CHAPTER 3 PROBLEM FORMULATION
3.4 Mathematical Model for Generalized CFP
As mentioned in the previous section, it is important and more practical to integrate the abovementioned factors simultaneously in the design of CMS. Cell formation, cell layout, and intracellular machine layout are three major steps in the design of CMS. Ideally, these steps should be addressed simultaneously in order to obtain the best results. However, this is not easy to do due to the NP-complete nature of each step and the limitations of traditional approaches. Moreover, intracellular machine layout is a detailed layout planning. It usually starts after the cell formation and cell layout decisions have been determined. Hence, a two-stage multi-objective mathematical programming model is formulated in this section to
integrate cell formation, inter-cell layout, and intracellular machine layout problem with considerations of alternative process routings, operation sequences, production volume, machine reliability, and different cellular layout type. The framework of the proposed two-stage model is given in Figure 3.7. The aim of stage I is to solve cell formation and inter-cell layout simultaneously and the primary work of stage II is to determine machine layout (sequence) in each cell based on the given cell formation determined in stage I.
Figure 3.7 The framework of the proposed two-stage model for generalized CFP
3.4.1 Assumptions
The mathematical model for generalized CFP in this research is formulated on the basis of the following assumptions:
(1) All parts are assigned to part families.
(2) All machines are assigned to machine cells.
(3) All machines are non-identical.
(4) The type of cellular layout and the distance moves between cells are known a priori.
(5) Operation requirements, including operation sequence, operation time, and production volume, are known.
(6) Inter-cell part transportation unit cost for each part, breakdown cost, and MTBF for each machine are known.
(7) The limitation of total number of machines in each cell is user-defined.
(8) The intra-cell move distances for each part are not considered.
Cell formation
& cell layout
Intracellular machine layout
Stage I Stage II
3.4.2 Mathematical formulation
By using the above notations and assumptions, the proposed two-stage multi-objective mathematical programming models are formulated, one for each stage, and are presented here.
3.4.2.1 Stage I: Cell formation and inter-cell layout
The aim of this stage is to solve cell formation and inter-cell layout simultaneously in terms of minimization of total inter-cell move cost (ICMC) and MBC. The multi-objective 0-1 integer programming model is given below.
Total ICMC:
The multi-objective function is as follows:
Min TC ICMC MBC= + (3.13)
In the above model, Eqs. (3.11) and (3.12) show the calculation of the total inter-cell part transportation cost and MBC, respectively. Eq. (3.13) is the objective function that seeks the minimization of total cost of inter-cell part transportation cost and machine
breakdown. Eq. (3.14) indicates that only one process routing will be assigned to each part, while Eq. (3.15) assigns the upper and lower limits of the cell size. Eq. (3.16) provides a restriction that each machine will be assigned to exactly one cell and Eq. (3.17) indicates that Ykl and Zij are 0–1 binary decision variables.
Obviously, the objective function is in a non-linear form and thus may require extensive computational efforts for current commercial solvers to obtain possibly local optimal solutions. A linearization approach (Jabal Ameli et al., 2008) for converting a non-linear model into linear form is adopted. The transformation equation is as follows.
ijklk l ij kl k l
The first three linearization constraints (Eqs. 3.19–3.21) ensure that if one of the primary binary variables has a zero value, then their corresponding new variables will take a zero value as well. The last constraint (Eq. 3.22) ensures that if all primary variables take unit values, then their corresponding new variables take unit values as well. We rewrite the objective function as follows:
1 ( )
This new form of the objective function is in a linear form. Thus, linear programming software, such as Lingo 8.0, can solve this model.
3.4.2.2 Stage II: Intracellular machine layout
The parts being transported from one machine to another within a cell are called intra-cellular flow. Intra-cellular part flows are usually rushed and short in distances. In CMS, these movements are very frequent, and the frequency directly affects the intracellular machine layout design. Based on the classification scheme of Aneke and Carrie (1986), intracellular flow can be classified into four categories (Figure 3.7): (1) repeat operation, R; (2) forward flows, FF; (3) by-pass movement, BP; and (4) reverse flows, RF.
The ideal material flow in a good layout design should be mostly consecutive forward flows (CFF). The CFF usually has the benefits of smaller flow distance, easier control of the production process, and easier material handling (Ho et al., 1993).
Figure 3.8 Intracellular part flows
Since the CFF is a good indicator of the goodness of the solution, Mahdavi and Mahadevan (2008) developed a flow matrix on the basis of the number of CFF between a
M1 M2 M3 M4
FF FF FF
BP
RF R
their method did not consider the effect of manufacturing volumes. As mentioned in Section 3.3, taking the effect of manufacturing volumes into account is more realistic when designing a performance measure for intracellular machine layout. A flow matrix with manufacturing volumes consideration is thus proposed here. The flow matrix ( F ) is re-defined as follows:
Based on the flow matrix, a CFF index (CFFI) for measuring intracellular machine layout is proposed in this section. The CFFI is defined as the ratio of total number of CFFs in all cells (Ncff) to the total number of flows (Ntf).
The primary goal of the second stage is to determine the machine layout (sequence) in each cell in terms of maximizing the CFFI based on the cell formation determined in stage one. The model is given below.
CFFI
In the above model, Eq. (3.29) is the objective function that seeks the maximization of CFFI. Eqs. (3.30) and (3.31) ensure that each position is assigned to one machine and each machine is assigned to exactly one position. Eq. (3.32) indicates that Xlbkis a 0–1 binary decision variable.
Due to the combinatorial nature of the above models, good heuristic approaches should be more appropriate than the exact method in terms of solution efficiency, especially for large-sized problems. Thus, in the next chapter, we develop two fast and effective two-stage approaches to solve these complex problems.
CHAPTER 4
PROPOSED ALGORITHMS
In the previous chapter, two mathematical models representing standard CFP and generalized CFP have been formulated. Due to the NP-hard nature of the presented mathematical formulations, solving these problems through a traditional optimization technique is difficult and impractical. Furthermore, as mentioned in chapter two, meta-heuristic algorithms such as SA, TS, and WFA, have been the most successful solution approaches to provide global or near-global optimal solutions within a reasonable computation time, and SCM-based methods are more flexible in incorporating various production data into the machine-part clustering process. Thus, two hybrid meta-heuristic algorithms based on SCM-based clustering algorithm and SA/TS/WFA are proposed to solve the complex problems.
Before proposed algorithms are described, some notations used in this chapter are introduced first.
α : Cooling rate
counter_iter : Number of iterations
counter_stag : Number of times the incumbent solution did not improve counter_mut : Number of times the mutation strategy has been implemented C* : Optimal number of cells
f(S) : Value of object function in solution S L : Markov chain length
Nmax : Maximum number of iterations NC : Number of cells
NF : Set of feasible solutions
NC : Set of solutions without violating cell cardinality constraints NT : Set of solutions in tabu status
NA : Set of solutions satisfying aspiration criterion
Stag_check : Maximum number of solution has not been improved S0 : Initial solution
S : Current solution
S N : Neighborhood solution
S * : Incumbent solution of current cell size
*
S * : Best solution found so far T0 : Initial temperature Tf : Final temperature
4.1 Proposed Algorithms for Standard CFP
Most algorithms designed to solve CFP attempt to obtain the machine-part groupings so that some decision objectives, such as grouping efficiency or grouping efficacy, can be maximized. However, without prior determination of the NC, the abovementioned objectives can hardly be achieved. It is given beforehand in a few cases, but is left to be determined as part of the decision in most. Usually, in the iterative solution process, the initial NC is set at two and is gradually increased by one unit. These algorithms are then repeatedly applied until the NC resulting in the best grouping efficiency/efficacy value becomes established. Thus, many computational efforts have to be exerted in order to obtain the optimal NC. Instead of using a beginning number as the starting point, identifying a good intermediate point for the NC at the very beginning should save plenty of run time when designing an algorithm to search for the optimal NC.
We present a test problem from literature (Carrie, 1973) as an example. The relationship between the NC and the resulting grouping efficacy is shown in Figure 4.1.
Grouping efficacy value increases as the NC increases, and the optimal/near-optimal value is achieved when cell size is nine. Afterwards, efficacy starts to decrease as the NC increases. Similar observations can be found in other test problems. Based on this, the NC can be automatically calculated and determined such that the best grouping efficacy may result in.
25 30 35 40 45 50
2 3 4 5 6 7 8 9 10 11 12 13 14
number of cells
grouping efficacy (%)
Figure 4.1 Relationship between grouping efficacy and number of cells
Based on the above discussion, we propose a two-stage hybrid algorithm HCFA to solve the standard CFP. The framework of the proposed two-stage HCFA is given in Figure 4.2. In the first stage, the SCM-based clustering algorithm is adapted to derive NC quickly.
NC value is then used as input to the second stage to search for the optimal/near-optimal solution through the proposed SA/TS/WFA algorithm. We anticipate that NC obtained in stage one can serve as a good lower boundary to start the solution process in stage two.
Hence, a considerable amount of computational efforts can be saved, especially when large-sized problems are solved. The procedures for both stages are described below.
Stage I of HCFA:
Step 1. Set NC = 2, f S( 0)= f S( *) 0= .
Step 2. Apply the SCM-based clustering algorithm to generate an initial solutionS0.
Step 3. If f( )S0 > f( )S* , then set S*←S0 , C =* NC , NC=NC+1, go to Step 2;
otherwise, report incumbent cell configuration found: S*, C*, and terminate stage one.
The solution obtained at the end of stage one, including the suggested NC (C*) and cell configurations (S*), is then used as the input in stage two to search for the
optimal/near-optimal solution through the proposed SA/TS/WFA procedure.
Stage II of HCFA:
Step 1. Read solutions from stage one, including C* andS*. Step 2. Set NC=C*,f S( 0)= f( ),S* f S( **) 0= , go to Step 4.
Step 3. Apply the SCM-based clustering algorithm to generate an initial solutionS0.
Step 4. Apply SA/TS/WFA procedure to improve S0 and generate an incumbent solutionS*.
Step 5. If f( )S* > f(S**), then set S**←S*, C =* NC , NC = NC+1, go to Step 3;
otherwise, report the current best cell configuration (S**) and NC (C*), and terminate stage two.
Figure 4.2 Two-stage approach: Hybrid Cell Formation Algorithm (HCFA) Set NC=2
Apply the SCM to generate an initial solutionS 0
IsS better than* generate an incumbent solutionS *
IsS better than *
Apply the SCM to generate an initial solutionS 0
The SCM-based clustering algorithm and SA/TS/WFA are primary algorithms that consist of HCFA. The details of them are described as follows.
4.1.1 SCM-based clustering algorithm
As mentioned in Section 2.3, SCMs are more flexible in incorporating various production data into the machine-part clustering process. Hence, this study proposes the use of an SCM-based clustering algorithm to generate quick initial solutions, which will then be later improved by SA/TS/WFA method. It is well known that decomposing an originally difficult problem into several sub-problems usually increases problem-solving efficiency.
Since the CFP considers the grouping of machines and parts, an intuitive solution approach is to decompose the entire problem into two sub-problems dealing with the assignment of machines and parts, respectively. In our construction of the initial solution, machine assignment is determined in the first stage, while the assignment of parts is achieved in the second stage.
Our approach for generating initial solutions consists of three steps: (1) computing similarity values between machine pairs and constructing a similarity matrix, (2) using a clustering rule to process the values in the similarity matrix and forming machine cells, and (3) assigning parts to machine cells using a parts assignment procedure. Details of them are described here.
(1) Machines assignment
As mentioned in Section 2.2, the Jaccard similarity coefficient is the most stable similarity coefficient. Hence, Jaccard’s similarity measure is used to evaluate similarity between machines as an important index for assigning machines to cells in this sub-problem.
The similarity measure, denoted by Sij, is defined as ij ij
ij ij ij
S a
a b c
= + + , where aij represents
by machine i but not by machine j; and cij is the number of parts processed by machine j but not by machine i. After calculating the similarity matrix for each pair of machines, we are able to generate the initial machines assignment by using the following greedy rule: the higher the similarity measure of a pair of machines, the higher priority they have for placement in the same cell. This process is repeated until all machines have been assigned to cells. For the sample machine-part matrix in Figure 4.3(a), the corresponding similarity matrix for machines is displayed in Figure 4.3(b). Assuming that two cells are to be formed, the largest coefficient in the matrix of Figure 4.3(b) is 0.67, indicating that machines 2 and 4 must be assigned to the same cell, e.g. cell 1. We proceed to the second largest coefficient in the matrix, 0.5, appearing in pairs (1, 3) and (1, 5). Since these three machines do not have any relationship with any machines in cell 1, they should be assigned together to the next cell, cell 2. Figure 4.4 shows the machines assignment using the proposed greedy rule.
P1 P2 P3 P4 P5 Figure 4.3 Machine-part matrix and corresponding similarity matrix for machines
P1 P2 P3 P4 P5
In this procedure, the parts are assigned to cells so that the number of voids and exceptional elements—major components comprising the formula of grouping
efficacy—are explicitly considered. It can be summarized as follows:
Step 1. Read the results of machines assignment.
Step 2. For each part, find the cell to which a part assignment will result in the least sum of number of exceptional elements and number of voids. If a tie happens, assign the part to a cell with the least number of voids.
Step 3. Repeat Step 2 until all parts have been assigned to cells.
Results of parts assignment shown in Figure 4.5 demonstrate this procedure. After calculating the sum of numbers of voids and exceptional elements for each part-cell combination, parts 2, 3, and 5 are assigned to cell 1, while parts 1 and 4 are assigned to cell 2. The initial solution matrix for this CFP can thus be obtained and the configuration for this initial solution can be represented by Figure 4.5.
P2 P3 P5 P1 P4
M2 1 1 1 0 0
Cell 1 M4 1 1 0 0 0
M1 0 0 0 1 1
Cell 2 M3 0 0 0 1 0
M5 0 0 0 0 1
Figure 4.5 Initial solution matrix obtained 4.1.2 SA/TS/WFA algorithms
When designing a heuristic search algorithm, several important considerations should be kept in mind. The first is to develop a mechanism for searching the neighborhood solutions for improvement. Since the neighborhood will be searched next, the choice of neighborhood function will strongly influence the direction of the search. Another consideration is the mechanism for allowing escape from local optima and for settling only in a global optimum. Based on these concepts, three algorithms, namely HSAM, HWFAM, and HTSM, are developed in this section.
(1) Configuration
string, whose size is equal to the number of machines/parts. The jth bit of the string stores the identifier of the cell to which the machine/part is assigned. For example, Figure 4.6 is the configurations for machine cells and part families. In such a configuration, the string (2, 1, 2, 1, 2) in Figure 4.6(a) indicates that machines 2 and 4 are assigned to cell 1, while machines 1, 3, and 5 are assigned to cell 2; the string (2, 1, 1, 2, 1) in Figure 4.6(b) represent that machines 2, 3, and 5 are assigned to cell 1, while parts 1 and 4 are assigned to cell 2.
Machine # 1 2 3 4 5 Cell # 2 1 2 1 2 (a) Configuration for machine cells
Part # 1 2 3 4 5 Cell # 2 1 1 2 1 (b) Configuration for part families Figure 4.6 Configuration of a feasible solution to the CFP
(2) Insertion-move operation
In this study, the insertion-move operation is applied as a mechanism for searching the neighborhood solutions for improvement. It moves a machine k from its current cell l (source cell) to a new cell l′ (destination cell). The new move is denoted as (l′, k). A move that results in the greatest improvement of the objective function value from the current solution is selected. That is,
( ', ) ( , )
( ) { l k - l k , , ' F, ' , }
Z l', k =Max obj obj ∀l l ∈N l ≠ ∀ ∈l k M (4.1)
where obj( , )l k is the objective function value; NF is the set of feasible solutions; and M is the set for machines.
(3) Mutation strategy
The mutation strategy of GA aims to increase the probability of finding more
“diversified” solutions in order to bring the searching process to a new and unexplored
solution space, thus ensuring that large areas of the space are searched. In this study, the mutation strategy mut_check is implemented when the number of moves has not been improved within a certain number of iterations. This performs an exchange of a machine to any cell other than the current one based on a prescribed probability β. That is, all machines have the probability of changing cell when machine mutation is applied. For each machine in the incumbent solution, a random number from (0, 1) is first drawn. If the value is greater than β, then the machine is exchanged with another randomly determined cell; otherwise, it stays in the current cell. Through this strategy, the search is able to explore a large solution space, thereby enhancing the possibility of finding the optimum solution in a very short time. The procedure of the mutation strategy in the pseudo-code format is shown in Figure 4.7.
Mutation_strategy ( β ) {
Let the current solution (S) equal to the current best solution (S*).
FOR each machine DO {
Generate a random number r U∈ (0,1). IF (r>β )
Exchange machine with any cells other than the current one.
ELSE
Stay machine in the current cell.
} }
Figure 4.7 Pseudo code of mutation strategy 4.1.2.1 SA-based algorithm (HSAM)
As mentioned in Section 2.6, the main disadvantages of SA are as follows: (1) high execution time, (2) ease of being trapped to local minima if the cooling speed is too fast or
solution if the search cannot reach the equilibrium state at each temperature. In this study, two types of mechanisms, the insertion-move and the mutation strategy of GA, are utilized to construct a hybrid SA method called HSAM to address these issues. Both mechanisms play different roles in the process of solution improvement. We use insertion-move as a primary tool for finding better neighborhood solution, while employing mutation strategy to increase the probability of finding more “diversified” solutions to bring the searching process to a new and unexplored solution space. The pseudo-code format of the proposed procedure HSAM is diagrammed in Figure 4.8 and described in detail below.
Algorithm HSAM
Step 1. Read initial solutionS0.
Step 2. Initialization: Let counter_MC = 0,T =T0,S ←S0,S*←S0.
Step 3. If counter_MC < L, then repeat Steps 3.1 to 3.5; otherwise, go to Step 4.
Step 3.1. If counter_mut ≥ mut_check, then apply the mutation strategy to generate a new current solution S and let counter_mut = 0.
Step 3.2. Generate a best solution SN
(
S ∈N NC)
in the neighborhood of S by performing the insertion-move operation.Step 3.2. Generate a best solution SN