A Fuzzy Approach to Model Evaluation of Project Complexity

Transcription

1 A Fuzzy Approach to Model Evaluation of Project Complexity EHSAN POURJAVAD, RENE V. MAYORGA Industrial Systems Engineering University of REGINA 3737 Wascana Parkway, Regina, SK, S4S 0A2 CANADA Abstract: Evaluating project complexity is the main means for successful projects and can play an important role in improving the project management of companies that take on multiple projects simultaneously. Despite the importance of this subject, there are few studies in this field and project managers have not paid a great deal of attention to it. In this paper, an exploratory approach has been taken to identify key project complexity criteria which have been narrowed down to the following; project size, project variety, project interdependency, and project context-dependence. Also, a Multiple-Criteria Decision-Making (MCDM) method based on a Fuzzy Analytical Hierarchy Process (FAHP) is proposed to evaluate project complexity using the chosen criteria. Employing the Fuzzy AHP method in group decision-making facilitates a consensus of decisionmakers and reduces uncertainty. An example of application is presented to display the utility of the model. Keywords: Project Complexity, Fuzzy AHP, Criteria, Evaluation, Project Management Introduction Nowadays, projects are dealing with a growing complexity, in both their structure and context. Project managers have to take into consideration a wide variety of parameters such as environmental, social, safety, security and a growing number of stakeholders, both inside and outside the project as well as the organizational and technical complexities to evaluate project complexity []. It should be noted that the project complexity has not been described obviously [2,3] but one subject that has been recognized by all project managers is, project complexity is one of the most important project characteristics that have effect on success of projects [4,5]. Recognizing sources of the project complexity and levels of project complexity has become an important subject to help modern project management [6]. Despite detecting the importance of project complexity to project management, few methods have been presented to evaluate the complexity of projects. Latva-Koivisto [7] assessed the complexity of business processes through the conversion of process charts to graphs. Vidal and Marle [8] analyzed project complexity and presented a project complexity model called ALOE to mange projects under conditions of complexity and to help project complexity understanding and management. Vidal et al. [6] identified multiple aspects of project complexity by the international Delphi study and presented the Analytical Hierarchy Process (AHP) method to evaluate project complexity. Xia and Xhan [9] used the Delphi method to identify complexity measures for building projects. Qureshi and Kang [0] suggested structural equation modelling to evaluate project complexity. A frequently used method to solve multi-criteria decision-making problems is the Analytical Hierarchy Process (AHP. In fact, AHP makes it possible to systematically structure and model a multi-criteria decision-making problem []. Vidal et al. [6] used this method to evaluate project complexity. One of the most important major advantages of AHP is its ability to use both qualitative and quantitative criteria. AHP needs exact numerical values (crisp numbers) for the pairwise comparison judgments, while decision-makers are often reluctant or unable to express judgments in crisp numbers in real-world situations, due to the complexity and uncertainty included [2]. To solve this shortcoming, Chang [22] proposed a Fuzzy AHP. In this paper, an integrated framework is proposed for modeling and analyzing project complexity to support and help project managers in multiple-project companies. Considering the fact that to evaluate project complexity there are several criteria to be considered, in this Paper the project complexity is characterized as a multi-criteria decision making problem. To solve the conventional AHP shortcomings, in this Paper a Fuzzy AHP is proposed. The ISBN:

2 proposed method is an adaptation and implementation of Chang s [22] method. The proposed method for project complexity evaluation is based (as in Chang s [22]) on using Triangular Fuzzy Numbers (TFNs) as a pair-wise comparison scale for taking human vagueness and uncertainty into account in decision-making. The reminder of this paper is organized as follows. Section 2 reviews the project complexity factors that have been used by other researchers. Section 3 introduces AHP fuzzy method. Section 4 depicts proposed methodology in this paper to evaluate complexity of projects based on fuzzy multi criteria decision making approach. Also, findings of case study are displayed in this section. The final section discusses reached results and makes conclusions and suggestions for the application of the proposed model. 2 Project Complexity Definition and Criteria Moral and Ramanujam [3] say complexity is described in different ways and different fields and this is reason why researchers could not reach a unified definition. Marle [4], Austin et al. [4] and Vidal et al. [8] define project complexity as following: project complexity is the property of a project which makes it difficult to understand, foresee and keep under control its overall behaviour, even when given reasonably complete information about the project system. This definition is considered as base definition of project complexity in this paper. The criteria that have been used by researchers to evaluate project complexity are presented in table briefly. Based on a review of the results of Vidal et al. [6], Vidal and Marle [8] and Qurshi and Kang, [0] a four category framework of project complexity consisting of Project size, project variety, project interdependencies and project context-dependence is considered in this study to measure the complexity of projects. 3 Proposed Fuzzy AHP The AHP approach was first presented by Saaty [] and used in different decision-making process. The basic supposition of AHP is the condition of functional independence of the upper part, of the hierarchy, from all its lower parts, and from the criteria or items in each level. It should be noted that AHP needs exact numerical values for the pair-wise comparison judgments, while decision-makers are often not able to express judgments in crisp numbers in real-world situations, due to the complexity and uncertainty involved [2]. To solve this shortcoming, Chang [22] proposed a Fuzzy AHP. The Chang s method uses Triangular Fuzzy Numbers (TFNs) as a pair-wise comparison scale for taking human vagueness and uncertainty into account in decision-making, which considerably eliminates the disadvantages of conventional AHP. In this Paper, the Chang [22] extent analysis method is adapted and implemented for project complexity evaluation. Table : The literature of project complexity criteria No Author Criteria expected project organization, type of structure, site constraints, method of Akintoye construction and construction techniques, [5] scale and scope of the project and complexity of design and construction Chan et al. [6] Sinha et al. [7] Leung [8] Vidal and marle [8] and Vidal et al. [6] Maylor et building structure & function, construction methods, urgency of the project schedule, project scale, neighboring environment, geological condition and repetition of similar type of projects workers, material and tools used in carrying out the project activity project duration, working spaces, contract sum, site area, type of structure, height of building, site location, client, usage of building and total floor area the project size, the project variety, interdependencies within the project system, context-dependence 6 mission, organization, delivery, stakeholders, al. [9] and team 7 Geraldi et structural, uncertainty, dynamics, pace and al. [20] socio-political complexity building structure & function; construction 8 Xia and method; the urgency of the project schedule; chan [9] project size; geological condition; and neighboring environment technological complexity, organizational 0 He et al., complexity, goal complexity, environmental [2] complexity, cultural complexity, information complexity Qureshi project size, project variety, elements of and kang, context, interdependencies in the project and [0] project complexity The steps of Chang [22] extent analysis approach are as follows: let X = {x,x 2,...,x n } be an object set, and U = {u,u 2,...,u m } be a goal set. According to the method of Chang [22] extent analysis, each objective is taken and extent analysis for each goal, g i, is performed, respectively. Therefore, m extent analysis values for each object can be obtained, with the following signs:, M, i=,2,.,n, () M, M Where all the M (j =,2,..,m) are TFNs. ISBN:

3 The steps of Chang s extent analysis can be given as in the following: Step : The value of fuzzy synthetic extent with respect to the ith object is defined as: S = M M M (2) To obtain, perform the fuzzy addition operation of m extent analysis values for a particular matrix such that: M = ( l, m, u ) (3) And to obtain M, perform the fuzzy addition operation of M (j =,2,..,m) values such that: M = l, m, u (4) and then compute the inverse of the vector in Eq. (4) such that M = u, m, l (5) Step 2: The degree of possibility of M 2 = (l 2,m 2,u 2 ) (l,m,u ) is defined as: V(M 2 M ) = sup [min(μ M (x), μ M2 (y))] (6) and can be equivalently expressed as follows: V(M 2 M ) = hgt (M M 2 ) = μ M2 (d)=, if m m 0, if l m (7), otherwise ( )( ) Where d is the ordinate of the highest intersection point D between μ M and μ M2 (see Fig. ). To compare M and M 2, we need both the values of V(M M 2 ) and V(M 2 M ). Step 3: The degree possibility for a convex fuzzy number to be greater than k convex fuzzy numbers M i (i =,2,...,k) can be defined by: V (M M, M 2,.M k ) = V[(M M ) and (M M 2 ) and..(m M k )] = min V (M M i ), i=,2,.,k. (8) Assume that d (A i ) = min (S i S k ). (9) For k =,2,...,n; k i. Then the weight vector is given by: W = d (A ), d (A ), d (A ), (0) Where A i (i =,2,...,n) are n elements. Step 4: Via normalization, the normalized weight vectors are: W = d (A ), d (A ), d (A ) () Where W is a non-fuzzy number. V(M 2 μ l M m l d Fig.. The intersection of M and M 2 4 Research Methodology and Findings Project complexity evaluation is one of the subjects that most project managers cope with. The difficulty in the evaluation of the project complexity is increased because of its intangible and multiple criteria structure. For this reason, a good method must be used that can handle this ambiguity. In this study, the Fuzzy AHP described model has been used to evaluate project complexity. The proposed methodology in this study is used to calculate complexity of the real case of a company s projects. This company works in the field of implementing industrial projects. The proposed methodology is composed of the following steps. 4. Finding Criteria to Evaluate Project Complexity The first step of the proposed model is finding appropriate criteria to evaluate the project complexity. vidal et al. [6], Vidal and Marle [8] and Qurshi and Kang, [0] researched and studied criteria used for project complexity and presented a complete classification of criteria which are capable of evaluating the complexity of all kinds of projects. They classified project complexity criteria to 4 clusters namely, project size, project variety, project interdependency and project context-dependence. In this research, these 4 criteria are used to assess complexity projects. It should be noted that considered company s experts chose 9 sub-criteria among the 70 sub-criteria to evaluate the complexity of their projects. The criteria and sub-criteria used in this research are shown in table Model Construction and Problem Structuring One of the most important steps in using Fuzzy AHP is laying the problem out in a hierarchy structure, explaining the problem clearly and exactly defining the dimensions of the problem. In this research, the problem structure is prepared with the u M m u M ISBN:

4 aid of brain storming and based on previous studies (Figure 2). The five projects of the considered company are alternatives of this model and are compared according to these criteria and subcriteria. Table 2: Criteria and Sub-criteria of project complexity Criteria Sub-criteria Project Size (PS) (C ) Project Variety (PV) (C 2 ) Project Interdependency (PI) (C 3 ) Project Context- Dependence (PCD) (C 4 ) Number of activities (Sc ) Duration of the project (Sc 2 ) Variety of activities (Sc 3 ) Variety of resources to be manipulated (Sc 4 ) Dependencies with environment (Sc 5 ) Interdependence between sites, departments and companies (Sc 6 ) Dependencies between schedule (Sc 7 ) Environment Complexity (Sc 8 ) New laws and regulations (Sc 9 ) this part are results of one of the company experts to paired comparison. Table 3. Linguistic scales for difficulty and importance Linguistic scale for importance Triangular fuzzy scale Triangular fuzzy reciprocal scale Just equal (,, ) (,, ) Equally Important (EI) (/2,, 3/2) (2/3,, 2) Weakly more Important (WMI) (, 3/2, 2) (/2, 2/3, ) Strongly more Important (SMI) (3/2, 2, 5/2) (2/5,/2, 2/3) Very strongly more Important (VSMI) (2, 5/2, 3) (/3, 2/5, /2) Absolutely more Important (AMI) (5/2, 3, 7/2) (2/7, /3, 2/5) 4.3. Paired Comparison of Criteria Considering Goal In the first step of pair-wise comparison, the importance of each criterion regarding the issue of evaluating complexity of projects is specified by paired comparison matrix. For example, the result of paired comparison of criteria related to the problem goal has been done by one of project managers (Table 4). μ.0 EI WMI SMI VSMI AMI Fig. 2. Framework for Fuzzy AHP process 4.3 Building Pair-wise Comparison Matrices Between Components/Attributes In this step, the importance of criteria and alternatives is calculated by paired comparison of criteria and alternatives. Paired comparison of criteria and alternatives are performed according to presented method in section 3. In fact, weight of criteria as well as alternatives is evaluated by Fuzzy AHP. The triangular fuzzy numbers are used to do paired comparison. The fuzzy scale regarding relative importance to measure the relative weights is given in Fig. 3 and Table 3. It should be mentioned that 24 experts and project mangers with considerable relevant experience were selected to perform paired comparisons. The displayed tables in /2 3/2 2 5/2 3 7/2 Fig. 3. The linguistic scale of the triangular numbers for relative importance (RI) The process of calculating criteria weight is made as follows. The value of fuzzy synthetic extent with respect to the any criterion is calculated: C = (3.6,4.5,6.5) * (0.04,0.06,0.08) = (0.3,0.27,0.52) C 2 = (2.56,3.6,4.66) * (0.04,0.06,0.08) =(0.0,0.9,0.37) C 3 = (2.66,3.66,5.5) * (0.04,0.06,0.08) = (0.,0.22,0.44) C 4 = (4,5.5,7)*(0.04,0.06,0.08) = (0.6,0.33,0.56) Then degree of possibility of each criterion related to other criteria is calculated. d (C ) = Min (V(C C 2 ), V(C C 3 ), V(C C 4 )) = Min (,,0.86) = 0.86 d (C 2 ) = Min (V(C 2 C ), V(C 2 C 3 ), V(C 2 C 4 )) = Min (0.75,0.89,0.6) = 0.6 d (C 3 ) = Min (V(C 3 C ), V(C 3 C 2 ), V(C 3 C 4 )) = Min (0.86,,0.72) = 0.72 d (C 4 ) = Min (V(C 4 C ), V(C 4 C 2 ), V(C 4 C 3 )) = Min (,,) = w = (d C, d C2, d C3, d C4) = (0.86,0.6,0.72,) RI ISBN:

5 Table 4: Paired comparison of criteria considering goal PS PV PI PCD PS (,, ) (,3/2,2) (/2,,3/2) (2/3,,2) PV (/2,2/3,) (,, ) (2/3,,2) (2/5,/2,2/3) PI (2/3,,2) (/2,,3/2) (,, ) (/2,2/3,) PCD (/2,,3/2) (3/2,2,5/2) (,3/2,2) (,, ) Normalizing w, the real weight of each criterion is evaluated. W presents weights of criteria. W = (d C,d C2,d C3,d C4 ) = (0.270,0.89,0.226,0.34) The results of paired comparison of criteria related to the goal problem according to all experts, opinion goes as follow. W criteria = (W C, W C2, W C3, W C4 ) = (0.2, 0.328, 0.28, 0.80) Paired Comparison of Sub-Criteria Related to Criteria As it was mentioned before, each criterion has some sub-criteria. In this step, importance of sub-criteria is calculated according 4 criteria Paired comparison of alternatives related to sub-criteria In the previous step, the importance of the subcriteria regarding the criteria was determined. In this step, the importance of the five projects with respect to the sub-criteria is measured Calculating the consistency ratio (CR) of comparison matrix In the next step, the consistency of every comparison is analyzed to assure a certain quality level of a decision. Saaty [] presented an index to measure consistency. In fact, this index is applied to calculate the consistency of the pair-wise comparison matrices. It should be noted that in Fuzzy AHP to calculate consistency, the fuzzy comparison matrices should be converted into crisp matrices. In this research, the method proposed by Chang et al. [23] is used to defuzzify the fuzzy numbers. With considering α as preference and as λ risk tolerance of decision-makers, the decision-makers can understand the uncertainties they face under different circumstances. A triangular fuzzy number indicated as a ij = (l ij, m ij, u ij ) can be defuzzified to a crisp number as follows: (a ) = λ. l + ( λ)u, 0 λ, 0 α, (2) Where l = (m ij l ij ) α + l ij, indicate the leftend value of α-cut for a ij, u = u ij (u ij m ij ) α indicates the right-end value of α-cut for a ij. It should be said, α can be considered as a stable or fluctuating condition, and is any value from 0 to. The decision-making environment stabilizes when increasing α. The degree of uncertainty is the highest when α = 0. Also, λ can be considered as the degree of a decision-maker s optimism and its range is between 0 and. When λ is 0, the decision-maker is highly optimistic. Conversely, when α is, the decision-maker is pessimistic [24]. Then, a comparison matrix is built by converting triangular fuzzy numbers to crisp numbers which is expressed as follows: (A ) = [(a ) ] = (a α 2 ) λ (a n (a α 2 ) λ (a α 2n ) λ. (3) (a α n ) λ (a α n2 ) λ The consistence index (CI) for a comparison matrix can be calculated with the use of the following equation. CI= (4) Where λ max is the largest eigenvalue of the comparison matrix, n is the dimension of the matrix. The consistency ratio (CR) is defined as a ratio between the consistency of a given evaluation matrix and consistency of a random matrix []. CR= (5) RI is a random index that depends on the size of matrix n, table 5 presents RI for different n. If the CR of a comparison matrix is equal or less than 0., it can be acceptable. Table 5: Random index (RI) of random matrices N RI α ) λ Final evaluation based on the results In this step, final weights of alternatives (projects) are calculated. The priority weight of each alternative can be calculated by multiplying the matrix of evaluation ratings by the vector of attribute weights and summing over all attributes. Table 6 shows final weights and ranking of projects. As can be observed from Table 6, project 3 has the highest final score with 0.272, and is therefore recognized as the most important project considering complexity criteria. The second place is awarded to project 5, with a final score of 0.207, followed by project and project 4, with a final score of 0.89 and 0.70 respectively. Project 2 received the lowest rank for this evaluation. ISBN:

6 Table 6: Priority weights of criteria and sub-criteria, and alternatives Subcriteria Criteria P P 2 P 3 P 4 P 5 C 0.2 C C C Sc Sc Sc Sc Sc Sc Sc Sc Sc Priority weights of projects Final ranking In fact, these results show that project managers in the considered company should pay more attention to project 3. The number of activities, variety of activities, dependencies between schedule, and environment complexity sub-criteria are the main reasons why this project was given first rank for project complexity. Project 3 had the highest weight for the criteria which have the most effect on complexity of projects. These parameters should be focused on during the planning and controlling of the project if project managers want to finish project 3 successfully. Also, the criteria of project complexity are evaluated and ranked using the proposed model in this paper. As can be seen from table 8, project variety plays the most important role for project complexity according to DMBT company experts opinions. In addition, project context-dependence criterion has the lowest degree of importance. 5 Discussion and Conclusion The aim of this study is to develop a model that could evaluate project complexity according to several criteria. Because of the multi-criteria nature of the project complexity problem, the proposed model in this study is built according to MCDM techniques. AHP is thought to be the most appropriate method for this problem considering the subjective nature of the criteria used to evaluate complexity of projects. However, it should be considered that most experts prefer natural language expressions rather than sharp numerical values in their evaluations, for this reason, the classical AHP may not yield a satisfactory result. In this paper, a Fuzzy AHP framework is presented to assess project complexity. While fuzzy AHP requires difficult computations, it is a more systematic method than the others, and it is more capable of taking into account a human s evaluation of ambiguity when complex multi-attribute decision-making problems are considered. In fact, pair-wise comparisons provide a flexible and realistic way to accommodate real-life data. One of the most important advantages of the proposed model is reliability, which means project managers and experts are able to be confident in the evaluation. Also, the proposed methodology is independent of the project models and is not related to the evaluation of the complexity of a given project model. In fact, this methodology can be used for all kinds of projects. The research results of this paper have some practical implications. Project managers are able to use the proposed model to compare and evaluate the complexity degrees of projects that they manage. In fact, results of this model help project managers to take appropriate management actions to reduce the potential risk and complete projects on time and properly. Also, the result of complexity information can be used to improve project planning and implementation. It is recommended that complexity measurement process must be done at the earliest possible lifecycle phase and then reviewed at the subsequent phases. Also project managers must prepare appropriate strategy and organization arrangements to respond to various types of complexity in a project which is resulted of changes in its environment. References [] Baccarini, D., Archer, R., The risk ranking of projects: a methodology, International Journal of Project Management, Vol. 9, No. 3, 200, pp [2] Williams, T.M., The need for new paradigms of project complexity, International Journal of Project Management, Vol. 7, No. 5, 999, pp [3] Bertelsen, S. and Koskela, L., Managing the three aspects of production in construction, Proceedings of IGLC-0, August, Gramado, Brazil, [4] Austin, S., Newton, A., Steele, J. and Waskett, P., Modeling and managing project complexity, International Journal of Project Management, Vol. 20, No. 3, 2002, pp [5] Chan, A.P.C., Scott, D. and Chan, A.P.L., Factors affecting the success of a construction project, Journal of Construction Engineering and Management, Vol. 30, No., 2004, pp [6] Vidal, Ludovic-Alexandre., Marle Franck, Bocquet, Jean-Claude., Using a Delphi process and the Analytic Hierarchy Process (AHP) to evaluate the complexity of projects, Expert ISBN:

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