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:

  • 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).

  • 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.

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. Şahin (2011) proposed a SA algorithm to solve the bi-objective 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 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

3.1 Description of the maintenance workshop

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.

Fig. 1
figure 1

The current facility layout for the maintenance workshop process

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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;

  • 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:

  1. I.

    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.

  2. II.

    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.

  3. III.

    Calculate quantitative performance indicators including distance, adjacency and shape ratio.

  4. IV.

    Apply fuzzy LP models for evaluation of qualitative performance indicators.

  5. V.

    Convert the fuzzy LP models to equivalent crisp ones using the possibilistic programming method proposed by Jiménez et al. (2007) and then evaluate the qualitative performance indicators including flexibility, accessibility, and maintenance.

  6. VI.

    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.

  7. VII.

    Develop the crisp simulation (CS) network model for each layout alternative.

  8. VIII.

    Fuzzify the required data.

  9. IX.

    Develop the simulation network model for each layout alternative with fuzzy inputs at different α-cut levels and then obtain their required outputs; i.e. average waiting time (AWT), average machine utilization (AMU), and average time in system (ATIS).

  10. X.

    Apply the questionnaire with fuzzy numbers to calculate the impact of health and environment indicators.

  11. XI.

    Apply FDEA model for assessment and ranking of the layout alternatives.

  12. XII.

    Incorporate distance, adjacency, shape ratio, flexibility, accessibility, maintenance, health, air pollution, tangible pollution and the fuzzy results of the FS model at α = 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.

  13. XIII.

    Convert the FDEA model to its equivalent crisp one using the possibilistic programming method proposed by Jiménez et al. (2007) to calculate the efficiency score for each layout alternative and finding the best one for maintenance workshop for different α-cut levels.

  14. XIV.

    Verify and validate the result of FDEA by FPCA at each α-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 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.

Table 1 Facilities sizes of the ten stages for maintenance workshop process
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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.

4.3 Fuzzy LP models

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.

Table 2 Saaty’s scale expressed in fuzzy numbers
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4.3.1 First stage: fuzzy linear program to establish the consistency bound

Suppose that \(\tilde{a}_{ij} = (a_{ij}^{p} ,a_{ij}^{m} ,a_{ij}^{o} )\) 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 ɛ ij is an error in the estimate of the relative preference \(\tilde{a}_{ij}\). If the decision maker is consistent, we have ɛ ij  = 0. The decision variables are given by w i  = weight of element i and error factor in estimating \(\tilde{a}_{ij}\). We use three transformed decision variables in LP model: x i  = ln w i , y ij  = ln ɛ ij , and z ij  = |y ij |. The modified fuzzy LP model for the first stage is as follows:

$${\text{Minimize }}\mathop \sum \limits_{\text{i = 1}}^{n - 1} \mathop \sum \limits_{j = i + 1}^{2} z_{ij}$$
(1)

s.t

$$x_{i} - x_{j} - y_{ij} = {\text{In}}\,\tilde{a}_{ij} ,i,j = 1,2, \ldots ,n;\,i \ne j,$$
(2)
$$z_{ij} \ge y_{ij} , i, j = 1, 2, \ldots , n; i < j,\,\,\,\,\,\, z_{ij} \ge y_{ji} , i, j = 1, 2, \ldots , n;\,i < j,$$
(3)
$$x_{1} = 0,\,\,\,\,\,\,\,\,\,x_{i} - x_{j} \ge 0,i,j = 1,2, \ldots ,n;\,\tilde{a}_{ij} \mathop > \limits_{\alpha } \tilde{1},$$
(4)
$$x_{i} - x_{j} \ge 0,i,j = 1,2, \ldots n;\tilde{a}_{ik} \mathop > \limits_{\alpha } \tilde{a}_{jk} \forall k,\tilde{a}_{iq} \mathop > \limits_{\alpha } \tilde{a}_{jq} \exists q,$$
(5)
$$z_{ij} \ge 0, i, j = 1, 2, \ldots , n\,\,\,\,\,\,\,\,x_{i} , y_{ij} \,unrestricted, \,i, j\, = \, 1, 2, \ldots , n.$$
(6)

Constraints (2) is the natural logarithm of \(\frac{{w_{i} }}{{w_{j} }} = \tilde{a}_{ij} \varepsilon_{ij}\) in comparison matrix A, if \(\tilde{a}_{ij}\) is overestimated at α degree (that is, the decision maker’s judgment of entry i versus entry j is greater than \(\tilde{1}\) at α degree), then \(\tilde{a}_{ji}\) is underestimated at this degree. We then have

$$\varepsilon_{ij} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\varepsilon_{ji} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\varepsilon_{ji} }$}},i,j = 1,2, \ldots ,n$$
(7)
$${\text{Or}}\,\,y_{ij} = - y_{ji} , i, j = 1, 2, \ldots , n.$$
(8)

By entering the grater of y ij and y ji 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 w i without loss of generality. This is done in constraint (5) by setting w 1 = 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 \(\tilde{a}_{ij} \mathop > \limits_{\alpha } \tilde{1}\) implies w i  ≥ w j . This property is known as ED or weak rank preservation. ED is preserved if \(\tilde{a}_{ij} \mathop > \limits_{\alpha } \tilde{1}\) implies w i  ≥ w j and this property is explicitly enforced through constraints (6). A solution method preserve ranks strongly if \(\tilde{a}_{ik} \mathop > \limits_{\alpha } \tilde{a}_{jk}\) for all k implies w i  ≥ w j . This property is known as RD or strong rank preservation and this property is explicitly enforced through constraints (7). In constraint (9) we point out that x i and y ij 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 ɛ ij  ≥ 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:

$$CI_{LP} = 2Z^{*} /n(n - 1)$$

CI LP is the average value of Z for elements above the diagonal in the comparison matrix. In preliminary computational experiments, CI LP 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 ɛ ij . 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 ɛ ij . The second-stage of fuzzy LP is given by the following:

$${\text{Minimize }}\,\,\,z_{ \hbox{max} }$$
(9)
$$s.t.$$
$$\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {z_{ij} } } = Z^{*}$$
(10)
$$x_{i} - x_{j} - y_{ij} = \ln \tilde{a}_{ij} ,i,j = 1,2, \ldots ,n;\,i \ne j,$$
(11)
$$z_{ij} \ge y_{ij} ,i,j = 1,2, \ldots ,n;\,i < j,$$
(12)
$$z_{ij} \ge y_{ji} ,i,j = 1,2, \ldots ,n;\,i < j,$$
(13)
$$z_{\hbox{max} } \ge z_{ij} ,i,j = 1,2, \ldots ,n;\,i < j,$$
(14)
$$x_{1} = 0,\,\,\,\,\,x_{i} - x_{j} \ge 0,i,j = 1,2, \ldots ,n;\,\tilde{a}_{ij} \mathop > \limits_{\alpha } \tilde{1},$$
(15)
$$x_{i} - x_{j} \ge 0,i,j = 1,2, \ldots ,n;\tilde{a}_{ik} \mathop > \limits_{\alpha } \tilde{a}_{jk} \forall k,\tilde{a}_{iq} \mathop > \limits_{\alpha } \tilde{a}_{jq} \exists q,$$
(16)
$$z_{ij} \ge 0,i,j = 1,2, \ldots x,n$$
(17)
$$x_{i} , y_{ij} \,unrestricted, \,i, j \, = \, 1, 2, \ldots , \,nz_{ \hbox{max} } \ge 0.$$
(18)

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 z max, the maximum value of the errors z. The objective function (9) minimizes z max. Constraint (18) is the non-negativity constraint for z max (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 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.

Table 3 Setup times, MTBPM and MTTPM for machines
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Table 4 Corrective maintenance time of each defective part in each stage
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Table 5 Time between arrivals of defective parts, (EXPON(X))
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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

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 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.

4.6 Evaluation of health and environment indicators

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 6 Fuzzy numbers
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4.7 FDEA for optimization of the maintenance workshop facility layout

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 output-oriented version for the maintenance workshop process can be formulated as:

$$\hbox{max} \theta$$
(19)
$$s.t.$$
$$\theta \tilde{y}_{r0} \le \sum\limits_{j = 1}^{75} {\lambda_{j} } \tilde{y}_{rj} ,r = 1,2, \ldots ,75,$$
(20)
$$\sum\limits_{j = 1}^{75} {\lambda_{j} } \tilde{x}_{rj} \le \tilde{x}_{r0,i = 1} \,\,\,\,\,\,\,\,\mathop \sum \limits_{j = 1} ^{ 7 5} \lambda_{j} = 1\lambda_{j} \ge 0, j = 1, 2, \ldots , 7 5$$
(21)

Dual fuzzy BCC model (output-oriented)

$$\hbox{min} \,\theta = v_{1} \tilde{x}_{10}$$
(22)
$$s.t.$$
$$\sum\limits_{r = 1}^{12} {u_{r} } \tilde{y}_{r0} + u_{13} = \tilde{1},\,\,\,\,\,\,\,\,\sum\limits_{r = 1}^{12} {u_{r} } \tilde{y}_{rj} - v_{1} \tilde{x}_{1j} + u_{13} \le \tilde{0},j = 1,2, \ldots ,75,\,\,\,\,\,\,\,\,u_{r} , v_{i} \ge \varepsilon > 0, r = 1, 2, \ldots , 1 2, i = 1.$$
(23)

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 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.

Fig. 2
figure 2

An example of convergence trend for crisp simulation model by AWT–Layout alternative #1

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Table 7 shows the fuzzy simulation results for all 75 layout alternatives at α = 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 α = 0 of the MTBA, MTTCM and MTTPM calculated in Sect. 4.6. The most possible values have been obtained from FS at α = 1. Note that since AWT and ATIS are undesirable outputs, their optimistic values (o) are less than their pessimistic values (p).

Table 7 An example of paired t-test–comparison between ATIS values obtained from fuzzy simulation for upper boundary at α = 0 and crisp simulation
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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 α-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 H 0:μ Difference  = 0 versus an alternative hypothesis such as H 1:μ Difference  ≠ 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 Anderson-Darling (A-D) Normality Test illustrated in Fig. 3 shows that, as an example, we are unable to reject the null hypothesis, H 0: data follow a Normal distribution versus H 1: data do not follow a Normal distribution, at the α = 0.05 significance level for differences between ATIS values obtained from FS for upper boundary at α = 0 and CS. This is because the p-value for the A-D test is 0.558, which is greater than 0.05.

Fig. 3
figure 3

An example of Anderson–Darling Normality Test–differences between ATIS values obtained from fuzzy simulation for upper boundary at α = 0 and crisp simulation

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Then, we performed a pairwise comparison between the results of FS at different α-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, μ Difference  = 0 at the α = 0.05 level of significance for ATIS values obtained from FS for upper boundary at α = 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 α-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 α-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.

Table 8 Scores and rankings of layout alternatives through FPCA
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Next, the results of FDEA are verified and validated by FPCA at each α-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. Using the \(r_{s} = 1 - \frac{{6\sum {d_{i}^{2} } }}{{N(N^{2} - 1)}}\) criteria, H 0 is tested for identifying 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 α-cut level especially at α = 0.4. Therefore, the ranking results obtained by FDEA and FPCA at each α-cut level are verified with relatively high degrees of confidence.

Table 9 Results of Spearman correlation experiment
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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 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, VIP-PLANOPT. 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/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.

Table 10 The features of the integrated fuzzy algorithm versus other methods/studies
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