Loop-based facility layout design under flexible bay structures

Accepted Manuscript Loop-based facility layout design under flexible bay structures Ardavan Asef-Vaziri, Hossein Jahandideh, Mohammad Modarres PII:

S0925-5273(17)30246-3

DOI:

10.1016/j.ijpe.2017.08.004

Reference:

PROECO 6785

To appear in:

International Journal of Production Economics

Received Date: 23 March 2016 Revised Date:

14 May 2017

Accepted Date: 3 August 2017

Please cite this article as: Asef-Vaziri, A., Jahandideh, H., Modarres, M., Loop-based facility layout design under flexible bay structures, International Journal of Production Economics (2017), doi: 10.1016/ j.ijpe.2017.08.004. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Loop-Based Facility Layout Design Under Flexible Bay Structures

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Ardavan Asef-Vaziri Systems and Operations Management College of Business and Economics California State University, Northridge [email protected]

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Hossein Jahandideh Anderson School of Management University of California, Los Angeles

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and Mohammad Modarres Industrial Engineering Sharif University of Technology, Tehran, Iran

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Abstract

We develop heuristic procedures for concurrent design of block layout, material flow network, and input and output stations. Unlike the common practice in the flexible bay structure layout design research, our objective function includes both loaded and empty flow between

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pairs of I/O stations. This measure of effectiveness is the main determinant of the fleet size of the vehicles, which in turn is the major driver of the total investment and operational costs in

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vehicle-based material handling. Our work is the first to design flexible bay structure layouts for loop-based material flow patterns. We first develop heuristic algorithms to configure the layout on the foundations of a unidirectional loop flow pattern. The loop and stations are then designed on the promising layouts. A set of improvement algorithms provide a better integration of the layout, loop, and stations.

Key Words and Phrases : flexible bay structures (FBS), loop based facility layout, vehicle based material handling, generalized traveling salesman problem (GTSP).

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Introduction

Since 1955, approximately 8% of the U.S. gross national product has been spent on new facilities. It is reasonable to assume that, in the U.S., over $300 billion will be spent annually on the planning and re-planning of facilities. Around 1/3 of the total operating expenses within manufacturing is attributed to material flow. It is generally agreed that effective

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facility layout and material flow network design can reduce these costs by about 20% (adopted from Tompkins et al.).

A block layout is a graphical representation of a set of not necessarily convex rectilinear polygons each defining the size, shape, and orientation of a workcenter. The flexible bay

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structure (FBS) restricts the shape of the polygons to rectangles arranged in parallel equilength bays that have flexible widths.

In the majority of the block layout design models, distances are measured between the

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centroids of the workcenters, on arbitrary free-flow rectilinear paths, which are also bidirectional. These assumptions may work particularly well for gantry cranes, but are far from reality for vehicle-based material handling. Since Montreuil (1990), the facility planning community has observed a trend towards integrating the block layout with other components of the material flow network. The four fundamental and interrelated components of the block layout and material flow network design are: (i) the size and shape of the work-

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centers and that of the block layout, (ii) the location and orientation of the workcenters in the block layout, (iii) a fully-connected material flow network on the boundary-lines defined as the boundaries of the workcenters, and (iv) the number and location of the input (I) and output (O) stations on the network (adopted from Kim and Goetschalckx, 2005).

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A loop flow pattern was recognized among the well-known flow patterns implemented in production systems by Apple (1977). Originating from the seminal work of Bartholdi and Platzman (1989), recently, there has been a surge in the research on loop-based material flow

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systems (Montoya-Torres et al., 2006 and De Koster and Yu, 2008). Ventura and Lee (2001 and 2003) and Ventura and Rieksts (2009) consider a circular flow path as a promising guide path for dispatching a fleet of automatic material transport vehicles. Nazzal and McGinnis (2007) and Zhang et al. (2016) consider the loop as the flow-path for automated material transport systems in semiconductor wafer fabrication industry. This accrued interest is paralleled in industry, as all the vehicle flow paths at Daifuku training facilities in Salt Lake City are loop-based (Asef-Vaziri and Laporte, 2009). This trend may have roots in the continual increase witnessed in the development of one-way streets, as well as the loop-based routes in public transportation. 1

ACCEPTED MANUSCRIPT Compared to the general unidirectional conventional configuration formed by all the boundary-lines in a block layout (Maxwell and Muckstadt 1982), a unidirectional loop covering at least one boundary-line of each workcenter occupies a smaller portion of the shop floor (and perhaps requires lower initial investment and operating costs). It also simplifies traffic management (there is no intersection), vehicle routing (there is only one route be-

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tween pairs of stations), and dispatching (a simple decentralized dispatching policy can be effectively implemented).

While the first paper on vehicle-based material handling (Maxwell and Muckstadt, 1982) addresses the empty vehicle flow by solving an empty vehicle transportation problem, until recently, the objective function of most of the facility layout models included only the loaded

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flow, and importance of the empty flow was overlooked. Loaded and empty flows are the main determinants of the fleet size of the vehicles. This in turn, is the major driver of

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the total investment and operational costs in vehicle-based material handling systems. The importance of the empty flow in the block layout design was demonstrated in Benjaafar (2002).

In the majority of the block layout models, the I/O stations are assumed to be located at the centroid of the workcenters. In a few models, a more realistic alternative is implemented where the stations are located on the nodes at the intersection of the boundary-lines of the workcenters. Goetschalckx and Palliyil (1994) proposed a set of intermediate nodes along the

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boundary-lines connecting the intersection nodes. According to Asef-Vaziri et al. (2007), the intermediate nodes are more flexible and lead to less congestion compared to the intersection nodes.

The block layout of this study will be developed under flexible bay structures (FBS) on

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the foundations of unidirectional loop flow pattern, where the not necessarily combined I/O stations are located on the intermediate nodes. The elements of the flow matrix correspond

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to the volume of the loaded flow between pairs of workcenters over a specific time horizon. The objective function includes both loaded and empty flow. The volumes of the empty flow are computed inside the model based on the design of the loop and sequencing of the stations. The distances are measured along the unidirectional loop from the O to I stations for loaded flow and from I to O stations for empty flow. Unit-load automatic vehicles perform the material transport tasks between pairs of workcenters. The remainder of this study is organized as follows. A literature review along with our contributions are provided in the next section. In section 3, we present a procedure for choosing the dimensions of the layout (if they are not predetermined by the available land)

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ACCEPTED MANUSCRIPT and the number of bays. We present an algorithm which generates random layouts that satisfy the aspect ratio constraint in Section 4. A genetic algorithm to design FBS layouts for loop-based material flow along with the loop and I/O stations is developed in Section 5. A loop-layout-station improvement procedure is presented in Section 6. Computational

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experiments are reported in Section 7. Conclusions follow in Section 8.

Literature Review

In the introduction section we referred to some literature to set the stage for our contributions. In this section we review additional literature on the FBS layout design, loop-based

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station sequencing, and loop and station design. Following crediting the previous works that are related to this study, we differentiate our contributions.

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Block Layout Design. Block layout design algorithms can be partitioned into discrete and continuous area allocation models. In a discrete model, an area-grid such as a 10 by 10 feet module is defined. Each workcenter is represented by a set of grids that are each connected by at least one side to others forming a not necessarily convex polygon. In continuous area allocation models, the constraint of the workcenters’ dimensions being an integer multiplier of the grids’ dimensions is relaxed. CRAFT (Buffa et al., 1964) and BLOCKPLAN (Donaghey, 1987) are early examples of discrete and continuous representations, respectively. There is a

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vast number of heuristics for the block layout design (see Heragu, 2008 and Tompkins et al., 2010). Meller et al. (1998), Sherali et al. (2003), and Castillo and Westerlund (2005) have developed optimization models for continuous area allocation. In the objective function of almost all the discrete and continuous area allocation optimization and heuristic models, flow

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includes only loaded flow, and distance is measured as a rectilinear bi-directional free-flow between the centroids of the workcenters.

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The FBS (Donaghey, 1987) and the slicing structure (Tam, 1992) restrict the shape of the workcenters to rectangles, and the arrangements of the rectangles to specific configurations. A block layout has a slicing structure if the workcenters are formed by adding a horizontal or vertical line to one of the existing rectangles in the block layout, thereby dividing the rectangle into two. The addition of horizontal and vertical lines continues until all the workcenters are formed. In an FBS layout, the rectangular workcenters are aligned in parallel horizontal or vertical bays. All the workcenters in the same bay have the same length in the direction perpendicular to the bays’ direction. Given the layout dimensions and required areas for the workcenters, an FBS layout is uniquely defined by a string S determining the number of bays and the order of workcenters inside the bays. For example, Figure 1 shows an 3

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Figure 1: Graphical representation of the string S = [1, 2, 3|4, 5, 6, 7, 8|9, 10]. FBS with three bays and a representative string S = [1, 2, 3|4, 5, 6, 7, 8|9, 10]. The translation

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of strings to layouts is discussed in further detail in Section 4.

The FBS was first introduced by Donaghey (1987), and its software (Donaghey and Pire,

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1990) was distributed as BLOCKPLAN90 for a price of $25 to support undergraduate student gatherings at the University of Houston. One may, however, find traces of the FBS layout in Micro-CRAFT (Hosni et al., 1980), where workcenters are arranged in bands having the same concept as bays. Nevertheless, Micro-CRAFT (i) had a discrete area allocation representation, (ii) the dimensions of all bands were the same, (iii) the workcenters were pseudo-rectangular, e.g., the sides perpendicular to the directions of the bands were not

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necessarily straight lines and could contain a partial row of grids, and (iv) the last workcenter in a band could overflow into the next band and form an L-shaped workcenter. Tong (1991), Goetschalckx (1992), Meller and Bozer (1996), Kochhar et al. (1998), Gau and Meller (1999), Kulturel-Konak et al. (2004), Tate and Smith (1995), Eklund et al. (2006), Kulturel-Konak et al. (2007), Scholz et al. (2009), Komarudin and Wong (2010), Wong

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and Komarudin (2010), Kulturel-Konak and Konak (2011a and 2011b), Ghaseminejad et al. (2011), Hernandez et al. (2011), Chang and Hsin (2012), Ulutas and Kulturel-Konak

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(2012), and Kulturel-Konak (2012) have developed genetic algorithm, simulated annealing, tabu search, and ant colony system heuristics for FBS layouts. The objective function of all these models is restricted to the loaded flow over rectilinear distances between the centroids of the workcenters. Norman et al. (2001) consider loaded flow between I/O stations along the boundary-lines.

The main challenging part in FBS layout design is satisfaction of the aspect-ratio constraints (AR-constraints), where the ratio of the long-side to the short-side of the workcenters do not exceed a predetermined threshold value. Most of the FBS layout design algorithms fail to handle the AR-constraints in a way that results in a compact rectangle shape for

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ACCEPTED MANUSCRIPT the block layout. In other words, the area of the rectangle enveloping all the workcenters is greater than the total required area. To adjust the final shape of the block layout, Tate and Smith (1995) and Kulturel-Konak and Konak (2011a and 2011b) assume dummy workcenters, Konak et al. (2006) allocate the empty space to one side of the block layout, and Wong and Komarudin (2010) and Kulturel-Konak and Konak (2010) allocate the empty space of

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each bay to the workcenters in that bay. Loop-Based Station Sequencing. Afentakis (1989) considers the problem of sequencing a set of dimensionless nodes around and outside of a unidirectional loop. He provides an exact formulation but solves the problem heuristically using an interchange procedure. In their

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seminal work, Kouvelis and Kim (1992) proved that the problem is NP-hard, and developed dominance relations to reduce the solution space. They provided a decomposition-based heuristic procedure and a branch-and-bound algorithm capable of solving small problems.

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The objective is to minimize the number of rounds the loop is traversed by unit-load vehicles to satisfy the material flow requirements. This is a novel objective function since it also includes the empty segment of the rotations around the loop. Kiran and Karabati (1993) have a quadratic assignment formulation for this problem. They proposed dominance relations, identified a polynomially solvable case, and developed approximate solution procedures for under 12-workcenter problems. Kaku and Rachamadugu (1992), Leung (1992), Cheng et al.

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(1996), Cheng and Gen (1998), Tansel and Bilen (1998), Potts and Whitehead (2001), Lee ¨ et al. (2001), Bennell et al. (2002), Malakooti (2004), Altinel and Oncan (2005), Nearchou ¨ (2006), Kumar et al. (2008), Oncan and Altinel (2008), Panahi et al. (2008), Kumar et al. (2009), and Zheng and Teng (2010) have developed solution procedures for the problem.

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In each of these models the workcenters are dimensionless, therefore, the outcome is not a block layout. In addition, the length of the loop cannot be computed since it will depend on the size, shape, orientation, and arrangement of the workcenters. In some models, instead of

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dimensionless workcenters, it is assumed that a set of equally-sized locations are positioned around a fixed loop. The objective of the model is to assign equally-sized workcenters to the equally-sized locations in order to minimize the number of rotations around the loop. From a modeling point of view, as long as the loop is fixed, the equal-area workcenter loop-based problem is reduced to a station sequencing problem. There exists one additional setting staying in between station sequencing and loop-based block layout design. That is when the rectangular workcenters can be located only outside the loop. In Banerjee and Zhou (1995), the objective function is the same as that of the traveling salesman problem. In Chae and Peters (2006), the objective function is to minimize 5

ACCEPTED MANUSCRIPT the loaded flow. While the combined I/O stations are positioned on a rectangular flow path, the distances are still measured in rectilinear form. Loop and Station Design. Tanchoco and Sinreich (1992) were the first to conceptualize the problem of designing a unidirectional loop to cover the workcenters of a block layout and to locate the I/O stations. Sinreich and Tanchoco (1993), Asef-Vaziri et al. (2001), Farahani

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et al. (2007a and b), and Caricato et al. (2007) have proposed optimization and heuristic models for this problem. The flow in the objective function of all these models includes only the loaded flow.

Asef-Vaziri et al. (2008), through computational experiments, showed that ignoring of

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empty flow leads to designs that are far from optimal when both loaded and empty flow are taken into account. Asef-Vaziri and Laporte (2009) demonstrated that a design based on shortest-trip-distance-first compared to a design based on first-come-first-served, leads to a

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better solution when it is implemented in the operations phase. Asef-Vaziri and Ortiz (2008) showed that the shortest loop on a block layout can be used as an effective heuristic scheme to achieve prosperous and robust solutions for the concurrent design of the loop and I/O stations under the total loaded and empty flow objective function. They develop alternative optimization formulations for the block layout shortest loop design problem. Eshghi and Kazemi (2006) provide a heuristic procedure.

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Contributions of Our Work. This paper has the following distinctions with the available literature.

(i) This paper is the first to design FBS layouts for unidirectional loop flow patterns.

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(ii) Unlike the majority of the block layout design models, the objective function of our model includes both loaded and empty flow. The distances are measured along the

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unidirectional loop from the O to I stations for loaded flow, and from I to O stations for empty flow.

(iii) In conformance with the literature and practice, the total facility size is pre-determined and is equal to the sum of the areas required by all the workcenters. Unlike the majority of the FBS layout design algorithms, which do not end up with a compact block layout, where the workcenters are fully packed in an external rectangle, our solutions are always compact- provided that a feasible solution exists. (iv) In most of the FBS layout algorithms, the dimensions of the block layout (length and width) are parameters. While our model can accept the dimensions as input 6

ACCEPTED MANUSCRIPT parameters, it is the first to offer the option of replacing these parameters by decision variables. This flexibility enables the user to expand the search space, in order to find the dimensions that are the best fit for the arrangement of the bays and alignment of the workcenters. This is particularly useful where the facility designers have the option of customizing a portion of a larger land as the site of the facility. For instance, in

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an Industrial/Logistics park, where the facility planners may have some flexibility to define the desired dimensions of appropriate parcel of land.

(v) We develop an algorithm that can check for availability of a feasible solution under the facility parameters and an AR-constraint. We then develop an algorithm to pro-

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duce random aspect-ratio feasible (AR-feasible) layouts. Tight AR-constraints often waste many generations of the available FBS evolutionary algorithms to reach a feasible solution. Our algorithms have solved the challenge of generating feasible layouts

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under tight AR-constraints. Our AR-feasible layout generation algorithm can produce promising feasible solutions for all the FBS-layout design problems in the literature (if a feasible solution exists).

(vi) Evolutionary algorithms generally require evaluating a large number of potential layouts. Hence, each layout is constructed based on a simple objective function requiring minimal computations. Given the complexity of our objective function, to acceler-

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ate the efficiency of the GA, we focus on simplifications to provide objective function approximations. We encode and decode the strings of GA chromosomes based on an intuition about how a loop may cover the workcenters, and then, identify surrogates for the optimal loop. While objective function approximations are developed to construct

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an initial layout, our improvement algorithms take the layout and design the loop and

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the layout under the the true objective function values.

Model Specifications Layout Characteristics

The layout we design must allocate a required area Ac to each workcenter c. We assume that the size of the facility is pre-determined and is equal to the sum of the Ac ’s. In most of the FBS layout design models, the dimensions of the block layout are parameters and the number of the bays is a variable to be determined inside the model. Our model can accept the dimensions both as input parameters or as decision variables. In addition, we do not treat the number of the bays as a decision variable but as a parameter. By enumerating over the 7

ACCEPTED MANUSCRIPT prosperous range of this parameter, we will reduce the computation times by magnitudes. We present an algorithm that simultaneously identifies the building dimensions and the number of bays (by enumerating over the prosperous range of this parameter). If the building dimensions are predetermined, the algorithm will only choose the number of bays.

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Notations : The set of the workcenters.

Ac : The area of workcenter c. A : The list of all Ac values. XYc

: A decision variable representing the ratio of the length of workcenters c on the X axis, to its height on the Y axis in a given layout.

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α : A parameter defining the AR-constraints; 1/α ≤ XYc ≤ α for all workcenter c on a feasible layout. : The number of bays in the iterations over the range of the prosperous number of bays. : The ratio of the X-side to the Y-side of the facility (could be a decision variable or a parameter).

Sb S

: The list of the workcenters assigned to bay b, ordered from left to right as they appear in the bay. : A string [S1 |S2 |...|SB ] showing the list of workcenters in the order they appear on the layout.

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B β

B − 1 separators in the string identify the bays and the workcenters assigned to them. The bays are horizontal and are ordered from top to bottom.

Note that given β and A, the string S uniquely defines the facility layout and dimensions of

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all bays and workcenters. (If the bays are aligned vertically, the terms height and width are switched in the following statements.) The width of all bays are equal in an FBS layout. The height of each bay will be proportional to the sum of the areas of the workcenters assigned to the bay. The height of all workcenters are the same in a bay, where the relative width

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of each workcenter is proportional to its area. For example, when all the workcenters are of the same size (Ac s are equal), the aspect-ratio of the building β = 1.75, and bays are

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horizontal, the string S = [1, 2, 3|4, 5, 6, 7, 8|9, 10] translates to the layout of Figure 1. In all of our algorithms (except for the GA) we operate on these strings and compute the implied workcenter dimensions.

A feasible loop is a loop along the boundaries of workcenters that has a common boundary with each and every workcenter, so that the material can be transported to/from each workcenter. I/O stations must be assigned to each workcenter along the boundaries they share with the loop.

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3.2

Objective Function

We now define the objective function for the layout/loop/I/O design problem. A matrix F defines the intensity of loaded flow between each pair of workcenters. The total material handling cost is proportional to the total loaded and empty flow required along the loop to transport material between pair of workcenters. The elements fij ∈ F are the number of

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loaded trips in a defined time horizon (an hour, a day, etc.) from workcenter i to workcenter j. Since the loop is unidirectional, any loaded trip on the busiest arcs will generate empty trips along some other arcs. Asef-Vaziri and Laporte (2009) show that in a deterministic optimal solution for a unidirectional loop design problem, there will be no empty flow on

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the arcs with the maximal loaded flow. The total loaded and empty flow may therefore be computed by identifying the arc with the highest intensity of loaded flow, and multiplying this flow (θ) by the length of the loop (L). Note that if an optimization or a heuristic

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procedure co-locates the O station of workcenter i and the I station of workcenter j, the flow fij is not added to the flow on any arc. It will be crossed out and does not affect the material transport cost. When the layout and loop are identified, the I/O stations are sequenced in order to minimize the intensity of the flow along the busiest arc. The objective function is then computed as (θ × L). By benefiting from Bartholdi and Platzman (1989) findings, Asef-Vaziri and Laporte (2009) show that the total stochastic loaded and empty flow in the

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operation phase is bounded up by 2θL. Therefore, minimization of the product of θ and L in a deterministic model in the design phase, also minimizes the total stochastic loaded and flow in the operation phase.

A directed loop can be uniquely defined by a sequence of arcs, and each arc on a directed

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loop can be identified by a single node. Without loss of generality, let the starting node of the loop be at the northwest corner of the layout. The directed-loop can be represented by a sequence of parameters enj , where enj = 1 if arc n is on the edge of workcenter j, and enj = 0

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otherwise. Note that each arc is shared by two workcenters. Let ℓ denote a loop, represented by a sequence of enj s, and L(ℓ) denote the length of this loop. The objective function of a loop ℓ is equal to L(ℓ)θ(ℓ), where θ(ℓ) is given by an optimization model locating the I/O stations to minimize the total loaded and empty flow. We define the following binary variables:

pnj = 1 if the output station of workcenter j is located on arc n, and 0 otherwise. dnj = 1 if the input station of workcenter j is located on arc n, and 0 otherwise. tnij = 1 if fij passes through arc n, and 0 otherwise.

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ACCEPTED MANUSCRIPT The optimization problem to determine θ(ℓ) is then formulated as (1a)

N X

pnj = 1, ∀ j

(1b)

dnj = 1, ∀ j

(1c)

n=1 N X n=1

pnj ≤ enj , ∀ n, j dnj ≤ enj , ∀ n, j n′ dn′ i + 1)(

N X

n′ pn′ j )(n −

N X

n′ pn′ j ) ≤ tnij U, ∀ n, i, j

(1d) (1e) (1f)

n′ =1

n′ =1

tnij fij ≤ θ ∀ n.

j

(1g)

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i

n′ dn′ i −

n′ =1

n′ =1

XX

N X

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(n −

N X

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s.t.

θ(ℓ) = min θ

Constraints (1b) and (1c) ensure that each workcenter has exactly one O station and one I station, respectively. Constraints (1d) and (1e) ensure that the I/O stations of each workcenter are chosen on its neighboring arcs. Constraint (1f) determines tnij by checking whether arc n is on the path between the O station of workcenter i and the I station of workcenter j. The fixed parameter U is chosen to be sufficiently large. This non-linear constraint can easily be partitioned into a series of linear constraints using standard binary programming tech-

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niques. Constraint (1g) sets θ to the total flow intensity on the arc with the largest loaded flow. The function θ(ℓ) can be computed using the algorithms presented in Asef-Vaziri and Ortiz (2008), and Asef-Vaziri and Laporte (2009). Let L(ℓ)θ(ℓ) denote the objective function of a layout/loop (a feasible loop in a given

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layout), and minℓ∈L L(ℓ)θ(ℓ) denote the objective function of an FBS layout, where L is the set of feasible loops in that layout. An approximation to minℓ∈L L(ℓ)θ(ℓ) is given in

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Section 5 to evaluate whether a layout potentially contains a near-optimal solution. For this approximation, L(ℓ∗ ) and θ(ℓ∗ ) are estimated separately without finding the optimal loop ℓ∗ . Similarly, an approximation to L(ℓ)θ(ℓ) is given in Section 6 which estimates θ(ℓ) without locating the I/O stations, but uses the exact value of L(ℓ). It is worth noting several observations. First, according to Asef-Vaziri and Ortiz (2008) shortest loop provides a promising approximation for the optimal loop, minimizing the total loaded and empty flow objective function. Second, we will show that the larger interior loop is a promising approximation for the shortest loop, i.e., it is a promising approximation for the optimal loop. Third, when the loop is known, the optimal location of the I/O stations minimizing the total loaded and empty flow objective function is reduced to a sequencing problem. Finally, these 10

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4

Feasible Layouts Algorithms

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The algorithms described in this section are independent of the flow matrix and material handling system, hence they are applicable to all FBS problems with other cost specifications. We first present Algorithm 1 to test for at least one feasible solution for a given β, α, and B, by returning a binary true/false value. We then implement Algorithm 1 iteratively in

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Algorithm 2 to find the β and B where the tightest value of α still contains at least one feasible solution. Algorithm 3 generates random AR-feasible solutions.

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Basic Intuitions and Simplifications. To obtain a feasible layout for a given [β, B, α], the intuition behind our algorithm relies on the assumption that FBS layouts have higher chances of feasibility if the workcenters in a bay do not significantly vary in size. For example, for the tightest aspect-ratio of 1, if there are 4 workcenters each of size of 9, six workcenters each of size of 4, and twelve workcenters each of size of 1, the only feasible solution satisfying the aspect-ratio of 1 is to place all workcenters of the same size in the same bay. That is 3 bays in a 12 by 6 block layout. Accordingly, our algorithm builds the string S by sorting

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the workcenters in a non-increasing order, and slides the separators around in search for a feasible solution (if it exists). While it is not difficult to show that the intuition does not always work, it was effective in our experimentation. For instance, the tightest α for a 4-bay layout for the 20-workcenter example of Armour and Buffa (1963) is 1.7046, and for a 3-bay

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layout for the 10-workcenter example of Van Camp et al. (1991) is 2.534. Our algorithm obtains a feasible layout even under such tight aspect-ratios.

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Given [β, B, α, Ac], Algorithm 1 searches for a feasible layout satisfying these parameters. Initially, we arrange the workcenters in non-increasing order of their size to form the string defining the layout. Next, we try to locate B − 1 separators to form B bays where 1/α ≤ XYc ≤ α ∀c ∈ C.

For the purpose of this algorithm and the next two algorithms, we define workcenter feasibility indices (W F Ic ∀c ∈ C) and bay feasibility indices (BF Ib : b = 1 to B).   if XYc > α 1 W F Ic = 0 if 1/α ≤ XYc ≤ α   −1 if XYc < 1/α. 11

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ACCEPTED MANUSCRIPT  1    0 BF Ib =  −1    2

if if if if

W F Ic = 0 or 1 ∀ c ∈ Sb , & ∃c ∈ Sb s.t. W F Ic = 1 W F I c = 0 ∀ c ∈ Sb W F Ic = 0 or − 1 ∀ c ∈ Sb , & ∃c ∈ Sb s.t. W F Ic = −1 ∃c ∈ Sb s.t. W F Ic = 1, & ∃c ∈ Sb s.t. W F Ic = −1.

(3)

1. For c : 1 → |C| and b : 1 → B. Set W F Ic = BF Ib = 0.

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Algorithm 1 Feasibility of a Given α and B 2. Build the underlying string of workcenters by sorting them in non-increasing order of Ac values. Assign approximately the same number of workcenters |C|/B to each bay b : 1 → B. 3. For b : 1 → B.

(a) For all workcenters c in bay b. If XYc > α, set W F Ic = 1. If XYc < 1/α, set W F Ic = −1.

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(b) If there is at least one W F Ic = 1 and no W F Ic = −1 in bay b, set BF Ib = 1. If there is at least one W F Ic = −1, and no W F Ic = 1, set BF Ib = −1. If there are both W F Ic = ±1, set BF Ib = 2.

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4. For b : 1 → B − 1.

(a) If BF Ib = 1, take workcenters from the left-side of the next bay and place them on the right-side of the current bay (i.e. move the bth separator, one workcenter to the right) until BF Ib 6= 1. If BF Ib ≤ 0, go to (4), otherwise, go to (4c). (b) If BF Ib = −1, take workcenters from the right-side of the current bay and place them on the left-side of the next bay until BF Ib 6= −1. If BF Ib < 2, go to (4); otherwise, go to (4c).

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(c) If b = 1 and BF I1 = 2, there is no AR-feasible layout under the current set of α, β, and B, stop. Otherwise, move one workcenter from the left-side of the next bay to the right-side of the current bay, and move one workcenter from the left-side of the current bay to the right-side of the previous bay. Repeat, until the index of the current bay is changed. i. BF Ib = 1: Go to (4a) if this bay has not already passed it, otherwise go to (4). ii. BF Ib = −1: Go to (4b) if this bay has not already passed it, otherwise, go to (4). iii. BF Ib = 0: Go to (4).

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5. Repeat (4) backwards from b : B → 2; giving the first workcenters or taking the last workcenters to or from bay b − 1 (i.e. move the b − 1th separator one workcenter to the right or left, respectively).

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6. If the W F Ic = 0 for all the workcenters, there is at least one feasible solution under the current set of α, β, and B.

Given the number of bays, an extension of Algorithm 1 is iteratively implemented to find the range of feasible β values for a given α. This is done by small increments in the value of β from a lower to an upper value. The smaller α (the aspect-ratio of the workcenters), the smaller the feasible range of β (the aspect-ratio of the building) values. Algorithm 2 iteratively implements Algorithm 1 to choose B and β values. If β is fixed (i.e. the building dimensions are already decided upon), it only iterates over the range of B values. For each potential B value, we also define a flexibility index α(B) as the tightest α for which a feasible solution still exists. The key intuition behind this definition is our 12

ACCEPTED MANUSCRIPT understanding that a B with the smallest flexibility index offers more flexibility to rearrange the workcenters. Hence, it enables us to find better solutions. In other words, B with the smallest flexibility index will lead to the expansion of the search space. If the facility dimensions are not already fixed, Algorithm 2 also obtains the β value. For each given B value, the algorithm finds the β value (over an allowable set S), for which the

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tightest α is feasible. This β value is referred to as β(B). If the dimensions are given, set S contains only a single β, and thus the algorithm finds the B ∗ using the single β value. After applying this algorithm to all the potential B values, the algorithm sets B ∗ = argminB (α(B)) and β ∗ = β(B ∗ ).

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Algorithm 2 Identifying the Dimensions of the Layout and the Number of Bays For each and every potential B

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1. Set p = 1 and q = α.

2. Set k = (p + q)/2 and run the extension of Algorithm 1 for α = k along set S to find a discrete set K for which feasible solutions exist. 3. If set K is empty, p = k, otherwise q = k.

4. If (p − q) ≤ 0.05 and set K contains only a single element, go to 5, otherwise go to 2. 5. Set α(B) = k and β(B) = K.

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One of the main challenges in the FBS research has been finding compact FBS structures to satisfy the AR-constraints. Often, under tight AR-constraints, many generations of the evolutionary algorithms are wasted in search of a feasible solution. In some cases the algorithms fail, and in many cases they stop at inferior solutions. The ability to generate

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a stream of AR-feasible layouts outside of the evolutionary algorithm is a contribution of this paper. AR-feasible layouts play three significant roles in our work. First, by seeding the GA with all-feasible solutions, we do not waste generations searching for feasible lay-

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outs. Together with the way we iterate on B values, they reduce the number of generations by magnitudes of over 100. Second, by generating a mass of the AR-feasible layouts, and identifying the most favorable, a set of benchmarks is provided to evaluate the effectiveness of our GA or any other FBS layout model. Third, the majority of the FBS layout design algorithms do not end up with a compact block layout. Provided that a feasible solution exists, our AR-feasible layouts are always compact, i.e., the workcenters are fully packed in an external rectangle. Basic Intuitions and Simplifications. Given the fact that all the bays have the same length in one direction (X direction in our examples), there are about |C|/B workcenters in 13

ACCEPTED MANUSCRIPT each bay. The height of each bay depends on the total area of the workcenters placed in the bay. Algorithm 3 usually starts with an infeasible layout and applies the following moves to drive towards feasibility. (i) Moving a workcenter to a wider or narrower (having a larger or smaller height in Y direction) bay to meet the AR-constraint. (ii) Moving a workcenter out of a bay with infeasible workcenters to make the bay narrower, and increase XYc of the

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remaining workcenters. (iii) Moving a workcenter to a bay with infeasible workcenters to make the bay wider and decrease XYc for all the workcenters in that bay.

Given [β, B, α and Ac ], a workcenter is infeasible if XYc or 1/XYc > α. After each iteration, XYc , W F Ic , and BF Ib are recalculated for all workcenters and all bays. The

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process continues until a feasible layout is generated - if a feasible layout exists. To avoid getting trapped in a degenerate circle of infeasible solutions, a randomized process rearranges

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the workcenters. The probability of each route in Algorithm 3 is denoted by P . Algorithm 3 Feasible FBS Layout Construction Case 1. BF Ib values are either all 1 or all -1.

P=2/3. Find the workcenter with the largest XYc , and the workcenter with the smallest XYc not in the same bay. Switch them.

Case 2. BF Ib = -1 or 1.

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P=1/3. Find the workcenter with the largest (if all BF Ib = 1) or smallest (if all BF Ib = −1) value of XYc . Switch it randomly with a workcenter in another bay.

Randomly select a workcenter from a BF Ib = −1 bay and move it to a BF Ib = 1 bay. Case 3. BF Ib = -1 or 2.

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P=1/2. Find a workcenter with largest XYc in a BF Ib = 2 bay. Find a workcenter with the smallest XYc in a BF Ib = −1 bay. Switch them. P=1/4. Find the workcenter with the smallest XYc , and switch it with a randomly selected workcenter.

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P=1/4. Find the workcenter with the smallest XYc , and move it to a randomly selected bay.

Case 4. BF Ib = 1 or 2.

P=1/2. Find the workcenter with the largest XYc in a BF Ib = 1 bay, and the workcenter with the smallest XYc in a BF Ib − 2 bay. Switch them.

P=1/4. Find the bay housing the workcenter with the largest XYc . Randomly select a workcenter and move it to this bay.

P=1/4. In the bay housing the workcenter with the largest XYc , find the workcenter with the minimal XYc . Switch it randomly with another workcenter in another bay. Case 5. BF Ib = 0 or -1. P=1/2. Move the workcenter with the smallest XYc to the bay housing the workcenter with the largest XYc .

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ACCEPTED MANUSCRIPT P=1/4. Move a randomly selected workcenter in a BF Ib = -1 bay to another bay. P=1/4. Randomly switch a workcenter in a BF Ib = -1 bay and a workcenter in a BF Ib = 0 bay. Case 6. BF Ib = 0 or 1. P=1/2. Randomly choose a BF Ib = 0 bay, and find the workcenter with the smallest XYc in that bay. Move this workcenter to a BF Ib = 1 bay.

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P=1/4. Find the bay housing the workcenter with the largest XYc , move a randomly selected workcenter to this bay. P=1/4. Switch the workcenter having the largest XYc with the workcenter having the smallest XYc not in the same bay. Case 7. More than one BF Ib = 2, all other BF Ib = 0.

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Randomly switch a W F Ic = 1 workcenter with a W F Ic = −1 workcenter. Case 8. One BF Ib = 2, all other BF Ib = 0.

5

The Genetic Algorithm

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Randomly switch the workcenter having the smallest XYc with another workcenter.

The growing trend in FBS layout design research is the use of evolutionary algorithms (Ulutas and Kulturel-Konak 2012). Most of these algorithms rely on a simplified objective function, such as the loaded flow over rectilinear distances between the centroids of the workcenters.

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Our objective function is more complex and requires designing a circular flow pattern on the FBS layout, locating the I/O stations, and evaluating the volume of loaded and empty flow. To allow for a computationally tractable GA, we implement simplified versions and approximations of the objective function. After identifying prosperous layouts, we first

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replace rough simplifications with better approximations. The actual objective function is finally implemented in an improvement algorithm.

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Basic Intuitions and Simplifications. One main contribution of our work is simplification and approximation of the GA’s objective function. We first simplify the objective function to the length of the loop multiplied by the intensity of the flow on the busiest arc. The objective function is then decomposed into shorten the length of the loop and lower the volume of the loaded flow on the busiest arc. We reduce the flow part of the objective function to the sequencing of dimensionless stations along a fixed loop. This sequence is then used to estimate the number of rotations around the loop. We assume that the sequence of the stations is defined by the first string of the GA chromosomes. We approximate the length of the optimal loop by the length of the shortest loop, and estimate the shortest loop by 15

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Figure 2: Decoding the chromosome [1,2,3,4,5,6,7,8,9,10][3,5,2] in our model. the larger interior loop (the largest loop that does not pass the boundaries of the block

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layout- not necessarily a feasible loop). Additional simplifications underlying our encoding, the scheme we use to fill the bays, and the objective function simplifications will be described

5.1

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in the subsequent sections.

Strings

We have benefited from Tate and Smith (1995) in the sense that the chromosomes in our GA are formed by two strings. The first string represents the order in which the workcenters are placed in the layout, and the second identifies the number of workcenters in each

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bay. However, there are significant differences between our approach and that of Tate and Smith (1995). These differences are reflected in the objective functions, the simplifications we make to approximate a complex objective function, and the encoding and decoding of the chromosomes. The arrangement of the bays and the alignment of the workcenters in

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our GA are on the foundations of a circular flow pattern. For example, the chromosome [1,2,3,4,5,6,7,8,9,10][3,5,2] is decoded as Figure 1 in Tate and Smith (1995), while it is decoded as Figure 2 in our GA. On the foundations of how an FBS layout is covered by a loop,

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we fill two bays at a time, alternating between them in such a way that they are formed almost simultaneously. For example, in decoding the [1,2,3,4,5,6,7,8,9,10][7,3] chromosome, workcenter 1 is placed in the first bay, workcenter 2 in the second bay, and workcenter 3 in the first bay. The ratio of the number of workcenters in the first bay to that of the second bay is 2/1 < 7/3. Therefore, the next workcenter (workcenter 4) is also placed in the first bay. Since 3/1 > 7/3, workcenter 5 is placed in the second bay resulting in a 3/2 ratio. Since 3/2 < 7/3, the next workcenter is placed in the first bay. The first pair of bays is filled from left to right, the second pair from right to left, the third from left to right, and so on, with the exception of the last bay, which always is filled from right to left. 16

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5.2

Crossover and Mutation Operators

Crossover. The crossover operator performs as follows.

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Crossover 1. The order string of the chromosome with the better fitness function remains unchanged, and that of the second chromosome shifts |C| times to the right. Each time, all workcenters shift one position to the right, and the last workcenter is moved to the beginning. At each shift, the workcenters that are in the same positions in the two strings are counted. The number of shifts Sm leading to the highest similarity between the two chromosomes, e.g. having the largest number of the workcenters in the same positions, is identified. The first string of the second chromosome is fixed at the state of Sm shifts. Crossover 2. Similar to Tate and Smith (1995), each position of the first string of offspring is filled randomly by one of the two workcenters in the same position as the two parents. Thus, some workcenters may take two positions while some others may have no assigned position. For the workcenters taking two positions, one position is randomly released and is randomly designated to an unassigned workcenter.

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Crossover 3. With a 0.7 probability, the first string of the offspring will be the one ordered in Crossover 1, which follows the first parent with a better fitness function. With a 0.3 probability, the string ordered in Crossover 1 is shifted Sm times to the left to follow the second less fitted parent.

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Crossover 4. With a 0.7 probability, the second string of the offspring will be copied from the first parent, and with a 0.3 probability from the second.

Mutation. One of the following operators, the first with 0.4, and the second and third with 0.3 probabilities, is applied at each mutation.

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Mutation 1. A random integer 1 ≤ Sm ≤ (|C| − 1)/2 is generated. The order of the workcenters in the first string is shifted Sm times, either to the left or to the right, with equal probabilities. p To allow a higher chance for smaller shifts, random real numbers are generated in the range of 0 to 3 (|C| − 1)/2), then are powered by 3 and rounded up. Mutation 2. A random integer 1 ≤ Sm ≤ |C| is generated, and Sm random switches are made in the order of workcenters in first string. Similar to Mutation 1, a higher probability is assigned to smaller numbers of switches.

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Mutation 3. One of the elements (bays) of the second string is randomly selected. With the same probabilities, the number of the workcenters in this bay is either decreased or increased by placing them into or taking them from the next bay. A random integer between one and one less than the number of workcenters inside the selected bay, is generated to identify the number of workcenters to withdraw or deposit. A higher probability is assigned to smaller numbers of changes. This mutation is not applied on the last bay, and is also not applied if the donor bay only has a single workcenter.

In many FBS heuristics (Tate and Smith, 1995, Kim-Goetschalckx, 2006, Kulturel-Konak and Konak, 2010, Kulturel-Konak and Konak, 2011a, and Kulturel-Konak, 2012) applying a mutation on the second string will change the number of bays. Mutation 3 does not change the number of bays in our algorithms. In addition, one of the challenges of FBS layout heuristic procedures is finding feasible solutions. While we start with all-feasible solutions, the adaptive penalty function of Tate and Smith (1995) of P (Li ) = (qi )k (Vf eas − Vall ) is applied to ensure that the promising infeasible solutions generated in the crossover and mutation processes are still examined. In this function, Li denotes the ith layout in the population, qi is the number of infeasible workcenters, k is a fixed integer that we set to 17

ACCEPTED MANUSCRIPT 2, where Vf eas and Vall are the best current objective function values for feasible layouts, and for all layouts, respectively. Under this penalty function, a layout with one infeasible workcenter and the best known objective function value would be as desirable as the best known feasible layout. We implement a population of 20 chromosomes. The number of generations, depending

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on the problem size, range from 1000 to 3000. To create a new generation, four copies of the three best solutions, three copies of the next four best solutions, two copies of the 7th to 10th best solutions, and one copy of the rest of the population (a total of 40 parents) are selected. They are then randomly paired, and each pair creates one offspring. The resulting 20 offspring are mixed with the best 10 chromosomes of the previous generation. Out of

Approximations of the Objective Function

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5.3

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these 30 chromosomes, the top 20 form the next generation.

As described in Section 3.2, given θ = Max {Tn =

P

i,j

fij tnij }, for all arcs on a unidirectional

loop, the whole loop is traversed θ rounds per time horizon. The total loaded and empty flow on a loop with length L is computed as θ × L. To evaluate the potential of a string, we ˆ separately approximate the intensity of the flow on the busiest arc of the optimal loop (θ), ˆ The product will serve as an approximation of the and the length of the optimal loop (L).

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objective function value in our GA. In order to verify reliability of comparing two layouts in the GA under this approximation, we generated a set of random layouts for each of the 8 problem instances taken from the literature in this manuscript Values of Z = minℓ∈L L(ℓ)θ(ℓ) ˆ were computed for all instances of each layout. We witnessed that, within ranand Zˆ = θˆL dom layouts of each problem instance, Z and Zˆ benefit from a strong correlation coefficient

5.3.1

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function.

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value of 96%, testifying that the approximation is a good substitute for the actual objective

Approximation of the Maximal Intensity of the Flow

At this stage, we treat the workcenters as dimensionless combined I/O stations, where their sequence is used to estimate θ. Consequently, the problem is reduced to sequencing a set of dimensionless nodes along a fixed loop (Afentakis, 1989). We assume that the sequence of the stations along the loop is defined by the first string of each chromosome (as shown in Figure 2). This can easily be considered as a greedy heuristic where a load is dropped as soon as logically possible. We further streamline the heuristic by using the string to approximate the length of the loop. These approximations are then directed to the actual objective function

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ACCEPTED MANUSCRIPT values by the interchanges in the improvement algorithm in the next section. 5.3.2

Approximation of the Length of the Loop

We assume that the length of the optimal loop in each layout is approximated by the length of the shortest loop in that layout. We further assume that the length of the shortest loop

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in a layout is approximated by the length of the larger interior loop (LIL) in that layout. We define LIL as the boundary of the polygon formed by all the workcenters not adjacent to the boundary of the layout. The LIL is not necessarily a feasible loop since one or more workcenters may not have a boundary-line on the loop. In addition, in special instances, for example where the number of bays are 1-2, or there are 1-2 workcenters in a bay, the

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layout has no LIL. These exceptions can be easily resolved. For example when B = 2, LIL is equal to the shortest loop, and is defined by one of the bays excluding its first and last

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workcenters, or when there are only two workcenters in a bay, the one with the smaller perimeter is included in the LIL. In cases of B = 3 or 4, when the middle bay(s) have more than 2 workcenters, the LIL is the shortest loop, if and only if the following LIL-conditions hold.

1. The length of the first workcenter in the first bay is larger than the length of the first workcenter in the second bay.

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2. The length of the last workcenter in the first bay is larger than the length of the last workcenter in the second bay. 3. The length of the first workcenter in the last bay is larger than the length of the first workcenter in one to the last bay. 4. The length of the last workcenter in the last bay is larger than the length of the last workcenter in one to the last bay.

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For the FBS layouts that are not larger than four bays, these conditions ensure that the LIL is the shortest loop. For larger number of bays, the LIL-conditions are still necessary

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for the existence of a feasible interior loop touching the boundaries of every workcenter. To substantiate this claim, we refer to Figure 3a, an FBS instance of the 11-workcenter example of Sinreich and Tanchoco (1993) designed by our GA, and Figure 3b, an LIL obtained by our GA for a larger problem. As observed in Figure 3a, if condition 1 did not hold (i.e. the length of top left workcenter did not exceed the workcenter below it), the interior loop would not be feasible. The same goes for Figure 3b and all the four conditions. For any FBS layout, our objective function encourages the reduction of the length of the LIL. This, together with the holding of LIL-conditions, increases the chances that the shortest loop to be an interior loop. As a result, the length of the LIL intuitively becomes a better estimate of the length of the shortest loop. The LIL-conditions are more significant in small problems 19

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Figure 3: Examples of the effectiveness of the LIL-conditions on small and large problems. such as Figure 3a. However, Figure 3b also illustrates that if the LIL-conditions do not hold, the interior loop is not a feasible loop.

To encourage satisfying LIL-conditions, we multiply the objective function value by 7.5y/|C| , where y = {0, 1, 2, 3, 4} is the number of unsatisfied LIL-conditions. The choice of 7.5 showed to be a promising value through a set of computational experiments. A pa-

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rameter of 7.5 led to better output layouts for the GA when the layouts were evaluated using the exact objective function.

6

Improving the Layout, Loop, and Station Locations

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In the previous steps, we developed a GA to construct promising FBS layouts, which we claim have high potential for designing a loaded and empty objective function value for

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circular flow patterns. In the improvement algorithms presented in this section, instead of the approximations used in the GA, we implement the true objective function. We first find a prosperous loop on a candidate layout proposed by the GA, and then identify the I/O station locations. In the next step, we implement a truncated interchange procedure to improve the concurrent loop-layout-station design. One may implement Asef-Vaziri et al. (2007) algorithm for the optimal concurrent design of the loop and stations. We have used Asef-Vaziri and Ortiz (2008) sequential exact algorithm to find the shortest loop, and then to locate the stations in order to minimize the total loaded and empty flow. We implement a truncated pairwise interchange algorithm to

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ACCEPTED MANUSCRIPT improve the layout and the loop. Side heuristics are developed to increase the likelihood of escaping from local optima. The truncation is performed by approximating the objective function of all candidate pairwise interchanges and screening the top 2|C| interchanges to be evaluated by the exact objective function. The procedure is described in Algorithm 4. Similar to the GA in Section

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5, workcenters are assumed to be dimensionless nodes on a loop. However, unlike the GA, we do not need to approximate the length of the loop using LIL, since we already have the exact loop for which the objective function is being defined. In Algorithm 4, when traversing the loop, each new boundary-line is defined by dropping one of the two workcenters defining the current boundary-line and replacing it by an unvisited workcenter. For the two

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workcenters adjacent to the first boundary-line, and for the pair of new workcenters visited at the intersection of 4 boundary-lines, the tie is broken arbitrarily. If the loop passes two

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non-adjacent boundary-lines of the workcenter, only the first line is considered. Algorithm 4 Objective Function Value Approximation on a Unidirectional Loop 1. Compute the length of the loop (L(ℓ)).

2. Start from the northwest corner of the loop, label the workcenters in the order they are covered by the loop. 3. Arrange the dimensionless workcenters along the loop in the sequence they are visited in (2). Compute ˆ the total loaded flow on the busiest arc, denote it by θ(ℓ).

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ˆ 4. Set the objective function approximated value to θ(ℓ)L(ℓ) .

The approximated objective function value is an upper bound for the actual objective function value. To substantiate this claim, we show that we can locate the I/O stations on the

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loop such that the total flow is equal to the value computed above. Therefore, when the I/O stations are located optimally, the actual flow will be less than or equal to the approximated

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value. In our numerical experiments, we realized that this approximation usually has less ˆ ≤ 1.2θ(ℓ)). The average gap was than a 20% gap with the actual objective function (θ(ℓ) less than 10%. We observed that within all possible triple-wise interchanges on a specific ˆ and θ(ℓ) had a 98% correlation coefficient. loop/layout, the values θ(ℓ) Let I/O stations of each workcenter to be collocated on the mid-point of its first boundaryline covered by the loop. If the stations of two adjacent workcenters are collocated, split the node in two. This is to avoid canceling the flow between the two adjacent workcenters due to the stationary transfer of load through the common node. Locate the I/O station of the second workcenter slightly after that of the first workcenter in the direction of the loop. The value of the objective function will be equal to the approximation defined above. 21

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Figure 4: Two states of a pairwise interchange. Algorithm 5 Truncated Pairwise Interchange

1. Prepare the list of top 2|C| interchanges candidates using Algorithm 4.

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2. Evaluate each interchange from the top-down list. If the new objective function is less than or equal to the previous one, apply the interchange, remove all interchanges in the list containing either of these two workcenters, and continue with the rest of the list.

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3. If the objective function has improved, return to (1). If the objective function has not improved, but the layout has changed, return to (1) unless this has happened for the 4th consecutive time. Stop.

In each pairwise interchange in Algorithm 5, if one of the two workcenters is inside and the other is outside the loop, there will be augmented interchanges. Consider the pairwise interchange of workcenters 7 and 4 in Figure 4a. The workcenters inside the loop after the interchange in Figure 4b remain the same as they were in Figure 4a, while they differ in

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Figure 4c. Accordingly, each element in the list of the top 2|C| interchange represents the two workcenters to be switched, as well as a binary decision of whether or not to change the loop, e.g., two interchanges.

The loop-based block layout design under FBS has three governing constraints of aspect-

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ratio, covering, and connectivity. The first constraint has its profound impact on the feasible FBS layout construction, as well as the improvement phase. The second and third constraints surface in the design of the loop and improvement of the layout, loop, and station locations.

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In each interchange, the approximated objective function is computed only if the new layout is feasible (AR-constraint is satisfied), and the loop is feasible (both covering and connectivity constraints are satisfied).

Algorithm 6 adds a complementary triplewise interchange to enhance Algorithm 5 in escaping local optima. Since the number of triplewise interchanges exponentially exceeds the number of pairwise interchanges, we again use the approximation of the objective function. Only the most favorable of the triplewise interchanges is then evaluated based on the exact objective function. Algorithm 6 Enhanced Truncated-Pairwise-Interchange. 22

ACCEPTED MANUSCRIPT 1. Iteratively perform Algorithm 5 until no further improvement is observed. Refer to the current best layout-loop-station as design-1. 2. Apply all triplewise interchanges inside each individual bay. For each interchange, approximate the objective function value as defined in Algorithm 4 for both cases of switching and not switching the workcenters inside the loop. Identify the best four triplewise interchanges. 3. Perform the next best triplewise interchange on the list. Refer to this layout-loop-station as design-2. 4. Iteratively perform Algorithm 5 on design-2 until no further improvement is observed.

Computational Considerations

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7

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5. If design-2 is better than design-1, replace it and go to (1), otherwise go to (3). If all triplewise interchanges on the list have already been examined with no further improvements, design-1 is the final layout-loop-station.

In this section we report the results of our extensive computational experiments on three

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sets of benchmarks. All the experiments were on an 8GB RAM computer with a core i7 CPU using MATLAB software. The results reported in Tables 1-4 are all based on the true objective function, not on the approximations we presented.

7.1

B and β Value Experiments

To evaluate the procedure proposed in Section 4, we applied it on a set of well-known test

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problems from the literature and identified the best B and β values. For each B and β value, we then apply our GA and layout improvement algorithms to design the layout and the loop. The same is done for several alternative choices of B and β to compare against the B and β chosen through our approach. The alternative designs are chosen by manually varying B

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and β in both directions to create reasonable benchmarks to evaluate our choice of B and β. To avoid biases that could result from the randomness of the GA, we ran the process 10 times for each alternative choice of B and β. The results are summarized in Table 1.

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Each design in Table 1 is identified by a triple of numbers. The first number is β, the second is B, and the third element is the objective function value. The objective function values are normalized by dividing them by the best objective function value obtained for the corresponding problem. The B and β values computed by our algorithms, in most cases, lead to lower objective function values compared to any other B and β values. In two problems, however, they are 4% and 9% larger than the minimal values. This is not a surprise, since the AR-feasible layout design algorithm does not take material flow into account, and is solely based on the area of the workcenters and feasibility of the aspect-ratios. While one may design a worst-case flow matrix to show the inefficiency of this design, as Table 1 shows, 23

ACCEPTED MANUSCRIPT it did not surface in the unbiased test problems from the literature. This is mainly because our design finds the dimensions and the number of bays in the direction of enlarging the feasible solution space. Thus, it increases the likelihood of finding quality solutions.

ML15 AB20 ML25 Ta30

8 0.8,4 1.39 2, 2 1.33 1.2,5 1.38 2,4 1.78 2,3 1.13 2,5 1.59 3,3 1.47

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Ba14

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ST11

Alternative Designs 1 2 3 4 5 6 7 1,3 1.5,3 1.5,2 1.8,2 2,2 0.5,4 0.7,4 1.30 1.37 1.40 1.57 1.66 1.51 1.41 1.3, 3 1.5, 3 1.8, 3 2, 3 1.4, 2 1.6, 2 1.8, 2 1.09 1.06 1.10 1.20 1.71 1.28 1.30 0.4,4 1.2,4 0.8,3 1.2,3 1.6,3 0.4,5 0.8,5 1.38 1.04 1.11 1 1 1.23 1.21 1,3 2,3 2,2 2.5,2 1,4 1.3,4 1.7,4 1.20 1.39 1.60 1.63 1.15 1.20 1.37 1.1,4 1.5,4 1,5 1.3,5 1.6,5 1.1,3 1.3,3 1.00 1.12 1.12 1.14 1.26 1.09 1.17 1.2,4 2,4 1.2,3 1.6,3 2,3 1.2,5 1.6,5 1.06 1.32 1.00 1.33 1.02 1.44 1.41 0.4,6 1,6 0.7,5 1,5 1,4 2,4 2,3 1.12 1.11 1.12 1.13 1 1.23 1.41

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Problem VC10

Our Design 1.2,3 1.00 1,3 1.00 0.8,4 1.02 1.7,3 1.00 1.3,4 1.04 1.6,4 1.09 0.7,6 1.04

7.2

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Table 1: The objective function values under the β and B determined by our algorithm compared to other likely values. The abbreviations used to refer to the test problems are described in the next section.

GA Experiments

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We now test the effectiveness of our GA by producing layouts that have the potential of designing a desirable circular flow pattern on them; we test the effectiveness of the objective function approximation used in the GA in evaluating the promising layouts. To test this,

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we find the shortest loop on the layout produced by the GA, and optimally locate the I/O stations on the shortest loop to minimize total loaded and empty flow. We then repeat the same experiment on a set of well-known layouts discussed in the literature. However, since our work is the first study to design a loop-based FBS layout under the objective function of loaded and empty flow, there is no literature baseline for comparison. We were compelled to provide a set of unfair (to the previous authors) benchmarks, as all these layouts have been designed to minimize either centroid-to-centroid, or station-tostation, bidirectional loaded flow. However, this is still a useful benchmark as it shows that, for the purpose of designing a circular flow pattern, a layout which minimizes the 24

ACCEPTED MANUSCRIPT objective function presented in Section 5 results in a better solution. This shows the ability of our proposed approximate objective function to evaluate the potential of a layout. We also consider a fair benchmark by randomly generating 20 AR-feasible layouts, designing a circular flow path on each, and choosing the best one. The layouts from the literature that we use as benchmarks include VC10 (the 10-

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workcenter example of Van Camp et al., 1991), Ba14 (the 14-workcenter example of Bazaraa, 1975), ML15 and ML25 (the 15- and 25-workcenter examples of Meller, 1992 and Liu and Meller, 2007 also exercised in several papers in the recent years), AB20 (the 20-workcenter example of Armour and Buffa, 1963 also implemented in many layout design papers in the past half a century), Ta30 (the 30-workcenter example of Tam, 1992), KG37 (the 37-workcenter

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example of Kim and Goetschalckx, 2005), De62 (the 62-workcenter example of Dunker et al., 2003). The best designs are either taken from the original papers, or from more recent

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papers of Ne01 (Norman et. al 2001) and KK11 (Kulturel-Konak and Konak, 2011a). The AR-feasible as well as the GA-generated layouts were designed under the same α values as the FBS layouts taken from the literature. The three alternatives for the initial loop-based FBS layout design; (i) the literature-based layouts, (ii) the best of the AR-feasible layouts, and (iii) GA-designed layouts were then compared by finding the shortest loop and locating the I/O stations. The total loaded and empty flow objective function values were then computed.

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The first three columns under ”Literature” in Table 2 show the reference to the original test problem, the reference to its last re-design in the literature, and the α value in the best design, respectively. The next single column represents the objective function value for the best design after designing the circular flow pattern. All the objective function values

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are normalized through dividing them by the best objective function value obtained for the corresponding problem. The three columns under ”Z* (AR-feasible)” show the best, mean,

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and worst objective function values for the best of the AR-feasible layouts. The solution times in seconds are shown in the next column. The same information for the design obtained by starting from GA design are shown on the last four columns. As it is shown, even the most unfavorable runs of our GA always provide the most favorable designs compared to the literature and most of the AR-feasible designs. The GA computation times are under two to five minutes for up to 37- and 62-workcenter examples, respectively. These times are, on average, about 10 times less than even the fastest evolutionary algorithms in the literature where the flow consists of only the loaded flow and distances were computed rectilinearly.

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ACCEPTED MANUSCRIPT Literature

Z* (AR-feasible)

Z* (GA)

Ref.

α

Z*

Best

Mean

Worst

T (s)

Best

Mean

VC10

N01

4

6.03

1

1.77

2.42

13

1.22

1.40

1.65

5

BA14

N01

1.63

1.99

2.25

21

1

1.25

1.51

8

ML15

KK11

4

3.64

2.12

2.58

2.92

27

1

1.43

2.81

7

AB20

N01

3

1.54

1.38

1.58

1.72

49

1

1.14

1.26

6

ML25

KK11

5

1.93

2.11

2.76

3.16

49

Ta30

KK11

5

1.74

1.32

1.53

1.74

71

KG37

KG07

3

2.07

1.45

2.37

2.97

81

De62

KK11

3

1.51

1.16

1.48

1.81

170

1

1.18

1.54

8

1

1.22

1.41

22

1

1.37

2.07

20

1

1.08

1.15

90

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3.61 2.57

Worst T (s)

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Prob.

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Table 2: Best, mean, and worst objective function values and solution times after GA and before the layout-loop-station improvement algorithm.

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Literature Z* Z*(AR-feasible) Prob- Refe- α Before After Before After Time lem rence B M W B M W Sec. VC10 N01 4 10.03 5.57 1.66 2.94 4.00 1.45 2.32 3.55 24 BA14 N01 3.61 2.67 2.01 1.69 2.07 2.33 1.27 1.72 1.85 42 ML15 KK11 4 5.19 1.76 3.02 3.68 4.16 1.80 2.22 2.88 60 AB20 N01 3 1.76 1.42 1.58 1.81 1.98 1.09 1.25 1.42 101 ML25 KK11 5 2.43 1.54 2.66 3.48 3.99 1.48 2.03 2.67 181 Ta30 KK11 5 2.21 1.64 1.68 1.95 2.22 1.31 1.57 1.75 471 KG37 KG07 3 3.02 1.43 2.11 3.45 4.33 1.25 2.06 2.63 507 De62 KK11 3 1.86 1.39 1.43 1.82 2.23 1.06 1.23 1.38 1487

Before B M W 2.03 2.33 2.74 1.04 1.29 1.57 1.43 2.03 4.01 1.15 1.31 1.45 1.26 1.48 1.94 1.27 1.56 1.80 1.46 2.00 3.02 1.23 1.33 1.42

Z*(GA) After B M W 1 1.40 2.16 1 1.21 1.42 1 1.35 1.57 1 1.09 1.13 1 1.25 1.44 1 1.15 1.31 1 1.45 1.84 1 1.17 1.27

Time Sec. 14 26 30 46 115 365 380 831

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Table 3: Quality of the solution, best (B), mean (M), worst (W), and solution time, before and after layout-loop-station improvement.

Improvement Algorithm Experiments

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The results of applying all the steps of our algorithms are shown in Table 3. The columns under ”Before” show the objective function value when the shortest loop is designed on an initial layout (before improving the layout-loop-station). These are the same results reported in Table 2, but the normalized values are different since they are now divided by the lowest objective function value obtained after the improvement step. The columns labeled ”After” show the final objective function value after the improvement step. As shown, for all problems, the quality of the GA layouts, even in the worst runs, is better than the layouts taken from the literature and adopted for loop-based flow patterns. As a result, the final layout-loop-station designs obtained by starting from the initial GA layouts, are 26

ACCEPTED MANUSCRIPT almost always better than those obtained by starting from a layout designed in the literature. Another benefit of starting from better initial layouts of GA is the vast reduction in solution times. If the initial layout is poor, even if the GA leads to significant improvement, it is achieved at a high computation cost. These claims are substantiated in Table 3. In spite of our more complex objective function, the total solution times when the GA is used in

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designing the initial layouts are less than those reported in the literature for all the FBS layout design algorithms. The main factors to our low solution times are treating the number of bays as a parameter, feeding the GA with all AR-feasible solutions, and approximating the objective function value.

The FBS loop-based layout for the 14-workcenter example of Bazaraa (1975) and the

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37-workcenter example of Kim and Goetschalckx (2005), after applying the GA, and after performing the layout-loop-station improvement process are shown in Figure 5.

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The last computational experiment is to compare our algorithm with a brute-force search procedure. We have found the optimal design for the VC10 problem by enumerating all potential feasible layouts under the α = 1.845. The optimal procedure of Asef-Vaziri et al. (2007) was implemented for concurrent design of the optimal loop and stations. The α = 1.845 enforces having 2 bays where workcenters 1, 3, 5, and 9 are in one bay. The optimal layout-loop-station design through complete enumeration was obtained in three hours. Table 4 shows the results of the best of 10 runs of our GA compared to the optimal solution. The ten

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runs took 14 seconds and the optimal solution was found in four runs. We then increased the aspect-ratio to 2.535 and the number of bays to 3. The AR-constraints enforce workcenters 1, 3, and 9 to be in one bay, workcenters 2, 5, and 10 in another bay, and the rest to share the third bay. This allows us to find the optimal solution using brute-force enumeration in

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two hours. Ten runs of our algorithms took 12 seconds and the optimal solution was found in five runs. Additionally, the sub-optimality of runs that did not obtain the optimal solution

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were small in all instances. Prob.

2-Bay 3-Bay

(Z*) Optimal Solution 36052 35128

Solution time 10775 7187

# of times Z* found 4 5

Coefficient of variations 0.05 0.01

Solution Time 14 12

% Gap with Z* Best Mean Worst 0% 5.9% 9.9% 0% 1% 2.1%

Table 4: Solution time and quality of the best, mean, and worst solution obtained by the heuristic compared to that of the optimal solution obtained through complete enumeration.

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Figure 5: The 14-workcenter example of Bazaraa (1975) and the 37-workcenter example of Kim and Goetschalckx (2005). (a) and (b) after applying the GA, (c) and (d) after performing layout-loop-station improvement algorithm.

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8

Conclusions

In this study we developed a model for loop-based facility layout design under flexible bay structures. Our model accounts for both loaded and empty flow between the I/O stations along unidirectional circular flow paths. Unlike all other FBS models, we treat dimensions as decision variable. We solved a major challenge in the literature, which was to determine

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whether or not feasible solutions exist under tight AR-constraints, and presented a algorithm that produces random AR-feasible solutions (if they exist) with little computational effort. Hence, unlike most of the GA heuristics, we are able to seed the algorithm with an all-feasible initial population.

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One of the contributions of our work is in developing an approximate objective function to evaluate the potential of a layout for a desirable circular flow pattern without actually designing it. This makes the algorithms time-efficient while they remain quality-effective.

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Through computational experiments, we showed that our approximate objective function is indeed an appropriate measure for the potential of a layout: minimizing this objective function leads to layouts where desirable circular flow patterns may be designed. Furthermore, we presented an iterative improvement algorithm which considers the final objective function in making these improvements, rather than the approximate objective function. Our computational experiments show that this algorithm is able to adjust the

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approximations made in generating a layout with the GA and significantly improves the final objective function. This improvement algorithm is fast because we truncate the potential improvement trials by first evaluating them with an approximate objective function requiring minimal computations.

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Our study is a starting point for future research on loop-based facility planning. The quality and time of our solutions provide benchmarks for future research. The directions of our future research is to work on better decompositions and integrations of the three

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components, layout, loop, and stations of this problem.

Acknowledgments

This paper is dedicated to the memory of Charles Donaghey, Ph.D., P.E. (1930-2016), Professor, University of Houston. We thank the two anonymous reviewers and the department editor for their constructive comments, which helped us to improve the manuscript. This work was partially supported by Research, Scholarship and Creative Activity Award, California State University, Northridge. This support is gratefully acknowledged.

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