Production, Manufacturing and Logistics
A water flow-like algorithm for manufacturing cell formation problems
Tai-Hsi Wu
a,*, Shu-Hsing Chung
b, Chin-Chih Chang
ba
Department of Business Administration, National Taipei University, 151, University Road, San Shia, Taipei 407, Taiwan
bDepartment of Industrial Engineering and Management, National Chiao Tung University, 1001, Ta Hsueh Road, Hsinchu 300, Taiwan
a r t i c l e
i n f o
Article history: Received 29 June 2008 Accepted 13 January 2010 Available online 18 January 2010 Keywords: Manufacturing Heuristics Cell formation Meta-heuristics
a b s t r a c t
Available research on the manufacturing cell formation problem shows that most solution approaches are either single- or multiple-solution-agent-based, with a fixed size of solution agents. Frequent prob-lems encountered during the process of solving the cell formation problem include solutions being easily trapped in local optima and bad solution efficiency. Yang and Wang [Yang, F.-C., Wang, Y.-P., 2007. Water flow-like algorithm for object grouping problems. Journal of the Chinese Institute of Industrial Engineers, 24 (6), 475–488] proposed the water flow-like algorithm (WFA) to overcome the shortcomings of single-and multiple-solution -agent-based algorithms. WFA has the features of multiple single-and dynamic numbers of solution agents, and its mimicking of the natural behavior of water flowing from higher to lower levels coincides exactly with the process of searching for optimal solutions. This paper therefore adopts the WFA logic and designs a heuristic algorithm for solving the cell formation problem. Computational results obtained from running a set of 37 test instances from the literature and newly created have shown that the proposed algorithm has performed better than other benchmarking approaches both in solution effectiveness and efficiency, especially in large-sized problems. The superiority of the proposed WFACF over other approaches from the literature should be attributed to the collaboration of the WFA logic, the proposed prior estimation of the cell size, and the insertion-move. The WFA is a novel heuristic approach that deserves more attention. More attempts on adopting the WFA logic to solve many other combinatorial optimization problems are highly recommended.
Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction
Manufacturing systems have become more efficient and pro-ductive with the application of group technology (GT) within man-ufacturing environments. GT groups parts with similar design characteristics or manufacturing characteristics into part families. One application of GT is cellular manufacturing (CM). By adopting GT in CM, benefits such as reduced inventory, reduced capacity, re-duced labor and overtime costs, shorter manufacturing lead times, and faster response to internal and external changes including ma-chine failures, product mix, and demand changes are realized. The entire production system is decomposed into production cells through CM. Machines are then assigned to these cells to process one or more part families so that each cell operates independently and intercellular movements are minimized or the number of part flows processed within cells is maximized.
Cell formation (CF) is one of the most important steps in CM. It becomes difficult to obtain optimal solutions in an acceptable amount of time, especially for problems with large sizes. Extensive
research has been devoted to CF problems, and various methods have been proposed to identify machine cells and part families. These methods fall into five categories: array-based, hierarchical/ non-hierarchical clustering, graph-based, mathematical program-ming, and heuristics/meta-heuristics.
Array-based methods obtain visible groupings of machines and parts until a satisfactory solution is found by repeatedly rearrang-ing the rows and columns of the machine–part matrix. These methods include the bond energy algorithm (BEA) byMcCormick et al. (1972), the rank order clustering (ROC) algorithm byKing (1980), the ROC2 byKing and Nakornchai (1982), the direct clus-tering algorithm (DCA) byChan and Milner (1982), and the close neighbor algorithm (CAN) byBoe and Cheng (1991). Hierarchical/ Non-hierarchical clustering methods in studies by McAuley (1972), Mosier and Taube (1985b), Seifoddini and Wolfe (1987), and Yasuda and Yin (2001)use a measure of similarity or dissimi-larity for the grouping of machines or parts.Yin and Yasuda (2006) discussed the similarity coefficients developed to date for use in solving the CF problem. In graph-based approaches, the process of forming manufacturing cells starts by collecting the problem data and then converting them into a weighted graph representa-tion. In these approaches, nodes represent machines and arcs rep-resent their relationships, defined as the value of total part flow
0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.01.020
*Corresponding author. Tel.: +886 2 86746574; fax: +886 2 86715912. E-mail addresses: [email protected] (T.-H. Wu), [email protected]. edu.tw (S.-H. Chung),[email protected](C.-C. Chang).
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between machines. Complicated features of the CF problems have been incorporated in models using mathematical programming ap-proaches, but these usually become computationally intractable especially for large-sized problems.Kusiak (1987)presented a p-median model, whileBoctor (1991)suggested a linear 0–1 formu-lation for the CF problem. However, the classical p-median model is limited to small-sized CF problems since it requires many binary variables.Won and Lee (2004)hence proposed two modified p-median formulations to resolve this difficulty.Chen and Heragu (1999)proposed two stepwise decomposition approaches to solve large-sized CF problems. Both approaches analyze the part–ma-chine relation, decompose the original problem into several large subsystems, and solve each subsystem using optimal solution technique.
Meta-heuristic algorithms such as tabu search, genetic algo-rithm (GA), simulated annealing (SA), and neural networks all comprise another class of search methods that have been adopted to solve the CF problem and its variants efficiently due to their excellent performance in solving combinatorial optimization prob-lems.Sun et al. (1995)presented a short-term tabu search-based algorithm for solving the CF problem which aimed to minimize the intercellular parts flows.Wu et al. (2004), on the other hand, maximized the parts flow within cells using a long-term tabu search-based algorithm.Cao and Chen (2004) developed a tabu search considering cell setup cost to solve the CF problem.Cheng et al. (1998)formulated a traveling salesman problem (TSP) to ad-dress the CF problem and proposed a solution methodology based on GA, while a hierarchical clustering approach based on genetic programming was presented by Dimopoulos and Mort (2001). Onwubulo and Mutingi (2001)developed a GA that accounts for intercellular movements and cell-load variation.Gonçalves and Re-sende (2004)reported promising results with a hybrid algorithm that combined a local search and a GA. Instead of the generally used simple machine encoding,Filho and Tiberti (2006)proposed a GA based on group encoding. The CF problem was addressed byJames et al. (2007)using a hybrid grouping GA that combines a local search with a standard grouping GA to form machine–part cells.Wu et al. (2008)proposed a simple yet effective simulated annealing-based algorithm (SACF), for solving the CF problems where singletons (cells having less than two parts or two ma-chines) are allowed.Yang and Yang (2008)proposed a modified ART1 neural learning algorithm, in which the vigilance parameter can be simply estimated by the data so that it is more efficient and reliable compared with previous neural network approaches.
Among the aforementioned heuristic algorithms, the SA and the tabu search are solution-agent-based algorithms. A single-agent-based heuristic algorithm searches the solution space step by step through the usage of systematic or random neighborhood exploration. Some of them employ adaptive memory, while some of them are memoryless. The GA, however, belongs to the group of multiple-solution-agent-based algorithms, which starts the opti-mization with a set of possible solutions, not only one possible solu-tion. The behavior of GAs is characterized by a balance between exploitation and exploration in the search space. The balance is strongly affected by strategy parameters such as maximum gener-ation and populgener-ation size, and GA’s performance depends very much on details of the settings of these parameters. Fixed parame-ters are usually used in most of the GA applications. Since the GA is an intrinsically dynamic and adaptive process, the use of constant parameters is thus in contrast to the general evolutionary spirit. Therefore, it is natural to try to modify the values of parameters during the run of the algorithm (Gen and Cheng, 2000).Yang and Wang (2007)stated that neither the single nor the multiple agent method is agile enough to conduct an efficient and effective solu-tion search. They hence proposed a methodology using a dynamic size of solution agents, the water flow-like algorithm (WFA).
The design of the WFA method was inspired by water flowing from higher to lower levels, where a flow will split into multiple subflows when it moves through uneven terrains. Conversely, sub-flows merge when they meet at the same level. Water flow ceases and stagnates at the lowest depressions, when momentum cannot expel water out of the depressions. Water flowing is analogous to problem solving. A flow is regarded as a solution agent, the solu-tion space of a problem is the geographical terrain, and the altitude of a flow represents the objective function value. Since the number of flows dynamically changes in this method, it is an agent popula-tion-varying method.
To our knowledge, the WFA has not been applied to solve any other combinatorial optimization problems aside from the bin packing problems studied inYang and Wang (2007). Besides, the solution process of searching for the optima is so analogous to water moving to a lower position. The above two points motivate our using WFA for solving the CF problem.
In this paper, we adopt the WFA logic and develop a heuristic algorithm (WFACF) for the CF problem. This approach combines a WFA using specifically tailored operations with a similarity coeffi-cient method constructed for generating quick initial solutions for later improvement, and two solution improving strategies for find-ing the best neighborhood solution. We test 37 problems from the literature and newly created, and provide comparisons against sev-eral algorithms from the literature. The WFACF is shown to per-form better than other benchmarking approaches both in solution effectiveness and efficiency, especially in large-sized prob-lems. According to our knowledge, this is the first time WFA has been used to successfully solve combinatorial optimization prob-lems aside from the bin packing probprob-lems. Apart from the above contribution, we further verify the effects of the two primary oper-ations of the WFA, namely, the evaporation and the precipitation, and conclude that they may not be as decisive as they were claimed in the original WFA.
The remainder of the paper is organized as follows. Section2 describes the problem definition while Section3details the pro-posed solution algorithm for the CF problem. Section4shows the computational results on test problems adopted from the literature and includes thorough analyses and discussions. The conclusions are laid out in Section5.
2. Cell formation problem
CF in a given 0–1 machine–part incidence matrix involves a rearrangement of rows and columns of the matrix to create part families and machine cells. In this research, we attempt to deter-mine a rearrangement minimizing intercellular movement and maximizing the utilization of the machines within a cell. Fig. 1 gives a sample solution matrix for a CF problem, in which two blocks can be observed along the diagonal of the solution matrix.
There have been several measures of goodness of machine–part groups in CM in the literature. Two measures frequently used are grouping efficiency (Chandrashekharan and Rajagopalan, 1986a) and grouping efficacy (Kumar and Chandrasekharan, 1990). Group-ing efficiency
g
is defined as follows:P2 P3 P5 P1 P4 Cel l 1 M2 1 1 1 0 0 M4 1 1 1 0 0 M1 0 0 0 1 1 Cel l 2 M3 0 1 0 1 1 M5 0 0 0 1 0
g
¼ qg
1þ ð1 qÞg
2;where
g
1is the ratio of the number of 1s in the diagonal blocks tothe total number of elements in the diagonal blocks of the final ma-trix,
g
2is the number of 0s in the off-diagonal blocks to the total number of elements in the off-diagonal blocks of the final matrix, and q is a weight factor. Any 1s outside the diagonal blocks are called ‘‘exceptional elements,” and any 0s inside the diagonal blocks are called ‘‘voids.”Although grouping efficiency has been widely used, critics ar-gue that in some cases, the size of the matrix impairs its discrimi-nation ability. To overcome this problem, Kumar and Chandrasekharan (1990)proposed another measure, grouping effi-cacyC, which is defined as:
C
¼e e0 e þ ev;
where e is the total number of 1s in the matrix, e0is the total
num-ber of exceptional elements, and ev is the total number of voids. Grouping efficacy ranges from 1 to 0, with 1 as the perfect grouping. As grouping efficacy has been widely accepted in recent studies regarding the CF problem, it is used as the performance measure for the proposed algorithm in this study.
3. Water flow-like approach for the CF problem
This paper adopts the WFA logic and designs a heuristic algo-rithm for solving the CF problem because the WFA has the features of multiple and dynamic numbers of solution agents. Adopted behaviors from fluid flows of the WFA are introduced in Section 3.1, followed by the proposed heuristic algorithm. Construction of the initial solutions is given in Section3.2, strategies and proce-dures for improving the solutions through neighborhood searching are presented in Section3.3, while the complete algorithm of the proposed WFACF is described in detail in Section3.4.
3.1. Water flow-like algorithm
The design of the WFA method (Yang and Wang, 2007) was in-spired by the natural behavior of water flowing from higher to low-er levels. On the earth’s surface, a flow will split into multiple subflows when rugged terrains are traversed. Subflows, however, will merge when they arrive at the same location. Governed by gravity and driven by fluid momentum, flows can run to higher levels or run over bumps to navigate various terrains. Water flow-ing will cease and stagnate at locally or globally lowest depres-sions, when the momentum left cannot expel the water out of the depressions. As the solution space of a problem can be mapped to the geographical terrain, and the objective value is mapped to the altitude, each flow can then be regarded as a solution agent. Water moving to a lower position can be considered as a solution searching for the optima. The solution search process is thus mod-eled as water flowing.
Yang and Wang (2007)have adopted several natural behaviors of water flow in presenting the WFA (Dougherty and Marryott, 1991). Their design ideas are summarized as follows:
1. Driven by gravity and governed by the energy conservation law, water will constantly flow to lower altitudes. Therefore, the solution search will recursively move from inferior to superior solutions.
2. Fluid momentum drives water moving forward through rough terrains. A flow will split into subflows when it encounters rug-ged terrain and when its momentum exceeds a base amount for splitting. WFA simulates this behavior as an agent forking oper-ation; that is, more than two agents are derived from a single
agent. A flow with larger momentum will generate more streams of subflows than one with less momentum. A flow with limited momentum will yield to the landform and maintain a single flow. Therefore, the fluid momentum of a flow is recalcu-lated to determine the number of subflows that can be forked after each move.
3. Water flows to lower altitudes and occasionally swells to higher altitudes as long as the kinetic energy is larger than the required potential energy. To avoid being trapped within a local mini-mum, WFA allows the water to flow to a worse location to broaden the exploration area, provided it has enough kinetic energy.
4. A number of flows merge into a single flow when they meet at the same location. WFA reduces the number of solution agents when multiple agents result in the same objective value to avoid redundant searches.
5. Water flows are subject to water evaporation in the atmo-sphere. The evaporated water will return to the ground during rainfall. In WFA, a part of the water flow is manually removed to mimic water evaporation. After evaporation, a precipitation operation is implemented in WFA to simulate natural rainfall and explore a wider solution area.
On the basis of the above idea, the computational flow of the WFA consists of four primary operations:(1) flow splitting and moving,(2)flow merging,(3)water evaporation, and(4) precipita-tion. Before proceeding to the descriptions of these four operations, we introduce some notations.
G iteration limit
W0 initial mass of original flow Wi mass of flow i
V0 initial velocity of original flow Vi velocity of flow i
T base momentum
n upper limit on the number of subflows split from a flow ni number of subflows forked from flow i
N total number of water flows in the current iteration Xi solution corresponding to flow i
Uik solution corresponding to subflow k split from flow i wik mass of subflow k split from flow i
lik
velocity of subflow k split from flow idik altitude drop from flow i to its subflow k; equivalently, changes in objective value from solution i to its neighbor-hood solution k
g gravitational acceleration
t a prescribed number of iterations a flow will be removed completely by evaporation
3.1.1. Flow splitting and moving operation
It is assumed that there is only one water flow in the beginning of the WFA, and that its location is randomly generated. Driven by fluid momentum and potential energy, the flow starts to move to new locations to explore the solution space for new and better solutions. Yang and Wang (2007)used constant-step movement to the best neighborhood solution when solving the object grouping problem in their paper. However, various flow-moving strategies can be de-signed and applied depending on the characteristics of different optimization problems.
In the WFA, flow splitting results from capable momentum, and a flow with higher momentum generates more subflows than a low-er one. The locations of the split subflows are dlow-erived from the neighboring locations of the original flow. When a flow does not split, it goes on as a single stream to the best feasible neighboring location. Let N be the number of water flows in the current iteration, the number of subflows ni forked from flow i ði ¼ 1; 2; . . . ; NÞ is
stays where it is and is considered a stagnant solution. A flow can split into subflows only when its momentum exceeds a predefined base momentum T. The number of subflows is determined by divid-ing its momentum by the base momentum T. If the momentum of a flow is between 0 and T, it is treated as a single stream moving to a new location without splitting. As the process of the WFA proceeds, it is possible that the number of subflows grows exponentially and exhausts the computational resource.Yang and Wang (2007) sug-gests imposing an upper limit, n, on the number of subflows forked from a flow at each iteration. The number of subflows split from a flow can thus be obtained through the formula below:
ni¼ min max 1; int
WiVi T ; n : ð1Þ
When the flow is split into subflows, its original mass has to be accordingly distributed to subflows based on the rule designed. Yang and Wang (2007)distribute the mass based on the ranks of the subflows, as shown in Eq.(2)
wik¼ niþ 1 k Pni r¼1 r 0 B B B @ 1 C C C AWi; k ¼ 1; 2; . . . ; ni: ð2Þ
For instance, if Wi¼ 5 and ni¼ 3, then
wi1¼ 3 1 þ 2 þ 3 5; wi2¼ 2 1 þ 2 þ 3 5; wi3¼ 1 1 þ 2 þ 3 5:
The velocity of each subflow is computed from the equation of en-ergy conservation.
l
ik, the velocity of subflow k split from flow i, is:l
ik¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V2 i þ 2gdik q ; if V2 i þ 2gdik>0; 0; otherwise; ( ð3Þwhere g is the gravitational acceleration, and dikis the altitude drop
from flow i to its subflow k, that is, the improvement of objective value moving from current solution i to its neighborhood solution k. When V2
i þ 2gdik<0, the momentum delivered to subflow k has
been used up, implying that this subflow will stagnate in its current location (solutions trapped in local optima) with no splitting and movement. Such stagnant flow will gradually evaporate into the atmosphere, returning to the ground by precipitation later on.
At the end of the splitting and moving operation, the original flow is hence discarded because subflows have been generated. Information regarding the current number of subflows and solu-tions sets will then be recorded.
3.1.2. Flow-merging operation
When more than two flows move to the same location, they will merge into one flow with a bigger mass and momentum. Whether a flow shares the same location with others in the WFA is thus sys-tematically examined. If a flow does share the same location, the latter flow is then merged into the former one. Assuming it is found that flows i and j share the same location, then flow j will be de-leted and the mass and velocity of flow i will be updated as follows: Wi¼ Wiþ Wj; ð4Þ Vi¼ WiViþ WjVj Wiþ Wj : ð5Þ
Using the flow-merging operation, the WFA reduces the number of solution agents when multiple agents result in the same objective value to avoid redundant searches.
3.1.3. Water evaporation operation
It is natural that water evaporates and returns to the ground through precipitation after possible movement from its original
location. Water evaporation and precipitation coincide with the ‘‘escaping from local optima” mechanism many heuristic algo-rithms nowadays use to avoid being trapped and to explore more solution spaces.
Each flow in the WFA is subject to water evaporation, where part of the water evaporates into the atmosphere. It is determined that a flow will be completely removed after a prescribed number of iterations, t; that is, the masses of all flows are decreased by the ratio of 1/t as shown in Eq.(6), every time evaporation occurs.
Wi¼ 1 1 t Wi; i ¼ 1; 2; . . . ; N: ð6Þ 3.1.4. Precipitation operation
When water vapor accumulates to a certain volume, it will re-turn to the ground in some form such as rain. In the original WFA, two types of precipitation are performed to simulate the nat-ural cycle of water, enforced and regular precipitation.
Enforced precipitation is applied when all flows are grounded with zero velocities. Under this circumstance, all flows are forced to evaporate into the atmosphere and then returned to the ground without changing the number of current flows. However, the loca-tions of these returned flows are deviated stochastically from the original ones. Mass of W0 is proportionally distributed to flows
based on their original mass with the same initial velocity. Conse-quently, the mass assigned to flow i, W0
i, can be determined using
Eq.(7) W0 i¼ Wi PN k¼1Wk ! W0: ð7Þ
Regular precipitation is applied periodically in balance with water evaporation. The regular precipitation operation is performed every t (same t value as in evaporation) iterations to pour down the evap-orated water. Note that the cumulated mass of the evapevap-orated water is W0PNk¼1Wk. Thus, instead of using Eq.(7), the mass
as-signed to flow i, W0i, is determined using Eq.(8)when applying
reg-ular precipitation. The newly poured flow joins the current solution set, thus increasing the number of current solutions. In addition, both the enforced and regular precipitation might generate several new flows on the same locations. A flow-merging operation will be executed to eliminate possible redundant flows
W0 i¼ Wi PN k¼1Wk W0 XN k¼1 Wk ! : ð8Þ
WFA has the features of multiple and dynamic numbers of solu-tion agents, and its mimicking of the natural behavior of water flowing from higher to lower levels coincides exactly with our pro-cess of searching for optimal solutions. This paper therefore adopts the WFA logic and designs a heuristic algorithm for solving the CF problem.
3.2. Construction of initial solutions
A large number of similarity coefficients methods (SCM) have been proposed for the CF problem. In this study, due to their sim-plicity and easy implementation, the SCM approach is used to gen-erate quick initial solutions for later improvement by the proposed algorithm. It is well known that decomposing an originally difficult problem into several subproblems usually increases problem-solv-ing efficiency. An intuitive solution approach is to decompose the entire problem into two subproblems that deal with the assign-ment of machines and parts, respectively, since the CF problem considers the grouping of machines and parts. In our construction of the initial solution, machines assignment is determined in the
first stage, while the assignment of parts follows in the second stage.
Our approach for generating initial solutions consists of three steps: (1) compute similarity values between machine pairs and construct a similarity matrix; (2) use a clustering rule to process the values in the similarity matrix and form machine cells; and (3) assign parts to machine cells using a parts assignment procedure.
3.2.1. Machines assignment
As in McAuley’s research (McAuley, 1972), Jaccard’s similarity measure is used to evaluate the similarity between machines as an important index for assigning machines to cells in this subprob-lem. The similarity measure, denoted Sij, is defined as Sij¼
aij
aijþbijþcij, where aijrepresents the number of parts processed by both
ma-chines i and j; while bijis the number of parts processed by
ma-chine i but not by mama-chine 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 gen-erate the initial machines assignment by using the following gree-dy rule: the higher the similarity measure a pair of machines has, the higher the priority its machines have for placement in the same cell. This process is repeated until all machines have been assigned to cells.
Considering the sample machine–part matrix inFig. 2a, the cor-responding similarity matrix for machines is displayed inFig. 2b. Assuming that two cells are to be formed, the largest coefficient in the matrix ofFig. 2b is 0.67, indicating that machines 2 and 4 must be assigned to the same cell, for instance cell 1. We proceed with the second-largest coefficient in the matrix, 0.5, appearing in pairs (1, 3) and (1, 5). As these three machines do not have any rela-tionship with any machines in cell 1, together they should be as-signed to the next cell, cell 2. Fig. 3 shows the machines assignment using the proposed greedy rule.
3.2.2. Parts assignment
In this procedure, the parts are assigned to machine cells so that the number of voids and exceptional elements – major compo-nents comprising the formula of grouping efficacy – are explicitly considered. It is 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 the 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 machines assignment shown in Fig. 3 are used to demonstrate the above procedure. After calculating the sum of voids and exceptional elements for each part-cell combination, it can be observed inFig. 4that parts 2, 3 and 5 are assigned to cell 1, while parts 1 and 4 are assigned to cell 2. The initial solution ma-trix for this CF problem can thus be obtained.
3.3. Searching neighborhood solutions (flow splitting and moving) The initial solution just obtained can be considered as the start-ing water flow, ready to split iteratively into subflows, traverse the solution space, and move toward the optimal solution through the WFA logic.
Among the four operations in the WFA, flow splitting and mov-ing operation is endowed with the mission of searchmov-ing for better neighborhood solutions and ultimately, the optimal solution. In solving the CF problem, we have designed two strategies, namely, machine shifting and insertion-move, for finding the best neigh-borhood solution of the current solution.
3.3.1. Machine shifting
In the WFA, the split subflows’ locations are derived from the neighboring locations of the original flow. In terms of the CF prob-lem, the initial solution matrix obtained in Section3.2is the origi-nal water flow. Before designing a strategy for finding a good neighborhood solution, it is crucial to identify and analyze factors influencing solution efficiency and quality. In Section3.2, after ma-chines assignment has been determined, a simple procedure for assigning parts follows. The procedure for assigning machines actually leads the procedure for assigning parts in this study. The solution quality of machines assignment thus plays a very critical role in the success of the entire solution quality. Hence, a ‘‘machine shifting” strategy is proposed to find a rough direction for the neighborhood solutions in the first stage promptly; the exact loca-tion for the best neighborhood soluloca-tion is then obtained through the ‘‘insertion-move” strategy in the second stage.
The machine-shifting strategy is implemented by reassigning a machine to any cells other than the current one based on a pre-scribed probability r. For each machine in the current solution, a random number from (0, 1) is first drawn. If the value is greater than r, then the machine is assigned to other cells; otherwise, it stays in the current cell. If singletons are not allowed, an additional check is performed after the machine shifting. The procedure of machine-shifting strategy in pseudo-code format is described in Fig. 5.
3.3.2. Insertion-move
The machine-shifting strategy only gives a quick and rough direction of the location of the neighborhood solutions. To assure that high-quality neighborhood solutions can be found at each iter-ation of the algorithm, a deliberate strategy has to be developed to
P1
P2
P3
P4
P5
M1
1
0
0
1
0
M2
0
1
1
0
1
M3
1
0
0
0
0
M4
0
1
1
0
0
M5
0
0
0
1
0
(a)
machine-part matrix
M1
M2
M3
M4
M5
M1
-
0
0.5
0
0.5
M2
-
0
0.67
0
M3
-
0
0
M4
-
0
M5
-
(b)
similarity matrix for machines
Fig. 2. Machine–part matrix and corresponding similarity matrix for machines.
P1 P2 P3 P4 P5 Cell 1 M2 0 1 1 0 1 M4 0 1 1 0 0 M1 1 0 0 1 0 Cell 2 M3 1 0 0 0 0 M5 0 0 0 1 0
find the best neighborhood solution. The neighborhood of a given solution is defined as the set of all feasible solutions reachable by a single move. This study implements an insertion-move, which is an operation that moves a machine i from its current cell k (source cell) to a new cell k0(destination cell). The new move is
de-noted (k0, i). For the insertion-move, a move that results in the most improvement in the objective function value from the current solu-tion is selected, that is:
Zðk0;iÞ ¼ maxfobjðk0;iÞ objðk;iÞ;
8
k; k02 C; k0–k;8
i 2 Mg;where C and M are the sets for cells and machines, respectively. Fig. 6demonstrates the splitting and moving operation we pro-posed for searching neighborhood solutions for the CF problem. As flow i splits into subflows, and the number of subflows niis
deter-mined by its momentum, say ni equals k. The machine-shifting
strategy is implemented to determine the rough directions for the k subflows, that is, we now have the locations of Xi1;Xi2; . . . ;Xik. The insertion-move is then performed to find the
best neighborhood solution around Xi1, that is, the Xi1. This is
re-peated until the best neighborhood solution for each of the sub-flows has been found. Iteration by iteration, these newly generated subflows may merge with others sharing the same loca-tion, proceed in a single stream, further split into more subflows at
later iterations, or stagnate in the current location until the stop-ping criteria of the algorithm is met.
The WFA procedure designed is described in detail as follows: Procedure WFA
Step 1. Initial WFA parameter settings: G; N; W0;V0;T.
Step 2. For each flow iði ¼ 1; 2; . . . ; NÞ, repeats Steps 3–6. Step 3. Calculate number of subflows nifor each flow i based on
Eq.(1).
Step 4. For each subflow k of flow i, find the best neighborhood solution through the machine-shifting and insertion-move strategies described in Section3.3.
Step 5. Distribute the mass of flow i to its subflows based on Eq. (2).
Step 6. Calculate the improvement in objective values and update the resulting velocity of subflow k split from flow i based on Eq.(3).
Step 7. Merge subflows with the same objective values and update the resulting mass and velocity based on Eqs. (4) and (5).
Step 8. Update the number of subflows for each flow i. Step 9. Update the total number of water flow:N PNi¼1ni.
Step 10. Perform evaporation operation and update the resulting mass and velocity for each water flow based on Eq.(6). Step 11. Check whether precipitation condition(s) is/are met. If
yes, perform Steps 12, 13, and 14; otherwise, go to Step 15.
Step 12. Perform machine-shifting strategy to the current best solution to generate new solutions deviated from the current ones.
Step 13. Distribute the mass to flows poured based on Eq.(7)or (8)depending on the type of precipitation.
Step 14. Check whether the new solution has the same objective value as the other solutions. If yes, merge it and update the resulting mass and velocity based on Eqs.(4) and (5), then update the total number of water flow N. Step 15. Check whether iteration limit G has been reached. If yes,
stop the algorithm; otherwise, increment the iteration counter by 1 and return to Step 2.
Note that in the machine-shifting strategy, if the threshold probability value is set at 0.7, it implies that each machine has a 30% probability of being assigned to other cells. In the above WFA procedure, the machine-shifting strategy is used in Steps 4 and 12, respectively, with different threshold probability values: 0.7 in Step 4 and 0.5 in Step 12. This is because the main purpose of Step 4 is to find some neighborhoods of the current solution, thus the probability of being assigned to other cells is set at a com-paratively low value. The purpose of Step 12, on the other hand, is to explore solutions of unvisited regions through the precipitation operation; thus, it becomes necessary to increase the probability of being assigned to other cells to find solutions more deviated from the current best.
When implementing the four operations in the WFA proce-dure, we made several changes based on our trials, experi-ences, and observations. First, in the flow splitting and moving operation, there is no upper limit on the number of subflows forked from a flow at each iteration since these num-bers are always within a controllable size during the problem-solving process.
Second, the mass of the subflows is determined based solely on the ranks of the subflows (as shown in the example of Section 3.1.1) without considering their respective performances in the original WFA. We hence design a new formula shown in Eq. (9) for assigning mass to each subflows based on the idea that sub-flows should compete for their masses. That is, subsub-flows with
bet-P2 P3 P5 P1 P4 Cel l 1 M2 1 1 1 0 0 M4 1 1 0 0 0 M1 0 0 0 1 1 Cel l 2 M3 0 0 0 1 0 M5 0 0 0 0 1
Fig. 4. Initial solution matrix obtained.
ter objective values should possess more masses and should stay longer in the water-flowing process.
wik¼ EðUikÞ Pni k¼1EðUikÞ ! Wi; ð9Þ
where EðUikÞ is the objective value (the grouping efficacy values in
the CF problem) of solution Uik.
Third, in addition to the fixed-ratio evaporation presented in the original WFA, another way of evaporation is presented and added in the procedure, the velocity-based evaporation. It can be observed from Eq. (3) that the higher the altitude drop (i.e., the larger improvement in objective value) of a subflow, the larger the velocity it will be rendered to. We define an evaporation ratio conversely related to improvement in velocity, where flows with smaller velocities should be evaporated more quickly than those with larger velocities. The formula is listed below: Wi¼ ð1
q
iÞWi; ð10Þ whereq
i¼ 1; ifl
ik¼ 0; 0; if lik Vi P1; 1 lik Vi; if 0 < lik Vi <1: 8 > < > :Lastly, two types of precipitation are performed in the original WFA, namely, enforced and regular precipitation. Enforced precipitation is applied when all flows are grounded with zero velocities.
How-ever, this has never happened in our solution process of the CF problem. Another precipitation is hence presented and added to the procedure, that is, the moist precipitation. Moist precipitation is used when the mass of the evaporated water flow reaches half of its original total mass, W0.
The proposed procedure WFA in pseudo-code format is shown inFig. 7.
3.4. Proposed algorithm WFACF for CF problem
Most of the algorithms designed to solve the CF problem at-tempt to obtain the machine–part groupings so that some decision objectives, such as the grouping efficiency or the grouping efficacy, can be maximized. However, without the prior determination of the ‘‘cell size,” the above objectives can hardly be achieved. It is gi-ven 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 cell size is set at two and is gradually increased by one. These algorithms are then repeatedly applied until a cell size resulting in the best grouping efficiency/efficacy value has been found. Thus, many computational efforts have to be exerted in or-der to obtain the optimal cell size. Instead of starting from a begin-ning number, identifying a good intermediate point for the cell size at the very beginning should save plenty of run time when design-ing an algorithm to search for the optimal cell size.
Take a test problem from the literature (Carrie, 1973) as an example. The relationship between the cell size and the resulting
grouping efficacy is shown inFig. 8. It is observed that the grouping efficacy value increases as the cell size increases, and the optima is found when cell size equals nine. After that, it starts to decrease as the cell size increases. Similar observations can be found in other test problems.
Hence, we propose a two-stage algorithm, WFACF, for solving the CF problem in this study. In the first stage, feasible solutions without an elaborate solution improvement are generated to de-rive a cell size quickly, which is then used as input to the second stage to search for the optimal/near-optimal solution through the proposed WFA. We anticipate that the cell size obtained in stage 1 can serve as a good lower bound to start the solution process
in stage 2. Hence, a considerable amount of computational efforts can be saved, especially when large-sized problems are solved. Be-fore we explain the solution procedure, additional notations are introduced.
NC number of cells (cell size)
p0 initial solution of parts assignment
p incumbent solution of parts assignment of current cell size p best solution of parts assignment so far
m0 initial solution of machines assignment
m incumbent solution of machines assignment of current cell size
m best solution of machines assignment so far
E0 grouping efficacy value of cell configuration ðm0;p0Þ E grouping efficacy value of cell configuration ðm;pÞ E grouping efficacy value of cell configuration ðm;pÞ
Procedures of both stages are described below. Stage 1 of WFACF:
Step 1. Read machine–part incidence matrix of test problems, and set NC ¼ 2; E0¼ E¼ 0.
Step 2. Compute the similarity values between machine pairs and construct a similarity matrix.
Step 3. Generate initial machines assignment, m0, by
performing the machines assignment procedure in Section3.2.1.
Step 4. Generate initial parts assignment, p0, based on m0 in
Step 3, by performing the parts assignment procedure in Section3.2.2.
Step 5. Calculate the grouping efficacy value E0of the resulting
cell configuration ðm0;p0Þ.
Step 6. If E0>E, then set ðm;pÞ ¼ ðm0;p0Þ; E¼ E0, NC = NC+1,
go to Step 3; otherwise, report current cell configuration ðm;pÞ and Eand terminate stage 1.
The solution obtained at the end of stage 1, including the sug-gested number of cells and cell configurations ðm;pÞ, is then used
as the input to stage 2 to search for the optimal/near-optimal solu-tion through the proposed WFA procedure in Secsolu-tion3.3.
Stage 2 of WFACF:
Step 1. Read solutions from stage 1, including number of cells, ðm;pÞ, and E. Set ðm;pÞ ¼ ðm;pÞ; E
¼ E. Step 2. Perform the proposed WFA procedure in Section 3.3,
obtain solution ðm;pÞ and the resulting grouping
effi-cacy value E.
Step 3. If E>E, then set E
¼ E;ðm;pÞ ¼ ðm;pÞ,
NC = NC + 1, go to Step 2; otherwise, report the current best cell configuration ðm;pÞ and Eand terminate
stage 2.
The procedures of stages 1 and 2 of the proposed WFACF in pseudo-code format are given inFigs. 9 and 10, respectively. After
25 30 35 40 45 50 2 3 4 5 6 7 8 9 10 11 12 13 14 cell size grouping ef ficac y (%)
Fig. 8. Relationship between grouping efficacy and cell size.
//Stage 1 of WFACF
READ
machine-part incidence matrix
and let cell number C*=NC=2
Calculate machine similarity matrix
DO
{
Create machine cells using machines assignment procedure
Assign parts to machine cells using parts assignment procedure
Calculate the grouping efficacy value of the cell configuration
If current solution is better than the best recorded so far, update C* = NC
Increment number of cell by 1: NC = NC+1
} WHILE the grouping efficacy value can be improved
Fig. 9. Procedure of stage 1 of WFACF.
//Stage 2 of WFACF
SET NC=C*
READ results from Stage 1 of WFACF
DO
{
Run Procedure WFA
IF current solution is better then the best recorded so far,
best solution = current solution
C* = NC
Increment cell number by 1: NC = NC+1
} WHILE the grouping efficacy value can be improved
SHOW best solution
intensive testing, parameters of the WFA, iteration limit G, initial mass W0, initial velocity V0, base momentum T and the number
of iterations for precipitation t are set at 100, 40, 15, 100, 20 respectively.
4. Computational results and discussion
Thirty-seven test instances from the literature and newly cre-ated are used to evaluate the computational characteristics of the proposed heuristic WFACF, the results of which are compared with those of algorithms reported in the literature. Some studies allow the existence of singletons in the solutions and some do not. We follow the studies that do not allow singletons in this research. The matrices of the test problems range from 5 7 to 50 100; they comprise well-structured and unstructured matrices. The first 35 test problems are widely adopted as the test instances in many studies (Gonçalves and Resende, 2004; James et al., 2007; Wu et al., 2008), while problems #36 and #37 are generated by this study. The proposed algorithm WFACF was coded in C and implemented on a Pentium III 933 MHz personal computer with 256 MB random access memory (RAM).
4.1. Computational results
Thirty-five test problems from the literature and two test prob-lems newly created by this study are used to examine the perfor-mance of the proposed WFACF algorithm. The results were then compared with the best results found in the literature, that is, the Hybrid-GA ofGonçalves and Resende (2004), where singletons are not allowed.Wu et al. (2008)proposed a simple yet effective simulated annealing-based algorithm (SACF), for solving the CF problems where singletons are allowed. The corresponding results are compared with several well-known algorithms published. Their comparative study shows that the SACF algorithm improves the grouping efficacy for 72% of the test problems. We adopted the SACF as a target for comparison with minor modifications to the program code of SACF to cope with the singletons restriction in this study.Table 1gives the computational results of these methods. Ten independent runs were performed for each test instance due to the stochastic features that SACF, Hybrid-GA, and WFACF might have.
For the 35 test problems directly adopted from the literature, Tables 1 and 2show that WFACF produced the best solutions in 34 out of 35 problems, while Hybrid-GA found 26 and SACF found 20. Solution qualities of the initial solutions (solutions obtained at the end of stage 1 of WFACF) are generally good. In 14 of 35 test problems, the initial solutions are exactly equal to the best solu-tions. In 25 out of 35 test problems, both the WFACF and the Hy-brid-GA obtain the same results, while WFACF produced better results than those of Hybrid-GA in nine problems (#16, #20, #21, #26, #29, #30, #31, #33, and #34) and Hybrid-GA has better re-sults than those of WFACF only in problem #19. It can thus be con-cluded that WFACF performs better than Hybrid-GA and SACF, especially in test problems with larger sizes. Similar conclusion can be made for test problems #36 and #37. All three methods give cell sizes resulting in the best grouping efficacy for each test problem. Cell sizes from the three methods are the same in most test instances, except in problems with larger sizes where minor differences can be seen.Fig. 11displays the error percentage to the best solutions of the SACF, Hybrid-GA, and WFACF methods, respectively, on 17 test instances, in which there are different grouping efficacies. The result ofFig. 11matches the observations inTables 1 and 2.
As for the comparison of run time, no significant difference is observed between the SACF and the WFACF when small or
medium-sized problems are solved. The WFACF, however, per-forms more efficiently than the SACF in large-sized test problems (#31, #32, #33, #35, and #37). Although this study has a lower computational environment (933 MHz) compared with the Hy-brid-GA (1.333 GHz), both SACF and WFACF take only a fraction of the run time of Hybrid-GA to obtain comparatively favorable solutions. The superiority in computational efficiency can be easily observed.
4.2. Analyses and discussion
One of the main objectives of this study is to examine in detail the various mechanisms of the WFA and their corresponding ef-fects to the overall solution efficiency and efficacy. Efef-fects of sev-eral strategies proposed in this study, together with the evaporation and precipitation operations, are therefore further analyzed in this section.
4.2.1. Effect of prior estimation of cell size
In the first stage of the proposed algorithm WFACF, feasible solutions without an elaborate solution improvement are gener-ated to derive a cell size quickly, which is then used as input to the second stage to search for the optimal/near-optimal solution. It is anticipated that the obtained cell size can serve as a good low-er bound to start the solution process in stage 2, saving a considlow-er- consider-able amount of computational efforts. An experiment without prior estimation of cell size (skipping stage 1) in the algorithm is con-ducted and compared with the WFACF. A comparison of results is shown inFig. 12. On average, the computer run time of WFACF (with prior estimation of the cell size) is 44% less than the case without prior estimation of the cell size. Additionally, the savings are even more significant in large-sized test problems. This finding justifies the effects of stage 1 of the proposed WFACF.
4.2.2. Effects of evaporation, precipitation, and insertion-move The water evaporation and precipitation operations coincide with the ‘‘escaping from local optima” mechanism many heuristic algorithms nowadays possess to avoid being trapped and to ex-plore more solution spaces. Additionally, the insertion-move pre-sented assures that high-quality neighborhood solutions can be found at each iteration of the algorithm. These three factors and their effect on solution qualities are thus considered in an experi-ment. The evaporation factor has three settings: no evaporation, fixed-ratio evaporation, or velocity-based evaporation; the precip-itation factor has three settings: no precipprecip-itation, regular precipita-tion, or moist precipitation; while the insertion-move factor has two options: with or without insertion-move. Legitimate combina-tions of the above three factors comprise 10 testing scenarios listed in Table 3. In each scenario, 35 test instances of Section 4.1are computed and the average grouping efficacies over the 35 in-stances are recorded and analyzed inFig. 13. The 10 scenarios have been separated into two groups. Scenarios adopting the insertion-move (scenarios #1, #3, #5, #7, and #9, same color in different line types inFig. 13) result in better objective values and obviously sur-pass the other group (scenarios #2, #4, #6, #8, and #10, same color in different line types inFig. 13), which does not include the inser-tion-move in the algorithm. As expected,Fig. 14shows that the group with the insertion-move spends more run time searching for the best solutions, especially in large-sized problems (problems #25–35). Insertion-move beats the other two factors and becomes the most significant and dominant factor in terms of solution qual-ity. This implies that the water evaporation and precipitation oper-ations may not be as significant as they were supposed to be in the solution process of CF problem, though considerable efforts have been spent in designing the contents of both operations. The importance of a good neighborhood-searching method, such as
Table 1
Performance of WFACF compared to SACF and Hybrid-GA.
Test instances SACF (10 replicates) Hybrid-GA (10 replicates) WFACF (10 replicates)
No. Source Size Max Cell size CPU time (s) Max Cell size CPU time
(s)
Initial Max Initial cell size
Cell size CPU time (s)
St. dev.
1 King and Nakornchai (1982) 5 7 73.68 2 0.00 73.68 2 0.53 73.68 73.68 2 2 0.01 0
2 Waghodekar and Sahu (1984) 5 7 62.50 2 0.00 62.50 2 0.47 52.00 62.50 2 2 0.00 0
3 Seifoddini (1989) 5 18 79.59 2 0.01 79.59 2 0.85 79.59 79.59 2 2 0.01 0
4 Kusiak and Cho (1992) 6 8 76.92 2 0.00 76.92 2 0.66 76.92 76.92 2 2 0.01 0
5 Kusiak and Chow (1987) 7 11 53.13 3 0.00 53.13 3 1.09 51.52 53.13 3 3 0.01 0
6 Boctor (1991) 7 11 70.37 3 0.01 70.37 3 1.35 70.37 70.37 3 3 0.01 0
7 Seifoddini and Wolfe (1986) 8 12 68.29 3 0.01 68.29 3 1.44 68.29 68.29 3 3 0.03 0
8 Chandrashekharan and Rajagopalan (1986a) 8 20 85.25 3 0.02 85.25 3 1.68 85.25 85.25 3 3 0.04 0 9 Chandrashekharan and Rajagopalan (1986b) 8 20 58.41 2 0.02 58.72 2 1.68 58.72 58.72 2 2 0.05 0
10 Mosier and Taube (1985a) 10 10 70.59 3 0.01 70.59 3 1.82 70.59 70.59 3 3 0.07 0
11 Chan and Milner (1982) 10 15 92.00 3 0.02 92.00 3 2.19 92.00 92.00 3 3 0.08 0
12 Askin and Subramanian (1987) 14 24 67.61 6 0.29 69.86 5 6.05 67.12 69.86 5 5 0.25 0
13 Stanfel (1985) 14 24 69.33 5 0.19 69.33 5 6.24 69.33 69.33 5 5 0.26 0
14 McCormick et al. (1972) 16 24 51.96 6 0.28 51.96 6 7.85 40.94 51.96 3 6 0.95 0
15 Srinivasan et al. (1990) 16 30 67.83 4 0.23 67.83 4 10.24 61.33 67.83 4 4 0.69 0
16 King (1980) 16 43 55.76 6 1.63 54.86 5 13.92 53.93 55.90 6 6 1.01 0.21
17 Carrie (1973) 18 24 53.64 6 0.56 54.46 6 11.34 44.53 54.46 5 6 1.17 0.15
18 Mosier and Taube (1985b) 20 20 42.34 5 0.24 42.96 5 10.89 35.19 42.96 2 5 1.53 0.14
19 Kumar et al. (1986) 20 23 49.65 5 0.40 49.65 5 12.03 44.27 49.61 6 6 0.85 0.11
20 Carrie (1973) 20 35 76.14 4 1.01 76.22 4 15.61 76.14 76.54 4 5 1.59 0.06
21 Boe and Cheng (1991) 20 35 56.04 4 0.94 58.07 5 16.38 55.06 58.15 5 5 1.10 0.09
22 Chandrasekharan and Rajagopalan (1989) 24 40 100.0 7 1.76 100.0 7 19.26 100.0 100.0 7 7 1.33 0 23 Chandrasekharan and Rajagopalan (1989) 24 40 85.11 7 1.91 85.11 7 25.28 85.11 85.11 7 7 2.18 0 24 Chandrasekharan and Rajagopalan (1989) 24 40 73.51 7 1.55 73.51 7 26.82 73.51 73.51 7 7 2.11 0 25 Chandrasekharan and Rajagopalan (1989) 24 40 51.88 9 4.76 51.97 9 26.48 44.10 51.97 6 10 4.58 0.1 26 Chandrasekharan and Rajagopalan (1989) 24 40 44.44 10 6.28 47.06 9 25.97 42.52 47.37 9 10 5.90 0.1 27 Chandrasekharan and Rajagopalan (1989) 24 40 44.24 9 5.10 44.87 9 26.06 36.27 44.87 8 10 3.05 0.05 28 McCormick et al. (1972) 27 27 53.19 3 0.31 54.27 4 25.90 32.38 54.27 2 4 2.56 0 29 Carrie (1973) 28 46 40.97 6 4.05 44.62 9 43.78 39.34 46.06 6 9 23.36 0.11
30 Kumar and Vannelli (1987) 30 41 58.58 10 8.08 58.48 11 43.00 53.67 59.52 10 10 5.41 0.12
31 Stanfel (1985) 30 50 60.00 12 14.95 59.66 12 52.45 54.55 60.00 9 12 10.26 0
32 Stanfel (1985) 30 50 50.51 11 20.42 50.51 11 48.97 43.86 50.51 8 11 10.17 0
33 King and Nakornchai (1982) 36 90 39.95 11 185.7 42.64 9 81.46 33.70 46.15 6 12 31.80 0.1
34 McCormick et al. (1972) 37 53 58.23 2 1.15 56.42 2 87.66 35.06 59.85 2 3 17.35 0.02 35 Chandrasekharan and Rajagopalan (1987) 40 100 84.03 10 161.0 84.03 10 152.1 84.03 84.03 10 10 19.20 0 36 This study 40 100 41.67 4 9.96 – – – 8.05 46.96 6 6 12.94 3.28 37 This study 50 100 43.89 7 65.07 – – – 5.08 44.20 7 7 22.58 0.53 T.-H. Wu et al. /European Journal of Operational Research 205 (2010) 346–360
the insertion-move we proposed, can never be overemphasized. This conclusion is applicable to any meta-heuristic algorithms. Although the water evaporation and precipitation operations may not be critical in solving the CF problem in this study, we still believe that the water-flow-like logic, with proper design and col-laboration in its internal operations, can be applied to solve other combinatorial optimization problems.
Table 3
Experimental testing scenarios.
Scenario # Evaporation setting Precipitation setting Insertion-move option Mean grouping efficacy (%)
1 Velocity based Moist With 64.2832
2 Velocity based Moist Without 59.1495
3 Velocity based Regular (t = 20) With 64.2816
4 Velocity based Regular (t = 20) Without 59.3005
5 Fixed ratio (0.05) Moist With 64.2954
6 Fixed ratio (0.05) Moist Without 59.3558
7 Fixed ratio (0.05) Regular (t = 20) With 64.2625
8 Fixed ratio (0.05) Regular (t = 20) Without 59.2771
9 No evaporation No precipitation With 64.2664
10 No evaporation No precipitation Without 59.4040
0 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
test instance number
run time (s)
.
without prior estimation of cell size with prior estimation of cell size
Fig. 12. Run time comparisons of with and without prior estimation of cell size. 0 2 4 6 8 10 12 14 16 9 12 16 17 18 19 20 21 25 26 27 28 29 30 31 33 34
test instance number
erro r to b est (% ) WFACF Hybrid-GA SACF
Fig. 11. Comparisons of error percentage to best solutions of SACF, Hybrid-GA, and WFACF. Table 2
Comparisons of solution ranking of three methods (for the first 35 test problems). Solutions ranking Ranked 1st Ranked 2nd Ranked 3rd
SACF 20 10 5
Hybrid-GA 26 6 3
5. Conclusions
The WFA uses a dynamic size of solution agents and is able to overcome the drawbacks of both single- and multiple-solution-agent-based algorithms. The WFA is a very new meta-heuristic algorithm, and to our knowledge, this is the first time the WFA is applied to solve a combinatorial optimization problem aside from the bin packing problems studied in the original paper ofYang and
Wang (2007). This research has adopted the WFA logic and pro-posed a heuristic algorithm, WFACF, for solving the CF problem to assure good solution quality while avoiding a heavy computa-tional burden.
The similarity coefficients method, together with the proposed machines assignment and parts assignment procedures, has been used to generate quick initial solutions for later improvement by the WFACF. Two strategies, the machine shifting and the
scenario1 scenario2 scenario3 scenario4 scenario5 scenario6 scenario7 scenario8 scenario9 scenario10 1 4 7 10 13 16 19 22 25 28 31 34
test instance number 0 20 40 60 80 100 run time (s)
Fig. 14. Comparison of mean run time for 10 testing scenarios.
scenario1 scenario2 scenario3 scenario4 scenario5 scenario6 scenario7 scenario8 scenario9 scenario10 1 4 7 10 13 16 19 22 25 28 31 34
test instance number
30 40 50 60 70 80 90 100 110 grouping efficacy (%)
insertion-move, have been presented for finding the best neighbor-hood solution around the current solution in the flow splitting and moving operations in the design of WFACF. In addition to the ori-ginal evaporation and precipitation operation of the WFA, veloc-ity-based evaporation and moist precipitation have been offered in this study.
The proposed WFACF is composed of two stages. In the first stage, feasible solutions without an elaborate solution improve-ment are generated to derive a cell size quickly, which is then used as input to the second stage in searching for the optimal/near-opti-mal solution. It is anticipated that the cell size obtained in stage 1 can serve as a good lower bound to start the solution process in stage 2, thus saving a considerable amount of computational ef-forts especially when large-sized problems are solved. The effect of this ingenious design has been justified through further experi-ment and analysis.
Computational results obtained from running a set of 35 test in-stances from the literature have shown that WFACF has produced best solutions in 34 out of 35 problems, while Hybrid-GA has found 26, and SACF has found 20. The proposed WFACF has performed bet-ter than the Hybrid-GA and the SACF, especially in test problems with larger sizes. As for the comparison of run time, WFACF has ta-ken only a small fraction of the run time of Hybrid-GA to obtain comparatively favorable solutions. Apart from the performance in solution effectiveness, the superiority of the proposed WFACF in solution efficiency over other approaches from the literature can be easily observed. Further analysis has been conducted to justify the effect of prior estimation of cell size in stage 1 of the WFACF. Computational result has demonstrated that the average run time of WFACF (case with prior estimation of the cell size) is 44% less than the case without prior estimation of the cell size. In addition, the savings are even more significant in large-sized test problems. We have further verified the effects of the evaporation, precip-itation operation, and insertion-move in the WFACF. Legitimate combinations of the above three factors have comprised 10 testing scenarios. In each scenario, 35 test instances were computed and the average grouping efficacies over the 35 instances were re-corded and analyzed. The insertion-move has beaten the other two factors and has become the most significant and dominant fac-tor in terms of solution quality. This implies that the water evapo-ration and precipitation opeevapo-rations may not be as decisive as they were supposed to be in the WFACF solution process of CF problem, though considerable efforts have been spent in designing the con-tents of both operations.
The contributions of this paper are summarized as follows: 1. We have applied the WFA for solving the CF problem with both
solution efficiency and effectiveness outperforming other benchmarking algorithms in the literature. This is the first time WFA has been used to solve combinatorial optimization prob-lems aside from the bin packing probprob-lems studied in the origi-nal paper.
2. Although the logic of WFA is adopted in this paper, we have specifically designed tailor-made WFA operations for solving the CF problems. For example, in the flow splitting and moving operation, both machine shifting and insertion-move strategies are proposed for finding the best neighborhood solution. In addition, a new formula is presented for making subflows with better objective values possess more masses and stay longer in the water-flowing process. Moreover, in the evaporation opera-tion, in addition to the fixed-ratio evaporation of the original WFA, the velocity-based evaporation is presented and added in the procedure. As for the precipitation operation, the moist precipitation is presented and added to the procedure when the mass of the evaporated water flow reaches half of its origi-nal total mass.
3. We have designed a very efficient solution algorithm. In order to find the best solution, the solution algorithms have to be repeatedly applied until a cell size resulting in the best group-ing efficiency/efficacy value has been found. Thus, many com-putational efforts have to be exerted in order to obtain the optimal cell size. In WFACF, prior estimation of cell size is implemented in stage 1 to serve as a good lower bound to start the solution process in stage 2, so that a considerable amount of computational efforts can be saved. Computational result has demonstrated that the average run time of WFACF (case with prior estimation of the cell size) is 44% less than the case with-out prior estimation of the cell size, and the savings are even more significant in large-sized test problems.
In conclusion, the superiority of the proposed WFACF both in solution effectiveness and efficiency over other approaches from the literature should be attributed to the collaboration of the WFA logic, the proposed prior estimation of the cell size, and the insertion-move. The WFA is a novel heuristic approach that de-serves more attention. More attempts on adopting the WFA logic to solve many other combinatorial optimization problems should be a good direction of future research.
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