unloading finished parts. Figure 1. A rotary transfer machine with turrets: a general view (PCI-SCEMM, France). Figure 2. A rotary table is used for part transfer ...
Combinatorial techniques to optimally customize an automated production line with rotary transfer and turrets
Abstract: A problem of design of complex rotary transfer machines with turrets is considered. Operations are partitioned into groups which are performed by spindle heads or by turrets. Constraints related to the design of spindle heads, turrets, and working positions, as well as precedence constraints related to operations, are given. The problem consists of minimizing the estimated cost of this automated production line, while reaching a given cycle time and satisfying all constraints. Two methods were proposed to solve the problem. The first uses a MIP formulation of the problem. The second method is based on its reduction to a constrained shortest path problem. An industrial example is presented. Keywords: Machining system, line design, line balancing, optimization, graph theory.
Rotary transfer machines with turrets offer an efficient solution for machining standardized parts for mass production (in grand series). Since standardization is one of the current trends in manufacturing, their role should even increase in the near future. Such machines present the advantage of being customizable and can be optimized for the specific part(s) to be produced, resulting in significant cost savings for the user. The customization is possible due to the use of standard spindle heads (modules or units) and turrets. The customization step considered here consists of selecting necessary standard modules, their order of activation and the cutting tools installed for each of them, in order to match technical and technological requirements expressed for a new part to be machined. This step does not include an in-depth reliability study considering that for standard modules the reliability problems were already solved and system reliability models will be used at the next step of line design. Here only major parameters are considered which are known at the preliminary design stage and which structure future design decisions and line cost. Several additional factors can be considered indirectly via compatibility constraints among operations introduced in the model proposed.
A general view of a rotary transfer machine with turrets provided by our industrial partner is given in Figure 1. Such a machine is multi-positional as shown in Figure 2, i.e. each part is sequentially machined on m working positions. The number m can be customized according to the part to be produced but it cannot exceed a technological limit of m0 positions. The first working position is exclusively used for loading new billets and unloading finished parts.
Figure 1. A rotary transfer machine with turrets: a general view (PCI-SCEMM, France)
Figure 2. A rotary table is used for part transfer
At each position, several machining modules (spindle heads) can be installed to process the operations assigned to this position. They are activated sequentially or simultaneously. Sequential activation is realized by the use of turrets. Simultaneous activation is possible if machining modules are related to the different sides of the part and work in parallel. The number of machining modules and the order of their activation at each working position are customizable. There are horizontal and vertical spindle heads and turrets to access different sides of parts at a working position. Figure 3 shows a horizontal turret with 3 machining modules (on the left) and a vertical turret with 2 machining modules (on the right).
Figure 3. A horizontal turret with 3 machining modules (on the left) and a vertical turret with 2 machining modules (on the right)
Figure 4. A horizontal turret with 5 machining modules, one of them has 2 tools
Finally, a machining module may process several machining operations. The tools to be installed are chosen according to the machining operations assigned to the module. Several cutting tools can be installed in the same
module; for example, Figure 4 presents a horizontal turret with 5 machining modules where a module has two cutting tools. In order to manage efficiently the customization of rotary transfer machines of this type, i.e. to find machine configurations capable of producing a given number of items of the same product at minimal cost, the machinery manufacturers need to have adapted decision support tools. However, to the best of the authors’ knowledge, this problem has not been addressed in the industrial engineering literature yet. System design via an adequate equipment selection was principally studied for serial production lines (Bard and Feo 1991, Bukchin and Tzur 2000, Kimms 2000, Özdemir and Ayağ 2011). Bukchin and Rubinovitz (2003) considered the case of station paralleling, but only identical equipment could be installed in parallel. The resource assignment in terms of workers has been also considered principally for serial assembly lines (Blum and Miralles, 2011; Yazgan et al., 2011). Series-parallel manufacturing systems were also studied in the literature, but without considering the customization problem explored in this paper. For example, Jin et al. (2010) and Li and Wang (2011) considered serial parallelmultistage manufacturing processes with the aim of optimizing the allocation of control charts. Interesting asymmetric configurations of assembly systems were studied by AlGeddawy and ElMaraghy (2010), Roemer and Ahmadi (2010), Li et al. (2011) and Hu et al. (2011). Few studies on rotary transfer machines (without turrets) have been published. Productivity of production lines with rotary transfer was evaluated by Usubamatov et al. (2008). Configuration of semi-automated systems with multi-turn rotary table was discussed in (Battini et al., 2007). Mathematical models of transfer machines with rotary or mobile tables were proposed in (Dolgui et al., 2008, 2009; Guschinskaya et al., 2007; and Battaïa et al., 2012a,b). Unfortunately, the principal advantage introduced by the use of turrets – the possibility of serial-parallel machining – cannot be taken into account by these models where only machining modules activated in series or in parallel are considered. A series-parallel execution of machining modules has been addressed only for the transfer lines and for the case where the machining modules were already known (Dolgui and Ihnatsenka, 2009). Therefore, the aim of this paper is to develop an adequate mathematical model and efficient optimization algorithms providing the machinery manufacturers with an efficient decision support for the problem of rotary transfer machine (automated line with rotary transfer) customization. The paper is organized as follows. Section 2 deals with the statement of the optimization problem. It also presents the relations between the present problem and other combinatorial optimization problems already studied in the literature. Section 3 presents a graph approach for
the considered problem. The proposed models are evaluated on a series of industrial examples in Section 4. Concluding remarks are given in Section 5.
2. PROBLEM STATEMENT 2. 1 Input data for the part to be machined The objective of the optimization problem is to customize the configuration of a rotary transfer machine (automated lines with rotary transfer) with turrets in accordance with the part(s) to be produced. Therefore, this problem has to take into account the data available about the part, such as the required machining operations and existing technological constraints. Let N be the set of machining operations defined for the machining process. For the considered machines, no part repositioning is possible. Therefore, the part machined has the same orientation at all working positions. As a consequence, all operations can be divided into two subsets N1 and N2 including operations to be executed by a horizontal or a vertical machining module, respectively. All operations p∈N are characterized by the following machining parameters: -
λ(p) is the length of the working stroke for operation p∈N, i.e. the distance to be run by the tool in order to complete operation p;
-
range [γ1(p), γ2(p)] of feasible values of feed rate which characterizes the machining speed.
These parameters are involved in calculating the effective machining time, considered here below in Subsection 2.3. All operations from set N have to be completed by the rotary machine. The aim of machine customization is to find the machine configuration capable of providing
them, with the minimal cost. Since the machine cost is mainly
defined by the costs of equipment used, this can be minimized by an optimal assignment of operations to working positions, turrets, and machining modules. To model this assignment, the following notations will be used: 1.
Let subset Nk, k=1,...,m contain the operations from set N assigned to k-th working position, where m is the total number of working positions used.
2. Let sets Nk1 and Nk2 be the sets of operations assigned to working position k that are concerned by vertical and horizontal machining, respectively.
3. Finally, let bkj be the number of machining modules of type j (vertical if j=1or horizontal if j=2) installed at k-th working position and respectively subsets Nkjl, l=1,...,bkj contain the operations from set Nkj assigned to the same machining module. This assignment has to respect the technological constraints that emanate from the machining process required. They can be grouped in three following families: 1. Inclusion constraints: they oblige the execution of some pairs of operations from N at the same working position, by the same turret or by the same machining module for each pair. Such relationships are modeled by undirected graphs GSB=(N,ESB), GST=(N,EST), and GSP=(N,ESP) where collections ESB, EST, and ESP of operations are used to define the pairs of operations (p,q) that must be executed by the same machining module (ESB), turret (EST), and position (ESP). 2. Exclusion constraints: they define the pairs of operations that cannot be performed together at the same working position, by the same turret or by the same machining module. Such restrictions are modeled by undirected graphs GDB=(N,EDB), GDT=(N,EDT), and GDP=(N,EDP) where collections EDB, EDT, and EDP of operations are used to define the pairs of operations (p,q) that cannot be executed by the same machining module (EDB), turret (EDT), and position (EDP). It should be noted that sometimes operations cannot be executed by a turret (due to distance constraints) but can be assigned to the same spindle head. 3. Precedence constraints: they define a partial order for processing of operations. A directed graph GOR=(N,DOR) is used to define them, where arc (p,q)∈DOR exists if and only if operation p has to be completed before operation q starts. It should be noted that if such operations p and q belong to different sides of the part then they cannot be executed at the same position without violating the precedence constraint. Let Pred(p) be the set of immediate predecessors of operation p in graph GOR.
2.2 Technological constraints related to the machine configuration The modular architecture of rotary transfer machines with turrets offers a combinatorial choice of their configurations. Nevertheless, several technical constraints exist and have to be taken into account by the customization. In this paper, the following assumptions are considered: 1. The number of working positions m cannot exceed m0. 2. The number of machining modules in a turret i.e. bkj cannot exceed b0.
3. If bkj=1, then a simple spindle head is installed instead of a turret (it has a lower cost). 4. At most one horizontal turret can be chosen per working position. 5. At most one vertical turret can be chosen for the machine (installed at one working position). If no vertical turret is used, then vertical machining on working positions can be performed by simple machining modules mounted in a common spindle head. 2.3 Cycle time constraint This constraint imposes that a finished part is unloaded every T0 time units, otherwise the required annual volume of parts cannot be provided by the machine. Since all working positions work in parallel, the time interval needed to provide a finished part or machine cycle time T(P) is calculated as follows: T(P)= max {Tk | k=1,…,m }, where Tk is the time the part spends at working position k. It is defined by the maximal time of vertical or horizontal machining at working position k, i.e.: Tk =τp+max{τg(bkj-1)+
bkj
∑t
b kjl
| j=1, 2}
(1)
l =1
where τg and τp are additional times for turret and table rotation, respectively, and t bkjl is the execution time of set of operations Nkjl by a common machining module. When set of operations Nkjl are executed simultaneously by a common machining module, the same parameters L(Nkjl) = max{λ(p) | p∈ Nkjl} and γkjl ∈ [Г1(Nkjl), Г2(Nkjl)] are applied for all operations from set Nkjl where Г1(Nkjl)=max {γ1(p)| p∈ Nkjl } and Г2(Nkjl)= min{γ2(p) | p∈ Nkjl}], i.e. the same working distance and feed rate. Similarly when all of operations from N1 are executed simultaneously at several working positions by a common spindle head, the same working distance Lk11 = L(N1) and feed rate γk11 ∈ [Г1(N1), Г2(N1)] are applied for all operations from set N1. In both cases the first parameter Lkjl has a fixed value and the second parameter γkjl may be chosen in different ways from the range of feasible values. As a consequence, the time required to execute Nkjl, denoted t bkjl , depends on this choice and is calculated as follows: b b t kjl =L(Nkjl) / γkjl +τ
where τb is an additional time for advance and disengagement of tools.
(2)
Obviously, the set N1 cannot be executed simultaneously at several working positions by a common spindle head if Г1(N1) > Г2(N1) or L(N1)/Г2(N1)+ τp +τb > T0
(3)
2.4 Objective function The objective of the machine customization is to minimize the cost of the machine capable of completing all given machining operations under existing constraints. The total machine cost is estimated by the cost of the equipment used such as working positions, turrets and machining modules. Let the following costs be given: C1,
cost of one working position,
C2,
cost of one turret,
C3,
cost of one machining module if installed in a turret,
C4,
cost of one machining module if installed in a spindle head.
Therefore, the cost of execution of the set of operations Nkj (at position k and with machining direction j – horizontal or vertical) depends on bkj, the number of machining modules of type j installed, and if a turret is required bkj > 1or not bkj > 0 . Obviously, if Nkj=∅, then bkj=0 and there is no cost implied. Let C(bkj) be the cost function assessing the cost of performing set of operations Nkj by bkj machining modules. It is defined as follows:
0 if bkj = 0 C 4 if bkj = 1 C(bkj) = C + C b if b > 1 kj 3 kj 2
(4)
The machine cost Q(P) is calculated as the total cost of all equipment used, i.e. m 2
Q( P) = C1m + ∑ ∑ C (bkj ) k =1 j =1
2.5 Mathematical model Taking into account the constraints and assumptions introduced above, the mathematical model of the optimization problem considered is formulated as follows. The following decision variables are involved in the model:
Xpkl is the decision variable used to define the assignment of operations, i.e. it is equal to 1 if operation p is assigned to the l-th machining module of k-th working position; Ykjl
auxiliary variable employed for counting the number of machining modules installed, i.e. it is equal to 1 if
at least one operation from Nj is executed by l-th machining module at k-th working position, j=1,2; auxiliary variable used for assessing the execution time required to perform operations from Nj by l-th
Fkjl
machining module at k-th working position, j=1,2; Zk
auxiliary variable for counting the number of working positions required, i.e. it is equal to 1 if at least one
operation from N is executed at k-th working position; minimal time necessary for execution of operations q and p in the same machining module, tqp = max(λ(q),
tqp
λ(p))/min(γ2(q),γ2(p))+τb. It is assumed that (p,q) ∈EDB if min(γ2(q),γ2(p)) < max(γ1(q),γ1(p)) or tqp +τp > T0. The objective function can be written as follows: Min Q ( P ) = C1
m0
m0 2
m0 2
m 0 2 b0
k =1
k =1 j =1
k =1 j =1
k =1 j = 1 l = 3
∑ Z k + C4 ∑ ∑ Ykj1 + ∑ ∑ Ykj 2 (C2 + 2C3 − C4 ) + C3 ∑ ∑ ∑ Ykjl
(5)
The problem constraints are as follows. Each operation is assigned to one machining module m0
b0
k =1
l =1
∑ ∑
X pkl = 1 ; p∈Nj; j=1,2
(6)
Each predecessor p of the operation q located at the same side of the part is assigned to the previous positions or the previous machining modules of the same turret m0
b0
k =1
l =1
∑ ∑
m0
((k − 1)b0 + l ) X pkl ≤ ∑ k =1
b0
∑ l =1
((k − 1)b0 + l − 1) X qkl ; (p,q)∈DOR; p,q∈Nj; j=1,2
(7)
Each predecessor p of the operation q located at the different side of the part is assigned to the previous positions m 0 b0
m 0 b0
k =1 l =1
k =1 l =1
OR j 3-j ∑ ∑ kX pkl ≤ ∑ ∑ ( k − 1) X qkl ; (p,q)∈D ; p∈N ; q∈N ; j=1,2
(8)
For operations p and q that have to be assigned to the same working position or to the same turret b0
∑ l =1
b0
X pkl = ∑ X qkl ; (p,q) ∈ESP∪EST ; k=1,…,m0 l =1
(9)
For operations p and q that have to be performed by the same machining module Xpkl=Xqkl ; (p,q) ∈ESB; k=1,…,m0; l=1,…,b0
(10)
For operations p and q that have to be assigned to different working positions b0
∑ l =1
b0
X pkl + ∑ X qkl ≤1, (p,q) ∈EDP; k=1,…,m0
(11)
l =1
For operations p and q that cannot be executed by a turret but can be assigned to the same spindle head b0
b0
l =1
l =1
DT j ∑ X pkl + ∑ X qkl + Ykj2 ≤ 2, (p,q) ∈E ; p,q∈N ; k=1,…,m0; j=1,2
(12)
For operations p and q that have to be assigned to two different machining modules Xpkl+Xqkl ≤1; p, q∈Nj; j=1,2; (p,q) ∈EDB; k=1,…,m0; l=1,…,b0
(13)
The time of each machining module cannot be less than the time of execution of any operation assigned to this machining module: Fkjl ≥ tppXpkl; p∈Nj; j=1,2; k=1,…,m0; l=1,…,b0
(14)
The time of each machining module cannot be less than the time of execution of any pair of operations assigned to this machining module: Fkjl ≥ tqp (Xpkl+Xqkl-1); p, q∈Nj; j=1,2; (p,q)∉ EDP∪EDB; p
