Advances in Fuzzy Systems

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Fuzzy Methods for Data Analysis

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Research Article | Open Access

Volume 2015 |Article ID 841485 |

Jonathan Davis, Margaret F. Shipley, Gary Stading, "A Fuzzy Supplier Selection Application Using Large Survey Datasets of Delivery Performance", Advances in Fuzzy Systems, vol. 2015, Article ID 841485, 14 pages, 2015.

A Fuzzy Supplier Selection Application Using Large Survey Datasets of Delivery Performance

Academic Editor: Ferdinando Di Martino
Received27 Aug 2014
Revised10 Oct 2014
Accepted13 Oct 2014
Published19 Mar 2015


A model is developed using fuzzy probability to screen survey data across relevant criteria for selecting suppliers based on fuzzy expected values. The values are derived from qualitative variables and expert opinion of membership in these variables found in industry survey data. The application is made to a supply chain management decision of supplier selection based upon delivery performance which is further divided into attributes that comprise this criterion. The algorithm allows multiple criteria to be considered for each decision parameter. Large sets of survey data regarding six suppliers in the electronic parts industry are gathered from over 150 purchasers and are analyzed through spreadsheet modeling of the fuzzy algorithm. The resulting decision support system allows supply chain managers to improve supplier selection decisions by applying fuzzy measures of criteria and associated beliefs across the dataset. The proposed model and method are highly adaptable to existing survey datasets, including datasets that have incomplete data, and can be implemented in organizations with low decision support resources, such as small and medium sized organizations.

1. Introduction

Selecting suppliers has become an area of increasing study due to its importance for establishing long-term channel relationships. Firms have become highly selective of their suppliers but suffer in this process because existing evaluative measures often do not rank suppliers on a relative basis. Thus, of importance is that the firm be able to identify suppliers through an effective evaluation process [1]. Yet this supplier selection evaluation process is a complex decision space with overlapping, complementary, and often contradictory selection criteria. Coupled with the ambiguity in selection criteria, uncertainty in such a decision exists since all relevant attributes may not be identified or even identifiable.

Many factors contribute to the difficulty of rank ordering in the supplier evaluation process. Selection criteria and scoring can occur at various levels of the organization, which leads to conflicts in scoring, particularly in systems built to reflect qualitative criteria. Assessors operating in these systems often experience inherent uncertainty regarding supplier performance due to lack of a proper anchoring or definition of numerical scores. This ambiguity means that a supplier’s score on a criterion may not be determinant. Assessors rely on statistical analysis that provides the highest probability of achieving a qualitatively defined membership, yet membership of a supplier in the category of “good,” for example, may vary in opinion over time and by different assessors.

Scoring models have been used in various aspects of supply chain management for many years because supplier selection lends itself to models incorporating management opinion [2, 3]. However, decision support systems can benefit from improvements that bring clarity to otherwise vague scoring criteria [46], because such improvements enlarge the available sets of useful data these systems can use. Given the benefit of using expert judgment from management and professionals, a fuzzy set based model that has a proven record of handling uncertainty of the decision maker and ambiguity of data while allowing these experts to assess membership across several criteria can be an effective evaluation tool.

Indeed, extant methods that use classical logic or statistics have been shown to be inadequate for effectively dealing with situations where limited information is available. Behavioral and expected utility theorists have argued that probability theory and other traditional quantitative techniques are not equipped to consider the uncertainty that exists in most personal judgments [7, 8]. Among alternative techniques, the Dempster-Shafer theory of evidence [912] provides useful measures for the evaluation of subjective certainty. Fuzzy set theory is among the most powerful tools for dealing with uncertainty where ambiguous terms are present [1316]. Of particular interest to the development of the algorithm and the ensuing application are fuzzy probability distributions [17] and the concept of fuzzy expected values [18].

This research furthers the use of fuzzy logic models in supplier selection decisions. Background information is presented relative to traditional supplier selection methods and the general advancement of fuzzy logic-based models into supply chain management with more recent models targeting supplier selection decisions. Additional background literature is presented that identifies the traditional criteria for supplier selection. Next, fuzzy logic notation and model development lead to a spreadsheet application based on a survey of responses from a sample of professional buyers. Guidelines are introduced to resolve problems with the dataset regarding data insufficiency and/or incomplete responses, and the dataset is filtered to create a subset of the original dataset that is sufficiently populated with data. Results are then presented and analyzed.

2. Materials and Methods

As noted in the introduction, supplier selection decisions are receiving increasing focus in industry, and the associated academic papers have no shortage of ideas about how selection decisions can be made. Alternatives range from simple expert opinion methods that are not reliably repeatable to quantitative methods that may not account for various criteria associated with standards for supplier performance.

2.1. Supplier Selection Models

Recent supplier selection models have focused on using the analytic hierarchy process (AHP) or providing case study illustrations of decision making processes utilizing expert opinions [1921]. Measuring the performance of suppliers [1] and stressing the importance of supplier selection criteria [2] for small firms in the United States [22] or large firms in Japan [23] reinforce that this initiative is not restricted to large firms and is global. Typical criteria used are price, delivery, and quality (PDQ) [1, 23] but Park and Krishnan [22] add several managerial criteria beyond PDQ including managerial forecasts, trust level, and organizational structure. Supplier selection research using decision support systems also extends supplier selection criteria in other dimensions including environmental considerations [6]. Weber et al. [24] considered other factors like facility location, capacity, and financial position. Ellram [25] discussed the importance of incorporating these and other decision criteria in the process of selecting suppliers, and Simpson et al. [1] provide an extensive study of criteria perceived as most crucial to the assessment.

Using the traditional PDQ criteria, Verma and Pullman [26] provide an extensive study of how purchasing managers evaluate trade-offs among the criteria. They point out that while the PDQ criteria are generally accepted in industry, delivery and quality lend themselves to decision support criteria models because the complexity and multiple dimensions of delivery and quality can confound a decision maker. Of these criteria, quality tends to be extensively studied by researchers [21], while supplier selection and evaluation are most often based entirely on price [27, 28]. Thus, by elimination, delivery does not have nearly the research support.

In response to this dearth of research and in recognition of the global importance of supplier selection, four separate focus groups representing international and national geographical areas were used to identify attributes of delivery that would confound a purchasing decision. While many attributes were identified by the focus groups, those used for the “delivery” example in this study include the following: inventory availability, on-time deliveries, shipping accuracy, and a supplier’s receptiveness to authorizing material returns (RMAs), as supported by Simpson et al. [1]. Given the previous justification of delivery as the focus criterion for this project with support by Vahdani and Zandieh [29] as to the strength of fuzzy set theory and outranking methods relative to the other decision support methodologies, fuzzy logic appeared appropriate for supplier selection decisions.

2.2. Fuzzy Logic in Supply Chain Management

As early as 1999, Liu [30] proposed a fuzzy model for partial backordering models. Little was done with inventory considerations until fully five years later when inventory discounting considered the buyer-seller relationships [31], and location aspects for inventory control became fuzzy considerations [32]. Supply chain decisions for integrated just-in-time inventory systems recognized the fuzzy nature of annual demand and production rates as being no longer statistically based. The supposition of known annual demand was considered by the authors to be unrealistic such that the proposed model included fuzzy annual demand and/or production rate, employing the signed distance, a ranking method for fuzzy numbers, to estimate fuzzy total cost of the JIT production in today’s supply chain environment. A fuzzy-set based method derived the optimal buyer’s quantity and number of lots from the vendor [33].

Some fuzzy set-based decision-making models appeared when fuzzy programming was used for the following: optimal product mix based on ABC analysis [34]; fuzzy multiobjective linear programming minimized total production and transportation costs; the number of rejected items and total delivery time as related to labor and budget constraints [35]; and fuzzy goal programming considered supply chain management from the perspective of activity-based costing with mathematically derived optimization for evaluating performance of the value-chain relationship [36]. Manufacturing processes as related to business logistics looked at the data itself as fuzzy in Quality Function Deployment’s relationship to customer service [37]. The attainment of goals such as quality further led to attempts to balance production processes of assembly lines. Fuzzy goals were used as an instrument and product for measuring, displaying, and controlling industrial process variables [34].

Considering different quality standards in a supply chain network a fuzzy neural approach was utilized to suggest adjustments of product quantity from various suppliers [38]. The Fuzzy Suitability Index (FSI) aggregated rankings and multiplied, by weight, each criterion [39]. With the same goal of ranking suppliers according to performance, a method was proposed whereby decision makers evaluated the performance of suppliers in criteria, rating the importance of the criteria in linguistic terms. Aggregation of the fuzzy expressions for importance weights and a fuzzy preference index led to rank ordering of the suppliers [40].

Supplier selection was developed from a rule-based perspective. The approach selected was fuzzy associated rule mining from the database for supplier assessment [14]. Sevkli [41] in a comparison of a recognized crisp ELECTRE model versus a fuzzy ELECTRE model reached the conclusion that using fuzzy sets for multicriteria supplier selection decisions is superior.

2.3. Theory

The model developed herein uses fuzzy set theory and extension principles with fuzzy probabilities that determine a belief-weighted score for each attribute of each criterion. A revision to an existing algorithmic process [12] is defined with presentation and justification specific to this supplier selection decision-making problem based upon delivery attributes.

2.3.1. Fuzzy Logic Approach

Fuzzy logic addresses the ambiguity of data and uncertainty in this decision making situation, where a fuzzy subset of a set is a function of into . For a brief foundation in the basics, see Zadeh [42], Bellman and Zadeh [13], Dubois and Prade [43], and Freeling [44]. While a new class of implication operators has been proposed [45], the more traditionally utilized fuzzy operations are used in this research. and denote two fuzzy sets, so the intersection, union, and complement are defined byand it is assumed that [4649].

Extension principles [18, 43, 50] often guide the computations when dealing with fuzzy sets. Letting be a function from into , with as any set and as above, then can be extended to fuzzy subsets of by

Thus, is a fuzzy subset of . In particular, if is a mapping from a Cartesian product such as to any set, , then can be extended to objects of the form where and are fuzzy subsets of and byA fuzzy set whose elements all lie on the interval can be expressed as a fuzzy probability.

2.3.2. Fuzzy Probability Distributions

Consider a set of fuzzy probabilities each having elements,where denotes the degree of belief that a possible value of is . Then constitutes a finite fuzzy probability distribution if and only if there are -tuples , , such that .

To qualify as a finite fuzzy probability distribution, each fuzzy probability in the distribution must have the same number of elements (some of the a’s may be zero), and these elements should be ordered in the sense that the sum of the elements in each specific position must equal one. So the -tuples , , form probability distributions in the crisp sense. This type of probability distribution can be transformed such that the resulting distribution has entropy at least as great as the original [17].

2.3.3. Fuzzy Expected Value

A version of fuzzy expected values was first used when Zebda [18] defined as the fuzzy probability that, from State and making Decision , reaches State . Associated with this are fuzzy benefits , where .

Then the averaged benefit is defined by where

2.3.4. Fuzzy Supplier Selection

In an earlier pilot study application [51], a fuzzy probability-based algorithm demonstrated a decision model based on a single rating for each supplier. But suppliers are usually not scored by only one kind of criterion, the attributes of which may vary in importance from one customer to another. Also, the application abbreviated the decision space considerably by disallowing suppliers that scored Below Average on an attribute, which works well for a single-attribute example but breaks down when applied across many attributes or criteria in a large dataset.

The proposed fuzzy algorithm recognizes that a supplier could score Below Average on an attribute that is relatively unimportant to the customer but still be rated highest overall if the supplier’s other scores are sufficiently high in areas of greater importance to the customer. Since the many respondents are not equally confident in the scores they give, the algorithm makes use of the value of fuzzy probability with respect to each supplier measure and limits the discarding of a great deal of valuable data.

2.4. Algorithm

The algorithm preserves information during the process of computing and evaluating fuzzy probabilities until a final weighted model collapses the results into an objective score.(0)Randomly partition the criteria dataset into subsets of equal size.(1)For each attribute of each supplier , subjectively assign scores . The supplier rating is then given by the equation for all where (; ; and ).(2)Define the fuzzy expected value, , for each attribute of each in terms of each as for all , where each represents belief in the probability that will be scored (; ; and ).(3)Group the probabilities into combinations such that for some set of ’s. for .(4)Across all partitions , compute if , otherwise 0 (; and = the distinct number of ; ).(5)For all find , where is the degree of belief that the expected value is .(6)Defuzzify the expected value for each attribute to find .

An illustration of this supplier selection modeling algorithm is presented next.

2.5. Algorithm Methods and Illustration

The application presents a real-world supplier selection decision-making problem based upon generation of data from a survey of purchasing professionals and partitioning of the resulting data to fit the algorithm detailed above.

2.5.1. Example Data

The algorithm example uses results from a survey instrument built with input from industry expert focus groups. The subsequent survey measures customer ratings of a group of suppliers for various variables including delivery attributes. The survey was distributed to about 3,000 companies that purchase semiconductors, passives, RF/microwaves, connectors and interconnects, and electromechanical devices from a small set of dominant suppliers. Representative industries included automotive, communications, contract design/engineering, power/electrical, medical/dental, computer, manufacturing, and military/aerospace. The survey queried each customer’s number of years of activity in the industry in designated ranges from less than two to 21 or more. Customers dealt with multiple suppliers and specified their firm’s annual sales revenue as under $5,000,000 to over $2,000,000,000. With 406 surveys received, the response rate was slightly under 15%.

The survey was scored on a 0–5 Likert scale for seven suppliers: Arrow, Avnet, Future, Insight, Kent, Pioneer, and TTI. Respondents used the same scale for satisfaction with each supplier individually on the service performance attributes.

For model application purposes, the survey provided performance measurements on each supplier, as well as measures of the importance of each criterion to the customer and the customer’s level of belief explicitly tied to the company’s annual amount of business conducted with the targeted group of suppliers. Survey questions relating directly to the importance of this fuzzy supplier selection application included a query of the amount of money the customer spends on electronic components in a year: <$100,000; $100,000–$499,999; $500,000–$999,999; $1,000,000–$9,999,999; $10,000,000–$24,999,999; and >$25,000,000. These ranges were used to identify a firm’s level of activity with the suppliers in question and, therefore, its expected level of confidence (interpreted as belief) in its assessments.

Given that real-world collection of survey data is, to varying degrees, imperfect, a threshold (10 data points) was established beyond which a respondent’s data cluster would be considered too incomplete and removed. Furthermore, one supplier, Kent, was removed from the set due to (a) low survey responses compared to the other suppliers and (b) no longer existing as an independent company, having been acquired by Avnet after the survey was conducted. The resulting dataset left a pool of 150 responses considered “useable” to be applied to the fuzzy algorithm. These responses each had between 0 and 10 missing data points, which correspond to individual survey questions that went unanswered by the respondent. It is necessary to accept and adapt to this imperfect data in order to have a sufficiently large dataset to draw useful conclusions, given the nature of survey data.

The 150 acceptable survey responses resulting from the filtering of the data from the original 406 surveys were randomly partitioned into two sets of 75 responses each in accordance with Step 0 of the model algorithm. These respondents evaluated suppliers on the delivery-specific attributes: on-time performance, availability of inventory, shipping accuracy, and RMAs.

2.5.2. Fuzzy Preference Algorithm

Using the partitioned datasets provides two inputs into the defined fuzzy probability distributions.

By Step 1 of the algorithm, attributes as defined above. Each of the four attributes is subjectively assigned a score by the respondent for each of the six suppliers (), equating to Poor, Below Average, Average, Above Average, and Excellent (). Supplier rating is then given by the equation for each supplier, , and, by Step 2, the fuzzy probability for each attribute of in terms of is for all . Each represents belief in the probability that will perform to the level of the assigned score (; ; ; and ).

The belief functions were populated based on a survey question indicating the amount of annual spending done by the respondent. Table 1 describes the scoring of respondent belief as proportional to total possible spending (conservatively assumed to be the low end of the top category, $25,000,000) as shown in Table 1.

SpendingDegree of belief


Beliefs were assigned for each respondent according to Table 1 and averaged across all respondents for each rating. After the assignment of belief, the algorithmic process for supplier shows four significant digits to make the method clear, although subsequent suppliers are rounded to two significant digits for readability and brevity. is as follows:For Arrow, .

In the case of 0.0 beliefs, the estimation of no likelihood (0.0) that the supplier’s on-time delivery will rate Poor is because no respondents in partition one scored this supplier as Poor. Since one respondent in partition two did rate the company’s delivery performance as Poor, and this respondent’s sales revenue volume was in the middle range, there is a 0.51 belief that there is a 0.04 probability that Arrow’s on-time delivery will be Poor. The highest beliefs (0.39 and 0.38) are for low probabilities (0.2267 and 0.2133) that the supplier has Average on-time performance. The highest probabilities (0.6133 and 0.3600) for Above Average have among the highest beliefs (0.3 and 0.35, respectively). While one group of respondents considered a 0.3067 probability of occurrence with 0.19 beliefs, the other group held an even higher belief for a low probability (0.1067) of Excellence in on-time delivery by Arrow.

Beliefs and corresponding probabilities are then defined as

According to Step 3 of the algorithm, all combinations of the five scores across both data partitions are considered for each outcome that sums one, which in this example yields

Possible -tuples are (0.0000, 0.0533, 0.2267, 0.6133, 0.1067) and (0.0400, 0.0800, 0.2133, 0.3600, 0.3067). Following Step 4, a “weighted average” probability for all is derived

The minimum degree of belief in the on-time delivery then assessed according to Step 5 considers only the cases where belief is greater than zero:

Step 6 defuzzifies the expected score such that ’s expected fuzzy score for on-time delivery performance is

Applying the algorithm (with significant digit rounding) to the second supplier (Avnet) yields

Again, according to Step 3 of the algorithm, all combinations are considered for each outcome that sums one, which in this example yields combinations:as for supplier (Arrow) but also

At this point, the spreadsheet organizes and solves the equations. The expected values are calculated based upon the algorithmic steps from the following respondent-provided fuzzy probabilities.

2.6. Illustrative Spreadsheet Example

While the algorithmic process is not overly complex, the transitioning of the algorithm into a realistic decision support system (DSS) capable of handling an extensive set of data has many challenges. Due to spreadsheet ease of use and availability to businesses, including small to mid-size enterprises, the focus of the research was to provide a fully functioning spreadsheet-based DSS. The illustration provided shows the resulting complexity of interpreting the algorithm with a standard spreadsheet approach.

An organized layout of the intermediate results of algorithm Steps 1 and 2 was created, as shown in Table 2, where 1–5 represent the qualitative variables defined as Poor, Below Average, Average, Above Average, and Excellent, respectively. Question 20 referred to the on-time delivery performance of the suppliers denoted as “a” through “f” for Arrow, Avnet, Future, Insight, Pioneer, and TTI, respectively.

SupplierOn-Time deliveryAvailability of inventoryShipping accuracyReturn authorizationOverall average

: Arrow3.803.773.933.803.82
: Avnet3.693.633.903.683.72
: Future3.713.603.813.303.61
: Insight3.653.333.833.663.62
: Pioneer3.573.273.703.613.54
: TTI3.893.714.063.683.83

The probability calculation is a simple counting of the number of responses of, for example, “2,” divided by a count of the number of responses in the appropriate partition. The function used is as follows:=COUNTIF([cell range of responses],  “=2)/COUNTIF([cell range of responses],  “>0)

The belief calculation relied upon the assumptions stated above and given in Table 1. Accordingly, the belief value for each score (1–5) is calculated using the following formula:=SUMIF([cell range of scores],  “=2,  [cell range of beliefs])/COUNTIF([cell range of scores],  “=2)

Rounding leads to the two-significant-digit 5-tuples shown in Table 3.


First partitioned dataProbability of score0.00000.01330.04050.02780.06760.0000
Weight of score (belief)0.00000.20000.63000.45000.26000.0000

Second partitioned dataProbability of score0.04000.02670.04110.01430.08110.0282
Weight of score (belief)0.25000.36000.73000.20000.31000.7000

Organizing the 5-tuples into sums of fuzzy probabilities is the next step, according to Step 3 of the algorithm. This was accomplished by creating a row/column spot for every possible combination of every element of every question, as shown in Table 3.

The numbers in Table 3 represent the possible combinations of beliefs summed for Question 20 for suppliers through ; 16 for each data partition per supplier. The shaded cells represent combinations that have fuzzy probabilities that sum 1.0 or that sum very close to 1.0, within the established threshold. Note that the top and bottom combinations always sum exactly to one since all possibilities are represented by the five scores (1–5) in any given partition.

With the 5-tuples identified that have fuzzy probabilities that sum 1, the resulting score is calculated for each of those 5-tuples by multiplying each fuzzy probability by its corresponding score. As an example, the overall score of the first 5-tuple is found using the following equation (rounded to four significant digits):The scores for all these first 5-tuples are shown in Table 4.


Raw sums of -tuples1.
0.991.010.911.01 1.001.03
0.970.971.091.01 0.990.94
0.990.961.050.921.040.96 1.011.06
1.010.991.090.99 1.000.97
1.010.991.090.99 1.000.97

Step 5 of the algorithm requires that the minimum belief for each usable 5-tuple is identified, discarding any case where the belief was zero.

Finally, each value in Table 4 is multiplied by each value in Table 5 and then summed down each column to form an expected score for each supplier on each question.


Fuzzy probabilities times values 1 through 53.77333.66673.66223.66673.51353.9254

Using the spreadsheet results, the first delivery attribute assessed (on-time delivery) is calculated from the fuzzy probability distributions for the remaining suppliers as