A Comparison of AHP and PROMETHEE Family Decision Making Methods for Selection of Building Structural System

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Suppose thatis the set of criteria. As seen in Equation (1), the result of pairwise comparison over n criteria is summarized in an n×n matrix where elements, a_ij (i,j=1,2,…,n), represent relative weights of criteria. In AHP, pairwise comparisons are made by more than one decision maker and all of these opinions are taken into account. In this case, geometric mean can be used as seen in Equation (2).

(1)

(2)

Where k is the number of decision makers. In the final step, the mathematical process begins with normalizing and finding relative weights for each matrix. Relative weights are defined through eigenvector:

(3)

The matrix A has the rank 1, if pairwise comparisons are completely consistent (). Here, weights can be obtained by normalizing any row or column. It should be noted that the quality of AHP output totally depends on the consistency among judgments made based on pairwise comparisons. Consistency is defined using the relation between inputs:. The Consistency Index (CI) is defined as

(4)

The Consistency Index Ratio (CIR) is used when evaluations are completely consistent:

(5)

where RI (from Table 2) is the Random Index.

Table 2. Random Consistency Indices (Adopted from [10])

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The value of 10 percent is acceptable as the upper limit for CIR. If CIR is larger than this value, assessment must be repeated to improve consistency. Measurements of consistency can be used to assess consistency among decision makers as well as consistency at different levels of hierarchy.

2.2. PROMETHEE Method

The PROMETHEE outranking method was adopted for this study to take the interests of various decision makers into consideration. This method is software-driven, user-friendly, provides direct interpretation of parameters, and analyzes sensitivity of results. The PROMETHEE family of outranking methods is one of the most recent MCDM methods which was developed by Brans ^[25] and further extended byBrans and Vincke ^[26].

2.2.1. PROMETHEE I

PROMETHEE I establishes a partial preorder among the alternatives. In PROMETHEE I, alternative a is preferred over alternative b, aPb, if alternative a has a greater leaving flow than alternative b and a smaller entering flow than the entering flow of alternative b:

(6)

PROMETHEE I allows indifference and incomparability situations. Therefore, sometimes partial rankings can be obtained. In the indifference situation (aIb), two alternatives a and b have the same leavingand entering flows:

(7)

Two alternatives are considered incomparable, aRb, if alternativeis better than alternative b in terms of leaving flow, while the entering flows indicate the reverse:

(8)

PROMETHEE I ensures creation of indifferent and incomparable alternatives. In some ranking problems, PROMETHEE I can give a complete ranking depending on the evaluation matrix values.

2.2.2. PROMETHEE II

PROMETHEE II establishes a complete preorder among the alternatives, and consists of building a valid outranking relation. The basic principle of PROMETHEE II is based on a pair-wise comparison of alternatives. The implementation of the PROMETHEE II requires the following two additional types of information ^[1]:

i. The Weights of Criteria: determination of the weights is an important step in most multi-criteria methods. PROMETHEE II assumes that a decision-maker is able to weight the criteria appropriately, at least when the number of criteria is not too large;

ii. The Preference Functions: for each criterion, the preference function translates the difference between the evaluations obtained by two alternatives into a preference degree ranging from zero to one. This function is used to compute the degree of preference associated with the best action in case of pairwise comparisons.

Weighting of the criteria considered during decision making and evaluation of these criteria via preference functions are performed simultaneously. In order to facilitate this process, six types of particular preference functions have been proposed by ^[27]: (1) usual, (2) U-shape, (3) V-shape, (4) level, (5) V-shape with indifference and (6) Gaussian. These six types are particularly easy to define. For each criterion, the value of an indifference threshold, q, the value of a strict preference threshold, p, and the intermediate value between p and q, s, has to be defined. One of the important steps in utilizing this method is to select the preference function which is very difficult for most of decision makers who do not have sufficient knowledge about each criterion. In the case of how to select the best preference function for one criterion, ^{[18, 28]} provided some suggestions. Some new preference functions have also been proposed by ^[29]. More details on these functions and how to select them for different types of criteria can be found in ^[18].

Figure 1 presents stepwise procedure for implementing PROMETHEE II. The procedure starts with determining deviations based on pair-wise comparisons where is the deviation of criterion for alternatives and . It is followed by using an associated preference function for each criterion in Step 2. For each alternative, belonging to the set of alternatives, is a global preference index of over which is calculated in Step 3. Step 4 shows the procedure for calculating the leaving and entering flows for each alternative and the partial ranking. The leaving flow is the measure of the outranking character of (how dominates all the other alternatives of ). Consequently the entering flow gives the outranked character of (how is dominated by all the other alternatives of ). The procedure comes to an end in step 5 by calculating the net outranking flow for each alternative and completing the ranking. The maximum amount of net flow denotes the best alternative.

2.2.3. PROMETHEE GAIA

PROMETHEEGAIA (Geometrical Analysis for Interactive Aid) is suitable for visualization of problem characteristics through geometrical interpretations ^[30]. Graphical GAIA displays the relative position of the alternatives, in terms of contributions to the various criteria. Principal Components Analysis (PCA) is applied to the matrix of “normed flows”, defined for alternative a and criterion j by:

(9)

where m is the number of alternatives, and this is used to generate a two-dimensional plot in which the alternatives and criteria are represented in the same plot.

The information relative to a decision problem, including n criteria, can be represented in an n-dimensional space. The GAIA plane is the principal plane obtained by applying PCA to the set of actions in this space. This plane is obtained by projecting this information on a plane so that the smallest amount of information is lost. Alternatives are represented by points and criteria by axes. The conflicting character of the criteria appears clearly; criteria expressing the similar preferences on the data are oriented in the same direction, while the conflicting criteria are pointing in opposite directions.

3. Technologies in Building Construction

Effective project control and quality control systems, reduction in energy consumption, optimal use of materials, reduction in production and construction costs, and improved quality of construction parts and components are among the most important advantages of using advanced technologies in building construction industry. These technologies can be classified into four categories:

1. New technologies used in production and application of building materials, including self-compacting concrete, high-strength concrete, and fire-resistant bricks.

2. Advanced technologies used in building elements including windows and glass, thermal and acoustic insulators, and indoor/outdoor drywalls.

3. New structural systems including Lightweight Steel Framing (LSF), 3D Panels, bearing wall systems with tunnel formwork.

4. Novel technologies used in building installations and equipment including heating/cooling systems, air conditioning, intelligent systems for controlling safety and energy use, and BMS (Building Management System).

Figure 1. Stepwise Procedure for PROMETHEE II

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Given the importance of industrialization of mass housing projects, this paper deals with the third category of novel structural systems. The case study of this research is a mass construction project of low-rise housings in Iran. The building structural systems which are listed below have been used in the same location for the first time in Iran which provided an opportunity for comparing them. The land area of this project is 18.3 hectare, the capitation is 64.64 unit per hectare, number of blocks is 197 in 3-story, and the average area of each unit is 78 square meters. In this case study, five systems including 3D Panel with light walls in building frames, LSF (Light Steel Frame), ICF (Insulating Concrete Formwork), Tunnel Formwork system, and Tronco approved by the Iranian Building and House Research Center (BHRC) have been selected. The most important features of these systems are light weight, ease of construction, high construction speed, high structural system stability, environmental friendliness, and ease of industrial production.

3.1. Alternatives

Among the new architectural technologies, we briefly examine five systems that have been approved by BHRC to select the best system. These systems have been chosen because of their prevalence in the mass construction projects in Iran. The main specifications of these systems are briefly introduced here.

3.1.1. Light Steel Frame (LSF)

Advantages of this structural system shown in Figure 2 include:

• Extraordinary improvement in quality as a result of 90 percent fabrication in factories,

• Sixty percent reduction in dead load compared to similar structures,

• Ten percent increase in useful space of buildings due to use of thinner walls compared to traditional structures,

• Creating joints and junctions at installation sites by using bolts and nuts, saving energy under national construction code,

• Constructability at any climate due to making use of dry sites,

• Creating economic value by reducing implementation time, minimized human resources required at site,

• No requirement for heavy machineries at the construction site,

• Life cycle time of over 50 years,

• Easy installation of mechanical and electric equipment,

• Easily changeable building plans,

• Excellent diversity in materials used in façade and interior finishes,

• Environmental friendliness, and

• Minimizing waste at construction sites (green industry).

Figure 2. Light Steel Frame (LSF) in Residential Buildings

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3.1.2. 3D Panels

3D Panel Building System shown in Figure 3 can be used for buildings with one or two stories. It provides light structures, no debris, resistance to heat, humidity and acoustic insulation, rapid transportation, easy installation, applicability in a wide range of buildings, increased useful space due to thinner walls, easy use for finishes, and impermeability. In addition, this system provides a clean waste-free site and a modified, ozone-friendly, fire-proof polystyrene core.

Figure 3. 3D Panel Building System

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3.1.3. Insulating Concrete Formwork (ICF)

ICF includes permanent formworks used for concrete work and making reinforced concrete walls (See Figure 4). Once concrete work is done, the formwork forms a part of walls. ICFs are usually made of expanded polystyrene which should be protected by the exterior and interior envelopes and finishes. ICFs are often composed of two 5cm thick polystyrene plates connected to each other through plastic or metal connectors. In this system, walls are the main elements for bearing and transferring gravity and lateral loads that are uniformly distributed over foundation through walls and slabs. ICF has a large degree of indeterminacy and therefore, is expected to exhibit acceptable seismic behavior.

Figure 4. Preparing Insulating Concrete Formwork (ICF)

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Figure 5. Tunnel Formwork System

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3.1.4. Tunnel Formwork Systems

Tunnel Formwork System shown in Figure 5is a novel structural system with bearing walls and concrete roof. In this system, walls and reinforced concrete roofs arereinforced, formed, and casted continuously and simultaneously. The name comes from the form of monolithic metal formworks used in walls and roofs. Loads are largely borne by bearing walls and flat slabs. Due to the absence of beams and columns, structural walls are the main elements for bearing and transferring gravity and lateral loads. This system has excellent seismic performance because of simultaneous concrete work over walls, slabs, and roofs, and as well as integrated joints and elements. Major characteristics of this structural system include monolithic structure, improved seismic behavior due to presence of a box mechanism,spread stress instead of node-concentrated stress due to transformation from beam-and-column system to wall-and-slab system, increased degree of indeterminacy, increased delay in forming plastic joints, rapid construction due to simultaneous construction of walls and roofs, reduced concrete consumption compared to conventional concrete frames, and reduced material wastes.

3.1.5. Tronco system

This system which is shown in Figure 6 is inspired by conventional methods. Tronco combines traditional methods with new ones in construction buildings with small number of stories. The main element of this system is pipe-shaped parts made out of galvanized steel prepared on site. Door and window frames and electric and mechanical installations are installed during the construction. Given the empty space within tubes, walls, and roofs, EPS panels are used to maximize energy saving and minimize weights in Tronco systems.

Figure 6. Tronco System using Galvanized Steel Pipe Elements

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3.2. Evaluation Criteria

In choosing the criteria for selection between alternatives, understanding and the acceptability of the method by the decision makers should be taken into account. The theory of multi-criteria evaluation should be understood by the decision makers in order to maximize the effectiveness of the decision making process and to avoid having the method become a “black box”. In this study, the five selected structural systems have been ranked based on the following criteria:

• Cost: This criterion includes costs per square meter of building, creating the production line, transportation, construction, and maintenance.

• Ease of Construction: Availability of special machineries, skilled labors, easy installation, and safety are some of the important aspects affecting this criterion.

• Energy saving: Energy parameters and indices include requirements of exterior walls and compliance with construction standards and codes.

• Dead load: to reduce damages resulting from earthquakes, buildings must be designed and constructed based on seismic provisions, latest standards, and minimum weights.

• Number of stories: number of stories and maximum allowable height are usually among the most important factors in mass construction projects.

• Life cycle time: permanency of materials, resistance to corrosion, and stability are among major concerns when evaluating life cycle time.

Some of these criteria are qualitative and should be transformed to quantitative criteria. The general method in evaluation of qualitative criteria is application of bipolar scale which is denoted in Table 3 for positive and negative indices.

Table 3. Bipolar Scale for Positive Indices

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Table 4. Pairwise Comparison Matrix

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4. Results and Discussion

Qualitative and quantitative reliability of decisions made through AHP and PROMETHEE methods highly depends on the quality of the questionnaires designed and the accuracy of the information provided by the experts, professionals, and staff of the organizations involved in the decision making process. The outputs of questionnaires in this study have been used along with other data from reliable sources to complete pairwise comparison tables. Questionnaires were completed by 10 contractors (two contractors for each system), five engineering consultants involved in designing projects, and five owners. The weights of criteria and preference functions were determined by the decision makers. An analysis of the findings revealed that consultants mainly have focused on the technical criteria including dead load and energy saving; contractors have emphasized on the ease of construction; and lastly, owners’ main concerns have been life cycle time and cost of the infrastructure.

4.1. AHP Ranking Results

Based on the criteria and alternatives, a hierarchical structure was designed (Figure 7). In order to compare criteria and alternatives, transforming relative concepts into quantities is needed. In the pairwise comparison matrix, each element represents relevance of the row criterion compared to the column criterion. Table 4 presents a pairwise comparison matrix. It should be noted that the matrix was constructed using geometric mean (Equation 2) of decision makers’ perspectives based on the assumption of equal decision making power. For a balanced comparison, weights must be assigned to the criteria. These weights were assigned by obtaining the weight vector for each matrix. Here we used an approximation method (arithmetic mean) for computing the matrix weights. First, elements of each column were normalized through division by sum of that column. Then, mean of the normalized values in each column was selected as the weight of the criterion. Table 4 shows the weights assigned to the criteria.

Figure 7. Hierarchical Structure for Alternatives and Criteria

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Pairwise comparison matrix is a reliable base for calculations only if it is consistent. Consistency means a condition in which ratios assigned for comparing criteria and options in a matrix (based on the pairwise connection between criteria) are mutually consistent. Table 5 presents consistency ratios based on the previous equations.

Table 5. Parameters in Calculating Consistency Ratio in Pairwise Comparison Matrix

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As mentioned earlier, for applicability of pairwise comparison matrix in AHP, inconsistency in the matrices must not exceed 10 percent, as is the case here. In addition, to determine the ratios for sub-criteria and alternatives, further comparison matrices were constructed. Consistency was determined in a similar way. For example, Table 6 presents the comparison matrix for the structural systems of buildings in terms of cost.

Table 6. Pairwise Comparison Matrix for Available Options in Terms of Cost

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Calculations showed an inconsistency coefficient of 4.38 percent which is smaller than the maximum allowable value (10%). In the same way, alternatives for other criteria were compared to determine coefficients of inconsistency which are summarized in Table 7. Finally, relative weights of criteria were used along with respective vectors for each option to determine scores used in raking shown in Table 8. According to the scores and ranking shown in this table, the best choice has been 3D Panel followed by Tronco.

Table 7. Coefficients of Inconsistency for Comparison of Alternative

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Table 8. Scores and Ranking of Alternatives

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4.2. PROMETHEE Ranking Results

Before using the PROMETHEE method to rank the alternatives, for each criterion, a specific preference function, with its thresholds was defined. Preference functions and threshold values have been defined by the same decision-making team consisting of contractors, consultants, and owners. Decision-making team has set these values by taking into consideration the features of alternatives and the project conditions. The preference functions and thresholds are provided in Table 9.

Table 9. Criteria Preference Functions and Thresholds

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After determining the evaluation matrix and preference functions, alternatives were evaluated by using the Decision Lab software. The leaving flow, entering flow and net flow values obtained from the evaluation process are shown in Table 10.

Table 10. PROMETHEE Flows

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By using the flow values in Table 10, firstly the partial ranking was determined via PROMETHEE I (Figure 8). PROMETHEE I used leaving and entering flow values to find the partial ranking based on strongly established preferences only. As a result, not all alternatives could be compared one-to-one with the others and some alternatives were simply incomparable. Highlighting incomparable alternatives was interesting for decision makers because it usually emphasizes on alternatives with quite different profiles.

Figure 8. PROMETHEE I Partial Ranking

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Tunnel Formwork system was determined as the worst alternative according to the PROMETHEE I partial ranking. LSF, 3D Panel, ICF, and Tronco systems were preferred to Tunnel Formwork system, and Tronco system was preferred to LSF system. On the other hand ICF, LSF, and Tronco systems were incomparable alternatives. As it is shown in Figure 8, PROMETHEE I selected3D Panel as the best alternative.

The net flow values, given in the last column of Table 10, were used in PROMETHEE II complete ranking to identify the best alternative (Figure 9). All the alternatives were ranked, leaving no incomparable pair of alternatives. This information is more straightforward and easier to use than the PROMETHEE I partial ranking, but it could be a reflection of less reliable preferences.

The 3D Panel was selected as the best alternative based on the provided information by PROMETHEE II, and the other ranked alternatives were ICF, Tronco, LSF, and Tunnel formwork system respectively.

Figure 9. PROMETHEE II Complete Ranking

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The GAIA plane provides the decision maker with comprehensive graphical image of the decision problem and is thus a descriptive complement to PROMETHEE rankings. The decision problem can be represented in the GAIA plane (Figure 10) where structural systems are represented by points and criteria by vectors. Through this approach, conflicting criteria may appear clearly. Criteria vectors expressing similar preferences are oriented in the same direction, while conflicting criteria are pointing in opposite directions. The length of each vector is a measure of its power in structural systems differentiation.

Figure 10. GAIA Plane

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This plane is the result of principal component analysis (PCA), projecting the 6-dimensional space of criteria onto a two-dimensional plane. As it is shown in Figure 10, the Delta-parameter is 96.65 percent, which means only 3.35 percent of the total information is lost by the projection. It can be seen from Figure 10 that the pi vector is in the direction of energy saving, maximum dead load, and cost and the closest alternative to the pi vector is 3D Panel. This result is consistent with the complete ranking of PROMETHEE II.

4.3. AHP Sensitivity Analysis

Sensitivity analysis over the criteria indicates the extent to which variations in criteria weights and decision makers’ perspective affect rankings obtained from AHP. The small difference between the scores obtained for LSF and Tronco indicates the sensitivity of these two options. Sensitivity analysis was carried out by utilizing Expert Choice 2000. The results are presented in Table 11. As seen in the table, the smallest variations are observed in the dead load criterion. It means that an increase by 25.7 percent in dead load exchanges LSF for Tronco (LSF goes down to the second place while Tronco moves up to the first rank). Therefore, dead load is the most sensitive criterion in determining the ranks of LSF and Tronco.

Table 11. Sensitivity Analysis Results on Ranks of LSF and Tronco

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4.4. PROMETHEE Sensitivity Analysis

For each criterion, a stability interval was computed. It indicates the range in which the weights of that criterion can be modified without affecting the PROMETHEE II complete ranking, provided that the relative weights of other criteria are not modified. This information is interesting for assessing the general robustness of the ranking. For the case of this research, it shows that if the weight of number of stories criterion changes in the range of 8.18 percent to 21.39 percent, the ranking does not change.

Table 12. Stability Intervals

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Table 13. Ranking of Alternatives for Each Method

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