ECONOMIC-STATISTICAL DESIGN APPROACH FOR A VSSI X-BAR CHART CONSIDERING TAGUCHI LOSS FUNCTION AND RANDOM PROCESS SHIFTS

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1 International Journal of Reliability, Quality and Safety Engineering Vol. 21, No. 2 (2014) (23 pages) c World Scientific Publishing Company DOI: /S ECONOMIC-STATISTICAL DESIGN APPROACH FOR A VSSI X-BAR CHART CONSIDERING TAGUCHI LOSS FUNCTION AND RANDOM PROCESS SHIFTS TONI LUPO Department of Chemical, Management Informatics and Mechanical Engineering Università deglistudidipalermo Palermo, Italy toni.lupo@unipa.it Received 7 June 2013 Revised 11 February 2014 Accepted 25 March 2014 Published 21 April 2014 Economic design approaches of control charts are commonly based on the assumption that various cost parameters values and the occurrence risk of assignable causes have to be aprioriknown with precision. However, in real operative contexts, such parameters can be really difficult to accurately estimate, especially considering costs arising from out-of-control conditions of the process. As consequence, pure economic design approaches can involve chart schemes with low statistical performance. To overcome such limitation, it is herein proposed a multi-objective economic-statistical design approach for an adaptive X-bar chart. In particular, such approach aims at the minimization of both the total quality related costs and the out-of-control average run length, in such a way assuring an optimal trade-off between economic and statistical performance of the related control procedure. Moreover, for a robust design approach, the mean shift is considered as a random variable. A mixed integer nonlinear constrained mathematical model is formulated to solve the treated problem, whereas the Pareto optimal frontier is described by the ε-constraint method. In order to show the employment of the proposed approach, an illustrative example is developed and the related considerations are given. Finally, some sensitivity analysis is also performed to investigate the effects of operative and costs parameters on chart parameters. Keywords: Statistical process control; adaptive X-bar control chart; multi-objective optimization problem; ε-constraint method. 1. Introduction Control charts are the widely used tool to perform Statistical Process Control (SPC) procedures. In particular, Shewhart charts, after more than 70 years of their introduction, are still the most considered control charts to monitor for variables criticalto-quality (CTQ) parameters of a manufacturing process outcome. Considering

2 T. Lupo such control charts, the R chart is widely used for the monitoring of the sample variability, whereas the X-bar chart is considered to monitor the sample mean. In order to design an X-bar chart, the values of three parameters have to be fixed: the sample size (n), the sampling interval (h) and the control limits width coefficient (k), i.e., the number of standard deviations of the sample mean between limits and the sample statistic mean. Commonly, such chart parameters cannot vary during SPC operations; in such circumstance, the related chart is referred as static or with fixed parameters chart. However, recent SPC strategies take advantages from the use of X-bar chart equipped with adaptive capability since, in such a way, the such enhanced chart schemes can be smarter than the related static ones in the sense that they can update themselves to fit well for different process situations. 1 Adaptive chart schemes for the X-bar chart, such as variable sample interval (VSI), variable sample size (VSS), variable sample size and interval (VSSI) and fully adaptive control chart (Vp), in which all the chart parameters are allowed to adaptively vary during SPC operations, have been extensively studied in the recent decade and several review works have been developed. For example, Tagaras 2 reviewed the development of adaptive charts until Another broad review presented by Woodall and Montgomery 3 covered some methods with variable sampling schemes. Zimmer et al. 4 compared the performance of Shewhart-type charts with adaptive sample sizes and/or sample intervals. The first attempt to statistically optimize an adaptive X-bar chart is attributed to Reynolds et al. 5 that introduced the concept of VSI chart scheme. In particular, the authors show that the proposed adaptive scheme, due to its intrinsic flexibility, is faster than the related static one in detecting moderate shifts in the process mean. Statistical enhancements for VSI chart schemes have been achieved also considering run rules 6 and a one side control limit. 7 Robustness studies for VSI chart schemes have been performed by Amin and Miller 8 and recently by Lin and Chon 9 by comparing the statistical performance of a VSI chart versus the related static one for controlling a non-normally distributed process mean. Also the statistical properties of VSS schemes have been widely studied in the literature. 10,11 In particular, as shown by Zimmer et al., 11 approaches using two sample sizes are mostly considered by researchers, since the simple dichotomous scheme of the sample size gains most of the benefits that can be reached by VSS schemes and, at the same time, they are relatively easy to implement in real operative contexts. The concept of adaptively varying in a combined way both the sampling interval and the sample size has been introduced in order to further improve the chart capability in detecting small shifts in the process mean. For example, Prabhu et al. 12 developed a statistical design approach for a VSSI X-bar chart. In particular, the authors by comparing the proposed adaptive chart with the related static chart, the VSS chart, and the VSI chart, show that substantial improvements in the statistical performance over these schemes can be obtained. More recent results for statistical optimizations of VSSI X-bar control charts are presented in Refs. 13 and 14, and auto-correlated processes variables are considered in Ref

3 Economic-Statistical Design Approach for a VSSI X-Bar Chart Taking into account the economic viewpoint to design an adaptive X-bar chart, to our knowledge, Das et al. 16 were the first researchers who attempted to design a VSI X-bar chart on the base of an economic criterion. More recently, Sim and Xie 17 developed an economic design approach for a VSS chart considering the process characterized by double assignable causes. More in detail, the authors show that a pure economic design approach can involve control chart schemes characterized by low statistical performance in terms of an excessive amount of false alarms. For such reason, in the literature economic design approaches under statistical constraints are frequently considered, even if such constraints can cause a substantial increase of SPC procedure related costs. For example, Prabhu et al. 18 proposed an economic design approach for a VSSI chart with double sample size and sample interval in which constraints on the in-control and out-of-control average run length (ARL) are considered to assure the required statistical performance of the chart. The authors highlight that, although the considered chart scheme outperforms the related static one, it can spend more resources since it uses rapid sampling rates and large average sample sizes to satisfy the statistical constraints. More recent results for economic optimizations of adaptive approach for X-bar chart are presented in Refs The basic assumption of economic design approaches of control charts is that various cost parameters and the occurrence risk of assignable causes have to be a priori known. However, in real operative contexts such parameters can be really difficult to accurately estimate. 20 For example, costs arising from the production of nonconforming parts when the process is out-of-control are really difficult to estimate with precision. Even costs arising from false alarms result in difficulty in evaluating or even immeasurable cost parts. 22 As a consequence, pure economic design approaches can involve chart schemes with low statistical performance in terms of false alarms number and also can imply low robustness of time intervals between samples, as well as imprecise chart parameters values. 19 In the light of the previous considerations, the purpose of the present paper is to propose a multi-objective economic-statistical design approach for an adaptive X- bar chart. In particular, the statistical optimization is herein considered to indirectly optimize the immeasurable or even hard to evaluate quality related costs part, since the latter is strongly related to the statistical performance of the chart. In fact, a statistically effective chart scheme determines a substantial reduction of both the false alarms number and the amount of the expected out-of control shift magnitude and so also for the immeasurable or even hard to evaluate quality related costs part. In addition, the economic optimization is even herein considered to also take care of the measurable quality related costs part. More in detail, in the case herein treated, the goal is to determine the parameters in VSSI X-bar chart to minimize both the hourly total quality related costs and the out-of-control ARL, ensuring a minimum allowable in control average run length. Moreover, in order to obtain more robust solutions, the shift magnitude of the controlled CTQ parameter is considered as a random variable modeled by a proper probability distribution function. A mixed

4 T. Lupo integer nonlinear constrained mathematical model is formulated to solve the treated multi-objective optimization problem whereas the ε-constraint method is used to describe the entire Pareto optimal frontier. The reminder of the present paper is organized as follows: in the next section, the main definitions for the proposed study are given. In Sec. 3, the model to evaluate the economic performance of the considered adaptive chart scheme is formulated. In Sec. 4, the mathematical optimization model is given and the considered resolution approach is described. In Sec. 5, an illustrative example is developed and the obtained results are interpreted and commented. Finally, the conclusions close the work. 2. Problem Definitions As mentioned before, it is herein considered a SPC procedure implementing a VSSI adaptive scheme of the X-bar chart with double sample size and sampling interval. The zone within the control limits is divided into three strips; two zones are defined: zone 1: the central zone within the warning limits [ w, w]; zone 2: the warning zone [ k, w) U (w, k]. If the point, representative of the tth sample mean, is plotted within the zone 1, then the next one will have a sample size of n 1 and will be taken h 1 time unit after the tth one. On the contrary, if the point representative of the tth sample mean is plotted within the zone 2, then the next one will have a sample size of n 2 >n 1 and will be taken h 2 <h 1 time unit after the tth one. Finally, if the point, representative of the tth sample statistic, is plotted outside the control limits (Action Zone), then the out-of-control signal of the chart has to be considered as a false alarm or as a consequence of the occurrence of an assignable cause. In the latter case, a corrective action has to be quickly undertaken by the quality expert team. The controlled CTQ parameter is assumed normally distributed with known mean µ 0 and standard deviation σ. Thus, the adaptive X-bar control chart can be plotted considering the standard normal distribution Z: Z i = X µ 0 σ/ n i, (1) where in Eq. (1), X is the observed sample mean of the under control CTQ parameter in the ith sample composed of n i units. Figure 1 shows the considered adaptive policy. Finally, it is assumed that the in-control time of the process is exponentially distributed with mean 1/λ and the out-of-control condition determines an instantaneous, persistent and random shift in the process mean to values µ δ = µ 0 + δσ without modifying the standard deviation value

5 Economic-Statistical Design Approach for a VSSI X-Bar Chart Fig. 1. VSSI X-bar chart scheme. The following basic nomenclature is hereafter considered: w Warning limit width k Control limit width n 1 Smaller sample size n 2 Larger sample size σ Process standard deviation µ 0 In-control process mean µ δ Out-of-control process mean h 1 Shorter sampling interval h 2 Longer sampling interval λ System failure rate δ Random process mean shift P in In-control period P out Out-of-control period T 0 Expected search time related to a false alarm T 1 Expected search time related to an out-of-control condition T 2 Expected time to restore the system C T Hourly total quality related costs C LR Expected labor resources cost C Q Expected hourly quality control costs C Q (δ) Hourly quality control costs related to the shift δ d 1 Dummy variable that takes 1 if production continues during assignable causes searches, 0 otherwise d 2 Dummy variable that takes 1 if production continues during the system repair, 0 otherwise ARL 0 In-control average run length

6 T. Lupo ARL δ LR 1 LR 2 Y W Out-of-control average run length Smaller amount of labor resources Higher amount of labor resources Costs associated with false alarms Costs associated with the removing of assignable causes. 3. Total Quality Related Costs C T The employment of a SPC procedure involves costs related to nonconforming products while the process is in-control and out-of-control, for false alarms, for assignable cause locations and system repairs and also for sampling and inspection activities. Moreover, if production ceases during the assignable cause searching, the system downtime cost also has to be considered. A solid approach to compute such costs has been proposed by Duncan. 23 Such approach has been successively generalized by Lorenzen and Vance 24 by developing an economic model that can be applied to several charts types. Considering such approach, the total quality related costs C T arising from the employment of a SPC inspection procedure over a system functioning cycle, can be evaluated by the following expression: C T = C LR + C Q, (2) where C LR and C Q are the hourly labor resource and quality control costs, respectively. In particular, the quality control costs C Q are related to the mean shift magnitude and the mentioned economic model 24 takes into account a single out-ofcontrol scenario characterized by a single mean shift magnitude δ. However, in a real operative context, different shift magnitudes can randomly occur with relation to failure modes related to different process out-of-control scenarios. Therefore, when true shift magnitudes are unknown or not constant, either assumptions have to be made regarding shift sizes or capable algorithms have to be designed to estimate such information. For example, in Refs. 25 and 26 a discrete number of process out-of-control scenarios is considered to design an X-bar chart. More recently, Wu et al. 27 proposed a robust economic design approach based on Taguchi s loss function in which it is assumed a random shift of the process mean modeled by means of the Rayleigh distribution: f(δ) = π δ 2µ 2 exp δ ( π δ2 4µ 2 δ ). (3) In particular, such distribution, where shape is related only to the expected value of the out-of-control process mean µ δ, can be conveniently used to fit well the randomness of the mean shift magnitude for a wide processes typology. 27 By considering such a assumption and by assuming that sequences of in-control/out-of-control system periods can be considered as a renewal stochastic process, the quality control costs C Q can be calculated by summing the quality control costs related to the shift

7 Economic-Statistical Design Approach for a VSSI X-Bar Chart magnitude δ, C Q (δ), over the process mean shift probability distribution: C Q = + 0 C Q (δ) f(δ)dδ. (4) In the next section, the system functioning cycle is defined and all the durations of its periods are evaluated, whereas the computations of the hourly labor resource costs C LR and quality control costs C Q (δ) are given in Secs. 3.2 and 3.3, respectively System functioning cycle time It is considered that the process follows functioning cycles constituted by the following periods (see Fig. 2): the in-control period P in ; the out-of-control period P out ; the time T 1 required to detect the assignable cause and the time T 2 to restore the system. The expected length of a system functioning cycle T (δ) represents the period between two successive starting of the manufacturing process, after detecting the occurred assignable cause and solving its related problems. 28 The in-control period length P in is given by: P in = 1 λ + (1 d 1) s T 0, (5) ARL 0 where λ is the failure rate of the manufacturing system, i.e., the number of occurrences of an assignable cause for time unit, while the second term takes into account false alarms effects over the in-control period being: d 1 a binary dummy variable that takes 1 if production continues during assignable causes searches and 0 otherwise, T 0 the expected search time associated to a false alarm, ARL 0 the in control average run length and s the expected number of samples taken during the incontrol period, which can be obtained as follows: s = 2 i=1 [ ] e λ hi Pr{zone = i δ =0}. (6) 1 e λ hi T( ) P in P out T 1 T 2 Fig. 2. System functioning cycle

8 T. Lupo P out Fig. 3. Out-of-control period P out. On the contrary, the out-of-control period length P out (Fig. 3) is given by the following equation: P out =(E δ (h) ζ)+e δ (h) (ARL δ 1) + E δ(n), (7) r IN where E δ (h) is the out-of-control expected value of sampling interval, ARL δ is the out-of-control average run length and the last term represents the time to inspect the sample providing the out-of-control signal, i.e., the last sample in the system functioning cycle, being E δ (n) the expected value of the sample size while the process is out-of-control and r IN the sample inspection rate. Finally, ζ is the expected time between the occurrence of an assignable cause and the previous sampling instant given by: ζ = 2 i=1 [ Pr{zone = i δ =0} 1 (1 + λ h ] i)e λ hi h i λ(1 e λ hi ) E 0 (h) in which E 0 (h) is the in-control expected value of the sampling interval. Therefore, according to Eqs. (5) and (7), the expected length of the system functioning cycle T (δ) assumes the following expression: T (δ) = 1 λ + (1 d 1) s T 0 ARL 0 (8) +ATS δ ζ + E δ(n) r IN + T 1 + T 2. (9) The functions of ARL 0,ARL δ,ats δ, E δ (n), E 0 (h) ande δ (h) aregivenwithmore details in the Appendix Labor resources cost C LR For the considered adaptive policy, the needed capacity of labor resources can vary at two levels, LR 1 and LR 2, according to the chart parameters stated at each sampling epoch. In particular, let i be the zone of the control chart that has been plotted the point representative of the (t 1)th sample. Under the assumption that the entire sampling interval h i is used to inspect the sample of size n i, the amount

9 Economic-Statistical Design Approach for a VSSI X-Bar Chart of the labor resources needed to inspect the next tth sample can be obtained by: LR i = n i for i =1, 2. (10) h i r IN Therefore, the expected amount of labor resources can be evaluated as the weighted sum of LR 1 and LR 2 by the related in-control probabilities that a point is plotted on the chart within the zone 1 or 2 of the chart. Thus, the latter can be written as: E(LR) = LR 1 Pr{zone = 1 δ =0} +LR 2 Pr{zone = 2 δ =0} (11) in which, the in-control probability that the point Z i, representative of the generic sample i, is plotted within the zone 1 or the zone 2 of the chart can be written respectively as: Pr{zone = 1 δ =0} = Pr{ w Z i w} Pr{ k Z i k} = Φ(w) Φ( w) Φ(k) Φ( k), (12) Pr{zone = 2 δ =0} = Pr{ k Z i < w w<z i k} Pr{ k Z i k} = 2[Φ(k) Φ(w)] Φ(k) Φ( k). (13) Consequently, the expected labor resources cost C LR can be computed by the following relationship: being c LR the hourly labor resources cost. C LR = E(LR) c LR (14) 3.3. Quality control costs C Q (δ) The hourly quality control costs related to the mean shift value δ, C Q (δ), can be obtained by dividing the costs arising from the quality control in a system functioning cycle C δ by the system cycle time T (δ) (Eq. (9)), that is: C Q (δ) = C δ T (δ). (15) InEq.(15),thecostsC δ can be computed by means of the following relationship 29 : where: C δ = C 1 + C 2 (δ)+c 3 + C 4 (δ), (16) C 1 and C 2 (δ) are the costs arising from the nonconforming products manufactured during the system in-control and out-of-control periods, respectively; C 3 is the cost due to the false alarms, the detection and the removing of assignable causes; C 4 (δ) is the cost arising from the sampling activities

10 T. Lupo The costs C 1 and C 2 (δ) represent the economic loss due to deviation of the CTQ characteristic from its target value T. As suggested by Wu et al., 27 such economic loss can be efficiently and effectively estimated by the quadratic representation of the quality loss function L(x): L(x) =c nc (x T ) 2. (17) Therefore, C 1 and C 2 (δ) can be calculated by using the following relationships: and C 1 = P in r PR = P in r PR C 2 (δ) =P out r PR + + c nc (x T ) 2 f(x)dx = P in r PR c nc [σ 2 +(µ 0 T ) 2 ] + c nc (x µ 0 + µ 0 T ) 2 f(x)dx c nc (x µ 0 δσ + µ 0 + δσ T ) 2 f(x)dx = P out r PR c nc [σ 2 +(µ 0 T ) 2 + δ 2 σ 2 2δσ(µ 0 T )], (18) (19) where in Eqs. (18) and (19), f(x) is the probability density function of the normal variable N(µ 0, σ), r PR is the production rate (units/h) and c nc is the loss coefficient, that is a constant depending on the external cost associated with the production of a nonconforming part. The cost C 3 is given by the following equation: C 3 = s Y + W (20) ARL 0 in which Y and W are the costs associated with false alarms and for the detection and the removing of assignable causes, respectively. Lastly, the sampling cost C 4 (δ) isgivenby: C 4 (δ) =(a + b E 0 (n)) s +(a + b E δ (n)) s, (21) where in Eq. (21), a and b are the cost sampling components, E 0 (n) the expected value of the sample size while the process is in-control and s and s the expected number of samples taken during the system in-control (Eq. (6)) and out-of-control periods, respectively. The latter can be obtained by the following equation: ( s = ARL δ + ) E δ (n) r IN + d 1 T 1 + d 2 T 2 E δ (h) in which d 2 is a dummy variable that takes 1 if production continues during the repair, 0 otherwise. Finally, the function of E 0 (n) is given in the Appendix (22)

11 Economic-Statistical Design Approach for a VSSI X-Bar Chart 4. Multi-Objective Problem Formulation and Resolution Approach As mentioned before, the aim of the proposed design approach is to find out the optimal parameters of a VSSI X-bar chart, i.e., the sample sizes n 1 and n 2,the sampling intervals h 1 and h 2 and the control limit widths w and k, inorderto minimize both the total quality related cost C T and the expected value of the outof-control average run length E(ARL δ ). In particular, the treated multi-objective problem can be formulated by the following mixed integer nonlinear constrained mathematical model: Subjected to: min C T (n 1,n 2,h 1,h 2, w, k), (23) min E(ARL δ )(n 1,n 2,h 1,h 2, w, k). (24) ARL 0 L, (25) LR min LR 1 LR 2 LR max, (26) n min n 1 <n 2 n max, (27) h min h 2 <h 1 h max, (28) k min w<k k max, (29) n 1,n 2 integer. (30) In particular, E(ARL δ ) function can be obtained by summing the out-of-control average run length related to the shift magnitude δ, ARL δ, over the process mean shift probability distribution: Moreover, the constraints: E(ARL δ )= + 0 ARL δ f(δ)dδ. (31) (25) ensures a minimum allowable in-control ARL; (26) implies that a minimum part of the available labor resource should be employed to perform SPC operations and, at the same time, that the labor resource do not exceed the available resource capacity; (27) (29) assure the adaptive capability of the chart scheme and limit the sampling effort; (30) assures that the variables n 1 and n 2 are integer variables. In order to solve the formulated mathematical model, since the considered objective functions contrast on each other, it is not possible to find out a single solution corresponding to the best result for all of the considered objectives. Thus, in multiobjective optimization problems (MOOPs), the concept of solutions dominance is

12 T. Lupo defined. Moreover, a solution X 1 of a MOOP dominates the solution X 2 if: The solution X 1 is nonworse than the solution X 2 for each objective; The solution X 1 is strictly better than the solution X 2 for at least one objective. In the literature, problems formulated by means of MOOPs have been basically solved by two different approaches. The first one requires a weights vector to construct a composite function converting the MOOP into a single-objective one. The poor practicability of such an approach is well known and it is due to the difficulty of normalizing the objective functions and of quantifying the related weights. A more recent approach consists in splitting the solution procedure into two phases: the first step consists in obtaining the set of nondominated trade-off solutions (Pareto optimal solutions) within the whole space of feasible ones. In the second step, solutions belonging to the Pareto frontier are evaluated and compared in order to select the best one. The two steps approach is preferable since particular aspects as further subjective information about the problem, or particular needs, or knowledge of practicable solutions, are provided to the decision maker only in a second phase. So, it will be easier to obtain the solution representing the best compromise for all objectives, choosing it from the restricted set of the nondominated ones. The hardest aspect of this approach is to obtain the Pareto frontier. 30 Different approaches to describe the Pareto frontier have been proposed in the literature. The simplest and probably the most widely used approach to describe the Pareto frontier is the weight sum method. 31 The greatest difficulty on applying this method consists in setting appropriate weight vectors in order to obtain the nondominated solutions. Moreover, this method is able to describe only the convex region of the frontier. The latter aspect is overcome by the ε-constraint method, which is able to find solutions also in the nonconvex region. Such method is considered in the present paper to describe the Paretofrontier. It consists in reformulating the MOOP considering only one of its objectives and restricting the remaining ones within userchosen values ε. By applying the ε-constraint method, the MOOP is reduced to a single-objective optimization problem (SOOP), and a set of nondominated solutions can be obtained in more optimization steps by iteratively varying the ε values within the ranges of interest. More in detail, in order to describe the Pareto frontier for the problem herein considered, the Lexicographic Goal Programming (LGP) method is initially used 31 to find out the extreme solutions of the Pareto frontier. This method separately considers the two objective functions, thereby reducing the multi-objective problem into a mono-objective one. For example, these sequential steps determine the extreme solution A of minimum total related quality costs (see Fig. 4): minimizing C T as a single objective problem obtaining C T,min, minimizing E(ARL δ ), by imposing the value of C T not greater than C T,min, obtaining E(ARL δ ) max

13 Economic-Statistical Design Approach for a VSSI X-Bar Chart C T C T,max C T,j B ( ) j C T,1 C T,min 1 A E(ARL ) E(ARL ) min E(ARL ) j E(ARL ) 1 E(ARL ) max Fig. 4. Multi-step optimization procedure. The procedure is analogously applied changing the objectives hierarchy to find the other two bounds C T,max and E(ARL δ ) min of the extreme solution B of maximum total related quality costs. Once the extreme points of the Pareto frontier are determined, the ε-constraint method is used to describe the whole Pareto optimal frontier. In particular, to describe the Pareto frontier a multi-step optimization procedure has to be applied: in the first step, by minimizing C T, and by imposing that the E(ARL δ ) function has to take a value less than E(ARL δ ) max previously obtained. In this way the optimal solution 1 (E(ARL δ ) 1 ; C T,1 ) belonging to the Pareto frontier can be found (see Fig. 4). In the next step, such procedure is repeated minimizing C T, and imposing that the E(ARL δ ) function has to take a value less to that one corresponding to the solution previously found. The procedure is repeated until the other Pareto frontier extreme solution is obtained. In such way, the ε-constraint method ensures the determination of the whole Pareto frontier also in the presence of nonconvex regions. 32 For more details on the considered multi-step optimization procedure, the reader may also see Refs. 33 and Illustrative Example In this section in order to show the employment of the proposed multi-objective design approach, an illustrative example is developed. In particular, in next section, a case study is solved and the related optimal Pareto frontier is described, whereas in Sec. 5.2 some sensitivity analysis and comparative study are performed Optimal Pareto frontier description A manufacturing process is considered in which SPC operations implement a VSSI X-bar chart to monitor the CTQ characteristic of the process outcome. The considered process and costs parameters are reported in Ref. 35: the fixed cost per sample a = 5($), the cost per unit sample b = 1 ($), the labor resource cost c LR =

14 T. Lupo ($/h), the cost to locate and repair the assignable cause W = 900($), the cost for false alarm Y = 300 ($), the inspection rate r IN = 20 (units/h), the expected search time when the signal is a false alarm T 0 = 2 (h), the expected time to discover the assignable cause T 1 = 2 (h), the expected time to repair the process T 2 = 0 (h), the labor resource level can vary within the range LR min =0.1, LR max =5 and when the process goes out-of-control, the expected mean shift magnitude µ δ is equal to 1.5. According to the quadratic loss function, c nc =1,σ 2 =1 and µ 0 = T. The production rate is equal to r PR = 300 (units/h) and on average the process remains in control for 100 (h). Moreover, the production continues during the search for the assignable cause and it ceases when the repair activity is performed: d 1 =1andd 2 = 0. Furthermore, the following limits are imposed on the decision variables: n min =1, n max = 50, h min =0.1(h), h max =50(h), (n 1 n 2 ) min =1, (h 1 h 2 ) min =0.1, w min =0.5, w max =5, k min = w +0.3 and the minimum required in-control average run length ARL 0 is equal to 215. To solve the tackled optimization problem, the nonlinear Generalized Reduced Gradient Algorithm (GRG) implemented on the solver of Microsoft Excel r has been adopted. In particular, the following setting options configuration has been considered: running time max: 100 s; max iteration number: 100; precision: ; tolerance: 5%; convergence: ; assume linear model: No; use automatic scaling: Yes. The obtained optimal chart parameters minimizing both the total quality related costs C T and the expected out-of-control average run length E(ARL δ )aregivenin Table 1, whereas the related optimal Pareto frontier is shown in Fig. 5. Note from Table 1 and Fig. 5, the optimal scheme from the economic viewpoint is represented by the solution 1 with C T = $ and E(ARL δ )=4.48. On the contrary, from the statistical viewpoint, the solution 10, with E(ARL δ )=2.51 and a C T = $, represents the related optimal chart scheme. In addition, Table 1 Table 1. Optimal adaptive chart parameters (ARL 0 215). Solution E(ARL δ ) Decision variable value n 1 n 2 h 1 h 2 w k

15 Economic-Statistical Design Approach for a VSSI X-Bar Chart shows that with the increase of the statistical performance required to the control chart schemes also increase both the optimal sample sizes (n 1 and n 2 )andthe sampling intervals (h 1 and h 2 ) with no severe change in the chart control limits w and k C T ($) E(ARL δ ) Fig. 5. Optimal Pareto frontier (ARL 0 215). Table 2. Sensitivity analysis data. Factor ( ) (+) c nc ($) 1 2 λ r IN (measures/h) 2 20 µ δ C T ($) C T ($) E(ARL δ ) E(ARL δ ) (a) (b) Fig. 6. Sensitivity analysis results. (a) λ =0.01,r IN = 20 units/h; (b) λ =0.01,r IN = 2 units/h; (c) λ =0.05,r IN = 20 units/h and (d) λ =0.05,r IN = 2 units/h

16 T. Lupo C T ($) C T ($) E(ARL δ ) E(ARL δ ) (c) (d) Fig. 6. (Continued ) 5.2. Sensitivity analysis In order to investigate on the influence of the several operating and costs parameters on chart parameters values, a sensitivity analysis is also herein carried out. Table 2 reports the factors values considered in such analysis. Figure 6 summarizes the obtained optimal Pareto frontiers. For each graph, the two iso-chromatic points sets are based on the value of c nc (red color: c nc =1;blue color: c nc = 2), whereas + denotes µ δ =1.5 and x µ δ =2.5. The optimal solutions with the minimum C T for the considered 16 runs of the sensitivity analysis are summarized in Table 3. The effects of the considered process and cost parameters are evaluated by analysis of variance. The obtained results are summarized in Table 4. Table 3. Solutions of the comparison study with minimum C T. λ r IN c nc µ δ E(ARL δ ) C T n 1 n 2 h 1 h 2 w k

17 Economic-Statistical Design Approach for a VSSI X-Bar Chart Table 4. Significant standardized effects in the analysis of variance. Response variable E(ARL δ ) C T n 1 n 2 h 1 h 2 w λ r IN c nc µ δ The obtained results confirm that both the optimal sample sizes n 1 and n 2,as well the sampling interval h 2 decrease with the increase of the expected shift in the process mean µ δ. The total quality related cost C T is meaningfully and positively influenced by the external cost associated with the production of nonconforming parts c nc, the failure rate λ and the expected shift in the process mean µ δ. Finally, the expected out-of-control average run length of the chart E(ARL δ ) is negatively influenced by both the expected shift in the process mean µ δ and the inspection rate r IN. 6. Conclusions In the present paper, a robust multi-objective design approach for the VSSI X-bar chart has been developed. In particular, such approach is able to find out the chart parameters values corresponding to the minimum of both the total quality related cost arising from the related SPC procedure and the expected out-of-control average run length, warranting a minimum allowable in-control average run length. The considered multi-objective problem has been formulated by a nonlinear mathematical programming model and solved by a macro of Microsoft Excel. More in detail, the ε-constraint method has been considered to obtain the nondominated solutions, namely the Pareto optimal frontier. The knowledge of such frontier supplies to the decision maker s suitable information about the analyzed problem: first of all costs and statistical performance ranges in which the optimal solutions will fall down. Moreover, if budget availability and/or statistical performance constraints are considered, the sub-set of feasible optimal solutions on which the decision maker has to restrict the selection can be immediately individuated. Furthermore, the frontier analysis could directly address the decision maker to discard some solutions that imply small cost reductions with meaningful decrease in statistical performance. Finally, the developed approach is simple to use, fully supported by an Excel macro and requires short computational time. Appendix A. The Markov Model for the Adaptive X-Bar Control Chart Some parameters characterizing the herein considered adaptive chart can be calculated adopting the Markov model described below. The process is considered to be in the specific state ith when the statistic Z i of the previous sample is plotted on the chart within the zone i. Therefore, the possible

18 T. Lupo p 21 p p 12 p 22 p 1A p 1A A 1 Fig. A.1. States for the considered VSSI X-bar chart policy. states in which the system can be in a generic time instant are (see Fig. A.1): (A) State 1, if the point representative of the statistic Z i, calculated in the previous instant, is plotted on the chart within the zone 1; (B) State 2, if the point representative of the statistic Z i, calculated in the previous instant, is plotted on the chart within the zone 2; (C) State A, if the point representative of the statistic Z i, calculated in the previous instant, is plotted outside of the control limits. Such a condition determines the end of a system functioning cycle. Once this state is reached, the system cannot make further transitions; thus the transition probability from this state to each other is equal to zero (absorption state). A.1. Transition probabilities matrix Q The full probability transition matrix Q is: p 11 p 12 p 1A Q = p 21 p 22 p 2A, where: (A.1) p ij (with i and j = 1 or 2) is the probability that the point representative of the current sample is plotted within the zone j, having the previous one been plotted within the zone i; p ia (with i = 1 or 2) is the probability that the point representative the current sample is plotted outside the control limits, having the previous one being plotted within the zone i. The transition probabilities matrix Q 0, when the process is in-control, is: [ ] p 0 Q 0 = 11 p 0 12 p 0 21 p 0, (A.2)

19 Economic-Statistical Design Approach for a VSSI X-Bar Chart where the probability p 0 ij is the probability that the point is plotted within the ith zone: p 0 i1 =Pr{ w 1 Z i w 1 } =Φ(w 1 ) Φ( w 1 ), (A.3) p 0 i2 =Pr{ k Z i < w w<z i k} =2[Φ(k) Φ(w)] (A.4) with i =1, 2. The transition probabilities matrix Q δ, when the process is out-of-control, is: [ ] p δ Q δ = 11 p δ 12 p δ 21 p δ, (A.5) 22 where the probability p δ ij is the probability that the point is plotted within the ith zone: p δ i1 =Pr{ w 1 δ n i Z i w 1 δ n i } =Φ(w 1 δ n i ) Φ( w 1 δ n i ), (A.6) p δ i2 =Pr{( k δ n i Z i < w δ n i ) (w δ n i <Z i k δ n i )} (A.7) =2[Φ(k δ n i ) Φ(w δ n i ) with i =1,2. A.2. Sample sizes The following relationships express the value of the sample size n based on the position of the point plotted on the adaptive X control chart: { n1 if w Z n = i w, (A.8) n 2 if k Z i < w w<z i k. The expected value of the sample size when the process is in-control E 0 (n), can be obtained as the weighted sum of the values n i of the sample size, when the point is plotted within the zone i, multiplied by the conditional probability that the point is plotted in the same zone, being the process in-control: 2 E 0 (n) = [Pr{zone = i δ =0} n i ]. (A.9) i=1 In order to calculate the expected value of the sample size when the process is out-of-control E δ (n), the Markov model has to be applied and it is calculated as follows: [ [1 0]Q δ [I Q δ ] 1 n1 ] [ ϑ1 ϑ ][ 2 n1 ] n det(i Q 2 δ ) n 2 E δ (n) = [1 0]Q δ [I Q δ ] 1 [ 1 1 ] = [1 0] 1 [1 0] 1 det(i Q δ ) [ ϑ1 ϑ ][ 1 1 = (n 1ϑ 1 + n 2 ϑ 2 ), (A.10) (ϑ 1 + ϑ 2 ) ]

20 T. Lupo where ϑ 1 and ϑ 2 have been introduced to simplify the above relationship: ϑ 1 = p δ 11 (1 pδ 22 )+pδ 12 pδ 21, ϑ 2 = p δ 11 pδ 12 + pδ 12 (1 pδ 11 ). (A.11) (A.12) A.3. Sampling intervals The following relationships express the value of the sampling interval h based on the position of the point on the adaptive control chart: { h1 if w Z i w, h = (A.13) h 2 if k Z i < w w<z i k. As in the case previously treated, the expected value of the sampling interval when the process is in-control E 0 (h), can be obtained as the weighted sum of the values h i of the sampling interval, when the point is plotted within the zone i, multiplied by the conditional probability that the point is plotted in the same zone, being the process in-control: 2 E 0 (h) = [Pr{zone = i δ =0} h i ]. (A.14) i=1 The expected value of the sampling interval when the process is out-of-control E δ (h) is calculated as follows [ [1 0]Q δ [I Q δ ] 1 h1 ] [ 1 ϑ1 ϑ ][ [1 0] 2 h1 ] h det(i Q 2 E δ (n) = [ 1 ] = δ ) h 2 [ [1 0]Q δ [I Q δ ] 1 1 ϑ1 ϑ [1 0] 2 1 ] 1 det(i Q δ ) (A.15)......][ 1 = (h 1ϑ 1 + h 2 ϑ 2 ), (ϑ 1 + ϑ 2 ) where ϑ 1 and ϑ 2 have the same meaning as above. A.4. Statistical performance parameters The in-control average run length ARL 0 is calculated as follows: [ 1 p 0 22 p 0 ] 12 ARL 0 =[10][I Q 0 ] 1[ 1 ] p p 0 [ 11 1 =[10] 1 det(i Q 0 ) 1] (A.16) 1 p 0 22 = + p0 12 (1 p 0 11 ) (1 p0 22 ) p0 12. (A.17) p0 21 Similarly, the out-of-control average run length ARL δ,is: ARL δ =[10][I Q δ ] 1[ 1 ] 1 p δ 22 = + pδ 12 1 (1 p δ 11 ) (1 pδ 22 ) pδ 12. (A.18) pδ

21 Economic-Statistical Design Approach for a VSSI X-Bar Chart The in-control average time to false alarm ATS 0 is calculated as follows: [ 1 p 0 22 p 0 ] 12 ATS 0 =[10][I Q 0 ] 1[ h ] 1 p p 0 [ 11 h1 ] =[10] h 2 det(i Q 0 ) h 2 (A.19) (1 p 0 = 22) h 1 + p 0 12 h 2 (1 p 0 11 ) (1 p0 22 ) p0 12. p0 21 Similarly, the out-of-control average time to out-of-control signal ATS δ is: ATS δ =[10][I Q δ ] 1[ h ] 1 (1 p δ 22 = ) h 1 + p δ 12 h 2 h 2 (1 p δ 11 ) (1 pδ 22 ) pδ 12. (A.20) pδ 21 References 1. D. C. Montgomery, Introduction to Statistical Quality Control (John Wiley & Sons, New York, 2013). 2. G. Tagaras, A survey of recent developments in the design of adaptive control charts, J. Qual. Tech. 30(3) (1998) W. H. Woodall and D. C. Montgomery, Research issues and ideas in statistical process control, J. Qual. Tech. 31(4) (1999) L. S. Zimmer, D. C. Montgomery and G. C. Runger, Guidelines for the application of adaptive control charting schemes, Int. J. Prod. Res. 38(9) (2000) M. R. Reynolds Jr., R. W. Amin, J. C. Arnold and J. A. Nachlas, X charts with variable sampling intervals, Technometrics 30(2) (1988) R. Cui and M. R. Reynolds Jr., X-charts with runs rules and variable sampling intervals, Commu. Stat. Simul. 17 (1988) M. R. Reynolds Jr., J. C. Arnold and J. W. Baik, Variable sampling interval XM charts in the presence of correlation, J. Qual. Tech. 28 (1996) R. W. Amin and R. W. Miller, A robustness study of X charts with variable sampling intervals, J. Qual. Tech. 25(2) (1993) Y. C. Lin and Y. C. Chou, On the design of variable sample size and sampling intervals X charts under non-normality, Int. J. Prod. Econ. 96 (2005) A. F. B. Costa, X charts with variable sample size, J. Qual. Tech. 26(3) (1994) L. S. Zimmer, D. C. Montgomery and G. C. Runger, Evaluation of a three-state adaptive sample size X control chart, Int. J. Prod. Res. 36(3) (1998) S. S. Prabhu, D. C. Montgomery and G. G. Runger, A combined adaptive sample size and sampling interval X control scheme, J. Qual. Tech. 26(3) (1994) D. S. Bai and K. T. Lee, Variable sampling interval X control charts with an improved switching rule, Int. J. Prod. Econ. 76(2) (2002) , doi: /S (01)00161-X. 14. A. F. B. Costa and M. S. De Magalhães, An adaptive chart for monitoring the process mean and variance, Qual. Reliab. Engrg. Int. 23(7) (2007) , doi: /qre C. A. Zou, Z. A. Wang and F. B. Tsung, Monitoring autocorrelated processes using variable sampling schemes at fixed-times, Qual. Reliab. Engrg. Int. 24(1) (2008) T. K. Das, V. Jain and A. Gosavi, Economic design of dual sampling-interval policies for X charts with and without run rules, IIE Trans. 29 (1997)

22 T. Lupo 17. S. B. Sim and M. Xie, On variable sample size X chart for processes with double assignable causes, Int. J. Reliab. Qual. Safety Engrg. 11 (2004) S. S. Prabhu, D. C. Montgomery and G. C. Runger, Economic-statistical design of an adaptive X chart, Int. J. Prod. Econ. 49 (1997) G. Nenes, A new approach for the economic design of fully adaptive control charts, Int. J. Prod. Econ. 131(2) (2011) C. Mortarino, Duncan s model for X-bar control charts: Sensitivity analysis to input parameters, Qual. Reliab. Engrg. Int. 26(1) (2010) A. F. B. Costa and M. S. D. Magalhães, Economic design of two-stage X charts: The Markov chain approach, Int. J. Prod. Econ. 95(1) (2005) 9 20, doi: /j.ijpe Y. K. Chen and H. C. Liao, Multi criteria design of an X-bar control chart, Com. Ind. Engrg. 46 (2004) A. J. Duncan, The economic design of X chart used to maintain current control of a process, J. Am. Stat. Assoc. 51 (1956) T. J. Lorenzen and L. C. Vance, The economic design of control charts: A unified approach, Technometrics 28(1) (1986) M. S. De Magalhães,E.K.EpprechtandA.F.B.Costa,EconomicdesignofaVpX chart, Eur. J. Oper. Res. 74 (2001) K. Linderman and A. S. Choo, Robust economic control chart design, IIE Trans. 34 (2002) Z. Wu, M. Shamsuzzaman and E. S. Pan, Optimization design of control charts based on Taguchi s loss function and random process shifts, Int. J. Prod. Res. 42(2) (2004) T. Lupo, Economic design approach for a SPC inspection procedure implementing the adaptive c chart, Qual. Reliab. Engrg. Int. (2013), doi: /qre T. Lupo, Comparing the economic effectiveness of various adaptive schemes for the c chart, Qual. Reliab. Engrg. Int. (2013), doi: /qre A. Certa, G. Galante, T. Lupo and G. Passannanti, Determination of Pareto frontier in multi-objective maintenance optimization, Reliab. Engrg. Syst. Safety 96 (2011) K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms (John Wiley & Sons Ltd, New York, 2002). 32. T. Lupo, A multi-objective design approach for the c chart considering Taguchi loss function, Qual. Reliab. Engrg. Int. (2013), doi: /qre A. Certa, M. Enea, G. Galante and T. Lupo, A multi-objective approach to optimize a periodic maintenance policy, Int. J. Reliab., Qual. Safety Engrg. 19(6) (2012) , doi: /S A. Certa, M. Enea and T. Lupo, ELECTRE III to dynamically support the decision maker about the periodic replacements configurations for a multi-component system, Decis. Support Syst. 55(1) (2013) , A. S. Safaei, R. B. Kazemzadeh and S. T. A. Niaki, Multi-objective economic statistical design of X-bar control chart considering Taguchi loss function, Int. J. Adv. Manuf. Technol. 59 (2012) About the Author Lupo Toni received the Master s degree in Mechanical Engineering from the Università degli Studi di Palermo (Italy), in 1996, and the Ph.D. degree in Industrial

23 Economic-Statistical Design Approach for a VSSI X-Bar Chart Engineering from the same university, in He is currently a Researcher at the Department of Chemical, Management, Informatics and Mechanical Engineering of the University of Palermo and he is a Professor of Service Quality Management (Division of Palermo) and Quality Industrial Management (Division of Agrigento). His research interests are mainly focused on optimization problems related with Statistical Process Control, Service Quality Management, Multi-Criteria Analysis and Maintenance Optimization Models. Dr. Lupo is currently a member of the Italian Association of Mechanical Technology (AITEM) and the Italian association of Quality Culture (AICQ-Sicilia)