Int J Syst Assur Eng Manag DOI 10.1007/s13198-014-0314-6
ORIGINAL ARTICLE
An integrated fuzzy algorithm approach to factory floor design incorporating environmental quality and health impact Ali Azadeh • Sahar Jebreili • Elizabeth Chang Morteza Saberi • Omar Khadeer Hussain
•
Received: 10 September 2014 The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2014
Abstract This paper presents an integrated algorithm based on fuzzy simulation, fuzzy linear programming (FLP), and fuzzy data envelopment analysis (FDEA) to cope with a special case of workshop facility layout design problem with ambiguous environmental and health indicators. First a software package is used for generating feasible layout alternatives and then quantitative performance indicators are calculated. Weights are estimated by LP for pairwise comparisons (by linguistic terms) in evaluating certain qualitative performance indicators. Fuzzy simulation is then employed for modeling different layout alternatives with uncertain parameters. Next, the impacts of environment and health indicators are retrieved from a standard questionnaire. Finally, FDEA is used for ranking the alternatives and consequently finding the optimal layout design alternatives. A possibilistic programming approach is used to modify the fuzzy DEA model to an equivalent crisp one. Moreover, fuzzy principal component analysis method is used to validate the results of FDEA model at various a-cut levels by Spearman correlation experiment. This is the first study that presents an integrated algorithm for optimization of facility layout with environmental and health indicators.
A. Azadeh S. Jebreili School of Industrial and Systems Engineering, College of Engineering, University of Tehran, Tehran, Iran
Keywords Facilities planning and design Data envelopment analysis Fuzzy sets Simulation optimization Environment Health
1 Introduction Facility layout means planning for the location of all machines, utilities, employee workstations, customer service areas, material storage areas, aisles, restrooms, lunchrooms, internal walls, offices, and computer rooms. It is also a planning for the flow patterns of materials and people around, into, and within buildings (Tompkins et al. 1996). Layout Optimization is one of the top issues for industrial facility planners in around the world. It has profound effects on organizational productivity and profitability. Optimal layouts reduce materials handling costs, help streamline all operations in a facility, and reduce energy bills. Selection of a useful, effective and manageable layout is the key in achieving the success of a particular manufacturing organization. Facility layout design is usually considered as a multiple-objective problem. According to Heragu et al. (1990) in the Facility layout design problems, minimizing material handling costs and providing a safe workplace for employees are the most important objectives. In summary, the most common objectives in facility layout design (FLD) problem can be summarized as follow:
E. Chang M. Saberi (&) O. K. Hussain School of Business, UNSW Canberra, Canberra, Australia e-mail:
[email protected]
•
M. Saberi Department of Industrial Engineering, University of Tafresh, Tafresh, Iran
•
Minimize the material handling costs, minimize overall production time, and minimize investment in equipment between the facilities (Yaman et al. 1999). Facilitate the traffic flow and minimize the costs of it (Heragu 1997).
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• •
•
Maximize the flexibility, accessibility and maintenance factors. Minimize the dimensional and form errors of products depending on the fixture layout (Prabhaharan et al. 2006). Maximize the layout performance.
Layout generation and evaluation is often a challenging and time consuming task due to its inherent multiple objective natures and its difficult data collection process. Different methodologies have been presented in the literature to deal with such problems. Algorithmic approaches usually simplify both design constraints and objectives in reaching a total objective to obtain the solution of the problems. These approaches lead to generation of efficient layout alternatives, especially, when commercial software is available. Nevertheless, the obtained quantitative results of these tools often do not capture all of the design objectives. On the other hand, procedural approaches are used in FLD processes which are able to incorporate both qualitative and quantitative objectives. To do so, the FLD process is divided into several steps to be sequentially solved. However, the success of this process strongly depends on the generation of quality design alternatives provided by an expert designer. Yang et al. (2000) showed that neither algorithmic nor procedural FLD methodology is necessarily effective in solving FLD problems. By applying the integrated proposed algorithm in this paper the under lying gap in the determined literature can be filled. Health is the level of functional or metabolic efficiency of a living being. In humans, it is the general condition of a person’s mind and body, usually meaning to be free from illness, injury or pain (as in ‘‘good health’’ or ‘‘healthy’’). The World Health Organization (WHO) defined health in its broader sense in 1946 as ‘‘a state of complete physical, mental, and social well-being and not merely the absence of disease or infirmity’’. ‘‘Proper design can often increase mental and physical health’’. On the other hand, environment indicator focuses on a facility layout design that improving environment’s quality. This paper presents an integrated fuzzy simulation-fuzzy LP-fuzzy DEA algorithm to solve facility layout design (FLD) problems. This integrated algorithm is applied for a special case of maintenance workshop of a large gas transmission unit in Qazvin, Iran. First VIP-PLANOPT software is applied to generate the feasible layout alternatives; VIP-PLANOPT is powerful general-purpose facility layout optimization software for engineers, industrial planners, and facility designers. While it has a host of capabilities for solving large real-world industrial facility layout design problems, it serves as an excellent teaching aid in facility layout design.
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The literature review of this problem is presented in Sect. 2. In Sect. 3, the proposed integrated algorithm is explained in detail. In Sect. 4, the implementation procedure of the algorithm has been investigated. Section 5 presents the computational results of this study and concluding remarks are given and discussed in Sect. 6. 2 Literature review Optimization of FLD’s problems has attracted many researchers in last decades. Various methodologies have been presented in the literature to deal with such problems. Since FLP’s are known as NP-hard problems, various meta-heuristics such as SA, GA, and ant colony are currently used to approximate the solution of very large FLP. The SA technique originates from the theory of statistical mechanics and is based upon the analogy between the annealing of solids and solving optimization problems. S¸ ahin (2011) proposed a SA algorithm to solve the biobjective facility layout problem, as well as a comparison of SA with the previous works is provided. In this article, a bi-objective facility layout problem (BOFLP) is considered by combining the objectives of minimization of the total material handling cost (quantitative) and the maximization of total closeness rating scores (qualitative), with the predetermined weights which are assigned to the respective objectives. GA gained more attention during the last decade than any other meta-heuristics; it utilizes a binary coding of individuals as fixed-length strings over the alphabet {0, 1}. GA iteratively search the global optimum, without exhausting the solution space, in a parallel process starting from a small set of feasible solutions (population) and generating the new solutions in some random fashion. Performance of GA is problem dependent because the parameter setting and representation scheme depends on the nature of the problem. Cheng and Gen (1996) applied GA for facility layout design under interflows uncertainty. They did not emphasize on performance measures such as cycle time. A comprehensive investigation of the FLP literature includes examining heuristics. Kumar et al. (1995) presented a constructive heuristic to solve the single row FLP with the objective of minimizing the materials handling cost. In this approach, the facilities with the highest frequency of parts between them and their adjacent locations were prior in adding to the solution sequence. Ho and Moodie (1998) proposed a two-phase heuristic procedure based on simulated annealing technique for solving a FLP within an automated manufacturing system. They took into consideration different evaluation criteria such as minimization of total flow distance and maximization number of in-sequence movements ‘fronting the optimal layout formation. Taghavi and Murat (2011) presented an efficient
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iterative heuristic procedure to solve the integrated layout design and product flow assignment problem. They proposed a novel integrated heuristic procedure based on the alternating heuristic, a perturbation algorithm, and sequential location heuristic.
3 The integrated algorithm
An actual maintenance workshop is used in this paper to illustrate the efficiency and effectiveness of the proposed integrated algorithm. The case is a gas transmission station in Qazvin, Iran. Defective parts are transferred to maintenance workshop in order to be repaired and returned to main unit in operating mode. These parts include shaft, three different plates, bar, nut, valve, and pipe. The maintenance facilities are Milling, Vertical Saw, Saw Lang, Wiredraw, Weld, Test Bench, Plasma Cutter, Lathe, Press and Drill. Figure 1 presents the existing layout of the ten facilities. Each defective part has separate job sequence. If the current layout is not efficient, the gas transmission station would like to know what layout alternatives are efficient. The proposed algorithm has the following assumptions:
•
Due to the low inventory cost of maintenance process, the most desirable layout is the one that maintains (repair) the most quantity of defective parts within a given period of time; Defective parts flows occur between the centers of facilities;
Fig. 1 The current facility layout for the maintenance workshop process
• •
•
3.1 Description of the maintenance workshop
•
•
The maintenance system is job shop which consists of ten stages (i.e. facilities); The defective parts flows is initiated from each stage; Time between arrivals of defective parts, corrective maintenance times for each defective part and preventive maintenance times for each machine has been obtained in accordance with both objective and subjective data; The setup times may be deterministic or stochastic obtained by statistical sampling methods.
3.2 General framework The proposed integrated algorithm is defined for the particular maintenance workshop process described above. However, this algorithm can be easily generalized to be implemented in other FLP’s by little modifications. The proposed integrated algorithm can be applied to the case of maintenance workshop process throughout the following steps: I.
II.
III. IV. V.
Collect the required data for designing the layout of the maintenance workshop such as the total space of the workshop, space of each machine etc. Generate different layout alternatives with respect to the collected data using a computer-aided layout planning tool and choose a number of alternatives according to expert’s judgment. Calculate quantitative performance indicators including distance, adjacency and shape ratio. Apply fuzzy LP models for evaluation of qualitative performance indicators. Convert the fuzzy LP models to equivalent crisp ones using the possibilistic programming method
Weld
Saw Lang
Milling
Wiredraw
Press
Drill
Vertical Saw
Test Bench
Plasma Cutter
Lathe
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VI.
VII. VIII. IX.
X.
XI. XII.
XIII.
XIV.
proposed by Jime´nez et al. (2007) and then evaluate the qualitative performance indicators including flexibility, accessibility, and maintenance. Collect the required data for the maintenance process such as time between arrivals of defective parts, corrective maintenance times for each defective part and preventive maintenance times for each machine, which can be obtained from both objective data from the history of the maintenance workshop and subjective data from the experts’ judgment. Develop the crisp simulation (CS) network model for each layout alternative. Fuzzify the required data. Develop the simulation network model for each layout alternative with fuzzy inputs at different acut levels and then obtain their required outputs; i.e. average waiting time (AWT), average machine utilization (AMU), and average time in system (ATIS). Apply the questionnaire with fuzzy numbers to calculate the impact of health and environment indicators. Apply FDEA model for assessment and ranking of the layout alternatives. Incorporate distance, adjacency, shape ratio, flexibility, accessibility, maintenance, health, air pollution, tangible pollution and the fuzzy results of the FS model at a = 0 including AWT, AMU, and ATIS as output indicators of the FDEA model. In this case, shape ratio, distance, air pollution, tangible pollution, AWT and ATIS are considered as undesirable output indicators and other indicators as desirable output of the DEA model. It must be noted that the proposed FDEA model has no inputs and so one dummy input is assumed for all DMU’s. Convert the FDEA model to its equivalent crisp one using the possibilistic programming method proposed by Jime´nez et al. (2007) to calculate the efficiency score for each layout alternative and finding the best one for maintenance workshop for different a-cut levels. Verify and validate the result of FDEA by FPCA at each a-cut level.
4 Experiments: algorithm implementation 4.1 Data collection for facility layout design It is critical to consider characteristics of all defective parts, entry volume of defective parts, corrective maintenance
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routing for each defective part and time in data collection in order to assure the validity of the input data at the design stage. Table 1 presents the facility sizes of the ten stages for maintenance workshop process. Hereafter, the qualitative and quantitative performance indicators can be defined as follows: •
Quantitative indicators: •
• •
•
Distance: the sum of the defective part flow volume and rectilinear distance between the centroid of two facilities, Adjacency score: the sum of all positive relationships between adjacent departments, Shape ratio: the maximum of the depth-to-width and width-to-depth ratio of the smallest rectangle which can completely surround the facility.
Qualitative indicators: •
• •
Flexibility: the capability of performing various tasks under various operating conditions and the sufficiency for future expansions, Accessibility: the ease of material handling and operator movement between facilities, Maintenance: the required space for maintenance actions and tool movements.
4.2 Generating Layout Alternatives VIP-PLANOPT is applied to efficiently generate a large number of layout alternatives. It is powerful general-purpose facility layout optimization software for engineers, industrial planners, facility designers. First 200 layout alternatives are generated and then 75 alternatives as the best choices are selected according to expert’s judgment. This software gives quantitative performance measures such as shape ratio and we are also able to calculate flow distance and adjacency score for each layout alternative.
Table 1 Facilities sizes of the ten stages for maintenance workshop process Size (m2)
No.
Name
Size (m2)
No.
Name
1
Saw lang
2.4
7
Plasma cutter
1
2
Lathe
3.6
8
Test bench
3.04
3
Milling
7
9
Drill
3.15
4
Wiredraw
5
10
Vertical saw
5.7
5
Weld
6
11
Workshop
77.5
6
Press
3.5
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4.3 Fuzzy LP models
x1 ¼ 0;
Over the years, several methods have appeared for estimating the weights from a matrix of pairwise comparisons such as LP approach that proposed by Chandran et al. (2005). In this paper, we utilized a fuzzy version of this LP approach for evaluation the qualitative performance indicators using Saaty’s scale. Table 2 presents triangular fuzzy numbers presented by Saaty. In other words we used a two-stage fuzzy LP approach for generating a priority vector of qualitative performance indicators. In the first stage of fuzzy LP approach, we formulate a fuzzy LP that provides a consistency bound for a specified pairwise comparison matrix. In the second stage, we use the consistency bound in a fuzzy LP whose solution was a priority, in fact we use fuzzy set theory to incorporate unquantifiable, incomplete and partially known information into the decision model. 4.3.1 First stage: fuzzy linear program to establish the consistency bound o Suppose that a~ij ¼ ðapij ; am ij ; aij Þ is a triangular fuzzy number that is entry for row i and column j in the matrix A, n is number of rows (columns) in this square matrix and eij is an error in the estimate of the relative preference a~ij . If the decision maker is consistent, we have eij = 0. The decision variables are given by wi = weight of element i and error factor in estimating a~ij . We use three transformed decision variables in LP model: xi = ln wi, yij = ln eij, and zij = |yij|. The modified fuzzy LP model for the first stage is as follows:
Minimize
n1 X 2 X
ð1Þ
zij
i¼1 j¼iþ1
s.t xi xj yij ¼ In a~ij ; i; j ¼ 1; 2; . . .; n; i 6¼ j;
ð2Þ
zij yij ; i; j ¼ 1; 2; . . .; n; i\j; zij yji ; i; j ¼ 1; 2; . . .; n; i\j;
ð3Þ
Table 2 Saaty’s scale expressed in fuzzy numbers Linguistic variables
Triangular fuzzy numbers
Inverse of Triangular fuzzy numbers
Equal importance
(1,1,1)
(1,1,1)
Low importance
(1/2,1,3/2)
(2/3,1,2)
Medium importance High importance
(3/2,2,5/2) (5/2,3,7/2)
(2/5,1/2,2/3) (2/7,1/3,2/5)
Very high importance
(7/2,4,9/2)
(2/9,1/4,2/7)
~ xi xj 0; i; j ¼ 1; 2; . . .; n; a~ij [ 1;
ð4Þ
a
xi xj 0; i; j ¼ 1; 2; . . .n; a~ik [ a~jk 8k; a~iq [ a~jq 9q;
ð5Þ
zij 0; i; j ¼ 1; 2; . . .; n xi ; yij unrestricted; i; j ¼ 1; 2; . . .; n:
ð6Þ
a
a
Constraints (2) is the natural logarithm of wwij ¼ a~ij eij in comparison matrix A, if a~ij is overestimated at a degree (that is, the decision maker’s judgment of entry i versus entry j is greater than 1~ at a degree), then a~ji is underestimated at this degree. We then have eij ¼ 1 eji ; i; j ¼ 1; 2; . . .; n
ð7Þ
Or yij ¼ yji ; i; j ¼ 1; 2; . . .; n:
ð8Þ
By entering the grater of yij and yji constraints (3) and (4) identify for each i and j the element that is overestimated. Since the solution set to constraints (2)–(4) is infinitely large, we can arbitrarily fix the value of any wi without loss of generality. This is done in constraint (5) by setting w1 = 1. Note that the final weights can be normalized to sum to one. There are two desirable properties of a pairwise comparison matrix—element dominance (ED) and row dominance (RD)—that we would like to model in our fuzzy linear program. A solution method preserves rank weakly if a~ij [ 1~ a
implies wi C wj. This property is known as ED or weak rank preservation. ED is preserved if a~ij [ 1~ implies a
wi C wj and this property is explicitly enforced through constraints (6). A solution method preserve ranks strongly if a~ik [ a~jk for all k implies wi C wj. This property is a
known as RD or strong rank preservation and this property is explicitly enforced through constraints (7). In constraint (9) we point out that xi and yij are unrestricted since they are logarithms of positive real numbers. The objective function (1) minimizes the sum of logarithms of positive errors in natural logarithm space. In the no transformed space, the objective function minimizes the product of the overestimated error eij C 1. Therefore, the objective function minimizes the geometric mean of all errors greater than 1. The notion of minimizing the geometric mean of errors fits well with the concept of multiplicative errors in the AHP. The objective function is, in some sense, a measure of the inconsistency in the pairwise comparison matrix, that is, the greater the value of the objective function, the more inconsistent is the matrix. We define the consistency index (CI) within the LP framework as follows: CILP ¼ 2Z =nðn 1Þ
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CILP is the average value of Z for elements above the diagonal in the comparison matrix. In preliminary computational experiments, CILP and CI seem to be highly correlated. 4.3.2 Second stage: fuzzy linear program to generate a priority vector When we solve the first-stage fuzzy LP, the solution set consists of all priority vectors that minimize the product of all errors eij. It is possible that there are multiple optimal solutions to the first-stage fuzzy model. In the second stage, we solve a linear program that selects from this set of alternative optima the priority vector that minimizes the maximum of errors eij. The second-stage of fuzzy LP is given by the following: ð9Þ
Minimize zmax s:t: n1 X
availability of data such as number of machines in each stage, setup time, preventive maintenance information for each machine, and corrective maintenance times for each defective part in each stage. The setup times are stochastic data analyzed by commercial curve fitting software, Easy Fit 5.5. The resulting distributions for each machine type are validated by either Chi square or Kolmogorov–Smirnov tests for their goodness of fit. Table 3 shows machine data for maintenance workshop process, Table 4 illustrates job sequences and corrective maintenance times for each defective part, and Table 5 demonstrates time between arrivals of defective parts to maintenance workshop. It is supposed that the flow of work-in-process (WIP) between stages has approximately 30 meter per hour velocity (considering all waste times). Thus, the time taken to transfer WIP between each two stages can be calculated by dividing the distance into the flow velocity. 4.5 Simulation network modeling
n X
zij ¼ Z
ð10Þ
i¼1 j¼iþ1
xi xj yij ¼ ln a~ij ; i; j ¼ 1; 2; . . .; n; i 6¼ j;
ð11Þ
zij yij ; i; j ¼ 1; 2; . . .; n; i\j;
ð12Þ
zij yji ; i; j ¼ 1; 2; . . .; n; i\j;
ð13Þ
zmax zij ; i; j ¼ 1; 2; . . .; n; i\j;
ð14Þ
~ x1 ¼ 0; xi xj 0; i; j ¼ 1; 2; . . .; n; a~ij [ 1; a
ð15Þ
xi xj 0; i; j ¼ 1; 2; . . .; n; a~ik [ a~jk 8k; a~iq [ a~jq 9q; a
a
In this study, Visual SLAM, as a fully object-oriented simulation language is used for modeling and simulating the maintenance workshop process (Pritsker and O’Reilly 1999). In the simulation network of maintenance workshop process, eight defective parts and ten machines are considered as entities and servers, respectively. Each defective part type (job type) has two attributes identifying its type and starting time and is emanated in network by a CREATE node. Starting time of each defective part is determined by second ATRIB in CREATE node and the type of is determined by an ASSIGN node positioned after the
ð16Þ zij 0; i; j ¼ 1; 2; . . .x; n
ð17Þ
Table 3 Setup times, MTBPM and MTTPM for machines
xi ; yij unrestricted; i; j ¼ 1; 2; . . .; nzmax 0:
ð18Þ
Stage name
Number of machines
Setup time (h)
MTBPMa (h)
MTTPMb (h)
Saw lang
1
Uniform (0.23,0.45)
4,320
exp (7)
Lath Milling
1 1
0.13 Uniform (0.205,0.7)
4,320 4,320
exp (4) exp (4)
Constraint (9) ensures that only those solution vectors that are optimal in the first-stage fuzzy linear program are feasible in the second-stage model. Recall that z is the optimal objective function value of the first-stage model. Constraints (14) find zmax, the maximum value of the errors z. The objective function (9) minimizes zmax. Constraint (18) is the non-negativity constraint for zmax (although this constraint is redundant). All other constraints in the second-stage fuzzy model are identical to the corresponding constraints in the first-stage fuzzy model. 4.4 Data collection for the manufacturing process To illustrate the efficiency of the proposed algorithm in evaluating the generated layout alternatives from operational viewpoints, a set of operational data from a large maintenance workshop is applied. It is required the
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Wiredraw
1
0. 24
4,320
exp (7.5)
Weld
1
0
4,320
exp (8)
Press
1
0
4,320
exp (7)
Plasma cutter
1
Uniform (0.31,0.7)
4,320
exp (9)
Test bench
1
Uniform (0.22,0.31)
4,320
exp (6.5)
Drill
1
0
4,320
exp (10)
Vertical SAW
1
Uniform (0.2,0.29)
4,320
exp (5)
a
Mean time between preventive maintenance
b
Mean time to preventive maintenance
Int J Syst Assur Eng Manag Table 4 Corrective maintenance time of each defective part in each stage Defective part
Shaft
Job sequence and corrective maintenance time
Linguistic variable
Triangular fuzzy numbers
Linguistic variable
Triangular fuzzy numbers
(0,0,2)
Medium good (MG)
(5,6.5,8)
Job sequence
Corrective maintenance time
Very poor (VP)
Saw lang-lathe-milling
(0.433–16–10)a
Poor (P)
(1,2,3)
Good (G)
(7,8,9)
Medium poor (MP)
(2,3.5,5)
Very good (VG)
(8,10,10)
Fair (F)
(4,5,6)
Plate 1
Saw lang-wiredraw
(0.21,0.25)
Bar
Saw lang-lathe-milling
(0.3–1–8)
Nut
Saw lang-lathe-drill-milling
(0.16–0.5–0.41–6)
Valve
Test bench
(12)
Plate 2
Drill-press
(0.33–0.5)
Pipe
Vertical saw-wiredrawmilling-weld
(0.45–0.1–7–12)
Plate 3
Plasma cutter-Drill
(0.33–0.083)
a
Table 6 Fuzzy numbers
4.7 FDEA for optimization of the maintenance workshop facility layout
corrective maintenance time, hour (EXPON (X))
CREATE node. The model has eight CREATE nodes and eight ASSIGN nodes due to the existence of eight defective part types. The time between arrivals of defective parts (MTBO) is presented in Table 5. A defective part (entity) is sent to original network. If the requisite machine repairing this defective part is available, then it is assigned to the machine for during the corrective maintenance time. Otherwise, this defective part must be awaited in the file number of the AWAIT node. This process is done with all requisite AWAIT nodes of each defective part. Corrective maintenance time of each defective part by each machine is considered as array for the activity duration. After finishing the repairing of each machine, the machine is freed by FREE node, and then it is returned to the network. In the exit node, the times in system are collected and a report is printed to an output file in a pre-defined format. After simulating network for one year (8,760 h) the simulation will be completed.
The data in the conventional CCR and BCC models assume the form of specific numerical values. However, the observed value of the input and output data are sometimes imprecise or vague. Sengupta (1992a, b) was the first to introduce a fuzzy mathematical programming approach in which fuzziness was incorporated into the DEA model by defining tolerance levels on both the objective function and constraint violations. In the maintenance workshop’s facility layout design of this paper, shape ratio, distance, air pollution, tangible pollution, AWT and ATIS are considered as undesirable output indicators and other indicators as desirable output of the FDEA model. This is for the reason that shape ratio, distance, air pollution, tangible pollution, AWT and ATIS are undesirable and have to be reduced to improve the performance whereas other indicators are desired to be increased. Besides, the FDEA model comprises of no input indicators and so one dummy input equal to 1 is assumed for all DMU’s. In the case of maintenance workshop process, there are 12 outputs, 1 input and 75 DMU’s. Therefore, the primal and its dual fuzzy BCC models in outputoriented version for the maintenance workshop process can be formulated as: ð19Þ
maxh
4.6 Evaluation of health and environment indicators
s:t: After filling the standard questionnaire and determining respond of each question for each layout alternative, each responds get a fuzzy number by averaging the achieved figures up, we reach another fuzzy number which the decision of health, tangible pollution, and air pollution level is made by this fixed numbers. To show certain degree of vagueness on replying to each question, we use linguistic terms represented by triangular fuzzy numbers. Table 6 presents the fuzzy numbers used in this paper. Table 5 Time between arrivals of defective parts, (EXPON(X))
h~ yr0
75 X
kj y~rj ; r ¼ 1; 2; . . .; 75;
ð20Þ
j¼1 75 X
kj x~rj x~r0;i¼1
j¼1
75 X
kj ¼ 1kj 0; j ¼ 1; 2; . . .; 75
j¼1
ð21Þ Dual fuzzy BCC model (output-oriented)
Defective part
Shaft
Plate 1
Bar
Nut
Valve
Plate 2
Pipe
Plate 3
X
30
40
30
25
10
20
20
10
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min h ¼ v1 x~10
ð22Þ
s:t: 12 X
~ ur y~r0 þ u13 ¼ 1;
r¼1
12 X
~ ur y~rj v1 x~1j þ u13 0;
r¼1
j ¼ 1; 2; . . .; 75;
ur ; vi e [ 0;
r ¼ 1; 2; . . .; 12; i ¼ 1:
ð23Þ
5 Computational results In this paper, an integrated fuzzy algorithm is proposed to cope with a special case of workshop facility layout design problem. 75 layout alternatives have been generated by a computer-aided layout planning tool, VIP-PLANOPT. Then, quantitative performance indicators including flow distance, adjacency and shape ratio for each layout alternative have been achieved. 5.1 Fuzzy LP results In this paper we used a modified version of the linear programming models using linguistic variables to show vagueness in pairwise comparisons matrix. The modified versions of the LP models are applied for maintenance workshop process and the priority of layout alternatives is determined for each qualitative indicator. 5.2 Fuzzy simulation results Machines priorities and corrective maintenance times for each stage are modeled and analyzed by simulation. Data entering for each model is carried out by suitable control statements. The control statements is a utility provided by Visual SLAM software to import information such as the modelers’ names, project name, date of developing the model, and number of runs. It is also used to equalize the variables used in the network to allowable variables in Visual SLAM. The required information about the defective part type dependent corrective maintenance times have to be read from array statement that make a table with for rows and eight columns. In this array statement the columns number is the same as the defective part number. First row shows corrective maintenance time for each defective part using Saw Lang, and the second to forth rows show corrective maintenance time for each defective part by milling, lath and Drill respectively. After defining the control statements, the simulation model will be ready to run. The simulation network has been modified based on the flow distances between each two stages obtained by VIP-PLANOPT software for all 75 layout alternatives. Each simulation network is run 100 times and the results of all runs are averaged. Despite
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consuming satisfactory computational time, the simulation models converge after overtaking just about half of the runs. Figure 2 present the convergence trend of crisp simulation model, as an example, with respect to AWT, AMU, and ATIS for first layout alternative, respectively. Table 7 shows the fuzzy simulation results for all 75 layout alternatives at a = 0. Symbols p, m, and o stand for the most pessimistic, the most possible, and the most optimistic values of each operational indicator, respectively. The pessimistic and optimistic values are obtained from FS using upper and lower boundaries at a = 0 of the MTBA, MTTCM and MTTPM calculated in Sect. 4.6. The most possible values have been obtained from FS at a = 1. Note that since AWT and ATIS are undesirable outputs, their optimistic values (o) are less than their pessimistic values (p). 5.3 Verification and validation of fuzzy simulation results Hereafter, the results of FS models must be verified and validated. To do so, the paired t test is employed to perform a pairwise comparison between the results of Fuzzy Simulation at different a-cut levels and CS models with respect each evaluation measure, i.e. AWT, AMU, and ATIS. The paired t-test procedure is used to compare the mean difference between two variables when we believe that some dependency exists. We may wish to test if the mean difference is significantly different from zero, i.e. test H0:lDifference = 0 versus an alternative hypothesis such as H1:lDifference = 0. An assumption for the paired t-test procedure is that the distribution in which the differences analyzed come from is Normal. Therefore, we created a column for the differences between the two variables, and investigated the distributional properties. The AndersonDarling (A-D) Normality Test illustrated in Fig. 3 shows that, as an example, we are unable to reject the null hypothesis, H0: data follow a Normal distribution versus H1: data do not follow a Normal distribution, at the a = 0.05 significance level for differences between ATIS values obtained from FS for upper boundary at a = 0 and CS. This is because the p-value for the A-D test is 0.558, which is greater than 0.05.
Fig. 2 An example of convergence trend for crisp simulation model by AWT–Layout alternative #1
Int J Syst Assur Eng Manag Table 7 An example of paired t-test–comparison between ATIS values obtained from fuzzy simulation for upper boundary at a = 0 and crisp simulation Paired T-Test and CI: FS, CS Paired T for FS, CS N
Mean
St. dev
SE mean
FS
75
965.78
24.74
2.86
CS
75
796.74
16.61
1.92
Difference
75
169.04
27.36
3.16
95 % CI for mean difference: (162.74, 175.33) T-Test of mean difference = 0 (vs. not = 0): T-Value = 53.51 p-value = 0.000
Then, we performed a pairwise comparison between the results of FS at different a-cut levels and CS models with respect each evaluation measure, i.e. AWT, AMU, and ATIS using the popular statistical software package, MINITAB. As the output in Table 7 exhibits, we are able to reject the null hypothesis, lDifference = 0 at the a = 0.05 level of significance for ATIS values obtained from FS for upper boundary at a = 0 and CS, as the p-value is less than 0.05. In fact, the evidence strongly suggests that there is a difference between the CS and FS models (the p-value is very low), and the nature of input data are considerably uncertain. As a result, the CS model unable to deal with the uncertainty associated with the FLP under study. On the other hand, the results of paired t-test certify that application of FS significantly enhances the precision of simulation with reference to implicit knowledge of decision makers regarding the process of maintenance workshop. 5.4 Evaluation health and environment indicators After filling the questionnaire and determining respond of each question for each layout alternative, each of respond get a fuzzy number by averaging the achieved figures up, we reach another fuzzy number which the decision of health and
environment level is made by this fixed numbers. As we mentioned above, symbols p, m, and o stand for the most pessimistic, the most possible, and the most optimistic values of each indicator, respectively. Note that since tangible pollution and air pollution are undesirable outputs, their optimistic values (o) are less than their pessimistic values (p). 5.5 Fuzzy DEA results The outputs of fuzzy simulation including AWT, ATIS and AMU in addition to qualitative, quantitative, health and environment indicators are directly imported to the FDEA model. Shape ratio, distance, air pollution, tangible pollution, AWT and ATIS are considered as undesirable output indicators and other indicators as desirable output of the fuzzy DEA model. Besides, one dummy input equal to 1 is assumed for all DMU’s. Therefore, fuzzy DEA is able to find the optimal alternative. The results of fuzzy DEA model including Ranks and efficiency scores of 75 DMU’s (layout alternative) for a-cuts 0, 0.2, 0.4, 0.6, 0.8. DMU No. 13 has the highest efficiency score amongst the entire layout alternatives at all a-cuts levels. It is distinguished as the optimal layout alternative. As observed, some DMU’s have taken efficiency scores more than one. This is due to imposing penalty functions to the slack and surplus variables in the objective function of the FDEA model. Li et al. (2007), thus all DMU’s with efficiency score equal to or more than one are efficient. 5.6 Verification and validation of fuzzy DEA results Hereafter, the results of Fuzzy DEA must be verified and validated. To do so, TFNPC Algorithm is used. To calculate centroid of triangular fuzzy numbers, we use MATLAB software in this study. Note that since the values of quantitative indicators are crisp the length of them are zero, therefore we neglect from corresponding columns. Table 8 shows the results of fuzzy PCA model including Ranks and Zi scores of 75 layout alternative.
Fig. 3 An example of Anderson–Darling Normality Test–differences between ATIS values obtained from fuzzy simulation for upper boundary at a = 0 and crisp simulation
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Int J Syst Assur Eng Manag Table 8 Scores and rankings of layout alternatives through FPCA
Layout
Zi
Rank
Layout
Zi
Rank
Layout
1
-0.1723
64
26
-0.1726
65
51
0.1649
8
2
0.1031
18
27
-0.0827
56
52
-0.2405
71 73
0.2403
4
28
0.0593
29
53
-0.2879
4
-0.0026
41
29
-0.1063
58
54
-0.3564
75
5
-0.1572
60
30
0.1211
16
55
0.0364
33
6
-0.1748
66
31
-0.2098
69
56
0.0518
32
7
-0.0643
50
32
-0.0602
48
57
0.0309
36
8
0.0836
23
33
0.0337
35
58
-0.2126
70
9
-0.0219
44
34
0.3254
2
59
0.1003
20
0.1416
13
35
0.0115
38
60
0.1794
7
11
0.2262
5
36
0.0520
31
61
-0.0649
51
12
-0.1625
61
37
-0.2803
72
62
0.0530
30
13
-0.1709
3
38
-0.0484
47
63
0.0730
27
14
0.0744
26
39
0.2457
63
64
-0.3127
74
15 16
-0.1821 0.1602
68 9
40 41
-0.0097 -0.0816
42 55
65 66
0.1527 0.4052
10 1
17
0.1820
6
42
-0.0848
57
67
-0.1171
59
18
0.1136
17
43
0.1430
11
68
0.0874
21
19
-0.0693
53
44
0.1014
19
69
0.1304
14
20
-0.0604
49
45
-0.0814
54
70
0.1427
12
21
0.1249
15
46
-0.1692
62
71
-0.1768
67
22
0.0215
37
47
0.0352
34
72
-0.0353
46
23
0.0029
40
48
-0.0678
52
73
0.0782
25
24
0.0814
24
49
0.0679
28
74
0.0849
22
25
-0.0123
43
50
0.0109
39
75
-0.0245
45
Next, the results of FDEA are verified and validated by FPCA at each a-cut level. To compare the results of FPCA and FDEA, a non-parametric method has been utilized. One of the methods used for testing the correlation of the ranking data is the Spearman non-parametric experiment. P d2
i Using the rs ¼ 1 NðN 2 1Þ criteria, H0 is tested for identi-
fying that the two stated methods are uncorrelated on a pairwise comparison basis. The spearman correlation scores indicate (Table 9) an acceptable high correlation among the ranks obtained by FPCA and FDEA at each acut level especially at a = 0.4. Therefore, the ranking results obtained by FDEA and FPCA at each a-cut level are verified with relatively high degrees of confidence.
6 Conclusions This paper proposed a novel algorithm based on fuzzy simulation, fuzzy linear programming models, and fuzzy DEA to solve a particular case of facility layout problem in a maintenance workshop. The time required for maintenance process could not be certainly identified, so this time
123
Rank
3
10
6
Zi
Table 9 Results of Spearman correlation experiment
a Values
Spearman correlation index
0
0.48
0.2
0.57
0.4
0.83
0.6
0.71
0.8
0.64
1
0.49
was defined according to the objective data as well as subjective data. The proposed algorithm is used for simulating and ranking a set of layout alternatives generated by a computer aided layout planning tool, namely, VIPPLANOPT. First, quantitative indicators including distance, shape ratio and adjacency were calculated. Second, fuzzy LP models were utilized to evaluate the qualitative indicators including flexibility, maintenance and accessibility. Third, health and environmental indicators were obtained by filling a standard questionnaire; fourth computer simulation was utilized to model the maintenance workshop process with respect to the operational data including AMU, ATIS and AWT. Finally, a new FDEA/
Int J Syst Assur Eng Manag Table 10 The features of the integrated fuzzy algorithm versus other methods/studies Feature Method
Practicability in real world cases
Evaluation of qualitative indicators using fuzzy LP models
Ambiguous and Vague Data
Optimization through FDEA
Consideration of environmental indicators
Consideration of health indicator
Verification and validation of results
The proposed algorithm
4
4
4
4
4
4
4
Yang and Kuo (2003)
4
Conventional simulation
4
Ertay et al. (2006)
4
Azadeh and Izadbakhsh (2008) Azadeh et al. (2011)
4
4
4
4
AR model was used to find the optimal layout design for maintenance workshop. The key point of the proposed fuzzy algorithm lies in multi-criteria decision making in FLP’s through integrating FS, FLP, and FDEA. The proposed methodology for estimating weights of pairwise comparisons as well as defuzzifying methodology for the FDEA model is unique in the literature. Moreover, proposed fuzzy algorithm provides a comprehensive and robust approach in solving real world FLD problems by considering various indicators especially health and environment, that it is concerned with finding the best layout alternative for maintenance workshop that implementing situation which promote health of workshop’s operators and also environment’s quality within maintenance workshop and around it. Also, it uses fuzzy set theory to incorporate unquantifiable, incomplete and partially known information into the decision model. The proposed fuzzy algorithm is compared with some of the relevant studies and methodologies in the literature in Table 10 to show its advantages and superiorities over previous models. The proposed fuzzy algorithm could be simply put into practice in other FLD problems by some minor changes.
References Azadeh A, Izadbakhsh HR (2008) A multivariate/multi-attribute approach for plant layout design. Intern J Indus Eng 15(2): 143–154 Azadeh A, Moghaddam M, Asadzadeh SM, Negahban A (2011) An integrated fuzzy simulation-fuzzy data envelopment analysis
4
algorithm for job-shop layout optimization: the case of injection process with ambiguous data. Eur J Oper Res 214(3): 768–779 Chandran B, Golden B, Wasil E (2005) Linear programming models for estimating weights in the analytic hierarchy process. Comput Oper Res 32(9):2235–2254 Cheng R, Gen M (1996) Genetic search for facility layout design under inter flows uncertainty. Jpn J Fuzzy Theo Syst 8: 335–346 Ertay T, Ruanb D, Tuzkayac UR (2006) Integrating data envelopment analysis and analytic hierarchy for the facility layout design in manufacturing systems. Inf Sci 176(3):237–262 Heragu S (1997) Facilities design. PWS Publishing Company, Boston Heragu SS, Kusiak A (1990) Machine layout: an optimization and knowledge based approach. Int J Prod Res 28(4):615–635 Ho YC, Moodie CL (1998) Machine layout with a linear single row flow path in an automated manufacturing system. J Manuf Syst 17:1–22 Jime´nez M, Arenas M, Bilbao A, Rodriguez MV (2007) Linear programming with fuzz parameters: an interactive method resolution. Eur J Oper Res 177:1599–1609 Kumar KR, Hadjinicola GC, Lin TI (1995) A heuristic procedure for the single row facility layout problem. Eur J Oper Res 87:65–73 Li S, Jahanshahloo GR, Khodabakhshi M (2007) A super-efficiency model for ranking efficient units in data envelopment analysis. Appl Math Comput 184(2):638–648 Prabhaharan G, Padmanaban KP, Krishnakumar R (2006) Machining fixture layout optimization using FEM and evolutionary techniques. Intern J Adv Manuf Technol 32:1090–1103 Pritsker AAB, O’Reilly JJ (1999) Simulation with Visual SLAM and AweSim. John Wiley and Sons Inc, New York S¸ ahin R (2011) A simulated annealing algorithm for solving the biobjective facility layout problem. Expert Syst Appl 38:4460– 4465 Sengupta JK (1992a) A fuzzy systems approach in data envelopment analysis. Comput Math Appl 24:259–266 Sengupta JK (1992b) Measuring efficiency by a fuzzy statistical approach. Fuzzy Sets Syst 46(1):73–80
123
Int J Syst Assur Eng Manag Taghavi A, Murat A (2011) A heuristic procedure for the integrated facility layout design and flow assignment problem. Comput Ind Eng 61:55–63 Tompkins JA, White JA, Bozer YA, Frazelle EH, Tanchoco JMA, Trevino J (1996) Facilities planning. Wiley, New York Yaman BR, Balibek E (1999) Decision making for facility layout problem solutions. Comput Ind Eng 37:319–322
123
Yang T, Kuo C (2003) A hierarchical AHP/DEA methodology for the facilities layout design problem. Eur J Oper Res 147(1): 128–136 Yang T, Su CT, Hsu YR (2000) Systematic layout planning: a study on semiconductor wafer fabrication facilities. Intern J Oper Prod Manag 20(11):1360–1372
