A hybrid optimization approach for unequal-sized dynamic facility layout problems under fuzzy random demands

Abstract

Dynamic facility layout problems involve devising the optimal layout for each different production period. This article studies unequal-area dynamic facility layout problems under fuzzy random environment to minimize the sum of the material handling costs and rearrangement costs. For a more general situation, a novel model of unequal-area dynamic facility layout problems is proposed on the basis of fuzzy random theory, in which uncertain demands are characterized by fuzzy random variables. Unequal-area dynamic facility layout problems are one of the non-deterministic polynomial-time hard problems. Therefore, a hybrid particle swarm optimization and simulated annealing algorithm is innovated to solve the proposed unequal-area dynamic facility layout problems model under fuzzy random environment, in which the shapes and areas of facilities are changed dynamically. Two facility-swapping methods and two local search methods help hybrid algorithm escape from local optima, allowing a more reliable solution. Moreover, a new shifting method is developed to prevent the spatial overlapping between adjacent facilities and save material handling costs. The performance of the hybrid algorithm is confirmed by some test problems available. Finally, the proposed method is extended to a facility layout planning of a new aircraft assembly shop floor. Computational results show that the efficiency and effectiveness of the proposed method, in sharp comparison with other approaches.

Introduction

Facility layout greatly influences the production system in terms of material handling costs, lead time, and work-in-process (WIP) inventory. ¹ It is a long-standing research topic involving the physical arrangements of a number of non-overlapping facilities including workstations, machines, and utilities. ² Material handling costs are one of the most crucial factors to measure the rationality of a layout since they account for 20%–50% of the overall operating costs and 15%–70% of the total manufacturing costs. ³ A rational facility can significantly improve the overall productivity and production efficiency and can reduce production cycles. ⁴ In turn, a poor facility layout may cause the accumulation of the WIP inventory, overloading of the material handling system, and longer queues. ⁵

Dynamic facility layout problems (DFLPs) are modeled by discretizing the time horizon into the multi-planning periods. The planning horizon can be defined in weeks, months, or years. ⁶ DFLPs, under uncertain environment, are caused by the elusive market fluctuation and changes in the product during each period. Exact approximation of certain product demand is difficult and, in some cases, impossible for such reasons as measurement errors and forecasting methods. ⁴ Essentially, demand requirements cannot be determined in advance due to the rapid changes in the market. Therefore, some researchers have considered facility layout problems (FLPs) as uncertain and imprecise nature of demand including fuzziness or randomness. Fuzzy logic has been used to cope with this imprecision within a certain range. Moreover, as to the fact that product demands fluctuate with process adjustment and quality tracing, probability theory has been introduced to deal with this kind of uncertainty. However, it is usually difficult to know precisely whether the demand requirements are fuzzy or random. This article applies fuzzy random variables (FRVs) to describe the two-fold uncertainty of demand for dealing with DFLPs. In addition, previous researches have focused on DFLPs with equal sizes and fixed orientations of facilities. The assumption that each facility has been assigned with the same size area and invariable shape is unreasonable in practical application and deduce the area utilization ratio. Therefore, this article focuses on unequal-area (UA) DFLPs under a fuzzy random environment where the unfixed shapes and free orientations of facilities are considered.

Due to the fact that FLPs are classified in non-deterministic polynomial-time (NP)-hard problems and combinatorial optimization problems, exact approaches (dynamic programming (DP), branch and bound method, cutting plane algorithm, etc.) cannot solve them because the computational time increases exponentially with the problem scale. Therefore, the heuristic and metaheuristic approaches are superior alternatives to exact approaches for solving FLPs, which satisfies certain constraints and provides sub-optimal solutions within a reasonable computational time. ⁷ It is vital to select a strong metaheuristic algorithm for finding a rational layout in an efficient manner. ⁸ Particle swarm optimization (PSO) has shown strong performance compared with other competing algorithms ⁹ owing to the simple implementation, few parameters setting, and effective global search. ¹⁰ However, PSO is prone to get stuck in local optimum when applied to the complex and combinatorial optimization problem. The exploration (global search across the entire search domain) and exploitation (local search) of PSO should be enhanced and balanced to obtain a better solution. In contrast to PSO, the simulated annealing (SA) algorithm is a stochastic searching method and has a mechanism to jump out of local minima. Therefore, a novel PSO and SA algorithm are proposed to solve UA DFLPs under fuzzy random demand, where the shapes and areas of facilities are unfixed. The proposed hybrid algorithm combines SA into the cycles of PSO that dynamically improve the global best-found solution in each PSO iteration. Moreover, two swapping methods, two local search methods, and a shifting method are used to improve the solution quality in each PSO iteration. The results show that the proposed hybrid algorithm can be effectively used for solving DFLPs under fuzzy random environment.

The general structure of this article is as follows: a literature review of UA DFLPs is introduced in section “Literature review.” Section “Problem formulation” presents dealing with FRVs and the objective function of UA DFLPs under total material handling costs minimization. In section “The proposed hybrid PSO and SA algorithm,” the proposed hybrid algorithm with PSO and SA is developed for solving UA DFLPs. In section “Performance evaluation,” the performance of the proposed algorithm is evaluated by some test problems. In section “Case study,” numerical experiments are implemented and computational results are discussed. The conclusions are demonstrated in section “Conclusion.”

Literature review

In previous studies, many researchers focused on the changes of material demands from period to period under deterministic environment.^11,12 With fuzzy production information and the uncertain results from the changes in product volume and varieties, material demands may change during each period. Therefore, some researchers paid attention to the non-deterministic nature of demands in DFLPs. The approaches for uncertain DFLPs can be categorized into two groups of probabilistic and fuzzy approaches. ⁶ For example, material demands between facilities are assumed as stochastic variables following the normal distribution, which could be predicted on the basis of historical information relating to the product.^6,13 Aiello et al. ¹⁴ assumed the uncertainty of material demands between facilities for each period as fuzzy numbers. Kaveh et al. ¹⁵ defined product demands (i.e. material demands) as fuzzy numbers and modeled in fuzzy programming. Decision-making process for the DFLPs often takes places in a hybrid uncertain environment, so it is difficult to know precisely whether the demand is fuzzy or random. Thus, FRVs are applied to describe hybrid uncertain demand. The concepts of FRVs were initially introduced to describe fuzzy random phenomenon simultaneously by Kwakernaak. ¹⁶ Afterwards, FRVs have expanded its application in many fields, such as resource allocation, ¹⁷ supply chain management, ¹⁸ inventory control, ¹⁹ and production planning. ²⁰ These studies have shown the effectiveness of FRVs in dealing with dual uncertain demand where fuzziness and randomness co-exist. Therefore, this article applies FRVs to deal with the two-fold uncertainty for material demands.

Various mathematical models, exact algorithms, and heuristic or metaheuristic methods have been developed in the past three decades. Rosenblatt ²¹ used DP and computerized relative allocation of facilities technique (CRAFT) to solve equal-sized DFLPs. Lacksonen and Emoryenscorejr ²² concluded that the cutting plane algorithm has better performance than CRAFT and proposed a DP algorithm for solving the DFLPs with the quadratic assignment problem (QAP) model. The exact approaches mentioned previously are effective approaches to obtain a satisfactory solution with small-sized DFLPs, but the subsequent intelligent algorithms are more suitable for solving large-sized DFLPs with high and low flow dominance, respectively. ²³ Baykasoğlu and Gindy ²⁴ used an SA algorithm to solve equal-sized DFLPs and proved that SA is more efficient than genetic algorithm (GA) in computational time. Şahin et al. ⁵ proposed an SA for the DFLPs with budget constraints. Rezazadeh et al. ²⁵ proposed discrete PSO for solving DFLPs with deterministic demand requirement. Moreover, Derakhshan Asl and Wong ²⁶ proposed a modified PSO algorithm with two local search methods to solve DFLPs, where facilities have fixed shapes and the demand requirement is considered as a certain value. Hosseini-Nasab and Emami ²⁷ proposed a hybrid PSO for equal-sized facility layout with complex discrete coding technique under a deterministic environment. A permutation-based GA algorithm was proposed to solve DFLPs with equal-sized facilities. ²⁸

The assumption that all facilities have fixed shaped or equal-sized during different periods is unreasonable in practical application. So far, there is no research focus on adopting PSO and SA in the field of UA DFLPs with uncertain demands, variation shapes, and free orientations. Therefore, this study proposes a novel model of UA DFLPs with fuzzy random demands and develops a hybrid PSO and SA algorithm to solve the proposed model within reasonable runtime, which makes planning and optimization of facility layout more comprehensive, realistic, and reasonable.

Problem formulation

A mixed integer nonlinear programming (MINLP) formulation is proposed for UA DFLPs under a fuzzy random environment where the shapes are unfixed throughout the time horizon, to create a continuous facility arrangement. That is, the objective function for UA DFLPs with fuzzy random demands is modeled to solve facilities arrangement in a continuous space. The model is formulated with the following assumptions: (1) there are multi-periods consisting of entire planning horizon, (2) the profile of department area is regarded as rectangle or square shapes ignoring its outline details, (3) the material demands between facilities are defined as FRVs, (4) each facility has different rearrangement cost, and (5) each facility has unfixed shapes and free orientations (i.e. the length and width of a facility can be exchanged).

Notations

The notations are given as follows:

Indexes

$j,k$  Index of facility

$t$   Index of period

Variables

$$r_{num}^t=\{\begin{array}{l}1\mathrm{if}\mathrm{rearrangment}\mathrm{of}\mathrm{facility}k\mathrm{happens}\mathrm{at}\mathrm{period}t\\ 0\mathrm{otherwise}\end{array}$$

$(x_{\mathrm{jt}},y_{\mathrm{jt}})$  Center coordinate of the facility $j$ in the period $t$

Parameters

$d_{\mathrm{jk}}^t$  Manhattan distance between facility $j$ and facility $k$

$d_l,d_u$  Material demands with maximum value and minimum value, respectively

$f_{{\tilde{\overline{D}}}_p^t\{\alpha ,\gamma \}}(x)$   Increasing functions

$g_{{\tilde{\overline{D}}}_p^t\{\alpha ,\gamma \}}(x)$  Decreasing functions

$x_j$  The centroid of the facility $j$ at the horizontal coordinate

$y_j$  The centroid of the facility $j$ at the vertical coordinate

${\tilde{D}}_{\mathrm{jk}}^t$  Demands between facility $j$ to facility $k$ in the period $t$

$\mathrm{EV}[{\tilde{\overline{D}}}_p^t\{\alpha ,\gamma \}]$  The expected value of material demands at the period $t$

${\tilde{F}}_{\mathrm{jk}}^t$  Material handling cost of moving facility $j$ to facility $k$ in period $t$ per unit distance

$L,W$  The length and the width of the shop floor, respectively

$\mathrm{MC}$  Total material handling costs

$N_{j,k}^t$  Number of the facility $j$ to the facility $k$ per transportation

$R_k^t$  Rearrangement cost of the facility $k$ at the period $t$

Dealing with FRVs

Considering the uncertainty nature of product demand, the material demand between facilities is defined as an FRV. The FRV ${\tilde{\overline{D}}}_{\mathrm{jk}}^t$ is introduced to describe the uncertain material demand in real production, which makes the entire problem become thornier. The expected method first transforms the FRVs into fuzzy variables and then into deterministic values. ¹⁷ The FRV ${\tilde{\overline{D}}}_{\mathrm{jk}}^t$ is converted into a trapezoidal fuzzy number ${\tilde{\overline{W}}}_{\mathrm{jk}}^t$ , where ${\tilde{\overline{D}}}_{\mathrm{jk}}^t=(d_l,\tau (d),d_u)$ , and $\tau (d)$ is estimated to follow a normal distribution $V({\mu }_0,{\delta }_0^2)$ . After that, the FRV is transformed into a $(\alpha ,\gamma )$ -level trapezoidal fuzzy variable ${\tilde{\overline{W}}}_{\mathrm{jk}}^t=(d_l,\underset{¯}{d},\overline{d},d_u)$ , where $\alpha$ is a given probability level of the fuzzy variable and $\gamma$ is a given probability level of the random variable. ²⁹ Based on the expected value operator method proposed by Heilpern, ³⁰ the fuzzy variable is transformed into a deterministic value $\mathrm{EV}[{\tilde{\overline{D}}}_{\mathrm{jk}}^t\{\alpha ,\gamma \}]$ as follows

$$\mathrm{EV}\left[{\tilde{\overline{D}}}_{\mathrm{jk}}^t\{\alpha ,\gamma \}\right]=\frac{1}{2}[(\underset{¯}{d}-\underset{d_l}{\overset{\underset{¯}{d}}{\int }}{f}_{{\tilde{\overline{D}}}_{\mathrm{jk}}^t\{\alpha ,\gamma \}}(x)\mathrm{dx})+(\overline{d}+\underset{\overline{d}}{\overset{d_u}{\int }}{g}_{{\tilde{\overline{D}}}_{\mathrm{jk}}^t\{\alpha ,\gamma \}}(x)\mathrm{dx})]$$

(1)

Objective function for total material handling costs minimization

Material handling cost is an important measure for the rationality of layout. In this model, two decisive parts are taken into account, including the material handling costs and the rearrangement costs, both of which are crucial when planning the DFLPs. The novel objective function of UA DFLPs model under fuzzy random environment for the total material handling costs minimization is described as follows

$$min\mathrm{MC}=\sum _{t=1}^T\sum _{j=1}^q\sum _{k=j+1}^q{\tilde{F}}_{\mathrm{jk}}^t{d}_{\mathrm{jk}}^t+\sum _{t=2}^T\sum _{j=1}^q{R}_k^t{r}_{\mathrm{num}}^t$$

(2)

Based on the transforming process and the objective function formula, the objective in equation (2) is transformed into the following expected value model

$$\begin{array}{c}\multicolumn{1}{c}{min\mathrm{MC}=\sum _{t=1}^T\sum _{k=1}^q\sum _{j=k+1}^q\frac{\mathrm{EV}\left[{\tilde{\overline{D}}}_{\mathrm{jk}}^t\{\alpha ,\gamma \}\right]}{N_{\mathrm{jk}}^t}{d}_{\mathrm{jk}}^t}\\ \multicolumn{1}{c}{+\sum _{t=2}^T\sum _{k=1}^q{R}_k^t{r}_{\mathrm{num}}^t}\end{array}$$

(3)

Subject to

$$F_{\mathrm{jk}}^t=\frac{D_{\mathrm{jk}}^t}{N_{\mathrm{jk}}^t}$$

(4)

$$d_{\mathrm{jk}}^t=|x_{\mathrm{jt}}-x_{\mathrm{kt}}|+|y_{\mathrm{jt}}-y_{\mathrm{kt}}|$$

(5)

It is notable that several constraints have to be met simultaneously to achieve satisfying solutions for DFLPs objective function, including spatial non-overlapping constraints, area constraints, logical constraints, and non-negative constraints. These constraints are explained as follows

$$x_{\mathrm{jt}}-l_{\mathrm{jt}}/2\ge 0$$

(6)

$$x_{\mathrm{jt}}+l_{\mathrm{jt}}/2\le L$$

(7)

$$y_{\mathrm{jt}}-w_{\mathrm{jt}}/2\ge 0$$

(8)

$$y_{\mathrm{jt}}+w_{\mathrm{jt}}/2\le W$$

(9)

$$(l_{\mathrm{jt}}+l_{\mathrm{kt}})/2\le |x_{\mathrm{jt}}-x_{\mathrm{kt}}|$$

(10)

$$(w_{\mathrm{jt}}+w_{\mathrm{kt}})/2\le |y_{\mathrm{jt}}-y_{\mathrm{kt}}|$$

(11)

$$|x_{\mathrm{jt}}-x_{\mathrm{jt}-1}|\ge K{r}_{\mathrm{num}}$$

(12)

$$|y_{\mathrm{jt}}-y_{\mathrm{jt}-1}|\ge K{r}_{\mathrm{num}}$$

(13)

The first set of area constraints (equations (6) and (9)) guarantees that each facility is assigned within the boundary shop floor, respectively. The second set of spatial non-overlapping constraints (equations (10) and (11)) ensures that there is no spatial overlapping between adjacent facilities. The third set of logic constraints (equations (12) and (13)) determines whether a facility needs to be re-arranged in the following period. It is worth noting that all parameters or variables in the objective function cannot take any negative values.

The proposed hybrid PSO and SA algorithm

The hybrid PSO and SA algorithm can combine each other’s merits for solving UA DFLPs. PSO is easy to trap in local optimum, but SA is known as a very fast algorithm among other local search-based metaheuristics. Moreover, SA has other advantages such as avoiding local minimum, accepting worse solution as a new solution with a certain probability, and less computational costs. Therefore, PSO is hybridized with a simple and fast SA to obtain a better solution. To further improve the quality of the solution, two swapping methods, two local search methods, and a shifting method are applied in this work. The detailed introduction is provided in following sub-sections.

Standard PSO

PSO was initially proposed by Kennedy and Eberhart. ³¹ Each member in a flock of birds determines its velocity and position on the basis of their personal experience as well as social experience obtained through interaction with other members of the flock. ²⁷ Supposing a swarm of $M$ particles in $d$ -dimensional continuous space is searching for the global best-found solution. The $X_i^t=(x_{i1}^t,x_{i2}^t,\dots ,x_{\mathrm{id}}^t)$ and $V_i^t=(v_{i1}^t,v_{i2}^t,\dots ,v_{\mathrm{id}}^t)$ denote the position and velocity of the particle $i$ at iteration $t$ , respectively. PSO defines $\mathrm{pbest}$ and $\mathrm{gbest}$ fitness value to update the current particles’ velocity and position in the next generation. The detailed update procedures are described as follows

$$v_i^t=\omega v_i^{t-1}+c_1{r}_1(p_{\mathrm{best}}-x_i^{t-1})+c_2{r}_2(g_{\mathrm{best}}-x_i^{t-1})$$

(14)

$$x_i^t=x_i^{t-1}+v_i^t$$

(15)

where $\omega$ denotes inertia weight, $r_1$ and $r_2$ are random numbers between 0 and 1, $c_1$ and $c_2$ denote the personal and global best-position acceleration constant, respectively.

Standard SA

SA was initially proposed by Kirkpatrick et al. ³² SA starts with an initial solution and then subsequently proceeds to neighboring solutions by a random move until the stopping criterion is achieved. ¹² SA algorithm begins with an initial temperature denoted by $T_0$ and slowly cools to final temperature denoted by $T_e$ . In addition, initial temperature is determined on the basis of the range of objective function value as follows

$$T_0=\frac{-\mathrm{\Delta }{f}_0}{{\ln }^{{\lambda }_0}}$$

(16)

where $\mathrm{\Delta }{f}_0$ denotes the deviation between objective function values of solution $s$ at iteration $t$ and improved solution $s\prime$ at iteration $t+1$ . In this study, the initial temperature is $\mathrm{gbest}$ . ${\lambda }_0$ denotes a constant which can be set by the user. The algorithm accepts a non-improving solution as a new solution with a probability obtained from Boltzmann distribution as follows

$$p(s,s\prime ,T_t)=\exp (-\frac{f(s\prime )-f(s)}{K\times T_t})$$

(17)

where $T_t$ is the temperature at iteration $t$ , and $K$ is the Boltzmann constant. In this article, $K=1$ . A linear cooling schedule is used, that is, updating the temperature at each iteration $t$ as $T_t=T_{t-1}\times \alpha$ , where $\alpha$ is the temperature ratio and belongs to (0,1). When reaching the maximum number of iteration, the SA algorithm terminates.

The proposed hybrid algorithm

Initial solutions

The proposed hybrid PSO algorithm generates $M$ random particles. Each particle is encoded as a matrix with multiple dimensions of the total number of facilities at different production periods. That is, if total facilities are indexed from 1 to $N$ , multi-periods are indexed from 1 to $p$ . The particle can be described as

$$\begin{array}{c}\multicolumn{1}{c}{x_i^t=}\\ \multicolumn{1}{c}{[x_{11}^t,x_{12}^t,\dots ,x_{1p}^t;x_{21}^t,x_{22}^t,\dots ,x_{2p}^t;\dots ;x_{N1}^t,x_{N2}^t,\dots ,x_{\mathrm{Np}}^t]}\\ \multicolumn{1}{c}{(i=1,2,\dots ,M)}\end{array}$$

(18)

where each element in the particle matrix represents the two-dimensional (2D) position of each facility at iteration $t$ . That is, each element represents the center coordinates of facilities along $x$ and $y$ axes including a sub-array of 2D positions. Therefore, the matrix consists of $N\times 2p$ elements. The row of matrix denotes that the same facility position vary with different production periods, while the column of matrix denotes different facility at the same production period.

The modified particle-updating process

Since the proposed algorithm employs $M$ particles to search feasible space, $M$ particles also need to update their positions and velocities along with algorithm iteration. In order to effectively balance global search and local search, the proposed PSO introduces linear descend inertial weight and constriction coefficient for updating each velocity of each particle $i$ at period $p$ and iteration $t$ , as follows

$$v_{\mathrm{id}}^t=\chi (\omega v_{\mathrm{id}}^{t-1}+c_1{r}_1(p_{\mathrm{best}}-x_i^{t-1})+c_2{r}_2(g_{\mathrm{best}}-x_i^{t-1}))$$

(19)

$$\omega ={\omega }_{max}-\frac{{\omega }_{max}-{\omega }_{min}}{T}\times t$$

(20)

$$\begin{array}{c}\multicolumn{1}{c}{\chi =\frac{2}{(c_1+c_2)-2+\sqrt{{(c_1+c_2)}^2-4(c_1+c_2)}};}\\ \multicolumn{1}{c}{c_1+c_2>4}\end{array}$$

(21)

where ${\omega }_{max}$ and ${\omega }_{min}$ are maximum and minimum inertia weight, respectively, which can be set by users, and $T$ is the total number of iterations. After updating the particles’ velocity and position, the algorithm continues to evaluate new position by fitness value. If necessary, $\mathrm{pbest}$ and $\mathrm{gbest}$ will also be updated.

Swapping method

In order to maintain the diversity of the population and avoid prematurity convergence, the swapping method is also applied for each particle in each iteration of the hybrid algorithm with a dynamic probability. The swapping method includes facility-swapping method and period-swapping method. The detailed introduction is provided as follows:

Dynamic swapping selection scheme

It is obvious that the probability for the swapping method is relatively low at early iteration. As iteration proceeds, the algorithm is easier to trap into local optimum. Therefore, the probability of swapping needs to gradient increase. Moreover, fixed iteration or probability of swapping scheme for the whole swarm may increase the disturbance and lead to algorithm divergence. This article proposes to use gradient sigmoid function to calculate the probability as follows

$$g=\frac{1}{1+e^{-{\alpha }_1(t-T)}}$$

(22)

where ${\alpha }_1$ denotes to shape factor with a value of 0.1. The shape factor value is set by 20 times running the proposed method.

Facility-swapping method

As mentioned before, the facility-swapping method is applied for each particle with a probability $g$ . The facility-swapping method includes the position and orientation swapping between each pair of facilities. That is, two different facilities in each particle exchange their positions, and each facility changes its length and width. ²⁶ For a period $t$ , facility $i$ and facility $j$ exchange their positions. Moreover, facility $i$ and facility $j$ exchange their length and width. Note that swapping method is applied at the same production period. The steps of facility-swapping method are explained in Table 1.

Table 1.

Steps of facility-swapping method.

Step number	Description
1	The position and orientation of all facilities are saved in the current particle, respectively.
2	Randomly select two different facilities (i.e. facility $i$ and facility $j$ ) in the current particle.
3	The orientation of facility $i$ exchange at a certain probability and the result is saved.
4	The orientation of facility $j$ exchange at a certain probability and the result is saved.
5	The orientation of facility $i$ and facility $j$ exchange at a certain probability and the result is saved.
6	The positions of facility $i$ and facility $j$ exchange at a certain probability and the result is saved.
7	The positions of facility $i$ and facility $j$ exchange at a certain probability. The orientation of facility $i$ changes simultaneously, and the result is saved.
8	The positions of facility $i$ and facility $j$ exchange at a certain probability, the orientation of facility $j$ changes simultaneously, and the result is saved.
9	The positions of facility $i$ and facility $j$ exchange at a certain probability, the orientations of facility $i$ and facility $j$ change at a probability simultaneously, and the result is saved.
10	The optimal result is selected and saved.

Period-swapping method

The period-swapping method is usually applied to the best global solution in DFLPs. When the number of iterations is no less than the subtraction of maximum iteration number to 50, this method is used for each pair of dynamic periods with a probability $g$ . To be specific, the center coordinates of facilities along $x$ and $y$ axes, and the orientation of facilities are exchanged for each pair of dynamic periods. Namely, all facilities change their positions from period $t$ to period $t+1$ and change their length and width from period $t$ to period $t+1$ , respectively. The steps of the period-swapping method are shown in Table 2.

Table 2.

Steps of period-swapping method.

Step number	Description
1	The position and orientation of all facilities are saved in the current particle, respectively.
2	Randomly select a facility (i.e. facility $i$ ) in the current particle.
3	Change the length and width of facility $i$ from period $t$ to period $t+1$ at a certain probability, and the result is saved.
4	Facility position $x_i^t$ at period $t$ and facility position $x_i^{t+1}$ at period $t+1$ exchange at a certain probability and the result is saved.
5	Change the length and width of facility $i$ from period $t$ to period $t+1$ at a certain probability. Simultaneously, facility position $x_i^t$ at period and $t$ facility position $x_i^{t+1}$ at period $t+1$ exchange at a certain probability and the result is saved.
6	The optimal result is selected and saved.

Local search method

Local search method 1

The local search method 1 is applied in each iteration for the best global solution of the proposed hybrid PSO and SA algorithm when the global solution of the next generation is worse than that of the previous generation. For each facility, the length and width are exchanged and movement along $x$ axis and $y$ axis are tested. Eight actions are used to shift facilities as shown in Table 3.

Table 3.

Eight actions of local search method 1.

Action number	Description
1	Generate a random number from 0 to 1 and shift a facility to the right along $x$ axis.
2	Generate a random number from 0 to 1 and shift a facility to the left along $x$ axis.
3	Generate a random number from 0 to 1 and shift a facility upward along $y$ axis.
4	Generate a random number from 0 to 1 and shift a facility downward along $y$ axis.
5	Generate two random numbers from 0 to 1, respectively, and shift a facility downward along $y$ axis and to the left along $x$ axis simultaneously.
6	Generate two random numbers from 0 to 1, respectively, and shift a facility downward along $y$ axis and to the right along $x$ axis simultaneously.
7	Generate two random numbers from 0 to 1, respectively, and shift a facility upward along $y$ axis and to the right along $x$ axis simultaneously.
8	Generate two random numbers from 0 to 1, respectively, and shift a facility upward along $y$ axis and to the left along $x$ axis simultaneously.

Local search method 2

The local search method 2 is utilized in each iteration for the best global solution of the proposed hybrid algorithm when their number of iterations is no less than the subtraction of maximum iteration number to 50. Before implementing, a permutation vector $\{1,2,\dots ,n\}$ of facility number is randomly generated. For every two sequential facilities, the length and width of a facility are exchanged and 20 actions of facilities along $x$ and $y$ axes are tested. Twenty actions are used to shift facilities as shown in Table 4.

Table 4.

Twenty actions of local search method 2.

Action number	Description
1	Generate a random number from 0 to 1 and shift facility $i$ and facility $i+1$ to the right along $x$ axis.
2	Generate a random number from 0 to 1 and shift facility $i$ and facility $i+1$ to the left along $x$ axis.
3	Generate a random number from 0 to 1 and shift facility $i$ and facility $i+1$ upward along $y$ axis.
4	Generate a random number from 0 to 1 and shift facility $i$ and facility $i+1$ downward along $y$ axis.
5	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ and facility $i+1$ downward along $y$ axis and to the left along $x$ axis simultaneously.
6	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ and facility $i+1$ downward along $y$ axis and to the right along $x$ axis simultaneously.
7	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ and facility $i+1$ upward along $y$ axis and to the right along $x$ axis simultaneously.
8	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ and facility $i+1$ upward along $y$ axis and to the left along $x$ axis simultaneously.
9	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ downward along $y$ axis and facility $i+1$ to the left along $x$ axis.
10	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ downward along $y$ axis and facility $i+1$ to the right along $x$ axis.
11	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ downward along $y$ axis and facility $i+1$ upward along $x$ axis.
12	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ upward along $y$ axis and facility $i+1$ to the left along $x$ axis.
13	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ upward along $y$ axis and facility $i+1$ to the right along $x$ axis.
14	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ upward and facility $i+1$ downward along $y$ axis, respectively.
15	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ to the right and facility $i+1$ to the left along $x$ axis, respectively.
16	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ to the right and facility $i+1$ to upward along $x$ axis, respectively.
17	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ to the right and facility $i+1$ to downward along $x$ axis, respectively.
18	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ to the left and facility $i+1$ to the right along $x$ axis, respectively.
19	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ to the left and facility $i+1$ downward along $x$ axis, respectively.
20	Generate two random numbers from 0 to 1, respectively, and shift facility $i$ to the left and facility $i+1$ upward along $x$ axis, respectively.

Shifting method for the overlapped facilities

The spatial overlapping between adjacent facilities is infeasible in facility layout. The previous study proposed a penalty for violent solution.^26,33 Jolai et al. ¹³ indicated that penalty cannot completely prevent spatial overlapping between adjacent facilities and proposed a shifting method with a single direction as well as repeated step length with multiple attempts. However, it may result in increased time consumption and efficiency decrease. This article develops a multi-direction shifting method with two combination directions. The two combination directions are classified as moving to the right side or left side along $x$ axis, and moving to upward or downward along $y$ axis. The shifting direction and distance depend on its relative position and orientation to a fixed facility. This method avoids unnecessary repeated shifting steps and significantly improves shifting efficiency. Moreover, it is worthy to note that the gap between adjacent facilities is reduced by the proposed shifting method, that is, the material handling costs are reduced because of the shorter delivery distance. Figure 1 shows a case of shifting method for overlapped facilities. For instance, suppose facility 2 is a fixed facility and $l_{\mathrm{AB}},l_{\mathrm{BC}},l_{\mathrm{DE}},l_{\mathrm{DF}}$ mean overlapping edge along the $x$ axis and $y$ axis, respectively. The detailed steps are demonstrated as follows:

• Step 1: Facility 1 is overlapped with facility 2. If $l_{\mathrm{AB}}>l_{\mathrm{BC}}$ , facility 1 moves upward along the y axis because of the lower shifting cost.

• Step 2: After Step 1, facility 1 may be overlapped with facility 4. Similarly, facility 4 moves to left along the x axis.

• Step 3: Facility 3 is overlapped with facility 2. If $l_{\mathrm{DF}}>l_{\mathrm{DE}}$ , facility 3 moves to right along the y axis for saving shifting cost.

• Step 4: Facility 5 is overlapped with facility 2. Similarly, facility 5 moves downward along the y axis.

• Step 5: Facility 6 is overlapped with facility 7 and facility 2. Facility 6 moves to left along the x axis. Facility 6 may also be overlapped with facility 7, and facility 7 moves downward.

Figure 1.

The illustrative example for shifting method.

The overall procedure for hybrid algorithm

The flowchart of the hybrid algorithm is shown in Figure 2.

The steps of the proposed algorithm are demonstrated as follows:

• Step 1: According to the mathematics method introduced in section “Dealing with FRVs,” convert the fuzzy random values into deterministic values before algorithm running.

• Step 2: Set PSO and SA relative control parameters as described in section “The proposed hybrid PSO and SA algorithm” (such as particles’ number $M$ , maximum iterations $T$ , maximum and minimum inertia weight ${\omega }_{max}$ and ${\omega }_{min}$ , acceleration constant $c_1,c_2$ , initial temperature $T_0$ , temperature decrease ratio $\alpha$ , final temperature $T_f$ , $f\%$ of the whole particles is replaced by $\mathrm{gbest}$ , et al.

• Step 3: Generate the particles’ random positions within the boundary of the shop floor. According to section “Standard PSO,” each particle is encoded in a multiple dimensions matrix with the total number of facilities at different production periods.

• Step 4: Evaluate each solution based on the calculation of fitness value and update the value of local best fitness value $\mathrm{pbest}$ and the global best fitness value $\mathrm{gbest}$ .

• Step 5: Run the SA for the $\mathrm{gbest}$ , and adopt roulette wheel selection to calculate the probability of accepts non-improving solution as new solutions.

• Step 6: According to equations (17) and (19), update each particles’ velocity and position.

• Step 7: Evaluate each particles fitness value and update the value of $\mathrm{pbest}$ and $\mathrm{gbest}$ .

• Step 8: Replace $f\%$ of worst particles with the global best particle.

• Step 9: If $\mathrm{gbest}(t)<\mathrm{gbest}(t-1)$ , turn to Step 10. Otherwise, turn to Step 15.

• Step 10: Apply the facility-swapping method at a certain probability $g$ , when the number of iteration satisfies dynamic swapping selection scheme.

• Step 11: Apply local search method 1 to enhance the global search region for $\mathrm{gbest}$ .

• Step 12: If the number of iterations is no less than the subtraction of maximum iteration number to 50, apply period-swapping method at a probability $g$ for $\mathrm{gbest}$ , otherwise, go to Step 14.

• Step 13: Use local search method 2 to enhance the global search region $\mathrm{gbest}$ .

• Step 14: Apply shifting method for the overlapped facilities.

• Step 15: Update particle, $\mathrm{pbest}$ and $\mathrm{gbest}$ .

• Step 16: If the iteration reaches the maximum, then stop the program. Return the values of $\mathrm{gbest}$ and its corresponding particle’s position and orientation. Otherwise, turn to Step 6.

Figure 2.

Flowchart of the proposed hybrid algorithm.

It is important to notice that applying the improved hybrid algorithm for the optimization objective, the pseudo code is generated. At each iteration of the hybrid algorithm, the current solution and the $\mathrm{pbest}$ of each particle can be obtained. Then the SA algorithm is used to search improved global solution. If the objective value is improved, the change will be accepted. Furthermore, the two swapping methods, two local search methods, and a shifting method are used to improve the solution quality. The complete pseudo code of the procedure is shown in Figure 3.

Figure 3.

The complete pseudo code of the procedure.

Performance evaluation

Based on our knowledge, the proposed model has not been studied so far, and no other solutions have been found to compare it with that of the proposed algorithm. To evaluate the performance of the proposed hybrid algorithm, some computational experiments have been discussed in the following sub-section.

Test problems

There are three metaheuristic methods (Dunker et al., ³⁴ Jolai et al., ¹³ Derakhshan Asl and Wong ²⁶ ) for DFLPs. To compare the attained results of the proposed algorithm with those of hybrid genetic algorithm (HGA), ³⁴ multiple objective particle swarm optimization (MOPSO), ¹³ and the modified PSO ²⁶ for two available test problems, the minimization material handling costs and rearrangement costs are taken as an optimization objective, and their related constraints are considered in our experiments.

Two test problem instances were taken from Dunker et al. ³⁴ Two test problems were described as follows: (1) DFLPs with six facilities and six dynamic periods (P6T6) and (2) DFLPs with 12 facilities and 4 dynamic periods (P12T4). According to basic information and data from Derakhshan Asl and Wong, ²⁶ all facilities have fixed areas and shapes, and plant floor is restricted to $30\times 30$ and $50\times 50$ in P6T6 and P12T4 problems, respectively. The aforementioned literature all employs the same problem instances.

Parameters control is significantly important to the proposed algorithm. To determine the values of parameters, extensive experiments are implemented with a different set of parameters (i.e. trial and error method). Parameter settings for the proposed method are presented in Table 5. The proposed algorithm is coded in MATLAB and executed on a Windows 7 operating system with 2.2 GHz CPU and 8GB RAM.

Table 5.

Parameters setting.

Parameter	Problems
	P6T6	P12T4	P10T5	P20T5
The population size of particles	50	100	100	150
Maximum iteration	500	600	–	–
Maximum inertia weight	0.75	0.9	0.9	0.9
Minimum inertia weight	0.4	0.4	0.4	0.4
Acceleration constant for $\mathrm{pbest}$	2.5	2.8	2.8	2.8
Acceleration constant for $\mathrm{gbest}$	2.5	2.8	2.8	2.8
Initial temperature	$\frac{-\mathrm{gbest}}{\log (0.95)}$	$\frac{-\mathrm{gbest}}{\log (0.95)}$	$\frac{-\mathrm{gbest}}{\log (0.95)}$	$\frac{-\mathrm{gbest}}{\log (0.95)}$
Temperature decrease ratio	0.9	0.97	0.97	0.97
Worst percent of the whole particles is replaced by $\mathrm{gbest}$	20	30	30	30

Results and discussion

To ensure a reasonable comparison, all solutions are obtained with the same number of iterations and population sizes. Each test problem was conducted 10 times for the proposed algorithm. The comparative results and runtime of three algorithms are given in Table 6.

Table 6.

The comparison of three algorithms for test problems P6T6 and P12T4.

	HGA 34		Modified PSO 26		MOPSO 13		Proposed hybrid PSO and SA
	MC	Runtime (s)	MC	Runtime (s)	MC	Runtime (s)	MC	Runtime (s)
P6T6	6906.5	1764	6763.8	2354.6	6659.4	504	6532.5	438.2
P12T4	29,098.5	9600	27,626.7	11,105.6	28,115.5	3030	27,197.3	2628.9

HGA: hybrid genetic algorithm; PSO: particle swarm optimization; MOPSO: multiple objective particle swarm optimization; SA: simulated annealing; MC: material handling cost.

Equation (23) is applied to compare the result of the proposed hybrid algorithm with three metaheuristic methods

$$\mathrm{dev}=\frac{(M{C}_{\mathrm{com}}-M{C}_{\mathrm{pro}})}{M{C}_{\mathrm{pro}}}\times 100$$

(23)

where $\mathrm{dev}$ is the relative deviation of $\mathrm{MC}$ , and $M{C}_{\mathrm{com}}$ and $M{C}_{\mathrm{pro}}$ denote the total of material handling costs of the compared method and proposed hybrid method, respectively.

As shown in Table 6, the best-found solution of the proposed hybrid algorithm is 6532.5. Note that the best-found solution is compared on condition that the number of iterations is equal. According to Table 2 and equation (23), the best-found solution of the proposed algorithm has improved P6T6 problems by 5.73%, 3.54%, and 1.94% in comparison with HGA, modified PSO, and MOPSO, respectively. Moreover, it can be seen that the best-found solution for the P6T6 problem using the proposed algorithm with minimum runtime is 438.2 s. Therefore, the proposed hybrid algorithm has improved solution quality and computation time for P6T6 problem. The center coordinates of the best-found layouts for P6T6 are shown in Table 7. Figure 4 presents the best-found layout of P6T6 for Period 6.

Table 7.

Center coordinates of facilities for the best-found layouts of P6T6.

Period	Facilities index	F1	F2	F3	F4	F5	F6
1	Center coordinate of facilities along x axis	18.5	21.4	24.0	23.5	18.5	24.1
	Center coordinate of facilities along y axis	11.0	21.0	7.5	15.0	15.0	11.5
	Length (m)	5.0	9.0	6.0	6.0	4.0	5.0
	Width (m)	4.0	8.0	5.0	4.0	4.0	3.0
2	Center coordinate of facilities along x axis	10.5	12.5	11.0	16.0	16.0	16.5
	Center coordinate of facilities along y axis	10.0	21.0	14.5	8.0	15.0	11.5
	Length (m)	5.0	9.0	6.0	6.0	4.0	5.0
	Width (m)	4.0	8.0	5.0	4.0	4.0	3.0
3	Center coordinate of facilities along x axis	10.5	12.5	16.0	15.0	10.0	10.5
	Center coordinate of facilities along y axis	11.0	21.0	10.5	15.0	15.0	7.5
	Length (m)	5.0	9.0	6.0	6.0	4.0	5.0
	Width (m)	4.0	8.0	5.0	4.0	4.0	3.0
4	Center coordinate of facilities along x axis	10.5	12.5	16.0	16.0	10.0	15.5
	Center coordinate of facilities along y axis	11.0	21.0	14.5	7.0	15.0	10.5
	Length (m)	5.0	9.0	6.0	6.0	4.0	5.0
	Width (m)	4.0	8.0	5.0	4.0	4.0	3.0
5	Center coordinate of facilities along x axis	10.5	18.5	11.0	11.0	15.0	16.5
	Center coordinate of facilities along y axis	14.0	21.0	18.5	23.0	12.0	15.5
	Length (m)	5.0	9.0	6.0	6.0	4.0	5.0
	Width (m)	4.0	8.0	5.0	4.0	4.0	3.0
6	Center coordinate of facilities along x axis	10.5	12.5	16.0	15.0	10.0	10.5
	Center coordinate of facilities along y axis	11.0	21.0	10.5	15.0	15.0	7.5
	Length (m)	5.0	9.0	6.0	6.0	4.0	5.0
	Width (m)	4.0	8.0	5.0	4.0	4.0	3.0

Figure 4.

The best-found layout of P6T6 for Period 6.

As shown in Table 6, the best solution of the proposed hybrid algorithm is 27,197.3 after 600 iterations. According to Table 6 and equation (23), the proposed algorithm has attained the improvement in P12T4 problems by 6.99%, 1.58%, and 3.38% in comparison with HGA, modified PSO, and MOPSO, respectively. Moreover, it can be seen that the best-found solution for P12T4 problem using the proposed algorithm with minimum runtime is 2628.9 s. Therefore, the proposed hybrid algorithm has improved solution quality and computation times for P12T4 problem. The center coordinates of the best-found layouts for P12T4 are shown in Tables 8 and 9. Figure 5 shows the best-found layout of P12T4 for Period 4.

Table 8.

Center coordinates of facilities F1-F6 for the best-found layouts of P12T4.

Period	Facilities	F1	F2	F3	F4	F5	F6
1	Center coordinate of facilities along x axis	21.1	16.7	16.2	22.2	27.2	26.7
	Center coordinate of facilities along y axis	22.0	17.5	27.5	31.0	31.0	26.0
	Length (m)	5.0	7.0	6.0	4.0	6.0	5.0
	Width (m)	4.0	5.0	5.0	4.0	6.0	4.0
2	Center coordinate of facilities along x axis	17.7	14.7	27.2	13.2	21.2	22.7
	Center coordinate of facilities along y axis	16.0	10.5	10.5	15.0	11.0	16.0
	Length (m)	5.0	7.0	6.0	4.0	6.0	5.0
	Width (m)	4.0	5.0	5.0	4.0	6.0	4.0
3	Center coordinate of facilities along x axis	23.8	14.7	21.2	19.3	27.2	14.8
	Center coordinate of facilities along y axis	15.0	19.5	19.5	15.0	20.0	15.0
	Length (m)	5.0	7.0	6.0	4.0	6.0	5.0
	Width (m)	4.0	5.0	5.0	4.0	6.0	4.0
4	Center coordinate of facilities along x axis	19.5	17.0	20.5	15.0	26.5	18.0
	Center coordinate of facilities along y axis	19.0	14.5	28.5	19.0	27.0	10.0
	Length (m)	5.0	7.0	6.0	4.0	6.0	5.0
	Width (m)	4.0	5.0	5.0	4.0	6.0	4.0

Table 9.

Center coordinates of facilities F7-F12 for the best-found layouts of P12T4.

Period	Facilities	F7	F8	F9	F10	F11	F12
1	Center coordinate of facilities along x axis	21.7	27.4	15.6	21.7	17.7	23.3
	Center coordinate of facilities along y axis	11.5	21.5	22.5	26.5	32.5	17.0
	Length (m)	10.0	7.0	6.0	5.0	5.0	6.0
	Width (m)	7.0	5.0	5.0	5.0	5.0	4.0
2	Center coordinate of facilities along x axis	19.0	28.8	27.4	15.9	26.5	21.4
	Center coordinate of facilities along y axis	26.5	15.5	20.5	20.5	25.5	20.0
	Length (m)	10.0	7.0	6.0	5.0	5.0	6.0
	Width (m)	7.0	5.0	5.0	5.0	5.0	4.0
3	Center coordinate of facilities along x axis	27.1	27.0	20.5	15.0	19.6	29.3
	Center coordinate of facilities along y axis	26.5	10.5	10.5	10.5	24.5	15.0
	Length (m)	10.0	7.0	6.0	5.0	5.0	6.0
	Width (m)	7.0	5.0	5.0	5.0	5.0	4.0
4	Center coordinate of facilities along x axis	27.0	15.0	23.5	15.0	21.0	26.5
	Center coordinate of facilities along y axis	16.5	23.5	10.5	28.5	23.5	22.0
	Length (m)	10.0	7.0	6.0	5.0	5.0	6.0
	Width (m)	10.0	7.0	6.0	5.0	5.0	6.0

Figure 5.

The best-found layout of P12T4 for Period 4.

It is clear that the proposed hybrid algorithm has better performance in sharp comparison with HGA, modified PSO, and MOPSO in the literature. That is because PSO is a classical swarm intelligent algorithm. The information among particles can be exchanged and shared while GA cannot, which is helpful to obtain a better solution. However, it is well known that PSO has weak local search ability, the introduction of SA can compensate for this defect because it has a stochastic mechanism for jumping out of local optimum. Moreover, to further improve the quality of the solution, two swapping methods, two local search methods, and a shifting method are applied. The facility-swapping method is utilized to avoid local optimum, and the period-swapping method is used only for DFLPs to improve the quality of layouts. Two local methods are applied to improve local search ability and quality of the solution. In addition, a shifting method can prevent spatial overlapping between adjacent facilities and save the total material handling costs. In short, the application of our hybrid algorithm to DFLPs is attractive in terms of both solution quality and computation time.

Case study

This section presents a practical application of the DFLPs in a new aircraft assembly shop floor to demonstrate the maneuverability and efficiency of the proposed method.

Case representation

The rationality of the facility layout for a new aircraft assembly shop floor planning is critically important for minimizing cost and raising production. It involves physical arrangements of many facilities to suitable sites, such as large assembly fixture storage area, material handling equipment, and machines. Due to the complex aeronautic assembly process and low production efficiency, the layout is implemented in phases. This article focuses on two typical stages including large parts docking and parts assembly. The partial shop floor schematic plan of a new aeronautic assembly shop floor is shown in Figure 6. It covers an area of $70\mathrm{m}\times 70\mathrm{m}$ and involves assigning 10 kinds of facilities and 20 kinds of facilities to suitable sites in two different stages, respectively. The area of each site had to be unequal, Its required length and width may vary with the implemented assembly process. Moreover, because of the long production schedule and multi-variety production, the typical two stages of production are divided into five periods, respectively. Each period requires various kinds of facilities and different product demands.

Figure 6.

The partial shop floor schematic plan of a new aeronautic assembly shop floor.

As mentioned before, the facility layout planning is usually drawn up before implementation. The demands between facilities can be approximately predicted by market statics at the planning stage, but they would fluctuate at real production within a certain range. The layout planner can only provide value ranges roughly on the basis of experience. Therefore, the uncertainty caused by market volatility can be described in fuzzy logic. In the production stage, since the enterprise information system has been widely applied, the material flow can be monitored. ³⁵ Therefore, the product demands between facilities can be predicted by history data such as enterprise resource planning (ERP) data, bill of materials (BOM), and quality data, which assume following a normal distribution. This article applies FRVs to describe the two-fold uncertainty of demand for dealing with DFLPs. Moreover, to further demonstrate the advantages of the proposed hybrid method under a fuzzy random demand, more computational results are shown using the hybrid PSO and SA algorithm to solve DFLPs under different environments and problem instances, respectively. Considering economic and production efficiency, the decision maker’s objectives are to save total material handling costs and prevent the overlap of facilities to avoid unnecessary damage or loss. Therefore, the proposed hybrid method is devoted to minimizing the total material handling costs. Parameter settings for the proposed method are presented in Table 5.

Based on our knowledge, the proposed method has not been considered on solving DFLPs under fuzzy random environment so far. To further illustrate the application of the proposed hybrid method under a fuzzy random demand, two problem instances are conducted as follows: (1) DFLPs under the fuzzy random environment with 10 facilities and 5 periods (P10T5) and (2) DFLPs under the fuzzy random environment with 20 facilities and 5 periods (P20T5). The previous two problem instances are derived from facility layout planning of a new aeronautic assembly shop floor.

Computational results

For DFLPs, if a facility needs rearrangement in the next period, the costs cannot be ignored. The rearrangement costs vary with different sites and periods. The rearrangement costs for each facility at each period are shown in Tables 10 and 11, respectively. According to the production schedule, the product demands between facilities for P10T5 problems at the first period are partly listed in Table 12 due to the short text. Using the method introduced in section “Dealing with FRVs,” the fuzzy random demand is transformed into a deterministic value.

Table 10.

Rearrangement costs for each facility at each period for P10T5 problem.

Facilities index	Period 1	Period 2	Period 3	Period 4	Period 5
F1	197	106	98	145	156
F2	78	183	153	204	93
F3	261	97	125	110	40
F4	128	293	276	100	78
F5	89	196	65	140	132
F6	203	85	164	37	126
F7	176	235	251	186	201
F8	152	74	122	137	92
F9	102	151	88	87	119
F10	95	126	46	132	76

Table 11.

Rearrangement costs for each facility at each period for P20T5 problem here.

Facilities index	Period 1	Period 2	Period 3	Period 4	Period 5
F1	197	106	98	145	156
F2	78	183	153	204	93
F3	261	97	125	110	40
F4	128	293	276	100	78
F5	89	196	65	140	132
F6	203	85	164	37	126
F7	176	235	251	186	201
F8	152	74	122	137	92
F9	102	151	88	87	119
F10	95	126	46	132	76
F11	150	130	67	82	90
F12	65	110	56	98	35
F13	105	84	46	39	87
F14	121	165	72	124	48
F15	35	183	92	185	64
F16	141	46	165	169	104
F17	79	106	94	124	90
F18	68	76	80	54	137
F19	54	62	39	82	71
F20	145	143	106	57	59

Table 12.

The fuzzy random material demands between any two facilities F1-F5 for Period 1.

Facility index	F1	F2	F3	F4	F5
F1	–	(50, ${\tau }_{12}$ ,70)	(42, ${\tau }_{13}$ ,65)	(7, ${\tau }_{14}$ ,10)	(35, ${\tau }_{15}$ ,46)
		${\tau }_{12}$ ∼N(62,10)	${\tau }_{13}$ ∼N(58,8)	${\tau }_{14}$ ∼N(8,3)	${\tau }_{15}$ ∼N(41,6)
F2	(50, ${\tau }_{21}$ ,70)	–	(10, ${\tau }_{23}$ ,25)	(20, ${\tau }_{24}$ ,32)	(5, ${\tau }_{25}$ ,10)
	${\tau }_{21}$ ∼N(62,10)		${\tau }_{23}$ ∼N(19,4)	${\tau }_{24}$ ∼N(28,5)	${\tau }_{25}$ ∼N(8,5)
F3	(42, ${\tau }_{31}$ ,65)	(10, ${\tau }_{32}$ ,25)	–	(20, ${\tau }_{34}$ ,32)	0
	${\tau }_{31}$ ∼N(58,8)	${\tau }_{32}$ ∼N(19,4)		${\tau }_{34}$ ∼N(28,5)
F4	(7, ${\tau }_{41}$ ,10)	(20, ${\tau }_{42}$ ,32)	(20, ${\tau }_{43}$ ,32)	–	0
	${\tau }_{41}$ ∼N(8,3)	${\tau }_{42}$ ∼N(28,5)	${\tau }_{43}$ ∼N(28,5)
F5	(35, ${\tau }_{51}$ ,46)	(5, ${\tau }_{52}$ ,10)	0	0	–
	${\tau }_{51}$ ∼N(41,6)	${\tau }_{52}$ ∼N(8,5)
F6	(5, ${\tau }_{61}$ ,8)	(4, ${\tau }_{62}$ ,7)	(15, ${\tau }_{63}$ ,25)	(6, ${\tau }_{64}$ ,9)	(2, ${\tau }_{65}$ ,7)
	${\tau }_{61}$ ∼N(6,2)	${\tau }_{62}$ ∼N(6,2)	${\tau }_{63}$ ∼N(19,4)	${\tau }_{64}$ ∼N(7,2)	${\tau }_{65}$ ∼N(4,2)
F7	(4, ${\tau }_{71}$ ,7)	(5, ${\tau }_{72}$ ,8)	(15, ${\tau }_{73}$ ,25)	(20, ${\tau }_{74}$ ,28)	(6, ${\tau }_{75}$ ,9)
	${\tau }_{71}$ ∼N(6,2)	${\tau }_{72}$ ∼N(6,2)	${\tau }_{73}$ ∼N(19,2)	${\tau }_{74}$ ∼N(23,5)	${\tau }_{75}$ ∼N(7,2)
F8	(90, ${\tau }_{81}$ ,120)	(45, ${\tau }_{82}$ ,70)	(55, ${\tau }_{83}$ ,75)	(25, ${\tau }_{84}$ ,42)	(5, ${\tau }_{85}$ ,10)
	${\tau }_{81}$ ∼N(110,20)	${\tau }_{82}$ ∼N(60,6)	${\tau }_{83}$ ∼N(65,7)	${\tau }_{84}$ ∼N(35,4)	${\tau }_{85}$ ∼N(35,4)
F9	(120, ${\tau }_{91}$ ,150)	(200, ${\tau }_{92}$ ,220)	(90, ${\tau }_{93}$ ,120)	(60, ${\tau }_{94}$ ,80)	(20, ${\tau }_{95}$ ,30)
	${\tau }_{91}$ ∼N(142,20)	${\tau }_{92}$ ∼N(217,10)	${\tau }_{93}$ ∼N(117,8)	${\tau }_{94}$ ∼N(75,6)	${\tau }_{95}$ ∼N(25,6)
F10	(7, ${\tau }_{101}$ ,10)	(5, ${\tau }_{102}$ ,8)	0	(5, ${\tau }_{104}$ ,8)	(4, ${\tau }_{105}$ ,7)
	${\tau }_{101}$ ∼N(8,3)	${\tau }_{102}$ ∼N(6,3)		${\tau }_{104}$ ∼N(6,3)	${\tau }_{105}$ ∼N(6,2)

To further assess the quality of the proposed hybrid algorithm, the number of iterations and computing time is increased. After running the program for over 10 times, the hybrid algorithm reaches a good solution after 800 iterations on P10T5 problems. Table 13 shows the convergence of the hybrid algorithm on P10T5 problem under fuzzy random environment. It can be found that the best found result for P10T5 problems is 86,217.9 with the runtime of 6231.5 s. The center coordinates of the best found layout for P10T5 at Period 5 are shown in Table 14. Figure 7 shows the best-found layout of P10T5 for Period 5.

Table 13.

The hybrid algorithm convergence for P10T5 and P20T5 problem.

	Iteration 500		Iteration 600		Iteration 800		Iteration 1000
	MC	Runtime (s)	MC	Runtime (s)	MC	Runtime (s)	MC	Runtime (s)
P10T5	104,844.2	1839.2	96,995.3	2458.4	86,218.7	3743.8	86,217.9	6231.5
P20T5	494,345.6	10,456.6	479,925.3	13,417.5	461,773.2	19,795.1	452,584.4	25,651.9

MC: material handling cost.

Table 14.

Center coordinates of facilities for the best-found layout of P10T5 at Period 5.

Facility index	Center coordinate of facilities along $x$ axis	Center coordinate of facilities along $y$ axis	Length (m)	Width (m)
F1	16.3	9.3	7.5	8.5
F2	17.2	19.4	6.4	3.9
F3	4.3	9.0	7.5	8.0
F4	9.5	17.3	9.0	7.5
F5	10.3	9.3	4.5	8.5
F6	26.3	18	8.5	10.0
F7	9.	25.6	8.0	9.2
F8	16.3	15.5	4.5	3.9
F9	17.5	23.6	9.0	4.5
F10	23.2	9.0	6.4	8.0

Figure 7.

The best-found layout of P10T5 for Period 5.

After running the program for over 10 times, the hybrid algorithm reaches a good solution after 1000 iterations on P20T5 problem. Table 13 shows the convergence of the hybrid algorithm on P20T5 problem under fuzzy random environment. It can be seen that the best-found result for P20T5 problems is 452,584.4 with the runtime of 25,651.9 s. The center coordinates of the best-found layout of P20T5 for Period 5 are shown in Table 15. Figure 8 shows the best-found layout of P20T5 for Period 5. By applying the proposed hybrid algorithm, the aforementioned results can provide decision makers with scientific and theoretical references.

Table 15.

Center coordinates of facilities for the best-found layouts of P20T5 at Period 5.

Facility index	Center coordinate of facilities along $x$ axis	Center coordinate of facilities along $y$ axis	Length (m)	Width (m)
F1	38.8	27.9	9.2	8
F2	29.5	32.4	9.0	3.9
F3	39.3	53.7	10.2	8.5
F4	19.8	27.2	10.5	6.4
F5	28.3	38.1	6.5	7.6
F6	35.3	38.1	7.5	7.5
F7	49.4	44.3	8.0	9.2
F8	28.3	18.3	6.5	7.6
F9	30.5	45.7	11.0	7.5
F10	58.7	40.7	10.5	11
F11	40.7	45.6	9.4	7.6
F12	29.6	26.1	9.2	8.0
F13	29.6	52.6	9.2	6.4
F14	38.2	21.6	8.0	4.5
F15	17.8	35.4	10.5	10.0
F16	47.2	35.6	7.6	8.3
F17	55.3	29.7	8.5	11.0
F18	7.9	29.2	9.4	8.3
F19	41.0	35.6	3.9	7.5
F20	47.2	28.2	7.6	6.4

Figure 8.

The best-found layout of P20T5 for Period 5.

Comparison with other models

Since the fuzzy random model in this article is formulated with some assumption, there are some possible modeling mistakes. However, the results can be used to help decision makers draw up a theoretical optimal plan. Further research should be done on the basis of the proposed model. To demonstrate the advantages of the proposed model under fuzzy random demand, more computational results are shown by using the same hybrid PSO and SA algorithm to solve DFLPs under deterministic and fuzzy modeling environment, respectively. The first is a DFLPs model with deterministic demand and other is a DFLPs model with fuzzy demand. Deterministic demand means the demand between facilities is a deterministic value. Fuzzy demand means the demand between facilities is a fuzzy value. To ensure its feasibility and reliability, each program runs over 10 times after 1000 iterations under the same MATLAB programming environment. In the deterministic environment, the mean value of each FRV in this case study has been calculated. For instance, for demand $d_{51}$ (i.e. $d_{51}=(35,{\tau }_{51},46)$ with ${\tau }_{51}~N(39,8)$ between facilities ${\mathrm{F}}_5$ and ${\mathrm{F}}_1$ , the deterministic demand value took the mean value of 35 and 46, equal to 40.5. Tables 16 and 17 show the computational results for P10T5 problem under deterministic environment. It is clear that the model including fuzziness and randomness factors performs better than the model without uncertainty. For P10T5 problem, the best solution, the worst solution, and the average result for the model without uncertainty are 30.2%, 26.3%, and 28.2% higher than the proposed solutions, respectively. For P20T5 problem, the best solution, the worst solution, and the average result for the model without uncertainty are 48.1%, 35.9%, and 25.8% higher than the proposed model respectively.

Table 16.

Model comparison in three different environments for P10T5.

Environments	Total material handling cost
	Best result	Worst result	Average result
Determined	112,234.4	116,164.3	114,199.4
Fuzzy	102,821.3	108,887.2	105,854.3
Fuzzy random	86,217.9	91,990.6	89,104.3

Table 17.

Model comparison in three different environments for P20T5.

Environments	Total material handling cost
	Best result	Worst result	Average result
Determined	670,235.6	709,102.7	689,669.2
Fuzzy	607,643.4	658,745.3	633,194.4
Fuzzy random	452,584.4	521,966.2	487,275.3

In the fuzzy environment, the fuzziness in material demand has been considered but the stochastic nature is neglected. The expectation method is used to convert the fuzzy demand into equivalent crisp demand. For instance, for demand $d_{51}$ (i.e. $d_{51}=(35,{\tau }_{51},46)$ ) with ${\tau }_{51}~N(39,8)$ between facilities ${\mathrm{F}}_5$ and ${\mathrm{F}}_1$ , the stochastic nature $d_{51}=(35,{\tau }_{51},46)$ is neglected, and the expectation value is used to replace the ${\tau }_{51}$ . Therefore, according to the expectation method, the material demand in the fuzzy environment was denoted as $(35,36,46)$ . Tables 16 and 17 show the computational results for P20T5 problem under fuzzy environment. It can be seen that the model that considers fuzziness and randomness simultaneously has more advantages than the model considering fuzziness. For P10T5 problem, the best solution, the worst solution, and the average result for the model in a fuzzy environment are 19.2%, 18.4%, and 18.8% higher than the proposed solution respectively. For P20T5 problem, the best solution, the worst solution, and the average result for the model in a fuzzy environment are 34.2%, 26.2%, and 29.9% higher than the proposed solution, respectively. This suggests that the model under the fuzzy random environment obtains a better solution than the model under fuzzy environment.

As the scale of facility increase, rough estimating the range of material demands on the basis of personal experience and taking an average under deterministic environment may produce accumulated errors in the long term. Moreover, according to the comparative results, ignoring the stochastic nature of material demand is infeasible, which expands a gap between the fuzzy environment, and the fuzzy random environment with the scale of the facility increased. Therefore, it can be concluded that the model under a fuzzy random environment is not only closer to reality but also has a better solution than the other two environments, which can minimize total material handling costs and avoid the overlap between facilities.

Conclusion

A new hybrid algorithm was proposed for UA DFLPs under fuzzy random environment. First, the main factors affecting the uncertainty demands are analyzed and the uncertain demands are presented based on fuzzy random theory. Second, a novel formulation of UA DFLPs mathematical model was developed under fuzzy random environment where the unfixed shapes and free orientations of facilities are considered throughout the time horizon. Subsequently, a hybrid PSO and SA algorithm was developed to solve the proposed model, which devotes to save the total material handling costs and prevents the spatial overlapping between adjacent facilities. Moreover, to improve the quality of solution and find a better global solution, this article applied five methods including two swapping methods (facility-swapping method and period-swapping method), two local search methods (local search method 1 and local search method 2) and a shifting method. The four different modified algorithms are compared in two test problems to highlight the performance of the proposed hybrid method. The application of our hybrid algorithm enabled better layouts in stark contrast to other algorithms. Finally, the proposed model and hybrid algorithm are applied to a new aircraft assembly shop floor. To verify the effectiveness of the proposed model, this article compares it with other models including the deterministic model and fuzzy model. The results show that the proposed hybrid algorithm is attractive in terms of both solution quality and computation time under fuzzy random environment.

For future work, this study is to be expanded in many directions. First, more objective is to be considered in DFLPs models such as operational cost, investment cost, human resource, inventory, and safety. Second, it is suggested that the multi-objective dynamic algorithm is to be used for solving DFLPs such as multi-objective imperialist competitive algorithm (MOICA), non-dominated sorting genetic algorithm (NSGAII), MOPSO, and so on. Third, it will further increase the size of facilities to study the accuracy of DFLPs model under fuzzy random environment in the long term.