Stochastic Modeling and Analysis of Generalized Kanban Controlled Unsaturated finite capacitated Multi-Stage Production System

Transcription

1 Stochastic Modeling and Analysis of Generalized anban Controlled Unsaturated finite capacitated Multi-Stage Production System Mitnala Sreenivasa Rao 1 orada Viswanatha Sharma 2 1 Department of Mechanical Engineering JNTU College of Engineering, Hyderabad, India 2 Centre for Energy Studies, JNTU College of Engineering, Hyderabad, India Abstract In recent years, variants of anban mechanisms are proposed to deal with different levels of uncertainties commonly exist in a typical production system. Among them, generalized anban control (Buzacott [1989]) mechanism gained popularity as it is able to deal with uncertainties effectively while maintaining pull philosophy. In literature, many models are developed to gain insights on the design of generalized anban control mechanism however they were developed under the assumption that external demand for finished products of a production system in its control is very large, nown as saturated condition. In the present globalization of maret, most of the products face finite rate of demand, nown as unsaturated condition and stochastic. In the present paper, Generalized anban Control unsaturated finite capacitated Multi-Stage Production of serial type (GCMP) system has been considered with uncertainties in external demand and processing times. A model of decomposition Continuous Time Marov Chain (CTMC) type is developed for analyzing the considered system for the performance measures, namely, fill rate, average inventory and throughput. The model is then validated with stochastic discrete event simulation model at 95 % confidence level. Insights on the issues relevant to deployment of safety stocs, such that specified fill rate is achieved with minimum average inventory, have been presented. It is found analytical results are in close agreement with simulation results and uniform increasing

2 anban allocation pattern is dominant over uniform allocation pattern for the specified criteria. (eywords: Multi-Stage Production System-Unsaturated finite capacitated-uncertainties- Decomposition based CTMC-various performance measures-validation- Discrete Stochastic Discrete event Simulation model at 95% C.F- Location & Sizing of buffer stocs) 1. Introduction The typical production systems are of multi-stage type, as most of the products require multiple operations. Among various mechanisms for controlling inventories in multi-stage production systems, pull mechanisms gained importance in recent years due to reason that Toyota manufacturing system and Hard Davidson Company could decrease inventories to minimum through implementation of them in their concerns. In pull mechanism, production is initiated at each stage of multi-stage production system as reaction to present demand of its downstream stage. Pull mechanisms are commonly implemented using anbans and hence they are alternatively nown as anban systems. anban is a Japanese word for a card or ticet, which is used as a manual information system. In reality, various sources of uncertainties, viz external demand uncertainty, processing times variations, yield uncertainty, worers absenteeism etc., do exist in any typical production system and anban systems are not exception to this, Often safety stocs are used to alleviate the ill effects of uncertainties. As per the philosophy of anban systems, uncertainties would be reduced rather than deployment of safety stocs however they are used as long as uncertainties exist. If safety stocs are used in multi-stage production as a buffer against uncertainties, their location and sizing are the related issues as deployment of safety stocs at

3 certain locations result more effective buffering than at other location. Further, larger the safety stocs higher the buffer against the uncertainties but at a cost of increased inventory of the system and longer flow times. In anban systems, inventories are clearly controlled and by addition or removal of anban from the system inventory in the system can be increased or decreased. Therefore, the problem of deployment of safety stocs in anban controlled production systems when uncertainties exist is equivalently treated as a problem of determining of number of anbans at each stage of the system such that specified level of performance is achieved with minimum inventory. It is noted that traditional anban systems cannot perform even with deployment of safety stocs when the level of uncertainties is high. For meeting high fluctuations while maintaining pull control, Generalized anban Controlled Production (GCP) system is devised (due to Buzacott [1989]). As compared to traditional anban controlled systems, in GCP system information about consumption of product by an external demand reaches the most upstream stage is fast and consequently a raw part enters into the system as a substitute for the consumed product. GCP systems can be operated either saturated or unsaturated conditions. In saturated condition, finished products have heavy external demand and essentially finished products are consumed as soon as they are available in the output buffer. On the other hand, in unsaturated condition, finished products face finite rate of demand however it can be stochastic and wait in the buffer for arrival of an external demand. It can be noted that in the present global competitive maret, most of products have finite rate of demand essentially unsaturated production environment prevails. Buzacott [1980] compared GCP system with MRP system and found that lie push systems former system faces uncertainties while maintaining the pull concept i.e., by effectively controlling inventories. Due to this, many

4 researchers have shown lot of interest in modelling various issues relevant to GCP system in recent years. 2. Review of literature Mitra & Mitrani [1990] considered saturated GCP system of serial type with processing times variations as sources of uncertainty and developed decomposition type evaluation model for the considered system. Later Mitra & Mitrani [1991] extended their model for considering stochastic demand case (unsaturated) under assumption that infinite numbers of bacorders are allowed. Tayur [1992] extended Mitra and Mitrani 1990] model for addressing the problem of deployment of safety stocs when processing times variations exist. He developed a heuristic based on the sample path analysis, which allocates anbans at each stage of system such that specified throughput (THT) is achieved with minimum average inventory (AI). Di Masocolo et al [1995] extended Mitra & Mitrani [1991] model to consider general production networ and they also decomposition type algorithm for evaluating the performance of the system. The basic difference between in the decomposition approach of Mitra & Mitrani [1990 & 1991] and Di Masocolo et al[1995] is each subsystem after decomposition of total system is analyzed by dealing with underlying Marov chain in the former system whereas in the later system, it is analyzed by product form approximation. However, both Mitra & Mitrani[1991] and Di Masocolo et al[1995] models do not consider the issues relevant to the problem of deployment of safety stocs in unsaturated GCP system. Ettl and Schwehm[94& 95] considered saturated Generalized anban controlled production lines and developed a heuristic based queuing networs and genetic algorithm for optimal portioning of the networ and allocation of anbans such that specified throughput is achieved with minimum average inventory Further, designing of the

5 system for specified off-the bin service to the external customer (alternatively nown as Customer Service Level (CSL)) is essentially needed so as to have competitive edge over others. In the present wor, unsaturated finite capacitated GCP system with sources of uncertainties in external demand and processing times is considered and analyzed for the various performance measures. Further, relevant issues of safety stocs viz, their location and sizing are also addressed. 3. Model The considered GCP system consists of number of stages arranged in series as shown in Fig.1. Each stage (=1,2,..) consist of queues, namely, anban post (P ), WorCentre (WC ) and Output Buffer (OB ) with their entities respectively as free anbans (anbans not attached with parts), anbans attached with raw parts (WIP) and anbans with finished products of stage. It is assumed that c number of anbans is associated with stage of the system. In the queue WC,(=12, ), there exist a machine M which process one part at a time. The processing time of M is random process and is assumed as exponentially distributed with mean service rate a µ. External demand occurs for finished products of stage and is assumed as Poisson process with mean arrival rate as λ. The difference in the number of entities in the queues P andob 1, the difference in number of entities in the queues P + 1 and OB at each stage and bacorder level in QD are used as state variables. Let b be the maximum number of bacorders allowed in the system.

6 3.1 Law of Motion The dynamics of the GCP system on occurrence of the events namely, arrival of external demand and service completion of machine M (=1,2,.) is called as law of motion of the system and it is as follows: When an external demand arrives, it is satisfied immediately if at least one finished product is available in the output buffer OB resulting in assembly of product with demand. The finished product leaves the system along with external demand after its associated anban is detached without any time delay. The detached anban joins the anban post products in P instantaneously. On the other hand, if there are no finished OB and the external demand is bacordered in the queueqd. It remains in that queue until a finished product joins the output bufferob. It is assumed that finite number of bacorders is allowed in the system. Let B be the maximum number of bacorders allowed in the system. Hence queues OB and QD can never be nonempty simultaneously although they can be empty simultaneously. The machine M ( = 1,2,..., ) after completion of operation on a part, the part along with its tagged anban is added to the output buffer OB without any time delay. The machine M ( = 1,2,..., ) processes the next part from WC if available, otherwise it becomes idle. When a part along with its tagged anban departs from M ( = 1,2,..., ) to output buffer OB, the part is detached from its associated anban and is attached with a free anban from the anban post P + 1 without any time delay if at least one free anban is available in that anban post. The part along with the attached anban moves to wor centerwc + 1. The anban, which is removed from the part, is transferred to P instantaneously. On the arrival of free anban to P, it is tagged to a part from the output buffer OB after removing its already associated anban of stage 1 without any time 1 delay if at least one part along with tagged anban is available in that buffer. The

7 removed anban joins P 1 instantaneously. The anban along with its attached part is moved to WC and it is loaded on to the machine M if idle, otherwise the part waits for processing in the worcentre. If OB 1 is empty, then free anban remains in the anban post, OB 1 cannot be non-empty simultaneously. P. This results in the queues P ( = 1,2,..., ) and 3.2 Mathematical statement of the problem The problem can now be stated as follows: Given allocation of anbans c 1, c 2,.. c at stage 1, 2.. respectively in the GCP of serial type with Poisson arrival process of external demand, and exponential processing times, it is needed to determine the Average Inventory (AI), Average Throughput (THT) and Customer Service Level (CSL). The underlying stochastic process of the system is Continuous Time Marov Chain (CTMC) due to the assumptions about external demand and processing times as Marovian processes. The underlying Marov chains of CTMC can be dealt and specified performance measures can be determined if the number of stages in the system is only two (Rao [2006]). When the number of stages in the system is more than two, it is computationally very difficult and hence decomposition type procedure is developed to evaluate the performance of the system. 4. Decomposition Procedure The GCP system is decomposed into number of subsystems, Ω ( ) ( = 1,2,..., ) as shown in Fig. 2. Each subsystem Ω() consists of a Wor Center WC, an upstream and a downstream synchronization stations IP and OP respectively. Upstream synchronization IP, taes place between the two queues viz., i) queue of raw parts, denoted as RP and ii)

8 queue of free anbans P. Similarly, in downstream station OP, synchronization taes place between the queue of demands DM and the queue of finished productsob. The sizes of RP and DM are chosen to coincide exactly with the capacities of OB 1 and P + 1 of stages -1 and +1 of the GCP system respectively. The number of anbans associated with Ω ( ) ( = 1,2,..., ) is equal to c of stage of the original system. It is assumed that each subsystem Ω () behaves in the same manner as that of stage in the original system. Each subsystem Ω ( ) ( = 1,2,..., ) is synchronized with two independent Poisson processes one at IP with parameter ρ and the other at OP withϖ. For analyzing the GCP system through decomposition algorithm, two issues are to be resolved namely, (i) analysis of subsystem Ω ( ) ( = 1,2,..., ) assuming that ρ ( = 1,2,..., ) and ϖ ( = 1,2,..., ) are given and (ii) determination of parameters ρ and ϖ for each subsystem Ω( ) ( = 1,2,..., ). 4.1 Analysis of Subsystem Ω( ) ( = 1,2,..., ) In this subsection, the procedure for analyzing Ω() is described with the assumption that ρ and ϖ are nown. Fig. 3 shows the schematic representation of Ω ( )( = 1,2,... ). As assumed earlier the service time of machine M follows exponential, arrival of raw parts and demands follow Poisson processes and c, c 1 and c + 1 are finite numbers, the underlying stochastic process involved with Ω () conforms to finite Continuous-Time Marov Chain(CTMC). Therefore CTMC modeling approach is followed for the analysis of Ω (). As per this, Let S (t) be the state of subsystem Ω () at given time t defined as

9 { I ( t), J ( t); c I ( t) c, c J ( t) c ; for =, c = b, t 0} S( t) = (1) where I (t) and J (t) represent difference of entities in the queues P and RP and the difference of entities in the queues OB and DM. at any given time t respectively. Let Q be the transition rate matrix of CTMC of Ω( ) ( = 1,2,3,... ). and can be written based on the net conservation flow (Viswanatham & Narahari [1995]). Let Π ( = 1,2,3,..., 1) be the row vector of steady-state probabilities of states of Ω ().and ( j) π be the j th component of the steady-state probability vector. Then Π can be determined by solving the following set of simultaneous linear global balance equations. Π Q = 0 (2) ( ) π j = 1 (3) j Let α be the throughput of the subsystem Ω ( ) ( = 1,2,3,... ).. Based conservation of flow of material, under steady-state the expression for α ( > 1) can be written as follows: α = ρ [ 1 P( I = c 1 )] for = 2,... (4) where P ( I = c 1 ) represents the steady-state probability that c 1 anbans accumulate in buffer RP. Alternatively in steady state as net inflow of parts to the buffer OB ( = 1,2,3,..., ) is equal to the effective outflow of demands that are accepted per unit time. Hence α ( 1) can be expressed as follows: α ϖ 1 P( J = c )] for = 1,2,..., 1 (5) = [ + 1

10 where P ( J = c + 1) is the probability that c + 1 number of free anbans accumulate in DM.. Using Π, the values of functions, viz, P ( I = c 1 ) and P ( J = c + 1) can be determined by proper summation of steady-state probabilities. 4.2 Estimation of parameters of Ω( ),( = 1,2,..., ) The -stage GCP system is approximated as a sequence of subsystems where a stream of raw parts and demands at Ω() is provided by the subsystem Ω( 1) and Ω( +1) respectively. On other hand, the last subsystem Ω () is non-typical system in which the demands for the finished products come from outside with Poisson distribution rate with parameter λ. The parameters of the subsystems are denoted as two different vectors viz, (i) ρ= ρ, ρ,..., ρ ) and (ii) ϖ= ϖ, ϖ,..., ϖ ). and are determined by following two ( 1 2 considerations respectively ( 1 2 i) Under steady-state, throughput of all subsystems must be the same and the system faces an average demand of λ ( P( J = b). Hence the throughput of each subsystem α = λ P( J b) and Eqns (4) and (5) can be written as ( J b) for 2,3 ρ [ 1 P( I = c 1 )] = λ P =,..., (6) ϖ 1 P ( J = c )] + = λ P( J b) for = 1,2,3,..., 1 (7) [ 1 Rearranging Eqs. (6) and (7), we get λ P( J b) ρ = for = 2,3,... (8) [1 P( I = c )] 1 ϖ λp( J = [1 P( J b) = c 1 )] for = 1,2,... 1 (9)

11 ii) As the input buffer RP of subsystem Ω () is equal to the output buffer of Ω( 1) and DM of subsystem Ω () is equal to P +1 of the subsystem Ω( +1) we get P( I = c 1 ) = P( J 1 = c 1 ) for = 2,3,... (10) P( J = c+ 1) = P( I + 1 = c + 1) for = 1,2,3,... 1 (11) Using Eqs. (10) and (11) Eqs.(8) and (9) can be rewritten respectively as λp( J b) ρ = for = 2,3,... (12) [1 P( J = c )] 1 1 λp( J b) ϖ = for = 1,2,... 1 (13) [1 P( I = c )] Rewriting, Eqs. (12) and (13) λp( J b) ρ + 1 = for = 1,2,3,..., 1 (14) [1 P( J = c )] λp( J b) ϖ 1 = for = 2,3,... (15) [1 P( I = c )] The equations (14) and (15) are nown as fixed-point equations. These equations are solved for ρ= [ ρ ρ ρ2,..., ρ ] and ϖ= ϖ, ϖ,..., ϖ ] iteratively by fixed point iterative algorithm 1 2, [ Fixed Point Iterative Algorithm The algorithm is initialized by setting the downstream parameters of all the subsystems to certain starting values asϖ = + ( = 1,2,..., 1) and P( J b) 1. As the µ 1 = number of bacorders allowed in the system is b ϖ = λ P( J b). The main loop consist of forward and bacward passes In the forward pass, for each subsystem Ω( ) ( = 2..., ),

12 the upstream parameters of ρ is updated according to Eq. (14) by analyzing the subsystem Ω( 1). In bacward pass, the downstream parameter, ϖ 1 of the subsystem Ω( 1) ( = 1,2,..., 1) is improved according to Eq.(15) by analyzing the subsystem Ω ( ) ( = 2,3,..., ). The algorithm iterates between the two passes until a convergence criteria is satisfied. Table.1 shows the summarization of fixed-point algorithm for estimating the vectors ρ andϖ. Once the convergence is achieved in the fixed-point algorithm, the whole system is decomposed into Ω ( 1), Ω(2),... Ω( ). These subsystems are analyzed in isolation independently and the performance measures of total system determined. 4.4 Performance measures Let (l, m) be the typical state of Ω ( ) ( = 1,2,... ) and its associated steady-state probability be π ( l, m). Using steady-state probabilities and other parameters, viz, c1, c2,... c, B expressions for the performance measures of original system namely, CSL, AI and THT are derived. CSL: It is defined as the probability that the arrival of external demand is met instantaneously and occurs only for finished products of stage in a multi-stage unsaturated GCP system. Based on the analysis of subsystem Ω () in isolation, the expression for CSL can be derived by proper summing of steady state probabilities as follows: CSL c = c j j= 1i i= c 1 π ( i, j) (16)

13 AI: It is sum of average inventory of each subsystemω ( ), = 1, 2,...,. The average inventory of each subsystem is the difference between the number of anbans associated with Ω () and the expected number of free anbans in P. Hence, expression for AI can be obtained as AI = c i = = 1 i= 1 j= c c 0 π ( i, j) (17) THT 1 0 = µ 1 π j= i= c ( c j, j) + π ( i, c ) 0 1 (18) To test the accuracy of the analysis, a simulation model is developed for GCP system. The results are compared with that obtained from the analysis. 5. Results and Discussion Decomposition type algorithm is developed for the evaluation of multi-stage unsaturated finite capacitated GCP system. Three types of data sets are considered, namely, identical processing set ( = ) type ( ) i < µ i+1 µ, non-identical processing set of type ( ) = i µ i + µ and of i > µ i+1 µ for validating the procedure developed and gaining insights on the issue of deployment of buffer stocs in the selected system under considered sources of uncertainties. In each type of data set, three types of external demand scenarios, viz., uniform, bowl, inverse bowl and random allocation patterns are considered (Siha [1994]). A comparison of the results obtained from decomposition procedure for the analysis of two stage unsaturated finite capacitated GCP system with that from CTMC and simulation models with 95% confidence level(c>f) is presented in Tables 2-4.

14 It may be observed from these tables that decomposition results are in close agreement with that of CTMC analysis under identical conditions. In addition, for validation of the algorithm developed for analyzing large system, six stage system is considered and is evaluated with uniform allocation, uniformly decreasing, uniformly increasing and random allocation patterns with varying demand rates, machine capacities and number of anbans and presented in Tables 5-9. From these tables, it can be noted that these algorithms yield results, which are in close agreement with that of simulation at 95% C.F further signifying their applicability to the analysis of multi-stage system. After validation of the algorithm developed, it is attempted to evaluate the five stage selected GCP system for gaining insights on the deployment of buffer stocs when the systems face uncertainties in external demand and machine processing times. The following sections present the effect of variations in external demand, effect of unbalances in machine capacities and the effect of allocation pattern on the performance of selected systems. 5.1 Effect of variation in external demand Figures 4 & 5 show the effect of increase of mean external demand rate on the CSL and AI respectively of unsaturated finite capacitated 5-stage GCP system under various types of allocation patterns namely, uniform, constantly increasing, constantly decreasing and random allocation patterns when machine capacities are balanced. It can be observed that as external demand increases both CSL and AI decreases however, rate of decrease of CSL is more as compared to AI. This is due to the fact that as external demand increases, congestion taes place which leads to longer flow times consequently rate of availability of finished products decreases and therefore CSL decreases. Further, more accumulation of free anbans occur in respective anban posts as external demand increases resulting in the decrease of AI. As per

15 the law of motion of considered GCP system that each stage, consists of input buffer WC, it results WIP as external demand increases therefore decrement in AI is less as compared that of CSL. Similar observations can be made from Figures 6 & 7 when machine capacities are decreasing from the most upstream stage to the most down stream stage and Figures 8 & 9 in which machine capacities increase from upstream stage to down stream stages 5.2 Effect of Variations in machine capacities and number of stages It may be observed from Figures 4, 6 and 8 that for a given allocation pattern and external demand scenarios balanced machine capacities results in better CSL compared to unbalanced machine capacities in the selected five-stage systems. In case of balanced capacities raw parts are made available to downstream stages at the required epochs of time. Unbalanced machine capacities results in either starvation or excess availability of parts in contrary to the demands raised by down stream stages. The presence of uniformly increasing machine capacities between the stages results in better CSL compared to the case of uniformly decreasing condition at low ( =0.3) and medium ( =0.5) rates of demand in the selected system. When the downstream stages are more capacitated the parts are processed at faster rate although upstream stages feed at lower rate resulting in availability of the finished products. 5.3 Effect of allocation pattern It can be noted from Figures 4-8 that in the GCP system at low rates of demand i.e., =0.3, the uniformly increasing allocation pattern dominates over other allocation pattern for a given total number of anbans. On the other hand that at higher rates of demand, > 0.3, no allocation pattern follow a set pattern, such that specified CSL is achieved with minimum AI.

16 It can be observed that at lower rates of external demand, exact allocation for the selected criteria is difficult to determine. Hence allocation of anbans among stages in CP systems is treated as a combinatorial problem. For determining the number of anbans at each stage in GCP system to the selected criterion namely achieving specified CSL with minimum AI is still an issue. 6. Conclusions A decomposition type procedure is developed for evaluating the performance of multi-stage unsaturated finite capacitated GCP system with uncertainties in processing times and external demand. Derived the expressions for performance measures namely, CSL,AI and THT. The accuracy of the procedure is tested with CTMC and simulation models. It is found that the results are in close agreement. Further, effect of variations in external demand rates, capacities of machines and allocation patterns on the performance of the system is studied. Based on this, the following conclusions can be drawn: o With an increase in external demand, both CSL and AI decrease. o Machines having higher capacities when located at downstream stages would result in more CSL and less AI in all demand scenarios compared to the machine located at upstream stages o No specific allocation pattern is perceptible for achieving a specified CSL with minimum AI in multi-stage system, unlie in two-stage GCP system. It can be categorized as a combinatorial problem. 7. References 1. Di Mascolo, M., Frein, Y., Baynat, B., and Dallery, Y.,(1993) Queuing networ modeling and analysis of generalized anban systems. European Control Conference. pp Di Mascolo, M., Frein, Y., Baynat, B., Dallery, Y. (1996) An analytical method for performance evaluation of anban controlled production systems. Operations Res. 44(1), pp

17 3. Ettl, M., and Schwehm, M. (1994) A design methodology for anban controlled production lines using queuing networs and genetic algorithms. Tenth International Conference on Systems Engineering, ICSE, 6-8 September, Coventry U.., pp Ettl, M., and Schwehm, M.(1995) Determining the Optimal Networ partition and anban Allocation in JIT production lines. Evolutionary Algorithms in Management Applications : Edited by Bielthah, J., and Nissen, V. Springer Verlag., pp Frein, Y. and Di Mascolo, M. (1995) On the Design of Generalized anban Control Systems. International Journal of Operations & Production Management, Vol.15, No.9, pp Matta, A., Dallery, Y. and Di Mascolo, M. (2005) Analysis of an assembly systems controlled with anbans. European Journal of Operations Research, Vol.166, pp Mitra, D.and Mitrani, I.(1990) Analysis of a anban discipline for cell coordination in production lines. Management Science, Vol.36, pp Mitra, D. and Mitrani, I.(1991) Analysis of a anban discipline for cell coordination in production lines, -II: Stochastic Demands. Operations Research, Vol.39, pp Rao, M.S. 2006, Analytical Investigations on anban Controlled Unsaturated Multi-Stage Production System. Ph.D thesis, JNT University, Hyderabad, India 10. Siha, S., 1994, The Pull Production System: Modeling and Characteristics. International Journal of Production Research. 32(32), pp Tayur, S. R. (1992) Properties of Serial anban Systems. Queuing systems. Vol.12, pp Tayur, S. R. (1993) Structural properties and a Heuristic for anban controlled Serial lines. Management Science, Vol.39, No. 11, pp Viswanadham, N. and Narahari, Y. (1994) Performance modeling of Automated Manufacturing Systems. Prentice-Hall India Ltd, pp Acnowledgement The authors than Dr. A. Seshu umar for the help rendered in the preparation of the paper. The financial support from Technical Education Quality Improvement Program (TEQIP) in the presentation of the article is acnowledged.

18 P1 0 0 OB0 External 1 1 WC1 1 Stage 1 M P2 P3 2 2 WC2 Stage 2 M2 2 2 WC Stage M P+1 WC Stage M QD Poisson demand with λ OB1 OB2 OB OB - Free anban of stage - anban with part of stage - External Demands Fig. 1 Schematic representation of multi-stage unsaturated finite capacitated GCP system

19 IP1 OP1 ϖ DM 1 1 M 1 P1 ρ 1 WC 1 RP OB 1 1 Ω(1) DM 2 ϖ 2 M 2 P2 ρ 2 WC 2 RP OB 2 2 Ω(2) IP OP ϖ DM M P ρ RP WC OB Ω() Fig. 2 Decomposition of -stage unsaturated finite capacitated GCP system

20 IP OP DM ϖ M P ρ RP WC Ω () OB Part with anban Free anban of stage Fig. 3 Isolated subsystems Ω() Customer Service Level (CSL) uniform constantly decreasing random µ i = 1.0, i = 1,2,...,5 constantly increasing Mean External Demand Rate (λ) Fig. 4 Variation of CSL with demand (µ i = 1.0, i=1,2,,5) in the case of five-stage unsaturated finite capacitated GCP system

21 20.0 µ i = 1.0, i = 1,2,..., Avaerage Inventory (AI) constantly decreasing constantly increasing random uniform Mean External Demand Rate (λ) Fig. 5 Variation of AI with demand (µ i = 1.0, i=1,2,,5) in the case five-stage unsaturated finite capacitated GCP system µ 1 = 1.0, µ 2 = 0.9, µ 3 = 0.8, µ 4 = 0.7, µ 5 = 0.6 Customer Service Level (CSL) random constantly decreasing uniform constantly increasing Mean External Demand Rate (λ) Fig. 6 Variation of CSL with demand (µ 1 =1.0, µ 2 =0.9, µ 3 =0.8, µ 4 =0.7, µ 5 =0.5) in the case of five-stage unsaturated finite capacitated GCP system

22 20.0 µ 1 = 1.0, µ 2 = 0.9, µ 3 = 0.8, µ 4 = 0.7, µ 5 = Average Iinventory (AI) random constantly increasing uniform constantly decreasing Mean External Demand Rate (λ) Fig. 7 Variation of AI with demand (µ 1 =1.0, µ 2 =0.9, µ 3 =0.8, µ 4 =0.7, µ 5 =0.6) in the case of five-stage unsaturated finite capacitated GCP system µ 1 = 0.6, µ 2 = 0.7, µ 3 = 0.8, µ 4 = 0.9, µ 5 = 1.0 Customer Service Level (CSL) random constantly decreasing constantly increasing uniform Mean External Demand Rate (λ) Fig. 8 Variation of CSL with demand (µ 1 =0.6, µ 2 =0.7, µ 3 =0.8, µ 4 =0.9, µ 5 =1.0) in the case of five-stage unsaturated finite capacitated GCP system

23 Average Inventory (AI) uniform constantly increasing µ 1 = 0.6, µ 2 = 0.7, µ 3 = 0.8, µ 4 = 0.9, µ 5 = 1.0 random constantly decreasing Mean External Demand Rate (λ) Fig. 9 Variation of AI with demand (µ 1 =0.6, µ 2 =0.7, µ 3 =0.8, µ 4 =0.9, µ 5 =1.0) in the case of five-stage unsaturated finite capacitated GCP system Table 1 Summary of Fixed point algorithm 1. INITIALILZATION: Set values for ρ 1 =, ϖ = λ and ϖ = µ + 1 ( = 1,2,..., 1) and P( J b) = 1 2. FORWARD PASS: In this pass, the upstream parameters ρ ( = 2,3,..., ) are updated. For this purpose, the subsystems are analyzed from Ω (1) to Ω( 1). In the analysis of Ω( ) = 1,2,..., 1, determine ρ + 1 using Eq BACWARD PASS: In this pass, the downstream parameters ( = 1,2,... 1) are modified. The subsystem, Ω( ) ( =, 1,...2), are analyzed for determining ϖ 1 using Eq CONVERGENCE: Repeat step 2 and 3 until the changes in parameters are small. ϖ

24 Table 2 Decomposition procedure for analysis of two-stage unsaturated GCP system and its validation with CTMC model and simulation for conditions µ 1 = µ 2 = 1.0, C = 6 and b = 100 Sl. No Ext. Demand rate Allocation GCP DECOM ` GCP CTMC Simulation with 95% C.F AI CSL AI CSL AI CSL ±0.002 ± ±0.002 ± ±0.002 ± ±0.002 ± ±0.004 ± ±0.005 ± ±0.005 ± ±0.004 ± ±0.002 ± ± ± ±0.002 ± ±0.002 ± ±0.002 ± ± ±0.002 ± ±0.008

25 Table 3 Decomposition procedure for analysis of two-stage unsaturated GCP system and its validation with CTMC model and simulation for conditions µ 1 =1.0 & µ 2 =0.8, C = 6 & b =100 Sl. No Ext. Demand rate Allocation GCP DECOM ` GCP CTMC Simulation with 95% C.F AI CSL AI CSL AI CSL ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.0075

26 Table 4 Decomposition procedure for analysis of two-stage unsaturated GCP system and its validation with CTMC model and simulation for conditions µ 1 =0.8 & µ 2 = 1.0, C = 6 & b =100 Sl. No Ext. Demand rate Allocation GCP DECOM ` GCP CTMC Simulation with 95% C.F AI CSL AI CSL AI CSL ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.009

27 Table 5 Decomposition procedure for analysis of six-stage unsaturated GCP system and its validation with simulation for conditions µ i = 1.0, i = 1, 2,,6 and b = 100 Sl. No Ext. Demand rate Allocation GCP DECOM Simulation with 95% C.F AI CSL AI CSL ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.0097

28 Table 6 Decomposition procedure for analysis of six-stage unsaturated GCP system and its validation with simulation for conditions µ 1 = 1.0, µ 2 = 0.9, µ 3 =0.8, µ 4 = 0.7, µ 5 = 0.6, µ 6 =0.5 and b = 100 Sl. No Ext. Demand rate Allocation GCP DECOM Simulation with 95% C.F AI CSL AI CSL ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.012

29 Table 7 Decomposition procedure for analysis of six-stage unsaturated GCP system and its validation with simulation for conditions µ 1 = 0.5, µ 2 = 0.6, µ 3 =0.7, µ 4 = 0.8, µ 5 = 0.9, µ 6 =1.0 and b = 100 Sl. No Ext. Demand rate Allocation GCP DECOM Simulation with 95% C.F AI CSL AI CSL ± ± ± ± ± ± ± ± X ± X ± X ± X ± ± ± ± ± ± ± ± ± ± ± X10-5 ± ±0.000