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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.16 No.3 pp.271-287
DOI : https://doi.org/10.7232/iems.2017.16.3.271

# A New Stochastic Time-Cost-Quality Trade-Off Project Scheduling Problem Considering Multiple-Execution Modes, Preemption, and Generalized Precedence Relations

Farahnaz Reza-Pour, Kaveh Khalili-Damghani*
Department of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran
June 9, 2016 July 3, 2017 August 3, 2017

## ABSTRACT

Project managers always try to make the best decisions in order to complete their projects in the shortest period of time, with the least amount of costs, and with the highest degree of quality. Therefore, the process of decision-making is conducted in a triangle of time, costs, and quality. This triangle is a crucial part of management process throughout the execution of the project. However, in all problems, some unpredictable situations may be faced. In such situations, some or all parameters of the problem are expressed by uncertain variables. In this paper, a stochastic time-costquality trade off project scheduling problem (STCQTP) considering multiple-execution modes, preemption, and generalized precedence relations is developed. In order to solve the STCQTP, a hybrid solution approach based on Stochastic Chance Constraint Programming (SCCP) and Goal Programming (GP) is proposed. SCCP and GP are used to handle the uncertain nature and multiple-objectives of STCQTP, respectively. Numerical example is solved in order to illustrate the applicability of proposed model and solution approach.

## 1.INTRODUCTION

Trade off project scheduling problems (TPSP) is a widely-discussed phenomena in the literature of project management. Since in TPSP, the objective functions are not consistent, the aim is to find non-dominated solutions on Pareto front of the problem. In classic project scheduling methods such as Critical Path Method (CPM), it is assumed that all activities are finished in a predefined normal time. In some cases, project managers may be interested to complete the project earlier. For this aim, the time of some activities should be reduced to crash time through putting more resources and cost on them. This will trace on project cost, and quality (Khalili-Damghani et al., 2015). Although, project quality is a crucial factor for managers, it has been largely neglected in the past studies. As mentioned, the shortening of project time may reduce the quality of activities (Amiri et al., 2013). Time, cost, and quality are the three main factors in every project. Considering time, cost, and quality concurrently forms a type of trade-off problem called time-cost-quality project scheduling problems (TCQTP). In TCQTPs, the managers attempt to complete the project in a situation in which a suitable balance of time, cost, and quality is met (Monghasemi et al., 2015). In the real world, project activities are subject to a lot of uncertainties (Herroelen and Leus, 2005).

The uncertainty of parameters of project is an unavoidable fact in the real world. In a real project, the actual progress of works is not usually compatible with the planned progress due to several uncertainties. Such unpredictable conditions prompted researchers to introduce project scheduling methods under uncertain conditions. Due to our best knowledge the literature of timecost- quality project scheduling trade-off problem is quite new and there are not several research works in this regards. Moreover, there has been reported no research concerning probabilistic and stochastic time-costquality project scheduling trade-off problem in literature of past work. In this paper, time-cost-quality trade-off project scheduling problem is formulated using a multiobjective mathematical model under probabilistic conditions. In the proposed stochastic time-cost-quality project scheduling trade-off problem (STCQTP), the time, cost, and quality of activities are considered to be uncertain modeled through random variables. The lag (lead) between activities are also considered to have probabilistic behavior. The activities are considered to have several execution modes while the generalized precedence relation between activities is considered. A suitable multiobjective stochastic method is presented to solve the associated STCQTP.

The following sections of this paper are organized as follows. In Section 2, the literature of past works are reviewed. In Section 3, the deterministic and probabilistic TCQTP are introduced and developed, respectively. In Section 4, the numerical example and results are presented and discussed, respectively. The conclusion remarks and future research direction are presented in Section 5.

## 2.REVIEW OF RELATED LITERATURE OF PAST WORK

The trade-off problems are consisted on a large and important class of problems in project management due to both practical and theoretical issues. Depending on the type of application and the conditions of problem in terms of objective function, types of activities, resources, and precedence relations, the trade-off problems can be classified. Also, because the trade-off problems are included in Non-deterministic Polynomial Time-hard (NP-hard) class (Demeulemeester et al., 1996), researchers have tried to find more effective solutions.

After reviewing past research works, TCTP and TCQTP problems were divided into five categories. This categorization has been presented in Table 1.

A quick review of methods employed to solve timecost- quality trade off problem shows that various optimization methods have been used to solve these problems. Rahimi and Iran-Manesh (2008) found that heuristic and meta- heuristic algorithms produced better results in small- and medium-sized trade off problems. Methods of solving discreet TCTP are divided into three groups: exact algorithms (such as linear programming, integer programming, dynamic programming, branch and bound algorithms, etc.), heuristic algorithms, and meta-heuristic algorithms. The classification of past researches based on solution procedures has been summarized in Table 2.

In real life project scheduling problems several issues may occur during implementation. For instance, the interactions and relations among the activities of a project is mainly defined using generalized precedence relations. There are multiple modes for execution of each activity in project. In each execution mode, the time, cost, and quality of activities are different. Activities of a project may interrupt during implementations. After interruption, the interrupted activity may be required to be started from the beginning, or its execution mode may change, or it can be continued from the interrupted point. Uncertainty of parameters of the project such as cost, time, quality, and lag (lead) times are other main concerns in real cases.

Due to our best knowledge, although the TCQTPs in real life projects usually are faced with uncertain parameters, generalized precedence, and preemptive activities, there is no model in the literature of project scheduling which can handle all of these issues concurrently. Moreover, the types of uncertainties considered in the past research works were often fuzzy, and stochastic TCQTP, in which random variables describe the time, cost, quality and other parameters of a project, have not been considered. In this paper, we are going to address the above mentioned shortcomings in the literature of TCQTP through proposing a new stochastic time-cost-quality trade-off project scheduling problem considering multiple-execution modes, preemption, and generalized precedence relations. Besides a suitable method of solution is proposed for the proposed new model.

## 3.PROPOSED MODEL

In this section the new stochastic time-cost-quality project scheduling problem considering multiple-modes, preemption of activities, and generalized precedence relations is proposed. First the model proposed by Tavana et al. (2013) is revisited briefly. This model is the basis of proposed model of this study. Then, the proposed model of this study is introduced and discussed in second sub-section.

### 3.1.Theoretical Bases: Tavana et al. (2013) Model

In this sub-section PGP-DTCQT model proposed by Tavana et al. (2013) is briefly revisited. The model has three objective functions that simultaneously minimizes completion time of project, minimizes total cost of the project, and maximizes quality of project. Generalized precedence relations among activities are considered. The activities may preempt during implementation. There also are possibility of implementing an activity through several execution modes. The aforementioned issues illustrate that the model proposed by Tavana et al. (2013) is one of the most practical models introduced in the literature of TCQTP. Equations (1)-(34) present the model proposed by Tavana et al. (2013).(2)(3)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)

$M i n T i m e = S n + 1 , 0$
(1)

$M i n C o s t = ∑ i = 1 i ∈ V n ∑ k = 1 r ( i ) x i k ⋅ c i k + ∑ P = 1 M a x n s u b i ∑ i = 1 i ∈ V 1 n ∑ k = 1 r ( i ) x i k , ​ P ⋅ c i k , P$
(2)

$a x Q u a l i t y = ∑ i = 1 i ∈ V 2 n ∑ k = 1 r ( i ) x i k ⋅ q i k + ∑ p = 1 M a x n s u b i ∑ i = 1 i ∈ V n ∑ k = 1 r ( i ) x i k , ​ p ⋅ q i k , p$
(3)

Subject to

$∑ k = 1 r ( i ) x i k = 1 ∀ i ∈ V 2$
(4)

$∑ k = 1 r ( i ) x i k , P = 1 ∀ i ∈ V 1 ; ∀ p$
(5)

$∑ i = 1 i ∈ V 2 n ∑ k = 1 r ( i ) x i k ⋅ c i k + ∑ p = 1 M a x n s u b i ∑ i = 1 i ∈ V 1 n ∑ k = 1 r ( i ) x i k , ​ p ⋅ c i k , p ≺ _ C$
(6)

$∑ i = 1 i ∈ V 2 n ∑ k = 1 r ( i ) x i k ⋅ q i k + ∑ P = 1 M a x n s u b i ∑ i = 1 i ∈ V 1 n ∑ k = 1 r ( i ) x i k , ​ p ⋅ q i k , p ≻ _ Q$
(7)

$S n + 1 ≺ _ T$
(8)

$F i + L i j ≺ _ S j ∀ i , j ∈ V 2 ; ∀ ( i , j ) ∈ E F S$
(9)

$F i , U i + L i j ≺ _ S j ∀ i ∈ V 1 ; ∀ j ∈ V 2 ; ∀ ( i , j ) ∈ E F S$
(10)

$F i + L i j ≺ _ S j , 1 ∀ i ∈ V 2 ; ∀ j ∈ V 1 ; ∀ ( i , j ) ∈ E F S$
(11)

$F i , U i + L i j ≺ _ S j , 1 ∀ i , j ∈ V 1 ; ∀ ( i , j ) ∈ E F S$
(12)

$S i + L i j ≺ _ F j ∀ i , j ∈ V 2 ; ∀ ( i , j ) ∈ E S F$
(13)

$S i , 1 + L i j ≺ _ F j ∀ i ∈ V 1 ; ∀ j ∈ V 2 ; ∀ ( i , j ) ∈ E S F$
(14)

$S i + L i j ≺ _ F j , U j ∀ i ∈ V 2 ; ∀ j ∈ V 1 ; ∀ ( i , j ) ∈ E S F$
(15)

$S i , 1 + L i j ≺ _ F j , U j ∀ i , j ∈ V 1 ; ∀ ( i , j ) ∈ E S F$
(16)

$S i + L i j ≺ _ S j ∀ i , j ∈ V 2 ; ∀ ( i , j ) ∈ E S S$
(17)

$S i , 1 + L i j ≺ _ S j ∀ i ∈ V 1 ; ∀ j ∈ V 2 ; ∀ ( i , j ) ∈ E S S$
(18)

$S i + L i j ≺ _ S j , 1 ∀ i ∈ V 2 ; ∀ j ∈ V 1 ; ∀ ( i , j ) ∈ E S S$
(19)

$S i , 1 + L i j ≺ _ S j , 1 ∀ i , j ∈ V 1 ; ∀ ( i , j ) ∈ E S S$
(20)

$F i + L i j ≺ _ F j ∀ i , j ∈ V 2 ; ∀ ( i , j ) ∈ E F F$
(21)

$F i + L i j ≺ _ F j , U j ∀ i ∈ V 1 ; ∀ j ∈ V 2 ; ∀ ( i , j ) ∈ E F F$
(22)

$F i + L i j ≺ _ F j , U j ∀ i ∈ V 2 ; ∀ j ∈ V 1 ; ∀ ( i , j ) ∈ E F F$
(23)

$F i , U i + L i j ≺ _ F j , U j ∀ i , j ∈ V 1 ; ∀ ( i , j ) ∈ E F F$
(24)

$ε i k ≺ _ t i k , p ≺ _ d i k ∀ i ∈ V 1 ; p = 1 , 2 , ⋯ , U i + 1 k = 1 , 2 , ⋯ , r ( i )$
(25)

$F i , P − S i , p = ∑ k = 1 r ( i ) x i k , p ⋅ t i k , p ∀ i ∈ V 1 ; p = 1 , 2 , ⋯ , U i + 1$
(26)

$F i − S i = ∑ k = 1 r ( i ) x i k ⋅ d i k ∀ i ∈ V 2$
(27)

$S i , P + 1 − F i , P ≺ _ = α i ∀ i ∈ V 1 ; p = 1 , 2 , ⋯ , U i$
(28)

$x i k , P + 1 ≺ _ x i k , P ∀ i ∈ V 1$
(29)

$∑ P = 1 M a x n s u b i ∑ k = 1 r ( i ) ( t i k , P d i k ) ⋅ x i k , p = 1 ∀ i ∈ V 1$
(30)

$F i , P , S i , p ≻ _ 0 ∀ i ∈ V 1 ; p = 1 , 2 , ⋯ , U i + 1$
(31)

$F i , S i ≻ _ 0 ∀ i ∈ V 2$
(32)

$F 0 , S 0 = 0$
(33)

$x i k , x i k , p ∈ { 0 , 1 } ∀ i , k , p$
(34)

The used notations of model proposed by Tavana et al. (2013) is presented in Table 3.

The detail description of Equations (1)-(34) are not presented here for sake of brevity. Interested readers can find the full description of model (1)-(34) in Tavana et al. (2013). In this paper, we are going to extend the proposed model by Tavana et al. (2013) into stochastic situation which is more realistic for real life projects. Moreover, the solution procedure is developed for proposed stochastic TCQTP.

### 3.2.Stochastic Chance Constraint Programming

In a stochastic programming, the aim is to find the impact of random variables on the results. In all stochastic programming models, the aim is to transform the model into and associate deterministic model. In order to solve multi-objective probabilistic problems, various methods have been suggested. The commonest method is Stochastic Chance Constraint Programming (SCCP).

SCCP was introduced by Charnes and Cooper (1963. In SCCP approach, the aim is to optimize the expected value of objectives, while a number of constraints are taken into account (Kim et al., 2012). Consider the deterministic multi-objective Model (35)-(39) and the way by which all objective functions are to be maximized.

$M a x z k = ∑ j = 1 n c k j ⋅ x j k = 1 , 2 , ⋯ , K$
(35)

Subject to:

$∑ j = 1 n a i j ⋅ x j ≺ _ b i i = 1 , 2 , ⋯ , m$
(36)

$x = ( x 1 , ⋯ , x n )$
(37)

$x ∈ S$
(38)

$x ≻ _ 0$
(39)

Model (35)-(39) has K objective functions that should be maximized, it has also m constraint in form of inequalities (36), the vector of decision variables is in form of (37), and the domain of variables are defined in form of (38)-(39). Where ckj is the benefit ration of decision variable j(j = 1, …, n) in equation of objective function k(k = 1, …, K), aij is the technology coefficient of decision variable j(j = 1, …, n) in constraint i, (i = 1, …, m), bi is the right hand side of constraint i, (i = 1, …, m).

In stochastic optimization problems, at least one of the parameters of cjk, aij, or bi might be defined in an uncertain manner parameterized through a random variable. In such a situation, the deterministic optimization procedures cannot be applied directly. In many cases, including the proposed method by Charnes and Cooper (1963), chance constraints are defined for the problem. For instance, the probability of meeting constraint i is defined using (40).

$P ( ∑ j = 1 n a ˜ i j ⋅ x j ≤ b ˜ i ) ≥ 1 − β i 1 = 1 , 2 , ⋯ , m$
(40)

where βi is the maximum permissible probability for not meeting constraint i, the βi is defined by decision maker (DM), and P(.) means the probability. On the other hand, Equation (4) checks if the inequality $∑ j = 1 n a ˜ i j ⋅ x j ≤ b ˜ i$ satisfies with probability greater than or equal to (1 - βk)%. If all parameters of Model (35)-(39), i.e., cjk, aij , and bi, are assumed to be random variables, then the Model (35)- (39) can be written as Model (41)-(45).

$M a x z k = ∑ j = 1 n c ˜ k j ⋅ x j k = 1 , 2 , ⋯ , K$
(41)

Subject to(42)(43)(44)

$∑ j = 1 n a ˜ i j ⋅ x j ≺ _ b ˜ i i = 1 , 2 , ⋯ , m$
(42)

$x = ( x 1 , ⋯ , x n )$
(43)

$x ∈ S$
(44)

$x ≻ _ 0$
(45)

where the ∼ shows the uncertainty of the parameters. All other definition are similar to those in Model (35)-(39). The Model (41)-(45) can be rewritten using probability rules as Model (46)-(50).

$M a x f k = E ( ∑ j = 1 n c ˜ k j ⋅ x j ≺ _ b ˜ i ) k = 1 , 2 , ⋯ , K$
(46)

Subject to(49)

$P ( ∑ j = 1 n a ˜ i j ⋅ x j ≺ _ b ˜ i ) ≻ _ 1 − β i i = 1 , 2 , ⋯ , m$
(47)

$x = ( x 1 , ⋯ , x n )$
(48)

$x ∈ S$
(49)

$x ≻ _ 0$
(50)

Set of Equation (46) maximize the expected value of objective functions. Set of constraint (47) check if the inequality $∑ j = 1 n a ˜ i j ⋅ x j ≤ b ˜ i$ satisfies with probability greater than or equal to (1 - βi)% for all constraints. It is clear that Model (46)-(50) cannot be optimized in its current form. It should be transformed into its associate deterministic form in a predefined level of (1 - βk)%, which is defined by DM.

#### 3.2.1.Deterministic Equivalence of SCCP

As mentioned, Model (46)-(50) should be transformed into a determinist model in order to be optimized. In order to achieve the associated equivalence of stochastic constraints (47), consider aij and bi follow independent normal Probability Density Function (PDF) with known means and standard deviations. Also, assume $M ˜ i ( x ) = ∑ j = 1 n a ˜ i j ⋅ x j − b ˜ i , i = 1 , ⋯ , m$. Therefore, $M ˜ i ( x )$ is also a random variable with normal distribution and has a mean of $E ( M ˜ i ( x ) )$ and a variance of $v a r ( M ˜ i ( x ) )$. The Set of constraints (47) can be rewritten as (51).

$P ( M ˜ i ( x ) ≤ 0 ) ≻ _ 1 − β i i = 1 , 2 , ⋯ , m$
(51)

So, normal decision variable $M ˜ i ( x )$ can be transformed into standard normal variable using (52)-(53).

$P ( M ˜ i ( x ) − E ( M ˜ i ( x ) ) v a r ( M ˜ i ( x ) ) ≺ _ − E ( M ˜ i ( x ) ) v a r ( M ˜ i ( x ) ) ) ≻ _ 1 − β i i = 1 , 2 , ⋯ , m$
(52)

$P ( Z ( x ) ≺ _ − E ( M ˜ i ( x ) ) v a r ( M ˜ i ( x ) ) ) ≻ _ 1 − β i i = 1 , 2 , ⋯ , m$
(53)

where Z(x) in (53) is standard normal variable. Based on definition in probability theory, the Cumulative Density Function (CDF) of a normal random variable is shown with ϕ(Z(x)) and can be calculated using (54)

$φ ( Z ( x ) ) = P ( Z ( x ) ≤ z )$
(54)

where, z is a real value.

Due to properties of inverse function, the constraints (53) is transformed into (55)-(56).

$− E ( M ˜ i ( x ) ) v a r ( M ˜ i ( x ) ) ≻ _ φ − 1 ( 1 − β i ) i = 1 , 2 , ⋯ , m$
(55)

$E ( M ˜ i ( x ) ) + φ − 1 ( 1 − β i ) ⋅ v a r ( M ˜ i ( x ) ) ≺ _ 0 i = 1 , 2 , ⋯ , m$
(56)

Replacing the value of $M ˜ i ( x ) = ∑ j = 1 n a ˜ i j ⋅ x j − b ˜ i , i = 1 , ⋯ , m$, the deterministic equivalence of (47) is as (57).

$E ( ∑ j = 1 n a ˜ i j ⋅ x j − b ˜ j ) + φ − 1 ( 1 − β i ) . v a r ( ∑ j = 1 n a ˜ i j ⋅ x j − b ˜ i ) ≺ _ 0 i = 1 , 2 , ⋯ , m$
(57)

The set of constraint (57) can easily be handled using properties of expected value and variance of random variable.

In order to achieve the deterministic equivalence of stochastic objective functions (46), assume that all profit coefficients in objective functions ($c ˜ k j$) are normal random variables. The following procedure is proposed in order to achieve the deterministic equivalence of stochastic objective functions (46).

First assume $c k j *$ is the maximum observable value of coefficient j in objective function k, so $c k j * = M a x c ˜ k j$. The value of $c k j *$ can easily be calculated through basic definition in normal distribution at confidence level (1 - βk)% using (58).

$c k j * = E ( c ˜ k j ) + z ( 1 − β k ) ⋅ v a r ( c ˜ k j )$
(58)

where, $E ( c ˜ k j )$ and $v a r ( c ˜ k j )$ are expected value and standard deviation of normal random variable $c ˜ k j$. On the other hand Equation (58) calculates the upper bound of normal random variable $c ˜ k j$ at confidence level (1 - βk)%.

Replacing $c k j *$ in all objective function (46), and solving k single objective optimization models will result in $f k + = M a x ( ∑ j = 1 n c k j * ⋅ x j )$, which is the maximum value of objective function k while taking into account the constraints of system. On the other hand k single objective optimization model in form of Model (59)-(61) are solved in order to achieve all upper bounds of objective functions (i.e., $f k + , k = 1 , ⋯ , K$) at confidence level (1 - βk)%.(61)

$f k + = M a x ∑ j = 1 n c ˜ k j * ⋅ x j$
(59)

Subject to

$x ∈ S$
(60)

$x ≻ _ 0$
(61)

It is obvious that the values of ($f k + , k = 1 , ⋯ , K$) are the upper bound of stochastic objective function k (i.e., $∑ j = 1 n c ˜ k j ⋅ x j$). So the following Equations (62)-(63) can be written.

$∑ j = 1 n c ˜ k j ⋅ x j ≺ _ f k + k = 1 , 2 , ⋯ , K$
(62)

$P ( ∑ j = 1 n c ˜ k j ⋅ x j ≺ _ f k + ) ≻ _ 1 − β k ∀ β k ∈ ( 0 , 1 ) ; k = 1 , 2 , ⋯ , K$
(63)

Defining the following variable exchange:(64)

$N ˜ k ( x ) = ∑ j = 1 n c ˜ k j ⋅ x j − f k +$
(64)

So the Equation (63) can be rewritten as Equation (65).

$P ( N ˜ k ( x ) ≺ _ 0 ) ≻ _ 1 − β k ∀ β k ∈ ( 0 , 1 ) ; k = 1 , 2 , ⋯ , K$
(65)

It obvious that $N ˜ k ( x )$ has normal distribution with a mean of $E ( N ˜ k ( x ) )$ and a variance of $v a r ( N ˜ k ( x ) )$. So, the deterministic equivalence of (65) can be achieved using (66)-(68).(67)

$P ( N ˜ k ( x ) − E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ≺ _ − E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ) ≻ _ 1 − β k ∀ β k ∈ ( 0 , 1 ) ; k = 1 , 2 , ⋯ , K$
(66)

$− E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ≻ _ φ − 1 ( β k ) ∀ β k ∈ ( 0 , 1 ) ; k = 1 , 2 , ⋯ , K$
(67)

$E ( N ˜ k ( x ) ) + φ − 1 ( 1 − β k ) ⋅ v a r ( N ˜ k ( x ) ) ≺ _ 0 k = 1 , 2 , ⋯ , K$
(68)

Replacing the value of $N ˜ k ( x ) = ∑ j = 1 n c ˜ k j ⋅ x j − f k +$, the deterministic equivalence of maximum objective functions (46) considering upper bound of objective values at confidence level (1 - βk)% is as (69).

$E ( ∑ j = 1 n c ˜ k j ⋅ x j − f k + ) + φ − 1 ( 1 − β k ) ⋅ v a r ( ∑ j = 1 n c ˜ k j ⋅ x j − f k + ) ≺ _ 0 k = 1 , 2 , ⋯ , K$
(69)

Second, assume $c k j *$ is the minimum observable value of coefficient j in cost objective function k, so $c k j * = M i n c ˜ k j$. The value of $c k j *$ can easily be calculated through basic definition in normal distribution at confidence level (1 - βk)% using (70).

$c k j * = E ( c ˜ k j ) − z ( 1 − β k ) ⋅ v a r ( c ˜ k j )$
(70)

where, $E ( c ˜ k j )$ and $v a r ( c ˜ k j )$ are expected value and standard deviation of normal random variable $c ˜ k j$. On the other hand Equation (70) calculates the lower bound of normal random variable $c ˜ k j$ at confidence level (1 - βk)%.

Replacing $c k j *$ in allobjective function (46), and solving k single objective optimization models will result in $f k − = M i n ( ∑ j = 1 n c k j * ⋅ x j )$ which is the minimum value of objective function k while taking into account the constraints of system. On the other hand k single objective optimization model in form of Model (71)-(73) are solved in order to achieve all lower bounds of objective functions (i.e., $f k − , k = 1 , ⋯ , K$)at confidence level (1 - βk)%.

$f k − = M i n ∑ j = 1 n c k j * ⋅ x j$
(71)

Subject to:(72)

$x ∈ S$
(72)

$x ≻ _ 0$
(73)

It is obvious that the values of ($f k − , k = 1 , ⋯ , K$) are the lower bound of stochastic objective function k (i.e., $∑ j = 1 n c ˜ k j ⋅ x j$). So the following Equations (74)-(75) can be written.

$∑ j = 1 n c ˜ k j ⋅ x j ≻ _ f k − k = 1 , 2 , ⋯ , K$
(74)

$∑ j = 1 n c ˜ k j ⋅ x j ≻ _ f k − k = 1 , 2 , ⋯ , K$
(75)

Defining the following variable exchange:(76)

$N ˜ k ( x ) = ∑ j = 1 n c ˜ k j ⋅ x j − f k −$
(76)

So the Equation (75) can be rewritten as Equation (77).

$P ( N ˜ k ( x ) ≻ _ 0 ) ≻ _ 1 − β k ∀ β k ∈ ( 0 , 1 ) ; k = 1 , 2 , ⋯ , K$
(77)

It obvious that $N ˜ k ( x )$ has normal distribution with a mean of $E ( N ˜ k ( x ) )$ and a variance of $v a r ( N ˜ k ( x ) )$. So, the deterministic equivalence of (77) can be achieved using (78)-(82).(79)(80)(81)

$P ( N ˜ k ( x ) − E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ≻ _ − E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ) ≻ _ 1 − β k ∀ β k ∈ ( 0 , 1 ) k = 1 , 2 , ⋯ , K$
(78)

$1 − P ( N ˜ k ( x ) − E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ≺ _ − E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ) ≻ _ 1 − β k ∀ β k ∈ ( 0 , 1 ) ; k = 1 , 2 , ⋯ , K$
(79)

$P ( N ˜ k ( x ) − E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ≺ _ − E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ) ≺ _ β k ∀ β k ∈ ( 0 , 1 ) ; k = 1 , 2 , ⋯ , K$
(80)

$− E ( N ˜ k ( x ) ) v a r ( N ˜ k ( x ) ) ≺ _ φ − 1 ( β k ) ∀ β k ∈ ( 0 , 1 ) ; k = 1 , 2 , ⋯ , K$
(81)

$E ( N ˜ k ( x ) ) + φ − 1 ( β k ) ⋅ v a r ( ( N ˜ k ( x ) ) ) ≻ _ 0 k = 1 , 2 , ⋯ , K$
(82)

Replacing the value of $N ˜ k ( x ) = ∑ j = 1 n c ˜ k j ⋅ x j − f k −$, the deterministic equivalence of objective functions (46) considering lower bound of objective values at confidence level (1 - βk)% is as (83).

$E ( N ˜ k ( x ) ) + φ − 1 ( β k ) ⋅ v a r ( ( N ˜ k ( x ) ) ) ≻ _ 0 k = 1 , 2 , ⋯ , K$
(83)

Based on the Equations (57), (69), and (83), the stochastic multi-objective chance constraint model (46)-(50) can be transformed into a deterministic multi-objective mathematical programming at confidence level (1 - βk)%.

Another issue is multiple-objectives of the model. In the following sub-section goal programming is proposed in order to resolve the aforementioned point.

### 3.3.Handling Multiple-Objectives through Generalized Scaled Weighted Goal Programming

Goal programming (GP) is widely-used method in multi-objective optimization. This method was initially proposed by Charnes and Cooper (1963). Using GP, we can simultaneously move toward several objectives, while those objectives might be even opposite. In this method, two separate types of constraint are defined: systemic constraints and goal constraints. For a goal constraint, deviation variables $( d k − , d k + )$ are defined. In contrast to linear programming, which minimizes or maximizes objectives, GP minimizes the deviations between goals of objectives and real results. Reconsidering multiple-objective Model (35)-(39). The goal programming Model (84)-(91) is proposed.

$M i n ∑ i = 1 n [ w 1 ⋅ ( d 1 + + d 1 − z 1 − − z 1 + ) + w 2 ⋅ ( d 2 + + d 2 − z 2 − − z 2 + ) + ⋯ + w k ⋅ ( d k + + d k − z k + − z k − ) ]$
(84)

Subject to:(87)(88)(89)(90)

$∑ j = 1 n a i j ⋅ x j ≺ _ b i i = 1 , 2 , ⋯ , m$
(85)

$z k ( x ) + ( d k − − d k + ) = z k * k = 1 , 2 , ⋯ , K z k * ∈ { z k − , z k + }$
(86)

$d k − ⋅ d k + = 0 k = 1 , 2 , ⋯ , K$
(87)

$d k − , d k + ≻ _ 0 k = 1 , ⋯ , K$
(88)

$x = ( x 1 , ⋯ , x n )$
(89)

$x ∈ S$
(90)

$x ≻ _ S$
(91)

where, $d k − , d k +$ are negative and positive deviations from goal k, respectively, wk is relative importance of objective function k, $z k −$ is anti-ideal value of objective function k, which is calculated through minimization of single objective k of Model (35)-(39), $z k +$ is ideal value of objective function k, which is calculated through maximization of single objective k of Model (35)-(39), and $z k *$ is the goal of objective function k.

Set of constraints (85) is the systematic constraints of original MODM model (35)-(39). Set of constraints (86) is the goal constraints associated with goals. It is notable that $z k ( x ) − z k + = d k + and z k ( x ) − z k − = d k −$.

Model (84)-(91) minimizes the scaled weighted sum of deviations from ideal and anti-ideal values of objective functions, while the goals are satisfied in set of constraint (86) as much as possible. The scaled term refers to the denominator of $d k + + d k − z k + − z k −$. As the term $z k + − z k −$ is the range of objective k, so the term $z k + − z k −$ is a scaled value. This eases the summation of deviations of several different objective function.

Model (84)-(91) is called generalized scaled weighted goal programming (GSWGP). It is notable that in practice the GSWGP is reduced. For instance, in the maximizing objective function negative deviations in the objective function of GSWGP are used, and in the minimizing objective function, positive deviations are considered in the objective function of GSWGP.

### 3.4.Proposed Stochastic Time-Cost-Quality Project Scheduling Model

In the previous sub-sections, a deterministic time-costquality trade-off project scheduling problem considering multiple-modes of activity, preemption, and generalized precedence relations, which was first was proposed by Tavana et al. (2013) was revisited. Then a stochastic multiobjective chance constraint mathematical programming procedure was developed in order to handle a multiobjective optimization problem in presence of random parameters. Afterwards, a generalized scaled weighted goal programming (GSWGP) was proposed to handle multipleobjective optimization considering different scales, ranges, and relative importance of objective functions.

In this sub-section, considering all above mentioned materials, we are going to develop the proposed model by Tavana et al. (2013) into a stochastic case, which is more preferable for real life project scheduling problems.

As mentioned, it is intended to investigate the proposed model of Tavana et al. (2013) in stochastic situation. For this aim, the parameters of time, cost, quality, and lag (lead) times between activities are considered to be uncertain parameterized through random distributions. Symbols and descriptions of these parameters are shown in Table 4.

As was mentioned earlier, for random parameters of the model, a defined PDF must be considered. In this study, it is assumed that all random parameters introduced in Table 4 have normal distribution. Normal distribution is one of the most-applied PDF in stochastic optimization and modeling in the real world. Main behavior of a normal random variable, x, is defined using mean (E(x)) and variance (var(x)). Having these two parameters for a normal random variable, the confidence interval of the variable can be calculated at desired confidence level.

The following Model (92)-(126) is proposed for Stochastic Time-Cost-Quality project Scheduling problem considering multiple-modes of activities, preemption of activities during implementation, generalized precedence relations among activities, and random nature of time, cost, quality, and lead (lag) time of activities.(93)(94)(95)(96)(97)(98)(99)(100)(101)(102)(103)(104)(105)(106)(107)(108)(109)(110)(111)(112)(113)(114)(115)(116)(117)(118)(119)(120)(121)(122)(123)(124)(125)

$M i n ∑ i = 1 n [ ( d 1 + z 1 − − z 1 + ) + ( d 2 + z 2 − − z 2 + ) + ( d 3 − z 3 + − z 3 − ) ]$
(92)

Subject to:

$S n + 1 , 0 − z 1 − − d 1 + = 0$
(93)

$∑ i = 1 i ∈ V 2 n ∑ k = 1 r ( i ) x i k ⋅ E ( c ˜ i k ) + ∑ p = 1 M a n x s u b i ∑ i = 1 i ∈ V 1 n ∑ k = 1 r ( i ) x i k , p ⋅ E ( c ˜ i k , p ) − f 2 + + φ − 1 ( β 2 ) ⋅ ∑ i = 1 i ∈ V 2 n ∑ k = 1 r ( i ) x i k ⋅ v a r ( c ˜ i k ) + ∑ p = 1 M a n x s u b i ∑ i = 1 i ∈ V 1 n ∑ k = 1 r ( i ) x i k , p ⋅ v a r ( c ˜ i k , P ) − d 2 + = 0$
(94)

$∑ i = 1 i ∈ V 2 n ∑ k = 1 r ( i ) x i k ⋅ E ( q ˜ i k ) + ∑ p = 1 M a n x s u b i ∑ i = 1 i ∈ V 1 n ∑ k = 1 r ( i ) x i k , p ⋅ E ( q ˜ i k , p ) − f 3 + + φ − 1 ( 1 − β 3 ) ⋅ ∑ i = 1 i ∈ V 2 n ∑ k = 1 r ( i ) x i k ⋅ v a r ( q ˜ i k ) + ∑ p = 1 M a n x s u b i ∑ i = 1 i ∈ V 2 n ∑ k = 1 r ( i ) x i k , p ⋅ v a r ( q ˜ i k , p ) + d 3 − = 0$
(95)

$F 1 + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ S j ∀ i , j ∈ V 2 ; ∀ ( i , j ) ∈ E F S$
(96)

$F i , U i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ S j ∀ i ∈ V 1 ; ∀ j ∈ V 2 ∀ ( i , j ) ∈ E F S$
(97)

$F i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ S j , 1 ∀ i ∈ V 2 ; ∀ j ∈ V 1 ∀ ( i , j ) ∈ E F S$
(98)

$F i , U i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ S j , 1 ∀ i , j ∈ V 1 ; ∀ ( i , j ) ∈ E F S$
(99)

$S i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ F j ∀ i , j ∈ V 2 ; ∀ ( i , j ) ∈ , E F S$
(100)

$S i , 1 + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ F j ∀ i ∈ V 1 ; ∀ j ∈ V 2 ∀ ( i j ) ∈ , E F S$
(101)

$S i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ F j , U j ∀ i ∈ V 2 ; ∀ j ∈ V 1 ∀ ( i j ) ∈ E F S$
(102)

$S i , 1 + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ F j , U j ∀ i , j ∈ V 1 ; ∀ ( i j ) ∈ E F S$
(103)

$S i , + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ S j ∀ i , j ∈ V 2 ; ∀ ( i j ) ∈ E S S$
(104)

$S i , 1 + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ S j ∀ i ∈ V 1 ; ∀ j ∈ V 2 ; ∀ ( i , j ) ∈ E S S$
(105)

$S i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ S j , 1 ∀ i ∈ V 2 ; ∀ j ∈ V 1 ; ∀ ( i , j ) ∈ E S S$
(106)

$S i , 1 + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ S j , 1 ∀ i , j ∈ V 1 ; ∀ ( i , j ) ∈ E S S$
(107)

$F i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ F j ∀ i , j ∈ V 2 ; ∀ ( i , j ) ∈ E F F$
(108)

$F i , U i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ F j ∀ i ∈ V 1 ; ∀ j ∈ V 2 ∀ ( i , j ) ∈ E F F$
(109)

$F i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ F j , U j ∀ i ∈ V 2 ; ∀ j ∈ V 1 ∀ ( i , j ) ∈ E F F$
(110)

$F i , U i + E ( L ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( L ˜ i j ) ≺ _ F j , U j ∀ i , j ∈ V 1 ; ∀ ( i , j ) ∈ E F F$
(111)

$t i k , p − E ( d ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ v a r ( d ˜ i j ) ≺ _ 0 ∀ i ∈ V 1 ; p = 1 , 2 , ⋯ , U i + 1 ; k = 1 , 2 , ⋯ , r ( i )$
(112)

$F i − S i − ∑ k = 1 r ( i ) x i k ⋅ E ( d ˜ i j ) + φ − 1 ( 1 − β i ) ⋅ ∑ k = 1 r ( i ) x i k ⋅ v a r ( d ˜ i k ) ≺ _ 0 ∀ i ∈ V 2$
(113)

$∑ k = 1 r ( i ) x i k ⋅ E ( d ˜ i k ) − F i + S i + φ − 1 ( 1 − β i ) ⋅ ∑ k = 1 r ( i ) x i k ⋅ v a r ( d ˜ i k ) ≺ _ 0 ∀ i ∈ V 2$
(114)

$∑ P = 1 M a n x s u b i ∑ k = 1 r ( i ) ( t i k , P E ( d ˜ i k ) ) ⋅ x i k , p + φ − 1 ( 1 − β i ) ⋅ ∑ P = 1 M a n x s u b i ∑ k = 1 r ( i ) ( t i k , P 2 v a r ( d ˜ i k ) ) ⋅ x i k , p ≻ _ 1 ∀ i ∈ V 1$
(115)

$∑ P = 1 M a n x s u b i ∑ k = 1 r ( i ) ( t i k , P E ( d ˜ i k ) ) ⋅ x i k , p − φ − 1 ( 1 − β i ) ⋅ ∑ P = 1 M a n x s u b i ∑ k = 1 r ( i ) ( t i k , P 2 v a r ( d ˜ i k ) ) ⋅ x i k , p ≺ _ 1 ∀ i ∈ V 1$
(116)

$ε i k ≺ _ t i k , p ∀ i ∈ V 1 ; p = 1 , 2 , ⋯ , U i + 1 ; k = 1 , 2 , ⋯ , r ( i )$
(117)

$∑ k = 1 r ( i ) x i k = 1 ∀ i ∈ V 2 ;$
(118)

$∑ k = 1 r ( i ) x i k , P = 1 ∀ i ∈ V 1 ; ∀ p$
(119)

$F i , P − S i , p = ∑ k = 1 r ( i ) x i k , P ⋅ t i k , p ∀ i ∈ V 1 ; p = 1 , 2 , ⋯ , U i + 1$
(120)

$S i , P + 1 − F i , P ≺ _ α i ∀ i ∈ V 1 ; p = 1 , 2 , ⋯ , U i$
(121)

$x i k , P + 1 ≺ _ x i k , P ∀ i ∈ V 1$
(122)

$F i , P , S i , p ≻ _ 0 ∀ i ∈ V 1 ; p = 1 , 2 , ⋯ , U i + 1$
(123)

$F i , S i ≻ _ 0 ∀ i ∈ V 2$
(124)

$F 0 , S 0 = 0$
(125)

$x i k , x i k , p ∈ { 0 , 1 } ∀ i , k , p$
(126)

The properties of Model (92)-(126) is briefly summarized as follows:

• It can produce suitable compromise solutions considering time, cost, and quality objectives of project.

• The preemption of activity is feasible during implementation.

• The generalized precedence relations are considered among activities of a project.

• There are multiple-modes and choices in order to accomplish each activity of the project.

• An activity can be continued from the point it was preempted, if and only if the interruption time does not exceed a predefined time.

• An activity should be re-scheduled from the beginning, if and only if the interruption time does exceeds a predefined time.

• An activity can be accomplished in its selected mode before preemption or even another execution-mode may be selected for an activity after preemption.

• Duration time of activities is assumed to be uncertain parameterized through normal probability density function with known mean and standard deviation.

• Direct cost of activities is assumed to be uncertain parameterized through normal probability density function with known mean and standard deviation.

• Level of quality of activities is assumed to be uncertain parameterized through normal probability density function with known mean and standard deviation.

• Time lag (lead) between activities of project is assumed to be uncertain parameterized through normal probability density function with known mean and standard deviation.

• The model can be run for a predetermined confidence level which can easily be set by project manager.

According to our best knowledge, there is no unique model in literature of trade-off project scheduling problems which can handle all above mentioned properties concurrently.

## 4.NUMERICAL EXAMPLE AND RESULTS

In order to illustrate the applicability of proposed model of this study a numerical example is introduced in this section and the results are discussed. The activity on node (AON) representation is selected in order to present the numerical example. The network of the numerical example has ten activity (node), twelve arcs (precedence relations). Some of the activities of the example may preempt into four sub-activities (pieces), and five execution modes are feasible for each activity. In this numerical example both type of preemptive and non-preemptive activities are exist. Nodes one, two, five, seven, and nine can be preempted (illustrated by white nodes in Figure 1 and other nodes cannot be preempted (illustrated by gray nodes in Figure 1. Precedence relations among activities and lag (lead) have been shown on arcs in Figure 1.

The value of parameters for numerical example is shown in Table 5, Table 6, and Table 7. It is notable that the values in Table 5, Table 6, and Table 7 are assumed to mean value of these parameters. The standard deviation are assumed to be 0.1*mean value for all parameters in Table 5, Table 6, and Table 7.

It is notable that the numerical example has been solved using proposed model of this study and the deterministic model proposed by Tavana et al. (2013). Under certainty conditions, all parameters are equal to their means. Proposed method of this study and the model proposed by Tavana et al. (2013) have been codified in GAMS software. Relative important of all objective functions has been set equally 0.333. The confidence level for all random parameters in all experiments is set 0.95.

First, single optimized value of all objective functions is achieved using both methods. The results presented in Table 8 shows that the stochastic model of this study is superior to deterministic model of Tavana et al. (2013) in cost and quality objective functions. The stochastic model of this study improves quality of project and reduces the costs in comparison to model of Tavana et al. (2013). However, it is clear that stochastic model of this study increases the finishing time of the project.

The results of objective functions in Table 8 reveals that the proposed model of this study not only handles the uncertainty of parameters in form of random variable but it also achieves qualified objective functions.

Then Model (92)-(126) is run for the numerical example. In goal programming the deviation form goals is the main concern. A sensitivity analysis has been accomplished on these deviations. The results are presented in Table 9.

The relative importance of deviations (i.e., w1, w2, and w3) has been chanced and the deviation values have been recorded. Four scenario based on priority of each objective function have been tested. It can be concluded from Table 9 that the proposed model of this study has a considerable sense subject to relative importance of objective functions. So, in real life project scheduling problem, this priority can be determined by project manager in order to generate desirable compromise solutions.

## 5.CONCLUSION REMARKS AND FUTURE RESEARCH DIRECTIONS

In this paper a preemptive time-cost-quality tradeoff project scheduling problem in presence of random parameters considering multi-mode activities, and generalized precedence relation was proposed. The time, cost, and quality of activities as well as lag (lead) between activities were assumed to be uncertain parameterized through normal random variables with known mean and standard deviation. Each activity can be accomplished in several modes with the associated random time, cost, and quality. The activity may be followed in a different mode before and after preemption. The activity may be rescheduled or continued form the preempted point based on duration of preemption. If the number of preemption of an activity exceeds the maximum number of allowed preemption, the activity should be rescheduled from the beginning.

Stochastic chance constraint programming (SCCP) was proposed to handle the uncertainty of the parameters of the proposed model. The SCCP procedure transformed the proposed model into its deterministic equivalence at a predefined confidence level. The multi-objective nature of the proposed model was handled through a weighted scaled goal programming approach. A numerical example was presented and solved using proposed model and the solution procedure in order to illustrate the applicability and efficacy of proposed method. The results were compromising and comparable with the result of similar model existed in literature of trade-off project scheduling problem. Sensitivity analysis on relative importance of objective functions was conducted. The results of sensitivity analysis showed that the proposed model has meaningful sense subject to relative importance of objective functions.

Form managerial point of view, the properties of proposed model and solution procedure of this study really fit with real projects. Due to our best knowledge, there was no unique model of time-cost-quality trade-off project scheduling problem, in which all of the preemption of activities, multiple-modes of execution, generalized precedence relation, and uncertainty of parameters of the project, were considered concurrently. All of the above mentioned issues were considered concurrently in the proposed model.

Other parameters of the projects such as maximum number of preemption which is an important factor in rescheduling of continuing an activity, number of execution modes, upper bound of cost of the project, upper bound of time of project, the type of generalized precedence relations can be assumed to have uncertainty in the future researches. Moreover, using the proposed procedure and mechanism of this study in facing with multiple objective, and handling the uncertain parameters, other type of random variable such as uniform, t-student, lognormal and Weibull can be considered and the proposed procedure can be extended for them in future research. Application of proposed model and solution procedure in real life projects may be another interesting direction for future research.

## Figure

Project network of numerical example.

## Table

Categorization of TCTPs and TCQTPs

Categorization of Methods of Solving TCTPs and TCQTPs

Sets, indices, parameters, and decision variables (Adapted from Tavana et al., 2013)

Random parameters

Values of parameters Ci,k,p and qi,k,p

Values of parameters di,k, Ci,k , and qi,k

Values of parameter Li,j

A comparison of objective function for numerical example

Sensitivity analysis of deviation values

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