Project crashing using a fuzzy multi-objective model considering time,cost,quality and risk under fast tracking technique: A case study

Abstract

Project managers normally investigate how to reduce the total completion time of projects undertaken subject to the pre-determined objectives. The purpose of this paper is compression of total projects’ duration subject to influencing factors such as cost, time, quality and risk. In the present study, by considering factors affecting project success such as cost, time, quality and risk, project crashing and fast tracking are both employed then a fuzzy multi-objective non-linear model is proposed. Each project is associated with uncertainty and the lack of consideration of these uncertainties might lead to project failure. In addition, a fuzzy approach has been employed for incorporating the uncertainties associated with activities. In the proposed model, crashing and fast tracking of projects were considered simultaneously for the first time. According to a real case study considered in this paper, after using the proposed model, the cost of compression is also reduced. In addition, we determined which activities have to be crashed or fast tracked in order to attain the objective. The proposed methodology can be practically applied through mega projects such as construction and “Engineering, Procurement, and Construction” (EPC) projects where the deadline closes to being achieved. As another contribution, a unique feature of the proposed model is its capability for taking both crashing and fast-tracking simultaneously into consideration.

Keywords

Fuzzy multi-objective model project crashing fast tracking time cost

1 Introduction

Nowadays, the knowledge of projects and how they are managed and planned has become highly crucial. As the projects are becoming larger, more complex and sensitive, experts are investigating in order to determine better and more effective tools and techniques for managing projects that have been undertaken. Project management is a structured process in which different tools and techniques are employed for effective management of the limited resources (time, cost, and human resource), leadership and management of the objectives. Project is a temporary effort which is being done for creating a product or a service with a unique result [1, 2]. The temporary nature of the project indicates that it has a specific beginning and end. The end of a project happens when the objectives are met or when, due to the objectives of a project, cannot be fulfilled enough, so the project will be terminated, or when there is no need to continue the project. There are six objectives for a project (time, cost, scope, quality, resources and risk) which can be of different importance based on project types [1]. The specific characteristics and conditions of each project can impact on the constraints so the project team needs to specifically notice this fact. In the relationship between these factors, if a factor changes, it will impact on at least one of the other factors. For example, if the duration of the project crashed, it is often necessary to increase the budget of the project as more resources are needed to complete the project. Therefore, factors affecting project success such as cost, time, quality and risk should be taken into account. Once the project has been initiated, project planning system should be then established. In most of the cases, after the schedule has been prepared by the project team, the finish date of the project takes longer than the client’s requirements and therefore, the project completion time has to be reduced. Hence, time-cost trade off problems is one of the important topic in project management. The main question of the research is “how can reduce the cost of compression of projects?” The Project Management Body of Knowledge (PMBOK5 Edition) standard has presented crashing and fast tracking techniques for resolving this issue. These techniques, due to reducing the project duration without changing the project scope, are being used for the fulfillment of the project deadline. Compression techniques include the following techniques, but are not limited to:

Crashing: a technique for reducing duration of the project with minimum possible cost and increasing some additional resources. Crashing is only applicable to activities where they are placed on the Critical Path (CP) where assigning more resources decreases the duration of the activity. Crashing may increases risks and costs [3].

Fast tracking: fast tracking is a compression technique in which activities or some phases of it are being done consecutively in parallel. Fast tracking is effective when the activities can be overlapped for reducing the duration of the project. Fast tracking increases the risk of project and multitasking of the resources [3].

The main contribution of the present research is taken into consideration crashing and fast tracing methods simultaneously. This can lead to the reduction of project compression costs. The mathematical model is used in AON network architecture. This model supports different relationships among activities and yields better performance of the proposed approach. In the present study, a comprehensive model for the use of project managers considering cost, time, quality and risk is presented.

2 Literature review

2.1 FMOLP

Decision-making in complex situations with considering multiple objectives and criteria is a major daily task of human beings. Several methods have therefore been presented in the existing literature for resolving such issues, including Multi-Attribute Utility Theory (MAUT) [4, 5], Bayesian theory methods [6] and goal programming methods [7]. Most problems require immediate decision-making in an environment characterized by the lack of certainty. Therefore, a large number of coefficients, objectives and constraints may not be estimated accurately and unequivocally. In practice, lack of access to sufficient samples or a fundamental statistical method would lead to inefficient statistical estimations. Hence, making use of a definite decision-making model in such a context would result in obtaining unreal responses [8]. In this context, the fuzzy set theory provides a theoretical and conceptual framework for managing such uncertainties.

The fuzzy sets theory was first applied by Zadeh [9] to the decision-making process. The concept of fuzzy mathematical programming was first introduced by Tanaka et al. [10] and Zimmerman [11] was the first to formulize fuzzy linear programming. Subsequently, many techniques were developed for solving fuzzy linear programming problems. These techniques fell under two broad categories, namely fuzzy linear programming and possibilistic linear programming. Dyson [12] believed that when the membership functions of fuzzy linear programming problems, such as the utility theory, are determined based on the concept of preference, this membership function does not differ from the utility function. However, this argument does not hold true for the distributions of probability in probabilistic linear programming. In each category of these models, some parts of the problem have been considered fuzzy or inaccurate and methods have been presented to solve such problems [13]. Due to simplicity, many of the presented models have considered the final solution of the problem in the form of crisp [14]. In other words, decisions have been made with certainty in a fuzzy environment. Therefore, the fuzzy aspect of the problem would be over-concentrated in the decision-making process. Thereafter, several models were proposed for resolving fuzzy or probabilistic linear programming problems with fuzzy decision-making variables, each of which have their advantages and disadvantages. Also, some have proposed fully fuzzy models [15 –19] in which all the problem parameters and simultaneous decision-making variables are considered to be fuzzy. The current study has adopted Alavidoost et al.’s approach [20] in order to solve the proposed model.

2.2 Project crashing

Project compression has been always a problem for project managers. Many studies have been conducted in this respect, and heuristic and meta-heuristic algorithms and mathematical models have been used. For instance, Babu and Suresh [21] presented a linear programming model in which cost, time and quality had been taken into consideration. They believed that project quality changes when project crashing is being done. The model presented by them, by specifying bounds for cost and time, determined the optimal value for the third index, i.e. quality. This model was very simple and therefore received some attention thereafter. Khang and Myint [22] implemented the model of Babu and Suresh [21] as a case study in a cement factory in Thailand and this model had some defects in terms of quality. Leu et al. [23] explored cost-time trade off, in order to use activity duration uncertainties, they used fuzzy theory and genetic algorithm was used for finding the optimal solution at different risk levels. Abbasi and Mukattash [24] presented a non-linear mathematical model for crashing project duration in Program Evaluation and Review Technique (PERT) network. The main aim of this study was minimizing the pessimistic duration in PERT network by more investment on critical path activities. Tareghian and Taheri [25], in order to make a balance among cost, time and quality, designed three interrelated integer programming models in a way that each model would achieve optimal solution by determining bounds for the other two models. In this study, it was assumed that project quality and time are discrete and, indirectly, from a non-renewable source. Yang [26] used the meta-heuristic algorithm Particle Swarm Optimization (PSO) for project cost-time tradeoff. The output of this model would present project cost curves which assists project managers in project duration and budget analysis. Ke et al. [27] presented two stochastic models for time-cost tradeoff problems in which chance-constrained programming and chance-dependent programing had been used for decision making. Also, by combining stochastic simulation algorithm and genetic algorithm, they designed an algorithm that would search for semi-optimal solutions under different decision making criteria. In another study, Ke et al. [28], in order to meet different management needs, presented triple stochastic models for time-cost tradeoff problems. The proposed models used an intelligent algorithm that was a combination of stochastic simulation and genetic algorithms. Stochastic simulation method was used for the assessment of random functions and the genetic algorithm was designed for finding optimal solution under different decision making criteria. Hazir et al. [29] explored robust optimization of cost-time tradeoff problems that are multi-mode scheduling problems with practical applications. For this purpose, they presented a two-stage robust scheduling algorithm. Sonmez and Halis [30], using a combinational strategy that employed genetic and simulated annealing algorithms, explored the optimization of cost-time tradeoff problems. In this study, ten costs-time tradeoff problems were tested using hybrid algorithm (HA) and acceptable solutions were obtained. Kim et al. [31] explored time-cost-quality tradeoff problems in projects using a mixed integer linear programming model. In their model the potential costs of quality had been taken into consideration. In this model the coefficient α was determined for considering risk of non-conformance by the project manager. This model was extremely simple and explored Activity On Arrow (AOA) networks. Tavana et al. [32] presented a multi-objective multi-mode model for solving cost-time-quality tradeoff that had the following characteristics: Simultaneous optimization of conflicting objectives, Activities can be split, Determining the general relationship among the project activities. Ke and Ma [33] presented three models for cost-time tradeoff problems which were applicable in complex environment with more than one type of uncertainty. The project environment of this model was described using random fuzzy theory. Multiple-Criteria Decision-Making (MCDM) approach can help decision-makers in selecting the most appropriate solution among the potential solutions. Monghasemi et al. [34] used Evidential Reasoning (ER) for the first time in the framework of project planning for the identification of the best Pareto solution for solving cost-time-quality tradeoff problems. In order to identify all optimal solutions, they tested a Multi-Objective Genetic Algorithm (MOGA) using Non-Dominated Sorting Genetic Algorithm II (NSGA-II) in a sample construction project for a highway. In another study, Aminbakhsh and Sonmez [35] used the Deterministic Particle Swarm Optimization (DPSO) algorithm for cost-time tradeoff. In this study, a project with 360 activities was discovered and acceptable results were then obtained. Salari et al. [36] proposed a new mechanism for solving cost-time tradeoff problems that was used for both the planning and re-planning of the project. The proposed mechanism includes monitoring the project performance during execution using Earned Value Management (EVM) and forecast the project’s future performance through statistical modelling. Saif et al. [37] presented a new meta-heuristic algorithm called Problem Data Based Optimization (PSBO) that had a high speed and provided an optimal or a near optimal solution. This algorithm is highly applicable for time-cost-quality tradeoff in software projects. Mahmoudi and Feylizadeh [38] explored tradeoff problems considering cost, time, quality and risk. In the proposed linear model, quality was taken into consideration based on conformance and non-conformance costs and, if quality was lower than the level acceptable by the project manager, preventive action would be taken in order to prevent the reduction of project quality. Also, project risk cost reduction or increase during tradeoff had been considered [39].

In the present study, using a fuzzy approach, a comprehensive model for project crashing is presented in which cost, time, quality and risk are simultaneously taken into consideration and this highly increases the model accuracy. Also, crashing and fast tracking can be done simultaneously in this model. To the best of our knowledge it is the first time that these two techniques are employed in a mathematical model. In order to present the real performance of the model, a case study is presented.

2.3 FMOLP in project crashing

Compression problems of projects are among the issues which require the simultaneous consideration of several objectives. At the beginning phases of almost all projects, members have only a low level of familiarity with the project and are therefore faced with many uncertainties. Hence, the fuzzy approach may prove effective in accounting for the uncertainties. Arikan and Gungor [40] used Fuzzy Goal Programming (FGP) approach for project crashing, aiming to minimize the time and costs of crashing. They also compared their proposed method with Fuzzy Linear Programming (FLP). Zhang and Xing [41] presented a model which simultaneously aimed at minimizing the time and costs as well as maximizing the quality. This model was called the time– cost– quality tradeoff (TCQT). In order to solve the presented model, they adopted the ‘Fuzzy-Multi-Objective Particle Swarm Optimization’ method. Bagherpour et al. [42] dealt with the crashing of material requirement planning problem. Their proposed model also aimed to minimize time-cost and maximize quality. In order to resolve the proposed model, Feylizadeh et al. [43] used Fuzzy Multi-Objective Programming approach, which establishes a relative balance among time-cost-quality objectives. Using the goal programming method, Jebaseeli and Dhayabaran [44] proposed a new algorithm for solving Fully Fuzzy Time-Cost Tradeoff Problems. They maintained that using this technique, the project manager would be able to easily determine the minimum costs and completion time of the project. Stakeholders are always concerned about the completion time of the project. Hossain et al. [45] proposed a multi-goal fuzzy model which simultaneously minimized the total costs of the project, completion time and failure costs (including direct and indirect costs). They believed that definite models in project management are never efficient, since the related parameters are fuzzy in nature.

Using a fuzzy multi-objective non-linear model and simultaneously considering costs, time, quality and risks parameters, the current study deals with the crashing and fast-tracking of projects. A unique feature of the proposed model is its capability for undertaking crashing and fast-tracking simultaneously.

3 Fuzzy approach

In this section, fuzzy approach and its applications are discussed. The fuzzy theory was introduced by Zadeh in 1965. Fuzzy theory is highly applicable in problems with high uncertainty [46 –49]. Project compression problems are among the problems whose parameters and variables have high uncertainties [50]. When system uncertainty is at a level that the system behavior and parameters cannot be accurately assessed, the concept of fuzzy can be used for modelling and analysis [51, 52]. Fuzzy systems are systems whose input information is not accurate. In other words, the input data of a fuzzy system is expressed in the form of a set of fuzzy sets or numbers. Linguistic variables can also be used in fuzzy theory [53, 54]. Linguistic variables refer to variables for which the acceptable values are linguistic words and sentences rather than numbers. Numerical variables must be used in mathematical calculations that need accurate values [55]. The linguistic variables used in the present study are shown in Table 1 [56].

Table 1
Fuzzy linguistic variables

Weights of criteria Fuzzy numbers Ratings of alternatives

Very Low (VL) (0.00, 0.00, 0.25) Worst (W)

Low (L) (0.00, 0.25, 0.50) Poor (P)

Medium (M) (0.25, 0.50, 0.75) Fair (F)

High (H) (0.50, 0.75, 1.00) Good (G)

Very High (VH) (0.75, 1.00, 1.00) Best (B)

Weights of criteria	Fuzzy numbers	Ratings of alternatives
Very Low (VL)	(0.00, 0.00, 0.25)	Worst (W)
Low (L)	(0.00, 0.25, 0.50)	Poor (P)
Medium (M)	(0.25, 0.50, 0.75)	Fair (F)
High (H)	(0.50, 0.75, 1.00)	Good (G)
Very High (VH)	(0.75, 1.00, 1.00)	Best (B)

4 Problem statement

Reducing the completion time of projects has always been a major concern of key stakeholders. Determining the activities which have to be crashed and the amount of time reduction for each activity fall under optimization issues. Reducing the completion time of projects has to be done in a way so as to impose the minimum amount of costs, risks and quality reduction on the project. Compression of the projects may be done using crashing and fast-tracking techniques. Such Compression has an impact on several factors including cost, time, quality, and risk. Any given project has to establish a balance among these factors. This balance also has to be created during project compression. For instance, the quality of activities should not be lower than a certain level and the costs of compression must not exceed a certain amount. To the best of our knowledge, up until now the proposed models have only focused on the crashing of projects. However, crashing alone may not result in the desired time reduction. Even if crashing results in reducing the completion time to a desirable extent, the costs of compression may undergo a significant increase. In addition, the executive capability of the plan may considerably reduce as a result of crashing.

Accordingly, the authors aim to present a comprehensive model which undertakes crashing and fast-tacking simultaneously and considers cost, time, quality and risk factors of the project.

Figure 1 displays the procedure of the current study.

Fig.1

The proposed approach.

First, a multi-objective fuzzy model is presented. Since the proposed model is non-linear, the next stage would be to linearize the proposed model. In the next step, the weight of each objective function would be calculated using the Z-number extension of fuzzy Analytical Hierarchy Process (AHP) approach. Then, the appropriate membership function is proposed for each objective function. Since the proposed model is multi-objective, it is converted into a single-objective model and then resolved. Ultimately, in order to evaluate the performance of the proposed model, a case study is done.

5 The proposed model

In most cases after the preparation of the initial project schedule, due to the existing requirements, the project duration needs to be reduced. The proposed model minimizes the project duration, cost and maximizes the quality. A comprehensive model has been presented considering cost, time, quality and risk simultaneously and AON network has been applied for the project planning. The model parameters and variables are defined in Table 2.

Table 2
The model parameters and variables

Sets

E The set of all activities

S The set of activities can be crashed

H The set of activities can be fast tracked

U The set of activities that need preventive actions

Indices

b represents the activity preceding activity

j represents activity

a represents the activity succeeding the activity j

n The number of the project activities

Variables and parameters

y_j The duration that needs to be reduced from activity j to achieve project dead line

c_j The cost per each unit reduction of the duration of activity j

yy_bj The duration added or reduced by the lag/lead the relationship bj

v_bj The possible duration that can be reduced from the relationship bj

P_bj The risk probability resulted from the project fast tracking in the relationship bj

P_j Risk probability resulted from crashing in the activity j

I_bj The effect of the risk resulted from fast tracking in the relationship bj

I_j The effect or risk resulted from crashing in activity j

q_j The effect of the quality of activity j on the overall project quality

d_j The primary duration needed to complete the activity j

w_j The binary variable that takes 1 if the activity quality becomes lower than the intended level, and takes 0 otherwise

ww_bj The binary variable that takes 1 if the activity of the relationship bj becomes lower than the intended level, and takes 0 otherwise

C_P-c The cost of preventive action for compensating for crashing quality reduction

C_P-f The cost of preventive action for compensating for fast tracking quality reduction

Xlf_j The latest time for the start of the activity j

Xes_j The earliest time for the start of the activity j

lag/lead_bj lag/lead between the activities b and j for Finish-to-Start (FS) relationship

q_f-threshold Quality threshold for the fast tracking relationship between activities

q_c-threshold Quality threshold for activity crashing

u_j The possible duration for the reduction of the duration of activity j

M Represents a big number

c_max Dead line of the project

Sets
E	The set of all activities
S	The set of activities can be crashed
H	The set of activities can be fast tracked
U	The set of activities that need preventive actions
Indices
b	represents the activity preceding activity
j	represents activity
a	represents the activity succeeding the activity j
n	The number of the project activities
Variables and parameters
y_j	The duration that needs to be reduced from activity j to achieve project dead line
c_j	The cost per each unit reduction of the duration of activity j
yy_bj	The duration added or reduced by the lag/lead the relationship bj
v_bj	The possible duration that can be reduced from the relationship bj
P_bj	The risk probability resulted from the project fast tracking in the relationship bj
P_j	Risk probability resulted from crashing in the activity j
I_bj	The effect of the risk resulted from fast tracking in the relationship bj
I_j	The effect or risk resulted from crashing in activity j
q_j	The effect of the quality of activity j on the overall project quality
d_j	The primary duration needed to complete the activity j
w_j	The binary variable that takes 1 if the activity quality becomes lower than the intended level, and takes 0 otherwise
ww_bj	The binary variable that takes 1 if the activity of the relationship bj becomes lower than the intended level, and takes 0 otherwise
C_P-c	The cost of preventive action for compensating for crashing quality reduction
C_P-f	The cost of preventive action for compensating for fast tracking quality reduction
Xlf_j	The latest time for the start of the activity j
Xes_j	The earliest time for the start of the activity j
lag/lead_bj	lag/lead between the activities b and j for Finish-to-Start (FS) relationship
q_f-threshold	Quality threshold for the fast tracking relationship between activities
q_c-threshold	Quality threshold for activity crashing
u_j	The possible duration for the reduction of the duration of activity j
M	Represents a big number
c_max	Dead line of the project

The model proposed for project duration reduction is presented in Equations (1) to (19):

Objective Functions: ${MinZ}_{1} = \sum_{j = 1}^{n} y_{j} \times c_{j}, j \in S$ (1) ${MinZ}_{2} = \sum_{j = 1}^{n} \frac{{yy}_{bj}}{v_{bj}} \times (P_{bj} \times I_{bj}), j \in H$ (2) ${MinZ}_{3} = {Xes}_{1} - {Xlf}_{n}$ (3) ${MaxZ}_{4} = \sum_{j} q_{cj} \times (\frac{d_{j} - y_{j}}{d_{j}}), j \in S$ (4) ${MaxZ}_{5} = \sum_{j} q_{fj} \times (\frac{v_{bj} - {yy}_{bj}}{v_{bj}}), j \in H$ (5) $\begin{matrix} {MinZ}_{6} = \sum_{j} C_{P - c} \times w_{j} \times y_{j} + \sum_{j} C_{P - f} \\ \times {ww}_{bj} \times {yy}_{bj}, j \in U \end{matrix}$ (6) ${MinZ}_{7} = \sum_{j = 1}^{n} \frac{y_{j}}{d_{j}} \times (P_{j} \times I_{j}), j \in E$ (7)

Subject to ${Xlf}_{n} \leq c_{\max}$ (8) ${Xlf}_{j} - {Xes}_{j} \geq d_{j} - y_{j}, j \in E$ (9) $\begin{matrix} {Xes}_{j} - {Xes}_{b} \geq (d_{b} - y_{b}) \\ + (lag / {lead}_{bj} - {yy}_{bj}), j \in E \end{matrix}$ (10) $\begin{matrix} {Xlf}_{a} - {Xlf}_{j} \geq (d_{a} - y_{a}) \\ + (lag / {lead}_{ja} - {yy}_{ja}), j \in E \end{matrix}$ (11) $y_{j} \leq u_{j}, j \in S$ (12) ${yy}_{bj} \leq v_{bj}, j \in H$ (13) $\begin{matrix} q_{f - threshold} \leq \frac{v_{bj} - {yy}_{bj}}{v_{bj}} + {ww}_{bj} \times M, \\ j \in U and j \in H \end{matrix}$ (14) $\begin{matrix} q_{c - threshold} \leq \frac{d_{j} - y_{j}}{d_{j}} + w_{j} \times M, \\ j \in U and j \in S \end{matrix}$ (15) ${yy}_{bj} \geq 0, j \in H$ (16) $y_{j} \geq 0, j \in S$ (17) $w_{j} = {0, 1}, j \in U$ (18) ${ww}_{j} = {0, 1}, j \in U$ (19)

Based on crashing technique, reducing the duration of activities increases resources and this increase in the resources causes an increase in the cost of activities. Objective function (1) considers the cost of reducing each unit of activities duration and minimizes the crashing costs. “Crashing typically increases project costs by some factor while fast tracking increases the risk of rework as activities are started before their initial predecessors are completed.” [57]. Therefore, fast tracking the projects increases their risks. In objective function (2), the project fast tracking cost is minimized based on the calculation of Expected Monetary Value (EMV) of risk. EMV is calculated using multiplication of probability by impact of risks in which the objective function (2) has been proposed based on EMV formula [57]. Reducing the duration of a project is one of the major objectives of any project manager. Objective function (3) indicates our willingness to reduce the project duration. In fact, this objective is considered for minimization of the overall project duration. Reducing the duration of activity directly impacts on the quality of that activity. Based on some previous research that was investigated in Section 2 a relationship exists between the quality of the activity and the duration of the activity which reduces the duration of the activities, and leads to a decrease in the quality of the activity [21]. Objective function (4) maximizes the project quality after crashing and Objective function (5) maximizes the project quality after project fast tracking. The cost of quality is divided into the following categories:

Cost of conformance

Cost of nonconformance

The costs of conformance have a preventive approach and prevent the reduction of product quality in future; such as cost of training the project team, inspection deliverables, etc. The costs of non-conformance are costs for modifying the deliverables. The proposed model has a preventive approach and, with considering conformance costs, will prevent of non-conformance costs in future. The nonlinear objective function (6) minimizes the costs of quality reduction of preventive action for compensation of crashing and fast tracking. Objective function (7) is for minimizing the project crashing risks based on EMV formula. The project team identifies these risks. Constraint (8) indicates the maximum project duration. In fact, if the project duration becomes greater than c_max, the primary requirement of the project team about time constraint is not fulfilled. In order to explore the performance of constraints (9) to (10), first consider the Fig. 2.

Fig.2

Modelling of the relationship among activities.

As seen in the Fig. (2), constraints (9) and (10) are to show the relationship among activities in an AON network. Other relationships among the activities in AON network can be transformed into FS relationship. Constraint (12) and (13) express the maximum possible amount of activity crashing and fast tracking for the relationships among the activities, respectively. Constraints (14) and (15) are for considering quality reduction in the model and if the activity is crashed or the relationships among the activities are fast tracked, quality reduction is considered in objective functions (4) and (5). The constraints (16) and (17) show the nonnegative variables and constraints (18) and (19) show the binary variables of the model.

As the proposed model is non-linear, the model linearization is done in the next section.

6 Linearization

Considering the proposed model, which is expressed in Equations (1) to (19), the objective function (6) causes the model to be non-linear. As the objective function is minimized, the variables w_j and ww_bj tend towards zero. The constraints (14) and (15) are related to project quality metrics. If the project quality becomes lower than the intended level, the variables w_j and ww_bj will be 1. These values impose the cost of preventive action for preventive quality reduction on the model. If the constraints (20) and (21) are added to the constraints of our proposed model, the binary variables will be considered. $y_{j} \leq M . w_{j}$ (20) ${yy}_{j} \leq M . {ww}_{bj}$ (21)

w_j and ww_bj can be eliminated from the objective function (6). This is because, whenever w_j becomes zero, y_j becomes zero too (considering the constraint (20)). As a result, there is no need for multiplication of w_j and y_j in the objective function (6). This is also true for constraint (21) and the variables ww_bj and yy_j. Therefore, constraints (20) and (21) are added to the proposed model, and the Equation (6) in the model is changed to Equation (22). ${MinZ}_{6} = \sum_{j} C_{P - c} \times y_{j} + \sum_{j} C_{P - f} \times {yy}_{j}, j \in U$ (22)

Therefore, the model becomes linear and can be solved using fuzzy linear methods.

7 Appropriate approach for solving the proposed model

After running the modeling procedure, an appropriate method for solving the model needs to be adopted. As projects face many uncertainties, we applied uncertainties approaches. Hence, fuzzy approach has been employed in the present study to consider non-deterministic parameters and variables. As the input parameters are fuzzy numbers, fuzzy approaches need to be used for solving this model. And in this study, the method by Alavidoost et al. [20] that is explained in the Section 7.5 has been used. Alavidoost et al.’s method [20] is one of the newest approaches which handles practical MOLP models and considers uncertainty and vagueness associated with MOLP models using linguistic variables. This method has an appropriate strategy which was proposed as a solution method to find an efficient compromise solution.

7.1 Defuzzification of fuzzy objective functions

Despite dealing with much fuzzy data in the real world, sometimes we need crisp data. For example, when we want to announce the duration of an activity to an executive team, we have to announce a crisp number. As the nature of project duration is fuzzy, we have to announce it in the form of a crisp number. In such cases, defuzzification methods need to be used for transforming fuzzy numbers into crisp numbers. As there are different deffuzzification methods, choosing an appropriate method is highly important. Therefore, at this stage, in order to solve fuzzy multi-objective linear models, all the parameters existing in the objective functions that have fuzzy numbers must be turned into crisp numbers using an appropriate method. Equation (23) has been used for defuzzification of triangular fuzzy numbers [58]. For $\tilde{A} = (a, b, c)$ :

D (\tilde{A}) = \frac{a + 2 b + c}{4}

(23)

7.2 Defuzzification of fuzzy constraints

This stage is similar to the previous stage and the only difference is that defuzzification of constraints are done rather than defuzzification of objective functions. In such problems some constraints may be crisp. Then, only the constraints that have fuzzy parameters are turned into crisp numbers using appropriate defuzzification methods.

7.3 Weight calculation for each objective

In multi-objective models, in most of the cases importance of the objective functions are not the same. Therefore, an appropriate weight for each objective function must be considered. Z-AHP method has been employed in this study to determine the weights of objective functions and this method is discussed in Sections 7.3.1 and 7.3.2.

7.3.1 Fuzzy AHP method

Conventional AHP method is not applicable for such problem much due to receiving crisp input parameters and does not provide suitable results. Fuzzy AHP method has been used for solving this problem in the present study. In fuzzy AHP method the paired comparison matrix is defined as Equation (24) [59]:

$\tilde{A} = [\begin{array}{l} 1 & {\tilde{a}}_{12} & ... & {\tilde{a}}_{1 n} \\ {\tilde{a}}_{21} & 1 & \dots & {\tilde{a}}_{2 n} \\ \dots & \dots & \dots & \dots \\ {\tilde{a}}_{n 1} & {\tilde{a}}_{n 2} & \dots & 1 \end{array}] = [\begin{array}{l} 1 & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ \frac{1}{{\tilde{a}}_{12}} & 1 & \dots & {\tilde{a}}_{2 n} \\ \dots & \dots & \dots & \dots \\ \frac{1}{{\tilde{a}}_{1 n}} & {\tilde{a}}_{2 n} & \dots & 1 \end{array}]$ (24)

7.3.2 Z-AHP method

Z-AHP method for obtaining the weight in each objective function is introduced in this section. Obtaining the appropriate weight for each objective in multi-objective models is considerably important. The concept of Z number was presented in 2011 for the first time [60]. This method can be used as follows [61]:

First, the reliability of each expert is turned into crisp number: $α = \frac{\int x μ_{\tilde{B}} (x) dx}{\int μ_{\tilde{B}} (x) dx}$ (25)

The reliability obtained in stage 1 is included in the view of each expert using Equation (26): ${\tilde{Z}}^{α} = {〈 x, μ_{{\tilde{A}}^{α}} (x) 〉 | μ_{{\tilde{A}}^{α}} (x) = α μ_{\tilde{A}} (x)}$ (26)

Using the Equation (27), Z-number weight is turned into normal fuzzy number: $\begin{matrix} {\tilde{Z}}^{'} = \sqrt{α} \times {\tilde{A}}^{α} = (\sqrt{α} \times a_{1}, \sqrt{α} \times a_{2}, \\ \sqrt{α} \times a_{3}, \sqrt{α} \times a_{4}) \end{matrix}$ (27)

Based on the opinion of experts and using Z-AHP, the favorable weight for each objective function is finally obtained.

7.4 Membership function for objective functions

Membership functions of objective functions can be discrete or continuous depending on the discreetness of continuousness of the variables. The definition of appropriate membership function is significant because if the membership function defined for a fuzzy set is not appropriate, all the subsequent analyses will be deviated. In the present study, appropriate membership functions are extracted for the objective functions by the project team and the overall structure needs to be as Equations (28) and (29) [20]:

If the objective function is of maximization type, Equation (28) will be used for the calculation of the membership function: $μ (x) = {\begin{matrix} 1 & if x \geq U_{x} \\ \frac{x - U_{x} \times (1 - D_{x})}{U_{x} \times D_{x}} & if U_{x} \times (1 - D_{x}) \leq x \leq U_{x} \\ 0 & if x \leq U_{x} \times (1 - D_{x}) \end{matrix}$ (28)

If the objective function is of minimization type, Equation (29) will be used for the calculation of the membership function: $μ (x) = {\begin{matrix} 1 & if x \leq U_{x} \\ \frac{- x + U_{x} \times (1 + D_{x})}{U_{x} \times D_{x}} & if U_{x} \times (1 + D_{x}) \geq x \geq U_{x} \\ 0 & if x > U_{x} \times (1 + D_{x}) \end{matrix}$ (29)

In Equations (37) and (38), the parameters are defined as follows:

x: Objective function

U_x: The favorable value for the objective function

D_x: The permissible deviation of the objective function from the favorable value that is determined by the linguistic variable.

7.5 Combining the objective functions

Multi-objective decision making is highly important and applicable. In the present study, using the approach of Alavidoost et al. [20] the fuzzy multi-objective model is transformed into a single objective model and solved. In a single objective model the importance of each objective function is specified using weight factor that is calculated using Z-AHP. The model presented by Alavidoost et al. [20] is as Equation (30): $\begin{matrix} \max λ (x) = λ_{0} + δ \sum_{k = 1}^{K} ϑ_{k} λ_{k} \\ s . t . \\ ϑ_{k} λ_{0} + λ_{k} \leq μ_{k} (x) k = 1, 2, \dots, K \\ λ_{k}, λ_{0} \in [0, 1], k = 1, 2, \dots, K \\ x \in F (x) \end{matrix}$ (30)

In model (30) the parameters and variables are as shown in Table 3:

Table 3

Variables & parameters of model (39)

Variables &parameters
λ₀	Minimum satisfaction level with the objective function
δ	A small positive number that is usually considered to be 0.01
ϑ_k	The weight of k^th objective function
λ_k	Auxiliary variable of k^th objective function
μ_k (x)	The membership function of k^th objective function
F (x)	A feasible region that includes all the constraints

Considering what has been pointed out, the model presented in Section 5 can be written as Equations (31) to (53): $\begin{matrix} \max λ (x) = λ_{0} + δ \times (ϑ_{1} λ_{1} + ϑ_{2} λ_{2} + ϑ_{3} λ_{3} \\ + ϑ_{4} λ_{4} + ϑ_{5} λ_{5} + ϑ_{6} λ_{6} + ϑ_{7} λ_{7}) \end{matrix}$ (31) Subject to $ϑ_{1} λ_{0} + λ_{1} \leq \frac{- Z_{1} + U_{Z_{1}} \times (1 + D_{Z_{1}})}{U_{Z_{1}} \times D_{Z_{1}}}$ (32) $ϑ_{2} λ_{0} + λ_{2} \leq \frac{- Z_{2} + U_{Z_{2}} \times (1 + D_{Z_{2}})}{U_{Z_{2}} \times D_{Z_{2}}}$ (33) $ϑ_{3} λ_{0} + λ_{3} \leq \frac{- Z_{3} + U_{Z_{3}} \times (1 + D_{Z_{3}})}{U_{Z_{3}} \times D_{Z_{3}}}$ (34) $ϑ_{4} λ_{0} + λ_{4} \leq \frac{Z_{4} - U_{Z_{4}} \times (1 - D_{Z_{4}})}{U_{Z_{4}} \times D_{Z_{4}}}$ (35) $ϑ_{5} λ_{0} + λ_{5} \leq \frac{Z_{5} - U_{Z_{5}} \times (1 - D_{Z_{5}})}{U_{Z_{5}} \times D_{Z_{5}}}$ (36) $ϑ_{6} λ_{0} + λ_{6} \leq \frac{- Z_{6} + U_{Z_{6}} \times (1 + D_{Z_{6}})}{U_{Z_{6}} \times D_{Z_{6}}}$ (37) $ϑ_{7} λ_{0} + λ_{7} \leq \frac{- Z_{7} + U_{Z_{7}} \times (1 + D_{Z_{7}})}{U_{Z_{7}} \times D_{Z_{7}}}$ (38) ${Xlf}_{n} \leq c_{\max}$ (39) ${Xlf}_{j} - {Xes}_{j} \geq d_{j} - y_{j}, j \in E$ (40) $\begin{matrix} {Xes}_{j} - {Xes}_{b} \geq (d_{b} - y_{b}) \\ + (lag / {lead}_{bj} - {yy}_{bj}), j \in E \end{matrix}$ (41) $\begin{matrix} {Xlf}_{a} - {Xlf}_{j} \geq (d_{a} - y_{a}) \\ + (lag / {lead}_{ja} - {yy}_{ja}), j \in E \end{matrix}$ (42) $y_{j} \leq u_{j}, j \in S$ (43) ${yy}_{bj} \leq v_{bj}, j \in H$ (44) $y_{j} \leq M . w_{j}, j \in U$ (45) ${yy}_{j} \leq M . {ww}_{bj}, j \in U$ (46) $\begin{matrix} q_{f - threshold} \leq \frac{v_{bj} - {yy}_{bj}}{v_{bj}} + {ww}_{bj} \times M, \\ j \in U and j \in H \end{matrix}$ (47) $q_{c - threshold} \leq \frac{d_{j} - y_{j}}{d_{j}} + w_{j} \times M, j \in U and j \in S$ (48) ${yy}_{bj} \geq 0, j \in H$ (49) $y_{j} \geq 0, j \in S$ (50) $w_{j} = {0, 1}, j \in U$ (51) ${ww}_{j} = {0, 1}, j \in U$ (52) $λ_{1}, λ_{2}, λ_{3}, λ_{4}, λ_{5}, λ_{6}, λ_{7}, λ_{0} \in [0, 1]$ (53)

8 Case study

In order to evaluate performance of the model presented in this research, a case study is presented in this section. The company is one of the biggest producers of refinery equipment which is located in south of Iran. The projects of this company include air-coolers, towers, shell and tube heat exchangers, plate exchangers, reactors, etc. The organizational structure of this company is project based and there are different project teams in the organization. This case study is the project of manufacturing a shell and tube heat exchanger. The project team has estimated completion time of this project 403 days. Considering the petrochemical overhaul of the client, the duration is not appropriate and needs to be reduced to 303 days. The project manager needs to reduce the project duration in the best and most logical method. The necessary data in the engineering phase of the study have been collected by experts in the engineering department and based on the data of previous projects. In the procurement phase, the duration spans and relations have been determined based on the company’s policy in selecting the suppliers.

As for the construction phase, the initial estimations were done according to the capacity of the factory and accessibility of resources and machineries. Ultimately, after holding a kickoff meeting with the client and making the necessary changes to the collected data, the data were finalized. The final data have been presented in Tables 4 and 5 and the initial time schedule of the project has been shown in Fig. 3. The information of this project was collected by the project team, which is shown in the Table 10 of Appendix A. Column C_j is calculated based on the Equation (54): $C_{j} = \frac{C_{crash} - C_{Original}}{d_{Original} - d_{crash}}$ (54)

Fig.3

The master time schedule for shell and tube heat exchanger manufacturing project.

Table 4

Fast tracking risks

Dependency	Risk	P _j	I _j on cost
9 -to -10 SS	Change drawing	H	M
11 -to -22 FS	Need to change purchase request	M	L
11 -to -26 FS	The need to purchase extra electrode and un-forecasted items	H	VL
18 -to -35 FS	Unavailability of rolling equipment	VH	H
35 -to -40 FS	Unavailability of the welding team	H	H

Table 5

Desirable values for objective functions

Goals	Target	Deviation in Linguistic
Crashing	3962646000	M
Fast Tracking	21887874.8	VH
Time	303	VL
Quality Of Crashing	15	L
Quality Of Fast Tracking	3	VL
Risk	209847530900	VH
Preventive Actions	1099575000	L

The information of the activity relationships is shown in Table 11 of Appendix A. The project team held a meeting for identifying the crashing risks and its data are shown in Table 12 of Appendix A. As the Subject-Matter Expert’s (SME) opinion, only the relationships in the Table 4 can be fast tracked.

After receiving input information based on Section5, the proposed model will be applied for the case study. The information shown in Table 5 is collected for considering the model uncertainties and fuzzy approach for membership functions of the objective functions.

In order to determine the weights of each objective function in the problem, the views of experts should be collected first, which are resulted as Table 6.

Table 6

The views of experts for comparing objective functions

Using Z-AHP method the weights of fuzzy objective functions, will be obtained as Table 7:

Table 7

The weight of each case study objective

Criteria	Weight
Crashing	0.26
Fast Tracking	0.19
Time	0.17
Quality Of Crashing	0.15
Quality Of Fast Tracking	0.10
Risk	0.10
Preventive Actions	0.04

Based on Section 7.5, the proposed model is solved for the case study using software LINGO 9 and the solutions for Crashing, Crashing-Fast tracking and Fast tracking obtained are as shown in Table 8.

Table 8

The solution of the proposed model for the case study

Variables	Crashing Fast tracking	Crashing	Fast tracking	Variables	Crashing Fast tracking	Crashing	Fast tracking
Variables	Value	Value	Value	Variables	Value	Value	Value
λ ₀	1.000	1.000	Not feasible	w ₈	1.000	1.000	Not feasible
λ ₁	0.411	0.388	Not feasible	w ₉	0.000	0.000	Not feasible
λ ₂	1.000	1.000	Not feasible	w ₁₀	0.000	0.000	Not feasible
λ ₃	0.829	0.829	Not feasible	w ₁₁	0.000	0.000	Not feasible
λ ₄	0.559	0.558	Not feasible	w ₁₂	0.000	0.000	Not feasible
λ ₅	1.000	1.000	Not feasible	w ₁₃	0.000	0.000	Not feasible
λ ₆	1.000	1.000	Not feasible	w ₁₄	0.000	0.000	Not feasible
λ ₇	1.000	1.000	Not feasible	w ₁₅	0.000	0.000	Not feasible
y ₆	35.000	35.000	Not feasible	w ₁₆	0.000	0.000	Not feasible
y ₇	23.000	23.000	Not feasible	w ₁₇	0.000	0.000	Not feasible
y ₈	23.000	23.000	Not feasible	w ₁₈	0.000	0.000	Not feasible
y ₉	0.000	0.000	Not feasible	w ₁₉	0.000	0.000	Not feasible
y ₁₀	0.000	0.000	Not feasible	w ₂₀	0.000	0.000	Not feasible
y ₁₁	0.000	0.000	Not feasible	w ₂₁	0.000	0.000	Not feasible
y ₁₂	3.980	5.000	Not feasible	w ₂₂	0.000	0.000	Not feasible
y ₁₃	0.000	0.000	Not feasible	w ₂₃	0.000	0.000	Not feasible
y ₁₄	0.000	0.000	Not feasible	w ₂₄	0.000	0.000	Not feasible
y ₁₅	0.000	0.000	Not feasible	w ₂₅	1.000	1.000	Not feasible
y ₁₆	0.000	0.000	Not feasible	w ₂₆	0.000	0.000	Not feasible
y ₁₇	0.000	0.000	Not feasible	w ₂₇	0.000	0.000	Not feasible
y ₁₈	14.000	14.000	Not feasible	w ₂₈	0.000	0.000	Not feasible
y ₁₉	0.000	0.000	Not feasible	w ₂₉	0.000	0.000	Not feasible
y ₂₀	0.000	0.000	Not feasible	w ₃₀	0.000	0.000	Not feasible
y ₂₁	0.000	0.000	Not feasible	w ₃₁	0.000	0.000	Not feasible
y ₂₂	0.000	0.000	Not feasible	w ₃₂	0.000	0.000	Not feasible
y ₂₃	0.000	0.000	Not feasible	w ₃₃	0.000	0.000	Not feasible
y ₂₄	0.000	0.000	Not feasible	w ₃₄	0.000	0.000	Not feasible
y ₂₅	6.000	6.000	Not feasible	w ₃₅	1.000	1.000	Not feasible
y ₂₆	0.000	0.000	Not feasible	w ₃₆	0.000	0.000	Not feasible
y ₂₇	0.000	0.000	Not feasible	w ₃₇	0.000	0.000	Not feasible
y ₂₈	0.000	0.000	Not feasible	ww ₆₇	0.000	0.000	Not feasible
y ₂₉	0.000	0.000	Not feasible	ww ₈₁₆	0.000	0.000	Not feasible
y ₃₀	0.000	0.000	Not feasible	ww ₈₂₀	0.000	0.000	Not feasible
y ₃₁	0.000	0.000	Not feasible	ww ₁₂₂₆	1.000	0.000	Not feasible
y ₃₂	0.000	0.000	Not feasible	ww ₂₆₃₀	0.000	0.000	Not feasible
y ₃₃	0.000	0.000	Not feasible	w ₈	1.000	1.000	Not feasible
y ₃₄	5.000	5.000	Not feasible	w ₉	0.000	0.000	Not feasible
y ₃₅	1.000	1.000	Not feasible	w ₁₀	0.000	0.000	Not feasible
y ₃₆	0.000	0.000	Not feasible	w ₁₁	0.000	0.000	Not feasible
y ₃₇	0.000	0.000	Not feasible	w ₁₂	0.000	0.000	Not feasible
yy ₆₇	1.019	0.000	Not feasible	w ₁₃	0.000	0.000	Not feasible
yy ₈₁₆	0.000	0.000	Not feasible	w ₁₄	0.000	0.000	Not feasible
yy ₈₂₀	0.000	0.000	Not feasible	w ₁₅	0.000	0.000	Not feasible
yy ₁₂₂₆	1.019	0.000	Not feasible	w ₁₆	0.000	0.000	Not feasible
yy ₂₆₃₀	0.000	0.000	Not feasible	w ₁₇	0.000	0.000	Not feasible
w ₆	1.000	1.000	Not feasible	w ₁₈	0.000	0.000	Not feasible
w ₇	1.000	1.000	Not feasible	w ₁₉	0.000	0.000	Not feasible

In Table 8, λ values indicate the fulfillment of each objective. For example, being λ₂, λ₅, λ₆ and λ₇ values equal to 1 indicate that objectives 2, 5 and 7 are completely fulfilled. According to Table 8 and Fig. 4, the fulfillment of each objective is equal or greater when the problem is solved in Crashing-Fast tracking mode. So, we can conclude that the project cost in crashing-fast tracking mode is less than crashing mode. As seen in Table 8, activities 6, 7, 8, 12, 18, 25, 34 (y₆ = 35, y₇ = 23, y₈ = 23, y₁₂ = 3.98, y₁₈ = 14, y₂₅ = 6, y₃₄ = 5) and 35 (y₃₅ = 1) need crashing and the relationships between activities 6 and 7 (yy₆₇ = 1.019) and activities 12 and 26 (yy₁₂₂₆ = 1.019) need fast tracking so that the project duration is adequately reduced.

Fig.4

Compare Different λ values in Crashing with Crashing-Fast tracking.

Also, it should be noted that, in order to prevent the reduction of quality of activities 6, 7, 8, 25 and 35, preventive action is needed. The relationship between activities 12 and 26 needs preventive action to prevent activities delay. The key factors of projects during compression in the proposed model are given as follows:

Taking preventive actions in order to prevent the reduction of quality of deliverable would be beneficial.

Considering the risk factor would ensure that the minimum possible risk is imposed on the project during compression.

Since crashing and fast-tracking are done simultaneously in the proposed model, the costs of compression are reduced and the best method for reducing the completion time of the project would be selected (either crashing or fast-tracking). Table 9 and Fig. 5 show the costs and times of Crashing, Fast tracking and Crashing-fast tracking for the case study. As can be seen, the cost of crashing is 89320000-unit price more than Crashing-Fast Tracking and finishing the project at the time 303 is impossible using only fast tracking.

Another capability of the proposed model is that certain factors may be considered based on the needs of the project manager, overlooking the irrelevant factors.

An additional advantage of the proposed model is that either crashing or fast-tracking may be performed independently.

Using Z-AHP method for determining the importance of each project objective, the results of the model would get closer to the needs of project stakeholders.

Fig.5

Compare Cost of Crashing Mode with Cost of Crashing-Fast tracking mode.

Table 9

Comparison between different scenarios

Scenario	Cost	Time
Crashing	16892060000	303
Crashing-Fast tracking	16802740000	303
Comparison between Crashing and Crashing-Fast tracking	89320000	0
Fast Tracking	In this case study, finishing project at the time 303 is impossible using only fast tracking

9 Conclusion

Reduction of project duration is always an important and challenging issue for project managers. In the present study, a comprehensive model for reducing project duration that considered cost, time, quality and risk was presented. Considering each of these criteria can significantly impact the decision made by the project manager. As the presented model supports AON network relationships, it has a high applicability. The model presented in this research simultaneously does crashing and fast tracking in the network which is a unique capability. As each project is unique, there are high levels of uncertainty in projects. In the present model, fuzzy approach was used for considering uncertainties. The model was applied through a real case study and the results showed when crashing and fast-tracking are done simultaneously in the proposed model, the costs of compression are then reduced. Combinational grey and fuzzy approaches can be used for reducing uncertainties and improving model performance in future studies. If the number of project activities is numerous, the proposed method is very timely, which is one of the limitations of the present research. Future studies can extend the proposed approach using meta-heuristic algorithms in mega projects.

Footnotes

Appendix A

Table 12

Crashing risks

Activity Name	Risk	p _{b _j}	i_{b _j} on cost
CALCULATION	Change of the design input data	M	H
TUBE &SPACER	Delay in reaching customs (due to transportation by ship, the ship may be wrecked or sea storms may occur)	H	VH
SHELL SEGMENTS PREPARATION(CUTTING BEVELING, ROLLING, FIT-UP, WELDING)	Segment deformation due to high rolling speed	H	H

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Project crashing using a fuzzy multi-objective model considering time,cost,quality and risk under fast tracking technique: A case study

Abstract

Keywords

1 Introduction

2 Literature review

2.1 FMOLP

2.2 Project crashing

2.3 FMOLP in project crashing

3 Fuzzy approach

Table 1 Fuzzy linguistic variables Weights of criteria Fuzzy numbers Ratings of alternatives Very Low (VL) (0.00, 0.00, 0.25) Worst (W) Low (L) (0.00, 0.25, 0.50) Poor (P) Medium (M) (0.25, 0.50, 0.75) Fair (F) High (H) (0.50, 0.75, 1.00) Good (G) Very High (VH) (0.75, 1.00, 1.00) Best (B)

7.1 Defuzzification of fuzzy objective functions

7.3 Weight calculation for each objective

7.3.1 Fuzzy AHP method

Footnotes

Appendix A

References

Table 1
Fuzzy linguistic variables

Weights of criteria Fuzzy numbers Ratings of alternatives

Very Low (VL) (0.00, 0.00, 0.25) Worst (W)

Low (L) (0.00, 0.25, 0.50) Poor (P)

Medium (M) (0.25, 0.50, 0.75) Fair (F)

High (H) (0.50, 0.75, 1.00) Good (G)

Very High (VH) (0.75, 1.00, 1.00) Best (B)