# Preorder of Factors Affecting Oil Prices: Fuzzy PROMETHEE Approach

Authors: Lazim Abdullah; Nurhatika Harun; Shalela Mohd Mahali; Naeem Jan; Ewa Rak
Journal: Journal of Mathematics
 (2022)

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## Abstract

Oil is a highly demanded worldwide product as oil prices can have a major economic and social impact. There are many factors affecting oil prices, but little is known about which factor mostly influenced oil prices. To understand what caused the fluctuation of oil prices, the investigation leading up to the identification of the factors must be constructed. Thus, this paper attempts to explore the factors that might have contributed to oil prices and analyze them using the fuzzy preference ranking organization method for enrichment evaluation (PROMETHEE) method. In particular, the objective of this study is to rank the influential factors affecting oil prices with the consideration of a few criteria. The fuzzy PROMETHEE model comprises four criteria and five factors as the input information. A group of four experts are invited to provide an evaluation on scales of 1 to 5 of the importance of factors with respect to criteria. In computation, the evaluations are treated as triangular fuzzy numbers in which the uncertainty in data is dealt with amicably. This approach combines the advantages of transformation equation and nonarithmetic mean of the defuzzification method that transforms triangular fuzzy numbers into crisp numbers. The results indicate that “supply and demand” is the most influential factor in determining oil prices. Comparative analysis and future research directions are also discussed at the end of this paper.


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## Body

## 1. Introduction

One of the most critical issues in the economic development of a country is global oil prices. The price of oil indirectly affects the cost of living of most people, since it is so directly attached to petrol costs. This implies that if global oil prices rise, so the price of petrol is rising as well, and this will affect people’s daily life. Oil prices are inconceivably sensitive, changing rapidly considering news cycles, arrangement changes, and fluctuation in the world’s business, manufacturing industries, and economic development sectors. For example, as oil prices go up, all the petrochemical-related industries such as plastic, rubber, paints, dyes, textile solvent, and so on will be affected directly [1]. It has been suggested that global oil price stands out among the most important floating prices in the world [2]. Oil prices are considered floating prices as there are many factors that can be attached to them. Oil prices changes are most likely to lead to inflation, and it is more intense when oil prices are on the rise. As indicated by the concept of supply and demand, the oil price should increase with increasing demand or decreasing supply [3]. Therefore, the oil price is not a straightforward issue as there are many other factors that concurrently attached to it. There is no quick-fix solution when it comes to finding the most influential factors that affect oil prices. Moreover, it shows that oil prices have a nonlinearly impact on inflation risk and other economic variables [4]. The presence of nonlinearity in the relationship between oil prices and their impacts and also factors that affect oil prices might influence the way oil prices are investigated.


### 1.1. Related Works

To further explore the issue of fluctuation and nonlinearity of oil prices, Kaufmann et al. [5] conducted a study that used a statistical model to assess casual relationships between factors to influence oil prices and monthly oil prices. A linear relationship could be a reasonable estimation under ordinary conditions. However, under extreme events, oil price may shift the market balance among supply and demand towards different types of markets in which prices are much more sensitive that will contain a small sample bias. Dees et al. [6] also show that a nonlinear effect of oil producers’ country’s capacity and condition in future markets are the factors of the oil price increase. The researchers developed a model to estimate the real average price of US crude oil. They expect the regression coefficient associated to be negative. However, there are a few weaknesses in the regression models that should be pointed out. Results from the econometric regression-based analysis are highly sensitive. Small changes in the data set and the model designs can give a huge impact on the outcomes [7]. Zakaria et al. [8] discussed further the nonlinearity of oil prices. They measured the influence of the world oil prices on inflation rates in South Asian countries using the value at risk, cointegration, and nonlinearity analysis. The results tell that cointegration holds among oil prices and inflation. Nonlinear causality tests stipulated that oil price causes inflation. A similar approach was also employed by Oloko et al. [9] who concluded that the monetary policy of oil-importing countries appears to accommodate oil price shocks. In a correlational study, Wu and Wang [10] examined the relationship between oil prices and corporate investment, conditional on market conditions. They discovered that oil prices are negatively correlated with corporate investment expenditure. However, they also found out that if market conditions are favorable, then corporate investment expenditure increases as oil prices rise.As to depart from regression- and correlation-based analysis where data employed in studies are very much sensitive, Wang et al. [11] used a different approach. They used an extensive variety of generalized Maclaurin symmetric mean (MSM) aggregation operators that multicriteria decision-making (MCDM) issues were solved. The criteria in MCDM problems were enhanced by considering weights to reflect the importance of the criteria. They provide a practical example and conduct a comparison analysis between the proposed method and other existing methods in order to verify the proposed method and demonstrate its validity. They also successfully applied the proposed methods to a real-life MCDM problem. Very recently, in application research to biodiesel feedstock, Anwar [12] conducted research on defining the best possible biodiesel feedstock using various MCDM processes. Four MCDM processes, namely the preference ranking organization method for enrichment evaluation (PROMETHEE) graphical analysis for interactive assistance [13, 14], weighted sum method [15, 16], weighted product method [17], and technique for order preference by similarity to the ideal solution [18], were implemented for this investigation.However, there are two pertinent matters that are normally attached to most of the MCDM methods. The matters are about their parametrization and its uncertainty. In fact, the feasibility of MCDM is directly connected to the importance and quality of the chosen parameters. One of the MCDM methods related to parametrization is the PROMETHEE. Unlike most of the MCDM method such as the analytic hierarchy process (AHP) that handle preference directly from decision-makers’ opinions, the PROMETHEE method heavily relies on the chosen parameters, which are normally known as preference functions. Preference function and pairwise comparison of alternatives and differences between two alternatives are the main advantages of PROMETHE that the final output is the partial preorder (partial ranking) and total preorder (complete ranking). Because of these advantages, the PROMETHEE method has specific application in decision-making and generally is utilized far and wide in a wide assortment of decision scenarios, in fields such as business, administrative establishment, health care, transportation, and education. The second matter is about uncertainty. As to deal with uncertainty in information, the fuzzy PROMETHEE was proposed. There is a fuzzy concept to the PROMETHEE method when dealing with uncertain and subjective information. The fuzzy set theory and the ranking technique were incorporated into the fuzzy PROMETHEE method. According to Brans and Mareschal [19], the fundamental advantage of this technique is the ease of use. Linguistic evaluations and the thought of fuzziness are the two notions that are germane to decision-making problem.


### 1.2. Motivation

A search of literature found that most of the researchers tend to conduct research using PROMETHEE rather than fuzzy PROMETHEE because the fuzzy PROMETHEE method is associated with the theory of fuzzy sets [20]. Another reason could be associated with heavy calculation in fuzzy PROMETHEE besides no readily available software to assist computation. However, in recent years, there has been an increasing interest in applications of fuzzy PROMETHEE not only as a single model but also integrated with other fuzzy-based decision-making models. For example, fuzzy PROMETHEE has been integrated with AHP in group decision-making [21]. Recent research also unveiled several new extensions of fuzzy PROMETHEEE such as m-polar fuzzy PROMETHEE [21], bipolar fuzzy PROMETHEE [22], and q-rung orthopair fuzzy PROMETHEE [23]. In application to health sciences, Maisaini et al. [24] conducted a study to break down the most common therapeutic techniques for lung cancer. In their venture, the fuzzy PROMETHEE method was utilized to investigate the therapeutic techniques that are dependent on six factors. The results demonstrated that the surgery among other techniques showed an incredible act on lung cancer treatment depending on the criteria, importance, and weights chosen. In addition, fuzzy PROMETHEE also has been integrated.Fuzzy PROMETHEE also shows that one can easily adjust the method by including more criteria and change their significance and weight relying upon the application. Afful-Dadzie et al. [25] had proposed a two-phase framework of a decision-making methods and sensitivity analysis to show how aid recipient countries can be assessed to heighten transparency, reasonableness, esteem for cash, and manageability of such guide programs. Utilizing the OECD set for subjective criteria for assessing aid programs, a numerical model precharacterized by linguistic terms parameterized by triangular fuzzy numbers was given to assess aid programs. Fuzzy PROMETHEE was utilized in the first phase to assess and rank aid recipients pursued by a comparative analysis with other fuzzy-based decision-making methods. The result demonstrates a framework that can be utilized in assessing the aid effectiveness of recipient countries. Very recently, Ziemba [26] applied the fuzzy PROMETHEE combined with the Monte Carlo method and elements of the stochastic multicriteria acceptability analysis method. They applied their proposed method for assessing vehicles under uncertain conditions. The approach allowed us to identify almost unambiguously the electric vehicle that is likely to gain the highest acceptance. As a result of the conducted research, it was found that the approaches to uncertainty based on fuzzy sets, outranking relations, and stochastic analysis complement each other, allowing the decision-maker to conduct a wider analysis of the imprecision of the obtained solution.


### 1.3. Innovative Contribution

Turning now to the fluctuations and nonlinearity of oil prices where understanding the importance of causal factors that affect oil prices is the main aim of this paper. The fuzzy PROMETHEE is used as a selection tool with the end goal is to identify the most influential factors that affect oil prices. The contributions of this paper are fourfold: (1) we propose to use a transformation equation in the computational steps of fuzzy PROMETHEE where information that is given in triangular fuzzy numbers is transformed into left-right fuzzy numbers. This transformation is meant to ease the computational burden and also to avoid the loss of information during the implementation. (2) We propose to introduce a nonarithmetic mean of defuzzification where a constant is used as a multiplying factor to the mid-values of the left-right triangular fuzzy numbers. This would enhance the dominance of mid-values of left-right triangular fuzzy numbers. (3) We propose partial preorder and total preorder of factors that affect oil prices. (4) We propose a comparative analysis of net value flows to validate the results obtained using the proposed method. These four contributions are embedded in the proposed fuzzy PROMETHEE where the partial preorder and total preorder of factors can indicate the most influential factors that affect global oil prices.


### 1.4. Framework of the Paper

The rest of this paper is organized as follows. Section2 briefly revisits the definitions of linguistic variables, triangular fuzzy numbers, and their algebraic operations. Section 3 presents the methodology of the research, which includes the evaluation model. In Section 4, computational procedures and results for the case study of factors affecting oil prices are described. Section 5 provides the comparative results of the final ranking and net flows. Finally, Section 6 concludes.


## 1.1. Related Works

To further explore the issue of fluctuation and nonlinearity of oil prices, Kaufmann et al. [5] conducted a study that used a statistical model to assess casual relationships between factors to influence oil prices and monthly oil prices. A linear relationship could be a reasonable estimation under ordinary conditions. However, under extreme events, oil price may shift the market balance among supply and demand towards different types of markets in which prices are much more sensitive that will contain a small sample bias. Dees et al. [6] also show that a nonlinear effect of oil producers’ country’s capacity and condition in future markets are the factors of the oil price increase. The researchers developed a model to estimate the real average price of US crude oil. They expect the regression coefficient associated to be negative. However, there are a few weaknesses in the regression models that should be pointed out. Results from the econometric regression-based analysis are highly sensitive. Small changes in the data set and the model designs can give a huge impact on the outcomes [7]. Zakaria et al. [8] discussed further the nonlinearity of oil prices. They measured the influence of the world oil prices on inflation rates in South Asian countries using the value at risk, cointegration, and nonlinearity analysis. The results tell that cointegration holds among oil prices and inflation. Nonlinear causality tests stipulated that oil price causes inflation. A similar approach was also employed by Oloko et al. [9] who concluded that the monetary policy of oil-importing countries appears to accommodate oil price shocks. In a correlational study, Wu and Wang [10] examined the relationship between oil prices and corporate investment, conditional on market conditions. They discovered that oil prices are negatively correlated with corporate investment expenditure. However, they also found out that if market conditions are favorable, then corporate investment expenditure increases as oil prices rise.As to depart from regression- and correlation-based analysis where data employed in studies are very much sensitive, Wang et al. [11] used a different approach. They used an extensive variety of generalized Maclaurin symmetric mean (MSM) aggregation operators that multicriteria decision-making (MCDM) issues were solved. The criteria in MCDM problems were enhanced by considering weights to reflect the importance of the criteria. They provide a practical example and conduct a comparison analysis between the proposed method and other existing methods in order to verify the proposed method and demonstrate its validity. They also successfully applied the proposed methods to a real-life MCDM problem. Very recently, in application research to biodiesel feedstock, Anwar [12] conducted research on defining the best possible biodiesel feedstock using various MCDM processes. Four MCDM processes, namely the preference ranking organization method for enrichment evaluation (PROMETHEE) graphical analysis for interactive assistance [13, 14], weighted sum method [15, 16], weighted product method [17], and technique for order preference by similarity to the ideal solution [18], were implemented for this investigation.However, there are two pertinent matters that are normally attached to most of the MCDM methods. The matters are about their parametrization and its uncertainty. In fact, the feasibility of MCDM is directly connected to the importance and quality of the chosen parameters. One of the MCDM methods related to parametrization is the PROMETHEE. Unlike most of the MCDM method such as the analytic hierarchy process (AHP) that handle preference directly from decision-makers’ opinions, the PROMETHEE method heavily relies on the chosen parameters, which are normally known as preference functions. Preference function and pairwise comparison of alternatives and differences between two alternatives are the main advantages of PROMETHE that the final output is the partial preorder (partial ranking) and total preorder (complete ranking). Because of these advantages, the PROMETHEE method has specific application in decision-making and generally is utilized far and wide in a wide assortment of decision scenarios, in fields such as business, administrative establishment, health care, transportation, and education. The second matter is about uncertainty. As to deal with uncertainty in information, the fuzzy PROMETHEE was proposed. There is a fuzzy concept to the PROMETHEE method when dealing with uncertain and subjective information. The fuzzy set theory and the ranking technique were incorporated into the fuzzy PROMETHEE method. According to Brans and Mareschal [19], the fundamental advantage of this technique is the ease of use. Linguistic evaluations and the thought of fuzziness are the two notions that are germane to decision-making problem.


## 1.2. Motivation

A search of literature found that most of the researchers tend to conduct research using PROMETHEE rather than fuzzy PROMETHEE because the fuzzy PROMETHEE method is associated with the theory of fuzzy sets [20]. Another reason could be associated with heavy calculation in fuzzy PROMETHEE besides no readily available software to assist computation. However, in recent years, there has been an increasing interest in applications of fuzzy PROMETHEE not only as a single model but also integrated with other fuzzy-based decision-making models. For example, fuzzy PROMETHEE has been integrated with AHP in group decision-making [21]. Recent research also unveiled several new extensions of fuzzy PROMETHEEE such as m-polar fuzzy PROMETHEE [21], bipolar fuzzy PROMETHEE [22], and q-rung orthopair fuzzy PROMETHEE [23]. In application to health sciences, Maisaini et al. [24] conducted a study to break down the most common therapeutic techniques for lung cancer. In their venture, the fuzzy PROMETHEE method was utilized to investigate the therapeutic techniques that are dependent on six factors. The results demonstrated that the surgery among other techniques showed an incredible act on lung cancer treatment depending on the criteria, importance, and weights chosen. In addition, fuzzy PROMETHEE also has been integrated.Fuzzy PROMETHEE also shows that one can easily adjust the method by including more criteria and change their significance and weight relying upon the application. Afful-Dadzie et al. [25] had proposed a two-phase framework of a decision-making methods and sensitivity analysis to show how aid recipient countries can be assessed to heighten transparency, reasonableness, esteem for cash, and manageability of such guide programs. Utilizing the OECD set for subjective criteria for assessing aid programs, a numerical model precharacterized by linguistic terms parameterized by triangular fuzzy numbers was given to assess aid programs. Fuzzy PROMETHEE was utilized in the first phase to assess and rank aid recipients pursued by a comparative analysis with other fuzzy-based decision-making methods. The result demonstrates a framework that can be utilized in assessing the aid effectiveness of recipient countries. Very recently, Ziemba [26] applied the fuzzy PROMETHEE combined with the Monte Carlo method and elements of the stochastic multicriteria acceptability analysis method. They applied their proposed method for assessing vehicles under uncertain conditions. The approach allowed us to identify almost unambiguously the electric vehicle that is likely to gain the highest acceptance. As a result of the conducted research, it was found that the approaches to uncertainty based on fuzzy sets, outranking relations, and stochastic analysis complement each other, allowing the decision-maker to conduct a wider analysis of the imprecision of the obtained solution.


## 1.3. Innovative Contribution

Turning now to the fluctuations and nonlinearity of oil prices where understanding the importance of causal factors that affect oil prices is the main aim of this paper. The fuzzy PROMETHEE is used as a selection tool with the end goal is to identify the most influential factors that affect oil prices. The contributions of this paper are fourfold: (1) we propose to use a transformation equation in the computational steps of fuzzy PROMETHEE where information that is given in triangular fuzzy numbers is transformed into left-right fuzzy numbers. This transformation is meant to ease the computational burden and also to avoid the loss of information during the implementation. (2) We propose to introduce a nonarithmetic mean of defuzzification where a constant is used as a multiplying factor to the mid-values of the left-right triangular fuzzy numbers. This would enhance the dominance of mid-values of left-right triangular fuzzy numbers. (3) We propose partial preorder and total preorder of factors that affect oil prices. (4) We propose a comparative analysis of net value flows to validate the results obtained using the proposed method. These four contributions are embedded in the proposed fuzzy PROMETHEE where the partial preorder and total preorder of factors can indicate the most influential factors that affect global oil prices.


## 1.4. Framework of the Paper

The rest of this paper is organized as follows. Section2 briefly revisits the definitions of linguistic variables, triangular fuzzy numbers, and their algebraic operations. Section 3 presents the methodology of the research, which includes the evaluation model. In Section 4, computational procedures and results for the case study of factors affecting oil prices are described. Section 5 provides the comparative results of the final ranking and net flows. Finally, Section 6 concludes.


## 2. Preliminary

In a similar way to MCDM methods, there is a fuzzy expansion of the PROMETHEE method when dealing with uncertain and subjective information. Fuzzy PROMETHEE has similarly observed enhancements in several variations such as PROMETHEE I, II, III, IV, V, and VI. The fuzzy PROMETHEE method has been used in solving numerous decision-making problems where the task of ranking alternatives and concurrently considering fuzzy information, fuzzy preferences, and fuzzy weights is indeed very challenging. This is partly due to interval data that would increase the complexity of computation. According to Brans and Mareschal [27], the fundamental advantage of this technique is ease of use in implementation because most of the decision-making problems can be solved by incorporating linguistic evaluations and the thought of fuzziness. Later, we revisit several notions related to fuzzy MCDM that will be used in the subsequent text.


### 2.1. Linguistic Variable [28]

Linguistic variables are used for system input and output and are represented by words or sentences instead of the number investigated by Zadeh [29, 30]. In explaining this, take the example of the linguistic variable “temperature.” Let T be a set of linguistic variable “temperatures” where T = {very cold, cold, moderate, warm, very hot}. The members of set T are called linguistic terms where gradations of T are defined. These linguistic terms are normally translated into triangular fuzzy numbers.


### 2.2. Triangular Fuzzy Numbers [31]

The triangular fuzzy number,μAx, is defined by its lower limit a, its upper limit b, and the modal value m so that a<m<b. It is written in the form of function A(x) as follows:(1)Ax=0,x≤a,x−a/m−a,x∈a,m,b−x/b−m,x∈b,m,1,x≥0.In this function, intervala to m is increasing function, and interval m to b is decreasing function.Forx<a or x>b, x does not belong to the set.Forx≤a≤b, the membership degree is indicated by membership function between 0 and 1.These fuzzy numbers and their basic arithmetic operations are widely used in the fuzzy MCDM method.


### 2.3. Basic Operations of Triangular Fuzzy Numbers [32]

LetA=a,b,c and B=d,e,f be two triangular numbers, then the basic arithmetic operations for addition, subtraction, multiplication, division, and scalar multiplication are defined as follows:(2)A+B=a,b,c+d,e,f=a+d,b+e,c+f,A−B=a,b,c−d,e,f=a−f,b−e,c−d,A×B=a,b,c×d,e,f=ad,be,cf,A÷B=a,b,c÷d,e,f=af,be,cd,nA=na,nb,nc.


### 2.4. Preference Function [27]

LetPja,b=Fjdja,b is a preference function of PROMEHEE, where Pja,b signify the function of the difference between the evaluations of alternative a with respect to alternative b on each criterion into a degree ranging from 0 to 1. The smaller number of functions indicates the indifference of the decision-maker. Conversely, closer to 1 indicates greater preference. If Pja,b the usual criteria function, then the indifference only occurs when fa=fb. The function of fx is given as follows:(3)px=0,forx≤0,1,forx>0.The above notions are used directly during the implementation of computational procedures of the fuzzy PROMETHEE. The output of computational procedures are partial ranking and complete ranking of factors affecting oil prices.


## 2.1. Linguistic Variable [28]

Linguistic variables are used for system input and output and are represented by words or sentences instead of the number investigated by Zadeh [29, 30]. In explaining this, take the example of the linguistic variable “temperature.” Let T be a set of linguistic variable “temperatures” where T = {very cold, cold, moderate, warm, very hot}. The members of set T are called linguistic terms where gradations of T are defined. These linguistic terms are normally translated into triangular fuzzy numbers.


## 2.2. Triangular Fuzzy Numbers [31]

The triangular fuzzy number,μAx, is defined by its lower limit a, its upper limit b, and the modal value m so that a<m<b. It is written in the form of function A(x) as follows:(1)Ax=0,x≤a,x−a/m−a,x∈a,m,b−x/b−m,x∈b,m,1,x≥0.In this function, intervala to m is increasing function, and interval m to b is decreasing function.Forx<a or x>b, x does not belong to the set.Forx≤a≤b, the membership degree is indicated by membership function between 0 and 1.These fuzzy numbers and their basic arithmetic operations are widely used in the fuzzy MCDM method.


## 2.3. Basic Operations of Triangular Fuzzy Numbers [32]

LetA=a,b,c and B=d,e,f be two triangular numbers, then the basic arithmetic operations for addition, subtraction, multiplication, division, and scalar multiplication are defined as follows:(2)A+B=a,b,c+d,e,f=a+d,b+e,c+f,A−B=a,b,c−d,e,f=a−f,b−e,c−d,A×B=a,b,c×d,e,f=ad,be,cf,A÷B=a,b,c÷d,e,f=af,be,cd,nA=na,nb,nc.


## 2.4. Preference Function [27]

LetPja,b=Fjdja,b is a preference function of PROMEHEE, where Pja,b signify the function of the difference between the evaluations of alternative a with respect to alternative b on each criterion into a degree ranging from 0 to 1. The smaller number of functions indicates the indifference of the decision-maker. Conversely, closer to 1 indicates greater preference. If Pja,b the usual criteria function, then the indifference only occurs when fa=fb. The function of fx is given as follows:(3)px=0,forx≤0,1,forx>0.The above notions are used directly during the implementation of computational procedures of the fuzzy PROMETHEE. The output of computational procedures are partial ranking and complete ranking of factors affecting oil prices.


## 3. Methodology

The methodology is a process used to collect information and data, which regularly includes interviews, data collection, and computational models. The purpose of this section is to have a discussion of the methodology used so that the objective of this study is met. This section begins by describing the type of data collection via experts, factors, and criteria that affect oil prices, followed by computational procedures of the fuzzy PROMETHEE method.


### 3.1. Experts and Linguistic Variable

In this research, data are collected via personal communication with a group of three experts. All experts have vast experiences in the oil and gas industry in which their details are summarized in Table1.Table 1
Biographical data of experts.
ExpertsDesignationYears of experienceCompany addressExpert 1 (E1)Project manager4Nuri Cerah Sdn Bhd (NCSB), Kota Kinabalu, Sabah MalaysiaExpert 2 (E2)Field engineer4Petronas MLNG Sdn. Bhd., Bintulu, Sarawak MalaysiaExpert 3 (E3)Instrument engineer6Petronas MLNG Sdn. Bhd., Bintulu, Sarawak MalaysiaIn personal communication, experts are requested to provide a rating of five factors that affect oil prices with respect to four criteria of oil prices using the linguistic variable shown in Table2.Table 2
Linguistic variable “influence” and corresponding fuzzy numbers.
Rating of factorsTriangular fuzzy numbersPoor(0.00, 0.00, 0.25)Average(0.00, 0.25, 0.50)Influence(0.25, 0.50, 0.75)Very influence(0.50, 0.75, 1.00)Extremely influence(0.75, 1.00, 1.00)The experts are also requested to provide a level of importance of each criterion using the linguistic variable of “importance.” This linguistic variable and its respective triangular fuzzy numbers are shown in Table3.Table 3
Linguistic variable “importance” and corresponding fuzzy numbers.
Weights of criteriaTriangular fuzzy scalesUnimportant(0.00, 0.00, 0.25)Little important(0.00, 0.25, 0.50)Moderately important(0.25, 0.50, 0.75)Important(0.50, 0.75, 1.00)Very important(0.75, 1.00, 1.00)


### 3.2. Factors and Criteria

Five factorsa1,a2,a3,a4, and a5 that represent supply, demand, report (OPEC, target pricing, and investment), political event and crises, and access markets, respectively, are the factors chosen in this study. Descriptions of factors are given in Table 4.Table 4
Descriptions of factors.
FactorsDescriptionsa1 (supply)The Organization of Petroleum Exporting Countries (OPEC) has been colossal on the world’s exchanging floors, with its oil-producing member nations cooperating to decide prices by boosting or decreasing oil production. OPEC as of now controls 40% of the world’s oil production; in this way, they still have a significant impact on the oil market [33]. Therefore, supply is one of the factors affecting oil prices.a2 (demand)Solid monetary development and modern generation will in general lift the demand for oil as reflected in changing demand patterns by non-OECD countries, which have grown quickly as of late. As indicated by the concept of supply and demand, the oil price should increase with increasing demand or decreasing supply [3].a3 (report)Report on creation figures, spare capacity, target pricing, and investments can be a critical factor in the setting of oil prices [34].a4 (political events and crises)War, natural disasters, political change, and new government pioneers are on the whole factors impacting crude oil pricing [35].a5 (access markets)The futures market of oil plays a role in influencing the spot price of oil [36]. An oil futures contract is a binding agreement that gives the privilege to buy oil by the barrel at a predefined price on a predefined date later on. Under a prospects contract, both the purchaser and the merchant are committed to satisfying their side of the exchange on the predetermined date.Through personal communication, the experts (E1,E2,E3) are requested to rank and evaluate the factors that affect oil prices with respect to four criteria. In this study, criteria are defined as a benchmark in determining oil prices. Benchmark crude or marker crude is raw petroleum that fills in as a source of the reference price for purchasers and dealers of raw petroleum. Benchmarks are utilized in light of the fact that there are various assortments and grades of unrefined petroleum. The benchmark makes it simpler for dealers, investors, experts, and others to decide the prices of multitudes of grades of unrefined petroleum [1]. In this study, the criteria are O1 (West Texas Intermediate, WTI), O2 (Brent), O3 (OPEC Reference Basket), and O4 (Western Canadian Select, WCS). Descriptions of the criteria are presented in Table 5.Table 5
Description of criteria.
CriteriaDescriptionsO1 (West Texas intermediate, WTI)WTI, otherwise called Texas light sweet, is a grade of crude oil used as a benchmark in oil pricing [37]. This grade is portrayed as medium crude oil in light of its generally low thickness and sweet because of its low sulphur contentO2 (Brent)Brent is also an important benchmark for crude oil pricing [38]. Brent contains around 0.37% of sulphur classifying it as sweet crude, yet not as sweet as WTI. Brent is reasonable for the production of petrol and middle distillates. Historically, price differences between Brent and other index crudes have been founded on physical contrasts in crude oil specifications and short-term variations in supply and demand.O3 (OPEC Reference Basket)Besides WTI and Brent, the OPEC Reference Basket is also a benchmark for oil pricing [39]. The OPEC Basket is a weighted average of prices for petroleum blends produces by OPEC members. It is utilized as an imperative benchmark for crude oil prices. OPEC has frequently endeavored to keep the price of the OPEC basket between upper and lower limits, by increasing and decreasing production. This makes the measure essential for market analysts.O4 (Western Canadian Select, WCS)WCS is one of North America’s biggest overwhelming crude oil streams. It is an alternative to the Canadian oil and gas industry, and individual firms’ equity prices react to oil price fluctuations [40]. It is an overwhelming mixed crude oil, made for the most part of bitumen mixes with sweet synthetic and condensate diluents and 25 existing streams of both ordinary and flighty. WCS is the benchmark for emerging heavy, high TAN (acidic).Evaluation of the criteria and the factors become the input data or information for computation using the fuzzy PROMETHEE. The output of the computations are partial ranking and complete ranking. Information processing is implemented using the fuzzy PROMETHEE where information of the input data are given in triangular fuzzy numbers. Detailed computational procedures are given as follows.


### 3.3. Computational Procedures of Fuzzy PROMETHEE Method

In most fuzzy MCDM frameworks, identifying alternatives, criteria, linguistic variables, and experts are required. In the following, we propose the computational procedures for the generic fuzzy PROMETHEE, which later can be used in the case of oil prices. These computational procedures are adopted from Abdullah et al. [41] where the steps of PROMETHEE with crisp numbers are the main designs. Several innovations are made in the proposed fuzzy PROMETHEE compared to the PROMETHEE [41]. Apart from the substitution of crisp numbers with triangular fuzzy numbers, the proposed fuzzy PROMETHEE method also includes a transformation process where triangular fuzzy numbers are transformed into left-right triangular fuzzy numbers. This transformation is needed as left-right fuzzy numbers can easily fit with the Yager index that will be employed in the subsequent step. Instead of using the arithmetic mean in defuzzification, this proposed method introduced coefficient 3 to the mid-value of left-right triangular fuzzy numbers. This will propel the dominance of the mid-value of left-right triangular fuzzy numbers without ignoring their left width and right width. Another innovation is the sequence of obtaining partial preorder and total preorder. In the proposed method, the partial preorder is obtained just before the step of transformation and defuzzification where this decision is made based on the values of triangular fuzzy numbers of inflows and outflows. This sequence is contrasting with the regular PROMETHEE where partial preorder and total preorder happen one after the other. The flowchart of the proposed method is shown in Figure 1.Figure 1
Flowchart of fuzzy PROMETHEE.The proposed nine-step computational procedures are as follows:Step 1.
Define factors, criteria, and experts.
Suppose in a decision-making problem, we havem alternatives, k criteria, and n experts.Step 2.
Define linguistic variables and linguistic terms in triangular fuzzy numbers.
Define the criteria weight of each criterion corresponding to the linguistic term. Thewj represents the weight of criterion based on linguistic terms assigned by experts. The weight of the criterion is expressed in triangular fuzzy numbers.Step 3.
Aggregate the evaluation.
The weights of the criteria and the ratings of factors are aggregated using the basic proportion equation as follows:(4)w∼jk=w˜jw˜k0≤w˜jk≤1,w˜k=maxw˜jk,j=1,2,…,n.Step 4.
Construct preference function.
The preference functionP∼jm,n represents the experts’ preference between pairs of factors. The preference function P∼ja,b for a criterion can be defined as follows:(5)P∼ja,b=0,xaj∼≤x∼bj,x∼aj−x∼bj,xaj∼>x∼bj,wherej=1,2,…,k.Step 5.
Compute the multifactor preference index.
Multicriteria preference index is used to choose the rate in outranking relation.(6)π∼a,b=∑j=1kwj∼p∼ja,b∑j=1kwj∼,where wj∼ denotes the important weight of the factor.
Ifπ∼a,b≈0, it implies a weak preference of a over b.
Ifπ∼a,b≈1, it implies a strong preference of a over b.Step 6.
Compute the positive and negative outranking flows for partial preorder.
For PROMETHEE I (partial ranking), the ranking of factors is made by calculating the positive and negative outranking flows as follows:(7)Positive outranking:φ∼+m=1n−1∑m≠lπ∼m,l,∀m,l∈A,(8)Negative outranking:φ∼−m=1n−1∑m≠lπ∼l,m,∀m,l∈A,where m and n denote the number of factors and experts, respectively.Step 7.
Transform triangular fuzzy numbers.
The triangular numberL,M,R needs to transform into the form M,a,b(9)wherea=M−Landb=R−M.
It is necessary to represent fuzzy numbers in form of left-right.Step 8.
Defuzzify.
The defuzzification method used in this study is the Yager index method. The average of left-right and middle of triangular fuzzy numbers is the defuzzification value. The defuzzification equation is(10)M,a,b=3M−a+b3.Step 9.
Establish total preorder
The complete or full ranking of factors is established based on the value of net flows. Negative net flows subtracted from positive net flows will give net flows.(11)Net flow:φ∼m=φ+∼m−φ−∼m,∀m∈A.
The above computational procedures are implemented in the case of factors affecting oil prices. Detailed computations and results are given in the following section.


## 3.1. Experts and Linguistic Variable

In this research, data are collected via personal communication with a group of three experts. All experts have vast experiences in the oil and gas industry in which their details are summarized in Table1.Table 1
Biographical data of experts.
ExpertsDesignationYears of experienceCompany addressExpert 1 (E1)Project manager4Nuri Cerah Sdn Bhd (NCSB), Kota Kinabalu, Sabah MalaysiaExpert 2 (E2)Field engineer4Petronas MLNG Sdn. Bhd., Bintulu, Sarawak MalaysiaExpert 3 (E3)Instrument engineer6Petronas MLNG Sdn. Bhd., Bintulu, Sarawak MalaysiaIn personal communication, experts are requested to provide a rating of five factors that affect oil prices with respect to four criteria of oil prices using the linguistic variable shown in Table2.Table 2
Linguistic variable “influence” and corresponding fuzzy numbers.
Rating of factorsTriangular fuzzy numbersPoor(0.00, 0.00, 0.25)Average(0.00, 0.25, 0.50)Influence(0.25, 0.50, 0.75)Very influence(0.50, 0.75, 1.00)Extremely influence(0.75, 1.00, 1.00)The experts are also requested to provide a level of importance of each criterion using the linguistic variable of “importance.” This linguistic variable and its respective triangular fuzzy numbers are shown in Table3.Table 3
Linguistic variable “importance” and corresponding fuzzy numbers.
Weights of criteriaTriangular fuzzy scalesUnimportant(0.00, 0.00, 0.25)Little important(0.00, 0.25, 0.50)Moderately important(0.25, 0.50, 0.75)Important(0.50, 0.75, 1.00)Very important(0.75, 1.00, 1.00)


## 3.2. Factors and Criteria

Five factorsa1,a2,a3,a4, and a5 that represent supply, demand, report (OPEC, target pricing, and investment), political event and crises, and access markets, respectively, are the factors chosen in this study. Descriptions of factors are given in Table 4.Table 4
Descriptions of factors.
FactorsDescriptionsa1 (supply)The Organization of Petroleum Exporting Countries (OPEC) has been colossal on the world’s exchanging floors, with its oil-producing member nations cooperating to decide prices by boosting or decreasing oil production. OPEC as of now controls 40% of the world’s oil production; in this way, they still have a significant impact on the oil market [33]. Therefore, supply is one of the factors affecting oil prices.a2 (demand)Solid monetary development and modern generation will in general lift the demand for oil as reflected in changing demand patterns by non-OECD countries, which have grown quickly as of late. As indicated by the concept of supply and demand, the oil price should increase with increasing demand or decreasing supply [3].a3 (report)Report on creation figures, spare capacity, target pricing, and investments can be a critical factor in the setting of oil prices [34].a4 (political events and crises)War, natural disasters, political change, and new government pioneers are on the whole factors impacting crude oil pricing [35].a5 (access markets)The futures market of oil plays a role in influencing the spot price of oil [36]. An oil futures contract is a binding agreement that gives the privilege to buy oil by the barrel at a predefined price on a predefined date later on. Under a prospects contract, both the purchaser and the merchant are committed to satisfying their side of the exchange on the predetermined date.Through personal communication, the experts (E1,E2,E3) are requested to rank and evaluate the factors that affect oil prices with respect to four criteria. In this study, criteria are defined as a benchmark in determining oil prices. Benchmark crude or marker crude is raw petroleum that fills in as a source of the reference price for purchasers and dealers of raw petroleum. Benchmarks are utilized in light of the fact that there are various assortments and grades of unrefined petroleum. The benchmark makes it simpler for dealers, investors, experts, and others to decide the prices of multitudes of grades of unrefined petroleum [1]. In this study, the criteria are O1 (West Texas Intermediate, WTI), O2 (Brent), O3 (OPEC Reference Basket), and O4 (Western Canadian Select, WCS). Descriptions of the criteria are presented in Table 5.Table 5
Description of criteria.
CriteriaDescriptionsO1 (West Texas intermediate, WTI)WTI, otherwise called Texas light sweet, is a grade of crude oil used as a benchmark in oil pricing [37]. This grade is portrayed as medium crude oil in light of its generally low thickness and sweet because of its low sulphur contentO2 (Brent)Brent is also an important benchmark for crude oil pricing [38]. Brent contains around 0.37% of sulphur classifying it as sweet crude, yet not as sweet as WTI. Brent is reasonable for the production of petrol and middle distillates. Historically, price differences between Brent and other index crudes have been founded on physical contrasts in crude oil specifications and short-term variations in supply and demand.O3 (OPEC Reference Basket)Besides WTI and Brent, the OPEC Reference Basket is also a benchmark for oil pricing [39]. The OPEC Basket is a weighted average of prices for petroleum blends produces by OPEC members. It is utilized as an imperative benchmark for crude oil prices. OPEC has frequently endeavored to keep the price of the OPEC basket between upper and lower limits, by increasing and decreasing production. This makes the measure essential for market analysts.O4 (Western Canadian Select, WCS)WCS is one of North America’s biggest overwhelming crude oil streams. It is an alternative to the Canadian oil and gas industry, and individual firms’ equity prices react to oil price fluctuations [40]. It is an overwhelming mixed crude oil, made for the most part of bitumen mixes with sweet synthetic and condensate diluents and 25 existing streams of both ordinary and flighty. WCS is the benchmark for emerging heavy, high TAN (acidic).Evaluation of the criteria and the factors become the input data or information for computation using the fuzzy PROMETHEE. The output of the computations are partial ranking and complete ranking. Information processing is implemented using the fuzzy PROMETHEE where information of the input data are given in triangular fuzzy numbers. Detailed computational procedures are given as follows.


## 3.3. Computational Procedures of Fuzzy PROMETHEE Method

In most fuzzy MCDM frameworks, identifying alternatives, criteria, linguistic variables, and experts are required. In the following, we propose the computational procedures for the generic fuzzy PROMETHEE, which later can be used in the case of oil prices. These computational procedures are adopted from Abdullah et al. [41] where the steps of PROMETHEE with crisp numbers are the main designs. Several innovations are made in the proposed fuzzy PROMETHEE compared to the PROMETHEE [41]. Apart from the substitution of crisp numbers with triangular fuzzy numbers, the proposed fuzzy PROMETHEE method also includes a transformation process where triangular fuzzy numbers are transformed into left-right triangular fuzzy numbers. This transformation is needed as left-right fuzzy numbers can easily fit with the Yager index that will be employed in the subsequent step. Instead of using the arithmetic mean in defuzzification, this proposed method introduced coefficient 3 to the mid-value of left-right triangular fuzzy numbers. This will propel the dominance of the mid-value of left-right triangular fuzzy numbers without ignoring their left width and right width. Another innovation is the sequence of obtaining partial preorder and total preorder. In the proposed method, the partial preorder is obtained just before the step of transformation and defuzzification where this decision is made based on the values of triangular fuzzy numbers of inflows and outflows. This sequence is contrasting with the regular PROMETHEE where partial preorder and total preorder happen one after the other. The flowchart of the proposed method is shown in Figure 1.Figure 1
Flowchart of fuzzy PROMETHEE.The proposed nine-step computational procedures are as follows:Step 1.
Define factors, criteria, and experts.
Suppose in a decision-making problem, we havem alternatives, k criteria, and n experts.Step 2.
Define linguistic variables and linguistic terms in triangular fuzzy numbers.
Define the criteria weight of each criterion corresponding to the linguistic term. Thewj represents the weight of criterion based on linguistic terms assigned by experts. The weight of the criterion is expressed in triangular fuzzy numbers.Step 3.
Aggregate the evaluation.
The weights of the criteria and the ratings of factors are aggregated using the basic proportion equation as follows:(4)w∼jk=w˜jw˜k0≤w˜jk≤1,w˜k=maxw˜jk,j=1,2,…,n.Step 4.
Construct preference function.
The preference functionP∼jm,n represents the experts’ preference between pairs of factors. The preference function P∼ja,b for a criterion can be defined as follows:(5)P∼ja,b=0,xaj∼≤x∼bj,x∼aj−x∼bj,xaj∼>x∼bj,wherej=1,2,…,k.Step 5.
Compute the multifactor preference index.
Multicriteria preference index is used to choose the rate in outranking relation.(6)π∼a,b=∑j=1kwj∼p∼ja,b∑j=1kwj∼,where wj∼ denotes the important weight of the factor.
Ifπ∼a,b≈0, it implies a weak preference of a over b.
Ifπ∼a,b≈1, it implies a strong preference of a over b.Step 6.
Compute the positive and negative outranking flows for partial preorder.
For PROMETHEE I (partial ranking), the ranking of factors is made by calculating the positive and negative outranking flows as follows:(7)Positive outranking:φ∼+m=1n−1∑m≠lπ∼m,l,∀m,l∈A,(8)Negative outranking:φ∼−m=1n−1∑m≠lπ∼l,m,∀m,l∈A,where m and n denote the number of factors and experts, respectively.Step 7.
Transform triangular fuzzy numbers.
The triangular numberL,M,R needs to transform into the form M,a,b(9)wherea=M−Landb=R−M.
It is necessary to represent fuzzy numbers in form of left-right.Step 8.
Defuzzify.
The defuzzification method used in this study is the Yager index method. The average of left-right and middle of triangular fuzzy numbers is the defuzzification value. The defuzzification equation is(10)M,a,b=3M−a+b3.Step 9.
Establish total preorder
The complete or full ranking of factors is established based on the value of net flows. Negative net flows subtracted from positive net flows will give net flows.(11)Net flow:φ∼m=φ+∼m−φ−∼m,∀m∈A.
The above computational procedures are implemented in the case of factors affecting oil prices. Detailed computations and results are given in the following section.


## 4. Computations and Results

Information about the evaluation becomes the input data for implementing the computation. All this information is given in triangular fuzzy numbers, and the execution of the computation is made according to the proposed computational procedures.Evaluation of criteria given by experts is summarized in Table6.Table 6
Fuzzy weight of criteria.
Oil benchmarkE1E2E3O1(0.50, 0.75, 1.00)(0.50, 0.75, 1.00)(0.25, 0.50, 0.75)O2(0.25, 0.50, 0.75)(0.75, 1.00, 1.00)(0.50, 0.75, 1.00)O3(0.25, 0.50, 0.75)(0.25, 0.50, 0.75)(0.75, 1.00, 1.00)O4(0.50, 0.75, 1.00)(0.75, 1.00, 1.00)(0.25, 0.50, 0.75)Aggregation of weight of criterion from experts is computed using equation (4).(12)w∼1=130.50,0.75,1.00⊕0.50,0.75,1.00⊕0.25,0.50,0.75=0.42,0.67,0.92,w∼2=130.25,0.50,0.75⊕0.75,1.00,1.00⊕0.50,0.75,1.00=0.5,0.75,0.92,w∼3=130.25,0.50,0.75⊕0.25,0.50,0.75⊕0.75,1.00,1.00=0.42,0.67,0.83,w∼4=130.50,0.75,1.00⊕0.75,1.00,1.00⊕0.25,0.50,0.75=0.5,0.75,0.92.Equation (12) shows the weights represent aggregated weights for oil benchmarks (criteria) Oii=1,2,3,4, respectively.Table7 shows the ratings of the factors with respect to criteria.Table 7
Fuzzy ratings of the factors.
CriteriaFactorsExpertsE1E2E3O1a1(0.25, 0.50, 0.75)(0.75, 1.00, 1.00)(0.25, 0.50, 0.75)a2(0.25, 0.50, 0.75)(0.75, 1.00, 1.00)(0.25, 0.50, 0.75)a3(0.25, 0.50, 0.75)(0.50, 0.75, 1.00)(0.00, 0.25, 0.50)a4(0.25, 0.50, 0.75)(0.50, 0.75, 1.00)(0.25, 0.50, 0.75)a5(0.25, 0.50, 0.75)(0.50, 0.75, 1.00)(0.00, 0.25, 0.50)O2a1(0.25, 0.50, 0.75)(0.75, 1.00, 1.00)(0.25, 0.50, 0.75)a2(0.25, 0.50, 0.75)(0.75, 1.00, 1.00)(0.50, 0.75, 1.00)a3(0.25, 0.50, 0.75)(0.50, 0.75, 1.00)(0.25, 0.50, 0.75)a4(0.00, 0.25, 0.50)(0.50, 0.75, 1.00)(0.25, 0.50, 0.75)a5(0.25, 0.50, 0.75)(0.50, 0.75, 1.00)(0.25, 0.50, 0.75)O3a1(0.00, 0.25, 0.50)(0.00, 0.25, 0.50)(0.50, 0.75, 1.00)a2(0.25, 0.50, 0.75)(0.00, 0.25, 0.50)(0.50, 0.75, 1.00)a3(0.50, 0.75, 1.00)(0.00, 0.25, 0.50)(0.25, 0.50, 0.75)a4(0.25, 0.50, 0.75)(0.50, 0.75, 1.00)(0.25, 0.50, 0.75)a5(0.00, 0.25, 0.50)(0.25, 0.50, 0.75)(0.25, 0.50, 0.75)O4a1(0.00, 0.25, 0.50)(0.00, 0.25, 0.50)(0.25, 0.50, 0.75)a2(0.25, 0.50, 0.75)(0.25, 0.50, 0.75)(0.50, 0.75, 1.00)a3(0.25, 0.50, 0.75)(0.00, 0.25, 0.50)(0.25, 0.50, 0.75)a4(0.25, 0.50, 0.75)(0.50, 0.75, 1.00)(0.25, 0.50, 0.75)a5(0.25, 0.50, 0.75)(0.25, 0.50, 0.75)(0.00, 0.25, 0.50)The above ratings are aggregated using equation (4) to obtain aggregated rating for each criterion. For example,(13)x∼11=130.25,0.50,0.75⊕0.75,1.00,1.00⊕0.25,0.50,0.75=0.42,0.67,0.83,x∼12=130.25,0.50,0.75⊕0.75,1.00,1.00⊕0.25,0.50,0.75=0.42,0.67,0.83,x∼13=130.25,0.50,0.75⊕0.50,0.75,1.00⊕0.00,0.25,0.50=0.25,0.5,0.75,x∼13=130.25,0.50,0.75⊕0.50,0.75,1.00⊕0.25,0.50,0.75=0.33,0.58,0.83,x∼13=130.25,0.50,0.75⊕0.50,0.75,1.00⊕0.00,0.25,0.50=0.33,0.58,0.83,Equation (13) shows part of the aggregated ratings for criteria. These results are summarized in Table 8.Table 8
Aggregated fuzzy rating of a factor with respect to criteria.
Oil benchmarkFactorsAggregated ratingO1a1(0.42, 0.67, 0.83)a2(0.42, 0.67, 0.83)a3(0.25, 0.50, 0.75)a4(0.33, 0.58, 0.83)a5(0.25, 0.50, 0.75)O2a1(0.42, 0.67, 0.83)a2(0.50, 0.75, 0.92)a3(0.33, 0.58, 0.83)a4(0.25, 0.50, 0.75)a5(0.33, 0.58, 0.83)O3a1(0.17, 0.42, 0.67)a2(0.25, 0.50, 0.75)a3(0.25, 0.50, 0.75)a4(0.33, 0.58, 0.83)a5(0.17, 0.42, 0.67)O4a1(0.08, 0.33, 0.58)a2(0.33, 0.58, 0.83)a3(0.17, 0.42, 0.67)a4(0.33, 0.58, 0.83)a5(0.17, 0.42, 0.67)This study uses the usual function (type 1) as a preference function (see equation (5)).For example,(14)p∼1a1,a2=0.42−0.83,0.67−0.67,0.83−0.42=−0.42,0,0.42,p∼1a1,a3=0.42−0.75,0.67−0.50,0.83−0.25=−0.33,0.17,0.58,p∼1a1,a4=0.42−0.75,0.67−0.58,0.83−0.33=−0.42,0.08,0.50,p∼1a1,a5=0.42−0.75,0.67−0.50,0.83−0.25=−0.33,0.17,0.58,p∼1a3,a1=0.25−0.83,0.50−0.67,0.75−0.42=−0.58,−0.17,0.33.=0,0,0,The initial part of the preference functions is shown in equation (14). Tables 9–13 show the fuzzy preference function of a1,a2,a3,a4,a5, respectively.Table 9
Fuzzy preference function,Pj˜a1,ai.
Oil benchmark,OiP∼ja1,a2P∼ja1,a3P∼ja1,a4P∼ja1,a5O1(−0.42, 0, 0.42)(−0.33, 0.17, 0.58)(−0.42, 0.08, 0.50)(−0.33, 0.17, 0.58)O2(0, 0, 0)(−0.42, 0.08, 0.50)(−0.33, 0.17, 0.58)(−0.42, 0.08, 0.50)O3(0, 0, 0)(0, 0, 0)(0, 0, 0)(−0.50, 0, 0.50)O4(0, 0, 0)(0, 0, 0)(0, 0, 0)(0, 0, 0)Table 10
Fuzzy preference function,Pj˜a2,ai.
Oil benchmark,OiPj˜a2,a1Pj˜a2,a3Pj˜a2,a4Pj˜a2,a5O1(−0.42, 0, 0.42)(−0.33, 0.17, 0.58)(−0.42, 0.08, 0.50)(−0.33, 0.17, 0.58)O2(−0.33, 0.08, 0.50)(−0.33, 0.17, 0.58)(−0.25, 0.25, 0.67)(−0.33, 0.17, 0.58)O3(−0.42, 0.08, 0.58)(−0.50, 0, 0.50)(0, 0, 0)(−0.42, 0.08, 0.58)O4(−0.25, 0.25, 0.75)(−0.33, 0.17, 0.67)(−0.50, 0, 0.50)(−0.33, 0.17, 0.67)Table 11
Fuzzy preference function,Pj˜a3,ai.
Oil benchmark,OiPj˜a3,a1Pj˜a3,a2Pj˜a3,a4Pj˜a3,a5O1(0, 0, 0)(0, 0, 0)(0, 0, 0)(−0.50, 0, 0.50)O2(0, 0, 0)(0, 0, 0)(−0.42, 0.08, 0.58)(−0.50, 0, 0.50)O3(−0.42, 0.08, 0.58)(−0.50, 0, 0.50)(0, 0, 0)(−0.43, 0.08, 0.58)O4(−0.42, 0.08, 0.58)(0, 0, 0)(0, 0, 0)(−0.50, 0, 0.50)Table 12
Fuzzy preference function,Pj˜a4,ai.
Oil benchmark,OiPj˜a4,a1Pj˜a4,a2Pj˜a4,a3Pj˜a4,a5O1(0, 0, 0)(0, 0, 0)(−0.42, 0.08, 0.58)(−0.42, 0.08, 0.58)O2(0, 0, 0)(0, 0, 0)(0, 0, 0)(0, 0, 0)O3(−0.33, 0.17, 0.67)(−0.42, 0.08, 0.58)(−0.42, 0.08, 0.58)(−0.33, 0.17, 0.67)O4(−0.25, 0.25, 0.75)(−0.50, 0, 0.50)(−0.33, 0.17, 0.67)(−0.33, 0.17, 0.67)Table 13
Fuzzy preference function,Pj˜a5,ai.
Oil benchmark,OiPj˜a5,a1Pj˜a5,a2Pj˜a5,a3Pj˜a5,a4O1(0, 0, 0)(0, 0, 0)(−0.50, 0, 0.50)(0, 0, 0)O2(0, 0, 0)(0, 0, 0)(−0.50, 0, 0.50)(−0.42, 0.08, 0.58)O3(−0.50, 0, 0.50)(0, 0, 0)(0, 0, 0)(0, 0, 0)O4(−0.42, 0.08, 0.58)(0, 0, 0)(−0.50, 0, 0.50)(0, 0, 0)The next step is calculating the multifactor preference index using equation (6).For example,(15)πa1,a2=∑−0.42,0,0.42∗0.42,0.67,0.920.42,0.67,0.92,=−0.1764,0,−0.38640.42,0.67,0.92.=0.42,0,0.42,πa1,a3=∑−0.33,0.17,0.58∗0.42,0.67,0.92,0.42,0.08,0.50∗0.5,0.75,0.920.42,0.67,0.92+0.5,0.75,0.92,=0.072,0.174,0.9940.92,1.42,1.84=0.078261,0.122535,0.540217,πa1,a4=∑−0.42,0.08,0.5∗0.42,0.67,0.92,−0.33,0.17,0.58∗0.5,0.75,0.920.42,0.67,0.92+0.5,0.75,0.92,=−0.3414,0.1811,0.99360.92,1.42,1.84=−0.371087,0.127535,0.54,πa1,a5=∑−0.33,0.17,0.58∗0.42,0,67,0.92,0.42,0.08,0.50∗0.5,0.75,0.92,−0.50,0,0.50∗0.42,0.67,0.380.42,0,67,0.92+0.5,0.75,0.92+0.42,0.67,0.38,−0.138,0.174,1.1841.34,2.09,2.22=−0.102985,0.083254,0.533333.Parts of the preference index values are shown in equation (15). It is summarized in Table 14.Table 14
Preference index value.
Factorsa1a2a3a4a5a1(0.42, 0, 0.42)(0.078261, 0.122535, 0.540217)(−0.371087, 0.127535, 0.54)(−0.102985, 0.083254, 0.533333)a2(−0.349348, 0.106021, 0.559490)(−0.368804, 0.200489, 0.596689)(−0.38831, 0.111106, 0.556667)(−0.350543, 0.148768, 0.606369)a3(−0.42, 0.08, 0.58)(−0.5, 0, 0.5)(0.42, 0.08, 0.58)(−0.484022, 0.018873, 0.509682)a4(−0.286522, 0.212254, 0.726615)(−0.463478, 0.037746, 0.523384)(−0.386418, 0.112297, 0.617297)(−0.358209, 0.141148, 0.632703)a5(−0.456522, 0.042254, 0.556615)(0, 0, 0)(−0.5, 0, 0.5)(−0.42, 0.08, 0.58)In the following, the positive outflows and negative outflows of each factor (PROMETHEE I partial ranking) can be determined using equations (7) and (8). Table 15 shows the positive outflows and negative outflows of factors.Table 15
Fuzzy PROMETHEE I flow.
Factorsϕ+aiϕ−aia1(0.024189, 0.333324, 2.03355)(−1.512392, 0.440529, 2.42272)a2(−1.457005, 0.566384, 2.319215)(−0.543478, 0.037746, 1.443384)a3(−0.984022, 0.178873, 2.169682)(−1.176961, 0.435321, 2.254203)a4(−1.494627, 0.503445, 2.49999)(−0.759397, 0.398641, 2.256667)a5(−1.376522, 0.122254, 1.636615)(−1.2957320, 0.392043, 2.282087)In fuzzy PROMETHEE I, the partial preorders of factors ofa1PIb2 are a1PIa2, a1PIa3,a1PIa4,a1PIa5,a2PIa3,a2PIa5,a3PIa5,a4PIa2,a4PIa3, and a4PIa5. The partial preorders of factors are shown in Figure 2.Figure 2
Fuzzy PROMETHEE I partial preorder value outranking graph.It can be seen that the dominant factor isa1 as this factor influences four other factors. However, PROMETHEE I could not tell the complete outranking of factors. The next analysis is PROMETHEE II where complete outranking can be obtained.Normalization equation (see equation (9)) is used to transform the triangular fuzzy numbers into left-right fuzzy numbers.For example,(16)ϕ+ai:a1:0.0242,0.3333,2.0336⟶0.3333,0.3333−0.0242,2.0336−0.333=0.3333,0.3091,1.7003,a2:−1.4570,0.5664,2.31922⟶0.56634,0.5664−−1.4570,2.31922−0.5664=0.5664,2.0234,17528.Two examples of left-right fuzzy numbers are shown in equation (16). Table 16 summarizes all flows written in form of left-right fuzzy numbers.Table 16
Fuzzy PROMETHEE flow in the form ofM,a,b.
FactorsLeaving flow,ϕ+aiEntering flow,φ−Aia1(0.3333, 0.3091, 1.7002)(0.4405, 1.9529, 1.9822)a2(0.5664, 2.0234, 1.7528)(0.0377, 0.5812, 1.4056)a3(0.0179, 1.0019, 2.1518)(0.4353, 1.6123, 1.8189)a4(0.5033, 1.998, 1.9966)(0.3986, 1.158, 1.858)a5(0.1223, 1.4988, 1.5144)(0.392, 1.6878, 1.89)The Yager index method is used as the defuzzification method in this study (see equation (10)). Table 17 presents the defuzzified value of flows.Table 17
Defuzzified fuzzy PROMETHEE I flow.
FactorsPositive outflows,ϕ+AiNegative outflows,φ−Aia10.7970.4503a20.47620.3126a30.40120.5042a40.50290.632a50.12740.4595Net flows (PROMETHEE II) of the factors are calculated using equation (11). The complete ranking is obtained by arranging net flows in descending order. Table 18 presents the net flow and complete preference ranking.Table 18
Net flow and ranking of factors.
FactorsNet flowRankinga10.346741a20.163652a3−0.1033a4−0.12914a5−0.3325The partial preorder of factors as shown in Figure2 dictates us to draw a complete preorder of factors in fuzzy PROMETHEE II. The complete preorder is(17)a1PIIb2,a2PIIb3,a3PIIb4anda4PIIb5.Equation (17) shows the general complete preorder of five factors. Figure 3 presents the complete order of factors.Figure 3
Complete preorder value outranking graph.As illustrated in Figure3, the most influential factor affecting oil prices is a1 (supply) followed by a2 (demand). The least influential factor affecting oil prices is a5 (access markets).The results are undergone further discussion and comparative analysis. The following section discusses the results from a comparative study perspective.


## 5. Comparative Analysis

It is known that the fuzzy PROMETHEE method is an extension of PROMETHEE in which the evaluation scales in PROMETHEE are substituted with triangular fuzzy numbers. The uses of fuzzy numbers are meant to capture uncertainty in the factors affecting oil prices. Therefore, in this section, a comparative analysis is presented in which the results obtained from fuzzy PROMETHEE are compared to the results obtained from PROMEHEE. The computational implementation of the PROMETHEE method is slightly straightforward as the computation only deals with crisp numbers. All computational procedures in Section3 are iterated with some modifications in the types of evaluation scales used.This comparative analysis is presented from two viewpoints. The first outlook is by seeing the respective values of net flows. Figure4 shows the comparison of fuzzy PROMETHEE and PROMETHEE methods based on their values of net flows.Figure 4
Net flow value between PROMETHEE and fuzzy PROMETHEE.It can be seen that all factors are consistently shared similar patterns for both methods. However, for the factor of political event and crises, fuzzy PROMEHEE provides negative net flows. On the other hand, the same factor provides positive net flows when the PROMETHEE is implemented. It shows that the fuzzy numbers used in the evaluation can capture wide-ranging values compared to crisp numbers in PROMETHEE.The second viewpoint in this comparative analysis is their final preference ranking. The comparative results of final ranking preference are summarized in Table19.Table 19
Ranking of Factors under PROMETHEE and fuzzy PROMETHEE.
FactorsPROMETHEE (crisp scales)Fuzzy PROMETHEE (triangular fuzzy number scales)a131a212a343a424a555It is apparent that the factora2 (demand) is ranked in the first position when the crisp scales are employed in evaluation. In contrast, the factor a1 (supply) is ranked in the first position when triangular fuzzy number scales are employed in evaluation. However, there is a similarity between the two methods. The factor a5 (access market) shares the weakest performance for both methods.The performance of these two methods is further analyzed using a statistical test. From the above analysis, the ranking of factors is obtained from the values of net flows. These values are seemed dissimilar due to the different methods used. However, these values can be hypothesized as having some extent of correlation between the methods. The hypotheses can be written as follows:(i)
H0:ρ=0(ii)
H1:ρ≠0 where is ρ an estimated correlation coefficientTest statistict=r/1−r2/n−2, with df = n – 2.Correlation coefficientr=∑XY−∑X∑Y/n/∑X2−∑X2/n∑Y2−∑Y2/n, where X and Y are net flows of fuzzy PROMETHEE and PROMETHEE, respectively, and n is the number of factors.From the calculation of the correlation coefficient, it is found thatr = 0.4775 and t = 0.9413. With a significant level α=0.05 and t is less than 2.306, this test indicates that there is no correlation between the net flows of PROMETHEE and fuzzy PROMETHEE. From this analysis, we can conclude that these two methods are independent because of the difference in the types of numbers used as input data. It is good to note that fuzzy PROMETHEE uses triangular fuzzy numbers as input data. In contrast, crisp numbers are the input data used in PROMETHEE. This comparison also tells us that the input values are very important in determining the outcome of ranking. In other words, the ranking results obtained from the fuzzy PROMETHEE are sensitive to the input data. Different types of fuzzy numbers used could give different ranking results.In addition to the comparative analysis between two PROMETHEE-based methods, it is also good to see the comparative analysis between PROMETHEE-based methods with other preference methods. Ziemba [42], for example, conducted a comparative study between fuzzy TOPSIS, fuzzy SAW, and NEAT fuzzy PROMETHEE II in the selection of electric vehicles. The selection problem was solved by these preference methods. The decision problem defined by the preference models gave three different solutions in the form of rankings. The rankings show that the best vehicle was awarded to A11-Volkswagen ID.3 Pro S, regardless of the preference method used. In another comparative analysis, ranking of contractors was conducted using q-ROF TOPSIS, q-ROF VIKOR, and q-ROF PROMETHEE methods [23]. They conclude that all these preference methods are close to the results of their proposed q-ROF PROMETHEE method where the optimal solution is the same for all methods


## 6. Conclusions

There are many factors that can contribute to global oil prices, and most of these factors are characterized by subjectivity, uncertainty, and variability. Investigation of factors affecting oil prices is therefore must be seen as a multifactor decision analysis, and more importantly, the concept of fuzziness and uncertainty must be embedded. In this paper, factors affecting oil prices have been investigated using the approach of fuzzy PROMETHEE where triangular fuzzy numbers are employed in expert evaluation. The fuzzy PROMETHEE used in this study is characterized by partial ranking and also complete ranking. The final preference ranking is indeed a modification to the partial ranking of positive outflow values. The final ranking result shows “supply” is the most influential factor in determining oil prices followed by “demand.” The current findings add significant knowledge to our understanding of the famous theory in the economic model of the price determination market. It is presumed that in a competitive market, the unit price for a specific cooperative attitude shifts until it settles at a point where the quantity demanded by buyers will equal the quantity supplied by producers. A research finding by Heakal [3] also pointed out that the oil price should increase with increasing demand or decreasing supply. Most researchers, for example [6, 7, 43], also concluded that supply and demand are the most influential factors in oil prices. Contrary to this finding, Chaw et al. [44] found that the “global economic rate” is the most influential factor. However, this research is subjected to some limitations. From the perspective of the evaluation model used, the weight assigned in this study could be improved by employing a more stable method that is germane to fuzzy knowledge such as entropy measures. Other than weight analysis, perhaps aggregation operators and sensitivity tools using PROMETHEE VI could be explored in the future. The latest aggregation methods such as Benforroni mean [45, 46], and Choquet integral [47] are among the potential candidates that could be explored as a future research direction. In future investigations, it might be possible to use different MCDM methods such as rough ELECTRE II [48] and D-TOPSIS [49], in which these methods can be applied to investigate the fluctuation of oil prices.



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