Construction of AOQ curve and calculate AOQL value based on limiting fraction

Description

AOQL_grab_A provides the AOQ curve and calculates AOQL value based on limiting fraction of contaminated increments.

Usage

AOQL_grab_A(c, r, t, d, N, method, plim)

Arguments

Details

Since P_{ND} is the probability of non-detection, p is the limiting fraction of contaminated increments and the outgoing contaminated proportion of primary increments is given by AOQ as the product pP_{ND}. The quantity AOQL is defined as the maximum proportion of outgoing contaminated primary increments and is given by

AOQL ={\max_{0\leq p\leq 1}}{pP_{ND}}

Value

AOQ curve and AOQL value based on on limiting fraction

See Also

prob_detect

Examples

  c <-  0
  r <-  25
  t <-  30
  d <-  0.99
  N <-  1e9
  method <- 'systematic'
  plim <- 0.30
  AOQL_grab_A(c, r, t, d, N, method, plim)

Construction of AOQ curve and calculate AOQL value based on average microbial counts

Description

AOQL_grab_B provides the AOQ curve and calculates AOQL value based on average microbial counts.

Usage

AOQL_grab_B(c, r, t, distribution,llim, K, m, sd)

Arguments

Details

Since P_a is the probability of acceptance, \lambda is the arithmetic mean of cell count and the outgoing contaminated arithmetic mean of cell count of primary increments is given by AOQ as the product \lambda P_a. The quantity AOQL is defined as the maximum proportion of outgoing contaminated primary increments and is given by

AOQL ={\max_{\lambda \geq 0}}{\lambda P_a}

Value

AOQ curve and AOQL value based on average microbial counts

See Also

prob_accept

Examples

  c <-  0
  r <-  25
  t <-  30
  distribution <- 'Poisson lognormal'
  llim <- 0.20
  AOQL_grab_B(c, r, t, distribution, llim)

Probability of detection or non detection versus fraction nonconforming curve

Description

This function allows comparison of different sampling schemes, which can be systematic and random sampling of primary increments or grab sampling of blocks of primary increments. A graphical display of the probability of detection P_D or probability of non detection P_{ND} versus fraction nonconforming p for up to four selected schemes will be produced.

Usage

compare_plans(d, N, plim, type, c1, r1, t1, method1, c2, r2, t2, method2,
                     c3, r3, t3, method3, c4, r4, t4, method4)

Arguments

Value

Probability of detection or non detection vs limiting fraction curves

Examples

c1 <- 0
c2 <- 0
c3 <- 0
c4 <- 0
r1 <- 1
r2 <- 10
r3 <- 30
r4 <- 75
t1 <- 750
t2 <- 75
t3 <- 25
t4 <- 10
d <- 0.99
N <- 1e9
method1 <- method2 <- method3 <- method4 <- 'systematic'
plim <- 0.10
compare_plans(d, N, plim, type ='D', c1, r1, t1, method1, c2, r2, t2, method2)
compare_plans(d, N, plim, type ='D', c1, r1, t1, method1, c2, r2, t2, method2,
                        c3, r3, t3, method3)
compare_plans(d, N, plim, type ='D', c1, r1, t1, method1, c2, r2, t2, method2,
                        c3, r3, t3, method3, c4, r4, t4, method4)
compare_plans(d, N, plim, type ='ND', c1, r1, t1, method1, c2, r2, t2, method2,
                        c3, r3, t3, method3, c4, r4, t4, method4)

Comparison based on OC curve

Description

This function produces overlaid Operating Characteristic (OC) curves for any three systematic/random sampling schemes for specified parameters.

Usage

compare_plans_oc(c1, c2, c3, r1, t1, r2, t2, r3, t3, distribution, K, m, sd)

Arguments

Value

overlaid OC curves

See Also

prob_accept

Examples

c1 <- 0
c2 <- 0
c3 <- 0
r1 <- 25
r2 <- 50
r3 <- 75
t1 <- 10
t2 <- 10
t3 <- 10
distribution <- 'Poisson lognormal'
compare_plans_oc(c1, c2, c3, r1, t1, r2, t2, r3, t3, distribution)

Serial correlation between grab samples

Description

This function calculates the resulting serial correlation between grab samples each having r primary increments with original serial correlation d.

Usage

correlation_grab(r, p, d)

Arguments

Details

The serial correlation between blocks (grab samples) is given by d_g as

d_g = [dp(1-p(1-d))^{r-1}]/p_d

where p_d is the probability of detection in any of the block (grab sample) which is calculated by using prob_detect_single_grab.

Value

Serial correlation between grab samples

See Also

prob_detect_single_grab

Examples

r <-  25
p <-  0.005
d <-  0.99
correlation_grab(r, p, d)

Construction of Operating Characteristic (OC) curve

Description

oc_plan provides the Operating Characteristic (OC) curve for known microbiological distribution such as lognormal. The probability of acceptance is plotted against mean log10 concentration.

Usage

oc_plan(c, r, t, distribution, K, m, sd)

Arguments

Details

Based on the food safety literature, mean concentration is given by \lambda = 10^{\mu+log(10)\sigma^2/2}.

Value

Operating Characteristic (OC) curve

See Also

prob_accept

Examples

  c <-  0
  r <-  25
  t <-  30
  distribution <- 'Poisson lognormal'
  oc_plan(c, r, t, distribution)

Probability of acceptance for grab sampling scheme

Description

This function calculates the overall probability of acceptance for given microbiological distribution such as lognormal.

Usage

prob_accept(c, r, t, mu, distribution, K, m, sd)

Arguments

Details

Based on the food safety literature, for given values of c, r and t, the probability of detection in a primary increment is given by, p_d=P(X > m)=1-P_{distribution}(X \le m|\mu ,\sigma) and acceptance probability in t selected sample is given by P_a=P_{binomial}(X \le c|t,p_d).

If Y be the sum of correlated and identically distributed lognormal random variables X, then the approximate distribution of Y is lognormal distribution with mean \mu_y, standard deviation \sigma_y (see Mehta et al (2006)) where E(Y)=mE(X) and V(Y)=mV(X)+cov(X_i,X_j) for all i \ne j =1 \cdots r.

The parameters \mu_y and \sigma_y of the grab sample unit Y is given by,

\mu_y =\log_{10}{(E[Y])} - {{\sigma_y}^2}/2 \log_e(10)

(see Mussida et al (2013)). For this package development, we have used fixed \sigma_y value with default value 0.8.

Value

Probability of acceptance

References

• Mussida, A., Vose, D. & Butler, F. Efficiency of the sampling plan for Cronobacter spp. assuming a Poisson lognormal distribution of the bacteria in powder infant formula and the implications of assuming a fixed within and between-lot variability, Food Control, Elsevier, 2013 , 33 , 174-185.

• Mehta, N.B, Molisch, A.F, Wu, J, & Zhang, J., 'Approximating the Sum of Correlated Lognormal or, Lognormal-Rice Random Variables,' 2006 IEEE International Conference on Communications, Istanbul, 2006, pp. 1605-1610.

Examples

  c <-  0
  r <-  25
  t <-  30
  mu <-  -3
  distribution <- 'Poisson lognormal'
  prob_accept(c, r, t, mu, distribution)

Probability of contaminated sample

Description

This function calculates the probability of exactly l contaminated samples out of t selected grab samples for given gram sample size r and serial correlation d at the process contamination level p for a production length of N.

Usage

prob_contaminant(l, r, t, d, p, N, method)

Arguments

Details

Let S_t be the number of contaminated samples and S_t=\sum X_t where X_t=1 or 0 depending on the presence or absence of contamination, then P(S_t=l) formula given in Bhat and Lal (1988), also we can use following recurrence relation formula,

P(S_t=l)=P(X_t=1;S_{t-1}=l-1) + P(X_t=0;S_{t-1}=l)

which is given in Vellaisamy and Sankar (2001). Both methods will be produced the same results. For this package development, we directly applied formula which is from Bhat and Lal (1988).

Value

Probability of contaminated

References

• Bhat, U., & Lal, R. (1988). Number of successes in Markov trials. Advances in Applied Probability, 20(3), 677-680.

• Vellaisamy, P., Sankar, S., (2001). Sequential and systematic sampling plans for the Markov-dependent production process. Naval Research Logistics 48, 451-467.

See Also

prob_detect_single_grab, correlation_grab

Examples

  l <-  1
  r <-  25
  t <-  30
  d <-  0.99
  p <-  0.005
  N <-  1e9
  method <- 'systematic'
  prob_contaminant(l, r, t, d, p, N, method)

Probability of detection under the grab sampling method

Description

This function gives the detection probability for t grab samples and given acceptance number under systematic or random sampling methods. This function is also used to calculate the detection probability for primary increments selection by setting the number of primary increments as one.

Usage

prob_detect(c, r, t, d, p, N, method)

Arguments

Details

The detection probability of entire selected grab samples is given by,

P_D=1-[P(S_t=0)+P(S_t=1)+\cdots +P(S_t=c)]

Value

Probability of detection in all seleceted grab samples

See Also

prob_contaminant

Examples

  c <-  1
  r <-  25
  t <-  30
  d <-  0.99
  p <-  0.005
  N <-  1e9
  method <- 'systematic'
  prob_detect(c, r, t, d, p, N, method)

Probability of detection in a single grab sample

Description

This function calculates the probability of detection in a single grab sample comprising of r primary increments for given serial correlation d.

Usage

prob_detect_single_grab(r, p, d)

Arguments

Details

The probability of detection in any of the grab sample is given by p_d as

p_d = 1-(1-p)(1-p(1-d))^{r-1}

Value

Probability of detection in a grab sample

Examples

   r <-  25
   p <-  0.005
   d <-  0.99
   prob_detect_single_grab(r, p, d)