1. Introduction
As the basis of quantitative research on schedule robustness, robustness measures are an effective tool to measure the anti-interference ability of schedules [
1]. There are two aspects of construction schedule robustness, namely, solution robustness and quality robustness, in measuring a schedule’s robustness. In detail, solution robustness refers to the deviation between the planned and the actual schedule, which measures the robustness from the perspective of project execution. Quality robustness refers to the deviation between the planned and the actual construction duration, which measures the robustness from the perspective of the project result. However, current research mostly focuses on a single robustness criterion, and few consider a composite robustness criterion, neglecting the bounded rationality of subjective weights and the inherent importance and nonlinear intercriteria correlations of objective weights. Therefore, it is of great importance to measure the schedule robustness comprehensively, which considers both solution robustness from project execution and quality robustness from project execution, in addition to the bounded rationality of subjective weights and the inherent importance and nonlinear intercriteria correlations of objective weights.
Studies of the robustness measure are mainly related to job shops, while a few studies on construction schedule robustness measures have been made. While current methods measuring robustness focus on using a single criterion of solution robustness, which is lacking the quality robustness criteria. Lambrechts [
2] measured the solution robustness using activity durations for project scheduling. Akkan [
3] used start time deviation to measure the solution robustness and the objective optimization function of job shop production. Sundström [
4] measured robustness by a solution robustness criterion, which calculates the average deviation of start time between the planned schedule and the actual schedule. Ansari [
5] took covariance as a solution robustness criterion and measured the robustness of the schedule obtained by simulations and the critical chain method. Liu [
6] established a robustness criterion using a time slack-based technique to deal with schedules with mechanical malfunction and new job arrivals. Xiao [
7] took the expected relative deviation between the planned and actual schedule as a solution robustness criterion and used the criterion to analyze the robustness of the stochastic job shop scheduling problem. Rahmani [
8] measured the schedule robustness of job shop production concerning mechanical malfunction through a solution robustness criterion. Pang [
9] proposed a start time deviation as the solution robustness criterion and used it as the optimization objective for robust project scheduling. Zhong [
10] carried out a construction schedule robustness measure of underground power stations by adopting a solution robustness criterion named ‘start time deviation’. Zhang [
11] proposed activity delay as the solution robustness criterion to deal with the materials ordering problem. Chang [
12] proposed the worst-case expected total flow time as the solution robustness criterion for the robust scheduling of a flowshop. Hu [
13] took the travel time as the solution robustness criterion in scheduling vehicle routing.
Studies on the quality robustness criterion are fewer than that on solution robustness criterion and concentrate on the robustness measure of air transport schedules. Liang [
14] presented the net present value as a quality criterion to investigate the robust resource-constrained project problem with stochastic activity durations. Novianingsih [
15] built a simulation model of the flight and the quotient of the number of iterations to find an optimal crew pairing, and the total number of iterations was taken as its criterion measuring quality robustness. Hussain [
16] generated various flight candidate schedules randomly, and their respective robustness was measured by taking quality robustness criteria into consideration, which included timeliness of delivery, amount of cargo moved, and cost. Detti [
17] took the quality robustness criterion of total completion times as the optimization objective and adopted a heuristic algorithm to solve the robust scheduling problem of a job shop. Additionally, some researchers proposed criteria concerning both solution robustness and quality robustness, while these criteria were not adopted comprehensively in measuring robustness. Lu [
18] proposed two criteria to measure the robustness of job shop production management—the expectation value of makespan as the quality robustness criterion and the expectation value of the total start time delay of all procedures as the solution robustness criterion. Shen [
19] defined the efficiency emphasized measurement as a solution robustness criterion and a cumulative distribution function inspired measurement as a quality robustness criterion, which were respectively applied in the optimization model. Lamas [
20] established a proactive management schedule in resource-restricted projects, where the start time deviation was adopted to measure the solution robustness, and the completion probability was considered as the criterion for quality robustness. Zhao [
21] and Cui [
22] proposed a linearly composite criterion of both solution robustness and quality robustness in job shop management concerning random malfunctions, but its weight was assigned as a subject to decision-makers’ preferences.
In summary, current research concerning robustness measures only focuses on solution robustness or quality robustness, which cannot measure the schedule robustness from both the project execution and the project result. Some criteria include both solution and quality robustness criteria, but their weights neglect the bounded rationality of subjective weights and the inherent importance and nonlinear intercriteria correlations of objective weights.
In this paper, the composite robustness criterion containing both solution robustness and quality robustness is proposed from both the project execution and the project result, considering the bounded rationality of subjective weights and the inherent importance and nonlinear intercriteria correlations of objective weights.
The remainder of this paper is organized as follows:
Section 1 provides an overview of the related work on weighting methods.
Section 2 describes the framework for this paper.
Section 3 describes the methodology of the construction schedule robustness measure. In
Section 4, a case study of an underground power station in China is presented to verify the consistency, representativeness, and the advantage of the proposed criterion and methods. The conclusion of this paper is highlighted at the end of the paper.
2. Related Work
Subjective weights are in good accordance with the basic cognition of experts to robustness criteria; however, the information contained in the robustness criteria cannot be considered by the experts when they are giving the subjective weights. Objective weights have strong data theoretical basis; however, they cannot take the experts’ experience and judgment into account. Therefore, the combination of subjective and objective weights can take advantage of and overcome the disadvantage of both weighting methods. In this section, the related work on subjective weighting methods, objective weighting methods, and methods of combining subjective and objective weights is reviewed and discussed.
Subjective weighting methods mainly include the analytic hierarchy process (AHP) [
23], the Delphi method [
24], and the expert-evaluation-based method [
25]. These methods are effective in reflecting the importance of different criteria because experts are well-experienced in this field, and they can make reasonable judgments according to actual situations. However, individual preferences are inevitably involved in subjective weighting. Thus, in the above methods, the experts are regarded as rational humans, and the evaluation curve is regarded as an expected utility curve. However, a large number of psychological experiments have demonstrated that the actual behaviors of humans do not correspond to the expected utility curve. Due to the bounded rationality of humans, there tends to be risk aversion when the profit probability is relatively high and risk-seeking when there is a rather high loss probability. At the same time, there tends to be risk-seeking when the profit probability is relatively low, and risk aversion when there is a relatively small chance to suffer losses [
26]. It can be concluded that human judgments have individual preferences. Prospect theory has established a successful model to simulate the psychological and behavioral characteristics of human beings, which has been widely applied in the field of engineering [
27,
28,
29], economics [
30,
31,
32], computer science [
33,
34,
35], and so on. This provides a convincing explanation of the fact that the results of subjective empowerment do not conform to reality. So, prospect theory is useful to modify the preliminary results obtained by subjective weighting so as to make the evaluation results become “rational” ones out of the “boundedly rational” ones.
By using objective weighting methods, including entropy weight method [
36], principal component analysis method [
37] as well as standard deviation method [
38], the weights of different criteria are calculated according to the amount of information they contain. The three robustness criteria proposed, namely, start time deviation, structural deviation, and completion probability, which can comprehensively evaluate the schedule robustness, are interrelated to some extent. Therefore, it is unreasonable to assign their respective weights without considering their intercriteria correlation. Criteria importance through intercriteria correlation (CRITIC) is an objective weighting method that measures a criterion by considering both the importance itself and the conflict caused by intercriteria correlations [
39]. In summary, in the traditional CRITIC method, the intercriteria correlation is determined by the Pearson correlation coefficient function, which reflects the linear correlation. The Copula function is very useful for the analysis of the correlation between variables when it cannot determine whether the correlation between variables is linear or nonlinear. Thus, it can effectively deal with the difficult intercriteria correlation among the three robustness criteria. Therefore, the CRITIC method is improved by introducing the Copula function to replace the Pearson correlation coefficient, and the objective weights that could incorporate both inherent importance and nonlinear intercriteria correlations can be obtained using the Copula-CRITIC method.
Methods of combining subjective and objective weights, including additive integration [
22], multiplicative integration [
40], and the eclectic method [
41], can balance the subjectivity of preferences with the objectivity of information. However, using these methods will cause an incomplete expression of subjective and objective weights. To solve this problem, Dempster and Shafer [
42] proposed the evidence reasoning theory and its corresponding method [
43] to combine multisource information without information loss. Liu [
44] calculated the combined weight of classifiers by adopting the evidence reasoning method, thus promoting the accuracy of data classification. Bao [
28] obtained the expectation values of different alternatives by using the evidence reasoning method, preventing the loss of decision-making information. Zhou [
45] carried out a multi-attribute decision-making process in which the subjective weights derived from experts and the objective weights obtained from the attributes were combined via the evidence reasoning method. In fact, the evidence reasoning method generates false information by replacing true information with the complement of true information sources, which is just one of the sources of false information. However, combining subjective and objective information based on this assumption will reduce the combination credibility of the results. Therefore, an information-entropy-based effective probability validation mechanism is introduced into the evidence reasoning method to improve the combination accuracy and the effectiveness of the results in this paper. By calculating the change in entropy when synthesizing the reliability function, the validity of the subjective and objective information is judged, and the invalid information is eliminated during their combination so that the validity of the schedule robustness measure results is guaranteed.
Aiming at the above problems, a construction schedule robustness measure based on improved prospect theory and the Copula-CRITIC method is proposed. Firstly, a composite measure criterion, including start time deviation rs, structural deviation rp, and completion probability rc, is proposed, so that both solution and quality robustness can be considered for measuring. Secondly, the subjective weights are assigned using improved prospect theory, which is improved by introducing an interval distance formula into it to overcome the shortcoming that prospect theory cannot deal with complete expert evaluation information, and the weights given by experts are transformed into a prospect value function so that bounded rationality can be considered. Thirdly, the Copula-CRITIC method, in which a Copula function is adopted to replace the original correlation method of the Pearson correlation coefficient for dealing with the difficult intercriteria correlation among the three robustness criteria, is proposed to determine the objective weights which could incorporate both inherent importance and intercriteria correlations. Finally, the subjective and objective weights are combined using the information-entropy-based evidence reasoning method in order to identify the validity of the weights during combination.