- Email: [email protected]
An integrated model based on statistical process control and maintenance
Computers & Industrial Engineering 61 (2011) 1245–1255
Contents lists available at SciVerse ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
An integrated model based on statistical process control and maintenance Zohreh Mehrafrooz ⇑, Rassoul Noorossana Industrial Engineering Department, Iran University of Science and Technology, Tehran 16846-13114, Iran
a r t i c l e
i n f o
Article history: Received 23 September 2010 Received in revised form 10 May 2011 Accepted 26 July 2011 Available online 11 August 2011 Keywords: Integrated model Statistical process control Maintenance Renewal reward process
a b s t r a c t The close relationship between quality and maintenance of manufacturing systems has contributed to the development of integrated models, which use the concept of statistical process control (SPC) and maintenance. Such models not only help to improve quality of products but also lead to lower maintenance cost. In this paper, an integrated model is presented which considers complete failure and planned maintenance simultaneously. This model leads to six different scenarios. A new procedure for calculating average cost per time unit is also presented. Finally, a numerical example is used to evaluate sensitivity of the model parameters and compare performance of the developed model to a planned maintenance model. Results indicate satisfactory performance for the developed model. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In today’s competitive arena, quality can play an important role in the business life of an organization. It is obvious that organizations that could fulfill this need effectively and efficiently will have a higher chance of survival. It is obvious that continuous reduction in variation of products and services is necessary for achieving and maintaining a desired level of quality. Control chart as a featured tool of statistical process control has been used extensively by practitioners to monitor and even reduce process variation by identifying and eliminating sources of variation (see Ho & case, 1994; Montgomery, 1980, 2001; Porteus & Angelus, 1997). One source of variation, which could ultimately affect product quality and increase process variation, is lack of proper maintenance of production equipment. Many researchers including Wang (2002), McKone and Weiss (1998), Valdez-Flores and Feldman (1989), Pierskalla and Voelker (1976) and McCall (1965) have reviewed maintenance policies. Since reduction in equipment performance leads to reduction in product quality, a suitable maintenance policy could reduce process variation and help to increase product quality. The close relationship between quality and maintenance has led researchers to develop integrated models, which are more realistic in practice. These models are developed with the aim of reducing total cost of quality. Ben-Daya and Duffuaa (1995) proposed two approaches for linking and modeling the relationship between quality and maintenance. The first approach is based on the idea that maintenance affects the failure pattern of equipment and the concept of imperfect ⇑ Corresponding author. E-mail address: [email protected] (Z. Mehrafrooz). 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.07.017
maintenance should be used to model this relationship. In the second approach, the deviation of product quality characteristics from their target value reduces when maintenance is performed. Rahim and Ben-Daya (2001) presented a survey of integrated models for production, quality control and maintenance. Many other researchers have investigated the relationship between quality and maintenance (see for example, Katter, Tu, & Monacelli, 1997; Kniele, Stephens, & Vasudeva, 1989; Lee & Rahim, 2001; Lochner, 1987; Tapiero, 1986). Cassady, Bowden, Liew, and Pohl (2000) performed a preliminary investigation on integrating maintenance and statistical process control. They proposed a coordinated strategy for the process that shifts to out-of-control condition due to failure of the equipment. They use a X control chart in conjunction with an age-replacement preventive maintenance (PM) policy. Rahim and Banerjee (1993) performed preventive maintenance actions when a shift takes place in the process. Yeung, Cassady, and Schneider (2008) suggested using the relationship between quality and maintenance in order to improve productivity of manufacturing processes. In addition, Rahim (1994), Ben-Daya (1999) and Ben-Daya and Rahim (2000) proposed an integrated model based on X control chart and preventive maintenance in which the in-control time follows a probability distribution with increasing hazard rate. Linderman, McKone-Sweet, and Anderson (2005) developed an integrated model to reduce cost. They assumed perfect maintenance and expressed three scenarios for their model. In two scenarios, process continues until planned maintenance time. At this time, performing planned maintenance when process is incontrol and performing preventive maintenance when process is out-of-control restores the process and brings it back to its original condition. In the third scenario, control chart detects
Linderman et al. (2005) Zhou and Zhu (2008) Panagiotidou and Nenes (2009) Charongrattanasakul and Pongpullponsak (2011) Our work
Adaptive Shewhart
Compensatory maintenance Preventive maintenance Planned maintenance EWMA Shewhart
6
Corrective maintenance
Driving expected value with focus on inspection intervals
Markov chain
Driving expected value with focus on scenario states
Exponential
Weibull
6
4 8
3
Complete failure Time to shift distribution Solution procedure Maintenance type Control Chart type Reference articles
out-of-control condition. They emphasized on adaptive maintenance in which maintenance is increased when process becomes unstable. Zhou and Zhu (2008) extended their model to include four scenarios by adding a scenario which incorporates false alarm of the control chart. Panagiotidou and Nenes (2009) also developed an integrated model based on Linderman et al. (2005) which applies an EWMA control chart. Consideration of warning limit led to six scenarios of in-control alert signal, outof-control alert signal, in-control no signal, out of control no signal, warning limit alert signal, and warning limit no signal. All of the above models consider just in-control and outof-control states and ignore the possibility of a complete equipment failure. In this paper, we develop a model that is an extension of Linderman et al. (2005) model to include the six scenarios. In addition to the false alarm of the control chart scenario, our model includes two scenarios that are equipment failure in the in-control and out-of-control conditions. Tagaras (1988) and Panagiotidou and Nenes (2009) and others studied integrated models based on quality and equipment failure. Tagaras (1988) develops an integrated model based on simultaneous use of process control and maintenance procedures for a production processes characterized by several quality states (a single incontrol state and multiple out-of-control state) and a single failure state. This model could be considered as an improvement and extension of one of the model proposed by Patè-Cornell, Lee, and Tagaras (1987). Panagiotidou and Nenes (2009) developed an integrated model for quality and maintenance. They applied an adaptive variable-parameter Shewhart control chart for monitoring process and compared their model with a fixedparameter control chart. Their model is the first model which combines PM actions with an adaptive SPC tool which could be considered as the main contribution of the paper. In the models that employ control charts to monitor the failure process of the equipment to improve reliability performance, the time between failures, both under exponential (Zhang, Xie, & Goh, 2005) and non-exponential distribution (Xie, Goh, & Ranjan, 2002; Zhang, Xie, Liu, & Goh, 2007), is studied. The time until the transition to an out-of-control state is usually assumed to be exponentially distributed (Duncan, 1956; Panagiotidou & Nenes, 2009; Tagaras, 1988) but other distributions have also been considered in the literature (Linderman et al., 2005; Zhou & Zhu, 2008). It is well known that exponential distribution has the most application in electronic equipments. Panagiotidou and Nenes (2009) assumed that the time to shifts in the quality level as well as the time to failure in both states is exponentially distributed random variables and the failure rate in the outof-control state is higher than the failure rate in the in-control state. Our assumptions are similar to Panagiotidou and Nenes (2009) but the main difference between two models is that Panagiotidou and Nenes (2009) do not consider planned maintenance in their model. In fact, our model could be considered an extension of Panagiotidou and Nenes (2009) model. A more detailed classification of the literature considering six characteristics of control chart type, maintenance type, solution procedure, time to shift distribution, complete failure, and number of scenarios is illustrated in Table 1. Due to the complexity of our model, the solution procedure used by Linderman et al. (2005) and Zhou and Zhu (2008) is not applicable here. The use of a Markov chain similar to what Panagiotidou and Nenes (2009) used is not possible for solving the proposed model since planned maintenance is considered. Therefore, a new solution procedure is developed. Our proposed method is based on all states related to each scenario and different distributions. The problem statement and assumptions are introduced in Section 2. Sections 3 and 4 present the integrated model and its cost analysis, respectively. An exact solution for the model is
Number of scenarios
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
Table 1 An overview of the existing models.
1246
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
presented in Section 5. Section 6 discusses the planned maintenance model and its cost analysis. Section 7 presents numerical analysis and sensitivity analysis. Our concluding remarks are provided in Section 8. 2. Problem statement and assumptions We consider a single product manufacturing system with specific and constant failure rate which currently is in-control. It is assumed that this system consists of single productive equipment. The process is subject to occurrence of an assignable cause which may shift the process to an out-of-control condition. In the outof-control condition, failure rate increases by a constant value. This deterioration of the equipment impacts on the quality of products and it is assumed that we could monitor the equipment by monitoring a key quality characteristic of products. Also, it is assumed that the quality characteristic is measurable on a continuous scale and is normally distributed with mean l0 and variance r2. As long as the distribution parameters remain constant, we could conclude that equipment is in its initial condition and failure rate has not changed. Once the system shifts from the in-control condition to the out-of-control condition, it is assumed that mean of the quality characteristic changes from l0 to l1 = l0 + r but process standard deviation remains unchanged. A control chart for mean is used to monitor quality characteristic or equivalently the system condition. A sample of size n is taken from the process at a regular interval of every h time units. The sample mean is then plotted on a X control chart with the following center line and lower and upper control limits:
UCL ¼ l0 þ krX CL ¼ l0 LCL ¼ l0 krX ; where k is the control chart constant showing the distance between the center line and control limits in standard deviation units. If a plotted sample mean on the control chart falls between LCL and UCL then we conclude that process is in-control. Otherwise, the Table 2 Model parameters. Parameter
Description
m
Number of inspections during time period of planned maintenance Time period of planned maintenance Interval between inspections Sample size Number of standard deviations between center line and upper and lower control limits Transition rate from the in-control condition to the out-ofcontrol condition Failure rate in the in-control condition Failure rate in the out-of-control condition Probability of signal when process is out-of-control Probability of no signal when process is out-of-control Probability of signal when process is in-control Probability of no signal when process is in-control Fixed cost of sampling Variable cost of sampling Cost of performing preventive maintenance Cost of performing corrective maintenance Cost of performing compensatory maintenance Cost of performing planned maintenance Cost of quality loss per time unit when process is in-control Cost of quality loss per time unit when process is out-of-control Expected time to perform preventive maintenance Expected time to perform corrective maintenance Expected time to perform compensatory maintenance Expected time to perform planned maintenance
T h n k k
c1 c2 Pd Pu Pf Pn CF CV CPM CC CU CP CIN COUT TPM TC TU TP
1247
process is assumed to be in an out-of-control condition. If the control chart does not signal after m samples, the planned maintenance is performed in the (m + 1)th inspection interval or equivalently at time T (T = (m + 1)h). At this time, if the quality characteristic is outof-control, the preventive maintenance is done instead of the planned maintenance. Traditionally, when the control chart signals an out-of-control condition a search begins to identify and eliminate source of the assignable cause. However, in our case an outof-control signal at any sampling interval is indicative of equipment deterioration. True signals require preventive maintenance and false alarms will lead to compensatory maintenance. The equipment could fail during an in-control condition or during an outof-control condition with relatively greater failure rate. Corrective maintenance is performed whenever process stops due to a failure. Any maintenance operation which is done out of the planed time unexpectedly renews the time for performing the planned maintenance. Hence, the new planned maintenance will be performed after T time units. Performed maintenance restores the system to the ‘‘good-as-new’’ condition and also renews a cycle. Table 2 provides a short description on the model parameters which are required in the proceeding sections.
3. The Integrated model In any cycle, time to shift to an out-of-control condition follows an exponential distribution with parameter k and time to equipment failure follows an exponential distribution with parameter c1. Once the process shifts to the out-of-control condition, the value of parameter c1 increases to c2 (c1 c2). Based on these assumptions, we develop our proposed model which consists of six different scenarios defined as follows. Fig. 1 shows these scenarios. Scenario 1: The equipment remains in-control but stops due to a failure before time T. A corrective maintenance restores the equipment to its in-control condition. Scenario 2: The equipment remains in-control until time T. Then the planned maintenance is done and the equipment is restored to the in-control condition. Scenario 3: The control chart generates a false out-of-control alarm in one of sampling intervals. After searching for an assignable cause and performing compensatory maintenance, the equipment is restored to its in-control condition. Scenario 4: The equipment shifts to the out-of-control condition. This state is not detected and the process fails by the increased failure rate. Corrective maintenance restores the equipment to its in-control condition. Scenario 5: The equipment shifts to the out-of-control condition. This state is not detected until the planned maintenance. At time T, preventive maintenance is done instead of planned maintenance in order to restore the process to its in-control condition. Scenario 6: The equipment shifts to the out-of-control condition But this state is detected by the control chart in one of the sampling intervals. At this time, the equipment is restored to the primary conditions by the preventive maintenance. Fig. 2 shows the timelines for the defined scenarios. In this figure, Bi indicates (i)th inspection interval. Fig. 2a illustrates the timeline for scenario1. As we can see, the process stops between (j)th and (j + 1)th sampling interval due to the occurrence of a failure. Performing the corrective maintenance takes the process to ‘‘good-as-new’’ condition. At this time, process cycle ends and the time of planned maintenance is renewed as shown in this
1248
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
Start
Equipment State
Control Chart Illustration
Scenarios
Failure
Corrective Maintenance
In-control
Planned Maintenance
In-control
Scenario 2 (S2)
Compensatory Maintenance
In-control
Scenario 3 (S3)
Failure
Corrective Maintenance
In-control
Preventive Maintenance
In-control
Scenario 5 (S5)
Preventive Maintenance
In-control
Scenario 6 (S6)
Scenario 1 (S1)
No signal in-control Alert Signal Start Monitoring
Scenario 4 (S4)
No signal out-of-control Alert Signal
Fig. 1. Integrated model scenarios.
figure. Also, Fig. 2b–f shows the timelines for scenarios 2 through 6, respectively.
4. Cost analysis for the integrated model In the cost analysis of the integrated model, total cost of inspection, maintenance, and quality is considered. As it was stated earlier, the proposed model is developed using the assumption of renewal-reward process. This assumption allows the process to be renewed after each time maintenance is performed. Therefore, the average cost per time unit for the integrated model can be expressed as the expected value of the cost per cycle divided by the expected value of time per cycle as follows (see Ross, 1996).
ESPC&PM ½cost per time unit ¼
ESPC&PM ðcycle costÞ ESPC&PM ðcycle timeÞ
ð1Þ
where
ESPC PM ½cycle cost ¼E½cycle costjS1 pðS1 Þ þ E½cycle costjS2 pðS2 Þ þ E½cycle costjS3 pðS3 Þ þ E½cycle costjS4 pðS4 Þ þ E½cycle costjS5 pðS5 Þ þ E½cycle costjS6 pðS6 Þ
5. The exact solution In order to calculate the exact value for probability, expected cycle time, and expected cycle cost associated with each scenario, we should consider all the states in which each scenario takes place. Table 3 shows the number of states for each scenario and also indicates how these states occur. We also assume that the random variables, time before transition to the out-of-control state (X), time before failure (Y), and time before failure in the out-of-control condition (Z) follow exponential distribution with parameters k, c1, and c2, respectively. It must be pointed out that X and Y are independent random variables and could take place independently. Whenever, an out-of-control condition takes place prior to a failure then Z starts. 5.1. Occurrence probability for each scenario Probability of occurrence associated with each scenario can be calculated by adding the probability for each individual state of the relevant scenario as shown in Table 3. Consider scenario 1 which has m + 1 states. Probability associated with the occurrence of the first state of this scenario or p(S11) can be calculated as follows.
and
ESPC PM ½cycle time ¼E½cycle timejS1 pðS1 Þ þ E½cycle timejS2 pðS2 Þ þ E½cycle timejS3 pðS3 Þ þ E½cycle timejS4 pðSÞ4Þ þ E½cycle timejS5 pðS5 Þ þ E½cycle timejS6 pðS6 Þ where p(S1), p(S2), p(S3), p(S4), p(S5), and p(S6) define the occurrence probability of scenario 1 through 6, respectively. To calculate the expected cost per time unit, we need the probability, the expected cycle time, and the expected cycle cost associated with each scenario. The exact solution for calculating these values are provided in the next section.
pðS11 Þ ¼ pðY h; Y XÞ ¼
Z 0
¼
h
Z y
1
ðkekX Þðc1 ec1 X Þdx dy
c1 ð1 eðc1 þkÞh Þ c1 þ k
ð2Þ
After calculating the probability of all states, the probability of scenario 1 is obtained by adding them up as follows.
pðS1 Þ ¼ pðS11 Þ þ pðS12 Þ þ þ pðS1ðmþ1Þ Þ
ð3Þ
The same analogy is used to calculate the probability associated with the occurrence of other scenarios.
1249
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255 failure occurrence, corrective maintenance
0
B1
Bj
B2
0
T
Bm
B(j+1)
T
B1
(a) scenario 1 planned maintenance
0
B1
Bm
B2
T
0
B1
T
Bm
(b) scenario 2 receiving an false alarm, compensatory maintenance
0
B1
Bj
B2
B(j+1)
0
Bm
T
B1
Bm
T
(c) scenario 3 transition to out-of-control
0
B1
Bj
B(j+1)
failure occurrence, corrective maintenance
B(j+i)
B(j+i+1)
Bm
T
B1
0
T
Bm
(d) scenario 4 transition to out-of-control
0
B1
Bj
preventive maintenance
Bm
B(j+1)
T
0
B1
Bm
T
(e) scenario 5 transition to out-of-control
0
B1
Bj
B(j+1)
out-of-control alarm preventive maintenance
B(j+i)
0
Bm
T
B1
Bm
T
(f) scenario 6 Fig. 2. Timelines for the events described in the six scenarios.
5.2. Calculation of the expected cycle time for each scenario The expected cycle time associated with each scenario can be calculated using the approach considered for the calculation of
the probability of occurrence for each scenario. The first scenario expected cycle time is computed by the expected value of Y. For the second and fifth scenarios, since process continues until the planned maintenance time, the expected cycle time would be
1250
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
Table 3 Number of states under each scenario. Scenario Scenario Scenario Scenario Scenario
1 2 3 4
Scenario 5 Scenario 6
Define states
Number of states
Process remains in-control, a failure occurs in interval (ih, (i + 1)h)i = 0 , . . . ,m Process remains in-control until T, no failure and no false alarm takes place Process remains in-control, a false alarm takes place at the ith sampling interval i = 1 , . . . ,m Process shifts to the out-of-control state in interval (jh, (j + 1)h)j = 0 , . . . ,m, and a failure occurs in interval (ih, (i + 1)h)i = j , . . . ,m
m+1 1 m
Process shifts to the out-of-control state in interval (jh,(j + 1)h) j = 0 , . . . ,m, and it is not detected until T Process shifts to the out-of-control state in interval (jh, (j + 1)h)j = 0 , . . . ,(m 1), and it is detected in the ith sampling interval i = (j + 1) , . . . ,m
m+1
ðmþ1Þðmþ2Þ 2 ðmÞðmþ1Þ 2
Table 4 Occurrence probability of each scenario in the integrated model for m = 1. Scenario
Symbol
Scenario 1
p(S11)
Scenario 2
p(S2)
ðe2hk Þðe2hc1 Þpn
Scenario 3
p(S3)
ðehk Þðehc1 Þpf
Scenario 4
p(S41)
kðe Þ ðkþc1 c2 Þh ð1 eðkþc1 Þh Þ kþ Þ c1 c2 ð1 e ð1 eðkþc1 c2 Þh Þðec2 h ec2 2h Þ c 2h 2 Þ ðkþc1 c2 Þh pn kþkc ðeðkþc1 Þh eðkþc1 Þ2h Þ pn kðe eðkþc1 c2 Þ2h Þ kþc c ðe
p(S12)
Probability of occurence c1 ðkþc1 Þh Þ kþc1 ð1 e pn kþc1c ðeðkþc1 Þh eðkþc1 Þ2h Þ 1
p(S42) p(S43)
c2 h
k kþc1
pu ðkÞ kþc1 c2 1
Scenario 5
p(S51) p(S52)
Scenario 6
p(S6)
equal to (m + 1)h. In the third and sixth scenarios, the expected cycle time is the time interval between the start of the cycle to the time when an alarm is generated. For scenario 4 in which the equipment shifts to the out-of-control condition before the cycle ends, expected value of the cycle time is calculated by adding up the expected value for random variables X and Z. For example, the expected value for the first state of scenario 1 is calculated as follows.
pðS1ðmþ1Þ Þ pðS11 Þ þ þ EðTjS1ðmþ1Þ Þ pðS1 Þ pðS1 Þ
2
and third scenarios, process does not shift to the out-of-control condition and the following equation can be used to calculate the expected cost of quality loss:
Eðcos t of quality lossÞ ¼ C IN EðTjSi Þ i ¼ 1; 2; 3
ð6Þ
In the fourth, fifth, and sixth scenarios in which the equipment is in-control for X time unit, the expected value for the cost of quality loss is calculated as follows:
Eðcos t of quality lossÞ ji X pðSir Þ ¼ ðC IN EðXjSir ÞÞ þ C OUT ðEðTjSir Þ EðXjSir ÞÞ ; pðSi Þ r¼1 ð4Þ
After calculating cycle time for each state of scenario 1, cycle time for this scenario is calculated as
EðTjS1 Þ ¼ EðTjS11 Þ
2
ð1 eðkþc1 c2 Þh Þ c ð2hÞ kðe 2 Þ pn ðeðkþc1 c2 Þh eðkþc1 c2 Þ2h Þ kþc c 1c h 2 kðe 2 Þ ðkþc1 c2 Þh Þ p d ð1 e kþc c 1
Rh R1 ðYÞðkekX Þðc1 ec1 X Þdx dy 0 y EðTjS11 Þ ¼ EðYjY h; Y XÞ ¼ R h R 1 ðkekX Þðc1 ec1 X Þdx dy 0 y 1 ðc1 þkÞh h þ c 1þk c1 þk e 1 ¼ ð1 eðc1 þkÞh Þ
1
kðec2 ð2hÞ Þpu kþc1 c2
ð5Þ
The same analogy is used to calculate expect cycle time for other scenarios. 5.3. Calculation of the expected cycle cost for each scenario The cycle cost consists of three important components namely the cost of quality loss, the cost of sampling, and the cost of investigating alarms and performing maintenance. It is assumed that time for sampling and plotting a sample on the control chart could be ignored. We assume that process stops during the time when a search begins for an assignable cause and performing maintenance. In the in-control condition, cost of quality loss is calculated by multiplying CIN by the expected value of in-control time. In the out-of-control condition, this cost is obtained by multiplying COUT by the expected value of out-of-control time. In the first, second,
i ¼ 4; 5; 6
ð7Þ
where ji is the state number for the ith scenario. The cost of sampling consists of two components namely the fixed cost of sampling and the variable cost of sampling. The fixed cost of sampling (CF) is the setup cost of test equipments and the variable cost of sampling (CV) is the inspection cost of each unit. Hence, the expected value for cost of sampling is calculated as follows.
Eðcost samplingjSi Þ ¼ ðC F þ C V nÞ Eðnumber of samplingjSi Þ i ¼ 1; . . . ; 6
ð8Þ
The expected number of samples for each scenario can be calculated using all the states of that scenario. If ji is the number of states in scenario i and Nir is the number of times sampling is done in the rth state of this scenario then the expected number of times sampling is performed in scenario i is calculated as follows:
Eðnumber of samplingjSi Þ ¼
ji X pðSir Þ Nir i ¼ 1; . . . ; 6 pðSi Þ r¼1
ð9Þ
The maintenance cost associated with scenarios 1 through 6 are equivalent to CC, CP, CU, CC, CPM, and CPM, respectively. This cost is
1251
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255 Table 5 Expected cycle time of each scenario in the integrated model for m = 1. Scenario
Symbol
Scenario 1
E(T|S11)
Expected time per cycle 1 ðkþc1 Þh h þ kþ1c =ð1 eðkþc1 Þh Þ þ T C kþc e 1
Scenario 2 Scenario 3 Scenario 4
1
E(T|S12)
ðhðeðkþc1 Þh Þ 2hðeðkþc1 Þ2h Þ þ kþ1c ðeðkþc1 Þh eðkþc1 Þð2hÞ ÞÞ=ðeðkþc1 Þh eðkþc1 Þð2hÞ Þ þ T C
E(T|S2) E(T|S3) E(T|S41)
2 h + TP h + TU 02
1
0 1 hþc1 kec2 h c h 1 2 ðkc2 Þh 1 1 kX @ Að1 eðkc2 Þh Þ @41 ekh h þ 1 kðe 2 Þ e h þ Þ þ ð1 e kc2 c2 kc2 k k kc2 kc2
c h c h 2 1 ðkc2 Þh h þ k1c =ð1 ekh Þ kðekc Þ ð1 eðkc2 Þh ÞÞ þ T C þ kekc2 kc2 e 2 2 2 h c h c 2h ðkc Þh 2 2 kðe e Þ 2 1 1e ðkc2 Þh h þ k1c þ kðec2 h Þ h þ c1 kðec2 2h Þ 2h þ c1 kc2 kc2 kc2 e 2 2 2 c 2h c h ðkc Þh kðe 2 e 2 Þ 2 1 ðkc2 Þh 1 c2 h c2 ð2hÞ 1e h þ kc e Þ =kðe Þ þ TC þ kc2 kc2 e kc2 2 h c 2h 2 hðekh Þ 2hðek2h Þ þ 1k ðekh ek2h Þ kðekc Þ hðeðkc2 Þh Þ 2hðeðkc2 Þ 2hÞ þ k1c 2 2 0 1 2hþc1 kec2 2h kðec2 2h Þ 2 ðkc2 Þh ðkc2 Þ2h 1 kh kð2hÞ ðk c Þh ðk c Þ2h A e 2 2 þ kc e ÞÞ þ c ðe e Þ@ e ðh ðeðkc2 Þh Þ ðe kc
E(T|S42)
E(T|S43)
2
2
2
2hðeðkc2 Þ2h Þ þ k1c ðeðkc2 Þh eðkc2 Þ2h Þ=ðekh ekð2hÞ Þ 2 c 2h 2 kðekc Þ ðeðkc2 Þh eðkc2 Þ2h ÞÞ þ T C 2
Scenario 5
E(T|S51) E(T|S52) E(T|S6)
Scenario 6
2 h + TPM 2 h + TPM h + TPM
Table 6 Expected cycle cost of each scenario in the integrated model for m = 1. Scenario
Symbol
Expected cost per cycle
Scenario 1
E(C|S1)
12 Þ C IN EðTjS1 Þ þ ðC F þ C V nÞ pðS pðS1 Þ þ C C
Scenario 2 Scenario 3 Scenario 4
E(C|S2) E(C|S3) E(C|S4)
CIN (2h) + (CF + CV n) + CP CIN E(T|S3) + (CF + CV n) + CU
Scenario 5
E(C|S5)
pðS51 Þ pðS5 Þ
Scenario 6
E(C|S6)
CIN E(X|S6) + COUT (h - E(X|S6)) + (CF + CV n) + CPM
pðS42 Þ 41 Þ ðC IN EðXjS41 Þ þ C OUT EðZjS41 ÞÞ pðS pðS Þ þ ðC IN EðXjS42 Þ þ C OUT EðZjS42 ÞÞ pðS4 Þ þ ðC IN EðXjS43 Þ þ C OUT EðZjS43 ÞÞ 4 pðS42 Þ pðS43 Þ 43 Þ pðS pðS4 Þ þ ðC F þ C V nÞ pðS4 Þ þ pðS4 Þ þ C C pðS52 Þ 52 Þ ðC IN EðXjS51 Þ þ C OUT ð2h EðXjS51 ÞÞÞ þ pðS pðS5 Þ ðC IN EðXjS52 Þ þ C OUT ð2h EðXjS52 ÞÞÞ þ ðC F þ C V nÞð pðS5 Þ Þ þ C PM
added to the cost of quality loss and cost of sampling for each scenario to obtain the expected cost for each scenario. The probability, the expected cycle time, and the expected cycle cost of each scenario for m = 1 with parameters k, c1, and c2 are presented in Tables 4–6. 6. The planned maintenance model In this model, planned maintenance is done on the equipment every T time units. Since no inspection is performed in this model, we can set zero for the value of Pd, Pf and one for the value of Pu, Pn in the integrated model. This eliminates the two scenarios associated with signals in the control chart. Finally, the model could be expressed as follow. Scenario 1: Equipment remains in-control but stops before time T due to failure. However, corrective maintenance restores the equipment to the in-control condition. Scenario 2: Equipment remains in-control until time T. Then the planned maintenance is performed and the equipment is restored to the in-control condition. Scenario 3: Equipment shifts to the out-of-control condition. In the out-of-control condition, process fails before time T due to increase in the failure rate. However, corrective maintenance brings the equipment to the in-control condition.
Scenario 4: Equipment shifts to the out-of-control condition. At time T, preventive maintenance is performed instead of planned maintenance in order to restore the process to its primary conditions. Since the model possesses the conditions for the renewalreward process, cost per time unit could be computed using the following equation:
EPM ½cost per time unit ¼
EPM ðcycle costÞ EPM ðcycle timeÞ
ð10Þ
where
EPM ½cycle cost ¼ E½cycle costjS1 pðS1 Þ þ E½cycle costjS2 pðS2 Þ þ E½cycle costjS3 pðS3 Þ þ E½cycle costjS4 pðS4 Þ EPM ½cycle time ¼ E½cycle timejS1 pðS1 Þ þ E½cycle timejS2 pðS2 Þ þ E½cycle timejS3 pðS3 Þ þ E½cycle timejS4 pðS4 Þ Exact solution method and assumptions of the integrated model holds here. Probability associated with each scenario, expected cycle time, and expected cycle cost per scenario for m = 1 are presented in Tables 7–9, respectively.
1252
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
Table 7 Occurrence probability of each scenario in the planned maintenance model for m = 1. Scenario
Symbol
Scenario 1
p(S11)
Probability of occurance c1
kþc1
p(S12) Scenario 2
p(S2)
Scenario 3
p(S31)
c2 h
kðe Þ ðkþc1 c2 Þh ð1 eðkþc1 Þh Þ kþ Þ c1 c2 ð1 e k ðkþc1 c2 Þh c2 h c2 2h Þðe e Þ kþc c ð1 e 1 2 c 2h kðe 2 Þ k ðkþc1 Þh ðkþc1 Þ2h ðkþc1 c2 Þh e Þ eðkþc1 c2 Þ2h Þ ðe ðe kþc1 c2 kþc 1c ð2hÞ 2 kðe Þ ð1 eðkþc1 c2 Þh Þ kþc c c1 ð2hÞ2 kðe 2 Þ ðeðkþc1 c2 Þh eðkþc1 c2 Þ2h Þ kþc c
k kþc1
p(S32) p(S33) p(S41)
Scenario 4
ð1 eðkþc1 Þh Þ
c1 ðkþc1 Þh eðkþc1 Þ2h Þ kþc1 ðe 2hk 2hc1 ðe Þðe Þ
p(S42)
1
2
Table 8 Expected cycle time for each scenario in the planned maintenance model for m = 1. Scenario
Symbol
Scenario 1
E(T|S11)
Expected time per cycle 1 ðkþc1 Þh h þ kþ1c =ð1 eðkþc1 Þh Þ þ T C kþc e 1
Scenario 2 Scenario 3
1
ðhðeðkþc1 Þh Þ 2hðeðkþc1 Þ2h Þ þ kþ1c ðeðkþc1 Þh eðkþc1 Þð2hÞ ÞÞ= eðkþc1 Þh eðkþc1 Þð2hÞ þ T C
E(T|S12)
1
E(T|S2) E(T|S31)
2 h + TP h c2 h 1 kh h þ 1k kðekc Þ k1c eðkc2 Þh h þ k1c þ c1 ð1 ekh Þ ke 2 2 2 2 0 1 c2 h 1 c h hþc ke c2 h 2 2 kðe Þ 1 ðkc2 Þh Að1 eðkc2 Þh Þ þ @ h þ k1c =ð1 ekh Þ kðekc Þ ð1 eðkc2 Þh ÞÞ þ T C kc kc kc e 2
2
E(T|S33)
kðec2 h ec2 2h Þ kc2
ðkc2 Þh
1 kc2
2
Scenario 4
2
2
1 kc2
2
ðkc Þh e hþ Þ h þ c1 kðec2 2h Þð2h þ c1 Þ 1ekc 2 þ kðe 2 2 2 c 2h c h ðkc Þh 2 2 Þ 1 ðkc2 Þh h þ k1c =kðec2 h ec2 ð2hÞ Þ 1ekc 2 Þ þ T C þ kðe ke c2 kc2 e 2 2 c 2h h 2 hðekh Þ 2hðek2h Þ þ 1k ðekh ek2h Þ kðekc Þ hðeðkc2 Þh Þ 2hðeðkc2 Þ2h Þ 2 0 1 2hþc1 kec2 2h 2 1 ðkc2 Þh ðkc2 Þ2h 1 kh kð2hÞ @ Aðeðkc2 Þh eðkc2 Þ2h Þ e ÞÞ þ c ðe e Þ þ kc ðe kc h
E(T|S32)
c2 h
2
2
c 2h 2 þ kðekc Þ ðhðeðkc2 Þh Þ 2hðeðkc2 Þ2h Þ þ k1c ðeðkc2 Þh eðkc2 Þ2h Þ=ðekh ekð2hÞ Þ kðekc Þ ðeðkc2 Þh eðkc2 Þ2h ÞÞ þ T C 2 2 2 2 h + TPM 2 h + TPM
E(T|S41) E(T|S42)
c2 2h
Table 9 Expected cycle cost for each scenario in the planned maintenance model for m = 1. Scenario
Symbol
Expected cost per cycle
Scenario 1
E(C|S1)
12 Þ C IN EðTjS1 Þ þ ðC F þ C V nÞ pðS pðS1 Þ þ C C
Scenario 2 Scenario 3
E(C|S2) E(C|S3)
CIN (2h) + (CF + CV n) + CP
Scenario 4
E(C|S4)
pðS32 Þ 31 Þ ðC IN EðXjS31 Þ þ C OUT EðZjS31 ÞÞ pðS pðS Þ þ ðC IN EðXjS32 Þ þ C OUT EðZjS32 ÞÞ pðS3 Þ þ ðC IN EðXjS33 Þ þ C OUT EðZjS33 ÞÞ 3 pðS33 Þ pðS32 Þ pðS33 Þ pðS3 Þ þ ðC F þ C V nÞ pðS3 Þ þ pðS3 Þ þ C C pðS41 Þ pðS42 Þ pðS42 Þ pðS4 Þ ðC IN EðXjS41 Þ þ C OUT ð2h EðXjS41 ÞÞÞ þ pðS4 Þ ðC IN EðXjS42 Þ þ C OUT ð2h EðXjS42 ÞÞÞ þ ðC F þ C V nÞ pðS4 Þ þ C PM
Table 10 Parameter value for the numerical example. Parameter
COUT
CV
CF
c2
c1
k
CIN
CU
CC
CPM
CP
TU
TC
TPM
TP
Low High
100 300
0.2 0.8
8 12
0.15 0.25
0.02 0.07
0.05 0.15
8 12
400 600
8000 12,000
1500 2500
800 1200
0.1 0.4
7 9
2 3
0.7 1.3
7. Numerical analysis and sensitivity analysis
7.1. Numerical example
In this section, the sensitivity of the integrated model parameters with the expected cost per time unit is investigated using a numerical example. Then, the integrated model is compared with the planned maintenance model.
Spot welding using welding gun is widely used in automotive industry. To ensure proper welding gun operation, planned maintenance at specified time is performed. However, due to some factors such as electrode erosion or input voltage drop, the welding
1253
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
99 E
95
J
Factor A B C D E F G H J K L M N O P
F
90
D
80
Percent
Effect Type Not Significant Significant
70 60 50 40 30 20 10 5
1 -50
0
50
100
150
Effect Fig. 3. Normal probability plot of the effects. Gama2
Gama1
560 520
Expected Cost
480 440 400 0.15
0.25
0.02
0.07
Lambda
Cc
560 520 480 440 400 0.05
0.15
8000
12000
(a) integrated model Gama2
600
Gama1
550
Expected Cost
500 450 0.15
0.25
0.02
Lambda
600
0.07 Cc
550 500 450 0.05
0.15
8000
(b) Planned maintenance model Fig. 4. Main effects plots.
12000
Name Cout Cv Cf Gama2 Gama1 Lambda Cin Cu Cc Cpm Cp Tu Tc Tpm Tp
1254
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
the integrated model has better economic behavior than the planned maintenance model.
800
Expected Cost
700 600
8. Conclusions
500
There exists limited number of models in the literature which use the concept of statistical process control and maintenance to develop integrated models. This paper considers a process with a single equipment to develop an integrated model. We assumed that monitoring quality of products is synonymous to monitoring status of the equipment and as the result an out-of-control signal would indicate deterioration of the equipment. It is presumed that performing any maintenance restores the equipment to the ‘‘goodas-new’’ condition. Through a numerical example, effects of the model parameters on the average cost are studied and it is shown that integrated model inquires lower cost than the planned maintenance model. For simplicity, exponential distributions for the times to failure are assumed. However, further research could be conducted on the subject assuming Weibull or a more general distribution for the time to failure or out-of-control condition. The issue of two or more out-of-control conditions and a multiple equipment manufacturing process is also an issue which requires further research.
400 300 200 100 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Integrated Model Planned M Model
Fig. 5. comparing cost of the integrated model to the planned maintenance model.
gun might shift to the out-of-control condition with higher failure rate. In order to control the welding gun condition, weld quality of products is monitored. To achieving this purpose, every three time units (h = 3), a sample of five products is taken from the process and the mean of spot welding resistances as a quality characteristic is calculated and plotted on a X control chart. Quality characteristic of interest in the in-control condition follows a normal distribution with l0 = 750 and r2 = 2500. In the out-of-control condition, process mean shifts to l1 = 800 while process variance remains unchanged. Time for transition from the in-control to the out-of-control condition is governed by an exponential distribution with parameter k. For the sake of simplicity, we assume that inspection is performed only once during the planned maintenance interval (m = 1). Equipment failure rate in the in-control and out-of-control conditions are equal to c1 and c2, respectively. Table 10 shows the high and low levels of the model parameters which are used in model sensitivity analysis.
7.2. Sensitivity analysis A designed experiment was used to study the effects of the model parameters on the expected cost per time unit. Employing a fractional factorial design with resolution IV requires 32 experimental runs for the 15 model parameters. Fig. 3 presents the normal plot of the effect estimates with expected cost per time unit as the response variable. These effects estimates are shown against the normal quintile error. The negligible effects lie close to the line while the large ones deviate from the line. This figure illustrates that CC, k, c1, and c2 have significant effects and increasing the value of each results in a significant increase in the cost.
7.3. Comparing the models To investigate the performance of the planned maintenance model with four parameters (CC, k, c1, and c2) and to compare the integrated model with the planned maintenance model, we employ a full factorial design. This design needs 16 experimental runs. The low and high values for these effective parameters are the same as the values in Table 10. For non-effective parameters, the mean value between the low and high values is set in the experiments. Fig. 4 shows main effects plots for the integrated and planned maintenance models. Hence, main effects of the integrated model seem effective in the planned maintenance model too. The cost of two models are compared in Fig. 5 which illustrate
References Ben-Daya, M. (1999). Integrated production maintenance and quality model for imperfect process. IIE Transactions, 31, 491–501. Ben-Daya, M., & Duffuaa, S. O. (1995). Maintenance and quality: The missing link. Journal of Quality in Maintenance Engineering, 1, 20–26. Ben-Daya, M., & Rahim, M. A. (2000). Effect of maintenance on the economic design of x-control chart. European Journal of Operation Research, 120, 131–143. Cassady, C. R., Bowden, R. O., Liew, L., & Pohl, E. A. (2000). Combining preventive maintenance and statistical process control: a preliminary investigation. IIE Transactions, 32, 471–478. Charongrattanasakul, P., & Pongpullponsak, A. (2011). Minimizing the cost of integrated systems approach to process control and maintenance model by EWMA control chart using genetic algorithm. Expert Systems with Applications, 38, 5178–5186. Duncan, A. J. (1956). The economic design of x-bar charts used to maintain current control of a process. Journal of the American Statistical Association, 52, 228–242. Ho, C., & Case, K. E. (1994). Economic design of control charts: A literature review for 1981–1991. Journal of Quality Technology, 26, 39–53. Katter, J., Tu, J., Monacelli, L. (1997). Gartner M. Predictive maintenance of an industrial laser using statistical process control charting. In Proceedings of the 1997 laser materials processing conference (pp. C57 ± C66). Kniele, R., Stephens, G., Vasudeva, K. (1989). Performance assessment and statistical process control. An approach to operation and maintenance cost reduction. In Proceeding of the IFAC/IFORS/IAEE symposium on energy, systems, management and economics (pp. 215 ± 221). Lee, B. H., & Rahim, M. A. (2001). An integrated economic design model for quality control, replacement, and maintenance. Quality Engineering, 13, 581–593. Linderman, K., McKone-Sweet, K. E., & Anderson, J. C. (2005). An integrated systems approach to process control and maintenance. European Journal of Operation Research, 164, 324–340. Lochner, R. (1987). Improved SPC using reliability concepts. In: 41st annual quality congress transactions (pp. 648 ± 653). McCall, J. J. (1965). Maintenance policies for stochastically failing equipment: A survey. Management Science, 11, 493–524. McKone, K., & Weiss, E. N. (1998). Total productive maintenance: Bridging the gap between practice and research. Production Operations Management, 7, 335–351. Montgomery, D. C. (1980). Economic design of control charts: A review of literature survey. Journal of Quality Technology, 12, 40–43. Montgomery, D. C. (2001). Introduction to statistical quality control (fourth ed.). New York: Wiley. Panagiotidou, S., & Nenes, G. (2009). An economically designed, integrated quality and maintenance model using an adaptive Shewhart chart. Reliability Engineering and System Safety, 94, 732–741. Patè-Cornell Lee, H. L., & Tagaras, G. (1987). Warning of malfunction: The decision to inspect and maintain production processes on schedule or on demand. Management Science, 33, 1277–1290. Pierskalla, W. P., & Voelker, J. A. (1976). A survey of maintenance models: The control and surveillance of deteriorating systems. Naval Research Logistics Quarterly, 23, 353–388. Porteus, E. L., & Angelus, A. (1997). Opportunities for improved statistical process control. Management Science, 43, 1214–1228. Rahim, M. A. (1994). Joint determination of quantity, inspection, schedule and control chart design. IIE Transactions, 26, 2–11.
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255 Rahim, M. A., & Banerjee, P. K. (1993). A generalized model for the economic design of x-bar control chart for production systems with increasing failure rate and early replacement. Navel Research Logistics, 40, 787–809. Rahim, M. A., & Ben-Daya, M. (2001). Integrated production, quality and maintenance models: An overview. In M. A. Rahim & M. Ben-Daya (Eds.), Integrated models in production planning, inventory, quality, and maintenance. Kluwer Academic Publishers. Ross, S. (1996). Stochastic Processes (second ed.). New York: Wiley. Tagaras, G. (1988). An integrated cost model for the joint optimization of process control and maintenance. The Journal of the Operational Research Society, 39, 757–766. Tapiero, C. S. (1986). Continuous quality production and machine maintenance. Naval research Logistics Quarterly, 33, 489–499. Valdez-Flores, C., & Feldman, R. M. (1989). A survey of preventative maintenance models for stochastically deteriorating single-unit systems. Naval Research Logistics Quarterly, 36, 419–446.
1255
Wang, H. (2002). A survey of maintenance policies of deteriorating systems. European Journal of Operation Research, 139, 469–489. Xie, M., Goh, T. N., & Ranjan, P. (2002). Some effective control chart procedures for reliability monitoring. Reliability Engineering & System Safety, 77, 143–150. Yeung, T. G., Cassady, C. R., & Schneider, K. (2008). Simultaneous optimization of xbar control chart and age-based preventive maintenance policies under an economic objective. IIE Transactions, 40, 147–159. Zhang, C. W., Xie, M., & Goh, T. N. (2005). Economic design of exponential charts for time between events monitoring. International Journal of Production Research, 43, 5019–5032. Zhang, C. W., Xie, M., Liu, J. Y., & Goh, T. N. (2007). A control chart for the Gamma distribution as a model of time between events. International Journal of Production Research, 45, 5649–5666. Zhou, W. H., & Zhu, G. L. (2008). Economic design of integrated model of control chart and maintenance management. Mathematical and computer Modelling, 47, 1389–1395.
Contents lists available at SciVerse ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
An integrated model based on statistical process control and maintenance Zohreh Mehrafrooz ⇑, Rassoul Noorossana Industrial Engineering Department, Iran University of Science and Technology, Tehran 16846-13114, Iran
a r t i c l e
i n f o
Article history: Received 23 September 2010 Received in revised form 10 May 2011 Accepted 26 July 2011 Available online 11 August 2011 Keywords: Integrated model Statistical process control Maintenance Renewal reward process
a b s t r a c t The close relationship between quality and maintenance of manufacturing systems has contributed to the development of integrated models, which use the concept of statistical process control (SPC) and maintenance. Such models not only help to improve quality of products but also lead to lower maintenance cost. In this paper, an integrated model is presented which considers complete failure and planned maintenance simultaneously. This model leads to six different scenarios. A new procedure for calculating average cost per time unit is also presented. Finally, a numerical example is used to evaluate sensitivity of the model parameters and compare performance of the developed model to a planned maintenance model. Results indicate satisfactory performance for the developed model. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In today’s competitive arena, quality can play an important role in the business life of an organization. It is obvious that organizations that could fulfill this need effectively and efficiently will have a higher chance of survival. It is obvious that continuous reduction in variation of products and services is necessary for achieving and maintaining a desired level of quality. Control chart as a featured tool of statistical process control has been used extensively by practitioners to monitor and even reduce process variation by identifying and eliminating sources of variation (see Ho & case, 1994; Montgomery, 1980, 2001; Porteus & Angelus, 1997). One source of variation, which could ultimately affect product quality and increase process variation, is lack of proper maintenance of production equipment. Many researchers including Wang (2002), McKone and Weiss (1998), Valdez-Flores and Feldman (1989), Pierskalla and Voelker (1976) and McCall (1965) have reviewed maintenance policies. Since reduction in equipment performance leads to reduction in product quality, a suitable maintenance policy could reduce process variation and help to increase product quality. The close relationship between quality and maintenance has led researchers to develop integrated models, which are more realistic in practice. These models are developed with the aim of reducing total cost of quality. Ben-Daya and Duffuaa (1995) proposed two approaches for linking and modeling the relationship between quality and maintenance. The first approach is based on the idea that maintenance affects the failure pattern of equipment and the concept of imperfect ⇑ Corresponding author. E-mail address: [email protected] (Z. Mehrafrooz). 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.07.017
maintenance should be used to model this relationship. In the second approach, the deviation of product quality characteristics from their target value reduces when maintenance is performed. Rahim and Ben-Daya (2001) presented a survey of integrated models for production, quality control and maintenance. Many other researchers have investigated the relationship between quality and maintenance (see for example, Katter, Tu, & Monacelli, 1997; Kniele, Stephens, & Vasudeva, 1989; Lee & Rahim, 2001; Lochner, 1987; Tapiero, 1986). Cassady, Bowden, Liew, and Pohl (2000) performed a preliminary investigation on integrating maintenance and statistical process control. They proposed a coordinated strategy for the process that shifts to out-of-control condition due to failure of the equipment. They use a X control chart in conjunction with an age-replacement preventive maintenance (PM) policy. Rahim and Banerjee (1993) performed preventive maintenance actions when a shift takes place in the process. Yeung, Cassady, and Schneider (2008) suggested using the relationship between quality and maintenance in order to improve productivity of manufacturing processes. In addition, Rahim (1994), Ben-Daya (1999) and Ben-Daya and Rahim (2000) proposed an integrated model based on X control chart and preventive maintenance in which the in-control time follows a probability distribution with increasing hazard rate. Linderman, McKone-Sweet, and Anderson (2005) developed an integrated model to reduce cost. They assumed perfect maintenance and expressed three scenarios for their model. In two scenarios, process continues until planned maintenance time. At this time, performing planned maintenance when process is incontrol and performing preventive maintenance when process is out-of-control restores the process and brings it back to its original condition. In the third scenario, control chart detects
Linderman et al. (2005) Zhou and Zhu (2008) Panagiotidou and Nenes (2009) Charongrattanasakul and Pongpullponsak (2011) Our work
Adaptive Shewhart
Compensatory maintenance Preventive maintenance Planned maintenance EWMA Shewhart
6
Corrective maintenance
Driving expected value with focus on inspection intervals
Markov chain
Driving expected value with focus on scenario states
Exponential
Weibull
6
4 8
3
Complete failure Time to shift distribution Solution procedure Maintenance type Control Chart type Reference articles
out-of-control condition. They emphasized on adaptive maintenance in which maintenance is increased when process becomes unstable. Zhou and Zhu (2008) extended their model to include four scenarios by adding a scenario which incorporates false alarm of the control chart. Panagiotidou and Nenes (2009) also developed an integrated model based on Linderman et al. (2005) which applies an EWMA control chart. Consideration of warning limit led to six scenarios of in-control alert signal, outof-control alert signal, in-control no signal, out of control no signal, warning limit alert signal, and warning limit no signal. All of the above models consider just in-control and outof-control states and ignore the possibility of a complete equipment failure. In this paper, we develop a model that is an extension of Linderman et al. (2005) model to include the six scenarios. In addition to the false alarm of the control chart scenario, our model includes two scenarios that are equipment failure in the in-control and out-of-control conditions. Tagaras (1988) and Panagiotidou and Nenes (2009) and others studied integrated models based on quality and equipment failure. Tagaras (1988) develops an integrated model based on simultaneous use of process control and maintenance procedures for a production processes characterized by several quality states (a single incontrol state and multiple out-of-control state) and a single failure state. This model could be considered as an improvement and extension of one of the model proposed by Patè-Cornell, Lee, and Tagaras (1987). Panagiotidou and Nenes (2009) developed an integrated model for quality and maintenance. They applied an adaptive variable-parameter Shewhart control chart for monitoring process and compared their model with a fixedparameter control chart. Their model is the first model which combines PM actions with an adaptive SPC tool which could be considered as the main contribution of the paper. In the models that employ control charts to monitor the failure process of the equipment to improve reliability performance, the time between failures, both under exponential (Zhang, Xie, & Goh, 2005) and non-exponential distribution (Xie, Goh, & Ranjan, 2002; Zhang, Xie, Liu, & Goh, 2007), is studied. The time until the transition to an out-of-control state is usually assumed to be exponentially distributed (Duncan, 1956; Panagiotidou & Nenes, 2009; Tagaras, 1988) but other distributions have also been considered in the literature (Linderman et al., 2005; Zhou & Zhu, 2008). It is well known that exponential distribution has the most application in electronic equipments. Panagiotidou and Nenes (2009) assumed that the time to shifts in the quality level as well as the time to failure in both states is exponentially distributed random variables and the failure rate in the outof-control state is higher than the failure rate in the in-control state. Our assumptions are similar to Panagiotidou and Nenes (2009) but the main difference between two models is that Panagiotidou and Nenes (2009) do not consider planned maintenance in their model. In fact, our model could be considered an extension of Panagiotidou and Nenes (2009) model. A more detailed classification of the literature considering six characteristics of control chart type, maintenance type, solution procedure, time to shift distribution, complete failure, and number of scenarios is illustrated in Table 1. Due to the complexity of our model, the solution procedure used by Linderman et al. (2005) and Zhou and Zhu (2008) is not applicable here. The use of a Markov chain similar to what Panagiotidou and Nenes (2009) used is not possible for solving the proposed model since planned maintenance is considered. Therefore, a new solution procedure is developed. Our proposed method is based on all states related to each scenario and different distributions. The problem statement and assumptions are introduced in Section 2. Sections 3 and 4 present the integrated model and its cost analysis, respectively. An exact solution for the model is
Number of scenarios
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
Table 1 An overview of the existing models.
1246
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
presented in Section 5. Section 6 discusses the planned maintenance model and its cost analysis. Section 7 presents numerical analysis and sensitivity analysis. Our concluding remarks are provided in Section 8. 2. Problem statement and assumptions We consider a single product manufacturing system with specific and constant failure rate which currently is in-control. It is assumed that this system consists of single productive equipment. The process is subject to occurrence of an assignable cause which may shift the process to an out-of-control condition. In the outof-control condition, failure rate increases by a constant value. This deterioration of the equipment impacts on the quality of products and it is assumed that we could monitor the equipment by monitoring a key quality characteristic of products. Also, it is assumed that the quality characteristic is measurable on a continuous scale and is normally distributed with mean l0 and variance r2. As long as the distribution parameters remain constant, we could conclude that equipment is in its initial condition and failure rate has not changed. Once the system shifts from the in-control condition to the out-of-control condition, it is assumed that mean of the quality characteristic changes from l0 to l1 = l0 + r but process standard deviation remains unchanged. A control chart for mean is used to monitor quality characteristic or equivalently the system condition. A sample of size n is taken from the process at a regular interval of every h time units. The sample mean is then plotted on a X control chart with the following center line and lower and upper control limits:
UCL ¼ l0 þ krX CL ¼ l0 LCL ¼ l0 krX ; where k is the control chart constant showing the distance between the center line and control limits in standard deviation units. If a plotted sample mean on the control chart falls between LCL and UCL then we conclude that process is in-control. Otherwise, the Table 2 Model parameters. Parameter
Description
m
Number of inspections during time period of planned maintenance Time period of planned maintenance Interval between inspections Sample size Number of standard deviations between center line and upper and lower control limits Transition rate from the in-control condition to the out-ofcontrol condition Failure rate in the in-control condition Failure rate in the out-of-control condition Probability of signal when process is out-of-control Probability of no signal when process is out-of-control Probability of signal when process is in-control Probability of no signal when process is in-control Fixed cost of sampling Variable cost of sampling Cost of performing preventive maintenance Cost of performing corrective maintenance Cost of performing compensatory maintenance Cost of performing planned maintenance Cost of quality loss per time unit when process is in-control Cost of quality loss per time unit when process is out-of-control Expected time to perform preventive maintenance Expected time to perform corrective maintenance Expected time to perform compensatory maintenance Expected time to perform planned maintenance
T h n k k
c1 c2 Pd Pu Pf Pn CF CV CPM CC CU CP CIN COUT TPM TC TU TP
1247
process is assumed to be in an out-of-control condition. If the control chart does not signal after m samples, the planned maintenance is performed in the (m + 1)th inspection interval or equivalently at time T (T = (m + 1)h). At this time, if the quality characteristic is outof-control, the preventive maintenance is done instead of the planned maintenance. Traditionally, when the control chart signals an out-of-control condition a search begins to identify and eliminate source of the assignable cause. However, in our case an outof-control signal at any sampling interval is indicative of equipment deterioration. True signals require preventive maintenance and false alarms will lead to compensatory maintenance. The equipment could fail during an in-control condition or during an outof-control condition with relatively greater failure rate. Corrective maintenance is performed whenever process stops due to a failure. Any maintenance operation which is done out of the planed time unexpectedly renews the time for performing the planned maintenance. Hence, the new planned maintenance will be performed after T time units. Performed maintenance restores the system to the ‘‘good-as-new’’ condition and also renews a cycle. Table 2 provides a short description on the model parameters which are required in the proceeding sections.
3. The Integrated model In any cycle, time to shift to an out-of-control condition follows an exponential distribution with parameter k and time to equipment failure follows an exponential distribution with parameter c1. Once the process shifts to the out-of-control condition, the value of parameter c1 increases to c2 (c1 c2). Based on these assumptions, we develop our proposed model which consists of six different scenarios defined as follows. Fig. 1 shows these scenarios. Scenario 1: The equipment remains in-control but stops due to a failure before time T. A corrective maintenance restores the equipment to its in-control condition. Scenario 2: The equipment remains in-control until time T. Then the planned maintenance is done and the equipment is restored to the in-control condition. Scenario 3: The control chart generates a false out-of-control alarm in one of sampling intervals. After searching for an assignable cause and performing compensatory maintenance, the equipment is restored to its in-control condition. Scenario 4: The equipment shifts to the out-of-control condition. This state is not detected and the process fails by the increased failure rate. Corrective maintenance restores the equipment to its in-control condition. Scenario 5: The equipment shifts to the out-of-control condition. This state is not detected until the planned maintenance. At time T, preventive maintenance is done instead of planned maintenance in order to restore the process to its in-control condition. Scenario 6: The equipment shifts to the out-of-control condition But this state is detected by the control chart in one of the sampling intervals. At this time, the equipment is restored to the primary conditions by the preventive maintenance. Fig. 2 shows the timelines for the defined scenarios. In this figure, Bi indicates (i)th inspection interval. Fig. 2a illustrates the timeline for scenario1. As we can see, the process stops between (j)th and (j + 1)th sampling interval due to the occurrence of a failure. Performing the corrective maintenance takes the process to ‘‘good-as-new’’ condition. At this time, process cycle ends and the time of planned maintenance is renewed as shown in this
1248
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
Start
Equipment State
Control Chart Illustration
Scenarios
Failure
Corrective Maintenance
In-control
Planned Maintenance
In-control
Scenario 2 (S2)
Compensatory Maintenance
In-control
Scenario 3 (S3)
Failure
Corrective Maintenance
In-control
Preventive Maintenance
In-control
Scenario 5 (S5)
Preventive Maintenance
In-control
Scenario 6 (S6)
Scenario 1 (S1)
No signal in-control Alert Signal Start Monitoring
Scenario 4 (S4)
No signal out-of-control Alert Signal
Fig. 1. Integrated model scenarios.
figure. Also, Fig. 2b–f shows the timelines for scenarios 2 through 6, respectively.
4. Cost analysis for the integrated model In the cost analysis of the integrated model, total cost of inspection, maintenance, and quality is considered. As it was stated earlier, the proposed model is developed using the assumption of renewal-reward process. This assumption allows the process to be renewed after each time maintenance is performed. Therefore, the average cost per time unit for the integrated model can be expressed as the expected value of the cost per cycle divided by the expected value of time per cycle as follows (see Ross, 1996).
ESPC&PM ½cost per time unit ¼
ESPC&PM ðcycle costÞ ESPC&PM ðcycle timeÞ
ð1Þ
where
ESPC PM ½cycle cost ¼E½cycle costjS1 pðS1 Þ þ E½cycle costjS2 pðS2 Þ þ E½cycle costjS3 pðS3 Þ þ E½cycle costjS4 pðS4 Þ þ E½cycle costjS5 pðS5 Þ þ E½cycle costjS6 pðS6 Þ
5. The exact solution In order to calculate the exact value for probability, expected cycle time, and expected cycle cost associated with each scenario, we should consider all the states in which each scenario takes place. Table 3 shows the number of states for each scenario and also indicates how these states occur. We also assume that the random variables, time before transition to the out-of-control state (X), time before failure (Y), and time before failure in the out-of-control condition (Z) follow exponential distribution with parameters k, c1, and c2, respectively. It must be pointed out that X and Y are independent random variables and could take place independently. Whenever, an out-of-control condition takes place prior to a failure then Z starts. 5.1. Occurrence probability for each scenario Probability of occurrence associated with each scenario can be calculated by adding the probability for each individual state of the relevant scenario as shown in Table 3. Consider scenario 1 which has m + 1 states. Probability associated with the occurrence of the first state of this scenario or p(S11) can be calculated as follows.
and
ESPC PM ½cycle time ¼E½cycle timejS1 pðS1 Þ þ E½cycle timejS2 pðS2 Þ þ E½cycle timejS3 pðS3 Þ þ E½cycle timejS4 pðSÞ4Þ þ E½cycle timejS5 pðS5 Þ þ E½cycle timejS6 pðS6 Þ where p(S1), p(S2), p(S3), p(S4), p(S5), and p(S6) define the occurrence probability of scenario 1 through 6, respectively. To calculate the expected cost per time unit, we need the probability, the expected cycle time, and the expected cycle cost associated with each scenario. The exact solution for calculating these values are provided in the next section.
pðS11 Þ ¼ pðY h; Y XÞ ¼
Z 0
¼
h
Z y
1
ðkekX Þðc1 ec1 X Þdx dy
c1 ð1 eðc1 þkÞh Þ c1 þ k
ð2Þ
After calculating the probability of all states, the probability of scenario 1 is obtained by adding them up as follows.
pðS1 Þ ¼ pðS11 Þ þ pðS12 Þ þ þ pðS1ðmþ1Þ Þ
ð3Þ
The same analogy is used to calculate the probability associated with the occurrence of other scenarios.
1249
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255 failure occurrence, corrective maintenance
0
B1
Bj
B2
0
T
Bm
B(j+1)
T
B1
(a) scenario 1 planned maintenance
0
B1
Bm
B2
T
0
B1
T
Bm
(b) scenario 2 receiving an false alarm, compensatory maintenance
0
B1
Bj
B2
B(j+1)
0
Bm
T
B1
Bm
T
(c) scenario 3 transition to out-of-control
0
B1
Bj
B(j+1)
failure occurrence, corrective maintenance
B(j+i)
B(j+i+1)
Bm
T
B1
0
T
Bm
(d) scenario 4 transition to out-of-control
0
B1
Bj
preventive maintenance
Bm
B(j+1)
T
0
B1
Bm
T
(e) scenario 5 transition to out-of-control
0
B1
Bj
B(j+1)
out-of-control alarm preventive maintenance
B(j+i)
0
Bm
T
B1
Bm
T
(f) scenario 6 Fig. 2. Timelines for the events described in the six scenarios.
5.2. Calculation of the expected cycle time for each scenario The expected cycle time associated with each scenario can be calculated using the approach considered for the calculation of
the probability of occurrence for each scenario. The first scenario expected cycle time is computed by the expected value of Y. For the second and fifth scenarios, since process continues until the planned maintenance time, the expected cycle time would be
1250
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
Table 3 Number of states under each scenario. Scenario Scenario Scenario Scenario Scenario
1 2 3 4
Scenario 5 Scenario 6
Define states
Number of states
Process remains in-control, a failure occurs in interval (ih, (i + 1)h)i = 0 , . . . ,m Process remains in-control until T, no failure and no false alarm takes place Process remains in-control, a false alarm takes place at the ith sampling interval i = 1 , . . . ,m Process shifts to the out-of-control state in interval (jh, (j + 1)h)j = 0 , . . . ,m, and a failure occurs in interval (ih, (i + 1)h)i = j , . . . ,m
m+1 1 m
Process shifts to the out-of-control state in interval (jh,(j + 1)h) j = 0 , . . . ,m, and it is not detected until T Process shifts to the out-of-control state in interval (jh, (j + 1)h)j = 0 , . . . ,(m 1), and it is detected in the ith sampling interval i = (j + 1) , . . . ,m
m+1
ðmþ1Þðmþ2Þ 2 ðmÞðmþ1Þ 2
Table 4 Occurrence probability of each scenario in the integrated model for m = 1. Scenario
Symbol
Scenario 1
p(S11)
Scenario 2
p(S2)
ðe2hk Þðe2hc1 Þpn
Scenario 3
p(S3)
ðehk Þðehc1 Þpf
Scenario 4
p(S41)
kðe Þ ðkþc1 c2 Þh ð1 eðkþc1 Þh Þ kþ Þ c1 c2 ð1 e ð1 eðkþc1 c2 Þh Þðec2 h ec2 2h Þ c 2h 2 Þ ðkþc1 c2 Þh pn kþkc ðeðkþc1 Þh eðkþc1 Þ2h Þ pn kðe eðkþc1 c2 Þ2h Þ kþc c ðe
p(S12)
Probability of occurence c1 ðkþc1 Þh Þ kþc1 ð1 e pn kþc1c ðeðkþc1 Þh eðkþc1 Þ2h Þ 1
p(S42) p(S43)
c2 h
k kþc1
pu ðkÞ kþc1 c2 1
Scenario 5
p(S51) p(S52)
Scenario 6
p(S6)
equal to (m + 1)h. In the third and sixth scenarios, the expected cycle time is the time interval between the start of the cycle to the time when an alarm is generated. For scenario 4 in which the equipment shifts to the out-of-control condition before the cycle ends, expected value of the cycle time is calculated by adding up the expected value for random variables X and Z. For example, the expected value for the first state of scenario 1 is calculated as follows.
pðS1ðmþ1Þ Þ pðS11 Þ þ þ EðTjS1ðmþ1Þ Þ pðS1 Þ pðS1 Þ
2
and third scenarios, process does not shift to the out-of-control condition and the following equation can be used to calculate the expected cost of quality loss:
Eðcos t of quality lossÞ ¼ C IN EðTjSi Þ i ¼ 1; 2; 3
ð6Þ
In the fourth, fifth, and sixth scenarios in which the equipment is in-control for X time unit, the expected value for the cost of quality loss is calculated as follows:
Eðcos t of quality lossÞ ji X pðSir Þ ¼ ðC IN EðXjSir ÞÞ þ C OUT ðEðTjSir Þ EðXjSir ÞÞ ; pðSi Þ r¼1 ð4Þ
After calculating cycle time for each state of scenario 1, cycle time for this scenario is calculated as
EðTjS1 Þ ¼ EðTjS11 Þ
2
ð1 eðkþc1 c2 Þh Þ c ð2hÞ kðe 2 Þ pn ðeðkþc1 c2 Þh eðkþc1 c2 Þ2h Þ kþc c 1c h 2 kðe 2 Þ ðkþc1 c2 Þh Þ p d ð1 e kþc c 1
Rh R1 ðYÞðkekX Þðc1 ec1 X Þdx dy 0 y EðTjS11 Þ ¼ EðYjY h; Y XÞ ¼ R h R 1 ðkekX Þðc1 ec1 X Þdx dy 0 y 1 ðc1 þkÞh h þ c 1þk c1 þk e 1 ¼ ð1 eðc1 þkÞh Þ
1
kðec2 ð2hÞ Þpu kþc1 c2
ð5Þ
The same analogy is used to calculate expect cycle time for other scenarios. 5.3. Calculation of the expected cycle cost for each scenario The cycle cost consists of three important components namely the cost of quality loss, the cost of sampling, and the cost of investigating alarms and performing maintenance. It is assumed that time for sampling and plotting a sample on the control chart could be ignored. We assume that process stops during the time when a search begins for an assignable cause and performing maintenance. In the in-control condition, cost of quality loss is calculated by multiplying CIN by the expected value of in-control time. In the out-of-control condition, this cost is obtained by multiplying COUT by the expected value of out-of-control time. In the first, second,
i ¼ 4; 5; 6
ð7Þ
where ji is the state number for the ith scenario. The cost of sampling consists of two components namely the fixed cost of sampling and the variable cost of sampling. The fixed cost of sampling (CF) is the setup cost of test equipments and the variable cost of sampling (CV) is the inspection cost of each unit. Hence, the expected value for cost of sampling is calculated as follows.
Eðcost samplingjSi Þ ¼ ðC F þ C V nÞ Eðnumber of samplingjSi Þ i ¼ 1; . . . ; 6
ð8Þ
The expected number of samples for each scenario can be calculated using all the states of that scenario. If ji is the number of states in scenario i and Nir is the number of times sampling is done in the rth state of this scenario then the expected number of times sampling is performed in scenario i is calculated as follows:
Eðnumber of samplingjSi Þ ¼
ji X pðSir Þ Nir i ¼ 1; . . . ; 6 pðSi Þ r¼1
ð9Þ
The maintenance cost associated with scenarios 1 through 6 are equivalent to CC, CP, CU, CC, CPM, and CPM, respectively. This cost is
1251
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255 Table 5 Expected cycle time of each scenario in the integrated model for m = 1. Scenario
Symbol
Scenario 1
E(T|S11)
Expected time per cycle 1 ðkþc1 Þh h þ kþ1c =ð1 eðkþc1 Þh Þ þ T C kþc e 1
Scenario 2 Scenario 3 Scenario 4
1
E(T|S12)
ðhðeðkþc1 Þh Þ 2hðeðkþc1 Þ2h Þ þ kþ1c ðeðkþc1 Þh eðkþc1 Þð2hÞ ÞÞ=ðeðkþc1 Þh eðkþc1 Þð2hÞ Þ þ T C
E(T|S2) E(T|S3) E(T|S41)
2 h + TP h + TU 02
1
0 1 hþc1 kec2 h c h 1 2 ðkc2 Þh 1 1 kX @ Að1 eðkc2 Þh Þ @41 ekh h þ 1 kðe 2 Þ e h þ Þ þ ð1 e kc2 c2 kc2 k k kc2 kc2
c h c h 2 1 ðkc2 Þh h þ k1c =ð1 ekh Þ kðekc Þ ð1 eðkc2 Þh ÞÞ þ T C þ kekc2 kc2 e 2 2 2 h c h c 2h ðkc Þh 2 2 kðe e Þ 2 1 1e ðkc2 Þh h þ k1c þ kðec2 h Þ h þ c1 kðec2 2h Þ 2h þ c1 kc2 kc2 kc2 e 2 2 2 c 2h c h ðkc Þh kðe 2 e 2 Þ 2 1 ðkc2 Þh 1 c2 h c2 ð2hÞ 1e h þ kc e Þ =kðe Þ þ TC þ kc2 kc2 e kc2 2 h c 2h 2 hðekh Þ 2hðek2h Þ þ 1k ðekh ek2h Þ kðekc Þ hðeðkc2 Þh Þ 2hðeðkc2 Þ 2hÞ þ k1c 2 2 0 1 2hþc1 kec2 2h kðec2 2h Þ 2 ðkc2 Þh ðkc2 Þ2h 1 kh kð2hÞ ðk c Þh ðk c Þ2h A e 2 2 þ kc e ÞÞ þ c ðe e Þ@ e ðh ðeðkc2 Þh Þ ðe kc
E(T|S42)
E(T|S43)
2
2
2
2hðeðkc2 Þ2h Þ þ k1c ðeðkc2 Þh eðkc2 Þ2h Þ=ðekh ekð2hÞ Þ 2 c 2h 2 kðekc Þ ðeðkc2 Þh eðkc2 Þ2h ÞÞ þ T C 2
Scenario 5
E(T|S51) E(T|S52) E(T|S6)
Scenario 6
2 h + TPM 2 h + TPM h + TPM
Table 6 Expected cycle cost of each scenario in the integrated model for m = 1. Scenario
Symbol
Expected cost per cycle
Scenario 1
E(C|S1)
12 Þ C IN EðTjS1 Þ þ ðC F þ C V nÞ pðS pðS1 Þ þ C C
Scenario 2 Scenario 3 Scenario 4
E(C|S2) E(C|S3) E(C|S4)
CIN (2h) + (CF + CV n) + CP CIN E(T|S3) + (CF + CV n) + CU
Scenario 5
E(C|S5)
pðS51 Þ pðS5 Þ
Scenario 6
E(C|S6)
CIN E(X|S6) + COUT (h - E(X|S6)) + (CF + CV n) + CPM
pðS42 Þ 41 Þ ðC IN EðXjS41 Þ þ C OUT EðZjS41 ÞÞ pðS pðS Þ þ ðC IN EðXjS42 Þ þ C OUT EðZjS42 ÞÞ pðS4 Þ þ ðC IN EðXjS43 Þ þ C OUT EðZjS43 ÞÞ 4 pðS42 Þ pðS43 Þ 43 Þ pðS pðS4 Þ þ ðC F þ C V nÞ pðS4 Þ þ pðS4 Þ þ C C pðS52 Þ 52 Þ ðC IN EðXjS51 Þ þ C OUT ð2h EðXjS51 ÞÞÞ þ pðS pðS5 Þ ðC IN EðXjS52 Þ þ C OUT ð2h EðXjS52 ÞÞÞ þ ðC F þ C V nÞð pðS5 Þ Þ þ C PM
added to the cost of quality loss and cost of sampling for each scenario to obtain the expected cost for each scenario. The probability, the expected cycle time, and the expected cycle cost of each scenario for m = 1 with parameters k, c1, and c2 are presented in Tables 4–6. 6. The planned maintenance model In this model, planned maintenance is done on the equipment every T time units. Since no inspection is performed in this model, we can set zero for the value of Pd, Pf and one for the value of Pu, Pn in the integrated model. This eliminates the two scenarios associated with signals in the control chart. Finally, the model could be expressed as follow. Scenario 1: Equipment remains in-control but stops before time T due to failure. However, corrective maintenance restores the equipment to the in-control condition. Scenario 2: Equipment remains in-control until time T. Then the planned maintenance is performed and the equipment is restored to the in-control condition. Scenario 3: Equipment shifts to the out-of-control condition. In the out-of-control condition, process fails before time T due to increase in the failure rate. However, corrective maintenance brings the equipment to the in-control condition.
Scenario 4: Equipment shifts to the out-of-control condition. At time T, preventive maintenance is performed instead of planned maintenance in order to restore the process to its primary conditions. Since the model possesses the conditions for the renewalreward process, cost per time unit could be computed using the following equation:
EPM ½cost per time unit ¼
EPM ðcycle costÞ EPM ðcycle timeÞ
ð10Þ
where
EPM ½cycle cost ¼ E½cycle costjS1 pðS1 Þ þ E½cycle costjS2 pðS2 Þ þ E½cycle costjS3 pðS3 Þ þ E½cycle costjS4 pðS4 Þ EPM ½cycle time ¼ E½cycle timejS1 pðS1 Þ þ E½cycle timejS2 pðS2 Þ þ E½cycle timejS3 pðS3 Þ þ E½cycle timejS4 pðS4 Þ Exact solution method and assumptions of the integrated model holds here. Probability associated with each scenario, expected cycle time, and expected cycle cost per scenario for m = 1 are presented in Tables 7–9, respectively.
1252
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
Table 7 Occurrence probability of each scenario in the planned maintenance model for m = 1. Scenario
Symbol
Scenario 1
p(S11)
Probability of occurance c1
kþc1
p(S12) Scenario 2
p(S2)
Scenario 3
p(S31)
c2 h
kðe Þ ðkþc1 c2 Þh ð1 eðkþc1 Þh Þ kþ Þ c1 c2 ð1 e k ðkþc1 c2 Þh c2 h c2 2h Þðe e Þ kþc c ð1 e 1 2 c 2h kðe 2 Þ k ðkþc1 Þh ðkþc1 Þ2h ðkþc1 c2 Þh e Þ eðkþc1 c2 Þ2h Þ ðe ðe kþc1 c2 kþc 1c ð2hÞ 2 kðe Þ ð1 eðkþc1 c2 Þh Þ kþc c c1 ð2hÞ2 kðe 2 Þ ðeðkþc1 c2 Þh eðkþc1 c2 Þ2h Þ kþc c
k kþc1
p(S32) p(S33) p(S41)
Scenario 4
ð1 eðkþc1 Þh Þ
c1 ðkþc1 Þh eðkþc1 Þ2h Þ kþc1 ðe 2hk 2hc1 ðe Þðe Þ
p(S42)
1
2
Table 8 Expected cycle time for each scenario in the planned maintenance model for m = 1. Scenario
Symbol
Scenario 1
E(T|S11)
Expected time per cycle 1 ðkþc1 Þh h þ kþ1c =ð1 eðkþc1 Þh Þ þ T C kþc e 1
Scenario 2 Scenario 3
1
ðhðeðkþc1 Þh Þ 2hðeðkþc1 Þ2h Þ þ kþ1c ðeðkþc1 Þh eðkþc1 Þð2hÞ ÞÞ= eðkþc1 Þh eðkþc1 Þð2hÞ þ T C
E(T|S12)
1
E(T|S2) E(T|S31)
2 h + TP h c2 h 1 kh h þ 1k kðekc Þ k1c eðkc2 Þh h þ k1c þ c1 ð1 ekh Þ ke 2 2 2 2 0 1 c2 h 1 c h hþc ke c2 h 2 2 kðe Þ 1 ðkc2 Þh Að1 eðkc2 Þh Þ þ @ h þ k1c =ð1 ekh Þ kðekc Þ ð1 eðkc2 Þh ÞÞ þ T C kc kc kc e 2
2
E(T|S33)
kðec2 h ec2 2h Þ kc2
ðkc2 Þh
1 kc2
2
Scenario 4
2
2
1 kc2
2
ðkc Þh e hþ Þ h þ c1 kðec2 2h Þð2h þ c1 Þ 1ekc 2 þ kðe 2 2 2 c 2h c h ðkc Þh 2 2 Þ 1 ðkc2 Þh h þ k1c =kðec2 h ec2 ð2hÞ Þ 1ekc 2 Þ þ T C þ kðe ke c2 kc2 e 2 2 c 2h h 2 hðekh Þ 2hðek2h Þ þ 1k ðekh ek2h Þ kðekc Þ hðeðkc2 Þh Þ 2hðeðkc2 Þ2h Þ 2 0 1 2hþc1 kec2 2h 2 1 ðkc2 Þh ðkc2 Þ2h 1 kh kð2hÞ @ Aðeðkc2 Þh eðkc2 Þ2h Þ e ÞÞ þ c ðe e Þ þ kc ðe kc h
E(T|S32)
c2 h
2
2
c 2h 2 þ kðekc Þ ðhðeðkc2 Þh Þ 2hðeðkc2 Þ2h Þ þ k1c ðeðkc2 Þh eðkc2 Þ2h Þ=ðekh ekð2hÞ Þ kðekc Þ ðeðkc2 Þh eðkc2 Þ2h ÞÞ þ T C 2 2 2 2 h + TPM 2 h + TPM
E(T|S41) E(T|S42)
c2 2h
Table 9 Expected cycle cost for each scenario in the planned maintenance model for m = 1. Scenario
Symbol
Expected cost per cycle
Scenario 1
E(C|S1)
12 Þ C IN EðTjS1 Þ þ ðC F þ C V nÞ pðS pðS1 Þ þ C C
Scenario 2 Scenario 3
E(C|S2) E(C|S3)
CIN (2h) + (CF + CV n) + CP
Scenario 4
E(C|S4)
pðS32 Þ 31 Þ ðC IN EðXjS31 Þ þ C OUT EðZjS31 ÞÞ pðS pðS Þ þ ðC IN EðXjS32 Þ þ C OUT EðZjS32 ÞÞ pðS3 Þ þ ðC IN EðXjS33 Þ þ C OUT EðZjS33 ÞÞ 3 pðS33 Þ pðS32 Þ pðS33 Þ pðS3 Þ þ ðC F þ C V nÞ pðS3 Þ þ pðS3 Þ þ C C pðS41 Þ pðS42 Þ pðS42 Þ pðS4 Þ ðC IN EðXjS41 Þ þ C OUT ð2h EðXjS41 ÞÞÞ þ pðS4 Þ ðC IN EðXjS42 Þ þ C OUT ð2h EðXjS42 ÞÞÞ þ ðC F þ C V nÞ pðS4 Þ þ C PM
Table 10 Parameter value for the numerical example. Parameter
COUT
CV
CF
c2
c1
k
CIN
CU
CC
CPM
CP
TU
TC
TPM
TP
Low High
100 300
0.2 0.8
8 12
0.15 0.25
0.02 0.07
0.05 0.15
8 12
400 600
8000 12,000
1500 2500
800 1200
0.1 0.4
7 9
2 3
0.7 1.3
7. Numerical analysis and sensitivity analysis
7.1. Numerical example
In this section, the sensitivity of the integrated model parameters with the expected cost per time unit is investigated using a numerical example. Then, the integrated model is compared with the planned maintenance model.
Spot welding using welding gun is widely used in automotive industry. To ensure proper welding gun operation, planned maintenance at specified time is performed. However, due to some factors such as electrode erosion or input voltage drop, the welding
1253
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
99 E
95
J
Factor A B C D E F G H J K L M N O P
F
90
D
80
Percent
Effect Type Not Significant Significant
70 60 50 40 30 20 10 5
1 -50
0
50
100
150
Effect Fig. 3. Normal probability plot of the effects. Gama2
Gama1
560 520
Expected Cost
480 440 400 0.15
0.25
0.02
0.07
Lambda
Cc
560 520 480 440 400 0.05
0.15
8000
12000
(a) integrated model Gama2
600
Gama1
550
Expected Cost
500 450 0.15
0.25
0.02
Lambda
600
0.07 Cc
550 500 450 0.05
0.15
8000
(b) Planned maintenance model Fig. 4. Main effects plots.
12000
Name Cout Cv Cf Gama2 Gama1 Lambda Cin Cu Cc Cpm Cp Tu Tc Tpm Tp
1254
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255
the integrated model has better economic behavior than the planned maintenance model.
800
Expected Cost
700 600
8. Conclusions
500
There exists limited number of models in the literature which use the concept of statistical process control and maintenance to develop integrated models. This paper considers a process with a single equipment to develop an integrated model. We assumed that monitoring quality of products is synonymous to monitoring status of the equipment and as the result an out-of-control signal would indicate deterioration of the equipment. It is presumed that performing any maintenance restores the equipment to the ‘‘goodas-new’’ condition. Through a numerical example, effects of the model parameters on the average cost are studied and it is shown that integrated model inquires lower cost than the planned maintenance model. For simplicity, exponential distributions for the times to failure are assumed. However, further research could be conducted on the subject assuming Weibull or a more general distribution for the time to failure or out-of-control condition. The issue of two or more out-of-control conditions and a multiple equipment manufacturing process is also an issue which requires further research.
400 300 200 100 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Integrated Model Planned M Model
Fig. 5. comparing cost of the integrated model to the planned maintenance model.
gun might shift to the out-of-control condition with higher failure rate. In order to control the welding gun condition, weld quality of products is monitored. To achieving this purpose, every three time units (h = 3), a sample of five products is taken from the process and the mean of spot welding resistances as a quality characteristic is calculated and plotted on a X control chart. Quality characteristic of interest in the in-control condition follows a normal distribution with l0 = 750 and r2 = 2500. In the out-of-control condition, process mean shifts to l1 = 800 while process variance remains unchanged. Time for transition from the in-control to the out-of-control condition is governed by an exponential distribution with parameter k. For the sake of simplicity, we assume that inspection is performed only once during the planned maintenance interval (m = 1). Equipment failure rate in the in-control and out-of-control conditions are equal to c1 and c2, respectively. Table 10 shows the high and low levels of the model parameters which are used in model sensitivity analysis.
7.2. Sensitivity analysis A designed experiment was used to study the effects of the model parameters on the expected cost per time unit. Employing a fractional factorial design with resolution IV requires 32 experimental runs for the 15 model parameters. Fig. 3 presents the normal plot of the effect estimates with expected cost per time unit as the response variable. These effects estimates are shown against the normal quintile error. The negligible effects lie close to the line while the large ones deviate from the line. This figure illustrates that CC, k, c1, and c2 have significant effects and increasing the value of each results in a significant increase in the cost.
7.3. Comparing the models To investigate the performance of the planned maintenance model with four parameters (CC, k, c1, and c2) and to compare the integrated model with the planned maintenance model, we employ a full factorial design. This design needs 16 experimental runs. The low and high values for these effective parameters are the same as the values in Table 10. For non-effective parameters, the mean value between the low and high values is set in the experiments. Fig. 4 shows main effects plots for the integrated and planned maintenance models. Hence, main effects of the integrated model seem effective in the planned maintenance model too. The cost of two models are compared in Fig. 5 which illustrate
References Ben-Daya, M. (1999). Integrated production maintenance and quality model for imperfect process. IIE Transactions, 31, 491–501. Ben-Daya, M., & Duffuaa, S. O. (1995). Maintenance and quality: The missing link. Journal of Quality in Maintenance Engineering, 1, 20–26. Ben-Daya, M., & Rahim, M. A. (2000). Effect of maintenance on the economic design of x-control chart. European Journal of Operation Research, 120, 131–143. Cassady, C. R., Bowden, R. O., Liew, L., & Pohl, E. A. (2000). Combining preventive maintenance and statistical process control: a preliminary investigation. IIE Transactions, 32, 471–478. Charongrattanasakul, P., & Pongpullponsak, A. (2011). Minimizing the cost of integrated systems approach to process control and maintenance model by EWMA control chart using genetic algorithm. Expert Systems with Applications, 38, 5178–5186. Duncan, A. J. (1956). The economic design of x-bar charts used to maintain current control of a process. Journal of the American Statistical Association, 52, 228–242. Ho, C., & Case, K. E. (1994). Economic design of control charts: A literature review for 1981–1991. Journal of Quality Technology, 26, 39–53. Katter, J., Tu, J., Monacelli, L. (1997). Gartner M. Predictive maintenance of an industrial laser using statistical process control charting. In Proceedings of the 1997 laser materials processing conference (pp. C57 ± C66). Kniele, R., Stephens, G., Vasudeva, K. (1989). Performance assessment and statistical process control. An approach to operation and maintenance cost reduction. In Proceeding of the IFAC/IFORS/IAEE symposium on energy, systems, management and economics (pp. 215 ± 221). Lee, B. H., & Rahim, M. A. (2001). An integrated economic design model for quality control, replacement, and maintenance. Quality Engineering, 13, 581–593. Linderman, K., McKone-Sweet, K. E., & Anderson, J. C. (2005). An integrated systems approach to process control and maintenance. European Journal of Operation Research, 164, 324–340. Lochner, R. (1987). Improved SPC using reliability concepts. In: 41st annual quality congress transactions (pp. 648 ± 653). McCall, J. J. (1965). Maintenance policies for stochastically failing equipment: A survey. Management Science, 11, 493–524. McKone, K., & Weiss, E. N. (1998). Total productive maintenance: Bridging the gap between practice and research. Production Operations Management, 7, 335–351. Montgomery, D. C. (1980). Economic design of control charts: A review of literature survey. Journal of Quality Technology, 12, 40–43. Montgomery, D. C. (2001). Introduction to statistical quality control (fourth ed.). New York: Wiley. Panagiotidou, S., & Nenes, G. (2009). An economically designed, integrated quality and maintenance model using an adaptive Shewhart chart. Reliability Engineering and System Safety, 94, 732–741. Patè-Cornell Lee, H. L., & Tagaras, G. (1987). Warning of malfunction: The decision to inspect and maintain production processes on schedule or on demand. Management Science, 33, 1277–1290. Pierskalla, W. P., & Voelker, J. A. (1976). A survey of maintenance models: The control and surveillance of deteriorating systems. Naval Research Logistics Quarterly, 23, 353–388. Porteus, E. L., & Angelus, A. (1997). Opportunities for improved statistical process control. Management Science, 43, 1214–1228. Rahim, M. A. (1994). Joint determination of quantity, inspection, schedule and control chart design. IIE Transactions, 26, 2–11.
Z. Mehrafrooz, R. Noorossana / Computers & Industrial Engineering 61 (2011) 1245–1255 Rahim, M. A., & Banerjee, P. K. (1993). A generalized model for the economic design of x-bar control chart for production systems with increasing failure rate and early replacement. Navel Research Logistics, 40, 787–809. Rahim, M. A., & Ben-Daya, M. (2001). Integrated production, quality and maintenance models: An overview. In M. A. Rahim & M. Ben-Daya (Eds.), Integrated models in production planning, inventory, quality, and maintenance. Kluwer Academic Publishers. Ross, S. (1996). Stochastic Processes (second ed.). New York: Wiley. Tagaras, G. (1988). An integrated cost model for the joint optimization of process control and maintenance. The Journal of the Operational Research Society, 39, 757–766. Tapiero, C. S. (1986). Continuous quality production and machine maintenance. Naval research Logistics Quarterly, 33, 489–499. Valdez-Flores, C., & Feldman, R. M. (1989). A survey of preventative maintenance models for stochastically deteriorating single-unit systems. Naval Research Logistics Quarterly, 36, 419–446.
1255
Wang, H. (2002). A survey of maintenance policies of deteriorating systems. European Journal of Operation Research, 139, 469–489. Xie, M., Goh, T. N., & Ranjan, P. (2002). Some effective control chart procedures for reliability monitoring. Reliability Engineering & System Safety, 77, 143–150. Yeung, T. G., Cassady, C. R., & Schneider, K. (2008). Simultaneous optimization of xbar control chart and age-based preventive maintenance policies under an economic objective. IIE Transactions, 40, 147–159. Zhang, C. W., Xie, M., & Goh, T. N. (2005). Economic design of exponential charts for time between events monitoring. International Journal of Production Research, 43, 5019–5032. Zhang, C. W., Xie, M., Liu, J. Y., & Goh, T. N. (2007). A control chart for the Gamma distribution as a model of time between events. International Journal of Production Research, 45, 5649–5666. Zhou, W. H., & Zhu, G. L. (2008). Economic design of integrated model of control chart and maintenance management. Mathematical and computer Modelling, 47, 1389–1395.