An Integrated Model of Production Scheduling, Maintenance and Quality for a Single Machine

Journal Pre-proofs An Integrated Model of Production Scheduling, Maintenance and Quality for a Single Machine Salih Duffuaa, Ahmet Kolus, Umar Al-Turki, Ahmed El-Khalifa PII: DOI: Reference:

S0360-8352(19)30708-9 https://doi.org/10.1016/j.cie.2019.106239 CAIE 106239

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Computers & Industrial Engineering

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20 May 2019 21 October 2019 18 December 2019

Please cite this article as: Duffuaa, S., Kolus, A., Al-Turki, U., El-Khalifa, A., An Integrated Model of Production Scheduling, Maintenance and Quality for a Single Machine, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie.2019.106239

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An Integrated Model of Production Scheduling, Maintenance and Quality for a Single Machine Salih Duffuaa1, Ahmet Kolus2, Umar Al-Turki3 and Ahmed El-Khalifa4 Department of Systems Engineering King Fahd University of Petroleum and Minerals Dhahran, 31261, Saudi Arabia [email protected] 1, [email protected], [email protected], [email protected]

Abstract It is well known that production scheduling, maintenance and process control decisions affect each other. Hence it is both essential and prudent to optimize them jointly. This paper develops a model that integrates and optimizes production, maintenance and process control decisions simultaneously for a single machine. Both types of maintenance, preventive and corrective, are taken into consideration. The methodology starts by developing an optimal preventive maintenance schedule. Then an integrated model that determines the decision variables and optimizes the total cost per unit time, resulting from production scheduling, inventory holding and maintenance and process control is developed. The utility of the model is demonstrated using an example from the literature. Systematic sensitivity analysis demonstrated that the model results are robust. Also; the model results are compared with the results of optimizing each system separately and with two-way integration. The comparisons indicate that the developed model results in a saving that ranged from 2.62 to 6.78 percent. The model provides a viable approach for optimizing production scheduling, maintenance and quality jointly. Keywords: production, maintenance, quality, integration and optimization 1. Introduction In today’s competitive environment organizations are facing difficult challenges due to globalization, shorter product life cycles and rapid technological and industrial developments.. To attain a competitive edge, organizations strive to improve the 1

performance of their production systems. Nowadays, modern production systems tend to optimize the functions (i.e., planning and scheduling) of their major elements, namely production, maintenance and quality. Maintaining production in good operating conditions, preserving high quality standards, and developing sound production planning are vital to meeting customer demands and adopting new management systems, (Crosby, 1984; Miltenburg, 1993). Understanding interactions and trade-offs between these elements is essential for sound decision-making (Ben-Daya and Rahim, 2001). Figure 1 shows the relationship between the major elements in a production system. Preventive maintenance (PM) minimizes the deterioration of equipment (by minimizing wear and breakdowns) but interrupts the production schedule resulting in a delay for customer orders. In order to meet customer demand, production schedulers may decide to utilize machines to their full capacity. This may increase productivity, but it will also decrease machine availability (due to machine failure) and product quality (due to machine degradation). Information related to the production process provides the knowledge to enhance process capability by removing the factors that cause the process to deteriorate. Also, having products with poor quality means time will be allocated for rework, which could be used otherwise for production.

Maintenance

Production system

Production

Quality

Customer Demand

Figure 1. Relationship between production, maintenance and quality The common practice in industry is to plan these elements independently through separate functional teams. This may result in an optimal plan for a specific function, but not for the 2

whole production system. Independent planning of major functions in the production system results in a suboptimal solution due to the dependency between its elements. Thus, there is a need to optimize the production system by considering the dependency between its elements. This justifies the need to develop a model that integrates and jointly optimizes the three elements. Integrated planning has gained the interest of researchers over the last three decades. Early attempts were made on the basis of the two-way integration of functions in production systems, such as: production and maintenance planning (Srinivasan and Lee, 1996; Goel et al., 2003; Ben-Daya and Noman, 2006; El-Ferik, 2008), production and maintenance scheduling (Cassady and Kutanoglu, 2003, 2005; Yulan et al., 2008; Berrichi et al., 2010), and maintenance and quality control (Banerjee and Rahim, 1988; Chiu and Huang, 1995; Ben-Daya and Rahim, 2000; and Linderman et al., 2005; Mehdi et al., 2010; Duan et al., 2018). Although plans based on two-way integration improve system performance, they are sub-optimal since they do not account for all major elements. A natural extension is to develop plans based on three-way integration. By the three-way integration of production, maintenance and quality, all major functions in the production system can be optimized simultaneously resulting in a global optimal solution. Integrated production models give organizations an opportunity to attain multiple objectives with a conflicting nature and hence maximize the performance of their production system. Some attempts can be found in the literature for the joint modeling of production, maintenance and quality (Huang and Chiu, 1995; Makis and Fung, 1995; Tseng et al., 1998; Rahim and Ben-Daya, 1998, 2001; Chelbi et al., 2008; and Pandy et al, 2011). Yet there are still major gaps in the literature and there is considerable room for improvement. The majority of the existing models assume a fixed cost for corrective maintenance (CM), which is unrealistic. In reality, the cost of CM does not depend only on the repair cost, but also on the cost of the downtime period during repair and the cost associated with the state of the machine after repair. Also, existing models assume that system failure results only in total shutdown, ignoring the fact that it may also result in performance degradation and hence quality reduction. Filling these gaps in the existing literature will enhance the level of integration of the production system elements and offer realistic integrated models. 3

This research is motivated by the growing need to improve the performance of production systems and reduce costs to stay competitive in the global marketplace. The objective of the paper is to develop a state-of-the-art cost model for the joint optimization of production scheduling, maintenance planning and quality control at the shop floor level for a single machine. The proposed model is expected to provide savings in operational costs, as well as improvements in product quality and resource utilization. The problem under consideration has three main components: maintenance (preventive and corrective) planning for the machine, scheduling the batches for production, and monitoring the production process for quality. The rest of the paper is organized as follows: section 2 reviews the related work on the integration of production, maintenance and quality. Section 3 presents the statement of the problem followed by the model development in Section 4. Section 5 presents the solution methodology of the integrated model. The results and sensitivity analysis are discussed in Section 6, followed by the conclusion and future directions for research in Section 7. 2. Literature review The triple integration of production, maintenance and quality has recently gained an increased interest since it accounts for major elements in the production system. Many modeling attempts considered the integration between production and quality under different PM policies (Huang and Chiu, 1995; Makis and Fung, 1995; Tseng et al., 1998). Huang and Chiu (1995) studied the impact of an imperfect production process on the optimal production cycle time. The production system was assumed to deteriorate during the production process and produce defective items. Two monitoring policies were developed for the planning of production, inspection and maintenance. The goal was to determine the perfect manufacture cycle duration while lowering the cost for the two policies (with/without PM). In 1995 Makis and Fung developed a model that jointly optimizes lot size, inspection interval and preventive replacement time for a production facility with random failure. The time when the system was in-control was exponentially distributed and defective items 4

were produced once the system was out of control. A periodic inspection was performed on the facility until it was replaced after a certain number of production runs. Makis and Fung (1998) extended this work to include the effect of machine failure. Tseng et al. (1998) determined the economic manufacturing quantity (EMQ) for a deteriorating production system under two imperfect maintenance policies: equally spaced and equally cumulative hazards. The optimal maintenance policy was evaluated using a numerical example. Rahim and Ben-Daya (1998) proposed a generalized model for the joint determination of production quantity, inspection schedule and control chart design under a continuous production process. Ben-Daya (1999) developed a model for the joint determination of the economic production quantity (EPQ), the optimal maintenance level and the economic design of the process mean control chart. He assumed that the production process was deteriorating and the in-control period followed a general probability distribution. Rahim and Ben-Daya (2001) investigated the impact of the deteriorating production process and products on the EPQ, control chart design parameters and inspection schedule. In another work they reviewed the literature related to the integration of production, maintenance and quality (Ben-Daya and Rahim, 2001). Chelbi et al. (2008) proposed a strategy for the joint determination of production lot size and PM schedule for an unreliable system with conforming and non-conforming items. They assumed that the production system has a single machine and needs to satisfy continuous and constant demands. After a random period of operation, non-conforming units were produced at a certain rate. Pandey et al. (2010) reviewed the literature addressing the integration of scheduling, maintenance and quality at the shop floor level. Also, a conceptual framework related to this integration was proposed for future development in this field. Rahim and Shakil (2011) considered the joint determination of EPQ, PM and the economic design of an x-bar control chart under non-uniform quality control parameters. They assumed a deteriorating process and a general probability distribution for in-control states. The Tabu search algorithm was used to determine the optimal quality control parameters for different PM levels. Pandey et al. (2011) proposed two models for the joint optimization of maintenance planning, process quality and scheduling. The first model jointly optimizes the PM level and control chart parameters that minimize cost per unit time. The second 5

model integrates the optimal PM interval and production schedule to determine the optimal batch sequence. Hadidi et al. (2011) reviewed the literature on the models addressing the relationship between production, maintenance and quality. They divided existing models into interrelated and integrated models. A theory to analyze the production of conforming items in a deteriorating production system undergoing PM was developed by Colledani and Tolio (2012). The results showed that joint analysis of production, maintenance and quality improves system performance. Haoues et al. (2013) proposed an integrated approach that considers production, quality and maintenance to identify the best outsourcing contractor in a system composed of a single manufacturer and multiple contractors. The proposed model was solved using a genetic algorithm based optimization. Tambe and Kulkarni (2014) developed a model that jointly optimizes production, maintenance and quality control policies to minimize the total operational cost. A simulated annealing technique was used to solve the integrated model. Fakher et al. (2014) integrated production planning with PM scheduling taking into consideration the quality of the production system. A mathematical model was developed and evaluated with respect to various cost components. Lu, Zhou and Lu, et al. (2016) jointly modeled EPQ maintenance and quality in a deteriorating single-machine production system. Nourelfath and et al. (2016) developed an integrated model of production, maintenance and quality in a multistate production system where machines could randomly degrade. Salmasnia et al. (2017) proposed a joint design of production run length, maintenance policy and control chart in the presence of multiple assignable causes. Fakher et al. (2017) integrated the planning of production, maintenance and process inspection in a multi-machine system that may degrade with a higher defective rate or to a failed state. The integrated model was solved using the genetic and Tabu search algorithms. 3. Statement of the problem Consider a production system with a single machine producing products of the same type in batches at a constant production rate on a continuous basis. The machine processes a set of n jobs as a batch of size N[i]. A batch i has a processing time p[i] and due date d[i]. The machine is subject to breakdown that can be divided into two failure types (Figure 2) 6

similar to the classification used by Lad and Kulkarni (2008): 1. Failure mode I (FMI): a machine failure that leads to immediate breakdown and stoppage of the machine, hence it can be detected immediately. A corrective maintenance will take place to restore the machine back to the operating condition, after which the disrupted job will be completed. This will result in an expected corrective maintenance cost (CMCFMI). 2. Failure mode II (FMII): a process failure that disturbs the functionality of the machine resulting in a shift in the process mean and hence in an increase in the process rejection rate. When the failure is detected, the process will be stopped immediately, and corrective actions will take place to restore the process back to the state before failure. This will result in an expected corrective maintenance cost (CMCFMII). In addition, the process may also deteriorate and shift to an out-of-control state due to external causes (E) such as environmental factors, operator's mistakes and incorrect tool usage. Similarly, when external causes are detected, the process will be restored back to the in-control state. The process time to failure is supposed to follow an exponential distribution as assumed in Duncan (1956).

Figure 2. Types of failure Identifying these types of failure (FMII and E) will take time since they do not result in immediate stoppage of the machine. Therefore, they are detected by monitoring the process 7

through a control chart mechanism. The X chart is used to monitor the quality characteristic of the finished product. The design variables of the chart are: 1. The time (length) between samples (h). 2. The sample size (n). 3. The number of standard deviations of the sample allocation that determines the distance between the middle of the chart and its limits (k). This will result in an expected total quality cost of process failure (TQC) owning to (FMII) and (E). Each failure mode of FMI and FMII will delay the completion times of successive batches by the needed time to perform corrective maintenance actions tCM (which are assumed to be constant) and will cause a fixed cost of performing repair actions (FCCM). Now, apart from the above corrective actions, preventive maintenance action is implemented to lower the frequency of failure occurrence. It is considered to be perfect which means it restores the machine to a new condition. This will result in an expected cost per preventive maintenance (PMC). Each PM batch we introduce will delay sequential batches by the time required to perform preventive maintenance actions tPM (which are assumed to be constant) and will cause a fixed cost of performing preventive maintenance action (FCPM). For all the batches produced its assumed that the raw materials are freed at the start of the sequence, and that the raw material inventory for a batch is at hold until it starts processing. Therefore, the inventory carrying (holding) cost for this period is computed based on the entire batch size and consists of the setup time and processing times of all the previous batches (if any), as well as the setup time of the current batch. After the processing of a batch starts, raw material will be consumed at a constant rate and accordingly the inventory carrying (holding) cost will be computed for this area based on half the batch size (average inventory). Each batch (i ≠ PM batch) waiting on the production line will cause a holding cost H per unit of time until its completion time i.e. E (C[i]). The problem under consideration is to schedule the batches for processing by integrating production scheduling, maintenance and quality control. Each of these operations (production scheduling, maintenance and quality) has a cost. The maintenance costs result from scheduling preventive maintenance and corrective maintenance cost. The quality cost results from the cost of appraisal, produced while the process is out of control and is a false 8

alarm. In addition, production scheduling may result in an inventory holding cost. The objective is to minimize the expected total cost per unit time for the three operations planned together. 4. Model development This section provides the details of the development of the integrated model. The first subsection states the notations employed for the model development followed by the model assumptions in subsection 4.2. Section 4.3 outlines the expected cost elements of the objective function for the integrated model. In subsections 4.4 -4.8 the derivations of the expected costs are derived and the complete integrated model is stated in subsection 4.9. 4.1

Notation

n

Number of jobs to be scheduled;

N[i]

Batch size;

xij

Job sequencing decision variable, this is a zero one variable;

p[i]

Processing time of ith job in the sequence;

d[i]

Due date of the ith job in the sequence;

PN[i]

Penalty of the ith job in the sequence;

C[i]

Completion time of the ith job in the sequence;

y[i]

PM batch decision variable;

θ[i]

Lateness of the ith job in the sequence;

η

Weibull scale parameter for probability distribution of T;

β

Weibull shape parameter for probability distribution of T;

tCM

Time required to perform corrective maintenance;

tPM

Time required to perform preventive maintenance;

τ

Preventive maintenance (PM) interval;

τ*

Optimal value of τ;

FCS

Fixed cost per sample;

FCJ

Fixed cost per job;

CR

Cost of rejection; 9

FCCM Fixed cost of corrective maintenance; Creset

Cost of resetting;

Cfalse Cost of searching for a false alarm per unit time; FCPM Fixed cost per preventive maintenance; Mean elapse time from the last sample before the assignable cause to the

𝜖

occurrence of assignable cause; LPC

Lost production cost;

LC

Labor cost per unit time;

PR

Production rate or production per unit time.

T0

Expected time consumed searching for a false alarm;

TS

Time to sample and chart a single component;

T1

Expected time to locate an assignable cause;

Treset

Expected time consumed to retest the process that moved to an out of control status to an in control state as a result of an external reason;

TSched Expected Scheduling period; Teval

Evaluation period;

4.2

Model assumptions The following are the assumptions of the model: 1. Machine failure leads to the following consequences: (i) Failure mode 1 (FM1) leads to immediate breakdown of the machine. (ii) Failure mode II (FM2) lead to reduction in process quality by shifting the process mean. 2. The quality characteristic measurement follows a normal random variable with mean μ and standard deviation σ. 3. The time between in control states is exponential. 4. The batch manufacturing time is the aggregate of the setup times and processing times of its member jobs. 5. A job cannot be preempted by another job. 10

6. The machine cannot be interrupted for PM until all the jobs in a batch are finished. 4.3 The objective function of the integrated model The objective function of the integrated model includes the following expected costs: 1. Expected cost of preventive maintenance denoted as PMC; 2. Expected cost of corrective maintenance denoted as 𝐶𝑀𝐶𝐹𝑀𝐼 3. Expected cost of the integrated production scheduling and maintenance dented as 𝑇𝑃𝐶𝑆𝑐ℎ𝑒𝑑𝑢𝑙𝑖𝑛𝑔 & 𝑀𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 4. Expected inventory holding cost denoted as 𝐻𝐶 5. Expected total cost of quality denoted as 𝐸(𝑇𝑄𝐶)𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 The objective function of the integrated model combines the above costs and is states in equation 1 below:

𝐸𝑇𝐶 =

𝑃𝑀𝐶 + 𝐶𝑀𝐶𝐹𝑀𝐼 + 𝐻𝐶 + 𝐸(𝑇𝑄𝐶)𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 + 𝑇𝑃𝐶𝑆𝑐ℎ𝑒𝑑𝑢𝑙𝑖𝑛𝑔 & 𝑀𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 𝑇𝑒𝑣𝑎𝑙

(1)

Next, the derivation of each component of the objective function of the integrated model is provided.

4.4 Expected cost of preventive maintenance Preventive maintenance (PM) is assumed imperfect, so the machine will be maintained to a better state but not as new. To estimate the expected cost of preventive maintenance the following cost parameters are considered: 1. The amount of time the machine is expected to be down each time the PM is performed 11

(tPM). 2. The down time cost during the PM estimated as a fixed cost plus the cost of lost production 3. The fixed cost of performing preventive maintenance (FCPM). 4. The number of preventive maintenances (NPM) during the evaluation period Teval. Therefore, the expected cost for preventive maintenance can be expressed as 𝑃𝑀𝐶 = 𝑁𝑃𝑀 ∗ [𝑡𝑃𝑀(𝐿𝑃𝐶 ∗ 𝑃𝑅 + 𝐿𝐶) + 𝐹𝐶𝑃𝑀]

(2)

4.5 Expected cost of corrective maintenance due to FMI To generate the expected cost of CM due to failure mode I denoted as FMI, the following parameters are considered: 1. The amount of time the machine is anticipated to be down every time CM is needed (tCM ). 2. The down time cost during the repair of the machine. 𝑡𝐶𝑀(𝐿𝑃𝐶 ∗ 𝑃𝑅 + 𝐿𝐶) 3. The fixed cost of performing corrective maintenance (FCCM). 4. The probability that the tool will break down due FMI , calculated as Since the failures are randomly distributed over the machine and the time to failure follows a two-parameter Weibull probability distribution having the shape and scale of B and η. The probability that the machine fails due to FMI in a given planning period Teval can be expressed as 𝑇𝑒𝑣𝑎𝑙 𝐵

𝑃𝐹𝑀𝐼 = 𝐹(𝑇𝑒𝑣𝑎𝑙;𝐵, 𝜂) = 1 ― 𝑒 12

(

𝜂

)

(3)

5. The number of failures during the period (0; Teval ), denoted by N(t), calculated for the Weibull distribution with the failure rate/hazard function 𝜏(𝑡)

𝜏(𝑡) =

𝐵 𝜂𝐵

𝑡𝐵 ― 1

Then, the expected number of machine failures during (0; 𝑇𝑒𝑣𝑎𝑙), is calculated as follows:

𝐸(𝑁(𝑃𝑀𝐼)) =

∫

𝑇𝑒𝑣𝑎𝑙

𝜏(𝑡) 𝑑𝑡 = 0

∫

𝑇𝑒𝑣𝑎𝑙 𝐵 0

𝐵―1

𝜂𝐵

𝑡

𝑑𝑡 =

𝑇𝑒𝑣𝑎𝑙

𝐵

( ) 𝜂

(4)

Therefore, considering the above parameters the CM cost due to FMI can be expressed as follows: 𝐶𝑀𝐶𝐹𝑀𝐼 = 𝑃𝐹𝑀𝐼 ∗ 𝐸[𝑁(𝑃𝑀𝐼)] ∗ [𝑡𝐶𝑀(𝐿𝑃𝐶 ∗ 𝑃𝑅 + 𝐿𝐶) + 𝐹𝐶𝐶𝑀]

(5)

4.6 Expected inventory holding cost Since we assumed that raw materials for every batch are freed to the shop at the beginning of the sequence, the raw material inventory for each batch is at hold until it begins processing. The manufacturer should consider the holding cost during the scheduling horizon. One batch of size N[i] consists of processing a set of jobs. Hence the batch size is 𝑛

𝑁[𝑖] =

∑𝑁 𝑥 𝑗

𝑖 = 1, 2, …, 𝑛

𝑖𝑗

𝑗=1

Two periods are there, one is before the processing of a batch in which the raw materials inventory for it are carried for the duration of the current batch setup time, as well as the setup and running times of all the previous batches (if any). Therefore the inventory holding cost will be calculated during this period for the whole batch size. While the batch is being processed, raw materials of the batch are consumed at a constant rate and accordingly the inventory carrying (holding) cost will be computed for this area based on half the batch 13

size (average inventory). Figure 3 shows the inventory holdings for batch i.

Figure 3: Inventory holdings for a single batch i Then, the average inventory quantity is calculated as

𝑁[𝑖] =

[

𝑁[𝑖] 𝐸(𝐶[𝑖]) ― =

[

[

[𝑝[𝑖] + 𝑡𝐶𝑀(𝑚(𝑎[𝑖]) ― 𝑚(𝑎[𝑖 ― 1][1 ― (𝑦[𝑖])]))] 2

]

𝐸(𝐶[𝑖])

= 𝑁[𝑖] 1 ―

= 𝑁[𝑖] 1 ―

𝑊𝑃 + 𝑃𝑃 𝐸(𝐶[𝑖])

[𝑝[𝑖] + 𝑡𝐶𝑀(𝑚(𝑎[𝑖]) ― 𝑚(𝑎[𝑖 ― 1][1 ― (𝑦[𝑖])]))] 2𝐸(𝐶[𝑖])

]

[𝑝[𝑖] + 𝑡𝐶𝑀(𝑚(𝑎[𝑖]) ― 𝑚(𝑎[𝑖 ― 1][1 ― (𝑦[𝑖])]))]

[

𝑖

2 ∑𝑘 = 1𝑡𝑃𝑀(𝑦[𝑘]) + (𝑝[𝑘]) + 𝑡𝐶𝑀[𝑚(𝑎[𝑘]) ― 𝑚(𝑎[𝑘 ― 1][1 ― (𝑦[𝑘])])]

14

]]

(26)

In conclusion, the model for the expected inventory holding cost is given as 𝑛

𝐻𝐶 = 𝐻 ∙

∑𝐸(𝐶

[𝑖])

𝑁[𝑖]

(27)

𝑖=1

4.7 Expected total cost of quality loss due to process failure In order to derive the expected total cost of quality per unit time, the process cycle length and the process quality cost expressions will be derived. The process mean can instantly shift due to (FMII) or (E) and create an out of control state in which the machine will be producing products with a lower quality or even defective items. When the process shifts to an out of control state, it can't come back to an in control state without interference. Since (FMII) and (E) cannot be directly detected, the reason of failure can't be specified without closing down the operation and carrying out a close inspection of the machine. A quality control chart 𝑋 is used to monitor the process behavior by calculating one key quality characteristic of the product. Let X be a normal random variable that indicates the estimation of this characteristic for a given product quality characteristic having a mean μ and a standard deviation σ. While being in control, the process mean is at its target value. Following a shift the process is considered to be out of the control limits and the updated mean is: 𝜇 = 𝜇0 +𝛿𝜎0 where 𝛿 is a nonzero real number. For the parameters of the 𝑋 chart (h; n and k) the resulting upper and lower control limits are:

𝑈𝐶𝐿 = 𝜇0 + 𝑘

𝜎 𝑛

,

𝐿𝐶𝐿 = 𝜇0 ― 𝑘

𝜎 𝑛

In the coming sections the process cycle length and quality cost expressions are developed.

15

4.7.1

The expected process cycle length

The expected process cycle length includes the process in the control period, the process in the out of control period and the repair time, illustrated as follows: 1. The process in-control period During this period the failure rate is constant. Therefore, we assume that the in control period follows an exponential distribution having a mean time to failure

1 𝜆

and a process

failure rate r (t) = 𝜆. The operation might break down due to machine deterioration or as a result of external reasons. So, let the failure rate as a result of machine deterioration (FMII) be (𝜆𝐹𝑀𝐼𝐼) and because of external reason (E) be (𝜆𝐸). Then,

𝜆𝐹𝑀𝐼𝐼 =

𝑃𝐹𝑀𝐼𝐼 ∗ 𝐸[𝑁(𝑃𝑀𝐼)] 𝑇𝑒𝑣𝑎𝑙

,

𝜆𝐸 =

1 𝑚𝑒𝑎𝑛 𝑡𝑖𝑚𝑒 𝑡𝑜 𝑓𝑎𝑖𝑙𝑢𝑟𝑒

Where, the probability that the machine fails due to FMII is 𝑇𝑒𝑣𝑎𝑙

𝑃𝐹𝑀𝐼𝐼

( ) = 𝐹(ℎ;;𝜆) = 1 ― 𝑒

𝐵

𝜂

(7)

Thus, the total process failure rate 𝜆 as a result of (FMII) and (E) is 𝜆 = 𝜆𝐹𝑀𝐼𝐼 + 𝜆𝐸

(8)

Now, the expected in control period composed of the following:

a. The mean time to failure

(1𝜆)

b. The expected time spent searching and inspecting for false alarms, which includes: i.

The expected number of samples (NS) taken while being in control, calculated as 16

(Lorenzen and Vance, 1986): Since, the probability density function 𝑓(ℎ, 𝜆) = 𝜆𝑒 ―𝜆ℎ ∞

𝑁𝑆 =

∑𝑖 𝑃𝑟(𝑎𝑠𝑠𝑖𝑔𝑛𝑎𝑏𝑙𝑒 𝑐𝑎𝑢𝑠𝑒 ℎ𝑎𝑝𝑝𝑒𝑛𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑖𝑡ℎ 𝑎𝑛𝑑 (𝑖 + 1) 𝑠𝑡 𝑠𝑎𝑚𝑝𝑙𝑒𝑠) 𝑖=0

∞

=

∑

𝑖 (𝑒

―𝜆ℎ𝑖

―𝑒

) = ― (1 ― 𝑒

―𝜆ℎ(𝑖 + 1)

𝑖=0

ii.

∞

𝑑 ) 𝑑(𝜆ℎ)

―𝜆ℎ

∑

𝑒 ―𝜆ℎ𝑖 =

𝑖=0

𝑒 ―𝜆ℎ

(1 ― 𝑒 ―𝜆ℎ)

The average run length while the process in control (ARLI), is calculated as

𝐴𝑅𝐿𝐼 =

1 𝛼

Where, 𝛼 = 𝑃𝑟 (𝑜𝑢𝑡 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑜𝑙│ 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑖𝑠 𝑖𝑛 𝑐𝑜𝑛𝑡𝑟𝑜𝑙) = 2𝐹( ― 𝑘) Then, the expected number of false alarms throughout this period, is calculated by

𝐸[𝑁𝑓𝑎𝑙𝑎𝑟𝑚] =

𝑁𝑆 𝐴𝑅𝐿𝐼

(9)

Thus, the expected amount of time spent searching and inspecting for false alarms is 𝑇0 𝐸[𝑁𝑓𝑎𝑙𝑎𝑟𝑚]

(10)

Therefore, the expected in-control period until the appearance of an assignable cause can be expressed as 1 𝐸[𝐼𝑇] = + 𝑇 𝐸[𝑁𝑓𝑎𝑙𝑎𝑟𝑚] 𝜆 0

2. The process out of control time 17

(11)

The expected out of control period consists of the following times: a. The expected time ahead of having a sample statistic falling beyond the control limit, is calculated as: Let 𝜖 be the mean time between the last sample prior to the assignable cause to the happening of the assignable cause where it takes place between the ith and (i + 1) st samples. Then 𝜖 can be calculated as in (Duncan, 1956) ℎ(𝑖 + 1)

𝜖=

𝜆(𝑥 ― ℎ𝑖)𝑒 ―𝜆𝑥𝑑𝑥

∫ℎ

ℎ(𝑖 + 1)

∫ℎ

𝜆𝑒 ―𝜆𝑥𝑑𝑥

[1 ― (1 + 𝜆ℎ)𝑒 ―𝜆ℎ] ℎ = = 2 [𝜆(1 ― 𝑒 ―𝜆ℎ)]

(12)

Now, the average run length after the process shifts to an out of control state (ARLO) is

𝐴𝑅𝐿𝑂 =

1 1―𝛽

Where, 𝛽 = 𝑃𝑟(𝑖𝑛 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑠𝑖𝑔𝑛𝑎𝑙 │ 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑖𝑠 𝑜𝑢𝑡 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑜𝑙) 𝛽 = 𝑃𝑟(𝐿𝐶𝐿 ≤ 𝑋 ≤ 𝑈𝐶𝐿 │ 𝜇 = 𝜇0 = 𝜇0 + 𝛿𝜎𝑝) Since that 𝜎2𝑝

( )

𝑋 ∼ 𝑁 𝜇,

𝑛

The upper and lower control limits will be

𝑈𝐶𝐿 = 𝜇0 + 𝑘

𝜎𝑝 𝑛

,

𝐿𝐶𝐿 = 𝜇0 ― 𝑘

𝜎𝑝 𝑛

Given that ɸ denotes the standard normal cumulative distribution function, then

18

𝛽=ɸ

(

) (

𝑈𝐶𝐿 ― 𝜇0 + 𝛿𝜎𝑝 𝜎𝑝 𝑛

―ɸ

)

𝐿𝐶𝐿 ― 𝜇0 + 𝛿𝜎𝑝 𝜎𝑝 𝑛

𝛽 = ɸ (𝑘 ― 𝛿 𝑛) ― ɸ ( ―𝑘 ― 𝛿 𝑛) Let the ARLO due to machine degradation (𝐹𝑀𝐼𝐼) be (𝐴𝑅𝐿𝑂𝐹𝑀𝐼𝐼), then

𝐴𝑅𝐿𝑂𝐹𝑀𝐼𝐼 =

=

1 1 ― 𝛽𝐹𝑀𝐼𝐼

1 1 ― [ɸ(𝑘 ― 𝛿𝐹𝑀𝐼𝐼 𝑛) ― ɸ( ―𝑘 ― 𝛿𝐹𝑀𝐼𝐼 𝑛)]

And due to external reasons (E) let it be (𝐴𝑅𝐿𝑂𝐸), then

𝐴𝑅𝐿𝑂𝐸 =

=

1 1 ― 𝛽𝐸

1 1 ― [ɸ(𝑘 ― 𝛿𝐸 𝑛) ― ɸ( ―𝑘 ― 𝛿𝐸 𝑛)]

Therefore, the expected time ahead of having a sample statistic fall beyond the control limit is

[(

𝜆𝐹𝑀𝐼𝐼

( )

ℎ 𝐴𝑅𝐿𝑂𝐹𝑀𝐼𝐼

𝜆

( ))]

+ 𝐴𝑅𝐿𝑂𝐸

𝜆𝐸 𝜆

―𝜖

b. The expected time to design and map a sample n TS c. The anticipated time to investigate the assignable cause occurrence T1 d. The anticipated time to renew the process, is calculated as 19

(13)

The restoration after detecting the assignable cause depends on the type of failure. The process is repaired as a result of FMII and restarted as a result of E, thus

[ ( )

𝐸[𝑇𝑟𝑒𝑠𝑡𝑜𝑟𝑒] = 𝑡𝐶𝑀

𝜆𝐹𝑀𝐼𝐼 𝜆

( )]

+ 𝑇𝑟𝑒𝑠𝑒𝑡

𝜆𝐸 𝜆

(14)

Therefore, the expected out of control time can be presented as

[(

( ) + 𝐴𝑅𝐿𝑂 ( ))] ―𝜖 + 𝑛𝑇

𝐸[𝑂𝑇] = ℎ 𝐴𝑅𝐿𝑂𝐹𝑀𝐼𝐼

𝜆𝐹𝑀𝐼𝐼 𝜆

𝜆𝐸 𝐸 𝜆

𝑆

+ 𝑇1 +𝐸[𝑇𝑟𝑒𝑠𝑡𝑜𝑟𝑒]

(15)

In conclusion, from equations (11) and equation (15), the model for the expected process cycle length is 𝐸[𝑇𝐶𝑦𝑐𝑙𝑒] = 𝐸[𝐼𝑇] + 𝐸[𝑂𝑇]

4.7.2

(16)

Expected process quality cost

The operation quality cost is composed of the costs generated during the in control period and the costs generated during the out of control period owning to false alarms, sampling the process, producing defective (non-conforming) units, searching for assignable alarms, restoring (repair or reset) the system, and the downtime cost. In this part we derive expressions for the expected cost of the process quality; which consists of the following costs expressions: 1. The expected cost of false alarms, is calculated as 𝐸[𝐶𝑓𝑎𝑙𝑠𝑒] = 𝐶𝑓𝑎𝑙𝑠𝑒(𝑇0𝐸[𝑁𝑓𝑎𝑙𝑎𝑟𝑚]) Where, Cfalse is the cost for inspecting a false alarm per unit time. 20

(17)

2. The expected cost of sampling per cycle, is calculated as Let FCS be the fixed cost per sample and FCJ be the fixed cost per job, then

𝐸[𝐶𝑆] =

[

(𝐹𝐶𝑆 + 𝑛 𝐹𝐶𝐽) ℎ

[(

( ))]

𝜆𝐹𝑀𝐼𝐼 𝜆𝐸 1 + 𝐴𝑅𝐿𝑂𝐸 + 𝑇0 𝐸[𝑁𝑓𝑎𝑙𝑎𝑟𝑚] + ℎ 𝐴𝑅𝐿𝑂𝐹𝑀𝐼𝐼 𝜆 𝜆 𝜆

( )

― 𝜖 + 𝑛 𝑇𝑆

]

(18) 3. The expected cost of nonconforming components (rejects) while the process is in control, is calculated as: Let RI be the proportion of non conforming components while the process in control. The type II error probability is given by 𝑅𝐼 = 1 ― ɸ(𝑘) ― ɸ( ― 𝑘) Therefore, the expected quality loss cost of non-conforming units when the process is incontrol is 𝐸[𝐶𝐼] = (𝑅𝐼 ∙ 𝐶𝑅 ∙ 𝑃𝑅)(𝐸 [𝐼𝑇])

(19)

4. The expected cost of non-conforming units (rejections) while running beyond control due to FMII , is calculated as follows. Let (𝑅𝛿)𝐹𝑀𝐼𝐼 be the proportion of non-conforming units when the process shifts 𝛿𝐹𝑀𝐼𝐼 to an out of control position owing to FMII. The process capability of the in control case is assumed to be 1. Thus the upper and lower quality limits will be at ± 3𝜎𝑃. Then the type II error probability will be 𝛽𝐹𝑀𝐼𝐼 = 𝐹(3 ― 𝛿𝐹𝑀𝐼𝐼) ― 𝐹( ―3 ― 𝛿𝐹𝑀𝐼𝐼) And the proportion of non-conforming components when the process shifts 𝛿𝐹𝑀𝐼𝐼 to an out of control state owning to FMII is given by 21

(𝑅𝛿)𝐹𝑀𝐼𝐼 = 1 ― 𝐹(3 ― 𝛿𝐹𝑀𝐼𝐼) ― 𝐹( ―3 ― 𝛿𝐹𝑀𝐼𝐼) Therefore, the expected cost of operating while being beyond control due to FMII is given as

𝐸[𝐶0]𝐹𝑀𝐼𝐼 =

([

(𝑅𝛿)𝐹𝑀𝐼𝐼

]

1 ― 𝛽𝐹𝑀𝐼𝐼

∙ 𝑃𝑅 ∙ 𝐶𝑅

[[ (

)(

𝜆𝐹𝑀𝐼𝐼 𝜆

𝜆𝐹𝑀𝐼𝐼

( )

ℎ 𝐴𝑅𝐿𝑂𝐹𝑀𝐼𝐼

𝜆

) ( ))]

+ 𝐴𝑅𝐿𝑂𝐸

𝜆𝐸 𝜆

]

― 𝜖 + 𝑛𝑇𝑆 + 𝑇1

(20)

5. The expected cost of non-conforming units (rejections) when the process moves to an out of control state due to E, is calculated as follows Let (𝑅𝛿)𝐸 be the proportion of non-conforming units when the process shifts 𝛿𝐸 beyond control owing to E. It is assumed that the procedure capability of the monitored state is 1. Thus, the upper and lower quality limits will be at ± 3𝜎𝑃. The type II error probability will be 𝛽𝐸 = 𝐹(𝑘 ― 𝛿𝐸 𝑛) ― 𝐹( ―𝑘 ― 𝛿𝐸 𝑛) And the proportion of non-conforming units when the process shifts 𝛿𝐸 to an out of-ofcontrol state owing to E is (𝑅𝛿)𝐸 = 1 ― 𝐹(3 ― 𝛿𝐸) ― 𝐹( ―3 ― 𝛿𝐸) Therefore, the expected quality cost of operating while being in an out-of-control state due to E is given as

𝐸[𝐶0]𝐸 =

(𝑅𝛿)𝐸

([ ] 1 ― 𝛽𝐸

∙ 𝑃𝑅 ∙ 𝐶𝑅

)(

𝜆𝐸 𝜆

) 22

[[ (

𝜆𝐹𝑀𝐼𝐼

( )

ℎ 𝐴𝑅𝐿𝑂𝐹𝑀𝐼𝐼

𝜆

( ))]

+ 𝐴𝑅𝐿𝑂𝐸

𝜆𝐸 𝜆

]

― 𝜖 + 𝑛𝑇𝑆 + 𝑇1

(21)

6. The expected cost of CM activity owing to FMII for locating and correcting the assignable cause, is calculated as

𝐶𝑀𝐶𝐹𝑀𝐼𝐼 = [𝑡𝐶𝑀(𝐿𝑃𝐶 ∗ 𝑃𝑅 + 𝐿𝐶) + 𝐹𝐶𝐶𝑀]

𝜆𝐹𝑀𝐼𝐼

( ) 𝜆

(22)

7. The expected cost of finding and resetting the assignable cause owing to E, is calculated as

𝐸[𝐶𝑟𝑒𝑠𝑒𝑡]𝐸 = [𝑇𝑟𝑒𝑠𝑒𝑡 ∙ 𝐶𝑟𝑒𝑠𝑒𝑡]

𝜆𝐸

()

(23)

𝜆

In conclusion, the expected process quality cost is 𝐸[𝑃𝑄𝐶] = 𝐸[𝐶𝑓𝑎𝑙𝑠𝑒] + 𝐸[𝐶𝑆] + 𝐸[𝐶𝐼] + 𝐸[𝐶0]𝐹𝑀𝐼𝐼 + 𝐸[𝐶0]𝐸 + 𝐶𝑀𝐶𝐹𝑀𝐼𝐼 + 𝐸[𝐶𝑟𝑒𝑠𝑒𝑡]𝐸

(24)

Therefore, the expected total cost of quality loss due to process failure for the evaluation period is

𝐸[𝑇𝑄𝐶]𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 = 𝐸[𝑃𝑄𝐶]

([

𝑇𝑒𝑣𝑎𝑙

𝐸 𝑇𝐶𝑦𝑐𝑙𝑒]

)

(25)

4.8 Integrated Productions Scheduling and Maintenance In order to obtain the optimal time after which PM should be performed on the machine, i.e. PM interval (PMI), we consider the time to perform PM as a batch to be inserted in the production sequence with its processing time as the expected time of performing PM (tPM) and its due date as the optimal ideal PM interval time (τ*). Since the machine may or may 23

not fail while processing a job, the completion time of a batch is strongly affected by the probability of machine failure. This probability is influenced by the machine’s age and the PM activities. The following assumptions are considered: 1. The batch manufacturing time is the aggregate of the setup times and processing times of its member jobs. 2. A job cannot be preempted by another job. 3. The machine cannot be interrupted for PM until all the jobs in a batch are finished. Let the age of the machine before starting the production schedule and performing the PM activities be a[0], the age of the machine immediately before carrying out the ith batch in the sequence (after PM batch, if any) be 𝑎[𝑖 ― 1] and the age of the machine immediately after the ith batch in the sequence be a[i]. Let the variable that restores the machine’s age after the PM activities (y[i]) be defined as:

𝑦[𝑖] =

{

1

𝑖𝑓 𝑡ℎ𝑒 𝑃𝑀 𝑏𝑎𝑡𝑐ℎ 𝑖𝑠 𝑠𝑐ℎ𝑒𝑑𝑢𝑙𝑒𝑑 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑖𝑡ℎ𝑏𝑎𝑡𝑐ℎ 𝑖 = 1, 2, …, 𝑛

0

𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Then, the machine’s age is: 𝑎[𝑖 ― 1] = 𝑎[𝑖 ― 1][1 ― (𝑦[𝑖])] 𝑎[𝑖] = 𝑎[𝑖 ― 1] + 𝑝[𝑖] The probability of machine failure is governed by a Weibull probability distribution, with T as the time until failure for a new machine and F (t) as the cumulative distribution function of T. Then,

[ ()]

𝑡 𝐹(𝑡) = 1 ― 𝑒𝑥𝑝 ― 𝜂

𝐹(𝑡) = 1 ― 𝐹(𝑡)

24

𝛽

Therefore, the probability that the machine fails while processing the ith batch [𝑎[𝑖 ― 1] < T < 𝑎[𝑖]] is determined as follows: 𝐹(𝑎[𝑖] = 𝑝[𝑖] + 𝑎[𝑖 ― 1]│ 𝑎[𝑖 ― 1]) = 𝑃𝑟{𝑇 ≤ 𝑝[𝑖] + 𝑎[𝑖 ― 1]│ 𝑇 > 𝑎[𝑖 ― 1]}

[

= 1 ― 𝑒𝑥𝑝 ―

𝑝[𝑖] + 𝑎[𝑖 ― 1]

(

) (

𝜂

[

= 1 ― 𝑒𝑥𝑝 ―

𝑎[𝑖]

𝛽

𝛽

( ) ( 𝜂

+

+

𝜂 𝛽

)]

𝑎[𝑖 ― 1] 𝜂

𝛽

)]

𝑎[𝑖 ― 1]

Define Φ[i] as: Φ[𝑖] = 𝐹(𝑎[𝑖]│ 𝑎[𝑖 ― 1]) Φ[𝑖] = 1 ― 𝐹(𝑎[𝑖]│ 𝑎[𝑖 ― 1]) , where i = 1, 2, …, n. While performing PM i.e. processing the PM batch, the probability of having machine failure during PM is: Φ[𝑖] = 0 Φ[𝑖] = 1 , where i = PM batch. Now, due to the maintenance activities, the completion time of the ith batch (C[i]) is a discrete random variable that depends on: 1. The age of the machine prior to manufacturing the batch (𝑎[𝑖 ― 1]). 2. The processing time of the batch and the completion time of previous batches. 3. Probability of machine failure while processing a batch and the associated repair time. Therefore,

25

(∑ ) 𝑖

𝐶[𝑖] =

𝑝[𝑖] + 𝑉[𝑖]

𝑖 = 1, 2, …, 𝑛

𝑖=1

Where, V[i] is a discrete random variable defined for considering machine failures and the time to correct them, i.e. it can take two possible values zero or tCM. Let Ni = {1, 2, …, i} and Ni,k denotes a subset of Ni containing k elements. Then, V[i] has the following probability mass function:

𝜋[𝑖,𝑘] = 𝑃𝑟{𝑉[𝑖] = 𝑘 ⋅ 𝑡𝐶𝑀} =

∑∏Φ ∏Φ [𝑖]

𝑁𝑖,𝑘 𝑙 ∈ 𝑁𝑖,𝑘

[𝑖],

𝑘 = 0, 1, 2, …, 𝑖

𝑙 ∉ 𝑁𝑖,𝑘

, where Φ[𝑖 = 𝑃𝑀 𝐵𝑎𝑡𝑐ℎ] = 0; Φ[𝑖 = 𝑃𝑀 𝐵𝑎𝑡𝑐ℎ] = 1 Thus, the batch completion time will be

(∑ ) 𝑖

𝐶[𝑖,𝑘] =

𝑝[𝑖] + 𝑘 ⋅ 𝑡𝐶𝑀 ,

𝑘 = 0, 1, 2, …, 𝑖; 𝑖 = 1, 2, …, 𝑛

𝑖=1

Then, the expected completion time of batch i in the schedule is given by: 𝑖

𝐸(𝐶[𝑖]) =

∑𝐶

[𝑖,𝑘]

𝜋[𝑖,𝑘]

𝑘=0

The PM interval for the machine is defined as the time completed prior to start of processing the PM batch and is given as: 𝑃𝑀𝐼 = 𝑎[0] + 𝐸(𝐶[𝑖 ― 1]),

𝑖 = 𝑃𝑀 𝑏𝑎𝑡𝑐ℎ

The number of PM intervals to be inserted in the sequence is an integer defined as:

26

𝑛

∑𝑖 = 1𝑝𝑖 , 𝑁𝑃𝑀 = 𝜏∗

𝑁𝑃𝑀 ≥ 0

The penalty cost will occur when a batch is delivered after its due date and when the PM batch is delivered before or after its due date (ideal PM interval). Let Θ[𝑖]be the tardiness of batch i. Note that Θ[𝑖] has (i + 1) possible values such as: 𝜃[𝑖,𝑘] = 𝑚𝑎𝑥(0, 𝐶[𝑖,𝑘] ― 𝑑[𝑖]),

𝑘 = 0, 1, 2, …, 𝑖

The earliness and tardiness for the PM batch are given by: 𝜃[𝑖 = 𝑃𝑀 𝐵𝑎𝑡𝑐ℎ, 𝑘] = |𝐶[𝑖 = 𝑃𝑀 𝐵𝑎𝑡𝑐ℎ, 𝑘] ― 𝜏 ∗ |,

𝑘 = 0, 1, 2, …, 𝑖

Thus, the expected lateness of the ith batch in the sequence is given by: 𝑖

𝐸(Θ[𝑖]) =

∑𝜃

[𝑖,𝑘]𝜋[𝑖,𝑘]

𝑘=0

Hence, the overall penalty cost associated with batch and maintenance delays can be calculated as 𝑛

𝑇𝑃𝐶𝑠𝑐ℎ𝑒𝑑𝑢𝑙𝑖𝑛𝑔 & 𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 =

∑𝑃𝑁

[𝑖]

𝐸(Θ[𝑖])

(28)

𝑖=1

4.9 The integrated production, maintenance and quality model The objective function of the integrated model shown in equation (29) consists of the expected cost of preventive maintenance, the expected cost of corrective maintenance, the expected inventory holding cost, the expected cost of quality and the expected cost of production scheduling and maintenance. The model has six sets of constraints. Constraints (30) and (31) are functional constraints. They ensure that each position in the schedule receives one job and that each job is assigned to one position in the schedule. The constraint set in equation (32) ensures the processing time of job ith is pi and the constraints set in 27

equation (33) compute the position of the ith job in the sequence. The set of constraints in equation (34) computes the expected lateness of the ith job and the set of constraints in equation (35) computes the expected completion time of the ith job.

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐸𝑇𝐶 =

𝑃𝑀𝐶 + 𝐶𝑀𝐶𝐹𝑀𝐼 + 𝐻𝐶 + 𝐸(𝑇𝑄𝐶)𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 + 𝑇𝑃𝐶𝑆𝑐ℎ𝑒𝑑𝑢𝑙𝑖𝑛𝑔 & 𝑀𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 𝑇𝑒𝑣𝑎𝑙

(29)

Subject to 𝑛

∑𝑖 = 1𝑥𝑖𝑗 = 1 𝑛

∑𝑗 = 1𝑥𝑖𝑗 = 1

𝑗 = 1, 2, …, 𝑛 𝑖 = 1, 2, …, 𝑛

𝑛

𝑝[𝑖] = ∑𝑗 = 1𝑝𝑗 𝑥𝑖𝑗 𝑛

𝑁[𝑖] = ∑𝑗 = 1𝑁𝑗 𝑥𝑖𝑗 𝑖

𝐸(Θ[𝑖]) = ∑𝑘 = 0𝜃[𝑖,𝑘] 𝜋[𝑖,𝑘]

𝑖 = 1, 2, …, 𝑛 𝑖 = 1, 2, …, 𝑛 𝑘 = 0, 1, 2, …, 𝑖

(30) (31) (32) (33) (34)

𝑖

𝐸(C[𝑖]) = ∑𝑘 = 1𝑡𝑃𝑀(𝑦[𝑘]) + (𝑝[𝑘]) + 𝑡𝐶𝑀[𝑚(𝑎[𝑘]) ― 𝑚(𝑎[𝑘 ― 1][1 ― (𝑦[𝑘])])] (35) 𝑥𝑖𝑗 𝑏𝑖𝑛𝑎𝑟𝑦

𝑖 = 1, 2, …, 𝑛

𝑦[𝑖] 𝑏𝑖𝑛𝑎𝑟𝑦

𝑖 = 1, 2, …, 𝑛

5. Solving the integrated problem Previous studies solved the problem of scheduling three batches of jobs on a single machine (Cassady and Kutanoglu, 2003, 2005; Laith et al., 2011). In this paper, we extend that to the possibility of scheduling n batches of jobs. The integrated problem is solved using complete enumeration. The following are the steps of the proposed solution algorithm: 1. For a given set of n batches find the number of PM batches to be inserted in the sequence and the optimal ideal PM interval 𝜏 ∗ then generate all the possible production schedules n!. Such as

28

𝜏∗ = 𝜂

[

]

𝑡𝑃𝑀

1 𝛽

𝑡𝐶𝑀(𝛽 ― 1)

𝑛

𝑁𝑃𝑀 =

∑𝑖 = 1𝑝𝑖 𝜏∗

2. Compute the costs of production and maintenance scheduling for each PM plan, which includes a. Penalty cost due to tardiness. b. Inventory holding cost. c. Corrective maintenance cost. d. Preventive maintenance cost. 3. Reduce the model to the quality cost and solve it using Maple 18 global optimization tool for the decision variables (n, h, and k). 4. Compute the total cost of integrating production, maintenance and quality for each production schedule. 5. Compare the total cost obtained from each schedule and select the minimum.

6.

Results and Analysis In this section we illustrate the use of the developed model through a numerical

example. The example was introduced by Cassady and Kutanoglu, (2003, 2005) and used by Hadidi et al., (2011). The results are analyzed and sensitivity analysis is conducted. 6.1

Numerical example Consider a production system with a single machine (production line), subject to

random failure following a two-parameter Weibull distribution as the normal shape and life parameter, respectively β = 2 and η = 100. The age of the machine at the beginning is a0 = 68. The expected time required to perform a corrective maintenance tCM = 15 units of 29

time and the repair is assumed to be minimal in which the machine will be repaired to its age before the failure and the restoration factor RFCM = 0. The expected time required to perform preventive maintenance tPM = 5 units of time, assuming a perfect PM in which the machine will be restored to its new state. The 𝑋 quality control chart is used to monitor the quality characteristic of the production process that produces items. The process in-control state quality characteristic, while the process is in an in-control state, is normally distributed with μ = 0 and process standard deviation of σ = 0.01 and will shift to an outof-control state due to random machine failure be 𝛿𝐹𝑀𝐼𝐼 = 0.8 or due to external reasons 𝛿𝐸 = 1, which will result in a shift of process mean from μ0 to (μ0 + δσ). The initial values for cost and time parameters used in the example are shown in Table 1. Table 1. Initial parameters Initial cost parameters Initial time parameters Cost Value Time Value parameters parameters (hours) FCS 40 T0 1 FCJ 10 TS 20/60 CR 5,000 TI 1 FCCM 10,000 Treset 2 Creset 1,500 0.8 FMII FCPM 800 1 E Cfalse 1,200 LPC 40 LC 500 PR 20

The machine will be processing a set of three batches having the following parameters (Table 2). Table 2. Parameters for processing the set of three jobs Job Batch Processing Due Penalty Release Inventory Set up size time date time holding time 1 500 23 67 10 0 1.71 3 2 500 28 114 2 0 1.71 1 3 500 43 65 5 0 1.71 2 The first step in our current solution is to determine the optimal ideal preventive maintenance intervals i.e. the due dates of the PM batches and the number of PM batches to be introduced to the sequence. 30

𝜏∗ = 𝜂

[

]

𝑡𝑃𝑀

1 𝛽

𝑡𝐶𝑀(𝛽 ― 1)

1

[

𝑛

𝑁𝑃𝑀 =

∑𝑖 = 1𝑝𝑖 𝜏∗

]

2 5 = 100 = 57.7 15(2 ― 1)

=

94 = 1.629 ≈ 1 57.7

Therefore, only one preventive maintenance action has to be performed on the machine as another batch is added to the schedule, with the production scheduling requirement as shown in Table 3. Table 3. Production and preventive maintenance requirement job 1 2 3 PM

Processing time 23 28 43 5

Due date 67 114 65 57.7

Penalty 10 2 5 0

The integrated scheduling model is solved using enumeration technique. The feasible sequences for this example are (n!) enumerated as shown in Table 4.

Table 4. Possible solution for the integrated model Batch sequence B1−B2−B3−PM B1−B2−PM−B3 B1−B3−B2−PM B1−B3−PM−B2 B1−PM−B2−B3 B1−B2−B3−PM B2−B1−B3−PM B2−B1−PM−B3 B2−B3−B1−PM B2−B3−PM−B1

PMI 222.9 133.3 222.9 154 96.4 96.4 229.3 133.3 229.3 161

h 5.34 4.69 5.24 4.74 4.67 4.67 5.34 4.68 5.22 4.75

k 3.49 3.57 3.49 3.55 3.61 3.61 3.49 3.57 3.5 3.55 31

n 32.84 27.66 32.34 28.61 25.85 25.85 32.87 27.51 32.24 28.73

ETC 3,210.67 2,132.21 3,133.28 2,346.78 1,816.96 1,895.97 3,216.64 2,132.06 3,121.75 2,403.74

B2−PM−B1−B3 B2−PM−B3−B1 B3−B1−B2−PM B3−B1−PM−B2 B3−B2−B1−PM B3−B2−PM−B1 B3−PM−B1−B2 B1−B2−B3−PM PM−B1−B2−B3 PM−B1−B3−B2 PM−B2−B1−B3 PM−B2−B3−B1 PM−B3−B1−B2 PM−B3−B2−B1

102.8 102.8 249 154 249 161 122.5 122.5 68 68 68 68 68 68

4.67 4.67 5.28 4.71 5.26 4.72 4.66 4.66 4.73 4.73 4.73 4.73 4.74 4.74

3.61 3.61 3.49 3.56 3.49 3.55 3.59 3.59 3.65 3.65 3.65 3.65 3.65 3.65

26.07 26.07 32.58 28 32.44 28.28 26.74 26.74 24.68 24.68 24.65 24.65 24.55 24.55

1,884.74 1,982.86 3,173.03 2,310.61 3,153.76 2,369.70 2,108.22 2,128.15 1466.62 1587.59 1492.15 1,651.55 1,661.27 1,699.01

Once this search has been executed for all sequences of the jobs, the tasks’ sequence with the lowest objective function value is determined as the global optimal solution. The optimal solution is ETC = 1,466.6 for this example. The optimal batch sequence is PMB1-B2-B3 with the PM plan performed prior to batch 1. 6.2

Sensitivity analysis and model comparisons In this subsection sensitivity analysis and comparisons between the model developed

in this paper and other scenarios of two way integration are presented. The data in the previous example is utilized to conduct the sensitivity analysis and the comparisons. .A systematic sensitivity analysis was conducted on cost, time and process shifts parameters and presented in table 5. The cost parameters considered in the sensitivity analysis are the fixed cost per sample FCS , fixed cost per job FCJ , cost of rejection CR , fixed cost of corrective maintenance FCCM , fixed cost of preventive maintenance FCPM, fixed cost of investigating false alarm Cfalse , lost production cost LPC, expected time to search for false alarm T0 , expected time to chart a sample Ts , expected time to locate assignable cause T1, expected time to reset the process that moved to an out of control state to in control state as a result of an external reason Treset , amount of shift due mode II failure δFMII and amount of shift due to external reasons

𝛿𝐸. The model is run keeping all

parameters fixed except the one under consideration and provided in the rows of table 5 . The parameters under consideration is changed twice. The first time changed by 10% and 32

the second time by 20% and presented in the columns titled level 2 and level 3 of table 5 respectively. The column titled ETC in table 5 contains the expected total cost (ETC) for the base set of data in the example. The column titled ETC 2 contains the ETC and its percentage change from the ETC for the base set of data when the parameters are increased by 10%. The column titled ETC 3 contains the ETC and its percentage from the ETC of the base set of data when the parameters are increased by 20%. For the 10% change in the parameters the absolute percentage change in the ETC ranges from 0 to 1.96 while it ranges from 0 to 3.90 percent in the case of the 20% changes in the parameters. The maximum percent change in ETC happened for changes made for the cost of rejection CR in both cases. Other changes in ETC that worth noting happened for changes in the cost of lost production LPC and the shift parameter in the process mean due to external reasons. The sensitivity analysis indicated that at most care should be taken in estimating the cost of rejection, cost of lost production and the amount of shift in the process in the process mean due to external reasons. However the percentage of change whether increase or decrease are not very high which is an indication that the developed model is robust.

Table 5. Sensitivity analysis for three levels of integration Parameters

Level 1

ETC

Level 2

ETC 2 ETC

Level 3 %

ETC 3

ETC

increase

increase

FCS

40

1,466.6

44

1,467.4

0.05

48

1,468.1

FCJ

10

1,466.6

11

1,471.5

0.33

12

1,476.2

CR

5,000

1,466.6

5,500

1,495.3

1.96

6,000

1,523.8

33

%

0.10

0.65 3.90

FCCM

10,000

1,466.6

11,000

1,468.8

0.15

12,000

1,471.1

FCPM

800

1,466.6

880

1,467.2

0.04

960

1,467.7

Cfalse

1,200

1,466.6

1,320

1,466.6

0.00

1,440

1,466.6

LPC

40

1,466.6

44

1,471.8

0.35

48

1,477.1

T0

1

1,466.6

1.1

1,466.6

0.00

1.2

1,466.6

T1

1

1,466.6

1.1

1,468.4

0.12

1.2

1,470.2

Treset

2

1,466.6

2.2

1,468.3

0.12

2.4

1,469.9

𝛿𝐹𝑀𝐼𝐼

0.8

1,466.6

0.88

1,463.5

-0.21

0.96

1,461.7

𝛿𝐸

1

1,466.6

1.1

1,460.8

-0.40

1.2

1,458.2

The developed model which is a three ways integration is compared with several two ways integration for the same data used in the example. The optimal solution obtained by the model developed in this paper is 1466.6 will be compared to four different scenarios of two integration and the percent increase in ETC in each case is computed. The increase can be interpreted as savings or profit made due to the developed model. The results are shown in table 6.

Table 6. Comparison between the integrated model and Other Scenarios of Integration Benchmark

ETC

Joint maintenance & quality + scheduling Joint scheduling & maintenance + quality Joint quality & scheduling + maintenance Separate consideration

1,505

% Savings due to the proposed integration 2.62

1,539

4.94

1,566

6.78

1,552

5.82

34

0.31 0.08 0.00 0.72 0.00 0.25 0.23 -0.33 -0.57

7.

Conclusion This paper proposes a model for integrating production (production scheduling and

inventory), maintenance planning and quality control decisions. The model allows the joint optimization of maintenance (prevent and corrective), job scheduling and control chart design parameters to decrease the expected total cost per unit time. Sensitivity analysis was conducted on various model parameters to investigate their impact on expected total cost of the system. The sensitivity analysis indicated that the model is robust and insensitive to changes in the model parameters. In addition comparisons with other integration schemes were made and the results indicated that the model provide a saving that ranges from 2.62 – 6.78 percent. A clear understanding of the relationship between the leading elements of the production system can help to overcome the conflicts arising between these elements in real life. The study shows that the least profitable case is when the schedule of the optimal jobs sequence is found first separately, without considering the preventive maintenance or quality control. Then, the preventive maintenance is linked with the quality control to find the optimal joint PM interval and quality control chart decisions. The most profitable case is when the optimal PM interval is found first separately, without considering the scheduling or quality control. Then, the jobs sequence is linked with the quality control to obtain the optimal joint jobs schedule and quality control chart parameters. It is clear that the quality control relationship with maintenance and scheduling, respectively, has the greatest impact on the model developed. A topic for the future research is to consider different objective functions such as maximizing the system availability, operation efficiency and Taguchi quadratic loss function. Multi-objective optimization offers an alternative approach for integrating maintenance, production and quality. The study offered in this paper is limited to a single machine system, however, it would be more reasonable and practical to extend the system to contain more than one machine and to add different flow patterns and sequence for dependent/independent setup times.

35

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Highlights An Integrated Model of Production Scheduling, Maintenance and Quality for a Single Machine    

A model that integrates production, maintenance and quality is developed. The model extends work in the literature especially work published in CAIE. Sensitivity analysis is conducted to exemplify the advantage of integration. The utility of the model is demonstrated using an example from the literature.

40

41