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Design of bi-criteria kanban system using simulated annealing technique
Computers & Industrial Engineering 41 (2002) 355±370
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Design of bi-criteria kanban system using simulated annealing technique P. Shahabudeen a,*, R. Gopinath b, K. Krishnaiah a a
Industrial Engineering Division, Anna University, Madras, India b Tata consultancy Services, Bangalore, India Accepted 30 July 2001
Abstract In the kanban system, the main decision parameters are the number of kanbans and lot size. In this paper, an attempt has been made to set the number of kanbans at each station and the lot size required to achieve the best performance using simulated annealing technique. A simulation model with a single-card system has been designed and used for analysis. A bi-criterion objective function comprising of mean throughput rate and aggregate average kanban queue has been used for evaluation. Different perturbation schemes have been experimented and compared. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Kanban; Simulated annealing
1. Introduction Just in time (JIT) production system is the manufacturing philosophy of producing what is needed at the right time and in right quantity (Hutchins, 1993). Kanban coupled with pull system of production is used as means of implementing JIT. Kanban means a `visible card', which serves as a planning and information tool to smoothen the ¯ow of material through the manufacturing and assembly process. The workstations located along the production lines only produce or deliver desired components when they receive a card and empty container, indicating that more parts will be needed in production. Each workstation will only produce enough components to ®ll containers and then stop. In addition, kanban limits the amount of inventory in the process by acting as an authorisation to produce. * Corresponding author. E-mail address: [email protected] (P. Shahabudeen). 0360-8352/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0 3 6 0 - 8 3 5 2 ( 0 1 ) 0 0 06 0 - 2
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The essential elements in the design of kanban production system are the number of kanbans needed to link processes together and the appropriate unit of lot size (Berkley, 1992a). 2. Literature review Kanban based operational planning and control issues have been tackled in a number of studies by means of analytical or simulation modelling. Berkley (1992a) has reviewed 50 papers in the area of kanban production control and organised them based on the type of system. He has also listed 24 vital operation design factors of kanban system. Price, Gravel and Nsakanda (1994) have reviewed optimisation models of kanban based systems. They have concluded that interesting direction for future development would be in the incorporation of models in decision support systems for production control. 2.1. Mathematical models Several researchers have attempted mathematical modelling approach. They have assumed deterministic demand and material handling as being carried out periodically and the periods correspond to ®xed withdrawal cycles. Kimura and Terada (1981) have developed a model for the kanban system, which can be considered to be a pioneering work in the area of kanban modelling. Their model has served as a reference for subsequent researchers. Bitran and Chang (1987) have extended the work of Kimura and Terada and offered a mathematical model for kanban system in a multi-stage production setting. Their deterministic model is designed to assist in the choice of the number of kanbans to use at each stage and thus to control the level of inventory. Bard and Golany (1991) have developed a mixed integer linear program to extend Bitran and Chang's model, by considering material shortage and the production of multiple parts at each stage. Fukukawa and Hong (1993) have developed a mixed integer programming model to determine the number of kanbans. Mitwasi and Askin (1994) have developed a non-linear model for a multi-item single stage kanban system. Price, Gravel, Nsakanda and Cantin (1995) have formulated an integer linear programming model for kanban based assembly shop. It is a multi-period model and the variables track the ¯ow of kanbans in the shop ¯oor, and in between time periods. Constraints are used to ensure conservation of ¯ow of kanbans, maximum number of kanbans, availability of the parts for an operation, to limit production to demand and the machine availability. The objective is to minimise the make span. Solution for ILP is possible for some parts. However, it cannot be applied for parts having complex assembly diagrams. They have concluded that a large number of cards are unlikely to reduce the makespan signi®cantly and it is certain to increase the work in progress. Gupta and Al-Turki (1998) have described a new kanban system called ¯exible kanban system. This system uses an algorithm based on mathematical model of the system to dynamically and systematically manipulate the number of kanban and starvation caused by stochastic factors. 2.2. Queuing and Markov chain models Formulating kanban-controlled lines as Markov chains has been a popular strategy to ®nd the optimal number of kanbans. In these models, researchers usually assume processing times to be exponential and give the state of the system by the number of full containers between each pair of stations. Deleersnyder,
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Hodgson, Muller and O'Grady (1989) have modelled a line with blocking by total queue size as a discrete time Markov chain to study the effect of kanban numbers, machine reliability, processing time and ®nished goods demand variability. They have mentioned the following as the sources of variability; the demand schedule, the processing time, machine breakdowns and the actual demand. If the processing time is longer than expected, fewer units are produced during the period. Berkley (1992b) has developed a decomposition approximation using imbedded Markov chains for two-card systems with periodic material handling and Erlang processing times. He has given several examples to show how the approximation could be used to ®nd the required number of kanbans, the required withdrawal cycle time, or both. Berkley (1994) has used a continuous time Markov chain to determine minimum performance levels required to obtain a desired production rate, independent of the average station process time. He has used ®ll rate, which is de®ned as the probability of meeting demand from stock, as the performance measure, in addition to average withdrawal kanban queue times. However, he prefers ®ll rate because operator himself can easily compute this statistics compared to queue statistics, which require data collection and computational devices. He has assumed that demand is in®nite, and handling time is zero. Nori and Sarker (1998) have modelled the kanban system using Markov Chain to determine the optimum number of kanbans between adjacent stations. Through a numerical example, they have shown that their procedure has developed closer boundaries than that of Wang and Wang (1991) and hence the search for optimum number of kanbans is more ef®cient. 2.3. Simulation based studies In the case of ®nding the required number of kanbans and lot size, simulation offers a number of advantages. One has to make a few assumptions about processing time and lead time distributions and one can run the models under the most current shop conditions. In a simulation study of a four station line with ®xed order points, Berkley (1993) has shown that order points have a signi®cant effect on average inventories and production rates. In another simulation study of a two-card system with ®xed withdrawal cycle times, Berkley and Kiran (1991) have found that average in process inventories are highly sensitive to withdrawal cycle times while ®nished goods withdrawal kanban queue time is much less sensitive. This suggests that material-handling frequency can be reduced to obtain signi®cant inventory savings at a small customer service cost. Gravel and Price (1988,1991) have developed kanban simulation models with the objective of reducing WIP and makespan. Abdou and Dutta (1993) have studied the effect of container size, number of kanbans and material handling frequency on the sum of set up, inventory holding and machine under utilisation costs. Simulation results for a multi-product, multiline two-card kanban system showed that the cost of minimising container size is off set both by increasing material handling frequency and increasing the number of kanbans. Savsar (1996) has presented a simulation study of JIT production control system and its performance under different operational conditions. Berkley (1996) has simulated a two-card kanban system with multiple part types to determine the effect of container size on average inventory and customer service levels in tandem so that total in process capacity remains the constant. Results show that smaller containers lead to smaller average inventories. He has also observed that smaller containers do not always lead to poor customer service levels. Andijani (1997) has used stochastic system simulation to generate and construct a set of kanban allocations, which maximise the average throughput rate and minimise the average system time.
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2.4. Performance measures used in previous studies Following is the summary of ®ndings from literature survey on the various performance measures used: (a) Average input inventory. It is the sum of average input material queues at all stations except the ®rst. (b) Average output inventory. It is the sum of average output material queues at all stations except the last. (c) Total WIP inventory. Sum of average input and output inventories. (d) Finished goods withdrawal kanban waiting time. It is the time from when a ®nished goods withdrawal kanban arrives at the stock point of last station to the time a container of ®nished part (same type) is available to withdraw. (e) Fill rate. The proportion of ®nished goods demand met from stock. (f) Finished goods order queue time. The time from when the order arrives until it can be completely satis®ed with ®nished parts. (g) Mean cumulative throughput rate. Ratio of total satis®ed demand to generated demand. (h) Mean utilisation of line. Mean utilisation of last station. (i) Mean total WIP length. Mean of all inprocess inventory lengths for all products including ®nished goods inventories. (j) Mean total waiting time. Mean waiting time of all products in all inprocess and ®nished good inventories. (k) Cycle time. The time between two consecutive outputs.
3. Present study 3.1. Bi-criteria objective It is found that, most of the researchers have used only single objective function as the performance measure. In this study, a bi-criteria objective kanban problem has been considered. The ®rst criterion is the maximisation of mean cumulative throughput rate, which is de®ned as the ratio of total satis®ed demand to the total generated demand. The other criterion is the minimisation of aggregate average kanban queue, which is the sum of average number of kanbans waiting in the queue at all the stations. In this paper, the following two parameters are considered to measure the performance of the system. 3.1.1. Mean throughput rate (MTR) Mean cumulative throughput rate is de®ned as the ratio of total satis®ed demand to the total generated demand n X
MTR
i1 n X i1
SDi 1 GDi
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where GDi is the generated demand during ith request, SDi the satis®ed demand during ith request and n is the number of demand request generated during the simulation. It is required to maximise this value. 3.1.2. Aggregate average kanban queue (AAKQ) This is the sum of average number of kanbans waiting in the queue during simulation at all the stations AAKQ
m X j1
Ij
2
where Ij is the average number of kanbans waiting at station j and m is the number of stations in the system. It is required to minimise this value. 3.2. Bi-criteria objective function The number of kanbans and lot size directly in¯uence the performance measures de®ned above. Since the above measures are of con¯icting nature, AAKQ expression is transformed to obtain a maximisation expression. The transformed form of the AAKQ expression (TAAKQ) is as follows: m X
TAAKQ
j1 m X j1
IUj 2
IUj 2
m X
j1 m X j1
Ij 3
ILj
where IUj is the maximum number of kanbans at station j and ILj is the minimum number of kanbans at station j. The combined objective function is as follows: n X
z maxf MTR; TAAKQ
i1 n X i1
m X
SDi 100 1 GDi
j1 m X j1
IUj 2
IUj 2
m X
j1 m X j1
Ij 100
4
ILj
The above constructed objective function has been used to determine the overall measure of the system. 3.3. Search technique The literature review reveals that search techniques have not been widely applied for setting the kanban system parameters. Hence, it is felt that some search heuristics can be attempted to handle such kind of problems. In the case of search techniques, simulated annealing technique is found to have frequent application in production related problems like group technology, scheduling, etc. for obtaining good results. In this study, simulated annealing technique has been used.
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4. Simulated annealing Simulated annealing is a search heuristic intensively used to solve optimisation problems and especially combinatorial ones. It was derived from an analogy to the physical annealing process and ®rst introduced by Cerny (1985) and Kirkpatrick, Gelatti and Vecchi (1983). Eglese (1989), Pirlot (1996) and Sridhar and Rajendran (1993) described simulated annealing and its implementation with some choice parameters. A generic simulated annealing algorithm for a maximization problem is given below: Step 1: Select an initial point (seed) S0, a termination criterion p , initial temperature ti, cooling rate a and ®nal temperature tf. Set n 0: Determine f0, the objective function value of solution S0 and update f p, the global maximum value of objective function and S p, the corresponding solution. Step 2: Determine the neighbourhood Sn11 using perturbation and compute fn11 Step 3: If d fn11 2 fn . 0 then set n n 1 1; else if r < exp 2d=tn where r is a random number in the range 0 to 1 then set n n 1 1: Step 4: Update f p and S p. Step 5: If termination condition p is reached then goto step 6 else reduce temperature by cooling rate a and goto step 2. Step 6: Report f p and S p. 4.1. Choice of parameters for the proposed simulated annealing algorithm Initial temperature (ti), ®nal temperature (tf) and cooling factor are determined by conducting some pilot studies as 60, 1 and 0.8 respectively. The other factors like accept limit, freeze limit, etc. are adopted from Sridhar and Rajendran (1993). 4.1.1. Length of plateau (L) This denotes the number of iterations at a temperature. Here instead of using L as a count, two counts ACCEPT and TOTAL are used. ACCEPT counts the number of accepted moves at a particular temperature and TOTAL counts the total number of moves at a particular temperature. Iterations at a particular temperature are terminated when ACCEPT is equal to accept limit (b ) or when TOTAL is equal to 4 times b . 4.1.2. Stopping rule The SAA is stopped when any one of the conditions is met with: (i) when the temperature falls to a speci®ed value (tf); (ii) when the freeze count reaches a pre-de®ned value. The freeze count is reset to zero whenever current solution is better than current best solution, thus hoping for better solutions. After iterations at
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Table 1 System parameters Part type
A B
Processing times at workstations (time units)
WS1
WS2
WS3
5^1 3^2
4^1 5^2
3^2 4^3
Demand qty (units)
Demand inter arrival time (time units)
200 ^ 50 250 ^ 40
30 ^ 20 40 ^ 10
a temperature, the ratio of ACCEPT to TOTAL is computed. If this ratio is less than or equal to 0.15, freeze count is incremented by one. This threshold value indicates that the chance of replacing the best solution is diminishing. Here when the freeze count reaches a value of 5, SA is terminated.
5. The kanban system model The system under study is dynamic kanban system with stochastic demand and processing times with the following assumptions. 5.1. Assumptions ² ² ² ² ² ² ² ² ² ²
Two types of jobs are produced. Each type of job has its own demand inter arrival distribution Each type of job has its own demand distribution. All the jobs must be processed in all the stations. Processing times of each job type are different. The demand is satis®ed on FCFS basis. There is in®nite supply of raw material at the input of the production system. Any kanban detached at the output of a stage is immediately available for up stream stage. The distance between the workstations is not substantial. Material handling resources are unrestricted.
It consists of a production line with three workstations handling two different products. When the demand arrives in the system, it is placed at the ®nal workstation and availability of the components to produce this product is checked and if available, the production at this stage starts, otherwise a request is issued to the previous stage for the required part. Upon receiving the request, the previous stage either starts production for the required parts or issues a request to its upstream stage. In this way, the production at a current stage is pulled from down stream processes. Only after the completion of processing, the kanban of the emptied container is sent to the preceding station's out bound inventory to be replaced with a full one, if available. Initial conditions for each run is such that each in process and
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Table 2 Objective function values for different lot sizes (number of kanbans at each station 5) Lot size
MTR
TAAKQ
Total (Z)
20 30 40 50 60 70 80 90 100 110 120 130
36.48 27.112 44.88 43.88 31.38 40.62 52.68 49.00 57.08 41.46 47.13 37.18
55.79 56.82 55.29 55.74 51.95 50.48 51.38 53.02 51.09 56.34 54.60 53.25
92.27 83.93 100.17 99.62 83.33 91.10 104.06 102.02 108.17 97.80 101.73 90.43
®nished goods inventories are full with containers. The various system parameters used in the model are given in Table 1. 6. Experimental design and results Experiments to decide the run length, to set a limit for number of kanbans and lot size are performed. The details are as follows. 6.1. Experiment 1 Ð effect of changes in lot size In this experiment, number of kanbans at each station has been considered to be 5 and lot size is varied from 20 to 130. Table 2 gives the objective function values obtained from this experiment. It is found that objective function values do not change signi®cantly above 70. Hence it has been decided to vary lot size from 20 to 80 only. 6.2. Experiment 2 Ð effect of changing number of kanbans at each station equally Here the number of kanbans has been varied from 1 to 13 maintaining the lot size at 50 (average of range obtained in experiment 1). It can be seen from Table 3 that AAKQ values rapidly decrease when the number of kanbans at each station is greater than 10. Hence it is decided to vary the number of kanbans from 1 to 10 in each station. 6.3. Experiment 3 Ð setting the run length With 50 units (mean of the range set by experiment 1) as lot size and number of kanbans as 5 (mean of the range set by experiment 2) at each station simulation has been performed with run lengths 1000, 1500 and 2000 h. Form ANOVA it is found that there is no signi®cant difference between run lengths and hence the run length has been ®xed at 1500 h. The details are given in Tables 4 and 5. Since already
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Table 3 Objective function values for different kanbans levels at each station No. of kanbans in station I
II
III
1 2 3 4 5 6 7 8 9 10 11 12 13
1 2 3 4 5 6 7 8 9 10 11 12 13
1 2 3 4 5 6 7 8 9 10 11 12 13
MTR
TAAKQ
Total (Z)
9.94 20.83 34.25 32.74 43.89 55.45 48.42 42.58 53.15 57.67 47.44 58.01 39.66
84.25 77.68 71.13 61.48 55.74 56.11 39.98 40.85 23.03 23.09 20.40 19.65 15.40
94.19 98.51 105.38 94.22 99.63 111.56 88.40 83.43 76.18 80.76 69.84 77.66 55.06
experiments 1 and 2 are performed with a run length of 2000 h there is no need to repeat these experiments and verify the values with 1500 h as run length. H0: no signi®cant difference in the measures of performance. H1: signi®cant difference exists. 6.4. Experiment 4 Ð setting the number of kanbans and lot size In determining the neighbourhood solution comprising of the number of kanbans at each station and lot size, the following two perturbation algorithms have been used. The limits set for each of the parameter is given in Table 6. Notations: K1: number of kanbans is workstation 1 1 # K1 # 10 K2: number of kanbans is workstation 2 1 # K2 # 10 K3: number of kanbans is workstation 3 1 # K3 # 10 LS: Lot size 10 # LS # 80 r1,r2: uniform random numbers between 0 and 1 Table 4 Objective function value for various run-lengths Run length
MTR
TAAKQ
Total (Z)
1000 1500 2000
36.71 31.38 33.82
55.42 51.95 50.21
92.13 83.33 84.03
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Table 5 Analysis of variance for run lengths Source of variation
Sum of squares
Degrees of freedom
Mean squares
F
Run length Error Total
63.96 4269.58 4333.54
2 6 8
31.98 711.59
0.0449 NS
Perturbation algorithm 1: Step 1: Consider the current solution vector. [K1, K2, K3, LS] Step 2: Set a tag for each of the above four parameters with equal probability. Step 3: Generate a uniform random number r1. Step 4: Based on r1 select the parameter to be changed. Step 5: Generate a random number r2 If r2 , 0:5 increment the selected parameter subject to upper limits else decrement the selected parameter subject to lower limits Step 6: Set the neighbour solution vector. Perturbation Algorithm 2: Step 1: Consider the current solution vector. [K1, K2, K3, LS] Step 2: Do the following for each of the four parameters: 2.1 Generate a random number r2. 2.2 Based on r2 perform any one of the following with equal probability for the parameter: (a) Increment the value subject to the upper limit (OR) (b) Decrement the value subject to the lower limit (OR) (c) No change in the value Step 3: Set the neighbour solution vector.
6.5. SAA based kanban search procedure To conduct the experiments a kanban simulation model is developed using GPSS. The simulated Table 6 Limits set for kanban and lot size
No. of kanbans Lot size
Minimum
Maximum
Increment
1 20
10 80
1 10
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annealing algorithm is coded using C language. These two models are interfaced and experiments are conducted. The steps involved in the search procedure are as follows. Notations used: ti: initial temperature tf: ®nal temperature acc: number of solutions accepted tot: total number of solutions generated F : freeze limit b : accept limit a : cooling rate S p: current best solution S0: initial solution vector Z: objective function value Z p: current maximum Z Step 1. Input SAA parameters: Set ti, tf, F , b , a ; n 0; Step 2. Input kanban model parameters: demand distribution parameters inter arrival distribution parameters for each of the p part types, process time distributions for each of the s work stations and material handling time distribution parameters, number of kanbans in each workstation and the lot size. Step 3. Generate initial solution S0 [5,5,5,60]. Step 4. Call the GPSS kanban system simulation module. Perform simulation for the duration of 1500 h. Compute Z p. Step 5. Set f0 Z p ; f p f0 ; Sp S0 ; Step 6. Set n n 1 1 Step 7. Using perturbation determine Sn (Ref. Section 6.4)
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Step 8. Call the GPSS kanban system simulation module. Perform simulation for the duration of 1500 h. Compute Z p. Step 9. Set fn Z p ; Sp S0 Step 10. Compute change in Z p value d fn 2 fn21 Step 11. If d . 0 then (accept the solution) goto step 12 else goto step 13 Step 12. If fn . f p then (New global value) set f p fn ; Sp Sn ; fz 0; goto step 16 else goto step 16 Step 13. Compute probability of accepting inferior solution at step n pan e 2d=tn Step 14. If r 0; 1 , pai then (accept inferior solution) goto step 17 else goto step 15 Step 15. tot tot 1 1; if tn < tf then goto step 23
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else goto step 7 Step 16. If tn < tf then goto step 23 else goto step 17 Step 17. acc acc 1 1; tot tot 1 1; Step 18. If tot . 4b or acc . b then goto step 19 else goto step 6 Step 19. Compute atr acc=tot Step 20. If atr < 0:15 then fz fz 1 1 Step 21. If fz > F then goto step 23 Step 22. tn11 tn a; acc 0; tot 0; goto 6 Step 23. End of SAA based kanban design parameters search algorithm. Report Z p and S p. STOP The initial seed for the search is considered to be 5 kanbans at each station and 60 units for lot size,
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Table 7 Comparison of objective function values perturbation algorithms 1 and 2 Trial no.
1 2 3 4 5 6 7 8 9 10 11 12
No.of iterations
Objective function value (Z p)
System parameters (K1,K2,K3,lot size)
Algo 1
Algo 2
Algo 1
Algo 2
Algo 1
Algo 2
271 179 274 176 254 183 284 326 224 187 248 217
328 190 191 235 219 184 182 276 143 311 191 215
117.04 128.24 118.64 127.92 123.41 118.63 117.81 115.38 139.42 131.28 118.01 114.82
133.24 133.24 123.41 121.35 131.30 115.45 128.21 118.06 125.05 121.96 123.40 133.24
3,6,4,60 3,4,6,70 2,3,4,50 5,3,6,60 4,5,6,40 2,3,4,50 3,4,3,60 3,8,9,70 2,4,5,80 2,3,3,60 5,6,5,40 6,8,8,70
4,2,4,70 4,2,4,70 4,5,6,40 9,3,4,80 2,4,6,80 3,3,7,70 8,6,8,40 4,5,8,70 2,3,6,50 2,9,5,70 4,5,6,40 4,2,4,70
based on average of range of parameters. Using the above two perturbation algorithms search has been performed for a number of trials and their respective objective function values are given in Table 7. ANOVA is performed to check the signi®cance between the objective function values obtained from the two algorithms. From the ANOVA results given in Table 8, it is concluded that there is no signi®cant difference between the two algorithms. H0: no signi®cant difference exist between the objective function value of two algorithms. H1: signi®cant difference exists. The best system parameter obtained from the above trials is the one with a maximum objective function value of 139.42 for the following con®guration: Station #
Number of kanbans
1 2 3 Lot size 80, Z p 139.42
2 4 5
7. Conclusion A simulation model of the single-card kanban system has been developed to determine the number of kanbans and lot size. A bi-criteria objective function consisting of throughput and aggregate kanban queue has been employed to obtain a solution that maximises the objective function value. Simulated annealing algorithm has been employed to search the solution space. Two types of perturbation schemes have been tried out. The results showed that there is no signi®cant difference between the two schemes.
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Table 8 Analysis of variance for objective function values obtained from perturbation algorithms 1 and 2 Source of variation
Sum of square
Degrees of freedom
Mean square
F
Perturbation algorithms Error Total
58.026 1051.005 1109.031
1 22 23
58.026 95.5459
0.6073 NS
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Kirkpatrick, S., Gelatti Jr, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220, 671±680. Mitwasi, M. G., & Askin, R. G. (1994). Production planning for a multi-item, single stage kanban system. International Journal of Production Research, 32, 1173±1195. Nori, V. S., & Sarker, B. R. (1998). Optimum number of kanbans between two adjacent stations. Production Planning and Control, 9, 60±65. Pirlot, M. (1996). General local search methods. European Journal of Operational Research, 92, 493±511. Price, W., Gravel, M., & Nsakanda, A. L. A. (1994). Review of optimisation models of kanban based production systems. European Journal of Operations Research, 75, 1±12. Price, W., Gravel, M., Nsakanda, A. L., & Cantin, F. (1995). Modelling the performance of a kanban assembly shop. International Journal of Production Research, 33, 1171±1177. Savsar, M. (1996). Effects of kanban withdrawal policies and other factors on the performance of JIT system Ð A simulation study. International Journal of Production Research, 34, 2879±2899. Sridhar, J., & Rajendran, C. (1993). Scheduling in a cellular manufacturing system: A simulated annealing approach. International Journal of Production Research, 31, 2927±2945. Wang, H., & Wang, H. (1991). Optimum number of kanbans between two adjacent workstations in a JIT system. International Journal of Production Economics, 22, 179±188.
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Design of bi-criteria kanban system using simulated annealing technique P. Shahabudeen a,*, R. Gopinath b, K. Krishnaiah a a
Industrial Engineering Division, Anna University, Madras, India b Tata consultancy Services, Bangalore, India Accepted 30 July 2001
Abstract In the kanban system, the main decision parameters are the number of kanbans and lot size. In this paper, an attempt has been made to set the number of kanbans at each station and the lot size required to achieve the best performance using simulated annealing technique. A simulation model with a single-card system has been designed and used for analysis. A bi-criterion objective function comprising of mean throughput rate and aggregate average kanban queue has been used for evaluation. Different perturbation schemes have been experimented and compared. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Kanban; Simulated annealing
1. Introduction Just in time (JIT) production system is the manufacturing philosophy of producing what is needed at the right time and in right quantity (Hutchins, 1993). Kanban coupled with pull system of production is used as means of implementing JIT. Kanban means a `visible card', which serves as a planning and information tool to smoothen the ¯ow of material through the manufacturing and assembly process. The workstations located along the production lines only produce or deliver desired components when they receive a card and empty container, indicating that more parts will be needed in production. Each workstation will only produce enough components to ®ll containers and then stop. In addition, kanban limits the amount of inventory in the process by acting as an authorisation to produce. * Corresponding author. E-mail address: [email protected] (P. Shahabudeen). 0360-8352/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0 3 6 0 - 8 3 5 2 ( 0 1 ) 0 0 06 0 - 2
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The essential elements in the design of kanban production system are the number of kanbans needed to link processes together and the appropriate unit of lot size (Berkley, 1992a). 2. Literature review Kanban based operational planning and control issues have been tackled in a number of studies by means of analytical or simulation modelling. Berkley (1992a) has reviewed 50 papers in the area of kanban production control and organised them based on the type of system. He has also listed 24 vital operation design factors of kanban system. Price, Gravel and Nsakanda (1994) have reviewed optimisation models of kanban based systems. They have concluded that interesting direction for future development would be in the incorporation of models in decision support systems for production control. 2.1. Mathematical models Several researchers have attempted mathematical modelling approach. They have assumed deterministic demand and material handling as being carried out periodically and the periods correspond to ®xed withdrawal cycles. Kimura and Terada (1981) have developed a model for the kanban system, which can be considered to be a pioneering work in the area of kanban modelling. Their model has served as a reference for subsequent researchers. Bitran and Chang (1987) have extended the work of Kimura and Terada and offered a mathematical model for kanban system in a multi-stage production setting. Their deterministic model is designed to assist in the choice of the number of kanbans to use at each stage and thus to control the level of inventory. Bard and Golany (1991) have developed a mixed integer linear program to extend Bitran and Chang's model, by considering material shortage and the production of multiple parts at each stage. Fukukawa and Hong (1993) have developed a mixed integer programming model to determine the number of kanbans. Mitwasi and Askin (1994) have developed a non-linear model for a multi-item single stage kanban system. Price, Gravel, Nsakanda and Cantin (1995) have formulated an integer linear programming model for kanban based assembly shop. It is a multi-period model and the variables track the ¯ow of kanbans in the shop ¯oor, and in between time periods. Constraints are used to ensure conservation of ¯ow of kanbans, maximum number of kanbans, availability of the parts for an operation, to limit production to demand and the machine availability. The objective is to minimise the make span. Solution for ILP is possible for some parts. However, it cannot be applied for parts having complex assembly diagrams. They have concluded that a large number of cards are unlikely to reduce the makespan signi®cantly and it is certain to increase the work in progress. Gupta and Al-Turki (1998) have described a new kanban system called ¯exible kanban system. This system uses an algorithm based on mathematical model of the system to dynamically and systematically manipulate the number of kanban and starvation caused by stochastic factors. 2.2. Queuing and Markov chain models Formulating kanban-controlled lines as Markov chains has been a popular strategy to ®nd the optimal number of kanbans. In these models, researchers usually assume processing times to be exponential and give the state of the system by the number of full containers between each pair of stations. Deleersnyder,
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Hodgson, Muller and O'Grady (1989) have modelled a line with blocking by total queue size as a discrete time Markov chain to study the effect of kanban numbers, machine reliability, processing time and ®nished goods demand variability. They have mentioned the following as the sources of variability; the demand schedule, the processing time, machine breakdowns and the actual demand. If the processing time is longer than expected, fewer units are produced during the period. Berkley (1992b) has developed a decomposition approximation using imbedded Markov chains for two-card systems with periodic material handling and Erlang processing times. He has given several examples to show how the approximation could be used to ®nd the required number of kanbans, the required withdrawal cycle time, or both. Berkley (1994) has used a continuous time Markov chain to determine minimum performance levels required to obtain a desired production rate, independent of the average station process time. He has used ®ll rate, which is de®ned as the probability of meeting demand from stock, as the performance measure, in addition to average withdrawal kanban queue times. However, he prefers ®ll rate because operator himself can easily compute this statistics compared to queue statistics, which require data collection and computational devices. He has assumed that demand is in®nite, and handling time is zero. Nori and Sarker (1998) have modelled the kanban system using Markov Chain to determine the optimum number of kanbans between adjacent stations. Through a numerical example, they have shown that their procedure has developed closer boundaries than that of Wang and Wang (1991) and hence the search for optimum number of kanbans is more ef®cient. 2.3. Simulation based studies In the case of ®nding the required number of kanbans and lot size, simulation offers a number of advantages. One has to make a few assumptions about processing time and lead time distributions and one can run the models under the most current shop conditions. In a simulation study of a four station line with ®xed order points, Berkley (1993) has shown that order points have a signi®cant effect on average inventories and production rates. In another simulation study of a two-card system with ®xed withdrawal cycle times, Berkley and Kiran (1991) have found that average in process inventories are highly sensitive to withdrawal cycle times while ®nished goods withdrawal kanban queue time is much less sensitive. This suggests that material-handling frequency can be reduced to obtain signi®cant inventory savings at a small customer service cost. Gravel and Price (1988,1991) have developed kanban simulation models with the objective of reducing WIP and makespan. Abdou and Dutta (1993) have studied the effect of container size, number of kanbans and material handling frequency on the sum of set up, inventory holding and machine under utilisation costs. Simulation results for a multi-product, multiline two-card kanban system showed that the cost of minimising container size is off set both by increasing material handling frequency and increasing the number of kanbans. Savsar (1996) has presented a simulation study of JIT production control system and its performance under different operational conditions. Berkley (1996) has simulated a two-card kanban system with multiple part types to determine the effect of container size on average inventory and customer service levels in tandem so that total in process capacity remains the constant. Results show that smaller containers lead to smaller average inventories. He has also observed that smaller containers do not always lead to poor customer service levels. Andijani (1997) has used stochastic system simulation to generate and construct a set of kanban allocations, which maximise the average throughput rate and minimise the average system time.
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2.4. Performance measures used in previous studies Following is the summary of ®ndings from literature survey on the various performance measures used: (a) Average input inventory. It is the sum of average input material queues at all stations except the ®rst. (b) Average output inventory. It is the sum of average output material queues at all stations except the last. (c) Total WIP inventory. Sum of average input and output inventories. (d) Finished goods withdrawal kanban waiting time. It is the time from when a ®nished goods withdrawal kanban arrives at the stock point of last station to the time a container of ®nished part (same type) is available to withdraw. (e) Fill rate. The proportion of ®nished goods demand met from stock. (f) Finished goods order queue time. The time from when the order arrives until it can be completely satis®ed with ®nished parts. (g) Mean cumulative throughput rate. Ratio of total satis®ed demand to generated demand. (h) Mean utilisation of line. Mean utilisation of last station. (i) Mean total WIP length. Mean of all inprocess inventory lengths for all products including ®nished goods inventories. (j) Mean total waiting time. Mean waiting time of all products in all inprocess and ®nished good inventories. (k) Cycle time. The time between two consecutive outputs.
3. Present study 3.1. Bi-criteria objective It is found that, most of the researchers have used only single objective function as the performance measure. In this study, a bi-criteria objective kanban problem has been considered. The ®rst criterion is the maximisation of mean cumulative throughput rate, which is de®ned as the ratio of total satis®ed demand to the total generated demand. The other criterion is the minimisation of aggregate average kanban queue, which is the sum of average number of kanbans waiting in the queue at all the stations. In this paper, the following two parameters are considered to measure the performance of the system. 3.1.1. Mean throughput rate (MTR) Mean cumulative throughput rate is de®ned as the ratio of total satis®ed demand to the total generated demand n X
MTR
i1 n X i1
SDi 1 GDi
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where GDi is the generated demand during ith request, SDi the satis®ed demand during ith request and n is the number of demand request generated during the simulation. It is required to maximise this value. 3.1.2. Aggregate average kanban queue (AAKQ) This is the sum of average number of kanbans waiting in the queue during simulation at all the stations AAKQ
m X j1
Ij
2
where Ij is the average number of kanbans waiting at station j and m is the number of stations in the system. It is required to minimise this value. 3.2. Bi-criteria objective function The number of kanbans and lot size directly in¯uence the performance measures de®ned above. Since the above measures are of con¯icting nature, AAKQ expression is transformed to obtain a maximisation expression. The transformed form of the AAKQ expression (TAAKQ) is as follows: m X
TAAKQ
j1 m X j1
IUj 2
IUj 2
m X
j1 m X j1
Ij 3
ILj
where IUj is the maximum number of kanbans at station j and ILj is the minimum number of kanbans at station j. The combined objective function is as follows: n X
z maxf MTR; TAAKQ
i1 n X i1
m X
SDi 100 1 GDi
j1 m X j1
IUj 2
IUj 2
m X
j1 m X j1
Ij 100
4
ILj
The above constructed objective function has been used to determine the overall measure of the system. 3.3. Search technique The literature review reveals that search techniques have not been widely applied for setting the kanban system parameters. Hence, it is felt that some search heuristics can be attempted to handle such kind of problems. In the case of search techniques, simulated annealing technique is found to have frequent application in production related problems like group technology, scheduling, etc. for obtaining good results. In this study, simulated annealing technique has been used.
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4. Simulated annealing Simulated annealing is a search heuristic intensively used to solve optimisation problems and especially combinatorial ones. It was derived from an analogy to the physical annealing process and ®rst introduced by Cerny (1985) and Kirkpatrick, Gelatti and Vecchi (1983). Eglese (1989), Pirlot (1996) and Sridhar and Rajendran (1993) described simulated annealing and its implementation with some choice parameters. A generic simulated annealing algorithm for a maximization problem is given below: Step 1: Select an initial point (seed) S0, a termination criterion p , initial temperature ti, cooling rate a and ®nal temperature tf. Set n 0: Determine f0, the objective function value of solution S0 and update f p, the global maximum value of objective function and S p, the corresponding solution. Step 2: Determine the neighbourhood Sn11 using perturbation and compute fn11 Step 3: If d fn11 2 fn . 0 then set n n 1 1; else if r < exp 2d=tn where r is a random number in the range 0 to 1 then set n n 1 1: Step 4: Update f p and S p. Step 5: If termination condition p is reached then goto step 6 else reduce temperature by cooling rate a and goto step 2. Step 6: Report f p and S p. 4.1. Choice of parameters for the proposed simulated annealing algorithm Initial temperature (ti), ®nal temperature (tf) and cooling factor are determined by conducting some pilot studies as 60, 1 and 0.8 respectively. The other factors like accept limit, freeze limit, etc. are adopted from Sridhar and Rajendran (1993). 4.1.1. Length of plateau (L) This denotes the number of iterations at a temperature. Here instead of using L as a count, two counts ACCEPT and TOTAL are used. ACCEPT counts the number of accepted moves at a particular temperature and TOTAL counts the total number of moves at a particular temperature. Iterations at a particular temperature are terminated when ACCEPT is equal to accept limit (b ) or when TOTAL is equal to 4 times b . 4.1.2. Stopping rule The SAA is stopped when any one of the conditions is met with: (i) when the temperature falls to a speci®ed value (tf); (ii) when the freeze count reaches a pre-de®ned value. The freeze count is reset to zero whenever current solution is better than current best solution, thus hoping for better solutions. After iterations at
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Table 1 System parameters Part type
A B
Processing times at workstations (time units)
WS1
WS2
WS3
5^1 3^2
4^1 5^2
3^2 4^3
Demand qty (units)
Demand inter arrival time (time units)
200 ^ 50 250 ^ 40
30 ^ 20 40 ^ 10
a temperature, the ratio of ACCEPT to TOTAL is computed. If this ratio is less than or equal to 0.15, freeze count is incremented by one. This threshold value indicates that the chance of replacing the best solution is diminishing. Here when the freeze count reaches a value of 5, SA is terminated.
5. The kanban system model The system under study is dynamic kanban system with stochastic demand and processing times with the following assumptions. 5.1. Assumptions ² ² ² ² ² ² ² ² ² ²
Two types of jobs are produced. Each type of job has its own demand inter arrival distribution Each type of job has its own demand distribution. All the jobs must be processed in all the stations. Processing times of each job type are different. The demand is satis®ed on FCFS basis. There is in®nite supply of raw material at the input of the production system. Any kanban detached at the output of a stage is immediately available for up stream stage. The distance between the workstations is not substantial. Material handling resources are unrestricted.
It consists of a production line with three workstations handling two different products. When the demand arrives in the system, it is placed at the ®nal workstation and availability of the components to produce this product is checked and if available, the production at this stage starts, otherwise a request is issued to the previous stage for the required part. Upon receiving the request, the previous stage either starts production for the required parts or issues a request to its upstream stage. In this way, the production at a current stage is pulled from down stream processes. Only after the completion of processing, the kanban of the emptied container is sent to the preceding station's out bound inventory to be replaced with a full one, if available. Initial conditions for each run is such that each in process and
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Table 2 Objective function values for different lot sizes (number of kanbans at each station 5) Lot size
MTR
TAAKQ
Total (Z)
20 30 40 50 60 70 80 90 100 110 120 130
36.48 27.112 44.88 43.88 31.38 40.62 52.68 49.00 57.08 41.46 47.13 37.18
55.79 56.82 55.29 55.74 51.95 50.48 51.38 53.02 51.09 56.34 54.60 53.25
92.27 83.93 100.17 99.62 83.33 91.10 104.06 102.02 108.17 97.80 101.73 90.43
®nished goods inventories are full with containers. The various system parameters used in the model are given in Table 1. 6. Experimental design and results Experiments to decide the run length, to set a limit for number of kanbans and lot size are performed. The details are as follows. 6.1. Experiment 1 Ð effect of changes in lot size In this experiment, number of kanbans at each station has been considered to be 5 and lot size is varied from 20 to 130. Table 2 gives the objective function values obtained from this experiment. It is found that objective function values do not change signi®cantly above 70. Hence it has been decided to vary lot size from 20 to 80 only. 6.2. Experiment 2 Ð effect of changing number of kanbans at each station equally Here the number of kanbans has been varied from 1 to 13 maintaining the lot size at 50 (average of range obtained in experiment 1). It can be seen from Table 3 that AAKQ values rapidly decrease when the number of kanbans at each station is greater than 10. Hence it is decided to vary the number of kanbans from 1 to 10 in each station. 6.3. Experiment 3 Ð setting the run length With 50 units (mean of the range set by experiment 1) as lot size and number of kanbans as 5 (mean of the range set by experiment 2) at each station simulation has been performed with run lengths 1000, 1500 and 2000 h. Form ANOVA it is found that there is no signi®cant difference between run lengths and hence the run length has been ®xed at 1500 h. The details are given in Tables 4 and 5. Since already
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Table 3 Objective function values for different kanbans levels at each station No. of kanbans in station I
II
III
1 2 3 4 5 6 7 8 9 10 11 12 13
1 2 3 4 5 6 7 8 9 10 11 12 13
1 2 3 4 5 6 7 8 9 10 11 12 13
MTR
TAAKQ
Total (Z)
9.94 20.83 34.25 32.74 43.89 55.45 48.42 42.58 53.15 57.67 47.44 58.01 39.66
84.25 77.68 71.13 61.48 55.74 56.11 39.98 40.85 23.03 23.09 20.40 19.65 15.40
94.19 98.51 105.38 94.22 99.63 111.56 88.40 83.43 76.18 80.76 69.84 77.66 55.06
experiments 1 and 2 are performed with a run length of 2000 h there is no need to repeat these experiments and verify the values with 1500 h as run length. H0: no signi®cant difference in the measures of performance. H1: signi®cant difference exists. 6.4. Experiment 4 Ð setting the number of kanbans and lot size In determining the neighbourhood solution comprising of the number of kanbans at each station and lot size, the following two perturbation algorithms have been used. The limits set for each of the parameter is given in Table 6. Notations: K1: number of kanbans is workstation 1 1 # K1 # 10 K2: number of kanbans is workstation 2 1 # K2 # 10 K3: number of kanbans is workstation 3 1 # K3 # 10 LS: Lot size 10 # LS # 80 r1,r2: uniform random numbers between 0 and 1 Table 4 Objective function value for various run-lengths Run length
MTR
TAAKQ
Total (Z)
1000 1500 2000
36.71 31.38 33.82
55.42 51.95 50.21
92.13 83.33 84.03
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Table 5 Analysis of variance for run lengths Source of variation
Sum of squares
Degrees of freedom
Mean squares
F
Run length Error Total
63.96 4269.58 4333.54
2 6 8
31.98 711.59
0.0449 NS
Perturbation algorithm 1: Step 1: Consider the current solution vector. [K1, K2, K3, LS] Step 2: Set a tag for each of the above four parameters with equal probability. Step 3: Generate a uniform random number r1. Step 4: Based on r1 select the parameter to be changed. Step 5: Generate a random number r2 If r2 , 0:5 increment the selected parameter subject to upper limits else decrement the selected parameter subject to lower limits Step 6: Set the neighbour solution vector. Perturbation Algorithm 2: Step 1: Consider the current solution vector. [K1, K2, K3, LS] Step 2: Do the following for each of the four parameters: 2.1 Generate a random number r2. 2.2 Based on r2 perform any one of the following with equal probability for the parameter: (a) Increment the value subject to the upper limit (OR) (b) Decrement the value subject to the lower limit (OR) (c) No change in the value Step 3: Set the neighbour solution vector.
6.5. SAA based kanban search procedure To conduct the experiments a kanban simulation model is developed using GPSS. The simulated Table 6 Limits set for kanban and lot size
No. of kanbans Lot size
Minimum
Maximum
Increment
1 20
10 80
1 10
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annealing algorithm is coded using C language. These two models are interfaced and experiments are conducted. The steps involved in the search procedure are as follows. Notations used: ti: initial temperature tf: ®nal temperature acc: number of solutions accepted tot: total number of solutions generated F : freeze limit b : accept limit a : cooling rate S p: current best solution S0: initial solution vector Z: objective function value Z p: current maximum Z Step 1. Input SAA parameters: Set ti, tf, F , b , a ; n 0; Step 2. Input kanban model parameters: demand distribution parameters inter arrival distribution parameters for each of the p part types, process time distributions for each of the s work stations and material handling time distribution parameters, number of kanbans in each workstation and the lot size. Step 3. Generate initial solution S0 [5,5,5,60]. Step 4. Call the GPSS kanban system simulation module. Perform simulation for the duration of 1500 h. Compute Z p. Step 5. Set f0 Z p ; f p f0 ; Sp S0 ; Step 6. Set n n 1 1 Step 7. Using perturbation determine Sn (Ref. Section 6.4)
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Step 8. Call the GPSS kanban system simulation module. Perform simulation for the duration of 1500 h. Compute Z p. Step 9. Set fn Z p ; Sp S0 Step 10. Compute change in Z p value d fn 2 fn21 Step 11. If d . 0 then (accept the solution) goto step 12 else goto step 13 Step 12. If fn . f p then (New global value) set f p fn ; Sp Sn ; fz 0; goto step 16 else goto step 16 Step 13. Compute probability of accepting inferior solution at step n pan e 2d=tn Step 14. If r 0; 1 , pai then (accept inferior solution) goto step 17 else goto step 15 Step 15. tot tot 1 1; if tn < tf then goto step 23
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else goto step 7 Step 16. If tn < tf then goto step 23 else goto step 17 Step 17. acc acc 1 1; tot tot 1 1; Step 18. If tot . 4b or acc . b then goto step 19 else goto step 6 Step 19. Compute atr acc=tot Step 20. If atr < 0:15 then fz fz 1 1 Step 21. If fz > F then goto step 23 Step 22. tn11 tn a; acc 0; tot 0; goto 6 Step 23. End of SAA based kanban design parameters search algorithm. Report Z p and S p. STOP The initial seed for the search is considered to be 5 kanbans at each station and 60 units for lot size,
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Table 7 Comparison of objective function values perturbation algorithms 1 and 2 Trial no.
1 2 3 4 5 6 7 8 9 10 11 12
No.of iterations
Objective function value (Z p)
System parameters (K1,K2,K3,lot size)
Algo 1
Algo 2
Algo 1
Algo 2
Algo 1
Algo 2
271 179 274 176 254 183 284 326 224 187 248 217
328 190 191 235 219 184 182 276 143 311 191 215
117.04 128.24 118.64 127.92 123.41 118.63 117.81 115.38 139.42 131.28 118.01 114.82
133.24 133.24 123.41 121.35 131.30 115.45 128.21 118.06 125.05 121.96 123.40 133.24
3,6,4,60 3,4,6,70 2,3,4,50 5,3,6,60 4,5,6,40 2,3,4,50 3,4,3,60 3,8,9,70 2,4,5,80 2,3,3,60 5,6,5,40 6,8,8,70
4,2,4,70 4,2,4,70 4,5,6,40 9,3,4,80 2,4,6,80 3,3,7,70 8,6,8,40 4,5,8,70 2,3,6,50 2,9,5,70 4,5,6,40 4,2,4,70
based on average of range of parameters. Using the above two perturbation algorithms search has been performed for a number of trials and their respective objective function values are given in Table 7. ANOVA is performed to check the signi®cance between the objective function values obtained from the two algorithms. From the ANOVA results given in Table 8, it is concluded that there is no signi®cant difference between the two algorithms. H0: no signi®cant difference exist between the objective function value of two algorithms. H1: signi®cant difference exists. The best system parameter obtained from the above trials is the one with a maximum objective function value of 139.42 for the following con®guration: Station #
Number of kanbans
1 2 3 Lot size 80, Z p 139.42
2 4 5
7. Conclusion A simulation model of the single-card kanban system has been developed to determine the number of kanbans and lot size. A bi-criteria objective function consisting of throughput and aggregate kanban queue has been employed to obtain a solution that maximises the objective function value. Simulated annealing algorithm has been employed to search the solution space. Two types of perturbation schemes have been tried out. The results showed that there is no signi®cant difference between the two schemes.
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Table 8 Analysis of variance for objective function values obtained from perturbation algorithms 1 and 2 Source of variation
Sum of square
Degrees of freedom
Mean square
F
Perturbation algorithms Error Total
58.026 1051.005 1109.031
1 22 23
58.026 95.5459
0.6073 NS
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