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Integrated model of quality inspection, preventive maintenance and buffer stock in an imperfect production system
Accepted Manuscript Integrated model of quality inspection, preventive maintenance and buffer stock in an imperfect production system Rodrigo Lopes PII: DOI: Reference:
S0360-8352(18)30489-3 https://doi.org/10.1016/j.cie.2018.10.019 CAIE 5458
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
1 March 2018 17 August 2018 8 October 2018
Please cite this article as: Lopes, R., Integrated model of quality inspection, preventive maintenance and buffer stock in an imperfect production system, Computers & Industrial Engineering (2018), doi: https://doi.org/10.1016/j.cie. 2018.10.019
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Integrated model of quality inspection, preventive maintenance and buffer stock in an imperfect production system Rodrigo [email protected] Department of Production Engineering, Federal University of Pernambuco, Rodovia BR-104, Km 59, s/n, Nova Caruaru, Caruaru - PE, CEP- 55002-970, Brazil
Highlights We propose an integrated quality control, preventive maintenance and buffer decision. We examined inspection errors, product warranty, and rework in an imperfect process. Numerical example is presented with managerial implications.
Abstract In this study, the influence of a quality inspection policy on an imperfect production system in which a percentage of the produced items is inspected is modelled. The inspection policy is imperfect, permitting type I and type II errors. Items are shipped for sale with a free minimal repair warranty policy. After each production cycle, preventive maintenance is performed. In case of failure in the production process before the cycleβs end, a minimal repair is indicated. A buffer stock is established to meet demand while preventive maintenance is performed. We also considered that defective items detected on inspection are sent for reworking. The model is formulated with the objective of minimizing the total cost expected per item while considering an average outgoing quality constraint. Subsequently, we present a numerical experiment to illustrate the proposed model. The results indicate that the probability of type I and type II inspection
errors has a significant impact on optimal production and the percentage of items inspected. Keywords: quality inspection, imperfect production system, integrated decision-making, imperfect inspection, type I and II inspection errors.
1.
Introduction
Quality inspection, production planning, and maintenance scheduling are the most fundamental elements of a production system. Integration between quality, production, and maintenance is vital for a company to be successful, and together, these elements contribute to improving the efficiency of a production system by decreasing costs (Bouslah et al., 2016). In most production systems, production is imperfect, resulting in a number of defective item; such items may result from insufficient process control, poorly planned maintenance, inadequate working instructions, or damage during production (Rad et al., 2014). Imperfect production results in items that do not conform to specifications, which can force the customer to return the defective item for an exchange or to request a refund. In e-commerce, the return of defective items can represent up to 25% of total sales (Krikke et al., 2004).
Therefore, to reduce the negative effects of defective items shipped to market, companies often invest in inspection processes to ensure their products conform to market expectations (Cavalcante et al., 2016). Both product and process quality should be defined (Muhammad et al., 2017). Consequently, the production manager must carefully consider the internal cost related to the inspection process and the post-sales costs of defective products (Juran and Gryna, 2016). The primary purpose of inspection is to identify product units that do not conform to product specifications. Nonconforming products can be discarded or reworked (Nourefath et al., 2016). Inspection operations may involve two types of error: the classification of a unit that conforms as nonconforming and the classification of a unit that does not conform as conforming. The first error type results in waste and low productivity. The second type results in poor quality, which can lead to warranty costs and loss of customers (Bouslah et al., 2016). In the literature, models have been presented that address the effects of quality inspection, maintenance, and buffer stock on an imperfect production process. Without being exhaustive, Ben-Daya (2002) derived an integrated model for joint determination of the amount of economic production and preventive maintenance in an imperfect process with an increasing rate of failure. Wang and Sheu (2003) determined an optimal policy of production, inspection, and maintenance under the effects of process inspection errors. Wang (2005) considered an inspection policy with two types of inspection error to determine the amount of economic production quantity. Darwish and Ben-Daya (2007) developed a stock model for an imperfect production process with errors in preventive maintenance and inspection. Lin and Yu (2009) investigated a system of imperfect production, maintenance, and production lot quality, with increasing rates of risk. Liao et al. (2009) developed preventive maintenance
programmes with an economic production model for an imperfect production process in which all defective items must be reworked. Chakraborty et al. (2009) state that nearly all integrated models do not simultaneously consider deterioration of the quality and reliability of equipment, and when both phenomena are observed, preventive maintenance will have the following two objectives: to restore the equipment and to improve the quality of the process. Hadidi et al. (2012) present a review of integrated models for production planning and scheduling, maintenance, and quality inspections. Bouslah et al. (2016) developed an integrated model for sampling quality control and the maintenance of deteriorating production systems; the model considered an average output quality constraint. Li et al. (2017) studied inventory and process deterioration in the economic production quantity (EPQ) model with reworking and backlogs. Kang and Subramaniam (2018) developed a dynamic maintenance policy integrated with production and opportunistic maintenance in a joint optimization problem. Cheng et al. (2018) presented a model based on simulation to jointly determine production, quality control, and maintenance for a single-unit production system that adopts a buffer stock. In this study, we extend the research of Sana (2012), who analysed an imperfect production system to determine optimal buffer levels and production runtime but did not determine the fraction of items inspected or consider imperfect inspection. We consider that the inspection is imperfect and that only a fraction of the items produced are inspected. More specifically, we construct a model that describes a production process in which at the beginning of the process the system is in an in-control state and produces few defective items. However, the system can shift into an out-of-control state at any time and start producing more defective items than in the in-control state. A fraction of items are inspected, and the inspection is imperfect. Defective items detected through inspection are reworked at a fixed cost. The items are sold with a minimal free
repair warranty (FRW). After each production cycle, preventive maintenance is performed, and in case of a failure in the production process before the cycleβs end, a minimal repair action is indicated. The manufacturer establishes a buffer stock to adjust to market demand. The buffer is consumed when preventive maintenance is performed. The model is formulated with the objective of minimizing the total cost expected per item while considering an average outgoing quality constraint. The remainder of this paper is organized as follows. The notations and assumptions are presented in Section 2. Section 3 presents the mathematical model. In Section 4, a numerical example is considered and a sensitivity analysis performed. Managerial implications are discussed in Section 5. Section 6 presents our conclusions.
2.
Notation and assumptions
2.1. Notation The following notations are used to develop the model: Decision variables S
buffer stock
i
noninspected fraction (0 β€ i β€ 1)
Parameters πΆπ
unit inspection cost
πΆπ
false rejection of nondefective item
K
setup cost per cycle
Ξ±
demand rate
p
production rate, where p>Ξ±
Cm
cost of material to produce an item
Co
cost per unit time for preventive maintenance
Cr
cost per item for each minimal warranty repair
Rc
rework cost per unit defective item
Ch
holding cost per unit item per unit time
Cs
shortage cost per unit item per unit time
C1
minimal repair cost of maintenance
Ο
a random variable, which is time-elapsed after which the production process shifts to the out-of-control state
m1
random variable representing type I error
m2
random variable representing type II error
f(Ο)
probability density function of Ο with mean 1/Β΅
ΞΈ1
probability of nonconforming items when the production process is in the in-control state
ΞΈ2
probability of nonconforming items when the production process is in the out-of-control state. Here, 0<ΞΈ1< ΞΈ2<1
T
production runtime per cycle
w
warranty period
h1(x)
failure rate function of conforming items
h2(x)
failure rate function of nonconforming items
t
preventive maintenance time, which is a random variable
Ο(t)
probability density function of t π΄ππ
average outgoing quality
ππππ₯
maximum percentage of defective items sent to the market
C(S, i)
expected total cost per item
2.2. Assumptions The following assumptions are considered while formulating the model:
1. Product inspection is performed to detect defective items. 2. Two types of errors, that is, type I and type II errors, occur during product inspection. 3. The inspection time is negligible. 4. This model assumes ΞΈ1< ΞΈ2; that is, the probable number of defective items in the incontrol state is less than the probable number of defective items in the out-of-control state. 5. Products are sold with a free minimal repair warranty. 6. The buffer stock is not subject to deterioration or obsolescence. 7. Minimal repair action does not affect the production rate. The first assumption is related to the inspection policy, according to which a fraction of items are not inspected. The second assumption is that the inspection process is not perfect and permits type I and type II errors. The third assumption implies that the time required to inspect the items is insignificant. The fourth assumption is related to an imperfect production system. The fifth assumption is related to the product warranty policy. The sixth assumption is that the items in the stock buffer are not subject to deterioration or obsolescence. The seventh assumption is related to the minimal repair action required when the system fails. This action requires an insignificant amount time such that the production rate is not affected. In the following section, the proposed model is formulated mathematically.
3.
Mathematical Model
3.1. System Description We consider a single process or equipment that produces a single product type or set of similar products at a particular rate of production p, whereby the process can be performed in two states: in-control and out-of-control. The in-control state produces
nonconforming items at rate ΞΈ1. At any point in time, this system can enter the out-ofcontrol state, in which the production rate of nonconforming items is higher: rate ΞΈ2. In this system, a product inspection policy is implemented, where i is the fraction of items not inspected. The inspection system is imperfect, and errors may occur. Type I error has the probability m1, whereas type II error has the probability m2. When type I error occurs, nondefective products are classified as defective, which results in cost πΆπ. However, when type II error occurs, certain defective items may be classified as nondefective and sent to the market, and these items increase the cost of the warranty. The inspection process has cost πΆπ. To meet consumer expectations, the system is subject to a constraint of defective items sent to the market ππππ₯. The average number of defective items AOQ (average outgoing quality) sent to the market must not exceed the maximum limit ππππ₯. After inspection, items classified as defective are sent for reworking at cost Rc. The products are sold with a minimal repair warranty policy, according to which each product failure is rectified by a free minimal repair in period w with cost Cr. The warranty policy considers the rate of failure of nonconforming items as greater than that of conforming items. In this system, to adjust production to demand, buffer stock S is maintained with cost Ch. This buffer stock is consumed at rate Ξ± while preventive maintenance is being performed. The system produces items during period T, after which, preventive maintenance is performed. Production equipment is restored to new condition, and the system returns to the in-control state. If there is a failure between [0,T], minimal repair actions are performed. This minimal repair action does not affect production rate p. During the preventive maintenance period, demand is met by buffer stock S. If the demand is greater than the buffer stock, this demand is lost at cost Cs. The buffer stock
consumption is described by the following equation: π = (π β πΌ)π, that is, π = π/ (π β πΌ). Our objective is to simultaneously find the optimal S and i, which minimizes the cost per unit produced considering a constraint of the defective items sent to the market (AOQ).
3.2. Cost structure For the described system, the cost of production includes the costs of quality inspection, material, maintenance, rework, setup, and warranty. Next, we discuss the composition of each cost in detail. First, as there are two types of error, the items can be divided into four cases: First case: This case is related to the expected number of items produced with defects that are inspected and false accepted according to type II error, as described by Equation 1.
{
π1 = π1 + (π β πΌ)
(
π2 β π1
)
π
(π βπ πΌ + π’)ππ’}(π(π βπ πΌ))π2(1 β π)
π
β«π β πΌπ’π 0
(1)
Second case: This case describes the expected number of items that are correctly classified as defective items, as described by Equation 2.
{
π2 β π1
(
π2 = π1 + (π β πΌ)
π
)β«
(π βπ πΌ + π’)ππ’}(π(π βπ πΌ))(1 β π2)(1 β π)
π πβπΌ
0
π’π
(2)
Third case: This case is related to the expected number of correctly classified nondefective items, as described by Equation 3.
{(
(
π3 = 1 β π1 + (π β πΌ)
π2 β π1 π
π πβπΌ
)β«0
(π βπ πΌ + π’)ππ’)}
π’π
(π( ))(1 β π1)(1 β π) π πβπΌ
(3)
Fourth case: This case represents the expected number of false rejections of nondefective items, as described by Equation 4.
{(
(
π4 = 1 β π1 + (π β πΌ)
(π( ))π1(1 β π) π πβπΌ
π πβπΌ
π2 β π1
)β«0
π
(
π’π
)
)}
π + π’ ππ’ πβπΌ
(4)
Next, we consider the inspection cost described by Equation 5, which relates to the cost of inspecting an item πΆπ multiplied by the number of inspected items.
( ( ))
πΆπΌ = πΆπ(1 β π) π
π πβπΌ
(5)
The cost of the false rejection of nondefective items is described by Equation 6, where πΆπ is multiplied by the number of erroneously rejected nondefective items. (6)
πΆπΈ = πΆππ4
The next cost is related to the cost of reworking, which can be obtained by multiplying the cost of each reworking Rc by the expected number of items sent for reworking, as described by Equation 7. (7)
π πΆ = π ππ2
The maintenance cost is the minimal repair cost C1 considering the failures that can occur during item processing [0,T] plus the cost per unit time for preventive maintenance, as described by Equation 8. π
β
ππ΄πΆ = πΆ1β«π0β πΌπ(π₯)ππ₯ + πΆπβ«0 π‘π(π‘)ππ‘
(8)
As the material (Eq 9), holding (Eq 10), and shortage (Eq 11) costs do not depend on the quality inspection, these costs were modelled following Sana (2012).
The material cost consists of the material cost per item multiplied by the number of items produced, as described by Equation 9.
( ( ))
ππΆ = ππ π
π πβπΌ
(9)
The cost of holding quantity S of the buffer stock is described by Equation 10. 2
π»πΆ =
(12)πΆβ(π β πΌ)πΌπ(π βπ πΌ)
(10)
The cost of shortage is described by Equation 11. π
β
(11)
πΆπ = πΆπ πΌβ«π (π‘ β πΌ)π(π‘)ππ‘ πΌ
The cost of the warranty is the cost of the defective products that as a result of type II error are accepted and sent to the market plus the cost of the products correctly classified as nondefective and sent to the market but that fail within period w plus the cost of the products that were not inspected and sent to the market and fail within period w, as described by Equation 12.
{
{
}
π€
π€
}
ππΆ = πΆπ π1β«0 β2(π₯)ππ₯ + πΆπ π3β«0 β1(π₯)ππ₯ + πΆπ
{{ {{ (
(π βπ πΌ + π’)ππ’}(π(π βπ πΌ))πβ« β2(π₯)ππ₯} + πΆπ π βπ π π1 + (π β πΌ)( π )β«0 π’π(π β πΌ + π’)ππ’)}πβ« β1(π₯)ππ₯ }
π1 + (π β πΌ) 1β
π2 β π1
(
π
)β« 2
π πβπΌ
π€ 0
π’π
0
1
π πβπΌ
π€ 0
(12)
Finally, the composition of the total cost is represented by a sum of the costs of inspection, rework, setup, warranty, holding, shortage, false rejection of nondefective items, materials, and maintenance divided by the quantity of items produced, as described by Equation 13.
πΆ(π,π) =
πΎ + πΆπΌ + πΆπΈ + π πΆ + ππΆ + ππ΄πΆ + π»πΆ + πΆπ + ππΆ
( )
π
π πβπΌ
(13)
To ensure average outgoing quality in the production process, the model uses the constraint of defective items shipped to market. This constraint consists of inspecting a set of defective items erroneously accepted as nondefective plus defective items that have not been inspected, as described by Equation 14. π΄ππ =
1
(
π
){
π πβπΌ
{(
{π1} + π1 + (π β πΌ)
π2 β π1
(
π
)
π
(π βπ πΌ + π’)ππ’(π(π βπ πΌ)))π}}
β«π β πΌπ’π 0
(14)
The optimization problem is to find the minimum cost that is subject to the AOQ constraint, as described by Equation 15.
(15)
πππππππ§π: πΆ(π,π) ππ’πππππ‘ π‘π π΄ππ β€ ππππ₯
0β€πβ€1 π>0 4.
Numerical Example
In this section, we use a numerical example to illustrate the proposed model. For illustrative purposes, we set πΆπ = $4, πΆπ = $20, K = $700, Ξ± = 500 units, p = 800 units, Cm = $150, Co = $140, Cr = $70, C1 = $20, Rc =$30, Ch =$2, Cs = $8, m1 =0.06, m2 = 0.02, π(π) = 0.6π β 0.6π, ΞΈ1 = 0.12, ΞΈ2 = 0.30, w = 4, β1(π₯) = =
1 2.5 2.5π₯2.5 β 1, 15
( )
2.5
(401 )
2.5π₯2.5 β 1, β1(π₯)
and ππππ₯ = 0.01. The distribution of process failure is described
by the Weibull distribution, πΉ(π₯) = exp [ β (ππ₯)2], with a mean time to failure E(x)=1 and π = Ξ(1 + 1/2). The time required for preventive maintenance follows distribution Ο(t)= 0.8π β 0.8π. Finding the optimal values by using Equation 15 results in buffer value S*=943.40, with a fraction of items not inspected as i*=0.059 and a production cycle of T*=3.145. This results in an expected cost of C = $161.235, with an inspection cost of
9.467ο΄103, a warranty cost of $522.792, a maintenance cost of $330.337, a cost of falsely accepted defective items of $2.476ο΄103, a holding cost of $4.747ο΄103, a shortage cost of 1.105ο΄103, a rework cost of $8.916ο΄103, and a total quantity produced of 2.516ο΄103. Fig. 1 (a)β(g) Sensitivity of the cost rate of the optimal (S,i) policy for various parameters: (a) cost of inspection πΆπ; (b) type I error m1; (C) cost of false rejection πΆπ; (d) nonconforming items in in-control state ΞΈ1; (e) error type II m2; (f) ππππ₯ number of defective items sent to market; (g) nonconforming items in out-of-control state ΞΈ2. In Fig. 2, a logic chart is presented to facilitate the process of implementing the proposed policy.
4.1. Sensitivity analysis A numerical experiment was designed to study the effects of changing the modelβs parameters. To study the impact of the variation of quality inspection parameters, Table 1 shows variations in the parameters (πΆπ, πΆπ, m2, m1, ΞΈ1, ΞΈ2, ππππ₯) with changes (β50%, β25%, +25%, +50%) while the other parameters are maintained fixed, as in the numerical example.
ο·
Variation of inspection cost πΆπ: When the cost of inspection decreases, the
fraction of items not inspected decreases. Thus, more items are inspected. The decrease in the cost of inspection also results in a lower cost per item produced. The cost of the erroneous rejection of nondefective items decreases, reflecting the increase in the number of items inspected. An increase in the cost produces the opposite effects. ο·
Variation false rejection of nondefective items πΆπ: When the value of πΆπ
decreases, the cost of the false rejection of nondefective items decreases. This cost is related to the number of nondefective items sent for reworking. Decreasing the value of πΆπ decreases the inspection error penalty. When the value of πΆπ increases, the cost of false rejection of nondefective items increases. ο·
Variation m1 representing type I error: As a result of changes in the value of
m1, the fraction of items not inspected increases, which can be explained by the fact that to reduce error in inspection, fewer items must be inspected. Consequently, the cost of inspection and the cost per item decrease. When the value of m1 increases, the fraction of items not inspected decreases, which indicates that more items should be inspected to satisfy the AOQ constraint. ο·
Variation m2 representing Type II error: With a decrease of 50% in the value
of m2, the cost of the false rejection of nondefective items decreases significantly. The number of items produced increases, reflecting an increase in the production run time. The cost per item produced decreases with a decrease in the value of m2. When the value of m2 increases, the cost per item produced increases. ο·
Variation of the parameter π½π: In case of an increase in the value of ΞΈ1, the
probability of nonconforming items increases when the production process is in the incontrol state. This model indicates that more items should be inspected. The increase in
ΞΈ1 results in a higher cost per item produced. When the value of ΞΈ1 decreases, the cost per item produced decreases. ο·
Variation of the parameter ΞΈ2: When considering a smaller value of ΞΈ2, the
fraction of items not inspected increases. This outcome can be explained by the fact that the system produces fewer defective items, which results in a lower inspection cost and a lower cost per item. However, when the value of ΞΈ2 increases, the number of items inspected increases to be able to meet the AOQ constraint. ο·
Variation of the maximum value of defective items sent to market ππππ₯:
When decreasing the value of ππππ₯, the fraction of items not inspected decreases, which reflects the need to satisfy the constraint of defective items sent to the market. When the value of ππππ₯ increases, the fraction of items not inspected increases.
4.1.2 Investing in higher-quality inspection and process improvements Inspection and process quality can be improved in a variety of ways, such as process technology investments and inspection training. To analyse the impact of joint improvements, we present Table 2, which highlights where investment may be most beneficial. Comparing the base case (line 1) with cases of joint improvement of ΞΈ1 with m1 (lines 2 and 3), we can observe that one results in a greater reduction in cost per unit produced than the reduction relative to ΞΈ2. An improvement in ΞΈ1 represents producing fewer nonconforming items during the period in which the process is in control. In practice, this improvement can be achieved by investing in process technology to reduce the production of nonconforming items. The reduction of m1 can be achieved by investing in quality inspection procedures. Another effect observed with the reduction of ΞΈ1 was longer production times and less need for quality inspection. According to line 6, joint improvement results in cost savings and a reduced need for inspection.
5.
Managerial implications
The quality control, maintenance, and buffer stock policy proposed in this study can be implemented for a single item or processing systems for similar items in which the inspection technique can be applied effectively. The cost of inspecting all items may be high, where the proposed model indicates the inspection of a proportion of items to meet the market demands constraints in relation to the number of defective items. The proposed policy can be applied in factories that produce electronic equipment, semiconductors, and military equipment. In addition, the imperfect production process is considered to result from imperfect inspection, which implies a comprehensive production system. The production manager must define this system in an integrated way and be aware that quality and the production rate contain information on the process that can be employed to make decisions regarding the quantity of items to inspect. Another factor considered in the model is the product warranty. Inspection can identify defective items, which are subsequently sent for reworking. Defective products sent to the market affect consumer satisfaction. As noted by Jack and Murth (2004), customer satisfaction with a product depends on the productβs performance under warranty and during the rest of its useful life. A rigorous inspection policy can lower the warranty cost and increase customer satisfaction. Conflicts between production and maintenance occur because of the need to meet demand. A buffer stock is established to meet demand while preventive maintenance is performed. Joint planning helps better meet demand and avoid conflicts between production and maintenance.
6.
Conclusion
A production system must satisfy expectations at the quality level established by the customer. In this study, we propose a holistic approach to a joint optimization of quality control, buffer, and maintenance scheduling while considering an output quality constraint for an imperfect production system. The suggested approach contributes to research on the integrated decision-making process. We investigated the characteristics of imperfect quality to demonstrate the relevance of quality information and imperfect inspection to the decision-making process. At the practical level, the operations manager must understand the relationships between quality control, production, and maintenance. A production system requires coordination between different elements to increase productivity and decrease costs. Through a numerical experiment, it was possible to analyse the effects of the modelβs parameters on the cost of the items produced.
Acknowledgement This research was supported by the Foundation for Science and Technology of the State of Pernambuco- Brazil [grant number: APQ-0228-3.08/15].
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Fig. 1. XXX Fig. 2. XXX
Table 1. Variation of parameters
Changes (in percent) -50 -25 πΆπ +25 +50 -50 -25 πΆπ +25 +50 -50 -25 m2 +25 +50 -50 -25 m1 +25 +50 -50 -25 ΞΈ1 +25 +50 -50 ΞΈ2 -25 +25 +50 -50 πmax -25 +25 +50
Optimal noninspec Optimal ted Optimal production Inspection Warranty Maintenance fraction buffer run time cost cost cost 0.0591 939.457 3.132 4.714x10Β³ 520.568 329.039 0.0592 941.431 3.138 7.086x10Β³ 521.679 329.687 4 0.0592 945.387 3.151 1.186x10 523.907 330.990 0.0592 947.360 3.158 1.426x104 525.019 331.642 0.0592 944.168 3.147 9.475x10Β³ 523.221 330.588 0.0592 943.787 3.146 9.471x10Β³ 523.007 330.463 0.0592 943.026 3.143 9.463x10Β³ 522.578 330.212 0.059 942.646 3.142 9.459x10Β³ 522.363 330.087 0.0687 943.342 3.144 9.371x10Β³ 523.033 330.316 0.0640 943.374 3.145 9.419x10Β³ 522.913 330.327 0.0544 943.440 3.145 9.516x10Β³ 522.671 330.348 0.0495 943.476 3.145 9.565x10Β³ 522.548 330.36 0.0592 944.159 3.147 9.474x10Β³ 536.93 330.585 0.0592 943.785 3.146 9.471x10Β³ 529.859 330.461 0.0592 943.031 3.143 9.463x10Β³ 515.732 330.213 0.0592 942.654 3.142 9.46x10Β³ 508.676 330.09 0.1264 1.008x10Β³ 3.363 9.401x10Β³ 594.293 352.658 0.0826 968.178 3.227 9.474x10Β³ 552.943 338.602 0.0445 923.730 3.079 9.414x10Β³ 496.604 323.925 0.0334 906.181 3.021 9.332x10Β³ 472.387 318.32 0.0635 860.111 2.867 8.592x10Β³ 479.876 304.118 0.061 901.657 3.006 9.03x10Β³ 501.2 316.893 0.0577 984.587 3.282 9.896x10Β³ 544.222 344.194 0.0565 1.0245x10Β³ 3.415 1.031x104 565.152 358.225 0.0193 938.2872 3.128 9.814x10Β³ 486.597 328.656 0.0392 940.8459 3.136 9.641x10Β³ 504.648 329.495 0.0791 945.9706 3.153 9.291x10Β³ 541.028 331.183 0.0991 948.5420 3.162 9.115x10Β³ 559.361 332.033
Table 2. Analysis of the benefit of joint improvements
Cost of false rejection of nondefecti ve items 2.466x10Β³ 2.471x10Β³ 2.481x10Β³ 2.487x10Β³ 1.239x10Β³ 1.858x10Β³ 3.094x10Β³ 3.711x10Β³ 2.451x10Β³ 2.464x10Β³ 2.489x10Β³ 2.502x10Β³ 1.239x10Β³ 1.858x10Β³ 3.094x10Β³ 3.711x10Β³ 2.624x10Β³ 2.561x10Β³ 2.381x10Β³ 2.279x10Β³ 2.264x10Β³ 2.37x10Β³ 2.581x10Β³ 2.683x10Β³ 2.567x10Β³ 2.522x10Β³ 2.43x10Β³ 2.384x10Β³
Rework cost 4.707x10Β³ 4.727x10Β³ 4.767x10Β³ 4.787x10Β³ 4.754x10Β³ 4.751x10Β³ 4.743x10Β³ 4.739x10Β³ 4.746x10Β³ 4.746x10Β³ 4.474x10Β³ 4.747x10Β³ 4.754x10Β³ 4.751x10Β³ 4.743x10Β³ 4.739x10Β³ 5.429x10Β³ 4.999x10Β³ 4.551x10Β³ 4.38x10Β³ 3.946x10Β³ 4.336x10Β³ 5.17x10Β³ 5.599x10Β³ 4.695x10Β³ 4.721x10Β³ 4.773x10Β³ 4.799x10Β³
Holdin cost 1.112x 1.109x 1.102x 1.098x 1.104x 1.104x 1.106x 1.107x 1.105x 1.105x 1.105x 1.105x 1.104x 1.104x 1.106x 1.106x 995 1.062x 1.141x 1.173x 1.263x 1.182x 1.035x 970 1.114x 1.11x 1.101x 1.096x
Joint Improvement 1 Base Case 2 ΞΈ1 and m1 3 ΞΈ1 and m2 4 ΞΈ2 and m1 5 ΞΈ2 and m2 6 ΞΈ1, ΞΈ2 and m1 and m2
ΞΈ1 0.12 0.01 0.01 0.12 0.12 0.01
ΞΈ2 0.30 0.30 0.30 0.15 0.15 0.15
m2 0.02 0.02 0.01 0.01 0.02 0.01
m1 0.06 0.02 0.06 0.06 0.02 0.02
Optimal noninspected fraction 0.059 0.812 0.834 0.073 0.064 0.842
Optimal Optimal production buffer runtime 943.40 3.145 1728.15 5.760 1785.09 5.950 860.12 2.867 860.27 2.867 1491.85 4.972
Cost per unit 161.23 154.97 155.09 160.95 160.35 154.41
S0360-8352(18)30489-3 https://doi.org/10.1016/j.cie.2018.10.019 CAIE 5458
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
1 March 2018 17 August 2018 8 October 2018
Please cite this article as: Lopes, R., Integrated model of quality inspection, preventive maintenance and buffer stock in an imperfect production system, Computers & Industrial Engineering (2018), doi: https://doi.org/10.1016/j.cie. 2018.10.019
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Integrated model of quality inspection, preventive maintenance and buffer stock in an imperfect production system Rodrigo [email protected] Department of Production Engineering, Federal University of Pernambuco, Rodovia BR-104, Km 59, s/n, Nova Caruaru, Caruaru - PE, CEP- 55002-970, Brazil
Highlights We propose an integrated quality control, preventive maintenance and buffer decision. We examined inspection errors, product warranty, and rework in an imperfect process. Numerical example is presented with managerial implications.
Abstract In this study, the influence of a quality inspection policy on an imperfect production system in which a percentage of the produced items is inspected is modelled. The inspection policy is imperfect, permitting type I and type II errors. Items are shipped for sale with a free minimal repair warranty policy. After each production cycle, preventive maintenance is performed. In case of failure in the production process before the cycleβs end, a minimal repair is indicated. A buffer stock is established to meet demand while preventive maintenance is performed. We also considered that defective items detected on inspection are sent for reworking. The model is formulated with the objective of minimizing the total cost expected per item while considering an average outgoing quality constraint. Subsequently, we present a numerical experiment to illustrate the proposed model. The results indicate that the probability of type I and type II inspection
errors has a significant impact on optimal production and the percentage of items inspected. Keywords: quality inspection, imperfect production system, integrated decision-making, imperfect inspection, type I and II inspection errors.
1.
Introduction
Quality inspection, production planning, and maintenance scheduling are the most fundamental elements of a production system. Integration between quality, production, and maintenance is vital for a company to be successful, and together, these elements contribute to improving the efficiency of a production system by decreasing costs (Bouslah et al., 2016). In most production systems, production is imperfect, resulting in a number of defective item; such items may result from insufficient process control, poorly planned maintenance, inadequate working instructions, or damage during production (Rad et al., 2014). Imperfect production results in items that do not conform to specifications, which can force the customer to return the defective item for an exchange or to request a refund. In e-commerce, the return of defective items can represent up to 25% of total sales (Krikke et al., 2004).
Therefore, to reduce the negative effects of defective items shipped to market, companies often invest in inspection processes to ensure their products conform to market expectations (Cavalcante et al., 2016). Both product and process quality should be defined (Muhammad et al., 2017). Consequently, the production manager must carefully consider the internal cost related to the inspection process and the post-sales costs of defective products (Juran and Gryna, 2016). The primary purpose of inspection is to identify product units that do not conform to product specifications. Nonconforming products can be discarded or reworked (Nourefath et al., 2016). Inspection operations may involve two types of error: the classification of a unit that conforms as nonconforming and the classification of a unit that does not conform as conforming. The first error type results in waste and low productivity. The second type results in poor quality, which can lead to warranty costs and loss of customers (Bouslah et al., 2016). In the literature, models have been presented that address the effects of quality inspection, maintenance, and buffer stock on an imperfect production process. Without being exhaustive, Ben-Daya (2002) derived an integrated model for joint determination of the amount of economic production and preventive maintenance in an imperfect process with an increasing rate of failure. Wang and Sheu (2003) determined an optimal policy of production, inspection, and maintenance under the effects of process inspection errors. Wang (2005) considered an inspection policy with two types of inspection error to determine the amount of economic production quantity. Darwish and Ben-Daya (2007) developed a stock model for an imperfect production process with errors in preventive maintenance and inspection. Lin and Yu (2009) investigated a system of imperfect production, maintenance, and production lot quality, with increasing rates of risk. Liao et al. (2009) developed preventive maintenance
programmes with an economic production model for an imperfect production process in which all defective items must be reworked. Chakraborty et al. (2009) state that nearly all integrated models do not simultaneously consider deterioration of the quality and reliability of equipment, and when both phenomena are observed, preventive maintenance will have the following two objectives: to restore the equipment and to improve the quality of the process. Hadidi et al. (2012) present a review of integrated models for production planning and scheduling, maintenance, and quality inspections. Bouslah et al. (2016) developed an integrated model for sampling quality control and the maintenance of deteriorating production systems; the model considered an average output quality constraint. Li et al. (2017) studied inventory and process deterioration in the economic production quantity (EPQ) model with reworking and backlogs. Kang and Subramaniam (2018) developed a dynamic maintenance policy integrated with production and opportunistic maintenance in a joint optimization problem. Cheng et al. (2018) presented a model based on simulation to jointly determine production, quality control, and maintenance for a single-unit production system that adopts a buffer stock. In this study, we extend the research of Sana (2012), who analysed an imperfect production system to determine optimal buffer levels and production runtime but did not determine the fraction of items inspected or consider imperfect inspection. We consider that the inspection is imperfect and that only a fraction of the items produced are inspected. More specifically, we construct a model that describes a production process in which at the beginning of the process the system is in an in-control state and produces few defective items. However, the system can shift into an out-of-control state at any time and start producing more defective items than in the in-control state. A fraction of items are inspected, and the inspection is imperfect. Defective items detected through inspection are reworked at a fixed cost. The items are sold with a minimal free
repair warranty (FRW). After each production cycle, preventive maintenance is performed, and in case of a failure in the production process before the cycleβs end, a minimal repair action is indicated. The manufacturer establishes a buffer stock to adjust to market demand. The buffer is consumed when preventive maintenance is performed. The model is formulated with the objective of minimizing the total cost expected per item while considering an average outgoing quality constraint. The remainder of this paper is organized as follows. The notations and assumptions are presented in Section 2. Section 3 presents the mathematical model. In Section 4, a numerical example is considered and a sensitivity analysis performed. Managerial implications are discussed in Section 5. Section 6 presents our conclusions.
2.
Notation and assumptions
2.1. Notation The following notations are used to develop the model: Decision variables S
buffer stock
i
noninspected fraction (0 β€ i β€ 1)
Parameters πΆπ
unit inspection cost
πΆπ
false rejection of nondefective item
K
setup cost per cycle
Ξ±
demand rate
p
production rate, where p>Ξ±
Cm
cost of material to produce an item
Co
cost per unit time for preventive maintenance
Cr
cost per item for each minimal warranty repair
Rc
rework cost per unit defective item
Ch
holding cost per unit item per unit time
Cs
shortage cost per unit item per unit time
C1
minimal repair cost of maintenance
Ο
a random variable, which is time-elapsed after which the production process shifts to the out-of-control state
m1
random variable representing type I error
m2
random variable representing type II error
f(Ο)
probability density function of Ο with mean 1/Β΅
ΞΈ1
probability of nonconforming items when the production process is in the in-control state
ΞΈ2
probability of nonconforming items when the production process is in the out-of-control state. Here, 0<ΞΈ1< ΞΈ2<1
T
production runtime per cycle
w
warranty period
h1(x)
failure rate function of conforming items
h2(x)
failure rate function of nonconforming items
t
preventive maintenance time, which is a random variable
Ο(t)
probability density function of t π΄ππ
average outgoing quality
ππππ₯
maximum percentage of defective items sent to the market
C(S, i)
expected total cost per item
2.2. Assumptions The following assumptions are considered while formulating the model:
1. Product inspection is performed to detect defective items. 2. Two types of errors, that is, type I and type II errors, occur during product inspection. 3. The inspection time is negligible. 4. This model assumes ΞΈ1< ΞΈ2; that is, the probable number of defective items in the incontrol state is less than the probable number of defective items in the out-of-control state. 5. Products are sold with a free minimal repair warranty. 6. The buffer stock is not subject to deterioration or obsolescence. 7. Minimal repair action does not affect the production rate. The first assumption is related to the inspection policy, according to which a fraction of items are not inspected. The second assumption is that the inspection process is not perfect and permits type I and type II errors. The third assumption implies that the time required to inspect the items is insignificant. The fourth assumption is related to an imperfect production system. The fifth assumption is related to the product warranty policy. The sixth assumption is that the items in the stock buffer are not subject to deterioration or obsolescence. The seventh assumption is related to the minimal repair action required when the system fails. This action requires an insignificant amount time such that the production rate is not affected. In the following section, the proposed model is formulated mathematically.
3.
Mathematical Model
3.1. System Description We consider a single process or equipment that produces a single product type or set of similar products at a particular rate of production p, whereby the process can be performed in two states: in-control and out-of-control. The in-control state produces
nonconforming items at rate ΞΈ1. At any point in time, this system can enter the out-ofcontrol state, in which the production rate of nonconforming items is higher: rate ΞΈ2. In this system, a product inspection policy is implemented, where i is the fraction of items not inspected. The inspection system is imperfect, and errors may occur. Type I error has the probability m1, whereas type II error has the probability m2. When type I error occurs, nondefective products are classified as defective, which results in cost πΆπ. However, when type II error occurs, certain defective items may be classified as nondefective and sent to the market, and these items increase the cost of the warranty. The inspection process has cost πΆπ. To meet consumer expectations, the system is subject to a constraint of defective items sent to the market ππππ₯. The average number of defective items AOQ (average outgoing quality) sent to the market must not exceed the maximum limit ππππ₯. After inspection, items classified as defective are sent for reworking at cost Rc. The products are sold with a minimal repair warranty policy, according to which each product failure is rectified by a free minimal repair in period w with cost Cr. The warranty policy considers the rate of failure of nonconforming items as greater than that of conforming items. In this system, to adjust production to demand, buffer stock S is maintained with cost Ch. This buffer stock is consumed at rate Ξ± while preventive maintenance is being performed. The system produces items during period T, after which, preventive maintenance is performed. Production equipment is restored to new condition, and the system returns to the in-control state. If there is a failure between [0,T], minimal repair actions are performed. This minimal repair action does not affect production rate p. During the preventive maintenance period, demand is met by buffer stock S. If the demand is greater than the buffer stock, this demand is lost at cost Cs. The buffer stock
consumption is described by the following equation: π = (π β πΌ)π, that is, π = π/ (π β πΌ). Our objective is to simultaneously find the optimal S and i, which minimizes the cost per unit produced considering a constraint of the defective items sent to the market (AOQ).
3.2. Cost structure For the described system, the cost of production includes the costs of quality inspection, material, maintenance, rework, setup, and warranty. Next, we discuss the composition of each cost in detail. First, as there are two types of error, the items can be divided into four cases: First case: This case is related to the expected number of items produced with defects that are inspected and false accepted according to type II error, as described by Equation 1.
{
π1 = π1 + (π β πΌ)
(
π2 β π1
)
π
(π βπ πΌ + π’)ππ’}(π(π βπ πΌ))π2(1 β π)
π
β«π β πΌπ’π 0
(1)
Second case: This case describes the expected number of items that are correctly classified as defective items, as described by Equation 2.
{
π2 β π1
(
π2 = π1 + (π β πΌ)
π
)β«
(π βπ πΌ + π’)ππ’}(π(π βπ πΌ))(1 β π2)(1 β π)
π πβπΌ
0
π’π
(2)
Third case: This case is related to the expected number of correctly classified nondefective items, as described by Equation 3.
{(
(
π3 = 1 β π1 + (π β πΌ)
π2 β π1 π
π πβπΌ
)β«0
(π βπ πΌ + π’)ππ’)}
π’π
(π( ))(1 β π1)(1 β π) π πβπΌ
(3)
Fourth case: This case represents the expected number of false rejections of nondefective items, as described by Equation 4.
{(
(
π4 = 1 β π1 + (π β πΌ)
(π( ))π1(1 β π) π πβπΌ
π πβπΌ
π2 β π1
)β«0
π
(
π’π
)
)}
π + π’ ππ’ πβπΌ
(4)
Next, we consider the inspection cost described by Equation 5, which relates to the cost of inspecting an item πΆπ multiplied by the number of inspected items.
( ( ))
πΆπΌ = πΆπ(1 β π) π
π πβπΌ
(5)
The cost of the false rejection of nondefective items is described by Equation 6, where πΆπ is multiplied by the number of erroneously rejected nondefective items. (6)
πΆπΈ = πΆππ4
The next cost is related to the cost of reworking, which can be obtained by multiplying the cost of each reworking Rc by the expected number of items sent for reworking, as described by Equation 7. (7)
π πΆ = π ππ2
The maintenance cost is the minimal repair cost C1 considering the failures that can occur during item processing [0,T] plus the cost per unit time for preventive maintenance, as described by Equation 8. π
β
ππ΄πΆ = πΆ1β«π0β πΌπ(π₯)ππ₯ + πΆπβ«0 π‘π(π‘)ππ‘
(8)
As the material (Eq 9), holding (Eq 10), and shortage (Eq 11) costs do not depend on the quality inspection, these costs were modelled following Sana (2012).
The material cost consists of the material cost per item multiplied by the number of items produced, as described by Equation 9.
( ( ))
ππΆ = ππ π
π πβπΌ
(9)
The cost of holding quantity S of the buffer stock is described by Equation 10. 2
π»πΆ =
(12)πΆβ(π β πΌ)πΌπ(π βπ πΌ)
(10)
The cost of shortage is described by Equation 11. π
β
(11)
πΆπ = πΆπ πΌβ«π (π‘ β πΌ)π(π‘)ππ‘ πΌ
The cost of the warranty is the cost of the defective products that as a result of type II error are accepted and sent to the market plus the cost of the products correctly classified as nondefective and sent to the market but that fail within period w plus the cost of the products that were not inspected and sent to the market and fail within period w, as described by Equation 12.
{
{
}
π€
π€
}
ππΆ = πΆπ π1β«0 β2(π₯)ππ₯ + πΆπ π3β«0 β1(π₯)ππ₯ + πΆπ
{{ {{ (
(π βπ πΌ + π’)ππ’}(π(π βπ πΌ))πβ« β2(π₯)ππ₯} + πΆπ π βπ π π1 + (π β πΌ)( π )β«0 π’π(π β πΌ + π’)ππ’)}πβ« β1(π₯)ππ₯ }
π1 + (π β πΌ) 1β
π2 β π1
(
π
)β« 2
π πβπΌ
π€ 0
π’π
0
1
π πβπΌ
π€ 0
(12)
Finally, the composition of the total cost is represented by a sum of the costs of inspection, rework, setup, warranty, holding, shortage, false rejection of nondefective items, materials, and maintenance divided by the quantity of items produced, as described by Equation 13.
πΆ(π,π) =
πΎ + πΆπΌ + πΆπΈ + π πΆ + ππΆ + ππ΄πΆ + π»πΆ + πΆπ + ππΆ
( )
π
π πβπΌ
(13)
To ensure average outgoing quality in the production process, the model uses the constraint of defective items shipped to market. This constraint consists of inspecting a set of defective items erroneously accepted as nondefective plus defective items that have not been inspected, as described by Equation 14. π΄ππ =
1
(
π
){
π πβπΌ
{(
{π1} + π1 + (π β πΌ)
π2 β π1
(
π
)
π
(π βπ πΌ + π’)ππ’(π(π βπ πΌ)))π}}
β«π β πΌπ’π 0
(14)
The optimization problem is to find the minimum cost that is subject to the AOQ constraint, as described by Equation 15.
(15)
πππππππ§π: πΆ(π,π) ππ’πππππ‘ π‘π π΄ππ β€ ππππ₯
0β€πβ€1 π>0 4.
Numerical Example
In this section, we use a numerical example to illustrate the proposed model. For illustrative purposes, we set πΆπ = $4, πΆπ = $20, K = $700, Ξ± = 500 units, p = 800 units, Cm = $150, Co = $140, Cr = $70, C1 = $20, Rc =$30, Ch =$2, Cs = $8, m1 =0.06, m2 = 0.02, π(π) = 0.6π β 0.6π, ΞΈ1 = 0.12, ΞΈ2 = 0.30, w = 4, β1(π₯) = =
1 2.5 2.5π₯2.5 β 1, 15
( )
2.5
(401 )
2.5π₯2.5 β 1, β1(π₯)
and ππππ₯ = 0.01. The distribution of process failure is described
by the Weibull distribution, πΉ(π₯) = exp [ β (ππ₯)2], with a mean time to failure E(x)=1 and π = Ξ(1 + 1/2). The time required for preventive maintenance follows distribution Ο(t)= 0.8π β 0.8π. Finding the optimal values by using Equation 15 results in buffer value S*=943.40, with a fraction of items not inspected as i*=0.059 and a production cycle of T*=3.145. This results in an expected cost of C = $161.235, with an inspection cost of
9.467ο΄103, a warranty cost of $522.792, a maintenance cost of $330.337, a cost of falsely accepted defective items of $2.476ο΄103, a holding cost of $4.747ο΄103, a shortage cost of 1.105ο΄103, a rework cost of $8.916ο΄103, and a total quantity produced of 2.516ο΄103. Fig. 1 (a)β(g) Sensitivity of the cost rate of the optimal (S,i) policy for various parameters: (a) cost of inspection πΆπ; (b) type I error m1; (C) cost of false rejection πΆπ; (d) nonconforming items in in-control state ΞΈ1; (e) error type II m2; (f) ππππ₯ number of defective items sent to market; (g) nonconforming items in out-of-control state ΞΈ2. In Fig. 2, a logic chart is presented to facilitate the process of implementing the proposed policy.
4.1. Sensitivity analysis A numerical experiment was designed to study the effects of changing the modelβs parameters. To study the impact of the variation of quality inspection parameters, Table 1 shows variations in the parameters (πΆπ, πΆπ, m2, m1, ΞΈ1, ΞΈ2, ππππ₯) with changes (β50%, β25%, +25%, +50%) while the other parameters are maintained fixed, as in the numerical example.
ο·
Variation of inspection cost πΆπ: When the cost of inspection decreases, the
fraction of items not inspected decreases. Thus, more items are inspected. The decrease in the cost of inspection also results in a lower cost per item produced. The cost of the erroneous rejection of nondefective items decreases, reflecting the increase in the number of items inspected. An increase in the cost produces the opposite effects. ο·
Variation false rejection of nondefective items πΆπ: When the value of πΆπ
decreases, the cost of the false rejection of nondefective items decreases. This cost is related to the number of nondefective items sent for reworking. Decreasing the value of πΆπ decreases the inspection error penalty. When the value of πΆπ increases, the cost of false rejection of nondefective items increases. ο·
Variation m1 representing type I error: As a result of changes in the value of
m1, the fraction of items not inspected increases, which can be explained by the fact that to reduce error in inspection, fewer items must be inspected. Consequently, the cost of inspection and the cost per item decrease. When the value of m1 increases, the fraction of items not inspected decreases, which indicates that more items should be inspected to satisfy the AOQ constraint. ο·
Variation m2 representing Type II error: With a decrease of 50% in the value
of m2, the cost of the false rejection of nondefective items decreases significantly. The number of items produced increases, reflecting an increase in the production run time. The cost per item produced decreases with a decrease in the value of m2. When the value of m2 increases, the cost per item produced increases. ο·
Variation of the parameter π½π: In case of an increase in the value of ΞΈ1, the
probability of nonconforming items increases when the production process is in the incontrol state. This model indicates that more items should be inspected. The increase in
ΞΈ1 results in a higher cost per item produced. When the value of ΞΈ1 decreases, the cost per item produced decreases. ο·
Variation of the parameter ΞΈ2: When considering a smaller value of ΞΈ2, the
fraction of items not inspected increases. This outcome can be explained by the fact that the system produces fewer defective items, which results in a lower inspection cost and a lower cost per item. However, when the value of ΞΈ2 increases, the number of items inspected increases to be able to meet the AOQ constraint. ο·
Variation of the maximum value of defective items sent to market ππππ₯:
When decreasing the value of ππππ₯, the fraction of items not inspected decreases, which reflects the need to satisfy the constraint of defective items sent to the market. When the value of ππππ₯ increases, the fraction of items not inspected increases.
4.1.2 Investing in higher-quality inspection and process improvements Inspection and process quality can be improved in a variety of ways, such as process technology investments and inspection training. To analyse the impact of joint improvements, we present Table 2, which highlights where investment may be most beneficial. Comparing the base case (line 1) with cases of joint improvement of ΞΈ1 with m1 (lines 2 and 3), we can observe that one results in a greater reduction in cost per unit produced than the reduction relative to ΞΈ2. An improvement in ΞΈ1 represents producing fewer nonconforming items during the period in which the process is in control. In practice, this improvement can be achieved by investing in process technology to reduce the production of nonconforming items. The reduction of m1 can be achieved by investing in quality inspection procedures. Another effect observed with the reduction of ΞΈ1 was longer production times and less need for quality inspection. According to line 6, joint improvement results in cost savings and a reduced need for inspection.
5.
Managerial implications
The quality control, maintenance, and buffer stock policy proposed in this study can be implemented for a single item or processing systems for similar items in which the inspection technique can be applied effectively. The cost of inspecting all items may be high, where the proposed model indicates the inspection of a proportion of items to meet the market demands constraints in relation to the number of defective items. The proposed policy can be applied in factories that produce electronic equipment, semiconductors, and military equipment. In addition, the imperfect production process is considered to result from imperfect inspection, which implies a comprehensive production system. The production manager must define this system in an integrated way and be aware that quality and the production rate contain information on the process that can be employed to make decisions regarding the quantity of items to inspect. Another factor considered in the model is the product warranty. Inspection can identify defective items, which are subsequently sent for reworking. Defective products sent to the market affect consumer satisfaction. As noted by Jack and Murth (2004), customer satisfaction with a product depends on the productβs performance under warranty and during the rest of its useful life. A rigorous inspection policy can lower the warranty cost and increase customer satisfaction. Conflicts between production and maintenance occur because of the need to meet demand. A buffer stock is established to meet demand while preventive maintenance is performed. Joint planning helps better meet demand and avoid conflicts between production and maintenance.
6.
Conclusion
A production system must satisfy expectations at the quality level established by the customer. In this study, we propose a holistic approach to a joint optimization of quality control, buffer, and maintenance scheduling while considering an output quality constraint for an imperfect production system. The suggested approach contributes to research on the integrated decision-making process. We investigated the characteristics of imperfect quality to demonstrate the relevance of quality information and imperfect inspection to the decision-making process. At the practical level, the operations manager must understand the relationships between quality control, production, and maintenance. A production system requires coordination between different elements to increase productivity and decrease costs. Through a numerical experiment, it was possible to analyse the effects of the modelβs parameters on the cost of the items produced.
Acknowledgement This research was supported by the Foundation for Science and Technology of the State of Pernambuco- Brazil [grant number: APQ-0228-3.08/15].
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Fig. 1. XXX Fig. 2. XXX
Table 1. Variation of parameters
Changes (in percent) -50 -25 πΆπ +25 +50 -50 -25 πΆπ +25 +50 -50 -25 m2 +25 +50 -50 -25 m1 +25 +50 -50 -25 ΞΈ1 +25 +50 -50 ΞΈ2 -25 +25 +50 -50 πmax -25 +25 +50
Optimal noninspec Optimal ted Optimal production Inspection Warranty Maintenance fraction buffer run time cost cost cost 0.0591 939.457 3.132 4.714x10Β³ 520.568 329.039 0.0592 941.431 3.138 7.086x10Β³ 521.679 329.687 4 0.0592 945.387 3.151 1.186x10 523.907 330.990 0.0592 947.360 3.158 1.426x104 525.019 331.642 0.0592 944.168 3.147 9.475x10Β³ 523.221 330.588 0.0592 943.787 3.146 9.471x10Β³ 523.007 330.463 0.0592 943.026 3.143 9.463x10Β³ 522.578 330.212 0.059 942.646 3.142 9.459x10Β³ 522.363 330.087 0.0687 943.342 3.144 9.371x10Β³ 523.033 330.316 0.0640 943.374 3.145 9.419x10Β³ 522.913 330.327 0.0544 943.440 3.145 9.516x10Β³ 522.671 330.348 0.0495 943.476 3.145 9.565x10Β³ 522.548 330.36 0.0592 944.159 3.147 9.474x10Β³ 536.93 330.585 0.0592 943.785 3.146 9.471x10Β³ 529.859 330.461 0.0592 943.031 3.143 9.463x10Β³ 515.732 330.213 0.0592 942.654 3.142 9.46x10Β³ 508.676 330.09 0.1264 1.008x10Β³ 3.363 9.401x10Β³ 594.293 352.658 0.0826 968.178 3.227 9.474x10Β³ 552.943 338.602 0.0445 923.730 3.079 9.414x10Β³ 496.604 323.925 0.0334 906.181 3.021 9.332x10Β³ 472.387 318.32 0.0635 860.111 2.867 8.592x10Β³ 479.876 304.118 0.061 901.657 3.006 9.03x10Β³ 501.2 316.893 0.0577 984.587 3.282 9.896x10Β³ 544.222 344.194 0.0565 1.0245x10Β³ 3.415 1.031x104 565.152 358.225 0.0193 938.2872 3.128 9.814x10Β³ 486.597 328.656 0.0392 940.8459 3.136 9.641x10Β³ 504.648 329.495 0.0791 945.9706 3.153 9.291x10Β³ 541.028 331.183 0.0991 948.5420 3.162 9.115x10Β³ 559.361 332.033
Table 2. Analysis of the benefit of joint improvements
Cost of false rejection of nondefecti ve items 2.466x10Β³ 2.471x10Β³ 2.481x10Β³ 2.487x10Β³ 1.239x10Β³ 1.858x10Β³ 3.094x10Β³ 3.711x10Β³ 2.451x10Β³ 2.464x10Β³ 2.489x10Β³ 2.502x10Β³ 1.239x10Β³ 1.858x10Β³ 3.094x10Β³ 3.711x10Β³ 2.624x10Β³ 2.561x10Β³ 2.381x10Β³ 2.279x10Β³ 2.264x10Β³ 2.37x10Β³ 2.581x10Β³ 2.683x10Β³ 2.567x10Β³ 2.522x10Β³ 2.43x10Β³ 2.384x10Β³
Rework cost 4.707x10Β³ 4.727x10Β³ 4.767x10Β³ 4.787x10Β³ 4.754x10Β³ 4.751x10Β³ 4.743x10Β³ 4.739x10Β³ 4.746x10Β³ 4.746x10Β³ 4.474x10Β³ 4.747x10Β³ 4.754x10Β³ 4.751x10Β³ 4.743x10Β³ 4.739x10Β³ 5.429x10Β³ 4.999x10Β³ 4.551x10Β³ 4.38x10Β³ 3.946x10Β³ 4.336x10Β³ 5.17x10Β³ 5.599x10Β³ 4.695x10Β³ 4.721x10Β³ 4.773x10Β³ 4.799x10Β³
Holdin cost 1.112x 1.109x 1.102x 1.098x 1.104x 1.104x 1.106x 1.107x 1.105x 1.105x 1.105x 1.105x 1.104x 1.104x 1.106x 1.106x 995 1.062x 1.141x 1.173x 1.263x 1.182x 1.035x 970 1.114x 1.11x 1.101x 1.096x
Joint Improvement 1 Base Case 2 ΞΈ1 and m1 3 ΞΈ1 and m2 4 ΞΈ2 and m1 5 ΞΈ2 and m2 6 ΞΈ1, ΞΈ2 and m1 and m2
ΞΈ1 0.12 0.01 0.01 0.12 0.12 0.01
ΞΈ2 0.30 0.30 0.30 0.15 0.15 0.15
m2 0.02 0.02 0.01 0.01 0.02 0.01
m1 0.06 0.02 0.06 0.06 0.02 0.02
Optimal noninspected fraction 0.059 0.812 0.834 0.073 0.064 0.842
Optimal Optimal production buffer runtime 943.40 3.145 1728.15 5.760 1785.09 5.950 860.12 2.867 860.27 2.867 1491.85 4.972
Cost per unit 161.23 154.97 155.09 160.95 160.35 154.41