PERFORMANCE ANALYSIS OF PRODUCTION SYSTEMS WITH REWORK LOOPS

Transcription

1 PERFORMANCE ANALYSIS OF PRODUCTION SYSTEMS WITH REWORK LOOPS Jingshan Li Enterprise Systems Laboratory General Motors Research & Development Center Mail Code Mound Road Warren, MI Abstract This paper is devoted to the study of production systems with rework loops. Such systems are often encountered in manufacturing plants and accurate performance analysis is needed for design and continuous improvement. However, no analytical methods to evaluate its production rate are available in the current literature. In this paper, we present an aggregation procedure using overlapping decomposition approach to approximate the production rate for systems with rework loops. The idea is to decompose the system into four serial production lines, with machines modified to accommodate the interactions with other machines and buffers. A recursive procedure which analyzes serial lines during each iteration is presented. The convergence of the iterative procedure and the uniqueness of the solution are justified analytically. The accuracy of the estimate is evaluated numerically with good results. Keywords: Production systems, rework loops, production rate.

2 1 INTRODUCTION 1.1 Motivation This paper studies production systems with rework loops. In many manufacturing plants, rework loops are often included for rework and multiple-pass processing of jobs. In the rework loops, defective parts are repaired and sent back to the production line for re-processing. The use of rework loops can significantly increase the system throughput and reduce scrap, cost, etc. To manage, operate, and improve the performance of such systems, modelling and analysis of production systems with rework loops are necessary and important. Performance evaluation of production system has attracted tremendous attentions in last 50 years see reviews [1], [2], monographs [3]-[5]. Due to machine breakdowns, the number of parts produced by the production system during a fixed time interval is a random variable. It can be characterized by its expectation, i.e., throughput. Often, this performance measure is normalized as system production rate PR, i.e., the number of parts produced by the production system per unit of time. The evaluation of system production rate has been one of the primary interests in the study of production systems with unreliable machines. Among these studies, serial production lines have been studied intensively. Exact analysis can be performed only for two-machine systems. Using two-machine line results, various aggregation and decomposition methods have been proposed to approximate the system performance measures for longer lines. By extending the results of serial lines, assembly/disassembly lines, parallel lines, etc., have been also studied see reviews [1], [2] and representative papers [6]-[13]. In practice, more complex production systems exist and need accurate analysis. For instance, in automotive paint shops, defective parts are repaired in the rework loop and sent back to the painting booth for re-painting. Analysis of production systems with rework loops 1

3 is still an open problem. Although some analytical methods have been developed to approximate the throughput of closed queueing networks with blocking see review [14], the study of closed loop production systems with unreliable machines and finite buffers is quite limited. Papers [15]-[17] have studied a closed loop serial production line with constant number of carriers, where parts are loaded and attached on the pallets at the first machine to undergo all the operations. Upon completion of these operations, finished parts are unloaded and the pallets are released and sent back to the first machine. In paper [15], an asymptotically reliable two-machine two-buffer closed serial line is analyzed. The closed loop line is reduced to an open production line where the effective buffer capacity depends on the relationship between the actual buffer capacity and the number of pallets. Expressions for production rate and work-in-process are derived and a case study in paint shop is described where the optimized choice of the number of carriers in the system and the capacity of the feedback buffer leads to substantial improvement in system s performance. In paper [16], a decomposition approach is presented to approximate the system production rate for homogeneous production lines i.e., machines have identical cycle times. The optimal number of carriers, which maximize system performance is investigated. Paper [17] extends the study and takes into account the correlation between number of pallets in the buffers. The algorithm can be applied to both small and large loops. Paper [18] develops a flexible decomposition framework and extends it to analysis of a work cell consisting of one workstation, one rework station and finite buffers feeded by a Poisson stream of discrete parts. All these contributions, however, have not addressed the production systems with reworks loop directly. To our best knowledge, there are no analytical methods available in the literature to analyze the performance of production systems with rework loops. The main contribution of this paper is to present a convergent recursive procedure to approximate the production rate of production systems with rework loops. 2

4 The remainder of this paper is structured as follows: Subsection 1.2 formulates the problem. An aggregation procedure to calculate production rate of serial line is described in Section 2. Section 3 presents the recursive procedure to estimate the production rate of production line with rework loop. Convergence of the procedure is justified analytically, and the accuracy is evaluated numerically. The relationship between system production rate and rework rate is discussed in Section 4. The conclusions are formulated in Section 5. All proofs are presented in the Appendix. 1.2 Problem Formulation The production system studied in this paper is shown in Figure 1. The circles represent the machines and squares are the buffers. m 1 m j-1 m j m j+1 mk-1 m k m B Bj-1 Bj Bk-1 Bk B 1 Main Line m M 1 +M 2 m M B B BM +1 B M 1+M2 M 1+M M1 Rework Loop Figure 1: Production System with Rework Loop m k+1 M 1 M 1-1 For convenience, the following notations are used throughout this paper: M 1 : M 2 : number of machines in the main line, i.e., the line without rework loop; number of machines in the rework loop; α : rework rate, i.e., probability that a part requires rework 0 < α < 1; m i : machine i, i = 1, 2,, M 1 + M 2 ; m k : machine after which the line splits into output main line and 3

5 rework line; m j : machine in front of which input main line and rework line joins; B i : buffer after machine i, i = 1,, M 1 1, M 1 + 1,, M 1 + M 2 ; B M1 : buffer in front of rework machine m M1 +1; 1/p i : average uptime of machine i, i = 1, 2,, M 1 + M 2 ; 1/r i : average downtime of machine i, i = 1, 2,, M 1 + M 2 ; N i : capacity of buffer B i, i = 1, 2,, M 1 + M 2. The following assumptions pertaining machines, buffers, and their interactions are introduced to model a production system with rework loop in this paper: i The system consists of a main production line machines m 1,, m M1, buffers B 1,, B M1 1 and a rework loop machines m M1 +1,, m M1 +M 2, buffers B M1,, B M1 +M 2. In the main line, M 1 machines are arranged serially and M 1 1 buffers separating each consecutive pair of machines. Machine m k and m j M 1 > k > j > 1 are the starting and ending points of the rework loop. In the rework loop, M 2 machines are also arranged serially, however, M buffers separate each consecutive pair of machines, including buffers B M1 and B M1 +M 2 separating machine pairs m k, m M1 +1 and m M1 +M 2, m j, respectively. ii Each machine m i, i = 1,, M 1 + M 2, has two states: up and down. When up, the machine is capable of producing with the rate 1 part per unit of time cycle; when the machine is down, no production takes place. Remark 1.1 Assumption ii on the identical cycle time is introduced to simplify the analysis. However, it is appropriate for many large volume manufacturing environment. 4

6 Typically, in systems with automated material handling, this assumption is valid. The system with unequal machine speeds will be addressed in future work. iii The uptime and the downtime of each machine m i, i = 1,, M 1 + M 2, are random variables distributed exponentially with parameters p i and r i, respectively. Remark 1.2 Due to assumption iii, the production rate in isolation of each machine i.e., the average number of parts produced per unit of time if no starvation or blockage takes place is P R isoi = r i p i + r i, i = 1,, M 1 + M The production rate in isolation is often referred to as the machine efficiency and is denoted as e i, i = 1,, M 1 + M 2, in this paper. iv Each buffer B i, i = 1,, M 1 + M 2, is characterized by its capacity, 0 N i <. v Machine m i is starved at time t if buffer B i 1 is empty at time t. Machine m 1 is never starved. Note that machine m j is starved if both B j 1 and B M1 +M 2 are empty. Remark 1.3 Assumptions vi and v describe the time dependent failure model, where machines can be down even if they are blocked or starved. In contrast, operation dependent failure model has been considered in [3], [4]. The time dependent failure model is assumed here to simplify the analysis. Reference [3] shows that the difference of production rate between time dependent and operation dependent failure models is usually small. vi Machine m i is blocked at time t if B i is full at time t. Machine m M1 is never blocked. In particular, machine m k is blocked by the main line if it produces a good part and 5

7 B k is full, while machine m k is blocked by the rework loop if it produces a defect part and B M1 is full. vii After processing by machine m k, a part is defective with probability α, 0 < α < 1, and needs to be repaired. A defective part is sent to B M1 if it is not full. The good part will be sent to B k with probability 1 α if B k is not full. Probability α is called the rework rate. Remark 1.4 Assumption vii does not have any constraint on the number of times that a defective part can be repaired. In other words, a part will continue to circulate in the system until it satisfies the quality requirements. This may not be the case in practice. For instance, in many paint shops in automotive assembly plants, a job usually can be repaired at most 3 times. However, since the rework rate, α, is usually small, the error introduced by this assumption is neglectable. For instance, if α = 0.2, a defective part has probability 0.04 to be repaired twice, and to be repaired three times. In such situations, this assumption works well. viii Machine m j can take one part each cycle either from B j 1 or B M1 +M 2. It is assumed that m j always takes part from B M1 +M 2 first if it is not empty. Remark 1.5 Assumption viii indicates that machine m M1 +M 2 in rework loop has higher priority than m j 1 in main line in releasing parts to machine m j. This assumption is introduced to avoid possible deadlock. Consider a situation that this assumption is relaxed. If machine m k is producing a defect part, and B M1 is full, then m k is blocked. If machine m j takes parts from B j 1 first and B j 1 is always non-empty, then finally m j will also be blocked and system will be in deadlock. Therefore, assumption vii is necessary. 6

8 A production line defined by i-viii is denoted as {p 1, r 1,, p M1 +M 2, r M1 +M 2, N 1,, N M1 +M 2, j, k, α}. The problem addressed in this paper is as follows: Given production system i-viii, develop a method for evaluating the production rate as a function of the system parameters. A solution to the problem is given in Section 3. 2 PERFORMANCE EVALUATION OF SERIAL LINES 2.1 Aggregation To evaluate the performance of a production system with a rework loop, an estimation of production rate of a serial line is needed. However, no closed form expression for P R of a serial line with more than two non-identical machines is available. Therefore, different approximation approaches are used based on aggregation and decomposition [6]-[9]. In this work, a novel aggregation procedure is considered. Compared to the aggregation approach in [9], this procedure modifies machine downtime parameters instead of uptime parameters to accommodating blocking and starving information. It can be shown that machine downtime is more sensitive to adjustment and more important in practice. In addition, the procedure provides the possibility to prove the convergence in aggregation procedures as well as the uniqueness of the solutions for both serial lines and systems with rework loops and, in addition, results in good accuracy. Consider a serial production line with M machines Figure 2 defined by assumptions i-vi with no rework loop. We aggregate the first two machines into a single machine, m f 2, with downtime parameter r f 2 defined as r f 2 = r 2 [1 Qp 1, r 1, p 2, r 2, N 1 ], 7

9 m 1 B 1 m 2 B 2... m M-1 B M-1 m M Figure 2: Serial Production Line and uptime parameter p f 2 selected so that r f 2 p f 2 + r f 2 = r 2 p 2 + r 2 [1 Qp 1, r 1, p 2, r 2, N 1 ], i.e., p f 2 = p 2 + r 2 Qp 1, r 1, p 2, r 2, N 1, where Q is the probability that machine m 2 is starved and is defined as follows [9]: 1 e 1 1 φ, if p 1 p 2, 1 φe βn r 1 r 2 Qp 1, r 1, p 2, r 2, N = p 1 p 1 + p 2 r 1 + r 2 p 1 + r 1 [p 1 + p 2 r 1 + r 2 + p 2 r 1 p 1 + p 2 + r 1 + r 2 N], 2.2 and if p 1 r 1 = p 2 r 2, e i = r i p i + r i, i = 1, 2, φ = e 11 e 2 e 2 1 e 1, 2.3 β = p 1 + p 2 + r 1 + r 2 p 1 r 2 p 2 r 1. p 1 + p 2 r 1 + r 2 Next, m f 2 is aggregated with m 3 to result in m f 3, with the parameters defined as above, and so on until all M machines are aggregated in a single one, m f M. This constitutes the forward aggregation superscript f is used to denote this fact. Then, in the backward aggregation, the last machine, m M, is aggregated with m f M 1 to result in mb M 1 and so on 8

10 until all machines are again aggregated in a single machine, m b 1. The procedure is repeated until a convergence criteria is satisfied. Formally, this process is represented as follows: Procedure 2.1 ri b s + 1 = r i r i Q p b i+1s + 1, ri+1s b + 1, p f i s, rf i s, N i, i = 1,, M 1, p b is + 1 = p i + r i Q p b i+1s + 1, ri+1s b + 1, p f i s, rf i s, N i, i = 1,, M 1, r f i s + 1 = r i r i Q p fi 1 s + 1, rfi 1 s + 1, pbis + 1, r bi s + 1, N i 1, 2.4 i = 2,, M, p f i s + 1 = p i + r i Q p fi 1 s + 1, rfi 1 s + 1, pbis + 1, r bi s + 1, N i 1, with boundary conditions and initial conditions where function Q is defined in Convergence p f 1s = p 1, r f 1 s = r 1, p b Ms = p M, r b Ms = r M, s = 0, 1, 2,, p f i 0 = p i, r f i 0 = r i, i = 2,, M 1, i = 2,, M, The question of convergence of the resulting sequences p b is, ri b s, p f i s, rf i s, i = 1,, M, s = 0, 1,, is answered in the following: 9

11 exist: Theorem 2.1 Recursive procedure 2.1 is convergent and, therefore, the following limits lim s pf i s = pf i, lim s rf i s = rf i, Moreover, the following relationship holds: lim s pb is = p b i, lim s rb i s = ri b, 2.5 i = 1, 2,, M. r f M p f M = rb p b 1 Corollary 2.1 The steady state equations of recursive procedure 2.1 has a unique solution. Proof: Similar to the proof of Theorem 3.1 in [9] with little modifications. Proof: See Appendix. The limits in 2.6 can be used to define estimates of performance measures for line i- vi. Indeed, since the last machine is not blocked and the first is not starved, production rate can be approximated as P Rp 1, r 1,, p M, r M, N 1,, N M 1 = r f M p f M + rf M = rb p b 1 + r1 b In addition, the probabilities that machine m i, i = 1,, M, is starved or blocked can also be calculated. Prob{m i is blocked} = Qp b i+1, r b i+1, p f i, rf i, N i, Prob{m i is starved} = Qp f i 1, rf i 1, pb i, r b i,, N i. 2.8 The accuracy of the estimate 2.7 is discussed next. 10

12 2.3 Accuracy The accuracy of the estimate 2.7 is explored numerically. We simulate dozens of systems defined by assumptions i-vi with various machine and buffer parameters assumed. Ten of them are shown in Table 1. In each simulation run, zero initial occupancy of all buffers has been assumed, and 5, 000 time slots of warm up period has been carried out. The next 50, 000 slots of stationary regime have been used to statistically evaluate PR. In Table 1, P R denotes the actual PR obtained by simulation, whereas P R denotes the estimate of PR calculated according to 2.7. As it can be seen from Table 1, the estimate results in relatively high precision of PR calculation. Table 1: Accuracy of PR Estimation in Serial Lines err% = P R P R P R 100% p i r i N i P R P R err% Remark 2.1 The confidence intervals have been calculated using the methodology of [19] with 20 runs. The t distribution was used. The same approach has been used in all 11

13 simulations throughout this paper. The 95% confidence intervals were consistently around ±0.0010, while the maximum was ± Similar numbers are obtained in Section 3. Estimate 2.7 is used below in the procedure to evaluate the production rate for production system with rework loop i-viii. 3 PERFORMANCE EVALUATION OF PRODUCTION SYSTEM WITH REWORK LOOP 3.1 Idea of Overlapping Decomposition Approach The idea of the approximation used in this paper is to decompose the production system i-viii into a set of serial production lines, as illustrated in Figure 3. Machines m 1 through m j and buffers B 1 through B j 1 constitute line 1; machines m j to m k and buffers B j to B k 1 constitute line 2; line 3 consists of machines m k to m M1 and buffers B k to B M1 1; Line 4 is represented by the rework loop, i.e., it consists of machines m k, m M1 +1 through m M1 +M 2, m j, and buffers B M1 through B M1 +M 2. Since machines m j and m k are included in more than one serial lines, this decomposition includes overlapping. Consider the serial production line consisting of the machines m k, m M1 +1,, m M1 +M 2, and the buffers B M1,, B M1 +M 2 Figure 3, i.e., line 4. A recursive procedure for evaluating the production rate of this line has been developed in Section 2. In order to use this procedure for the serial line at hand, assume that the parameters of machines m k and m j are modified so as to account for the existence of other machines and buffers. Specifically, introduce the fictitious machines, denoted as m k and m j i.e., the first and last machines of line 4 with parameters p k, r k, p j and r j defined as r k = r k α1 Prob{m k is starved}, p k = p k + r k [1 α1 Prob{m k is starved}], 12

14 !! # $!!! "!! " %!!!! " Figure 3: Overlapping Decomposition of System i-viii into Lines

15 r j = r j 1 Prob{m j is blocked}, p j = p j + r j Prob{m j is blocked}, where p k and r k are selected such that e k = e j = r k r k + p k r j r j + p j = e k α1 Prob{m k is starved}, = e j 1 Prob{m j is blocked}. If probabilities Prob{m k is starved} and Prob{m j is blocked} are known, the recursive procedure 2.1 would result in the production rate of line 4. Analogously, if we know the probabilities that machine m k is starved by B k 1, by modifying machine m k, and using the recursive procedure 2.1, we can calculate the production rate of line 3. The production rate of line 1 can be calculated similarly if the probability that machine m j is starved by B M1 +M 2 and the probability that m j is blocked by B j are known. Finally, the production rate of line 3 depends on the probabilities that machine m k is blocked by B k and B M1 and the probabilities that machine m j is starved by B k 1 and B j 1, respectively see Figure 3. The system production rate is equal to that of line 3. Then, the production rate of the system which equals the production rate of line 3 can be calculated. Since these probabilities are unknown, we introduce iterations, as described below. At the first step, assume Prob{m k is starved} and Prob{m j is blocked} are 0 and 1, respectively. First consider line 4: modify machines m k and m j to m k and m j with parameters defined as above, using the recursive procedure 2.1, calculate probabilities Prob{m k is blocked by B M1 } and Prob{m j is starved by B M1 +M 2 }. Next, consider line 3, modify machine m k to m k r k = r k 1 α[1 Prob{m k is starved}], with parameters defined as p k = p k + r k [1 1 α1 Prob{m k is starved}], 14

16 and e k = e k 1 α[1 Prob{m k is starved}]. Using procedure 2.1, probability Prob{m k is blocked by B k } is calculated. Similarly, consider line 1, modify machine m j to m j with parameters and r j = r j Prob{m j is starved by B M1 +M 2 }1 Prob{m j is blocked}, p j = p j + r j [1 Prob{m j is starved by B M1 +M 2 }1 Prob{m j is blocked}], e j = e j Prob{m j is starved by B M1 +M 2 }1 Prob{m j is blocked}. Again, using procedure 2.1, calculate the probability Prob{m j is starved by B j 1 }. as Finally, for line 2, introduce fictitious machines m j and m, with parameters modified k and r j = r j 1 Prob{m j is starved by B j 1 } Prob{m j is starved by B M1 +M 2 }, p j = p j + r j Prob{m j is starved by B j 1 } Prob{m j is starved by B M1 +M 2 }, r k = r k [1 αprob{m k is blocked by B M1 } 1 αprob{m k is blocked by B k }], p k = p k + r k [αprob{m k is blocked by B M1 } + 1 αprob{m k is blocked by B k }], e j = e j 1 Prob{m j is starved by B j 1 } Prob{m j is starved by B m1 +M 2 }, e k = e k [1 αprob{m k is blocked by B M1 +M 2 } 1 αprob{m k is blocked by B k }], and using the recursive procedure 2.1 again, the probabilities Prob{m j is starved} and Prob{m j is blocked} are calculated. Use now these probabilities for the second iteration in analysis of line 4 and continue this process, alternating among lines 1 to 4. As it is shown below, the iterations are convergent and result in the estimate of system production rate. 15

17 Remark 3.1 Similar approach can be generalized to apply to production systems with multiple rework loops. In this case, more than 4 serial production lines would need to be considered. 3.2 Recursive Procedure For simplification, introduce the following notations: b j = Prob{machine m j is blocked}, s k = Prob{machine m k is starved}, b k1 = Prob{machine m k is blocked by B k }, b k2 = Prob{machine m k is blocked by B M1 }, s j1 = Prob{machine m j is starved by B j 1 }, s j2 = Prob{machine m j is starved by B M1 +M 2 }, pr i = production rate estimation of line i, i = 1, 2, 3, 4. Introduce also operators Φ 1 and Φ 2. Consider a serial production line with machines m i, i = 1,, M. Operator Φ 1 is used to calculate the probability that the first machine is starved, whereas operator Φ 2 is used to calculate the probability that the last machine is blocked. In other words, b 1 = Φ 1 p 1, r 1,, p M, r M, N 1,, N M 1, s M = Φ 2 p 1, r 1,, p M, r M, N 1,, N M 1. Operators Φ 1 and Φ 2 are defined in 2.8 by recursive procedure 2.1. Remark 3.2 When r 1 = 0, obviously b 1 = Φ 1 = 0. Analogously, if r M = 0, s M = Φ 2 = 0. 16

18 Formally, using P R defined in 2.7, and Φ 1 and Φ 2 defined in 2.8, the recursive procedure is: Procedure 3.1 Line 4 r kn + 1 = r k α[1 s k n], p kn + 1 = p k + r k 1 α[1 s k n], r jn + 1 = r j [1 b j n], p jn + 1 = p j + r j b j n, 3.9 pr 4 n + 1 = P R p kn + 1, r kn + 1, p M1 +1, r M1 +1,, p M1 +M 2, r M1 +M 2, p jn + 1, r jn + 1, N M1,, N M1 +M 2, b k2 n + 1 = Φ 1 p kn + 1, r kn + 1, p M1 +1, r M1 +1,, p M1 +M 2, r M1 +M 2, p jn + 1, r jn + 1, N M1,, N M1 +M 2, s j2 n + 1 = Φ 2 p kn + 1, r kn + 1, p M1 +1, r M1 +1,, p M1 +M 2, r M1 +M 2, Line 3 p jn + 1, r jn + 1, N M1,, N M1 +M 2. r kn + 1 = r k [1 s k n]1 α, p kn + 1 = p k + r k 1 1 α[1 s k n], 3.10 pr 3 n + 1 = P R p kn + 1, r kn + 1, p k+1, r k+1,, p M1, r M1, N k,, N M1 1, b k1 n + 1 = Φ 1 p kn + 1, r kn + 1, p k+1, r k+1,, p M1, r M1, N k,, N M1 1, Line 1 r j n + 1 = r j s j2 n + 1[1 b j n], 17

19 p j n + 1 = p j + r j 1 s j2 n + 1[1 b j n], 3.11 pr 1 n + 1 = P R p 1, r 1,, p j 1, r j 1, p j n + 1, r j n + 1, N 1,, N j 1, s j1 n + 1 = Φ 2 p 1, r 1,, p j 1, r j 1, p j n + 1, r j n + 1, N 1,, N j 1, Line 2 r j n + 1 = r j [1 s j1 n + 1s j2 n + 1], p j n + 1 = p j + r j s j1 s j2 n + 1, r k n + 1 = r k [1 αb k2 n αb k1 n + 1], p k n + 1 = p k + r k [αb k2 n αb k1 n + 1], 3.12 pr 2 n + 1 = P R p j n + 1, r j n + 1, p j+1, r j+1,, p k 1, r k 1, p k n + 1, r k n + 1, N j,, N k 1, b j n + 1 = Φ 1 p j n + 1, r j n + 1, p j+1, r j+1,, p k 1, r k 1, p k n + 1, r k n + 1, N j,, N k 1, s k n + 1 = Φ 2 p j n + 1, r j n + 1, p j+1, r j+1,, p k 1, r k 1, p k n + 1, r k n + 1, N j,, N k 1, n = 0, 1, 2,, with initial conditions s k 0 = 0, b j 0 = 1. Remark 3.3 The initial values of s k and b j are used to prove the convergence only. The convergence takes place for all s k and b j between 0 and 1. 18

20 3.3 Convergence Theorem 3.1 Under assumptions i-viii, recursive procedure 3.1 is convergent, i.e., lim pr in = pr i, i = 1, 2, 3, 4, n lim b jn = b j, n lim b k 1 n = b k1, n lim s j 1 n = s j1, n lim s k n = s k, n lim b k2 n = b k2, 3.13 n lim s j2 n = s j2. n Proof: See Appendix. Using the limits in 3.13, the system production rate can be evaluated as: P R = pr In addition, the following relationships are hold: Corollary 3.1 Under assumptions i-viii, pr 1 = pr 3, pr 2 = pr 1 + pr Proof: See Appendix. Corollary 3.2 The steady state equations of recursive procedure 3.1 has a unique solution. Proof: See Appendix. 19

21 3.4 Accuracy The accuracy of the estimate 3.14 is investigated numerically. Dozens of systems defined by assumptions i-viii with various machine and buffer parameters setting are simulated. Fifteen of them are shown in Table 2. As it is shown in Table 2, the estimates of PR have relatively high precision and are comparable to that of [6]-[12]. Recursive procedure 3.1 has been applied in several paint shops in automotive assembly plants. Experimental results show that the method works well using actual operation data and results in high precision in production rate estimation. Moreover, it has been used to guide design of continuous improvement projects. 4 PRODUCTION RATE VS. REWORK RATE Obviously the system production rate, P R, is a monotonically decreasing function of the rework rate, α. Figure 4 illustrates the relationship between P R defined by 3.14 and α using Examples 1-4 of Table 2. It shows that P R decreases almost linearly with respect to α, except when α is very small. Figure 4 shows that reducing rework rate could significantly improve system production rate in nearly a linear manner. Using linear approximation, we can rewrite P R as a function of α. P Rα = αx + y, where x and y are constants shown in Figure 4. Consider a production system without rework loop, and assume that all the defect parts are scrapped after last operation. We calculate the system production rate using procedure 2.1. In other words, the production rate of system without rework loop can be approximately calculated as follows and is illustrated in Figure 4: P R without rework α = P R α=0 1 α, 20

22 Table 2: Accuracy of PR Estimation in Production System with Rework Loop err% = P R P R P R 100% Ex. Main line Rework line j k α P R P R err% # p i r i N i p i r i N i

23 1 0.8 Ex.1: x= , y= : Recursive procedure...: Linear approximation Ex.2: x= , y= < PR : Without rework < PR α α 1 Ex.3: x= , y= Ex.4: x= , y= < PR < PR α α Figure 4: Production Rate as a Function of Rework Rate 22

24 where P R α=0 is the production rate of the main line without rework calculated by procedure 2.1. To investigate the effect of the rework loop, Figure 5 illustrates the improvement of production rate by introducing the rework loop as a function of α for the same examples shown in Figure 4, i.e., Improvement% = P R with rework P R without rework P R without rework 100%. As it is shown in Figure 5, the improvement of production rate is monotonically increasing with respect to rework rate α. The system production rate can be improved significantly % provided that rework loop is introduced. Remark 4.1 It is easy to understand that when α is small, the improvement rate is high, and it saturates at mid-range of α. However, when α approximates to 1, the improvement becomes high again. This can be explained as follows: from Figure 4, when α 1, P R with rework decreases almost linearly with higher rate than P R without rework. Let P R with rework α = αx + y, where y > x. Then, the improvement rate can be written as follows: Improvement lim α 1 α = lim α 1 x P R α=0 1 α + αx + y =: lim P R α=0 1 α 2 α 1 C 1 α 2, where C > 0. Therefore, the improvement rate is higher when α approximates to 1. 5 CONCLUSIONS Modelling and analysis of production systems with rework loops are important for design and continuous improvement of such systems. However, the current literature does not offer any analytical methods to evaluate its performance, for instance, its production rate. This paper presents an aggregation procedure to calculate the production rate of production systems 23

25 25 Example 1 20 Example 2 Improvement % Improvement % α α 12 Example 3 25 Example Improvement % Improvement % α α Figure 5: Improvement of Production Rate 24

26 with rework loops when machines have identical cycle times. The convergence of the procedure and the uniqueness of the solution are proved analytically with good accuracy found by numerical examples. It is shown that the accuracy is comparable to that of production rate calculation in serial and assembly production lines. The method developed can be used for design and analysis of production systems with rework loops. In addition, this approach has been applied in a paint shop to evaluate its production rate and provide recommendations for continuous improvements. Fairly good results have been obtained. APPENDIX: PROOFS Proof of Corollary 2.1: By contradiction: Consider the steady state equations of recursive procedure 2.1 are as follows: Define ri b = r i r i Q p b i+1, ri+1, b p f i, rf i, N i, i = 1,, M 1, p b i = p i + r i Q p b i+1, ri+1, b p f i, rf i, N i, i = 1,, M 1, r f i = r i r i Q p fi 1, rfi 1, pbi, r bi, N i 1, i = 2,, M, A.1 p f i = p i + r i Q p fi 1, rfi 1, pbi, r bi, N i 1, i = 2,, M, p f 1 = p 1, r f 1 = r 1, p b M = p M, r b M = r M. e f i = e b i = r f i r f i + pf i r b i, i = 1,, M, ri b +, i = 1,, M. A.2 pb i From [9], we obtain ] e b i = e i [1 Qp b i+1, ri+1, b p f i, rf i, N i, 1 i M 1, 25

27 ] e f i = e i [1 Qp f i 1, rf i 1, pb i, ri b, N i 1, 2 i M, A.3 e f 1 = e 1, e b M = e M, where function Q is defined in 2.2. Introduce M 1 two machine one buffer production line L i, i = 1,, M 1, where the first machine has the uptime parameter p f i and the downtime parameter r f i, the second pb i+1 and r b i+1, and the buffer capacity is N i. The following properties hold: Let P R i be the production rate of line L i, i = 1,, M 1, and let P R M = e f M. Then from [9], P R i = ef i eb i e i, i = 1,, M. A.4 Moreover, P R i = P R j, i j. A.5 Assume that along with the solution S agg = [p f 1, r f 1,, p f M, rf M, pb 1, r1, b, p b M, rb M ]T to equations A.1, there exists another solution denoted by S agg = [p f 1, r f 1,, p f M, rf M, pb 1, r b 1,, p b M, r b M] T. Suppose that p b 1 > p b 1, it follows from A.1 that r1 b = r 1 + p 1 p b 1, r b 1 = r 1 + p 1 p b 1, which leads to r b 1 < r1. b By Theorem 2.1, P R j > P R j, j = 1,, M. A.6 In addition, from r1 b = r 1 r 1 Qp b 2, r2, b p 1, r 1, N 1 > r b 1 = r 1 r 1 Qp b 2, r b 2, p 1, r 1, n 1, it immediately follows that Qp b 2, r b 2, p 1, r 1, N 1 < Qp b 2, r b 2, p 1, r 1, n 1. A.7 26

28 Then the following property follows: p b 2 < p b 2, r b 2 > r b 2. This property is proved by contradiction. Assume that p b 2 p b 2, it follows from 2.5 that r b 2 r b 2. By monotonicity of Q see [9] for details we have Qp b 2, r b 2, p 1, r 1, N 1 > Qp b 2, r b 2, p 1, r 1, n 1, which is a contradiction to A.7. Therefore we obtain p f 2 = p 2 + r 2 Qp 1, r 1, p b 2, r b 2, N 1 > p 2 + r 2 Qp 1, r 1, p b 2, r b 2, N 1 = p f 2, and r f 2 < r f 2. Now proceed inductively. Assume p f j > p f j. The base case j = 2 has already been established. By equations A.1, r f j < rf j, which implies ef j < ef j. From A.6, we must have p b j < p b j, r b j > r b j, i.e., Qp b j+1, rj+1, b p f j, rf j, N j < Qp b j+1, r b j+1, p f j, rf j, N j. It implies that Qp f j, rf j, pb j+1, rj+1, b N j > Qp f j, rf j, pb j+1, r b j+1, N j, which leads to p f j+1 > pf j+1. 27

29 Thus, the inductive hypothesis is established, and p f j > p f j, 2 i M. In particular, p f M > pf M. From A.6, it implies that P R M < P R M, which contradicts equation A.6. We therefore conclude that p b 1 p b 1. Assuming that p b 1 > p b 1, and proceeding analogously, yields p b 1 p b 1. Therefore p b 1 = p b 1. The equality of the remaining components of S agg = S agg will be shown by induction. Note that p f 1 = p 1 = p f 1, r f 1 = r 1 = r f 1, and that p b 1 = p b 1 and r b 1 = r b 1. Assume that p b j = p b j, p f j = pf j, which implies rb j = r b j and r f j = rf j. In addition, Qp b j+1, r b j+1, p f j, rf j, N j = Qp b j+1, r b j+1, p f j, rf j, N j. It follows that p b j+1 = p b j+1, r b j+1 = r b j+1, which leads to p f j+1 = pf j+1, rf j+1 = rf j+1. The inductive hypothesis is established. It may conclude that there is a unique solution to the steady state equations of recursive procedure 2.1. To prove Theorem 3.1, the following 3 lemmas are needed. Lemma A.1 shows that in serial line i-vi, if p 1 is increased and r 1 is decreased, then the probability of first machine to be blocked, b 1, is decreased, and vice versa. Lemma A.2 shows that in serial line i-vi, if p M is increased and r M is decreased, then the probability of last machine to be starved, s M, is decreased, and vice versa. Lemma A.3 shows that in procedure 3.1 if s k n > s k n 1 and b j n < b j n 1, then s k n + 1 > s k n and b j n + 1 < b j n. 28

30 Lemma A.1 In serial line i-vi, if p 1 is increased and r 1 is decreased, then the probability of first machine to be blocked, b 1, is decreased. Proof: Assume there exists another serial line denoted by {p 1, r 1, p 2, r 2,, p M, r M, N 1,, N M 1 }, with production rate P R, and aggregation parameters p f i, pb i and r f i, rb i, i = 1,, M, where p 1 < p 1, r 1 > r 1. Let P R be the production rate of the original line. It is obvious that P R < P R. A.8 In other words, Qp f M 1, rf M 1, p M, r M, N M 1 > Qp f M 1, rf M 1, p M, r M, N M 1. A.9 and Qp M, r M, p f M 1, rf M 1, N M 1 < Qp M, r M, p f M 1, rf M 1, N M 1. A.10 Then the following property follows: p f M 1 > pf M 1, rf M 1 < rf M 1. In addition, From A.10, we have e f M 1 < ef M 1. r b M 1 = r M 1 [1 Qp M, r M, p f M 1, rf M 1, N M 1] > r M 1 [1 Qp M, r M, p f M 1, rf M 1, N M 1] = rm 1, b 29

31 and p b M 1 < p b M 1. Repeatly performing the similar analysis, finally we have r b 2 > r b 2, p b 2 < p b 2. It implies that b 1 = Qp b 2, r b 2, p 1, r 1, N 1 < Qp b 2, r b 2, p 1, r 1, N 1 = b 1. Therefore, b 1 is decreased if p 1 is increased and r 1 is decreased. This proves the first part. By doing similar analysis, the second part is readily proved. Lemma A.2 In serial line i-viii, if p M is increased and r M is decreased, then the probability of last machine to be starved, s M, is decreased. Proof: By reversibility, Lemma A.2 is proved. Lemma A.3 In Procedure 3.1 if s k n > s k n 1 and b j n < b j n 1, then s k n+1 > s k n and b j n + 1 < b j n. Proof: If s k n > s k n 1, b j n < b j n 1, A.11 30

32 from 3.9, we have r kn + 1 < r kn, r jn + 1 > r jn, p kn + 1 > p kn, p jn + 1 < p jn. It follows from Lemmas A.1 and A.2 that b k2 n + 1 < b k2 n, s j2 n + 1 > s j2 n. A.12 Similarly, from 3.10 we have and from Lemma A.1, r kn + 1 < r kn, p kn + 1 > p kn, b k1 n + 1 < b k1 n. A.13 From 3.11 and A.12, it implies that By Lemma A.2, it leads to r kn + 1 < r kn, p kn + 1 > p kn. s j1 n + 1 < s j1 n. A.14 From 3.12 and A.12 - A.14, it follows that From Lemmas A.1 and A.2, we obtain r j n + 1 < r j n, p j n + 1 > p k n, r k n + 1 > r k n, p k n + 1 < p kn. b j n + 1 < b j n, s k n + 1 > s k n. A.15 Lemma A.3 is proved. 31

33 Proof of Theorem 3.1: By induction. For n = 0, s k 0 = 0, b j 0 = 1. For line 4, from 3.9 we have r k1 = r k α, r j1 = 0, which implies that From 3.10, It follows that For line 1, by 3.11 we have and Finally, for line 2, from 3.12, b k2 1 = 1, s j2 1 = 0. r k1 = r k 1 α. 0 < b k1 1 < 1. r j 1 = 0, s j1 1 = 0. r j 1 = r j, r k 1 = r k [1 α 1 αb k1 ] = r k 1 α1 b k1 > 0. It follow that 0 < b j < 1, 0 < s k < 1. 32

34 Therefore we obtain s k 1 > s k 0, b j 1 < b j 0. Assume now n > 0, s k n > s k n 1, b j n < b j n 1. Then from Lemma A.3 we obtain s k n + 1 > s k n, b j n + 1 < b j n. Therefore, s k n and b j n are monotonically increasing or decreasing, respectively. Since they are bounded by 0 and 1 [9], they are convergent. Theorem 3.1 is proved. Proof of Corollary 3.1: pr 1 + pr 4 = e j s j2 1 b j 1 s j1 + e j 1 b j 1 s j2 = e j 1 b j s j2 s j1 s j2 + 1 s j2 = e j 1 b j 1 s j1 s j2 = pr 2, where e j = Rewrite pr 4 and pr 2 in terms of e k where e k = r j p j + r j. r k p k +r k, we have pr 1 = pr 2 pr 4 = e k [1 αb k2 1 αb k1 ]1 s k e k α1 s k 1 b k2 = e k 1 s k [1 αb k2 1 αb k1 α1 b k2 ] 33

35 = e k 1 s k 1 α1 b k1 = pr 3. Corollary 3.1 is proved. Proof of Corollary 3.2: By Contradiction. The steady state equations of recursive procedure 3.1 are as follows: Line 4 r k = r k α[1 s k ], p k = p k + r k 1 α[1 s k ], r j = r j [1 b j ], p j = p j + r j b j, A.16 pr 4 = P R p k, r k, p M1 +1, r M1 +1,, p M1 +M2, r M1 +M2, p j, r j, N M1,, N M1 +M2, b k2 = Φ 1 p k, r k, p M1 +1, r M1 +1,, p M1 +M 2, r M1 +M 2, p j, r j, N M1,, N M1 +M 2, s j2 = Φ 2 p k, r k, p M1 +1, r M1 +1,, p M1 +M 2, r M1 +M 2, p j, r j, N M1,, N M1 +M 2, Line 3 r k = r k [1 s k ]1 α, p k = p k + r k 1 1 α[1 s k ], pr 3 = P R p k, r k, p k+1, r k+1,, p M1, r M1, N k,, N M1 1, A.17 b k1 = Φ 1 p k, r k, p k+1, r k+1,, p M1, r M1, N k,, N M1 1, Line 1 r j = r j s j2 [1 b j ], p j = p j + r j 1 s j2 [1 b j ], pr 1 = P R p 1, r 1,, p j 1, r j 1, p j, r j, N 1,, N j 1, A.18 s j1 = Φ 2 p 1, r 1,, p j 1, r j 1, p j, r j, N 1,, N j 1, Line 2 r j = r j [1 s j1 s j2 ], p j = p j + r j s j1 s j2, 34

36 r k = r k [1 αb k2 1 αb k1 ], p k = p k + r k [αb k2 + 1 αb k1 ], pr 2 = P R p j, r j, p j+1, r j+1,, p k 1, r k 1, p k, r k, N j,, N k 1, A.19 b j = Φ 1 p j, r j, p j+1, r j+1,, p k 1, r k 1, p k, r k, N j,, N k 1, s k = Φ 2 p j, r j, p j+1, r j+1,, p k 1, r k 1, p k, r k, N j,, N k 1. Let S agg = [s k, s j1, s j2, b k1, b k2, b j ] T be the solution to equations A.16 - A.19. Assume that along with this solution, there exists another solution denoted by S agg = [s k, s j1, s j2, b k1, b k2, b j ] T. Suppose that s k > s k, it follows from A.17 that which leads to pr 3 < pr 3, and b k1 < b k1. r k < r k, p k > p k, From Corollary 3.1 we must have pr 1 < pr 1, which is equivalent to It follows from A.18 that and From A.16 we obtain r j < r j, p j > p j. s j1 < s j1, s j2 1 b j < s j2 1 b j. r k < r k, r j < r j. The latter is proved by contradiction. Suppose we have b j b j, which implies r j r j. It follows that s j2 > s j2, and, in addition, s j2 1 b j > s j2 1 b j, which is a contradiction. Therefore, we must have b j > b j. This leads to r j < r j and pr 4 < pr 4. From Corollary 3.1 we have pr 2 < pr 2. There exists four possible combinations of relationship between r k and r k, r j and r j, respectively. The following shows that all of them lead to contradictions to the above analysis. 35

37 Case 1: r k r k, r j r j. It follows that pr 2 pr 2, which is a contradiction. Case 2: r k Case 3: r k Case 4: r k r k, r j < r j. It implies that b j b j, which is a contradiction. < r k, r j r j. It leads to s k < s k, which again is a contradiction. < r k, r j < r j. It follows that s j1 s j2 > s j1 s j2, αb k2 + 1 αb k1 > αb k2 + 1 αb k1, which implies that By using recursive procedure 2.1, we have s j2 > s j2, b k1 > b k1. s j2 = Qp f M 1 +M 2, r f M 1 +M 2, p j, r j, N M1 +M 2 > Qp f M 1 +M 2, r f M 1 +M 2, p j, r j, N M1 +M 2 = s j2. From monotonicity of Q [9], we must have p f M 1 +M 2 > p f M 1 +M 2, r f M 1 +M 2 < r f M 1 +M 2, i.e. In addition, e f M 1 +M 2 < e f M 1 +M 2. r b M 1 +M 2 = r M1 +M 2 r M1 +M 2 Qp j, r j, p f M 1 +M 2, r f M 1 +M 2, N M1 +M 2 > r M1 +M 2 r M1 +M 2 Qp j, r j, p f M 1 +M 2, r f M 1 +M 2, N M1 +M 2 = r b M 1 +M 2, p b M 1 +M 2 < p b M 1 +M 2, 36

38 i.e., e b M 1 +M 2 > e b M 1 +M 2. Recall that r f M 1 +M 2 = r M1 +M 2 r M1 +M 2 Qp f M 1 +M 2 1, rf M 1 +M 2 1, pb M 1 +M 2, r b M 1 +M 2, N M1 +M 2 1 < r f M 1 +M 2 = r M1 +M 2 r M1 +M 2 Qp f M 1 +M 2 1, rf M 1 +M 2 1, pb M 1 +M 2, r b M 1 +M 2, N M1 +M 2 1, it follows that rm b 1 +M 2 1 = r M1 +M 2 1 r M1 +M 2 1Qp b M 1 +M 2, rm b 1 +M 2, p f M 1 +M 2 1, rf M 1 +M 2 1, N M 1 +M 2 1 > r M1 +M 2 1 r M1 +M 2 1Qp b M 1 +M 2, r b M 1 +M 2, p f M 1 +M 2, r f M 1 +M 2, N M1 +M 2 = r b M 1 +M 2 1, p b M 1 +M 2 1 < p b M 1 +M 2 1, i.e., e b M 1 +M 2 1 > e b M 1 +M 2 1. Then from A.5, e f M 1 +M 2 e b M 1 +M 2 < e f M 1 +M 2 e b M 1 +M 2. It implies that e f M 1 +M 2 1 < ef M 1 +M 2 1, i.e., p f M 1 +M 2 1 > pf M 1 +M 2 1, rf M 1 +M 2 1 < rf M 1 +M 2 1. Proceed repetitively we obtain r b i > r b i, p b i < p b i, 37

39 r f i < r f i, pf i > p f i, i = M 1 + 1,, M 1 + M 2, A.20 i.e., e b i > e b i, e f i < e f i, i = M 1 + 1,, M 1 + M 2. A.21 Analogously, from b k1 = Qp b M 1 +1, r b M 1 +1, p k, r k, N M1 > Qp b M 1 +1, r b M 1 +1, p k, r k, N M1 = b k1. we have r b M 1 +1 < r b M 1 +1, p b M 1 +1 > p b M 1 +1, and e b M 1 +1 < e b M 1 +1, which is a contradiction to A.21. Therefore, we conclude that s k s k. Assuming that s k < s k, and proceeding analogously, yields s k s k. Therefore s k = s k. From A.17, we have b k1 = b k1, pr 3 = pr 3. By Corollary 3.1, pr 1 = pr 1. Then, from A.18, we have s j1 = s j1, s j2 1 b j = s j2 1 b j. By contradiction, we show that b j = b j. Assume b j > b j, then s j2 > s j2. From A.16, we have r j < r j, and, b k2 < b k1, s j2 < s j2, 38

40 which is a contradiction. Therefore, b j b j. Analogously, assuming b j < b j leads to b j b j. To conclude we must have b j = b j. It follows that b k2 = b k2, s j2 = s j2. Hence, S agg = S agg, and p f i = p f i, pb i = p b i, r f i = r f i, rb i = r b i, i = 1,, M 1 + M 2. Therefore, we conclude that there is a unique solution the steady state equations of recursive procedure 3.1. References [1] Y. Dallery and S.B. Gershwin, Manufacturing Flow Line Systems: A Review of Models and Analytical Results, Queueing Systems, Vol. 12 pp. 3-94, [2] H.T. Papadopoulos and C. Harvey, Queueing Theory in Manufacturing Systems Analysis and Design: A Classification of Models for Production and Transfer Lines, European Journal of Operational Research, Vol. 92, pp. 1-27, [3] J.A. Buzacott and J.G. Shanthikumar, Stochastic Models of Manufacturing Systems, Prentice Hall, [4] S.B. Gershwin, Manufacturing Systems Engineering, Prentice Hall, [5] T. Altiok, Performance Analysis of Manufacturing Systems, Springer,

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