1 Introduction

In recent years, sustainable operations management has attracted attention from both academics and practitioners. The concept of ‘sustainable operations management’ has gained serious considerations due to scarce natural resources and rapid change in climate and increasing social inequality, which forced enterprises to revisit their operations management practices to address 3Ps, that is, planet, people, and profit (Drake and Spinler 2013). Since the 1980s Kleindorfer and Kunreuther (1994) have argued how operations management practices can contribute towards sustainability. Since then, over three decades, work on sustainable operations is still in its infancy. The sustainable operations management field has been rapidly replaced by the holistic term “sustainable supply chain management (SSCM)” (see Govindan and Cheng 2015). Still, sustainable operations decisions and in particular facility layout are important and need to be guided by low cost and environmental related regulatory norms (Bayraktar et al. 2007; Subramoniam et al. 2009).

In this paper we are concerned with facility layout decision in sustainable operations. In recent years it has been noted that most of the manufacturing units have been moved to low labor cost country and weak regulatory norms. There is a rich body of literature on facility layout problems that focuses on cost, but research on facility layout design from a sustainability point of view is scant (Sacaluga and Frojan 2014). Hence, we argue that to offer holistic solutions to current problems, the 3 pillars of sustainability—economic, social and environment must be aligned in finding a desirable facility layout which is shown in Fig. 1.

Fig. 1
figure 1

3-Ps of sustainability

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A typical facility layout problem involves optimum placement of facilities by minimizing the material handling cost. However, due to fluctuation in economic and political situations and seasonal changes the production rates inevitably fluctuate. A stochastic dynamic facility layout model incorporates these variations as an expression of demand variability in the facility layout. These are expressed as probability distribution function. This argument is formulated as a mathematical expression with the aim to minimize the material handling and rearrangement cost (quantitative factors) and is known as stochastic dynamic facility layout problem (SDFLP). This model, however, ignores social and environmental factors such as ease of maintenance, waste disposal, ease of working, and job creation. These characteristics can be expressed as qualitative parameters, and when associated with the SDFLP model provides a sustainable SDFLP model, which can be solved to get a sustainable layout. The framework of proposed sustainable SDFLP is shown in Fig. 2.

Fig. 2
figure 2

3-P’s framework of sustainable SDFLP

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In the paper, a novel method is proposed which solves sustainable SDFLP considering both qualitative and quantitative factors under stochastic product demand flow over multi time period, using the hierarchical framework of-meta heuristic, multiple attribute decision making (MADM) techniques and consensus ranking method. The proposed methodology integrates meta-heuristics techniques viz. SA, CSA, Hybrid FA/CSA to generate layouts followed by applying DEA to identify efficient layouts among the generated ones, and finally applying MADM approaches such as TOPSIS, IRP and AHP in association with aggregate ranking methods viz. Borda–Kendall and integer linear programming (ILP) considering six different criteria i.e. material handling cost, flow distance, rearrangement cost, accessibility, maintenance and waste management to design SSDFLP. Our contribution lies in addressing the FLP problem from a sustainability perspective (investigating economic, social, and environmental perspectives) (Yang et al. 2013; Sacaluga and Frojan 2014; Lieckens et al. 2015) while incorporating both quantitative and qualitative criteria (Moslemipour and Lee 2011; Garcia-Hernandez 2013; García-Hernández 2015; Yang et al. 2013; Tayal and Singh 2014).

The paper is organized as follows. Section 2 reviews the past literature and underlines the research gaps. Section 3 discusses the mathematical formulation of SDFLP, the qualitative and qualitative parameters of sustainability and formulates the Sustainable SDFLP model. Section 4 elucidates the methodology to identify the optimum layout. Section 5 provides the numerical illustration using problem size, \(\hbox {N}=12\), time period, \(\hbox {T}=5\) and Gaussian distribution product demand. Section 6 discusses our results in light of the literature, whereas Sect. 7 summarizes our research findings.

2 Literature review

2.1 Sustainable operations management

Elliott (2001) has argued the role of operations management in sustainability, whereas in a later study (Drake and Spinler 2013) have argued that the future role of operations management needs to address issues related to the 3Ps, that is, planet, people and profit. Gupta (1995) have discussed the need for aligning environmental strategy with operations strategy. To address environmental problems Gupta and Sharma (1996) have proposed the term ‘environmental operations management’ (EOM), defined as the integration of environmental management with operations management principles. Sarkis (2001) further attempted to extend the EOM definition by focusing on tools such as: design for environment (DOE), green supply chains, total quality environmental management (TQEM), and reverse logistics. However, the most notable contribution towards the emerging field of sustainable operations management (SOM) was by Kleindorfer et al. (2005), who have identified the scope for operations management surrounding around three Ps (planet, people, and profit) in three areas: (1) Green product and process development, (2) Lean and green operations management and (3) Remanufacturing and closed-loop supply chains. Linton et al. (2007) underlined the implications of sustainability for supply chains, whereas Nunes and Bennett (2010) have noted the importance paid by manufacturers to issues related to green buildings, eco-design, green supply chains, reverse logistics and innovation. In a recent study Yu and Ramanathan (2015) have investigated two dimensions of green operations (i.e. internal green practices and green product/ process design) on environmental performance under the influence of stakeholder’s pressures.

Within sustainable operations, facility layout design has been identified as having an essential impact on the operations performance, especially within manufacturing systems (Yang et al. 2013), and is explicated in the next section.

2.2 Facility layout design

Layout design is a strategic issue (Timothy 1998; Yang et al. 2013) and has a significant impact on the performance of a manufacturing or service industry (Canen and Williamson 1998; Yang et al. 2013). Engineers, workers, and decision makers have attempted to obtain the best layout with the view to optimize material flow distance, total product produced, cycle time, waiting time, facility utilization, etc. According to Tompkins et al. (2003), total MHC is an appropriate measure to evaluate the efficiency of the layout and forms 20–50 % of the total manufacturing cost. Researchers classified the facility layout problem into static and stochastic facility layout problem. In today’s manufacturing environment product flow is uncertain over multi time period hence the facility layout needs to be adept to these changes. This type of facility layout problem is referred to as stochastic dynamic facility layout problem (SDFLP). SDFLP is a combinatorial optimization and non-deterministic polynomial complete problem (for FLP see O’brien and Abdel 1980; Tompkins et al. 1996; Kusiak and Heragu 1987; Rosenblatt and Lee 1987; Singh and Sharma 2006; Singh and Singh 2010). McKendall et al. (2006) have addressed the need for building dynamic facility layout problem (DFLP) due to demand uncertainty and supply uncertainty. Balakrishnan and Cheng (2009) have further argued to develop DFLP algorithms so that demand uncertainty does not influence the algorithms performance. Lieckens et al. (2015) have argued the need for sustainable aspect, which includes moral hazards while locating the maintenance services with remanufacturing unit location and its layout design. Recently, Akash and Singh (2016) applied big data analytics to optimize stochastic dynamic facility layout problem.

However, the majority of the literature on stochastic FLP literature uses mostly quantitative criteria including shape ratio, material handling cost and rearrangement cost, adjacency score, and space demand as well as qualitative criteria such as flexibility and quality (Les and Fariborz 1998; Albert et al. 2010; Moslemipour and Lee 2011; Yang et al. 2013; Tayal and Singh 2014) but apart from few exceptions focusing mainly on energy-efficient facility layouts (Yang et al. 2013; Sacaluga and Frojan 2014), literature has not yet fully discussed social and environmental issues which are key to sustainable operations management, and, has not looked into the generation of aggregate ranking to obtain a desirable layout that has a highest degree of satisfaction for quantitative and qualitative sustainability parameters. To address these gaps, this study proposes a sustainable SDFLP model that considers both qualitative and quantitative criteria under stochastic product demand flow over multi time period, using the hierarchical framework of-meta heuristic, multiple attribute decision making (MADM) techniques and consensus ranking method. The model is discussed in the next sections. More details of facility layout can be seen from Borda (1781), Kendall (1962), Rosenblatt (1979), Dutta and Sahu (1982), Kirkpatrick et al. (1983), Fortenberry and Cox (1985), Hajek (1988), Khare et al. (1988a, b), Zouein and Tommelein (1999), Sinuany-Stern et al. (2000), Yang and Kuo (2003), Kulturel-Konak et al. (2004), Bruglia et al. (2004, 2005), Kulturel-Konak (2007), Tavana et al. (2007), Kia et al. (2012), Moslemipour et al. (2012), and Date et al. (2014).

3 Sustainable SDFLP formulation

The various aspects of sustainable SDFLP formulation—mathematical equations, quantitative and qualitative factors of sustainability, are discussed in the next sub-sections.

3.1 Mathematical formulation of SDFLP

FLP was modeled as quadratic assignment problem (QAP) by Koopmans and Beckman (1957), given in Eqs. (1)–(4). Balakrishnan et al. (1992) provided the QAP mathematical model for dynamic facility layout problem (DFLP), including the rearrangement cost, is given in Eqs. (5)–(9).

$$\begin{aligned} C\left( \pi \right) =\mathop {\sum }\nolimits _{{i=1}}^{N} \mathop {\sum }\nolimits _{{j=1}}^{N} \mathop {\sum }\nolimits _{{l=i}}^{N} \mathop {\sum }\nolimits _{{q=1}}^{N} f_{ij} d_{lq} x_{il} x_{jq} \end{aligned}$$
(1)

Notations

Description

i, j

Index for facilities \(\left( {i,j=1,2,\ldots N} \right) ;i\ne j\)

l, q

Index for locations \(\left( {l,q=1,2,\ldots N} \right) ;l\ne q\)

\(f_{ij} \)

Flow of material between facilities i to j

\(f_{tij} \)

Flow of material between facilities i to j in time period t

\(d_{lq} \)

Distance between locations l and q

N

Number of facilities

\(C\left( \pi \right) \)

Total MHC for layout \(\pi \)

\(E\left( \pi \right) \)

Expected value of a \(\pi \)-th layout

\(Var\left( \pi \right) \)

Variance of a \(\pi \)-th layout

\(Pr(\pi )\)

Probability of a \(\pi \)-th layout

Zp

Standard Z (random variable) value for percentile p

\(U\left( {\pi ,p} \right) \)

Maximum value upper bound of \(C\left( \pi \right) \)with confidence level p

K

Index for parts (k = 1, 2,..., K)

\(M_{ki} \)

Operation number for the operation done on part k by facility i

\(D_{kt} \)

Demand for part k in period t

\(B_k \)

Transfer batch size for part k

\(C_{tk} \)

Cost of movements for part k in period t

Z

Random variable

\(a_{tilq} \)

Fixed cost of shifting facility i from location l to location q in period t

\(R_c \)

Rearrangement cost

MHC

Material handling cost

\(\mu _{ij} \)

Mean of product demand

\(\sigma _{ij}^2\)

Variance of product demand

Subject to:

$$\begin{aligned}&\mathop {\sum }\nolimits _{i=1}^N x_{il} =1;\qquad \quad \forall l \end{aligned}$$
(2)
$$\begin{aligned}&\mathop {\sum }\nolimits _{l=1}^N x_{il} =1;\qquad \quad \forall i \end{aligned}$$
(3)
$$\begin{aligned}&x_{il} =\left\{ {{\begin{array}{ll} 1,&{}\quad {\textit{if facilities i is assigned to location l}} \\ 0,&{}\quad {\textit{otherwise}} \\ \end{array} }} \right. \end{aligned}$$
(4)

Dynamic FLP is modeled as shown below:

$$\begin{aligned} \hbox {Minimize}\,\hbox {C}\left( \pi \right)= & {} \mathop {\sum } \nolimits _{\mathrm{t=1}}^\mathrm{T} \mathop {\sum }\nolimits _{\mathrm{i=1}}^\mathrm{M} \mathop {\sum }\nolimits _{\mathrm{j=1}}^\mathrm{M} \mathop {\sum }\nolimits _{\mathrm{l}=1}^\mathrm{M} \mathop {\sum }\nolimits _{\mathrm{q=1}}^\mathrm{M} f_{tij} \hbox {d}_{\mathrm{lq}} x_{til} x_{tjq} \nonumber \\&+\,\mathop {\sum } \nolimits _{\mathrm{t=2}}^\mathrm{T} \mathop {\sum }\nolimits _{\mathrm{i=1}}^\mathrm{M} \mathop {\sum }\nolimits _{\mathrm{l}=1}^\mathrm{M} \mathop {\sum }\nolimits _{\mathrm{q=1}}^\mathrm{M} \hbox {a}_{\mathrm{tilq}} \hbox {y}_{\mathrm{tilq}} \end{aligned}$$
(5)

Subject to:

$$\begin{aligned}&\mathop {\sum }\nolimits _{\mathrm{i}=1}^\mathrm{M} x_{til} =1;\qquad \quad \forall \hbox {t},\hbox {l} \end{aligned}$$
(6)
$$\begin{aligned}&\mathop {\sum }\nolimits _{\mathrm{l}=1}^\mathrm{M} x_{til} =1;\qquad \quad \forall \hbox {t},\hbox {i} \end{aligned}$$
(7)
$$\begin{aligned}&x_{til} =\left\{ {{\begin{array}{ll} 1,&{}\quad { if\,facilities\,i\,is\,assigned\,to\,location\,l\,in\,period\,t} \\ 0,&{}\quad { otherwise} \\ \end{array} }} \right. \end{aligned}$$
(8)
$$\begin{aligned}&\hbox {y}_{\mathrm{tilq}} =x_{(\mathrm{t}-1){\mathrm{il}}} \times x_{\mathrm{tiq}} \end{aligned}$$
(9)

The product flows between facilities are generally an expression of demand, which could be static, dynamic or uncertain. Rosenblatt and Kropp (1992) first proposed an analytical formulation of static stochastic facility layout problem (SFLP). The uncertainty treatment in the facility layout has gained prominence in the present scenario where the product demand or the product mix is not known deterministically but stochastically. DFLP mathematical model can be modified for the Stochastic DFLP model by assuming product demand to be random variable and is expressed as probability distribution function (PDF) with known mean and variance. Equation (5) is modified for stochastic process and \(C(\pi )\) becomes a function of random variables. Here, \(f_{{tij}}\) is changed to stochastic variable due to uncertainty of demand with mean \(\upmu _{\mathrm{ij}}\) and variance \(\upsigma _{\mathrm{ij}}^{2}\). Objective function for SDFLP includes MHC and \(\hbox {R}_{\mathrm{c}}\) and given in Eq. (10) (Moslemipour and Lee 2011).

$$\begin{aligned} \hbox {Minimize}\left\{ {{\begin{array}{l} {\left[ {{\begin{array}{l} {\mathop {\sum }\nolimits _{t=1}^T \mathop {\sum }\nolimits _{i=1}^N \mathop {\sum }\nolimits _{j=1}^N \mathop {\sum }\nolimits _{k=1}^N \frac{E(D_{tk} )}{B_k }C_{tk} \mathop {\sum }\nolimits _{l=1}^N \mathop {\sum }\nolimits _{q=1}^N d_{lq} x_{til} x_{tjq} } \\ {+Z_p \sqrt{\mathop {\sum }\nolimits _{{t=1}}^T \mathop {\sum }\nolimits _{{i=1}}^{N} \mathop {\sum }\nolimits _{{j=1}}^{N} \mathop {\sum }\nolimits _{k=1}^K \frac{Var(D_{tk} )}{B_k^2 }C_{tk}^2 \left( {\mathop {\sum }\nolimits _{l=1}^N \mathop {\sum }\nolimits _{{q=1}}^{N} d_{lq} x_{til} x_{tjq} } \right) ^{2}}} \\ \end{array} }} \right] } \\ {\quad +\left[ {\mathop {\sum }\nolimits _{{t=2}}^T \mathop {\sum }\nolimits _{{i=1}}^{N} \mathop {\sum }\nolimits _{l=1}^N \mathop {\sum }\nolimits _{{q=1}}^{N} a_{tilq} x_{\left( {t-1} \right) il} x_{tiq} } \right] } \\ \end{array} }} \right\} \end{aligned}$$
(10)

Subject to:

$$\begin{aligned}&\mathop {\sum }\nolimits _{\mathrm{i}=1}^\mathrm{N} x_{til} =1;\qquad \quad \forall \hbox {t},\hbox {l} \end{aligned}$$
(11)
$$\begin{aligned}&\mathop {\sum }\nolimits _{\mathrm{l}=1}^\mathrm{N} x_{til} =1;\qquad \quad \forall \hbox {t},\hbox {i} \end{aligned}$$
(12)
$$\begin{aligned}&x_{til} =\left\{ {{\begin{array}{ll} 1,&{}\quad \text {if facilities is assigned to location in period t} \\ 0,&{}\quad \text {otherwise} \\ \end{array} }} \right. \end{aligned}$$
(13)
$$\begin{aligned}&\left| {\hbox {M}_{\mathrm{ki}} -\hbox {M}_{\mathrm{kj}} } \right| =1 \end{aligned}$$
(14)

3.2 Quantitative and qualitative attributes for sustainability

A preliminary review of the literature and experts’ opinion was conducted to determine the quantitative and qualitative design attributes of the model, as well as the sustainability pillars to be included. The quantitative attributes included material handling cost (MHC), flow distance and rearrangement cost. Qualitative attributes included accessibility, maintenance, and waste management. The economic, social, and environmental pillar of sustainability were included as follows: (i) for the economic pillar the model included MHC, Rearrangement cost \((R_{c})\) and flow distance. Material handling cost (MHC), is calculated as product of flow of material between the facilities and travelled distance between the locations. Due to change in product demand there is a change in flow of materials from one time period to next. Rearrangement Cost \((\hbox {R}_\mathrm{c})\), is the variable cost of moving facility i in time period t to facility j in time period \(t+1.\) Flow distance, is equal to the sum of the products of flow volume and rectilinear distance between the centroids of two departments. (ii) For the social pillar the model included maintenance and accessibility. Maintenance is related to a number of activities like upgradation of the existing facility, recycling, waste disposal in the built-in environment so as to reduce the level of hazards, pollution and consumption of environmental resources.Accessibility involves the required space for material handling path, personal flow (operator path), information flow and equipment flow. (iii) For the environmental pillar the model included waste management. Waste management involves all those activities or actions required to manage waste from its inception to its disposal. Waste flow time is the time required for the movement of waste between two departments (machines).

3.3 Sustainable SDFLP formulation

The sustainable SDFLP involves assigning facilities to location to satisfy the multiple quantitative and qualitative parameters. For sustainable facility layout design problem in a stochastic demand, MHC, \(\hbox {R}_{\mathrm{c}}\), flow distance and waste are minimized while accessibility and maintenance are maximized. Figure 3 presents the flow chart to model sustainable stochastic dynamic facility layout problem (SSDFLP) and shows major stages involved in modeling the proposed SSDFLP. The methodology to solve the proposed sustainable SDFLP is discussed in the following section.

Fig. 3
figure 3

Flow chart of SSDFLP

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4 Methodology to solve sustainable SDFLP

Malakooti (1989) presented three methodologies for solving MO-FLP problem which are described below:

  1. 1.

    Generating a set of efficient layout alternatives by varying the weights assigned to the objective functions and presenting it to the decision maker,

  2. 2.

    Assessing the decision-maker’s preferences first, then generating the best layout alternative, and

  3. 3.

    Using an interactive method to find the best layout alternative.

This paper adds to the aforementioned methodologies by proposing a fourth methodology, that is, ranking a pool of layouts using expert’s opinion and MADM techniques to find a practical facility layout satisfying the qualitative and quantitative criteria. The proposed approach includes three steps: (1) generating pool of optimal layouts, (2) ranking the layout using expert opinion and various MADM techniques, and (3) subjectivity reduction in ranks using aggregate ranking method. To generate the pool of optimal layouts either meta-heuristic techniques or computer aided software can be used. The layouts are assessed by the experts based on the 3Ps of sustainability.

Evaluating and analyzing a pool of layout is a challenge for any expert therefore a reduced set of layouts was needed. According to Tompkins et al. (2003), total MHC (sum of material handling cost and rearrangement cost) forms 20–50 % of the total manufacturing cost, hence the layouts were evaluated on material handling cost, rearrangement cost and flow distance which forms the Profit factor of sustainable SDFLP. Data envelopment analysis (DEA) technique was applied. This reduced set of layout need to be ranked for which experts were involved for computing the weights of criteria’s (3P’s). Both MCDM techniques and expert opinions were applied to get the rank of the layouts. Ranking of conflicting quantitative and qualitative criteria’s of 3Ps is highly subjective; to overcome subjectivity, aggregate ranking is applied. The description of the methodology to solve proposed sustainable SDFLP is presented in Fig. 4. The detailed description of each step is provided in the following paragraphs.

Fig. 4
figure 4

Methodological framework of proposed SSDFLP

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Step 1: SDFLP layout generation

This step uses either commercial computer-aided planning tools such as Spiral, ALDEP, BLOKPLAN or metaheuristic techniques (SA, CSA, Hybrid FA/CSA) to generate layout alternatives, as well as a collection of quantitative performance data. The techniques SA, CSA and Hybrid FA/CSA are used to generate a pool of layouts and its data for quantitative parameters is collected as shown in Figs. 5, 6, and 7, respectively. Detailed description on meta-heuristic techniques for solving SDFLP can be found in Tayal and Singh (2015), Tayal and Singh (2016a, b).

Fig. 5
figure 5

Simulated annealing for solving SDFLP

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Fig. 6
figure 6

Chaotic simulated annealing for solving SDFLP

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Fig. 7
figure 7

Hybrid FA/CSA for solving SDFLP (Tayal and Singh 2016c)

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Step 2: identify efficient SDFLP layouts using DEA

Data envelopment analysis (DEA) is applied to identify set of efficient layouts among all possible layouts obtained in Step 1. DEA is a non-parametric approach in operations research that does not require any assumptions about the functional form for the estimation of production frontiers. Assume that there are n decision-making units (DMUs) to be evaluated. Each DMU consumes varying amount of m different inputs to produce s different outputs. Following are the notations used in the DEA.

Notations

Description

\(\hbox {DMU}_\mathrm{k} \)

\(\hbox {k}{\mathrm{th}}\) decision making unit (DMU), \(\hbox {k}=1,2,\ldots ,\hbox {n}\)

\(\hbox {X}_{\mathrm{ik}} \)

\(\hbox {i}{\mathrm{th}}\) input for the \(\hbox {k}{\mathrm{th}}\) DMU, \(\hbox {i}=1,2,\ldots ,\hbox {m}\) and \(\hbox {k}=1,2,\ldots ,\hbox {n}\)

\(\hbox {Y}_{\mathrm{rk}} \)

\(\hbox {r}{\mathrm{th}}\) output for the \(\hbox {k}{\mathrm{th}}\) DMU, \(\hbox {r}=1,2,\ldots ,\hbox {s}\) and \(\hbox {k}=1,2,\ldots ,\hbox {n}\)

\(\hbox {v}_\mathrm{i} \)

Associated weight for the \(\hbox {i}{\mathrm{th}}\) input, \(\hbox {i}=1,2,\ldots ,\hbox {m}\)

\(\hbox {u}_\mathrm{r} \)

Associated weight for the \(\hbox {r}{\mathrm{th}}\) output, \(\hbox {r}=1,2,\ldots ,\hbox {s}\)

\(\hbox {h}_\mathrm{k} \)

Efficiency score (\(\hbox {h}_\mathrm{k} \le 1)\)

Specifically, \(\hbox {DMU}_\mathrm{k} \) consumes amount \(\hbox {X}_{\mathrm{ik}} \) of input i and produces amount \(\hbox {Y}_{\mathrm{rk}}\) of output r, that can be incorporated into an efficiency measure—the weighted sum of the outputs divided by the weighted sum of the inputs \(\hbox {h}_\mathrm{k} =\sum \hbox {u}_\mathrm{r} \hbox {Y}_{\mathrm{rk}} /\sum \hbox {v}_\mathrm{i} \hbox {X}_{\mathrm{ik}} \). This definition requires a set of factor weights \(\hbox {u}_\mathrm{r} \) and \(\hbox {v}_\mathrm{i} \) which are the decision variables. Each \(\hbox {DMU}_\mathrm{k} \) is assigned the highest possible efficiency score (\(\hbox {h}_\mathrm{k} \le 1)\) by choosing optimal weights for the outputs and inputs. DEA often generates several 100 % efficient frontiers among the DMU’s resulting in discrepancy to identify the top choice.

The data from Step 1 is taken as DMU’s with 3 inputs (material handling cost, rearrangement cost and flow distance) and 1 output (set equal to 1) for identifying efficient layouts using DEA.

Step 3: compute weights for quantitative and qualitative criteria

Qualitative and quantitative criteria may be complex and conflicting, hence weight importance is provided by experts using analytic hierarchy process (AHP) (Saaty 1980). AHP is a popular technique that has been employed to model subjective decision-making processes based on multiple criteria. However, the importance of each criterion is not necessarily equal. To resolve this problem, Saaty uses the eigenvector method to determine the relative importance (weights) among the various criteria based on the pairwise comparison matrix in AHP.

If \(A=\left[ {a_{ij} } \right] \) is a positive reciprocal matrix, then the geometric mean of each row \(r_i =\left( {\mathop \prod \nolimits _{j=1}^n a_{ij} } \right) ^{1/n}\). Saaty defined \(\lambda _{\mathrm{max}} \) as the largest eigenvalue of A, and the weights \(w_i \) as the components of the normalized eigenvector corresponding to \(\lambda _{\mathrm{max}} \), where \(w_i =r_i /\left( {r_1 +r_2 +\cdots +r_n } \right) \).

The decision maker has to redo the ratios when the comparison matrix fails to pass the consistency test, because the lack of consistency in decision-making can lead to inconsistent results. Hence, a consistency index to ensure that AHP’s pairwise comparison method is consistent needs to be calculated. The consistency index is given in Eq. (15):

$$\begin{aligned} CI=\frac{\lambda _{\mathrm{max}} -n}{n-1} \end{aligned}$$
(15)

where \(\lambda _{\mathrm{max}} \) denotes the maximal eigenvalue of the matrix R. When matrix R is consistent then \(\lambda _{\mathrm{max}} =n\) and \(CI = 0\). Consistency ratio \(({=}CI/RI(n))\) is the ratio of the consistency index to the corresponding random index. Following Saaty (1980), a consistency ratio (CR) of 0.1 or less is acceptable. Hence, weights for the 6 criteria, quantitative attributes (material handling cost, the rearrangement cost and flow distance) and qualitative attributes (accessibility, maintenance and waste management), were obtained using AHP.

Step 4: ranking of Layouts using MADM methods

MADM deals with the problem of choosing an option from the set of alternatives, which are characterized in terms of their attributes. Here, we provide a conceptual description of MADM techniques used in this paper.

  • TOPSIS—Euclidian and Manhattan

A multi criteria decision making (MCDM) problem can be expressed in a matrix format, in which columns indicate attributes rows list the competing alternatives. Alternatives are represented by \((\hbox {A}_1,\hbox {A}_2,\ldots \hbox {A}_{\mathrm{m}})\) and criteria by \((\hbox {C}_1,\hbox {C}_2,\ldots \hbox {C}_{\mathrm{n}})\). An element \(x_{ij}\) of the matrix indicted the performance rating of the \(\hbox {i}{\mathrm{th}}\) alternatives, \(A_i\), with respect to the \(\hbox {j}{\mathrm{th}}\) criteria, \(C_{j}\), as shown in Eq. (16):

$$\begin{aligned} {\begin{array}{ccccc} \hbox {D}&{}{{\begin{array}{ccccc} =&{}{{\begin{array}{ccccc} &{}{{\begin{array}{ccccc} {\hbox {C}_{1}}&{}\quad {{\begin{array}{ccccc} {\hbox {C}_{2}}&{}\quad {\hbox {C}_{3} }&{}\quad {{\begin{array}{ccccc} {\ldots }&{}\quad {\hbox {C}_\mathrm{n} } \\ \end{array} }} \\ \end{array} }} \\ \end{array} }} \\ {{\begin{array}{ccccc} {\hbox {A}_1 } \\ {{\begin{array}{ccc} {\hbox {A}_2 } \\ {\hbox {A}_3 } \\ {{\begin{array}{ccc} \vdots \\ {\hbox {A}_\mathrm{m} } \\ \end{array} }} \\ \end{array} }} \\ \end{array} }}&{}\quad {\left[ {{\begin{array}{ccccc} {\hbox {x}_{11} }&{}{{\begin{array}{ccccc} {\hbox {x}_{12} }&{}\quad {\hbox {x}_{13} }&{}\quad {{\begin{array}{ccccc} \ldots &{}\quad {\hbox {x}_{1\mathrm{n}} } \\ \end{array} }} \\ \end{array} }} \\ {{\begin{array}{ccc} {\hbox {x}_{21} } \\ {\hbox {x}_{31} } \\ {{\begin{array}{ccc} \vdots \\ {\hbox {x}_{\mathrm{m}1} } \\ \end{array} }} \\ \end{array} }}&{}{{\begin{array}{ccc} {{\begin{array}{ccc} {\hbox {x}_{22} } \\ {\hbox {x}_{32} } \\ {{\begin{array}{ccc} \vdots \\ {\hbox {x}_{\mathrm{m}2} } \\ \end{array} }} \\ \end{array} }}&{}{{\begin{array}{ccc} {\hbox {x}_{23} } \\ {\hbox {x}_{33} } \\ {{\begin{array}{ccc} \vdots \\ {\hbox {x}_{\mathrm{m}3} } \\ \end{array} }} \\ \end{array} }}&{}{{\begin{array}{ccc} {{\begin{array}{ccc} \ldots \\ \ldots \\ {{\begin{array}{ccc} \ddots \\ \ldots \\ \end{array} }} \\ \end{array} }}&{}{{\begin{array}{ccc} {\hbox {x}_{\mathrm{2n}} } \\ {\hbox {x}_{3\mathrm{n}}} \\ {{\begin{array}{ccc} \vdots \\ {\hbox {x}_{\mathrm{mn}} } \\ \end{array} }} \\ \end{array} }} \\ \end{array} }} \\ \end{array} }} \\ \end{array} }} \right] } \\ \end{array} }} \\ \end{array} }} \\ \end{array} } \end{aligned}$$
(16)

Hwang and Yoon (1981) developed TOPSIS based on the concept that the chosen alternative should have the shortest distance from the positive ideal solution and the longest distance from the negative ideal solution. The terms used are defined as follows:

Criteria: attributes \((\hbox {C}_\mathrm{j}, \hbox {j}=1,2, \ldots ,\hbox {n})\) should provide a means of evaluating the levels of an objective. For SDFLP attributes are MHC, rearrangement cost, flow distance, accessibility, maintenance and waste management.

Alternatives: these are synonymous with ‘options’ or ‘candidates’. Alternatives \((\hbox {A}_\mathrm{i},\,\hbox {i}=1,2 \ldots \hbox {m})\). Alternatives are the efficient layouts obtained from Step 2.

Criteria weights: weight values \((\hbox {w}_\mathrm{j})\) represent the relative importance of each attribute to the others. \(\hbox {W}=\left\{ {\left. {\hbox {w}_\mathrm{j} } \right| \hbox {j}=1,2,\ldots ,\hbox {n}} \right\} \). Attributes weights are obtained from Step 3.

Normalization: normalization seeks to obtain comparable scales, which allows attribute comparison. The vector normalization approach divides the rating of each attribute by its norm to calculate the normalized value of \(\hbox {x}_{\mathrm{ij}} \) as defined in Eq. (17):

$$\begin{aligned} \hbox {r}_{\mathrm{ij}} =\frac{\hbox {x}_{\mathrm{ij}} }{\sqrt{\mathop \sum \nolimits _{\mathrm{i}=1}^\mathrm{m} \hbox {x}_{\mathrm{ij}}^2 }}, \quad \hbox {i}=1,\ldots ,\hbox {m};\hbox {j}=1,\ldots ,\hbox {n}. \end{aligned}$$
(17)

Figure 8 provides the pseudo code of TOPSIS based on Euclidian and Manhattan distance for ranking the layouts.

Fig. 8
figure 8

Pseudo code of TOPSIS method for ranking layouts

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  • AHP

AHP is also applied to rank the layouts. For each of the criteria a pair wise comparison matrix of the efficient layouts is formulated and consistency index is computed. Given the information of the relative importance i.e. weights of each criteria (obtained in Step 3) and preferences, mathematical procedure is used to synthesize the information and provide priority ranking of all alternatives (layouts). The overall priority of each decision alternative is obtained by summing the product of the criteria priority i.e. weights times the priority of the decision alternative with respect to the criteria.

  • IRP

To overcome the limitations of intuitive process and rational choice process, interpretive ranking process (IRP) proposed by (Sushil 2009) is applied. This technique uses the strengths of both the processes of decision making and complementing the limitations of each one by the other. Steps of IRP methods are shown in Fig. 9.

Fig. 9
figure 9

Steps of IRP method for ranking layouts

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TOPSIS (Euclidian and Manhattan distance), AHP, and IRP are applied for ranking efficient layouts obtained by the DEA approach in Step 2 taking into account the quantitative and qualitative factors along with their weights.

Step 5: consensus ranking method

To obtain the ranking of multiple decision makers regarding the layouts aggregation techniques need to be used. There are several techniques such as Borda–Kendall, Integer linear model for rank aggregation, Beck and Lin (1983), Cool and Kress to yield a compromise or aggregate ranking. In this paper, we used 2 techniques—(1) Borda–Kendall (Cook and Seiford 1982; Cook and Kress 1985) and (2) Integer linear model for rank aggregation (Kaur et al. 2017).

  1. 1.

    Borda–Kendall (BAK) technique it is the most widely used to formulate and solve consensus ranking from various MADM algorithms. In this method, we calculate the positional mean value of the ranks for each project (layout) over all decision makers (MADM algorithms). The project with the lowest combined score is most preferred and the project with the highest combined score is least preferred.

  2. 2.

    Integer linear Programming (ILP) for rank aggregation let there be n number of efficient facility, which are ranked according to m different MADM techniques. An integer linear model for rank aggregation ranks different MADM techniques into consensus ranking is explained below: following are the notations used,

\(Y_{i}\) :

Final aggregated ranking of facility i

\(X_{ij}\) :

Rank of facility i using \(j\hbox {th}\) multi-attribute decision making (MADM) technique

n :

Number of facilities

m :

Number of MADM techniques

Objective function

$$\begin{aligned} \hbox {Min}\, Z=\mathop {\sum }\limits _{i=1}^n \mathop {\sum }\limits _{j=1}^m \left| {X_{ij} -Y_i } \right| \end{aligned}$$
(18)

Subject to

$$\begin{aligned}&1\le Y_{i} \le n\qquad \forall i\,\left( {1,2,\ldots ,n} \right) \end{aligned}$$
(19)
$$\begin{aligned}&Y_i \ne Y_k \quad \forall \,i,\quad k\,{ such}\,{ that}\qquad i\ne k \end{aligned}$$
(20)
$$\begin{aligned}&Y_{i}\, { is}\,{ integer}\qquad \forall i\,\left( {1,2,\ldots ,n} \right) \end{aligned}$$
(21)

The objective function of the model as shown in Eq. (18) minimizes the deviation of the final ranking from individual rankings from various MCDM techniques. Equation (19) restricts the ranking of n suppliers from 1 to n only. Equation (20) ensures that no two suppliers are given same rank; hence every supplier is given a different rank. Integer value of the rank is ensured by Eq. (21).

5 Numerical illustration

To validate the sustainable SDFLP formulation and its solution methodology, the SDFLP example considered has the product demand to be Gaussian distribution for facility (machine) size, \(\hbox {N}=12\), (U-shaped layout is shown in Fig. 10) and multiple time periods, \(\hbox {T}=5\). The data set has been taken from (Moslemipour and Lee 2011).

Fig. 10
figure 10

U-shaped facility layout for \(\hbox {N}=12\)

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The adjacency matrix, separation matrix and waste flow time matrix are empirically generated (refer “Appendix 1”). The efficient layout along with adjacency, separation and waste flow time matrix are used by the experts to compute the quantifiable values of accessibility, maintenance and waste management parameters, which form a pool of sustainable layouts. The flow chart shown in Fig. 11 presents the entire methodology to solve SSDFLP for the numerical illustration considered. Figure 11 also shows various tables i.e. from Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 generated while applying the proposed methodology to solve SSDFLP. Table 1 shows the pool of thirty layouts generated applying step 1.

Fig. 11
figure 11

Flow chart to solve SSDFLP of the numerical illustration

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Table 1 Pool of layouts
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DEA using CCR (Charnes et al. 1978) model is applied to 30 layouts (as independent DMU’s with 3 inputs (material handling cost, rearrangement cost and flow distance) and 1 output (set equal to 1) for identifying the efficient layouts, Table 2 extrapolates the efficiency scores of the layouts.

Table 2 Efficiency of layouts using DEA
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Weights (sum of weights equal to 1) for each attribute were computed using AHP (preferences of expert) as given in Tables 3 and 4. It can be seen that the experts have given importance to MHC (profit—economic pillar) then Maintenance (people—social pillar) and waste management (planet—environmental pillar). This shows that all 3Ps of sustainability are important for designing a sustainable SDFLP.

Finally, 9 efficient layouts were identified, which are considered as alternatives \((\hbox {A}_1,\hbox {A}_{2},\ldots , \hbox {A}_9)\) for ranking based on six attributes—namely, MHC, rearrangement cost, flow distance, accessibility, maintenance and waste management—using TOPSIS—Euclidian distance, TOPSIS—Manhattan Distance, AHP and IRP methods. Four different rankings of the 9 layouts are obtained which are summarized in Tables 5, 6, 7, 8 and 9. The rankings of the layout are based on the weights given to 6 criteria and the expert opinion, which changes as preferences or weights assigned to the criteria are varied. The rankings of layout are not unique therefore aggregate ranking methods need to be applied to find the optimum (and most suitable) layout. Borda–Kendall (BAK) method and ILP were applied to obtain the consensus ranking as shown in Table 10.

Table 3 Decision matrix for criteria using AHP
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Table 4 Weights of the six criteria obtained from AHP
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Table 5 Ranking using TOPSIS (Euclidian)
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Table 6 Ranking using TOPSIS (Manhattan)
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Table 7 Ranking using AHP
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Table 8 Dominating interaction matrix of IRP
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Table 9 Ranking using IRP
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Table 10 Consensus ranking
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Table 11 Ranked layouts using BAK
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Table 12 Ranked layouts using ILP
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Table 11 gives the ranking of layout based on BAK method and Table 12 gives the ranking of layout based on ILP. “Layout 29” (BAK) and “Layout 20” (ILP) gets an aggregate rank score “1”. The corresponding parameter values of both layouts are very close, thus, giving the best trade-off balancing all the three pillars of sustainable operations. Hence, the proposed methodology facilitates the decision maker in identifying an optimal SDFLP which satisfy the sustainability factors. Data for the numerical illustration is provided in “Appendix 1” (from Table 13 to Table 15). All the nine efficient facility layouts are shown in “Appendix 2” (from Tables 16, 17, 18, 19, 20, 21, 22, 23 and 24).

6 Discussion

Our interest in investigating the stochastic dynamic facility location problem was triggered by three gaps within facility layout design problem literature: firstly, the inherent uncertainties in demand and supply, which are widely noted in operations management literature (Balakrishnan and Cheng 2007, 2009; Dubey et al. 2015); secondly, the lack of studies that look into the FLP from a sustainability point of view, apart from exceptions (Yang et al. 2013; Sacaluga and Frojan 2014; Lieckens et al. 2015) and thirdly, the lack of studies in the stochastic FLP literature that use both quantitative and qualitative criteria apart from notable exceptions (Moslemipour and Lee 2011; Garcia-Hernandez 2013; García-Hernández 2015; Yang et al. 2013; Tayal and Singh 2014). We are in agreement with Yang et al. (2013) who suggest that simplifying practical FLP (and in our case, SSDFLP) in mathematical models or simulation models for objective optimization (Ertay et al. 2006; Yang and Hung 2007) needs to be complemented by qualitative criteria. Even though there are studies using qualitative criteria in conjunction with quantitative ones, they are not focusing on sustainability, rendering thereby our paper one of the first studies, if not the first, to look into the FLP problem from a sustainability perspective.

Therefore, our contribution lies in addressing these gaps; we propose and provide a sustainable stochastic dynamic facility layout problem (SDFLP) that uses both qualitative and quantitative factors under stochastic product demand flow over multi time period for the three pillars of sustainability (economic, social, and environmental), using the hierarchical framework of metaheuristic, MCDM techniques and consensus ranking method. Our methodology attempts to integrates metaheuristics (SA, CSA, hybrid Fa/CSA), DEA (to get efficient layouts), TOPSIS, AHP and IRP (for MCDM) and aggregate ranking (Borda–Kendall method and ILP) for six criteria i.e. MHC, flow distance, rearrangement cost, accessibility, maintenance and waste management.

7 Conclusion

The layout design problem is a strategic issue and has significant impact to the efficiency of a manufacturing system. The paper proposes a novel method to design and solve facility layout problem considering both qualitative and quantitative factors under stochastic product demand flow over multi time period is proposed, using hierarchical framework of-meta heuristic, MADM techniques and consensus ranking method. The proposed methodology for sustainable layout integrates meta-heuristics techniques viz. SA, CSA, Hybrid FA/CSA to generate layouts followed by applying DEA to identify an efficient layouts among the generated ones, and finally applying MADM approaches such as TOPSIS, IRP and AHP in association with aggregate ranking methods viz. Borda–Kendall and integer linear programming (ILP) considering six different criteria.

The effective systematic decision-making described in this paper help the facility designer to reduce the risk of choosing a poor layout design. Thus, the 3 pillars of sustainability were addressed for facility layout operations. The current research provides new insights for designing sustainable stochastic layouts. The proposed methodology is different from conventional methods where the environment and social outcomes are dealt as corrective action after designing the layout. Here, an inclusive approach is undertaken to design SSDFLP.