Mathematical Problems in Engineering

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Building Mathematical Models for Multicriteria and Multiobjective Applications 2020

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Volume 2021 |Article ID 8833250 | https://doi.org/10.1155/2021/8833250

Chien-Chou Yu, Xiang Li, Hui Lu, "A Novel Procedure to Pursue Aspired Procurement Negotiation Outcomes Using a Combined MADM Model", Mathematical Problems in Engineering, vol. 2021, Article ID 8833250, 17 pages, 2021. https://doi.org/10.1155/2021/8833250

A Novel Procedure to Pursue Aspired Procurement Negotiation Outcomes Using a Combined MADM Model

Academic Editor: Danielle Costa Morais
Received07 Sep 2020
Revised19 Dec 2020
Accepted24 Dec 2020
Published11 Feb 2021

Abstract

In the modern global economy, public and private organizations frequently procure goods and services from external suppliers. As such, negotiations are essential to reach procurement agreements and thus achieve organizational objectives and meet criteria in a timely and economically efficient manner. However, numerous relevant studies have revealed that suboptimal agreements frequently occur in procurement negotiation (PN) settings, which negatively affect the realization of business objectives and criteria. This study proposes the addition of a novel procedure that integrates a combined multiple attribute decision-making (MADM) model into the PN framework to identify, measure, and depict suboptimal situations in the context of an influential network relation map (INRM). This approach enables visualized and systematic information to be continuously provided, thus helping to determine possible improvement initiatives for transitioning suboptimal agreements to aspired levels. A real numerical case study is used to demonstrate the practical application of the proposed procedure. The results reveal that by employing the combined MADM model, the proposed procedure can provide managers with a practical foundation for early identification of the critical factors/dimensions for continuous improvement of a negotiated agreement regardless of how or why a suboptimal agreement initially occurs.

1. Introduction

In the modern cooperative business environment, functions are rarely performed entirely in-house. Public and private organizations must procure goods and services from external suppliers to achieve organizational objectives and criteria in a timely and economically efficient manner [1]. However, managing multiple procurement tasks can be complicated. Procurement typically involves an unclear scope, unforeseen costs, or a long lifespan that requires buyers and sellers to engage in trade-offs of technical, financial, and commercial factors [2]. In such situations, negotiations are commonly used to reach final agreements [3, 4]. Given that procurement is essential to businesses’ success, such negotiations play a critical role in corporate management [58].

However, although some procurement decision-makers successfully apply such negotiations to clarify trade terms in formulating mutually beneficial agreements [9], numerous studies have indicated that other negotiated agreements are unable to fully support all aspects of procurement in achieving business objectives [1012]. For example, in most procurement negotiation (PN) settings, negotiators typically hide information to gain advantages [1]. This opacity of information prevents managers from reaching agreements that ensure optimized business objectives [13]. Additionally, most procurement agreements have a defined lifespan. A time constraint is typically imposed on PN completion [1], which can create pressure and reduce a manager’s motivation or ability to fully process, evaluate, and determine all possible alternatives in the pursuance of optimization [14, 15]. Instead, managers are forced to finalize negotiations to achieve suboptimal agreements that create variations, including changed orders and claims, during their implementation [4, 5]. These studies have implied that suboptimal agreements occur frequently in the PN setting.

A practical survey by Turner and Keegan [12] indicated that an agreement is merely a tentative settlement, which by its nature is incomplete; further improvements are always required later during the process of implementation. More comprehensively, Ertel [11] analyzed hundreds of complex negotiation projects and concluded that “countless deals that were signed with optimism fall apart during implementation, despite the care and creativity with which their terms were crafted.” His study emphasized how decisions made in a negotiated agreement can be affected by future trends, and he also highlighted the need for continuous improvement to facilitate optimal results for the business objectives and criteria. Kujala et al. [3] examined negotiated agreements from the perspectives of sales and implementation and highlighted the importance of improving joint decisions between buyers and sellers over a transaction’s lifecycle. Yang et al. [16] argued that negotiated agreements cause the most substantial project delays, which in turn create serious conflicts during and after implementation of such agreements. These studies have suggested that an efficient approach is required to improve the implementation of negotiated agreements.

The relevant literature on how to settle suboptimal negotiated agreement problems can be traced back to 1985, when Raiffa [17] argued that the majority of negotiated agreements have the potential for improvement. His study also proposed a postsettlement settlement concept that encourages negotiators to use the negotiated agreement as a foundation to seek additional value. Based on Raiffa’s concepts, Susskind [18] argued that pos-settlement settlement can be applicable under cooperative negotiation, in which the negotiating parties treat each other as partners and share information. William [19], who adopted a macroperspective, suggested that monitoring and evaluation of negotiated agreements should be viewed as an essential part of the negotiation process. Smolinski and Xiong [20] concluded that postsettlement settlement is a critical negotiation competency to manage conflicts in the increasingly complex modern business environment. These studies have introduced several helpful ideas to address the problems that are associated with negotiated agreements. However, based on this study’s literature review, these ideas have not been developed into an operational procedure for practical use.

This study proposes a novel procedure that integrates a hybrid multiattribute decision-making (HMADM) model into the PN framework to increase the ability of negotiated agreements to obtain optimal PN outcomes during their implementation. The HMADM model was originally introduced by Tzeng to solve decision problems in interdependent situations [21]. The HMADM model provides theoretical suggestions for how to continuously improve decision implementation toward aspired levels [2225], and thus, its use is appropriate for this study. The proposed procedure enables the identification, measurement, and depiction of suboptimal agreements in a context of the influence network relation map (INRM). The INRM provides managers with visualized and systematic information to easily analyze index gaps among factors, dimensions, and the overall agreement to pursue the aspired PN outcomes. This study uses a numerical example to demonstrate the functionalities of the proposed procedure. The results of the HMADM model revealed that the proposed procedure can provide managers with a critical foundation for the early identification of the critical factors and dimensions of a negotiated agreement, which are needed to continuously improve outcomes, regardless of how and why a suboptimal agreement initially occurs. The remainder of this study is arranged as follows. Section 2 reviews the literature on the PN process. Section 3 introduces the HMADM model comprising the DEMATEL method, the DEMATEL-based analytic network process, and the modified VIKOR method. Section 4 discusses the proposed procedure. Section 5 demonstrates the operation of the proposed procedure by examining a real-world numerical example and discusses the results. Section 6 provides conclusions and further remarks.

2. Literature Review

In practice, negotiations can be characterized differently depending on specific business situations. PNs can occur between buyers and sellers at numerous points in the procurement process. However, this study emphasizes how to improve negotiated agreements during the implementation of such agreements and when two parties have multiple issues in play. Additionally, this study assumes that the parties involved in PN intend to obtain and implement favorable agreements to produce aspired outcomes and thus fulfil their procurement objectives and criteria.

Typically, PN follows a three-phase framework adopted by procurement professionals worldwide [1] comprising prenegotiation, meeting, and postnegotiation (Figure 1).

Prenegotiation starts with issues that the negotiating parties disagree about, such as time, cost, scope, and quality [26]. In the PN environment, if any of these issues or dimensions change, at least one other issue or dimension is likely to be influenced. For example, if the procurement time is reduced, costs often rise due to the additional resources needed to complete the same scope of procurement in a shorter time. If a budget increase is not possible, both procurement scope and desired quality may be reduced. Negotiators must consider these mutually interdependent situations to prepare a set of alternatives to finalize negotiations in each PN phase [1].

Two types of negotiating strategies are most common: win-lose and win-win. In a win-lose situation, each party seeks a maximized share of a fixed amount of resources. In a win-win negotiation, one party’s gain does not necessarily come at the other’s expense [13]. In most PN settings, a win-lose strategy typically emerges early in the negotiation process, but communication and information sharing can transform this into a win-win situation [27]. During the meeting phase, negotiators present their initial offers and then decide whether to make concessions based on a package or separately [3]. Determining and offering the right options can provide flexibility and expedite the negotiation process. A range of decision-making methods have therefore been developed to streamline option selection in negotiations [5].

Keeney and Raiffa [28] proposed a utility function to model negotiator preferences. Rubinstein [29] proposed the use of alternating-offer bargaining in his analysis of negotiation game outcomes. Following these studies, many researchers have proposed negotiation decision aids [30]. Teich et al. [31] classified these decision-aid models into four categories: (a) value function-based and concession-based models; (b) value function-based and Pareto-improvement-seeking models; (c) interactive concession-making models; and (d) interactive Pareto-improvement models. Most decision-aid models emphasize the determination of a Pareto-optimal solution through concessions [32] under limited resources (which are presumably subject to constraints). However, limited resources impose difficulties in determining the Pareto frontier to seek optimal solutions, resulting in suboptimal agreements that deliver inferior outcomes in the conventional approach [20]. To fulfil the shortcomings, Tzeng and Huang [33] proposed a hybrid multiple attribute decision-making (HMADM) model to pursue aspired decision outcomes through continuous improvements.

The HMADM model employs a decision-making trial and evaluation laboratory (DEMATEL) technique [34], which allows for interinfluential effects to be identified between the decision factors/dimensions (or objectives/criteria) on an influential network relations map (INRM). Second, this model applies an analytic network process (ANP) [35] to transform the DEMATEL interinfluential values into influential weights (IWs), so as to be able to better prioritize the decision-making criteria, a procedure called a DEMATEL-based ANP (DANP). Third, this model employs an “aspiration level” principle to modify the multicriteria optimization and compromise solution method, named “ViseKriterijumska Optimizacija I Kompromisno Rešenje” in Serbian (called the “modified VIKOR method”) [36], to avoid “choosing the best among inferior options/alternatives,” [33]. The aspiration level concept was proposed by Simon [37], who argued that actual human decision-making behavior is a sequential process, and a decision-maker is satisfied when a selected alternative aligns with an aspiration level criterion, which is the highly desirable outcome level. Therefore, the modified VIKOR method replaces the traditional max/min approach (choosing relatively good solutions from existing alternatives) with an aspiration level, which enables a shift from ranking and selection when determining the most preferable alternative to attaining performance improvements based on the INRM [36]. By combining these aforementioned concepts and procedures, the HMADM model can provide managers with systematic visual information to allow for the identification of the critical factors needed to enact improvement strategies that would better enable the aspired outcomes to be reached [21].

Currently, behaviors, models, and support systems associated with multiple criteria group decision-making and negotiations remain a prominent research topic for a wide range of applications [38, 39]. For example, Frej et al. [40] proposed a decision support tool based on the FITradeoff method to expertise agreement achievement for multiple criteria group decision-making with situations of partial/incomplete/imprecise information. Their study has also indicated the usefulness of graphical visualization techniques in collective decision-making processes. With these advanced studies, negotiators enable to apply new methods and tools to reduce the level of uncertainties in evaluating, ranking, and selecting alternative offers and counteroffers in obtaining group consensus at different stages within a negotiation process.

In the postnegotiation phase, both parties implement and administer the negotiated agreement in an attempt to achieve the aspired outcomes and to satisfy the determined performance objectives and criteria [11]. However, suboptimal negotiated agreements frequently occur, and the parties involved typically hold meetings to discuss related issues such as performance gaps, possible changes, and improvement requirements [17, 19]. Generally, these meetings can foster closer relationships between the parties and increase mutual trust through information sharing [13]. Additionally, implementing such agreements often takes a long time. In the cloud-computing era, new technologies and management mechanisms emerge on an almost daily basis [41]. Accordingly, in contrast to traditional negotiations that follow a suboptimal agreement, the modern postnegotiation phase provides opportunities to create additional value.

Research has indicated that adding value to a negotiated agreement requires additional effort to evaluate problems and to generate a comprehensive list of potential options from which workable solutions can be selected [13]. Additionally, the negotiating parties must constantly assess the implementation of the agreement and maintain proactive communications to determine possible improvements [16]. As such, the next section proposes a novel procedure to analyze and improve on suboptimal outcomes in the postnegotiation phase within the PN setting.

3. A Novel Procedure to Pursue Aspired PN Outcomes

This section first briefly introduces the essential concepts and computational equations related to the DEMATEL method, DANP, and the modified VIKOR method. Next, it explains the proposed procedure.

3.1. The DEMATEL Method

The DEMATEL method was developed by the Battelle Geneva Institute in 1972 for assessing and solving complex groups of problems. This method uses Boolean operations and Markov processes [34] to measure cause and effect relationships in each dimension or criterion within a system (or subsystem). Quantitative measurements are then mapped onto an INRM-representing problem structure with a visual rout exhibiting the degree and direction in which each dimension or criterion influences the overall system performance. This method has been widely applied to help managers easily obtain valuable information for practical improvements in fields such as security systems and aerospace services [26, 33]. The computational steps for the DEMATEL method are described as follows.

Step 1. Obtain the initial average influence relation matrix A. This step uses a team comprising E experts to identify the number of factors or criteria, n, in a system. Each expert measures the degree of influence that factor i has on factor j in achieving system objectives. Typically, the measurement scale ranges from 0 to 4, with 0 representing “absolutely no influence,” 1 representing “low influence,” 2 representing “medium influence,” 3 representing “high influence,” and 4 representing “very high influence.” Through pairwise comparisons, the results of all expert measurements are denoted as matrix , where , and E is the number of experts. By averaging the matrix , the initial average influence relation matrix A can be obtained, as illustrated bywhere is the measurement by eth expert in He.

Step 2. Obtain the normalized influence relation matrix D. By using A [ ], the normalized influence relation matrix D can be obtained as shown inwhere .

Step 3. Obtain the total influence relation matrix T. Through the matrix operation of D, the total influence relation matrix T can be obtained as shown inwhere I is an identity matrix, and at least one column or one row of summation but not every column or row equals one; then, can be guaranteed.

Step 4. Obtain the INRM. Based on the matrix T, the INRM can be constructed using the following substeps (SSs):SS 4-1. Define each row sum and column sum of matrix T as a respective vector, as shown in the following equations:where indicates the total influence that factor i has on the all other factors and indicates the total influence that the all other factors have on factor j for i, j = 1, 2, …, n; the superscript ′ denotes the transpose.SS 4-2. Compute and , . When , provides an index representing the strength of the total influence that each factor exerts on and receives from the others; that is, indicates the centrality of the role that factor i plays in the system. In addition, indicates the net influence that factor i contributes to the system. If is positive, factor i influences the other factors, and if is negative, factor i is influenced by the other factors.SS 4-3. Plot the dataset into the INRM to visualize the structure of the interrelationship among all factors related to the system performance; this plot reveals valuable information for problem solving.

3.2. DANP

DANP can be described as DEMATEL-based ANP. ANP was proposed by Saaty [35] to address interdependence and feedback among the factors, dimensions, or alternatives associated with a decision-making problem. By applying the basic concept of ANP to formulate the influence relation matrix obtained from the DEMATEL method, DANP can derive the influence weights (IWs) among a set of interrelated factors for superior communication of real interdependent problematic situations, improvement alternatives, and decisions [33]. DANP’s operational steps are described as follows.

Step 5. Obtain an unweighted supermatrix. This step involves the following three SSs:SS 5-1. Classify all factors in the total influence relation matrix T (see Step 3) into the appropriate dimensions (clusters) to form a new matrix, which is referred to as the total influence relation matrix of factors, TC, as shown in equation (6), where , , and is an matrix. SS 5-2. Normalize each dimension (cluster) of criteria by its total degree of effect to obtain a matrix, which is referred to as the normalized total influence relation matrix of factors, as demonstrated inwhere is calculated as shown in equation (8). The other elements, , can be obtained using the same method.where .SS 5-3. Transpose into an unweighted supermatrix W according to the dependent relationship in the dimension (cluster), as shown inwhere W11 is calculated as indicated in equation (10). The other Wij can be obtained using the same method. If a blank space or a zero appears in the matrix, the dimension or factor is independent.

Step 6. Obtain the weighted supermatrix. This involves the following three SSs:SS 6-1. Based on equation (6) in SS 5-1, each dimension in TC is grouped as the total influence relation matrix of dimensions TD by usingSS 6-2. Normalize each dimension in TD with respect to the total degree of the effects and obtain the normalized total influence relation matrix of dimensions , as shown inwhere .SS 6-3. Based on equations (9) and (12), the weighted supermatrix can be obtained using equation (13), where is a scalar and .

Step 7. Limit the weighted supermatrix. This step involves multiplying the weighted supermatrix by itself, denoted as , until converges to some other value that is the same for all instances of j. Using , the IWs of factors can be obtained as follows: .

3.3. Modified VIKOR

The VIKOR method was proposed by Opricovic [42] to solve problems that involve incommensurable and conflicting factors. Originally, this method focused on analyzing a set of alternatives and selecting a compromise solution closest to the ideal state [36, 43]. The ideal state was defined as a set of maximum or minimum values related to the benefit or cost criteria for all alternatives, if performance values exhibited k alternatives in the jth criterion, where higher is better, and then and in the conventional approach (i.e., “max-min” as the benchmark). However, these traditional compromises can lead to situations of “choosing the best among inferior options/alternatives”; hence, Tzeng and Huang [33] proposed the modified VIKOR method to replace the maximum/minimum approach with the “aspired worst” method, in which and are set as the aspired level and the worst level, respectively, for criterion j, if the questionnaire scores range from 0 (“complete dissatisfaction/bad”) to 10 (“extreme satisfaction/good”). Recently, this method has been used to aid decision-makers in identifying critical gaps in need of further improvement [44]. Details regarding the operational steps of the modified VIKOR method are presented as follows.

Step 8. Obtain an aspired or tolerable level. Assume that a problem with K alternatives denoted as is evaluated using n factors. The performance value of alternative with respect to the jth factor is denoted as , and the IW of the jth factor is denoted as obtained using DANP, where . Then, the best value (aspired level) and worst value are determined for all criteria , and the original ratings are transformed into normalized gap-ratings, as shown in

Step 9. Compute the mean of group utility Sk (average value) and maximal regret Qk (priority improvement). In this step, Sk indicates the synthesized gaps based on the weighted average operations from each factor into specific dimensions and overall. Qk indicates the maximal gap of k alternatives in each dimension and the overall for the priority improvement. The values of Sk and Qk are computed as presented in the following equations:where are the influential weights of the criteria from DANP.

Step 10. Compute the index value. The index values are computed using the relation as shown inwhere , and setting “ and ” and “ and ”. We can also assume that and (when all criteria are achieved up to the aspiration level, completely and without gaps as the target) and that and (i.e., all criteria are as in the worst situations); is presented as the weight of the strategy for paying attention to the average gap Sk (i.e., the maximum group utility in how the gap nears zero), or paying attention to punishing gap Qk which is the maximized gap criterion (i.e., the individual regret/gap) that should be prioritized for improvement. As such, equation (17) can be rewritten as equation (18) to measure the gaps with dual ranking and selecting criteria.where , , , and .

Step 11. Rerank or improve the alternatives for a compromise solution. In this step, the values for are sorted in decreasing order, and a compromise solution is presented based on the alternatives by applying equation (17). Finally, the performance gaps in each factor, each dimension, and the overall system that correspond to the compromise solution are measured using equations (15) and (16).

3.4. Development of the Proposed Procedure

This section introduces how the MADM method is integrated into the PN setting to evaluate the dependency relationships and performance gaps in a negotiated agreement for obtaining the aspired PN outcomes. Figure 2 is a graphical representation of the proposed procedure. As presented in Figure 2, the proposed procedure first organizes a team of experts familiar with the procurement project under negotiation. The experts then identify and determine strategies to improve all factors that may impair the realization of the procurement objectives, dimensions, and criteria during the postnegotiation stage. More specifically, the proposed approach comprises four main stages: (1) identification of factors (criteria) to be improved using the expert team; (2) evaluation of interinfluential effects at all tiers using the DEMATEL method to reveal the degree and direction of each factor’s influence upon the aspired objectives or criteria; (3) computation of the IW of each factor or dimension using DANP to determine at which priority level they help reduce the performance gaps; and (4) measurement of the gap indices using the modified VIKOR method. We then integrate the measurements with the weights on the INRM, thus providing managers with systematic tools for determining how to implement continuous improvement. The next section demonstrates the proposed procedure in practice.

4. Application of the Proposed Procedure to a Real Numerical Example: Results and Discussion

This section examines a logistics service project operating in both the public and private sectors to demonstrate the proposed procedure’s functionality in a PN setting. To preserve project information confidentiality, all data are transformed into equivalent units.

4.1. Background

The case study organization (hereafter referred to as the buyer) has implemented a logistics transformation policy by outsourcing part of its organic workload to domestic private contractors. This policy requires potential contractors to ensure that support systems are reliable, available, and maintainable by the end users. Due to the limited number of suppliers in the market, such logistics service projects are executed through sole-source procurement by negotiation. One example is a three-year logistics support service for a TA helicopter that the buyer purchases from Company B (hereafter referred to as the seller) in a PN context. During the meeting phase of the negotiation, the seller disagrees with the supportability deployment level of several subsystems requested by the buyer. To meet the end user’s deadline, the buyer accepts the seller’s offered supportability. However, the compromise-level agreement contains performance gaps that could negatively impact the buyer’s operations. Therefore, the buyer asks the seller to implement improvements over the 3-year service period. The seller claims that because this necessitates additional resources, they cannot guarantee improvements to all subsystems. Meanwhile, the seller asks the buyer to submit a proposal for further consideration. The buyer applies this study’s proposed procedure to obtain a proposal to satisfactorily settle the problems that result from the suboptimal negotiated agreement.

4.2. Application of the Proposed Procedure

This section describes how the buyer follows the proposed procedure to pursue its aspired PN outcomes.

4.2.1. Identification of Factors to be Improved Using an Expert Team

The buyer forms a team of experts from its departments in charge of logistics (two experts), procurement (one expert), finance (one expert), end users (two experts), and project management (two experts), all of whom have considerable backgrounds in this project. The team members analyze the negotiated agreement and the pertinent information, determining that three main systems (dimensions) of the 12 subsystems (factors or criteria) contain performance gaps that require improvement in the deployment level of supportability (Table 1). Table 1 illustrates the three different levels of supportability under the buyer’s consideration on each respective subsystem in terms of its operations. The threshold is the lowest level of supportability (minimized criteria) required by the buyer, the deployment is the highest level of supportability (maximized criteria) that the buyer expects to obtain, and the negotiated level of supportability is what the buyer and the seller agree on. Additionally, the gaps to be improved are the differences between the negotiated level and the deployment level that the buyer requests from the seller during implementation of the agreement.


Dimensions and factors/criteriaSupportability (%)
Main systems (dimensions)Subsystems (factors/criteria)ThresholdDeploymentNegotiatedGaps to be improved

Airframe (7)Landing gear (LG)7685787
Hydraulics system (HS)7886806
Fuel system (FS)8088835
Assistant power units (APU)7884804
Structure system (SS)7886797
Transmission system (TS)7684786
Flight control system (FCS)7583776
Electrics (2)Communication system (CS)8084831
Radio system (RDS)8287852
Weapons (3)Missile system (MS)7981801
Rocket system (RS)8588862
Gun system (GS)8489872

4.2.2. Evaluation of Interinfluential Effects Using the DEMATEL Method

As illustrated in Table 1, the team members aggregate the analytical results in the 12-by-12 matrix of initial average influence relations A (Table 2) by using equation (1). The initial average influence relation matrix is further normalized to obtain matrix D (Table 3) by using equation (2). Subsequently, the team members calculate a total influence relation matrix T by using equation (3). They then use equation (6) to classify the 12 factors and three dimensions to obtain a total influence relation matrix of factors or criteria , after which equation (11) is used to obtain a total influence relation matrix for dimensions . The results are summarized in Table 4. In matrix T, the inconsistency rate of the evaluation results from the experts is only 1.16%, which is much lower than 5%. This result implies that the inclusion of an additional expert would not influence the findings and that the results are significant at a confidence level of 98.84%.


ALDHSFSAPUSSTSFCSCSRDSMSRSGS

LD0.00001.12501.62501.62503.00002.00001.12500.37500.37500.37500.37500.3750
HS3.62500.00003.87503.75002.00003.50003.50001.75001.75003.12503.00002.8750
FS2.62503.62500.00001.37501.25001.25001.37501.25001.25001.25001.25001.1250
APU3.50003.00002.87500.00002.00002.87503.12501.87501.87502.87502.00001.0000
SS2.37501.87502.12503.00000.00002.87502.25002.00001.87502.87502.87502.7500
TS2.00002.75002.75002.87502.00000.00002.87501.00001.00001.87501.87501.8750
FCS2.00002.12502.00003.00001.00002.00000.00003.00003.87503.12503.00002.8750
CS1.25000.37501.00000.37501.87501.00003.12500.00004.25002.62502.62503.0000
RDS1.12500.25000.87500.25001.75000.87503.00003.75000.00002.87502.87502.8750
MS0.62501.75000.87501.75001.75001.00001.87501.00001.00000.00002.50002.5000
RS0.62501.00000.87500.87501.75001.00001.00001.00001.00002.62500.00002.8750
GS0.62500.87500.87500.87501.87501.00001.12500.87501.00002.62503.12500.0000


DLDHSFSAPUSSTSFCSCSRDSMSRSGS

LD0.00000.03440.04960.04960.09160.06110.03440.01150.01150.01150.01150.0115
HS0.11070.00000.11830.11450.06110.10690.10690.05340.05340.09540.09160.0878
FS0.08020.11070.00000.04200.03820.03820.04200.03820.03820.03820.03820.0344
APU0.10690.09160.08780.00000.06110.08780.09540.05730.05730.08780.06110.0305
SS0.07250.05730.06490.09160.00000.08780.06870.06110.05730.08780.08780.0840
TS0.06110.08400.08400.08780.06110.00000.08780.03050.03050.05730.05730.0573
FCS0.06110.06490.06110.09160.03050.06110.00000.09160.11830.09540.09160.0878
CS0.03820.01150.03050.01150.05730.03050.09540.00000.12980.08020.08020.0916
RDS0.03440.00760.02670.00760.05340.02670.09160.11450.00000.08780.08780.0878
MS0.01910.05340.02670.05340.05340.03050.05730.03050.03050.00000.07630.0763
RS0.01910.03050.02670.02670.05340.03050.03050.03050.03050.08020.00000.0878
GS0.01910.02670.02670.02670.05730.03050.03440.02670.03050.08020.09540.0000


T (T C)LDHSFSAPUSSTSFCSCSRDSMSRSGS

LD0.06600.09580.11120.11310.14710.12080.10640.06610.06930.09060.08790.0833
HS0.24580.13980.24900.24810.20180.23610.26460.17520.18410.27040.26190.2485
FS0.16080.18050.08420.12550.12130.11960.13890.10900.11390.14270.13980.1311
APU0.22180.20180.20210.12380.18020.19980.23160.16130.16900.23600.20830.1743
SS0.18440.16640.17550.20150.11960.19430.20320.16110.16510.23460.23060.2193
TS0.16680.18200.18450.19000.16310.10440.20560.12320.12990.19070.18650.1786
FCS0.17430.17130.17170.19910.15300.17030.14470.19540.22580.24800.24080.2299
CS0.11890.09170.11090.09740.14570.11160.19500.08780.21030.19840.19710.2025
RDS0.11070.08440.10300.08980.13790.10390.18600.18560.09000.19890.19780.1940
MS0.09380.12140.09980.12620.12670.10300.14280.09760.10250.10190.17050.1651
RS0.07990.08920.08650.08950.11580.09020.10470.08620.09010.16070.08530.1619
GS0.08050.08670.08710.09030.11990.09090.10860.08360.09070.16190.17370.0823

TDAirframe (D1)Electrics (D2)Weapons (D3)

Airframe (D1)0.16860.14630.1921
Electrics (D2)0.12050.14340.1981
Weapons (D3)0.10160.09180.1404

Note., where denote the average influence of the ith criterion on the jth criterion by the experts (samples) for and , respectively, and denotes the number of factors or criteria. Thus, the results are significant at a confidence level of 98.84%, which is greater than the 95% level used to test for significance.

Table 4 illustrates the degree of total influence that a factor exerts on the other factors using equation (4) and the degree of total influence received by a factor from the other factors using equation (5). The results are summarized in Table 5, which also presents role centrality () and net influence () for the factors and dimensions. Based on these values, the buyer establishes an INRM illustrating the degrees and directions of interinfluential effects between the 12 factors in the three dimensions associated with the supportability deployment level for the mission operations (Figure 3).


Dimensions/factorsriciri + ciri − ci

Airframe (D1)0.89780.1163
 LD (C1)1.15751.70382.8613−0.5463
 HS (C2)2.72521.61114.33641.1141
 FS (C3)1.56731.66553.2328−0.0981
 APU (C4)2.30991.69434.00420.6157
 SS (C5)2.25561.73223.98780.5234
 TS (C6)2.00551.64503.65050.3606
 FCS (C7)2.32442.03214.35650.2923

Electrics (D2)0.84360.0805
 CS (C8)1.76741.53213.29960.2353
 RDS (C9)1.68201.64073.32270.0413

Weapons (D3)0.86430.1968
 MS (C10)1.45142.23483.6862−0.7834
 RS (C11)1.24012.18023.4203−0.9400
 GS (C12)1.25622.07093.3271−0.8148

The values in bold are used to distinguish the total influence between dimensions and factors.

For example, in the airframe dimension (D1; upper right in Figure 3), the x-coordinate is the role centrality (), and the y-coordinate is the net influence (). First, with reference to Table 5, we determine the values for the airframe (D1), electrics (D2), and weapon (D3) dimensions, which are (0.8978, 0.1163), (0.8436, 0.0805), and (0.8643, −0.1968), respectively. We then determine their directions based on the degree of total influence of the dimensions according to Table 4, which indicates that the total influence degree of the D1 on D2 is 0.1436; conversely, the total influence degree of D2 on D1 is 0.1205. The directional arrow is then drawn from D1 to D2 because 0.1436 is greater than 0.1205. The influence directions between all dimensions and factors are similarly determined and presented in Figure 3. As Figure 3 illustrates, the interinfluential relationships between the three dimensions are as follows: (1) D1 influences D2 and D3, and (2) D2 influence D3. When adopting the same approach, the interinfluential effects visualized on the different tiers of the INRM reveal a structure that allows for analysis of the factors and dimensions that require enhanced scrutiny when determining strategies for improvement.

4.2.3. Computation of IWs Using DANP

After the establishment of the INRM, the expert team normalizes the total influence relation matrix of factors and dimensions by using equations (7), (8), (11), and (12), as exhibited in Table 6. Additionally, the expert team transposes the matrices in Table 6 to an unweighted supermatrix by using equations (9) and (10) and then uses equation (13) to obtain a weighted supermatrix, as illustrated in Table 7. Finally, the buyer multiplies the weighted super matrix until it converges into a steady-state condition, with the IWs for the factors and dimensions, as presented in Table 8.


LDHSFSAPUSSTSFCSCSRDSMSRSGS

LD0.08690.12600.14630.14870.19340.15890.13990.48800.51200.34600.33590.3180
HS0.15500.08820.15710.15650.12730.14890.16690.48760.51240.34630.33540.3183
FS0.17280.19390.09040.13480.13030.12850.14920.48910.51090.34490.33800.3171
APU0.16300.14830.14850.09100.13240.14680.17010.48830.51170.38150.33670.2818
SS0.14810.13360.14100.16180.09610.15610.16320.49390.50610.34280.33690.3203
TS0.13940.15210.15420.15880.13640.08730.17190.48680.51320.34310.33550.3214
FCS0.14720.14460.14500.16810.12920.14380.12220.46390.53610.34510.33500.3199
CS0.13650.10530.12720.11180.16720.12810.22390.29450.70550.33170.32960.3386
RDS0.13570.10350.12620.11010.16910.12740.22800.67330.32670.33670.33480.3285
MS0.11530.14920.12270.15510.15570.12660.17550.48780.51220.23290.38970.3774
RS0.12190.13610.13190.13650.17660.13750.15960.48910.51090.39400.20920.3969
GS0.12120.13060.13120.13610.18060.13690.16350.47980.52020.38740.41560.1970

Airframe (D1)Electrics (D2)Weapons (D3)

Airframe (D1)0.33260.28860.3788
Electrics (D2)0.26080.31040.4288
Weapons (D3)0.30440.27500.4206


WLDHSFSAPUSSTSFCSCSRDSMSRSGS

LD0.02790.04970.05540.05230.04750.04470.04720.03190.03170.03350.03540.0352
HS0.04040.02830.06220.04760.04290.04880.04640.02460.02420.04330.03950.0380
FS0.04690.05040.02900.04760.04520.04950.04650.02970.02950.03560.03830.0381
APU0.04770.05020.04320.02920.05190.05090.05390.02610.02570.04510.03970.0395
SS0.06200.04080.04180.04250.03080.04370.04140.03900.03950.04520.05130.0525
TS0.05100.04780.04120.04710.05010.02800.04610.02990.02970.03680.03990.0398
FCS0.04490.05350.04790.05460.05240.05510.03920.05230.05320.05100.04640.0475
CS0.15020.15010.15060.15030.15200.14990.14280.10060.22990.14460.14500.1422
RDS0.15760.15770.15730.15750.15580.15800.16500.24090.11150.15180.15150.1542
MS0.12850.12860.12810.14170.12730.12740.12820.14100.14310.09620.16270.1600
RS0.12480.12460.12550.12500.12510.12460.12440.14010.14230.16090.08640.1717
GS0.11810.11820.11780.10470.11900.11940.11880.14390.13960.15580.16390.0814

LDHSFSAPUSSTSFCSCSRDSMSRSGS

LD0.03790.03750.03730.03730.03750.03770.03760.03640.03640.03750.03730.0373
HS0.03720.03740.03640.03700.03710.03700.03700.03600.03610.03670.03700.0372
FS0.03740.03730.03780.03730.03750.03730.03740.03650.03650.03750.03720.0373
APU0.03840.03830.03850.03880.03820.03830.03820.03760.03770.03820.03860.0386
SS0.04390.04450.04460.04450.04460.04440.04450.04460.04450.04480.04440.0443
TS0.03770.03780.03810.03780.03770.03820.03780.03710.03720.03800.03780.0378
FCS0.05030.05000.05010.05000.05000.05000.05020.04990.04980.04980.05010.0500
CS0.15280.15270.15270.15270.15240.15270.15370.16140.14470.15220.15220.1527
RDS0.16120.16120.16130.16130.16150.16120.16020.15300.16980.16090.16090.1603
MS0.13700.13700.13700.13590.13720.13710.13710.13830.13790.14030.13580.1362
RS0.13530.13530.13520.13510.13520.13530.13530.13700.13650.13430.13990.1327
GS0.13110.13100.13120.13230.13100.13090.13090.13210.13300.12990.12880.1356


LDHSFSAPUSSTSFCSCSRDSMSRSGS

LD0.03710.03710.03710.03710.03710.03710.03710.03710.03710.03710.03710.0371
HS0.03670.03670.03670.03670.03670.03670.03670.03670.03670.03670.03670.0367
FS0.03710.03710.03710.03710.03710.03710.03710.03710.03710.03710.03710.0371
APU0.03820.03820.03820.03820.03820.03820.03820.03820.03820.03820.03820.0382
SS0.04450.04450.04450.04450.04450.04450.04450.04450.04450.04450.04450.0445
TS0.03760.03760.03760.03760.03760.03760.03760.03760.03760.03760.03760.0376
FCS0.05000.05000.05000.05000.05000.05000.05000.05000.05000.05000.05000.0500
CS0.15260.15260.15260.15260.15260.15260.15260.15260.15260.15260.15260.1526
RDS0.16110.16110.16110.16110.16110.16110.16110.16110.16110.16110.16110.1611
MS0.13750.13750.13750.13750.13750.13750.13750.13750.13750.13750.13750.1375
RS0.13590.13590.13590.13590.13590.13590.13590.13590.13590.13590.13590.1359
GS0.13170.13170.13170.13170.13170.13170.13170.13170.13170.13170.13170.1317

IWsLDHSFSAPUSSTSFCSCSRDSMSRSGS

Local0.03710.03670.03710.03820.04450.03760.05000.15260.16110.13750.13590.1317
GlobalAirframeElectricsWeapons
LDHSFSAPUSSTSFCSCSRDSMSRSGS
0.28120.31370.4051
0.13190.13050.13190.13580.15830.13370.17780.48650.51350.33940.33550.3251

The values in bold are used to distinguish the IWs between dimensions and factors.