A Novel Opportunity Losses-Based Polar Coordinate Distance (OPLO-POCOD) Approach to Multiple Criteria Decision-Making

Te ability to make decisions is crucial for achieving success in any feld, particularly in areas that involve managing extensive information and knowledge. Te process of decision-making in real-world scenarios often involves considering numerous factors and aspects. It can be challenging to make decisions in such complex environments. In this paper, we present a new technique that solves multicriteria decision-making (MCDM) problems by considering opportunity losses-based polar coordinate distance (OPLO-POCOD). MCDM is a subdiscipline of operations research in which some alternatives are evaluated concerning some criteria to choose the most optimal alternative(s). Opportunity loss is a fundamental concept in economics and management, which can be used as a basis for determining the value associated with information. Te authors emphasize that the technique incorporates the concept of opportunity losses and uses distance vectors in polar coordinates, making it a compelling approach. By considering opportunity losses, decision-makers gain a better understanding of the trade-ofs involved in selecting alternatives, enabling them to make more informed decisions. Finally, the proposed method is exhibited through the use of numerical an example to illustrate its process. Additionally, a comparative sensitivity analysis is conducted to evaluate the outcomes of OPLO-POCOD and compare them with existing MCDM methods. Te OPLO-POCOD method is found to have high reliability compared to other methods, as indicated by Spearman’s correlation coefcient, which is greater than 0.9. Te method shows a correlation of over 98.5% with TOPSIS, COPRAS, ARAS, and MCRATmethods, demonstrating its robustness and efectiveness. Tese analyses show the efciency of the proposed method and highlight the stability of the results.


Multicriteria Decision Making (MCDM).
Te process of decision-making is a complex mental program that aims to solve problems by considering multiple aspects and determining the most desirable outcome.Te process is impacted by a variety of factors, including physiological, biological, cultural, and social.Tis process can be either rational or irrational, and depending on the complexity of the problem, it may require implicit or explicit assumptions.Te decision-making process can be classifed as structured or unstructured [1].Today, complex decision problems can be solved using mathematical equations, statistical models, mathematics, econometrics, and computing devices that facilitate the automatic calculation and estimation of solutions to decision problems.Decision analysis examines the problem of decision research, which comprises procedures and methods for decision-making [2].Decision analysis is strongly related to the term decision support, which refers to providing aid in fnding solutions to concerns raised by the decision-maker throughout the decision-making process [3].
Decision support approaches can be categorized into single-criteria and multicriteria, precisely according to the decision problems that they are used to solve [4].
In contrast to multiple-criteria approaches, singlecriteria methods focus on optimizing the solution to a problem.Te steps used to solve discrete problems are outlined in [5].
Real-world problems typically require complex and sophisticated analysis.Even simple daily decisions taken by an individual can lead to an incredibly complicated process [6].
One of the most accurate methods of decision-making is multicriteria decision-making (MCDM), which has been considered a revolution in this feld [7,8].MCDM techniques are used to evaluate and select options from multiple criteria after a decision has been made [9].
Benjamin Franklin published one of the earliest research papers on multicriteria decision-making about the moral algebra concept.
Since the 1950s, numerous researchers have been working on MCDM methods to evaluate their mathematical modeling capabilities.Tis framework has provided a structure for decision-making problems and generates preferences from various options.Based on the research literature, philosophers and mathematicians, such as Ramon Llull (1232-1316) and Nicolaus Cusanus (1401-1464) (Spanos, 2004), are the pioneers of MCDM techniques.
In 1944, John von proposed and developed the formulation of the "utility theory".Roy [10] developed the theory of outranking relations for dealing with multidimensional decision-making problems.
After 1970, there was a signifcant increase in scientifc research and practical applications related to multi-criteria decision-making at both theoretical and practical levels.
Table 1 outlines some of the most essential multicriteria decision-making techniques that researchers have explored.
Roy categorized models into three main groups: unique synthesis, relationships of superiority, and interactive approach [31].
According to Siskos, criteria models can be categorized into two groups: compensatory and noncompensatory models.Furthermore, criteria synthesis and models may be classifed into three categories: functional methods, relational methods, and analytical methods based on calculus [32].
Pardalos et al. [31] classifed the models concerning the forming process (multiobjective programming, multiattribute decision making, utility theory, outranking relations theory, and analytical synthetic).An examination of the relevant literature indicates that diferent MCDM methods can be used to solve the same decision problems.Selecting the right MCDM method for a decision-making process is vital to guarantee that the fnal solution accurately refects the preferences of the decision-maker [33].
Te primary distinction between these approaches is regarding the complexity of the algorithms, the weighting of criteria, the means of displaying the priority assessment criteria, the extent of certainty or uncertainty in the data, and fnally the type of data aggregation [34].
Te purpose of this research is to introduce and present a new method called the OPLO-POCOD method for handling multicriteria decision-making (MCDM) problems.Te researchers aim to address some limitations of existing MCDM methods and propose a new approach that takes into account the concept of opportunity losses and converts them into distances in the polar coordinate system.Te hypothesis is that the OPLO-POCOD method can provide a more comprehensive and accurate assessment of alternatives in MCDM problems by considering the concept of opportunity losses.Te researchers also aim to validate the efciency of the new method through a numerical example and provide a conclusion based on their fndings.
Te article also describes the proposed methodology and demonstrates its application through a numerical example.It emphasizes the unique features of this method, particularly its use of opportunity losses converted into distances in the polar coordinate system.Te lower distances indicate less lost opportunity and are considered more desirable in alternative selection.
Additionally, the article validates the efciency of the OPLO-POCOD method and concludes with a discussion of its fndings.
Te rest of this paper is organized as follows.Section 2 explains opportunity cost, opportunity losses, and polar coordinate distance.Section 3 describes the new proposed methodology.In Section 4, we employ a numerical example to explain the process of the OPLO-POCOD method.In tabular and graphical forms, in Section 5, we validate the efciency of the new method, and fnally, the conclusion is discussed in Section 6.

Te Concept of Opportunity Cost and Opportunity Losses.
Opportunity cost is a fundamental principle in the realms of decision-making and economics [35,36].Given the limited resources available, individuals endeavor to make the most suitable decisions.Consequently, individuals are worried about the efcient utilization of limited resources.Te scarcity of resources necessitates optimal decisionmaking.
Economists use the term "opportunity cost" to describe what must be sacrifced to acquire something desired.Tis concept is an essential principle of economics, as it asserts that any choice has an accompanying opportunity cost.Te concept of opportunity cost implies that the cost of one item is the lost opportunity to do or consume something else.
What is the consequence of decision-making?What is the result of good or bad decision-making?To answer these questions, managers use the concept of opportunity lost (losses).Te best decision is chosen based on the least lost opportunity.Understanding the value of an alternative chosen requires knowledge of the value of opportunity lost based on the best alternative.Briefy, opportunity lost is the value of the next best alternative.
Te opportunities losses have diferent forms according to the type of business.Manufacturing, service, and transportation organizations have completely diferent opportunities for losses.Numerous factors may lead to lost opportunities in a manufacturing organization described as follows: (i) Products are not being made (ii) Income is lost (iii) Upcoming income is impacted (iv) Inventory levels are afected (v) Orders cannot be flled (vi) Morale negatively impacts Lost proft analysis is a widely utilized approach to estimating the intangible losses resulting from risks related to systems [37].
Te analysis begins with an assessment of information risks and then progresses to analyzing their implications in the business domain.Business process analysis should be focused on losses resulting from information availability problems that have diferences in terms of assumptions about the nature of business losses, information risks, and the content of analysis methods.
Several methods are included such as labor cost analysis, lost proft analysis [37,38], information asset value analysis [39], business process analysis [40][41][42], and stock market reaction analysis [43].Te most important potential business losses include operative business losses, competitive losses, shareholder losses, company image losses, customer service processes, or losses resulting from legal processes.

The Concept of Distance and Its Measurement
Te dimension of distance per unit of time and space has specifc meanings.In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference.
To convert a point from the polar coordinate system to the Cartesian coordinate system, the functions sine and cosine are used to convert the radius and angle of the point circle into its corresponding x and y components.

Polar Coordinates Distance.
Te Cartesian coordinates (rectangular coordinates) of a point are a pair or a triplet of numbers (in two or three dimensions) that state signed distances from the coordinate axis.Te Cartesian coordinates of a point in the two-dimensional plane are represented by (x, y) as shown in Figure 1, while threedimensional space requires (x, y, z).
Instead of Cartesian coordinates, polar coordinates specify the location of a point P in the plane by its distance r from the origin and an angle θ measured from the horizontal axis anticlockwise to the line r, giving coordinates (r, θ).
Te polar coordinates (r, θ) of a point P and the Cartesian plane are illustrated in Figure 2.
Each angle can be represented on a grid so that its position relative to other angles can be better understood [44,45].
As θ ranges from 0 to 2π and r ranges vary from 0 to infnity, the point P (r, θ) covers every point in the plane.
Terefore, to more clearly illustrate the angle between two vectors, we can utilize a grid illustrated as follows.Let us explore the process of determining the distance between two polar points on the grid in Figures 3 and 4.
If we have two points as (4,105 °) and (3,225 °) in Figure 3, we plotted a graph to show two points on the polar grid.Te x-axis represents the length of the vector and the other axis (y-axis) represents the angle of the vector, as shown in Figure 4, and the angle between the two points is shown in Figure 5.
To calculate the angle of two points (4,105 °) and (3,225 °) assuming that we have the angle of each point concerning the x-axis, frst we must rotate 105 °and 225 °counterclockwise to get from the polar axis.Te angle between the two points is obtained by subtracting the larger angle minus the smaller angle 225 °− 105 °� 120 °, as shown in Figure 6.
Figure 7 shows the distance between the frst and second points which is a line segment from the pole to (4,105 °) and another from the pole to (3,225 °).
On the other hand, according to the law of cosines (also called the cosine rule), it can be stated as follows: Equation ( 1) helps us solve some triangle problems.For example, if two known sides will be (b: (4, (https://greenemath.com/Trigonometry/43/Polar-Equations-Graphs-IILesson.html).
In general, if we have two polar points indicated by (r 1 , θ 1 ) and (r 2 , θ 2 ), with r 1 and r 2 representing the two known sides, then the angle between those two sides can be determined as (θ2 − θ1).
We can use the law of cosines, substituting our two known sides as r 1 and r 2 , along with the angle between them (θ 2 − θ 1 ), to solve for our unknown side. (3) Instead of the unknown side length, we will use d for the distance between two known sides, according to equations ( 4) and ( 5): 2.2.Cosine Similarity.Te cosine similarity calculation starts by computing the cosine of two nonzero vectors.
Cosine similarity is an indication of the similarity between two nonzero vectors.Tis can be derived using the Euclidean dot product formula (equation ( 6)) which can be written as follows [46,47]: Te cosine similarity, cos (θ), between vectors A and B is calculated using the dot product and magnitude (equation ( 7)) which are defned as follows: where the angle θ represents the angle between the vectors A and B, and A.B represents the dot product between A and B.

A.B � A T B � 􏽘
and ‖A‖ represents the L2 norm or magnitude of the vector which is calculated as follows:

The Proposed Methodology
Te OPLO-POCOD method is concerned with structuring problems involving multiple criteria.Tis approach is based on the concept of selecting the alternative with the minimum opportunity losses and shortest distance in the polar coordinate distance.Tis is a method of compensatory aggregation that evaluates and ranks a set of alternatives.Te fundamental notion behind compensatory methods is that they permit trade-ofs between criteria, where an inadequate outcome in one criterion may be ofset by superior performance in another.
Te criteria used in this method include qualitative and quantitative criteria that should be quantifed in the process of implementing the technique.

Te Executive Steps of the OPLO-POCOD Method
Step 1. Formation of the initial matrix Construct an initial matrix composed of m alternatives and n criteria, denoted as x ij (m × n) or x m×n .Tis matrix is generated from the information received from the decision maker, as outlined in the following equation: where x ij represents the element of the decision matrix for the ith alternative in jth criteria.
Numerous methods of rating scales have been developed to measure attitudes directly.Te Likert scale (1932) is perhaps the most widely used scientifc instrument.According to the Likert scale, an individual can express a particular statement in the form of a fve (or seven) point scale.Each response has a numerical value equal to which would be used to measure the attitude under the survey (Table 2) [48].
Step 2. Constitution of the opportunity loss matrix.Te opportunity loss or regret values for each state of criteria can be determined by subtracting the payof values associated with each state of criteria from their respective maximum payofs in the case of a proft or gain (or, conversely, from the minimum payout in cases of cost or loss).

Opportunity loss � best value for each action − value course of action,
Here, for each state, x best equals the minimum number for negative criteria and the maximum number for positive criteria. ( According to equations ( 13) and ( 14), we create the opportunity losses (OPL) matrix, based on the following equation: Step 3. At this stage, the X pair (ordered pair matrix) should be formed.Te components of the ordered pair are x ij (equation ( 12)) and opportunity losses opl ij (equation ( 14)).
Step 4. Calculating the distance matrix in polar coordinates.In this step, the distance of each point from the best point for that criterion should be calculated.Here, a point is (x ij , opl ij ) and a point is the best column value corresponding to the point (x best ,opl xbest ) where opl xbest is equal 0. Te distance between these two points is obtained here using the following equation: where A � (x ij , pl ij ) and B � (x best , opl x best ) and (A, θ 1 ) and (B, θ 2 ).

A.B � ‖A‖‖B‖ cos θ
As mentioned, in this equation, the cosine similarity according to equation ( 7) can be used to calculate the cosine of the angle diference between two vectors.Based on equation (1), matrix D should be formed.
Step 5. Creating a weighted distance matrix Since the criteria may have diferent values, appropriate weight should be assigned to each criterion.Terefore, the weighted matrix D w (equation ( 22)) should be obtained using equation (21): Step 6. Calculating the total distance

Journal of Mathematics
In this step, the sum distance obtained for each alternative (each row) is calculated according to the following equation: Here, S T is the total distance.
Step 7. Calculating the degree of opportunity loss and achievement opportunity Te degree of opportunity loss for each option is determined by the following equation: Here,  m i�1 DOL i � 1.

Percentage of opportunity achievement POA
Te value of DOL i lies in the range between 0 and 1, with a closer value of 0 indicating lower opportunities for the best-ranked alternative and a closer value of 1 implying more opportunities for the lowest-ranked alternative.In the explanation of the value of the POA i , it should be noted that a value close to zero indicates a lower level of opportunity achievement for that alternative (the lowest rank), while any value close to one implies greater opportunity achievement (the highest rank).
Tis method ofers several benefts.First, it considers the concept of opportunity losses, providing a more comprehensive evaluation of alternatives by taking into account potential benefts or opportunities that are foregone when choosing one alternative over another.Second, the method utilizes the polar coordinate system to represent the distances between alternatives, allowing for a more intuitive understanding of the relative positions and distances between alternatives.Tird, it provides a comprehensive assessment of alternatives by considering multiple criteria simultaneously, capturing the complexity of real-world decision problems.Fourth, the OPLO-POCOD method introduces unique features, such as the conversion of opportunity losses into distances in the polar coordinate system, ofering a novel approach to decisionmaking.Finally, the method aims to select the alternative that minimizes opportunity losses, resulting in a more accurate representation of the decision-makers' preferences and leading to more informed and efective decision-making.Overall, the OPLO-POCOD method ofers valuable benefts and is a valuable technique in the feld of multicriteria decisionmaking.

Schematic Diagram.
A schematic diagram of the developed OPLO-POCOD method for determining the priorities of alternatives is provided in Figure 8.

Illustrative Example
To illustrate the process of the OPLO-POCOD method, we use an example in this section.Te example "material selection of break booster valve body in a vehicle" is a suitable example for testing our proposed method that was investigated in Moradian's study.Tis example has also been solved using several MCDM methods by Abdulaal and Bafail [9].Now, to solve the material selection problem with the proposed technique, we need to do the following steps: Step 1.We created the initial table, which consists of 4 criteria and 16 alternatives, according to Table 3. Te weight of the criteria was determined based on a pairwise comparison based on the expert's judgment.It is given in the last row of Table 3.
Step 2. In this step, we can calculate the opportunity loss for each column and then the table DOL, by specifying the best criteria in each column of Table 4.For c 3 and c 4 , the best value is the smallest element in the corresponding column, and for other criteria, the largest element is the best value.Based on equations ( 11) and ( 14), the value of the opportunity losses table is obtained (Table 5).
Step 4. We calculate the distance table for the ordered pairs obtained in the x pair table and create the D table using equations ( 18) and (20) (Table 7).
Step 5. Calculating the weighted distance table.Based on the weights expressed by the experts, we should obtain the weighted distance table (Table 8).
Step 6.Using equations ( 23)-( 26), we calculate the sum of distances, degree opportunity loss (DOL), and percentage of opportunity achievement (POA) for each alternative.Based on this, an alternative that has fewer opportunity losses or more percentage of opportunity achievement (POA) can certainly be a better option to choose.Terefore, based on the DOL and POA indexes, the ranking of the rooms is given in Table 9.
According to the DOL index results, we conclude that A 2 is in the frst rank with 0.0001% and is the best option.After that, A 8 with 0.0051% is in the second rank and then A 6 and A 4 with 0.0387% and 0.0458% are in the third and fourth ranks, respectively.
Based on the POA results, we can say A 2 with 0.9999 achievement opportunities as the frst rank, and after that, the alternatives A 8 , A 6 , and A 4 , with 0.9949, 0.9613, and 0.9542 achievement opportunities are ranked second, third, and fourth.
According to the philosophy of the technique, which is based on the concept of lost opportunity, it can be said that the alternative that has the least lost opportunity is the best.Since the lost opportunity has been converted into the distance dimension according to the proposed method, Figure 9 shows the total amount of opportunity losses based on the weighted distance of the alternatives.

8
Journal of Mathematics    Te chart consists of 4 criteria and 16 alternatives.We can see immediately there were substantial diferences between A 2 and A 11 .As can be seen from the chart, A 2 has the least lost opportunity and A 11 has the most.
In more detail, as shown in Table 10, it can be stated that A2 has the best performance so that opportunity loss in the criteria c1 and c2 is zero; it is also slightly far from the best in c3 and c4 with 0.074 and 0.142.In contrast, A11 has the highest opportunity loss along with the amount of 120.008 for c1 and c2 with 45.362.

Validity Test of the Novel Method
To validate the new method, it is necessary to compare its performance with other MCDM methods.Te example solved in this article is taken from Moradian et al. [49] who used Journal of Mathematics MCDM methods to evaluate the material selection of break booster valve body in a vehicle.Tis example has also been solved by Abdulaal and Bafail [9] using several MCDM methods.For the same numerical example, the OPLO-POCOD method was applied and the results were compared to those obtained from a variety of other MCDM methods.Table 11 provides a summary of the rankings yielded by these methods.Te Spearman's rank correlation method was utilized to determine the correlation coefcients between methods.As illustrated in Table 12, there is a correlation degree between the novel methods and other methods for this numerical example.
To compare the ranking results obtained from the different methods, Spearman's rank correlation coefcient (r) is used.Tis is a suitable coefcient when we have ordinal variables or ranked variables.Table 12 represents the   12 Journal of Mathematics correlation coefcients that show the association between the results of the proposed method and the selected MCDM methods.If this correlation coefcient is greater than 0.8, the relationship between variables is very strong.As can be seen in Table 12, all values of r are greater than 0.8.Terefore, we can confrm the validity and stability of the results of the OPLO-POCOD method.
Te OPLO-POCOD method has the least correlation of 93.2% than the RAPS and SAW methods, as indicated in Table 12.Te OPLO-POCOD method has more than 95% correlation with other methods.Te proposed method has demonstrated its ability to efectively rank alternatives.

Result and Conclusion
In recent times, there has been a growing application of multicriteria decision-making in tackling a wide range of real-world problems.Additionally, researchers have made signifcant progress in suggesting and refning numerous methods and techniques for this purpose.
In this paper, we proposed a new MCDM method, namely, the OPLO-POCOD method.To assess the alternatives on multiple criteria, it evaluates the alternatives based on opportunity losses as a fundamental concept, and polar coordinate distance, a seven-step procedure, was used for the OPLO-POCOD method.A numerical example has been used to illustrate the OPLO-POCOD method.Moreover, we have performed a comparative sensitivity analysis to demonstrate the validity and stability of the proposed method.In this analysis, ten sets of criteria weights are simulated and the results of the OPLO-POCOD method have been compared with the results of some existing MCDM methods.According to the results of this analysis, we can say that the proposed method is efcient in dealing with MCDM problems.
Te example of material selection of a brake booster valve body in a vehicle was investigated by several researchers with diferent techniques.Abdulaal and Bafail [9] used several MCDM methods for selecting the best alternative out of 16 available alternatives with four selection criteria.
Te efectiveness of the proposed technique OPLO-POCOD was compared to the other MCDM methods, including ARAS, SAW, TOPSIS, VIKOR, WASPAS, MOORA, RAMS, RAPS, and MCRAT.Te results of the OPLO-POCOD method show that A 2 is the best alternative and has a minimum opportunity or distance and the fnal ranking is A We also utilize Spearman's correlation coefcient to analyze the correlation between the results of the OPLO-POCOD method and other methods.
Te results of this analysis are shown in Table 12.According to Table 12, all correlation coefcients are greater than 0.9; thus, this indicates that our technique is highly reliable compared to other methods.Te OPLO-POCOD method demonstrated a correlation of over 98.5% with each of the TOPSIS, COPRAS, ARAS, and MCRAT methods.Considering the ranking of the evaluation criteria, we demonstrated that the OPLO-POCOD method yields the same result and has salient features based on the opportunity losses concept and the distance vector in polar coordinates, which make it a robust and compelling method.
Te fnal ranking from the OPLO-POCOD method is highly reliable as it provides more detailed information with DOL and POA indexes being more understandable results to managers and decision-makers compared to other MCDM methods.Also, the proposed new technique provides a more detailed analysis with high accuracy of each alternative based on diferent criteria.

Te Advantages of the OPLO-POCOD Method.
Te beneft of the OPLO-POCOD method is that it introduces a new approach to handling multiple criteria decision making (MCDM) problems.It incorporates the concept of opportunity losses and converts them into distances in the polar coordinate system.Tis allows for a more comprehensive assessment of alternatives.
One of the advantages of this method is that it considers opportunity losses, which are often overlooked in other MCDM methods.By taking into account the potential losses associated with each alternative, decision-makers can make more informed choices.
Additionally, the use of polar coordinates allows for a more intuitive representation of distances.Lower distances in the polar coordinate system indicate less lost opportunity, making those alternatives more desirable.
Overall, the OPLO-POCOD method provides a unique perspective on MCDM problems by incorporating opportunity losses and utilizing a polar coordinate distance approach.Tis can lead to more accurate and efective decision-making in various domains.
Te advantages of the OPLO-POCOD method are as follows: (1) Incorporation of opportunity losses: this method introduces the concept of opportunity losses into the MCDM framework.Opportunity losses refer to the potential benefts that are forgone when selecting a particular alternative.By considering these losses, the method provides a more comprehensive assessment of alternatives and helps decision-makers make informed choices.
(2) Unique features: the OPLO-POCOD method ofers features that are not found in other MCDM methods.Tese unique features provide a fresh perspective on decision-making and can potentially lead to more accurate and efective outcomes.(3) Conversion into polar coordinates: the method converts opportunity losses into distances in the polar coordinate system.Tis conversion allows for a more intuitive representation of distances, where lower distances indicate less lost opportunity.Tis makes it easier for decision-makers to understand and compare alternatives.(4) Enhanced alternative selection: by considering opportunity losses and utilizing polar coordinate distances, the OPLO-POCOD method helps in identifying alternatives that have lower lost opportunities.Tese alternatives are considered more desirable and can result in improved decisionmaking.
Overall, the advantages of the OPLO-POCOD method lie in its incorporation of opportunity losses, unique features, intuitive representation of distances, and the potential for enhanced alternative selection.Tese advantages make it a valuable technique for handling MCDM problems.

Future Research Directions for Tis Article
(1) Validation and application in real-world scenarios: the authors have provided a numerical example to demonstrate the efectiveness of the OPLO-POCOD method.However, further research could involve applying the technique to real-world decisionmaking problems in various industries and evaluating its performance and reliability in diferent contexts.(2) Comparison with additional MCDM methods: the authors have compared the OPLO-POCOD method with eleven diferent MCDM methods.However, there are numerous other MCDM methods available.Future research could involve comparing the OPLO-POCOD method with additional methods to further validate its superiority and robustness.
(3) Sensitivity analysis: conduct sensitivity analysis to examine the impact of changes in criteria weights and values on the fnal rankings obtained by the OPLO-POCOD method.Tis would help in understanding the stability and sensitivity of the technique and provide insights into its applicability in diferent decision-making scenarios.
(4) Extension to dynamic decision-making: the OPLO-POCOD method is currently applied to static decision-making problems.Future research could explore the extension of this technique to dynamic decision-making scenarios, where criteria and alternatives may change over time.Tis would involve developing a framework that incorporates 14 Journal of Mathematics time-dependent information and updating the rankings accordingly.(5) Incorporation of uncertainty and risk: decisionmaking often involves dealing with uncertainty and risk.Future research could focus on developing extensions of the OPLO-POCOD method that incorporate uncertainty and risk analysis techniques, such as fuzzy logic, Monte Carlo simulation, or Bayesian inference.Tis would provide decisionmakers with a more comprehensive understanding of the potential outcomes and associated risks when selecting alternatives.(6) User-friendly software implementation: develop user-friendly software or tools that implement the OPLO-POCOD method and provide decisionmakers with an intuitive interface for inputting criteria and alternatives, calculating opportunity losses, and obtaining the fnal rankings.Tis would enhance the practical applicability of the technique and make it more accessible to a wider range of users.

igure 6 :igure 7 :
Te distance between two points on the grid.Geometric interpretation of the distance.

Figure 8 :
Figure 8: Schematic diagram of the steps of the proposed methodology.

Figure 9 :
Figure 9: Te total amount of opportunity losses (based on weighted distance) for each alternative.

Table 2 :
Measurement of attitudes using a seven-point scale.

Table 3 :
Te initial table with 4 criteria and 16 alternatives.

Table 3 :
Continued.C4-cost of the product (minimized negative); here, criteria c 3 and c 4 are negative and the other criteria are positive.

Table 4 :
Best values for each criterion.

Table 5 :
Te value opportunity losses for each alternative.

Table 7 :
Te distance calculation for each criterion.

Table 9 :
Ranking alternatives based on DOL and POA indexes.

Table 10 :
Degree opportunity loss for best and worst alternatives.

Table 11 :
Comparing rankings from various MCDM methods.

Table 12 :
Correlation coefcients between the ranking results of the OPLO-POCOD and the other methods.