Ibrahim Krayem

Romain Mercier

Cédric Killian

Angeliki Kritikakou

Daniel Chillet

Data and Fault Aware Routing Algorithm for NoC Based Approximate Computing

Keywords: Network-on-Chip, Adaptive Routing Algorithm, Fault Mitigation, Approximate Computing, Bit-Shuffling

Due to transistor shrinking and core number increasing in System-on-Chip (SoC), fault tolerance has become a critical concern. Given the amount of data communications on such architectures, Network-on-Chips (NoCs) lead a crucial role in terms of performance. Even if fault correction approaches have been developed, they cannot efficiently address several permanent faults on NoC, due to their high hardware costs and correction limitations. In parallel, Approximate Computing domain considers applications that can tolerate errors, hence allowing fault mitigation instead of correction. This latter brings the opportunity of low implementation cost techniques to improve the reliability of SoC.

In this work, we propose a routing technique which selects a path between cores according to data type and permanent fault positions. Error tolerant data are able to cross faulty paths by using a bit-shuffling error mitigation technique. Critical data circumvent faulty paths or are duplicated and shuffled in case there is no other correct path available. Results show that our routing technique allows to maintain all the communication paths within the NoC for a large amount of permanent errors. To further evaluate the behavior of the proposed technique, we performed a comprehensive analysis of the technique on the packet latency and saturation injection rate with respect to the number of faults and traffic type.

I. INTRODUCTION

Since several decades, technology evolution enabled high transistor density, reaching today billions of transistors per chip. While frequencies and chip density met the power limit, performance increase has been reached by inserting more cores into a single chip, giving birth to System-on-Chips (SoCs), multi-and many-cores. However, the increase in number of cores induced more data transfers, which cannot be handled by conventional communication means. To address this gap, Network-on-Chip (NoC) appeared as a scalable solution for communications between a large number of cores [START_REF] Xu | A Methodology for Design, Modeling, and Analysis of Networks-on-chip[END_REF].

In this technology era, NoC became more sensitive to permanent faults. As the transistor size reaches 10 nm and below [START_REF] Bohr | Logic Technology Scaling to Continue Moore's Law[END_REF], the engraving thinness causes more and more hardware defects, due to manufacturing process [START_REF] Dubrova | Fault-Tolerant Design[END_REF]. During system operation, aging defects [START_REF] Kundu | Network-on-Chip: The Next Generation of System-on-Chip Integration[END_REF], like electromigration, Negative-Bias Temperature Instability (NBTI), Hot Carrier Injection (HCI), Time-Dependent Dielectric Breakdown (TDDB), and radiations [START_REF]Space Product Assurance: Techniques for Radiation Effects Mitigation in AASIC and FPGAs Handbook[END_REF] become additional sources of permanent faults. Manufacturing and aging defects are well known sources of Single Bit Upsets (SBUs), whereas radiations can lead to SBUs and Multiple Bit Upsets (MBUs) [START_REF] Mutuel | Single Event Effects Mitigation Techniques Report[END_REF]. While SBUs are composed by several Single Event Upsets (SEUs), which affect several bits in the same word, MBUs, also called burst faults, are induced by a single SEU, which affects several adjacent bits of the same data word.

In this context, fault tolerant techniques are applied to the NoC to correct/mitigate permanent faults [START_REF] Khalil | Self-Healing Hardware Systems: A Review[END_REF]. These techniques are based on i) mitigation through routing algorithms [START_REF] Fu | ZoneDefense: A Fault-Tolerant Routing for 2-D Meshes Without Virtual Channels[END_REF], ii) hardware reconfiguration through spare resources or default backup paths [START_REF] Ebrahimi | Minimal-Path Fault-Tolerant Approach Using Connection-Retaining Structure in Networks-on-Chip[END_REF], iii) correction through circuit replication [START_REF] Dubrova | Fault-Tolerant Design[END_REF], and iv) information redundancy [START_REF] Liva | Code Design for Short Blocks: A Survey[END_REF]. Although the aforementioned approaches are efficient for single permanent fault, they are less appropriate for multiple permanent faults.

Besides correction techniques, approximate computing, that considers applications able to tolerate errors, brings the possibility to mitigate faults instead of correcting them. It offers new opportunities to make circuits more resilient, with solutions that are less area hungry, and the possibility of still using the hardware with the presence of faults. For instance, a run-time approach, named Bit-Shuffling (BiSu) [START_REF] Mercier | BiSuT: A NoC-Based Bit-Shuffling Technique for Multiple Permanent Faults Mitigation[END_REF], reduces the impact of multiple permanent faults in NoC datapaths, by shuffling the bits inside a flit and transferring the faults on the Least Significant Bits (LSBs) of each data. However, the impact on the latency has not been evaluated for this technique neither the impact on the performance with respect to traffic type, e.g. critical or error tolerant data.

To deal with both critical and error-tolerant data in a costeffective way, we propose a routing algorithm for NoC-based computing architectures. The proposed algorithm, named BiRoPro, run-time decides the transmission path based on the data type and the position of faults. As long as the data are error-tolerant, they are routed on faulty paths applying bit-shuffling to reduce the fault impact. Critical data use an adaptive circumventing path without faults. In case such a path does not exist, the critical data are spread on duplicated flits and shuffled for a correct transmission. Moreover, we propose a comprehensive study on the latency with respect to the permanent fault density within the NoC and the traffic type.

The rest of the paper is organized as follows: Section II presents the related work. Section III describes the required background and BiRoPro technique. Considered architecture and experimenal setup are presented in Section IV, while the evaluation results of the proposed approach are presented in Section V. Hardware costs are given in Section VI, and finally, Section VII concludes this work.

II. RELATED WORK AND POSITIONING

Regarding correction/mitigation of permanent faults in NoC, several approaches have been proposed. Existing methods are based on reconfiguration using spare resources [START_REF] Khalil | Self-Healing Hardware Systems: A Review[END_REF] or defaultbackup paths with re-transmission [START_REF] Ebrahimi | Minimal-Path Fault-Tolerant Approach Using Connection-Retaining Structure in Networks-on-Chip[END_REF]. However, under multiple permanent faults, the number of required spare resources and re-transmission latency is prohibitively increased. Other techniques use spatial redundancy, i.e., Triple Modular Redundancy (TMR) [START_REF] Dubrova | Fault-Tolerant Design[END_REF], and information redundancy, i.e., Error-Correcting Code (ECC) [START_REF] Sánchez-Macián | Hamming SEC-DAED and Extended Hamming SEC-DED-TAED Codes Through Selective Shortening and Bit Placement[END_REF], to protect NoC architectures. When multiple permanent faults need to be addressed, their induced hardware costs are drastically increased.

Regarding NoC routing algorithms, deterministic approaches, such as XY, provide high performance in terms of latency and bandwidth. However, they cannot handle permanent faults, potentially leading to a high number of unavailable paths between Intellectual Properties (IPs). As depicted in Fig. 1a, there is no XY available path for communication from IP 3 to IP 5 and from IP 0 to IP 2 (permanent faults are illustrated by red lightnings and unavailable paths with red dotted arrow), while communication from IP 3 to IP 7 is routed through the fault-free path (depicted by the green arrow). Therefore, fault-tolerant adaptive routing algorithms [START_REF] Fernandes | Using Smart Routing for Secure and Dependable NoC-Based MPSoCs[END_REF] are proposed to circumvent faulty paths in NoC architectures, using only the healthy resources [START_REF] Fu | ZoneDefense: A Fault-Tolerant Routing for 2-D Meshes Without Virtual Channels[END_REF]. This behavior is depicted by the fault-free path (green arrow) which circumvents the permanent fault in the communication from IP 3 to IP 5 in Fig. 1b. Typical techniques include rules to avoid congestion and deadlock during packet transmissions. Although these methods handle permanent faults with lower hardware costs, the latency, throughput and network congestion increase and IPs can become unreachable, such as IP 2 in Fig. 1b.

Last, but not least, fault tolerant methods based on approximate computing recently offered techniques to protect communications on the NoCs. A bit-shuffling hardware mechanism, called (BiSu) [START_REF] Mercier | BiSuT: A NoC-Based Bit-Shuffling Technique for Multiple Permanent Faults Mitigation[END_REF], exploits the position of permanent faults and changes the bit organization inside flits to reduce the fault impact. This opens new opportunities in a faulty NoCs, with respect to the error tolerance of data, hence calling for specific routing algorithm. However, for application domains where not only error tolerant data, but also critical data are required, such approaches that reduce the fault impact, without correcting the faults, are less appropriate.

To deal with this limitation, we propose a Bit-shuffling based Routing Protocol (BiRoPro) for NoCs dedicated for applications with both critical and error-tolerant data. The main idea is to associate bit-shuffling technique with adaptive algorithm offering the possibility to route critical data on fault-free paths. Error-tolerant data are routed on a faulty path following a bit-shuffling approach to reduce the fault impact, e.g., through the faulty path illustrated with a dashed blue arrow from IP 3 to IP 5 . On the contrary, critical data will benefit from fault-free paths, highlighted in green. Note that, the proposed algorithm allows to maintain every path on the architecture, even from IP 0 to IP 2 , hence outperforming classic adaptive routing algorithms. In the case of isolated IP, error-tolerant data benefit from the mitigation technique, while critical data are sent with double flits as illustrated with a double yellow arrow [START_REF] Mercier | BiSuT: A NoC-Based Bit-Shuffling Technique for Multiple Permanent Faults Mitigation[END_REF]. Our method is more detailed in the next section.

III. PROPOSED APPROACH

In this section, we present the Bit-shuffling Routing Protocol (BiRoPro). First, Section III-A explains the target domain and assumptions of the proposed method. Then, the principle of the BiSu technique is presented in Section III-B. Finally, the proposed BiRoPro routing protocol is detailed in Section III-C.

A. Target Domain and Assumptions

BiRoPro mitigates multiple permanent faults, which can occur in NoC, and especially on the datapath part of the interconnect affecting data flits which transit on the NoC. As illustrated by the red lightnings in Fig. 1 and Fig. 2, faults can be located in i) the interconnections between routers, and ii) the buffers and the crossbar within each router. Due to nanoscale technologies and power scaling, devices and components become more susceptible to multiple permanent faults [START_REF] Mutuel | Single Event Effects Mitigation Techniques Report[END_REF]. As the buffers and the crossbar are the largest components of a router [START_REF] Deorio | A Reliable Routing Architecture and Algorithm for NoCs[END_REF], they have higher probability of accumulating faults due to radiation effects, manufacturing defects or other intrinsic failures. For the same reasons, interconnections are often impacted by permanent faults, usually expressed as stuck-at, short or bridge faults [START_REF] Dally | Principles and Practices of Interconnection Networks[END_REF]. Such faults are managed in a similar way by the proposed method.

We assume that the positions of the faults are known and provided by methods such as Built-In Self-Test (BIST) techniques [START_REF] Mohammed | Tolerating Permanent Faults in the Input Port of the Network on Chip Router[END_REF], which diagnose faults in interconnections and routers using Test Pattern Generator (TPG) and Output Response Analyzer (ORA) blocks. Other techniques available in the literature can be used to diagnose the permanent faults in a NoC [START_REF] Xiang | A Unified Test and Fault-Tolerant Multicast Solution For Network-on-Chip Designs[END_REF]. As these techniques are largely studied in the literature, they are not detailed further. We consider a source routing guaranteeing the routing algorithm to be deadlock and live-lock free.

B. Error mitigation technique:Bit-Shuffling (BiSu)

The BiSu technique extends a NoC router with extra hardware blocks, i.e. Shuffler (S) and De-shuffler (D) blocks, as shown in Fig. 2-(a). The flits of size S F bits, which transit through the NoC, are divided into N SF blocks of bits, called Subflit (SF). Each SF consists of S SF bits, which are reorganized by shuffler blocks to minimize fault impacts.

For instance, in Fig. 2, packets are organized with three 8bit flits (SF = 8), and each flit is decomposed in 2-bit subflits (S SF = 2). If two faults appear on bits 6 and 7 of the router, each flit will be corrupted and the impact on data (e.g. on an 8 bits uchar) will be very high. Before crossing the faulty router, the Shuffler block (S) re-organizes the flits by shuffling subflits SF 0 and SF 3 to reduce the impact of the faults, as The BiSu technique is applied to mitigate errors on the interconnection bus and inside the router. One extra D block is added in the routing controller (CTRL) to read the routing information of the shuffled header and forward the current packet towards the expected output. Header flits are critical, i.e., cannot tolerate any fault. If sufficient number of bits are unused in the header flit (localized on the LSBs positions), then BiSu can exploit these unused bits to protect the header bits. Otherwise, a spreading technique is used where the header is divided in two parts. Each part is placed into the Most Significant Bits (MSBs) on a flit. In this way, unused bits are created and thus BiSu is able to protect the header bits. The flit-spreading technique could also be used for critical data, e.g., instructions, where each flit is split on two flits, at the cost of more buffer occupancy. C. Bit-shuffling Routing Protocol (BiRoPro) Fig. 3 presents the overview of BiRoPro. First, the proposed approach determines if the XY path to reach the destination is impacted by any fault. If the path is fault-free, the packet is sent through the NoC using XY routing algorithm. This behavior is illustrated through the communication from IP 3 to IP 7 in Fig. 1-(c). If the path is faulty, BiRoPro determines the nature of the data contained in the packet. If the data is not critical (error-tolerant), then the packet is sent through the NoC following XY path applying the BiSu approach, where the header is spread on two flits to mitigate faults. This behavior is depicted in Fig 1-(c) for the communication from the IP 3 to IP 5 depicted by the dashed blue arrow.

S SF SF 0 SF 1 SF 2 SF 3 SF 0 SF 1 SF 2 SF 3 SF 0 SF 1 SF 2 SF 3 SF 0 SF 1 SF 2 SF 3 SF 0 SF 1 SF 2 SF 3 SF 0 SF 1 SF 2 SF 3 SF 3 SF 1 SF 2 SF 0 SF 3 SF 1 SF 2 SF 0 SF 3 SF 1 SF 2 SF 0 Shuffling De-Shuffling X X X X X X X X X X X X
Regarding critical packets, i.e., packets that cannot tolerate faults, an adaptive routing algorithm is used to forward them until the destination IP by circumventing faulty elements in the NoC. This behavior is depicted in Fig 1-(c) through the communication from the IP 3 to IP 5 represented by the green arrow. In the case where the destination IP is isolated from the source IP, the packets are sent through the NoC following the XY path using BiSu approach to protect data. This is illustrated in Fig. 1-(c) through the communication from IP 0 to IP 2 represented by the doubled yellow arrow. As these flits contain critical data, they are spread on two flits in order to enable data protection.

IV. EXPERIMENTAL SETUP A. Architecture and fault model

For our experiments, we consider an 8 × 8 NoC with a mesh topology. The router service time is set to 2 clock cycles, and the router is considered without virtual channel. Packets include 64 bytes of data organized in 16 flits plus one header flit of 32 bits, and are injected according to different traffic models, such as random, perfect shuffle and transpose traffic patterns. The proposed BiRoPro approach is implemented and is compared to i) the XY routing algorithm and ii) an adaptive routing algorithm searching for the shortest fault-free path between source and destination IPs [START_REF] Fernandes | Using Smart Routing for Secure and Dependable NoC-Based MPSoCs[END_REF].

Faults are randomly injected in the NoC datapath, i.e., routers and interconnections. To avoid fault masking due to data values, we consider a permanent fault by simulating a bit-flip on the data. In our experiments, we express the number of injected faults by using the fault density metric, which corresponds to the average number of faults per router. For example, 10 injected faults within an 8 × 8 NoC correspond to a fault density of 0.16. Note that it is possible to have several faults on a same router.

B. Simulator

In this paper, given the large exploration space regarding the fault positions, we use an analytical model based on queuing theory, offering effective NoC-performance analyzing [START_REF] Ogras | An Analytical Approach for Network-on-Chip Performance Analysis[END_REF]. This model provides different performance metrics, i.e., average buffer utilization and average packet latency. The latter enables to compute the saturation packet injection rate used to evaluate our contribution. Each experiment is performed 10, 000 times and box plots are used to represent the results through the minimum, median and maximal values and the first and third quartiles.

V. RESULTS

A. Correct Packet Transmission and Latency

We study the packet transmission rate and latency of the proposed BiRoPro method. We first compute the percentage of packets that are transmitted through the NoC without error, considering a random traffic pattern. Results are compared with XY and adaptive routing algorithms. Then, we compute the latency overhead by taking as reference the best case, i.e., a XY routing algorithm with fault free NoC, i.e. the shortest path. Results are respectively presented in Fig. 4a and Fig. 4b according to the NoC fault density.

In Fig. 4a, we can observe that the XY and adaptive routing algorithm drastically reduce the transmission rate of correct packets, when the fault number increases in the NoC. On the contrary, the proposed BiRoPro technique maintains the transmission rate constant even in presence of multiple faults. For example, considering a fault density equal to 0.09 fault per router, i.e., 6 faults in the 8 × 8 NoC, the percentage of correct packet transmission is, on average, 54.98% and 82.64%, for XY and the adaptive routing algorithm, respectively. BiRoPro maintains the correct packet transmission at 100%. Moreover, we can deduce that the percentage of correct packet transmissions decreases, when the fault density increases.

In the following we only highlight results on BiRoPro which is the only capable of routing any pair of IPs in presence of faults. Indeed, as shown in Fig. 4a, in presence of faults, XY and adaptive routing algorithms are not able to maintain all the paths and some IPs are isolated. This will lead to less injected packets as our method. Moreover, it will also reduce the amount of available IPs, hence decreasing the system performance. This latter is out of scope or the proposed paper.

As the proposed BiRoPro is the only approach to achieve 100% under multiple faults, we further analyse its behavior regarding latency overhead in Fig. 4b. The critical packets circumvent the faulty routers or are doubled in case of an isolated IP, which naturally impacts the latency compared to a fault-free execution. Note that, section V-C further analyses the latency overhead considering error-tolerant packets. From the obtained results, we observe that with higher fault density, the gap between the best and worst case latency increases. For instance, with a fault density of 0.16 (i.e., 10 faults), the latency overhead varies from 17% to 27% with an average of 23%. Such variation is due to the fault positions, potentially leading to long paths that circumvent the faulty routers.

B. Saturation Packet Injection Rate

In this part, we study the congestion by the proposed method considering i) Random, ii) Transpose and iii) perfect Shuffle traffic patterns, considering only critical packets. In Random pattern, the source and the destination are randomly generated. In Transpose pattern, some pairs of cores have a higher communication probability than others, whereas the rest of the traffic is uniform random. Given a vector of N elements, the perfect Shuffle pattern is a communication pattern that divides the elements of this vector into two equal pieces, and then, combines these pieces by shuffling them [START_REF] Stone | Parallel processing with the perfect shuffle[END_REF]. The values of the Saturated Packet-Injection Rate (SP IR) are computed according to the analytical model presented in Section IV-B with respect to the fault density in the NoC. The SP IR is considered reached for a P IR providing an average latency higher than 10× the latency of a zero-load NoC, with a very low packet injection rate 1.562 × 10 -5 packet/cycle/router.

The obtained results are shown in the Fig. 5. We can first observe that the Random traffic pattern provides more congestion in the NoC, i.e., the packet injection rate that provokes the saturation of the NoC is lower. Transpose and Shuffle patterns are almost similar regarding the SP IR. For instance, with a fault-free NoC, the SP IR are 1.588×10 -4 , 5.344×10 -3 , and 3.778 × 10 -3 packet/cycle/router, respectively for Random, Transpose and perfect Shuffle traffic patterns. In terms of impact on the SP IR, the perfect Shuffle traffic pattern is the less sensitive to fault, with only a SP IR reduction equal to 41% for a fault density of 0.16 compared to its fault-free value. Regarding the Transpose traffic pattern, the reduction of the S P IR is 49%, due to proportion of random traffic Error-tolerant and critic packet proportion included which bring more contention. Finally, the random traffic pattern is the more sensitive with a reduction of 62% on the S P IR , which is reasonable regarding the amount of injected faults, i.e., 10 for an 8 × 8 NoC.

C. Packet-Type Impact

In this part, we explore the impact of the ration between error-tolerant and critical packets, which transit the NoC, on the latency overhead and the SP IR. Due to page limitations, for each packet ratio we represent one box plot considering a fault density from 0 to 0.16.

Fig. 6a presents the latency overhead of BiRoPro according to ET x /C y , with x and y the percentage of error-tolerant and critical packets, respectively. Results are normalized taking as basis a fault-free NoC with an XY routing algorithm. Overall, we observe wider box plots compared to the ones presented in Fig. 4b. Indeed, each box represents a fault density ranging from 0 to 0.16. For instance, ET 0 /C 100 in Fig. 6a integrates all the results from Fig. 4b. In Fig. 6a, we can observe that the latency overhead of BiRoPro decreases when the proportion of error-tolerant packets increases. This is due to the fact that error-tolerant packets directly use faulty paths instead of circumventing faults, or by duplicating the flits to reach isolated IPs. Finally, with only error-tolerant packets, the proposed algorithm has only 2% of latency penalty compared to a fault-free NoC. Fig. 6b presents the SP IR of BiRoPro for the Random, Transpose and perfect Shuffle traffic patterns according to the ratio of packet type. We observe that the SPIR increases (higher is better) with the increase of error-tolerant packets. Moreover, it almost reaches the fault-free SP IR for a traffic with only error-tolerant packets. Indeed, the SP IR for ET 100 /C 0 are 1.539 × 10 -4 , 3.512 × 10 -3 and 5.821 × 10 -3 packet/cycle/router, respectively for Random, Transpose, and perfect Shuffle traffic patterns. Error-tolerant packets can be directly go through faulty routers only with duplicated spread header, then maintaining packet latency low.

VI. HARDWARE COST STUDY

In this section, we evaluate the hardware cost induced by the BiRoPro approach in terms of area and power overheads. The extra hardware blocks used in the BiRoPro approach are synthesized on 28 nm FDSOI technology through a High Level Synthesis (HLS) tools of Mentor Graphic by targeting a 1 GHz clock frequency.

Table I presents the hardware costs of the proposed technique which impacts both router and NI architectures. We show the result for two sizes of flits, 32-and 64-bits, respectively with S SF equal to 4 and 8 bits. This ratio of flit and S SF bring a good trade-off between error mitigation and area overhead [START_REF] Mercier | BiSuT: A NoC-Based Bit-Shuffling Technique for Multiple Permanent Faults Mitigation[END_REF]. We can see that, for 32-bit flits, the area overhead if only 7% for the routers. Regarding the NI the area overhead is 147% as it includes packets reorganization with the possibility to spread each flit in two. However, it has to be noticed that NIs are only small part of the while NoC. Indeed, the total area overhead for the NoC is 24.8% and 27.6%, for 32-and 64-bit flits, respectively. Regarding the power overhead, at the NoC scale, it requires 21.1% and 19.9% more power, or 32-and 64-bits flits, respectively.

VII. CONCLUSION

In this paper, we present BiRoPro, an adaptive routing algorithm with bit-shuffling for NoC based Approximate Computing. Our proposal is able to manage communications in a faulty NoC with respect to permanent fault positions and data type by distinguishing fault-tolerant and critical packets. The results demonstrate that the proposed approach allows to maintain all the communication paths within a faulty NoC where classic routing algorithm such XY or adaptive algorithm cannot. Moreover, we performed a comprehensive analysis to illustrate the robustness of our algorithm with respect to the number of faults injected by exploring the packet latency and saturation packet injection rate (S P IR ). We show that our proposed algorithm performs better while the number of faulttolerant packets increases, but also performs well with critical data. Finally, in terms of overheads, the proposed solution only requires 27.6% and 19.9% area and power overheads, respectively, for a 64-bit NoC.

Fig. 1 :

 1 Fig. 1: Illustration of the behavior of state-of-the-art routing algorithms and the proposed BiRoPro approach on a faulty NoC highlighting 3 communications: IP 3 to IP 7 , IP 3 to IP 5 and IP 0 to IP 2 . (a) XY routing algorithm. (b) Adaptive routing algorithm. (c) Proposed BiRoPro approach.

  NoC extended with the BiSu method.

Fig. 2 :

 2 Fig. 2: NoC extended with the BiSu method.

Fig. 3 :

 3 Fig. 3: BiRoPro diagram.

  Latency overhead compared to a fault-free XY routing algorithm.

Fig. 4 :

 4 Fig.4: NoC performance evaluations using BiRoPro.

Fig. 5 :

 5 Fig. 5: Saturation Packet Injection Rate of BiRoPro technique for different traffic patterns.

Fig. 6 :

 6 Fig. 6: Impact of the approximate-packet proportion ET x /C y with x and y respectively the percentage of error-tolerant and critical packets considering 8 × 8 NoC.

TABLE I :

 I Hardware overhead for BiRoPro at the Network Interface (NI) and NoC level considering 8 subflits per flit.

ACKNOWLEDGMENT

This work is supported by the ANR SHNoC project, grant ANR-18-CE25-0006 of the French Agence Nationale de la Recherche.


Salah Haridy
email: sharidy@sharjah.ac.ae

Ling Nger
email: ngerlingc@gmail.com

Chong

Michael B C Khoo

Mohammad Shamsuzzaman
email: mshamsuzzaman@sharjah.ac.ae

Philippe Castagliola
email: philippe.castagliola@univ-nantes.fr

Synthetic control chart with curtailment for monitoring shifts in fraction nonconforming

Keywords: control chart, synthetic chart, curtailment, fraction nonconforming, average number of nonconforming units

The integration of the curtailment method with control charts considerably improves the detection speed by signaling an out-of-control condition prior to the inspection of the whole sample. To date, few research works have focused on the incorporation of the curtailment method to improve the performance of control charts. Thus, this paper incorporates the curtailment approach with the synthetic chart to propose a synthetic control chart with curtailment (Curt_Syn) for detecting upward shifts in the fraction nonconforming, p. We compared the newly developed Curt_Syn chart with the synthetic, exponentially weighted moving average (EWMA), cumulative sum (CUSUM), EWMA with curtailment (Curt_EWMA), and CUSUM with curtailment (Curt_CUSUM) charts. From an overall perspective, the results reveal that the Curt_Syn chart surpasses the synthetic chart by 38% under various conditions. For all p shifts, the Curt_Syn chart outperforms the CUSUM and EWMA charts. When the p shift is large, the Curt_Syn chart is superior to the Curt_CUSUM and Curt_EWMA charts. To demonstrate the implementation of the Curt_Syn chart, an illustrative example is provided.

Introduction

Throughout the years, Statistical Process Control (SPC) which is a technique that aids in monitoring the behavior of a process has been widely adopted to attain a competitive advantage and an economic benefit due to the increasing competition in the manufacturing and service industries. Most companies emphasize on continuous improvement of quality, meeting the requirements of customers and reduction of cost.

Hence, a control chart which is the most powerful SPC tool is a common application to monitor the stability of a process. A process can be monitored by two types of control charts, i.e. variable and attribute charts. The type of the control chart to be used is mainly decided based on the nature of the quality characteristic to be monitored. When the quality characteristic is a quantitative measure, variable charts are used. On the other hand, attribute charts are widely used for monitoring quality characteristics that cannot be represented quantitatively or not easily expressed on a numerical scale. Each item can be categorized as either conforming or nonconforming, depending on whether the specification of the product is met. The fraction nonconforming p is defined as the ratio of the number of nonconforming items to the total number of items in a population (Montgomery, 2019). The np chart is used to monitor the number d of nonconforming units in a sample of size n and it is used as an alternative to the p chart when n is constant for each sample.

The conforming run length (CRL) chart was developed by [START_REF] Bourke | Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection[END_REF], where the CRL value changes with shifts in p. The CRL refers to the number of inspected samples between two consecutive nonconforming samples, including all conforming samples in between as well as the ending nonconforming sample [START_REF] Wu | A generalized conforming run length control chart for monitoring the mean of a variable[END_REF]. The concept of the CRL control chart is quite similar to the system proposed by [START_REF] Chen | A surveillance system for congenital malformations[END_REF] for monitoring congenital malformations which is based on the number of consecutive births occurring between the birth of an infant with the specific malformation being monitored and the birth of the next infant with that malformation. Such a group of consecutive births is defined as a set. Both the CRL and the size of the set proposed by [START_REF] Chen | A surveillance system for congenital malformations[END_REF] are random numbers and follow a geometric distribution.

The synthetic control chart was proposed by Wu et al. (2001) who combined the CRL and np charts to study upward shifts in fraction nonconforming. The synthetic chart uses information on the number of conforming samples between two consecutive nonconforming samples, in contrast to the np chart that only uses information on the number of nonconforming units in the last sample [START_REF] Chong | Synthetic double sampling np control chart for attributes[END_REF]. Thus, it was found that the synthetic chart significantly outperforms its standard counterparts (CRL and np charts) in detecting p shifts. Researchers have incorporated the synthetic feature with various attribute and variable control charts. Wu and Yeo (2001) computed the average time to signal (ATS) of the synthetic control chart for attribute data. [START_REF] Bourke | Performance comparisons for the synthetic control chart for detecting increases in fraction nonconforming[END_REF] monitored the increases in fraction nonconforming with the synthetic control chart. A synthetic-np chart was developed by [START_REF] Haridy | A combined synthetic and np scheme for detecting increases in fraction nonconforming[END_REF] and they discovered that the synthetic-np chart outperforms the synthetic and np charts by 31% and 73%, respectively. [START_REF] Celano | A synthetic control chart for monitoring the ratio of two normal variables[END_REF] monitored the ratio of two normal variables using a synthetic control chart. [START_REF] Haq | New synthetic control charts for monitoring process mean and process dispersion[END_REF] proposed a synthetic control chart that detects shifts in mean and dispersion. [START_REF] Khoo | A synthetic double sampling control chart for the process mean[END_REF] They observed that the proposed charts have attractive zero-state and steady-state properties and outperform the existing traditional synthetic double sampling chart and all other competing charts in many situations.

The performance of a control chart can be improved through the incorporation of various approaches. An effective method to enhance a control chart's performance is by integrating the curtailment feature which has been extensively adopted in acceptance sampling plans. When the number of nonconforming units surpasses an acceptance number, practitioners stop inspecting the sample and reject the corresponding lot (Montgomery, 2019). According to Montgomery (2019), curtailment substantially reduces the average sample number used in an acceptance sampling plan. In fact, the advantage of incorporating the curtailment with control charts has been shown by various research publications. [START_REF] Wu | Optimal np control chart with curtailment[END_REF] integrated the curtailment approach with the np chart and found that its performance improved significantly by decreasing the out-ofcontrol average time to signal (ATS) by almost half in comparison with the standard np chart. In addition, [START_REF] Haridy | Binomial CUSUM chart with curtailment[END_REF] developed the CUSUM chart with curtailment which surpasses its counterpart without curtailment by 30%. While the rate of false alarm is maintained at a specified level, the incorporation of the curtailment feature into the EWMA chart by [START_REF] Haridy | EWMA chart with curtailment for monitoring fraction nonconforming[END_REF] has also shown a considerable improvement in the overall detection speed. The EWMA chart with curtailment was compared with the standard EWMA chart under various settings.

To the best of the authors' knowledge, a synthetic chart that incorporates the concept of curtailment is not present in the literature. Given the significant improvement of control charts through the incorporation of the curtailment approach, this paper fills this gap by proposing a synthetic control chart with curtailment to monitor p shifts. For brevity, the synthetic control chart with curtailment will be referred to as the Curt_Syn chart. This research focuses only on upward p shifts as it is more important to detect a deterioration in quality rather than a process improvement (downward p shifts). In this paper, it is assumed that d follows a binomial distribution and the in-control value of p which is denoted as p0 is known. The Curt_Syn chart is compared with the conventional synthetic chart, in terms of the performance measures ATS and average number of nonconforming units (ANC). The organization of this paper hereafter is as follows:

Section 2 provides an overview of the synthetic control chart while the implementation of the Curt_Syn chart is outlined in Section 3. In Section 4, a discussion of the statistical measures of performance is provided while Section 5 explains the optimal design of the Curt_Syn chart. Section 6 provides the numerical analysis and an illustrative example that shows the implementation of the Curt_Syn chart is given in Section 7. Lastly, Section 8 completes the paper with conclusions and suggestions for further research.

Synthetic control chart

Wu et al. ( 2001) combined the operations of the CRL and np charts to develop the synthetic control chart. The CRL refers to the number of samples between two successive nonconforming samples, with the inclusion of the nonconforming sample at the end. The implementation of the synthetic chart is as follows:

(1) Determine the sample size n, warning limit w (of the np sub-chart) and lower control limit L (of the CRL sub-chart).

(2) Inspect a sample of n units and determine the number of nonconforming units d present in the sample.

(3) If dw  , the sample is conforming and the control flow returns to Step 2 for taking the next sample. Otherwise, if dw  the sample is nonconforming and the control flow proceeds to the next step.

(4) Determine the value of CRL. If CRL L  , the process is in-control and the control flow returns to Step 2. Otherwise, if CRL < L , the process is out-of-control and the control flow proceeds to the next step.

(5) Stop the process and take corrective actions to identify and remove the assignable cause(s)

Curt_Syn chart

Like the synthetic chart, the implementation of the Curt_Syn chart involves the parameters n, w and L. For the synthetic chart, the status of a process can only be decided after inspecting all the units in a sample of size n. On the other hand, the Curt_Syn chart enables the detection of an out-of-control condition prior to the inspection of all n units in the sample. The implementation of the Curt_Syn chart can be described as follows:

(1) Identify the sample size n, warning limit w (of the np sub-chart) and lower control limit L (of the CRL sub-chart).

(2) Inspect a sample of n units one by one and increase the cumulative number c of detected nonconforming units by one whenever a nonconforming unit is found.

(3) If cw  up to the end of the sample, the sample is conforming and return to Step 2 for taking the next sample. Otherwise, if cw  at any moment, the sample is nonconforming and proceed to the next step.

(4) Determine the value of CRL. If CRL L  , the process is in-control and return to

Step 2. Otherwise, if CRL < L , the process is out-of-control and proceed to the next step.

(5) Stop the process and take the necessary actions to eliminate the assignable cause(s). Subsequently, the control flow returns to Step 2.

It is worth mentioning that when d is larger than w (in the case of the synthetic chart) or c exceeds w (in the case of the Curt_Syn chart), the sample is considered as nonconforming but an out-of-control signal is not issued immediately. According to Wu et al. (2001), the sample is marked as a nonconforming one and an out-of-control signal is produced only if the CRL is less than L. In both synthetic and Curt_Syn charts, the CRL is defined as the number of samples between two consecutive nonconforming samples (including the ending nonconforming sample) as mentioned earlier.

Statistical measures of performance

In this section, the performance measures ATS and ANC are discussed. ATS is a widely adopted performance indicator to measure the speed of signaling an out-of-control condition. When the process is in-control, a larger ATS indicates a lower Type-I error or false alarm rate in comparison to other charts. On the other hand, a smaller out-of-control ATS indicates that the chart signals faster and is more sensitive to process shifts in comparison to other charts. In other words, a chart with a lower out-of-control ATS has a superior shift detection ability. Generally, ATS can be calculated as follows [START_REF] Haridy | Binomial CUSUM chart with curtailment[END_REF]:

ATS ARL h = (1)
where h refers to the sampling interval while the average run length (ARL) is the expected number of samples until an out-of-control condition is signaled. According to [START_REF] Montgomery | Introduction to Statistical Quality Control[END_REF], an attribute control chart based on 100% inspection of all process output is common due to the simplicity of the inspection of attribute data. Hence, 100% inspection is used in the discussions of this paper (similar to [START_REF] Wu | Optimal np control chart with curtailment[END_REF][START_REF] Haridy | Binomial CUSUM chart with curtailment[END_REF][START_REF] Haq | New synthetic CUSUM and synthetic EWMA control charts for monitoring the process mean using auxiliary information[END_REF].

The value of h is equivalent to the product of n and the time (t) required for inspecting a unit with 100% inspection. Hence, ATS is given as [START_REF] Haridy | Binomial CUSUM chart with curtailment[END_REF]

) ATS ARL ARL h n t =  =   (2)
If we assume t = 1 time unit, we simply have [START_REF] Haridy | EWMA chart with curtailment for monitoring fraction nonconforming[END_REF] ATS ARL n =

(3) There are two types of ATS, i.e. zero-state and steady-state ATS. The zero-state ATS is the expected time from the beginning of the process to the time when the chart signals an out-of-control condition. On the other hand, the steady-state ATS is defined as the expected time from the occurrence of an assignable cause to the time an out-of-control signal is issued by the chart. In this paper, the out-of-control ATS is computed using the steady-state mode. This is because the steady-state mode leads to more realistic results as, during the inspection of the sample, the p shift can occur randomly at any time [START_REF] Wu | Optimal np control chart with curtailment[END_REF]. During the occurrence of a process shift, the out-of-control value of p is defined as follows [START_REF] Haridy | EWMA chart with curtailment for monitoring fraction nonconforming[END_REF]:

0 pp  = (4)
where δ measures the size of an upward shift in p, in terms of p0. Note that max 1   such that max  is the maximum shift. When δ = 1 (p = p0), the process is in the in-control state. Meanwhile, when max 1   , the process is out-of-control with p at its maximum max max 0 pp  = .

To obtain an overall measure of the performance of the charts, the ANC which is the expected number of nonconforming units produced in various out-of-control cases, for a range of p shifts, is used. When comparing several charts, the one with a smaller ANC is more efficient over various values of the shift δ. In fact, the ANC is a weighted average of ATS, where the weight is δ. The ANC is given by [START_REF] Haridy | EWMA chart with curtailment for monitoring fraction nonconforming[END_REF]

) max 0 1 ANC N ATS( ) ( ) d p f      =     ( 5
)
where N is the number of units produced per time unit, () f  represents the probability density function of δ, while ATS( )  refers to the ATS value for the shift δ. As N in Equation ( 5) does not affect the performance of the charts, we may assume N=1 and;

hence, Equation (5) simplifies to

max 0 1 ANC ATS( ) ( ) d p f      =     (6)
In general, there is no closed-form for the ANC and the value of the integral can only be obtained using a numerical method like, for instance, the Legendre-Gauss Quadrature.

Additionally, note that Equation ( 6) (without N) can be used for the design and comparison of the charts. However, to compute the actual ANC value, Equation ( 5) should be used.

Typically, it is assumed that the shift δ for fraction nonconforming p follows a probability distribution. In this paper, we assume that δ follows a Rayleigh distribution.

For more information on the Rayleigh distribution, readers can refer to [START_REF] Haridy | Binomial CUSUM chart with curtailment[END_REF][START_REF] Haridy | EWMA chart with curtailment for monitoring fraction nonconforming[END_REF]. If we assume a Rayleigh distribution for the shift δ, the probability density

function of δ is 2 22 ( 1) ( 1) ( ) exp 2( 1) 4( 1) f           -- =- -- (7)
and its cumulative distribution function is

2 2 ( 1) F( ) 1 exp 4( 1)      - = - -  -  (8)
where   is the mean of δ. The historical data corresponding to the out-of-control cases can be used to estimate 𝜇 𝛿 . Let 𝑝̂𝑖 be the value of p obtained during the follow-up investigation after the control chart signals an out-of-control condition, then an estimate of the sample shift ˆi  is 𝑝̂𝑖 𝑝 0 ⁄ . Suppose that there are k records of ˆi  , i =1,...,k, then an estimation ˆ  of   is [START_REF] Haridy | Binomial CUSUM chart with curtailment[END_REF])

1 k i i k    = =  (9)
Note that the value of δmax in Equation ( 6) can be obtained from Equation ( 8) such that the probability of δ > δmax is negligible (lower than 0.001). Consequently, F(δmax) in Equation ( 8) will be equal to 0.999 and δmax can be computed from   as follows:

2 max 2 2 max 2 2 max 2
( 1) 0.999 1 exp 4( 1)

( 1) exp 0.001 4( 1)

( 1) ln 0.001 4( 1)

          - = - -  -   - -=  -  - -= -

Optimal design of the Curt_Syn chart

This section outlines the optimal design of the Curt_Syn chart with the objective of minimizing the ANC. In this paper, the design algorithm minimizes the ANC instead of the ATS to obtain a better overall detection effectiveness over a range of process shifts.

It has been shown by [START_REF] Haridy | Binomial CUSUM chart with curtailment[END_REF] that a control chart that uses the ANC as the objective function has a better overall performance compared to its counterpart that minimizes the ATS. Prior to the optimal design of the Curt_Syn chart, three specifications should be determined:

(1) τ: minimum allowable value of the in-control ATS (ATS0),

(2) p0: in-control fraction nonconforming,

(3) 𝜇 𝛿 : mean of the shift δ.

With reference to the false alarm rate, the quality engineer can determine the value of τ. When managing false alarms is costly, a larger value of τ can be used to lower the frequency of false alarms. When the process is in-control, 𝑝 0 can be estimated from the Phase I data. Additionally, Equation ( 9) can be used to estimate the value of 𝜇 𝛿 from the historical data of the out-of-control cases.

The optimal design of the Curt_Syn chart is based on the following model: A two-level search is used to implement the optimal design of the Curt_Syn chart as follows:

(1) Specify the values of τ, p0 and 𝜇 𝛿 .

(2) Initialize a very large number (i.e. 10 7 ) as the minimum value of ANC (ANCmin).

(3) For the first level, the optimal value of n is determined by trying all possible values with an increment of 1 from the initial value n = 1, until the ANC cannot be further reduced.

(4) For the second level, using the value of n obtained from the first level, search for the optimal value of w in the range 0 ≤ w ≤ n. For each set of (n, w) values,

• Determine the value of L such that it satisfies the constraint 0 ATS   .

• Given that all the values of the three charting parameters n, L and w have been determined, compute the ANC using Equation ( 6).

• If the computed ANC has a smaller value than ANCmin, then substitute the latter with the former and store the current n, L and w values as a temporary optimal solution.

At the end of the two-level search, the optimal Curt_Syn chart that produces the minimum ANC value and meets the constraint 0 ATS   is obtained. At the same time, the finalized optimal parameters n, L and w are determined.

The grid search used in this optimization procedure can be considered as a global one because of the discrete nature of attributes which allows all possible values of the two independent integer variables (n and w) to be explored. A code has been written in C language to compute the optimal parameters and to calculate the performance measures, ATS and ANC, of the Curt_Syn chart by simulation using 10,000 replications. This code can be obtained from the authors upon request.

Numerical analysis

In this section, the sensitivity of the synthetic and Curt_Syn charts in detecting upward p shifts is compared. The design of both charts follows the model in Equations ( 11) and ( 12) where the objective function is to minimize the ANC while meeting the constraint 0 ATS   .

Comparison under one case

This section compares the synthetic and Curt_Syn charts under one case in which the design specifications are as shown below:

𝑝 0 = 0.01, τ = 50/𝑝 0 , 𝜇 𝛿 = 7
where τ is expressed in terms of 𝑝 0 (Wu et al., 2001[START_REF] Wu | Optimal np control chart with curtailment[END_REF]. The optimal parameters of both charts are:

Synthetic chart: n = 67, L = 20, w = 2 Curt_Syn chart: n = 71, L = 16, w = 2
Table 1 shows the ATS values of both charts. The in-control ATS values are obtained when δ = 1 and the out-of-control ones are obtained when δ = {2,3,...,18}. Note that the value of max  can be determined using Equation ( 10). For this case, max  is approximately equal to 18. With reference to Table 1, it can be seen that ( 1) The synthetic and Curt_Syn charts have 0 ATS   when the process is in-control.

Hence, a common ground for comparison is provided as both charts have similar false alarm rates.

(2) For all shifts, the Curt_Syn chart has lower ATS values compared to the synthetic chart. Thus, the Curt_Syn chart is superior in detecting increasing p shifts as the out-of-control condition is detected earlier. From an overall viewpoint, adapting curtailment improves the effectiveness of the synthetic chart.

(3) For both charts, the ATS value decreases as δ increases. This indicates that the charts become more sensitive in detecting p shifts as δ becomes larger which is justified by the need to detect large shifts that lead to a significant loss of quality swiftly.

(4) It can also be seen from Case 0 in Table 2 that ANCSynthetic/ANCCurt_Syn = 1.43.

This reveals that the Curt_Syn chart outperforms the synthetic chart by 43% from an overall standpoint over the entire range of shifts (1 < δ ≤ 18). This is justifiable because when δ is large, the curtailment mechanism will come into play very effectively and gives an out-of-control signal much earlier before all of the n units in the sample are inspected.

[Please insert Table 1 here]

[Please insert Figure 2 here]

Comparison under more cases

In this section, the synthetic and Curt_Syn charts are compared under different conditions.

As shown in Table 2 (in cases 1 to 35), there are three input factors (p0, τ and 𝜇 𝛿 ). p0 has 5 levels, while τ and 𝜇 𝛿 are varied at 3 levels. The levels of p0 and τ are decided with to those commonly used by many authors [START_REF] Reynolds | The SPRT chart for monitoring a proportion[END_REF], Wu et al. 2001[START_REF] Bourke | Performance comparisons for the synthetic control chart for detecting increases in fraction nonconforming[END_REF][START_REF] Haridy | An optimization design of the combined np-CUSUM scheme for attributes[END_REF]. Case 0 corresponds to the specific case in ) values are enumerated in Table 2 for the 36 cases. It can be seen that, for all the 36 cases, the Curt_Syn chart outperforms the synthetic chart. This is because the ANC of the synthetic chart is larger than the ANC of the Curt_Syn chart for all cases. Additionally, the normalized ANC of the synthetic chart for all cases is larger than one, indicating that the Curt_Syn chart surpasses the synthetic chart. This is especially so in case 5, where the normalized ANC of the synthetic chart is at its maximum at 1.9476. In other words, the outperformance of the Curt_Syn chart compared to the synthetic chart is 95%.

In order to obtain a complete view of the performance of the charts, a grand average of the normalized ANC for the synthetic chart in all the 36 cases shown in , where the Curt_Syn chart outperforms the synthetic chart by 38%, on average in detecting p shifts. From a comprehensive point of view, the Curt_Syn chart is considerably superior to the synthetic chart. This clearly reflects the substantial contribution of the curtailment method in improving the performance of the synthetic chart.

[Please insert Table 2 here]

Comparison with CUSUM and EWMA charts with and without curtailment

Lastly, the performance of the Curt_Syn chart is compared with the CUSUM and EWMA charts, as well as their respective counterparts with curtailment, denoted as Curt_CUSUM chart and Curt_EWMA chart, respectively, for detecting increasing p shifts. We selected four different cases from [START_REF] Haridy | Binomial CUSUM chart with curtailment[END_REF][START_REF] Haridy | EWMA chart with curtailment for monitoring fraction nonconforming[END_REF] and studied each case in terms of the ANC. Table 3 shows the design specifications, optimal parameters and results of each case. Note that H is the upper control limit of the CUSUM and EWMA charts, k is the reference parameter of the CUSUM chart and λ is the weighting parameter of the EWMA chart.

[Please insert Table 3 here]

Based on Table 3, the Curt_Syn chart outperforms the CUSUM and EWMA charts for all cases as the CUSUM and EWMA charts' normalized ANC values are more one. In fact, the Curt_Syn chart significantly outperforms the CUSUM and EWMA charts when p0 is large (p0 = 0.03). The Curt_Syn chart surpasses the synthetic chart for all cases. Meanwhile, the Curt_Syn chart also outperforms the Curt_CUSUM and Curt_EWMA charts when p0 is large.

It can also be seen that, generally, the Curt_CUSUM and Curt_EWMA charts are slightly superior to the Curt_Syn chart when p0 is small (p0 = 0.005). However, as the synthetic chart without curtailment has larger ANC values compared to the Curt_CUSUM and Curt_EWMA charts for all cases (except for case 4 where the Curt_CUSUM chart has a larger ANC value), the incorporation of the curtailment method has improved the performance of the synthetic chart significantly. To illustrate, in case 2, the synthetic chart has ANC = 11.564 which decreases to ANC = 7.496 for the Curt_Syn chart, which is closer to the performance of the Curt_CUSUM (ANC = 7.865) and Curt_EWMA (ANC = 7.459) charts. It can also be observed in case 3 that the addition of the curtailment approach to the synthetic chart reduced its ANC from 5.147 to 3.286; hence outperforming the Curt_CUSUM (ANC = 4.441) and Curt_EWMA (ANC = 4.332) charts.

Illustrative example

The implementation of the Curt_Syn chart in a company that manufactures golf balls is [Please insert Table 5 here]

[Please insert Figure 4 here]

Conclusions

In this paper, we propose a new synthetic chart with curtailment which is abridged as the Curt_Syn chart. The performance of the Curt_Syn chart is comprehensively studied in this paper for various conditions, in terms of ATS and ANC to provide readers with an overall view of the performance of the chart. Additionally, the implementation, optimal design and performance evaluation of the Curt_Syn chart are explained to assist practitioners in using the chart.

The curtailment method can be easily applied and has significantly improved the performance of the synthetic chart by 38%, in terms of ANC. The Curt_Syn chart also outperforms the CUSUM and EWMA charts, in terms of ANC. In addition, the Curt_Syn chart surpasses the Curt_CUSUM and Curt_EWMA charts for large p shifts. When p shifts occur, an out-of-control condition will be signaled by the Curt_Syn chart before the inspection of all the n units in a sample; consequently, an improvement in the speed of detecting p shifts is attained.

As

  developed a synthetic double sampling control chart by integrating the synthetic and double sampling charts to monitor the process mean while the synthetic double sampling chart was proposed by[START_REF] Chong | Synthetic double sampling np control chart for attributes[END_REF] for attribute data.Additionally,[START_REF] Haq | New synthetic CUSUM and synthetic EWMA control charts for monitoring the process mean using auxiliary information[END_REF] studied the synthetic exponentially weighted moving average (EWMA) and synthetic cumulative sum (CUSUM) charts with auxiliary information. In the presence of measurement errors,[START_REF] Hu | The effect of measurement errors on the synthetic 𝑋 ̅ chart[END_REF] evaluated the performance of the synthetic 𝑋 ̅ chart. The performance of the synthetic 𝑋 ̅ chart was studied by[START_REF] Hu | Guaranteed in-control performance of the synthetic 𝑋 ̅ chart with estimated parameters[END_REF] when the parameters are estimated. As a considerable amount of Phase I data is required to reduce variability but is limited in practice, they used a bootstrap approach to adjust the parameters of the synthetic 𝑋 ̅ chart. In terms of the median run length,Lee and Khoo (2016a) showed the outperformance of the synthetic np chart compared to the basic np chart. Meanwhile,Lee and Khoo (2016b) studied the performance of the multivariate synthetic |s| control chart. In a multivariate normally distributed process, the synthetic |s| control chart outperforms the standard |s| chart in detecting shifts in the covariance matrix.Shongwe and Graham (2017) combined the synthetic and runs-rules charts with the 𝑋 ̅ chart.[START_REF] Haq | A new nonparametric synthetic EWMA control chart for monitoring process mean[END_REF] proposed a nonparametric synthetic EWMA control chart while[START_REF] Shongwe | A modified side-sensitive synthetic chart to monitor the process mean[END_REF] modified the sidesensitive synthetic chart.[START_REF] Amdouni | One-sided run rules control charts for monitoring the coefficient of variation in short production runs[END_REF] proposed a new efficient method to monitor the coefficient of variation in a short production.[START_REF] Haghighati | Recognition performance of imputed control chart patterns using exponentially weighted moving average[END_REF] studied the performance of control chart pattern recogniser with incomplete data and investigated the effectiveness of the exponential smoothing in restoring the patterns in order to enhance the recognition accuracy. Malela-Majika and Rapoo (2019) developed two new synthetic double sampling charts for monitoring the location process parameter.

Figure 1

 1 Figure 1 illustrates three samples of the CRL, with CRL1 = 2, CRL2 = 3 and CRL3 = 4.

  of the model is to determine the optimal parameters n, L and w that minimize the ANC while adhering to the constraint 0 ATS   . Note that the value of ATS0may not be exactly equal to τ due to the variability of the simulation results and the discrete characteristic of attribute data. Nevertheless, ATS0 should be approximately equal to τ.

Figure 2

 2 Figure2shows the curves of the normalized ATS (ATS/ATSCurt_Syn) for both charts. The normalized ATS of the synthetic chart is more than 1 for all δ; hence for all p shifts, the Curt_Syn chart outperforms the synthetic chart. The detection ability of the synthetic

  Section 6.1. The factors varied at different levels are shown below: p0: 0.005, 0.03, 0.05, 0.1, 0.15 τ: 10/𝑝 0 , 30/𝑝 0 , 100/𝑝 0 𝜇 𝛿 : 4, 6, 10 Note that τ is expressed in terms of p0 as indicated in the previous section. To illustrate, if p0 = 0.005, then τ = 10/p0 = 2000. Along with their respective charting parameters, the ANC and normalized ANC ( Curt_Syn ANC / ANC

  shown in this example. A golf ball is considered nonconforming if the label on the golf ball is printed wrongly. As the quality engineer is only interested in process deterioration, only upward shifts in p are monitored. Based on the Phase I dataset, the p0 value is estimated as 0.01. The mean value 𝜇 𝛿 of the random shift δ is estimated as 5, based on investigation records of the out-of-control cases. The quality engineer has also set the allowable minimum τ = 6000. The value of max  is 12 when 𝜇 𝛿 = 5 according to Equation (10). Using the optimization program, the optimal parameters, ANC and normalized ANC of the two charts are as follows: Synthetic chart: n = 67, L = 16, w = 2, ANC = 6.3380, ANCSyn/ANCCurt_Syn = 1.2283 Curt_Syn chart: n = 62, L = 23, w = 2, ANC = 5.1598 [Please insert Table 4 here] [Please insert Figure 3 here]It can be seen from Table4that the Curt_Syn chart is superior to the synthetic chart for all shifts. From an overall point of view, the Curt_Syn chart outperforms the synthetic chart by 23%. Additionally, Figure3shows that the curve of the Curt_Syn chart is lower than the synthetic chart; thus the Curt_Syn chart has lower ATS values and it is more sensitive for detecting p shifts. Using the Curt_Syn chart, 20 samples, each with size n = 62 are selected. Table5shows the cumulative number of detected nonconforming units and the status of the samples while Figure4shows the implementation of the Curt_Syn chart. For each sample, the 62 units are inspected one by one and c is increased by one when a nonconforming unit is found. At sample #17, c = 5 exceeds w = 2, thus, the sample is nonconforming. As CRL = 17 does not exceed L = 23, an out-of-control condition is signaled at sample #17. Hence, the quality engineer stops the process to take corrective actions.

Figure 1 :

 1 Figure 1: Example of conforming run length (CRL)

Table 2

 2

	, denoted by	ANC	syntheti	c	/ ANC	Curt_Sy n	, is computed. We found that
	ANC	synthetic	/ ANC	Curt_Syn	=	1.3785

Table 1 :

 1 the Curt_Syn chart is studied based on 100% inspection in this paper, further research can investigate the Curt_Syn chart's performance for uniform or random sampling inspection. In addition, the performance of the adaptive (i.e. variable sampling interval (VSI), variable sample size (VSS) and variable sample size and sampling (VSSI)) charts can be improved with the addition of curtailment. Further research can also be done on the incorporation of the curtailment technique with multiattribute charts. ATS values of the synthetic and Curt_Syn charts for the specific case

			ATS
	δ	Synthetic	Curt_Syn
	1	5123.611	4901.927
	2	507.787	495.920
	3	210.200	183.360
	4	133.934	104.543
	5	102.719	73.772
	6	82.491	56.530
	7	71.430	45.623
	8	62.653	38.800
	9	57.232	33.983
	10	52.757	29.987
	11	48.817	26.953
	12	45.956	25.013
	13	43.236	22.847
	14	41.621	21.181
	15	39.799	20.067
	16	39.062	18.435
	17	38.412	17.442
	18	37.615	16.487

Table 2 :

 2 Comparison of the synthetic and Curt_Syn charts under different levels of p0, τ and 𝜇 𝛿

	Case	p0	τ	𝜇 𝛿	Chart	n	L	w	ANC ANC/ANCCurt_Syn
	0	0.01 50/p0 7 Synthetic 67 20	2	5.6048	1.4295
					Curt_Syn 71 16	2	3.9207	1.0000
	1	0.005 10/p0 4 Synthetic 1	22	0	4.2733	1.3025
					Curt_Syn 70 27	1	3.2808	1.0000
	2	0.005 10/p0 10 Synthetic 1	22	0	2.7678	1.2094
					Curt_Syn 75 21	1	2.2886	1.0000
	3	0.005 10/p0 6 Synthetic 1	22	0	3.4452	1.2790
					Curt_Syn 77 20	1	2.6936	1.0000
	4	0.005 100/p0 4 Synthetic 49	5	1 12.7656	1.6430
					Curt_Syn 112 19	2	7.7698	1.0000
	5	0.005 100/p0 10 Synthetic 38	9	1	7.5626	1.9476
					Curt_Syn 111 19	2	3.8830	1.0000
	6	0.005 100/p0 6 Synthetic 49	5	1	9.2602	1.7490
					Curt_Syn 111 19	2	5.2946	1.0000
	7	0.005 30/p0 4 Synthetic 40 29	1	7.6711	1.6349
					Curt_Syn 162 17	2	4.6921	1.0000
	8	0.005 30/p0 10 Synthetic 40 29	1	4.3554	1.2698
					Curt_Syn 40 29	1	3.4299	1.0000
	9	0.005 30/p0 6 Synthetic 40 29	1	5.6335	1.5286
					Curt_Syn 185 10	2	3.6853	1.0000
	10	0.03 10/p0 4 Synthetic 12 28	1	3.5987	1.2811
					Curt_Syn 108 4	3	2.8090	1.0000
	11	0.03 10/p0 10 Synthetic 1	4	0	2.6398	1.2205
					Curt_Syn 13 21	1	2.1628	1.0000
	12	0.03 10/p0 6 Synthetic 12 28	1	3.3596	1.2825
					Curt_Syn 13 21	1	2.6195	1.0000
	13	0.03 100/p0 4 Synthetic 39 18	3	8.3399	1.2792
					Curt_Syn 109 4	5	6.5194	1.0000
	14	0.03 100/p0 10 Synthetic 22 11	2	5.5922	1.5480
					Curt_Syn 20 16	2	3.6126	1.0000
	15	0.03 100/p0 6 Synthetic 20 16	2	6.3430	1.2546
					Curt_Syn 40 16	3	5.0559	1.0000
	16	0.03 30/p0 4 Synthetic 30 12	2	5.6705	1.2953
					Curt_Syn 105 4	4	4.3777	1.0000
	17	0.03 30/p0 10 Synthetic 25 28	2	5.1487	1.8023
					Curt_Syn 31 10	2	2.8567	1.0000
	18	0.03 30/p0 6 Synthetic 30 12	2	5.1858	1.4672
					Curt_Syn 31 10	2	3.5345	1.0000

Table 4 :

 4 ATS values of the Synthetic and Curt_Syn charts corresponding to the golf ball manufacturing process

	δ	Synthetic	Curt_Syn
	1	6108.4034	6053.5346
	2	593.2122	535.6702
	3	217.5566	201.5670
	4	135.7697	114.9303
	5	101.8074	80.9814
	6	83.4829	58.9744
	7	71.3157	47.9615
	8	63.9055	40.1599
	9	57.1117	34.7150
	10	52.2140	30.3919
	11	48.8171	27.3797
	12	45.6547	24.5865

Table 5 :

 5 Phase II dataset from the golf ball manufacturing process

	Sample	c	Status
	1	1	Conforming
	2	2	Conforming
	3	1	Conforming
	4	0	Conforming
	5	1	Conforming
	6	1	Conforming
	7	1	Conforming
	8	0	Conforming
	9	1	Conforming
	10	2	Conforming
	11	2	Conforming
	12	1	Conforming
	13	1	Conforming
	14	0	Conforming
	15	0	Conforming
	16	0	Conforming
	17	5	Nonconforming
	18	0	Conforming
	19	1	Conforming
	20	0	Conforming

Acknowledgements

The authors sincerely thank the editor and reviewers for their constructive suggestions and valuable comments that led to a substantial improvement of the paper. This research is supported by the University of Sharjah, UAE, under Competitive Research Project No.

18020405112.