⩽ C ⩾ β > > : x is an integer vector
where α is a predetermined confidence level, (T1, T2) is a given time range of uncertain completion time, T(x, ξ) is the uncertain completion time defined by (8), C(x, ξ) is the uncertain total cost defined by (9), and C0 is a given budget of total cost. In uncertain optimization models established above, there exist three kinds of uncertain functions with uncertain variables. (i) The inequality MfCðx; ξÞ ⩽ C0 g ⩾ α means that the uncertain measure of total cost controlled in a given budget is not less than a predetermined level. The inequality MfT ðx; ξÞ ⩽ T0 g ⩾ α denotes that the uncertain measure of completion time constrained in a given due date is not less than a given level. (ii) The term min C j M C ðx; ξÞ ⩽ C ⩾ β indicates the pessimistic value of project total cost. Similarly, the term min T j M T ðx; ξÞ ⩽ T ⩾ α means the pessimistic value of completion time.
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(ii) For i > 1, j < n + 1, the uncertain starting time of activity ðnode i; node jÞ 2 A could be written as
(iii) The term MfT1 ⩽ T ðx; ξÞ ⩽ T2 g denotes the uncertain measure of completion time constrained in a predetermined time range.
Tij ðx; ξÞ ¼ xij _
The cases (i) and (ii) could be calculated directly by using inverse uncertainty distribution. In the next section we convert these uncertain functions with uncertain variables into their crisp forms by making use of properties in uncertainty theory. However, the case (iii) could not be transformed to its crisp form directly by using classical method. We will introduce the uncertain simulation to approximate it in Section 6. Generally, a predetermined confidence level is provided with a number in the interval (0, 1). That is, 0 < α < 1 and 0 < β < 1.
Ψij- 1 ðx; αÞ ¼ xij _
we have Ψij- 1 ðx; αÞ ¼ xij _
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ðnode k; node iÞ2A
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(ii) For i > 1, j < n + 1, the inverse uncertainty distribution of starting time Tij(x, ξ) can be written as -1 max Ψki- 1 ðx; αÞ Ψij ðx; αÞ ¼ xij _
TH
α - αkil ykil + 1 - ykil + ykil + αlki+ 1 - αkil
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max
-1 Ψi;n + 1 ðx; αÞ
l α - αi;n +1
l yli;n+ 1+ 1 - yi;n +1
l αli;n+ 1+ 1 - αi;n +1
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ykil
Ψki- 1 ðx; αÞ
α - αkil ylki+ 1 - ykil + αlki+ 1 - αkil
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(iii) For j = n + 1, note that ði; n + 1Þ 2 A is one of the ending activities of project. The completion time of total project is the maximum of all ending activities, written as: T ðx; ξÞ ¼ max Ti;n + 1 ðx; ξÞ + ξi;n + 1 ðnode i;node n + 1Þ2A
By Theorem 1, we have the inverse uncertainty distribution of T(x, ξ) -1 -1 Ψ - 1 ðx; αÞ ¼ max Ψi;n + 1 ðx; αÞ + Φi;n + 1 ðx; αÞ ðnode i; node n + 1Þ2A
where ðnode k; node iÞ 2 A is the foregoing activity of Tij (x, ξ); (iii) For j = n + 1, the inverse uncertainty distribution of completion time T(x, ξ) can be written as ðnode i; node n + 1Þ2A
+
max
ðnode k;node iÞ2A
C
Ψ1j- 1 ðx; αÞ ¼ x1j ;
α - αkil ylki+ 1 - ykil ; αkil + 1 - αkil
ðnode k; node iÞ 2 A
Theorem 4 Let ξij be an uncertain activity duration time with inverse empirical uncertainty distribution Φij− 1(α). We can obtain the inverse uncertainty distribution Ψij- 1 ðx; αÞ of uncertain starting time Tij(x, ξ).
l + @yi;n +1 +
ðnode k; node iÞ2A
-1 Ψki ðx; αÞ + Φki- 1 ðx; αÞ
Since
In this section, objective and constraint functions with uncertain variables just like MfC ðx; ξÞ ⩽ C0 g ⩾ α and min C j M Cðx; ξÞ ⩽ C ⩾ β are transformed into their crisp forms by making use of properties in uncertainty theory.
0
max
Φki- 1 ðx; αÞ ¼ ykil +
(i) For i = 1, the inverse uncertainty distribution of the starting time T1j(x, ξ) of activity ðnode 1; node jÞ 2 A can be written as
ðTki ðx; ξÞ + ξki Þ
where ðnode k; node iÞ 2 A is the foregoing activity of Tij(x, ξ). By Theorem 1, we have the inverse uncertainty distribution
5. Equivalent transformation
Ψ - 1 ðx; αÞ ¼
max
ðnode k; node iÞ2A
11 AA
Since the inverse empirical uncertainty distribution is l l+1 l y α - αi;n y +1 i;n + 1 i;n + 1 -1 l Φi;n + 1 ðx; αÞ ¼ yi;n + 1 + l+1 l α αi;n +1 i;n + 1 we have the inverse uncertainty distribution of completion time -1 max Ψi;n Ψ - 1 ðx; αÞ ¼ + 1 ðx; αÞ ðnode i;node n + 1Þ2A
Proof (i) For i = 1, the starting time T1j(x, ξ) of activity ðnode 1; node jÞ 2 A is
0 l + @yi;n +1 +
l α - αi;n +1
T1j ðx; ξÞ ¼ x1j ; ðnode 1; node jÞ 2 A we have Ψ1j- 1 ðx; αÞ ¼ x1j .
The proof is completed. □
l yli;n+ 1+ 1 - yi;n +1
l αli;n+ 1+ 1 - αi;n +1
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Chunxiao Ding and Yuanguo Zhu—Two empirical uncertain models for project scheduling problem
Theorem 5 Let ξij be an uncertain activity duration time with empirical uncertainty distribution. We can obtain the inverse uncertainty distribution ϒ − 1(x, α) of total cost C(x, ξ): ϒ - 1 ðx; αÞ ¼
X
cij ð1 + rÞdΨ
-1
It follows from Theorem 4 and Theorem 5 that 8 > min min w1 C + w2 T > > x > > > > > > subject to > > > > > > > -1 > > max > ðnode i;node n + 1Þ2A Ψi;n + 1 ðx; αÞ > > > < !! ðα - αli;n + 1 Þðyli;n+ 1+ 1 - yli;n + 1 Þ > l > ⩽T + yi;n + 1 + > > > αli;n+ 1+ 1 - αli;n + 1 > > > > > > > -1 P > > cij ð1 + rÞdΨ ðx; βÞ - xij e ⩽ C > > > > ðnode i;node jÞ2A > > > > : x is an integer vector (17)
ðx;αÞ - xij e
ðnode i; node jÞ2A
where Ψ − 1(x, α) is the inverse uncertainty distribution of completion time T(x, ξ) obtained in Theorem 4, cij is a fixed cost of activity ðnode i; node jÞ 2 A, and xij is loan time of activity ðnode i; node jÞ 2 A. Proof The conclusion can be obtained directly from Theorem 2 and Theorem 4. The uncertain programming models could be transformed to their crisp forms. □ Theorem 6 The uncertain multi-objective pessimistic value model (13) is equivalent to
cij ð1 + rÞdΨ
-1
ðx;βÞ - xij e
⩽C
(15) ⩽T
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ðnode i; node jÞ2A
x is an integer vector
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Proof Let the uncertainty distribution of T(x, ξ) be Ψ(x, t) and the uncertainty distribution of C(x, ξ) be ϒ(x, c). It follows from the definition of uncertain distribution (Definition 2) that M T ðx; ξÞ ⩽ T ⩾ α is equivalent to Ψ x; T ⩾ α. Then we can obtain Ψ - 1 ðx; αÞ ⩽ T by using Definition 3 and Theorem 1. In the same way, M C ðx; ξÞ ⩽ C ⩾ β could be written as ϒ - 1 ðx; βÞ ⩽ C. Therefore the multi-objective pessimistic value model (13) is equivalent to 8 min min w1 C + w2 T > > x > > > > > > < subject to > > > > > > > > :
8 max MfT1 ⩽ T ðx; ξÞ ⩽ T2 g > > x > > > > > subject to < -1 P > cij ð1 + rÞdΨ ðx;αÞ - xij e ⩽ C0 > > > ðnode i; node jÞ2A > > > : x is an integer vector
Ψ - 1 ðx; αÞ ⩽ T
ϒ - 1 ðx; βÞ ⩽ C x is an integer vector
(16)
(18)
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!!
αli;n+ 1+ 1 - αli;n + 1
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Theorem 7 The uncertain time range measure optimization model (14) is equivalent to
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ðα - αli;n + 1 Þðyli;n+ 1+ 1 - yli;n + 1 Þ
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+
yli;n + 1
TH
> > > > > > > > > > > > > > > > > > > > > > :
□
PY
8 > min min w1 C + w2 T > > x > > > > > > subject to > > > > > > > -1 > > max Ψi;n + 1 ðx; αÞ > > ðnode i; node n + 1Þ2A > >
: 1;
t - ði + j + 1Þ ; 4
if i + j + 1 ⩽ t > > < t - ði + jÞ
ðijÞ
PY
otherwise, return L1 ¼ 0:5, L2 ¼ 0:5, ðnode i; node jÞ 2 A: Step 7. If a ¼ minðnode i; node jÞ 2 A LðijÞ 1 >0:5; then L = a; ðijÞ
¼ð30; 40; 60; 70; 80; 100; 110Þ
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if b ¼ minðnode i; node jÞ 2 A L2 >0:5; then L = 1 − b; otherwise, L = 0.5.
7.1. Example 1
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7. Numerical examples
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Take the project in Figure 1 for example, we consider a project with six milestones and seven activities. The vector of uncertain activities is
TH
ξ ¼ fξij ; ðnode i; node jÞ 2 Ag
¼ ðξ12 ; ξ13 ; ξ24 ; ξ34 ; ξ35 ; ξ46 ; ξ56 Þ
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The inverse empirical uncertainty distribution is obtained from expert’s experimental data as follows. We invite five experts to provide experimental data 3 2 ði + j; 0Þ ði + j + 1; 0:3Þ ði + j + 3; 1Þ 7 6 6 ði + j; 0Þ ði + j + 1; 0:7Þ ði + j + 3; 1Þ 7 7 6 7 6 6 ði + j; 0Þ ði + j + 1; 0:4Þ ði + j + 3; 1Þ 7 7 6 7 6 6 ði + j; 0Þ ði + j + 1; 0:6Þ ði + j + 3; 1Þ 7 5 4 ði + j; 0Þ ði + j + 1; 0:5Þ ði + j + 3; 1Þ From the method provided in Section 3.3, we can obtain the empirical uncertainty distribution of uncertain activity duration time ξij εði + j; 0; i + j + 1; 0:5; i + j + 3; 1Þ
If we want to minimize the total cost, as well as to finish the project as quickly as possible, two goals should be achieved simultaneously. Hence a multi-objective pessimistic value model introduced in (13) is needed. Assume that w1 = w2 = 0.5. Then the uncertain multi-objective pessimistic value model (13) is 8 min min 0:5C + 0:5T > > x > > > > > subject to > < (20) M T ðx; ξÞ ⩽ T ⩾ 0:9 > > > > > M Cðx; ξÞ ⩽ C ⩾ 0:9 > > > : x is an integer vector It follows from Theorem 6 that the above model is equivalent to 8 min min 0:5C + 0:5T > > x > > > > > > < subject to > > > > > > > > :
Ψ - 1 ðx; 0:9Þ ⩽ T
ϒ - 1 ðx; 0:9Þ ⩽ C x is an integer vector
Chunxiao Ding and Yuanguo Zhu—Two empirical uncertain models for project scheduling problem
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where Ψ - 1 ðx; 0:9Þ ¼
max
ðnode k; node 6Þ2A
-1 -1 Ψk;6 ðx; 0:9Þ + Φk;6 ð0:9Þ
-1 -1 -1 -1 -1 ð0:9Þ _ x24 + Φ24 ð0:9Þ _ x13 + Φ13 ð0:9Þ _ x34 + Φ34 ð0:9Þ _ x46 + Φ46 ð0:9Þ x12 + Φ12 -1 -1 -1 ð0:9Þ _ x35 + Φ35 ð0:9Þ _ x56 + Φ56 ð0:9Þ _ x13 + Φ13 (" # l + 1 l 0:9 - αl24 ðyl24+ 1 - yl24 Þ ð0:9 - αl12 y12 - y12 l l ¼ x12 + y12 + _ x24 + y24 + l+1 αl12+ 1 - αl12 - αl24 α24 0:9 - αl13 yl13+ 1 - yl13 0:9 - αl34 yl34+ 1 - yl34 l l _ x34 + y34 + _ x46 _ x13 + y13 + αl13+ 1 - αl13 αl34+ 1 - αl34 ð0:9 - αl46 Þ yl46+ 1 - yl46 0:9 - αl13 yl13+ 1 - yl13 l l _ x13 + y13 + _ x35 + y46 + l+1 α46 αl13+ 1 - αl13 - αl46 0:9 - αl35 yl35+ 1 - yl35 0:9 - αl56 yl56+ 1 - yl56 l l _ x56 + y56 + + y35 + αl35+ 1 - αl35 αl56+ 1 - αl56 ¼
(21)
¼ f½ðx12 + 4:6Þ _ x24 + 7:6 _ ½ðx13 + 5:6Þ _ x34 + 8:6 _ x46 + 11:6g
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_f½ðx13 + 5:6Þ _ x35 + 9:6 _ x56 + 12:6g
ðnode i; node jÞ
ðx;0:9Þ - x12 e
+ c34 ð1 + 0:053ÞdΨ
-1
ðx;0:9Þ - x34 e
ðx;0:9Þ - x12 e
+ 70 ´ ð1 + 0:053ÞdΨ
-1
ðx;0:9Þ - x34 e
-1
ðx;0:9Þ - x13 e -1
+ 40 ´ ð1 + 0:053ÞdΨ
+ c24 ð1 + 0:053ÞdΨ
ðx;0:9Þ - x46 e
-1
-1
-1
ðx;0:9Þ - x24 e
+ c56 ð1 + 0:053ÞdΨ
ðx;0:9Þ - x13 e
+ 80 ´ ð1 + 0:053ÞdΨ
-1
ðx;0:9Þ - x56 e
+ 60 ´ ð1 + 0:053ÞdΨ
ðx;0:9Þ - x35 e
-1
(22)
ðx;0:9Þ - x24 e
+ 100 ´ ð1 + 0:053ÞdΨ
-1
ðx;0:9Þ - x46 e
ðx;0:9Þ - x56 e
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+ 110 ´ ð1 + 0:053ÞdΨ
-1
+ c46 ð1 + 0:053ÞdΨ
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¼ 30 ´ ð1 + 0:053ÞdΨ
-1
+ c13 ð1 + 0:053ÞdΨ
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-1
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¼ c12 ð1 + 0:053ÞdΨ
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ϒ - 1 ðx; 0:9Þ -1 P ¼ cij ð1 + 0:053ÞdΨ ðx;0:9Þ - xij e
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and
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We suppose that the generation number is 600, the population size is 30, the probability of mutation is pm = 0.3, and the probability of crossover is pc = 0.2. By Genetic Algorithm (GA), we obtain an optimal schedule for loan time of (20) x* ¼ðx12 ; x13 ; x24 ; x34 ; x35 ; x46 ; x56 Þ ¼ð1; 3; 7; 8; 4; 15; 14Þ and C ¼ 1271:8, T ¼ 26:8:
7.2. Example 2 In practice, for example in road construction, blocking the whole road is impossible. So we have no choice but to build the
road piecewise in order to minimize the inconvenience. Thus the durations of road blocking is vital in project scheduling. The uncertain time range measure optimization model (14) can be used to measure the uncertain confidence level that the project is completed on schedule. Assume that time range is [26, 30] and budget of total project is C0 = 1300. The uncertain time range measure optimization model (14) is 8 max Mf26 ⩽ T ðx; ξÞ ⩽ 30g > > x > > > < subject to > > MfCðx; ξÞ ⩽ 1300g ⩾ 0:9 > > > : x is an integer vector
(23)
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It follows from Theorem 7 that the above model is equivalent to 8 max Mf26 ⩽ T ðx; ξÞ ⩽ 30g > > > x > > < subject to > > ϒ - 1 ðx; 0:9Þ ⩽ 1300 > > > : x is an integer vector where the ϒ − 1(x, 0.9) can be obtained in (22). And Mf26 ⩽ T ðx; ξÞ ⩽ 30g is calculated by uncertain simulation provided in Section 6 with parameters N = 1000, n = 7. We suppose that the generation number is 600, the population size is 40, the probability of mutation is pm = 0.3, and the probability of crossover is pc = 0.2. GA yields that an optimal schedule for loan time of (23) is x* ¼ðx12 ; x13 ; x24 ; x34 ; x35 ; x46 ; x56 Þ ¼ð3; 0; 7; 7; 6; 16; 8Þ and the optimal uncertain measure is
PY
L ¼ Mf26 ⩽ T ðx; ξÞ ⩽ 30g ¼ 0:84637
Chen X and Ralescu DA (2011). A note on truth value in uncertain logic. Expert Systems with Applications 38(12): 15582–15586. Freeman RJ (1960). A generalized network approach to project activity sequencing. Institute of Radio Engineers Transactions on Engineering Management 7(3): 103–107. Gao Y (2011). Shortest path problem with uncertain arc lengths. Computers and Mathematics with Applications 62(6): 2591–2600. Gao Y (2012). Uncertain models for single facility location problem on networks. Applied Mathematical Modelling 36(6): 2592–2599. Gao Y, Wen M and Ding S (2013). (s, S) policy for uncertain single period inventory problem. International Journal of Uncertainty, Fuzziness and Knowledge -Based Systems 21(6): 945–953. Ke H and Liu B (2005). Project scheduling problem with stochastic activity duration times. Applied Mathematics and Computation 168(1): 342–353. Ke H and Liu B (2007). Project scheduling problem with mixed uncertainty of randomness and fuzziness. European Journal of Operational Research 183(1): 135–147. Ke H and Liu B (2010). Fuzzy project scheduling problem and its hybrid intelligent algorithm. Applied Mathematical Modelling 34(2): 301–308. Kelley JE (1961). Critical-path planning and scheduling: Mathematical basis. Operations Research 9(3): 296–320. Kelley JE (1963). The critical-path method: Resources planning and scheduling. Industrial Scheduling 13: 347–365. Liu B (2007). Uncertainty Theory. 2nd edn. Springer-Verlag: Berlin. Liu B (2008). Fuzzy process hybrid process and uncertain process. Journal of Uncertain Systems 2(1): 3–16. Liu B (2009a). Theory and Practice of Uncertain Programming. 2nd edn. Springer-Verlag: Berlin. Liu B (2009b). Some research problems in uncertainty theory. Journal of Uncertain Systems 3(1): 3–10. Liu B (2010). Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Springer-Verlag: Berlin. Peng J and Yao K (2011). A new option pricing model for stocks in uncertainty markets. International Journal of Operations Research 8(2): 18–26. Peng Z and Iwamura K (2010). A sufficient and necessary condition of uncertainty distribution. Journal of Interdisciplinary Mathematics 13(3): 277–285. Prade H (1979). Using fuzzy set theory in a scheduling problem: A case study. Fuzzy Sets and Systems 2(2): 153–165. Rong L (2011). Two new uncertainty programming models of inventory with uncertain costs. Journal of Information & Computational Science 8(2): 280–288. Sheng Y and Yao K (2012). Fixed charge transportation problem and its uncertain programming model. Industrial Engineering and Management Systems 11(2): 183–187. Yang L and Zhou X (2014). Constraint reformulation and a Lagrangian relaxation-based solution algorithm for a least expected time path problem. Transportation Research Part B: Methodological 59: 22–44. Yao K (2014). Sine entropy of uncertain set and its applications. Applied Soft Computing 22: 432–442. Yao K and Li X (2012). Uncertain alternating renewal process and its application. IEEE Transactions on Fuzzy Systems 20(6): 1154–1160. Zhang X and Chen X (2012). A new uncertain programming model for project scheduling problem. Information: An International Interdisciplinary Journal 15(10): 3901–3910. Zhu Y (2010). Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems: An International Journal 41(7): 535–547. Zhu Y (2012). Functions of uncertain variables and uncertain programming. Journal of Uncertain Systems 6(4): 278–288.
8. Conclusion
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In this paper, we studied project scheduling problems with uncertain activity durations, whose uncertainty distributions were collected by experts’ experimental data. Under the environment of uncertainty, for the purpose of minimizing the total cost and finishing the project as soon as possible, an uncertain multi-objective pessimistic value model was given and solved by using inverse uncertainty distribution. In order to maximize the uncertain measure of the event that the completion time belongs to a given time range, an uncertain time range measure optimization model was established and approximated by the uncertain simulation. The feasibility and effectiveness of the proposed models and algorithm were shown by two numerical examples. The approach for modelling uncertain project scheduling problems is also suitable to deal with other engineering problems.
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Acknowledgements —This work is supported by the National Natural Science Foundation of China (No. 61273009).
References Charnes A and Cooper WW (1957). Chance-constrained programming. Management Science 6(1): 73–79. Charnes A, Cooper WW and Thompson GL (1964). Critical path analysis via chance constrained and stochastic programming. Operational Research 12(3): 460–470. Chen X, Kar X and Ralescu DA (2012). Cross-entropy measure of uncertain variables. Information Sciences 201(15): 53–60.
Received 6 January 2014; accepted 21 October 2014 after one revision
Journal of the Operational Research Society (2014), 1–10
www.palgrave-journals.com/jors/
Two empirical uncertain models for project scheduling problem Chunxiao Ding and Yuanguo Zhu* Nanjing University of Science and Technology, Nanjing, China The project scheduling problem with uncertain activity durations is considered, and two types of models for uncertain project scheduling problems are established according to different management requirements. These models are transformed to their crisp forms, which may be solved by classical optimization methods. For the models that could not be transformed to their crisp forms, an uncertain simulation is employed to approximate uncertain functions. Finally, two numerical examples are given to illustrate the usefulness of proposed models. Journal of the Operational Research Society advance online publication, 26 November 2014; doi:10.1057/jors.2014.115 Keywords: project scheduling; uncertain theory; uncertain programming; genetic algorithm; uncertain simulation
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Project scheduling problem is to determine the schedule of allocating resources so as to balance the total cost and the completion time. This question has been studied by many researchers. In 1961, Kelley presented a function relationship between the total costs and activity durations, which established the mathematical foundation for project scheduling problem. Following that, Charnes and Cooper (Charnes and Cooper, 1957; Charnes et al, 1964) studied project scheduling problem via chance constrained programming. Kelley (1963) formulated an approach to model project scheduling problem with the objective of minimizing the total cost. All studies mentioned above worked in crisp environments. It is generally known that non-deterministic factors exist in project scheduling problem. The non-determinacy in activity durations was regarded as randomness by Freeman (1960). From then on, the stochastic project scheduling problem was studied widely using probability theory. Ke and Liu (2005) built three types of models for project scheduling problem with stochastic activity durations. Although probability theory has been successfully applied into dealing with project scheduling problem and got lots of attentions, it was unreasonable to treat activity durations as random variables in some cases. For instance, the statistical data may be insufficient, or some probability distributions of activity durations may be unknown and so on. The activity durations in project scheduling problem were assumed to fuzzy variables by Prade (1979). Then Ke and Liu (2007, 2010) presented three types of fuzzy models for
project scheduling problem and provided a hybrid intelligent algorithm to solve them. In order to deal with human non-determinacy, uncertainty theory, a branch of axiomatic mathematics was founded by Liu in 2007 (Liu, 2007) and refined in 2010 (Liu, 2010). Nowadays, uncertainty theory has been applied to uncertain programming (Liu, 2009a; Gao, 2011; Rong, 2011; Gao, 2012; Sheng and Yao, 2012; Gao et al, 2013; Yang and Zhou, 2014), uncertain logic (Chen and Ralescu, 2011), uncertain entropy (Yao, 2014), uncertain process (Liu, 2008; Yao and Li, 2012), uncertain calculus (Liu, 2009b), uncertain differential equations (Chen et al, 2012; Liu, 2008; Peng and Yao, 2011), uncertain optimal control (Zhu, 2010), and so on. Uncertain project scheduling problem was proposed by Liu (2010) in uncertainty theory. In 2012, Zhang and Chen (2012) proposed a new expected model of uncertain project scheduling problem. On the basis of previous studies, this paper presents two new empirical uncertain models for uncertain project scheduling problem. Influenced by human behaviour, it is more reasonable to let activity durations be uncertain variables, whose uncertainty distributions are analysed by expert’s experimental data. In addition to inverse uncertainty distribution, an uncertain simulation (Zhu, 2012) will be employed to deal with uncertain programming models. The rest frame is organized as follows. In next section, some concepts and properties in uncertainty theory are reviewed. In Section 3, we state the uncertain project scheduling problem, and obtain the empirical uncertainty distributions of activity durations. In Section 4, two new programming models, uncertain multi-objective pessimistic value model and uncertain time range measure optimization model, are presented. In Section 5, some theorems are proposed to convert these uncertain optimization models into their equivalent crisp forms. In Section 6, an uncertain simulation is introduced to approximate uncertain
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1. Introduction
*Correspondence: Yuanguo Zhu, Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China. E-mail: [email protected]
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measure of the completion time constrained in a given time range. In Section 7, two numerical examples are suggested to illustrate the usefulness of the models and the efficiency of the algorithm. At last a brief summary is given.
Let ξ be an uncertain variable. Assume that we have obtained a set of expert’s experimental data ðx1 ; α1 ; x2 ; α2 ; ¼ ; xn ; αn Þ
2. Preliminary
that meet the following consistence condition (perhaps after a rearrangement) x1 < x2 < <xn ; 0 > < i + 1 - xi Þ ; if αi ⩽ α ⩽ αi + 1 ; 1 ⩽ i > > : if αn > > < ΦðxÞ ¼ αi + > > > : 1;
k¼1
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Axiom 4: (Product measure axiom) Let Γk be non-empty sets on which Mk are uncertain measures, k = 1, 2, … , respectively. Then the product uncertain measure M is an uncertain measure on the product σ-algebra L1 ´ L2 ´ ¼ , satisfying ( ) 1 Y 1 Λk ¼ ^ Mk fΛk g M k¼1
That is, for each event Λ ∈ L, we have
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8 > > > >
> > > :
sup
for all α ∈ [0, 1]. Theorem 2 (Liu, 2010) Let ξ1, ξ2, …, ξn be independent uncertain variables with uncertainty distributions Φ1, Φ2, …, Φn, respectively. If f(x1, x2, …, xn) is strictly increasing with respect to x1, x2, …, xm, and strictly decreasing with respect to xm + 1, xm + 2, …, xn, then ξ = f(ξ1, ξ2, …, ξn) has
min Mk fΛk g
if
Λ1 ´ Λ2 ´ Λ 1 ⩽ k0:5
Λ1 ´ Λ2 ´ Λ 1 ⩽ k > > > :
inf c
MfAi g;
1 P
if
MfAi g; if
B ∪ Ai i¼1
0:5;
1 P
inf
B ∪ Ai i¼1
inf c
Definition 6 (Zhu, 2012) An uncertain vector ξ = (ξ1, ξ2, … , ξn) is common if every uncertain variable ξi is common for i =1, 2, …, n.
MfAi g < 0:5
1 P
B ∪ Ai i¼1
3
Each element Ai emerging in the sequel is in ϱ. The uncertainty distribution of f(ξ) for a common uncertain variable ξ or a common uncertain vector is as follows.
MfAi g < 0:5
otherwise
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(4)
Theorem 3 (Zhu, 2012)
1 P
MfAi g;
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inf
ff ðξÞ ⩽ xg ∪ Ai i¼1
1> > > > > > :
1 P
inf
ff ðξÞ>xg ∪ Ai i¼1
R
ΨðxÞ ¼ Mf f ðξÞ ⩽ xg ¼
8 > > > > > >
xg ∪ Ai i¼1
MfAi g < 0:5
MfAi g < 0:5
(5)
otherwise
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0:5;
if
inf
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(ii) Let f : 0:5 > > 1⩽k⩽n > Λ1 ´ Λ2 ´ ´ Λn Λ 1 ⩽ k ⩽ n > < Λ1 ´ Λ2 ´ ´ Λn Λ ¼ 1min Mk fΛk g if sup min Mk fΛk g > 0:5 sup > Λ1 ´ Λ2 ´ ´ Λn Λc 1 ⩽ k ⩽ n Λ1 ´ Λ2 ´ ´ Λn Λc 1 ⩽ k ⩽ n > > > : 0:5; otherwise
where Λ = f − 1(− ∞, x), and each Mk fΛk g is derived from 8 > > > > > > < MfΛk g ¼
inf
1 P
Λk ∪ Ai i¼1
1> > > > > > :
inf c
MfAi g;
1 P
Λk ∪ Ai i¼1
0:5;
MfAi g;
if if
inf
1 P
Λk ∪ Ai i¼1
inf c
1 P
Λk ∪ Ai i¼1
otherwise
MfAi g < 0:5 MfAi g < 0:5
(7)
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of activity ðnode 1; node jÞ 2 A is its loan time, denoted by
Generally, a project scheduling problem could be depicted by a directed acyclic network graph G ¼ ðV; AÞ; in which nodes set V ¼ fnode 1; node 2; ¼ ; node ðn + 1Þg corresponds to milestones and arcs set A ¼ fðnode i; node jÞ j ðnode i; node jÞ 2 Ag stands for activities (see Figure 1). In order to model project scheduling problem with uncertain activity durations, we introduce the following indices and parameters:
The starting time of the whole project is the minimal value of T1j (x, ξ). The starting time Tij (x, ξ) of activity ðnode i; node jÞ 2 A is Tij ðx; ξÞ ¼ xij _
max
ðTki ðx; ξÞ + ξki Þ
ðnode k;node iÞ2A
By a recursive process, we can obtain the completion time T(x, ξ) of total project is T ðx; ξÞ ¼ max Tk;n + 1 ðx; ξÞ + ξk;n + 1 (8) ðnode k; node n + 1Þ2A
3.2. Uncertain total cost The total cost of project could be presented as follows: X
Cðx; ξÞ ¼
cij ð1 + rÞdT ðx; ξÞ - xij e
(9)
ðnode i; node jÞ2A
Where dae represents the minimal integer greater than or equal to a, and T(x, ξ) is the completion time of project mentioned in (8).
3.3. Expert’s experimental data for uncertain activity durations The expert’s experimental data are collected by questionnaire survey and uncertain statistics. We invite m domain experts to provide experimental data of uncertain activity durations.
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uncertain duration of activity ðnode i; node jÞ 2 A; ξij : Φij (t): uncertainty distribution of ξij; inverse uncertainty distribution of ξij; Φij−1(α): ξ ¼ fξij ; ðnode i; node jÞ 2 Ag: vector of uncertain durations; xij: decision variable representing loan time of activity ðnode i; node jÞ 2 A; x ¼ fxij ; ðnode i; node jÞ 2 Ag: vector of decision variables; uncertain starting time of activity ðnode i; Tij(x, ξ): node jÞ 2 A; Ψij− 1(x,α): inverse uncertainty distribution of Tij(x, ξ); T(x, ξ): uncertain completion time of total project; Ψ − 1(x,α): inverse uncertainty distribution of T(x, ξ); cij: a given loan amount of activity ðnode i; node jÞ 2 A; c ¼ fcij ; ðnode i; node jÞ 2 Ag: the vector of cost variables; r: interest rate; C(x, ξ): uncertain total cost of project; ϒ - 1 ðx; αÞ: inverse uncertainty distribution of C(x, ξ).
T1j ðx; ξÞ ¼ x1j ; ðnode 1; node jÞ 2 A
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3. Uncertain project scheduling problem
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In mathematics, the lexicographic or lexicographical order (also known as lexical order, dictionary order, alphabetical order, or lexicographical product) is a generalization of the way that the alphabetical order of words is based on the alphabetical order of their component letters (http://en.wikipedia.org/wiki/ Lexicographical_order). Assume that the unit of loan time is month. So xij is a non-negative integer. We rank ξij in ξ, xij in x, and cij in c with lexicographical (alphabetical) order of ij for all (node i, node j) in A, respectively.
2
y1ij ; α11 ij
6 6 6 1 21 6 yij ; αij 6 6 6 .. 6 . 6 4 1 yij ; αm1 ij
y2ij ; α12 ij y2ij ; α22 ij
.. . y2ij ; αm2 ij
ynij ; α1n ij
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7 7 7 7 ynij ; α2n ij 7 7 7 .. .. 7 . . 7 5 n mn yij ; αij
3.1. Uncertain completion time
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In this section we formulate the completion time and the total cost of uncertain project scheduling problem. The starting time
where y1ij, y2ij … ynij mean predetermined possible data of uncertain activity duration ξij, and αijkl is the kth expert’s belief degree for that uncertain activity duration ξij is less than or equal to a predetermine duration yijl, k =1, 2, … , m, l =1, 2, … , n, ðnode i; node jÞ 2 A. The expected value of m experts’ experimental data is considered to be the experimental data of uncertain duration time ξij, written as
Figure 1 A project.
y1ij ; α1ij ; y2ij ; α2ij ; ¼ ; ynij ; αnij
P kl where αijl ¼ m k¼1 αij =m; l ¼ 1; 2; ¼ ; n. On the basis of the experts’ experimental data and Definition 4, the empirical uncertainty distribution of uncertain activity duration ξij could
Chunxiao Ding and Yuanguo Zhu—Two empirical uncertain models for project scheduling problem
be written as 8 0; if t < y1ij > > > > > > l+1 l l > > α α t y > ij ij ij < l αij + ; Φij ðtÞ ¼ ylij+ 1 - yijl > > > > > if yijl ⩽ t ⩽ ylij+ 1 ; l ¼ 1; 2; ; n - 1 > > > > : 1; if t > ynij
(10)
and its inverse uncertainty distribution is 8 1 yij ; if α < α1ij > > > > > > > ðα - αijl Þðylij+ 1 - yijl Þ > < yl + ; ij αijl + 1 - αijl (11) Φij- 1 ðαÞ ¼ > > > l l+1 > if αij ⩽ α ⩽ αij ; l ¼ 1; 2; ; n - 1 > > > > : n yij ; if α > αnij
5
where α and β are two given confidence levels, T(x, ξ) is the uncertain completion time defined by (8), and C(x, ξ) is the uncertain total cost defined by (9). The most common approach to multi-objective optimization is the weighted sum method, in which all objective functions are combined to form a single function. Also weighted sum criterion is a type of utility function in which parameters are used to describe the preferences and keep balance between two objectives. Thus the uncertain multi-objective pessimistic value model (12) could be transformed into the following model, 8 min min w1 T + w2 C > > > x > > > > > < subject to (13) M T ðx; ξÞ ⩽ T ⩾ α > > > > > M Cðx; ξÞ ⩽ C ⩾ β > > > : x is an integer vector where w1 and w2 are the weights of total cost and completion time.
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4.2.Uncertain time range measure optimization model In some practical projects, a project may be required to be completed in a predetermined time range. Decision makers tend to maximize the uncertain measure of the event that completion time belongs to a given time range, under the condition that the uncertain measure of total cost controlled in a given budget is not less than a predetermined confidence level. In this case, the uncertain time range measure optimization model is formulated as follows. 8 max MfT1 ⩽ T ðx; ξÞ ⩽ T2 g > > x > > > < subject to (14) > > M C ð x; ξ Þ ⩽ C ⩾ α f g 0 > > > : x is an integer vector
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In many projects, especially large-scale ones, the costs are obtained by loans. So how to design the schedule of allocating loans is always a significant problem to decision makers. Because the models with uncertain variables cannot be calculated as crisp ones, uncertain measure is employed to formulate the uncertain models in this paper. In this section, two kinds of methods are presented to design the optimal schedule of allocating loans according to different management goals.
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4. Uncertain programming models
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4.1. Uncertain multi-objective pessimistic value model
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In order to constrain uncertain measure in a given confidence level, uncertain pessimistic value is defined as follows.
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Definition 7 (Liu, 2007) The pessimistic value of an uncertain variable ξ is defined as minfrjMfξ ⩽ rg ⩾ αg
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where α is a predetermined confidence level.
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In practice, decision makers may want to minimize the total cost, as well as to finish the project as soon as possible. We optimize the schedule of allocating loans in order to minimize the hybrid uncertain pessimistic value of total cost and completion time under uncertain measure constraints. The multiobjective pessimistic value model can be written as 8 minfðmin T; min CÞg > > > x > > > > > < subject to (12) M T ðx; ξÞ ⩽ T ⩾ α > > > > > > M C ðx; ξÞ ⩽ C ⩾ β > > : x is an integer vector
where α is a predetermined confidence level, (T1, T2) is a given time range of uncertain completion time, T(x, ξ) is the uncertain completion time defined by (8), C(x, ξ) is the uncertain total cost defined by (9), and C0 is a given budget of total cost. In uncertain optimization models established above, there exist three kinds of uncertain functions with uncertain variables. (i) The inequality MfCðx; ξÞ ⩽ C0 g ⩾ α means that the uncertain measure of total cost controlled in a given budget is not less than a predetermined level. The inequality MfT ðx; ξÞ ⩽ T0 g ⩾ α denotes that the uncertain measure of completion time constrained in a given due date is not less than a given level. (ii) The term min C j M C ðx; ξÞ ⩽ C ⩾ β indicates the pessimistic value of project total cost. Similarly, the term min T j M T ðx; ξÞ ⩽ T ⩾ α means the pessimistic value of completion time.
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(ii) For i > 1, j < n + 1, the uncertain starting time of activity ðnode i; node jÞ 2 A could be written as
(iii) The term MfT1 ⩽ T ðx; ξÞ ⩽ T2 g denotes the uncertain measure of completion time constrained in a predetermined time range.
Tij ðx; ξÞ ¼ xij _
The cases (i) and (ii) could be calculated directly by using inverse uncertainty distribution. In the next section we convert these uncertain functions with uncertain variables into their crisp forms by making use of properties in uncertainty theory. However, the case (iii) could not be transformed to its crisp form directly by using classical method. We will introduce the uncertain simulation to approximate it in Section 6. Generally, a predetermined confidence level is provided with a number in the interval (0, 1). That is, 0 < α < 1 and 0 < β < 1.
Ψij- 1 ðx; αÞ ¼ xij _
we have Ψij- 1 ðx; αÞ ¼ xij _
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ðnode k; node iÞ2A
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(ii) For i > 1, j < n + 1, the inverse uncertainty distribution of starting time Tij(x, ξ) can be written as -1 max Ψki- 1 ðx; αÞ Ψij ðx; αÞ ¼ xij _
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α - αkil ykil + 1 - ykil + ykil + αlki+ 1 - αkil
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max
-1 Ψi;n + 1 ðx; αÞ
l α - αi;n +1
l yli;n+ 1+ 1 - yi;n +1
l αli;n+ 1+ 1 - αi;n +1
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ykil
Ψki- 1 ðx; αÞ
α - αkil ylki+ 1 - ykil + αlki+ 1 - αkil
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(iii) For j = n + 1, note that ði; n + 1Þ 2 A is one of the ending activities of project. The completion time of total project is the maximum of all ending activities, written as: T ðx; ξÞ ¼ max Ti;n + 1 ðx; ξÞ + ξi;n + 1 ðnode i;node n + 1Þ2A
By Theorem 1, we have the inverse uncertainty distribution of T(x, ξ) -1 -1 Ψ - 1 ðx; αÞ ¼ max Ψi;n + 1 ðx; αÞ + Φi;n + 1 ðx; αÞ ðnode i; node n + 1Þ2A
where ðnode k; node iÞ 2 A is the foregoing activity of Tij (x, ξ); (iii) For j = n + 1, the inverse uncertainty distribution of completion time T(x, ξ) can be written as ðnode i; node n + 1Þ2A
+
max
ðnode k;node iÞ2A
C
Ψ1j- 1 ðx; αÞ ¼ x1j ;
α - αkil ylki+ 1 - ykil ; αkil + 1 - αkil
ðnode k; node iÞ 2 A
Theorem 4 Let ξij be an uncertain activity duration time with inverse empirical uncertainty distribution Φij− 1(α). We can obtain the inverse uncertainty distribution Ψij- 1 ðx; αÞ of uncertain starting time Tij(x, ξ).
l + @yi;n +1 +
ðnode k; node iÞ2A
-1 Ψki ðx; αÞ + Φki- 1 ðx; αÞ
Since
In this section, objective and constraint functions with uncertain variables just like MfC ðx; ξÞ ⩽ C0 g ⩾ α and min C j M Cðx; ξÞ ⩽ C ⩾ β are transformed into their crisp forms by making use of properties in uncertainty theory.
0
max
Φki- 1 ðx; αÞ ¼ ykil +
(i) For i = 1, the inverse uncertainty distribution of the starting time T1j(x, ξ) of activity ðnode 1; node jÞ 2 A can be written as
ðTki ðx; ξÞ + ξki Þ
where ðnode k; node iÞ 2 A is the foregoing activity of Tij(x, ξ). By Theorem 1, we have the inverse uncertainty distribution
5. Equivalent transformation
Ψ - 1 ðx; αÞ ¼
max
ðnode k; node iÞ2A
11 AA
Since the inverse empirical uncertainty distribution is l l+1 l y α - αi;n y +1 i;n + 1 i;n + 1 -1 l Φi;n + 1 ðx; αÞ ¼ yi;n + 1 + l+1 l α αi;n +1 i;n + 1 we have the inverse uncertainty distribution of completion time -1 max Ψi;n Ψ - 1 ðx; αÞ ¼ + 1 ðx; αÞ ðnode i;node n + 1Þ2A
Proof (i) For i = 1, the starting time T1j(x, ξ) of activity ðnode 1; node jÞ 2 A is
0 l + @yi;n +1 +
l α - αi;n +1
T1j ðx; ξÞ ¼ x1j ; ðnode 1; node jÞ 2 A we have Ψ1j- 1 ðx; αÞ ¼ x1j .
The proof is completed. □
l yli;n+ 1+ 1 - yi;n +1
l αli;n+ 1+ 1 - αi;n +1
1 A
Chunxiao Ding and Yuanguo Zhu—Two empirical uncertain models for project scheduling problem
Theorem 5 Let ξij be an uncertain activity duration time with empirical uncertainty distribution. We can obtain the inverse uncertainty distribution ϒ − 1(x, α) of total cost C(x, ξ): ϒ - 1 ðx; αÞ ¼
X
cij ð1 + rÞdΨ
-1
It follows from Theorem 4 and Theorem 5 that 8 > min min w1 C + w2 T > > x > > > > > > subject to > > > > > > > -1 > > max > ðnode i;node n + 1Þ2A Ψi;n + 1 ðx; αÞ > > > < !! ðα - αli;n + 1 Þðyli;n+ 1+ 1 - yli;n + 1 Þ > l > ⩽T + yi;n + 1 + > > > αli;n+ 1+ 1 - αli;n + 1 > > > > > > > -1 P > > cij ð1 + rÞdΨ ðx; βÞ - xij e ⩽ C > > > > ðnode i;node jÞ2A > > > > : x is an integer vector (17)
ðx;αÞ - xij e
ðnode i; node jÞ2A
where Ψ − 1(x, α) is the inverse uncertainty distribution of completion time T(x, ξ) obtained in Theorem 4, cij is a fixed cost of activity ðnode i; node jÞ 2 A, and xij is loan time of activity ðnode i; node jÞ 2 A. Proof The conclusion can be obtained directly from Theorem 2 and Theorem 4. The uncertain programming models could be transformed to their crisp forms. □ Theorem 6 The uncertain multi-objective pessimistic value model (13) is equivalent to
cij ð1 + rÞdΨ
-1
ðx;βÞ - xij e
⩽C
(15) ⩽T
O
ðnode i; node jÞ2A
x is an integer vector
A
U
Proof Let the uncertainty distribution of T(x, ξ) be Ψ(x, t) and the uncertainty distribution of C(x, ξ) be ϒ(x, c). It follows from the definition of uncertain distribution (Definition 2) that M T ðx; ξÞ ⩽ T ⩾ α is equivalent to Ψ x; T ⩾ α. Then we can obtain Ψ - 1 ðx; αÞ ⩽ T by using Definition 3 and Theorem 1. In the same way, M C ðx; ξÞ ⩽ C ⩾ β could be written as ϒ - 1 ðx; βÞ ⩽ C. Therefore the multi-objective pessimistic value model (13) is equivalent to 8 min min w1 C + w2 T > > x > > > > > > < subject to > > > > > > > > :
8 max MfT1 ⩽ T ðx; ξÞ ⩽ T2 g > > x > > > > > subject to < -1 P > cij ð1 + rÞdΨ ðx;αÞ - xij e ⩽ C0 > > > ðnode i; node jÞ2A > > > : x is an integer vector
Ψ - 1 ðx; αÞ ⩽ T
ϒ - 1 ðx; βÞ ⩽ C x is an integer vector
(16)
(18)
O
!!
αli;n+ 1+ 1 - αli;n + 1
P
Theorem 7 The uncertain time range measure optimization model (14) is equivalent to
C
+
ðα - αli;n + 1 Þðyli;n+ 1+ 1 - yli;n + 1 Þ
R
+
yli;n + 1
TH
> > > > > > > > > > > > > > > > > > > > > > :
□
PY
8 > min min w1 C + w2 T > > x > > > > > > subject to > > > > > > > -1 > > max Ψi;n + 1 ðx; αÞ > > ðnode i; node n + 1Þ2A > >
: 1;
t - ði + j + 1Þ ; 4
if i + j + 1 ⩽ t > > < t - ði + jÞ
ðijÞ
PY
otherwise, return L1 ¼ 0:5, L2 ¼ 0:5, ðnode i; node jÞ 2 A: Step 7. If a ¼ minðnode i; node jÞ 2 A LðijÞ 1 >0:5; then L = a; ðijÞ
¼ð30; 40; 60; 70; 80; 100; 110Þ
C
O
if b ¼ minðnode i; node jÞ 2 A L2 >0:5; then L = 1 − b; otherwise, L = 0.5.
7.1. Example 1
R
7. Numerical examples
O
Take the project in Figure 1 for example, we consider a project with six milestones and seven activities. The vector of uncertain activities is
TH
ξ ¼ fξij ; ðnode i; node jÞ 2 Ag
¼ ðξ12 ; ξ13 ; ξ24 ; ξ34 ; ξ35 ; ξ46 ; ξ56 Þ
A
U
The inverse empirical uncertainty distribution is obtained from expert’s experimental data as follows. We invite five experts to provide experimental data 3 2 ði + j; 0Þ ði + j + 1; 0:3Þ ði + j + 3; 1Þ 7 6 6 ði + j; 0Þ ði + j + 1; 0:7Þ ði + j + 3; 1Þ 7 7 6 7 6 6 ði + j; 0Þ ði + j + 1; 0:4Þ ði + j + 3; 1Þ 7 7 6 7 6 6 ði + j; 0Þ ði + j + 1; 0:6Þ ði + j + 3; 1Þ 7 5 4 ði + j; 0Þ ði + j + 1; 0:5Þ ði + j + 3; 1Þ From the method provided in Section 3.3, we can obtain the empirical uncertainty distribution of uncertain activity duration time ξij εði + j; 0; i + j + 1; 0:5; i + j + 3; 1Þ
If we want to minimize the total cost, as well as to finish the project as quickly as possible, two goals should be achieved simultaneously. Hence a multi-objective pessimistic value model introduced in (13) is needed. Assume that w1 = w2 = 0.5. Then the uncertain multi-objective pessimistic value model (13) is 8 min min 0:5C + 0:5T > > x > > > > > subject to > < (20) M T ðx; ξÞ ⩽ T ⩾ 0:9 > > > > > M Cðx; ξÞ ⩽ C ⩾ 0:9 > > > : x is an integer vector It follows from Theorem 6 that the above model is equivalent to 8 min min 0:5C + 0:5T > > x > > > > > > < subject to > > > > > > > > :
Ψ - 1 ðx; 0:9Þ ⩽ T
ϒ - 1 ðx; 0:9Þ ⩽ C x is an integer vector
Chunxiao Ding and Yuanguo Zhu—Two empirical uncertain models for project scheduling problem
9
where Ψ - 1 ðx; 0:9Þ ¼
max
ðnode k; node 6Þ2A
-1 -1 Ψk;6 ðx; 0:9Þ + Φk;6 ð0:9Þ
-1 -1 -1 -1 -1 ð0:9Þ _ x24 + Φ24 ð0:9Þ _ x13 + Φ13 ð0:9Þ _ x34 + Φ34 ð0:9Þ _ x46 + Φ46 ð0:9Þ x12 + Φ12 -1 -1 -1 ð0:9Þ _ x35 + Φ35 ð0:9Þ _ x56 + Φ56 ð0:9Þ _ x13 + Φ13 (" # l + 1 l 0:9 - αl24 ðyl24+ 1 - yl24 Þ ð0:9 - αl12 y12 - y12 l l ¼ x12 + y12 + _ x24 + y24 + l+1 αl12+ 1 - αl12 - αl24 α24 0:9 - αl13 yl13+ 1 - yl13 0:9 - αl34 yl34+ 1 - yl34 l l _ x34 + y34 + _ x46 _ x13 + y13 + αl13+ 1 - αl13 αl34+ 1 - αl34 ð0:9 - αl46 Þ yl46+ 1 - yl46 0:9 - αl13 yl13+ 1 - yl13 l l _ x13 + y13 + _ x35 + y46 + l+1 α46 αl13+ 1 - αl13 - αl46 0:9 - αl35 yl35+ 1 - yl35 0:9 - αl56 yl56+ 1 - yl56 l l _ x56 + y56 + + y35 + αl35+ 1 - αl35 αl56+ 1 - αl56 ¼
(21)
¼ f½ðx12 + 4:6Þ _ x24 + 7:6 _ ½ðx13 + 5:6Þ _ x34 + 8:6 _ x46 + 11:6g
PY
_f½ðx13 + 5:6Þ _ x35 + 9:6 _ x56 + 12:6g
ðnode i; node jÞ
ðx;0:9Þ - x12 e
+ c34 ð1 + 0:053ÞdΨ
-1
ðx;0:9Þ - x34 e
ðx;0:9Þ - x12 e
+ 70 ´ ð1 + 0:053ÞdΨ
-1
ðx;0:9Þ - x34 e
-1
ðx;0:9Þ - x13 e -1
+ 40 ´ ð1 + 0:053ÞdΨ
+ c24 ð1 + 0:053ÞdΨ
ðx;0:9Þ - x46 e
-1
-1
-1
ðx;0:9Þ - x24 e
+ c56 ð1 + 0:053ÞdΨ
ðx;0:9Þ - x13 e
+ 80 ´ ð1 + 0:053ÞdΨ
-1
ðx;0:9Þ - x56 e
+ 60 ´ ð1 + 0:053ÞdΨ
ðx;0:9Þ - x35 e
-1
(22)
ðx;0:9Þ - x24 e
+ 100 ´ ð1 + 0:053ÞdΨ
-1
ðx;0:9Þ - x46 e
ðx;0:9Þ - x56 e
U
+ 110 ´ ð1 + 0:053ÞdΨ
-1
+ c46 ð1 + 0:053ÞdΨ
TH
¼ 30 ´ ð1 + 0:053ÞdΨ
-1
+ c13 ð1 + 0:053ÞdΨ
R
-1
O
¼ c12 ð1 + 0:053ÞdΨ
C
ϒ - 1 ðx; 0:9Þ -1 P ¼ cij ð1 + 0:053ÞdΨ ðx;0:9Þ - xij e
O
and
A
We suppose that the generation number is 600, the population size is 30, the probability of mutation is pm = 0.3, and the probability of crossover is pc = 0.2. By Genetic Algorithm (GA), we obtain an optimal schedule for loan time of (20) x* ¼ðx12 ; x13 ; x24 ; x34 ; x35 ; x46 ; x56 Þ ¼ð1; 3; 7; 8; 4; 15; 14Þ and C ¼ 1271:8, T ¼ 26:8:
7.2. Example 2 In practice, for example in road construction, blocking the whole road is impossible. So we have no choice but to build the
road piecewise in order to minimize the inconvenience. Thus the durations of road blocking is vital in project scheduling. The uncertain time range measure optimization model (14) can be used to measure the uncertain confidence level that the project is completed on schedule. Assume that time range is [26, 30] and budget of total project is C0 = 1300. The uncertain time range measure optimization model (14) is 8 max Mf26 ⩽ T ðx; ξÞ ⩽ 30g > > x > > > < subject to > > MfCðx; ξÞ ⩽ 1300g ⩾ 0:9 > > > : x is an integer vector
(23)
Journal of the Operational Research Society
It follows from Theorem 7 that the above model is equivalent to 8 max Mf26 ⩽ T ðx; ξÞ ⩽ 30g > > > x > > < subject to > > ϒ - 1 ðx; 0:9Þ ⩽ 1300 > > > : x is an integer vector where the ϒ − 1(x, 0.9) can be obtained in (22). And Mf26 ⩽ T ðx; ξÞ ⩽ 30g is calculated by uncertain simulation provided in Section 6 with parameters N = 1000, n = 7. We suppose that the generation number is 600, the population size is 40, the probability of mutation is pm = 0.3, and the probability of crossover is pc = 0.2. GA yields that an optimal schedule for loan time of (23) is x* ¼ðx12 ; x13 ; x24 ; x34 ; x35 ; x46 ; x56 Þ ¼ð3; 0; 7; 7; 6; 16; 8Þ and the optimal uncertain measure is
PY
L ¼ Mf26 ⩽ T ðx; ξÞ ⩽ 30g ¼ 0:84637
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8. Conclusion
A
U
TH
O
R
C
In this paper, we studied project scheduling problems with uncertain activity durations, whose uncertainty distributions were collected by experts’ experimental data. Under the environment of uncertainty, for the purpose of minimizing the total cost and finishing the project as soon as possible, an uncertain multi-objective pessimistic value model was given and solved by using inverse uncertainty distribution. In order to maximize the uncertain measure of the event that the completion time belongs to a given time range, an uncertain time range measure optimization model was established and approximated by the uncertain simulation. The feasibility and effectiveness of the proposed models and algorithm were shown by two numerical examples. The approach for modelling uncertain project scheduling problems is also suitable to deal with other engineering problems.
O
10
Acknowledgements —This work is supported by the National Natural Science Foundation of China (No. 61273009).
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Received 6 January 2014; accepted 21 October 2014 after one revision
