In construction project scheduling, if it is required to finish the project before a specified due time, the duration of performing some activities must be decreased. To achieve this goal, it is necessary to either increase some resources or change the execution methods of the activities, which will result in increased costs or altered quality. The time-cost-quality trade-off aims to select a set of activities for crashing and an appropriate execution method for each activity to minimize the project's cost and time. In contrast, the total project quality is maximized.

In most of the time cost trade-off models, the relationship between the decreased duration and the increased amount of cost is assumed to be linear. Furthermore, the aim is to finish the project before the due time and minimize the total cost. Numerous methods have been proposed [1]-[6] to solve such a linear model. Nonlinear models have also been proposed for the time cost trade-off problem [7]-[10]. The Discrete Time-Cost Trade-Off Problem (DTCTP) applies to real-world applications. Unlike linear models, there are few studies regarding DTCTP. Therefore, a discrete time-cost relationship is more relevant to real-world projects since resources, execution methods, and technology types in real-world projects are represented by discrete values.

The solution methods of DTCTP are classified as exact and heuristic. None of the precise techniques can solve real-world projects with many activities. DTCTP is known as an NP-hard problem [11]. Thus, no exact solution method can be found to have the required efficiency for solving DTCTP. However, many approaches are based on heuristic algorithms for solving DTCTP [12]-[16].

The total project quality is affected by the project crashing. Thus, it is necessary to include the quality factor in the time-cost trade-off problem, leading to the time-cost- quality trade-off problem. This problem is represented by a Discrete-Time Cost Quality Trade-off Problem (DTCQTP) for discrete cases. In this problem, each project activity can be executed in one of many modes. The execution mode of any activity is related to the resources, execution methods, and execution technology used in the activity.

The time-cost-quality trade-off optimization problem aims to complete the project simultaneously, considering the minimal cost, duration, and maximum quality. Not many studies on this optimization problem can be found in the literature. By relaxing some non-realistic assumptions, more realistic methods must be presented. In this paper, we consider the problem in the discrete case (i.e., DTCQTP) and relax the assumption of the linear relationship between time-cost and time-time-quality [17]-[22].

In the operations research literature, many scholars and researchers incorporate fuzzy variables into their models due to the uncertain nature of real-world problems [23]-[26]. In this paper, the quality of each mode is a linguistic variable. Linguistic variables are not numbers but words or sentences in a natural or artificial language. Fuzzy set theory is applied to manage uncertain and linguistic data. Fuzzy numbers are a particular type of fuzzy set, which are normal and convex. Although these numbers can be described using many special methods and shapes, triangular and trapezoidal shapes are widely best used for solving practical applications. The α-cut is a commonly used method to connect the principles of fuzzy sets with a collection of crisp sets, which can be fed into most of the existing systems. The α–cut level set of A is defined as follows:

Aα = {(x, μA(x)) ≥ α ⋮ xϵX}, for all α ∈ [0,1], where, X: the range of possible values.

μA(x): the membership function takes values within the [0,1] range, which specifies the degree to which x belongs to the fuzzy set A.

A triangular fuzzy number is shown as A ̃ = (a , a , a ) with the following membership function:

1 2 3

What makes this study different from the previous ones is that it considers uncertainty in the quality of each activity and introduces an innovative approach to solving the problem. The high speed of the algorithm and quick convergence of the solutions make this approach suitable for large projects with large numbers of activities.

The quality of each mode and the acceptable quality of a project cannot be measured precisely by the manager. They are linguistic variables and are presented by fuzzy numbers. Therefore, in this paper, fuzzy set theory is employed to consider the effects of uncertainty on total project quality.

The degree of acceptable risk by the managers is represented as α. The project manager's selection of an appropriate value of α influences the project's outcome significantly. As a result, identifying the relationship between the value of α (0 to 1) and the corresponding risk level (no risk to full risk) by collecting managers' risk attitudes would be indispensable. By setting α equal to zero, no risk is acceptable, and the total quality is considered, while α equal to 1 results in accepting full risk, and project quality is transformed from a fuzzy value into a crisp one.

This paper is organized as follows. In Section 2, we present the problem definition and the problem formulation. In Section 3, a solution procedure is introduced. We developed an algorithm called NHGA. Some examples are presented in Section 4 to illustrate the proposed approach.

A project is represented by a direct acyclic graph G=(V, E), with m nodes and n arcs where V={ 1,2,3,….,m} is the set of node and E={ (i,j),…,(l,m) } is the set of direct arcs. Arcs and nodes represent activities and events, respectively.

Each project activity says (i, j) ∈ E can be executed by a set of modes, Mij. Each kϵMij needs an execution time of tijk, and a cost of cijk, while its quality is represented by q ̃ ijk. Let k and r be two modes for activity (i, j) and k<r, then, it is assumed tijk>tijr, cijk<cijr, and q ̃ijk≠q ̃ijr . Although in the literature, it is assumed that any decreased activity time leads to decreased activity quality, it is noteworthy that, in real-world projects, this is not always the case. For instance, a new technology can be employed to reduce the required time, while an increase in quality and cost can accompany this reduction.

This research aims to obtain the optimal combination (tijk, cijk, q ̃ ijk) of each activity for crashing the project network, such that along with reducing the total project duration, the total project cost (direct plus indirect) is minimized. In contrast, the total fuzzy quality of the project does not fall below the desired level. Notations used for the problem formulation are as follows:

CI: project indirect cost per unit of time.

Mij: set of available execution modes for activity ij; ij ∈ E.

cijk: direct cost of activity ij if performed by execution mode k.

tijk: duration of activity ij if performed by execution mode k.

q ̃ ijk: fuzzy quality of activity ij if performed by execution mode k.

wij: quality weight of activity ij in the project ∑ij∈E wij = 1.

Q ̃ : lower bound for fuzzy quality of the project.

1, if mode k is assigned to activity ij, yijk = [ o, otherwise.

xi = earliest time of event i, i = {1,2, … … … , m}.

C: total project cost (direct plus indirect).

T: total project duration.

The problem of DTCQTP is formulated using the mixed integer programming model.

Minimize C = (∑ ∑ cijk ∗ yijk) + CI ∗ ( − 1). (1)

ij∈E k∈Mij

Minimize T = xm − x1. (2)

s. t.

∑ w ∑ q ̃ ∗ y ≥ Q ̃ .

ij ijk ijk L (3)

ij∈E k∈Mij

xj − xi ≥ ∑ tijk ∗ yijk, ij ∈ E, i, j ∈ V.

k∈Mij

∑ yijk = 1, ij ∈ E.

k∈Mij

xi ≥ 0, i ∈ V. (6) yijk ∈ {0, 1}, ij ∈ E, k ∈ Mij. (7)

Eq. (1) and Eq. (2) minimize the total project cost and duration. Eq. (3) enforces that the fuzzy total project quality does not fall below the acceptable level. Eq. (4) preserves the precedence relations between project activities. In Eq. (5), only one execution mode is assigned to each activity.

In the DTCQTP, there are a number of execution modes to select for each activity. If the number of project activities is n and there are k execution modes for each activity to choose from, then there are kn solutions, which results in a very large search space. Therefore, it is necessary to develop an efficient evolutionary algorithm.

A New Hybrid Genetic Algorithm (NHGA) is used to obtain the optimal (or near-optimal) solution [21]. The high speed of the algorithm and quick convergence of the solutions make it suitable for solving the above problem. NHGA is developed with some modifications in GA (Fig. 1).

As an example, a project that includes nine activities is presented in this section, Fig. 2.

The network matrix is presented in Fig. 1. It is an adjacency matrix used to transfer networks to a matrix. Different execution modes of each activity, with associated time, cost, and fuzzy quality, are presented in

The model is programmed in Microsoft Excel using Visual Basic Application (VBA). The project data are entered: Fig. 1, Table 1, CI= 20, L = [82, 83, 84], and α =1. In this example, there are 1500000 solutions. The proposed model was used to obtain the optimal solution. Since the number of activities is small in the example, a combination of a two-point crossover, a uniform crossover with pre-specified weights, and a two-point mutation were used.

NHGA parameters are set as follows:

G=80, N=50, two-point crossover rate=0.6, uniform crossover rate=0.2, mutation rate=0.2, hill climbing rate =0.15, Wexp=0.4, Wline= 0.6.

The program is running, and the chromosome [4,2,2,1,1,5,1,4,4] is obtained. The output was obtained as its corresponding time, cost, and fuzzy quality (T=34, Cd=1440, = [82.9, 84.5, 86.7]). Also, total project cost was obtained (C=2120). The project manager then obtained other optimum solutions by increasing Q ̃L and changing the value for the level of risk (α-cut level), which are presented in the results in Table 2.

The project manager can make decisions by using the information in Table 2 and analyzing the internal and external conditions of the project.

This section evaluates the performance of our proposed NHGA and the classical GA. It is used the Relative Deviation Index (RDI) as a common performance measure to compare these algorithms that are computed by Calg − Cexa Talg − Texa RDI = + wt ∗ , Cexa Texa where Calg and Talg are the total project cost and duration for a given algorithm, respectively, and Cexa and Texa are the solutions obtained using the exact methods for the given small example. The NHGA and GA are implemented with the same parameters twenty times. Their results are analyzed via the analysis of variance (ANOVA) method. The main hypotheses, containing normality, homogeneity of variance, and independence of residuals, are checked, and there is no bias for questioning the experiment's validity. The means plot and Least Significant Difference (LSD) interval for the NHGA and GA are shown in Fig. 4. It demonstrates that the NHGA gives better outputs than the GA statistically.

The weight of the exponential function (Wexp) and the weight of the linear function (Wline) is an essential parameter of the decline rate function (F(h)). The considered levels of these parameters (Wexp, Wline) are as (0,1),(0.2,0.8), (0.4,0.6), (0.5,0.5),(0.6,0.4),(0.8,0.2), and (1,0). The NHGA is implemented twenty times for each level of the Wexp. The results are analyzed using the ANOVA method. The means plot and LSD interval for the levels of the Wexp are shown in Fig. 4. This figure demonstrates that Wexp = 0.4 or (Wexp, Wline) = (0.4,0.6) results in better outputs than the other (Wexp, Wline) levels statistically.

Many project managers perform project crashes intending to reduce total cost and time and achieve a desirable quality. A linguistic variable expresses the quality of each activity. The model proposed in this paper addresses real-world projects. In order to be as realistic as possible, the problem was considered in the discrete case (DTCQTP), and the quality of activities was presented using triangular fuzzy numbers. In this problem, each project activity can be executed in one of several modes. Associated with each execution mode of any activity, there are specific resources, execution methods, and technology. For each mode of an activity, there is a triple combination (t,c,q ̃) indicating the time, cost, and fuzzy.

Solving the problem gave an optimal solution, including the time, cost, and project quality for each α-cut level. The α represents the risk the managers are prepared to take. An optimal solution is obtained for each acceptable project quality and α-cut level. The α represents the degree of risk the managers are prepared to accept. Project managers can make effective decisions with these optimal solutions and by analyzing the environmental conditions.

A hybrid algorithm, NHGA, was also developed to solve the problem. The proposed algorithm's high speed and quick convergence make it desirable for large projects with many activities. Furthermore, using the ANOVA method, we used the RDI measure to compare the performance of the NHGA and GA. We have also demonstrated that (Wexp, Wline) = (0.4,0.6) has resulted in statistically better output than the other (Wexp, Wline) levels.

Considering uncertainty in duration time, cost factors, and even simultaneous uncertainty in more than one factor, this model can be extended to such cases, which makes it more realistic.