CHAPTER 4 PROPOSED ALGORITHMS
4.2 Proposed Algorithms for Generalized CFP
4.2.2 Proposal algorithms for stage II
This stage consists of two procedures: initial solution construction and solution improvement (i.e., SA, TS, and WFA). The SA/TS/WFA procedure is the same as discussed in Section 4.1.2. Initial solution construction and some elements comprising the proposed algorithms are described below.
(1) Initial solution construction
The initial solution of the sequence of machines in each cell can be generated by the following procedure:
Step 1. Read the machine cells determined in stage one.
Step 2. Arrange machine cells by cell number in an ascending order.
Step 3. Arrange the sequences of machines in each cell in an ascending order.
(2) Configuration
A three-dimensional array is used to represent the configuration of a feasible solution of the sequences of machines within each cell. Figure 4.13 shows an example where machine #3 was assigned to the first sequence of cell #1.
Cell # 1 2 3
Sequence # 1 2 3 1 2 3 1 2 3 4
Machine # 3 7 8 2 4 6 1 10 9 5
Figure 4.13 Configuration of an initial solution to sequence of machines
(3) Neighborhood solution searching
In this stage, the neighborhood of a given solution is defined as the set of all feasible solutions reachable by an exchange-move. The exchange-move is an operation that exchanges any pair of machines within the same cell. If we exchange machine k with machine k′, then the new move is denoted as (k, k′). The move that results in the most improvement in CFFI value from the current solution is selected; that is,
( , ) ( , )
( ) { k k - k k , , F }
Z k, k′ =Max obj ′ obj ′ ∀k k′∈M and ∈N and k k≠ ′ (4.5)
where obj( , )k k′ is the objective function value; M is the set for machines; and NF is the set of feasible solutions.
(4) Mutation strategy
When the number of moves has not been improved within a certain number of iterations, the mutation strategy (mut_check) is implemented by exchanging any pair of machines within the same cell based on a prescribed probability β. For each machine in the same cell, a random number from (0, 1) is first drawn. If the value is greater than β, then the machine sequence is exchanged with another randomly determined machines sequence within the same cell; otherwise, it remains in the current sequence. The procedure of machine sequence mutation strategy is presented in pseudo-code format in Figure 4.14.
Mutation_strategy ( β ) {
Let the current solution (S) equal to the best solution (S*).
FOR each machine in the same cell DO {
Generate a random number r U∈ (0,1). IF (r>β )
Exchange machine sequence with the other machines.
ELSE
Stay machine in the current sequence.
} }
Figure 4.14 Pseudo code of mutation strategy (5) Tabu list
In the TS procedure, a two-dimensional array (m×m) TL[ ][ ]m m ,where m is the number of machines, is used as a tabu list to check if a move from a solution to its neighborhood is forbidden or allowed. If a pair of machines k and k′are exchanged, then the exchanging of machine k′ and k will be forbidden for a certain number of iterations, which is equal to the tabu list size tls (e.g.,TL[ ][ ]k k′ =tls).
CHAPTER 5
NUMERICAL ILLUSTRATIONS
As mentioned in Chapter 4, two hybrid meta-heuristic algorithms integrating SCM-based clustering algorithm and SA/TS/WFA are proposed to solve standard CFP and generalized CFP, respectively. To illustrate the effectiveness of our developed algorithms, two test examples are demonstrated in this chapter. Example #1 includes 10 machines and 10 parts that comprise a simple CFP with a 0-1 machine-part incidence matrix. Example #2 consists of 10 machines and 10 parts that form a generalized CFP. The proposed algorithms were coded in C++ using Microsoft Visual Studio 6.0 and implemented on an Intel(R) 1.66GHz PC with 1GB RAM. Computation results for both types of CFP are shown and discussed separately in this chapter.
5.1 An Illustrative Example for Standard CFP
The 0-1 machine-part incidence matrix for example #1 is given in Figure 5.1. The minimum number of machines in each cell (Lm) is limited to 2 (i.e., singletons are not allowed). The objective function aims to determine machine cells and part families in which grouping efficacy can be maximized. The implementation of the proposed method for standard CFP is described as follows:
M\P P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
M1 0 1 0 0 1 0 0 1 0 0
M2 1 0 1 0 0 0 1 0 0 0
M3 0 0 0 1 0 1 0 0 1 1
M4 1 0 0 0 0 0 1 0 0 0
M5 0 1 0 0 1 0 0 1 1 0
M6 1 0 0 0 0 0 1 0 0 0
M7 0 0 1 1 0 1 0 0 1 1
M8 0 0 1 1 0 1 0 0 1 1
M9 0 1 0 0 0 0 0 1 0 0
M10 0 1 0 0 1 0 0 0 0 0
Figure 5.1 0-1 machine-part matrix of example #1
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.
As mentioned in Section 4.1.1, the SCM-based clustering algorithm consists of three steps: (1) computation of similarity values between machine pairs and construction of a similarity matrix, (2) utilization of a clustering rule to process the values in the similarity matrix and formation of machine cells, and (3) assignment of parts to machine cells using a parts assignment procedure. They are described as follows:
(1) Calculation of machine similarity matrix
The corresponding similarity matrix for machines can be obtained by using Eq. (2.1) and is shown in Table 5.1.
Table 5.1 Similarity matrix for machines in example #1
Machine 1 2 3 4 5 6 7 8 9 10
1 -
2 0 - 3 0 0 -
4 0 0.67 0 -
5 0.75 0 0.14 0 -
6 0 0.67 0 1.00 0 -
7 0 0.14 0.80 0 0.13 0 -
8 0 0.14 0.80 0 0.13 0 1.00 -
9 0.67 0 0 0 0.50 0 0 0 -
10 0.67 0 0 0 0.50 0 0 0 0.33 -
(2) Formation of machine cells
The similarity matrix shows that the largest coefficient in the matrix is 1, appearing in pairs (7, 8) and (4, 6). Since pairs (7, 8) and (4, 6) do not have any relationship, they are assigned to cell #1 and cell #2, respectively. The second largest coefficient in the matrix (0.8) appears in pairs (3, 7); because machine 7 has been assigned to cell #1, machine 3 is assigned to the same cell. Next in line is pair (1, 5); machines 1 and 5 have not been
assigned to any cell so they should be assigned to cell #3. However, the initial NC is 2, so that machines 1 and 5 are assigned to cell #2. Using the same logic, we can assign machines 2, 9, and 10 to cell #2. Thus, machines 3, 7, and 8 are assigned to cell #1, while machines 1, 2, 4, 5, 6, 9, and 10 are assigned to cell #2, as shown in Figure 5.2.
Cell No. M\P P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
M3 0 0 0 1 0 1 0 0 1 1
1 M7 0 0 1 1 0 1 0 0 1 1
M8 0 0 1 1 0 1 0 0 1 1
M1 0 1 0 0 1 0 0 1 0 0
M2 1 0 1 0 0 0 1 0 0 0
M4 1 0 0 0 0 0 1 0 0 0
2 M5 0 1 0 0 1 0 0 1 1 0
M6 1 0 0 0 0 0 1 0 0 0
M9 0 1 0 0 0 0 0 1 0 0
M10 0 1 0 0 1 0 0 0 0 0
Figure 5.2 Assignment of machines (3) Formation of part families
After calculating the sum of voids and exceptional elements for each part-cell combination (Figure 5.3), it became apparent that parts 3, 4, 6, 9, and 10 should be assigned to cell #1 and that parts 1, 2, 5, 7, and 8 should be assigned to cell #2 because this arrangement results in the least sum of voids and exceptional elements. Thus, the initial machine-part incidence matrix has been generated with a total grouping efficacy (Γ) of 57.69%, as shown in Figure 5.4.
Cell No. M\P P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
Sum of voids and exceptional elements
Cell #1 6 7 2 0 6 0 6 6 1 0
Figure 5.4 Solution configuration for NC=2
Step 3. Since 0.5769 0> , then set S*←S0, Let C =* 2, NC = +2 1. Repeat Steps 2 and 3 untilf( )S0 < f( )S* .
The relationship between the NC and the resulting grouping efficacy ( f S( )0 ) is shown in Figure 4.1. It is observed that f S( )0 increases as NC increases, and the optimal/near-optimal is achieved when NC=3. After that, efficacy starts to decrease as NC
increases. The suggested number of cells (C*=3) and the cell configurations (S*) are shown in Figure 5.6.
0 20 40 60 80 100
2 3 4
Number of cell(NC )
Grouping efficacy(%) .
Figure 5.5 Relationship between grouping efficacy and number of cells for example #1
Cell No. M\P P1 P7 P3 P4 P6 P9 P10 P2 P5 P8
M2 1 1 1 0 0 0 0 0 0 0
1 M4 1 1 0 0 0 0 0 0 0 0
M6 1 1 0 0 0 0 0 0 0 0
M3 0 0 0 1 1 1 1 0 0 0
2 M7 0 0 1 1 1 1 1 0 0 0
M8 0 0 1 1 1 1 1 0 0 0
M5 0 0 0 0 0 1 0 1 1 1
3 M9 0 0 0 0 0 0 0 1 0 1
M10 0 0 0 0 0 0 0 1 1 0
M1 0 0 0 0 0 0 0 1 1 1
Grouping efficacy ( Γ )=85.71%
Figure 5.6 Solution configuration for NC=3 Stage II of HCFA:
Step 1. Read solutions from stage one, including number of cells (C* ) and cell configurations (S*).
Step 2. Let S0←S*, NC C= *.
Step 3. Apply HSAM algorithm (as mentioned in Section 4.1.2.1) to improve S0 (Figure 5.6). Using C*=3, a total grouping efficacy ( Γ ) of 85.71% can be obtained. Note that the solution is the same as the initial solution found in stage I. This implies that our proposed SCM-based clustering algorithm can produce a good initial solution.
To evaluate the performance of our proposed HCFA, the mathematical model described in Section 3.2.2 is solved using Lingo 8.0 software. The Lingo solver status for example #1 is shown in Figure 5.7. It took about 32 seconds to obtain the optimal solution (0.857143). In contrast, our proposed HCFA was able to find the optimal solution in 0.14 seconds, thus illustrating the superiority of HCFA in solution efficiency. We believe this superiority will be even more significant as the problem size increases.
Figure 5.7 Lingo solver status for example #1 5.2 An Illustrative Example for Generalized CFP
Example #2 consists of 10 machines, 10 parts, and 18 process routings. The production data (production volumes, alternative process routings, and processing times) are
summarized in Table 5.2. In this table, entry (j, k) means the order of machine k in part route j, while the number in parentheses refers to production time. For instance, the order of machine #1 (M1) in routing #1 (R1) for part #1 (P1) is 1. The production time of machine
#1 (M1) in routing #1(R1) for part #1 (P1) is 2 minutes. Machine reliability information is shown in Table 5.3. For instance, the MTBF for machine #1 is 2241 minutes and breakdown cost is 1300. The maximum number of machines in each cell (Um) is limited to 4 and the minimum number of machines in each cell (Lm) is 2. Intercellular movement unit cost is assumed to be 5. The linear single-row layout is chosen. Implementation of the proposed HGCFA for cell formation, cell layout, and intracellular machine layout simultaneously with considerations of alternative routing, production volume, and machine reliability is described as follows:
Table 5.2 Initial machine-part matrix of example #2
PN P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
PV 150 95 130 80 120 95 135 145 100 150
RN R1 R2 R1 R2 R1 R1 R2 R1 R1 R2 R1 R2 R1 R2 R1 R2 R1 R2 M1 *1(2) 1(6) 1(2) 1(4) 1(5) 1(3) 1(3) 1(5)
M2 2(5) 2(2) 1(2) 2(6) 1(6) 2(6)
M3 1(6) 1(6) 3(3) 1(3) 1(6) 1(4) 2(2) 1(5)
M4 3(6) 1(4) 2(4) 3(4)
M5 3(5) 4(6) 3(5) 5(2) 3(3) 4(5)
M6 3(3) 3(4)
M7 2(4) 3(3) 3(5) 3(6) 3(6) 3(4) 3(5)
M8 1(6) 2(4) 2(3) 2(5) 2(2) 3(4) 2(3) 2(6) 2(6)
M9 3(2) 3(3) 4(5) 2(5) 3(4)
M10 2(6) 2(2) 2(5)
PV: Production Volume; PN: Part Number; RN: Routing Number; * Process Sequence (Production Times)
Table 5.3 Machine reliability information for example #2
(1) Determination of the initial number of cells and selection of cell layout
The initial number of cells is calculated: NC =
⎡
10/4⎤
=3. Then, cells are arranged in a linear single-row cellular layout, as shown in Figure 5.8.Cell #1 (1,1) Cell #2 (1,2) Cell #3 (1,3) Figure 5.8 Initial linear single-row layout
(2) Calculation of machine similarity matrix
The corresponding similarity matrix for machines can be obtained by using Eq. (4.4) and is presented in Table 5.4.
Table 5.4 Similarity matrix for machines in example #2
Machine 1 2 3 4 5 6 7 8 9 10
(3) Assignment of machines to cells
Three cells are to be formed initially. The largest coefficient in the similarity matrix is 1, indicating that machines 4 and 6 must be assigned to cell #1. The second largest coefficient in the matrix (0.82) appears in pair (7, 8). Since machines 7 and 8 have not been assigned to any cell, they are assigned to cell #2. Pair (5, 10) is considered next, and since machines 5 and 10 have not been assigned to any cell, they are assigned to cell #3. The next choice is pair (3, 7); machine 3 is assigned to cell #2 as well because machine 7 has been assigned there. Next is pair (2, 4); machine 2 is also assigned to cell #1 because machine 4 has been assigned there. Applying the same logic, we can determine that machine 1 should be assigned to cell #3, while machine 9 should be assigned to cell #3. Thus, machines 2, 4, and 6 are assigned to cell #1; machines 3, 7, and 8 are assigned to cell #2; and machines 1, 5, 9, and 10 are assigned to cell #3, as shown in Table 4.
Table 5.5 Formation of machine cells for numerical example #2
PN P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
PV 150 95 130 80 120 95 135 145 100 150
Cell No.
RN R1 R2 R1 R2 R1 R1 R2 R1 R1 R2 R1 R2 R1 R2 R1 R2 R1 R2
M2 2(5) 2(2) 1(2) 2(6) 1(6) 2(6)
1 M4 3(6) 1(4) 2(4) 3(4)
M6 3(3) 3(4)
M3 1(6) 1(6) 3(3) 1(3) 1(6) 1(4) 2(2) 1(5)
2 M7 2(4) 3(3) 3(5) 3(6) 3(6) 3(4) 3(5)
M8 1(6) 2(4) 2(3) 2(5) 2(2) 3(4) 2(3) 2(6) 2(6) M1 1(2) 1(6) 1(2) 1(4) 1(5) 1(3) 1(3) 1(5)
3 M5 3(5) 4(6) 3(5) 5(2) 3(3) 4(5)
M9 3(2) 3(3) 4(5) 2(5) 3(4)
M10 2(6) 2(2) 2(5)
(4) Selection of routings for each part
Routing 2 is selected by parts 1, 4, 6, 8, 9 and 10; while routing 1 is selected by parts 2, 3, 5 and 7, because these selections will result in the least total ICMC and MBC. Thus, Table 5.5 is rearranged as Table 5.6.
Table 5.6 Part routing assignment for numerical example #2
PN P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Total inter-cell move cost (ICMC) and the machine breakdown cost (MBC)
ICMC 1500 0 475 0 650 400 0 0 1425 0 0 1350 1450 0 500 500 750 0 MBC 839 581 610 1766 397 575 495 874 555 397 776 917 2584 1347 899 835 789 906
TC 2339 581 1085 1766 1047 975 495 874 1980 397 776 2267 4034 1347 1399 1335 1539 906
(5) Formation of part families
After calculating the sum of voids and exceptional elements for each part-cell combination (Figure 5.6), we observed that parts 1 and 7 should be assigned to cell #1, parts 2, 3, 4, 6, 9 and 10 should be assigned to cell #2, and parts 5 and 8 should be assigned to cell #3 because this arrangement results in the least sum of voids and exceptional elements.
Thus, the initial machine-part incidence matrix has been generated with total ICMC and MBC of 8843, as shown in Figure 5.9.
PN P1 P7 P2 P3 P4 P6 P9 P10 P5 P8
Total inter-cell move cost (ICMC) and the machine breakdown cost (MBC)=8843
ICMC 0 0 475 650 0 0 500 0 0 0
MBC 581 776 610 397 495 397 835 906 874 1347 TC 581 776 1085 1047 495 397 1335 906 874 1347
Sum of voids and exceptional elements
Cell #1 0 0 6 4 6 6 6 6 6 6
Figure 5.9 Initial solution of stage I for example #2
(6) Improvement of ICMC through HGCFA algorithm in stage I
The initial solution generated in Figure 5.9 with total ICMC and MBC of 8843 can be improved through the HGCFA algorithm in stage I. After 0.64 seconds CPU time, the final solution with a total ICMC and MBC of 8843 can be obtained. Note that the final solution is similar to the initial solution. This means that our proposed generalized SCM-based clustering algorithm can produce a good initial solution. So far, cells have been formed and cell layout has been determined. Solution regarding the machine layout (sequence) for each
In order to get the optimal solution, a pure integer liner model described in Section 3.4.2.1 is solved using a branch and bound (B&B) algorithm with the Lingo 8.0 software.
The Lingo solver status for example #2 is shown in Figure 5.10. The optimal solution (8843.22) is obtained in 4 seconds. In contrast, our proposed HGCFA was able to find the optimal solution in 1 second, thus implying the superiority of HGCFA in solution efficiency.
Similarly, we believe this superiority will be even more significant as problem size increases.
Figure 5.10 Lingo solver status for stage I (cell formation and inter-cell layout) Stage II of HGCFA:
(1) Generation of initial solution in stage II
The initial solution of the sequence of machines in each cell can be generated by the following procedure:
Step 1. Read the machine cells determined in stage one.
Step 2. Arrange the machine cells by cell number in an ascending order.
Step 3. Arrange the sequences of machines in each cell in an ascending order.
Using the above procedure, Figure 5.11, showing the initial machine sequences within each cell, can be constructed.
Cell # 1 2 3
Sequence # 1 2 3 4 1 2 3 1 2 3 Machine # 1 5 9 10 3 7 8 2 4 6
Figure 5.11 Initial configuration of machine sequence for example #2
Based on the solution from stage I and the initial configuration of machine sequence, the corresponding flow matrix for machines can be obtained by using Eq. (3.25). This is presented in Table 5.7. Using Eq. (3.26), the initial CFFI can be calculated as follows:
CFFI= 270 11.25%
(2) Improvement of CFFI through HGCFA algorithm in stage II
Through the proposed HTSCF in stage II, the CFFI can be improved to 65.21(%) after 0.19 seconds CPU time. The final corresponding configuration for the cell formation, cell layout, and intracellular machine layout is displayed in Figure 5.12.
PN P1 P7 P2 P3 P4 P6 P9 P10 P5 P8
Total inter-cell move cost (ICMC) and the machine breakdown cost (MBC)=8843
ICMC 0 0 475 650 0 0 500 0 0 0
MBC 581 776 610 397 495 397 835 906 874 1347 TC 581 776 1085 1047 495 397 1335 906 874 1347
Consecutive forward flow index (CFFI) = 65.21(%)
Cell #1 (1,1)
M4, M2, M6 Cell #3 (1,3)
M1, M9, M5, M10 Cell #2 (1,2)
M3, M8, M7
Figure 5.12 Final solution of stage II (cell formation, inter-cell layout and intra-cell layout)
To evaluate the performance of our proposed HGCFA, the mathematical model described in Section 3.4.2.2 is solved using Lingo 8.0 software. The Lingo solver status is shown in Figure 5.13. The optimal solution (0.6521) is obtained in 1 second. In contrast, our proposed HGCFA was able to find the optimal solution in 0.2 seconds, thus illustrating the superiority of HGCFA in solution efficiency. We believe this superiority will be even more significant as problem size increases.
Figure 5.13 Lingo solver status for stage II (intra-cell layout)
CHAPTER 6
COMPUTATIONAL RESULTS AND COMPARISONS
Two types of CFP are addressed in this thesis. Standard CFP is represented by a binary machine-part incidence matrix, while the generalized CFP had layout design and machine reliability considerations. In order to solve these problems quickly and effectively, two hybrid algorithms merging an SCM-based clustering algorithm and SA/WFA/TS meta-heuristics are presented in this thesis. The proposed algorithms were coded in C++
using Microsoft Visual Studio 6.0 and implemented on an Intel(R) 1.66 GHz PC with 1 GB RAM. Since the proposed methods might have stochastic features, five independent runs were performed for each test. The computation results for both types of CFP are shown and discussed separately in this chapter.
6.1 Computational Results for Standard CFP
To validate the quality of the solutions provided by the proposed algorithms for standard CFP, 35 test instances, each represented by a binary machine-part incidence matrix, from literature (Table 6.1) are used to evaluate the computational characteristics of our proposed HCFA. The matrices of the test problems range from 5×7 to 40×100, and consist of both well-structured and unstructured matrices. The parameters values for HCFA-HSAM, HCFA-HWFAM, and HCFA-HTSM throughout all runs are described in Table 6.2. Some studies in the literature allowed the existence of singletons in the solutions, while some did not. To make comparisons fair and meaningful, the computational results are shown and discussed separately in two subsections.
Table 6.1 Test instances from the literature for standard CFP
8 Chandrasekharan and Rajagopalan (1986a) 8×20 9 Chandrasekharan and Rajagopalan (1986b) 8×20
10 Mosier and Taube (1985a) 10×10
11 Chan and Milner (1982) 10×15
12 Askin and Subramanian (1987) 14×23
13 Stanfel (1985) 14×24
14 McCormick et al.(1972) 14×24
15 Srinivasan et al.(1990) 16×30
16 King (1980) 16×43
22 Chandrasekharan and Rajagopalan (1989) 24×40 23 Chandrasekharan and Rajagopalan (1989) 24×40 24 Chandrasekharan and Rajagopalan (1989) 24×40 25 Chandrasekharan and Rajagopalan (1989) 24×40 26 Chandrasekharan and Rajagopalan (1989) 24×40 27 Chandrasekharan and Rajagopalan (1989) 24×40
28 McCormick et al. (1972) 27×27
29 Carrie (1973) 28×46
30 Kumar and Vannelli (1987) 30×41
31 Stanfel (1985) 30×50
32 Stanfel (1985) 30×50
33 King and Nakornchai (1982) 30×90
34 McCormick et al. (1972) 37×53
35 Chandrasekharan and Rajagopalan (1987) 40×100
Table 6.2 Parameters setting for HCFA-HSAM, HCFA-HWFAM, and HCFA-HTSM Algorithm Parameter settings
HCFA-HSAM Initial temperature (T0): 80 Final temperature (Tf): 0.002 Cooling rate (α): 0.7
Markov chain length (L): 30 Mutation probability (β): 0.8 HCFA-HWFAM Iteration limit (Nmax) : 100
Initial mass (W0 ): 40 Initial velocity (V0 ): 15 Base momentum (Tm) : 100 Mutation probability (β): 0.8
HCFA-HTSM Maximum number of iterations (Nmax): 3000
Maximum number of solution has not been improved (Nstag ): 1000 Tabu list size ( tls ): 7
Mutation probability (β): 0.8
6.1.1 Solutions allowing singletons
The HGA (Tariq et al., 2009) and the GA (Mahdavi et al., 2009) are compared in order to demonstrate the power of the proposed algorithm given that singletons are allowed (i.e., Lm=1). The computational results are summarized and compared in Table 6.3. The best values for grouping efficacy (Γ) achieved by our proposed HCFA-HSAM, HCFA-HWFAM, and HCFA-HTSM are the same. In addition, our proposed algorithms are better than or equal to previously reported results except in problems #18 and #27. To be more specific, our proposed algorithms were able to improve the best values for grouping efficacy compared with the HGA method for 23 problems (#1, #5, #6, #7, #10, #12, #13, #14, #15,
#16, #17, #19, #20, #25, #26, #27, #28, #29, #30, #31, #32, #33, and #34); while for 11 problems, our proposed algorithms obtained grouping efficacy values that are equal to the best results of the HGA method. Compared with the GA method, our proposed algorithms improved the best values for grouping efficacy for 5 problems (#16, #18, #20, #25, and #26);
while for 16 problems, our proposed algorithms obtained grouping efficacy values that are equal to the best results of the GA method. Thus, it can be concluded that our proposed algorithms perform better than HGA and GA, especially in test problems with larger sizes.
One noteworthy observation is that all the best solutions can be found in less than 63 seconds regardless of the problem size, thus illustrating the superiority of our proposed algorithms in solution efficiency.
6.1.2 Solutions not allowing singletons
In order to demonstrate the power of the proposed algorithms when singletons are not allowed (i.e., Lm=2), comparisons against the KHMCF (Unler and Gungor, 2009) were performed. The computational results are summarized and compared in Table 6.4. The best values for grouping efficacy achieved by our proposed HCFA-HSAM, HCFA-HWFAM, and HCFA-HTSM methods are similar and our proposed algorithms yielded values better than or equal to those previously reported except in problem #32. To be more specific, HCFA improved the best grouping efficacy values of the KHMCF method for 9 problems (#12, #14, #16, #17, #18, #20, #26, #31, and #34); while for 10 problems, our proposed algorithms obtained grouping efficacy values that are equal to the best results of the KHMCF method. However, it should be noted that our proposed algorithms can achieve the best solutions in less than 56 seconds regardless of the problem size, thereby illustrating the superiority of our proposed algorithms in solution efficiency.
Table 6.3 The computational results in the case where singletons are allowed (Lm=1)
Test instances HGA GA a HCFA-HSAM b HCFA-HWFAM b HCFA-HTSM b
No. Source Size
(m×p) Γ(%) NC Γ(%) CPU
time(s) NC Γ(%) CPU
time(s) NC Γ(%) CPU
time(s) NC Γ(%) CPU time(s) 1 King and Nakornchai (1982) 5×7 73.68 - - - 3 75.00+ 0.17 3 75.00+ 0.02 3 75.00+ 0.14 2 Waghodekar and Sahu (1984) 5×7 69.57 2 69.57 0.01 2 69.57 0.13 2 69.57 0.01 2 69.57 0.12 3 Seifoddini (1989) 5×18 79.59 2 79.59 0.02 2 79.59 0.19 2 79.59 0.03 2 79.59 0.16 4 Kusiak and Cho (1992) 6×8 76.92 2 76.92 0.01 2 76.92 0.15 2 76.92 0.02 2 76.92 0.13 5 Kusiak and Chow (1987) 7×11 58.62 5 60.87 0.02 5 60.87 0.28 5 60.87 0.03 5 60.87 0.22 6 Boctor (1991) 7×11 70.37 4 70.83 0.03 4 70.83 0.24 4 70.83 0.03 4 70.83 0.20 7 Seifoddini and Wolfe (1986) 8×12 68.30 - - - 4 69.44+ 0.29 4 69.44+ 0.03 4 69.44+ 0.22 8 Chandrasekharan and Rajagopalan (1986a) 8×20 85.25 3 85.25 0.03 3 85.25 0.29 3 85.25 0.07 3 85.25 0.22 9 Chandrasekharan and Rjagopalan (1986b) 8×20 58.72 2 58.72 0.03 2 58.72 0.28 2 58.72 0.03 2 58.72 0.21 10 Mosier and Taube (1985a) 10×10 70.59 5 75.00 0.03 5 75.00 0.42 5 75.00 0.09 5 75.00 0.32 11 Chan and Milner (1982) 10×15 92.00 3 92.00 0.03 3 92.00 0.29 3 92.00 0.09 3 92.00 0.22 12 Askin and Subramanian (1987) 14×23 70.83 - - - 7 73.13+ 1.03 7 73.13+ 0.38 7 73.13+ 0.79 13 Stanfel (1985) 14×24 70.51 7 71.83 0.57 7 71.83 1.12 7 71.83 0.40 7 71.83 0.79 14 McCormick et al.(1972) 14×24 51.96 - - - 8 53.26+ 2.13 8 53.26+ 1.65 8 53.26+ 1.78 15 Srinivasan et al.(1990) 16×30 67.83 - - - 6 68.99+ 1.72 6 68.99+ 1.24 6 68.99+ 1.24 16 King (1980) 16×43 54.86 7 56.13 1.53 8 56.85+ 2.74 8 56.85+ 1.30 8 56.85+ 2.26 17 Carrie (1973) 18×24 54.95 - - - 9 57.73+ 2.25 9 57.73+ 2.24 9 57.73+ 2.00 18 Mosier and Taube (1985b) 20×20 43.45 5 42.94 0.62 5 43.36- 1.83 5 43.36- 1.73 5 43.36- 1.68 19 Kumar et al. (1986) 20×23 49.65 - - - 7 50.81+ 1.27 7 50.81+ 0.97 7 50.81+ 1.12
20 Carrie (1973) 20×35 76.14 5 77.91 1.25 5 78.40+ 1.58 5 78.40+ 0.99 5 78.40+ 1.12 21 Boe and Cheng (1991) 20×35 58.38 - - - 5 58.38 1.14 5 58.38 1.11 5 58.38 1.19 22 Chandrasekharan and ajagopalan (1989) 24×40 100.00 7 100.00 1.60 7 100.00 1.73 7 100.00 0.73 7 100.00 1.26 23 Chandrasekharan and Rajagopalan (1989) 24×40 85.11 7 85.11 1.92 7 85.11 1.74 7 85.11 2.14 7 85.11 1.29 24 Chandrasekharan and Rajagopalan (1989) 24×40 73.51 7 73.51 1.48 7 73.51 1.76 7 73.51 1.56 7 73.51 1.28 25 Chandrasekharan and Rajagopalan (1989) 24×40 52.50 10 52.87 3.26 11 53.29+ 6.23 11 53.29+ 7.05 11 53.29+ 5.63 26 Chandrasekharan and ajagopalan (1989) 24×40 46.84 12 48.85 6.24 12 48.95+ 6.30 12 48.95+ 6.15 12 48.95+ 7.48 27 Chandrasekharan and Rajagopalan (1989) 24×40 44.85 12 47.26 11.23 12 46.26- 5.76 12 46.26- 5.98 12 46.26- 6.54 28 McCormick et al. (1972) 27×27 54.31 - - - 5 54.82+ 2.57 5 54.82+ 6.21 5 54.82+ 2.29 29 Carrie (1973) 28×46 46.43 - - - 11 47.23+ 6.57 11 47.23+ 24.57 11 47.23+ 9.76 30 Kumar and Vannelli (1987) 30×41 60.74 - - - 14 62.86+ 7.88 14 62.86+ 13.91 14 62.86+ 9.15 31 Stanfel (1985) 30×50 59.66 13 60.12 19.30 13 60.12 9.11 13 60.12 12.78 13 60.12 7.84 32 Stanfel (1985) 30×50 50.51 14 50.83 22.21 14 50.83 11.68 14 50.83 18.21 14 50.83 13.37 33 King and Nakornchai (1982) 30×90 44.67 - - - 16 47.85+ 22.81 16 47.85+ 63.89 16 47.85+ 32.82 34 McCormick et al. (1972) 37×53 59.60 - - - 3 60.50+ 2.20 3 60.50+ 22.97 3 60.50+ 2.33 35 Chandrasekharan and Rajagopalan (1987) 40×100 84.03 10 84.03 99.63 10 84.03 8.34 10 84.03 17.43 10 84.03 5.67
a Run on a Pentium IV, 2.1 GHz PC.
b Run on a Pentium IV, 1.6 GHz PC.
+: Solutions obtained by the proposed approach are superior to best solutions found in the literature.
+: Solutions obtained by the proposed approach are superior to best solutions found in the literature.