CHAPTER 3 PROBLEM FORMULATION
3.3 Problem Description for Generalized CFP
In the standard CFP, the binary machine-part incidence matrix is the main production data. Some real-life production factors, such as alternative process routings, operation sequences, production volume, machine reliability, and cellular layout, are not addressed in the design of CMS, thereby limiting the practical applied in a real CMS environment. Hence,
a generalized CFP that incorporates the abovementioned factors in the design of CMS is introduced. These factors are described in detail next.
(1) Alternative process routings
A process routing for a given part is the set of machines passed by this specific part. In most CF methods, parts are assumed to have a unique part process plan. However, it is well known that alternatives may exist in any level of a process plan. In some cases, there may be many alternative process plans for making a specific part, especially when the part is complex (Qiao et al., 1994). In the case shown in Figure 3.2(a), part #1 has three process routings: R1, R2, and R3. When introducing alternative process routings to CFP, the grouping of parts can be more effective due to the flexibility of the routes. However, it leads to a more complex problem than the standard CFP. Under this circumstance, not only the formation of part families and machine cells must be determined but also the selection of routings for each part to achieve decision objectives, such as the minimization of intercellular movement. For instance, Figure 3.2 (b) provides a feasible solution to the sample problem of Figure 3.2 (a) where routing #2 is selected by all parts.
PN P1 P2 P3 P4 P5 PN P1 P3 P2 P4 P5
PV 50 30 20 30 20 PV 50 20 30 30 20
RN R1 R2 R3 R1 R2 R1 R2 R1 R2 R1 R2 RN R2 R2 R2 R2 R2
M1 2 2 2 1 1 1 M2 1 2
M2 1 1 1 2 M4 2 1
M3 2 2 1 2 1 M1 2 1 1
M4 1 2 1 1 2 2 M3 1 2
(a) Problem data (b) Final solution Figure 3.2 Cell formation with alternative process routings
(2) Cellular layout
In CMS, different cellular layout type and cellular layout sequence will affect the inter-cell move distance (ICMD). They are described as follows.
Cellular layout is represented in Figure 3.3, where r is the number of rows, c is the number of columns, and (Xr,Yc) is the coordinate of cell cr. The cellular layout type can be determined, while the value of r is set by the layout designer. For example, when r=1, the cellular layout type is linear single-row layout (Figure 3.4 (a)); and when r=2, the cellular layout type is linear double-row layout (Figure 3.4 (b)), where NC is the number of cells.
1 2 … c
Figure 3.4 Two typical cellular layouts: (a) linear single-row layout (r=1) (b) linear double-row layout (r=2)
(b) Inter-cell move distance
There are two popular methods for measuring ICMD between a pair of cells l and l′(Tam and Li, 1991). They are:
Cartesian method: Dl l,′=⎡⎣
(
Xl′−Xl) (
2+ Y Yl′− l)
2⎤⎦ , 1/ 2 (3.9) Manhattan method: Dl l,′= Xl′−Xl +Yl′−Yl , (3.10)where
(
X Yl, l)
and(
X Yl′, l′)
are the coordinates of the measuring points of cells l and l′.In terms of measuring points, we can use either: (a) the centroid of a cell site or (b) the nearest point between adjacent cells. In this thesis, the Cartesian method was chosen and the centroid of a cell site will be used for calculating ICMD. Thus, the ICMD between cells 1 and 2 in Figure 3.4(a) is equal to1=⎡⎣
( ) ( )
1 1− + −2 2 1 2⎤⎦ . 1/ 2(c) Effects of cellular layout type
Different cellular layout types will result in different ICMD. Figure 3.5 shows that the ICMD between cells 1 and 3 in Figure 3.5(a) will be twice the distance moved between cells 1 and 3 in Figure 3.5(b). Hence, the cellular layout type is an important issue in CMS design.
Cell 1 Cell 2 Cell 3 Cell 1 Cell 3 (1,1) (1,2) (1,3) (1,1) (1,2)
Cell 2 (2,1)
(a) linear single-row layout (r=1) (b) linear double-row layout (r=2) Figure 3.5 Two typical cellular layouts (NC=3)
(d) Effects of cellular layout sequence
We present an example to illustrate the effects of cellular layout sequence. If the NC is equal to three and a linear single-row layout (r=1) is considered as shown in Figure 3.5(a), then ICMD between cells 1 and 3 will be twice the distance moved between cells 1 and 2 or between cells 2 and 3. When a linear double-row layout (r=2) is considered as shown in
Figure 3.5(b), the corresponding ICMD between cells 2 and 3 will be 2 times the distance moved between cells 1 and 2 or between cells 1 and 3. Hence, the cellular layout is an important issue in CMS design.
(3) Operation sequence and production volume
The operation sequence and production volume of each part affects the machine cell formation significantly. Therefore, both operation sequence and production volume of each part should be incorporated in the analysis of CM systems. For example, a simple CFP consists of four machines (M1, M2, M3, M4) and two parts (P1, P2) with part routes (M2, M1) for P1 and (M4, M1, M3) for P2. Suppose that the annual demands of P1 and P2 are 40 units and 60 units, respectively. Two cell formation results are shown in Figure 3.6, where the number in each entry indicates the visiting order of part to machine. If we do not consider machine sequence in calculating inter-cell movement, the solution in Figure 3.6 (a) is better than that in Figure 3.6(b) because there are 60 inter-cell movements in Figure 3.6 (a) and 100 inter-cell movements in Figure 3.6 (b). However, if the machine sequence is considered, the solution in Figure 3.6(b) is better, because the sum of inter-cell movements in Figure 3.6(a) is 120 (60 × 2) compared with 100 (40 + 60) in Figure 3.6(b). If we do not consider production volume in calculating inter-cell movement, the solutions of Figure 3.6(a) and (b) are the same with 2. However, if the manufacturing volumes are considered, the solution in Figure 3.6(b) is better, because the sum of inter-cell movements in Figure 3.6(a) is 120 (60 × 2) compared with 100 (40 + 60) in Figure 3.6(b).
(a) P1 P2 (b) P1 P2
PV 40 60 PV 40 60
M1 2 2 M2 1
M2 1 M3 3
M3 3 M1 2 2
M4 1 M4 1
Figure 3.6 An example for the affect of operation sequence
(4) Machine reliability
A number of previous works assumed that all machines are 100% reliable. However, this is not always the case. Machines are key elements in manufacturing systems and oftentimes it is not possible to handle their collapse as quickly as production requirements dictate. Their collapse can dramatically affect system performance measures and bring about detrimental effects on due date performance. Hence, machine reliability should be taken into account during the design of CMS to improve the overall performance of the system (Jeon et al., 1998).
A common way of dealing with machine reliability in the design phase of a manufacturing system is by the evaluation of the quantities of the mean time between failures (MTBF). MTBF can be obtained by taking the reciprocal of λ , where λ is the machine failure rate. As long as the breakdown cost for each machine is known in advance, the cost caused by machine unreliability can be acquired after simple calculation. Jabal Ameli and Arkat (2008) have presented a mathematical approach to calculate machine breakdown cost (MBC) that involves dividing production time by MTBF and then multiplying this quantity by the unit MBC (Eq. (3.12)).