Inherently safer facility layout with multiple floors

Diego Alejandro Sierra

Abstract

In a process plant, operation and safety factors are affected by the positioning of equipment among space. Asmathematical and optimization methods have started to aid expert criterion for layout distribution, interesthas been drawn towards efficient formulations and methods which include uncertainty treatment and riskassessment tools in both objective functions and constraints. The current work proposes a MINLP formulationfor a 3D facility layout problem based on a multi-floor approach, which accounts for both operation and safetyaspects. The Safety Weighted Hazard Index designed by Khan, Husain, and Abbasi is used to measure theseaspects. A metaheuristic algorithm known as Bacterial Foraging Optimization (BFO) algorithm solves theproblem. The proposed methodology is applied to a facility example case used in the literature to evaluatethe effectiveness and efficiency of the current approach.

1 Introduction

Process plant design is a discipline that comprises several steps of increasing complexity to establisha chemical facility [1]. One of these critical steps is facility layout, which is a crucial factor when itcomes to the performance of any chemical process [2]. The positioning of equipment units within agiven space can impact the behavior of any system in safety, operation, maintenance, and economicconsiderations, a reason for which the interest on layout design has risen in recent times [3].

The layout design problem can be stated as the decision-making process that includes the orien-tation, positioning, and elevation of equipment units within a specific area [2]. This problem presentsconsiderable complexity due to the number of variables that must be taken into account to devisea good layout; namely, acceptable levels of hazard within the site and to the public, efficient plantoperation, convenient access for maintenance and emergencies, cost-effective and appropriate use ofspace, among others [4]. Furthermore, changing layout decisions after they have been taken may becostly in economic terms, highlighting the importance of making the right choices in the early stepsof design [2].

Due to the importance of the problem, plant layout is usually expert-driven. In previous decades,it was relevant to aid these experts with guidelines that compiled lessons learned and experiencefrom different kinds of plants worldwide [4]. In the last three decades, however, there has been agrowing interest in mathematical modeling and optimization techniques, which aim at enhancing theperformance of plant layouts [5].

At first, initial facility layouts were focused on process intensification, which required closenessbetween equipment; because of this, designs were usually enclosed and compact. Jayakumar andReklaitis [6] proposed an optimization model for laying out enclosed both single-floor and multi-level facilities with aisles. Subsequent approaches, such as the model proposed by Georgiadis andMacchietto [7], allocate process equipment in grid spaces based on the associated costs. In thissense, layout problems were economic-driven, and researchers considered distances as a factor forrising capital costs. These studies shed light on the necessity of efficient formulations that includedboth integer and continuous variables and integrated separation distances between equipment withina safety perspective [2].

1

1 Introduction 2

Initially, safety inclusion started with Penteado and Ciric [8], who proposed a mixed integernon-linear programming (MINLP) model for optimizing both safety (based on exposure radii) andeconomic aspects. By that time, safety indices had started to rise for assessing loss, due to acci-dents in process plants. To prevent such accidents, the Chemical Center for Process Safety (CCPS)presented the Guidelines for Facility Layout and Siting in 2003 [9]. Such guidelines allowed opti-mization models to include prescriptive-based safety distances between equipment as constraints tobe fulfilled.

Patsiatzis, Knight, and Papageorgiou [10] developed a continuous linear model, with an accuratecomputation of both safety aspects and economic loss as proposed by the Dow’s Fire and ExplosionIndex. Guirardello and Swaney [11] took a different approach and introduced an optimization modelfor 3D-layouts based on routing and piping costs, constraining safety distances between equipment.Because of the available tools, safety considerations in the literature used to be prescriptive-basedand not performance-based: they relied on heuristic concepts rather than evaluation on the layoutsafety [10].

These limitations made clear that prescriptive-based approaches could not certainly avoid cascad-ing effects [12]. To prevent such events, plant layout had to include performance-based metrics (suchas risk assessments) in optimization models, although there was a particular difficulty when handlingthe uncertainty associated with unwanted events [13]. Tugnoli et al. [14] proposed a Domino HazardIndex which accounts for specific undesired events and allows for evaluation of layout safety . Sucha feature was later exploited by both De Lira-Flores et al. [15] and Latifi, Mohammadi, and Khakzad[5], who integrated the Domino Hazard Index to optimization models. Neghabi and Ghassemi Tariproposed a novel adjacency model to evaluate distances as safety and process features [16]. ToufiqurRahman, Salim, and Syeda suggested a risk score based on toxic gas release and vapor cloud ex-plosions for safety quantifying [17], while Han, Cho, and Yoon proposed additional constraints forsocial safety distances[18]. Brunoro Ahumada, Quddus, and Mannan [19] used a quantitative riskanalysis to refine safety considerations within an optimization model, although they use a grid-basedpositioning to reduce the search space of the algorithm. Ejeh et al.[20] proposed a linear formulationfor 3D modeling of equipment layout, and Park, Shin, and Won [21] presented a variation of suchapproach by focusing on Liquified Natural Gas (LNG) cases and adding the Fire and Explosion Indexto take into account safety measures. As safety features have been incipiently incorporated to math-ematical formulations of layout designs, these problems have become more demanding in aspectssuch as nature (usually stochastic and mixed-integer) and size, opening the possibility for differenttypes of solution algorithms. Xu and Papageorgiou [22] proposed an improvement-type algorithmfor solving a layout problem relying on connection costs, while Caputo et al. [23] used a geneticalgorithm (GA) to obtain layouts based on annual loss costs. Also, because of the complexity of theapproached model, Latifi, Mohammadi, and Khakzad[5] solve their MINLP formulation with the Batmetaheuristic.

The Domino Hazard Index presented by Tugnoli et al. allows for a new possibility regardingsafety concerns: safety indices. These indices calculate hazard scores based on the most processconditions, the classification of substances and the possible undesired events. Although mainlythought for assessing hazard in already settled plants [24], these tools offer a practical methodof understanding and simplifying propagation phenomena and uncertainty of unwanted events, asuitable characteristic for optimization problems addressing layout design. Even with the reductionof uncertainty to deterministic calculations because of the safety indices, a big issue remains in thehigh non-linearity and mixed integer nature of these problems. Metaheuristic methods are then seenas a possible option for solving this kind of problems, as used in various works in the literature suchas Caputo et al. [23] and Latifi, Mohammadi, and Khakzad [5].

In general terms, it could be said that expert criterion still guides the most of layout design,although recent research has proven its worthiness by aiding experts on evaluating candidate layouts.

2 BFO approach to the layout design problem 3

In this sense, experts become critical decision makers who must assess the fitness of a specificequipment arrangements. The current work focuses on obtaining good candidate designs that fulfillcertain layout conditions to aid experts in deciding the best positioning for equipment.

As for the rest of the article, Section 2 describes the methodology used, and Section 3 introducesa case study that serves the purpose of showing the applicability and effectiveness of the proposedapproach, together with the analysis of the obtained results. Finally, Section 4 provides conclusionsand recommendations for future work.

2 BFO approach to the layout design problem

The following section introduces the proposed methodology. First, Section 2.1 presents a generaldescription of the problem. Section 2.2 explains the Safety Weighted Hazard Index, the tool usedfor assessing risk in this work, and Section 2.3 proposes a mathematical formulation for the problem.Finally, Section 2.4 explains the metaheuristic algorithm used, and Section 2.5 details the solutionscheme as a whole.

2.1 Problem descriptionThe layout design problem presented in this work consists of deciding about the best location ofequipment on several floors so to obtain the minimal cost associated to area, loss, piping and safetydevices’ costs. The problem has the following assumptions: the irregularity of equipment shapes isenclosed in Cartesian volumes, the rectilinear distances between connection points are assumed asthe piping length, and installation costs of equipment are neglected. The problem can be stated asfollows:

Given:

1. A set of equipment with their respective dimensions in the x,y and z coordinates;

2. The total number of levels for the layout;

3. Required connectivity between equipment units according the process flow (Process Flow Di-agram or Piping and Instrumentation Diagram);

4. A set of possible safety devices involving different layers of protection;

5. Specific conceptual engineering and detailed design information about the conditions of theprocess; and

6. Data about capital costs of piping, equipment, area usage and implementation of safety devices;

Determine:

1. The spatial arrangement of process units; and

2. The decisions about implementation of safety devices in all the equipment pieces;

To minimize the costs associated to:

1. The area of each floor;

2. The usage of each safety devices;

3. The piping of the layout; and

4. The possible loss associated with undesired events.

2 BFO approach to the layout design problem 4

2.2 Safety Weighted Hazard IndexThe Safety Weighted Hazard Index (SWeHI) is a safety-based index whose value is the radius ofa circular area under moderate hazard [25]. Its primary purpose is to assess risk related to bothinternal (process-related) and external factors within a process unit or equipment. According to Royet al. [24], SWeHI is a comprehensive safety index for process design, improving over the previouslydeveloped HIRA (Hazard Identification and Ranking) by including integration of safety barriers toreduce hazard radii. SWeHI also provides specificity referring to different types of units, meaningthat it can apply to any chemical process [25].

SWeHI calculation involves two main sub-indices: the B sub-index, related to hazard identificationand assessment, and the A sub-index, related to safety measures.

2.2.1 Sub-index B

The B sub-index expresses the radius of a circular area under 50% probability of damage. It can becalculated from Equation 1:

B = max(B1, B2) (1)

where B1 and B2 are sub-indices for damage due to fire and explosion and toxic release, respectively.In this sense, the index B takes the worst-case scenario between a fire or explosion accident, and atoxic release affecting people nearby. The focus of this work is to observe the possible damage of anunwanted event from one unit to its neighbors, for which the toxic release scenarios, which cannotaffect other equipment, will not be taken into consideration.

B1 calculation differs from unit to unit. For instance, Khan, Husain, and Abbasi has groupedthe process units into five categories:

1. Storage units;

2. Units related to physical operations, like mass transfer, heat exchanging, separation processes,pumping, and compression;

3. Equipment related to chemical reactions;

4. Transportation units; and

5. Other units, mainly equipment working with direct fire for heating.

There are three factors, common to all groups, that must be computed before further calculations.They are obtained from the following equations:

F1 = 0.1 ·M · Hc

K(2)

F2 = 1.304× 10−3 · PP · V (3)

F3 = 1× 10−3 · 1T + 273 · (PP · V P )2 · V (4)

where:

1. M is the mass of the most hazardous chemical involved or the mass release rate, in kgs ;

2. Hc is the heat of combustion of the most hazardous chemical involved, in kJmol ;

2 BFO approach to the layout design problem 5

3. K is a constant;

4. PP is the operating pressure of the equipment unit, in kPa;

5. V is the volume of the equipment unit, in m3;

6. T is the operating temperature of the equipment unit, in řC; and

7. V P is the vapor pressure at the operating conditions of the equipment, in kPa;

A fourth factor is included for units involving chemical reactions:

F4 = M · Hrxn

K(5)

where Hrxn is the heat of reaction of the primary reaction taking place in the equipment, in kJmol .

Once the energy factors are found, eight penalties are calculated and multiplied with these factorsto calculate a hazard potential. SWeHI takes into account information from the process in severalaspects: penalties 1 to 4 are based on process parameters, penalties 5 and 6 are based on equipmentpositioning and penalties 7 and 8 are based on external events. Because these two last penaltiesare dependant on siting decisions, they will not be taken into account for SWeHI calculation in thiswork. Although their values differ according to the unit type, the penalties relate to the followingaspects:

1. Effect of temperature, quantified in the penalty through the operating temperature T , theflash point FlP , the fire point FP and the autoignition temperature AIT ;

pn1 = f1(T, F lP, F irP,AIT ) (6)

2. Effect of pressure, quantified through the operating pressure PP , the atmospheric pressureAP , the vapor pressure V P , and the NFPA [26] rankings for flammability NF and reactivityNR. Comparison between pressures will discard either factor F2 or F3, while the penaltyis obtained from the ratio between the NFPA rankings and the graph shown in Figure 1 (orsimilar, depending on the unit type);

pn2 = f2(PP, V P,NF,NR) (7)

3. Effect of the quantity of substance handled ChQ, according to the NFPA [26] rankings forflammability NF and reactivity NR and the graph shown in Figure 2 (or similar, dependingon the unit type);

pn3 = f3(ChQ,NF,NR) (8)

4. Effect of the chemical characteristics (reactivity and flammability), quantified through theNFPA [26] rankings for flammability NF and reactivity NR;

pn4 = f4(NF,NR) (9)

5. Effect of the distance to the nearest hazardous unit N quantified through the graph shown inFigure 3;

pn5 = f5(N) (10)

2 BFO approach to the layout design problem 6

Fig. 1: Penalty vs. Operating pressure, obtained from [25].

Fig. 2: Penalty vs. Chemical quantity, obtained from [25].

6. Effect of the density of units D quantified through an equation for equipment density withina radius of 30 meters;

pn6 = f6(D) (11)

Reaction units and transportation units have additional penalties associated with the type ofreactions involved (main and side reactions) and the physical state of the substance transported,respectively. As mentioned before, these penalties are related to the energy factors to obtain a

2 BFO approach to the layout design problem 7

Fig. 3: Penalty vs. Distance to the nearest hazardous unit, obtained from [25].

hazard potential. The equation for this potential also depends on the unit type, although it can beexpressed, in general terms, as shown in Equation 12.

H = (F1 · pn1 + F2 · pn2) ·n∏

i=1pni (12)

where H is the hazard potential, pn are the calculated penalties, and n is the index of the lastpenalty. Inclusion in Equation 12 of either F2, F3 or both depends on the pressure penalty. Oncethe hazard potential is obtained, the following relation is used to calculate the B1 factor:

B1 = 4.76 ·H 13 (13)

2.2.2 Subindex A

SWeHI takes into account the possibility of installing safety measures within the process plant.These measures aim at reducing risk potential of the plant at a particular time, either by reducingthe probability of occurrence of an unwanted event, or by mitigating the effects of such event onceit has already happened. In this sense, SWeHI accounts for the installation and usage of barriers indifferent layers of protection.

Possible safety measures are grouped into eight categories:

1. Emergency resource planning (ERP);

2. Disaster management plan (DMP);

3. Damage control measures;

4. Process control system;

5. Detection devices;

2 BFO approach to the layout design problem 8

Control measure Credit factor (cr3)Foam in adequate quantity 0.25

Water with separate pumps and pipeline 0.2Fire extinguishing gases in sufficient amount 0.25

Flame arrestor 0.2Inert gases for dilution of explosive gases 0.35

Credits for the different control systems (cr4)

Control type Advanced control(computer-based) Analog control Manual control

Temperature control 0.8 0.5 0.2Pressure control 0.8 0.4 0.1

Level control 0.8 0.5 0.2Flow control 0.7 0.4 0.1

Tab. 1: Credit factors for the control measures and the process control systems , obtained from [25]

6. Emergency control measures;

7. Human error; and

8. Equipment reliability.

Credit factors are assigned to each one of these categories, according to the hazard controlmeasures applied. These credits work similarly to the penalties in Section 2.2.1, yet these creditfactors take values from 0 to 1 and do not work with specific functions like those in Figures 1, 2 and3, but are rather discrete and assigned based on expert criterion. Devices are not exclusive, whichmeans that it is possible to implement multiple safety devices for each category, and the final creditfactor is the summation of the partial credits.

Detecting devices Credit factor (cr5)Flammable chemical detection device at the current unit 0.8

Explosive mixture formation detection device at current unit 0.8Flame arrestor at current unit 1

Tab. 2: Credit factors for the detecting devices, obtained from [25]

Because measures related to procedural safety (ERP and DMP) and human error cannot bedetermined before the plant is in operation, an average value is chosen for calculation of the sub-index A throughout the work. Tables 1,2, and 3 show the possible safety measures and their creditfactors.

2 BFO approach to the layout design problem 9

Emergency control measure Credit factor (cr6)Shut down measures 0.8

Interlocks 0.8Showers 1

Safety vents 0.2

Probability of failure (per year) Credit (cr8)p >{}10−3 0.1

10−4< p <10−3 0.210−5< p <10−4 0.410−6< p <10−5 0.610−7< p <10−6 0.8

p<10−7 1

Tab. 3: Credit factors for emergency control measures and equipment reliability, obtained from [25]

Once the credits are calculated for all of the categories, the following equation is used to find theA sub-index:

A = 0.15 ·8∏

i=1cri (14)

where cri is the credit factor for the i group. Finally, and after the B and A sub-indices have beenfound, the resulting SWeHI can be found as follows:

SWeHI = B

A(15)

2.3 Mathematical formulationThis section presents the mathematical formulation of the problem. The following seven subsectionspropose the different constraints and the objective function. The formulation is based in the workproposed by Ejeh et al. [20].

2.3.1 Floor constraints

In the proposed multi-floor approach, the height of each equipment e ∈ E (being E the set of allequipment units) is used to obtain a parameter Me. This parameter represents the number of floorsthat each equipment unit occupies. In the optimization model, this is represented by the followingconstraint: ∑

f∈F

pef = Me ∀e ∈ E (16)

where F is the set of available floors and pef is a binary variable that takes the value of 1 in casethe equipment e belongs to the floor f . For following constraints, it is necessary to define a binaryvariable Neif that measures if units e and i are on the same floor f , with the following constraint:

Neifpef · pif ∀e, i ∈ E f ∈ F (17)

2 BFO approach to the layout design problem 10

This definition allows Nei to take the value of 1 in case that units e and i are on the same floorand the value of 0 in case they are not.

2.3.2 Orientation constraints

Rotation of equipment is supposed to happen only in the xy plane, with two asymmetric possibilitiesof orientation. Because equipment dimensions are input parameters, length and width variables arecreated and defined as follows.

le = αe ·Oe + βe · (1−Oe) ∀e ∈ E (18)

de = αe + βe − le ∀e ∈ E (19)

where le and de are the length and width variables previously mentioned, αe and βe are equipmentdimensions as specified to the problem, and Oe is a binary variable that defines the orientation ofequipment e.

2.3.3 Multi-floor constraints

Because equipment can occupy several floors, binary variables for starting floors are defined andconstrained as follows:

− pef + pe,f−1 + Ssef ≥ 0 ∀e ∈ E f ∈ F, f > 1 (20)

− pef + Ssef ≥ 0 ∀e ∈ E f = 1 (21)

where Ssef is a binary variables that take the value of 1 in case the equipment e starts in the floor

f . Additional constraints to ensure each equipment starts only on one floor are also presented.∑f∈F

Ssef = 1 ∀e ∈ E (22)

In Equation 16, equipment units are constrained to take their respective Me floors, althoughthese floors are not necessarily consecutive. The following equation uses the information from thestart floor and forces the following floor variables to take the value of 1 to fulfill with the successivefloors constraint:

f+Me−1∑k=f

pek ≥Me · Ssef ∀e ∈ E f ∈ F (23)

2.3.4 Distance constraints

In this work, rectilinear distances are considered as connecting distances from one equipment toanother, although for following constraints, it is relevant to define rectilinear distances betweenequipment in each axis.

DXei = |xe − xi| ∀e, i ∈ E (24)

DYei = |ye − yi| ∀e, i ∈ E (25)

2 BFO approach to the layout design problem 11

DZei = FH ·∑f∈F

(f − 1) · (Ssef − Ss

if ) +OPei − IPei ∀e, i ∈ E (26)

where xe and ye are the coordinates in the x and y-axes of the center of equipment e, OPei andIPei are relative height positionings of output and input nozzles for equipment e and i, and FH isthe floor height. Although distances in the x and y-axes are straightforward, distance in the z-axistakes into account the chosen elevation in Section 2.3.3, the floor height, and the positioning ofinput and output nozzles in the equipment. Euclidean distances are also of importance due to theexposure radii calculated by the SWeHI, and their subsequent inclusion in the objective function.These distances are found as:

Ecei =√

(xe − xi)2 + (ye − yi)2 ∀e, i ∈ E (27)

2.3.5 Non-overlapping constraints

Non-overlapping prevents positioning of equipment one over another, and it must be checked forpairs of equipment on the same floor. As such, and given that the multi-floor approach works withthe discretization of the elevation of equipment units, overlapping in the z-axis is prevented withoutthe need of any additional constraints. The overlapping, however, can happen in the xy plane, andthe constraints to prevent it are stated as follows:

DXei +BM · (1−Nei) ≥le + li

2 ∀e, i ∈ E (28)

DYei +BM · (1−Nei) ≥de + di

2 ∀e, i ∈ E (29)

where BM is a large number. These disjunctive constraints allow the coordinates of the equipmentunits to satisfy a minimal distance if they are in the same floor and to remain without constrainingin case they are not.

2.3.6 Safety constraints

As stated in Section 2.2, a SWeHI value can be calculated for each equipment. This value willrepresent the radius of the area under a 50% probability of fatality or damage, resulting in a completeloss of whatever is included in the area of exposure. Process parameters listed in Section 2.2 areconsidered as input parameters, although they are not affected by the layout, and for such reason,they are addressed in Figure 4. However, the distance to the nearest hazardous unit and the densityof units are variables that are be defined as:

NHe = mini∈E,i6=e

Ecei ∀e ∈ E (30)

DHe =∑

i∈E|i 6=e,Ecei<30

αi · βi/(π · 302) ∀e ∈ E (31)

These variables change their penalties in the hazard potential presented in Equation 12 (andconsequently, in the B sub-index), although a series of penalties (those referring to the processparameters) remain unchanged. As such, a new hazard potential can be defined as:

He = Ste · pn5e · pn6e ∀e ∈ E (32)

2 BFO approach to the layout design problem 12

where Ste is the non-changing term of Equation 12 (i.e., the multiplication of the terms relatedto penalties 1 to 4), and pn5e and pn6e are found according to De,Ne and Equations 10 and 11explained in Section 2.2. As for the sub-index A, a set of binary decision variables are created tomodel the possibility of choosing safety devices to reduce exposure radii. Also, not all of the devicesare eligible for all of the equipment, so this is modeled by modifying Equation 14 to:

Ae = 0.15 ·∏

h∈H

∑b∈Bh

crb · qbe · vbe ∀e ∈ E (33)

where H is the set of groups, Bh is the subset of devices in group h, crb is the partial credit ofdevice b, qbe is the binary variable that takes the value of 1 if the device b is installed in equipmente, and vbe is the eligibility of device b for equipment e. Credit 8 (equipment reliability) does notchange, because there is no possible choice to modify the probability of failure. After finding bothsub-indices, SWeHI is calculated for each equipment with Equation 15.

Now that exposure radii are calculated, it is necessary to account for equipment loss in case two(or more) equipment are closer to one another. In this work, the loss is accounted for in two ways:equipment loss, which is asset loss occurring when the radius of one equipment compromises otherunits, and area loss, which is the total loss of the area defined by the exposure radius. Equipmentloss can be expressed by the following constraint:

ele = CCe +∑

i∈E|i 6=e

CCi · wei ∀e ∈ E (34)

where ele is the total equipment loss for equipment e, CCe is the capital cost of equipment e, andwei is a binary variable that takes the value of 1 if equipment i is included within the exposure radiusof equipment e. This variable is also useful for defining the total loss of area, defined as follows:

loe = π · SR2e +

∑i∈E|i 6=e

π · SR2e · wei ∀e ∈ E (35)

where SR is the SWeHI radius of equipment e.

2.3.7 Area constraints

Because of the multi-floor approach, each floor has a specific area value. Such value is dependent onthe limits of each floor, which can be defined as the maximum and minimum equipment coordinatesin both axes. This constraint can be seen as:

Arf = (Xf −¯Xf ) · (Yf −

¯Yf ) ∀f ∈ F (36)

where,Xf = max

e∈E|pef =1xe + le

2 ∀f ∈ F (37)

¯Xf = min

e∈E|pef =1xe −

le2 ∀f ∈ F (38)

Yf = maxe∈E|pef =1

ye + de

2 ∀f ∈ F (39)

¯Yf = min

e∈E|pef =1ye −

de

2 ∀f ∈ F (40)

2 BFO approach to the layout design problem 13

2.3.8 Objective function

The main focus of this work is to minimize capital costs associated with piping installation, areausage, device installation, and losses. An expression comprising such terms is presented as theobjective function of the problem:

(41)minCP ·

∑e ∈E

∑i ∈E,|i6=e

cei(DXe +DYe +DZe) + CA

∑f∈F

Arf +∑e∈E

∑h∈H

∑b∈Bh

cdb · qbe · vbe

+∑e∈E

ele + CA ·∑e∈E

loe

where CP is the unitary piping cost, cei is a binary connection parameter that represents thatequipment e and i are connected due to the process, CA is the unitary area cost, and cdb is the costof installing device b.

2.4 Bacterial foraging optimizationInspired in the foraging strategy of E. coli bacteria, the BFO algorithm is a metaheuristic methodthat searches good solutions in randomly generated directions [27]. BFO has been used for problemsinvolving manufacturing layout cells, assembly line balancing [28] and multiple sequence alignment[29], although it has not been used in problems related to layout distributions of chemical facilities.This specific type of metaheuristic was chosen for its flexibility in constraint management, whichallows an straightforward representation of multiple optimization facets, and for its computationalefficiency [30]. The metaheuristic generates and evolves populations of bacteria. Each bacteriarepresents a solution for the optimization problem. Bacteria are selected for reproduction accordingto the value they have in terms of the objective of the problem, which is mathematically expressed asan objective function. The method, which is illustrated in the form of pseudo-code in Algorithm 1,starts with the generation of an initial population of bacteria, each representing a possible solutionfor the specific optimization problem at hand. These solutions start to vary in randomly generateddirections, moving in what Nouri calls a biased random walk [31]. Once a suitable direction has beenfound, bacteria will explore new solutions in this direction either a fixed number of times (calledswimming steps) or until there is no improvement. This exploration process is repeated for a givennumber of times and is known as chemotaxis [32].

Algorithm 1 BFOAGeneration of initial bacterial population θ = {θ1, θ2, . . . , θNb}Evaluation of objective function J(θi), i = 1, . . . , Nb

for t = 1 to Ned dofor u = 1 to Nre do

for v = 1 to Nc dofor l = 1 to Nb doChemotaxis(θl)

end forend forReproduction(θ)

end forEliminationDispersion(θi), i = 1, . . . , Nb

end for=0

2 BFO approach to the layout design problem 14

In the pseudo-code above:

θ : is the bacteria population.

Nb : is the size of the bacteria population.

J(θl) : is the value of the objective function for the individual θl.

Ned : is the maximum number of elimination and dispersion cycles.

Nre : is the maximum number of reproduction cycles.

Nc : is the maximum number of chemotactic cycles.

After the bacteria have finished (swimming), the algorithm progresses to the cycle of bacteriareproduction, which can be seen as multiplication of the best bacteria and elimination of the worstones. The proportion between the number of bacteria that reproduce and the number of bacteriathat are discarded must be an integer number R, so that, at the end of the reproduction cycle, thebest bacteria have reproduced exactly R times [33]. Each time the population reproduces, a newbacterial generation is formed. In this sense, reproduction discards unfavorable solutions in favor ofpromising ones. The pseudo-code of the reproduction step is provided in Algorithm 2.

Algorithm 2 Reproduction

Subset of bacteria with the worst objective function θW = {θ1W , θ2

W , . . . , θSnr

W }Subset of bacteria with the best objective function θB = {θ1

B , θ2B , . . . , θ

Sr

B }Bacterial population θ = {θW ,θB}Ratio counter for replication rep = 1for a = 1 to Snr doθa

W = θa/repB

if a = Sr thenrep = rep+ 1

end ifend for=0

In the pseudo-code above:

θW : is the part of the bacterial population with the worst objective functions.

θB : is part of the bacterial population with the best objective functions.

Snr : is the size of the subset θW , i.e., the number of bacteria that will not reproduce.

Snr : is the size of the subset θB , i.e., the number of bacteria that will reproduce.

rep : is a counter designed for cyclic reproduction of the best bacteria over the worst.

Finally, elimination-dispersal takes place. These events mimic the death of bacteria by externalcauses and the appearance of new bacteria that sample random locations in the solution space. Byeliminating and dispersing, bacteria are forced to avoid local optima, and the search space is widenedbecause of solution movement to unexplored regions. The pseudo-code for elimination-dispersal isprovided in Algorithm 3.

2 BFO approach to the layout design problem 15

Algorithm 3 Elimination-dispersalE = random variate from Unif(0, 1)if E ≥ Ped then

θW = θN

end if=0

In the pseudo-code above:

E : is a random number between 0 and 1.

Ped : is a fixed probability parameter to which E compares for determining if elimination-dispersaloccurs.

θW : is a fixed subset of the worst bacteria in the current population. The size of this subset isuser-defined and is constant throughout the algorithm.

θN : is a fixed set of new bacteria that have either appeared or are moved from their originallocation. Sizes of θW and θN must be equal.

Hierarchically, elimination-dispersal is located in the top of the algorithm. Elimination-dispersalintroduces randomness in the BFOA metaheuristic scheme, as each element of the population isselected for elimination (and subsequent replacement) according to random sampling from a discreteprobability distribution of occurrence; specifically, a Bernoulli distributed variable of parameter Ped

[27]. Finally, an important term must be introduced. Given that chemotactic cycles model bacteriallife, each time one of this cycles finishes, a new generation is born. The total bacterial generationsis then defined by the multiplication of chemotactic, reproduction and elimination-dispersal cycles.

2.5 BFO for the layout design problemIn Section 2.4, the general scheme for the BFO algorithm was presented, although it does not includespecific modifications referring to the solution search for the plant layout problem. As such, thissection describes the tailoring of the general metaheuristic BFO scheme presented in the previoussection to the specific problem of the plant layout design.

The proposed solution approach, shown in Figure 4, requires a previous detailed design stepwhere the following parameters must have been calculated for each equipment unit: mass of releasedchemical, heat of combustion, operating pressure, operating temperature, volume of current unit, firepoint, flash point, autoignition temperature and NFPA rankings NF and NR for the most hazardoussubstance within the unit, chemical quantity involved, vapor pressure of the substances handled, andapplicable safety devices. These parameters were previously mentioned in Section 2.2 for SWeHIcalculation.

With such values, an initialization method is performed to generate the first population of bac-teria (i.e., the first set of candidate solutions). Two kinds of variables are defined: primary decisionvariables, which are independent and not constrained by definition, and intermediate variables, whichare mostly calculated from equality constraints involving primary decision variables and other inter-mediate variables. In the first group, the variables pef , xe, ye and qbe are defined, while the rest ofthe variables found in Section 4 are defined as intermediate variables. Initialization procedures mustdecide about equipment positioning (both in the xy plane and the floor they are on) and the selec-tion of possible safety devices. Compliance with constraints related to primary decision variables ispreferred for the algorithm not to waste unnecessary time trying to restrict these variables. Because

2 BFO approach to the layout design problem 16

Fig. 4: Main elements of the proposed BFO solution scheme

3 Case study 17

of this, a random (yet possible) selection scheme for safety devices is selected. As for equipmentpositioning, the following five possible initialization procedures were proposed and evaluated:

1. U-shaped initialization, which uses initial exposure radii for locating the units along a U-shape,with each unit exposure radius being tangent to the radii of adjacent units;

2. Two-lined positioning, which arranges units along two different lines with a fixed space betweeneach pair;

3. Linear proximity, which positions equipment in a single line according to the logical orderdefined by the process flow;

4. Random allocation, which scatters equipment units across the plane;

5. Same-place arrangement, which puts all process equipment on the same spatial coordinates.

Notice that these initialization procedures might generate layouts that are not feasible, i.e.,that do not respect all the constraints specified in Section 2.3. Also, the last two initializationmethods can generate layouts that are physically meaningless, as units may overlap. Once the primarydecision variables have been defined, all the equality constraints are used to define the intermediatevariables, so that the algorithm can directly calculate several variables without additional variation.Furthermore, this will help the algorithm comply with the vast majority of constraints stated inSection 2.3. However, there are constraints that cannot be fulfilled with the definition of otherdecision variables, for which it is necessary for the algorithm to understand the problem of solutionsthat do not satisfy these restrictions. The original BFO, as shown in Section 2.4, does not takeinto account feasibility when generating possible solutions to the optimization problem. Unfeasiblesolutions must be penalized when their value is assessed by the objective function. Penalization isincluded in this function as the sum of an unfavorable term (highly positive for minimization casesand highly negative for maximization cases). This term is mainly included for the non-overlappingconstraints stated in Section 2.3.5, and it is a function of the overlapping area. In this sense, ifthe algorithm reduces overlapping, the penalty function will be closer to zero. Once the algorithmhas started, it must be remarked that possible solution directions mentioned in Section 2.4 do notchange entirely in every chemotactic step. Specifically, the variables pef and qbe are instructed tochange only in the first cycle, with the intention of allowing the algorithm to explore enough spatialconfigurations in the xy plane before changing equipment floors to reduce piping costs or select safetydevices for radii reduction.

Also, in this BFO implementation, the swimming stage is made adaptive: more alternative layoutsare explored in the first generations, while the algorithm becomes more conservative as the number ofgenerations advances. In this way, the metaheuristic widens the search space in early generations ofthe algorithm and tries to limit excessive and unnecessary variability in further steps. The definitionof a suitable search stopping criterion is another essential aspect of a metaheuristic scheme. Forthe proposed BFO implementation, a minimal threshold for improvement is set between any twoconsecutive reproduction cycles, which if not met, stops the computation. When the stoppingcriterion is met, the BFO algorithm terminates and returns the best solution found for the positionsof the equipment units and the selected safety devices, along with performance measures such ascomputation time and solution quality.

3 Case study

To provide an example of the application of the proposed BFO scheme, the case study proposed byEjeh et al.[20], previously used by [10], is used. The process involves the oxidation of ethylene to

3 Case study 18

obtain ethylene oxide. The process plant must be laid out in two different floors; all the units occupyonly one floor, except for the towers T-101 and T-102. Specific information about equipment sizingcan be found in the aforementioned works, while the rest of the input information (related to processparameters) was obtained from Turton[34]. The process flow diagram of the process is found inFigure 5. The most hazardous substances involved in the process are the ethylene and the ethyleneoxide, due to their flammability and reactivity; they represent the center of hazard evaluation forNFPA rankings and safety parameters used in Section 2.2.1. As for the economic values for theprocess equipment, Patsiatzis, Knight, and Papageorgiou obtained an initial approximation for theyear 2004; escalation had to be done with the Chemical Engineering Plant Cost Index (CEPCI)CECPI to obtain current values.

Installation costs of safety devices is based in the costing performed in Patsiatzis, Knight, andPapageorgiou [10]. Failure frequencies for calculation of Credit 8 are obtained from the CCPSGuidelines for Process Equipment Reliability Data ccpsfailure.

Fig. 5: Process flow diagram for the ethylene oxide production [35]

The algorithm was run with the following parameters: 10 swimming cycles, 8 bacteria, 4 chemo-tactic cycles, 4 reproduction cycles, and 4 elimination-dispersal cycles. Figures presented in thefollowing subsections can be seen in higher resolution in Appendix A.

3.1 InitializationThe whole bacteria population started in the same solution so that results could be compared oncethe algorithm had finished. From the initialization methods presented in Section 2.5, two-linedpositioning was chosen as the best due to the compliance of the initial solution with the non-overlapping constraints summed to the low connection costs it presented. This first solution for bothfloors is presented in Figure ??.

As stated previously, both absorption towers span two floors, while the rest of the equipmentoccupies only one floor. The initialization procedure does not take into account floor positioning,which explains the gap between E-101 and V-101 in the first floor. The performance of this solutionis also shown in Figure ??, where a cost breakdown with percentages of the five costs comprising theobjective function, is presented. The total objective function for this initial solution was $3′048.493, 1.

Because of the low distances between equipment, loss costs and area loss costs are the majorpart of the layout cost. Connection and area costs are highly reduced in comparison to the following

3 Case study 19

(a) first floor (b) second floor

(c) 3D-view (d) cost breakdown

Fig. 6: Initial layout and cost breakdown

3 Case study 20

solution, mainly because of the compactness of the layout. This solution is presented as an unbal-anced one because of the high disproportion between percentages of costs, and it is also expectedto be enhanced in terms of total layout cost.

3.2 BFO resultsAlgorithm solutions vary according to generated directions, which are random, although they tendto reach similar results after a considerable amount of exploration. The solution proposed in thissection, graphically seen in Figure 7 is the best bacteria of the last generation.

The difference between this solution and the one obtained in Figure ?? is the increase of thedistance between equipment to reduce loss costs and area loss costs. Specifically, the algorithmreduces this last cost from 61 % to 38.2%. The loss costs were also reduced, keeping the samepercentage as the initial solution; this reduction can be seen in the total value of the function: theobjective function for this solution was $781.964, 1, resulting in a 74,34 % reduction because of thealgorithm exploration. As expected, area and connection costs rose due to the higher distancingbetween equipment, and the safety devices’ cost decreased in order to avoid credits which werenot necessary for the function to reduce. Higher distances, because of the dynamic nature of theexposure radii, resulted in lower area loss costs.

3.3 Comparison with literature resultsThe multi-floor solution obtained by Ejeh et al.[20] was replicated and compared with the obtainedsolution in this work. Their proposed solution was evaluated with the objective function stated inSection 2.3.8, while the safety devices’ selection was assumed as the one presented in Section 3.2.The difference between layouts is the compactness and closeness of equipment units, as can be seenin Figures 18, 19 and 20.

Initialization BFO solution Ejeh et al.[20]Area costs 53.663,3 115.911 36.229Safety devices’ costs 71.000 64.000 64.000Loss costs 1.051.450 272.400 1.309.250Piping costs 12.732 28.371 12.704Area loss costs 1.859.648 301.282 1.013.646Total 3.048.493,3 781.964 2.435.829

Tab. 4: Cost comparison between different solutions

Each of the presented costs are compared in Table 4. The total layout cost for the solution is$2′435.829, 1.The BFO solution presented in Section 3.2 obtained a 67.89 % reduction over thisvalue, due to the reduction of both loss costs and area loss costs. The distribution of costs can becompared in Figures 6 and 7.

The formulation presented by Ejeh et al. does not consider area costs within the objective function,but rather constraints the area to a singular value with specified x and y coordinates as limitsfor the layout. Also, the current work includes the safety constraints stated in Section 2.3.6, adifference that impacts the value of the objective function. Because of this, the layout is verysimilar to the initialization solution proposed in Figure 6. Furthermore, the focus of the objectivefunction presented by Ejeh et al. is operation-based, a significant difference with the capital cost-based approach presented in this work.

3 Case study 21

(a) first floor (b) second floor

(c) 3D-view (d) cost breakdown

Fig. 7: BFO layout and cost breakdown

3.4 Algorithm performanceFigure 8 shows the evolution of the objective function of the best bacteria within the generations. Abig leap is performed in the first stages of the algorithm, although this reduction is less pronouncedas generations increase. As explained in Section 2.5, step length is adaptive and changes betweengenerations to avoid unnecessary variability in higher generations. The stopping criteria stated inthe same section is not fulfilled, although it is clear that solution quality has not been enhanced in

4 Conclusions 22

the last cycles, prompting the algorithm to stop searching.

Fig. 8: Evolution of the best objective function within the generations

Fig. 9: Evolution of the best and worst objective function within the generations

Figure 9 shows a comparison of the best bacterial function with the worst. Maximum valuesrise as the algorithm performs the elimination-dispersal cycle, resulting in under-performing solutions(compared to the ones the algorithm had before). Although it is highly probable that these solutionswere further discarded in the reproduction cycles, the algorithm is verifying that there are not solutionsin other parts of the solution space that may be better than the solutions brought by the generalexploration. The similarity of maximum and minimum solutions in specific parts of the explorationshows how the worst solutions tend to get discarded in favor of promising ones, and the positiveeffect of such method to drive the algorithm to better solutions.

4 Conclusions

The approach presented in this work proposes a layout design problem formulation and solves it witha BFO algorithm. Safety considerations are represented by the Safety Weighted Hazard Index, aconvenient tool for hazard assessing that takes into account layout features such as distances between

4 Conclusions 23

equipment and unit density to find exposure radius for each equipment. By definition, this tool alsopresents a fixed probability of fatality or damage of 50%, meaning that the loss costs associated tothe layout take into account both the possibility and the consequence of an unwanted event. A casestudy is solved to demonstrate the usefulness of the metaheuristic for this kind of problems, showingthe improvement of the obtained solutions over the run time of the algorithm, and comparing solutionquality with other approaches that worked with the same case study. Domino effect considerationsare not directly constrained into the problem, yet the results presented in Section 3 confirm thatpropagation of critical events is reduced considerably— as the algorithm advances. In this sense,the approach provides a starting point for inherently safer designs based on simple techniques thatseek to produce robust layouts without a considerable amount of computational effort. The goal ofthese computer-generated layouts is to strengthen expert criteria in the decision-making process ofthe first stages of plant layout. BFO characteristics, such as flexibility in constraint handling andexploration based on initial solutions, allow applications for other types of cases. Further research interms of specific domino-effect functions for different types of fires and explosions could be beneficialto the solution quality.

Acknowledgments

The author would like to thank professor Felipe Munoz Giraldo and the Department of ChemicalEngineering of Universidad de los Andes for all the support in the development of this work. Also,thanks to professor Faisal Khan for the constant aid and help with the current master thesis.

Nomenclature

SetsE Set of equipment units.F Set of available floors.H Set of groups of safety devices.Bh Subset of devices in group h ∈ H.

ParametersMe Number of consecutive floors that equipment e ∈ E must occupy.αe Given length of equipment e ∈ E.βe Given width of equipment e ∈ E.FH Floor height.OPei Relative height of output nozzle for piping going from equipment e ∈ E to

equipment i ∈ E|e 6= i.IPei Relative height of input nozzle for piping going from equipment e ∈ E to equip-

ment i ∈ E|e 6= i.BM Large number used for disjunctive constraints.Ste Static term for calculation of hazard potential in equipment e ∈ E.vbe Partial credit for device b ∈ Bh, h ∈ H.vbe Binary parameter for eligibility of device b ∈ Bh, h ∈ H in equipment e ∈ E.CCe Capital cost of equipment e ∈ E.CP Unitary cost of piping.cei Binary parameter for process connectivity between equipment e ∈ E and i ∈

E|i 6= e.

REFERENCES 24

CA Unitary cost of area.cdb Cost of installing device b ∈ Bh, h ∈ H.

Variablespef Binary variable that takes the value of 1 if equipment e ∈ E belongs to the floor

f ∈ F .nei Binary variable that takes the value of 1 if equipment e ∈ E and i ∈ E|i 6= e

are on the same floor.Wf Binary variable that takes the value of 1 if floor f ∈ F exists.le Variable for the length of equipment e ∈ E.de Variable for the width of equipment e ∈ E.le Binary variable that defines the orientation of equipment e ∈ E.Ss

ef Binary variable that takes the value of 1 if equipment e ∈ E starts in the floorf ∈ F .

Sfef Binary variable that takes the value of 1 if equipment e ∈ E ends in the floor

f ∈ F .xe Position in the x coordinate for the center of equipment e ∈ E.ye Position in the y coordinate for the center of equipment e ∈ E.DXei Rectilinear distance in the x-axis between centers of equipment e ∈ E and

i ∈ E|i 6= e.DYei Rectilinear distance in the y-axis between centers of equipment e ∈ E and

i ∈ E|i 6= e.DZei Rectilinear distance in the z-axis between nozzles of equipment e ∈ E and

i ∈ E|i 6= e.Ecei Euclidean distance in the xy plane between centers of equipment e ∈ E and

i ∈ E|i 6= e.NHei Distance to the nearest hazardous unit for equipment e ∈ E.DHei Density of units in a 30 m radius for equipment e ∈ E.He Hazard potential for equipment e ∈ E.Ae Sub-index A for equipment e ∈ E.qbe Binary variable that determines if device b ∈ Bh, h ∈ H is installed in equipment

e ∈ E.ele Total equipment loss involving equipment e ∈ E.wei Binary variable that takes the value of 1 if the radius of equipment e ∈ E includes

equipment i ∈ E|i 6= e.loe Total area loss involving equipment e ∈ E.SRe Calculated SWeHI radius for equipment e ∈ E.Arf Area of floor f ∈ F .Xf Highest x coordinate of occupied equipment in floor f ∈ F .Xf Lowest x coordinate of occupied equipment in floor f ∈ F .Yf Highest y coordinate of occupied equipment in floor f ∈ F .Yf Lowest y coordinate of occupied equipment in floor f ∈ F .

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Appendix A: Results

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Fig. 10: Layout of the first floor for the initial solution

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Fig. 11: Layout of the second floor for the initial solution

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Fig. 12: 3D layout for the initial solution

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Fig. 13: Cost breakdown for the initial solution

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Fig. 14: Layout of the first floor for the BFO solution

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Fig. 15: Layout of the second floor for the BFO solution

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Fig. 16: 3D layout for the BFO solution

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Fig. 17: Cost breakdown for the BFO solution

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Fig. 18: Layout of the first floor for the solution presented by Ejeh et al.[20]

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Fig. 19: Layout of the second floor for the solution presented by Ejeh et al.[20]

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Fig. 20: 3D layout for the solution presented by Ejeh et al.[20]

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Fig. 21: Cost breakdown for the solution presented by Ejeh et al.[20]