CC BY
10
+ 0.372 • [1.75]2] - [1.89]2 = [1.97]2, c(3) @ 1.97, m(3) = p1 m (1)(4) + p 2 m (2)(4) + p3 m (3)(4)
+ p 4 m(4) (4) + p5 m (5)(4) + p6 m (6)(4),
@ 0.145 • 2.27 + 0.098 • 2.27 + 0.004 • 1.49 + 0.007 • 1.49 + 0.374 • 1.43 + 0.372 • 1.49 = 1.66,
[c (4)]2 @ 2[0.145 • [2.27]2 + 0.098 • [2.27]2
+ 0.004 • [1.49]2 + 0.007 • [1.49]2 + 0.374 • [1.43]2
+ 0.372 • [1.49]2] - [1.66]2 = [1.73]2, c (4) @ 1.3.
The mean values of the system lifetimes in the particular safety states, by (34), are
m(1) = m(1) - m(2) = 0.44,
m(2) = m(2) - m(3) = 0.33,
m(3) = m(3) - m(4) = 0.23,
m (4) = m (4) = 1.66.
If the critical safety state is r = 2, then the system risk function, according to (6), is given by
R(t) = 1 - ^(¿,2)
= 0.145 • exp[-0.31t] + 0.098 • exp[-0.31t]
+ 0.004 • exp[-0.49t] + 0.007 • exp[-0.49t]
+ 0.3.74 • exp[-0.55t] + 0.372 • exp[-0.51t] for t > 0.
Hence, the moment when the system risk function exceeds a permitted level, for instance 5 = 0.05, from (7), is
t = r_1(5) @ 0.11 years.
9. Conclusion
In the paper the multi-state approach to the safety analysis and evaluation of systems related to their variable operation processes has been considered. Theoretical definitions and preliminary results have been illustrated by the example of their application in the safety evaluation of a ship transportation system with changing in time its operation states. The ship safety structure and its safety subsystems characteristics are changing in different states what makes the analysis more complicated but also more precise than the analysis performed in [2]. However, the varying in time ship safety structure used in the application is very general and simplified and the subsystems safety data are either not precise or not real and therefore the results may only be considered as an illustration of the proposed methods possibilities of applications in ship safety analysis. Anyway, the obtained evaluation may be a very useful example in simple and quick ship system safety characteristics evaluation, especially during the design and when planning and improving her operation processes safety and effectiveness.
The results presented in the paper suggest that it seems reasonable to continue the investigations focusing on the methods of safety analysis for other more complex multi-state systems and the methods of safety evaluation related to the multi-state systems in variable operation processes [9], [10] and their applications to the ship transportation systems [5].
References
[1] Aven, T. (1985). Reliability evaluation of multistate systems with multi-state components. IEEE Transactions on Reliability 34, 473-479.
[2] Dziula, P., Jurdzinski, M., Kolowrocki, K. & Soszynska, J. (2007). On multi-state approach to ship systems safety analysis. Proc. 12th International Congress of the International Maritime Association of the Mediterranean, IMAM 2007. A. A. Balkema Publishers: Leiden -London - New York - Philadelphia - Singapore.
[3] Grabski, F. (2002). Semi-Markov Models of Systems Reliability and Operations. Warsaw: Systems Research Institute, Polish Academy of Science.
[4] Hudson, J. & Kapur, K. (1985). Reliability bounds for multi-state systems with multi-state components. Operations Research 33, 735- 744.
[5] Jurdzinski, M., Kolowrocki, K. & Dziula, P. (2006). Modelling maritime transportation systems and processes. Report 335/DS/2006. Gdynia Maritime University.
[6] Kolowrocki, K. (2004). Reliability of large Systems. Elsevier: Amsterdam - Boston - Heidelberg - London - New York - Oxford - Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo.
[7] Lisnianski, A. & Levitin, G. (2003). Multi-state System Reliability. Assessment, Optimisation and Applications. World Scientific Publishing Co., New Jersey, London, Singapore , Hong Kong.
[8] Meng, F. (1993). Component- relevancy and characterisation in multi-state systems. IEEE Transactions on reliability 42, 478-483.
[9] Soszynska, J. (2005). Reliability of large seriesparallel system in variable operation conditions. Proc. European Safety and Reliability Conference, ESREL 2005, 27-30, Tri City, Poland. Advances in Safety and Reliability, Edited by K. Kolowrocki, Volume 2, 1869-1876, A. A. Balkema Publishers: Leiden -London - New York - Philadelphia - Singapore.
[10] Soszynska, J. (2006). Reliability evaluation of a port oil transportation system in variable operation conditions. International Journal of Pressure Vessels and Piping, Vol. 83, Issue 4, 304-310.
[11] Xue, J. & Yang, K. (1995). Dynamic reliability analysis of coherent multi-state systems. IEEE Transactions on Reliability 4, 44, 683-688.
Elleuch Mounir Ben Bacha Habib Masmoudi Faouzi
National school of engineering, Sfax, Tunisia
Improvement of manufacturing cells with unreliable machines
Keywords
manufacturing cell, intercellular transfer, markov chains, availability, simulation, performances Abstract
The performance of cellular manufacturing (CM) is conditioned by disruptive events, such as failure of machines, which randomly occur and penalize the performance of the cells and disturb seriously the smooth working of the factory. To overcome the problems caused by the breakdowns, we develop a solution, based on the principle of virtual cell (VC) and the notion of intercellular transfer that can improve performances of the system. In this context, we use an analytical method based on Markov chains to model the availability of the cell. The found results are validated using simulation. The proposed solution in this paper confirmed that it is possible to reduce the severity of breakdowns in the CM system and improve the performances of the cells through an intercellular transfer. Simulation allowed a validation of the analytical model and showed the contribution of the suggested solution.
1. Introduction
Group technology (GT) is a manufacturing philosophy that has attracted a lot of attention because of its positive impacts in the batch-type production. Cellular manufacturing (CM) is an application of GT to manufacturing. It has emerged in the last two decades as an innovative manufacturing strategy that collects the advantages of both product and process oriented system for a medium-volume and medium-variety production. By applying the GT concept and CM system, manufacturing companies can achieve many benefits including reduced set-up times, reduced work-in-process, less material handling cost, higher throughput rates.
The performance of a cellular manufacturing system is conditioned by disruptive events (e.g., failures of machines) that randomly occur and penalize the performance of the system. Therefore, equipment that falls in breakdown generates eventually the interruption of the whole cell. Consequently, failure of the machine implies total loss of cell capabilities and it leads to the partial deterioration of the performance of the total system [1]. Therefore, the application of an efficient strategy against these perturbations permits to improve the performance of those production systems.
Few researches can be found related to the effect of the failure on the operation of CM system. Some of these discussed the efficient maintenance politics to improve the performance of the cellular manufacturing [1], [4] and [5]. Others developed new coefficients of similarity that consider a number of alternative ways during the machine breakdown [3]. This paper is concerned with problems of the availability of production cells facing random event due to an internal disruption of breakdown-machine type. It uses intercellular transfer as a policy to surmount this type of disruption. The proposed solution is based on the external routing flexibility: the ability to release parts to alternative cell. This policy is assessed through modelling of the production cell and its simulation with Arena software. The remainder of this paper is organized as follows. In section 2 and 3, we formulate a comprehensive idea of intercellular transfer policy and we give the method for modelling the availability of the cell. Section 4 presents a comprehensive simulation model to validate the analytical model and evaluate the policy of intercellular transfer. Finally, we recapitulate in section 5 the main conclusion of this work and we make recommendations for future research.
2. Description of the manufacturing cell
The shop consists of several machines that are grouped into different group technology cells operating in a static environment. Each cell is characterized by a classically structured flow line with (m) machines in series. These machines are unreliable, with operation dependent failures, and have a constant failure and repair rates 1, and m, (i = 1- m). In this section, we introduce the parameters that characterize the behaviour of machines in a material flow model. Each machine Mi is characterized by three parameters:
- Average utilization rate Tu,: This is the rate at which material flows gets processed through the machine Mi in the absence of failure.
- Average failure rate 1,: This is the rate at which machine Mi fails when working at its maximum processing rate (100%).
- Average repair rate m,: This is the rate at which machine Mi gets repaired if it is down for a failure. Using theses parameters Tu,, 1, and m, we can define the basic parameters of an isolated machine Mi:
- There is the average failure rate 1i.Fc: This is the rate at which machine Mi fails when working at its utilization rate Tu,.
l i .Fc = l i ' F c
where
Fc = Tu.
(1)
(2)
- Isolated efficiency A,(<»): that is the average proportion of time during which machine Mi would be operational. It is equal to the steady-state availability. Ai(<x) is defined in terms of parameters as follows:
m ,
Ai (») =
i i.Fc + m i
(3)
If the single machine, whose parameters are defined as above, were part of a production cell, additional parameters would be needed to characterize it. These parameters are given below.
- Average utilization rate Tui : in a production cell environment, a cell allows the manufacturing of family products that made of several types, which forms various batches, having different sequences of operations. We regard Tui.k as the utilisation rate of the machine Mi during the manufacture of the batch k. Therefore to determine the machine utilisation rate, it is enough to calculate its utilisation rate during the
manufacturing of family products (various batches allocated to cell) without taking account of the effect of the breakdowns of the other machines. This rate is given by the following expression:
XTU- .k •ti
Tu i --
with:
(4)
tt i
- ti.k: the time put by the machine (i) to manufacture the batch k,
- tti: the total time to manufacture family products on the machine (i),
- n: number of batches treated in manufacturing cell.
Taken account of the condition which says that the breakdown of a machine generates the interruption of the cell, i.e. that we do not have a breakdown at the same time inside a cell; the utilization rate will be approximated by the following expression:
Tu, - Tu, • E,
(5)
- where E is the efficiency of the cell given by the following expression [2]
E = -
1
i+1
Kf, m,
(6)
- Failure rate l, Fc*: This is the rate at which machine Mi fails when it work in a cell.
l
i .Fc *
l, • Fc
where
Fc* = Tui •-
(7)
(8)
1+X
m
j =1 1 j Fc j *•
with: m represents the number of machine in the cell. Then the stationary availability will be given by the expression (9).
k =1
i-1
1
•4» (¥)=-
1+1
1
(9)
i m
3. Problem definition and solution technique
The failure of only one machine in the cellular manufacturing system can disrupt the product flow in the whole system. Indeed, this failure is going to generate the interruption of different machine in the corresponding cell. It implies a reduction of the machine utilization rate, a reduction of the production capacity and a dissatisfaction of customers. In this study we are interested in a solution based on the external routing flexibility. This solution has the tendency to apply a strategy that permits to reduce the severity of the failure by the application of an intercellular transfer policy in case of breakdown of a machine of the cell. For a production cell treating a type of product, the breakdown of a machine doesn't imply the interruption of the production in this cell. Sometimes the continuity of the production will be assured by the transfer of the product flow toward a neighbouring cell admitting an inactive machine capable to treat this product type. By this action, it will be possible to continue the process of manufacturing in presence of the fault. The cell will be formed by machines of the first cell and the standby machine of the second cell. The creation of this intercellular transfer is the origin of the formation of the virtual cells. Then virtual cells are created periodically, for instance at machine breakdown, depending on the presence of the failure and the standby machine. It is necessary to note that the realization of intercellular transfer can bring advantages at the level of performance of the system taking into account the transfer duration, the inactivity delay of the standby machine and the repair duration. Therefore, for the studied system the strategy consists in applying the intercellular transfer of the cell (a) toward the cell (b) in case of failure of one of machines of the first cell (see Figure 1).
The production cell can be assimilated to a repairable system operating according to a set structure composed of (M) independent modules. The number of modules is equal to the number of machines constituting the studied cell (a). Then the block diagram of cell reliability (a) is given by Figure 2. We study the system in steady state; that is where the probability of the system being in a given state does not depend on the initial conditions. In the case of an application of a transfer policy, the availability of the cell is determined from different module availability. The availability of module (2) will be determined with the process of Markov chains.
L___J
Cell a
Principal cell
Virtual cell
Cell b
Intercellular transfer
Figure 1. Manufacturing system with intercellular transfer
11a, mi
Ml
Module 1
12a, m 2
M2a
Pt2b
^- M2b
5 2a2b
Module 2
13a, m 3
M3a
Module 3
Figure 2. Function block diagram of cell reliability (a)
For this raison, we are considering the module represented by both machines: the main machine and the standby one. The main machine Mi belonging to the studied cell, exhibits breakdown and repair rates 11 and m respectively. The standby machine M2 is characterized respectively by breakdown and repair rates 12 and m2. In case of a breakdown of the main machine, the probability of transfer toward the replacement machine M2 is equal to Pt2 with a transfer rate equal to S12.
The Markov process describing the evolution of the stochastic behaviour of the module is given by the state diagram depicted in Figure 3.
Given a set of differential equations developed from the diagram, we can determine the module availability represented in the following operational states i, 2i, and 22. Therefore, the stationary availability is valued by the expression (10).
A M
¿IMo&de V >
5,2-(m,-m 2-py
-m-2 + 12
)+1, ■ Pt 2 ■ (m, ■ I2 + m, ■ m 2+1, - m 2 + m 2'
1, -12-5,2 ■ Pt2 (m, + 1,)+5,2 -m 2 -(1, -(m 2 + 1,)+12 -(m, + 1,)) + m, ■ m2'1, ■ Pt 2 ' (12 +m2 + m, + 1,)+ m, ' y-2 '5,2 ■ (m, + 2-1, + m 2)0
(10)
1
ll. Pt
1;(1-Pt2j
^^ Operational state (f^j) Transitory state
Not-operational state
8 12
h2 -(mi+ii+ m2+i2)
(11)
m1 +2 -m1 -i1+I12 +m1-m 2+ m2-1
AModule W =
(m 1 -m 2-(m 1
11 +m2+12)+i 1 - Pt 2 -lm 1 -12 + m 1 -m2 + 11 -m2 + m2
%r 12- Pt2 (m 1 + 11)+m 2-(1r (m 2+11)+12-(m 1+11 + m r m 2-(m 1+211 +m 2)
(12)
4. Simulation of manufacturing cells
Figure 3. Diagram of module state
Besides, the expression (10) shows that intercellular transfer in case of failure permits to improve the stationary availability of the module, only in the case where the transfer rate is superior to a limit value (d12*). This value is given by the expression (11).
With the help of a good team of maintenance arranging some necessary logistical means, the time of preparation of an intercellular transfer can be reduced. In these conditions, times of transfer preparation can be disregarded in front of times between failings and repair machine times. Therefore, the expression of the availability of the module will be given by the expression (12).
To conduct our simulation, we defined first the problem and stated our objectives. The problem facing cellular manufacturing is the effect of breakdown machines. The objectives of this simulation were to examine the performance of the system with and without the policy of intercellular transfer in the event of breakdowns and to validate the analytical model. We consider the system shown in the Figure 4. The cell (a) is constituted of three different machines dedicated to the manufacturing of two product types. The cell (b) is formed of four different machines capable to manufacture three product types.
Cell a
Cell b
Figure 4. Manufacturing system with intercellular transfer
Table 1 summarizes the routing and the processing time information of each part
Table 1. The routing and processing times of each part
Product type Batch size Machine (Processing times)
Cell a 1 19 M1(19) ^ M2(12) ^ M3(13)
2 30 M2(10) ^ M3(9)
Cell b 3 16 M1(14) ^ M2(11) ^ M(13)
4 25 M1(16) ^ M3(14) ^ M4(11)
5 20 M2(17) ^ M3(13) ^ M(7)
The mean time to failure (MTTF) and the mean time to repair (MTTR) for all machines are shown in Table 2.
Table 2. MTTF and MTTR of each machine in the system
Machine M1a M2a M3a Mb1 Mb2 Mb3 Mb4
MTTF 6000 3500 3000 5500 4000 3500 6500
MTTR 500 420 350 400 400 360 380
In this paper, the batches arrival is considered cyclic. Indeed, the manufacturing of a new batch is only permitted if the previous batch is finished. In addition, products are generated in a cyclic manner. For a given batch, the time between the arrivals of two products is equal to the time of execution of the first operation.
We perform discrete flow simulation using simulation software called ARENA. We simulate the system during a 12 years horizon; during the first 60000 minutes, statistics are not collected. This warm-up period (the first 60000 minutes) is used to avoid transient effects on the final results.
2
