Classification of the Monolithic Columns Produced in Troad and Mysia Region Ancient Granite Quarries in Northwestern Anatolia via Soft Decision-Making

Ay and Tolun [An Archaeometric Approach on the Distribution of Troadic Granite Columns in the Western Anatolian Coasts. Journal of Archaeology & Art, 156, 2017, 119-130 (In Turkish)] have analysed the distribution of the monolithic columns produced in the ancient granite quarries, located in Troad Region and Mysia Region in Northwestern Anatolia, by archaeometric analyses. Moreover, they have achieved some results by interpreting the prominent data obtained therein. In this study, we propose a novel soft decision-making method, i.e. Monolithic Columns Classification Method (MCCM), constructed via fuzzy parameterized fuzzy soft matrices (fpfs-matrices) and Prevalence Effect Method (PEM). MCCM provides an outcome by interpreting all the results of the analyses mentioned above. We then apply the method to the problem of monolithic columns classification. Finally, we discuss the need for further research.


Introduction
In the Roman Imperial Period, Troad Region and Mysia Region are two essential regions contained ancient granite quarries (Figure 1. a.) (Galetti et al., 1992;Williams-Thorpe and Thorpe, 1993;Williams-Thorpe and Henty, 2000) such as Koçali (Figure 1 (Ponti, 1995;Ay, 2017;Ay and Tolun, 2017a, b) are located in Ezine/Çanakkale, Kozak ancient granite quarry in Mysia Region  is located in Bergama/Izmir. However, there are not exist a sufficient number of an archaeological document about some subjects such as the exportation of the columns produced in these centres located in Troad and Mysia Region.
For this reason, to locate the source of a column considered in an ancient city, the method commonly used is to compare some archaeological samples taken from this city and some geological samples taken from the granite quarries by using mineralogical-petrographic and geochemical analyses (Williams-Thorpe and Thorpe, 1993;Williams-Thorpe and Henty, 2000;Potts, 2002;Williams-Thorpe, 2008;Ay, 2017;Ay and Tolun, 2017b).
The mineralogical-petrographic analyses are an examination of the samples in a microscopic environment using their thin sections. These analyses carry out to determine the types, quantities, sizes, and shapes of the minerals forming the rock types, main and secondary components of the samples (Galetti et al., 1992;Williams-Thorpe, 2008;Ay, 2017;Ay and Tolun, 22 2017b). The geochemical analyses perform in determining the type and number of major elements contained in the samples (Galetti et al., 1992;Potts, 2002;Williams-Thorpe, 2008).
Recently, Ay and Tolun have examined the distribution in Northwestern Anatolia of the monolithic columns produced in the ancient granite quarries, located in Troad Region and Mysia Region, by using archaeometric methods (Ay, 2017;Ay and Tolun, 2017b). For this aim, by using the qualitative mineralogicalpetrographic and geochemical analyses, they have compared the geological samples taken from Koçali-Akçakeçili ancient quarries in Troad Region and Kozak ancient quarry in Mysia Region with the archaeological samples taken from Smintheion (Smintheion 1, Smintheion 2), Pergamon Red Hall/Serapeion, Smyrna Agora (Smyrna Agora 1, Smyrna Agora 2), Tlos Stadium, Tlos Theatre, and Side Theatre.
Moreover, Ay and Tolun have divided the samples into two groups as ancient granite quarries and ancient city ( Figure 2 (De Vecchi et al., 2000) The concept of soft sets was introduced by Molodtsov (1999) to cope with uncertainty and have been applied to many areas from analysis to decision-making problems (Maji et al., 2001;Çağman et al., 2011a;Çağman and Deli, 2012;Deli and Çağman, 2015;Enginoğlu and Demiriz, 2015;Enginoğlu and Dönmez, 2015;Karaaslan, 2016;Şenel, 2016;Zorlutuna and Atmaca, 2016;Atmaca, 2017;Bera et al., 2017;Çıtak and Çağman, 2017;Şenel, 2017;Çıtak, 2018;Enginoğlu and Memiş, 23 2018a, b, c, d;Enginoğlu et al., 2018a, b, c, d;Gulistan et al., 2018;Mahmood et al., 2018;Şenel, 2018;Ullah et al., 2018). Recently, some soft decisionmaking methods constructed by fuzzy parameterized fuzzy soft matrices (fpfs-matrices) have enabled data processing in many problems containing uncertainty. Being one of these methods, Prevalence Effect Method (PEM) (Enginoğlu and Çağman, In Press) has been applied to a performance-based value assignment to some methods used in noise removal so that the methods can be ordered in terms of performance. We use this method for classification the monolithic columns mentioned in (Ay, 2017;Ay and Tolun, 2017b). The results show that Monolithic Columns Classification Method (MCCM) is successfully model the monolithic columns classification (MCC) problem. Here, fpfs-matrices have a row consisting of the significance degrees (membership degrees) of the parameters. These values are usually determined by consulting an expert.  (Ay and Tolun, 2017b) In this study, we have identified the values, that is, the weights of archaeometric and geochemical parameters, concerning the opinions mentioned in (Ay, 2017;Ay and Tolun, 2017b). Moreover, Ay and Tolun have considered of more effective the geochemical data than the archaeometric data. Therefore, we set a higher value to geochemical data than archaeometric data in the final decision step.
In Section 2 of the present study, we present the concept of fpfs-matrices and PEM. In Section 3, we give all the results of the qualitative mineralogical-petrographic and geochemical analyses provided in (Ay, 2017;Ay and Tolun, 2017b). In Section 4, we propose a new method, i.e. MCCM. In section 5, we apply MCCM to the MCC problem. Finally, we discuss the need for further research.

Preliminaries
In this section, we first present the concept of fuzzy soft matrices (fs-matrices) (Çağman and Enginoğlu, 2012). Throughout this paper, let be universal set, be a parameter set, ( ) be the set of all fuzzy sets over , and ∈ ( ). Here, a fuzzy set is denoted by { ( ) ∶ ∈ }.
Definition 2.1. (Çağman et al., 2011b) Let be a universal set, be a parameter set, and be a function from to ( ). Then, the set {( , ( )): ∈ } being the graphic of is called a fuzzy soft set (fs-set) parameterized via over (or briefly over ).
In the present paper, the set of all fs-sets over is denoted by ( ). In ( ), since the graphic of ( ℎ( )) and generate each other uniquely, the notations are interchangeable. Therefore, as long as it does not cause any confusion, we denote an fs-set ℎ( ) by .   Secondly, we present the concept of fpfs-matrices.
In the present paper, the set of all fpfs-sets over is denoted by ( ). In ( ), since the ℎ( ) and generate each other uniquely, the notations are interchangeable. Therefore, as long as it does not cause any confusion, we denote an fpfs-set ℎ( ) by .
Herein, the set of all fpfs-matrices parameterized via over is denoted by [ ].

The Qualitative Mineralogical-Petrographic and Geochemical Analyses Results
In this section, we give tables of the results of the qualitative mineralogical-petrographic and geochemical analyses provided in (Ay, 2017;Ay and Tolun, 2017b). The qualitative mineralogicalpetrographic analyses result from Koçali and Akçakeçili are the same, and the geochemical analyses results are close to each other. Since Koçali and Akçakeçili ancient quarries are the same structure, Ay and Tolun have compared eight samples with two sources: Bergama Kozak and Koçali-Akçakeçili in (Ay, 2017;Ay and Tolun, 2017b). Therefore, in the next section, we use the mean results from Koçali and Akçakeçili.

Research Method
In this section, we first present MCCM and which also uses the abilities of PEM (Enginoğlu and Çağman, In Press).  Ay and Tolun (2017b) consider more effective the geochemical data than archaeometric data. Therefore, we set 0.25 and 0.75 values to these data as weights, respectively, in the final decision stage.

Pre-process Steps for Archaeometric Data
Step  where ∅ denotes empty fuzzy set. Here, for brevity, the notation 3 has been used instead of 3 1 . Also, the elements such 3 0 and ( 4 , ∅) have not been shown in the sets containing them.
The fs-matrices corresponded to the fs-sets and , respectively, are as follows: Step 3.

Main Steps for Archaeometric Data
Step 1. Step 2.

Output Steps
Step

Conclusion
We, in this paper, proposed a novel method MCCM to model an MCC problem. We then applied MCCM to the data provided in (Ay, 2017;Ay and Tolun, 2017b). The results affirmed those obtained by archaeometric analyses. Since this method is the first, it could not be compared with other methods for now. Soon, however, if another soft decision-making method that differs from PEM is applied to this problem, then a comparison of these methods can be given.