Transcript 4th AEGEAN INTERNATIONAL CONFERENCE ON “ANALYSIS OF

FIFTH INTERNATIONAL CONFERENCE ON
“ANALYSIS OF MANUFACTURING SYSTEMS-PRODUCTION MANAGEMENT”
MAY 20-25 2005, ZAKYNTHOS ISLAND, GREECE
“On the optimization of Server allocation in large reliable production lines
with exponential processing times”
A. C. Diamantidis ([email protected])
UNIVERSITY OF THE AEGEAN
DEPARTMENT OF PRODUCT AND SYSTEMS DESIGN ENGINEERING
SYROS ISLAND, GREECE
C. T. Papadopoulos ([email protected])
ARISTOTLE UNIVERSITY OF THESSALONIKI
DEPARTMENT OF ECONOMICS
THESSALONIKI, GREECE
PRESENTATION OUTLINE
• PROBLEM DEFINITION
• ASSUMPTIONS OF THE MODEL
• THE PERFORMANCE EVALUATION TECHNIQUE
• THE OPTIMIZATION TECHNIQUE
• NUMERICAL RESULTS FOR THE SERVER
ALLOCATION PROBLEM
• FINDINGS
• FURTHER RESEARCH
• REFERENCES
2
PROBLEM DEFINITION
• This study examines a constrained optimisation
problem in designing large production lines with
reliable multiple identical parallel machine
workstations.
• The problem is how to allocate a total number of
servers among all workstations in order to
maximize the throughput of the production line.
This problem is known as the sever allocation
problem (SAP).
3
A production line with reliable multiple identical
parallel machine workstations
Figure 1: A flow line with N parallel-machine workstations and N-1
intermediate buffers.
4
ASSUMPTIONS OF THE MODEL
1.
Each workstation Mi in figure 1, consists of Si reliable
and identical parallel machines. At each workstation,
each one of the Si parallel machines has exponentially
distributed service time with mean 1/μi, i=1,…,N.
2.
The parallel machines of different work stations are not
necessarily identical, viz., station processing times are
assumed to be exponentially distributed with non
identical mean service rates.
3.
It is also assumed that when any one of the Si parallel
machines at workstation Mi completes a part, that part is
placed in the buffer Bi downstream of the workstation
immediately, provided the buffer is not full.
5
LITERATURE REVIEW
• There is a relatively scarce literature concerning the
buffer allocation and server allocation problem for flow
lines with multiple parallel-machine workstations.
• Magazine and Stecke (1996) considered small flow lines
with two and three workstations consisting of parallel
machines. They examined how the throughput of such
systems may be improved if specific parameters of the
system such as the allocation of machines among the
workstations, allocation of workload to the workstations
and buffer allocation between workstations vary.
Considering the (SAP) for lines with three workstation
they found that the middle station needs to be more
efficient, so it takes the largest number of the available
servers.
6
• Hillier and So (1989) examined how the assignment of extra servers to
small (up to eight workstations) production lines with small or no
buffers maximizes the throughput of such systems. With a fixed
number of extra servers over an equal allocation to all stations they
focused on the question of where to place these extra servers in order
to maximize throughput . The main conclusion was that the interior
stations (especially the center stations) should be given priority over
the end stations for allocating an extra server. They also presented
numerical results for Erlang, exponential and two-stage Coxian service
time distributions.
• Hillier and So (1995) considered tandem queueing systems that could
be formulated as a continuous time Markov-chain. They considered
various constrained optimization problems such as buffer allocation
problem (BAP), SAP and the workload allocation problem (WAP).
Each decision variable was examined either in isolation or
simultaneously with each one of the other two variables. They
considered production lines where the maximum number of
workstations is up to eight and the buffer capacities between the
workstations are one or zero. For the SAP their main finding was that
the extra servers (above a uniform allocation) should go to the centre
workstations rather than the first or last workstation.
7
• Hillier and So (1996) studied the problem of the simultaneous
optimization of server and work allocation of small serial
production lines. The most important finding of their study is
the L-phenomenon according to which every station receives
just a single server except for one of the two end stations which
receives all the extra servers.
• The optimal allocation of parallel servers with different nonexponential service time distributions at each workstation was
considered by Futamura (2000). The effect of the coefficient of
variation (cv) of the service time distribution on the throughput
of systems, where cv varies from one workstation to another,
was examined
8
EVALUATION AND OPTIMIZATION TECHNIQUES
PERFORMANCE EVALUATION
TECHNIQUES
1) MARKOV CHAIN
2) DECOMPOSITION
3) SIMULATION
4) PETRI NET
5) OTHER METHODS
OPTIMIZATION
TECHNIQUES
(GENERATIVE)
1) ENUMERATION
2) HEURISTIC
3) SIMULATED ANNEALING
4) GENETIC ALGORITHMS
5) TABU SEARCH
6) OTHER METHODS
9
PERFORMANCE EVALUATION TECHNIQUE:
THE DECOMPOSITION METHOD
• The use of Markovian analysis for the numerical computation
of the performance measures of large systems like the one
depicted in Figure 1 is almost impossible due to the enormous
resulting state space.
• In this study, the classical decomposition method, proposed by
Gershwin (1987) and Dallery et al (1988) was applied in order
to evaluate the performance measures of production lines like
the one depicted in figure 1. The decomposition equations
were modified to take into account the existence of parallel
machines at each workstation.
10
• Ancelin and Semery (1987) described a method that replaces each parallelmachine workstation by an equivalent single machine workstation. The
processing rate of the equivalent workstation equals the sum of the
processing rates of all parallel machines in the workstation.
• Burman (1995) presented a method that replaces each parallel server
workstation by a single equivalent workstation for the case of continuous
flow of material. He assumes that the equivalent workstation has a
maximum processing rate which equals the sum of the processing rates of
the parallel machines.
• Patchong and Willaeys (2001) presented a technique that replaces each
parallel machine workstation by an equivalent single machine workstation
for the case of serial production lines. The sets of equations that are
necessary for this replacement are derived.
• The difference of the proposed approach and those of Ancelin and Semery
(1987), Burman (1995) and Patchong and Willaeys (2001), is that each
parallel-machine workstation is not replaced by an equivalent workstation.
That is, the decomposition approach is applied directly to each one of the
parallel machines for each workstation without using replacement
11
techniques.
SOLUTION OF LARGE SYSTEMS WITH PARALLEL
MACHINES AT EACH WORKSTATION
• In order to develop the decomposition method, for each buffer
Bi between two workstations Mi and Mi+1 with Si and Si+1parallel
machines respectively , a virtual upstream workstation Mu(i) that
represents the flow of material into this buffer has to be introduced.
• Similarly a virtual downstream workstation Md(i) that represents
the flow of material out of buffer has to be introduced.
• Workstation Mu(i) consists of Si virtual machines while workstation
Md(i) consists of Si+1virtual downstream machines.
• The service times of the Si parallel machines of pseudo workstation
Mu(i) are exponentially distributed with mean 1/μu(i), while the
service times of the Si+1 parallel machines of pseudo workstation
Md(i) are also exponentially distributed with mean 1/μd(i),
i=1,…,N-1.
12
See figure2
THE SETS OF THE DECOMPOSITION EQUATIONS
• Since, in our model, all Si parallel machines of each workstation
Mi in the real line L, are reliable, only two sets of decomposition
equations are derived. These are: the conservation of flow
equations and the flow rate idle time equations.
• CONSERVATION OF FLOW EQUATIONS
The fact that the flow is conserved because there is no mechanism
for the creation or destruction of material, leads to the
conservation of flow equations.
• FLOW RATE IDLE TIME EQUATIONS
These sets of equations are used to calculate the parameters μu(i)
and μd(i-1) , i=2,…,N-1.
13
See the decomposition equations
• An algorithm that simultaneously solves all the
derived sets of the decomposition equations in
order to evaluate the performance measures of
large
flow
lines
with
parallel-machine
workstations was developed.
• The numerical results indicate that the
decomposition algorithm is very accurate. The
average percentage error of the throughput
obtained from the proposed decomposition
algorithm and simulation (in Arena) for lines with
up to 100 stations is less than 1.2%, whereas the
results for lines with up to 1000 stations indicate
that the percentage error is less than 2.5%.
14
SIMULATED ANNEALING
• Simulated annealing is an adaptation of the
simulation of physical thermodynamic
annealing principles described by Metropolis
et al. (1953) to the combinatorial optimization
problems (Kirkpatrick et al. 1983, Cerny
1985).
• The simulated annealing method starts with a
non-optimal initial configuration (which may
be chosen at random) and works on
improving it by selecting a new configuration
using a suitable mechanism (at random in the
simulated annealing case) and calculating the
corresponding cost differential (ΔR). If the cost
15
is reduced, then the new configuration is
• Unfortunately, such methods can become
`trapped’ in a local optimum that is far from
the global optimum. Simulated annealing
avoids this problem by allowing`uphill’ moves
based on a model of the annealing process in
the physical world.
16
NUMERICAL RESULTS FOR THE SERVER
ALLOCATION PROBLEM
• For all cases examined in this study the unit of
time is normalized by setting the average of the μj,
j=1,…,N, values equal to 1.
• All the buffer capacities qi, i=1,…,N-1 are
assumed equal to zero. The throughput of the
system is denoted PR(s).
• The optimal server allocation among the N
workstations is denoted by the vector s= (s1,…,sN).
• The parameters of the numerical experiments are
summarized in the following two tables
17
Parameter
Range
N even
6,8,..,30(2)
36,40,46,56 and 50,…,100(10)
S
Varies from 7 to 101
qi
0 for all i=1,…,N-1
E
1
μj
1 for all j=1,…,N
Parameter
Range
N odd
7,9, 15,…,29(2) 35,…,55(10)
41,…61(10)
S
Varies from 8 to 62
qi
0 for all i=1,…,N-1
E
1
μj
1 for all j=1,…,N
18
• Similarly to Hillier and So (1989) we define n and E
as follows:
n is the smallest integer that is less than or equal to
S/N and
E=S-n N
i.e., the feasible values of E are E=0,1,…,N-1.
Quantity E actually represents the number of “extra”
servers that are available for allocation beyond a
uniform allocation of n servers to each station.
19
See the numerical results
FINDINGS
• The numerical experiments have pointed out that
PR(s1,s2,…,sN)PR(sN,sN-1,…,s1).
• Considering the results given in Table 1, for E=1 and N
even, the optimal solution assigns the one extra server to
one of the two center stations. The only difference between
this design rule and the one presented by Hillier and So
(1989) is that the optimal solution is not always the one
that assigns the extra server to the back center station.
• Considering the results given in Table 2 for E=1 and N odd
the throughput is maximized by assigning the one extra
server to the center station. This design rule is identical to
the one presented in Hillier and So (1989) for small
production lines.
20
CONCLUSIONS
• In this study the server allocation problem for large
production lines with multiple parallel machine
workstations was examined.
• An extension of the original decomposition method that is
applicable for large production lines with multiple parallel
machine workstations developed by Diamantidis,
Papadopoulos and Heavey (2005) was used as evaluative
technique, whereas simulated annealing was used as
generative technique.
• The design rules presented by Hillier and So (1989) for
small lines (up to eight stations) with multiple parallel
machine workstations are extended to large (up to 100
stations) production lines of the same type.
21
FURTHER RESEARCH
• The authors are currently working on the
buffer allocation problem and the
simultaneous optimization of the server and
buffer allocation for large production lines
with multiple parallel machine
workstations.
22
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
Ancelin, B. and Semery, A. (1987), ``Calcul de la
productivité
d'une ligneintégrée de fabrication: CALIF, une méthode analytique
industrielle’’, RAIRO APII, 21(3), 209--238.
Burman, Mitchell H. (1995), “New Results in Flow Line
Analysis”,
Ph.D.
Thesis,
Operations
Research
Center,
Massachusetts Institute of Technology, USA.
Cerny,V., 1985, Thermodynamical approach to the traveling
salesman problem: an efficient simulation algorithm. Journal of
Optimization Theory and Applications, 45, 41-51.
Dallery, Y., David, R. and Xie, X. (1988), “An
efficient
algorithm for the analysis of transfer lines with unreliable
machines and finite buffers”, IIE Transactions, 20(3), 280– 283.
Futamura, K. (2000), “The multiple server effect: Optimal
allocation of servers to stations with different service – time
distributions in tandem queuing networks”, Annals of Operations
Research, 93, 71–90.
Gershwin, S.B. (1987), “An efficient decomposition algorithm for the
approximate evaluation of tandem queues with finite storage space
and blocking”, Operations Research, 35, 291–305.
23
[7]
[8]
[9]
[10]
[11]
[12]
[13]
Hillier, Frederick S. and So, Kut C. (1989), “The Assignment of
Extra Servers to Stations in Tandem Queueing Systems with Small or
No Buffer”, Performance Evaluation, 10, 219-231.
Hillier, Frederick.S.and So, Kut.C (1995), “On the optimal design
of tandem queueing systems with finite buffers”, Queueing Systems
21, 245-266.
Hillier, Frederick S. and So, Kut C. (1996), “On the simultaneous
optimization of server and work allocations in production line systems
with variable processing times”, Operations Research, Vol. 44, No. 3,
435-443.
Kirkpatrick, S., Jr.,Gelatt,C.D., and Vecchi,M. P., 1983,
Optimization by simulated annealing. Science, 220, 671- 679.
Magazine, M.J. and Stecke, K.E. (1996), “Throughput for
production lines with serial workstations and parallel service
facilities”, Performance Evaluation, 25, 211–232.
Metropolis,N.,Rosenbluth,A.N.,Rosenbluth,M.N.,Teller,A.H., and
Teller,E.,1953, Equation of state calculation by fast computing
machines. Journal of Chemical Physics, 21, 1087- 1092.
Patchong, A. and Willaeys, D. (2001), “Modelling and analysis
of an unreliable flow line composed parallel – machine stages”,
IIE Transactions, 33, 559–568.
24