Uncertain mean-variance and mean-semivariance models for optimal project selection and scheduling

Knowledge-Based Systems 93 (2016) 1–11

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Uncertain mean-variance and mean-semivariance models for optimal project selection and scheduling Xiaoxia Huang∗, Tianyi Zhao, Shamsiya Kudratova Donlinks School of Economics and Management, University of Science and Technology Beijing, Beijing 100083, China

a r t i c l e

i n f o

Article history: Received 7 April 2015 Revised 27 October 2015 Accepted 30 October 2015 Available online 10 November 2015 Keywords: Uncertain programming Project selection Project scheduling Genetic algorithm Investment analysis

a b s t r a c t This paper discusses a joint problem of optimal project selection and scheduling in the situation where initial outlays and net cash inflows of projects are given by experts’ estimates due to lack of historical data. Uncertain variables are used to describe these parameters and the use of them is justified. A new meanvariance and a mean-semivariance models are proposed considering relationship and time sequence order between projects. In order to solve the complex problems, the methods for calculating uncertain lower partial semivariance and higher partial semivariance values are introduced and a hybrid intelligent algorithm which integrates genetic algorithm with cellular automation is provided. In addition, two examples are presented to illustrate the application and significance of the new models, and numerical experiments are done to show the effectiveness of the proposed algorithm.

1. Introduction The original project selection refers to selecting an appropriate combination of projects among available ones to obtain the maximal total profit within budget limitation. A main contribution to the problem was made by Weingartner [38] who first introduced mathematical programming method into the field. Since then, a variety of models were developed to increase applicability of the proposed models to the real life, as discussed by Dickinson et al. [5], Gutjahr et al. [11], Liu and Wang [26], Padberg and Wilczak [31], Xiao et al. [39], etc. These studies treated the project parameters as exact values, yet it is usually difficult to get the exact numbers of them because of the complexity of real world. Therefore, scholars studied the project selection problem with imprecise project parameter values. Traditionally, people employed probability theory to handle the problems. For example, De et al. [4], Keown and Martin [18], Keown and Taylor [19] initiated chance-constrained programming methods to deal with the project selection problem with random inflows and outlays. Medaglia et al. [29] put forward a new evolutionary method for solving linearly constrained project selection problems. Shakhsi-Niaei et al. [34] employed Monte Carlo simulation to propose a two phases framework for project selection problem under randomness and subject to realworld constraints. Although probability theory is powerful for handling indeterminacy, its use is only suitable when people have sufficient historical

∗

Corresponding author. Tel.: +86 10 82376260; fax: +86 10 62333582. E-mail address: [email protected] (X. Huang).

http://dx.doi.org/10.1016/j.knosys.2015.10.030 0950-7051/© 2015 Elsevier B.V. All rights reserved.

© 2015 Elsevier B.V. All rights reserved.

data such that the probability distributions can be obtained. However, there are situations in real life where there are scarce or no historical data. This is especially true for the selection of R & D projects whose initial costs of research and the incomes brought about by the new products have no observed data and can only be estimated by the experts. A great deal of evidence shows that people usually include a wider range of values in their estimation of an indeterminant number than it may really take. Then in that situation if we still employ probability theory to help make decisions, counterintuitive results may appear. The examples can be found in Liu [24] and Zhang et al. [41]. Especially, Zhang et al. [41] shows that in this situation if we inappropriately used probability theory in project selection, a budget exceeding event which is sure to happen would be judged the event that will surely not happen. Considering that people will not take action for an event that will never happen, this mistaken result may bring great loss to the investors. To deal with men’s estimates, scholars have studied employing fuzzy set theory and have applied fuzzy set theory in different fields [27,42]. To solve project selection problems, scholars have developed a variety of fuzzy models, eg., Bhattacharyya et al. [1], Huang [12], Karsak and Kuzgunkaya [17], Tsao [36], Zhang et al. [43] etc. These researches opened a new perspective for dealing with project selection problems with parameters given by experts’ estimations. However, recently it was found that paradoxes will occur if we use fuzzy variables to describe the subjective estimations of project parameters [40,41]. In order to model human being’s imprecise estimations toward indeterminant quantities, Liu [25] founded an uncertainty theory based on four axioms and refined it in Liu [23]. If we use uncertainty theory to model human being’s imprecise

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X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11

estimations, no paradoxes appear. In fact, Liu [25] has shown that human beings’ estimations expressed by belief degrees satisfy the four axioms of the uncertainty theory, which implies that we can use uncertainty theory to model human uncertainty. So far, uncertainty theory has been used to handle many optimization problems concerning human uncertainty. For example, Liu [21] proposed a spectrum of uncertain programming models and applied them to solve machine scheduling, vehicle routing and project scheduling problems [23]. In 2010, Huang [13] first employed uncertainty theory to propose a theory of uncertain portfolio selection. Based on the risk measurement of variance, Zhang et al. [45] discussed two uncertain portfolio selection models. Applications of uncertainty theory can also be found in many other optimization fields, e.g., shortest path problem [7], facility location problem [8], Chinese postman problem [44], single-period inventory problem [32], uncertain aggregate production planning problem [30], and multi-product newsboy problem [6], etc. In the area of project selection, Zhang et al. [40] first applied uncertainty theory to solve a multinational project selection problem. Zhang et al. [41] later proposed a profit risk index and a cost overrun risk index and developed an uncertain mean-risk index domestic project selection model. In this paper we will explore using uncertainty theory to solve a joint problem of optimal project selection and scheduling with initial outlays and net cash inflows given by experts’ estimates. Different from previous uncertain project selection studies [40,41], in our problem, not only the projects need to be selected, but also the start times of the selected projects need to be scheduled to ensure effective use of budget. The logical relations such as independent, dependent, exclusive, and time sequence order among the candidate projects will be considered. The objective is to get the maximum profit under the capital exceeding risk and profit risk control. With an essential innovation, a new mean-variance and mean-semivariance models for project selection and scheduling in different situations will be developed. To solve the proposed complex mixed integer programming problem, a hybrid intelligent algorithm will be presented. The experimental test results will show that the algorithm can improve the convergence speed and effectively solve the problem. The paper proceeds as follows. In Section 2 we will review some fundamentals of uncertainty theory which will be used in the paper. In Section 3 we will develop a mean-variance and a meansemivariance models for a project selection and scheduling problem taking different interactions among candidate projects into account. In Section 4 we will provide a hybrid intelligent algorithm for solving the proposed problem. To illustrate the modeling idea and to show the effectiveness of the proposed algorithm, we will present two numerical examples and experiments in Section 5. Finally, in Section 6 we will give some concluding remarks.

2. Fundamentals of uncertainty theory Uncertainty theory is developed based on the below four axioms. Definition 1. Let L be a σ -algebra over a nonempty set  . Every element  ∈ L is called an event. If a set function M{} satisfies the following four axioms, we call it an uncertain measure [22,25]: (i) (Normality) M{ } = 1. (ii) (Duality) M{} + M{c } = 1. (iii) (Subadditivity) For every countable sequence of events {k }, we have



M

∞ 

k=1



k

≤

∞ 

M{k }.

k=1

The triplet ( , L, M) is called an uncertainty space.

Φ(r) 1.0 α

a

0.0

b

c

r

Fig. 1. A zigzag uncertain variable ξ = Z (a, b, c, α).

(iv) (Product Axiom) Let (k , Lk , Mk ) be uncertainty spaces for k = 1, 2, . . . , The product uncertain measure is



M

∞ 



k

=

k=1

∞ 

Mk {k },

k=1

where k are arbitrarily chosen events from Lk for k = 1, 2, . . . , respectively. Definition 2 [25]. An uncertain variable is a measurable function ξ from an uncertainty space ( , L, M) to the set of real numbers. It has been proved that for any events 1 ⊂ 2 , we have

M{1 } ≤ M{2 }. In application, an uncertain variable is characterized by an uncertainty distribution function which is defined as follows: Definition 3 [25]. The uncertainty distribution :  → [0, 1] of an uncertain variable ξ is defined by

(r) = M{ξ ≤ r}. For example, if an uncertain variable has the following normal uncertainty distribution, we call it a normal uncertain variable:





π (μ − r) (r) = 1 + exp √ 3σ

−1

,

r ∈ ,

where μ and σ are real numbers and σ > 0. We denoted it in the paper by ξ ∼ N (μ, σ ). We call an uncertain variable a zigzag uncertain variable if it has the following zigzag uncertain distribution (Please also see Fig. 1):

(r) =

⎧ 0, ⎪ ⎪ ⎨α(r − a)/(b − a),

(r − b − α r + α c)/(c − b), ⎪ ⎪ ⎩1,

if if if if

r ≤ a, a < r < b, b ≤ r < c, r ≥ c,

where a, b, c and α are real numbers and a < b < c and α ∈ [0, 1]. The variable is denoted in the paper by ξ ∼ Z (a, b, c, α). When the uncertain variables ξ1 , ξ2 , . . . , ξn are represented by uncertainty distributions, the operational law is given by Liu [23] as follows: Theorem 1 [23]. Let ξ1 , ξ2 , . . . , ξn be independent uncertain variables with uncertainty distributions 1 , 2 , . . . , n . Let f (r1 , r2 , . . . , rn ) be strictly increasing with respect to r1 , r2 , . . . , rn . Then

ξ = f (ξ1 , ξ2 , . . . , ξn ) is an uncertain variable with inverse uncertainty distribution function −1 −1 −1 (α) = f (−1 1 (α), 2 (α), . . . , n (α)),

0 < α < 1.

(1)

Theorem 2 [23]. Let ξ1 , ξ2 , . . . , ξn , ξn+1 , . . . , ξn+m be independent uncertain variables with uncertainty distributions 1 , 2 , . . . , n , n+1 , . . . , n+m . Let g(r1 , r2 , . . . , rn , rn+1 ,. . . , rn+m ) be strictly increasing with respect to r1 , r2 , . . . , rn and strictly decreasing with respect to rn+1 , rn+2 , . . . , rn+m . Then

η = g(ξ1 , ξ2 , . . . , ξn , ξn+1 , . . . , ξn+m )

X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11

3

is an uncertain variable with inverse uncertainty distribution function −1 −1 −1 −1 (α) = g(−1 1 (α), . . . , n (α), n+1 (1−α), . . . , n+m (1−α)), (2) 0 < α < 1.

To tell the size of an uncertain variable, Liu defined the expected value of uncertain variables. Definition 4 [25]. The expected value of an uncertain variable ξ is defined by

E[ξ ] =



∞

0

M{ξ ≥ r}dr −



0

−∞

M{ξ ≤ r}dr

(3)

on condition that at least one of the two integrals is finite. It can be calculated that the expected value of the normal uncertain variable ξ ∼ N (μ, σ ) is

E[ξ ] = μ

(4)

and the expected value of the zigzag uncertain variable ξ ∼ Z (a, b, c, α) is

E[ξ ] = α(a − b) + α 2 (c − a) + (b + c − 2α c)/2(1 − α).

(5)

Theorem 3 [23]. Let ξ be an uncertain variable with continuous uncertainty distribution . If its expected value exists, then

E[ξ ] =



1

0

−1 (α)dα .

(6)

Definition 5 [25]. Let ξ be an uncertain variable that has finite expected value e. Then the variance of ξ is defined by

V [ξ ] = E[(ξ − e)2 ]. When an uncertain variable ξ has a continuous uncertainty distribution , then

V [ξ ] = =



+∞

M{(ξ − e)2 } ≥ rdr

0  +∞

 ≤

+∞

0

 =

 =

M{(ξ ≥ e +

0

√ √ r) ∪ (ξ ≤ e − r)}dr √

√

(M{ξ ≥ e + r} + M{ξ ≤ e − r})dr

+∞ 0 +∞ e

V [ξ ] = 2

+∞ e

2(r − e)(1 − (r) + (2e − r))dr.

(r − e)(1 − (r) + (2e − r))dr.

xi : the decision variables defined by:



xi =

1, 0,

if projects i are selected, otherwise,

i = 1, 2, . . . , n, respectively. si : another type of decision variables which represent the investment start time of projects i; when si = 0, the projects i start at present time; otherwise, the projects i start at the end of month si ; τ i : investment length of projects i; D: deadline of the whole projects investment; Ti : the operation lifetime of projects i; Iit : net cash inflows of projects i at time t, which are given by experts’ estimations and are treated as uncertain variables; Oit : initial outlays of projects i at time t, which are offered by experts and are treated as uncertain variables; Wt : the company’s available capital at time t; r: hurdle rate of the capital or the required profit rate;

si +τi +Ti

NPVi = (7)

It can be calculated that the variance value of a normal uncertain variable ξ ∼ N (μ, σ ) is

V [ξ ] = σ 2 .

The objective of the company is to obtain the maximum investment return with risk control. To ensure effective use of budget, the company takes flexibility of projects start time into account. The problem is how to select and schedule the projects. For the sake of discussion convenience, we use the below symbols:

In project selection problem it is usually assumed that initial outlays happen at the beginning of a month and net cash flows occur at the end of a month. Furthermore, we assume that each project starts to yield net cash inflow at the end of the project investment period. Then the profits of projects i in terms of net present value (NPV) is

√ √ (1 − (e + r) + (e − r))dr

In this case, it is always assumed that the variance is



Fig. 2. Cash flows of project 1.

(8)

3. Uncertain project selection and scheduling models

i = 1, 2, . . . , n.

For example, suppose we have project 1 with s1 = 1, τ1 = 2, and T1 = 4. Then the cash flows of the project 1 can be shown in Fig. 2 where downward arrows represent investment outlays, and upward arrows net cash inflows. It is easy to see that the NPV of the project 1 is

NPV1 =

7  t=3

3.1. Uncertain mean-variance model Consider a company that plans to launch suitable R & D projects from n numbers of candidate projects. Among these candidates, some are independent, some are exclusive, and some are interdependent. In addition, there is investment sequence order requirement for some interdependent projects. Since there are no historical data, all project parameter values are estimated by experts and treated as uncertain variables. No salvage value is considered when each project dies. The company faces budget limitations over the investment periods. It must make investment decision at the beginning of the first month.

si +τi −1  Iit Oit − , t ( 1 + r ) ( 1 + r)t t=si +τi t=si



 O1t I1t − . t (1 + r) (1 + r)t 2

t=1

Generally, the company’s profit from investment in the selected project portfolio can be calculated as follows: τi +Ti n si +   i=1

τi −1 n si +   Iit Oit xi − xi . t ( 1 + r ) ( 1 + r)t t=si +τi i=1 t=si

The objective of the company is to obtain the maximum NPV value from the investment. Considering that the objective function includes uncertain variables, we cannot directly maximize it. However, we can employ the expected value of the NPV as the investment

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X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11

return. Then the objective value can be expressed mathematically as follows:



E



τi +Ti n si +   i=1

τi −1 n si +   Iit Oit x − xi . t i ( 1 + r ) ( 1 + r)t−1 t=si +τi i=1 t=si

Variance was first proposed by Markowitz [28] to measure investment risk. Huang [14] extended the idea to use uncertain variance for measuring the security investment risk with returns given by experts. In the paper, we retain the spirit to measure the project investment risk. Then if the investors preset a tolerable return risk level a, the return risk constraint can be expressed as follows:



τi +Ti n si +  

V

i=1

τi −1 n si +   Iit Oit xi − xi t ( 1 + r ) ( 1 + r)t t=si +τi i=1 t=si



≤ a,

where V is the variance operator of the uncertain variables. Let Ht represent the set of projects which are being invested at time t. To control the budget exceeding risk, the company can require that the expected value of total investment outlay at each investment time t should not exceed the available capital Wt and further ask that the variance value of the total investment outlay at each investment time t should not exceed the preset tolerable variance level γt , t = 0, 1, . . . , D − 1, respectively. To express the idea mathematically we have



E





xi Oit

≤ Wt ,

t = 0, 1, . . . , D − 1,

≤ γt ,

t = 0, 1, . . . , D − 1.

i∈Ht

 V



 xi Oit

i∈Ht

Since there are different independent, dependent and exclusive relations between projects, we should take these relationships into account when making decision. The below constraint (9) requires that projects in the set Uq are mutually exclusive, constraint (10) means that projects in the set Vq are interdependent, and constraint (11) says that projects in the set Vq are one-way dependent, i.e., if project i is selected, project j must be selected; however, if project j is selected, project i is not necessarily selected.



xi ≤ 1.

(9)

i∈Uq

xi = x j ,

i = j,

i, j ∈ Vq .

(10)

xi ≤ x j ,

i = j,

i, j ∈ Vq .

(11)

If one project is the successor of another one, it should be started after the completion of the preorder project when both of the projects are selected. Let project j be the successor of project i. Then the time sequence order between them can be expressed as follows,

s j ≥ si + τ i ,

i = j,

i, j ∈ Mq ,

where Mq is the set of the projects that have succession relationship. Though the selected projects can be started at different time, the whole investment period should not be longer than the deadline D, which we can express mathematically as follows:

si + τi ≤ D,

i = 1, 2, . . . , n.

Thus, taking flexibility of start time into account, to obtain the maximum expected investment return with the profit risk and budget exceeding risk control, we can select the projects and schedule their start time according to the following model:

 s +τ +T  ⎧ τi −1 n i n si + i i    ⎪ Iit Oit ⎪ max E x − x ⎪ ⎪ ⎪ (1 + r)t i i=1 t=s (1 + r)t i ⎪ i=1 t=si +τi ⎪ i ⎪ ⎪ ⎪ ⎪ subject to: ⎪ ⎪  s +τ +T  ⎪ ⎪ τi −1 n i n si + ⎪ i i    ⎪ Iit Oit ⎪ ⎪ V x − x ≤a ⎪ ⎪ (1 + r)t i i=1 t=s (1 + r)t i ⎪ ⎪ i=1 t=si +τi i ⎪ ⎪   ⎪ ⎪ ⎪  ⎪ ⎪ x O ≤ Wt , t = 0, 1, . . . , D − 1 E ⎪ i it ⎪ ⎪ ⎪ i∈Ht ⎪ ⎨   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩



V

xi Oit

≤ γt ,

t = 0, 1, . . . , D − 1

(12)

i∈Ht



(i )

xi ≤ 1

i∈Uq

xi = x j ,

i = j,

i, j ∈ Vq

xi ≤ x j ,

i = j,

i, j ∈ Vq

s j ≥ si + τi ,

i = j,

(ii) (iii)

i, j ∈ Mq

si + τi ≤ D xi ∈ {0, 1}, si , integers.

It has been proven [3] that the normal uncertain variable has the maximum entropy value among all the uncertain variables that have the same expected and variance values. Since it is most difficult to predict the occurrence of a specific number for the uncertain variable that has the maximum entropy, for the safety reason of decision, when we get the expected and variance values and no other information, we can use normal uncertain variables to describe the project parameters. According to Liu [23], the sum product of scalar numbers and normal uncertain variables is still a normal uncertain variable, i.e., for    ξi ∼ N (μi , σi ), we have ni=1 ki ξi ∼ N ( ni=1 ki μi , ni=1 |ki |σi ). Since for a normal uncertain variable ξ ∼ N (μ, σ ), we have E[ξ ] = μ and V [ξ ] = σ 2 . Then when net cash inflows Iit and initial outlays Oit are independent normal uncertain variables Iit ∼ N (μit , σit ) and Oit ∼ N (μ it , σit ), respectively, model (12) can be changed into the following form. ⎧ τi −1 n si +τi +Ti n si +   μ it ⎪ μit ⎪ max   ⎪ xi − x ⎪ t ⎪ (1 + r ) (1 + r)t i ⎪ i=1 t=si +τi i=1 t=si ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ subject to: ⎪ ⎪ τi +Ti τi −1 ⎪ n si + n si + ⎪     √ σit ⎪ σit ⎪ ⎪ x + xi ≤ a i ⎪ t ⎪ ( 1 + r ) ( 1 + r)t ⎪ i=1 t=si +τi i=1 t=si ⎪ ⎪ ⎪  ⎪ ⎪ xi μ it ≤ Wt , t = 0, 1, . . . , D − 1 ⎪ ⎪ ⎪ ⎪ i∈Ht ⎪ ⎨  √ xi σit ≤ γt , t = 0, 1, . . . , D − 1 (13) ⎪ ⎪ i∈H t ⎪ ⎪  ⎪ ⎪ ⎪ xi ≤ 1 (i ) ⎪ ⎪ ⎪ i∈Uq ⎪ ⎪ ⎪ ⎪ xi = x j , i = j, i, j ∈ Vq (ii ) ⎪ ⎪ ⎪ ⎪ ⎪ xi ≤ x j , i = j, i, j ∈ Vq (iii ) ⎪ ⎪ ⎪ ⎪ ⎪ s ≥ s + τ , i =  j, i, j ∈ M ⎪ q j i i ⎪ ⎪ ⎪ ⎪ si + τi ≤ D ⎪ ⎪ ⎪ ⎪ ⎩ xi ∈ {0, 1}, si , integers 3.2. Uncertain mean-semivariance model The use of uncertain mean-variance model (12) implies that the uncertainty distributions of project parameters are symmetrical. Otherwise, when eliminating variance value, both higher deviations and lower deviations from the expected value are eliminated equally. However, higher deviation from the expected NPV value implies the potential high return of the investment and the lower deviation from the expected initial outlay implies lower initial cost, which are both welcomed by the investors. Thus, when the uncertainty distributions

X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11

of project parameters are asymmetrical, it is unsuitable to use variance as a risk measurement. Therefore, we use lower partial semivariance which is abbreviated to LSV to measure investment return risk and higher partial semivariance which is abbreviated to HSV to measure budget exceeding risk. The definitions of LSV and HSV are as follows: Definition 6 [14]. Let ξ be an uncertain variable with finite expected value e. Then the lower partial semivariance of ξ is defined by



LSV [ξ ] = E[[(ξ − e) ] ], where − 2

(ξ − e) = −

ξ − e, if 0, if

ξ ≤ e, ξ > e. (14)

When the uncertain variable ξ has continuous uncertainty distribution , then

LSV [ξ ] =



+∞ 0

 =

+∞

0

 =

M{((ξ − e)− )2 ≥ t }dt

e

−∞

M{ξ ≤ e −

√ t }dt

(15)

2(e − t )(t )dt.

Definition 7. Let ξ be an uncertain variable with finite expected value e. Then the higher partial semivariance of ξ is defined by



HSV [ξ ] = E[[(ξ − e)+ ]2 ], where

(ξ − e)+ =

ξ − e, if ξ ≥ e, 0, if ξ < e. (16)

When the uncertain variable ξ has continuous uncertainty distribution , then

HSV [ξ ] =



 =

+∞ 0

0

 =

+∞

e

+∞

M{((ξ − e)+ )2 ≥ t }dt √

(1 − M{ξ ≤ e + t })dt

(17)



 2 1 − (t ) (t − e)dt.

  ⎧ τi +Ti τi −1 n si + n si +     ⎪ Iit Oit ⎪ x − x ⎪max E ⎪ ⎪ (1 + r)t i i=1 t=s (1 + r)t i ⎪ i=1 t=si +τi i ⎪ ⎪ ⎪ ⎪ ⎪ subject to: ⎪ ⎪   ⎪ ⎪ τi +Ti τi −1 n si + n si + ⎪     I O ⎪ it it ⎪ x − x ≤b LSV ⎪ ⎪ (1 + r)t i i=1 t=s (1 + r)t i ⎪ ⎪ i=1 t=si +τi i ⎪ ⎪   ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ xi Oit ≤ Wt , t = 0, 1, . . . , D − 1 E ⎪ ⎪ ⎪ ⎪ i∈Ht ⎨    ⎪ ⎪ xi Oit ≤ ωt , t = 0, 1, . . . , D − 1 HSV ⎪ ⎪ ⎪ i∈Ht ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ xi ≤ 1 (iv) ⎪ ⎪ ⎪ i∈Uq ⎪ ⎪ ⎪ xi = x j , i = j, i, j ∈ Vq (v) ⎪ ⎪ ⎪ ⎪ ⎪ xi ≤ x j , i = j, i, j ∈ Vq (vi) ⎪ ⎪ ⎪ ⎪ ⎪ s ≥ s + τ , i =  j, i, j ∈ M q j i i ⎪ ⎪ ⎪ ⎪ si + τi ≤ D ⎪ ⎪ ⎪ ⎩ xi ∈ {0, 1}, si , integers

4. Hybrid intelligent algorithm There are two types of decision variables for each project in the mean-variance models (12) and (13) and the mean-semivariance model (18). Then when the number of candidate projects increases, the number of decision variables will increase rapidly. As a result, traditional methods can hardly solve them effectively. Genetic algorithm (GA) is a good way for solving complex optimization problems and has successfully solved many complex problems [2,20], especially project selection [33] and project scheduling problems [9,10]. Huang and Zhao [16] has used GA to solve a project selection and scheduling problem when risk index is applied to measure cost overrun risk. However, premature convergence may appear in the process of evolution, and the result may be trapped in local optimal solution. Cellular automation (CA) was first proposed by Von Neumann [37] and has gained popularity in optimization searching because it can help diversify the population and avoid being trapped in local optimal solution. Huang et al. [15] hybridized cellular automation with particle swarm optimization to solve a project adjustment and selection problem. In this paper, we will design a hybrid intelligent algorithm (HIA) which integrates CA into the GA to improve the convergence speed and avoid falling into locally optimal solution. Below we will introduce the method to calculate lower partial semivariances and higher partial semivariances, and then the the hybrid intelligent algorithm. 4.1. Calculating LSV and HSV via inverse uncertainty distribution We can calculate expected and variance values through the methods introduced by Huang [14]. Here, we will introduce a way for calculating lower partial semivariances and higher partial semivariances so that the results can be integrated into our designed hybrid intelligent algorithm to solve the proposed problems in general cases. Let ξ i denote uncertain variables with uncertainty distribution functions

i , ηi uncertain variables with uncertainty distribution functions i , and xi decision variables, i = 1, 2, . . . , n, respectively. In order to solve the proposed model problems, we must handle the following two types of uncertain functions:

U1 : U2 :

Then we build the mean-semivariance model as follows:

5

n  i=1 n 

xi ξ i −

n 

xi η i .

i=1

xi η i .

i=1

According to the Eq. (2) in Theorem 2, we can calculate the inverse uncertainty distribution function of uncertain function U1 as follows:

U−1 (α) = 1

n 

xi −1 (α) − i

i=1

n 

xi i−1 (1 − α).

i=1

According to the Eq. (1) in Theorem 1, we can calculate the inverse uncertainty distribution function of uncertain function U2 as follows: (18)

where b is the tolerable lower partial semivariance level towards return risk and ωt the tolerable higher partial semivariance levels towards budget exceeding risk.

U−1 (α) = 2

n 

xi i−1 (α).

i=1

Then according to the Eqs. (15) and (17), the lower partial semivariance of U1 can be calculated via



LSV [U1 ] = 2

e

−∞

(e − t ) U1 (t )dt,

and the higher partial semivariance of U2 can be calculated via



HSV [U2 ] = 2

e

+∞

(1 − U2 (t ))(t − e)dt.

We set up a step length λ, the value of which should be determined according to the problem and precision requirement. Then we design the process for computing LSV[U1 ] and HSV[U2 ] as follows:

6

X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11

Method 1: Step 1. Use the method introduced in Huang [14] to calculate E[U1 ], and set e = E[U1 ]. Step 2. Set lsv = 0. Step 3. Set j = 1. Step 4. Let y = 2 × λ · j(e − U−1 (λ · j))( U−1 (λ · ( j + 1) − U−1 (λ · 1

1

1

j)). Step 5. Let lsv = lsv + y. Step 6. If U−1 (λ · j) < e, let j = j + 1. Then go back to step 4. Other1

wise, return lsv, where lsv is the approximation of LSV[U1 ]. Method 2: Step 1. Use the method introduced in Huang [14] to calculate E[U2 ],

and set e = E[U2 ]. Step 2. Use bisection search method to find the β value such that

U−1 (β) = e . 2

Step 3. Set hsv = 0. Step 4. Set j = 1.

Step 5. Let y = 2(1 − β − λ · j)( U−1 (β + λ · j) − e )( U−1 (β + λ · 2

j) − U−1 (β + λ( j − 1))). 2

2

Step 6. Let hsv = hsv + y . Step 7. If β + λ · j < 1, let j = j + 1. Then go to step 5. Otherwise, return hsv, where hsv is the approximation of HSV[U2 ]. 4.2. Representation structure in GA Since the solution of each model consists of xi which should be 0 or 1, and si which are integer numbers, we use an integer vector x = (x1 , x2 , . . . , xn , xn+1 . . . , x2n ) to represent a solution to the optimization problems, where n is the number of candidate projects, x1 , x2 , . . . , xn ∈ {0, 1}, xn+1 , xn+2 , . . . , x2n are integer numbers and xn+i ∈ {0, 1, . . . , D − τi }, i = 1, 2, . . . , n. Then the solution x = (x1 , x2 , . . . , xn , xn+1 , . . . , x2n ) can be encoded as the chromosome C = (c1 , c2 , . . . , cn , cn+1 , . . . , c2n ), where ci ∈ {0, 1} and cn+i ∈ {0, 1, . . . , D − τi }, i = 1, 2, . . . , n, respectively. The mapping between a solution and a chromosome is x ≡ C. In the following processes, we divide C = (c1 , c2 , . . . , cn , cn+1 , . . . , c2n ) into two parts because of the characteristic of x = (x1 , x2 , . . . , xn , xn+1 . . . , x2n ), i.e., the front n genes in one part and the rear n in the other part.

4.3. Initialization process In each model there are eight kinds of constraints, of which the three project relationship constraints are called “Constraint A” and the rest “Constraint B”. That is, for model (12), Constraint A consists of constraints (i), (ii), (iii), for model (13), Constraint A consists of constraints (i ), (ii ), (iii ), and for model (18), Constraint A consists of constraints (iv), (v), (vi). The reason we divide the constraints in this way is that relationship constraints are only related to the front n decision variables and we can increase efficiency of acquiring feasible chromosomes by the following steps. Step 1. Generate an integer vector (c1 , c2 , . . . , cn ) from integer set {0,1} randomly. If (c1 , c2 , . . . , cn ) does not pass Constraint A, generate another (c1 , c2 , . . . , cn ) from integer set {0,1} until the vector passes Constraint A. For example, for both models (12) and (18), the passing of Constraint A is checked as follows:

If



xi > 1,

return 0;

l∈Uq

I f xi = x j ,

i = j,

i, j ∈ Vq ,

I f xi > x j ,

i = j,

i, j ∈ Vq ,

return 1;

return 0; return 0;

in which 1 means the front n genes (c1 , c2 , . . . , cn ) has passed Constraint A, and 0 otherwise. Step 2. Remain the front genes (c1 , c2 , , cn ) got from the first step unchanged and generate cn+i from integer set {0, 1, . . . , D − τi } where i = 1, 2, . . . , n, to produce a chromosome C = (c1 , c2 , . . . , cn , cn+1 , cn+2 , . . . , c2n ). Then use the Methods 1 and 2 introduced in Section 4.1 and the way to calculate expected and variance values in Huang [14] to calculate the objective, variance, lower partial semivariance and higher partial semivariance values and check if the chromosome can pass Constraint B. If it passes Constraint B, a feasible initial chromosome is obtained. Otherwise, generate (cn+1 , cn+2 , . . . , c2n ) from integer set {0, 1, . . . , D − τi } to produce a new chromosome C = (c1 , c2 , . . . , cn , cn+1 , cn+2 , . . . , c2n ) until the chromosome is feasible or a preset repetition times is reached. The reason why we set a repetition times is to avoid the condition where with the front genes (c1 , c2 , . . . , cn ) unchanged, even when we generate (cn+1 , cn+2 , . . . , c2n ) a great deal of times, the chromosome C = (c1 , c2 , . . . , cn , cn+1 , cn+2 , . . . , c2n ) cannot pass Constraint B. If we still cannot obtain a feasible initial chromosome when the preset repetition times is reached, go back to step 1 until we get a feasible initial chromosome. Repeat the processes for obtaining a feasible initial chromosome pop_size times, then pop_size feasible chromosomes C1 , C2 , . . . , C pop_size are produced. For example, the checking of passing of Constraint B for the model (12) is done as follows:

I f s j < si + τi ,

i = j,

I f si + τi > D,

return 0;



return 0;

τi +Ti n si +  

If V

i=1

 If E

i, j ∈ Mq ,



τi −1 n si +   Iit Oit x − xi i t ( 1 + r ) ( 1 + r)t t=si +τi i=1 t=si

xi Oit

 If V

> a,

return 0;



i∈Ht





> Wt , t = 0, 1, . . . , D − 1,

return 0;

> γt , t = 0, 1, . . . , D − 1,

return 0;

 xi Oit

i∈Ht

return 1; in which 1 means the chromosome C = (c1 , c2 , . . . , cn , cn+1 , cn+2 , . . . , c2n ) has passed Constraint B, and 0 otherwise. The checking of passing of Constraint B for the model (18) is done as follows:

I f s j < si + τi ,

i = j,

I f si + τi > D,

return 0;



I f LSV

If E





τi −1 n si +   Iit Oit x − xi > b, return 0; i t ( 1 + r ) ( 1 + r)t t=si +τi i=1 t=si



xi Oit

> Wt , t = 0, 1, . . . , D − 1,

i∈Ht



I f HSV

return 0;

τi +Ti n si +   i=1



i, j ∈ Mq ,



return 0;

 xi Oit

> ωt , t = 0, 1, . . . , D − 1,

return 0;

i∈Ht

return 1; in which 1 means the chromosome C = (c1 , c2 , . . . , cn , cn+1 , cn+2 , . . . , c2n ) has passed Constraint B, and 0 otherwise. Here, since the rear genes (cn+1 , cn+2 , . . . , c2n ) are generated from the set {0, 1, . . . , D − τi } in Initialization process, the chromosome C = (c1 , c2 , . . . , cn , cn+1 , cn+2 , . . . , c2n ) can surely pass the constraint si + τi ≤ D in Initialization. However, in Crossover and Mutation operations, the constraints si + τi ≤ D will not surely be satisfied. Therefore, we give above the general way for checking the feasibility of the chromosomes.

X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11

4.4. Selection process The selection of chromosomes is done by spinning the roulette wheel such that the better chromosomes will have more chances to produce offspring. We employ the Rank-based evaluation function to assign a probability of reproduction to each chromosome C. Since the process is similar to the selection process in Huang [12], we omit it here. 4.5. Crossover operation We define a crossover probability Pc to select parents to do crossover operation, and we get the parents in the following way. Randomly generate a real number a from the interval [0,1] repeatedly for pop_size/2 times. Generate two chromosomes randomly as parents for crossover operation if a < Pc . The crossover operation is done as follows: Step 1. Randomly generate some gene positions from the front part of one parent. Then give the genes of the parent on these positions to a child and fill empty positions of the front part with front part genes of another parent by a left-to-right scan. Check if the child’s front part genes (c1 , c2 , . . . , cn ) passes Constraint A. If it doesn’t pass Constraint A, repeat the step until we get a child whose front part genes (c1 , c2 , . . . , cn ) passes Constraint A. Step 2. Do the crossover process for the rear part genes in the similar way to that of the front part genes. Then check if the obtained chromosome C = (c1 , c2 , . . . , cn , cn+1 , cn+2 , . . . , c2n ) can pass Constraint B. If it passes Constraint B, we obtain a feasible chromosome. Otherwise, remain the front n genes which we have obtained in step 1 unchanged and generate cn+i from integer set {0, 1, . . . , D − τi } where i = 1, 2, . . . , n, until the produced new chromosome C = (c1 , c2 , . . . , cn , cn+1 , cn+2 , . . . , c2n ) can pass Constraint B or a preset repetition times is reached. If the child is still infeasible when the preset repetition times is reached, go back to step 1 until the child is feasible. We set a repetition times for the same reason as in Initialization. Since a chromosome with bigger ordinal number is a worse one, we take the place of the ( pop_size/2 + i)th parent chromosome by the child if the ith feasible child chromosome is obtained.

7

trapped in local optimal solution and the cellular automata implementation aims at the current optimal chromosome C∗ of each generation. So the C∗ produced after mutation operation is treated as the cell and the cell space includes the cell itself. To find suitable neighbors for the cell, we analyze the proposed models. It can be found that if the selected projects in the C∗ keep unchanged, but one of the selected projects can start earlier, the changed chromosome can surely pass the Constraint A. If it can also pass the Constraint B, then this changed chromosome is feasible and a bigger objective value, i.e., the expected NPV, can be obtained. Therefore, using C∗ = (c1 , c2 , . . . , cn , cn+1 , cn+2 , . . . , c2n ) as an illustration, we build the neighborhood as follows. First, keep c1 , c2 , . . . , cn unchange. Next, for each i = 1, 2, . . . , n, if ci = 1 and cn+i − 1 ≥ 0, check the feasibility of the new chromosome C = (c1 , c2 , . . . , ci , . . . , cn , cn+1 , cn+2 , . . . , cn+i − 1, . . . , c2n ). If C is feasible, take it as a neighbor of the chromosome C∗. Let Nj be the neighbors of the current optimal chromosome C∗ at each generation. The transition rule of the cell in the paper is

f (C∗ ) ∨ f (N1 ) ∨ f (N2 ) ∨ · · · ∨ f (NJ ).

(19)

where J denotes the number of neighbors of the chromosome C∗, and f( · ) the predefined fitness function. In this paper, for each proposed model, f( · ) is the objective function. Then the implementation of CA at each generation is summarized as follows: Step 1. Build the neighborhood of the C∗. Step 2. Calculate the objective function values of C∗ and all its neighbors. Step 3. Use transition rule (19) to replace the C∗. 4.8. Hybrid intelligent algorithm procedure A new population is produced after selection, crossover, mutation and cellular automata implementation. New circles of evolution will continue until a given number of cyclic repetitions is reached. Then we regard the best chromosome as the solution of the proposed model problems.

4.6. Mutation operation

5. Numerical examples

For this, a mutation probability Pm is selected for selection of parents to mutate. Then we generate a real number b from the interval [0,1] at random repeatedly for pop_size times. The ith chromosomes are selected as parents for mutation if b < Pm at the ith times. The mutation process is similar to the one process in Huang and Zhao [16], we omit it here. Nevertheless, we would like to point out that in our mutation, when a child is checked to be feasible, it has to be compared with its parent. If the child is superior to its parent, the child replaces its parent. Otherwise, the parent is kept.

To illustrate the modeling idea and demonstrate the effectiveness of the designed HIA, we present two numerical examples here. One is for mean-semivariance model and the other is for mean-variance model. We use a personal computer, FOUNDER, 3.20 GHz CPU, 1.96 GB memory, and set the parameters in the GA as follows: the population size pop_size = 30, the probability of crossover Pc = 0.6, the probability of mutation Pm = 0.6, and the parameter in the rank-based evaluation function ν = 0.05. The step length is set at λ = 0.0001. Suppose in each example the company requires that all the selected projects should be completed at the end of the twelfth month. The discount rate is r = 5.25%.

4.7. Cellular automata implementation A standard CA includes four basic components: cell state, cell space, neighborhood and transition rule. The cell state is the distinct state number that a single cell can be in. A cell space describes how cells are associated with each other. The neighborhood is the set of all the neighbors of a given cell. Transition rule tells how the next state of the cell will be according to the current state of the given cell and the states of its neighbors in its neighborhood. The idea of CA can be simply summarized as “In a preassigned cell space, every cell updates its current cell state governed by a transition rule according to its own state and the states of cells in the neighborhood at discrete time steps”. The application of CA does not necessarily need rigorous mathematical reasoning and is relatively simple, yet it can turn out to be very powerful [35]. In this paper, we employ CA to avoid being

Table 1 Relationship among projects in Example 1. Project

Interdependent

1 2 3 4 5 6 7 8 9 10

7 3 2

One-way dependent

Exclusive

Successor

8 6

3

10 8

9

9 2

1 1 5

8

X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11 Table 2 Investment length, operation lifetime and net cash inflows of projects in Example 1 (time unit: month; capital unit : thousand dollar). P.

1 2 3 4 5 6 7 8 9 10

τi

3 4 5 4 5 5 4 4 6 3

Ti

61 76 64 74 67 73 62 72 60 77

Oit

Iit

Z (112, 120, 125, 0.9) Z (110, 115, 120, 0.9) Z (111, 115, 119, 0.9) Z (126, 140, 150, 0.9) Z (130, 141, 148, 0.9) Z (122, 130, 148, 0.9) Z (129, 138, 143, 0.9) Z (138, 150, 158, 0.9) Z (111, 129, 140, 0.9) Z (124, 134, 138, 0.9)

si +τi ≤ t ≤ si + τi + 11

si + τi + 12 ≤ t ≤ si + τi + 23

si + τi + 24 ≤ t ≤ si + τi + Ti

Z (25, 27, 28, 0.9) Z (27, 34, 39, 0.9) Z (30, 38, 42, 0.9) Z (28, 31, 36, 0.9) Z (28, 33, 38, 0.9) Z (29, 31, 39, 0.9) Z (31, 36, 38, 0.9) Z (26, 30, 35, 0.9) Z (28, 34, 39, 0.9) Z (26, 32, 37, 0.9)

Z (26, 28, 31, 0.9) Z (23, 26, 30, 0.9) Z (27, 29, 33, 0.9) Z (26, 30, 32, 0.9) Z (25, 28, 34, 0.9) Z (27, 30, 33, 0.9) Z (29, 31, 35, 0.9) Z (24, 29, 32, 0.9) Z (26, 30, 33, 0.9) Z (25, 27, 29, 0.9)

Z (31, 34, 38, 0.9) Z (29, 33, 37, 0.9) Z (31, 35, 38, 0.9) Z (30, 33, 36, 0.9) Z (31, 34, 40, 0.9) Z (32, 36, 39, 0.9) Z (33, 35, 39, 0.9) Z (30, 33, 36, 0.9) Z (32, 35, 38, 0.9) Z (29, 32, 36, 0.9)

Table 3 The optimal project selection and scheduling plan of Example 1. Selected projects

1

10

Start times (month) Objective value (thousand dollar)

0 0 299.57

2

7

3

3

3

7

Table 4 Relationships between projects in Example 2. Project

Interdependent

1 2 3 4 5 6 7 8 9 10

6

One-way dependent

Exclusive

Successor

5 10

9

10

9

7 1 1

3 3

Example 1. Assume that a company is considering selecting some R & D projects among 10 candidate ones. The relationships between the projects are presented in Table 1. The investment length, operation lifetime, monthly initial outlays and monthly net cash inflows of the projects at different periods are given in Table 2. The company can provide 300 thousand dollars for investment each month. Since the project parameters are not symmetrically distributed, the company requires that the lower partial semivariance of investment return should not be greater than 1102 and higher partial semivariance of budget exceeding each month should not exceed 30. Then the company can use the model (18) to obtain the maximum expected NPV under profit and budget exceeding risk control. Running the proposed hybrid intelligent algorithm with 2000 generations indicates that the company should select the projects 1, 2, 3, 7, 10 and arrange them according to Table 3. The maximum expected value of NPV is 299.57 thousand dollars. Example 2. Suppose now the relationships of the 10 projects and time sequence requirements between them are what in Table 4, and the investment length and the operation lifetime of the projects are what in Table 5. The monthly initial outlays and monthly net cash inflows of the projects at different periods are now normal uncertain variables. Since the distributions of project parameters are symmetrical, we can use mean-variance model (13) to help select and schedule the projects. Suppose the company can provide 350 thousand dollars each month. It requires that variance of investment return should not be greater than 802 and the variance of total monthly outlay should not be greater than 202 . Then the company can make decision according to the following model:

⎧

τi +Ti τi −1 10 si + 10 si +     ⎪ μit μit ⎪ ⎪ max x − x i ⎪ ⎪ (1 + 0.0525)t (1 + 0.0525)t i ⎪ i=1 t=si +τi i=1 t=si ⎪ ⎪ ⎪ ⎪ subject to: ⎪ ⎪ ⎪ ⎪

τi +Ti τi −1 10 si + 10 si + ⎪     ⎪ σit σit ⎪ ⎪ x + x ≤ 80 ⎪ i ⎪ (1 + 0.0525)t (1 + 0.0525)t i ⎪ ⎪ i=1 t=si +τi i=1 t=si ⎪  ⎪

⎪ ⎪ xi μit ≤ 350, t = 0, 1, . . . , 11 ⎪ ⎪ ⎪ i∈Ht ⎪ ⎪  ⎪

⎪ ⎪ xi σit ≤ 20, t = 0, 1, . . . , 11 ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

i∈Ht

x1 + x5 ≤ 1 x3 + x10 ≤ 1 x 1 = x6 x 3 = x9 x2 ≤ x10 x4 ≤ x7 s9 ≥ s3 + 3 si + τi ≤ 12, i = 1, 2, . . . , 10 xi ∈ {0, 1},

si , integers,

i = 1, 2, . . . , 10. (20)

Running the proposed algorithm with 2000 generations, we get that the company should select the projects 1, 3, 6, 9 and arrange them according to Table 6. The maximum expected NPV is 320.90 thousand dollars. To show the significance of the project selection and scheduling models, we compare the decision results of the optimal project selection and scheduling with pure project selection. We select the projects in examples 1 and 2 again requiring that all the projects that are not successors of any other projects must be started simultaneously at the beginning of the first month. The corresponding optimal solutions are shown in Tables 7 and 8, respectively. It can be seen that if no start time flexibility is considered, in the case of Example 1, only three projects are selected because of lack of capital at early months though the company has 300 thousand dollars at the rest of later months, and in the case of Example 2, only two projects are selected because of lack of capital at early months though the company has 350 thousand dollars at the rest of later months. The expected NPV of pure project selection in Example 1 is 158.17 thousand dollars, which is smaller than 299.57 thousand dollars when start time flexibility is considered, and the expected NPV of pure project selection in Example 2 is 285.42 thousand dollars, which is smaller than 320.90 thousand dollars when start time flexibility is allowed. To test the effectiveness of the proposed HIA, we change the parameter values of GA and proceed the experiments. First, we change parameter values in the GA (population size, crossover probability,

X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11

9

Table 5 Investment length, operation lifetime and monthly net cash inflows of projects in Example 2 (time unit: month; capital unit: thousand dollar). P.

1 2 3 4 5 6 7 8 9 10

τi

4 6 3 5 5 6 6 5 4 5

Oit ∼ N (μit , σit )

Ti

N (108, 2) N (100, 4) N (99, 2) N (103, 4) N (108, 4) N (98, 2) N (109, 4) N (105, 5) N (96, 2) N (103, 5)

32 31 30 29 28 30 27 30 27 29

Iit ∼ N (μit , σit ) si + τ i ≤ t ≤ si + τi + 7

si + τi + 8 ≤ t ≤ si + τi + 23

si + τi + 24 ≤ t ≤ si + τi + Ti

N (38, 2) N (35, 2) N (36, 1) N (40, 2) N (32, 2) N (39, 2) N (41, 2) N (38, 2) N (37, 1) N (40, 2)

N (36, 1) N (32, 2) N (35, 1) N (36, 1) N (30, 1) N (37, 1) N (39, 1) N (36, 1) N (34, 1) N (37, 2)

N (41, 1) N (44, 2) N (40, 1) N (42, 2) N (39, 1) N (42, 1) N (45, 2) N (43, 2) N (40, 1) N (44, 1)

Table 6 Optimal project selection and scheduling plan of Example 2. Projects

3

1

9

6

Start times (month) Objective value (thousand dollar)

0 1 320.90

5

6

Table 7 Pure optimal project selection plan for Example 1. Project

2

10

Start time (month) Objective value (thousand dollar)

0 0 158.17

3 7

Table 10 Optimal objective values with different parameter values of the GA for Example 2. Pop_size

Pc

Pm

Generation for finding the optimal solution

Objective value (thousand dollar)

30 30 30 30 30 30 30 30 30 30

0.6 0.4 0.6 0.6 0.4 0.6 0.4 0.6 0.4 0.6

0.01 0.02 0.04 0.1 0.2 0.3 0.6 0.6 0.7 0.8

336 288 120 113 140 43 56 29 82 64

320.90 320.90 320.90 320.90 320.90 320.90 320.90 320.90 320.90 320.90

Table 8 Pure optimal project selection plan for Example 2. Project

3

9

Start time (month) Objective value (thousand dollar)

0 3 285.42

Table 9 Optimal objective values with different parameter values of the GA for Example 1. Pop_size

Pc

Pm

Generations

Objective value (thousand dollar)

30 30 30 50 50 50 100 100 100

0.4 0.6 0.3 0.4 0.3 0.6 0.4 0.6 0.6

0.01 0.02 0.04 0.1 0.4 0.2 0.6 0.4 0.6

2000 2000 2000 2000 2000 2000 2000 2000 2000

299.57 299.57 299.57 299.57 299.57 299.57 299.57 299.57 299.57

mutation probability) for Example 1 and provide the results in Table 9. It is seen from Table 9 that when the parameters change, we can still find the optimal solutions within 2000 generations and the optimal solutions remain unchanged, which implies that the proposed algorithm is quite robust. Next, we keep population size and change crossover probability and mutation probability for Example 2 and present the results in Table 10. The results in Table 10 show that though the generations for finding the optimal solution at different parameter settings are different, the proposed GA can find the optimal solution fairly well at all parameter settings, which further implies that our algorithm is robust and good for solving the problem. It is also seen from Table 10 that when Pm values becomes bigger but smaller than 0.6, the generations for finding the optimal solution be-

come smaller either at Pc = 0.4 or Pc = 0.6. The reason may be that in our mutation process, only when the feasible children are superior to their parents can they replace their parents; Otherwise, the parents are kept and the children are abandoned. Therefore, moderately big mutation rate can on one hand keep fairly big numbers of good chromosomes from crossover operation and on the other hand offer opportunities to produce a few good children that can replace their parents. So the generation for finding the solution is small. However, when mutation rate is too big (bigger than 0.6), the number of good chromosomes from crossover becomes much small, so the generation for finding the solution becomes big again. Furthermore, we compare the proposed HIA with the GA [16] and the enumeration method. To do so, we use the three algorithms to solve the project selection and scheduling problem with 6, 8, 10, 12 candidate projects, respectively. For convenience, suppose there is no relationship among projects in the four situations. The parameter values of the projects at different periods are shown in Table 11. Since the distributions of the project parameters are all normal uncertain variables, the decision is made based on the model (13). The values of a, Wt and γ t in four situations are given in Table 12. We run the three algorithms and show the results in Tables 13 and 14. We can see from the two tables that both HIA and GA can find the optimal solutions which are the same as those obtained by enumeration in all the four situations. It is seen from Table 13 that when the number of candidate projects is small (i.e., when there are 6 and 8 candidate projects), enumeration costs less time than GA and HIA for finding the optimal solution. However, when the project number increases, enumeration costs far more time than the other two methods. And the more number of candidate projects is, the more advantage GA and HIA show in solution time. When the number of candidate projects reaches 12, enumeration needs 427329 s to find the optimal solution, which is almost 5 days. Yet, GA can find the optimal solution within 71 s, and HIA can find the optimal solution within only 24 s. In addition, it is seen

10

X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11 Table 11 Investment length, operation lifetime and monthly net cash inflows of projects for further comparison (time unit: month; capital unit: thousand dollar). P.

1 2 3 4 5 6 7 8 9 10 11 12

τi

5 5 3 4 5 3 5 3 4 3 4 6



Oit ∼ N (μit , σit )

Ti

N (104, 3) N (101, 4) N (98, 5) N (102, 4) N (105, 3) N (99, 2) N (104, 4) N (105, 6) N (97, 2) N (101, 4) N (108, 2) N (100, 4)

30 29 28 27 27 29 26 29 26 27 32 31

Iit ∼ N (μit , σit ) si +τi ≤ t ≤ si + τi + 7

si + τi + 8 ≤ t ≤ si + τi + 23

si + τi + 24 ≤ t ≤ si + τi + Ti

N (36, 2) N (34, 2) N (37, 2) N (40, 2) N (33, 2) N (37, 2) N (40, 2) N (36, 2) N (34, 1) N (40, 1) N (38, 2) N (35, 2)

N (37, 2) N (32, 2) N (34, 1) N (35, 1) N (30, 2) N (34, 1) N (37, 1) N (35, 1) N (32, 1) N (33, 2) N (36, 1) N (32, 2)

N (40, 2) N (43, 2) N (40, 1) N (41, 2) N (37, 2) N (41, 1) N (42, 2) N (43, 1) N (41, 1) N (42, 1) N (41, 1) N (44, 2)

Table 12 Parameters of constraints in four situations (capital unit: thousand dollar). Number of candidate projects

a

Table 16 Parameters of constraints in the situation of 20 projects (capital unit: thousand dollar).

γt

Wt

Number of candidate projects 20

602 832 852 1202

6 8 10 12

102 222 252 302

180 260 300 300

a 160

2

Wt

γt

700

402

Table 17 Solution results by HIA and GA in case of 20 candidate projects (time unit: month; capital unit: thousand dollar). Algorithm

Selected projects

Start time

Optimal solution

Generation for optimal solution

Error rate

GA HIA

3,6,8,10,11,13 3,6,8,10,11,13

1,0,1,1,6,0 0,0,1,1,7,0

1034.4 1040.6

95 26

0.6%

Table 13 Time for finding the optimal solutions by enumeration, GA and HIA (time unit: s). Number of candidate projects

Objective value of optimal solution

Time by enumeration

Time by GA

Time by HIA

6 8 10 12

371.53 525.04 564.78 746.25

1.468 111.093 4484.27 427329

25.359 185.953 520.265 70.265

21.437 154.625 280.671 23.562

Table 14 Generation of optimal solution by GA and HIA. Number of candidate projects

Objective value of optimal solution

Generation of optimal solution by GA

Generation of optimal solution by HIA

6 8 10 12

371.53 525.04 564.78 746.25

6 17 77 104

2 4 29 5

Next, we add another 8 candidate projects to let the number of the candidate projects up to 20. The parameters of the added 8 projects are presented in Table 15 and the values of a, Wt and γ t are given in Table 16. In this situation, enumeration cannot find the optimal solution in an acceptable period of time. So we only perform the HIA and GA and provide the results in Table 17. It is seen from Table 17 that HIA can find the optimal solution faster than GA and that HIA can find better solution than GA. The objective value of the optimal solution by HIA is 1040.6 thousand dollars while the objective value by GA is 1034.4 thousand dollars. The relative error is 0.6%. The results show that the proposed HIA does performs better than GA. 6. Conclusions

from Tables 13 and 14 that HIA finds the optimal solution faster than GA in all the four situations in terms of both time and generations.

This paper has discussed a joint problem of optimal project selection and scheduling with initial outlays and net cash inflows of projects given by experts’ estimates. Uncertain variables are used to describe these parameters, and a new mean-variance

Table 15 Investment length, operation lifetime and monthly net cash inflows of added 8 projects (time unit: month; capital unit: thousand dollar). P.

13 14 15 16 17 18 19 20

τi

3 5 5 6 6 5 6 5

Ti

30 29 28 30 27 30 27 29



Oit ∼ N (μit , σit )

N (99, 2) N (103, 4) N (108, 4) N (98, 2) N (109, 4) N (105, 5) N (96, 2) N (103, 5)

Iit ∼ N (μit , σit ) si +τi ≤ t ≤ si + τi + 7

si + τi + 8 ≤ t ≤ si + τi + 23

si + τi + 24 ≤ t ≤ si + τi + Ti

N (36, 1) N (40, 2) N (32, 2) N (39, 2) N (41, 2) N (38, 2) N (37, 1) N (40, 2)

N (35, 1) N (36, 1) N (30, 1) N (37, 1) N (39, 1) N (36, 1) N (34, 1) N (37, 2)

N (40, 1) N (42, 2) N (39, 1) N (42, 1) N (45, 2) N (43, 2) N (40, 1) N (44, 1)

X. Huang et al. / Knowledge-Based Systems 93 (2016) 1–11

and a mean-semivariance models are proposed taking relationship and time sequence order among projects into account. Since the proposed problem is too complicated to solve by traditional methods, we have provided a hybrid intelligent algorithm to solve the problems. In the proposed algorithm, we have integrated genetic algorithm and cellular automation to produce a hybrid intelligent algorithm, in which the objective and constraint values are calculated via the inverse uncertainty distributions of project parameters. The results of the numerical experiments and the comparison with GA and enumeration show that the proposed algorithm is effective for solving the proposed problems. In addition, the example results also show that the proposed project selection and scheduling models can help investors better use the available capital and gain more profits than pure project selection approach. In reality, there are situations where some project parameters have to be given by experts’ estimates due to lack of historical data. Therefore, our approach provides a useful way for companies to optimally select projects and schedule them in these situations. Further research may involve taking more real life settings into account and applying the proposed method to real case problems. Acknowledgments This work was supported by National Natural Science Foundation of China Grant No. 71171018 and Specialized Research Fund for the Doctoral Program of Higher Education No. 20130006110001. References [1] R. Bhattacharyya, P. Kumar, S. Kar, Fuzzy R&D portfolio selection of interdependent projects, Comput. Math. Appl. 62 (2011) 3857–3870. [2] J. Bobadilla, F. Ortega, A. Hernando, J. Alcalá, Improving collaborative filtering recommender system results and performance using genetic algorithms, Knowl. Based Syst. 24 (8) (2011) 1310–1316. [3] X.W. Chen, W. Dai, Maximum entropy principle for uncertain variables, Int. J. Fuzzy Syst. 13 (2011) 232–236. [4] P.K. De, D. Acharya, K.C. Sahu, A chance-constrained goal programming model for capital budgeting, J. Oper. Res. Soc. 33 (1982) 635–638. [5] M.W. Dickinson, A.C. Thornton, S. Graves, Technology portfolio management: Optimizing interdependent projects over multiple time periods, IEEE Trans. Eng. Manag. 48 (2001) 518–527. [6] S. Ding, Y. Gao, The (σ , S) policy for uncertain multi-product newsboy problem, Expert Syst. Appl. 41 (2014) 3769–3776. [7] Y. Gao, Shortest Path Problem with Uncertain Arc Lengths, Comput. Math. Appl. 62 (2011) 2591–2600. [8] Y. Gao, Uncertain Models for Single Facility Location Problems on Networks, Appl. Math. Model. 36 (2012) 2592–2599. [9] M. Gen, R.W. Cheng, Genetic Algorithms and Engineering Optimization, John Wiley & Sons, New York, 2000. [10] P. Ghoddousi, E. Eshtehardian, S. Jooybanpour, A. Javanmardi, Multi-mode resource-constrained discrete time-cost-resource optimization in project scheduling using non-dominated sorting genetic algorithm, Autom. Constr. 30 (2013) 216–227. [11] J. Gutjahr, S. Katzensteiner, P. Reiter, C. Stummer, M. Denk, Multi-objective decision analysis for competence-oriented project portfolio selection, Eur. J. Oper. Res. 205 (2010) 670–679. [12] X. Huang, Chance-constrained programming models for capital budgeting with NPV as fuzzy parameters, J. Comput. Appl. Math. 198 (2007) 149–159. [13] X. Huang, Portfolio Analysis: From Probabilistic to Credibilistic and Uncertain Approaches, Springer-Verlag, Berlin, 2010. (Chapter 4)

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