DETERMINATION OF OPTIMAL DESIGN PARAMETERS FOR X CONTROL CHART WITH TRUNCATED WEIBULL IN- CONTROL TIMES

Transcription

1 International Journal of Production Tecnology and Management (IJPTM) Volume 7, Issue 1, Jan June 016, pp , Article ID: IJPTM_07_01_001 Available online at ttp:// Journal Impact Factor (016): (Calculated by GISI) ISSN Print: and ISSN Online: IAEME Publication DETERMINATION OF OPTIMAL DESIGN PARAMETERS FOR X CONTROL CHART WITH TRUNCATED WEIBULL IN- CONTROL TIMES Anand. Ayyagari Scool of Management Studies, G.V.P. College for Degree and P.G. Courses (Autonomous) Visakapatnam, INDIA Srinivasa Rao.Kraleti Department of Statistics, Andra University Visakapatnam, INDIA Laksminarayana. Jayanti Department of Statistics, Andra University Visakapatnam, INDIA ABSTRACT Control carts play an important role in monitoring te quality of a production process. Usually in designing te control carts it is assumed tat te in-control times (te time to occurrence of an assignable cause) follow exponential or two parameter Weibull distribution. Te tree parameter Weibull distribution includes exponential, two parameter Weibull distributions as limiting cases. Wit suitable cost considerations te optimal values of te design parameters viz. sample size n, te sampling interval and control limits k subject to constraints on type I and type II error probabilities are obtained. Te sensitivity analysis of te statistical economic design of X control cart wit respect to model parameters and costs is also studied. It was found tat te economic design of X cart is greatly influenced by te distributional assumptions. Tis design is also extended to te case of random out of control times. Key words: Quality Control, In-Control Time, Left Truncated Weibull Distribution, X Control Cart, Expected Cost per a Unit Time. ttp:// 1 editor@iaeme.com

2 Anand. Ayyagari, Srinivasa Rao.Kraleti and Laksminarayana. Jayanti Cite tis Article: Anand. Ayyagari, Srinivasa Rao.Kraleti and Laksminarayana. Jayanti, Determination of Optimal Design Parameters For X Control Cart wit Truncated Weibull In-Control Times, International Journal of Production Tecnology and Management (IJPTM), 7(1), 016, pp ttp:// 1. INTRODUCTION To maintain te statistical control of a production process control carts are effectively utilized. Tey are also used to estimate te capability of te production process. Muc work as been reported about designing of control carts. Among various control carts te X control cart plays a dominant role, as it as te capability of controlling te process mean tat as a vital bearing on productivity. For large expected sifts in process mean, te X cart is used in practice (Saniga et al (1). Prajapati and Maapatra [ () ave studied te design of X and R cart to monitor te process mean and standard deviation. Duncan,A.J. (3) pioneered te economic X control cart design. Montgomery, (4) as presented a review on economic control cart design. Later Lorenzen and Vance (5) presented a unified model tat establises a common notation and flexible metodology for a variety of situations. Al-Oraini and Raim (6) ave developed an economic statistical design of X control cart wit te assumption tat te in-control time follows Gamma (λ,) distribution... Biscak and Trietsc (7) argue tat te rate of false signals (RFS) concept is more appropriate tan te average run lengt concept wen used wit carts aving estimated limits. Nenes and Tagaras (8) developed a model for te economic design of an X cart using te Bayesian sceme. Teyaracakul, et al. (9) considered te problem of finding te limits for statistical process control of process mean wen te process distribution is unknown using te bootstrap metod. Sasibusan and Digambar (10) derived a compreensive cost function for a modified variable sample size and sampling interval X cart. Dias (11) presented results in economic statistical quality control. Vommi, and Seetala, (1) developed an economic design of X cart in wic te input parameters are expressed as ranges. Costa. and De Magalaes, (13) considered te adaptive non-central ci-square statistic cart. Taga. (14) studied te problem of monitoring a process in wic te observations can be presented as a first order auto regressive model following a eavy tail distribution. Te economic statistical design metod for te two of two runs and te two of tree runs rule sceme utilizing te Markov cain approac is given by Kim et al. (15). McWilliams (16), Cen and Ceng (16) ave utilized two parameter Weibull distribution considering increasing, decreasing and constant failure times of te process. However tey ignored te significance of minimum in-control times of te process. Recently Neelufur et al (17) ave studied te economic and statistical design of X control cart wit Pareto in-control times and fixing te type I and type II errors associated wit te process at fixed levels. Teir major consideration for utilizing Pareto in-control times is tat te in-control times of te process ave a minimum in-control time i.e. te failure of te process occurs only after a certain specified period of time and as a long upper tail. Tis assumption is valid only wen te rate of failure of te process is inversely proportional to time. ttp:// editor@iaeme.com

3 Determination of Optimal Design Parameters For X Control Cart Wit Truncated Weibull In-Control Times In many practical situations arising at places like, cement, cemical, fertilizer, puncing and packaging industries te in-control times ave a minimum guaranteed period once te process is kept for operation eiter after installation or repair and te rate of failure need not be inversely proportional to time. Tat is te failure of te process will appen only after a certain specified period of time. Te in-control time takes values only after a certain value can be caracterized by truncation of te incontrol time distribution. Ignoring te effect of left truncation on te distribution of in-control time leads to falsification of te cost model and te optimal design parameters derived may lead to deviation from optimality. Hence, in tis paper a statistical economic design of X control cart is developed and analyzed wit te assumption tat te in-control times of te process are random and follow a left truncated Weibull distribution. Te left truncated Weibull distribution caracterizes more flexible nature of te failure times, i.e. it includes increasing, constant and decreasing failure rates. It also includes te non-truncated Weibull distribution as a particular case wen te location parameter tends to zero. Wit suitable cost considerations te expected cost per unit time is derived under constraints on te type I and type II error probabilities. Te optimal design parameters, namely, te sample size n, te time interval between successive samples and te number of standard deviations to make te process in-control k are derived troug non-linear programming. Te sensitivity of te optimal design parameters wit respect to te model parameters and costs is also studied. Tis design is also extended to statistical economic design of X cart wit random out of control times wic follow two parameter Gamma Distribution. Section of tis article deals wit assumptions and notation used for designing te X control cart. In section 3 te expected cycle time is derived, section 4 deals wit expected cost per cycle. Te optimization problem for determining te optimal design parameters using non-linear programming is given in section 5. A case study in a cement plant is discussed in section 6. Te sensitivity analysis of te design parameters wit respect to te costs and model parameters is studied in section 7. Section 8 deals wit Statistical Economic Design of X control cart wit random out of control times. Te results are summarized wit conclusions in section 9.. ASSUMPTIONS AND NOTATION.1. Assumptions Te following assumptions are made for developing te cost model. 1. Te quality caracteristic (X) follows a Normal Distribution wit mean μ and variance σ, as a result of it te process mean also follows a normal distribution.. Te in-control times of te process are independently identically distributed (i.i.d.) random variables following a left truncated Weibull distribution wit probability density function ( 1) ( t ) f ( t) ( t ) e, for 0, 0, t =0,oterwise. Tis assumption is made because wen once te process is kept for operation eiter after maintenance or installation, it will take some minimum period of time for te process to go out of control and also wen te process is operated it is subjected to (1) ttp:// 3 editor@iaeme.com

4 Anand. Ayyagari, Srinivasa Rao.Kraleti and Laksminarayana. Jayanti various random factors like, tool wear, input quality, defective raw material, operator s efficiency, etc., ere η is te truncation point. 3. Te control carting tecnique generally involves taking a sample of n observations from te process output every units of time. A searc for an assignable cause is undertaken if a calculated process measurement suc as sample mean exceeds certain control limits. Tese control limits can be specified in terms of k te number of standard deviations above or below te process central line. 4. Te production process is assumed to start in an in-control state were te mean of quality measurements is μ 0 and standard deviation σ/ n. Wen an assignable cause occurs, te process mean sifts from μ 0 to μ 0 +Δσ, were Δ is a real number. 5. Wen te process goes out of control due to te occurrence of an assignable cause te process mean is assumed to sift to a known state and can not return to an in-control state witout intervention. Tis intervention requires searcing for te assignable cause and its removal. 6. Tere are costs associated wit collecting te sample, searcing for te assignable cause and repairing te process. Tese costs are assumed to be known and can be estimated. 7. It is furter assumed tat time value of money remains constant trougout te period under consideration... Notation Te following notation is used in developing te design. S: Expected number of samples taken wile te process is in-control. ARL 0 : Average Run Lengt wile te process is in-control. ARL 1 : Average Run Lengt wile te process is out of control. k: Number of Standard deviations from Control Limits to te Central Line. Δ: Number of standard deviations slipped wen te process went out of control. n: Size of te sample selected. E: Expected time for selecting one sample observation. δ 1: An indicator variable indicating weter production continues during searc for assignable cause. δ 1= 1, if production continues during searc operation.δ 1= 0, if production stops during searc operation. δ : Anoter indicator variable indicating weter production continues during repair of te process. δ = 1, if production continues during repair of te processδ = 0, if production stops during repair of te process. T 0 : Expected assignable cause searc time for a false alarm. T 1 : Expected time to identify te assignable cause. T : Expected time to repair te process. a: Fixed cost of selecting a sample. b: Variable cost of selecting a sample. C 0 : Cost of non-conformities produced per unit time wen te process is in-control. C 1 : Cost of non-conformities produced per unit time wen te process is not incontrol. C : Cost per false alarm. ttp:// 4 editor@iaeme.com

5 Determination of Optimal Design Parameters For X Control Cart Wit Truncated Weibull In-Control Times W: Cost of locating and removing te assignable cause. α: Probability tat X falls outside te control limits wile te process is in-control. β: Probability tat X falls witin te control limits wile te process is not in-control E(C): Expected Cycle Cost. E (T): Expected Cycle Time. 3. EXPECTED CYCLE TIME A production process in wic te process mean is monitored by using X control cart is considered. In general te control carting tecnique involves taking a sample of n observations from te output every units of time and carting some process measurement suc as sample mean. Wen observations are independent te Average Run Lengts ARL 0 and ARL 1 are related to α and β as ARL 0 = 1/α. were, α = Probability [ X μ 0 -kσ/ n or X μ 0 +kσ/ n wen μ = μ 0 ] α = Ф (-k). () and ARL 1 = 1/ (1-β) were, β = Probability [μ 0 -kσ/ n X μ 0 +kσ/ n wen μ = μ 0 +Δσ] β = Ф (k- Δ n ) Ф (-(k+ Δ n )) (3) Were, Ф denotes te cumulative distribution function of Standard Normal Distribution Wen te in-control times in eac cycle are assumed to be identically and independently distributed te expected cost per unit time E(C/T) is equal to te ratio of Expected Cycle Cost E(C) to te Expected Cycle Time E (T). Te expected cycle time consists of four parts, (1) te time elapsed before te assignable cause occurred, () te expected time between te occurrence of an assignable cause and te next out of control signal, (3) te expected time T 1 required to locate te assignable cause and (4) te expected time to repair te process. Te expected time tat elapsed until an assignable cause as occurred is = Mean of in-control times + time spent searcing during false alarms. Te expected time spent searcing during false alarms = T 0. (Number of false alarms). Expected Number of false alarms = S/ARL 0 = S α were, S is expected number of samples taken wile te process is in-control. S i0 i.prob( Assignable cause occured between i t and (i+1) t samples) S i exp( ( i ) ) exp( (( i 1) ) ) i1 S exp( ( i ) ) i1 (4) (5) ttp:// 5 editor@iaeme.com

6 Anand. Ayyagari, Srinivasa Rao.Kraleti and Laksminarayana. Jayanti were, [η/] is te greatest integer contained in η/. If te production is stopped during searc operations ten te expected time until 1 an assignable cause occurs equals to 1 ST0 (6) 1 Expected time until an assignable cause occurs is 1 1 (7) (1 1) ST0 1 were, δ 1 is an indicator variable. δ 1= 1 if production continues during searc for assignable cause and δ 1 =0 if production stops. Total Number of samples taken = Number of samples taken wile te process is in control + Number of samples taken wile te process is out of control. Total number of samples taken =(S+ARL 1 ). If is te time interval between collection of successive samples, ten te time required for taking (S + ARL 1 ) samples is 1 S (8) (1 ) If production continues during searc operations, ten samples are being taken after every units of time irrespective of te fact weter te process is in control or not. If E is te expected time for sampling and carting te result for one unit and a sample of size n is taken, ten te total expected time between te occurrence of an assignable cause and te next out of control signal is S ne (9) 1 (1 ) If T 1 is te expected time to identify te assignable cause and T is te expected time to repair te process ten te Expected Cycle Time is 1 E( T) 11 ST0S ne T1 T (1 ) (10) ttp:// 6 editor@iaeme.com

7 Determination of Optimal Design Parameters For X Control Cart Wit Truncated Weibull In-Control Times 4. EXPECTED COST PER CYCLE Te Expected Cost per Cycle includes 1) Cost of non-conformities produced, ) Cost of false alarms, 3) Cost of sampling and carting te result, and 4) Cost of repairing te process. Let C 0 be te cost of non-conformities produced per unit time wile te process is in in-control and C 1 be te corresponding cost wen te process is not in-control, were C 1 >C 0. Assuming tat production continues during searc and repair of te process te expected cost of non-conformities per cycle is C 0 C 1 S ne 1T 1 T 1 1 (1 ) (11) were, δ 1 = 1 if production continues during searc operations and δ 1 = 0 if production stops. Likewise δ = 1 if production continues during repair and δ = 0 if production stops during repair of te process. If C is te cost per false alarm ten expected cost of false alarms is = C S α. Te fixed cost of sampling is a and variable cost of sampling is b. Te expected cost of sampling and carting te result is ( a bn) 1 S ne 1T 1 T (1 ) So te Expected Cost per Cycle is 1 1 ( a bn) 1 E( C) C0 C1 W C 1 S C1 S ne 1T 1 T (1 ) Te Expected Cost per unit time is Z = E(C)/ E (T). 1 1 ( a b. n) 1 ( C0 C1 ) W C 1 S C1 S n E 1 T1 T ( ) (1 ) Z 1 (1 1) S T0 S n E T1 T (1 ) (1) (13) (14) ttp:// 7 editor@iaeme.com

8 Anand. Ayyagari, Srinivasa Rao.Kraleti and Laksminarayana. Jayanti 5. OPTIMIZATION PROBLEM In tis section we discuss te determination of optimal design parameters namely sampling interval, sample size n and control limit coefficients k. Tese decision variables are muc important for designing te X control cart in monitoring te process mean. We concentrate on minimizing te expected cost per a unit time subject to constraints on te probabilities of type I and type II errors, te type I and type II errors are te parameters caracterizing te producer s risk and consumer s risk wic sould not exceed a tresold value. Te tresold values can be determined based on te nature of te product and te quality specifications specified by te producer as well as te customer. Let p 1 be te maximum limit of allowable type I error and p be te maximum limit of type II error. Wit tese considerations te optimization problem for designing X control cart is Determine te optimal design parameters n, and k suc tat te above Expected Cost Per Unit Time Z is minimum subject to te conditions tat α p 1 and β p. i.e. Min Z=E(C)/E (T), wic is given in equation number 14 above, subject to te conditions α p 1, β p, >0, n>0, k>0 and n is an integer. Since te objective function is non-linear and multi-modal te optimal values of, n and k can be obtained by using non-linear integer programming wit grid searc metod using LINGO 8.0 computer package for given values of te model parameters and costs. 6. CASE SUDY In tis section a case study is presented to validate te solution procedure of te economic statistical design of X cart wit left truncated Weibull in-control times. A cement plant wic produces cement wit clinkers as a crucial intermediate product wose quality as a direct bearing on te quality of te final product is considered. Te quality of tis caracteristic is measured troug te concentration of lime in terms of weigt percentages. Te manufacturer uses X cart to monitor te process. Based on te analysis of Quality control personnel s salaries, cost of te equipment used for testing it is estimated tat te fixed cost of taking a sample is 0.50 units wile te variable cost of sampling is 0.10 units. Te cost of non-conformities produced wile te system is in control i.e.c 0 is estimated to be per our wile te cost of non-conformities produced wile te system is out of control C 1 is estimated to be per our. Te cost per false alarm C was found to be and te cost of locating te assignable cause and removing it W is found to be Te expected time for searcing a false alarm is 1 our, te expected time for identifying te assignable cause is ours and te expected time to remove te assignable cause is 1 our and te expected time for sampling and carting te result for one unit is 0.01 ours. Te concentration of lime follows a normal distribution. Te restrictions on type I error and type II error probabilities are α 0.05 and β 0.10 respectively. From te records of te manufacturer it is estimated tat te in-control times of te process follow a truncated Weibull distribution wit parameters ttp:// 8 editor@iaeme.com

9 Determination of Optimal Design Parameters For X Control Cart Wit Truncated Weibull In-Control Times θ= , γ=.0000, and η= Te goodness of fit of te distribution is examined by probability plotting. Te non-linear programming problem associated wit tis case is ( n) 15.00S Min Z S. 0.01n 3.00 (1 ) suc tat 0.05, 0.10, > 0, k > 0, n > 0 and n is an integer. (16) were, z k e dz ( k n) z 1 e ( k n) dz (17) (18) S i 1 e ( i ) Te optimal values of te decision variables, n and k are determined using non-linear programming tecnique namely grid searc metod under constrained optimization i.e. te separable programming tecnique given in LINGO 8.0 is used.te code for solving te above non-linear programming problem is developed and is given in appendix A. For te estimated costs and time parameters te optimal values of, n and k are found to be ours, 16 and.7464 wit an expected minimum cost of units. Using tese values of, n and k, a suitable control cart is designed and suggested for optimal operating policy of te manufacturer. A comparative study of te developed design is carried out wit te design of X control cart wit two parameter Weibull distribution given by Mc Williams in Te difference between te suggested model and te design suggested by Mc Williams is tat in our design we ave introduced a truncation parameter along wit upper bounds on type I and type II error probabilities. It is observed tat ignoring te goodness of fit for te in-control time data (wic gives goodness of fit to tree parameter Weibull distribution at 5% level of significance) te design parameters are obtained and sown in table 1. (19) ttp:// 9 editor@iaeme.com

10 Anand. Ayyagari, Srinivasa Rao.Kraleti and Laksminarayana. Jayanti Table 1 Comparative study of two designs. Design Two parameter Weibull incontrol times Tree parameter Weibull incontrol times (proposed design) Optimal values of design parameters. n k Minimum Cost Z From table 1 it is observed tat te developed design wit tree parameter Weibull in-control times is aving less optimal cost compared to te earlier design proposed by Mc Williams(1989), i.e. tis design out performs te earlier design for normal quality variates wit skewed in-control times. Te k value of te proposed model is more compared to te earlier existing model wic indicates improvement of quality of te product. Te reduction in cost per a unit time indicates tat te proposed design is more optimal. 7. SENSITIVITY ANALYSIS Sensitivity analysis as been performed to investigate ow te parameters of incontrol time distribution and cost parameters effect te optimal values of te design parameters viz. sample size n, sampling interval te number of standard deviations from central line to te control limits k, te probability of false alarm α, te probability of not getting an out of control signal wen te process is out of control β and te optimum cost Z. Te values of te parameters of te cost model are set as δ 1= δ = 1, a= 0.50, b= 0.10, T 0 = 1our, T 1 = ours, T =1our, E=0.01ours, C 0 =10.00, C 1 =15.00, C =50.00, W= Using te equations in 16 to 19, te optimal values of te design parameters are obtained for various values of te model parameters and costs and are presented in table and 3. Te relationsips of te design parameters are sown in figure 1. ttp:// 10 editor@iaeme.com

11 Optimal Design Parameters Optimal Design Parameters Optimal Design Parameters Optimal Design Parameters Determination of Optimal Design Parameters For X Control Cart Wit Truncated Weibull In-Control Times Location Parameter Vs. Optimal Design Parameters Scale Parameter Vs.Optimal Design Parameters n k n k Location Parameter Scale Parameter Sape Parameter Vs.Optimal Design Parameters Sift Vs Optimal Design Parameters n k n k Sape Parameter Sift Figure 1 Effect of in-control time distribution parameters and optimal design values. Table Optimal values of, n, k, α, β and Z for given values of η, θ, γ, Δ, p 1, p (a=0.50, b=0.10, C 0 =10.00, C , C =50.00 and W=100.00) η θ γ Δ p1 p Mean n k α β Z ttp:// 11 editor@iaeme.com

12 Anand. Ayyagari, Srinivasa Rao.Kraleti and Laksminarayana. Jayanti TABLE 3 Optimal values of, n, k, α, β and Z for given values of a, b, C 0, C1, C and W (η=50.00, θ=0.0001, γ=.00, Δ=1.00, p 1 =0.05 and p =0.10) a b C0 C1 C W Mean n k α β Z It is observed tat as te mean of in-control time distribution increases te optimal sampling interval also increases wile tere is decrease in te optimum cost. It is furter observed tat as te sift in te level of te process becomes larger tere is a decrease in te sampling interval and te sample size. However tere is an increase in te value of k. It is observed tat te restrictions on te type I and type II error probabilities ave considerable effect on te optimal design parameter values and te optimum cost. Furter it is observed tat te various costs involved in te design ave considerable effect on te optimal design parameter values. 8. X CONTROL CHART WITH RANDOM OUT OF CONTROL TIMES. In te previous section T 1, te time required to detect te assignable cause and T, te time required to repair te process are assumed to be deterministic. In tis section te out-of control times are assumed to be random and follow two parameter Gamma distribution. Let T 1 be a random variable following a Gamma distribution wit sape parameter υ 1 and scale parameter λ 1.Its probability density function is 1 ( 11) 1t 1 t e f1( t), for 1 0, 1 > 0, t > 0 ( ) (0) = 0, Oterwise. 1 T be anoter independent random variable following anoter Gamma distribution wit sape parameter υ and scale parameter λ. Its probability density function is ttp:// 1 editor@iaeme.com

13 Determination of Optimal Design Parameters For X Control Cart Wit Truncated Weibull In-Control Times t e f ( t), for 0, > 0, t > 0 ( 1) t ( ) = 0, Oterwise. Te expected cycle time is 1 1 E( T) (1 1) S T0S n E. () (1 ) 1 (1) Te expected cost per cycle is 1 1 ( a bn) 1 1 E( C) C0 C1 W +C 1 S + C1 S n E 1 (1 ) 1 (3) Te expected cost per unit time is ten 1 1 ( a bn) 1 1 ( C0 C1 ) W C 1 S C1 S ne 1 (1 ) 1 Z 1 1 (1 1) S T0S n E (1 ) 1 To obtain te optimal design parameters one as to minimize Z subject to te constraints α p 1 and β p for given values of model parameters and costs. For various values of te model parameters and costs te optimal design parameters, n, k and Z are obtained using non-linear integer programming tecniques and sown in table 4 and 5. Table 4 Optimal values of, n, k, α, β and Z for given values of ν 1, λ 1, ν, λ, η (θ=0.0001, γ=.00, Δ=1.00, p 1 =0.05, p =0.10, a=0.50, b=0.10, C 0 =10.00, C 1 =15.00, C =50.00 and W=100.00) (4) ν 1 λ 1 ν λ η Mean n k α β Z ttp:// 13 editor@iaeme.com

14 Anand. Ayyagari, Srinivasa Rao.Kraleti and Laksminarayana. Jayanti Table 4 continued Optimal values of, n, k, α, β and Z for given values of θ, γ, Δ, p 1, p (ν 1 =1.60, λ 1 =0.80, ν =0.80, λ =0.80 η=50.00, a=0.50, b=0.10, C 0 =10.00, C , C =50.00 and W=100.00) θ γ Δ p1 p Mean n k α β Z Table 5 Optimal values of, n, k, α, β and Z for given values of a, b, C 0, C 1, C and W (ν 1 =1.60, λ 1 =0.80, ν =0.80, λ =0.80 η=50.00θ=0.0001, γ=.00, Δ=1.00, p 1 =0.05 and p =0.10) a b C0 C1 C W Mean n k α β Z It is observed tat te sape and scale parameters of te out of control time distribution ave considerable influence on te sampling interval te oter design parameters are not tat muc affected. Wit respect to te influence of in-control time distribution parameters and costs, tere is no difference between random out of control times and fixed out of control times. However, it is observed tat te random out of control times are more suitable for alternate analysis since tey include te fixed out of control times also as a particular case wen te out of control time distribution degenerates to a fixed value. ttp:// 14 editor@iaeme.com

15 Determination of Optimal Design Parameters For X Control Cart Wit Truncated Weibull In-Control Times Te developed design for X control cart for normal quality caracteristic is capable of portraying various patterns of in-control times and out of control times. Tat is te tree parameter Weibull distribution wic is considered for te process in-control times is includes exponentially distributed in-control times, two parameter Weibull distributed in-control times, increasing failure rate in-control times, decreasing failure rate in-control times for specific values of te in-control time distribution parameters. Te minimum guarantee period of process in-control times eiter after repair or installation as a significant influence on te optimal values of te design parameters and also ave tremendous managerial implications in monitoring process mean. Te assumption on random out of control times is also anoter important consideration wit practical utility in designing te X control cart. It is quite natural tat te time required for detecting te cause of failure and te time for repair are influenced by various factors. By suitably identifying te associated probabilities for process out of control times leads to ave optimal design parameters wit realistic postulates. By fixing te optimal design parameters te sop floor people can effectively monitor and control te process and avoid wastage and rework due to non-conformity of te product. 8. CONCLUSIONS Te statistical economic design of X control cart as been developed assuming tat te quality caracteristic X follows normal distribution and te in-control time follows a left truncated Weibull distribution. By fixing te upper bounds of type I and type II errors of production te quality of te product can be maintained easily and effectively. Te assumption on te in-control time distribution as a significant effect on te effectiveness of te X cart in identifying te assignable cause. Minimizing te expected cost per a unit time te optimal design parameter sample size, time, interval between successive samples and control limits are derived. Te numerical illustration presented indicates te utility of te proposed design. Sensitivity analysis carried out indicated tat te optimal design parameters and te cost per a unit time are more sensitive towards te cost parameters tan oter parameters. Tese carts are very useful for quality control of several manufacturing process like cement, paints, edible oils were te in-control time of te process can be caracterized by a left truncated tree parameters Weibull distribution. It is also possible to develop statistical economic design of X control cart for multiple types of assignable causes wic will be taken up. REFERENCES [1] Saniga. E, McWilliams T. Davis D, Lucas, J., Economic control cart policies for monitoring variables, International Journal of Productivity and Quality Management, 006, 1(1/): [] Prajapati. D.R., Maapatra. P. B. An effective joint X and R cart to monitor te process mean and variance, International Journal of Productivity and Quality Management, 008, (4): [3] Duncan, A.J. Te Economic design of X carts used to maintain current control of a process, Journal of American Statistical Association, 1956; 66: [4] Montgomery. D.C. Te economic design of control carts: a review and literature survey, Journal of Quality Tecnology, 1980, Vol. 1:75 87 ttp:// 15 editor@iaeme.com

16 Anand. Ayyagari, Srinivasa Rao.Kraleti and Laksminarayana. Jayanti [5] Lorenzen. T.J., Vance. L. C. Te economic design of control carts: a unified approac, Tecnometrics, 1986, Vol. 8:3 10 [6] Al-Oraini. H.A, Raim, M.A., Economic statistical design of X control carts for systems wit gamma (λ, ) in-control times, Journal of Applied Statistics, 003, Vol. 30: [7] Biscak, D.P, Triestsc. D. Te rate of false signals in X control carts wit estimated limits, Journal of Quality Tecnology, 007; Vol. 39(1):54 65 [8] Nenes. G, Tagaras. G Te Economically designed two sided Bayesian X control cart, European Journal of Operations Researc, 007 Vol. 183(1): [9] Teyaracakul, S., Cand, S., Tang, J. Estimating te limits for statistical process control carts: a direct metod improving upon te Bootstrap. European Journal of Operations Researc, 007, Vol.178(): [10] Sasibusan, B.M., Digambar, T.S. Economic design of a modified variable sample size and sampling interval X cart, Economic Quality Control, 007, Vol. ():73 93 [11] Dias. J.R. New results in economic statistical quality control. Economic Quality Control, 007, Vol. (1):41-54 [1] Vommi, V.B., Seetala, M.S.N. A new approac to robust economic design of control carts, Applied Soft Computing, 007, Vol. 7(1):11 8 [13] Costa. A.F.B, De Magalaes, M.S. An adaptive cart for monitoring te process mean and variance, Quality Reliability Engineering International, 007, Vol. 3(7): [14] Taga. K. Control cart or auto correlated processes wit eavy tailed distributions, Economic Quality Control, 008, Vol. 3(): [15] Kim, Y.B., Hong, J.S., Lie, C.H. Economic statistical design of -of- and -of- 3 runs rule sceme. Quality and Reliability Engineering International, 009, Vol. 5():15-8 [16] McWilliams. T.P. Economic control cart designs and te in-control time distribution: a sensitivity study, Journal of Quality Tecnology, 1989, Vol. 1: [17] Cen.H, Ceng. Y Non-normality effects on te economic statistical design of X carts wit Weibull in-control time, European Journal of Operations Researc, (007) Vol. 176(): [18] Dwyia S. Hassun, Constructing A New Family Distribution From Tree Parameters Weibull Using Entropy Transformation, International Journal of Advanced Researc in Engineering and Tecnology, 5(6), 014, pp [19] Neelufur, Srinivasa Rao, K. and Venkata Subbaia, K. Optimal Design of X control cart wit Pareto in-control times. International Journal of Advanced Manufacturing Tecnology, 010, Vol. 48(9): ttp:// 16 editor@iaeme.com

17 Determination of Optimal Design Parameters For X Control Cart Wit Truncated Weibull In-Control Times APPENDIX-A LINGO COMPUTER CODE FOR DETERMINIG THE OPTIMAL DESIGN sets: NUMBERS /1..00/; end sets DATA: PARAMETERS c0=10.00;c1=15.00;c=50.00;e=0.01;d1=1; D=1;a=0.50;b=0.10;T0=1;T1=.00;T=1.0;w=100; t=0.0001; L=50.00; g=.0; DELTA=1.00; End Data Min=((((c0- c1)*mean)+w)+((c1*+(a+b*n)+(c*alpa))*s)+((c1+((a+b*n)/))*((/(1- Beta))+n*E+D1*T1+D*T)))/(((((1-D1)*T0*Alpa)+)*s)+((/(1- Beta))+n*E+T1+T)); s=(@sum(numbers(im):@exp(-t*(((im)*)-l)^g)));!@gin(n); (*@psn(-k))<=0.05; (@psn(k-delta*(n^0.5))-@psn(-k-delta*(n^0.5)))<=.10; mean=(l+(m/(t^(1/g)))); alpa=(*(@psn(-k))); beta=(@psn(k-delta*(n^0.5))-@psn(-k-delta*(n^0.5))); end ttp:// 17 editor@iaeme.com