# Vapour – Liquid Equilibrium (VLE): An Algorithm Inspired by the Chemical Phenomenon for Solving a New Patient Bed Assignment Problem

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1. Introduction

In Chemistry, the vapor-liquid equilibrium process describes the distribution of chemical species combining two essential phases [1]: vapor phase and liquid phase. Using a binary system of compounds is possible to simulate a search process based on the equilibrium between both phases [2]. Understanding the distribution of chemical components in both the vapor and liquid phase, called vapor-liquid equilibrium is essential to the design, operation, and analysis of many engineering processes [3].

2. Why we study vapor liquid equilibrium?

Many Chemical and environmental processes involve Vapor – Liquid Equilibrium (VLE) undertaken the bulk of industrial separation processes, particularly in distillation, drying and evaporation processes [4].

2. Inspiration of VLE

VLE algorithm is inspired by the chemical process described by the liquid-vapor equilibrium phenomenon. The bubble point is the temperature at which a liquid mixture, at a certain pressure, begins to boil [5]. The exact moment for this phase occurs when the first vapor bubble appears. On the contrary, the dew point is the temperature at which a mixture of vapors, at a certain pressure, begins to condense. The phase begins when the first drop of liquid appears. These equilibrium conditions were modeled and embedded in our algorithm to solve optimization problems.

3. Analysis and Optimization of Patient Bed Assignment Problems using VLE

The VLE approach is applied for solving the operating room scheduling and assignment problem [6] we will integrate this solution to a platform in order to optimize the assignment process and reduce the workload of the clinical staff [7].

4.Numerical Expression of VLE Algorithm

Initially use a two-state strategy model the exploration process through the bubble point or dew point, respectively.

L 1 and V 1 represent molar fractions in the liquid phase and vapor phase, respectively. The vapor pressure of chemical species at time T and E1 describes the liquid-vapor equilibrium ratio [8]. A linear transfer functions from the real domain (lower bound and upper bound) towards a real auxiliary domain.

S n search range to decision variables. An inverse linear transfer function that takes values of the auxiliary domain and transform them into molars fraction as defined below,

While the defined stop criterion is not achieved the current solutions are updated. New solutions are generated based on the chemical phenomenon .A restart parameter is evaluated in order to update the solutions

The relations of the phase equilibrium are evaluated   E is the values of chemical species are computed by the equation as below, this formula indicates that the equilibrium relationship has been established based on Raoult’s law

VP denotes the vapor pressure of the chemical species [10].These equations allow obtaining the necessary expressions to calculate the bubble and dew points of the binary mixtures.

6. Flowchart of VLE

7. VLE Algorithm

8. Application of VLE

Reference

[1]C. Taramasco, B. Crawford, R. Soto, E. Cortés-Toro and R. Olivares, “A new metaheuristic based on vapor-liquid equilibrium for solving a new patient bed assignment problem”, 2020. .

[2]S. Araghi, A. Khosravi and D. Creighton, “Intelligent cuckoo search optimized traffic signal controllers for multi-intersection network”, Expert Systems with Applications, vol. 42, no. 9, pp. 4422-4431, 2015. Available: 10.1016/j.eswa.2015.01.063.

[3]R. Aringhieri, P. Landa, P. Soriano, E. Tànfani and A. Testi, “A two level metaheuristic for the operating room scheduling and assignment problem”, Computers & Operations Research, vol. 54, pp. 21-34, 2015. Available: 10.1016/j.cor.2014.08.014.

[4]R. Ben Bachouch, A. Guinet and S. Hajri-Gabouj, “An integer linear model for hospital bed planning”, International Journal of Production Economics, vol. 140, no. 2, pp. 833-843, 2012. Available: 10.1016/j.ijpe.2012.07.023 [Accessed 27 August 2020].

[5]R. Bekker and P. Koeleman, “Scheduling admissions and reducing variability in bed demand”, Health Care Management Science, vol. 14, no. 3, pp. 237-249, 2011. Available: 10.1007/s10729-011-9163-x [Accessed 27 August 2020].

[6]T. Dokeroglu, E. Sevinc, T. Kucukyilmaz and A. Cosar, “A survey on new generation metaheuristic algorithms”, Computers & Industrial Engineering, vol. 137, p. 106040, 2019. Available: 10.1016/j.cie.2019.106040 [Accessed 27 August 2020].

[7]K. Hussain, M. Mohd Salleh, S. Cheng and Y. Shi, “Metaheuristic research: a comprehensive survey”, Artificial Intelligence Review, vol. 52, no. 4, pp. 2191-2233, 2018. Available: 10.1007/s10462-017-9605-z [Accessed 27 August 2020].

[8]A. Kavousi-Fard, H. Samet and F. Marzbani, “A new hybrid Modified Firefly Algorithm and Support Vector Regression model for accurate Short Term Load Forecasting”, Expert Systems with Applications, vol. 41, no. 13, pp. 6047-6056, 2014. Available: 10.1016/j.eswa.2014.03.053 [Accessed 27 August 2020].

[9]K. Szwak and A. Sacchetti, “Droperidol Use in Pediatric Emergency Department Patients”, Pediatric Emergency Care, vol. 26, no. 4, pp. 248-250, 2010. Available: 10.1097/pec.0b013e3181d6d9f2.

[10]N. Nethercote, P. Stuckey, R. Becket, S. Brand, G. Duck and G. Tack, “MiniZinc: Towards a Standard CP Modelling Language”, Principles and Practice of Constraint Programming – CP 2007, pp. 529-543. Available: 10.1007/978-3-540-74970-7_38 [Accessed 27 August 2020].

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