Int. Journal of Business Science and Applied Management, Volume 7, Issue 3, 2012
Performance evaluation of a merge supply system with a
distribution centre, two reliable suppliers, one buffer and Erlang
lead times
Michael Vιdalis
University of the Aegean, Department of Business Administration
Michalio Bldg, Michalon 8, 82100, Chios, Greece
Telephone: +30 22710 35143
Email: mvid@ath.forthnet.gr
Stelios Koukoumialos
TEI of Larisa, Department of Business Administration
Peripheral Rd., 411 10, Larissa, Greece
Telephone: +30 24210 53692
Email: skoukoum@teilar.gr
Despoina Ntio
University of the Aegean, Department of Business Administration
Michalio Bldg, Michalon 8, 82100, Chios, Greece
Telephone: +30 24650 21948
Email: dntio@teikoz.gr
Georgios Varlas
University of the Aegean, Department of Business Administration
Michalio Bldg, Michalon 8, 82100, Chios, Greece
Telephone: +30 22710 77024
Email: gvarlas@hotmail.com
Abstract
In this work a two echelon merge supply chain is examined. Specifically, two non identical reliable suppliers
feed a distribution centre with a shared buffer. The first echelon consists of the distribution centre and the shared
buffer, the second echelon includes two non identical reliable suppliers. There is an unlimited supply of
materials to suppliers and an unlimited capacity shipping area after the distribution centre. In other words,
suppliers are never starved, and the distribution centre is never blocked. The materials are processed by
suppliers with rates following the Erlang distribution. The distribution centre has a reliable machine that pushes
material with service times following the Erlang distribution. Blocking appears when one or more suppliers
finish their process and try to feed the buffer that is full. The supply network is modelled as a continuous time
Markov process with discrete states. The structures of the transition matrices of those systems are explored and
a computational algorithm is developed. Our aim is to generate stationary distributions for different values of
system’s parameters so as the various measures of the system can be estimated. Finally, for the mathematical
programming model and the rest of the calculations the Matlab software is used.
Keywords: supply chain management, merge systems, performance measures, Markov processes
Acknowledgements: The work in this paper was partly supported by grant MIS 379526 “Odysseus: A holistic
approach for managing variability in contemporary global supply chain networks,” which was co-financed by
the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program
“Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding
Program: THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation.
Int. Journal of Business Science and Applied Management / Business-and-Management.org
43
1 INTRODUCTION
A supply chain consists of several possible stages, where goods are produced, transformed, assembled and
distributed in order to cover the customer demand. Supply chain networks are distinguished by the pull and push
processes. Supply chain management (SCM) is the management of flows (products, capital and information)
among the stages of the supply chain, aiming to maximize the total expected profitability.
Supply network designing is a strategic issue that affects the whole supply chain performance. There is a
close connection between the design and the profitability of a supply chain. In reality these networks are quite
complex, they are structured with many facilities and intermediary nodes that are costly. Some times a rational
elimination or merging of some of those facilities contributes to cost reduction and improves the total
performance of the system. The design of a supply network requires decisions regarding parameters such as the
number of suppliers, their capacities, the capacity of production - distribution centre and the capacity of the
buffers in order to control the efficient flow of raw materials, in process inventory and finished goods
(throughput).
In a supply chain push and pull processes are usually performed. In a push-based supply chain, products are
pushed through the supply network from the suppliers to the customers, while in a pull-based supply chain the
product flow is based on the consumption of goods in the downstream stages. The interface between the push-
based processes and the pull-based processes is known as the push–pull boundary.
The performance of a supply system is affected by the randomness of the stochastic processes that take
place such as the supplier’s service rates and the production centre service rate, and by system characteristics
such as the number of suppliers and their capacities, the production centre, and the buffer capacity.
The random nature of the processes involved in the behaviour of the system under consideration renders it
difficult to obtain important performance measures without a stochastic evaluative model. Evaluative or
descriptive models assume a given set of input data and decision variables of the system under study and
subsequently the performance measure(s) of the system are evaluated. Such models could be used as generative
or prescriptive models (see Figure 1).
Figure 1: The synergistic relationship between evaluative and generative models (Papadopoulos et al.,
2010)
In this study a two echelon discrete material, merge supply network is examined. In this network the same
part type, after its elaboration, is shipped to a production-distribution centre by two reliable suppliers. The first
echelon consists of the distribution centre and a shared buffer, the second echelon includes two non identical
reliable suppliers. The paper is organized as follows; section two presents a literature review of the subject. In
sections three and four system assumptions and model formulation are laid out. Section five presents illustrative
numerical results. Finally section six summarises conclusions and further research.
2 LITERATURE REVIEW
A first variable we can use to classify supply networks is their structure. Based on this there are three
generic types of supply networks.
• Linear, if each node-stage receives goods from a single upstream node (supplier) and ships goods to a
single downstream node (internal customer).
• Distributive or, divergent or arborescent, if each node receives goods from a single source at the most,
but can ship goods to more downstream nodes.
• Assembly or convergent, if each stage-node delivers to a single source but it can receive goods from
more than one sources.
• To our best knowledge, a few analytical methods are available in the current literature. Our review
starts with references that pertain to discrete material flow systems.
• Altiok and Perros (1986) develop an approximation procedure to decompose split and merge
configurations of open networks of queues with blocking.
Evaluative
model
Generative
model
Michael Vιdalis, Stelios Koukoumialos, Despoina Ntio and Georgios Varlas
44
• Hyo-Seong and Pollock (1989) analyze a merge configuration of open queuing networks with
exponential service times and finite buffers. They provide an iterative algorithm to decompose the queuing
network into individual queues and analyses each individual queue in isolation.
• Gopalan and Kumar (1994) analyzed a merge production system which has two parallel stations in the
first stage followed by a single station in the second stage. The transient behaviour of the system is analyzed
and various measures of system performance are evaluated.
• Helber (1998, 1999), deals with merge operations on production systems with a limited buffer capacity
and random processing times using the decomposition approach.
• Papadopoulos and Vidalis (2004) analyze a discrete material flow system consisting of three unreliable
machines and one buffer of limited capacity. Diamantidis and Papadopoulos (2006) allow the machines to fail
not only when they are operational but also when are either blocked or starved.
• Macgregor and Cruz (2005) use an approximation formula for the Buffer Allocation Problem (BAP) in
series, merge, and splitting topologies of finite buffer queuing networks.
• Li and Huang (2005) examine a two-product split-merge system consisting of one common main line
and two dedicated lines for two part types. The authors develop a method (overlapping decomposition) for
evaluating the throughput as a function of the system parameters.
• Bulgak (2006) presents a new approach in optimal interstage buffer allocation problem (BAP) of split-
and-merge unpaced open assembly systems. A simulation model developed is used in conjunction with genetic
algorithms (GA) to find optimal interstage buffer configurations yielding a maximum production rate.
Alternatively an artificial neural network (ANN)-GA approach is used for the same optimization problem
(BAP).
• Liu and Li (2009) investigate discrete time split and merge systems with unreliable machines. Buffers
of finite capacity are interposed among the machines. They adopt three scheduling policies: circulate, priority
and percentage, and present analytical methods to approximate the system production rates of split and merge
systems.
• Liu and Li (2010) provide analytical methods to obtain performance analysis of split and merge
production systems with exponential machine reliability models, operating under circulate, strictly circulate,
priority, and percentage split/merge policies.
For continuous material flow systems we refer to the works of Tan (2001), Helber and Mehtrens (2003),
Helber and Hanifa (2004) and Tan and Gershwin (2009). Tan examines a system consisting of two upstream
unreliable machines that serve a shared buffer in front of the third machine. He assumes that the combined speed
of the machines upstream the buffer is lower than the speed of the downstream machine. Helber and Mehtrens
(2003) develop an exact algorithm to compute the throughput for a similar but more general system, which
allows for arbitrary deterministic processing times and exponentially distributed failure and repair times. To
cope with this, they introduce a priority unblocking rule. Helber and Hanifa (2004) extend the previous
approach to more complicated flow lines with unreliable machines and limited buffer capacity. Tan and
Gershwin (2009), consider a two stage continuous material flow system separated by a finite capacity buffer.
The system is modelled as a continuous time, continuous-discrete state space stochastic process and the steady-
state distribution is determined.
In the reported above studies the service or breakdown times are supposed to be exponentials.
Our research is closer to Diamantidis and Papadopoulos (2004). The system under investigation has the
same structure but processing times here are non identical and assumed to be stochastic with Erlang-K for all
machines. The contribution of the present work to the research of merge systems is the adoption of Erlang
processing times in both echelons.
3 DESCRIPTION OF THE SYSTEM
In this study, we examine a two echelon discrete material flow merge supply network that consists by two
suppliers, one buffer and one production-distribution (DC) (see Figure 2). The same type item is shipped by the
suppliers to the DC. The actual processing + lead time that the materials are processed by the two suppliers are
non identical and follow the Erlang distribution with p
s
phases (p
s
≥ 2) with average rate μ
1
and μ
2
respectively.
The production-distribution centre performs another operation on the materials collected in the buffer and the
finished products flow outside the system. The processing time in DC is assumed also to follow the Erlang
distribution with P
DC
phases (P
DC
≥ 2). The first echelon consists of the distribution centre and the shared buffer,
the second echelon includes two suppliers. This structure can represent a wide range of real systems.
Additionally, there is an unlimited supply of materials to suppliers and an unlimited capacity shipping area
after the distribution centre. In other words, suppliers are never starved, and the production-distribution centre is
never blocked. Blocking appears when one or more suppliers have finished their process and try to feed the
buffer that is full (blocking after service, BAS). The blocking remains until the machine finishes service at DC
and one unit from the buffer enters the machine. In case of more than one blockage, on unblocking the supplier
with the smallest index has priority. That is suppliers are ranked based on their unblocking priority.
Int. Journal of Business Science and Applied Management / Business-and-Management.org
45
Figure 2: A two stage merge supply network, with two suppliers, a shared buffer and a DC with one
machine.
The system under investigation is fully described by the numbers:
K: Number of merged stations (here K=2).
N: Capacity of distribution centre (i.e. number of machines here N=1).
B: Capacity of shared buffer (number of slots).
P
s
: The number of Erlang phases for suppliers (here P
s
=2).
P
DC
: The number of Erlang phases for DC (here P
DC
=2)..
μ
1
: Average processing rate of Supplier
1
.
μ
2
: Average processing rate of Supplier
2
.
μ: Average processing rate of DC.
The main objective of this work is to investigate the influence of each parameter (i.e. P
s
, P
DC
, B, μ
1
, μ
2
and
μ) on system performance measures. The main contribution of this work is the presentation of an exact
evaluative model to calculate the performance measures of a merge push system with Erlang replenishment and
processing times. This model can be used as a generative one to determine the values of the parameters that
optimize the behaviour of the system given an objective function.
4 METHODOLOGY
Many business problems or procedures can be described by Markov Chain, which stand for modelling
uncertainty of adequate real-world dynamic systems. Due to Markovian property the future behaviour of the
process depends only on the present state of the process and it is not influenced by its past history.
The Markov process {X(t), t≥0 } is completely determined by the probability distribution of the initial state
X
0
and the one-step transition probabilities p
ij
(dt). In applications of Markov chains the art is (Tijms, 2003):
(a) to choose the state variable(s) such as the Markovian property holds,
(b) to determine the one-step transition probabilities p
ij
.
In simple words, to model a system as a Markov process the following steps must be implemented: first
identify the state space and the possible number of the states, second, create the transition matrix by determining
the one-step transition probabilities p
ij
. Third, derive the steady state probability vector and finally by the steady
state probabilities calculate the performance measures of the system.
The merge supply network is analyzed as a continuous time Markov process with a finite number of states.
More specifically the system under consideration may be viewed as a birth-death stochastic process. Births
correspond to inputs to the buffer and DC and deaths correspond to outputs from the DC. As consequence, the
transition matrix is a tri-diagonal matrix. The transition matrix structure is affected by the system parameters:
number of suppliers (K), capacity of distribution centre (N) and capacity of shared buffer (B). This structure can
be exploited. This leads to the development of a computational algorithm that generates transition matrices for
any value of K, B and N. The stationary distribution is used to compute the system’s performance for different
system characteristics.
The Erlang distribution is selected as a research tool over exponential distribution because it offers lower
processing variability, i.e. the variability of processing times diminishes as the number of phases P
s
or P
DC
increases.
S
1
S
2
Buffer
μ
DC
Suppliers
μ
1
μ
2
Michael Vιdalis, Stelios Koukoumialos, Despoina Ntio and Georgios Varlas
46
4.1 Solution procedure
The steps of the solution method for solving the system under consideration are similar to those applied in
Papadopoulos (1989), Papadopoulos and O’Kelly (1989), Papadopoulos, Heavey and O’Kelly (1989a, 1989b),
Heavey, Papadopoulos and Browne (1993), Vidalis (1998), Vidalis and Papadopoulos (1999) and Vidalis and
Papadopoulos (2001). These steps are described below:
• Step 1: Calculate the dimension of the transition matrix as a function of parameters K, P
s
, P
DC
, B and N
(here K=2, B=1 and N=1)
• Step 2: Generate the transition matrix
• Step 3: Calculate the steady-state probability vector and
• Step 4: Compute the selected performance measures.
4.2 Illustrative Example
In order to be more understandable, an illustrative example is given for the simple case of a system with
one supplier, buffer capacity equal to one and one machine at the DC. The processing times at both echelons are
Erlang with 2 phases (P
s
=P
DC
=2). At first we identify the state space and the possible number of the states. The
number of states of a system with K suppliers, B slots in buffer and N identical machines in distribution centre
is denoted by
,,K B N
S
. The number of the states is given by the relationship:
,,K B N
S
= P
s
K
+ [P
s
K
* B + (P
s
+ 1)
K
] * P
DC
.
The number of states for the system (K=1, B=1, N=1), i.e. the
1,1,1
S
= 2
1
+[2
1
*1+(2+1)
1
]*2= 12 possible
states for the system. Symbols are used, so as the physical condition of the system is represented. Table 1 shows
the interpretation of those symbols. For instance, “b” symbolizes the supplier’s blocking when the buffer is full
and the DC is busy, “1” or “2” means that the supplier or the DC is in the first or second phase of material
processing respectively.
Table 1: System’s state representation symbols
Supplier 1or 2
Buffer
DC
1st erlang phase
0: empty
0: idle
2nd erlang phase
1: full
1st erlang phase
b: blocked
-
2nd erlang phase
Int. Journal of Business Science and Applied Management / Business-and-Management.org
47
Table 2: the possible states of a two stage inventory system with K=1 supplier, B=1 slot, N=1 machine at
DC and Erlang -2 processing times at supplier and DC.
DC
B
S
1
states
0
0
1
100
2
200
b
-
1
1
-
2
-
b
-
1
0
1
101
2
201
b
-
1
1
111
2
211
b
b11
2
0
1
102
2
202
b
-
1
1
112
2
212
b
b12
The states of the system are symbolised by a three digit vector (see table 2). The first number indicates the
supplier’s Erlang phase or blocking (S
1
=1, 2 or b), the second number presents the buffer’s level (B=0,1) and
the third number indicates the machine condition on DC, if it is idle or is in the first or second phase of Erlang
processing time (DC=0,1,2). Table 2 presents all the possible combinations between these variables. The 100
state denotes that the supplier is in the 1
st
phase, the buffer is empty and the DC machine is idle. The b12 state
represents the situation that the supplier is blocked, the buffer is full and the DC machine is in the 2
nd
phase. The
supplier can not be blocked while the buffer is empty. So states b00, b01 and b02 are not feasible. Also the DC
machine can not be idle while the buffer is full. So the states 110, 210, b10 are also excluded.
Thus, for the merge system K=1, B=1, N=1, the transition matrix has dimension 12x12 (see table 2).
Changes in the state of the system at each transition step are caused by the occurrence of one of the following
events:
• Completion of the 1
st
phase at supplier S1 at time Δt with probability 2μ
1
Δt
• Completion of the 2
nd
phase at supplier S1 at time Δt with probability 2μ
1
Δt
• Completion of the 1
st
phase at DC at time Δt with probability 2μ
Δt
• Completion of the 2
nd
phase at DC at time Δt with probability 2μ
Δt
To be more specific, if the system is in state 100, in the next time Δt the supplier’s processing phase 1 may
be finished with probability rate 2μ
1
Δt, so the system jumps to state 200 or remains at the initial state 100
with probability rate (1-2μ
1
)
Δt respectively. The transition matrix is completed following the same concept
for the remaining states.
Michael Vιdalis, Stelios Koukoumialos, Despoina Ntio and Georgios Varlas
48
Table 3: The transition matrix of a two stage push merge system K=1, B=1,N=1and Ps=P
DC
=2.
100
200
101
201
111
211
b11
102
202
112
212
b12
100
-2μ1
2μ1
200
-2μ1
2μ1
101
-2μ1-2μ
2μ1
2μ
201
-2μ1-2μ
2μ1
2μ
111
-2μ1-2μ
2μ1
2μ
211
-2μ1-2μ
2μ1
2μ
b11
-2μ
2μ
102
2μ
-2μ1-2μ
2μ1
202
2μ
-2μ1-2μ
2μ1
112
2μ
-2μ1-2μ
2μ1
212
2μ
-2μ1-2μ
2μ1
b12
2μ
-2μ
A computational algorithm in MatLab is developed to generate the transition matrices for different values
of buffer capacities. The proposed algorithm then solves the linear system of steady state equations and
calculates the steady state probability vector. Once we have calculated the steady-state probabilities, all the
performance measures of the system can be estimated. The most important performance measures are the
average inventory on system WIP
system
, the mean output rate or throughput of the system and the mean flow time
on system..
The average inventory on system is the mean number of machines that are occupied on DC, the mean
number of slots occupied on buffer, and the number of suppliers K.
WIPsystem = Number of suppliers + Mean number of Occupied machines on DC + Mean Buffer Level.
The throughput or mean output rate of the system is given by the relationship:
THR = μ ∙ Pr [ DC is busy ]
The WIP
system
is the mean number of flow units in the system. High levels of WIP
system
imply increasing
costs and vice versa. On the other hand, Throughput is the production performance measure under
consideration. Thus, high productivity rates require high Throughput Rates.
5 VALIDATION OF THE MODEL AND NUMERICAL RESULTS
To validate the model, a simulation model in Arena 12.0 has been created to compare the numerical results
given by the analytical model. Table 4, shows the results of simulation and the analytical model, which indicates
negligible differences between analytical and simulation results. Consequently, the analytical model is verified
by simulation, thus the accuracy of numerical results that are given in this section, is ensured.
In this section, some numerical results are presented; thus, the need for validation of the analytical model
and the precision of the given results emerges. For that purpose, a simulation model in Arena was created, in
order to compare the numerical results.
Int. Journal of Business Science and Applied Management / Business-and-Management.org
49
Table 4: Comparison of results for a system K=2, B=1, N=1, P
s
, P
DC
=2-11 and μ
1
=μ
2
=μ=1.
Simulation
Analytical
Simulation
Analytical
P
s
,
P
DC
THR
THR
Difference
WIP
system
WIP
system
Difference
2
0,98663
0,98665
0,00002
3,91145
3,91145
0,00000
3
0,99515
0,99515
0,00000
3,94855
3,94855
0,00000
4
0,99809
0,99807
0,00002
3,96762
3,96755
0,00007
5
0,99919
0,99918
0,00001
3,97846
3,97839
0,00007
6
0,99964
0,99964
0,00000
3,98509
3,98502
0,00007
7
0,99983
0,99983
0,00000
3,98938
3,9893
0,00008
8
0,99991
0,99992
0,00001
3,99225
3,99217
0,00008
9
0,99995
0,99996
0,00001
3,99425
3,99417
0,00008
10
0,99997
0,99998
0,00001
3,99566
3,99559
0,00007
11
0,99998
0,99999
0,00001
3,9967
3,99663
0,00007
Figures 3, 4 represent the values for Throughput and WIP
system
respectively from analytical and simulations
models. There is almost perfect matching, that indicates the accuracy of the analytical model.
Figure 3: Comparison of throughput for the system K=2, B=1, N=1, P
s
, P
DC
=2-11 and μ1=μ2=μ=1.
0,97500
0,98000
0,98500
0,99000
0,99500
1,00000
1,00500
1 2 3 4 5 6 7 8 9 10
Ps, Pdc
THR
Simulation THR
Analytical THR
Figure 4: Comparison of WIP
system
for the system K=2, B=1, N=1, P
s
P
DC
=2-11 and μ
1
=μ
2
=μ=1.
3,86000
3,88000
3,90000
3,92000
3,94000
3,96000
3,98000
4,00000
4,02000
1 2 3 4 5 6 7 8 9 10
Ps, Pdc
WIP
Simulation WIPsys
Analytical WIPsys
Michael Vιdalis, Stelios Koukoumialos, Despoina Ntio and Georgios Varlas
50
6 NUMERICAL RESULTS
In this Section, some sample numerical results, are provided. In Sub-section 6.1 the effect of the number of
phases P
s
and P
DC
on Throughput and WIP
system
is examined, while in Sub-section 6.2 the effect of supplier's
processing rates μ
1
and μ
2
on performance measures is investigated. Finally Sub-section 6.3 analyzes the effect
of DC’s processing rate on system performance measures.
6.1 Effect of the number of phases P
s
and P
DC
on Throughput and WIP
system
Firstly, the effect of Supplier's and DC's erlang processing times on performance measures is examined.
Table 5 and Figure 5 represent the evolution of Throughput and WIP
system
, when P
s
and P
dc
scale from 2 to 11,
while supplier’s rate are μ
1
=μ
2
= 0.5, and DC rate is μ=1 that is the system is balanced. The inbound rate to the
buffer is equal to the outbound rate. As the number of phases (P
s
and P
DC
) increases in both replenishment times
and service times so Throughput and WIP
system
also increase in a diminished rate. The throughput increases
tending to its upper bound due to the reduction of processing time variability. Also WIP
system
increases but the
effect of the increment of phases P
s
and P
DC
is smaller comparing with throughput’s augmentation.
Table 5: Throughput and WIP
system
for a system K=2, B=1, N=1, P
s
, P
DC
=2-11 , μ
1
,μ
2
=0.5
and μ=1.
P
s
, P
DC
THR
WIP
system
2
0,83511
3,39475
3
0,86529
3,42911
4
0,88459
3,45164
5
0,89824
3,46766
6
0,90851
3,47966
7
0,91658
3,48899
8
0,92312
3,49644
9
0,92854
3,50252
10
0,93314
3,50759
11
0,93708
3,51187
Figure 5: Throughput and WIP
system
for the system K=2, B=1, N=1, P
s
=P
DC
=2-11, μ
1
=μ
2
=0.5
and μ=1.
2
3
4
5
6
7
8
9
10
11
THR
0
1
2
3
4
# of phases
Throughput and WIPsystem for the system K=2, B=1, N=1, Ps=PDC=2-11,
μ1=μ2 =0.5 and μ=1.
THR
WIPsystem
A question is placed: is there any relationship between throughput and WIP
system
’s increment in a balanced
merge system? The scatter diagram in Figure 6 below indicates that there is a connection between the two
variables.
Int. Journal of Business Science and Applied Management / Business-and-Management.org
51
Figure 6: Scatter diagram for WIPsystem and Throughput data
3.38
3.4
3.42
3.44
3.46
3.48
3.5
3.52
0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96
WIPsystem
Throughput
Scatter diagram for WIP and Throughput data
Regression Analysis describes as closely as possible the relationship between throughput and WIP
system .
So
we can predict what value the dependent variable (WIP
system
) will assume, given specific value for the
independent variable (throughput). For the values from Table 5 regression analysis gives the relationship
WIP
system
=1,153921
Throughput + 2,431009 (*)
e.g. for throughput =0,92312 we have by (*)
WIP
system
=1,153921 *0,92312 +2,431009 =3,496217
The R
2
statistic is 0,99997. Because this value is close to the maximum possible R
2
value (1) this statistic
indicates that the regression function we have estimated fits our data well. Also the standard error S
e
is equal to
0,00029 indicating a good accuracy of the prediction obtained from the regression model.
Figure 7: WIP
system
evolution as a function of Throughput
3.38
3.4
3.42
3.44
3.46
3.48
3.5
3.52
0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96
WIPsystem
Throughput
WIPsystem
Estimated WIP
6.2 Effect of supplier's processing rate μ
1
and μ
2
on performance measures
Additionally, the effect of supplier's processing rate μ
1
and μ
2
on performance measures is investigated (see
table 6, Figure 8). The processing rates of suppliers, μ
1
and μ
2
increase from 0.2-1.8, while the processing rate at
DC remains stable, μ=1, and the number of phases P
s
and P
DC
are constant (equal to 4). As the processing rates
μ
1,
μ
2
are growing, throughput is growing too. When μ
1
end μ
2
reach the value 0.8, Throughput catches a slow
growing rate. The explanation is that in this value the inbound rate is greater than the outbound rate so the
system’s output tends to its upper value that is that is the processing rate of the bottleneck station (here the DC)
WIP
system
follows a high growing rate for all values of μ
1
, μ
2
from 0,2-0,8.
Table 6: Throughput and WIPs
ystem
for a system K=2, B=1, N=1, P
s
=P
DC
=4, μ
1
, μ
2
=0.2-1.8 and μ=1.
Michael Vιdalis, Stelios Koukoumialos, Despoina Ntio and Georgios Varlas
52
μ
1
,μ
2
THR
WIP
system
0,2
0,39962
2,45932
0,4
0,76495
3,11731
0,6
0,94927
3,68973
0,8
0,99073
3,90499
1,0
0,99807
3,96755
1,2
0,99952
3,98738
1,4
0,99986
3,99455
1,6
0,99995
3,99744
1,8
0,99998
3,99872
Figure 8: Throughput and WIP
system
for a system K=2, B=1, N=1, P
s
, P
DC
=4, μ
1
, μ
2
=0.2-1.8 and μ=1,5
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8
μ
1
, μ
2
THR, WIPsystem
THR
WIPsystem
Figure 9: WIP
system
evolution as a function of throughput
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
WIPsystem
Throughput
WIPsystem as a function of Thr
Figure 9 indicates that WIP
system
’s increment is greater than the corresponding of Throughput.
6.3 Effect of DC's processing rate on performance measures
Finally, the impact of DC's processing rate on performance measures is examined. The value of μ scales
from 0.2 to 1.8, while μ
1
, μ
2,
P
s
and P
DC
are constant and equal to 0.5 (table 7, Figure 10). In that case, as μ value
increases, Throughput increases therefore, the productivity of the systems increases too. On the other hand, as μ
value increases, WIP
system
declines so, the holding cost declines as well. The explanation is: for the values of μ,
0.2≤ μ≤ 0.8 the bottleneck station is the DC (the inbound supply rate is greater than the outbound rate of the
DC). At the point of μ=1 the system becomes balanced. For the rest values of μ, 1.2≤ μ≤ 1.8 the DC outbound
rate is greater than the supply inbound rate. As DC rate increases also throughput increases until its upper limit
Int. Journal of Business Science and Applied Management / Business-and-Management.org
53
(=1) because blocking is decreased. WIP
system
decreases because as DC becomes faster blocking and buffer
utilization are decreased.
Table 7: Throughput and WIP
system
for the system K=2, B=1, N=1, P
s
=P
DC
=4, μ
1
=μ
2
=0,5 and μ=0,2-1,8.
μ
THR
WIP
system
0,20
0,1999
3,9998
0,40
0,3998
3,9898
0,60
0,5957
3,9213
0,80
0,7672
3,7327
1,00
0,8845
3,4516
1,20
0,9467
3,1772
1,40
0,9752
2,9618
1,60
0,9879
2,8042
1,80
0,9937
2,6889
Figure 10: Throughput and WIP
system
for a system K=2, B=1, N=1, P
s,
P
DC
=4, μ
1
, μ
2
=0,5 and μ=0,2-1,8.
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8
μ
THR, WIPsystem
THR
WIPsystem
7 CONCLUSIONS AND FURTHER RESEARCH
In this work, two non identical reliable suppliers feed a DC with a shared buffer. The DC has a reliable
machine that proceeds the material. The system was analyzed as a continuous time Markov process with discrete
states. The structure of the transition matrix was investigated and a computational algorithm is developed to
generate stationary distribution for two suppliers (K=2), buffer capacity equal to one (B=1) and one machine in
DC (N=1). The input rates (active processing time + replenishment times) are non identically random variables
erlangian distributed. The algorithm is used to explore the influence of each or combination of parameters on
system performance measures (Throughput and WIP
system
).
The main findings of this work may be summarized as follows:
• As the number of phases (P
s
and P
DC
) increases in both replenishment times and service times so
throughput and WIP
system
also increase in a diminished rate in a balanced merge system.
• As the number of phases (P
s
and P
DC
) increases in both replenishment times and service times
throughput’s increment is more intense.
• Regression Analysis describes a strictly linear relationship between throughput and WIP
system
in a
balanced merge system.
• As the processing rates μ
1,
μ
2
are growing, throughput is growing too. When μ
1
end μ
2
reach the value
0,8, Throughput catches a slow growing rate. WIP
system
follows a high growing rate for all values of μ
1
, μ
2
from
0,2-0,8
• As DC rate μ increases, Throughput increases therefore, the productivity of the systems increases too.
On the other hand, as μ value increases, WIP
system
declines.
As possible areas for further research are suggested:
• The system may be extended according to the number of suppliers, the buffer capacity and number of
machines in DC.
• Suppliers may have different capacities (more than one machine)
Michael Vιdalis, Stelios Koukoumialos, Despoina Ntio and Georgios Varlas
54
• Active processing time + replenishment time may follow another phase type distribution
• Suppliers may be unreliable
• Machines at DC may be unreliable
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