Fuzzy evaluation model for attribute service performance index

1 Introduction

Chen et al. [1] pointed out that store owners can conduct survey questionnaires about customers’ satisfaction as well as identify service items which require improvement using the performance evaluation matrix so as to raise the percentage of customer satisfaction and in-store consumption [2–7]. As suggested by some researchers, the number of customers who enter a store can be offered to help a store measure its performance because it can instantly affect business profits and operating costs [8–10]. Some researchers pointed the number of customers who enter a store is a Poisson process [11–15]. Let N _Lt be the minimal required number of customers’ arrival at a store in the t unit of time. The attribute service performance index built on the Poisson process proposed by Chen et al. [1] is rewritten as follows: S λ t = λ t N Lt ,(1)

where λt is the Poisson process mean. It is obvious that the Poisson process mean λt is larger than the minimal number of customers N _Lt in the t unit of time, and the value of attribute service performance index S _λt is larger than 1. As the Poisson process mean λt is large, the value of service performance index S _λt is large as well. Obviously, service performance index S _λt will reduce to S _λ, proposed by Chen et al. [1]. At the same time, N _Lt will fall to N _L and λt will fall to λ.

OPEN IN VIEWER

Fig. 1

Architecture diagram.

Since the service performance index has unknown parameters, Chen et al. [1] found that the Uniformly Minimum Variance Unbiased Estimator (UMVUE) of the service performance index S _λt is built on random samples. Also, the UMVUE is viewed as the test statistic measuring whether the operating performance of a store can meet the performance requirements.

In addition, some researchers have indicated that as the Internet of Things (IoT) gets more and more prevalent and well-developed [16–18], a variety of technologies for business data collection also constantly improve. Effective data analysis and applications can support businesses to make wise and quick decisions within a short time [19–21]. In order to grasp the timeliness of evaluation, Chen et al. [1] reviewed the rules of statistical testing and put forward a fuzzy testing model for index S _λt on the basis of the confidence interval to improve the testing accuracy of small-sized samples. This fuzzy testing method is based on the confidence interval and then uses the derived critical value as the basis for the fuzzy testing. In order to be more convenient and closer to the application in the industry, this paper directly adopts the fuzzy one-tailed testing method by means of the upper confidence limit of the index to evaluate whether the operation performance of the store meets the required level and quickly grasp the opportunity for improvement. After evaluation, when the decision is that improvement is necessary, the two-tailed improvement testing of dual fuzzy indicators is directly carried out with the confidence interval of the indicators before and after the improvement, so as to confirm that the improvement effect can meet the required level. This is the motivation and purpose of the two fuzzy testing models proposed in this paper.

In summary, the architecture diagram, regarding the attribute service performance index used for evaluating store performance and the fuzzy evaluation model based on the confidence interval of this index, is displayed as follows:

Obviously, the advantages of the fuzzy testing model proposed in this paper are:

The proposed fuzzy evaluation model is built on the confidence interval of the index, so that the wrong judgment resulting from sampling error can be minimized.

Concerning other sections in this paper, they are arranged as follows. In Section 2, the central limit theorem is adopted to gain the confidence interval for the service performance index, and the fuzzy one-tailed testing model of index S _λt is re-examined as well. In Section 3, a fuzzy two-tailed testing model of two indices based on the confidence interval is put forward to verify whether the improvement has a significant effect. For the application of the industry, this paper will provide a case to demonstrate the application of the dual-index two-tailed fuzzy testing model presented by this study in Section 4. Last but not least, conclusions are made in Section 5.

It is assumed that random variable X represents the number of customers’ arrival the store in a time unit. As described above, we let random variable X be distributed as a Poisson distribution with mean λt. Let X₁, . . . , X _j , . . . , X _n be a random sample retrieved from random variable X. The probability density function of random variable X can be shown as follows: f ( x ) = e - λ t ( λ t ) x x ! , x = 0 , 1 , 2 , . . . . .(2)

As noted by Chen et al. [1], the uniformly minimum variance unbiased estimator of index S _λt is expressed as follows: S λ t * = λ * t N Lt ,(3) where λ * t = ( ∑ j = 1 n X j ) / n . Then, ∑ j = 1 n X j is distributed as Poisson distribution with mean nλt, E [ S λ t * ] = E [ λ t * N Lt ] = n λ t nN Lt = S λ t(4) and Var [ S λ t * ] = E [ ( S λ t * ) 2 ] - E 2 [ S λ t * ] = S λ t nN Lt(5)

Let Z = S λ t * - S λ t S λ t * / nN L ,(6) then Z is approaching the standard normal distribution when the sample size is large according to the Central Limits Theorem (CLT). Therefore, we have 1 - α = p { - z α / 2 ⩽ ( S λ t * - S λ t ) S λ t * / nN L ⩽ z α / 2 } = p { S λ t * - z α / 2 × S λ t * nN Lt ⩽ S λ t ⩽ S λ t * + z α / 2 × S λ t * nN Lt }(7) where z _α/2 is the upper α/ 2 quintiles of N (0, 1). The 100 (1 - α)% confidence interval (CI) of S _λt is denoted as: CI = [ S λ t * - z α / 2 × S λ t * nN Lt ⩽ S λ t ⩽ S λ t * + z α / 2 × S λ t * nN Lt ] .(8)

Let x₁, . . . , x _j , . . . , x _n be the observed value of X₁, . . . , X _j , . . . , X _n . The observed value of s λ t * is expressed as: s λ t * = λ 0 * t N Lt ,(9) where λ 0 * = ( ∑ j = 1 n x j ) / n is the observed value of λ^*. Based on Chen et al. [22] and Li et al. [23], the α - cuts of the triangle-shaped fuzzy number S ˜ λ t is expressed as: S ˜ λ t [ α ] = { [ S λ t 1 ( α ) , S λ t 2 ( α ) ] , 0 . 01 ⩽ α ⩽ 1 [ S λ t 1 ( 0.01 ) , S λ t 2 ( 0.01 ) ] , 0 ⩽ α ⩽ 0.01 ,(10) where S λ t 1 ( α ) = s λ t * - z α / 2 × s λ t * nN Lt(11) and S λ t 2 ( α ) = s λ t * + z α / 2 × s λ t * nN Lt .(12)

Therefore, the triangle-shaped fuzzy number is S ˜ λ t = Δ ( S L , S M , S R ) , where S L = s λ t * - z 0.005 × s λ t * nN Lt = s λ t * - 2.576 × s λ t * nN Lt ,(13) S M = s λ t * ,(14) and S R = s λ t * + z 0.005 × s λ t * nN Lt = s λ t * + 2.576 × s λ t * nN Lt .(15)

Furthermore, the membership function of S ˜ λ t is expressed below: η ( x ) = { 0 , if x < S L 2 × ( 1 - Φ ( s λ t * - x s λ t * / nN Lt ) ) , if S L ⩽ x < S M 1 , if x = S M 2 × ( 1 - Φ ( x - s λ t * s λ t * / nN Lt ) ) , if S M < x ⩽ S R 1 , if S R < x(16)

Based on the above triangle-shaped fuzzy number and membership function, this study presented a one-tailed fuzzy testing model with the service performance index to evaluate the performance of a store. Suppose R should be the performance requirement value of the service index S _λt of a store. The condition of the hypothesis test is considered below: H 0 : S λ t ⩾ R ; H 1 : S λ t < R .

Subsequently, according to Equation (16), the membership functions diagram of η (x) with vertical line x = R is displayed in Fig. 2.

OPEN IN VIEWER

Fig. 2

The membership function diagram of η (x) with vertical line x = R.

Therefore, suppose A _T should be the area in the graph of η (x) as follows: A T = { ( x , α ) | S λ t 1 ( α ) ⩽ x ⩽ S λ t 2 ( α ) , 0 ⩽ α ⩽ 1 } .(17)

As noted by Chen et al. [24] and Yu et al. [25], the calculation of A _T is so complicated that this study replaced A _T with d _T , the length of the bottom of A _T . Then, d _T  = S _R  - S _L can be expressed as follows: d T = 2 × z 0.005 × s λ t * nN Lt = 5.512 × s λ t * nN Lt .(18)

Additionally, suppose A _R should be the area to the right of the vertical line x=R in the graph of η (x), displayed as follows: A R = { ( x , α ) | R ⩽ x ⩽ S λ t 2 ( α ) , 0 ⩽ α ⩽ a } ,(19) where a = 2 [ 1 - Φ ( R - s λ t * s λ t * / nN Lt ) ] ,(20) such that S_λt2 (a) = R. Therefore, d _R  = S _R −R is expressed as follows: d R = s λ t * + 2.576 × s λ t * nN Lt − R .(21)

Also, the value of d _R / d _T can be received as follows: d R d T = 2.576 × s λ t * / nN Lt + s λ t * - R 5.152 × s λ t 0 * / nN Lt .(22)

Note that we let 0 < φ₁ < φ₂ < 0.5[26], in which the values of φ₁ and φ₂ can be received according to the production data accumulated in the past or professionals’ experience [27–31]. As noted by Yu et al. [32], the fuzzy testing rules are inferred as follows:

If d _R / d _T  ⩽ φ₁, then reject H₀ and conclude S _λt  < R.

As mentioned earlier, a confidence-interval-based fuzzy testing model for service index S _λt was put forward by this study to evaluate whether a store can meet the requirements of service performance and seize the opportunities for improvement. Next, this paper used the fuzzy testing model to verify the improvement effect. Let random variable X′ represent the number of customers who enter the store in a unit of time after improvement. As described above, we assume that random variable X′ is distributed as a Poisson distribution with mean λ′t. Let X 1 ′ , . . . , X j ′ , . . . , X n ′ ′ be a random sample of random variable X′ and x 1 ′ , . . . , x j ′ , . . . , x n ′ ′ be the observed value of this random sample. Then the service performance index estimation formula is expressed as follows: S λ ′ t * = λ ′ * t N Lt .(23)

The observed value of S _λ′t^* is s λ ′ t * = λ 0 ′ * t N Lt ,(24) where λ 0 ′ * = ( ∑ j = 1 n ′ x j ′ ) / n ′ is the observed value of λ^′*. Like S ˜ λ t , the α - cuts of triangle-shaped fuzzy number S ˜ λ ′ t is expressed as follows: S ˜ λ ′ t [ α ] = { [ S λ ′ t 1 ( α ) , S λ ′ t 2 ( α ) ] , 0.01 ⩽ α ⩽ 1 [ S λ ′ t 1 ( 0.01 ) , S λ ′ t 2 ( 0.01 ) ] , 0 ⩽ α ⩽ 0.01 ,(25) where S λ ′ t 1 ( α ) = s λ ′ t * - z α / 2 × s λ ′ t * n ′ N Lt(26) and S λ ′ t 2 ( α ) = s λ ′ t * + z α / 2 × s λ ′ t * n ′ N Lt .(27)

Therefore, the triangle-shaped fuzzy number is S ˜ λ ′ t = Δ ( S L ′ , S M ′ , S R ′ ) , where S L ′ = s λ ′ t * - z 0.005 × s λ ′ t * n ′ N Lt = s λ ′ t * - 2.576 × s λ ′ t * n ′ N Lt ,(28) S M ′ = S λ ′ t 0 * ,(29) and S R ′ = s λ ′ t * + z 0.005 × s λ ′ t * n ′ N Lt = s λ ′ t * + 2.576 × s λ ′ t * n ′ N Lt(30)

Accordingly, the membership function of S ˜ λ ′ t is: η ′ ( x ) = { 0 , if x < S L ′ 2 × ( 1 - Φ ( s λ ′ t * - x s λ ′ t * / n ′ N Lt ) ) , if S L ′ ⩽ x < S M ′ 1 , if x = S M ′ 2 × ( 1 - Φ ( x - s λ ′ t * s λ ′ t * / n ′ N Lt ) ) , if S M ′ < x ⩽ S R ′ 1 , if S R ′ < x(31)

Based on the above-mentioned, to test the outcome of the improvement, the condition of the hypothesis test is taken into account as follows: H 0 : S λ t = S λ ′ t and H 1 : S λ t ≠ S λ ′ t .

Figure 3 below presents a diagram corresponding to membership functions η (x) and η′ (x).

OPEN IN VIEWER

Fig. 3

Membership functions η (x) and η′ (x).

As shown in Fig. 3 and based on Equations (12, 26), α = a′, such that { S λ ′ t 1 ( a ′ ) = S λ t 2 ( a ′ ) = c for S λ ′ t ⩾ S λ t S λ t 1 ( a ′ ) = S λ ′ t 2 ( a ′ ) = c for S λ ′ t < S λ t(32)

Suppose set A _T should be the area in the graph of η (x) shown in Equation (17), and d _T  = S _R  - S _L is shown in Equation (18). Conversely, suppose set A R ′ should be the area but to the right of the vertical line, x=c, in the graph of η (x), displayed as follows: A R ′ = { ( x , α ) | c ⩽ x ⩽ S λ t 2 ( α ) , 0 ⩽ α ⩽ a ′ } ,(33) where η (x) is shown in Equation (33). Therefore, d R ′ = S R − c is displayed as follows: d R ′ = s λ t * + 2.576 × s λ t * nN Lt - c(34)

Then, the value of d R ′ / d T can be expressed below: d R ′ d T = 2.576 × s λ t * / nN Lt + s λ t * - c 5.152 × s λ t 0 * / nN Lt(35)

As same as Section 2, we let 0 < φ < 0.5, where the value of φ can be determined by means of the previously accumulated production data or expert experience. As noted by Yu et al. [32], the following fuzzy testing rules can be inferred:

If d R ′ / d T ⩽ φ , then reject H₀ and conclude S _λ′t > S _λt , showing that the improvement has been effective.

As noted by Chen et al. [1], the number of customers who enter a store is distributed as a Poisson process with the mean λt = λ for t = 1. Suppose the store should set the minimal required number of customers who enter the store as N _Lt  = N _L  = 10 in a unit of time. The performance requirement value of the index is larger than or equal to 1 (S _λt  ⩾ 1). Therefore, the condition of the hypothesis test is demonstrated as follows: H 0 : S λ t ⩾ 1 ; H 1 : S λ t < 1 .

The observed value of the random sample is x₁, . . . , x _j , . . . , x₂₅ with sample size n = 25, then λ 0 * = ( ∑ j = 1 25 x j ) / 25 = 9.0 and S λ 0 * = λ 0 * N L = 0.90

Therefore, the triangle-shaped fuzzy number is S ˜ λ t = Δ (0.745, 0.90, 1.055), and the values can be expresses as follows: S L = s λ t * - 2.576 × s λ t * nN Lt = 0.90 - 2.576 × 0.90 250 = 0.745 , S M = s λ t * = 0.90 , and S R = s λ t * + z 0.005 × s λ t * nN Lt = 0.90 + 2.576 × 0.90 250 = 1.055 .

Accordingly, the membership function of S ˜ λ t is: η ( x ) = { 0 , if x < 0.745 2 × ( 1 - Φ ( 0.90 - x 0.06 ) ) , if 0.745 ⩽ x < 0.90 1 , if x = 0.90 2 × ( 1 - Φ ( x - 0.90 0.06 ) ) , if 0.90 < x ⩽ 1.055 1 , if 1.055 < x

Then, d _T  = S _R  - S _L  = 1.055−0.745 = 0.31 and d _R  = S _R −R = 1.055−1 = 0.055. Thus, we have d _R / d _T  = 0.055/0.31 = 0.177. According to previously accumulated production data and experts’ experience, we have φ₁ = 0.2 and φ₂ = 0.4. In the one-tailed fuzzy testing rules, since d _R / d _T  = 0.17 < 0.2, then reject H₀ and conclude that S _λt  < 1.

As mentioned earlier, because the testing result is S _λt  < 1, it must be improved. Based on the abovementioned, to examine the improvement effect, the problem of the hypothesis test is considered as follows: H 0 : S λ t = S λ ′ t and H 1 : S λ t ≠ S λ ′ t .

It is assumed that x 1 ′ , . . . , x j ′ , . . . , x 30 ′ is the observation values of a set of random variables after improvement, denoted as follows: x 1 ′ = 11 , x 2 ′ = 10 , x 3 ′ = 11 , x 4 ′ = 14 , x 5 ′ = 11 , x 6 ′ = 12 , x 7 ′ = 10 , x 8 ′ = 15 , x 9 ′ = 11 , x 10 ′ = 10 , x 11 ′ = 11 , x 12 ′ = 10 , x 13 ′ = 12 , x 14 ′ = 11 , x 15 ′ = 12 , x 16 ′ = 14 , x 17 ′ = 11 , x 18 ′ = 13 , x 19 ′ = 11 , x 20 ′ = 13 , x 21 ′ = 14 , x 22 ′ = 13 , x 23 ′ = 10 , x 24 ′ = 14 , x 25 ′ = 11 , x 26 ′ = 13 , x 27 ′ = 12 , x 28 ′ = 12 , x 29 ′ = 10 , x 30 ′ = 14 .

Then, λ 0 ′ * = ( ∑ j = 1 30 x j ′ ) / 30 = 12.0 , and S λ ′ 0 * = λ 0 ′ * N Lt = 1.20 .

Therefore, the triangle-shaped fuzzy number is S ˜ λ t = Δ ( 0.75 , 0.90 , 1.05 ) , where S L ′ = s λ ′ t * - 2.576 × s λ ′ t * n ′ N Lt = 1.20 - 2.576 × 1.20 300 = 1.038 , S M ′ = s λ ′ t * = 1.20 , and S R ′ = s λ ′ t * + 2.576 × s λ ′ t * n ′ N Lt = 1.20 + 2.576 × 1.20 300 = 1.362 .

Then, the membership function of S ˜ λ ′ t is: η ′ ( x ) = { 0 , if x < 1.038 2 × ( 1 - Φ ( 1.20 - x 0.063 ) ) , if 1.038 ⩽ x < 1.20 1 , if x = 1.20 2 × ( 1 - Φ ( x - 1.20 0.063 ) ) , if 1.20 < x ⩽ 1.362 1 , if 1.362 < x

Figure 4 below presents a diagram corresponding to membership functions η (x) and η′ (x).

OPEN IN VIEWER

Fig. 4

The membership functions η (x) and η′ (x) for practical application.

Based on Equation (32–35), we have a′ = 0.014, such that c = S _λ′t1 (0.014) = S_λt2 (0.014) =1.045 . Therefore, d R ′ = S R − c = 1.055 − 1.045 = 0.01 and d R ′ / d T = 0.01 / 0.31 = 0.032 . Considering the previously accumulated production data and experts’ experience, we take φ = 0.2. In the fuzzy testing rules, since d R ′ / d T = 0.032 < φ = 0.2 , then reject H₀ and conclude that S _λ′t > S _λt , indicating that the improvement has been effective.

Chen et al. [1] came up with an attribute service performance index on the basis of the Poisson process. This paper re-examined this attribute service performance index and put forward a confidence-interval-based one-tailed fuzzy testing model to evaluate whether the operation performance of a store can reach the required level and judge whether it needs improvement. Next, a duel-index two-tailed fuzzy testing model on the basis of the confidence interval was proposed to identify whether the improvement has a significant effect. Since these two fuzzy testing models were built on the confidence interval of the index, the risk of misjudgment caused by sampling error could be diminished. In addition, the design of these two fuzzy testing models can incorporate the past data or expert experience, so even when a sample size is small, the accuracy of evaluation can still remain unchanged. As a result, they can satisfy the needs of enterprises for rapid response and reduce the evaluation cost. In the cases provided by this paper, the first one-tailed fuzzy test assessed the need for improvement, showing that it can grasp the opportunity for improvement; the second two-tailed fuzzy test not only can verify the effectiveness of improvement but also can check whether the method of improvement is correct. The latter case also makes readers more aware of the application of the model proposed in this paper.