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M.Elleuch, B.Ben, F.MasmoudiImprovement of manufacturing cells with unreliable machines - RTA # 3-4, 2007, December - Special IssueAcell( )∞ =11λm+∑i=1*i.Fcµi.3. Problem definition and solution technique(9)The failure of only one machine in the cellularmanufacturing system can disrupt the product flow inthe whole system. Indeed, this failure is going togenerate the interruption of different machine in thecorresponding cell. It implies a reduction of themachine utilization rate, a reduction of the productioncapacity and a dissatisfaction of customers. In thisstudy we are interested in a solution based on theexternal routing flexibility. This solution has thetendency to apply a strategy that permits to reduce theseverity of the failure by the application of anintercellular transfer policy in case of breakdown of amachine of the cell. For a production cell treating atype of product, the breakdown of a machine doesn'timply the interruption of the production in this cell.Sometimes the continuity of the production will beassured by the transfer of the product flow toward aneighbouring cell admitting an inactive machinecapable to treat this product type. By this action, it willbe possible to continue the process of manufacturing inpresence of the fault. The cell will be formed bymachines of the first cell and the standby machine ofthe second cell. The creation of this intercellulartransfer is the origin of the formation of the virtualcells. Then virtual cells are created periodically, forinstance at machine breakdown, depending on thepresence of the failure and the standby machine. It isnecessary to note that the realization of intercellulartransfer can bring advantages at the level ofperformance of the system taking into account thetransfer duration, the inactivity delay of the standbymachine and the repair duration.Therefore, for the studied system the strategy consistsin applying the intercellular transfer of the cell (a)toward the cell (b) in case of failure of one of machinesof the first cell (see Figure 1).The production cell can be assimilated to a repairablesystem operating according to a set structure composedof (M) independent modules. The number of modulesis equal to the number of machines constituting thestudied cell (a). Then the block diagram of cellreliability (a) is given by Figure 2.We study the system in steady state; that is where theprobability of the system being in a given state doesnot depend on the initial conditions. In the case of anapplication of a transfer policy, the availability of thecell is determined from different module availability.The availability of module (2) will be determined withthe process of Markov chains.Figure 1. Manufacturing system with intercellulartransferFigure 2. Function block diagram of cell reliability (a)For this raison, we are considering the modulerepresented by both machines: the main machine andthe standby one. The main machine M 1 belonging tothe studied cell, exhibits breakdown and repair rates λ 1and µ 1 respectively. The standby machine M 2 ischaracterized respectively by breakdown and repairrates λ 2 and µ 2 . In case of a breakdown of the mainmachine, the probability of transfer toward thereplacement machine M 2 is equal to Pt 2 with a transferrate equal to δ 12 .The Markov process describing the evolution of thestochastic behaviour of the module is given by the statediagram depicted in Figure 3.Given a set of differential equations developed fromthe diagram, we can determine the module availabilityrepresented in the following operational states 1, 2 1 ,and 2 2 . Therefore, the stationary availability is valuedby the expression (10).AModuleM 1aλ 1a, µ1a( ∞)M 2aM 3aM 2b M 1bCell aPrincipal cellVirtual cellM1aModule 1Pt 2b2( µ ⋅ µ ⋅( µ + λ1+ µ + λ 2) + λ1⋅Pt 2 ⋅ ( µ ⋅λ2+ µ ⋅ µ + λ1⋅µ + µ)δ 12 ⋅1 2 1=⎛λ1⋅λ2 ⋅δ12 ⋅ Pt 2⎜⎝+µ1⋅µ2⋅λ1⋅Pt2λ 2a,µ 2aM2aλ 2b,µ 2bM2bδ2a2bModule 2M 3b1Cell b( µ + ) + ⋅ ⋅( ⋅( + ) + ⋅( + ))1 λ1δ 12 µ2 λ1µ2 λ1λ 2 µ1 λ1⎞⋅( λ + µ + µ + λ ) + µ ⋅ µ ⋅δ⋅( µ + ⋅ λ + µ ) ⎟⎟ 2 2 2 1 1 1 2 12 12 1 2 ⎠(10)12M 4bIntercellular transferλ3a,µ 3aM3aModule 322- 56 -
M.Elleuch, B.Ben, F.MasmoudiImprovement of manufacturing cells with unreliable machines - RTA # 3-4, 2007, December - Special Issueδ12λ1. Pt212Pt2µ 2µ 122λ121µ 2λ231M 1aM 2aM 3aM 2b M 1bM 3b M 4bλ1.(1-Pt2)Figure 3. Diagram of module stateBesides, the expression (10) shows that intercellulartransfer in case of failure permits to improve thestationary availability of the module, only in the casewhere the transfer rate is superior to a limit value(δ 12 *). This value is given by the expression (11).δ*122µ1⋅( µ1+ λ1+µ2+ λ 2)= . (11)µ212+ 2⋅µ ⋅λ+ λ + µ ⋅ µ + µ ⋅λ1111With the help of a good team of maintenance arrangingsome necessary logistical means, the time ofpreparation of an intercellular transfer can be reduced.In these conditions, times of transfer preparation canbe disregarded in front of times between failings andrepair machine times. Therefore, the expression of theavailability of the module will be given by theexpression (12).AModule( ∞)=22( µ ⋅ µ ⋅( µ + λ 1 + µ + λ 2) + λ 1⋅Pt 2 ⋅( µ ⋅λ2 + µ ⋅ µ + λ 1⋅µ + µ)12µ1321⎛λ1⋅λ2 ⋅ Pt⎜⎝+µ ⋅ µ ⋅1 222112 ( µ1+ λ 1) + µ2⋅( λ 1⋅( µ2+ λ 1) + λ 2 ⋅( µ1+ λ 1))( ) ⎟ ⎞µ + 2⋅λ1 +1µ2⎠(12)4. Simulation of manufacturing cellsTo conduct our simulation, we defined first theproblem and stated our objectives. The problem facingcellular manufacturing is the effect of breakdownmachines. The objectives of this simulation were toexamine the performance of the system with andwithout the policy of intercellular transfer in the eventof breakdowns and to validate the analytical model.We consider the system shown in the Figure 4. Thecell (a) is constituted of three different machinesdedicated to the manufacturing of two product types.The cell (b) is formed of four different machinescapable to manufacture three product types.µ1Operational stateTransitory stateNot-operational state1222Figure 4. Manufacturing system with intercellulartransferTable 1 summarizes the routing and the processingtime information of each partTable 1. The routing and processing times of each partCellaCellbCell aProducttypeBatchsizeCell bMachine (Processing times)1 19 M 1 (19) M 2 (12) M 3 (13)2 30 M 2 (10) M 3 (9)3 16 M 1 (14) M 2 (11) M 4 (13)4 25 M 1 (16) M 3 (14) M 4 (11)5 20 M 2 (17) M 3 (13) M 4 (7)The mean time to failure (MTTF) and the mean time torepair (MTTR) for all machines are shown in Table 2.Table 2. MTTF and MTTR of each machine in thesystemMachine M 1a M 2a M 3a M b1 M b2 M b3 M b4MTTF 6000 3500 3000 5500 4000 3500 6500MTTR 500 420 350 400 400 360 380In this paper, the batches arrival is considered cyclic.Indeed, the manufacturing of a new batch is onlypermitted if the previous batch is finished. In addition,products are generated in a cyclic manner. For a givenbatch, the time between the arrivals of two products isequal to the time of execution of the first operation.We perform discrete flow simulation using simulationsoftware called ARENA. We simulate the systemduring a 12 years horizon; during the first 60000minutes, statistics are not collected. This warm-upperiod (the first 60000 minutes) is used to avoidtransient effects on the final results.- 57 -
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ISSN 1932-2321RELIABILITY:Theory &
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