A teaching–learning based optimization approach for economic design of X-bar control chart

Applied Soft Computing 24 (2014) 643–653

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A teaching–learning based optimization approach for economic design of X-bar control chart Abhijeet Ganguly, Saroj Kumar Patel ∗ Department of Mechanical Engineering, National Institute of Technology, Rourkela, Odisha 769008, India

a r t i c l e

i n f o

Article history: Received 8 March 2013 Received in revised form 16 July 2014 Accepted 11 August 2014 Available online 23 August 2014 Keywords: Analysis of variance Economic design Fractional factorial design Loss cost Teaching–learning based optimization X-bar chart

a b s t r a c t Being simple to use X-bar control chart has been most widely used in industry for monitoring and controlling manufacturing processes. Measurements of a quality characteristic in terms of samples are taken from the production process at regular interval and the sample means are plotted on this chart. Design of a control chart involves the selection of three parameters, namely the sample size (n), the sampling interval (h) and the width of control limits (k). In case of economic design, these three control chart parameters are selected in such a manner that the total cost of controlling the process is the least. The effectiveness of this design depends on the accuracy of determination of these three parameters. In this paper, a new efficient and effective optimization technique named as teaching–learning based optimization (TLBO) has been used for the global minimization of a loss cost function expressed as a function of three variables n, h and k in an economic model of X-bar chart based on unified approach. In this work, the TLBO algorithm has been modified to simplify the tuning of teaching factor. A MATLAB computer program has been developed for this purpose. A numerical example has been solved and the results are found to be better than the earlier published results. Further, the sensitivity analysis using fractional factorial design and analysis of variance have been carried out to identify the critical process and cost parameters affecting the economic design. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Control charts are generally used to establish and maintain statistical control of a process. The prime objective of a control chart is to help management to detect process shift by distinguishing between two different sources of variation in a process. As per Deming’s philosophy these variations are called as special and common causes of variability [1]. The control charts are broadly categorized into two groups, i.e., charts for variables and charts for attributes. Xbar is a type of variable control chat and it is most widely used due to its simplicity. The major purpose of these charts is to detect the occurrence of assignable causes so that the necessary and corrective action may be taken before a large number of nonconforming products are manufactured. These are also effective tools to estimate the process parameters and analyze process capability. The use of a control chart requires the selection of three parameters, namely sample size n, sampling interval h, and width of control

∗ Corresponding author. Tel.: +91 661 2462516; fax: +91 661 2472926. E-mail address: [email protected] (S.K. Patel). http://dx.doi.org/10.1016/j.asoc.2014.08.022 1568-4946/© 2014 Elsevier B.V. All rights reserved.

limits k for the chart. The selection of these three parameters is called the design of a control chart. Traditionally, control charts have been designed with respect to statistical criteria where the two statistical errors, namely Type-I error (˛) and Type-II error (ˇ) are kept at minimum values. However, the design of a control chart has economic significance in the activities like sampling and testing, investigating out-of-control signals, correcting the out-of-control process, the loss of company’s goodwill on the delivery of nonconforming units to the consumer, etc., which are significantly affected by the choice of the control chart parameters. Therefore, the design of control chart from an economic view point has received much attention over the recent years [2]. The economic design is a mathematical model which finds parameters of a control chart by minimizing an expected cost function, which considers economic aspects such as costs of sampling and testing, costs associated with detecting and investigating outof-control signals and possibly correcting assignable cause(s), and costs of allowing nonconforming units to reach the customers. Although the optimal control rules in their model are too complex to have practical utility, their work provided the basis for most of the cost-based models in control chart designs. Duncan [3] first

A. Ganguly, S.K. Patel / Applied Soft Computing 24 (2014) 643–653

proposed an economic model for design of X-bar chart assuming a random shift in process mean due to single assignable cause and the failure mechanism from in-control to out-of-control state has exponential distribution. Panagos et al. [4] treated two distinct situations in economic design viz., (i) the process continues in operation while searches for the assignable cause are made and (ii) the process must be shut down during the search. Exhaustive literature reviews have been made by Montgomery [2], Svoboda [5] and Ho and Case [6] on the economic design of control charts where it is observed that majority of the researchers have considered X-bar chart and Duncan’s [3] single assignable cause model where the loss cost is expressed as a function of three variables n, h and k. Selecting these parameters on economic criteria, referred to as economic design, is becoming more and more popular because of its capability of maintaining the process under statistical control at lower cost [7–13]. The effectiveness of economic design depends on how accurately this loss cost function is minimized to determine the values of the three design variables. Various optimization techniques have been used for its minimization [4–6,14]. Recently, a few non-traditional global optimization techniques like Genetic Algorithm [15], Particle Swarm Optimization [16,17], Simulated Annealing [18] have also been tried for the same purpose. Non-traditional optimization methods have also been used for economic design of charts other than X-bar chart [19,12]. Although recently a new effective non-traditional optimization technique called teaching–learning based optimization (TLBO) has been used for solving wide variety of industrial optimization problems, it has not been tried so far for minimization of loss cost function in economic design of X-bar control chart. The objective of this work is to develop a MATLAB computer program based on TLBO to select optimum values of parameters for the economic design of X-bar chart and to identify the critical factors that affect this design with the help of analysis of variance (ANOVA). The remainder of the paper is organized in five more sections. An example of X-bar chart has been illustrated in Section 2. Section 3 briefly describes economic model of X-bar chart leading to the formulation of loss cost function. It also includes description of all the notations used. The TLBO and its algorithm are briefly explained in Section 4. A numerical example has been solved and the results obtained are compared with the earlier published results in Section 5. Sensitivity analysis with the help of ANOVA on the results of economic design has been presented in Section 6 and finally conclusions are listed in Section 7.

2. X-bar chart Usually, the control charts are made up of three quantities: the center line (CL) which represents the average significance of the quality characteristic corresponding to in-control state, and the other two lines, called upper control limit (UCL) and lower control limit (LCL), which are chosen to assure that if the process is in-control, nearly all of the sample points will fall between them. As long as the point plots within the control limits, the process is assumed to be in-control, the condition when the outputs of the process have low variability around their target values. Whenever, any point that falls outside of the control limits is interpreted as evidence that the process is out-of-control, the condition when the outputs of the process have high variability around their target values, and investigation and corrective action are required to find and eliminate the assignable cause (or causes) responsible for this behavior [2]. An example of X-bar chart has been illustrated below for the benefit of the readers who might be unfamiliar with the subject. Let a machine is producing screws on a continuous basis and the process is known to be under control initially with a process mean of 35 mm and standard deviation of 12 mm for screw diameter. As

Table 1 Measurement of screw diameters. Sample number

Diameters of screws (mm) Xi

1 2 3 4 5 6 7 8 9 10

1

2

3

4

33 38 35 34 37 41 30 40 36 53

36 42 40 37 31 39 30 32 42 58

42 32 33 40 34 36 31 34 38 55

40 43 34 32 34 37 31 36 35 56

Total



Xi

151 155 142 143 136 153 122 142 151 222

Mean X¯i

37.75 38.75 35.50 35.75 34.00 38.25 30.50 35.50 37.75 55.50

60

UCL

50 40

Diameter

644

CL

30 20

LCL

10 0

1

2

3

4

5

6

7

8

9

10

Sample Number Fig. 1. X-bar control chart.

the process continues, it is desired to check whether the process is under control or not. Therefore, let samples of four screws each are drawn at an interval of 1 h and the screw diameters are measured. For each sample the mean X¯i is calculated. As an example, 10 sets of such data are listed in Table 1. Each time a sample is taken, its sample mean X¯i is plotted on X-bar chart against its corresponding sample number as shown in Fig. 1. There are three horizontal lines on this chart. The center line (CL) is drawn at process mean value. Two control limits, i.e., UCL and LCL are drawn symmetrically on either side of CL at a distance of three-sigma (k value is usually taken as 3). Thus, the values of the three lines are calculated as CL = 0 = 35 mm  12 UCL = 0 + k √ = 35 + 3 √ = 35 + 18 = 53 mm n 4  12 LCL = 0 − k √ = 35 − 3 √ = 35 − 18 = 17 mm n 4 Fig. 1 shows that the process was running in-control for the first nine sample and it has gone out of control at 10th sample as this point falls outside the control limits (i.e., here above UCL). Similarly, the process would have gone out of control if a point has fallen below LCL. Then, necessary action is taken for the finding out assignable cause and its removal to bring back the process to in-control state. 3. Formulation of loss cost function The loss cost function formulated by van Deventer and Manna [7] based on the economic model of Lorenzen and Vance [14] has been considered for optimization by TLBO in the present work. A brief description of the formulation is given below. Initially, the process is assumed to be in in-control state and the quality

A. Ganguly, S.K. Patel / Applied Soft Computing 24 (2014) 643–653

Cycle starts

Last sample before assignable cause

645

Sample with out-of-control signal Out-of control signal detected

Assignable cause occurs

Assignable cause found Assignable cause eliminated

τ 1

λ

h −τ (1− β )

+ (1 − ξ1 ) sZ 0α

In-control state

gn

Z2

Z1

Out-of-control state Production cycle Fig. 2. Cycle time for the process.

characteristic of the process is assumed to be normally distributed with mean 0 and variance  2 . At random, let the process be disturbed due to occurrence of an assignable cause at a rate of  as per exponential distribution. If sample size is n and k is the width of the control limits for the X-bar chart, the center line will be at 0 √ and the two control limits will be at 0 ± k/ n. When the process is in-control, there may be occurrences of false alarm at a rate of



∞

˛=2

(1)

(z) dz k

where (z) is the standard normal density. At random, let the process be disturbed due to occurrence of an assignable cause at a rate of  as per exponential distribution. If the shift in process mean is ı, the probability that the shift will be detected on any subsequent sample is





√ (−k−ı n)

1−ˇ =

(z) dz +

∞

√ (k−ı n)

−∞

(2)

(z) dz

The expected number of samples, s, taken while in control is s=

∞ 

iP (assignable cause occurs between the ith and

i=0

(i + 1)st sample) =

∞ 

i(ehi − eh(i+1) ) =

i=0

1 eh

−1

E(T ) =

=

ih

(t − ih)e−t dt

 (i+1)h ih

e−t dt

=

1 − (1 + h)e−h (1 − e−h )

h 1 = −  eh − 1

A production cycle consists of five periods: (i) the in-control period, 1/ + (1 −  1 )sZ0 ˛, (ii) time to generate out-of-control signal, h/(1 − ˇ) − , (iii) the time to take a sample and interpret the results, gn, (iv) the time to discover the assignable cause, Z1 and (v) the time to eliminate assignable cause, Z2 . The entire cycle is represented in Fig. 2. Adding all the above five components, the expected cycle time is

(3)

where = indicator variable 1 ⎧ ⎪ ⎨ 1 when production continues during search for assignable cause = , ⎪ ⎩ 0 when production ceases during search for assignable cause Z0 = expected search time when the signal is a false alarm, and g = time to sample and chart one item. The cost model includes the fixed cost a and variable cost b of sampling, the cost of searching for the assignable cause when it exists, the repair cost and the cost of searching for an assignable cause that does not exist. (a) The expected cost when the process is in-control Q0 /, (b) the expected cost when the process is out-of-control Q1 [h/(1 − ˇ) −  + gn +  1 Z1 +  2 Z2 ], (c) the expected cost during search period due to a false alarm sY˛, (d) the expected cost for search and repair of true alarm W, (e) the expected cost due to fixed and variable cost of sampling (a + bn)[{1/ + h/(1 − ˇ) −  + gn +  1 Z1 +  2 Z2 }/h]. Thus the total quality cost per cycle is E(C) =



h Q0 + Q1 −  + gn + 1 Z1 + 2 Z2  (1 − ˇ)



+ sY˛ + W + (a + bn)

and the average time of occurrence of the assignable cause within the ith and (i + 1)st interval is given by

 (i+1)h

1 h + (1 − 1 )sZ0 ˛ + −  + gn + Z1 + Z2  (1 − ˇ)



×

1/ + h/(1 − ˇ) −  + gn + 1 Z1 + 2 Z2 h

(4)

where Q0 = quality cost per hour while producing in-control, Q1 = quality cost per hour while producing out-of-control 2  = indicator variable 1 when production continues during repair = , 0 when production ceases during repair Y = cost per false alarm, and W = cost to locate and repair the assignable cause. Finally, the expected loss cost per unit time is calculated as

E(L) =

E(C) (Q0 /) + Q1 [(h/(1 − ˇ)) −  + gn + 1 Z1 + 2 Z2 ] + sY˛ + W + (a + bn)[(1/) + (h/(1 − ˇ)) −  + gn + 1 Z1 + 2 Z2 /h] = E(T ) (1/) + (1 − 1 )sZ0 ˛ + (h/(1 − ˇ)) −  + gn + Z1 + Z2

(5)

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A. Ganguly, S.K. Patel / Applied Soft Computing 24 (2014) 643–653

The above cost E(L) is a function of the three quality control chart parameters such as the sample size n, the sampling frequency h, and the control limit width parameter k. In economic design of Xbar control chart, this function is minimized. Thus, it is an example of multi-variable unconstrained minimization problem including discrete-continuous nonlinear, nondifferentiable objective function.

4.1. Teacher phase It is the first part of the algorithm where learners learn through the teacher. During this phase, a teacher tries to increase the mean result of the class in the subject taught by him depending on his capability. Assume that there are J number of subjects each of which representing a design variable and K number of learners in the class representing the population. Thus, the population size is K. At any iteration i, if Mij is the mean result of the class of K learners in a particular subject j (j = 1, 2, 3, . . ., J), then

4. Teaching–learning based optimization (TLBO)

Step 1:

1 Xijk K K

Nature inspired population based algorithms mimic different natural phenomena to solve a wide range of problems. Recently, a new effective and efficient population based teaching–learning based optimization (TLBO) algorithm was proposed by Rao et al. [20–23,25,26] and Pawar and Rao [24]. It simulates the teaching–learning process of the classroom for finding the global optimal solution. It does not have the trouble of adjusting any algorithm-specific parameters. It requires only common controlling parameters like population size and number of generations for its working. In this algorithm a group of learners is considered as population and different subjects taught to the learners are considered as different design variables of the optimization problem. A learner’s overall result is analogous to the value of the objective function. The working of TLBO comprises two phases, namely teacher phase and learner phase.

Mij =

where Xijk is the result of learner k in subject j in iteration i. Out of all the learners (i.e., k = 1, 2, 3, . . ., K) one would have the best overall result taking all the subjects into consideration and let it be k-best. For example, in case of minimization problem, for the k-best solution the value of objective function would be the lowest. The difference between the existing mean result for each subject j and the corresponding result of the best learner k-best in the same subject at any iteration i is given by dmij = ri (Xijk−best − TF Mij )

(i.e., the design variables). Select maximum number of iterations I. Set iteration counter i = 0. Step 2:

Generate a random population of results X and calculate corresponding f(X).

Step 3:

Calculate mean result in each subject as calculated in Eqn. 6.

Step 4:

Identify the best learner k-best.

Step 5:

Calculate Update

Step 6:

)

dmij = ri X ijk -best - TF M ij for each j.

X ijk

= X ijk +

dmij for all j and k.

If f(X ) gives better result Accept X else Retain X as X .

Step 7:

Randomly select two learners K1 and K 2 such that

Step 8:

If f(XK1) is better than f(XK2)

(

)

(

)

f ( X )K1

X ijK1 = XijK1 + ri X ijK1 - XijK 2 , else

X ijK 1 = XijK 1 + ri X ijK 2 - XijK 1 . Step 9:

If f(X ) is better Accept X else Retain X as X

Step 10:

If i

(7)

where ri is the uniformly distributed random number in the range [0,1]; Xijk-best the result of the best learner k-best in the subject j

Initialize the number of learners K (i.e., the population size), number of subjects J taught to the learners

(

(6)

k=1

I Terminate and X is solution

else i = i+1, X = X , go to Step 3 Fig. 3. TLBO algorithm.

f ( X )K 2

A. Ganguly, S.K. Patel / Applied Soft Computing 24 (2014) 643–653

at any iteration i; TF the teaching factor which decides the value of mean to be changed and TF value is taken as 1 in the present work. Since there are a total of J subjects, there will be J different values of dmij at any iteration i. For each subject j (j = 1, 2, 3,. . ., J) the results of all the learners (k = 1, 2, 3, . . ., K) are updated by adding the value of dmij as expressed below:  Xijk = Xijk + dmij

(8)

 is the updated value of X . Accept X  if it gives better where Xijk ijk ijk overall result, i.e., better function value. Otherwise retain Xijk as  . If the vector X = (X , X , X , . . ., X )T is a solution, then f(X) is Xijk J 1 2 3

647

the value of objective function representing the overall result. The  become the input to the learner phase. values of Xijk 4.2. Learner phase Another way of improving the knowledge is through mutual discussion among the learners of the class. Randomly select two learners K1 and K2 such that the f (X  )K1 = / f (X  )K2 where,   f (X )K1 and f (X )K2 are the updated values of overall results of K1 and K2, respectively at the end of teacher phase at any iteration i. The solutions are updated using either of the following rules:

Fig. 4. The flowchart of TLBO algorithm.

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A. Ganguly, S.K. Patel / Applied Soft Computing 24 (2014) 643–653

Table 2 Economic design results. n

van Deventer and Manna

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

TLBO

h

k

˛

ˇ

E(L)

h

k

˛

ˇ

E(L)

0.7 0.7 0.9 0.9 1.1 1.3 1.3 1.5 1.6 1.6 1.7 1.9 1.9 2 2.1 2.2 2.2 2.3 2.4 2.4

2.1 2.3 2.3 2.4 2.4 2.4 2.5 2.5 2.5 2.6 2.6 2.6 2.7 2.7 2.7 2.7 2.8 2.8 2.8 2.9

0.3570 0.0214 0.0214 0.0164 0.0164 0.0164 0.0124 0.0124 0.0124 0.0093 0.0093 0.0093 0.0069 0.0069 0.0069 0.0069 0.0051 0.0051 0.0051 0.0037

0.8634 0.8120 0.7149 0.6554 0.5651 0.4803 0.4421 0.3713 0.3085 0.2870 0.2368 0.1938 0.1826 0.1488 0.1204 0.0968 0.0929 0.0746 0.0595 0.0580

19.22 17.36 16.43 15.87 15.51 15.28 15.11 14.99 14.92 14.87 14.85 14.84 14.85 14.86 14.89 14.92 14.96 15.01 15.06 15.11

0.5969 0.6856 0.8104 0.9424 1.0746 1.2044 1.3277 1.4453 1.5560 1.6597 1.7567 1.8474 1.9324 2.0122 2.0873 2.1584 2.2258 2.2899 2.3512 2.4102

2.1562 2.2854 2.3471 2.3879 2.4202 2.4484 2.4763 2.5037 2.5314 2.5600 2.5894 2.6196 2.6505 2.6819 2.7139 2.7463 2.7790 2.8120 2.8453 2.8785

0.0319 0.0222 0.0193 0.0174 0.0156 0.0135 0.0134 0.0128 0.0127 0.0105 0.0096 0.0088 0.0080 0.0073 0.0067 0.0061 0.0054 0.0051 0.0046 0.0040

0.8734 0.8086 0.7284 0.6472 0.5726 0.5084 0.4318 0.3678 0.3063 0.2736 0.2336 0.1991 0.1701 0.1446 0.1232 0.1048 0.0898 0.0748 0.0640 0.0552

19.2036 17.3518 16. 4219 15.8697 15.5131 15.2725 15.1071 14.9940 14.9185 14.8714 14.8461 14.8381 14.8438 14.8608 14.8870 14.9207 14.9607 15.0059 15.0555 15.1087

Bold represents the optimal solution for the economic design after exploring at various values of sample size n.

 XijK1

=

 XijK1

 + ri (XijK2

 − XijK1 )



if f (X )K2 is better

(9)

Economic design of control chart using TLBO

(10)

 if it gives better function value f(X ). Accept XijK1 In the present work, a computer program has been developed in MATLAB language based on this algorithm whose steps are as summarized in Fig. 3. The flowchart of TLBO algorithm is given in Fig. 4.

5. Numerical example A numerical example recently considered by van Deventer and Manna [7] for designing an X-bar control chart has been selected for the present work where the data given are:  = 0.01, ı = 1, g = 0.05, a = 0.50, b = 0.10, Y = 50, W = 25, Q0 = 10, Q1 = 100, Z0 = 0, Z1 = 2, Z2 = 0, ␰1 = 1 and ␰2 = 1.

19.2036

19

18 Loss cost E(L)

    XijK1 = XijK1 + ri (XijK1 − XijK2 ) if (X )K1 is better

17.3518

17 16.4219

16 15.8697

15.5131 15.2725

15

14.9940 14.8714

15.1071 14.9185

In the present work, the loss cost function E(L)has been minimized by running the MATLAB computer program which was developed based on TLBO algorithm. Out of three variables, sample size n is integer whereas other two, i.e., sampling interval (h) and width of control limits (k) are taken as real values on continuous scale. The search domain selected for searching the optimum solution are 1–20 for n, 0.1–5.0 for h and 0.1–5 for k. The optimum values of h and k along with corresponding minimum values of expected loss cost E(L) obtained for all integer values of n varying from 1 to 20 have been listed in Table 2. This table also presents the corresponding values of the two errors ˛ and ˇ. As shown in this table, the optimum value of cost E(L) decreases as n value increases from 1 to 12 and then increases at higher values of n. This trend is also graphically shown in Fig. 5. The rate of reduction of loss cost E(L)is observed to be very large as the sample size increases in the beginning and then the rate gradually diminishes till the cost becomes minimum. Thus, the minimum possible cost is found to be E(L) = 10.8381 and this occurs at n = 12. The corresponding values of h and k at optimum solution are 1.8474 h and 2.6196, respectively. The same trend of relationship between E(L)and n has also been reported by van Deventer and Manna [7] as shown in Table 2. They have also obtained the most minimum cost at n = 12. On comparison of results, it is clear that at all values of sample size; the optimum

2

4

6

8

14.8461

14.8438

14.8608

12

14

15.0059

16

15.1087

15.0555

14.9607

14.8870

14.8381

10

14.9207

18

20

Sample size n Fig. 5. Variation of loss cost with sample size.

costs obtained by TLBO are lower than that of van Deventer and Manna [7]. 6. Sensitivity analysis The economic model of X-bar chart used in the present work has nine process and cost parameters as listed in Table 3. Each of these parameters is called a factor in the terminology of design of experiments. Accordingly, let all the nine parameters be identified as factors from A to I as shown in this table. Each of these factors operates at two levels. The low and high values of all the nine parameters considered by Chen and Tirupati [27] are also shown in Table 3. Since each factor is present at two levels, 2k–p fractional factorial design of resolution IV can be used to examine the effects of these parameters on n, h, k and E(L). The use of resolution IV design ensures that no main effects are aliased with each other,

A. Ganguly, S.K. Patel / Applied Soft Computing 24 (2014) 643–653 Table 3 Factors and their levels used in the experimental design. S. no.

Factors

Parameters

1 2 3 4 5 6 7 8 9

A B C D E F G H I

 ı a b Y W Q0 Q1 g

Level Low

High

0.01 1.00 0.50 0.10 50.00 25.00 10.00 50.00 0.00

0.05 2.00 5.00 1.00 500.00 50.00 50.00 1000.00 0.05

but that two factor interactions are aliased with other two-factor interaction. The details of 2k–p design is given by Montgomery [28]. A 29−4 factorial design with 32 runs was chosen for the model IV with generators I = ABCF, I = ABDG, I = ABEH and I = ACDI. Since the experimental design is based on resolution IV, it has been considered to concentrate on estimating the main effects of each of the factors. A large number of additional experimental runs would have been required to separate the effects of two-factor interactions. At each run, the values of nine parameters are taken as per the fractional factorial design 29–4 as shown in Table 4 and the loss cost function E(L)is minimized by running the same MATLAB program developed on the basis of TLBO algorithm. Thus, X-bar chart is designed from economic view point at each of the 32 runs and the respective optimal values of design parameters n, h and k are listed in the same table along with corresponding loss cost E(L). Analysis of variance has been done on the optimum values of E(L), n, h and k obtained out of 32 experimental runs at a

649

significance level of 0.05 and the results obtained are presented in Tables 5–8. Table 5 reveals that the loss cost of process control is significantly affected by three parameters , Q0 and Q1 . All the parameters are observed to have positive effects. All these parameters have p value less than the predetermined significance level of 5%. A positive effect implies that changing the factor from low to high level increases the response, while a negative effect implies that changing the factor from low to high level decreases the response. As the rate of occurrence of the assignable cause per hour  increases, so does the loss cost of the process E(L). The same is true for the hourly costs for operating in the in-control state Q0 and in the out-of-control state Q1 . Table 6 presents an analysis of variance on the sample size n. The sample size is primarily determined by ı, the size of the shift. As ı gets smaller, the optimum sample size increases. The second most significant factor is Q0 , i.e., the cost of production during in-control state. It has also the negative effect like ı. The cost of production during out-of-control period Q1 also affects n but it has the positive effect. The effect of factor g, i.e., the time to acquire and analyze each observation is comparable to that of Q1 in terms of p value. However, unlike Q1 the factor g has effect in negative aspect. A large g results in smaller sample. The least significant factor is b, i.e., the variable cost of sampling. For smaller b value, a larger sample would be required. Table 7 displays an analysis of variance on the sampling interval h. The large values of  and Q1 get smaller value of h. Factors b and Q0 are also significant. A large value of variable cost of sampling b or hourly cost for operating in the in-control state Q0 results in a larger value of time between sampling. Table 8 presents an analysis of variance on the control limit width k. The most significant factors are Q0 andQ1 , i.e., the costs of production during in-control and out-of-control periods,

Table 4 Optimum economic designs for X-bar chart. Runs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Parameters

Optimum values



ı

a

b

Y

W

Q0

Q1

g

n

h

k

E(L)

0.01 0.01 0.05 0.05 0.01 0.01 0.01 0.05 0.01 0.01 0.01 0.05 0.05 0.01 0.05 0.05 0.01 0.05 0.05 0.05 0.01 0.05 0.05 0.01 0.01 0.05 0.01 0.05 0.01 0.05 0.05 0.01

1 2 2 1 2 1 2 2 2 2 2 2 2 1 2 1 1 1 1 2 1 1 1 1 1 2 2 1 2 2 1 1

0.5 0.5 0.5 0.5 0.5 5.0 0.5 0.5 0.5 5.0 5.0 5.0 0.5 0.5 5.0 5.0 5.0 5.0 0.5 5.0 0.5 0.5 0.5 5.0 5.0 0.5 5.0 5.0 5.0 5.0 5.0 0.5

1 1 1 0.1 0.1 1 1 1 0.1 0.1 0.1 0.1 0.1 0.1 1 1 1 0.1 1 1 1 1 0.1 0.1 0.1 0.1 1 0.1 1 0.1 1 0.1

50 500 500 50 500 500 50 50 50 500 50 500 50 500 50 500 50 500 50 500 500 500 500 500 50 500 50 50 500 50 50 50

25 25 25 50 50 25 50 50 25 25 50 25 25 25 25 25 50 50 25 50 50 50 25 50 25 50 25 25 50 50 50 50

10 50 10 10 10 10 10 50 50 50 10 10 10 10 10 50 50 10 50 50 50 10 50 50 10 50 50 50 10 50 10 50

50 50 1000 50 1000 1000 1000 50 50 1000 50 50 1000 50 50 50 50 1000 1000 1000 1000 50 1000 50 1000 50 1000 50 50 1000 1000 1000

0.05 0.05 0.00 0.00 0.05 0.05 0.00 0.05 0.00 0.00 0.05 0.05 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.00 0.05 0.05 0.05 0.00 0.00 0.05 0.05 0.00 0.00 0.05 0.05

6 1 5 16 5 12 3 1 1 9 6 7 3 20 4 1 1 20 8 4 15 12 9 1 20 1 3 1 5 8 5 8

5.0000 5.0000 0.5011 1.6464 0.4132 1.7580 0.8150 5.0000 5.0000 1.1369 5.0000 2.8048 0.1753 3.4333 3.5464 5.0000 5.0000 0.5696 0.6437 0.6519 1.7549 4.2987 0.1806 5.0000 1.2090 5.0000 1.2993 5.0000 5.0000 0.5481 0.6965 0.4381

1.9062 5.0000 3.1928 2.7185 3.5780 2.6198 2.4301 5.0000 5.0000 3.7338 2.9176 3.4454 2.8023 3.3766 2.3523 5.0000 5.0000 3.1401 1.8908 2.9279 2.7789 2.6239 3.1528 5.0000 2.6574 5.0000 2.2040 5.0000 3.2254 3.1878 1.5565 2.5084

14.6316 50.3066 125.7537 18.8579 37.9443 59.6375 40.3235 50.3135 50.1266 79.4351 13.5133 19.6570 118.9882 12.7072 20.4615 51.2002 51.2003 129.6758 172.3390 178.9066 89.9013 24.6556 178.3865 51.0203 41.9006 50.1335 84.6588 51.0202 14.5394 160.1233 156.6716 81.4054

650

A. Ganguly, S.K. Patel / Applied Soft Computing 24 (2014) 643–653

Table 5 Analysis of variance for loss cost E(L). Source

Effect

DF

Seq. SS

Adj. SS

Adj. MS

F

P

 ı a b Y W Q0 Q1 g Residual error Total

45.868 −5.627 2.928 5.663 1.708 1.123 36.285 74.482 3.940

1 1 1 1 1 1 1 1 1 22 31

16 831.2 253.3 68.6 256.5 23.3 10.1 10 532.8 44 380.1 124.2 15 156.6 87 636.7

16 831.2 253.3 68.6 256.5 23.3 10.1 10 532.8 44 380.1 124.2 15 156.6

16 831.2 253.3 68.6 256.5 23.3 10.1 10 532.8 44 380.1 124.2 688.9

24.43 0.37 0.10 0.37 0.03 0.01 15.29 64.42 0.18

0.000* 0.551 0.755 0.548 0.856 0.905 0.001* 0.000* 0.675

*

Significant at 5%.

Table 6 Analysis of variance for sample size n. Source

Effect

DF

Seq. SS

Adj. SS

Adj. MS

F

P

 ı a b Y W Q0 Q1 g Residual error Total

−0.687 −5.562 −0.437 −3.063 2.063 0.062 −4.813 3.312 −3.312

1 1 1 1 1 1 1 1 1 22 31

3.78 247.53 1.53 75.03 34.03 0.03 185.28 87.78 87.78 361.94 1084.72

3.781 247.531 1.531 75.031 34.031 0.031 185.281 87.781 87.781 361.937

3.781 247.531 1.531 75.031 34.031 0.031 185.281 87.781 87.781 16.452

0.23 15.05 0.09 4.56 2.07 0.00 11.26 5.34 5.34

0.636 0.001* 0.763 0.044* 0.164 0.966 0.003* 0.031* 0.031*

*

Significant at 5%.

Table 7 Analysis of variance for sampling interval h. Source

Effect

DF

Seq. SS

Adj. SS

Adj. MS

F

P

 ı a b Y W Q0 Q1 g Residual Error Total

−0.687 0.016 0.308 0.526 0.093 0.009 0.612 −3.621 0.119

1 1 1 1 1 1 1 1 1 22 31

3.778 0.002 0.757 2.210 0.069 0.001 2.993 104.902 0.114 9.508

3.778 0.002 0.757 2.210 0.069 0.001 2.993 104.902 0.114 9.508

3.778 0.002 0.757 2.210 0.069 0.001 2.993 104.902 0.114 0.432

8.74 0.01 1.75 5.11 0.16 0.00 6.93 242.73 0.26

0.007* 0.944 0.199 0.034* 0.693 0.969 0.015* 0.000* 0.612

*

124.333

Significant at 5%.

respectively. The large value of Q0 gets larger value for k, whereas larger value of Q1 results smaller k. The increase in the cost of investigating a false alarm Y results in a larger width of the control limit k. It is further observed from Tables 5–8 that the cost to search and eliminate assignable cause W and the fixed cost of sampling a

have no significance on any of the responses n, h, k and E(L). The relative significance of parameters in case of all the responses n, h, k and E(L) are also graphically displayed by means of the normal plots of standardized effects as shown in Figs. 6–9, respectively, at significance level of 5%.

Table 8 Analysis of variance for control limit width k. Source

Effect

DF

Seq. SS

Adj. SS

Adj. MS

F

P

 ı a b Y W Q0 Q1 g Residual error Total

−0.0591 0.3167 0.0630 −0.4694 0.5415 0.0162 1.1151 −1.1378 −0.1526

1 1 1 1 1 1 1 1 1 22 31

0.0279 0.8025 0.0318 1.7626 2.3455 0.0021 9.9475 10.3564 0.1863 10.8976 36.3601

0.0279 0.8025 0.0318 1.7626 2.3455 0.0021 9.9475 10.3564 0.1863 10.8976

0.0279 0.8025 0.0318 1.7626 2.3455 0.0021 9.9475 10.3564 0.1863 0.4953

0.06 1.62 0.06 3.56 4.74 0.00 20.08 20.91 0.38

0.815 0.216 0.802 0.073 0.041* 0.949 0.000* 0.000* 0.546

*

Significant at 5%.

A. Ganguly, S.K. Patel / Applied Soft Computing 24 (2014) 643–653

651

Normal Plot of the Standardized Effects (response is EL, Alpha =0.05) 0 99

Effect Type Not Significant Significant

95 Q1

90 mbda Lam

Percent

80 Q0

70 60 50 40 30 20 10 5 1

-2

0

2 4 Standardized Effect

6

8

Fig. 6. Normal plot for response of loss cost E(L).

Normal Plot of the Standardized Effects (response is n, Alpha = 0.05) 99 Effect Type Not Significant Significant

95 Q1

90

Percent

80 70 60 50 40 30

b g

20

Q0

10

Delta

5 1

-4

-3

-2

-1 0 1 Standardized Effect

2

3

Fig. 7. Normal plot for response of sample size n.

Normal Plot of the Standardized Effects (response is h, Alpha = 0.05) 99

Effect Type Not Significant Significant

95 Q0

Percent

90 b

80 70 60 50 40 30 20 10

Lambda Q1

5 1

-15

-10

-5

0

Standardized Effect Fig. 8. Normal plot for response of sampling interval h.

5

652

A. Ganguly, S.K. Patel / Applied Soft Computing 24 (2014) 643–653

Normal Plot of the Standardized Effects (response is k, Alpha = 0.05) 99

Effect Type Not Significant Significant

95 Q0

90 Y

Percent

80 70 60 50 40 30 20 10

Q1

5 1

-5.0

-2.5

0.0

2.5

5.0

Standardized Effect Fig. 9. Normal plot for response of control limit width k.

7. Conclusions

References

In this paper, a computer program in MATLAB language has been developed using TLBO algorithm for optimal economic design of X-bar control chart based on Lorenzen and Vance [14] economic model. To verify the performance of the TLBO based economic design methodology the numerical example of van Deventer and Manna [7] is illustrated, and the results obtained are found to be superior. The best economic design for this chart is obtained with sample size 12, sampling interval 1.8474 h and width of control limits 2.6196 in this numerical example. There are three major contributions in this work. Although TLBO has been applied successfully to large number of benchmarking as well as various types of optimization problems since its inception in the year 2011 [20,22], it has not yet been applied for economic design of X-bar control chart to find the optimal solution. The first contribution of this work is that TLBO can also be successfully employed in the field of statistical quality control for minimizing an unconstrained non-linear objective function representing the loss cost function in the economic design of X-bar control chart, and is found to be very competitive compared to the results reported earlier. In recent years, this new algorithm is gaining more popularity because unlike other nature inspired nontraditional optimization algorithms, it does not require any algorithm specific parameters. Other algorithms require proper tuning of algorithm-specific parameters in addition to the tuning of common controlling parameters and the effectiveness of such algorithms very much depend on the correctness of tuning. The algorithm proposed in the present work is also simpler compared to the earlier versions of TLBO [20–22]. It takes only one value for teaching factor TF (i.e., TF = 1) in all iterations whereas earlier it was required to be selected as 1 in some iterations and 2 in others at random. Thus, the second contribution of this work is that the modified version of TLBO proposed in this work is completely free from the task of tuning any factors. In the present algorithm, the proposed procedure is able to obtain a more near optimal design rather than the approximate designs of Duncan [3] and other subsequent researchers. Lastly, the results of sensitivity analysis would be helpful to the quality engineers in identifying the significant cost and process parameters, and accordingly take utmost care in using their values while designing X-bar chart on economic consideration.

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