Asymptotic variance rate of the output in

Annals of Operations Research 93 (2000) 385–403

385

Asymptotic variance rate of the output in production lines with finite buffers Barıs¸ Tan Graduate School of Business, Koc¸ University, Istinye 80860, Istanbul, Turkey E-mail: [email protected]

Production systems that can be modeled as discrete time Markov chains are considered. A state-space-based method is developed to determine the variance of the number of parts produced per unit time in the long run. This quantity is also referred to as the asymptotic variance rate. The block tridiagonal structure of the probability matrix of a general twostation production line with a finite buffer is exploited and a recursive method based on matrix geometric solution is used to determine the asymptotic variance rate of the output. This new method is computationally very efficient and yields a thousand-fold improvement in the number of operations over the existing methods. Numerical experiments that examine the effects of system parameters on the variability of the performance of a production line are presented. The computational efficiency of the method is also investigated. Application of this method to longer lines is discussed and exact results for a three-station production line with finite interstation buffers are presented. A thorough review of the pertinent literature is also given.

1.

Introduction

Performance modeling of production systems has been subject to numerous studies [2]. Most of these studies focus on the mean performance measures such as the production rate or the average Work-In-Progress (WIP) inventory levels. Recent advances in manufacturing management techniques, such as agile manufacturing, made variability an important design criterion in order to ensure predictability and dependability of production systems. Gershwin [5] reports that his numerical and simulation experimentation and factory observations indicate that the standard deviation of weekly production can be over 10% of the mean. This means that it is very likely that customer requirements cannot be met exactly on time due to this variability. Since meeting customer requirements on time and shortening this time are becoming more and more important every day, there is an increasing need for analytical models to study the variability of the output. These models can be used both to design production systems that cope with variability more effectively and also to operate production systems, e.g., to set the due dates and the optimal production quotas, etc. This study is motivated by the need of developing analytical models for the variance of the output that are to be used in design and control of manufacturing  J.C. Baltzer AG, Science Publishers

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B. Tan / Asymptotic variance rate of the output

systems. The focus of this study is to obtain a long run variability measure that is the variance of the number of parts produced per unit time in the long run referred to as the asymptotic variance rate V . It is shown that the distribution of the number of parts produced in the long run is normal. Therefore the production rate E and the asymptotic variance rate V determine the distribution of the number of parts to be produced in a production system in the long run. For large t, the number of parts to be produced during [0, t) is normally distributed with mean Et and variance V t. One can utilize this distribution to answer questions like “what is the probability that a given due date is to be met on time”, or “what is the due date to be quoted according to a given service level”, etc. If the time period of interest is short, then the above results will be rather crude approximations. Then determining the mean and variance, or the distribution of the number of parts to be produced during [0, t), is of interest. This study is a first step towards the analysis of output variability in production systems. Following this work, Tan [23] gives a method to determine the exact mean and variance of the number of parts to be produced during [0, t) and approximate mean and variance of the time to produce a given number of parts. Tan [21] gives the exact distribution of both the number of parts to be produced during [0, t) and also the distribution of the time to produce a given number of parts. The organization of the remaining part of this study is as follows: in section 2, we present a thorough review of the studies on variability of the output in manufacturing systems. The model we consider and the methodology are explained in section 3. An efficient method to solve the equations to obtain the variance of the output from a general two-station production line with a limited buffer is presented in section 4. Numerical results for a specific two-station production line with a finite buffer are given in section 5. Extension of this method to multistation production systems is discussed in section 6. Finally, conclusions are given in section 7. 2.

Past work

The number of studies on variability in manufacturing systems is limited. The ones that are closest to the one presented in this study are tabulated in table 1 following a classification similar to [16]. In this table, the models are first classified according to the type of material flow. Most of these studies assume discrete material flow, which is of more interest in a manufacturing setting. The models that assume continuous material flow can be used to analyze process industries or to approximate discrete material flow production systems. The models are then grouped according to the arrangement of workstations. The majority of the studies are on production lines, i.e., on a series arrangement of workstations. There are two studies on cyclic production lines, by Duenyas and Hopp [3] and Duenyas et al. [4], and one study on a series-parallel system with no buffers [20]. After stating the arrangement of workstations, the number of stations and buffer capacities are also given. In order to give more information about the computational requirements of the methodologies presented in these studies, the largest model, de-

Table 1 Summary of models and their assumptions.

Arrangement of workstations series series-parallel cyclic series

Tan [19]

disca

cont cont

cont disc

√

√

disc

√

disc √

disc

–

√

disc

disc

√

√

disc

disc. disc disc disc

√

√

3 7

8 ∞

6e − 3 0–7

1d , 10 –, 50

8–6 0–8

6 ∞

5 3

2 10

Processing time distribution

det., geo.

expo.

expo.

det.

det.

det.

√

√

sxp. unf. erl. √

√

geo. geo.

Solution Analytical Numerical Exact Approximate

√ √

√ √ 38%

√

–

√

No. of stationsb Buffer capacityc

Reliable stations √ Unreliable stations Repair time geo. distribution Failure time geo. distribution

√

disc

√ √

√

√ √ √ √ 17%g

√ √

√

*

√

1 –

40 0

40f , 4 20 0 0

2–3 400–20

det. det. det. det.

det.

det.

det.

det.

det.

√

√

√

√

√

√

√

expo.

det

geo. geo. geo. det.

expo.

expo. expo. expo. geo.

expo.

geo.

geo. geo. geo. geo.

expo.

expo. expo. C2:b

√

√ √

√

√

√

√

√

√ √ 7%

√

2 10 200 0

√

√ √

√

√ √

5 3

√

√ √

√

√ √

geo.

B. Tan / Asymptotic variance rate of the output

Material flow

Miltenburg Duenyas Hendrics Gershwin Hendrics Duenyas, Jacobs Tan Tan Tan Li and Kim Tan Tan [14] and Hopp [8] [5] and McClain Hopp and and Meerkov [21] [23] [22] Meerkov and Alden [18] [20] [3] [9] Spearman [10] [13] [12] [4]

√ √

11%

387

388

Miltenburg Duenyas Hendrics Gershwin Hendrics Duenyas, Jacobs Tan Tan Tan Li and Kim Tan Tan Tan * [14] and Hopp [8] [5] and McClain Hopp and and Meerkov [21] [23] [22] Meerkov and Alden [18] [20] [19] [3] [9] Spearman [10] [13] [12] [4] Measuresh √ Asy. variance rate i Transient Var[N (t)] Var[Tn ]j Due-date perfor.k

√

√

√ √

√

√

√ √

√ √ √ √

√

√

√ √

√ √

√

√

√

√

√

Notes: ∗ Model and methodology presented in this study. a Abbreviations used in the table: disc. discrete, cont. continuous, geo. geometric, exp. exponential, det. deterministic, sxp. shifted exponential, unf. uniform, erl. erlang, C2:b balanced mean coxian, stan. dev. standard deviation. b The largest number reported in the study. c The largest buffer capacity reported in the study (if the buffer capacity is not limited, ∞). d Gershwin [5] presents an exact analytical result for a station and then uses this result in approximation. e Largest systems are a 6-machine line with no buffers and a 3-machine line with buffer capacities equal to 7. f Largest system is a 40 station in series and 4 stations in parallel system. g Largest relative error in standard deviation of the number of parts produced in a fixed time interval. h Performance measures determined in the study (only the ones that are closely related to this study are reported). i Variance of the number of parts produced in [0, t). j Variance of the time to produce n items. k Probability measures related to the due-date performance.

B. Tan / Asymptotic variance rate of the output

Table 1 (continued)

B. Tan / Asymptotic variance rate of the output

389

fined by the number of stations and the capacities of the buffers, presented in these studies are reported in the table. Some of these models assume reliable stations and random processing time while others assume deterministic processing time and random failure and repair times. Almost all of the distributions used in these models are geometric or exponential. Therefore the resulting models are Markovian. Only Hendrics and McClain [9] use a semi-Markov model and thus allow uniform, and shifted exponential distributions. When we consider the methodologies presented in these studies, we first classify the models according to whether an analytical result or a numerical method is presented and whether an approximation or the exact solution is obtained. Then the performance measures determined in these studies are considered. That is, we indicate whether the study yields asymptotic results for the variability of the output, namely the asymptotic variance rate of the output, the variance of the output during a short time interval, time to produce a given number of parts, or due-date performance measures. Only the performance measures that are discussed in this study are considered. Numerical methods are mostly based on state-space representation of the model. Miltenburg [14] uses the results developed for the sojourn time in Markov chains to determine the asymptotic variance rate. This method uses the fundamental matrix, which is obtained from the inverse of a matrix of same size as the state-transition matrix. Tan [22,23] use the results developed for Markov reward systems. These studies are also state-space based. The methods presented in these studies are modified for a general model of a two-station model with a finite buffer. Hendrics [8] and Hendrics and McClain [9] use an extended state-space representation of the system. More specifically, the state-space is further expanded to distinguish the states that yield production of a part. Most of the analytical methods yield exact results. Tan [22] presents a closedform expression for a discrete material flow production line with no interstation buffers and cycle-dependent failures. Gershwin [5] presents a closed-form expression for the variance of the output from an unreliable station during a time interval and also for the asymptotic variance rate. This method is based on solution of a set of difference equations that describes the system. Carrascosa [1] extends Gershwin’s results for twostation no buffer production lines and presents numerous simulation results related to variability of the output from two-station production lines with a finite buffer. Tan [18–20] present closed-form expressions for the asymptotic variance rate of the output from different continuous material flow production lines with no interstation buffers. These results are based on a general result derived for the mean and variance of the total sojourn times in continuous time Markov chains. Duenyas and Hopp [3] and Duenyas et al. [4] present expressions that approximately give the variance of the output from a cyclic production line with m stations and n jobs in the context of CONWIP systems. Gershwin [5] uses the exact results derived for a single station to determine the variance of the output from a multistation production line approximately with decomposition. For the approximate methods, the largest relative errors in standard deviation of the number of parts produced dur-

390

B. Tan / Asymptotic variance rate of the output

ing a fixed time interval are also reported. Note that since the models analyzed are not the same, these error figures cannot be interpreted as measures to compare the approximation methods. Kim and Alden [12] consider a single station with deterministic processing time and exponentially distributed failure and repair times. They give an analytic approximation of the density function and variance of the time to produce a fixed lot size. Jacobs and Meerkov [10] study due time performance of lean and mass manufacturing environments. They measure due time performance with the frequency with which a production system meets production commitments. This measure is closely related to variance of the output. Li and Meerkov [13] consider the problem of production variance, the problem of constant demand satisfaction, and the random demand satisfaction. They assume Bernoulli statistics of machines and provide bounds for the variability of the output. The summary of the methodology and the model presented in this study is provided in the rightmost column of the table. The main contribution of this study is to present an efficient state-space-based method to determine the asymptotic variance rate of output from a general production line with finite buffers. The method uses a method to determine the variance of Markov reward systems introduced by Grassman [7]. After choosing the appropriate reward function, the equations that determine the variance of total reward are solved by a recursive method that exploits the special structure of the probability matrix. This method is similar to matrix-geometric solution used to determine the steady-state probabilities and thus it is computationally very efficient. It is observed that when this method is applied to a two-station production line with a finite buffer, it yields nearly a thousand-fold improvement in the number of operations compared to Miltenburg’s method. 3.

Determining the asymptotic variance rate of the output

We consider Markov chain models of discrete material flow production systems. We are interested in the number of parts produced in this production system during [0, t), N (t). One of the most commonly used performance measures is the production rate E. Production rate is the expected number of parts produced per unit time in the long run given by E[N (t)] . t→∞ t In this study, the asymptotic variance rate of N (t) denoted by V and defined as E = lim

(1)

Var[N (t)] (2) t→∞ t is of interest. It is known that the distribution of N (t) is asymptotically normal [14,18]. Thus once E and V are determined, the distribution of N (t) can be approximated with a normal distribution with mean Et and variance V t. V = lim

B. Tan / Asymptotic variance rate of the output

391

In this study we utilize the results given for the mean and variance of the total reward connected with a Markov reward process to determine V in an efficient way after modeling a production system as a Markov reward system. Let Xt denote the state of the system at time t, Xt ∈ S where S is the state space. The time-homogeneous probability matrix of the Markov chain {Xt , t = 0, 1, . . .} is Q. We assume that the process is ergodic and thus the steady-state probabilities exist. The steady-state probability of state i is denoted by πi , i ∈ S and π = [πi ]. Let g : S → R be the reward function. Let Yt = g(Xt ) be the reward at time t and g = [g(i)]. Then the expected total reward m is given as X g(i)πi = πg. (3) m= i∈S

Grassman [7] presents an efficient method to determine the variance of the total reward during [0, t) in a Markov reward system. Assuming that X0 has invariant distribution, then the variance of the total reward in steady state V is given as P X
2 Var[ t−1 k=0 Yk ] g(i) − m πi + 2βg, (4) = V = lim t→∞ t i∈s

where the vector β is the solution of the following set of equations: β(Q − I) = −αQ, βu = 0,

(5) (6)

where I is the identity matrix of appropriate size, and α is a row vector whose ith element is πi (g(i) − m). The above method can be used to determine the asymptotic variance rate of N (t) by defining the reward function appropriately. Let us consider the last station in a production system where a part leaves the system after this last process. If this station is operational, i.e., up and operating and not starved, in state i at the beginning of time t, a part will be produced and leave the system by the beginning of time t + 1. Therefore we define the reward function g as n 1 if the last station is up and operating (not starved) in state i, g(i) = (7) 0 otherwise. Then the mean number of products produced at the last station per unit time in the long run, i.e., the production rate, is E = m = πg as given in equation (3). The asymptotic variance rate can now be determined from equations (4), (5), and (6). Note that this method yields V with the same number of computations required to determine the steady-state probabilities of a Markov process. However, using the results for the asymptotic variance of the total sojourn time in Markov chains [11,14] requires inversion of the fundamental matrix, thus it requires a much larger number of computations than the proposed method. This method combines flexibility of directly working with state-transition matrix and computational efficiency. Once the probability matrix of the Markov chain model

392

B. Tan / Asymptotic variance rate of the output

of a production system is available, equations (4), (5), and (6) can directly be used to determine the mean and variance by using UL decomposition or other numerical methods developed for the solution of large Markovian models, e.g., see [17]. Similar to all state-space-based methods, this method is also affected by the rapid growth of states in the state space. In the next section we exploit the special structure of the state-transition matrix of a production line with two stations and a finite buffer and present a recursive method that uses a matrix-geometric argument to solve the necessary equations. This method does not require generating and storing the state-transition matrix and is thus computationally more efficient. Note that when the number of stations in the line increases, although the matrix-geometric method cannot be used, one can still obtain the variance by using UL decomposition. 4.

A matrix-geometric method for the variance of output from a general two-station production line with a limited buffer

The probability matrix of a two-station production line with a finite buffer of size M is in block tridiagonal form [24,15] as shown below:   D0 A0   C1 D A     C D A     C D A ,  (8) Q=  · · ·     C D A    C D AM −1  CM DM where the submatrices D0 , D, DM , C1 , C, CM , A0 , A, and AM −1 describe the transitions among the states at a given buffer level. Note that the sizes of these submatrices do not depend on the buffer size, but depend on the assumptions of the model. This special structure of the probability matrix is exploited in numerous studies to obtain computationally efficient solution procedures to determine the steady-state probabilities and other first order performance measures such as the production rate and the average WIP levels [15,24,18,25]. Once the probability matrix Q, or equivalently, the submatrices D0 , D, DM , C1 , C, CM , A0 , A, and AM −1 , are given, we first determine the steady-state probability vector π and the production rate E by using the above mentioned methods. Next, the reward function g is defined according to the state of the second station. Namely, g(i) is 1 if the second station is up and not starved in state i, i ∈ S, and 0 otherwise. Note that g can also be partitioned according to the partitioning of Q: g = [g0 , g1 , g2 , . . . , gM −1 , gM ], where gi = g1 , i = 2, 3, . . . , M − 2. Given g(i), πi , and E, the row vector α can now be formed with its ith element given as πi (g(i) − E), i ∈ S.

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393

The computational efficiency is realized by observing the repeating structure of Q and exploiting this structure in the solution of equation (5). Let equation (5), β(Q − I) = −αQ, be partitioned according to the partitioning of Q. Then  T   β0 B0 A0  β  C B A   1   1       ·   C BA       ·    C B A          ·    · · ·      ·    CBA         C B AM −1   βM −1   βM CM BM  T   α0 D0 A0  α  C D A   1   1       ·    C DA      ·    C DA     = − (9)   ,  ·    · · ·      ·    CDA         C D AM −1   αM −1   αM CM DM where B0 = D0 − I, B = D − I, B0 = D0 − I, BM = DM − I and I is the identity matrix of the appropriate size and T represents transpose operation. This equation can be rewritten as the following set of equations: β0 B0 + β1 C1 = −α0 D0 − α1 C1 , β0 A0 + β1 B + β2 C = −α0 A0 − α1 D − α2 C, βi−1 A + βi B + βi+1 C = −αi−1 A − αi D − αi+1 C, i = 2, . . . , M − 2, βM −2 A + βM −1 B + βM CM = −αM −2 A − αM −1 D − αM CM , βM −1 AM −1 + βM BM = −αM −1 AM −1 − αM DM .

(10) (11) (12) (13) (14)

Note that the set of equations (10)–(14) is very similar to the set of equations to determine the steady-state probabilities. Namely, the only difference is the right-hand side of these equations. If the right-hand sides of these equations were set to zero, these equations would yield the steady-state probabilities after normalization. Similar to the recursive determination of the steady-state probabilities, the subvectors βi can be obtained recursively starting with β0 as βi = βi−1 Φi + Γi ,

i = 1, . . . , M ,

where the matrices Φi and the vector Γi are to be determined.

(15)

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B. Tan / Asymptotic variance rate of the output

Starting with i = M , the matrices Φi and Γi can be determined recursively as −1 , ΦM = −AM −1 BM

−1 ΓM = −λM BM , −1

ΦM −1 = −A(B + ΦM CM )

,

Φi = −A(B + Φi+1 C)−1 , Φ1 = −A0 (B + Φ2 C)−1 ,

(16)

ΓM −1 = −(λM −1 + ΓM CM ) × (B + ΦM CM )−1 , (17) −1 Γi = −(λi + Γi+1 C)(B + Φi+1 C) , (18) Γ1 = −(λi + Γ2 C)(B + Φ2 C)−1 , (19)

where λ0 = α0 D0 +α1 C1 ; λ1 = α0 A0 +α1 D +α2 C; λi = αi−1 A+αi D +αi+1 C, i = 2, . . . , M −2; λM −1 = αM −2 A+αM −1D+αM CM ; and λM = αM −1 AM −1 +αM DM . Furthermore, it can be shown that the matrices Φi are identical to the matrices used in determination of the steady-state probabilities, i.e., π i = π i−1 Φi , i = 1, 2, . . . , M , where π i is obtained from partitioning of π according to the partitioning of Q. Yeralan and Muth [24] show that if rank (A) = 1 or rank (C) = 1 then the matrix-geometric property holds. That is, Φi = Φ, i = 2, . . . , M − 2. The matrix Φ satisfies the following quadratic matrix equation: A + ΦB + Φ2 C = 0.

(20)

This matrix quadratic equation is well studied and there exist various algorithms to determine Φ [15,17]. Once β0 is obtained, equation (15) is used to determine βi recursively. In order to determine β0 the remaining equation (6) is used: M X

βi 1 = 0,

(21)

i=0

where 1 is a column vector of ones with an appropriate size. The above sum for M > 3 yields the following equation for β0 : ! M −3 X Φk 1 + Φ1 ΦM −3 ΦM −1 (1 + ΦM 1) β0 1 + Φ1 k=0

= −ΓM −1 (1 + ΦM 1) − ΓM 1 ! M −2 M −i X X k M −2−i − Γi Φ 1+Φ ΦM −1 (1 + ΦM 1) . i=1

(22)

k=0

Note that when M < 3, the probability matrix can easily be written. Thus the results for Markov reward systems can be applied directly. Furthermore, if (I − Φ)−1 exists, the above equation can be further simplified as

β0 1 + Φ1 (I − Φ)−1 I − ΦM −2 1 + Φ1 ΦM −3 ΦM −1 (1 + ΦM 1) = −ΓM −1 (1 + ΦM 1) − ΓM 1

B. Tan / Asymptotic variance rate of the output

−

M2 X

395



Γi (I − Φ)−1 I − ΦM −i+1 1 + ΦM −2−i ΦM −1 (1 + ΦM 1) . (23)

i=1

In equations (22) and (23), 1’s are again column vectors of ones with appropriate sizes. The solution of these equations yields β0 . Note that the number of equations in (22) and (23) does not depend on the buffer size but on the sizes of the submatrices, and thus on the assumptions of the model. Now, since β0 is available, βi can be obtained recursively by using equation (15). Once β is obtained, equation (4) yields the desired variance. This completes the solution procedure. The solution procedure presented in this section uses only the submatrices. Thus it is not necessary to generate and to store the state-transition matrix. Furthermore, it is not necessary to invert large-size matrices. Therefore the solution procedure is very efficient and the buffer size does not increase the number of computations extensively. In the following section we investigate the numerical efficiency of the solution procedure compared to Miltenburg’s method. 5.

Numerical results for a specific two-station production line

The procedure presented in the preceding section can be used to analyze a wide range of two-station production lines with a finite buffer. In order to present numerical results, we consider a specific two-station production line with a finite buffer and station breakdown. The model studied here is model 1 of [24] and also very similar to the model analyzed in [1]. It is assumed that the first station is never starved and the second station is never blocked. The system is observed at the end of each cycle. The service times are equal to the cycle time. It is assumed that station i, i = 1, 2, working on an item may break down at the end of the cycle with probability pi . The item a station has been working on is passed on at breakdown provided that the station is not blocked, in which case the item is held until blocking is removed. Similarly, the probability that the broken down station i is repaired at the end of a cycle is ri . Blocked and starved stations may not break down. Let the state of the system be denoted by the vector (i, j, k), where i is the state of station 1, j is the number of items in the buffer, and k is the state of station 2. Let U , D, S, B, and H denote that the station is Up, Down, Starved, Blocked, and Blocked&Down. Let the state space of this system be ordered as {(U 0S), (D0S), (U 0U ), (U 0D), (D0U ), (D0D), (U 1U ), (U 1D), (D1U ), (D1D), . . . , (U M U ), (U M D), (DM U ), (DM D), (BM D), (HM D)}. According to this ordering of states, the submatrices D0 , D, DM , C, C1 , CM , A0 , A, AM −1 are explicitly given in the appendix. In this section we investigate the effects of system parameters on the performance of production lines. Balance of the line and the size of the buffer are the two main sources that affect the performance of production lines [24]. Thus a number of

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Figure 1. Production rate as a function of the buffer size. Two-station production line (table 1).

Figure 2. Asymptotic variance rate as a function of the buffer size. Two-station production line (table 1).

numerical experiments is conducted to study the effects of system parameters on the performance of the system. Here only a number of representative cases are presented. Figures 1 and 2 depict the expected production rate and the asymptotic variance rate for a production line with a finite buffer and station breakdown as a function of the buffer size for six different cases. These cases are given in table 2. Figures 3 and 4 also show the asymptotic variance rate as a function of the buffer size. These two cases are examined in [1] by using Miltenburg’s method. As also mentioned in [1], using Miltenburg’s method is computationally very expensive since it requires inversion of a dense matrix. Finally, we investigate the complexity of the solution procedure numerically and compare it to Miltenburg’ s method. For this purpose, Matlab 4.2’s flops function is used. This function counts the number of major floating point operations. Figure 5 shows the comparison of the number of operations of the proposed method versus

B. Tan / Asymptotic variance rate of the output

397

Table 2 The parameters of pi , ri , i = 1, 2. Case 1 2 3 4 5 6

p1

r1

ε1

p2

r2

ε2

0.010 0.010 0.010 0.020 0.020 0.020

0.090 0.090 0.090 0.180 0.180 0.180

0.900 0.900 0.900 0.900 0.900 0.900

0.010 0.010 0.010 0.020 0.020 0.020

0.085 0.090 0.095 0.170 0.180 0.190

0.895 0.900 0.905 0.895 0.900 0.905

Figure 3. Asymptotic variance rate as a function of the buffer size. Two-station production line: p2 = 0.1, r2 = 0.1.

Figure 4. Asymptotic variance rate as a function of the buffer size. Two-station production line: p2 = 0.02, r2 = 0.1.

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B. Tan / Asymptotic variance rate of the output

Figure 5. Number of floating point operations as a function of the buffer size: Two-station production line (p1 = 0.1; r1 = 0.8; p2 = 0.2; r2 = 0.75).

those of Miltenburg’s method for a specific case. As can be seen in the figure, the number of operations needed in Miltenburg’s method is almost a thousand times more than in the proposed method. Furthermore, the proposed method becomes more and more efficient compared to Miltenburg’s method as the buffer size increases. When the computation times are compared, the proposed method is almost a hundred times faster than Miltenburg’s method. 6.

Extensions to longer lines

The method presented in section 3 can be applied to a wide range of production systems given that the probability matrix and the reward function are available. Once the probability matrix of the Markov chain model of a production system is available, equations (4), (5), and (6) can directly be used to determine the mean and variance by using UL decomposition or other numerical methods developed for the solution of large Markovian models, e.g., see [17]. Furthermore, the states of the system can be ordered in such a way that the probability matrix has a repeating block tridiagonal structure to improve the efficiency of the numerical methods. An algorithm has recently been developed to generate the state-transition matrix automatically for discrete material manufacturing systems when the system parameters, production system structure and operational rules are given. This algorithm takes the system structure, i.e., how the stations are interconnected, where the buffers are located, etc., the system parameters, i.e., buffer capacities, the number of stations, failure, repair probabilities, etc., and the operational rules, i.e., the blocking mechanism used, failure mechanism, the conventions used to update the system when a station fails, when the buffer level changes, etc, as inputs. Given this information, the algorithm

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Figure 6. Exact production rate and the asymptotic variance rate of a three-station production line vs. the interstation buffer capacity: p1 = p3 = 0.01, p2 = 0.03, ri = 0.09, Mi = M , i = 1, 2, 3.

Figure 7. The maximum production rate, the minimum asymptotic variance rate, the minimum total expected WIP, and the maximum probability of meeting a customer order on time, P [N(T ) > Q], T = 100, Q = 0.8E · T for the optimal buffer allocation of the total buffer capacity. p1 = p3 = 0.01, p3 = 0.03, ri = 0.09, Mi = M , i = 1, 2, 3.

generates the state space, the state-transition probability matrix, and the reward function automatically. In this section, we present preliminary results for a three-station transfer line with interstation buffers. By exploiting the repeating block tridiagonal structure of the state-transition matrix, the exact production rate and the asymptotic variance rate are obtained.

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Figure 6 shows the exact production rate and the asymptotic variance rate as a function of the interstation buffer capacity for a three-station production line. Figure 7 investigates the effects of buffer allocation on four performance measures; the production rate, the asymptotic variance rate, total expected WIP inventory, and the probability of meeting a customer order on time. For each of the performance measures, the optimal buffer allocation that optimizes the corresponding objective is determined by total enumeration. Determining a desired throughput by minimum total buffer capacity, or determining the maximum throughput by a given total buffer capacity attracted some interest in recent years. As an addition to that finding the best buffer allocation that maximizes the due-date performance is also of interest. In figure 7, the due-date performance is measured by the probability of meeting a customer order on time. Let the order be for Q units and the due date T time units and let us assume that there are no other orders, this order will be met on time if the number of parts produced during [0, T ) is greater than or equal to Q. Thus the probability of this event is P [N (T ) > Q]. Since the distribution of N (t) is approximately normal with mean Et and V t, this probability can be determined approximately. This approximation will be accurate if t is large compared to the processing time of the stations. In figure 7, Q is set to 80% of the expected number of parts that will be produced during [0, T ), i.e., it is assumed that the capacity of the production system is adequate to meet the orders. Figure 7 shows that as the total buffer capacity increases, the due-date performance also increases and it is possible to achieve a due-date performance of 84.5% with the optimal allocation of the total buffer capacity. Further research is required on exact analysis of variability in longer production lines and other arrangements of workstations and also buffer allocation problems to optimize the due-date performance of production systems. 7.

Conclusions

In this study, an efficient method to determine the asymptotic variance rate of the number of units is presented. The method is state-space based and once the probability matrix is given the asymptotic variance rate is obtained with the same order of computations required to determine the steady-state probabilities. For general two-station production lines with a finite buffer, this method is further improved by exploiting the special structure of the state-transition matrix and by using a matrixgeometric recursive method. In this method, it is not necessary to generate and to store the state-transition matrix. Therefore it yields a thousand-fold improvement in the number of computations over the previous result [14] where the state-transition matrix is generated and a dense matrix of the same size is inverted. Some preliminary results for a three-station production line are also presented. However, since the method is state-space based, it suffers from rapid increase of the number of states when the number of stations and the buffer capacities increase. Therefore, the main objective of applying this method to multistation production systems is not to evaluate the performance of, say 50-station, multistation production systems,

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but to derive some insights from the results of manageable-size more complicated production systems. One example is a three-station production line, which is the next more complicated system after two-station single buffer production lines. The performance of this system can be analyzed quite efficiently and it allows us to study buffer allocation problems. To evaluate the variability performance of more complicated systems, approximation methods, probably based on decomposition, should be developed. The accuracy of these approximation methods can be tested with respect to the exact results.

Appendix The submatrices D0 , D, DM , C, C1 , CM , A0 , A, AM −1 for the specific twostation production line are  0 0 p1 0 p1 0  r r 0 0 0 0   1  1    0 0 p1 p2 0 p1 p2 0   D0 =   0 0 p r 0 p r 0 , 1 2   1 2    r1 p2 r1 p2 0 r1 p2 0 r1 p2  r1 r2 r1 r2 0 r1 r 2 0 r1 r2   p1 p2 0 p1 p2 0 p1 p2 p1 p2 p r 0 p r 0 p r p r   1 2 1 2 1 2 1 2    0 r1 p2 0 r 1 p2 0 0  ,  DM =  0    0 r1 r2 0 r 1 r 2 0   r r 0 r r 0 r r r r  1 2 1 2 1 2 1 2 r2 0 0 0 0 r 2   p1 p2 0 p1 p2 0 p r 0 p r 0  1 2  1 2  D= ,  0 r1 p2 0 r 1 p2  0 r1 r2 0 r 1 r 2 

(24)

(25)

(26)

A = [p2 r 2 0 0]T [0 p1 0 p1 ],

(27)

T

C = [0 0 p2 r2 ] [r1 0 r1 0],  T A0 = 0, AT , C1 = [0, C],

(28) AM −1 = [A, 0],

CM = [C T , 0]T ,

where the zeros denote arrays of zeros of suitable dimension and x = 1 − x.

(29)

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