# A mixed-integer linear programming formulation for the modular layout of three-dimensional connected systems

Journal article

O’Neill, Sam, Wrigley, Paul and Bagdasar, Ovidiu 2021. A mixed-integer linear programming formulation for the modular layout of three-dimensional connected systems. Mathematics and Computers in Simulation. https://doi.org/10.1016/j.matcom.2021.09.019
Authors O’Neill, Sam, Wrigley, Paul and Bagdasar, Ovidiu Given the considerable complexity of process plants, there has been a great deal of research focused on aiding the design of plant layout through mathematical optimisation, i.e. optimising the positioning of the equipment in the plant for space and cost efficiency. Recently, the use of modular approaches within the construction industry, whereby work is performed off-site before being assembled on-site, has become a popular and powerful way of reducing build schedules and costs. Modular approaches have many other real applications where items must be packed to minimise the connections between them (e.g. piping, wiring, modular office and factory layouts) and consider the modular layout of the system. In this paper, we provide a formulation of the problem that, in addition to the standard layout problem, considers a modular block layout to allow modular construction and transportation of the plant. The problem is represented as a directed network, with the aim to pack the items into predefined containers and minimise the rectilinear distance between the connected items. We propose mixed-integer linear programming (MILP) models for the 2-dimensional and 3-dimensional problems and solve them using the state-of-the-art mathematical programming solver, Gurobi. Because of the combinatorial nature of the problem, solutions involving a large number of items may not converge and a suboptimal solution must be considered. However, our results suggest that even in the case of optimising a large number of items, the suboptimal solutions found after a reasonable number of iterations where deemed, by a domain expert, to be a good enough starting point to continue the design process, especially in the early concept phase. Applied Mathematics; Modelling and Simulation; Numerical Analysis; General Computer Science; Theoretical Computer Science 2021 Mathematics and Computers in Simulation Elsevier BV 0378-4754 https://doi.org/10.1016/j.matcom.2021.09.019 http://hdl.handle.net/10545/626095 https://www.elsevier.com/tdm/userlicense/1.0/ hdl:10545/626095 02 Oct 2021 10 Nov 2021, 15:57 21 Sep 2021 University of Derby license.txtFile Access LevelOpen ICMMCMSE_2020___Connected_Bin_Packing__Revised_.pdf

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Bagdasar, Ovidiu and Popa, Ioan-Lucian 2016. On the geometry of certain periodic non-homogeneous Horadam sequences. Electronic Notes in Discrete Mathematics. https://doi.org/10.1016/j.endm.2016.11.002
##### New results and conjectures on 2-partitions of multisets.
Bagdasar, Ovidiu and Andrica, Dorin 2017. New results and conjectures on 2-partitions of multisets. Proceedings of the 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO). https://doi.org/10.1109/ICMSAO.2017.7934928
##### Module layout optimization using a genetic algorithm in light water modular nuclear reactor power plants.
Wrigley, P., Wood, Paul, Stewart, Paul, Hall, Richard and Robertson, D. 2018. Module layout optimization using a genetic algorithm in light water modular nuclear reactor power plants. Nuclear Engineering and Design. https://doi.org/10.1016/j.nucengdes.2018.10.023
##### Qualitative case study methodology: Automatic design and correction of ceramic colors
Bagdasar, O., Birluţiu, A., Chen, M. and Popa, I.-L. 2017. Qualitative case study methodology: Automatic design and correction of ceramic colors. IEEE. https://doi.org/10.1109/icstcc.2017.8107118
##### New results and conjectures on 2-partitions of multisets
Bagdasar, O. and Andrica, D. 2017. New results and conjectures on 2-partitions of multisets. 7th International Conference on Modeling, Simulation, and Applied Optimization, ICMSAO 2017. IEEE Computer Society. https://doi.org/10.1109/icmsao.2017.7934928
Bagdasar, Ovidiu D., Larcombe, Peter J. and Bagdasar, O. 2017. ON THE MASKED PERIODICITY OF HORADAM SEQUENCES: A GENERATOR-BASED APPROACH. Fibonacci Quarterly.
##### LOCAL MAXIMIZERS OF GENERALIZED CONVEX VECTOR-VALUED FUNCTIONS
Bagdasar, Ovidiu, Popovici, Nicolae and Bagdasar, O. 2017. LOCAL MAXIMIZERS OF GENERALIZED CONVEX VECTOR-VALUED FUNCTIONS. Journal of Nonlinear and Convex Analysis.
##### On certain series expansions of the sine function: Catalan numbers and convergence
Larcombe, Peter J., O'Neill, Sam T. and Fennessey, Eric J. 2014. On certain series expansions of the sine function: Catalan numbers and convergence. Fibonacci Quarterly.
##### On the number of complex horadam sequences with a fixed period
Bagdasar, Ovidiu and Larcombe, Peter J. 2013. On the number of complex horadam sequences with a fixed period. Fibonacci Quarterly.
##### On the characterization of periodic complex Horadam sequences
Bagdasar, Ovidiu and Larcombe, Peter J. 2013. On the characterization of periodic complex Horadam sequences. Fibonacci Quarterly.
##### On a result of Bunder involving Horadam sequences: A proof and generalization
Larcombe, Peter J. and Bagdasar, Ovidiu 2013. On a result of Bunder involving Horadam sequences: A proof and generalization. Fibonacci Quarterly.
##### Influence discovery in semantic networks: An initial approach
Trovati, M. and Bagdasar, O. 2014. Influence discovery in semantic networks: An initial approach. IEEE. https://doi.org/10.1109/uksim.2014.48
##### Complementary iInequalities involving the Stolarsky Mean
Bagdasar, Ovidiu 2010. Complementary iInequalities involving the Stolarsky Mean. Journal of Inequalities and Applications. https://doi.org/10.1155/2010/492570
##### Concise computer mathematics: tutorials on theory and problems
Bagdasar, Ovidiu 2013. Concise computer mathematics: tutorials on theory and problems. Springer.
##### On the characterization of periodic generalized Horadam sequences
Bagdasar, Ovidiu and Larcombe, Peter J. 2014. On the characterization of periodic generalized Horadam sequences. Journal of Difference Equations and Applications. https://doi.org/10.1080/10236198.2014.891022
##### Local maximum points of explicitly quasiconvex functions
Bagdasar, Ovidiu and Popovici, Nicolae 2014. Local maximum points of explicitly quasiconvex functions. Optimization Letters. https://doi.org/10.1007/s11590-014-0781-3
##### A horadam-based pseudo-random number generator
Bagdasar, O.D., Chen, M. and Bagdasar, O. 2014. A horadam-based pseudo-random number generator. IEEE. https://doi.org/10.1109/uksim.2014.55
##### On the structure of periodic complex Horadam orbits
Bagdasar, Ovidiu D., Larcombe, Peter J., Anjum, Ashiq and Bagdasar, O. 2016. On the structure of periodic complex Horadam orbits. Carpathian Journal of Mathematics.
##### Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems
Popa, I.-L., Ceauşu, T., Bagdasar, O. and Agarwal, R.P. 2016. Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems. Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica. 27 (2), pp. 153-166. https://doi.org/10.2478/auom-2019-0025
##### On some functions involving the lcm and gcd of integer tuples
Bagdasar, Ovidiu 2014. On some functions involving the lcm and gcd of integer tuples. Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics. https://doi.org/10.5937/SPSUNP1402091B
##### ON A RESULT OF BUNDER INVOLVING HORADAM SEQUENCES: A NEW PROOF
Larcombe, Peter J., Bagdasar, O. and Fennessey, Eric J. 2014. ON A RESULT OF BUNDER INVOLVING HORADAM SEQUENCES: A NEW PROOF. Fibonacci Quarterly. pp. 175-177.