Informatica

A Multiple Attribute Decision Making problem (MADM) is a procedure to find the best alternative that has the highest degree of satisfaction from a finite set of alternatives characterized with multiple criteria (Biswas et al., 2019). MADM problems have been established as an effective methodology for solving a wide variety of decision making issues. As shown in equation (1), a MADM matrix is generally displayed by m alternatives ${A_{1}},\dots ,{A_{m}}$, and n criteria ${C_{1}},\dots ,{C_{n}}$, such that entry ${x_{ij}}$ indicates the performance measure of alternative ${A_{i}}$ for criterion ${C_{j}}$. The weight attached to criterion ${C_{j}}$ is represented by ${w_{j}}$ in a way that $0\leqslant {w_{j}}\leqslant 1$ and ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$. A criterion is either quantitative or qualitative. Quantitative criteria are stated in numerical performance measures, but qualitative criteria are expressed in ordinal qualitative data. There are many instructions for conversion of qualitative data into numerical values. Anyway, from now on, we assume that all of performance measures ${x_{ij}}$ are numerical. Moreover, a criterion is either benefit type or cost type. The former receives high performance measures for preferable alternatives, and the latter receives low performance measures for desirable alternatives. Let’s, ${x_{j}^{\max }}={\max _{i=1,\dots ,m}}{x_{ij}}$ and ${x_{j}^{\min }}={\min _{i=1,\dots ,m}}{x_{ij}}$, thus for a benefit criterion the former is the ideal performance measure, and the latter shows the anti-ideal performance measure. Absolutely, these terms are vice versa for a cost criterion.

In the literature, there are several methods to evaluate the alternatives in a MADM problem. These methods can be broadly classified into two major types: compensatory and non-compensatory (Hwang and Yoon, 1981). The former assumes compensation among the criteria, while the latter rejects this assumption. The compensatory methods are classified as scoring (or weighting or aggregating), compromising (or ideal point) and concordance (or outranking) (Hwang and Yoon, 1981). The concentration point of the current paper is the scoring methods. In the scoring methods, an aggregate score for each alternative is calculated (from now on, for short, we call it score), indicating total performance of that alternative. Generally, an alternative that has the largest score is selected. In Section 2, the existent scoring methods are reviewed. Let’s explain two motivations of the current research, as follows:

The aim of the current paper is to extend scoring methods to evaluate the alternatives in an MADM problem, by taking the two above notions into consideration. The paper is organized as follows. Section 2 reviews the existent scoring methods in the related literature. Section 3 defines the YT concept, and justifies why any MADM evaluation method has to take this concept into account. Section 4 explicates the IT concept, and enhances it by giving a new definition. In sections 5 and 6, two proposed extended scoring methods are explained. Section 7 is to present the procedure and results of a simulation study carried out to compare the ranking structures of the alternatives by different scoring methods. Section 8 illustrates a real-world study case performed in the oil industry. Finally, discussion and conclusion are pointed out in the last section.

The seminal and traditional scoring method is Weighted Sum Model (WSM) (Churchman and Ackoff, 1954). In this method, after normalization of the performance measures by the linear normalization (equations (2) and (3) respectively for benefit and cost criteria), the score for alternative ${A_{i}}$ is calculated by equation (4). The WSM formula is defined in additive form. The multiplicative pair for the WSM is called Weighted Product Model (WPM) (Miller and Starr, 1969). Equation (5) shows the WPM function to calculate the scores. The normalization technique of the WPM is just like the WSM. In Additive Ratio Assessment (ARAS) (Zavadskas and Turskis, 2010a), firstly, a hypothetical alternative, namely, and ideal alternative including the ideal performance measures is added to the MADM matrix at row zero $(i=0)$. Then, the method benefits from the linear normalization sum-based, as follows from equations (6) and (7) respectively for benefit and cost criteria. Next, equation (8) is used to work the primary scores of the alternatives out. As it can be seen from this equation, the ARAS score calculation is like the WSM. At the end, final scores are determined by comparison of the primary scores with the primary score of the ideal alternative, as follows from equation (9). Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) was recently introduced by Stevic et al. (2020). This method is a generalized version of the ARAS. The MARCOS starts with adding two hypothetical alternatives1 to the MADM matrix: ideal (denoted by AI) and anti-ideal (denoted by AAI). The ideal alternative contains the ideal performance measures, and vice versa. The MARCOS scores represent the position of the alternatives with regard to ideal and anti-ideal alternatives. This method employs the WSM formula (equation (4)), to calculate its scores, as follows from equation (10).

In the methods WSM, WPM, ARAS, and MARCOS, the relevant procedures to compute the scores do not distinguish between benefit criteria and cost criteria, and therefore, during the normalization step, cost criteria must be converted into benefit criteria. On the contrary, in the procedure of some other methods, benefit criteria are differentiated from cost criteria. The seminal method by such a procedure is Complex Proportional Assessment (COPRAS), proposed by Zavadskas and Kaklauskas (1996). Equation (14) depicts the COPRAS primary formula, where ${S_{i}^{+}}$ and ${S_{i}^{-}}$ are determined by equations (12) and (13). Both these equations use only equation (11) to get the weighted normalized performance measures. Lastly, the COPRAS score is calculated by equation (15).

Multi-Objective Optimization on the basis of Ratio Analysis (MOORA) was introduced by Brauers and Zavadskas (2006). In the ratio system part of this method, it makes use of equation (16), called vector normalization, to normalize all the MADM matrix entries. Then, equations (17) and (18) are applied to calculate the MOORA score as equation (19). A minor variant of the MOORA is called Multi-Objective Optimization on the basis of Simple Ratio Analysis (MOOSRA) (Das et al., 2012) in which equation (20) is used instead of equation (19). Brauers and Zavadskas (2010) suggested the multiplicative form of the MOORA (called MULTIMOORA or MMOORA). The normalization step is similar to the MOORA. Then, ${S_{i}^{+}}$ and ${S_{i}^{+}}$ are determined by equations (21) and (22). As it can be seen, these elementary functions act in a similar way to the WPM method. Finally, equation (23) shows the MMOORA function to reach the scores. There are some methods which take a strategy to mix additive and multiplicative forms (often the WSM and the WPM). A convex linear combination of the WSM and WPM formulas is Weighted Aggregated Sum Product Assessment (WASPAS) proposed by Zavadskas et al. (2012). In the WASPAS function (equation (24)) μ is an adjusting parameter between 0 and 1. Usually $\mu =0.5$ is chosen by the Decision Maker (DM). Combined Compromise Solution (COCOSO) (Yazdani et al., 2018) includes an idea to mix additive and multiplicative forms of the scores. Firstly, the COCOSO uses the linear max-min normalization (equations (25) and (26) respectively for benefit and cost criteria) to convert the performance measures into dimensionless values. Secondly, three appraisal formulas as equation (27) to equation (29) are used to get the mixtures. Equation (29) contains a parameter μ that similarly to the WASPAS is an adjusting parameter on interval $[0,1]$, with default value of 0.5. Finally, equation (30) is the proposed COCOSO model to determine the scores. Equation (31) was used by Keshavarz Ghorabaee et al. (2021) to determine the alternative scores. This formula was employed in a criteria weighting model called Method based on the Removal Effects of Criteria (MEREC), in which ${r_{ij}}$ are dimensionless values of the performance measures obtained by linear normalization, i.e. equations (2) and (3). It seems equation (31) originates in the famous Entropy method (Hwang and Yoon, 1981) and a logarithmic normalization formula suggested by Zavadskas and Turskis (2008). Since the MEREC disregards the criteria weights, it may be inapplicable in many situations.

Reviewing the scoring methods displays that different methods employ various techniques to remove the difference in scale of each criterion. For review of different kinds of normalization techniques, see Brauers and Zavadskas (2006), Zavadskas and Turskis (2008), and Jahan and Edwards (2015).

The Aspiration Level (AL) idea was introduced by Lotfi et al. (1992). For a given criterion, the AL is a value that represents the minimum satisfactory outcomes for the DM. Lotfi et al. (1992) have stated that the AL for a criterion should not exceed the ideal performance measure of that criterion, and should be less than the anti-ideal performance measure of that criterion. For example, in a house selection MADM problem with three alternatives, if areas are 60, 40, and 80 square meters, the DM’s AL for house area should be a number between 40 to 80. A drawback to such assumption is that the DM’s willingness actually may be beyond this range, e.g. 200 square meters. In fact, confining the DM’s desire to performance measure range is not reasonable. To overcome the AL limitation, this paper proposes the Yearning Threshold (YT) concept as follows.

Now, having the above YT definition, let us discuss this fact that incorporation of the YT concept into the MADM evaluation models is not an option, but rather a requirement to ensure the accuracy of the decisions. Again, consider the example of house selection problem by the following MADM matrix. The criteria are (${C_{1}}$:) area and (${C_{2}}$:) the ratio of the area of windows to the house area. The YT is 200 for ${C_{1}}$, and 0.4 for ${C_{2}}$. Without loss of generality, we assume that the criteria have identical weights, and we use the linear normalization and the WSM to calculate the scores as follows:

The suggested solution is ${A_{3}}$. Is this decision accurate? Regarding ${C_{1}}$, all performance measures (60, 40, and 80) are very far from the DM’s YT (200 square meters), and thus all the alternatives are rather undesirable for the DM. Regarding ${C_{2}}$, ${A_{2}}$ exactly meets the DM’s YT. After applying any dimensionless rule, a reasonable expectation is that all “1” in the matrix must indicate the best preferred value for the DM, i.e. entries ${r_{31}}$ and ${r_{22}}$ in the above matrix. Entry ${r_{22}}=1$ is valid, because ${A_{2}}$ with 0.4 is completely welcome for the DM. Conversely, ${A_{3}}$ with 80 square meters is far from the minimum desired level of the DM, consequently, entry ${r_{31}}$ must adjust a lot less than 1. In order to resolve this issue, we suggest normalizing the MADM matrix by using the YTs in spite of the ideal performance measures (e.g. in the following matrix, at column ${C_{1}}$ we would have $60/200=0.3$, $40/200=0.2$ and $80/200=0.4$):

In the above matrix, we see that the normalized performance measures are scaled down in proportion to the relevant YTs, and the proper solution is ${A_{2}}$. We conclude that using the YTs to normalize the MADM matrix in any alternative evaluation method is a necessary condition to guarantee a proper decision. The YT is “null” when ${\mathrm{YT}_{j}}\leqslant {x_{j}^{\max }}$ for benefit criteria, and when ${\mathrm{YT}_{j}}\geqslant {x_{j}^{\min }}$ for cost criteria. Moreover, if the DM has no idea about the YT, it can be considered as ${\mathrm{YT}_{j}}={x_{j}^{\max }}$ for benefit criteria, and ${\mathrm{YT}_{j}}={x_{j}^{\min }}$ for cost criteria.

We know that in mathematics, $|Y|$ shows the absolute value of Y, such that $|Y|=\max \{Y,-Y\}$. In fact, a number and its opposite have the same absolute value. This operator is defined in additive form of numbers. Now, let’s generalize it into multiplicative form, by the following definition.

In what follows, two definitions are introduced. The former depicts the preliminary and traditional statement of the IT concept, and the latter is a novel statement for the IT concept that is firstly introduced in the current study.

Notably, the DIT values are not homogeneous (because dimensional unit of ${\mathrm{DIT}_{j}}$ is the same as dimensional unit of criterion ${C_{j}}$); on the contrary, all the RIT values range from 0 to 1, thus the RIT has an advantage over the DIT from this point of view.

The underlying idea of the proposed EWPM stands on the relative relations between the alternatives. This idea leads us to a pairwise comparison matrix whose rows and columns are the alternatives. Let’s portray the notion using a simple numerical example as follows, including 3 alternatives ${A_{1}}\hspace{2.5pt}{A_{2}}$, and ${A_{3}}$, and two criteria ${C_{1}}$ (benefit type) and ${C_{2}}$ (cost type) weighted as 0.7 and 0.3, respectively.

If the analyst wishes to compare two alternatives ${A_{1}}$ and ${A_{2}}$ from the point of view of criterion ${C_{1}}$, then the fraction $6/2$ can be taken. This fraction is called individual importance of ${A_{1}}$ relative to ${A_{2}}$. Similarly, the individual relative importance of ${A_{1}}$ to ${A_{2}}$ in view of ${C_{2}}$ is $(\frac{1}{8})/(\frac{1}{3})$ or $3/8$ (Note that ${C_{2}}$ is a cost criterion). Now, we need to combine 6/2 and 3/8. These two numbers are ratios; therefore, to combine them the WPM form can be employed. Thus, the combined relative importance of ${A_{1}}$ to ${A_{2}}$ would be ${(6/2)^{0.7}}\times {(3/8)^{0.3}}$. This calculation is sometimes called dimensionless analysis because its mathematical structure removes any units of measure (Triantaphyllou, 2000). By performing similar calculations for the other pairs of the alternatives, the result is the following matrix called Objective Multiplicative Pairwise Comparison (OMPC):

The OMPC matrix is actually in a multiplicative preference relation format, but in spite of preference judgment by the DM, the objective relative importance is replaced. Thus, each entry ${a_{ik}}$ $(i,k=1,\dots ,m)$ of the OMPC matrix represents the objective importance of alternative ${A_{i}}$ relative to alternative ${A_{k}}$ in terms of a multiplicative reasoning. Equation (36) shows a general formula to determine the entries of the OMPC matrix. In this function, ${t_{j}}$ is determined by equation (37):

According to the following Proposition 1, the OMPC matrix is multiplicative inverse (i.e. reciprocity rule ${a_{ik}}\times {a_{ki}}=1$) and full consistent (i.e. transitivity rule ${a_{ik}}\times {a_{kl}}={a_{il}}$) for all entries of the matrix.

Now we deal with the famous problem of deriving the scores from a pairwise comparison matrix. To extract the alternative scores from the OMPC matrix, with regard to the multiplicative form the entries, the Simple Geometric Mean (SGM) (Crawford and Williams, 1985) is utilized here. In the current situation, this method has been recommended by many researchers (Zhang et al., 2004; Hovanov et al., 2008). Therefore, equation (38) represents the formula for calculating the score of alternative ${A_{i}}$. The scores in the above example are 1.2286, 0.7642, and 1.0651, respectively.

In the previous section, we saw that the EWPM is founded on a multiplicative structure. There is a pair for the EWPM, founded on the additive structure. Suppose the same numerical example of previous section: In the additive form, we can say that in view of ${C_{1}}$, the importance of ${A_{1}}$ compared with ${A_{2}}$ is $6-2=4$. Indeed, regarding ${C_{1}}$, the distance between the importance of ${A_{1}}$ and ${A_{2}}$ is 4. In the same way, the relative importance of ${A_{1}}$ to ${A_{2}}$ from the point of view of ${C_{2}}$ is $(1/8)-(1/3)=-5/24$ (Note that ${C_{2}}$ is a cost criterion). The two numbers 4 and $-5/24$ are distances, thus the WSM rule directs the current method to additively combine the relative importance. Because the distances may differ in dimensional unit, distance $6-2$ is transformed to ($6-2)/6$ in which denominator 6 is the ideal performance measure of ${C_{1}}$. In the same way, distance $(1/8)-(1/3)$ is transformed to $((1/8)-(1/3))/(1/3)$ where denominator $1/3$ is the reciprocal of the ideal performance measure of ${C_{2}}$. The combined relative importance of ${A_{1}}$ and ${A_{2}}$ is written as $0.7\times (\frac{6-2}{6})+0.3\times (\frac{(1/8)-(1/3)}{(1/3)})$. By running the same calculations for the other pairs of the alternatives, the following matrix is obtained. We name it Objective Additive Pairwise Comparison matrix (OAPC): The OAPC matrix is in additive preference relation format. In this matrix, each entry ${b_{ik}}$ $(i,k=1,\dots ,m)$ is interpreted as the objective importance degree of alternative ${A_{i}}$ over alternative ${A_{k}}$ in terms of an additive reasoning. Equation (43) displays a general formula to obtain the entries of the OAPC matrix, in which ${t_{j}}$ is determined like the EWPM (i.e. ${t_{j}}$ is +1 for benefit criteria, and is −1 for cost criteria). In equation (43), ${x_{j}^{\ast }}$ refers ${x_{j}^{\max }}$ for benefit criteria, and ${x_{j}^{\min }}$ for cost criteria. It can be simply demonstrated that the OAPC matrix is additive inverse (opposition rule), i.e. ${b_{ik}}+{b_{ki}}=0$, and full consistent (transitivity rule), i.e. usually ${b_{ik}}+{b_{kl}}\hspace{2.5pt}=\hspace{2.5pt}{b_{il}}$.

We omit the proof here, since the proof is straightforward and similar to that in Proposition 1 (In this point, the reader is advised to see Appendix A).

Because the entries of the OAPC matrix are in the additive form, the Simple Column Sum (SCS) method (Saaty, 1980) is a proper way to derive the scores. Equation (44) shows the relevant formula. The scores for alternatives ${A_{1}}$, ${A_{2}}$, and ${A_{3}}$ in the above example are 0.1483, −0.1308, and −0.0175, respectively, such that their sum is equal to 0.

This section is to answer two questions. Whether a given scoring method selects the same best alternative as another given scoring method? And, how much is similarity between the overall rank structures of the alternatives obtained by two given scoring methods? To these, we examine a kind of simulation process that is a broadly accepted framework to compare the performance of the methods (Ahn, 2017; Sarabando and Dias, 2010; Ahn and Park, 2008; Barron and Barrett, 1996). The idea is to repeatedly (N times, namely simulation run number) generate a random MADM matrix and random criteria weights, and investigate how well the result of a given method matches another one, in terms of an efficacy measure. To answer the above first question, a measure called Hit Rate (HR) is used as equation (49). where N is a simulation run number, and B is the number of runs in which a method selects the same best alternative as another method does. The HR value ranges from 0 to 1, such that 1 indicates two methods select identical best alternative through all simulation runs. To answer the second question, Rank order Correlation (RC) is employed here. The RC shows the similarity of the overall rank structures of the alternatives constructed by two given methods. The RC is calculated using Kendall’s formula (Winkler and Hays, 1985) as equation (50). where A is the sum of alternative numbers through all simulation runs, and V indicates the number of pairwise preference violations in all simulation runs. The RC value ranges from $-1$ to $+1$, where $+1$ means perfect correspondence between the two rank orders in all simulation runs.

The simulation is designed with two levels of the alternatives: Low $(3\leqslant m\leqslant 6)$ and High $(9\leqslant m\leqslant 12)$, and two levels of the criteria: Low $(3\leqslant n\leqslant 6)$ and High $(9\leqslant n\leqslant 12)$. For each combination of the levels (i.e. combination 1: low level of the alternatives and low level of the criteria, combination 2: low level of the alternatives and high level of the criteria, combination 3: high level of the alternatives and low level of the criteria, and combination 4: high level of the alternatives and high level of the criteria), the following 10-step procedure is performed.

Step 1: Set $N=10000$ (N shows simulation run number), $k=0$, and $c=1$ (c indicates combination type, i.e. c could be 1, 2, 3, or 4).

Step 2: Set $k=k+1$, $A=0$, ${B_{pq}}=0$ and ${V_{pq}}=0$ $(p,q=1,\dots ,12)$. Where “12” is the number of the methods to be compared (the WSM, WPM, ARAS, MARCOS, COPRAS, MOORA, MOOSRA, MMOORA, WASPAS, COCOSO, null EWPM, and null EWSM). Thus, $p=1$ indicates the WSM, $p=2$ indicates the WPM method, and so on.

Step 3: Based on the combination type, generate a random integer number for the number of alternatives $(m)$, and an independent random integer number for the number of criteria $(n)$. Set $A=A+m$.

Step 4: Generate a random MADM matrix from the independent uniform distribution on $(0,1]$.

Step 5: Transform the MADM matrix to different normalized forms, depending on different methods mentioned in Step 2.

Step 6: Generate a random weight vector from the independent uniform distribution on $(0,1]$. Then normalize the weights to add up to 1.

Step 7: Determine the ranking structure of the alternatives by employing each of the 12 methods.

Step 8: Make HR pairwise comparison between each two methods. For two given methods p and q, if method p selects the same best alternative as method q, then set ${B_{pq}}={B_{pq}}+1$.

Step 9: Draw RC pairwise comparison between each two methods. For two given methods p and q, and for each alternative, if the methods p and q do not assign a similar rank to that alternative, then set ${V_{pq}}={V_{pq}}+1$.

Step 10: If $k=c\ast N/4$ then calculate the average HR for each two given methods p and q by ${\mathrm{HR}_{pq}^{c}}={\mathrm{B}_{pq}}/N$, calculate the average RC for these two methods by ${\mathrm{RC}_{pq}^{c}}=1-(2{V_{pq}}/A)$, set $c=c+1$, and set $k=0$. In this point, if $k=N$, calculate the overall HR and overall RC by ${\mathrm{HR}_{pq}}={\textstyle\sum _{c=1}^{4}}{\mathrm{HR}_{pq}^{c}}/4$ and ${\mathrm{RC}_{pq}}={\textstyle\sum _{c=1}^{4}}{\mathrm{RC}_{pq}^{c}}/4$, and stop the process, otherwise go to Step 2.

The above simulation experiment is conducted using the Visual Basic for application in the Excel’s programming language on a personal computer. The simulation runs are made in 5 rounds. Finally, the mean of 5 rounds are considered to compare the results. The individual results of 4 combinations are presented in Appendix B and Appendix C. In a broader sense, 5 values (for the HR and RC and for each combination) are obtained from 5 rounds. Each of these 5 values is the mean of 10.000 values resulted from 10.000 simulation runs. The averages of the presented values in Appendix B and Appendix C are shown in Table 1 and Table 2.

Looking at the above tables, some considerable facts are found out. The null EWSM and the MARCOS generate ranking structures of the alternatives similar to the WSM. Moreover, ranking structures of the alternatives by the null EWPM and the MMOORA are similar to that by the WPM. We name these kinds of correlations full HR (i.e. HR = 1) and full RC (i.e. RC = 1). By assuming high HR values as $1\gt \mathrm{HR}\geqslant 0.85$, the following pairs fall into this category: the WASPAS and WSM, the WASPAS and WPM, the WASPAS and MARCOS, the WASPAS and MMOORA, the null EWPM and COCOSO, and the null EWSM and WASPAS. Similarly, by assuming high RC by $1\gt \mathrm{RC}\geqslant 0.35$, the following pairs fall into this category: the WASPAS and WSM, the WASPAS and MARCOS, and the WASPAS and null EWSM. Taking together, we conclude that the WASPAS results are the most apt to be similar to the other methods.

By reviewing Tables 7–14 in Appendix B and Appendix C, an important fact is also clarified. As a remarkable outcome, for most pair of the methods, as the number of alternatives/criteria increases, both the HR and RC values decrease. In this regard, the impact of the number of alternatives is more than that of the number of criteria. As an example, for pair of the WSM and null EWPM, for low level of the alternatives and criteria, the HR equals 0.8589, for low level of the alternatives and high level of the criteria, the HR becomes 0.7953, for high level of the alternatives and low level of the criteria, the HR decreases to 0.7649, and for high low level of the alternatives and criteria, the HR decreases to 0.6825. Such a downward trend is obviously visible in most analogous cells in sequence Tables 7 > Table 8 > Table 9 > Table 10 and, in most cells, in sequence Table 11 > Table 12 > Table 13 > Table 14, except the cells indicating full HR or full RC.

One of the most fundamental challenges in oil industries is the need of Exploration and Production (E&P) companies for selecting and performing portfolios which have enough capacity from the view of technical, financial, and economic aspects. A portfolio is a collection of projects, programmes, and operations handled as a pack to achieve strategic objectives. In a real-world problem, the EWPM was used to select the best portfolio under Iranian financial regimes. In this case, a Delphi analysis was done to identify key criteria, with participation of sixteen experts. The participants for this analysis were chosen among a variety of project managers and subject matter experts of E&P companies, with more than 12 years of working experience on average. Fifteen criteria were identified through Delphi method:

(${C_{1}}$) Size of reservoir: The larger the reservoir, the higher potential for oil recovery, thus this is a benefit criterion.

(${C_{2}}$) Average distance from resources/infrastructures: Oil field that is close to pipelines, export systems, storage capacities, refineries, power stations, processing facilities, water resources, and so on, is preferred as it needs less investment to dispose of its oil and gas.

(${C_{3}}$) Net return on investment: This economic indicator is a function of development cost analysis and benchmarking international oil prices. Obviously, the higher this criterion results in, the better the development portfolio.

(${C_{4}}$) Security and political stability: A politically stable and safe region and country of investment is more favourable in the oil field.

(${C_{5}}$) Quality of upstream oil field development plan: Clearly, a development plan that accurately addresses key issues is considered as a favourable plan. The key items refer to a variety of things, including, but not limited to, project management guidelines, production challenges, separation systems, logistics, fiscal policy, budget constraints, water supplies, gas capturing and the number of wells to be drilled.

(${C_{6}}$) Cooperation opportunities: It indicates capabilities for making benefits from local communities, local businesses and subcontractors.

(${C_{7}}$) Reservoir characteristics: It refers properties such as oil/water level, oil movement behaviour, reservoir pressure, etc. The relevant data includes a variety of data acquired by logging, coring, and testing for each well drilled.

(${C_{8}}$) Plateau production rate: This is the major field output, and refers to the level of production rate at the plateau phase of oil production.

(${C_{9}}$) Plateau production period: This criterion depicts the fact that longer duration of plateau production justifies employment of installed capacities, decreases capital cost per unit of production, and causes minimal spare capacity maintenance cost.

(${C_{10}}$) Subsurface complexity: This criterion stands for several rock properties such as porosity, permeability, hydrocarbon layer continuity, and water saturation. A less subsurface complexity means optimal rock properties.

(${C_{11}}$) Human resources: Some zones have qualified human resources in abundance, while they are scarce in others. Oilfields that are located in areas where qualified human resources are available could be considered preferably.

(${C_{12}}$) Safe location: The safer the oilfield is during natural disasters, such as flooding and quakes or from damages to existing protective structures such as dams and bunds, the better.

(${C_{13}}$) Type of contract: The common types of current E&P agreements include Service Contract (SC) and Production Sharing Agreement (PSA). Preferred type would be based on whether the cash flow will ensure highest returns for the investor.

(${C_{14}}$) Environmental protection: A project with potential for optimal reduction of footprint is preferred. Footprint reduction indicators include (1) protecting surface and subsurface sources of fresh water, (2) minimizing effluents, (3) treating and re-injecting produced water, and (4) using land that is already degraded rather than agricultural land.

(${C_{15}}$) Management capacity: An upstream development program with existing capacity to efficiently and effectively supervise implementation, track commitments and avoid delays and penalties could be considered more favourably.

Later, the criteria were assigned numerical weights using Improved Rank Order Centroid or IROC method (Hatefi, 2023). For employing the IROC method, firstly, each expert was asked to individually rank the $n=15$ criteria. The ranks are integer number ranges from 1 to 15. Then, the Kendall’s coefficient of concordance (Kendall and Gibbons, 1990) was used in order to measure the degree of agreement among the expert’ judgments. Equation (51) represents this coefficient, i.e. a ratio between 0 (no agreement) and 1 (full agreement), where $E=16$ shows the number of experts, and ${R_{j}}$ displays the sum of given ranks by the experts for criterion ${C_{j}}$. An interpretation of T is that the consensus is very week, weak, fair, strong, very strong, and complete, if it is in intervals $[0.0,0.1)$, $[0.1,0.3)$, $[0.3,0.5)$, $[0.5,0.7)$, $[0.7,0.9)$, and $[0.9,1.0]$, respectively. If T indicates a strong or higher consensus (i.e. not less than 0.5), then we can accept the ranks as valid data. In the study case, $T=0.607$ was obtained, indicating a strong agreement. After that, we prioritized the criteria, such that the bigger value of ${R_{j}}$ indicates the lower concordant rank of criterion ${C_{j}}$. After ranking, the IROC formula (equation (52)) was used to estimate the criteria weights. In this formula ${\varphi _{\mathrm{yn}}}$ are constant parameters of the IROC model, reported in Hatefi (2023). These constants for $n=15$ are 0.04916, 0.06070, 0.06690, 0.07044, 0.07264, 0.07378, 0.07352, 0.07305, 0.07161, 0.06962, 0.06781, 0.06597, 0.06366, 0.06155, and 0.05959. Table 3 represents the obtained ranks and weights for the 15 criteria.

Checking the 15 weights, the most crucial criterion weight is 0.2057, while the least important equals 0.0040. This means that criteria with weights less than 0.1028 ($\cong 0.2057/2$) are not important, thus we eliminate them from the model. As a matter of fact, the fast and frugal concept (Katsikopoulos and Fasolo, 2006) leads us to concentrate on just the important criteria. Consequently, only four criteria are essential, i.e. ${C_{3}}$: Net return on investment (in percents), ${C_{9}}$: Plateau production period (in years), ${C_{1}}$: Size of reservoir (in billion barrels), and ${C_{2}}$: Average distance from resources/infrastructures (in kilometers).

Table 4 presents the MADM matrix and some additional data for the study case. All the performance measures (${x_{ij}}$) for all the criteria (${C_{1}}$, ${C_{2}}$, ${C_{3}}$ and ${C_{9}}$) are received from the relevant technical documents and reports. As mentioned above, the criteria weights are determined using the IROC method. In Table 4, the IROC weights for the four criteria (0.1265, 0.1043, 0.2057, and 0.1573) are normalized to add up to one. Moreover, the RIT and YT values are given by the E&P company’ owners.

The EWPM scores are obtained as 0.8553, 1.0992, 0.9700, 1.0391, 0.9711, and 1.0866 for ${A_{1}}$ to ${A_{6}}$, respectively. Correspondingly, by setting $\pi =0.05$, the portfolios ${A_{2}}$ and ${A_{6}}$ are identified as strong alternatives; among them, ${A_{2}}$ is proposed as the best portfolio to be taken.

This paper started by discussing the fact that, in evaluation of the alternatives in MADM, the existent scoring methods do not take the two momentous concepts into account, i.e. the Indifference Threshold (IT), and the Yearning Threshold (YT). The paper, firstly, enhanced the IT and the YT concepts. As a matter of fact, along the traditional IT definition (Distance-based IT or DIT), a new definition called Ratio-based IT or RIT was developed. Additionally, the traditional idea of Aspiration Level (AL) was revised and redefined to develop the new YT concept. Accordingly, to incorporate the IT and YT concepts into the scoring methods, the paper proposed two extended methods called Extended Weighted Product Model (EWPM) and Extended Weighted Sum Model (EWSM). Both the methods try to establish a pairwise comparison matrix extracted from the MADM matrix. As a remarkable advantage of the proposed methods over the existent methods, the calculation relations in both the methods allow the Decision Maker (DM) to incorporate his/her ITs and YTs into the models. Table 6 draws a brief comparison between the EWPM and EWSM methods in view of their structures and scores.

Several advantages of the proposed methods over the existent methods in the literature (reviewed in Section 2) are worthwhile emphasizing as follows:

The paper carried out a simulation experiment to compare the outputs of different scoring methods. Both the simulation experiment and theoretical analysis presented that if the ITs and YTs are null, the WPM and the EWPM gave the same results. This fact is true of the WSM and the EWSM. Nevertheless, the DM is advised to use the EWPM and the EWSM, because these extended methods can classify the alternatives, while the WPM and the WSM do not have such an advantage. When there are effective IT values or YT values, application of the EWPM or the EWSM is confidently the decisive and main recommendation, because the other methods disregard the IT and YT values.

Further works can be undertaken in some areas. The first idea is mixing the proposed EWPM and EWSM. In this case, approaches of the mix methods such as the WASPAS and the COCOSO can be applied. The major focus point of the current paper was incorporation of the IT and YT concepts into the scoring methods. In such a way, future studies can be performed to address the question of how the IT and YT values can be incorporated into the compromising and concordance methods. Another idea, for future studies, is to extend the EWPM and EWSM in uncertain environments, e.g. grey numbers (Deng, 1982; Zavadskas and Turskis, 2010b) and fuzzy numbers (Zadeh, 1965; Aydogdu et al., 2023).

Author contributions: The author confirms sole responsibility for study conception and design, data collection, analysis and interpretation of results, and manuscript preparation.

Conflict of interest: The author declares that there is no conflict of interests regarding the publication of this paper.

Human participants and/or animals: This article does not contain any studies with human participants or animals performed by the author.

Ethical approval: There is no ethical approval, due to this study is not with human or animal subjects, or where private, protected, or culturally significant research locations were used.