# An Application of an Unequal-Area Facilities Layout Problem with Fixed-Shape Facilities
## 5 Conclusions  
The UA-FLP is considered in this paper and is defined as the problem of locating rectangular facilities, with fixed dimensions, on a rectangular floor space such that facilities do not overlap while minimizing the total weighted distance (TWD). This research was motivated by the investigation of layout options for a production area at the Toyota Motor Manufacturing West Virginia (TMMWV) plant in Buffalo (WV, USA). A mixed integer linear programming (MILP) model and a genetic algorithm (GA) is presented for locating facilities on a continuous plant floor. The proposed GA consists of the boundary search heuristic (BSH), a linear programming (LP) model, and the dual simplex method. The GA is used to obtain permutations of the facilities, which specify the order in which facilities should be placed on the plant floor. From these permutations, BSH is used to construct the layouts. Then the layouts are improved using the LP model and the dual simplex method. Instead of solving the problem from scratch, from one LP model to another, some of the right-hand side values of the LP model were updated/modified, and the model is quickly solved using the dual simplex method.  
The computational results show that the proposed GA is effective for solving problems with 50 or less facilities, as in the UA-FLP encountered at the TMMWV plant. However, as stated previously, in some cases the simple GA may have difficulty converging to the local optimum. More specifically, the GA is a stochastic heuristic, which is good for obtaining diverse solutions but may lack in intensification (i.e., the ability to thoroughly explore the regions of these diverse solutions). As a result, for future research, the tabu search (TS) heuristic can be used to search for the local optimum for each chromosome during each generation (i.e., intensify the search). This would require the development of a memetic heuristic (e.g., hybrid GA and TS heuristic) for the UA-FLP. Otherwise, a hybrid heuristic could be developed, which uses the diversification strategies of simulated annealing and intensification strategies of tabu search. These heuristics can be compared to the proposed GA.  
Conceptualization, A.M. and A.H.; methodology, A.M. and A.H.; software, A.H.; validation, A.M.; writing--original draft preparation, A.M.; writing--review and editing, A.M. and A.H.; supervision, A.M. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Min TWD** & **Mean TWD** & **Std. Dev.** & **\% Improvement** & **CP Time** \\ \hline Proposed GA & 71,031.24 & 71,401.19 & 424.29 & 0.36 & 60 m \\ TS/BSH [18] & **71,291.4** & - & - & 0 & 2.1 h \\ \hline \hline \end{tabular}
\end{table}
Table 6: Summary of results for test problem P50.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Min TWD** & **Mean TWD** & **Std. Dev.** & **\% Improvement** & **CP Time** \\ \hline Proposed GA & 1147 & 1152.08 & 5.64 & 0.38 & 15 m \\ TS/BSH [18] & **1151.4** & – & – & 0 & 2.9 h \\ MS-SA [16] & 1218.75 & 1270.55 & 20.09 & \(-\)5.85 & – \\ \hline \hline \end{tabular}
\end{table}
Table 5: Summary of results for test problem P20.  
This research received no external funding.  
Not applicable.  
We would like to thank [PERSON] and [PERSON] of Toyota Motor Manufacturing West Virginia (TMMWV) plant in Buffalo, WV for giving us the opportunity to work on TMMWV's facility layout problem. Also, we would like to thank the anonymous reviewers for their constructive comments.  
The authors declare no conflict of interest.