ASSESSMENT OF A COMPLEX MACHINE MIX PROBLEM
BY INTEGRATED SIMULATION AND AHP MODELING
A. Azadeh*, M. Haghnevis and Y. Khodadadegan
Department of Industrial Engineering and Department of Engineering Optimization
Faculty of Engineering, University of Tehran
P.O. Box 11365-4563, Tehran, Iran
Phone: +9821-8021067 Fax: +9821-8013102
[email protected] or [email protected]
ABSTRACT
The objective of this study is to introduce an integrated simulation AHP model for modeling,
assessment and improvement of a machine mix problem with complex queue priorities and service
times. Furthermore, there is no closed form expression for such situations, as previous studies show
no mathematical models for machine mix with integrated service and queue priorities. The production
lines of a powder coating manufacturing system with three parallel machines and queue and service
time priorities was considered. The machines produce eight different spray coatings. Production
operation starts as soon as an order is received. Spray coatings numbers 1 and 2 are handled by
machine 1, spray coatings numbers 3 to 6 are handled by machine 2 and spray coatings numbers 7
and 8 are handled by machine 3. Priority is with the arriving entity (new order) which has identical
number as the entity being serviced. Also, if the arriving entity has higher number than entity being
serviced and there is no identical order as the entity being serviced in the queue, then the arriving
entity occupies the first place in the queue. Service time priority means that a machine is only allowed
to service identical products (spray coatings) up to a limited period that is defined as ta. For example,
if machine 2 has been servicing product number 6 for the last two hours due to high demand, the
remaining allotted time if required would be ta - 2. In addition, the services turn over moves clockwise.
A new operation is initiated according to queue priority after completion of the operation for product
number 6, which would be greater than or equal to ta. The reader should note that closed form
expression for such production systems is impossible due to effects of consolidated service time
priority and queue priority. The unique feature of this study is integrated simulation AHP modeling of
a production system with consolidated service time and queue priorities. The simulation approach of
this study for such complex settings may be applied for other similar production systems.
KEY WORDS: Queuing, complex, service, priority, simulation, AHP
1. Introduction
Queuing systems require controlling and maintaining a list of events to determine next occurrences.
This list is representative of various event times related to each entity in the system. Most of queuing
problems can be modeled and solved by closed form mathematical methods (Gross & Harris, 1984;
Kleinrock, 1975; White, Schmidt and Bennett, 1975; Ross, 1983; Jaiswal, 1968) but in few cases
because of severe complexity and variety of constraints does not have closed form expressions.
Therefore, computer simulation approach can be used in these rare cases (Shannon, 1975).
Priority queues have been studied since the early days of queuing theory Stidham (2002). The
monograph on queues by Cox and Smith (1961) gives a concise summary of the early work. The book
by Jaiswal (1986) provides a compendium of known result as of the late `60s. Most of the research
until then concerned single-server queues with fixed priorities. Operating under preemptive or
nonpreemptive disciplines.
Various studies have been presented about priority queues. Mean delays in standalone M=M=m
systems with preemptive or nonpreemptive priorities have been studied by Cobham (1954), Mitrani
*
Corresponding Author
and King (1981), and Buzen and Bondi (1983). Schaack and Larson (1986) derive explicit
expressions for the Laplace transforms of the delays in an M=M=m system with identical servers for
the significantly different case of nonpreemptive cuto9 priorities. Infinite buffer priority queues with
Poisson arrivals are considered in detail in Jaiswal. (1968). In Khamisy (1992) & Takine (1996),
head-of-line priority (HOL) queues have been analyzed with a wide variety of arrival and service time
distributions. Seal (1995) demonstrates the application of spreadsheets in simulation queuing systems
with arrivals from a finite population. Wagner (1995) considers a continuous queue with Markov
modulated arrivals and infinite buffers and obtains the moment generating function of the waiting
times for both the priority classes. Huarng & Lee (1996) explains how a computer simulation model
was developed to study how changes in the appointment system, staffing policies and service units
would affect the observed bottleneck. Leavens & Bruneel (1998) analyze the performance of discretetime multiserver queues that use a priority structure in the scheduling of departures. Such systems are
often encountered in the context of ATM and in voice-data multiplexing. At many instances, discretetime multiserver queues have been successfully used in the performance evaluation of computer and
communication systems in various contexts, such as TDM Chang & Harn (1992) and Chu (1970)
voice-data integration Li & Mark (1988), deflection routing Bignell (1990) and more recently, ATMswitching technology Bruneel & Kim (1993) and Bruneel et.al (1994). A lot of applications have in
common that, in order to achieve certain quality-of-service standards, some cells (or messages,
customers, etc.) should get preferential treatment over others. Kim et.al. (1999) study analyzes the
admission-and-discharge processes of one particular ICU, that of a public hospital in Hong Kong, by
using queuing and computer simulation models built with actual data from the ICU. The results
provide insights into the operations management issues of an ICU facility to help improve both the
unit's capacity utilization and the quality of care provided to its patients. Boucherie (2000) consider an
extension of the model in Nunez-Queija et.al. (1999) assuming that the high-priority customers can be
buffered, and propose a numerical approach to calculate the performance parameters of interest.
Nunez-Queija (2000) considers an M/M/1-PS model with an ON/OFF server, and derives closed-form
expressions for several sojourn-time statistics. In particular, a closed-form expression for the limiting
sojourn-time distribution under heavy traffic assumptions was developed. Nunez-Queija et al. (1999)
consider a multi-server model with two priority classes, where the high-priority customers may be
blocked when all servers are busy, whereas the low-priority customers utilize the remaining service
capacity in a PS fashion.
Steady state simulation results illustrate the use of the model for a "what if" optimization approach to
the berth planning problem. Dupon et al (2002) basic model consists of five machines, each preceded
by a buffer with infinite capacity. In a first part, they rely on the first in, first out (FIFO) priority rule,
common in queuing theory. The results of this study indicate that in this setting sequence changes
have no significant impact on the average product lead time but do increase the variance of the lead
time and thus influence customer service. Walraevens et al (2002) consider a discrete-time queuing
system with head-of-line priority. They give some general results on a GI-1-1 queue with priority
scheduling and effect of head-of-line (HOL) (or non-preemptive) delay-priority scheduling is
considered. Van der mei et al (2003) study mean sojourn times in a multi-server processor sharing
system with two priority classes and with general service-time distributions. For high-priority
customers, the mean sojourn time follows directly from classical results on symmetric queues. For
low-priority customers, in the absence of exact results, they propose a simple and explicit
approximation for the mean sojourn time. Tham et.al. (2002) extend the average delay analysis of the
probabilistic priority (pp) scheduling discipline proposed to the multi-class. Juiz & Puigjaner (2002)
propose new analytical approximations for basic and priority pools. Basic pool modeling is based on
semaphore queues whereas priority pool modeling is inspired in classical non-preemptive priority
queues. Afeche (2003) evaluates the delay performance of an open multi-class stochastic processing
network of multi-server resources with Preemptive-resume priority service. This analysis includes a
derivation of these explicit expressions for an isolated M=M=m system with identical servers.
This study considers an actual queuing system of a production system with complex service times and
queue priorities. This is due to attributes and limitations of the actual production system. A computer
simulation methodology is used to model the limitations and complexities since a closed form
expression could not be developed. The sequence of the queuing system is determined with regard to
the entity that has just received service. Furthermore, the priority is with entities which have the same
order as the entity being serviced. Otherwise, if the waiting entities have different orders than the
entity being serviced, the priority is with the entity with the highest rank. The arriving entities (orders)
are ranked numerically from one to eight depending on the color of spray coating (order) required. To
prevent severe waiting times of entities, the maximum time allotted by the management for a
particular entity (order) is defined as ta for a = 1,...,8 representing the eight spray coatings.
Furthermore, if ta is reached, the entity (order) with higher ranking is prepared for the proper machine
immediately after service completion.
The above phenomena are used as production planning and scheduling of a spray coatings
manufacturer. The manufacturer is capable of producing eight different spray coatings (from faint
color to dark color) with three parallel machines. The faintest and darkest colors are ranked one and
eight, respectively. There is a set-up time if a new order must be started for any of the machines.
Hence, there is no set-up time as long as the same order is prepared by any of the parallel machines.
The objective of this study is to develop a simulation model for the complex mix queuing system
described in this paper. The simulation model is then used as production planning and scheduling tool
to identify the most optimum alternative (lowest set-up and waiting times).
2. The Queuing System
The system being studied is the production lines of a powder coating manufacturing system. It has
three parallel machines with complex mix queuing and service disciplines. The machines produce
eight different spray coatings (number 1 to 8). The schematic diagram of the queuing effects of the
powder coating manufacturing is shown in Figure1. Production operation starts as soon as an order is
received. Spray coatings numbers 1 and 2 are handled by machine 1, spray coatings numbers 3 to 6
are handled by machine 2 and spray coatings numbers 7 and 8 are handled by machine 3. Priority is
with the arriving entity (new order) which has identical number as the entity being serviced. Also, if
the arriving entity has higher number than entity being serviced and there is no identical order as the
entity being serviced in the queue, then the arriving entity occupies the first place in the queue. For
example, if machine 2 is servicing product number 5 and product numbers 4, 4 and 3 are waiting in the
queue, respectively and order number 6 is received, order number 6 occupies the first place (priority)
in the queue and queue priority would be 6, 4, 4, 3. Otherwise, FIFO discipline is used for as long as
arriving order has lower number than the entity being serviced. If the order of service changes the
queue discipline would be changed according to above format. Service time priority means that a
machine is only allowed to service identical products (spray coatings) up to a limited period that is
defined as ta. For example, if machine 2 has been servicing product number 6 for the last two hours
due to high demand, the remaining allotted time if required would be ta - 2. In addition, the services
turn over moves clockwise. A new operation is initiated according to queue priority after completion
of the operation for product number 6, which would be greater than or equal to ta.
The reader should note that closed form expression for such production systems is impossible due to
effects of consolidated service time priority and queue priority. Furthermore, there is no evidence of
such mathematical modeling developments in previous literatures. The unique feature of this study is
simulation modeling of a production system with consolidated service time and queue priorities. The
simulation approach of this study for such complex settings may be easily extended to other similar
production systems. Next sections describe the simulation approach of this study together with
verification and validation of the complex queuing model.
Complex priorities
i
Order type i where
i = 1…8
Arrival
of
orders
2
2
4
4
7
6
3
7
1
8
1
6
8
5
8
Higher order priority discipline
FIFO discipline
Limited buffer size (15)
1
Machine
a
1
5
Machine
b
5
8
Machine
c
8
Customer
Service time limitation discipline (ta)
Figure 1: The schematic diagram of the powder coating manufacturing
3. The Simulation Model
The simulation model was developed by Visual SLAM (Pritsker, Oreilly & LaVal, 1997; Pritsker,
1990; Pritsker, Sigal, & Hammesfahr, 1989). The network model of one the parallel machines are
shown in Figure 2. It begins with CREATE node "a" which models the time between receiving of
orders from management and then an entity travels to GOON node "a-a". This would avoid looping
error. If the priority of a new entity (order) is greater than or equal to entity which is being serviced
(ATRIB[8] >= XX[4]) the entity goes to ASSIGN node for placement of production time in XX[10].
This is done to avoid error in the next stages. Then, the entity is sent to AWAIT node (file 25) which
is basically the buffer of production with the capacity of 15 entities. The entities are balked to
COLLECT node q1 to collect the number of balked entities and for the purpose of production planning
if the buffer capacity is exceeded. The duration of production activity is identified as " Fa" (activity
number 15). In addition, the entity is saved in LTRIB[1] of AWAIT node to be freed later by FREE
node at the end of the production process. Next, XX[4] is used to save color coating of the in-process
order and XX[7] is used to save the color coating of the manufactured orders.
If (ATRIB[8] < XX[4]) i.e., the priority of the incoming order is smaller than the order being
processed, then the incoming orders is placed at the end of the queue because there are only two types
of orders (1 and 2) processed by machine a.
After production activity, the entity (completed order) is transferred to GOON node labeled as a1. If
the completed order has the same rank as the previous order (XX[7] = XX[8]), the production time of
a particular order (XX[9]) is set to XX[9] + XX[10] where XX[10] is the production time of present
order (order being processed). Then, the entity is transferred to the GOON node goon a. Otherwise, if
XX[7] is not equal to XX[8] then XX[8] = XX [7] and the production time of a particular order XX [9]
is set to its pre-assigned production time XX[10]. The machine needs to be cleaned at this stage with
the time TC (activity number 16). The entity is then routed to GOON node labeled as goon a.
Total production time of (XX[9]) a particular coating is controlled next. Furthermore, if a particular
order is continuously produced because it has higher priority than other orders, its total production
time may not exceed 600 minutes. If XX[9] is les than or equal to 600 minutes then the machine is
freed at the FREE node labeled as KIA with LTRIB[0] and hence production process of existing order
is ended and a new order is initiated. If XX[9] > 600 the production priority will be given to the next
prioritized order or XX[4] = XX[4] + 1. If the order is changed, the machine needs to be cleaned
which takes TC minutes. Then, a FINDER node is used to remove all entities in file 25 (AWAIT
node) to be re-ordered for prioritization in the queue. The entities are routed to GOON node labeled
as aa.
Figure 2: Visual SLAM network of machine a
Production behavior of machines b and c are similar as machine a with the following exceptions:
1234567-
XX[5] and XX[6] are used as variables defining color priority, respectively.
Color codes are identified with XX[11] and XX[15] instead of XX[7]
Previous color codes are identified with XX[16] and XX[12] instead of XX[8]
XX[13] and XX[17] have been placed for period making of a proper code instead of XX[9]
The substitution of XX[10] is both XX[14] and XX[18] as production's time of "Fb" and "Fc".
ATRIB[8] is respectively defined as 4 and 2 units for machine b and c.
LTRIB[2] and LTIRB[3] are used respectively, instead of LTRIB[1].
4. Verification and Validation
The movements of entities in the simulation were traced with MONTOR statement. It was then
compared with the movement of orders in the actual production system and hence it is verified
that the simulation has the same behavior as the actual system. To validate the simulation model
against the actual system the most important performance measure of the production system which
is number of completed orders in a two weeks period was selected and verified by the
management. The simulation was ran for two working weeks and replicated twenty times. The
results of the twenty runs were then compared with 20 random samples of the actual system.
Independent t-test has been used to compare the production system with simulation model with
respect to total number of completed orders. The null hypothesis tested H0: µ1 = µ2 at α = 0.01
level of significance. It is concluded that with respect to total number of completed orders the
systems and simulation model have the same performance (Table 1). In addition, the equality of
variance was tested prior to the t-test and the null hypothesis of H0: σ12= σ22 was accepted at α =
0.05 (Montgomery, 1991; Ross, 1970). Table 1 shows a P-value of 0.297, which is considerably
an attractive result of t-test.
Table 1: The results of independent t-test for simulation and production system
No. of completed
orders in two weeks
Alternative
Mean
Standard
deviation
33.55
5.33
Simulation
6.72
Actual system 31.15
to
P-value
1.07
0.297
5. Conclusion
A simulation model was developed for an actual production system with complex mix priorities and
service disciplines. The simulation model has the following features:
• The bottlenecks and sensitive points of the system are identified.
• Sensitivity analysis could be easily performed.
• Customer satisfaction may be enhanced due to introduction of an optimum model with lowest
set-up and waiting times.
The verified and validated simulation model was used as production planning and scheduling tool to
identify the optimum alternatives (lowest set-up and waiting times). Table 2 illustrates the results of
six distinct production planning and scheduling alternatives tested by the simulation model.
Furthermore, each alternative shows percent improvement increase with respect to the existing
production system. A complete description of the alternatives in particular and the paper in general
will be given in an extended article in the near future.
Performance
measure
Table 2: The results of production planning alternatives by the simulation model
a
b
c
d
e
f
1
25.62
8.82
42.99
1.38
46.99
66.34
2
27.02
29.1
54.91
8.43
58.53
50.84
3
8.27
3.6
11.64
2.26
8.92
11.27
4
13.55
2.16
16.92
5.99
15.47
27.54
5
5.57
1.73
16.86
2.39
18.39
15.97
6
26.6
14.68
21.02
18.39
25.22
19.92
Considered conditions (alternatives)
In summary, a comparative study is performed by Analytic Hierarchical Process (AHP) shown in
Table 3. AHP developed by Saaty (1977) has been applied in more than 20 diverse areas to rank,
select, evaluate, and benchmark decision alternatives. In the AHP, the decision maker models a
problem as hierarchy of criteria, sub criteria, and alternatives. After the hierarchy is constricted, the
decision maker assesses the importance of each element at each level of the hierarchy. This is
accomplished by generating entries in a pair wise comparison matrix where elements are compared to
each other. For each pair wise comparison matrix, the decision maker typically uses the eigenvector
method (EM) to generate a priority vector hat gives the estimated, relative weights of the elements at
each level of hierarchy. Weights across various levels of the hierarchy are then aggregated using the
principle of hierarchic composition to produce a weight for each alternative (Chandrana et al, 2005).
Table 3: AHP rankings of alternatives
AHP score
AHP rank
a
b
C
d
e
14.67
7.33
19.47
6.74
20.17
3
4
2
5
1
The results of AHP analysis show that alternatives are divided into two categories with respect to
performance: high performance alternatives a, c and e and low performance alternatives b and d. The
reader should note there is large gap between AHP scores of high and low performance alternatives.
By noting value performance = (worth)/ (cost) in context of value engineering (Parker 1998
and Shellito 1992), value performance of each alternative is shown in Table 4. Low
performance alternatives are discarded from further considerations and the results of value analyze
analysis shows the superiority of alternative "c" over all other alternatives.
Table 4: Value analysis of alternatives
High performance
Parametric Cost
Value performance
Value rank
Low performance
a
c
e
b
d
6α
2.446
2
7α
10α
α
3α
2.781
1
2.017
3
7.333
1
2.247
2
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