R-SWARA and COPRAS Integration for Evaluating Third-Party Logistics Providers

2.1 Phase 1: determining rough criteria weights using the R-SWARA method

SWARA, introduced by ⁴², allows experts to assign weights to criteria based on their relative importance. Unlike the AHP, SWARA is efficient for group decision-making, requiring experts to rank criteria and estimate their comparative significance. However, SWARA’s reliance on precise expert judgments can introduce subjectivity and uncertainty ⁴³. To address this issue, ⁴³ developed R-SWARA by integrating rough set theory, which uses interval data with upper and lower bounds to handle vague or conflicting subjective opinions. Their study validated R-SWARA in selecting wagons for internal transport, demonstrating a strong correlation with the rough best-worst method (BWM) and rough AHP in sensitivity analysis, thus confirming its robustness.

The R-SWARA method consists of the following steps ⁴³:

Step 1.1: defining the criteria

Let a set of n criteria be defined as:

Step 1.2: ranking the criteria

A group of k experts then provides rankings of the criteria from most to least important. Let r _ij ​ denote the ranking value assigned by expert i to criterion C _j ​.

Step 1.3: constructing a rough group matrix

Since experts may provide different values, each criterion is represented by a rough number (RN) as an interval rather than a single value. For each score r _ij ​, two limits are calculated: (i) the lower limit, which is the average of all scores that are lower than or equal to r _ij , and (ii) the upper limit, which is the average of all scores that are greater than or equal to r _ij . This is shown in Eqs. ¹ and ², respectively.

where N _L and N _U are number of scores lower than or equal to r _ij and the number of scores greater than or equal to r _ij , respectively. Therefore, for criterion C _j ​, the rough number is

Step 1.4: normalizing comparative importance

Once the rough numbers RN(C _j ) have been obtained for each criterion C _j , they are converted into comparative importance values s _j ​. Formally:

• For the most important criterion (ranked first), the comparative importance is fixed as s ₁ = (1, 1). This means that the top-ranked criterion is treated as the reference baseline.

• For every other criterion j > 1, the comparative importance is obtained by adding 1 to the rough number interval: s _j = RN(C _j ) + 1.

Step 1.5: recalculating weights

After obtaining the comparative importance intervals s _j , the next step is to compute the recalculated coefficients q _j as follows:

• For the most important criterion (the reference baseline), the recalculated coefficient is fixed as q ₁ = (1, 1).

• For every subsequent criterion j > 1, the coefficient is computed iteratively by dividing the previous coefficient by the current comparative importance: q _j = q _j-1 /s _j .

Step 1.6: calculating the relative weights

Once the recalculated coefficients q _j ​ have been obtained, the final step is to normalize them to produce the relative rough weights w _j ​. This is formally expressed in Eq. ³:

where both the numerator and the denominator are intervals, so the division is again performed using interval arithmetic. The normalization ensures that the weights sum to 1. Each is the final rough weight interval for criterion C _j ​, representing its relative importance after accounting for expert disagreement and uncertainty. These rough weights are then applied as inputs in the COPRAS method to evaluate and rank the alternatives under consideration.

2.2 Phase 2: evaluation of alternatives using COPRAS

The COPRAS method, proposed by ⁴⁴, ranks alternatives by comparing their performance across multiple criteria, distinguishing between beneficial (maximizing) and cost (minimizing) criteria. The integrated R-SWARA-COPRAS method involves the following steps:

Step 2.1: defining alternatives and criteria

Let the alternatives be A = {A ₁, A ₂, …, A _m } and the criteria be C = {C₁, C₂, …, C _n }. The criteria are partitioned into a beneficial set J ⁺ ⊆ {1, 2, …, n} and a non-beneficial set J ^- ⊆ {1, 2, …, n}, where J ⁺ ∩ J ^- = ∅ and J ⁺ ∪ J ^- = {1, 2, …, n}.

Step 2.2: establishing the decision matrix

A decision matrix is created in which each alternative A _i is evaluated against each criterion C _j . The matrix entries x _ij ​ represent expert-assigned performance scores for alternative A _i under criterion C _j . These scores can be individual expert values or aggregated through rough number logic, if multiple expert evaluations per entry are available. In this study, the individual expert scores were aggregated via the simple average.

Step 2.3: normalizing the decision matrix

The decision matrix is normalized to eliminate the influence of differing units across criteria. For each criterion j, the normalized value is computed using Eq. 4.

This ensures that each value is expressed as a proportion of the total performance under that criterion.

Step 2.4: applying the criterion weights

Let w _j be the weight of criterion C _j ​ from R-SWARA. Next, the weighted normalized matrix D = (d _ij ) is defined as shown in Eq. (5)

Step 2.5: Computing maximizing and minimizing indices

The next step is to separate the contributions of beneficial criteria (the higher the better) and non-beneficial criteria (the lower the better).

Eq. ⁶ is used for the beneficial criteria J ⁺:

Eq. ⁷ is used for the non-beneficial criteria J ^-:

Step 2.6: calculating the relative significance

Once the maximizing and minimizing indices have been calculated, COPRAS combines them into a single measure of performance for each alternative called the relative significance Q _i ​, as defined by Eq. (8)

Step 2.7: computing the utility degree and ranking the alternatives

Finally, the utility degrees U _i are computed by Eq. (9)

where is the best (maximum) relative significance among all alternatives. The alternative with the highest utility score (i.e., U _i = 100%) is considered to be the most preferred. Other alternatives are scaled proportionally relative to the most preferred one.

The proposed framework provides structured decision support, but it involves several assumptions and limitations. First, the weighting stage depends on expert judgement; although the method is designed to structure and aggregate these judgements, different expert panels may yield different weights and rankings. Second, the results are sensitive to the definition and operationalization of criteria (including the way in which benefit and cost criteria are specified) and to the quality of the input assessments. Third, the analysis reflects the context of the case and the set of evaluated alternatives. Consequently, the ranking should not be interpreted as universally generalizable beyond the study setting. Finally, as with many MCDM approaches, the method does not fully capture dynamic changes over time (e.g., performance variability or market shocks) unless updated inputs are provided.

3.1 Defining the criteria

The company aims to select a 3PL provider from six candidates (A, B, C, D, E, F) based on criteria tailored to cold chain logistics requirements. Based on the company’s operational needs and prior literature ⁹-¹¹, seven criteria were identified for 3PL evaluation:

• Security (C1) measures the provider’s ability to ensure safe handling and protection of perishable goods against theft, damage, or contamination.

• Punctuality (C2) evaluates the provider’s reliability in meeting delivery schedules, critical for minimizing spoilage in cold chain logistics.

• Cost (C3) assesses the total logistics cost, including transportation and handling fees, as a minimizing criterion.

• Delivery area (C4) gauges the provider’s coverage and ability to serve diverse geographic markets, including remote regions.

• Customer service (C5) reflects the provider’s responsiveness and communication quality in addressing client needs and issues.

• Assurance (C6) measures the provider’s compliance with quality standards (e.g., ISO certifications, cold chain protocols) and reliability guarantees.

• Logistics experience (C7) evaluates the provider’s expertise and track record in refrigerated transportation for perishable goods.

3.2 Collecting the data

Three experts with extensive experience in fishery logistics were engaged to assess the evaluation criteria and alternative vendors. Although the number of experts was limited, it remained within the range of commonly reported in previous research works, involving panels of three ³⁶, ⁴⁵, ⁴⁶, four ⁴⁷, or five experts ⁴⁸-⁵⁰. The three experts in the study possess experience and knowledge related to cold logistic providers (Table I), ensuring a reliable assessment. The data collection process ensured a diversity of perspectives while maintaining consistency through structured questionnaires, and it facilitated discussions to resolve ambiguities. The questionnaire was divided into two main sections. The first section aimed to determine the relative importance of each criterion. The experts ranked the criteria, with a lower score representing the most important and higher score indicating the least important. The result is shown in Table II. The second section focused on the performance evaluation of each 3PL parties. The experts scored the six 3PL providers (A, B, C, D, E, F) regarding each criterion via a 0-100 scale, based on historical performance data, service agreements, and site visits. The result is shown inTable III.

Table I: Expert background

Table II: Result of the ranking

Table III: 3PL provider scores

3.3 Result and discusión

The R-SWARA method was applied to determine the weights of the seven criteria identified. Using the expert ranks from Table I, the criteria were first ranked by averaging the ranks to establish a strict order. The order from the most to the least preferred can be presented as follows: C2​ ≻ (C4 and C5; tie) ​≻ (C1, C3, C6; tie) ​≻ C7​. Using Eqs. ¹ and ², the rough numbers for each criterion are calculated. For instance, criteria C4 = {2,1,1} has and . Thus, and Therefore, and . Using the same procedure, the rough numbers for each criterion are as follows: RN(C4) = (1.11,1.55), RN(C2) = (1.00,1.00), RN(C5) = (1.11,1.55), RN(C1) = (1.45,1.89), RN(C3) = (1.45,1.89), RN(C6) = (1.45,1.89), RN(C7) = (2.00,2.00). The normalization of the matrix yielded the following rough numbers for each criterion: RN(C2) = (1, 1), RN(C4) = RN(C5) = (0.55, 0.775), RN(C1) = RN(C3) = RN(C6) = (0.725, 0.945).

Afterwards, the comparative importance (s ₂) was set to (1, 1) for C2 and computed as s _j + 1 for others. For instance, RN(C4) = (0.55, 0.775) becomes RN(S4) = (1.55, 1.775). The recalculated weight q _j and the relative weights are then computed using Eq. ³. For C4, and Thus, RN(Q₄) = (0.563,0,645). With the same equation, each criterion can be calculated. The result is shown in Table IV. The COPRAS method was then applied to rank the six 3PL providers using the averaged scores from Table III, with cost (C3) as a minimizing criterion and the others as beneficial ones. The normalized decision matrix is given in Table V. The results of the weighted normalized decision matrix, relative significance, and utility degrees are presented in Tables VI and VII.

Table IV: Result of the R-SWARA method

Table V: Normalized decision matrix

Table VI: Weighted Normalized Decision Matrix

Table VII: Relative significance and utility degrees

As summarized in Table VIII, the findings align with prior studies on 3PL selection, particularly in emphasizing criteria like punctuality and security for perishable goods logistics, in order to ensure product freshness and reliability. The high weight assigned to punctuality underscores its importance in maintaining strict delivery schedules to prevent spoilage, while security reflects the necessity of protecting perishable goods from contamination or damage. For instance, ⁹ highlighted on-time delivery and vehicle quality as key factors in FMCG logistics, which share similarities with fisheries’ cold chain requirements. Similarly, ¹⁰ identified performance and quality assurance as critical in the oil and gas industry, supporting the relevance of criteria like assurance (C6) and customer service (C5) in this study. However, unlike the aforementioned studies, which used single-value weights (e.g., via AHP or TOPSIS), the integration of R-SWARA addresses expert subjectivity by incorporating rough set intervals, offering a methodological advancement over traditional approaches. This is consistent with ⁴³, who demonstrated R-SWARA’s robustness in handling uncertainty regarding transport-related decisions. The use of COPRAS further complements R-SWARA by providing a comprehensive ranking that balances trade-offs across all criteria, including the minimizing criterion cost (C3), which aligns with the emphasis placed by ¹¹ on cost efficiency in FMCG logistics. The close utility degrees suggest competitive performance among vendors, but the leading provider’s dominance reflects its balanced excellence across all criteria, particularly those critical to cold chain logistics.

Table VIII: Comparative table of related works

3.4 Sensitivity Analysis

To evaluate the robustness of the integrated R-SWARA and COPRAS methods for selecting 3PL providers for a mid-sized fishery in Jakarta, Indonesia, a sensitivity analysis was conducted by varying the weight of the highest-ranked criterion, punctuality (C2), established with a rough weight interval of (0.142, 0.170). Punctuality was selected due to its critical role in cold chain logistics, as timely delivery is essential to prevent the spoilage of perishable seafood. The midpoint of C2’s weight, calculated as (0.142 + 0.170) / 2 = 0.156, was adjusted by ± 20%, resulting in a reduced weight of 0.125 (80% of 0.156) and an increased weight of 0.187 (120% of 0.156). To maintain the normalization requirement (criterion weights must sum to 1), the weight difference of 0.031 was redistributed equally across the six remaining criteria: security ((0.113, 0.175)), cost ((0.126, 0.192)), delivery area ((0.080, 0.110)), customer service ((0.124, 0.195)), assurance ((0.126, 0.192)), and logistics experience ((0.122, 0.166)). In the reduced case, each criterion’s midpoint weight increased by approximately 0.005 (0.031/6), whereas, in the increased case, each decreased by 0.005. The COPRAS method was then reapplied, recomputing the weighted normalized matrix, the maximizing and minimizing indices, the relative significance scores, and the utility degrees for the six 3PL providers (A, B, C, D, E, F).

In the reduced weight scenario (C2=0.125), the recalculated utility degrees were approximately as follows: Vendor A at 100%, Vendor D at 99.0%, Vendor C at 98.6%, Vendor F at 98.5%, Vendor E at 98.0%, and Vendor B at 96.9%. The ranking order remained stable (A > D > C > F > E > B), with slight increases in utility degrees for non-top vendors, as the reduced emphasis on punctuality, where Vendor A scored highly with an average of 91.33, slightly narrowed the performance gap. In the increased weight scenario (C2=0.187), the utility degrees were approximately as follows: Vendor A at 100%, Vendor D at 98.7%, Vendor C at 98.3%, Vendor F at 98.2%, Vendor E at 97.6%, and Vendor B at 96.5%. The ranking order was unchanged, with slightly decreased utility degrees due to the greater emphasis on punctuality, amplifying Vendor A’s advantage. The tie between Vendors C and F persisted in both scenarios, reflecting their closely matched performance across criteria, particularly in customer service and delivery area.

Figure 2: Sensitivity analysis

To further validate robustness, an additional sensitivity analysis varied the weights of customer service ((0.124, 0.195)) and delivery area ((0.080, 0.110)) by ± 20% simultaneously, redistributing the weight changes across the remaining criteria. The results consistently maintained the ranking order, with Vendor A at 100%, followed by D, C, F, E, and B, though utility degree gaps narrowed slightly (e.g., D at 99.1%, C and F at 98.5% in the reduced case), as shown in Fig. 2. This stability, driven by R-SWARA’s use of rough set intervals to capture expert uncertainty, aligns with the findings of ⁴³, who demonstrated R-SWARA’s reliability in transport-related decisions. However, extreme weight changes, such as ± 50% for punctuality, could potentially disrupt the tie between Vendors C and F, indicating sensitivity to significant shifts in critical criteria.