# An Improved Genetic Algorithm Based Robust Approach for Stochastic Dynamic Facility Layout Problem

Authors: Yunfang Peng; Tian Zeng; Lingzhi Fan; Yajuan Han; Beixin Xia
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

This paper deals with stochastic dynamic facility layout problem under demand uncertainty in terms of material flow between facilities. A robust approach suggests a robust layout in each period as the most frequent one falling within a prespecified percentage of the optimal solution for multiple scenarios. Mont Carlo simulation method is used to randomly generate different scenarios. A mathematical model is established to describe the dynamic facility layout problem with the consideration of transport device assignment. As a solution procedure for the proposed model, an improved adaptive genetic algorithm with population initialization strategy is developed to reduce the search space and improve the solving efficiency. Different sized instances are compared with Particle Swarm Optimization (PSO) algorithm to verify the effectiveness of the proposed genetic algorithm. The experiments calculating the cost deviation ratio under different fluctuation level show the good performance of the robust layout compared to the expected layout.


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## Body

## 1. Introduction

Nowadays, the increasingly fierce market competition and the growing variable demands of customers have gradually made the production mode shift from high volume and repetitiveness to high mix and low volume. How to meet the variant production requirements is thought to be an important objective for new intelligent manufacturing system, which is an advanced manufacturing mode of next generation. Designing agile facilities makes much sense to satisfy these requirements [1]. Dynamic changes, such as fluctuations in product quantity, varieties in product mix, introduction of new products, and discontinuation of existing products, are frequently taking place in high-mix and low-volume production environment. As a result of these changes, the previous layout becomes less efficient, which makes material handling cost increased. The uncertainty of demand has brought great challenges to design a suitable facility layout. Therefore, it is of great theoretical and practical significance to study the facility layout problem under dynamic environment [2].The research on facility layout problem (FLP) started in the 1950s. Koopmans and Beckmann [3] initially defined the facility layout problem as the assignment of facilities to discrete locations with the objective of minimizing the material handling cost. Static facility layout is obtained according to the deterministic material flow when the demand is constant. But when the demand changes frequently with time, static layout becomes no more suitable for various periods in the planning horizon. Dynamic facility layout problem (DFLP) divides the planning horizon into several discrete time periods, with different product demand. Designing flexible layout and designing robust layout are two approaches to deal with DFLP [4]. Flexible layout optimization is to design an optimal layout for each period in the multiperiod planning horizon so that the total material handling and rearrangement cost is minimized. The facility layout is changed according to the production demand in different periods. On the other hand, the robust approach is to design a fixed robust layout to minimize the total material handling cost over the entire time planning horizon [5]. Although the robust layout is not an optimal layout for a particular time period, its performance is good in each period. In a traditional DFLP, the demand in each period is determined by demand forecasting. But, in reality, the product demand in one period is difficult to forecast accurately. Therefore, it would be more valuable to study DFLP, assuming the product demand is stochastic in each period. Considering uncertainty of the product demand in each period leads to stochastic dynamic facility layout problem (SDFLP) [6].


## 2. Literature Review

The approaches for the DFLP can be classified into four categories: exact methods, heuristics, metaheuristics, and hybrid approaches [1, 4]. Rosenblatt [7] who is the pioneer in DFLP presented a dynamic programming (DP) formulation. Based on this formulation, both optimal and heuristic procedures are developed. Lacksonen and Enscore [8] extended five algorithms to solve the DFLP, which is modeled as a modified Quadratic Assignment Problem (QAP) formulation. And the cutting plane algorithm was illustrated to be the best for all the test problems. To avoid the computational complexity of the DP and QAP formulations, Urban [9] proposed a new heuristic algorithm based on the steepest-decent pairwise-interchange procedure, which performs well in most of situations. Recently, more specifically metaheuristic and hybrid approaches, such as genetic algorithm (GA), tabu search (TS), and simulated annealing (SA), have been widely applied for DFLP. Kaku and Mazzola [10] defined a TS heuristic utilizing a dynamic tabu list for dynamic plant layout problem. Madhusudanan et al. [5] applied a robust layout to minimize the total material handling cost over all periods. They proposed a mathematical model for robust approach and designed a SA algorithm to solve the model. Tayal and Singh [11] presented mathematical formulation for multiobjective stochastic dynamic facility layout problem and solved it by SA and chaotic simulated annealing (CSA) metaheuristics. The experiment results observed that CSA performs better than SA. GA has been proven to be effective to generate suboptimal solutions for large-scale dynamic facility layout problems. Fazlelahi et al. [12] devised a customized permutation-based robust genetic algorithm in dynamic manufacturing environments, which is expected to be generating a unique robust layout for all the manufacturing periods.Kulturel-Konak surveyed recent developments in designing robust and flexible facilities layout under uncertainty [1]. Webster and Tyberghein [13] measured the flexibility of a layout as the ability to react to disturbances caused by future change. They analyzed the annual material handling costs to measure the flexibility. Gupta [14] solved the FLP by Monte Carlo simulation to randomly generate the flow between all pairs of departments. Chan and Malmborg [15] also used Monte Carlo simulation to empirically search for robust solutions for dynamic line layout problem. Rosenblatt and Lee [16] solved the single period plant layout problem under stochastic demand by a robustness approach. They defined the robustness of a layout as the frequency it falls within a prespecified percentage of the optimal solution for various sets of scenarios. The robustness approach searches for a reliable layout for all scenarios but not the optimal layout for any given scenario. Besides the scenario-based robust optimization to deal with uncertainty, some other methods such as stochastic programming or fuzzy programming are widely used. Stochastic programming employs probabilistic models and describes the uncertainty by probability distributions. Moslemipour and Lee [6] considered the randomly changing product demands as independent normally distributed random variables with known probability density function. SA metaheuristic algorithm was utilized to solve the mathematical model. Fuzzy programming models uncertain parameters with fuzzy numbers and establishes constraints using fuzzy sets and membership functions. Considering the uncertainty of material flows, Cheng et al. [17] introduced fuzzy numbers to represent the material flows between department pairs. Then, GA was applied to solve this hard fuzzy combinatorial problem. Kaveh et al. [18] modeled the DFLP as fuzzy programming and solved the models by a hybrid intelligent algorithm including GA, simulated annealing, and fuzzy simulation.Although previous studies have significantly improved FLP with uncertainty, most of articles assumed that the demand is in exact probability distribution or is defined by fuzzy numbers. In fact, the information about the uncertainty is sometimes lacking and its behavior is difficult to predict. Therefore, a scenario-based method is applied in this paper to describe the demand uncertainty. Designing flexible layout and designing robust layout are two approaches to cope with dynamic layout problem. In this article, these two approaches are combined. In each period, a robust layout inspired from Rosenblatt and Lee [16] for DFLP considering the assignment of transport devices under uncertain demands is present. The robust layout is the most frequent layout falling within a prespecified percentage of the optimal solution for different sets of scenarios generated by Mont Carlo simulation. To improve the search speed of finding the robust layout, an improved adaptive genetic algorithm with population initialization strategy is proposed to reduce the search space and improve the efficiency of solving the model.The rest of this paper is organized as follows. In the next section, the dynamic facility layout problem that considers the assignment of transport devices is modelled. Then, a robust approach based on Monte Carlo simulation is proposed to deal with material flow uncertainty in Section3. After that, an improved genetic algorithm is developed to solve the mathematical model. Some numerical results are compared and the advantages of the robust layout are illustrated in Section 5. Finally, the conclusions are given.


## 3. Dynamic Facility Layout Problem

DFLP considers material flow over multiple time periods. The material flow between facilities changes over time. But traditional dynamic layout optimization is studied under the condition that the demand in each period is constant; it cannot effectively solve the problem with demand fluctuation. Therefore, on the basis of proposing an improved adaptive genetic algorithm, Monte Carlo simulation method is used to describe the effect of demand fluctuation on the material flow. In our research, the assignment of transport devices (such as conveyor, AGV, and tow train) is an important decision because of different unit material handling cost for each transport device. Although we consider the fluctuation of material flow in each period, the problem becomes a determined dynamic facility problem when one scenario is generated.The mathematical model for DFLP is discussed as follows and Table1 gives the notations used.Table 1
Notations.
Indices: i,j Indices for facilities wherei, j=1,...,I, I= number of facilities in the layout t Index for periods wheret=1,...,T, T=number of periods in the planning horizon k,h Index for locations of facility wherek, h=1,...,L, L=number of locations in the layout z Indices for transport device types wherez=1...,G, G=number of transport device types Parameters: D kh The distance between locationk and location h A kh The cost of rearrangement from locationk to location h P khz The cost of transporting unit material from positionk to position h using device z Q tij The random variable of the total material flow from facilityi to facility jin periodt Variables: x tik Binary variable:1, if facilityi is placed at location k in period t; 0 otherwise y tzij Binary variable:1, if using devicezto transport material between facility i and facility j during period t; 0 otherwiseThe objective (3) is to minimize the total material handling cost (see (1)) and the rearrangement cost (see (2)). Constraints (4) ensure that each facility should be placed in exactly one location in each period. Constraints (5) indicate that a location can place at most one facility in each period. Constraints (6) ensure that only one transport device can be used between two facilities in each period.(1)MHC=∑t=1T∑z=1Z∑i=1I∑j=1I∑k=1L∑h=1Lxtik∙xtjh∙ytzij∙Qtij∙Pkhz∙Dkh(2)AC=∑t=1T-1∑i=1I∑k=1L∑h=1Lxtik∙xt+1ih∙Akh(3)min⁡TC=MHC+AC(4)∑k=1Lxtik=1,∀t,i(5)∑i=1Ixtik≤1,∀t,k(6)∑z=1Gytzij=1,∀t,i,j


## 4. Robust Approach Based on Mont Carlo Simulation

Considering demand uncertainty in terms ofQtij, Monte Carlo simulation method is used to generateN different scenarios. In each period, we find a robust layout, which is the most frequent layout falling within a prespecified percentage of the optimal solution for different sets of production scenarios. Robust layout constraints (7) and (8) are defined to ensure that the gap of cost between robust layout and optimal layout in each scenario falls within a prespecified percentage.(7)RCCn=TCn-TCnoptTCnopt(8)RCCn≤MIn these constraints,n is the index for scenarios wheren=1,..,N.RCCn is the robust control coefficient of thenth scenario. TCnopt is optimal cost of thenth scenario, whileTCn denotes the total cost of the robust layout of thenth scenario.M is the prespecified percentage that is suggested to be less than 15% [19]. The detailed process to find the robust layout is as follows (Figure 1).Figure 1
The robust layout generation process based on MCS.Step 1.
Initially setn=1 and candidate solution setSn=Φ,.Step 2.
Generate the expected scenario with the material flowQtij setting to the expected value.Step 3.
Obtain the optimal layoutLayout(n) and minimum cost TCnopt by genetic algorithm.Step 4.
Generate the initial population and improve the solution through evolution process until the solution satisfies the robust layout constraints. Add current layout into the candidate solution setSn.Step 5.
If the number of elements inSn is less thanNmax, go back to the fourth step; otherwise, output setSn and go to next step.Step 6.
If the number of simulations is less thanN,n=n+1. Generate a new scenario corresponding to the random numbers Qtij in a certain range and go back to Step 2. Else, go to the next step.Step 7.
Output the most frequent layout (robust layout) contained inSn wheren=1..N. If more than two layouts are satisfied, search for the robust layoutLayout' with minimal cost deviation from the optimal cost.


## 5. An Improved Adaptive Genetic Algorithm Design

Genetic algorithm (GA) is an excellent heuristic algorithm, which is a random optimization algorithm for the simulation of biological evolution in nature. Because of its strong practicability and robustness, it is widely used in the field of facility layout optimization, scheduling, and transportation. But genetic algorithm has the shortcomings of easy falling into local optimal and premature convergence. Therefore, in view of the above shortcomings, combined with the characteristics of the facility layout problems, an improved genetic algorithm is developed.


### 5.1. Population Initialization

Because the decision variables of the model are all 0-1 variables, binary coding is adopted. Each cell containsT chromosomes. The first I×L position in a chromosome defines the layout of the facility, and the latter G×I×I position denotes the assignment of transport devices. In order to improve the efficiency of the algorithm and combine the constraint conditions of the model, a new population initialization strategy is proposed on the premise of ensuring diversity, which is as follows.A sample of two facilities located in three alternative locations with three transport devices in two periods is shown in Figure2. In the initialization of chromosome, the front-end part, due toI<L, generates a matrixA withL×L dimensions, which has only one element with value 1 in each row and each column, such asA=[0,1,0; 1,0,0; 0,01]. After that, we randomly selectI rows to compose matrixB;B=[0,1,0; 0,0,1], thus generating the layout gene positionChrom1x=[0,1,0,1,0,0]. The back-end part needs to be initialized according to the transport devices in each period. If Qtij is 0, then directly generate 1×G dimension zero vector. If Qtij is not 0, generate the 1×G dimension row vector with only one element of 1 and the rest is 0. Then it is converted to 1×(G×I×I) gene positionChrom1y=[0,0,0,1,0,0,0,1,0,0,0,0]. Finally, it is combined withChrom1x, so an initialization chromosomeChrom1=[0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0] is formed. According to the total number of time periodsT, a cell is composed ofT chromosomes.Figure 2
An example of a cell.


### 5.2. Selection Operation

Selection operations are the process of eliminating individuals with low adaptation from an old population and selecting excellent individuals with high fitness. Compared to the common genetic algorithm, the selection process is divided into two steps, namely, cell selection and chromosome selection, due to the particularity of this problem. We use traditional roulette wheel selection method to select the cell and chromosome.


### 5.3. Mutation and Crossover Operations

Crossover is a random selection of individuals' genes and changes them to produce evolutionary operations for new individuals. Mutation operation is a crucial step in genetic algorithm, which ensures the diversity of population and avoids premature convergence. However, for different optimization problems, it is necessary perform tests repeatedly to determine cross probabilityPc and mutation probabilityPm, which is a very tedious task, and it is difficult to find the best value for each problem. So the idea of adaptive genetic algorithm is applied to determine the probability of cross and mutation.(9)P=max⁡popfitness-min⁡popfitnessmax⁡popfitnessBecause binary coding is adopted in this problem, the mutation and cross operations are combined together. During cross and mutation operations, it is necessary to fragment the chromosomes so as to find the corresponding facilities, locations, and types of transport devices. For the front-end partChrom1x=[0,1,0; 0,0,1], as shown in Figure 3, assuming the mutation position is 2, the corresponding value 1 indicates that facility 1 is placed on location 2. Then value changes from 1 to 0, and the remaining two locations are randomly selected to place facility 1. If it is placed on location 1, there is no need for adjustment. If it is placed on position 3, we found that location 3 has already been placed by facility 2. Then, facility 2 is adjusted to the original variant location (location 2). If the mutation position is 3 corresponding to value 0, which means facility 1 is not placed on location 3, then change the value from 0 to 1, and meanwhile change the value 1 on position 2 to 0. After that change, we find that facilities 1 and 2 are both placed on location 3; then we randomly place facility 2 on the remaining locations. As a result, in condition 1, the values in positions 2 and 3 are crossed, and the values in positions 5 and 6 are exchanged. Meanwhile, in condition 2, the values in positions 1 and 2 are exchanged.Figure 3
An example of mutation and cross operation of front-end part.For the variation of the back-end part, as shown in Figure4, if the mutation position is 2 corresponding to value 0, because this position represents no transport devices between facility 1 and facility 1, there is no need to do any operation. If the mutation position is 4 corresponding to value 0, we change value 0 to 1 and change the remaining position to 0. If the variation position is 5 corresponding to value 1, we change value 1 to 0. The residual position is randomly selected as 1. As a result, in conditions 2 and 3, the values on positions 4 and 5 are exchanged.Figure 4
An example of mutation and cross operation of back-end part.


### 5.4. Termination Criteria

The algorithm is terminated when the generation reached the maximal numberM, or the best value did not improve for 0.05×M generations.


## 5.1. Population Initialization

Because the decision variables of the model are all 0-1 variables, binary coding is adopted. Each cell containsT chromosomes. The first I×L position in a chromosome defines the layout of the facility, and the latter G×I×I position denotes the assignment of transport devices. In order to improve the efficiency of the algorithm and combine the constraint conditions of the model, a new population initialization strategy is proposed on the premise of ensuring diversity, which is as follows.A sample of two facilities located in three alternative locations with three transport devices in two periods is shown in Figure2. In the initialization of chromosome, the front-end part, due toI<L, generates a matrixA withL×L dimensions, which has only one element with value 1 in each row and each column, such asA=[0,1,0; 1,0,0; 0,01]. After that, we randomly selectI rows to compose matrixB;B=[0,1,0; 0,0,1], thus generating the layout gene positionChrom1x=[0,1,0,1,0,0]. The back-end part needs to be initialized according to the transport devices in each period. If Qtij is 0, then directly generate 1×G dimension zero vector. If Qtij is not 0, generate the 1×G dimension row vector with only one element of 1 and the rest is 0. Then it is converted to 1×(G×I×I) gene positionChrom1y=[0,0,0,1,0,0,0,1,0,0,0,0]. Finally, it is combined withChrom1x, so an initialization chromosomeChrom1=[0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0] is formed. According to the total number of time periodsT, a cell is composed ofT chromosomes.Figure 2
An example of a cell.


## 5.2. Selection Operation

Selection operations are the process of eliminating individuals with low adaptation from an old population and selecting excellent individuals with high fitness. Compared to the common genetic algorithm, the selection process is divided into two steps, namely, cell selection and chromosome selection, due to the particularity of this problem. We use traditional roulette wheel selection method to select the cell and chromosome.


## 5.3. Mutation and Crossover Operations

Crossover is a random selection of individuals' genes and changes them to produce evolutionary operations for new individuals. Mutation operation is a crucial step in genetic algorithm, which ensures the diversity of population and avoids premature convergence. However, for different optimization problems, it is necessary perform tests repeatedly to determine cross probabilityPc and mutation probabilityPm, which is a very tedious task, and it is difficult to find the best value for each problem. So the idea of adaptive genetic algorithm is applied to determine the probability of cross and mutation.(9)P=max⁡popfitness-min⁡popfitnessmax⁡popfitnessBecause binary coding is adopted in this problem, the mutation and cross operations are combined together. During cross and mutation operations, it is necessary to fragment the chromosomes so as to find the corresponding facilities, locations, and types of transport devices. For the front-end partChrom1x=[0,1,0; 0,0,1], as shown in Figure 3, assuming the mutation position is 2, the corresponding value 1 indicates that facility 1 is placed on location 2. Then value changes from 1 to 0, and the remaining two locations are randomly selected to place facility 1. If it is placed on location 1, there is no need for adjustment. If it is placed on position 3, we found that location 3 has already been placed by facility 2. Then, facility 2 is adjusted to the original variant location (location 2). If the mutation position is 3 corresponding to value 0, which means facility 1 is not placed on location 3, then change the value from 0 to 1, and meanwhile change the value 1 on position 2 to 0. After that change, we find that facilities 1 and 2 are both placed on location 3; then we randomly place facility 2 on the remaining locations. As a result, in condition 1, the values in positions 2 and 3 are crossed, and the values in positions 5 and 6 are exchanged. Meanwhile, in condition 2, the values in positions 1 and 2 are exchanged.Figure 3
An example of mutation and cross operation of front-end part.For the variation of the back-end part, as shown in Figure4, if the mutation position is 2 corresponding to value 0, because this position represents no transport devices between facility 1 and facility 1, there is no need to do any operation. If the mutation position is 4 corresponding to value 0, we change value 0 to 1 and change the remaining position to 0. If the variation position is 5 corresponding to value 1, we change value 1 to 0. The residual position is randomly selected as 1. As a result, in conditions 2 and 3, the values on positions 4 and 5 are exchanged.Figure 4
An example of mutation and cross operation of back-end part.


## 5.4. Termination Criteria

The algorithm is terminated when the generation reached the maximal numberM, or the best value did not improve for 0.05×M generations.


## 6. Computational Analysis

Because the transport device is not mentioned in most of articles, there are no standard benchmark instances. 20 determined instances with different size as article [20] are randomly generated. The size of instance is denoted byT_I_L_G, whereT is the number of periods,I is the number of facilities,L is the number of locations, andG is the number of transport device types. We compare the improved genetic algorithm with the Particle Swarm Optimization (PSO) algorithm proposed in [21]. The parameters are set as follows: the number of populations is 20 and the maximal number of generations is 100. In order to eliminate the influence of the random factors, the average values of the 20 simulation results are compared. A personal computer with 2.7GHz Intel Core i5, 8G RAM, is used to run the algorithms that are coded by matlab2014b. The results are shown in Table 2. It is found that the improved genetic algorithm is superior to the PSO in both solving speed and solution quality in almost all the instances except for instances 8_24_30_10 and 14_32_36_18. Especially for large-scaled instances, the improved genetic algorithm presented in this paper has more advantages.Table 2
Comparison of algorithms.
Instance scale Improved genetic algorithm PSO T_I_L_G Time(s) Total cost Time(s) Total cost 4_8_12_4 40.4572 140 47.4022 140 4_10_16_6 58.7843 178 68.4300 178 6_14_20_6 94.8301 312 142.0803 314 8_20_26_8 180.2430 478 277.0293 480 8_24_30_10 210.4790 648 312.0019 642 10_24_30_8 260.4341 750 377.0341 750 10_30_36_14 380.4339 1078 500.0391 1090 14_32_36_18 1009.2215 1977 1228.0012 1975 14_38_42_22 1900.0034 2130 2277.1014 2134 20_48_52_32 7708.9988 4701 8792.1044 4710 28_60_68_48 15366.4222 7999 18722.0924 8073 35_70_80_60 23177.7788 9876 26718.8819 9915 40_80_90_70 32421.6644 13077 37877.0701 13203 45_90_100_80 40500.7721 18093 45722.3307 18210 50_100_110_90 56783.4451 24088 60421.7701 24191 55_105_115_95 69803.1111 34330 81445.7769 34700 60_110_120_100 77099.2299 40099 89001.7721 40280 60_115_125_105 84123.7798 46788 93100.1109 46989 65_120_130_110 93401.1194 52103 101012.1981 52708 65_125_135_115 99088.7711 58021 107662.0061 58810In order to illustrate the effectiveness of the robust layout proposed in this paper, we compare it with the traditional expected layout (the optimal layout in each period with expected material flow). Cost departure ratio denoted byH is defined to describe the deviation ratio between the total cost of the evaluated layout and the optimal cost in each scenario.Hr represents the robust layout cost deviation ratio, whileHd represents expected layout cost deviation ratio.(10)H=∑n=1NTCn-∑n=1NTCnopt∑n=1NTCnoptFor the stochastic variable material flowQtij, assume that the expected average value isa, and the fluctuation level isb. It means that the material flow is in the rage of [(1-b)∗a,(1+b)∗a]. We randomly generateN=200 scenarios, and the prespecified percentageM in robust constraint is 15%. The expected layout is obtained in the scenario when Qtij is equal toa. The cost deviation ratiosHr andHd are calculated under different fluctuation level in three instances. The results are shown in Table 3.Table 3
Comparison of cost deviation ratio.
Fluctuation level 6_14_20_6 20_48_52_32 40_80_90_70 H r H d H r H d H r H d 1.00% 4.11% 10.75% 2.71% 13.82% 1.72% 15.90% 1.45% 4.08% 10.88% 2.90% 14.50% 1.80% 16.98% 1.90% 4.22% 11.03% 3.11% 15.70% 1.83% 17.88% 2.35% 4.27% 11.77% 3.23% 16.10% 1.88% 19.09% 2.80% 4.29% 12.09% 3.29% 16.77% 1.91% 20.12% 3.25% 4.66% 12.80% 3.30% 17.23% 1.93% 21.98% 3.70% 4.88% 12.91% 3.45% 17.46% 1.94% 22.10% 4.15% 4.91% 13.70% 3.56% 17.56% 1.96% 22.70% 4.60% 4.97% 13.92% 3.71% 18.10% 1.99% 22.90% 5.05% 4.99% 14.10% 3.77% 18.22% 2.01% 23.11% 5.50% 5.01% 14.45% 3.89% 18.28% 2.09% 23.34% 5.95% 5.09% 14.67% 3.91% 18.45% 2.21% 23.56% 6.40% 5.03% 14.72% 3.94% 18.66% 2.32% 23.88% 6.85% 5.21% 14.92% 3.99% 18.88% 2.26% 23.99% 7.30% 5.27% 14.98% 4.01% 19.10% 2.45% 24.67% 7.75% 5.29% 15.21% 4.03% 19.23% 2.48% 24.99% 8.20% 5.33% 15.35% 4.07% 19.45% 2.29% 25.16% 8.65% 5.35% 15.62% 4.11% 19.57% 2.38% 25.37% 9.10% 5.66% 16.11% 4.13% 19.82% 2.44% 25.99% 9.55% 5.70% 16.32% 4.22% 20.03% 2.51% 26.81% 10.00% 5.72% 16.78% 4.25% 20.70% 2.68% 27.32%From Table3, we can see that, with the increase of fluctuation level, the cost deviation ratio will increase for both robust layout and expected layout. We can conclude that the greater the fluctuation is, the bigger difference between robust/expected layout cost and minimum cost is. Moreover, for all these three different sized instances, the cost deviation ratio of robust layout is much less than expected layout. As the problem size increases, the cost deviation ratio of expected layout raises, while the cost deviation ratio of robust layout reduces. It is obviously concluded that the robust approach proposed in this paper is effective to deal with dynamic layout problem under demand uncertainty, especially for large sized problem.


## 7. Conclusions

In this paper, an improved genetic algorithm is designed to solve dynamic facility problem considering the assignment of transport devices. The developed algorithm has given near-optimal solutions to different sized case studies randomly generated. In addition, a robust layout is suggested with uncertain material flow in each period. The robust layout is the most frequent one falling within a prespecified percentage of the optimal solution for multiple scenarios generated by Mont Carlo simulation. Although robust layout may not be optimal in any scenario, the deviation from the best cost is small. Its performance is much better than the expected layout through sensitivity analysis.More uncertain factors except for material flow can be investigated in our future research, such as the unit material handling cost and the flexible routing. Moreover, unequal-sized facilities can be considered and more efficient metaheuristic algorithm needs to be developed in the future.



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