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On the search of workstations arrangement in pull production systems
Available online at www.sciencedirect.com
Computers & Industrial Engineering 54 (2008) 613–623 www.elsevier.com/locate/dsw
On the search of workstations arrangement in pull production systems q Hsu-Tung Lee a
a,*
, Michael H. Wang
b
Department of Business Administration, National Taipei University, 151, University Road, San Shia, Taipei 237, Taiwan, ROC b Department of Industrial and Manufacturing Systems Engineering, University of Windsor, 401 Sunset Avenue, Windsor, Ont., Canada N9B 3P4 Received 7 July 2006; received in revised form 20 September 2007; accepted 20 September 2007 Available online 29 September 2007
Abstract This research presents a model to determine the workstation arrangement and kanban number for pull production systems. Several practical production line characters are considered, and evolutionary algorithms are utilized to obtain the optimal/near-optimal result. The proposed system is a flow-shop, pull production line with N unreliable workstations and N 1 inter-stage buffers. Workstations may be allocated to any stage in the production line and have a different processing rate at each stage. Workstations will breakdown/recover at fixed rates, and the operating-breakdown-recovery process is formulated as a Markov process in this model. The size of the inter-stage buffers varies according to the arrangements of workstations. A model is established on the objective of minimizing the total production cost per item of the finished product. The costs considered in the system contain the allocation of workstations, operations, inventory, and production shortage costs. An evolutionary algorithm, genetic algorithm, is used to obtain the allocation of workstations in the production line. 2007 Elsevier Ltd. All rights reserved. Keywords: Pull production system; Unreliable machine; Evolutionary algorithm; Workstation arrangement
1. Introduction Any production system is subject to stoppage. This scenario includes machine breakdown, unscheduled maintenance, job interruption, and so on. The unanticipated stoppage has a significant impact on the performances of manufacturing systems, especially when breakdown happens at the bottleneck workstation. Manufacturers usually use inter-stage buffers to neutralize the influence of breakdown and other similar situations. Thus, to study the effect of buffer and to determine the size of buffer is an important issue since the space of a plant is limited. El-Rayah (1979) studies the behavior of production lines with different inter-stage buffer q *
This manuscript was processed by Area Editor Prof. Y.M. Dessouky. Corresponding author. E-mail address: [email protected] (H.-T. Lee).
0360-8352/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.09.011
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assignments. He proves that assigning larger inter-stage buffers capacities to the middle workstation will be more efficient while unbalancing the production line in terms of buffer capacities. The buffer-size-determination becomes more complicated when machine breakdown is taken into consideration. The phenomenon of machine breakdown is best described with stochastic models. However, the state space will increase dramatically when the status of machine is involved. For instance, the state space of a two-machine, one-buffer (capacity = 3) system will increase from 4 to 16. Gershwin and Schick (1983), Choong and Gershwin (1987), D’angelo et al. (1988), and Glassey and Hong (1993) study the behavior of production line with unreliable workstations. Choong and Gershwin develop a decomposition method to reduce the state space in this problem and prove it to be accurate. Hillier and So (1991) study the effects of machine breakdowns and interstage buffer capacity on performance of the production line. Kouikoglou and Phillis (1994) introduce a newer version of unreliable production line problem by using a discrete model with limited repair resources. Watanabe, Hashimoto, Nishikawa, and Tokumaru (1995) and McMullen and Frazier (1998) use search algorithms, e.g. genetic algorithms and simulated annealing, respectively, to obtain the optimal solutions to line balancing problems. The Just-in-time (JIT) with kanban production system is a common philosophy in manufacturing process nowadays. This philosophy aims at reducing work-in-process inventory and increasing production. Price, Gravel, and Nsakanda (1994) review some of the kanban-based optimization models. Villeda, Dudek, and Smith (1988) seek the solution to increase the production rate in the JIT environment by examining various unbalancing line techniques and show that the ‘‘high-medium-low’’ arrangement of workstation provides better output rate in the JIT system. Wang and Wang (1990) develop a method to determine the optimum number of inter-stage kanban in one-station-to-one-station, multiple-station-to-one-station, and multiple-station-tomultiple-station cases based on cost minimization. In Wang and Wang’s study, a method for unreliable workstations is presented. Fan (1994) carries out Wang and Wang’s study numerically. This study further extends Wang and Wang and Fan’s study by formulating the problem more efficiently and considers more characters of unreliable production line. 2. Problem statement A flow shop type pull production line with finite inter-stage buffers is considered. Workstations can be located in different positions with position dependent performance subject to breakdown. The system has N unreliable workstations and N 1 inter-stage buffers. The buffer capacity is applied to neutralize the possible breakdown and the capacity (number of kanban) is dependent on workstation assignment. The objective of this research is to determine the optimal arrangement of workstations and the inter-stage buffer size associated with workstation allocation. Some of the characters for this production line are listed as follows: 1. All items enter at the first workstation and leave at the last workstation. 2. The preceding/succeeding of sequence of job exists. Thus, some workstations have to be placed at a particular position. 3. Blockage (starvation) exists when the machine in the upstream (downstream) is ready but there’s no kanban to (can) authorize production activity. This situation may happen when the downstream (upstream) machine breaks down and the buffers are full (empty). Breakdown cannot occur at a starved or blocked workstation. 4. There is an infinite source for the production system, and an infinite storage capacity at the end. In other words, the first machine never starves, and the last machine can never be blocked. 5. The process time, failure rates, and recovery rates of workstations and its endurance are formulated as a Poisson process. 2.1. Notations N lij
number of workstations. mean processing rate of workstation i in stage j, which is formulated as a Poisson distribution.
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Fi Ri Hi Si lsj Fsj Rsj D THR INVj K EFj IEFj HCsj SCsj TC C1
615
failure rate of workstation i, which is formulated as a Poisson distribution. recovery rate of workstation i, which is formulated as a Poisson distribution. holding cost of workstation i, per item per unit time. shortage cost of workstation i. mean processing rate of workstation in stage j, after position assignment. failure rate of workstation in stage j, after position assignment. recovery rate of workstation in stage j, after position assignment. demand rate of finished products. throughput rate of the system. inventory cost of buffer j, after position assignment. number of kanban at particular stage. the effective processing rate of workstation in stage j. the ineffective processing rate of workstation in stage j. inventory holding cost per item per unit time in stage j. shortage cost of WIP item in stage j. total cost of production per item. operation cost per unit time, which includes labor, machine, maintenance, etc.
2.2. Methodology The decomposition method introduced by Gershwin is utilized in this research to reduce the state space of the stochastic problem. Due to the nature of pull production system, some of the characters, e.g. throughput rate and demand rate, are closely related, and the relationship can be utilized to further reduce the complexity of this problem. Before mathematical formulation is derived, some fundamental information of this production line should be determined first. It includes the system’s process rate, the throughput rate, and the optimal number of kanban. 2.2.1. The process rate According to Buzacott (1967), the effective processing rate for workstation in stage jðls0j Þ due to the unreliable behavior of workstations can be determined by: ls0j ¼ lsj
Rsj Fsj þ Rsj
for j ¼ 1–N
ð1Þ
2.2.2. The throughput rate The throughput rate should be equal to the demand rate in an ideal JIT manufacturing system, and it can be shown as: THR ¼ MinfD; ls01 ; ls02 ; ls03 ; . . . ; ls0n g Here the throughput rate is the minimum of the effective processing rate mand rate.
ð2Þ (ls0j )
of all workstations and the de-
2.2.3. The optimum kanban number Wang and Wang’s study demonstrated an effective way to determine the number of kanban, also known as the buffer capacity in this research. There are five major steps to determine the optimum kanban number. The method is briefly described below: I. Define the state space E for the production system Suppose the number of kanban in a decomposed system (a pair of workstations) is K. The state space of this production system can be described as a Markov process with NS number of states. NS ¼ ðK þ 1Þ 2 2
ð3Þ
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This equation calculates the number of states for state space E, with K number of kanban. The number of states NS is equal to K + 1 (possible kanban level from zero to K) multiplied by two of the possible status (operating/under repair) of the preceding workstation and two of the possible status of the succeeding workstation. Similarly, each component of the state space can be represented as (n, I1, I2), where n (0 6 n 6 K) is the number of fulfilled kanban; I1 and I2 represent the status of preceding and succeeding workstations respectively (1 representing the workstation is operating properly; 0 representing the workstation is under repair). For a three kanban case, the state space is E = (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1 1 ), (2, 0, 0), (2, 0, 1), (2, 1, 0), (2, 1, 1), (3, 0, 0), (3, 0, 1), (3, 1, 0), (3, 1, 1). II. Define the transition matrix For state (0, 0, 1), the workstation ahead is down and there’s no job waiting in the buffer. Based on the starved-machine-never-breakdown rule, the only possible state in the next is (0, 1, 1). Similarly, the only consequence of (K, 1, 0) is (K, 1, 1) according to the blocked-machine-never-breakdown rule. The transition matrix (QK) for this Markov process of all scenarios can be derived as shown in Table 1. For those states with empty containers (the first four rows in Table 1), there is no production activity at the succeeding workstation due to the void buffer. Thus the kanban levels will either stay the same (preceding machine breakdown) or go up a level. For states with full containers (the last four rows in Table 1), there is no production activity at the preceding workstation. The succeeding workstation is either working properly to consume parts or it is broken down with the buffer staying full. The kanban levels will either stay the same (succeeding machine breakdown) or go down one level. For all other states (the middle four rows in Table 1), the kanban level can go up a level, stay the same, or go down one level as described in the previous situation. An example illustrating a two-buffer system with kanban level changes is shown in Table 2. The steady state of this Markov process can be achieved by calculating the unique solution of the transition matrix (Q). To further reduce the calculation for the problem, the transition matrix can be rewritten in terms of kanban levels, and it is shown in Eq. (4). 3 2 P4 P5 0 0 0 7 6 0 7 6 P1 P2 P3 0 7 6 60 P1 P2 P3 0 7 7 6 6. .. 7 .. .. .. .. 7 6 . ð4Þ QK ¼ 6 . . 7 . . . . 7 6 60 P1 P2 P3 0 7 7 6 7 6 0 P1 P2 P3 5 40 0 0 0 P6 P7
Table 1 Possible state changes Current state
Next possible states
(0, 0, 0) (0, 1, 0) (0, 0, 1) (0, 1, 1) (n, 0, 0) (n, 1, 0) (n, 0, 1) (n, 1, 1) (K, 0, 0) (K, 1, 0) (K, 0, 1) (K, 1, 1)
(0, 0, 1), (0, 1, 0) (1, 1, 0), (0, 1, 1), (0, 0, 0) (0, 1, 1) (1, 1, 1), (0, 0, 1) (n, 0, 1), (n, 1, 0) (n + 1, 1, 0), (n, 1, 1), (n, 0, 0) (n, 1, 1), (n, 0, 0), (n 1, 0, 1) (n + 1, 1, 1), (n, 0, 1), (n, 1, 0),(n 1, 1, 1) (K, 1, 0), (K, 0, 1) (K, 1, 1) (K, 0, 0), (K, 1, 1) , (K 1, 0, 1) (K, 1, 0), (K 1, 1, 1)
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Table 2 Kanban level changes in a two buffer system Current state (0, 0, 0) (0, 1, 0) (0, 0, 1) (0, 1, 1) (1, 0, 0) (1, 1, 0) (1, 0, 1) (1, 1, 1) (2, 0, 0) (2, 1, 0) (2, 0, 1) (2, 1, 1)
Down a kanban level
(0, 0, 1) (0, 1, 1)
(1, 0, 1) (1, 1, 1)
Same kanban level (0, 0, 1), (0, 1, 1), (0, 1, 1) (0, 0, 1) (1, 0, 1), (1, 1, 1), (1, 1, 1), (1, 0, 1), (2, 1, 0), (2, 1, 1) (2, 0, 0), (2, 0, 1),
(0, 1, 0) (0, 0, 0)
Up a kanban level (1, 1, 0) (1, 1, 1)
(1, 1, 0) (1, 0, 0) (1, 0, 0) (1, 1, 0) (2, 0, 1)
(2, 1, 0) (2, 1, 1)
(2, 1, 1) (2, 1, 0)
In Eq. (4), p1 and p6 are the probabilities of going to the state with a lower kanban level; p2, p4 and p7 are the probabilities of staying in the current state; and p3 and p5 are the probabilities of going to the state with a higher kanban level. The necessary condition for increasing one kanban level is that the preceding workstation works properly, while decreasing one kanban level is that the succeeding workstation works properly. The probability to stay in the same kanban level is equal to 1 (probability of one kanban level up) (probability of one kanban level down). All these probabilities can be determined by the following equations: EF j1 EF j1 þ IEF j1 þ EF j þ IEF j IEF j1 þ IEF j ¼ EF j1 þ IEF j1 þ EF j þ IEF j EF j ¼ EF j1 þ IEF j1 þ EF j þ IEF j IEF j1 ¼ EF j1 þ IEF j1 EF j1 ¼ EF j1 þ IEF j1 EF j ¼ EF j þ IEF j IEF j ¼ EF j þ IEF j
p1 ¼
ð5Þ
p2
ð6Þ
p3 p4 p5 p6 p7
ð7Þ ð8Þ ð9Þ ð10Þ ð11Þ
The effective processing rate EFj and the ineffective processing rate IEFj can be determined according to the research by Buzacott. Rsj EF j ¼ ls0j ¼ lsj for j ¼ 1–N Rsj þ Fsj Rsj IEF j ¼ lsj 1 for j ¼ 1–N Rsj þ Fsj
ð12Þ ð13Þ
III. Determine the limiting distribution The limiting distribution for the Markov process can be determined according to Wang and Wang’s study. 1. Determine the unique solution of m = mQK, where m = (m0, m1, m2, . . . ,mK2, mK1, mK). 2. Determine the exponential distribution for staying in each state (k). The expected number of visits to other states i between two visits to a certain state l is m(i)/m(l) and the expected time of staying in state l is 1/k(l), where k = (k0, k1, k2, . . . , kK2, kK1, kK), which is the parameter of exponential distribution for staying in each state.
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3. The expected time spent in state l, which is the limiting distribution, can be determined by the following equation: mðlÞ kðlÞ
pðlÞ ¼ PK
mðjÞ j¼0 kðjÞ
for l ¼ 1–K
ð14Þ
IV. Calculate the expected cost for the production system with K kanbans The cost factor for each state can be determined as follows: 1. For 0 kanban level, the cost factor C0 is SCsj (the shortage cost). 2. For maximum kanban level K, the cost factor CK is HCsj · K 3. For kanban level l, where 0 < l < K, the cost factor Cl is HCsj · l. 4. The expected total cost can be calculated as: Expected total cost of this stage ¼
K X
pK ðlÞ C l
ð15Þ
l¼0
V. Determine the optimal number of kanban The optimum kanban number can be determined by repeating the previous steps to select the kanban number with minimum expected total cost. 2.2.4. The model A mathematical model is established to evaluate the workstation assignments and determine the optimal workstation allocation. The model uses the total cost of production per item (TC) as the performance measure of the system. The total cost consists of: operation cost per unit time (C1) and inventory holding cost plus shortage cost of WIP parts for each workstation per unit time (INVj) The objective function can be written in the following format: Minimize TC ¼ C 1
N 1 1 X INV j þ THR D j¼2
Subject to: N X
P ij ¼ 1
For j ¼ 1–N
ð16Þ
P ij ¼ 1
For i ¼ 1–N
ð17Þ
i¼1 N X j¼1
lsj ¼
N X
P ji lij
For j ¼ 1–N
ð18Þ
P ji F i
For j ¼ 1–N
ð19Þ
P ji Ri
For j ¼ 1–N
ð20Þ
i¼1
Fsj ¼
N X i¼1
Rsj ¼
N X i¼1
From Eq. (2) THR ¼ MIN fD; ls01 ; ls02 ; ls03 ; . . . ; ls0n g For n ¼ 1–N
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HCsj ¼
N X
P ji H i
For j ¼ 1–N
619
ð21Þ
i¼1
SCsj ¼
N X
P ji S i
For j ¼ 1–N
ð22Þ
i¼1
INV j ¼
K X
pK ðlÞ C jl
For j ¼ 2–N
ð23Þ
l¼0
Pij represents the assignment matrix where j is the machine number and i is the position assignment. Eqs. (16) and (17) represent that there is only one workstation assigned to each position and each workstation is assigned to only one position respectively. Eqs. (18)–(20) convert the process rate, failure rate, and recovery rate to the position assigned process rate, failure rate, and recovery rate. Eqs. (21) and (22) convert holding cost and shortage cost to the position assigned cost, and Eq. (23) is used to calculate the inventory cost for a specific kanban number. 3. Genetic algorithms In this research, genetic algorithm (GA) is used as a search algorithm to obtain the optimal or near-optimal solution to this workstation arrangement problem. The GA, including three fundamental operations, reproduction, crossover, and mutation, is a robust search algorithm that can render near optimum solution to combinatorial problems through evolutionary processes. In this model, the decision variable needed to be determined is the allocation order of workstations and it can be coded as a string with N digits, where the value 1 to N is assigned to each digit to represent the workstation numbers. With no repeated value allowed, each string is a depiction of workstation sequence. An example of a coding string with five workstations is illustrated in Table 3. The position of workstation 1 (PW1) is 2, which means machine 1 is allocated at the second position. The procedures to apply genetic algorithm to the workstation allocation problem could be listed as follows. 1. Input the number of workstations N, the processing rates for each workstation in different positions, the failure rate and recovery rate for each workstation. 2. Generate a set of strings by Random Keys Representation method to avoid any illegal representation. 3. Evaluate the performance of each string by checking the TC value with the mathematical model. 4. Apply genetic algorithm operations, e.g. reproduction, partially mapped crossover (PMX), and mutation to generate new strings. 5. Repeat steps 3 and 4 until no further improvement is achieved. The allocation restrictions and the preceding-succeeding relationship of workstation must be checked if any of them occurs. A preset termination criterion, while no further improvement can be achieved, is used to stop the search process. 4. Example of five workstations allocation problem The mathematical model is applied to an example of a five-workstation production line (N = 5). The inventory holding costs Hj and WIP shortage costs Sj, are presented in Table 4. The failure rate (Fi) and recovery rate (Ri) are shown in Table 5. The processing rates (lij) for the five workstations in each position are shown in Table 6.
Table 3 Allocation string with five workstations
String
PW1
PW2
PW3
PW4
PW5
2
1
3
5
4
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Table 4 Inventory holding cost and WIP shortage cost for workstations Workstation
1
2
3
4
5
Hj Sj
15 175
20 200
20 110
35 155
50 180
Table 5 Failure rates and recovery rates for workstations Workstation
1
2
3
4
5
Failure Rate Recovery Rate
0.25 4
0.5 6
0.25 4
0.1 2
0.8 10
Table 6 Processing rates of 5-workstation example Workstation
Processing rate at stage 1
Processing rate at stage 2
Processing rate at stage 3
Processing rate at stage 4
Processing rate at stage 5
1 2 3 4 5
50 45 35 40 55
20 40 24 45 35
35 30 30 30 25
45 25 20 25 30
24 35 25 35 45
Unit: item per hour.
Other given variables are shown below. There is no restriction for workstation allocation in this example. • C1 = $240/h • D = 20 4.1. Application of the model With the given information, the mathematical model is illustrated numerically. The procedure to determine the expected total cost of production (TC) is presented in the following nine steps. Step 1: Randomly assign the allocation of workstations. For example, PW = (1, 2, 3, 4, 5), the machine assignment matrix would be: 3 2 1 0 0 0 0 7 6 60 1 0 0 07 7 6 7 pij ¼ 6 60 0 1 0 07 7 6 40 0 0 1 05 0 0 0 0 1 Step 2: Convert the processing, failure, and recovery rates with Eqs. (18)–(20). lsj ¼ ð50; 40; 30; 25; 45Þ for j ¼ 1–5 Fsj ¼ ð0:25; 0:5; 0:25; 0:1; 0:8Þ for j ¼ 1–5 Rsj ¼ ð4; 6; 4; 2; 10Þ for j ¼ 1–5 Step 3: Determine the effective processing rate of all workstations by Eq. (1), and then calculate the throughput rate of the production system using Eq. (2). ls0j ¼ ð47:06; 36:92; 28:24; 23:81; 41:67Þ
for j ¼ 1–5
THR ¼ MINf20; 47:06; 36:92; 28:24; 23:81; 41:67g ¼ 20
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Step 4: The buffer size and the associated inventory holding and WIP shortage (INV) cost can be determined buffer by buffer. Initially, set the first buffer K2 (buffer between workstations 1 and 2) as 1. The number of states (NS) is equal to 8, which is defined by Eq. (3). The state space is E = (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1). Step 5: Before calculating the transition matrix, the probabilities p1, p2, p3, p4, p5, p6, and p7 need to be determined. The effective processing rate EFj and the ineffective processing rate IEFj are required for the computation of the probabilities. They are determined by the aforementioned Eqs. (12) and (13). EF j ¼ ls0j ¼ ð47:06; 36:92; 28:24; 23:81; 41:67Þ for j ¼ 1–5 IEF j ¼ ð2:94; 3:08; 1:76; 1:19; 3:33Þ
for j ¼ 1–5
For the first buffer K2, the probabilities are defined using Eq. (5)–(11). P1 = 0.52; P2 = 0.07; P3 = 0.41; P4 = 0.06; P5 = 0.94; P6 = 0.92; P7 = 0.08 With the probabilities established, the transition matrix Q1 for the buffer in stage 2 can be determined. P4 P5 0:06 0:94 Q1 ¼ ¼ 0:92 0:08 P6 P7 Step 6: Solving m = m QK, the value of mj is obtained. mð0Þ ¼ 0:49; mð1Þ ¼ 0:51 The exponential distribution for staying in each state is kð0Þ ¼ 50; kð1Þ ¼ 40 Finally, from Eq. (14), the limiting distribution is determined. pð0Þ ¼ 0:43; pð1Þ ¼ 0:57 Step 7: With the inventory holding cost HCsj and WIP shortage cost SCsj converted from Table 4, the cost factor for each is calculated as follows. C 0 ¼ SCs2 ¼ 200 C 1 ¼ HCs2 K ¼ 20 1 ¼ 20 From Eq. (15), the expected cost INV2 could be determined according to the following result. INV 2 ¼ 0:43 200 þ 0:57 20 ¼ 97:4 Step 8: Repeat steps (4)–(7) for different kanban levels of K = 2, 3, . . ., until there is no further improvement in expected cost. The expected cost for different values of K is determined and showed in Table 7. From Table 7, the lowest expected cost for this inter-stage is $86.39 per hour with three kanbans. It is now necessary to repeat the procedures for the rest of the buffers. The results are presented in Table 8. Step 9: With all required data gathered, the expected production cost for each finished product (TC) can be determined as: 1 1 TC ¼ 240 þ ð86:39 þ 57:7 þ 87:66 þ 110:91Þ 20 20 TC ¼ 29:13 Table 7 Number of kanban and expected cost for stage 2 K2
1
2
3
4
INV2
97.4
89.47
86.39
86.69
622
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Table 8 Number of kanban and expected cost for stages Stage j
2
3
4
5
Kj INVj
3 86.39
2 57.70
2 87.66
2 110.91
As a result, the expected total cost TC for allocation {1, 2, 3, 4, 5} is $29.13 per item. The optimum arrangement of workstation can be found by exploring all possible workstation sequences to find the best one with minimum TC. However, it is computationally exhausting with the increasing number of workstation. Search algorithms, like genetic algorithms and simulated annealing, which provide systematical search procedures, is preferred in solving this combinatorial problem. 4.2. Application of genetic algorithm to the example A computer program is developed to solve the problem of workstation allocation by the genetic algorithm. The value of the termination factor (TF) used for this numerical example is 15, which means the search will terminate after 15 more trials without any improvement with the objective function (expected total cost of production TC). In this problem, the initial population of allocation strings is set at seven, and the total costs of the seven strings are listed in Table 9. The mutation rate is set at 0.1%. The optimal (near-optimal) allocation of workstations is obtained in the fifth generation and the search is terminated at the 20th trial, with an expected total cost of production of $28.57 per item. It is not a very impressive result at the first glance. However, the total cost in this example consists of two parts: operation Table 9 Initial set of strings for the numerical example String
Allocation strings (PWi)
Total cost (TC)
1 2 3 4 5 6 7
(2, 5, 1, 3, 4) (4, 3, 1, 5, 2) (3, 5, 2, 1, 4) (5, 3, 4, 1, 2) (1, 2, 3, 4, 5) (4, 5, 1, 3, 2) (1, 4, 3, 2, 5)
31.58 29.5 30.26 29.99 29.13 29.56 29.3
Average cost: 29.9
Table 10 Allocation of workstation for 5-workstation example Workstation
Allocation (PWi)
1 2 3 4 5
5 3 1 4 2
Table 11 Number of kanban for 5-workstation problem Buffer (j)
Number of Kanban (Kj)
2 3 4 5
4 2 2 2
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cost and holding cost. If we deduct the operation cost which is fixed (=12) from total cost, in average, the GA model improves the holding cost by 8% (from 17.9 to 16.57). The result of workstation allocation is listed in Table 10 and the number of kanban for this arrangement is listed in Table 11. 5. Conclusions The workstation allocation problem is a combinatorial optimization problem. The objective is to determine the optimal allocation of a finite number of unreliable workstations with finite number of inter-stage buffer capacities in a pull production line. To determine the allocation, the optimal number of kanban for each work stage has to be defined first. There existed many production line balancing techniques, e.g. bowl phenomenon or unbalancing buffer size method, to improve production rate. In this study, Genetic algorithm is applied to reduce the calculation efforts in the production line with unreliable workstations. With Random Keys Representation method to prevent illegal representations, and Partially Mapped Crossover method to ensure correct crossover, GA is much efficient than traditional optimization methods. With the model developed, the unreliable-workstation-line-balancing problem can be easily adopted without much of the efforts in calculation. References Buzacott, J. A. (1967). Automatic transfer lines with buffer stocks. International Journal of Production Research, 5(3), 182–200. Choong, Y. F., & Gershwin, S. B. (1987). A decomposition method for the approximate evaluation of capacitate transfer lines with unreliable machines and random processing tmes. IIE Transactions, 19(2), 150–159. D’angelo, H., Caramanis, M., Finger, S., Mavretic, A., Phillis, Y. A., & Ramsden, E. (1988). Event-driven model of unreliable production lines with storage. International Journal of Production Research, 26(7), 1173–1182. El-Rayah, T. E. (1979). The effect of inequality of inter-stage buffer capacities and operation time variability on the efficiency of production line systems. International Journal of Production Research., 17(1), 77–89. Fan, F. (1994). Arrangement of workstations in pull production lines: A search algorithm approach. Master Thesis, University of Windsor. Gershwin, S. B., & Schick, I. C. (1983). Modeling and analysis of three-stage transfer lines with unreliable machines and finite buffers. Operations Research, 31(2), 354–380. Glassey, C. R., & Hong, Y. (1993). Analysis of behaviour of an unreliable n-stage transfer line with (n 1) inter-stage storage buffers. International Journal of Production Research., 31(3), 519–530. Hillier, F. S., & So, K. C. (1991). The effect of machine breakdowns and inter-stage storage on the performance of production line systems. International Journal of Production Research, 29(10), 2043–2055. Kouikoglou, V., & Phillis, Y. (1994). Discrete event modeling and optimization of unreliable production lines with random rates. IEEE Transactions on Robotics and Automation, 10(2), 153–159. McMullen, P. R., & Frazier, G. V. (1998). Using simulated annealing to solve a multiobjective assembly line balancing problem with parallel workstations. International Journal of Production Research, 36(10), 2717–2741. Price, W., Gravel, M., & Nsakanda, A. L. (1994). A review of optimisation models of kanban-based production systems. European Journal of Operational Research, 75(1), 1–12. Villeda, R., Dudek, R., & Smith, M. L. (1988). Increasing the production rate of a just-in-time production system with variable operation times. International Journal of Production Research, 26(11), 1749–1768. Wang, H., & Wang, H. P. (1990). Determining the number of kanbans: A step toward non-stock-production. International Journal of Production Research, 28(11), 2101–2115. Watanabe, T., Hashimoto, Y., Nishikawa, I., & Tokumaru, H. (1995). Line balancing using a genetic evolution model. Control Engineering Practice, 3(1), 69–76.
Computers & Industrial Engineering 54 (2008) 613–623 www.elsevier.com/locate/dsw
On the search of workstations arrangement in pull production systems q Hsu-Tung Lee a
a,*
, Michael H. Wang
b
Department of Business Administration, National Taipei University, 151, University Road, San Shia, Taipei 237, Taiwan, ROC b Department of Industrial and Manufacturing Systems Engineering, University of Windsor, 401 Sunset Avenue, Windsor, Ont., Canada N9B 3P4 Received 7 July 2006; received in revised form 20 September 2007; accepted 20 September 2007 Available online 29 September 2007
Abstract This research presents a model to determine the workstation arrangement and kanban number for pull production systems. Several practical production line characters are considered, and evolutionary algorithms are utilized to obtain the optimal/near-optimal result. The proposed system is a flow-shop, pull production line with N unreliable workstations and N 1 inter-stage buffers. Workstations may be allocated to any stage in the production line and have a different processing rate at each stage. Workstations will breakdown/recover at fixed rates, and the operating-breakdown-recovery process is formulated as a Markov process in this model. The size of the inter-stage buffers varies according to the arrangements of workstations. A model is established on the objective of minimizing the total production cost per item of the finished product. The costs considered in the system contain the allocation of workstations, operations, inventory, and production shortage costs. An evolutionary algorithm, genetic algorithm, is used to obtain the allocation of workstations in the production line. 2007 Elsevier Ltd. All rights reserved. Keywords: Pull production system; Unreliable machine; Evolutionary algorithm; Workstation arrangement
1. Introduction Any production system is subject to stoppage. This scenario includes machine breakdown, unscheduled maintenance, job interruption, and so on. The unanticipated stoppage has a significant impact on the performances of manufacturing systems, especially when breakdown happens at the bottleneck workstation. Manufacturers usually use inter-stage buffers to neutralize the influence of breakdown and other similar situations. Thus, to study the effect of buffer and to determine the size of buffer is an important issue since the space of a plant is limited. El-Rayah (1979) studies the behavior of production lines with different inter-stage buffer q *
This manuscript was processed by Area Editor Prof. Y.M. Dessouky. Corresponding author. E-mail address: [email protected] (H.-T. Lee).
0360-8352/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.09.011
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H.-T. Lee, M.H. Wang / Computers & Industrial Engineering 54 (2008) 613–623
assignments. He proves that assigning larger inter-stage buffers capacities to the middle workstation will be more efficient while unbalancing the production line in terms of buffer capacities. The buffer-size-determination becomes more complicated when machine breakdown is taken into consideration. The phenomenon of machine breakdown is best described with stochastic models. However, the state space will increase dramatically when the status of machine is involved. For instance, the state space of a two-machine, one-buffer (capacity = 3) system will increase from 4 to 16. Gershwin and Schick (1983), Choong and Gershwin (1987), D’angelo et al. (1988), and Glassey and Hong (1993) study the behavior of production line with unreliable workstations. Choong and Gershwin develop a decomposition method to reduce the state space in this problem and prove it to be accurate. Hillier and So (1991) study the effects of machine breakdowns and interstage buffer capacity on performance of the production line. Kouikoglou and Phillis (1994) introduce a newer version of unreliable production line problem by using a discrete model with limited repair resources. Watanabe, Hashimoto, Nishikawa, and Tokumaru (1995) and McMullen and Frazier (1998) use search algorithms, e.g. genetic algorithms and simulated annealing, respectively, to obtain the optimal solutions to line balancing problems. The Just-in-time (JIT) with kanban production system is a common philosophy in manufacturing process nowadays. This philosophy aims at reducing work-in-process inventory and increasing production. Price, Gravel, and Nsakanda (1994) review some of the kanban-based optimization models. Villeda, Dudek, and Smith (1988) seek the solution to increase the production rate in the JIT environment by examining various unbalancing line techniques and show that the ‘‘high-medium-low’’ arrangement of workstation provides better output rate in the JIT system. Wang and Wang (1990) develop a method to determine the optimum number of inter-stage kanban in one-station-to-one-station, multiple-station-to-one-station, and multiple-station-tomultiple-station cases based on cost minimization. In Wang and Wang’s study, a method for unreliable workstations is presented. Fan (1994) carries out Wang and Wang’s study numerically. This study further extends Wang and Wang and Fan’s study by formulating the problem more efficiently and considers more characters of unreliable production line. 2. Problem statement A flow shop type pull production line with finite inter-stage buffers is considered. Workstations can be located in different positions with position dependent performance subject to breakdown. The system has N unreliable workstations and N 1 inter-stage buffers. The buffer capacity is applied to neutralize the possible breakdown and the capacity (number of kanban) is dependent on workstation assignment. The objective of this research is to determine the optimal arrangement of workstations and the inter-stage buffer size associated with workstation allocation. Some of the characters for this production line are listed as follows: 1. All items enter at the first workstation and leave at the last workstation. 2. The preceding/succeeding of sequence of job exists. Thus, some workstations have to be placed at a particular position. 3. Blockage (starvation) exists when the machine in the upstream (downstream) is ready but there’s no kanban to (can) authorize production activity. This situation may happen when the downstream (upstream) machine breaks down and the buffers are full (empty). Breakdown cannot occur at a starved or blocked workstation. 4. There is an infinite source for the production system, and an infinite storage capacity at the end. In other words, the first machine never starves, and the last machine can never be blocked. 5. The process time, failure rates, and recovery rates of workstations and its endurance are formulated as a Poisson process. 2.1. Notations N lij
number of workstations. mean processing rate of workstation i in stage j, which is formulated as a Poisson distribution.
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Fi Ri Hi Si lsj Fsj Rsj D THR INVj K EFj IEFj HCsj SCsj TC C1
615
failure rate of workstation i, which is formulated as a Poisson distribution. recovery rate of workstation i, which is formulated as a Poisson distribution. holding cost of workstation i, per item per unit time. shortage cost of workstation i. mean processing rate of workstation in stage j, after position assignment. failure rate of workstation in stage j, after position assignment. recovery rate of workstation in stage j, after position assignment. demand rate of finished products. throughput rate of the system. inventory cost of buffer j, after position assignment. number of kanban at particular stage. the effective processing rate of workstation in stage j. the ineffective processing rate of workstation in stage j. inventory holding cost per item per unit time in stage j. shortage cost of WIP item in stage j. total cost of production per item. operation cost per unit time, which includes labor, machine, maintenance, etc.
2.2. Methodology The decomposition method introduced by Gershwin is utilized in this research to reduce the state space of the stochastic problem. Due to the nature of pull production system, some of the characters, e.g. throughput rate and demand rate, are closely related, and the relationship can be utilized to further reduce the complexity of this problem. Before mathematical formulation is derived, some fundamental information of this production line should be determined first. It includes the system’s process rate, the throughput rate, and the optimal number of kanban. 2.2.1. The process rate According to Buzacott (1967), the effective processing rate for workstation in stage jðls0j Þ due to the unreliable behavior of workstations can be determined by: ls0j ¼ lsj
Rsj Fsj þ Rsj
for j ¼ 1–N
ð1Þ
2.2.2. The throughput rate The throughput rate should be equal to the demand rate in an ideal JIT manufacturing system, and it can be shown as: THR ¼ MinfD; ls01 ; ls02 ; ls03 ; . . . ; ls0n g Here the throughput rate is the minimum of the effective processing rate mand rate.
ð2Þ (ls0j )
of all workstations and the de-
2.2.3. The optimum kanban number Wang and Wang’s study demonstrated an effective way to determine the number of kanban, also known as the buffer capacity in this research. There are five major steps to determine the optimum kanban number. The method is briefly described below: I. Define the state space E for the production system Suppose the number of kanban in a decomposed system (a pair of workstations) is K. The state space of this production system can be described as a Markov process with NS number of states. NS ¼ ðK þ 1Þ 2 2
ð3Þ
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This equation calculates the number of states for state space E, with K number of kanban. The number of states NS is equal to K + 1 (possible kanban level from zero to K) multiplied by two of the possible status (operating/under repair) of the preceding workstation and two of the possible status of the succeeding workstation. Similarly, each component of the state space can be represented as (n, I1, I2), where n (0 6 n 6 K) is the number of fulfilled kanban; I1 and I2 represent the status of preceding and succeeding workstations respectively (1 representing the workstation is operating properly; 0 representing the workstation is under repair). For a three kanban case, the state space is E = (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1 1 ), (2, 0, 0), (2, 0, 1), (2, 1, 0), (2, 1, 1), (3, 0, 0), (3, 0, 1), (3, 1, 0), (3, 1, 1). II. Define the transition matrix For state (0, 0, 1), the workstation ahead is down and there’s no job waiting in the buffer. Based on the starved-machine-never-breakdown rule, the only possible state in the next is (0, 1, 1). Similarly, the only consequence of (K, 1, 0) is (K, 1, 1) according to the blocked-machine-never-breakdown rule. The transition matrix (QK) for this Markov process of all scenarios can be derived as shown in Table 1. For those states with empty containers (the first four rows in Table 1), there is no production activity at the succeeding workstation due to the void buffer. Thus the kanban levels will either stay the same (preceding machine breakdown) or go up a level. For states with full containers (the last four rows in Table 1), there is no production activity at the preceding workstation. The succeeding workstation is either working properly to consume parts or it is broken down with the buffer staying full. The kanban levels will either stay the same (succeeding machine breakdown) or go down one level. For all other states (the middle four rows in Table 1), the kanban level can go up a level, stay the same, or go down one level as described in the previous situation. An example illustrating a two-buffer system with kanban level changes is shown in Table 2. The steady state of this Markov process can be achieved by calculating the unique solution of the transition matrix (Q). To further reduce the calculation for the problem, the transition matrix can be rewritten in terms of kanban levels, and it is shown in Eq. (4). 3 2 P4 P5 0 0 0 7 6 0 7 6 P1 P2 P3 0 7 6 60 P1 P2 P3 0 7 7 6 6. .. 7 .. .. .. .. 7 6 . ð4Þ QK ¼ 6 . . 7 . . . . 7 6 60 P1 P2 P3 0 7 7 6 7 6 0 P1 P2 P3 5 40 0 0 0 P6 P7
Table 1 Possible state changes Current state
Next possible states
(0, 0, 0) (0, 1, 0) (0, 0, 1) (0, 1, 1) (n, 0, 0) (n, 1, 0) (n, 0, 1) (n, 1, 1) (K, 0, 0) (K, 1, 0) (K, 0, 1) (K, 1, 1)
(0, 0, 1), (0, 1, 0) (1, 1, 0), (0, 1, 1), (0, 0, 0) (0, 1, 1) (1, 1, 1), (0, 0, 1) (n, 0, 1), (n, 1, 0) (n + 1, 1, 0), (n, 1, 1), (n, 0, 0) (n, 1, 1), (n, 0, 0), (n 1, 0, 1) (n + 1, 1, 1), (n, 0, 1), (n, 1, 0),(n 1, 1, 1) (K, 1, 0), (K, 0, 1) (K, 1, 1) (K, 0, 0), (K, 1, 1) , (K 1, 0, 1) (K, 1, 0), (K 1, 1, 1)
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Table 2 Kanban level changes in a two buffer system Current state (0, 0, 0) (0, 1, 0) (0, 0, 1) (0, 1, 1) (1, 0, 0) (1, 1, 0) (1, 0, 1) (1, 1, 1) (2, 0, 0) (2, 1, 0) (2, 0, 1) (2, 1, 1)
Down a kanban level
(0, 0, 1) (0, 1, 1)
(1, 0, 1) (1, 1, 1)
Same kanban level (0, 0, 1), (0, 1, 1), (0, 1, 1) (0, 0, 1) (1, 0, 1), (1, 1, 1), (1, 1, 1), (1, 0, 1), (2, 1, 0), (2, 1, 1) (2, 0, 0), (2, 0, 1),
(0, 1, 0) (0, 0, 0)
Up a kanban level (1, 1, 0) (1, 1, 1)
(1, 1, 0) (1, 0, 0) (1, 0, 0) (1, 1, 0) (2, 0, 1)
(2, 1, 0) (2, 1, 1)
(2, 1, 1) (2, 1, 0)
In Eq. (4), p1 and p6 are the probabilities of going to the state with a lower kanban level; p2, p4 and p7 are the probabilities of staying in the current state; and p3 and p5 are the probabilities of going to the state with a higher kanban level. The necessary condition for increasing one kanban level is that the preceding workstation works properly, while decreasing one kanban level is that the succeeding workstation works properly. The probability to stay in the same kanban level is equal to 1 (probability of one kanban level up) (probability of one kanban level down). All these probabilities can be determined by the following equations: EF j1 EF j1 þ IEF j1 þ EF j þ IEF j IEF j1 þ IEF j ¼ EF j1 þ IEF j1 þ EF j þ IEF j EF j ¼ EF j1 þ IEF j1 þ EF j þ IEF j IEF j1 ¼ EF j1 þ IEF j1 EF j1 ¼ EF j1 þ IEF j1 EF j ¼ EF j þ IEF j IEF j ¼ EF j þ IEF j
p1 ¼
ð5Þ
p2
ð6Þ
p3 p4 p5 p6 p7
ð7Þ ð8Þ ð9Þ ð10Þ ð11Þ
The effective processing rate EFj and the ineffective processing rate IEFj can be determined according to the research by Buzacott. Rsj EF j ¼ ls0j ¼ lsj for j ¼ 1–N Rsj þ Fsj Rsj IEF j ¼ lsj 1 for j ¼ 1–N Rsj þ Fsj
ð12Þ ð13Þ
III. Determine the limiting distribution The limiting distribution for the Markov process can be determined according to Wang and Wang’s study. 1. Determine the unique solution of m = mQK, where m = (m0, m1, m2, . . . ,mK2, mK1, mK). 2. Determine the exponential distribution for staying in each state (k). The expected number of visits to other states i between two visits to a certain state l is m(i)/m(l) and the expected time of staying in state l is 1/k(l), where k = (k0, k1, k2, . . . , kK2, kK1, kK), which is the parameter of exponential distribution for staying in each state.
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3. The expected time spent in state l, which is the limiting distribution, can be determined by the following equation: mðlÞ kðlÞ
pðlÞ ¼ PK
mðjÞ j¼0 kðjÞ
for l ¼ 1–K
ð14Þ
IV. Calculate the expected cost for the production system with K kanbans The cost factor for each state can be determined as follows: 1. For 0 kanban level, the cost factor C0 is SCsj (the shortage cost). 2. For maximum kanban level K, the cost factor CK is HCsj · K 3. For kanban level l, where 0 < l < K, the cost factor Cl is HCsj · l. 4. The expected total cost can be calculated as: Expected total cost of this stage ¼
K X
pK ðlÞ C l
ð15Þ
l¼0
V. Determine the optimal number of kanban The optimum kanban number can be determined by repeating the previous steps to select the kanban number with minimum expected total cost. 2.2.4. The model A mathematical model is established to evaluate the workstation assignments and determine the optimal workstation allocation. The model uses the total cost of production per item (TC) as the performance measure of the system. The total cost consists of: operation cost per unit time (C1) and inventory holding cost plus shortage cost of WIP parts for each workstation per unit time (INVj) The objective function can be written in the following format: Minimize TC ¼ C 1
N 1 1 X INV j þ THR D j¼2
Subject to: N X
P ij ¼ 1
For j ¼ 1–N
ð16Þ
P ij ¼ 1
For i ¼ 1–N
ð17Þ
i¼1 N X j¼1
lsj ¼
N X
P ji lij
For j ¼ 1–N
ð18Þ
P ji F i
For j ¼ 1–N
ð19Þ
P ji Ri
For j ¼ 1–N
ð20Þ
i¼1
Fsj ¼
N X i¼1
Rsj ¼
N X i¼1
From Eq. (2) THR ¼ MIN fD; ls01 ; ls02 ; ls03 ; . . . ; ls0n g For n ¼ 1–N
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HCsj ¼
N X
P ji H i
For j ¼ 1–N
619
ð21Þ
i¼1
SCsj ¼
N X
P ji S i
For j ¼ 1–N
ð22Þ
i¼1
INV j ¼
K X
pK ðlÞ C jl
For j ¼ 2–N
ð23Þ
l¼0
Pij represents the assignment matrix where j is the machine number and i is the position assignment. Eqs. (16) and (17) represent that there is only one workstation assigned to each position and each workstation is assigned to only one position respectively. Eqs. (18)–(20) convert the process rate, failure rate, and recovery rate to the position assigned process rate, failure rate, and recovery rate. Eqs. (21) and (22) convert holding cost and shortage cost to the position assigned cost, and Eq. (23) is used to calculate the inventory cost for a specific kanban number. 3. Genetic algorithms In this research, genetic algorithm (GA) is used as a search algorithm to obtain the optimal or near-optimal solution to this workstation arrangement problem. The GA, including three fundamental operations, reproduction, crossover, and mutation, is a robust search algorithm that can render near optimum solution to combinatorial problems through evolutionary processes. In this model, the decision variable needed to be determined is the allocation order of workstations and it can be coded as a string with N digits, where the value 1 to N is assigned to each digit to represent the workstation numbers. With no repeated value allowed, each string is a depiction of workstation sequence. An example of a coding string with five workstations is illustrated in Table 3. The position of workstation 1 (PW1) is 2, which means machine 1 is allocated at the second position. The procedures to apply genetic algorithm to the workstation allocation problem could be listed as follows. 1. Input the number of workstations N, the processing rates for each workstation in different positions, the failure rate and recovery rate for each workstation. 2. Generate a set of strings by Random Keys Representation method to avoid any illegal representation. 3. Evaluate the performance of each string by checking the TC value with the mathematical model. 4. Apply genetic algorithm operations, e.g. reproduction, partially mapped crossover (PMX), and mutation to generate new strings. 5. Repeat steps 3 and 4 until no further improvement is achieved. The allocation restrictions and the preceding-succeeding relationship of workstation must be checked if any of them occurs. A preset termination criterion, while no further improvement can be achieved, is used to stop the search process. 4. Example of five workstations allocation problem The mathematical model is applied to an example of a five-workstation production line (N = 5). The inventory holding costs Hj and WIP shortage costs Sj, are presented in Table 4. The failure rate (Fi) and recovery rate (Ri) are shown in Table 5. The processing rates (lij) for the five workstations in each position are shown in Table 6.
Table 3 Allocation string with five workstations
String
PW1
PW2
PW3
PW4
PW5
2
1
3
5
4
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Table 4 Inventory holding cost and WIP shortage cost for workstations Workstation
1
2
3
4
5
Hj Sj
15 175
20 200
20 110
35 155
50 180
Table 5 Failure rates and recovery rates for workstations Workstation
1
2
3
4
5
Failure Rate Recovery Rate
0.25 4
0.5 6
0.25 4
0.1 2
0.8 10
Table 6 Processing rates of 5-workstation example Workstation
Processing rate at stage 1
Processing rate at stage 2
Processing rate at stage 3
Processing rate at stage 4
Processing rate at stage 5
1 2 3 4 5
50 45 35 40 55
20 40 24 45 35
35 30 30 30 25
45 25 20 25 30
24 35 25 35 45
Unit: item per hour.
Other given variables are shown below. There is no restriction for workstation allocation in this example. • C1 = $240/h • D = 20 4.1. Application of the model With the given information, the mathematical model is illustrated numerically. The procedure to determine the expected total cost of production (TC) is presented in the following nine steps. Step 1: Randomly assign the allocation of workstations. For example, PW = (1, 2, 3, 4, 5), the machine assignment matrix would be: 3 2 1 0 0 0 0 7 6 60 1 0 0 07 7 6 7 pij ¼ 6 60 0 1 0 07 7 6 40 0 0 1 05 0 0 0 0 1 Step 2: Convert the processing, failure, and recovery rates with Eqs. (18)–(20). lsj ¼ ð50; 40; 30; 25; 45Þ for j ¼ 1–5 Fsj ¼ ð0:25; 0:5; 0:25; 0:1; 0:8Þ for j ¼ 1–5 Rsj ¼ ð4; 6; 4; 2; 10Þ for j ¼ 1–5 Step 3: Determine the effective processing rate of all workstations by Eq. (1), and then calculate the throughput rate of the production system using Eq. (2). ls0j ¼ ð47:06; 36:92; 28:24; 23:81; 41:67Þ
for j ¼ 1–5
THR ¼ MINf20; 47:06; 36:92; 28:24; 23:81; 41:67g ¼ 20
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Step 4: The buffer size and the associated inventory holding and WIP shortage (INV) cost can be determined buffer by buffer. Initially, set the first buffer K2 (buffer between workstations 1 and 2) as 1. The number of states (NS) is equal to 8, which is defined by Eq. (3). The state space is E = (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1). Step 5: Before calculating the transition matrix, the probabilities p1, p2, p3, p4, p5, p6, and p7 need to be determined. The effective processing rate EFj and the ineffective processing rate IEFj are required for the computation of the probabilities. They are determined by the aforementioned Eqs. (12) and (13). EF j ¼ ls0j ¼ ð47:06; 36:92; 28:24; 23:81; 41:67Þ for j ¼ 1–5 IEF j ¼ ð2:94; 3:08; 1:76; 1:19; 3:33Þ
for j ¼ 1–5
For the first buffer K2, the probabilities are defined using Eq. (5)–(11). P1 = 0.52; P2 = 0.07; P3 = 0.41; P4 = 0.06; P5 = 0.94; P6 = 0.92; P7 = 0.08 With the probabilities established, the transition matrix Q1 for the buffer in stage 2 can be determined. P4 P5 0:06 0:94 Q1 ¼ ¼ 0:92 0:08 P6 P7 Step 6: Solving m = m QK, the value of mj is obtained. mð0Þ ¼ 0:49; mð1Þ ¼ 0:51 The exponential distribution for staying in each state is kð0Þ ¼ 50; kð1Þ ¼ 40 Finally, from Eq. (14), the limiting distribution is determined. pð0Þ ¼ 0:43; pð1Þ ¼ 0:57 Step 7: With the inventory holding cost HCsj and WIP shortage cost SCsj converted from Table 4, the cost factor for each is calculated as follows. C 0 ¼ SCs2 ¼ 200 C 1 ¼ HCs2 K ¼ 20 1 ¼ 20 From Eq. (15), the expected cost INV2 could be determined according to the following result. INV 2 ¼ 0:43 200 þ 0:57 20 ¼ 97:4 Step 8: Repeat steps (4)–(7) for different kanban levels of K = 2, 3, . . ., until there is no further improvement in expected cost. The expected cost for different values of K is determined and showed in Table 7. From Table 7, the lowest expected cost for this inter-stage is $86.39 per hour with three kanbans. It is now necessary to repeat the procedures for the rest of the buffers. The results are presented in Table 8. Step 9: With all required data gathered, the expected production cost for each finished product (TC) can be determined as: 1 1 TC ¼ 240 þ ð86:39 þ 57:7 þ 87:66 þ 110:91Þ 20 20 TC ¼ 29:13 Table 7 Number of kanban and expected cost for stage 2 K2
1
2
3
4
INV2
97.4
89.47
86.39
86.69
622
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Table 8 Number of kanban and expected cost for stages Stage j
2
3
4
5
Kj INVj
3 86.39
2 57.70
2 87.66
2 110.91
As a result, the expected total cost TC for allocation {1, 2, 3, 4, 5} is $29.13 per item. The optimum arrangement of workstation can be found by exploring all possible workstation sequences to find the best one with minimum TC. However, it is computationally exhausting with the increasing number of workstation. Search algorithms, like genetic algorithms and simulated annealing, which provide systematical search procedures, is preferred in solving this combinatorial problem. 4.2. Application of genetic algorithm to the example A computer program is developed to solve the problem of workstation allocation by the genetic algorithm. The value of the termination factor (TF) used for this numerical example is 15, which means the search will terminate after 15 more trials without any improvement with the objective function (expected total cost of production TC). In this problem, the initial population of allocation strings is set at seven, and the total costs of the seven strings are listed in Table 9. The mutation rate is set at 0.1%. The optimal (near-optimal) allocation of workstations is obtained in the fifth generation and the search is terminated at the 20th trial, with an expected total cost of production of $28.57 per item. It is not a very impressive result at the first glance. However, the total cost in this example consists of two parts: operation Table 9 Initial set of strings for the numerical example String
Allocation strings (PWi)
Total cost (TC)
1 2 3 4 5 6 7
(2, 5, 1, 3, 4) (4, 3, 1, 5, 2) (3, 5, 2, 1, 4) (5, 3, 4, 1, 2) (1, 2, 3, 4, 5) (4, 5, 1, 3, 2) (1, 4, 3, 2, 5)
31.58 29.5 30.26 29.99 29.13 29.56 29.3
Average cost: 29.9
Table 10 Allocation of workstation for 5-workstation example Workstation
Allocation (PWi)
1 2 3 4 5
5 3 1 4 2
Table 11 Number of kanban for 5-workstation problem Buffer (j)
Number of Kanban (Kj)
2 3 4 5
4 2 2 2
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cost and holding cost. If we deduct the operation cost which is fixed (=12) from total cost, in average, the GA model improves the holding cost by 8% (from 17.9 to 16.57). The result of workstation allocation is listed in Table 10 and the number of kanban for this arrangement is listed in Table 11. 5. Conclusions The workstation allocation problem is a combinatorial optimization problem. The objective is to determine the optimal allocation of a finite number of unreliable workstations with finite number of inter-stage buffer capacities in a pull production line. To determine the allocation, the optimal number of kanban for each work stage has to be defined first. There existed many production line balancing techniques, e.g. bowl phenomenon or unbalancing buffer size method, to improve production rate. In this study, Genetic algorithm is applied to reduce the calculation efforts in the production line with unreliable workstations. With Random Keys Representation method to prevent illegal representations, and Partially Mapped Crossover method to ensure correct crossover, GA is much efficient than traditional optimization methods. With the model developed, the unreliable-workstation-line-balancing problem can be easily adopted without much of the efforts in calculation. References Buzacott, J. A. (1967). Automatic transfer lines with buffer stocks. International Journal of Production Research, 5(3), 182–200. Choong, Y. F., & Gershwin, S. B. (1987). A decomposition method for the approximate evaluation of capacitate transfer lines with unreliable machines and random processing tmes. IIE Transactions, 19(2), 150–159. D’angelo, H., Caramanis, M., Finger, S., Mavretic, A., Phillis, Y. A., & Ramsden, E. (1988). Event-driven model of unreliable production lines with storage. International Journal of Production Research, 26(7), 1173–1182. El-Rayah, T. E. (1979). The effect of inequality of inter-stage buffer capacities and operation time variability on the efficiency of production line systems. International Journal of Production Research., 17(1), 77–89. Fan, F. (1994). Arrangement of workstations in pull production lines: A search algorithm approach. Master Thesis, University of Windsor. Gershwin, S. B., & Schick, I. C. (1983). Modeling and analysis of three-stage transfer lines with unreliable machines and finite buffers. Operations Research, 31(2), 354–380. Glassey, C. R., & Hong, Y. (1993). Analysis of behaviour of an unreliable n-stage transfer line with (n 1) inter-stage storage buffers. International Journal of Production Research., 31(3), 519–530. Hillier, F. S., & So, K. C. (1991). The effect of machine breakdowns and inter-stage storage on the performance of production line systems. International Journal of Production Research, 29(10), 2043–2055. Kouikoglou, V., & Phillis, Y. (1994). Discrete event modeling and optimization of unreliable production lines with random rates. IEEE Transactions on Robotics and Automation, 10(2), 153–159. McMullen, P. R., & Frazier, G. V. (1998). Using simulated annealing to solve a multiobjective assembly line balancing problem with parallel workstations. International Journal of Production Research, 36(10), 2717–2741. Price, W., Gravel, M., & Nsakanda, A. L. (1994). A review of optimisation models of kanban-based production systems. European Journal of Operational Research, 75(1), 1–12. Villeda, R., Dudek, R., & Smith, M. L. (1988). Increasing the production rate of a just-in-time production system with variable operation times. International Journal of Production Research, 26(11), 1749–1768. Wang, H., & Wang, H. P. (1990). Determining the number of kanbans: A step toward non-stock-production. International Journal of Production Research, 28(11), 2101–2115. Watanabe, T., Hashimoto, Y., Nishikawa, I., & Tokumaru, H. (1995). Line balancing using a genetic evolution model. Control Engineering Practice, 3(1), 69–76.