BERNOULLI SERIAL PRODUCTION LINES WITH QUALITY CONTROL DEVICES : THEORY AND APPLICATION

This paper develops the procedure for the analysis of the production systems with quality control devices. The evaluation of the production system requires an expression for the system performance measures as functions of the machine and buffer parameters. This paper presents a method for evaluating these functions and illustrates their practical utility using a case study at a production plant.


Introduction
Serial production lines with quality control (QC) devices are sets of processing machines, QC devices and material handling devices arranged in the consecutive order so as to produce high-quality products.The QC devices are installed to check whether the parts being processed meet a required specification or not [8,13].By doing this, a high-quality product can be produced to satisfy the customer.Thus, the manufacturing enterprise can be successful and everlasting in the competing market.Therefore, the goal of this paper is to provide an engineering method for estimating the performance of a serial production line with QC devices.Such a method can be used to improve an existing production line or to design a new production line so that it can perform in a highly efficient manner and have capability of producing high-quality products.
The processing machines and the QC devices are subject to random breakdowns, which lead to unscheduled downtime and loss of production.Two basic models of machines reliability have been discussed in the literature: Markovian [4][5][6][7] and Bernoulli [2,3,[10][11][12].Usually, the machine in the Markovian model needs two parameters to describe its behavior: one for the uptime and the other for the downtime.The parameters are conditional probabilities which are used to determine the state of the machine in a cycle time (i.e., the time necessary to process a part), given the condition being the state of the machine in the previous cycle.This is a Markov process, which is why the term "Markovian" is used.In the Bernoulli model, the state of a machine in each cycle is determined by the process of Bernoulli trials.Each of the models has its domain of applicability.Markovian models seem to be more appropriate for machining operations where the machine stops due to a technical malfunction and may stay in this state until a repair is carried out.Bernoulli models seem to be appropriate for assembly operations where the machines are interrupted for typically short periods of time, comparable with the cycle time.In this paper, the Bernoulli model is used to describe the machine reliability.
Two reasons for adopting the Bernoulli model are as follows.First, the simplicity of the model allows us to carry out the more complete analysis for the production line.Second, in the picture tube production lines, where we test the applicability of our results, a machine being down is often due to the pallet jam on the operation conveyor.To correct for this problem, a short period of time is required.The duration of the "breakdown" is the order of the cycle, and, therefore, the Bernoulli model is more appropriate.
Performance analysis of production lines with quality control devices was initiated in [9].Only the simplest case of two-machine one-buffer asymptotically reliable serial production line was addressed.Since in most applications more than two machines are involved in the production line, development of performance analysis of production lines with more than two machines and QC devices is desirable.In [1], a model of a long serial production line with quality control devices is introduced and analyzed.However, only highly reliable machines are considered.Therefore, the purpose of this research is to develop performance analysis of a long serial production line with unreliable processing machines and QC devices, and demonstrate its applicability using a case study at a production plant.
The model used in this paper is summarized in the following: the machines in the production line operate asynchronously with a fixed cycle time, and machine i fails during any cycle with a very small probability.A nondefective part is made defective by machine i with a very small probability.A defective part can be identified by the quality control device, Q i , with probability 0 ≤ d i ≤ 1.The above assumptions with 0 ≤ d i ≤ 1, for all i, and unreliable machines (not necessary to be asymptotically reliable) are considered in the model of this paper.
The outline of this paper is as follows: in Section 2, system model of a production line with QC devices is introduced, and the problem addressed is stated.Section 3 is devoted to the development of numerical approach for the system, while Section 4 is devoted to the development of analytical approach.The case study is described in Section 5. Finally, the conclusions are formulated in Section 6.All of the proofs are given in the appendix.

System model
Consider a serial production line with QC devices defined by the following assumptions (see Figure 2.1).
(ii) The machines (including QC devices) have identical cycle time t c .The time axis is slotted with the slot duration t.Machines begin operating at the beginning of each time slot, but QC devices begin checking at the ending of each time slot.(iii) Each buffer is characterized by its capacity,

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(iv) A nondefective part is made defective by machine m i with probability d i , i = 1,...,M − 1.A defective part can be detected by QC device Q i and discarded from the system.Parameters d i , i = 1,...,M − 1, are referred to as bad part ratios.
(v) Machine m i is starved during a time slot if buffer B i−1 is empty at the beginning of the time slot.QC device Q i is starved during a time slot if machine m i fails to produce a part at the beginning of the time slot.Machine m 1 is never starved.
(vi) Machine m i is blocked during a time slot if QC device Q i is blocked or is down at the time slot.QC device Q i is blocked during a time slot with the conditional probability (1 − d i ), given that buffer B i has N i parts at the beginning of the time slot and machine m i+1 fails to take a part during the time slot.Machine m M is never blocked.
(vii) Machine m i and QC device Q i , being neither blocked nor starved during a time slot, produces a part with probability p i and operates to check a part with probability c i , respectively, and fails to do so with probability (1 − p i ) and (1 − c i ), respectively.Parameters p i and c i are referred to as the production rate of machine m i and working rate of QC device Q i in isolation, respectively.
A few remarks concerning this model are in order.
Remark 2.1.To gain the computation simplicity of the performance evaluation for the production line, we assume that there is no QC device after machine m M in assumption (i).If there is a QC device installed after the last machine on the factory floor, one virtual large buffer (e.g., N M = 20) and one virtual perfect machine (i.e., p M+1 = 1) can be added to accommodate this situation.Since the performance of the modified production line is close to the performance of the original production line, the former one can be analyzed by using the method developed in this paper to evaluate the performance of the latter one.
Remark 2.2.If some c i = 1 and d i = 0 in assumptions (i)-(vii), then machine m i can be viewed as a machine without a QC device after it.That is, there is no quality inspection in the stage i, where machine m i is involved.In the case of some p i = 1, it means that there is only quality inspection involved in the stage i.
Remark 2.3.In the case of c i = 1 and d i = 0, i = 1,...,M − 1, in assumptions (iv) and (vii), the model is reduced to the model of a production line without QC devices, as defined in [10].Therefore, this model is a more general model for Bernoulli production lines.
Manufacturing systems, defined by (i)-(vii), are called serial production lines with QC devices.The performance measures addressed in this paper are as follows.
(a) Production rate, PR, the average steady-state number of parts produced by the machine m M during a cycle time t c .
(b) Average steady-state buffer occupancy E[b i ], i = 1,...,M − 1, where b i are the steadystate occupancy of buffer B i .
Remark 2.4.Throughout, symbols with the "-" denote the exact values of the appropriate quantity.The respective approximations, to be introduced below, are denoted by the same symbols but without the "-." Unfortunately, no tractable exact methods for calculation of PR and E[b i ], i = 1,..., M − 1, for line (i)-(vii) are at present known.The goal of this paper is to derive the estimates PR and E[b i ] and to evaluate their accuracy.The problem of such a line, addressed in this paper, can be described by the following characteristics.

Numerical approach
Before deriving the analytical performance evaluation, the numerical procedure of the problem is developed.Based on the numerical results of the procedure, the accuracy of the analytical performance evaluation can be analyzed.
Consider the line (i)-(vii) with M machines, (M − 1) QC devices, and (M − 1) buffers.Let K = [k 1 ,...,k M−1 ] be the state of the system where the consecutive buffers B 1 ,...,B i ,..., B M−1 contain k 1 ,...,k i ,...,k M−1 parts, respectively, 0 ≤ k i ≤ N i .It follows that k ∈ {{0, ..., N 1 }, ...,{0,...,N M−1 }}.Let Y (K, j) denote the joint probability of buffers B i , i = 1,..., M − 1, in the state K at time slot j.Since the line can be described by Markov chain, and by total probability theorem, then we write where G(K | K ) denotes the transition probability of the state changing from K to K, and Shu-Yin Chiang 5 where Note that Y ( j) and Y ( j + 1) are column vectors with M−1 i=1 (N i + 1) elements, and T is a square matrix with M−1 i=1 (N i + 1) 2 elements.Then, (3.2) can be expressed in terms of the initial value as follows: where Y (0) is the initial probability distribution.Since the line (i)-(vii) is an ergodic system, we have (3.6) ..,M − 1, be the steady state probability of buffer B i containing b i parts.Then Based on (3.7), the numerical solutions of the problem are obtained as fo1lows.
(a) Since the production rate is equal to the average steady-state number of parts produced by the line during a cycle, the last machine must be operational, p M , and not starved, which is equal to the absence of emptiness of the last buffer, [1 − X M−1 (0)].Therefore, the production rate can be expressed as The average steady-state buffer occupancy can be calculated as Unfortunately, there is no closed-form expression for the transition matrix T in a long production line.However, it is possible to find out for a computation procedure.The input data for the algorithm are given in the following.
Rule 3.1.The transition probability from the state K to K is given as follows: where where A procedure is developed by using the above rule to calculate all elements of the matrix T.

Numerical procedure
Step 3.2.Establish a program function to check whether there is a transition from the state K to K under the condition [α,β,γ].Then, the transition probability G(K | K ) can be calculated by (3.10) and (3.11).
Step 3.3.For each state from K to K, using the function established in Step 3.2, calculate G(K | K ).Then continue the same process until the elements in matrix T are obtained, then stop.

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A useful justification for the correctness of the computer program finding the elements in the matrix T is provided as (3.13) Step 3.4.Computing the steady-state probability distribution of consecutive buffers B 1 , ...,B M−1 , Y .From (3.4), by using T and the initial value Y (0), let where V is a vector of 1's M−1 i=1 (N i + 1) dimensions.Then, The following equation can be used to justify the correctness of the resulting Y :

Analytical approach
This section is devoted to develop the performance evaluation of the production line with QC devices.The performance measures of interest in this paper are the production rate, PR, and the average steady-state buffer occupancy, E[b i ], i = 1,...,M − 1.However, these functions cannot be calculated in closed form for the long production line.Therefore, we derive their estimates, PR and E[b i ], i = 1,...,M − 1, below and then evaluate their accuracy.The accuracy analysis is based on the results obtained in Section 3.
To characterize the steady-state behavior of production line (i)-(vii), introduce the function where which is a monotonically increasing function of p 1 , c 1 , p 2 , and N 1 , and decreasing of d 1 .
Proof.See the appendix.
Proof.See the appendix.
These two lemmas play a critical role for developing the analytic performance evaluation for a production line with M > 2.
Next, consider now a line with M machines, (M > 2), (M − 1) QC devices, and (M − 1) buffers.Based on Lemma 4.1 and an aggregating technique consisting of two principal components, a forward and a backward aggregation, a recursive procedure is developed.
In the forward aggregation procedure, the first two machines, and the middle QC device and buffer, being repeatedly replaced by a single machine, such that reducing the length of the line, until the overall line has been reduced to a single machine.That is, where parameter p f i is the machine parameter of a single machine replacing the first i machines and (i − 1) QC devices and (i − 1) buffers.
It follows from above that the blockage of the aggregated machine results in the blockage of every machine involved in the aggregation which is not the same amount of blockage in the real system.Therefore, to eliminate this undesirable phenomenon, let us aggregate the line going from the back (backward aggregation) taking the results of the forward aggregation into account.In other words, in the backward aggregation procedure, the last two machines, and the middle QC device and buffer, being repeatedly replaced by a single machine until the overall line has again been reduced to a single machine.Here the Shu-Yin Chiang 9 blockage probabilities are calculated from the forward aggregation.Specifically, we write where parameter p b i is the machine parameter of the single machine replacing machines m j , j = i,...,M, QC devices Q j , and buffers B j , j = i,...,M − 1.
Now we iterate the process by constructing a new forward aggregation based on this backward aggregation, and so on.As a result, we obtain the following recursive procedure: with initial condition where F(x, y,N) is defined by (4.1).The question of convergence of the resulting p f i (s) and p b i (s), s = 1,2,3,..., is answered in the following.
(4.13)Then, we have the following numerical fact.

Numerical fact. For serial production line defined by assumptions (i)-(vii),
Numerical justification.The result is justified by extensive numerical experimentation, which is developed in Section 3. As it follows from the experimentation, the value of δ is always small.An illustration is given in Tables 4.1 and 4.2 for several lines with c i = 0.95, d i = 0.05, and N i = 3, i = 1,2,3.Although we do not have an analytical proof that δ 1, on account of this evidence, we conclude that numerical fact holds.
Proof.See the appendix.
Proof.See the appendix.
Remark 4.6.The meaning of Theorem 4.5 is that the average buffer occupancy for B i equals (1 + 2 + 3 + ••• + N i ), which is equal to N i (N i + 1)/2.Therefore we have the normalized estimated error which is equal to the difference between real buffer occupancy and the estimated buffer occupancy divided by the average buffer occupancy.
Although these estimates are just numerical experimentation (since δ is not an asymptotic parameter), illustrated in Tables 4.1 and 4.2, they show that the proportionality constant in O(δ) is quite small, and the estimate (4.11) and (4.12) results in high accuracy.
The following theorem is a useful property of production system.
Proof.See the appendix.
In addition, the monotonic properties of the line production rate can help us to determine which machines, QC devices, or buffers should be improved so that we will have the best possible performance improvement.

Case study
The performance analysis described above has been used in the continuous improvement project of a production line at a picture tube plant.The structure of the production line is shown in Figure 5.1.It consists of 8 operations, 6 processing machines (m 1 − m 5 and m 8 ), and 2 QC devices (Q 6 and Q 7 ), and a conveyor in-process material handling system.The material handling system is structured as a set of palletized conveyors connecting each pair of consecutive operations and providing buffering capacities.Parameters of the machines and buffers identified are given in Table 5.1.F i and R i , listed in Table 5.1, refer to the mean time to failure and mean time to repair of machine m i or QC device Q i , respectively.Thus, the isolated machine production rate, or the isolated working rate, can be calculated as (5.1) To make the system shown in Figure 5.2 fit the model defined in Section 2, consider the rework ratios as the bad part ratios.The capacity of the physical buffer was taken as Shu-Yin Chiang 13 the difference of the actual number of pallets between two operations and the number of pallets necessary to sustain a continuous operation if no break down occurs.Then, by a transfer technique, we have the buffer capacity from the physical buffer size to the Bernoulli one in [11].The transformation is based on the idea of choosing the buffer size in Bernoulli model to be the minimum value between the average downtime of upstream machine and that of the downstream one.Thus, the model of the system amenable to the performance evaluation techniques developed in Section 4 is identified.The corresponding Bernoulli parameters are listed in Table 5.2, and the simplified system is given in Figure 5.2.The reasons for neglecting the feedback effect of the rework parts are as follows.First, at present, we do not have a theory for analyzing such production lines, nor the current literature offers such a method.Second, some of the rework parts are discarded from the system, and some of them go to m 3 or m 4 to be reworked.The former term, in fact, can be viewed as the bad parts, and contribute to the bad part ratios.The latter term can cause the decrease in starvation probabilities of m 3 and m 4 , and increase in blockage probabilities of m 2 , m 3 , and m 4 .We assume the effects of starvations and blockages are neutralized mutually.Although this assumption leads to a loss in accuracy, we believe the loss would be bounded in small quantity.Thus, the simplified system, shown in Figure 5.2, is appropriate for the analysis of the actual production system.The estimate of the average production rate of the production system, PR, has been calculated, using the recursive procedure (4.7), and the result has been compared with the actual average production rate PR.The results are as follows: (5.2) Thus, based on the analytical results, the production rate estimated matches the actual system with an error 2.67%.Therefore, we accept the model identified as validated by the actual performance.
According to the technique of performance analysis results, two improvement measures are performed as follows.
(I) Since all machines of the system are highly reliable, it is difficult to obtain a significant improvement by improving machines' reliability.Therefore, we focus on the reconstruction of the material handling system.To obtain some ideas on how to reconstruct the material handling system, the average buffers occupancy of the system are calculated by using (4.12), and the calculated results are listed in Table 5.3.It follows from this result and Table 4.2, that buffer 7 has low buffer occupancy and large buffer capacity and, on the contrary, buffers 2 and 3 have high buffer occupancies and low buffer capacities.Usually, the factory space is limited, and therefore, the total capacities of all buffers are considered to be fixed.Under this constraint, we reallocate two units of the capacity of buffer 7 to buffers 2 and 3 (each one obtains one unit of capacity).The production rate of the above modified system is increasing 1 part/hour.
(II) From the discussion with the engineer on the factory, we have the following statement: the effort of reducing the bad part ratio of m 4 (d 6 ) by 50% is closed to the effort of reducing the bad part ratio of m 3 (d 7 ) by 50%.To know which reduction will gain a better improvement, the production rates of the systems using the technique of the performance analysis developed in this paper.The resulting production rate increase of reducing the bad part ratio of m 4 (d 6 ) by 50% is 7 parts/hour, and the production rate increase of reducing the bad part ratio of m 3 (d 7 ) by 50% is 10 parts/hour.
Based on the above results, the improvement effort should be directed to m 3 to obtain the best possible improvement.The plant management has accepted this recommendation.

Conclusion
The performance evaluation of production lines with Bernoulli machines and QC devices is obtained in a numerical form and a closed form within an acceptable error bound.These two results are justified each other, and are in a good agreement.Some important properties of the production line are also obtained in this work as follows.The estimated production rate is monotonically increasing in machines' production rate and QC devices working rates in isolation, and decreasing in bad part ratios.The results can be utilized for analysis and design of serial production lines with QC devices.Since the best test of a theoretical result is practice, it has been used in a project carried out at a picture tube plant.It is hoped that this approach will prove to be useful for other practitioner and theoreticians alike.The extension of the result to production lines with Markovian machines is the subject of the future work.
After some algebra, this simplifies to For the line to produce a part during a cycle, the second machine must be operational and not starved.Therefore, the production rate, PR, can be calculated as follows: where F(x, y,N) is defined by (4.1).From (A.3) and (A.6), after simplifying, we have Since the first machine produces a part if it is operational and not blocked.However, the parts produced by the line added to the bad parts equal the parts produced by the first machine.Therefore, (A.9) The monotonicity of the production rate in p 1 , c 1 , p 2 , and N 1 (or d 1 ) follows directly form (A.7), (A.9), and properties (a)-(d) of function F(x, y,N).
Proof of Lemma 4.2.By (A.3), the average steady-state buffer occupancy of buffer B 1 is (A.10)

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After some algebra and by (4.1), we can derive where Functions F(x, y,N) and Z are given in (4.1).
The proof of Lemma 4.3, in addition to the above properties of function F, we also require the following fact.
Lemma A.1.Consider p f i (s) and p b i (s), i = 1,...,M, defined by procedure (4.7).Sequences p f i (s) and p b i (s), s ∈ N, are monotonically increasing and decreasing, respectively.Proof.By induction, for s = 1, due to the properties of function F(x, y,N), we have (A.12) Assume that for s > 1, Then from the recursive procedure (4.7), for j = 2, by (A.13), Then for j = M − 1, by (A.16), These probabilities play a crucial role in the proof of Theorem 4.4.Specifically, we show below (Lemma A.3) that if p f i and p b i+1 are known, then the stationary probability distribution of buffer occupancy, X i (•), can be calculated with the error O(δ).Further, Lemma A.5 shows that p f i and p b i+1 can be calculated from the steady state of recursive procedure (4.7), with the error O(δ).Therefore, since the production rate can be calculated from (3.8), PR = p M [1 − X M−1 (0)], the claim of Theorem 4.4 will follow.
Lemma A.2.Under numerical fact, the conditional probabilities p f i , p b i take the following forms: Prob n i = 0,K i | B i does not input a part, m i+1 produce, n i = 0,K i . (A.30)

Ki
Prob Consider now the first term of the right-hand side of (A.32): When buffer B i−1 contains at least one part, machine m i is not starved, and when buffer B i contains zero parts, machine m i is not blocked.Therefore, the probability in the first term on the right-hand side of (A.33) is equal to [(1 When buffer B i−1 contains zero parts, machine m i is starved, and the probability in the second term on the right-hand side of (A.33) is equal to one.Consequently, (A.34) Shu-Yin Chiang 23 Using numerical fact, this can be rewritten as (A.35) By Lemma A.2, we finally obtain (A.36) Analysis of the second term on the right-hand side of (A.32) proceeds analogously and results in Similar arguments can be used to obtain equations for X i ( j), j = 1,...,N i .As a result, we obtain the following set of equations: These equations can be written in matrix form as where and ΔA is a diagonal matrix with diagonal elements all of the order O(δ), and therefore ΔA ∼ O(δ).It follows from (A.2) of Lemma A.1, the equilibrium distribution of parts X i (•) of line L i is described by X = AX i , where A is given in (A.40).Since A is the state transition matrix of an ergodic Markov chain, λ = 1 is an eigenvalue of multiplicity 1 of A. Therefore, using the perturbation theory, we obtain Our next goal is to show how p f i and p b i , the steady state values determined via recursive procedure (4.7), can be used to determine p f i and p b i .Before we do so, we will need a few preliminary results.
As it follows from Lemma 4.3, the steady state equation of recursive procedure (4.7), that is, T .We prove below that this solution is in fact unique.To accomplish this we introduce (M − 1) two-machine one-buffer serial production lines, L i , i = 1,...,M − 1, where the first machine has the isolation production rate p f i , the second p b i+1 , and the buffer capacity is N i .The following properties hold.
Lemma A.4.Let PR i be the production rate of line This proves the first statement of the lemma.Moreover, which establishes the inductive hypothesis.Lemma A.3 showed that if the conditional probabilities p f i and p b i , i = 1,...,M, are known, then it is possible to determine, approximately, the steady-state buffer occupancy probability distributions X i (•), i = 1,...,M − 1.Therefore, the following lemma shows that they are given, approximately, by recursive procedure (4.7).
Lemma A.6.The following relationships hold: where p f i and p b i are given by Lemma 4.3 and δ is defined in (4.13).Proof.Let X i (•) be the equilibrium probability distribution of buffer occupancy of line L i , i = 1,...,M − 1, as described earlier, and let X i (•) be the equilibrium probability distribution of buffer occupancy for buffer i of line (i)-(vii).Let the conditional probabilities p
Prob machine i produces a part | machine i is not blocked , p b i ≈ Prob machine i produces a part | machine i is not starved .
) and p b j (s), s = 1,2,3,..., are monotonically increasing and decreasing, respectively.Proof of Lemma 4.3.Since the sequences p f j (s) and p b j (s), 1 < j < M, are monotonic (Lemma A.1) and bounded (result in (d)), so they are convergent.The proof of Theorem 4.4 requires the following lemmas.Introduce the following conditional probabilities: one solution P agg = [p f 1 ,..., p f M , p b 1 ,..., p b M ]

Table 4 .
1. Estimation error of PR. − 1) two machines, one QC device, and one buffer line L i , i = 1,..., M − 1, where the first machine is defined by p f i , the second by p b i+1 , and the buffer is of capacity N i .From Lemma 4.2, the average steady-state buffer occupancy of buffer B i , i = 1,...,M − 1, can be calculated as

Table 4 .
2. Estimation error of E[b The relationship between the exact production rate, PR, and the calculated one, PR, and the relationship between the exact steady-state buffer occupancy, E[b i ], and the calculated one, E[b i ], i = 1,...,M − 1, are given by Theorems 4.4 and 4.5, respectively.

Table 5 .
2. The corresponding parameters of the case study.
Figure 5.2.Implementation of the system model.