Application of Multi-Criteria Decision-Making in Achieving the Right Mix Asphalt Mixtures

methodologies are share almost ABSTRACT: Marshall Test is one of the tests used for asphalt mixture. This test has six parameters, including stability, unit weight, flow, void in mineral aggregate, voids in total mix, and voids filled with asphalt. Further, the additional cost of various additives to the asphalt mixture for building one kilometer of the road is a negative parameter. However, using additives improves the technical characteristics of the asphalt mixture. There-fore, making a proper choice among the alternatives would be a hard task. This paper tries to provide an appropriate method for mixing the design of asphalt mixtures to select the best alternative by considering all technical criteria. For this purpose, the concordance analysis method, which is one of the compensation models of the Multi-Criteria Decision-Making (MCDM), was used. The results showed that the concordance analysis method using a pairwise comparison strategy combines the different criteria to prioritize alternatives and to propose the best alternative. Therefore, this method can be used as a superior method for mixing the design of asphalt mix compared to conventional methods.

Various methods, such as Marshall Test, have been proposed for mixing the design of asphalt mix. In this test, different alternatives are tested with different additives to achieve a high-quality asphalt mix, and its results are expressed based on six parameters. On the other hand, adding the appropriate additives to the mixture raises concerns about the increase in the construction cost (negative parameter). A combination of the different criteria, which may be opposing, to select the best alternative seems to be a challenging task.
In this paper, an attempt has been made to provide an appropriate method for mixing the design of asphalt mixtures to select the best alternative by considering all technical criteria. For this purpose, 25 alternatives and seven different criteria were introduced. Then, the evaluation of proposed alternatives using concordance analysis has been intro-duced. Finally, given the quantitative amounts of each criterion and determining the criteria weight, the best alternative is selected.

LITERATURE REVIEW
According to Hwang and Yoon 1981 classification, the Multi-Criteria Decision-Making (MCDM) is divided into multiple attribute decision-making (MADM) and multiple objective decision-making (MODM) [18]. MADM is used to evaluate discrete variables. In addition, this is an a priori process. Experts take part in the initial stage of the process, giving the weightings of the criteria, or assessing any attribute of the problem. Finally, the best solution is obtained. MODM allows for the obtainment of a continuous set of solutions regarding two or more criteria, called Pareto front. These solutions are characterized by each being considered equally good. The experts also take part in the end stage of the process, choosing one among the many solutions [19].
There are various techniques of MCDM to conduct multi-criteria decision analysis. There is no better or worse technique because the method's appropriateness depends on the specific decision situation [20]. Different MCDM techniques have been developed to tackle the different problems under different circumstances and fields of application [21]. MCDM methodologies are very similar and share almost similar steps of organization and decision matrix construction, but each methodology synthesizes information differently [22]. A brief description of some standard methodologies is presented in table 1. In an article by Jato-Espino et al. 2014, the Multi-Criteria Decision Analysis (MCDA) approach was chosen as a branch of operations research. The application of 22 different methods belonging to the construction field, which is classified into 11 groups, was examined. The most significant methods were briefly discussed, and their main strengths and limitations were stated. A great variety of MCDM methods have been developed to solve such problems in terms of practical features, [23].
For more detailed reviews of the literature on MCDA for the evaluation of transportation and construction projects, readers are referred to Macharis and Bernardini 2015 [21] and Jato-Espino el al. 2014 [23], respectively. Also, in this section, a brief review of MCDA is presented (table 1). Saphira and Goldenberg (2005) presented a selection model based on the Analytical Hierarchy Process (AHP), a multi-attribute decision analysis method. Their model can handle a significant number of different criteria in a way that genuinely reflects the complex reality, incorporates the context and unique conditions of the project, and allows for the manifestation of user experience and subjective perception. The problem was divided into four criteria and eighteen sub-criteria hierarchy, which addressed three perspectives: cost evaluation benefits evaluation and total evaluation [24]. Chou (2008) proposed a Case-Based Reasoning (CBR) estimation method model that compares and retrieves the most similar instance across the case library. Four CBR approaches were presented and assessed in terms of their mean absolute prediction error rates. The similarity between current and previous cases was measured after establishing pairwise comparisons through the AHP technique [25]. Rahman et al. (2012) developed the Knowledgebased Decision Support system for roofing Material Selection and cost estimating (KDSMS) system, a knowledge-based decision support system for the selection of optimal Materials for building design. The system uses product cost modeling techniques and the MCDM technique of TOPSIS for optimal materials selection and has been implemented as a prototype system for optimal roofing material selection and cost modeling [26].
Şimşek et al. (2013) applied a multi-response Taguchi method to investigate the ranking of the different factor levels and the best possible mix proportions of high strength self-compacting concrete (HSSCC). They proposed a hybrid Taguchi method and Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) to solve the multiresponse optimization problem in a ready-mixed concrete plant. In fact, the TOPSIS method was used to transform the multi-response problem into a single-response problem. The results showed that the produced concrete samples satisfied the expected properties of the HSSCC [27]. The dynamic modulus of the mixtures was compared and analyzed. The unit costs defined the inputs, whiles the outputs were a series of environmental factors [28].
Various methods have been proposed by researchers for the optimal design of the asphalt mix.
Lim et al. introduced a new design method for asphalt mixes containing additives. The laboratory results showed that careful selection of the additive dose is required to balance the cracking and groove performance of asphalt mixtures [29].
Other methods have been proposed for the optimal design of the type of concrete pavement. In another study, a method was proposed to determine the optimum amount of water for roller compacted, steel fiber-reinforced, and polymer-modified bonded concrete overlays [30].
Jato-Espino et al. (2014) proposed a new multicriteria model based on the combination of several existing decision-making tools for the selection of urban pervious pavements. They developed a Multicriteria approach based on the Integrated Value Model for Sustainable Assessments (MIVES) method and improve the model by including some auxiliary complements such as Monte Carlo simulations, fuzzy sets and the AHP method [31].
There is also some successful application of MCDM in prioritizing highway safety improvement projects [32, 33, 34, and 35].
Yu and Liu (2012) proposed a method based on fuzzy AHP to tackle the difficulties of multi-criteria decision-making environment of prioritizing highway safety improvement projects and considering the effect of using uncertain data. Their research indicated that AHP could be used as an efficient tool in selecting and prioritizing the most beneficial projects given budget constraints [32].
Dadashi and mirbaha (2019) proposed a new multi-criteria methodology based on the integration of DEA and Monte-Carlo simulation to consider existing uncertainties of the problem. Results indicate how their proposed methodology can be useful for detecting sensitive decision-making units and providing a more comprehensive view for decisionmakers to allocate a limited budget to the most efficient safety improvement projects [35].
According to the discussions, the application of the MCDM method has been widely used in engineering sciences. However, the use of this method in the mixing design of asphalt pavement has not been investigated. Hence, in this article, for determining the best mixing design, the concordance analysis method was used to select the best alternative from 25 different potential alternatives.

METHODOLOGY
In this paper, the selection of the appropriate mixture was investigated by using the results of the Marshall Test and using the concordance analysis method. Concordance analysis is one of the most critical methods of evaluating the cases whose criteria do not fit together and cannot be converted to each other. The concordance analysis method using a pairwise comparison strategy combines the different criteria to prioritize alternatives and to propose the best alternative. This method ranks the alternatives using a concordance and non-concordance sets, through pairwise comparison of alternatives.
Concordance and non-concordance indices can be calculated for each weighting system for various purposes. The concordance index shows the dominance of one alternative over the other alternatives, and the non-concordance index indicates the dominance of other alternatives on the desired alternative. The domination index is constructed by Using concordance and non-concordance indices, which is used to determine the dominance of either alternative. The alternatives that are better than average are called non-dominated alternatives. We assume that the evaluation of i alternatives to be considered based on j criteria [36]: If j is a positive measure: Concordance and Non-Concordance series are as follows: The concordance set C ′ ،is a set of all criteria where the i th alternatives are better than the i' option.
The non-concordance set D ′ ،is a set of all criteria where the i th alternatives are worse than the i' option. The concordance index is as follows: Where Wj is jth criterion weight than other criteria.
Therefore, it can be said [9]: Non-dominated alternatives refer to those alternatives that, for a specific weighting system, operate better than average. By changing the weighting system, non-dominated alternatives change. Conventional alternatives in the non-dominated set can be considered in a set, called competitor alternatives. This set operates better than average for any weighting system. The selection of any of the alternatives of the competitor set is the proper solution for the problem. Finally, using the Giuliano method, various options using a concordance and non-concordance net ranking have been compared, and the results were evaluated [36].

DATA ANALYSIS
Here, the alternatives examined based on the results of the Marshall test as their criteria. By creating concordance and non-concordance sets according to rij parameters, which are calculated for different weighting systems, concordance, and nonconcordance indices are determined, leading to the identification of the non-dominated alternatives and finally determining a set of competitor alternatives. Here, the top alternatives are better asphalt mixture. The following tables (Table 2, Table 3, and Table 4) show the properties of bitumen and materials that have been used [37][38][39]:  The amount of stretch at 25℃, in cm --More than 100

Data inputs (raw data)
In this paper, six parameters of Marshall Test including as well as the cost of the construction of one kilometer of asphalt were selected(unlike previous research [12]) as evaluation criteria in the concordance analysis. Therefore, seven factors entered in the calculations as a factor (j).
The cost criteria were calculated as the additional cost of adding additives to the mixture for each kilometer of a highway [1,2] and [14,15]. The following table shows raw Pij data (results of Marshall) used in the Concordance analysis [37][38][39]. Note that the costs of additives were high in this project, and using additives with reasonable prices can help reduce the costs in various alternatives. Also, different alternatives included different percentages for adding additives to the asphalt mixture, and thus, 25 alternatives (i) were entered in calculations and were compared. The percentage of polyester fibers and nano-carbon black added to the asphalt mixtures are presented in Table 5 [37-39].

Weighting system
Different pavement experts cooperated on the current study by providing their opinions on the importance of each of the items in the objective function based on a scoring method. According to experts' opinions, a group of weights was provided for objective functions. Each category is called a weighting system.
Based on the weighting system, the net values of concordance and non-concordance indices were calculated for each mixture, and accordingly, the proper mixture is introduced. In this paper, eight weighing systems were considered in table 6. As can be seen, in one of the weight systems, the weight of all the criteria is considered equal, and in 7 other weight systems, stability and price of asphalt mixture due to extremely high importance have had higher weight than other indicators.

Data normalization
Normalization of various criteria was performed in this way. For criteria with a better maximum and minimum values, it should be done as in section 2, but for some criteria which the regulations restrict them periodically, it was done in such a way that the middle of the interval is considered as the maximum value and equal to 1. The further we get from the middle of the interval, the lower the number. This approach is summarized as follows: Suppose the interval of the regulation is (a, b), and the obtained numbers are maximum and minimum for a specific criterion. The middle of the interval is defined as follows: And also, So, the VTM, FLOW, and VFA Criteria, were normalized as follows:   As seen in table 7, the alternatives 2, 8, 14, 15, and  18 compared with the first alternative (do nothing) have participated in all eight weight designs, and have priority. Also, the alternatives 3, 9, 10 and 13 with the first option (do nothing) were involved in seven weight design, and any of the four alternatives can be used rather than the first option (do nothing). Due to the high expense or mechanical properties, other alternatives have had the lowest frequencies in different weighting designs and, therefore, are out of the competition with other alternatives. The following table shows the results of the Concordance analysis.
On the other hand, as shown in

CONCLOSION
For the first time in this study, an alternative method was used to mix the design of asphalt mixture. For this purpose, the concordance analysis method was used to rank, and thus acquire the right asphalt mix.
The concordance analysis method with the possibility of involving different weighted combinations in the calculations reduces the effect of error in weighting. As mentioned in the previous section, alternatives 2, 8, 14, 15, and 18 compared with the first option (do nothing) have participated in all eight weight designs and are in priority.
On the other hand, alternative 2 has a lower average rating than the first alternative (do nothing), and alternatives 8 and 18 were ranked as third and fourth. Comparing the results from both methods, alternative two is better than the first alternative (do nothing), and alternatives 8 and 18 due to better technical properties were second and third. Table 9 lists the top 10 alternatives in order of priority based on the results of both methods.
The summary of the results is as follows: • If the Concordance method is used, alternatives 2, 8, and 18 recommended.
• If this method is not used, with evaluation and comparison of different alternatives in table 5, and with this assumption that we can apply all alternatives (for establishing a balance between this case and the case using concordance analysis), the alternative 18 would have better state than other alternatives, and alternative 20 would be the second one (because of high stability).
• It seems that concordance analysis proposes better alternatives than the conventional method. Therefore, the use of this method in the mixing design of the asphalt mix has resulted. Also, the use of this method can be used to obtain the optimal bitumen percentage.