Journal Identification = IFS
Article Identification = 0684
Date: January 11, 2013
Time: 5:11 pm
1
Journal of Intelligent & Fuzzy Systems xx (20xx) x–xx DOI:10.3233/IFS-120684 IOS Press
Sequential sampling plan using fuzzy SPRT
2
Ezzatallah Baloui Jamkhaneha,∗ and Bahram Sadeghpour Gildehb
3
a Department
4
b Department
7 8 9
or P
6
of Statistics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran of Statistics, Faculty of Basic Science, University of Mazandaran, Babolsar, Iran
Abstract. In this paper we introduce a new sequential sampling plan based on sequential probability ratio test for fuzzy hypotheses testing. In order to deal with uncertainty happened, the triangular fuzzy number (TFN) is used to express the fuzzy phenomenon of the sampling plan’s parameters. In this plan the acceptable quality level (AQL) and the lot tolerance percent defective (LTPD) are TFN. For such a plan, we design a decision criterion of acceptance and rejection for every arbitrary λ-cut and a particular table of rejection and acceptance is calculated. This plan is well defined, since, if the parameters are crisp, it changes to a classical plan.
uth
5
roo f
1
Keywords: Sequential sampling plan, fuzzy number, acceptable quality level, lot tolerance percent defective
11
1. Introduction
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Item by item sequential sampling plan, in traditional form, is based on the sequential probability ratio test (SPRT) with exact hypotheses. However, there are many different situations in which the parameters are imprecise. In the design sequential plan two levels quality AQL and LTPD, are crisp. In the other hand, sometimes the available qualities are not precise. To overcome this problem, first we are developed SPRT for fuzzy hypotheses where the imprecise parameters are as fuzzy numbers. Then item by item SSP is defined and extended based on fuzzy SPRT. Testing fuzzy hypotheses was discussed by Arnold [2] and [3], Delgado et al. [17], Watanabe and Imaizumi [18], Taheri and Behboodian [19], Torabi and Behboodian [10], Torabi and Behboodian [11]. SPRT for the fuzzy hypotheses were discussed by Torabi and Mirhosseini [12], Torabi and Behboodian [9] and Akbari [16]. Sadeghpour Gildeh et al. [4] and Baloui Jamkhaneh et al. [6–8] considered acceptance sampling plan under the conditions of the fuzzy parameter. We organize the matter in the following way: The next section introduces SPRT for fuzzy hypotheses. In the third section we design the fuzzy sequential sampling plan, considered broadly, and its limiting lines are computed. The results are summarized in the conclusions.
cte
15
rre
14
co
13
One of the most widely used statistical quality control (SQC) tools is the lot acceptance sampling plan (LASP), which can be applied in a variety of ways. In LASP area, several sampling schemes are available for the application of attribute quality characteristics, namely, single, double, multiple, chain and sequential sampling plan (SSP). In sequential sampling, we take a sequence of samples from the lot and allow the number of samples to be determined by the results of the sampling process. Under sequential sampling, samples are taken, one at the time, until a decision is made on the lot to accept, reject or continue sampling (take another sample and then repeat the decision process). If the inspected sample size at each sequence of them is greater than one, the plan is called group sequential sampling, but if at each stage it is one, the plan is called item by item sequential sampling [5]. Sequential sampling was developed in1943 for use in rapid quality inspection in war research and production [15]. SSP has been studied by many researchers, and thoroughly elaborated on by Hardeo Sahai et al. [20].
Un
12
dA
10
∗ Corresponding
author. Ezzatallah Baloui Jamkhaneh, Department of Statistics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran. E-mail: e
[email protected].
1064-1246/12/$27.50 © 2012 – IOS Press and the authors. All rights reserved
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
Journal Identification = IFS
Article Identification = 0684
E.B. Jamkhaneh and B.S. Gildeh / Sequential sampling plan using fuzzy SPRT
62 63 64 65
where “Hj : θ is Hj (θ), j = 0, 1” implies that θ is a fuzzy set in (the parameter space) with membership function Hj (θ) i.e. a function from to [0, 1]. Note that the crisp hypothesis Hj (θ) : → [0, 1], j = 0, 1 is a fuzzy hypothesis with membership function Hj (θ) = 1, θ ∈ j , and zero otherwise. Let random variable X have the fuzzy probability ˜ where θ˜ is a fuzzy density function (FPDF) f˜ (x; θ) parameter. Under the hypotheses Hj , j = 0, 1, the λ-cut of FPDF is as follows f˜ j (x, θ˜ j )[λ] = {fj (x, θj ) θj ∈ θ˜ j [λ]} (2) = [fjL (x)[λ], fjU (x)[λ]], j = 0, 1, where
(3)
Lθ˜ 0 (x1 , ..., xn ) ˜ ˜ n (x) = R θj ∈ θj [λ], j = 0, 1 , Lθ˜ 1 (x1 , ..., xn ) ˜ ˜ n f0 (xi , θ0 ) ˜ = i=1 θj ∈ θj [λ], j = 0, 1 , f˜ 1 (xi , θ˜ 1 )
ni=1 f0L (xi , θ0L ) ni=1 f0U (xi , θ0U ) , = , ni=1 f1U (xi , θ1U ) ni=1 f1L (xi , θ1L )
L Lθ˜ LU ˜ θ 0 = , L0 , Lθ˜ LU ˜θ 1
=
[RL n (x) ,
1
RU n (x)],
where
(4)
fjL (xi , θjL ) = min{fj (xi , θj ) θj ∈ θ˜ j [λ]}, j = 0, 1, fjU (xi , θjU ) = max{fj (xi , θj ) θj ∈ θ˜ j [λ]}, j = 0, 1.
For fixed k0 and k1 satisfy 0 < k0 < k1 , adopt the following procedure: Take observation x1 and com˜ 1 ; if RU ≤ k0 , reject H0 ; if RL ≥ k1 ; accept pute R 1 1 U H0 ; and if RL 1 < k1 and R1 > k0 , take observation x2 , ˜ 2 ; if RU ≤ k0 , reject H0 ; if RL ≥ k1 , and compute R 2 2 U accept H0 ; and if RL 2 < k1 and R2 > k0 , take observation x3 , etc. The idea is to continue sampling as long as U U RL n < k1 and Rn > k0 and stop as soon as Rn ≤ k0 L U or Rn ≥ k1 , rejecting H0 if Rn ≤ k0 and accepting H0 if RL n ≥ k1 . Consequently, the critical region of the described SPRT for fuzzy hypotheses testing of Equation (1) is C = ∪∞ n=1 Cn , where
cte
(For more details see [13]). Let X = (X1 , X2 , ..., Xn ) is a random sample from ˜ and (X) is a test funca population with the f˜ (x; θ), tion, it is the probability of rejecting H0 provided that X = x is observed. The probability of type I and II errors for the testing of fuzzy hypothesis of Equation (1) are ˜ 0 ((X)) and β˜ = 1 − E ˜ 1 ((X)) respectively, α˜ = E ˜ j ((X)) is the fuzzy expected value of (X) in which E ˜ , j = 0, 1 (For more details over the FPDF of f˜ j (x, θ) see [13]). ⎧ ⎫ ⎨ ⎬ ˜ j ((X))[λ] = E (x)fj (x, θj )dx θj ∈ θ˜ j [λ] ⎩ ⎭
rre
66
fjL (x)[λ] = min{fj (x, θj ) θj ∈ θ˜ j [λ]} , fjU (x)[λ] = max{fj (x, θj ) θj ∈ θ˜ j [λ]},
x
co
61
= [EjL ((X))[λ], EjU ((X))[λ]], j = 0, 1, where EjL = min
⎧ ⎨ ⎩
EjU = max
x
⎧ ⎨ ⎩
x
Un
60
roo f
In fuzzy hypothesis testing (FHT) with crisp data, fuzzy hypotheses are H0 : θ˜ = θ˜ 0 H0 : θ is H0 (θ), (1) or H1 : θ is H1 (θ), H1 : θ˜ = θ˜ 1
Now we propose the sequential probability ratio test for FHT. Let X1 , X2 , ... denote a sequence of the iid ˜ random variables, from a population with FPDF f˜ (x; θ). ˜ ˜ First compute sequentially R1 , R2 , ... for an arbitrary λ -cut, where
or P
2. SPRT for fuzzy hypothesis testing
uth
59
Time: 5:11 pm
dA
2
Date: January 11, 2013
⎫ ⎬ (x)fj (x, θj )dx θj ∈ θ˜ j [λ] , ⎭ ⎫ ⎬ (x)fj (x, θj )dx θj ∈ θ˜ j [λ] . ⎭
Cn = {(x1 , ... xn ) RL j < k1 and
(5)
U RU j > k0 , j = 1, ..., n − 1, Rn ≤ k0 }.
Similarly, the acceptance region can be as A = ∪∞ n=1 An , where An = (x1 , ...xn ) RL j < k1 and L RU j > k0 , j = 1, ..., n − 1, Rn ≥ k1
(6)
Journal Identification = IFS
Article Identification = 0684
Date: January 11, 2013
Time: 5:11 pm
E.B. Jamkhaneh and B.S. Gildeh / Sequential sampling plan using fuzzy SPRT
β˜ =
72 73 74 75
n=1C
n
˜ ≤
0
LL θ˜
1
0
0
1
then α˜ =
∞
˜ θ˜ (X)dX ≤ ˜ L 0
n=1C
∞ n=1C
n
n
˜ = k0 (1 − β)
αU ≤ k0 1 − βU
(8)
If RL i ≥ k1 then null hypothesis accepted, it means that in acceptance region we have 0
LU θ˜
∞
˜ θ˜ (X)dX, L 0
n=1A
n
∞ n=1A
U ˜ θ˜ (X)dX = 1 − α β˜ , k1 L 1 βU
(12)
n
˜ ( 1 − α˜ )βU . ⇒ (1 − αU ) β˜ ≤
By using Equations (11) and (12) we have
therefore
LL θ˜
˜ αU (1 − β˜ ), ⇒ α˜ (1 − βU ) ≤
˜ ≥
˜ θ˜ (X)dX k0 L 1
(11) αU (1 − β˜ ), U 1−β
n
1 − α˜ =
1
˜ θ˜ (X)dX = k0 L 1
and ˜˜ ≤ ˜˜ ˜ k0 L ≤ k0 ⇒ LU ≤ k0 LL ⇒ L θ˜ θ θ θ˜
∞ n=1C
Proof. If RU i ≤ k0 then null hypothesis reject, it means that in critical region we have LU θ˜
(10)
According to Equation (10) proof is obvious. Lemma 2. If α˜ and β˜ are the error size of the SPRT defined by k0 and k1 , then α U + β U ≤ αU + βU for arbitrary λ-cut. Proof: ∞ ˜ θ˜ (X)dX, α˜ = L 0
dA
71
0
Ln 1−β ZiL < U , and i=1
U
XA = s(l) n − h(l) (Acceptance line),
(27)
XR = s(u) n + h(u) (Rejection line),
(28)
here, we have Ln s
=
s
=
1−pL 0 1−pU 1
k(u) 1−pU 0 1−pL 1
Ln (l)
kl
,k
(u)
h
=
Ln 1−α βU kl
,h
= Ln
,k
(l)
U
(l)
113
114
Using rejection and acceptance regions, and the ˜ p˜ 1 ), α˜ and β˜ the SSP using parameters p˜ 0 , p˜ 1 (p˜ 0 < fuzzy SPRT is determined by the acceptance and rejection lines given as follows:
(u)
112
i=1
Ln 1−α . βU
∀ p ∈ (0, 1), j = 0, 1. Finally by using of Equations (4) and (22) for arbitrary λ-cut, we obtain U 1−xi xi ni=1 (pL 0 ) (1 − p0 ) , U xi n 1−xi i=1 (p1 ) (1 − pL 1) (23) xi L pL 1 − pU n 0 0 (1 − p1 ) = i=1 , U 1 − pL pU 1 1 (1 − p0 )
n
i=1 n
cte
H0 : p is H0 (p), H1 : p is H1 (p),
RL n (x) =
or P
= Xi Ln
L pL 0 (1 − p1 ) U pU 1 (1 − p0 )
then at the nth stage of sampling,
We propose to design item by item SSP with fuzzy AQL and fuzzy LTPD. Such a plan is based on the concept of the SPRT for fuzzy hypotheses. The sampling distribution of the number of nonconforming items in every stage (Xi ) is the Bernoulli distribution with fuzzy parameter p˜ (incoming quality level), because each item inspected is either defective or not. That is, Xi = 1, if the ith inspected item is defective, otherwise Xi = 0. In FHT with crisp data, fuzzy hypotheses are
j
ZiL
(20)
H1 : p ≈ p1 .
j
If
ZiU
H0 : p ≈ p0 ,
roo f
H0 : p = p0 ,
When designing an item by item sequential sampling plan, four parameters of the AQL, the producer’s risk α (the probability of rejecting a lot with AQL quality), LTPD and the consumer’s risk β (the probability of accepting a lot with LTPD quality) must be determined prior to determining the acceptance and rejection lines. In the design SSP two levels of quality are crisp, but sometimes these are not exact and certain. So, we face a fuzzy hypothesis as follows
L 1−xi xi ni=1 (pU 0 ) (1 − p0 ) , U n L i=1 (p1 )xi (1 − p1 )1−xi xi (24) U pU 1 − pL n 0 (1 − p1 ) 0 = i=1 . L pL 1 − pU 1 (1 − p0 ) 1
RU n (x) =
H1 : p = p1 .
and
dA
5
(u)
=
= Ln Ln 1−β αU ku
L pL 1 (1 − p0 ) , U pU 0 (1 − p1 )
U pU 1 (1 − p0 ) , L pL 0 (1 − p1 )
U
. (29)
Journal Identification = IFS
Article Identification = 0684
116 117 118 119 120 121 122 123 124
E.B. Jamkhaneh and B.S. Gildeh / Sequential sampling plan using fuzzy SPRT
i. k(l) > 0, k(u) > 0, s(u) > 0, s(l) > 0. ii. For low values of p˜ 0 and p˜ 1 , it can be concluded that k(l) ≥ k(u) , and hence we have s(l) ≤ s(u) . iii. The parameters uncertainty is less slop of rejection and acceptance lines are closer together. If the parameters are crisp, then the lines are parallel. ˜ p˜ 0 and p˜ 1 , it can be con˜ β, iv. For low values of α, cluded that h(l) > 0 and h(u) > 0. Instead of using the graph to decisions about of the lot, one can resort to generating tables of acceptance number and rejection number. Both acceptance numbers and rejection numbers must be integers. The acceptance number is the next integer less than or equal to XA , and the rejection number is the next integer greater than or equal to XR . For example, suppose we will be to find a SSP for which α [0] = 0.06, p˜ 0 = (0.005,0.01,0.015), βU [0] = 0.11, p˜ 1 = (0.055, 0.06, 0.065), = 1.3615, s
= 0.0457, h
= 1.9808,
(u)
k(l) = 2.6029, s(l) = 0.01593, h(l) = 0.8242, therefore the acceptance and rejection lines are XA = 0.01593n − 0.8242,
130 131 132
133 134
135
136 137 138 139
(i) Accept lot, if the number of nonconforming items is less than or equal to 1. (ii) Continue sampling, if the number of nonconforming item is in (1, 8). (iii) Reject lot, if the number of nonconforming items is greater than or equal to 8. Table 1 shows that the lot cannot be accepted until at least 51 items have been tested.
4. Conclusions
Re. Number
0 ... 0 0 ... 0 1 ... 1 1 ... 1 1 ... 1 1 1 2
7 ... 7 8 ... 8 8 ... 8 9 ... 9 10 ... 10 11 11 11
a-means that acceptance not possible, b-means that rejection not possible.
cte
rre
129
Now for n = 120, according to Equation (30) we obtain
co
128
Ac. Number
88 ... 109 110 ... 114 115 ... 131 132 ... 153 154 ... 175 176 177 178
References
Un
127
n
b b 3 3 ... 3 4 ... 4 5 ... 5 5 ... 5 6 ... 6
140 141 142 143 144
145
(30)
XR = 0.0457n + 1.9808. 126
Re. Number
a a a a ... a a ... a a ... a 0 ... 0 0 ... 0
dA
k
(u)
Ac. Number
1 2 3 4 ... 22 23 ... 44 45 ... 51 52 ... 66 67 ... 87
comprehensive since, if the AQL and LTPD are crisp, it changes to traditional SSP. The parameters uncertainty is less slop of rejection and acceptance lines are closer together. If the parameters are crisp, then the lines are parallel.
then, with using Equation (29) we have: (u)
n
uth
U
125
Table 1 Sequential sampling plan p0 ≈ 0.01, αU [0] = 0.06, p1 ≈ 0.06, βU [0] = 0.11
Remarks:
roo f
115
Time: 5:11 pm
or P
6
Date: January 11, 2013
In this paper, a new approach for SPRT based on fuzzy hypothesis was presented. Then, we have proposed a method to design SSP with fuzzy AQL and fuzzy LTPD based on our fuzzy SPRT. This plan is
[1] [2]
[3] [4]
[5] [6]
[7]
[8]
[9]
A. Wald, Sequential analysis, John Wiley, New York, 1947. B.F. Arnold, Statistical tests optimally meeting certain fuzzy requirements on the power function and on the sample size, Fuzzy Sets and System 75(2) (1995), 365–372. B.F. Arnold, An approach to fuzzy hypotheses testing, Metrika 44 (1996), 119–126. B. Sadeghpour Gildeh, Gh. Yari and E. Baloui Jamkhaneh, Acceptance double sampling plan with fuzzy parameter, 11th Joint Conference on Information Sciences, Atlantis Press, China, 2008. D.C. Montgomery, Introduction to statistical quality control, Wiley, New York, 1991. E. Baloui Jamkhaneh and B. Sadeghpour Gildeh, Sequential sampling plan by variable with fuzzy parameters, Journal of Mathematics and Computer Science 1(4) (2010), 392–401. E. Baloui Jamkhaneh, B. Sadeghpour Gildeh and Gh. Yari, Inspection error and its effects on single sampling plans with fuzzy parameters, Structural and Multidisciplinary Optimization 43(4) (2011), 555–560. E. Baloui Jamkhaneh, B. Sadeghpour Gildeh and Gh. Yari, Acceptance single sampling plan with fuzzy parameter, Iranian Journal of Fuzzy Systems 8(2) (2011), 47–55. H. Torabi and J. Behboodian, Sequential probability ratio test for fuzzy hypotheses testing with vague data, Austrian Journal of Statistics 34(1) (2005), 25–38.
146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170
Journal Identification = IFS
Article Identification = 0684
Date: January 11, 2013
Time: 5:11 pm
E.B. Jamkhaneh and B.S. Gildeh / Sequential sampling plan using fuzzy SPRT
177 178 179
[13]
180 181 182 183 184 185
[14]
[18] [19] [20]
roo f
[12]
or P
176
[17]
uth
175
[16]
K. Suey-Sheng, Sequential sampling plan for insect pests, Phytopathologist & Entomologist, NTU, 1984, pp. 102–110. M.Gh. Akbari, Sequential test of fuzzy hypotheses, American Open Journal of Statistics 1 (2011), 87–92. M. Delgado, M.A. Verdegay and M.A. Vila, Testing fuzzy hypotheses: A Bayesian approach, in: M.M. Gupta et al. (Eds), Approximate Reasoning in Expert Systems, North-Holland Publishing Co, Amsterdam, 1985, pp. 307–316. N. Watanabe and T. Imaizumi, A fuzzy statistical test of fuzzy hypotheses, Fuzzy Sets and Systems 53 (1993), 167–178. S.M. Taheri and J. Behboodian, A Bayesian approach to fuzzy hypotheses testing, Fuzzy Sets and Systems 123 (2001), 39–48. S. Hardeo, K. Anwer and A. Muhammad, A visual basic program to determine Wald’s sequential sampling plan, Rev Invest Oper 25(1) (2004), 84–96.
dA
[11]
[15]
cte
174
rre
173
H. Torabi, J. Behboodian and S.M. Taheri, Neyman-Pearson lemma for fuzzy hypotheses testing with vague data, Metrika 64(3) (2006), 289–304. H. Torabi and J. Behboodian, Likelihood ratio test for fuzzy hypotheses testing, Statistical Papers 48(3) (2007), 509–522. H. Torabi and S.M. Mirhosseini, Sequential probability ratio tests for fuzzy hypotheses testing, Applied Mathematical Sciences 3(33) (2009), 1609–1618. J.J. Buckley, Fuzzy probability and statistics, Springer-Verlage Berlin, Heidelberg, 2006. K. Dumicic, V. Bahovec and N.K. Zivadinovic, Studying an OC curve of an acceptance sampling: A statistical quality control tool, Proceeding of the 7th WSEAS International Conference on Mathematics & Computers in Business & Economics, cavtat, croatia, JUNE 13-15, 2006, pp. 1–6.
co
[10]
172
Un
171
7
186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
