Mathematical Problems in Engineering

Volume 2018, Article ID 9246372, 11 pages

https://doi.org/10.1155/2018/9246372

## Joint Economic Design of CUSUM Control Chart and Age-Based Imperfect Preventive Maintenance Policy

^{1}College of Economics and Management, Nanjing Forestry University, Nanjing, China^{2}Department of Industrial Engineering & Management, Shanghai Jiao Tong University, Shanghai, China

Correspondence should be addressed to Yaping Li; nc.ude.ufjn@ilgnipay

Received 8 March 2018; Accepted 14 May 2018; Published 12 June 2018

Academic Editor: Neale R. Smith

Copyright © 2018 Yaping Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The close relationship between statistical process control and maintenance has attracted lots of researchers to focus on the jointly economic design of control chart (a main tool of statistical process control) and preventive maintenance policy, and much progress has been made in this field. However, in the existing literatures, the chart is used most, and other charts are rarely considered. In this paper, the economic design of CUSUM chart and age-based imperfect preventive maintenance policy is presented. The process is considered as a multiphase system, and a recursive algorithm is used to model each phase. Besides, a sampling policy under the non-Markovian deterioration assumption is employed, and an age-based imperfect preventive maintenance policy is used. An optimization model with the objective of minimizing the expected cost per unit time is constructed to obtain the near-optimal solution of decision variables: the age of the machine for maintenance, the number of age-based maintenances, sample size, sampling intervals, and the decision interval coefficient and reference value coefficient of CUSUM chart. The solution procedure of the model is provided. Also, sensitivity analysis is performed on the decision variables for each of the various parameters.

#### 1. Introduction

##### 1.1. Prior Literature and Motivation

Control chart, as an effective tool of statistical process control, has been used extensively by practitioners to monitor production process and help to identify and eliminate assignable causes. Since Duncan [1] first proposed an economic design method of charts to maintain current control of a process; the economic design of control charts has been one import issue in the quality control field. Also optimizing maintenance strategy is a hot issue in the reliability field. To study the two problems separately is necessary and reasonable; however, in most cases, the two problems are relevant and interact on each other. In the course of statistical process control, planned/preventive maintenances need to be carried out to decrease the failure rate of the machine and reduce product variation. Similarly, corrective maintenances need to be performed to restore an out-of-control state back to an in-control state and thus have an impact on the failure mode of the machine which ultimately leads to a reduction in quality shift [2, 3] and further change the process control requirement. The close relationship between quality and maintenance has led lots of researchers to focus on the integrated model of control chart and maintenance which are more realistic in practice.

There has been much research on the integrated optimization for control chart and maintenance. Lochner [4], Kneile, Stephens and Vasudeva [5], and Katter et al. [6] preliminary studied the relationships between quality and maintenance. Tagaras [7] first put forward the cost model for statistical process control and maintenance. Ben-Daya and Rahim [8] developed a joint optimization model to determine preventive maintenance (PM) level and the design parameters of the control chart, namely, inspection intervals, sample size, and control limit, in which inspection intervals are different and the failure mechanism follows a general distribution with the increasing hazard. Later, Lee and Rahim [9] considered replacement cost and the remaining value of the machine in the economically integrated model in which maintenance cost is the function of machine age. Cassady et al. [10] proposed an integrated model of control chart and age-limited PM strategy, which captures the costs associated with product inspection, process downtime, and poor quality. Same to Cassady et al. [10], considering control chart and age-limited PM strategy, Yeung et al. [11] used discrete Markov process to get the approximately optimal joint strategy. Based on the research of Linderman et al. [12], Zhou and Zhu [13] studied the economic design of the integrated model of control chart and maintenance management. The grid-search approach was used to search the optimal solution. Chan and Wu [14] used CCC (cumulative count of conforming chart) and planned maintenance policy to optimize the integrated model. Mehrafrooz and Noorossana [15] presented an integrated model which considered complete failure and planned maintenance simultaneously. Six scenarios in production process are analyzed and a procedure for calculating average cost per time unit was proposed too. Yin et al. [16] developed an integrated model which considered the delayed monitoring and ten scenarios were analyzed in the paper. Charongrattanasakul and Pongpullponsak [17] proposed integrated approach for process control and maintenance by EWMA control chart. Genetic algorithm is used to find the optimal values of variables in the model. Later, Ardakan [18] studied the economic design of multiple variable EWMA control chart and maintenance strategy. Shrivastava et al. [19] proposed jointly optimal design of PM and CUSUM control chart with consideration of minimal corrective maintenance (CM) and imperfect maintenance strategy. The aim of the model is to minimize the cost per unit time to get the values of variables. Li et al. [20] considered machine health condition in jointly optimizing predictive maintenance policy and X-bar control chart. Markov method gets popular in recent years. Ho and Quinino [21] developed a Markov model with several control zones to analyze the maintenance performance. Xiang [22] proposed a model for a production process that deteriorates according to a discrete-time Markov chain and further provided a breakthrough in designing an efficient solution algorithm in obtaining analytical results. Liu et al. [23] used a five-state continuous time Markov chain for a two-unit series system to find the optimal control chart parameters by minimizing the average maintenance costs. Zhang et al. [24] proposed a delayed maintenance policy, estimated the state probabilities during the delayed period by Bayesian theory, and used Markov method to model the monitoring-maintenance process.

If defining that a new phase starts after a PM and that a cycle may contain many phases, we find that the above-mentioned literature simplified the model by discussing a single phase without considering the repetitiveness of maintenance, or a multiphase process in which a PM was performed at every sampling, or a multiphase process with equivalent sampling intervals in the cycle. Moreover, in previous literature sampling number was usually set as a decision variable, as well as sampling interval, and thus the age of the machine can be calculated by them when PM was performed. We find that the age of the machine varies with the age-based PM actions, which caused unnecessary or absent maintenance actions with a high probability due to ignoring the actual age condition of the machine. In addition, in most references discussed above control chart is used most, but other control charts like CUSUM chart and EWMA chart are studied by only few researchers. To our knowledge, so far there is only one paper [19] about the joint optimization of CUSUM chart and PM policy. However, its researchers calculated the expected cycle cost and the expected cycle time by a direct analysis on different production scenarios, which cannot work very well for a more complex production process. This is because a direct analysis simplifies the process greatly due to its limitation that it has no ability to cover all possible scenarios.

The motivation of this paper is based on these observations from the literatures. We consider a multiphase production system in which sampling for quality control inspection is carried out with unequal intervals and age-based PMs are performed in parallel. To guarantee the rationality of PM actions, we propose an age-based imperfect PM policy by setting the age of the machine for maintenance as a decision variable. We use a CUSUM chart to jointly model with PM policy by a recursive algorithm ([8, 9] Nourelfath, Nahas, and Ben-Daya, 2016)). The objective of the paper is to minimize the expected cost per unit time to obtain the near-optimal solutions of the age of the machine for maintenance, the number of age-based maintenances, sample size, sampling intervals, and the decision interval coefficient, and reference value coefficient of CUSUM control chart.

##### 1.2. Contributions and Outline

The paper develops an integrated model to simultaneously optimize the design parameters of CUSUM chart and PM policy and presents the following important characteristic.

First, a jointly economic design of CUSUM control chart and age-based imperfect PM policy is studied. To make up the deficiency of the direct analysis method in the literature [19], a recursive algorithm, modified from previously mentioned literatures, is proposed to model the problem. Using recursive algorithm can consider much more scenarios than using the direct analysis and thus obtain a relatively accurate solution for decision variables. For the characteristic of the objective function that it behaves similar to a convex function of (the number of age-based maintenances), a solution procedure is developed to find a near-optimal policy which is also effective on sensitivity analysis.

Second, a multiphase process constituted by multiple PMs and multiple samplings in each PM interval is studied. Inequivalent sampling intervals are used, and an age-based imperfect preventive maintenance policy is employed. If the machine functions without a detected failure for time units, then PM is performed, and each PM restores the machine to a state between as good as new and as bad as old. The age of the machine after a PM will be reduced to a certain value, not zero because of the imperfectness of maintenance. , as the age of the machine for maintenance, is designed to be a decision variable. As a result, our policy will reduce the probability of unnecessary or absent maintenances.

The remainder of this paper is organized as follows. Section 2 provides problem statement and model assumptions, in which quality control, imperfect PM, and sampling interval are discussed. Section 3 introduces how to obtain the expected phase cost and expected phase time using the recursive method. On the basis of this, the expected cycle cost and expected cycle time are calculated by taking a consideration of the repetitiveness of the sampling and maintenance. Then to minimize the expected cost per unit time, the joint model is built; further model features and solution algorithm are reported. Next, in Section 4 a numerical case is used to verify the effectiveness of the model and perform the sensitivity analysis on the decision variables for each of the various parameters. At last, conclusions and possible research in the future are presented in Section 5.

#### 2. Problem Statement and Assumptions

The nomenclature is defined as Table 1.