- Email: [email protected]
Single facility siting involving allocation decisions
Journal Pre-proof
Single Facility Siting Involving Allocation Decisions Alan T. Murray , Richard L. Church , Xin Feng PII: DOI: Reference:
S0377-2217(20)30084-9 https://doi.org/10.1016/j.ejor.2020.01.047 EOR 16301
To appear in:
European Journal of Operational Research
Received date: Accepted date:
5 December 2018 21 January 2020
Please cite this article as: Alan T. Murray , Richard L. Church , Xin Feng , Single Facility Siting Involving Allocation Decisions, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.01.047
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Highlights
Reviews classic Weber problem for industrial facility location Formulates extended interpretation of Weber problem Develops an exact solution algorithm based on the use of GIS Empirical applications demonstrate utility of developed approach
Single Facility Siting Involving Allocation Decisions Alan T. Murray Richard L. Church Xin Feng
Department of Geography University of California at Santa Barbara Santa Barbara, CA 93106, USA
(Email: [email protected])
December 27, 2019
1
Abstract
Single facility siting is often viewed as the most basic of location planning problems. It has been approached by many researchers, across a range of disciplines, and has a rich and distinguished history. Much of this interest reflects the general utility of single facility siting, but also the mathematical and computational advancements that have been made over past decades to support better decision making. This paper discusses a recently rediscovered form of Weber’s classic single facility location problem that is both important and relevant in contemporary planning and decision making. This form of the Weber problem involves locating a production plant where there are multiple sources of each needed raw material (input) distributed throughout a region. This means that the selection of a given raw material source may vary depending on the plant location. In essence, this makes the problem non-convex, even when locating only one production plant. We review elements of the Weber problem that have been addressed in the literature along with proposed solution techniques. In doing so, we highlight elements of the problem originally noted by Weber, but to date have not been operationalized in practice – allocation selection among multiple sources of given raw material inputs. A problem formulation involving allocation decisions for this generalization is derived and an optimal solution approach is developed. Application results demonstrate the significance of addressing important planning characteristics and the associated nuances that result.
Keywords: Location; Weber; spatial optimization; Voronoi diagram; GIS
2
Introduction
There are two basic forms of location models that have been proposed in the literature: those defined on a network and those involving continuous space (e.g., Cartesian plane, sphere, etc.). Although both problem domains are important, location science (or analysis or modeling) can trace its roots to formalisms first defined on a Cartesian plane specifically involving the Euclidean distance measure. Approaches used to solve continuous space problems and network based problems rarely intersect due to significant differences in representation, though there are exceptions such as the noted mixed use of continuous and discrete solution approaches by Brimberg et al. (2014). One of the true classic location problems recognized in operations research, economics and production and operations management is referred to as the Weber problem, although others have also been credited with its introduction, such as Fermat, Torricelli, Cavalieri, and Simpson (see Wesolowsky 1993). As proposed by Weber (1909), the problem is defined on a continuous plane and involves finding the best location for a factory or production plant so as to minimize associated transportation costs in shipping raw materials or components to the factory and in shipping the final product to retail centers, where distances are measured using the Euclidean metric. Accordingly, the transportation costs are viewed as critically important under conditions where land acquisition and labor costs are constant across a region. What has made this basic problem of such great interest is that economic efficiency is formalized, and that the importance of spatial location is inherently central to notions of efficiency. Further, the simplicity of this problem along with mathematical specificity make clear that it reflects many siting and planning contexts, and can be extended in different ways, as demonstrated in Francis et al. (1992), Carrizosa et al. (1998), Berman et al. (2007), Bischoff and Klamroth (2007), Canbolat and Wesolowsky (2010, 2012) and Venkateshan et al. (2017), among others.
Of great significance is that the work of Weber (1909) remains relevant today, arguably more so than it ever has been. The problem has been portrayed as one where a factory is to be sited 3
given two localized raw material inputs and one market to which the final products are to be shipped. This is commonly referred to as the Weber locational triangle (see for example Isard 1960, Wesolowsky 1993, Capello 2014, among others). Weber (1909) actually referred to this as a locational figure as he purposely made his drawings as simple as possible to convey the issue at hand, in this case highlighting the need for locating one of more facilities dependent on localized raw materials in order to serve retail centers. Accordingly, the simplest form could involve just three points, two raw material sources and one market, to illustrate the basic concept. His descriptions of this problem clearly indicate that each raw material may be sourced from a number of discrete locations and that the selection of a given needed raw material would be from that location which would involve the least transport cost in bringing the necessary amounts to the facility. Hence, raw materials would be allocated in order to minimize transport costs, and those allocations would be a function of the positioning of the plant. Whereas, the literature has treated the single plant location problem of Weber (1909) as focusing only on location (no material allocations), the real intent was to describe a true location-allocation problem. This is the subject of this paper. Surprisingly given the widespread interest in Weber (1909), this interpretation to address resource allocation was only recently uncovered in a review by Church (2019).
In the following sections we formulate what many describe as the classic Weber problem. We follow this with a discussion of how this problem has been solved, review its relationship to the center of gravity problem, and highlight the use of associated models in the fields of business and productions and operations management. This is followed by an overview of how the Weber problem has been extended to add relevant features for specific applications. We then develop a model formulation that includes the allocation of possible raw material or component sources in selecting a location for a production facility in order to serve a set of demands and markets. A solution approach is then derived that exploits spatial properties using Voronoi diagrams. Application results are presented to highlight the capabilities of both the new model formulation as well as the solution approach. The paper ends with discussion and concluding comments.
4
Background
Consider the illustrative example of the problem attributed to Weber (1909): find the production plant location, (
), that minimizes total costs, where costs are a function of the
amount of materials shipped and the distance it must be transported. This simple problem can be expressed mathematically as:
Minimize )
√( √(
)
(
(
)
√(
)
(
)
)
(1)
where there are two required raw materials (or components) and one market. For example, point 1 is the source location of raw material 1 at ( second needed raw material at (
), point 2 is the source location for the
), and point 3 is the location of the market (or point of
demand) at (
). Associated with each point is attribute information, such as the amount to
be shipped ( ,
and
market, (
), (
) from/to the factory and the location of the raw materials and ) and (
), respectively. Thus, what remains is finding the optimal
location for siting the production plant, (
). One option is the physical analog frame
attributed to Varignon (1687). A second approach is the use of isotims (lines of equal cost from a given source or to a given market) and isodapanes (lines of equal total transport cost from the associated sources and to the markets), as proposed by Weber (1909) and employed by others (e.g., Isard 1956). A third is an iterative approach outlined in Weiszfeld (1937) obtained through calculus using derivatives. This has been widely used, with one slight modification to handle the case when a trial solution or a final solution coincides with one of the input/output points (Wesolowsky and Love 1972, Ostresh 1978). Worthy of note is that the Weiszfeld algorithm has been rediscovered since its original appearance by Miehle (1958), Kuhn and Kuenne (1962), Cooper (1963), and Vergin and Rodgers (1967), among others.
5
Related to the simple form of the Weber problem, (1), is the following approach which minimizes weighted squared distances:
Minimize
((
)
(
) )
((
)
(
) )
((
)
(
) )
The difference in (2) is the squaring of Euclidean distance, which effectively removes the radicals. This is the center of gravity problem. Through calculus using derivatives, a closed form solution is possible. The result is an optimal solution for (2) as follows:
(3a) (3b)
Although most transport cost functions are reasonably approximated by the weighted Euclidean distance in (1), this is not true for weighted squared distances in (2). Plastria (2016a) and others have noted the significant problems associated with cost approximation using squared distances (see also Church and Murray 2009). Nevertheless, the closed form solution in (3) has been appealing to use for many due to its simplicity and ease of application. Unfortunately (and incorrectly!), some call (1) a center of gravity problem (e.g., Ivanov et al. 2019), which means that there appears to be some confusion between these two models (see also Plastria 2016a). Of course, mathematically they are not the same nor will they identify the same location as an optimal solution (Wesolowsky and Love 1972, Church and Murray 2009, Plastria 2016a). Interestingly, AIMMS recently released a “Center of Gravity Navigator” as a “quantitative method for locating distribution facilities based upon distance and the weight of demand/or supply.” 1 It is unclear as to whether they actually support solving the center gravity model or
1
https://aimms.com/english/news-archive/aimms-launches-center-gravity-app/
6
(2)
rather the Weber problem. Whichever is the case, both the center of gravity and the Weber approaches are central models in production and operations management, with most textbooks describing how to solve such models along with examples of how they can be applied in an industrial setting (see Stevenson 2018, Bozarth and Handfield 2019, Ivanov et al. 2019, among others). Given that the Weber problem is virtually always noted in distribution system design (e.g., Klose and Drexl 2005), described as a principal approach in warehouse, production, and distribution design in textbooks (e.g., Stevenson 2018, Bozarth and Handfield 2019, Ivanov et al. 2019), and that support tools have been designed to solve it with interesting map interfaces (e.g., Murray 2018), it is clear that it is a de facto analytical approach to site a production facility and/or adjust a location within a supply chain.
Worth highlighting as well is that even though most of the applications of single facility siting have focused on the use of Euclidean distance, a number have proposed the use of other metrics, including rectilinear or Manhattan distance (Wersan et al. 1962, Vergin and Rodgers 1967, Love et al. 1988. Francis et al. 1992). In fact, many textbooks in production and operations management cover the use of both Euclidean and rectilinear distance measure in the application of the Weber problem (e.g., Reid and Sanders 2016, Ivanov et al. 2019). Research on solution and extension of the Weber problem has been extensive. Work continues to explore solution properties, and ways to make methods more computationally efficient (e.g., Vardi and Zhang 2001, Gorner and Kanzow 2016). However, problem nuances, such as regret (Drezner and Guyse 1999), regional representation (Yao and Murray 2013, 2014), demand uncertainty (Plastria 2016b), and multi-facility extension (Irawan et al. 2017, Drezner et al. 2018), among others, continue to be addressed in order to better reflect the realities of encountered planning contexts. The Weber problem is indeed a fundamental approach utilized across many different disciplines and applied in a range of planning contexts.
One of the key elements raised by Weber (1909) was that there could be several sources of a given raw material or product component, and that it would be most efficient to use that source
7
which offered the closest transport. That is, Weber (1909) noted that his location construct would require allocation decisions when multiple sources of a given needed input material were present. This is an important aspect of the original work that appears to have been completely overlooked until recently (see Church 2019). Most researchers have viewed Weber as focusing on location only, creating a cost function that was convex and solvable using a descent algorithm like that of Weiszfeld (1937) and Ostresh (1978). However, these approaches in and of themselves cannot completely address the situation when multiple sources of a given raw material are present as this creates a cost service that is non-convex. The reason is that allocation decisions must be simultaneously addressed as well.
Mathematical Model
The locational triangle characterization of the Weber problem was mathematically structured in (1). This problem can be generalized for any number of raw material inputs or market outputs. Consider the following notation:
index of resource (raw material)/demand (market) points total amount to be shipped to/from point (
(
)
)
geographic coordinates of point
location to site production facility
Again, of particular importance is (
), the problem decision variables. These indicate the
location, as a coordinate pair, where the production facility is to be sited. With this notation, the traditionally conceived version of the Weber problem is:
8
Minimize
∑
√(
)
(
)
(4) (5)
The objective, (4), seeks the minimum total weighted distance from the sited facility to all resource and demand points. This function reflects costs associated with the transportation of materials and goods. Constraints (5) formalize that the decision variables could be positive or negative, depending on the coordinate system for the region being evaluated.
The model formulation, (4)-(5), is a necessary step in structuring this problem, but in order for it to be of practical use and significance, a solution method is needed. In this particular case, methods for solving (4)-(5) do exist, and will be discussed in the next section.
Explicitly structured in (4)-(5) is that all resources and markets interact with the sited production facility, impacting where it should be located. That is, points representing input and output destinations all contribute to the objective function, (4), adding to the total transportation costs structured as weighted distance. A situation arises, however, when a particular resource has multiple options. If we think about an individual resource type, then perhaps there are alternative possibilities. For example, say that resource type 1 corresponds to raw material 1. There may be many vendors capable of supplying raw material 1. The production facility has need for only one vendor, the one that is least costly, assuming no difference in quality, pricing, etc. However, the one that is the most cost efficient depends on the location of the production facility because transportation costs are the primary concern. The implication is that in addition to the location decision, (
), there is also an allocation decision for each resource input. The
traditionally conceived formulation, (4)-(5), is therefore incomplete without taking into account alternative input options.
9
Consider this additional notation:
set of resource (input) points set of demand (market) points index of resource types (entire set ) set of points providing resource of type number of type
assignments to be made
{
The index refers to resource (raw material) and/or demand (market) points. Some of the points are resource inputs, , while others are demand (markets),
. Further, the set of
resources can be partitioned by types, where for any resource type ⋃
we have
. Without loss of generality, it can be further stipulated that where
resource
. The introduced variable,
and for all
⋂
, represents the allocation decision for each
. This notation allows for a more general formulation that includes multiple input
options for each type of resource.
The multiple-source, raw material allocation form of the Weber problem is introduced and formulated as follows:
Minimize
∑
Subject to
∑
√(
)
(
)
∑
√(
)
(
)
(6) (7)
10
(8) {
}
(9)
The objective, (6), remains consistent with the specification in (4)-(5), and (1). However, it now includes allocation decision variables so that only utilized points of actual service are included in the measurement of total cost, and subsequently locational siting. Constraints (7) indicate a prespecified number of members in each resource (input) type. Decision variable conditions are stipulated in constraints (8) and (9).
To illustrate the nuances of this new formulation, (6)-(9), let us return to something akin to the Weber location triangle example. In this case, assume that the number of types, | |, is two, where
1 (raw material 1, e.g., timber) and
one point from each type is needed, e.g.,
2 (raw material 2, e.g., coal), and that only for each . Figure 1 depicts a representative
situation, where there are two options for raw material 1 ( material 2 (
) and three options for raw
). There are also two markets to be served. The location and associated
amounts to be transported are known in advance. The additional complexity means possible choices for each raw material. Not all potential resource destinations will actually be utilized once the facility has been sited, so it is important to include in the decision making process the fact that there are alternatives to select from. This selection, or allocation, will necessarily impact what will ultimately be the best location for siting the production facility.
11
Figure 1. Extension of the Weber locational triangle.
Formalizing the example, it is possible to visualize the decision variables, unknowns, in this particular case. Figure 2 includes all decision variables for the model that would result, where the lines represent the allocations, many of which are potential allocation variables. The optimal production facility location, (
), would depend on which input points are included in the
system. The efficiency criteria in objective (6) provides a specification for identifying the best location for the facility, with the associated allocations reflecting only those points that are interacting with the facility. This represents the total transportation costs. If only one supplier of raw material 1 is necessary, then the associated since
1 type constraint (7) would be
. Similarly, if one supplier of raw material 2 is needed, then the associated
type constraint (7) would be
, 2
. Collectively, this means that two resource
(input) points are only to be considered in facility siting as ∑
, but which two is part of
the overall decision making process. Further, it is not just any two, but rather one from the set of type 1 (raw material 1) and one from the set of type 2 (raw material 2). In addition, the two markets are to be served.
12
Figure 2. Visualization of model decision variables.
Consistent with the classic Weber problem, if we assume that resource supplies are unlimited along with indistinguishable pricing among alternatives, then the least cost resource of each type would be selected based upon the location of the production facility.
Solution
The formal mathematical specification of the problem is clearly important, and highlights the nuances of production facility siting that are overlooked in classical interpretations of the Weber problem. Of course, the issue of being able to solve this problem to identify the best production facility location is also critical. It is likely that the classical interpretation of the Weber problem has been appealing precisely because it is amenable to exact (optimal) solution. The Weber locational triangle, or that reflected in (4)-(5), can be solved using a physical analog (or apparatus), the so called Varignon frame (see Varignon 1687, Wesolowsky 1993, Bjelland et al. 2014). Another possibility is isotims and isodapanes detailed in Weber (1909). Commercial and open source software for nonlinear optimization too may be an option. The primary approach, 13
however, for solving the Weber problem exactly has been that of Weiszfeld (1937) (see Wesolowsky and Love 1972, Ostresh 1978, Wesolowsky 1993, Church and Murray 2009, Weiszfeld and Plastria 2009). This method is based on the derivatives of (4) with respect to and with respect to . This information can be employed to structure an iterative scheme using an initial estimate/guess of the production facility location, ( be revised, or updated, to (
[ ]
[ ]
[ ]
[ ]
). This estimate can then
) and continues to be updated until it converges to the
optimal location.
The iterative scheme of Weiszfeld (1937) for solving (4)-(5) is based on the derivatives of (4) as follows:
∑
(
[
]
[
]
)
(
∑ √( [ ]
)
( [ ]
∑ )
( [ ]
Equation (10) relies on the estimate ( [
]
[
]
)
( [ ]
)
)
( [ ]
)
∑ √( [ ]
(
√( [ ]
)
). In theory
√( [ ]
)
[ ]
[ ]
)
(10)
) to then revise/update the next iteration,
, but in practice this process terminates when two consecutive
estimates are essentially the same. That is, √(
[
]
[ ])
(
[
]
[
])
, where
is a tolerance value representing zero.
The Weiszfeld solution approach reflected in equation (10), or its variants and physical analogs, assume that all inputs and outputs are to be serviced. However, the multiple-source, raw material allocation form of the Weber problem structured in (6)-(9) recognizes that some inputs will not be used, and therefore should not influence the production facility location selected. This means that the Weiszfeld approach cannot be directly applied in this case, unless of course the input and output points to be served are known in advance. Unfortunately this is not the case. However, the solution approach developed below exploits the partitioning of space in such a manner so as to reflect different service configurations. 14
Before presenting the developed solution approach, some preliminary methods used to delineate and define continuous space are introduced, namely Voronoi diagram and vector overlay. The Voronoi diagram has proven to be an efficient approach for allocation and partitioning of space, representing access, service assignments and trade areas (Feng and Murray 2018). A formal mathematical presentation of the Voronoi diagram may be found in Okabe et al. (2000). The Voronoi diagram is a widely relied upon method in computational geometry, location modeling and GIScience for partitioning space. Given a set of generator points , an individual Voronoi polygon associated with each generator point is defined in set theoretic terms as:
̂
{(
)
|
where any point, (
(
)(
)
(
), in the region
)(
)
(
)}
(11)
is contained within the Voronoi polygon of the closest
generator point. Further conditions are that the combination of all Voronoi polygons is equal to the region and that no two Voronoi polygons overlap. Formally, the conditions are:
⋃ ̂ ̂ ⋂̂
(12) (
)
(13)
This can be relevant for solving the multiple-source, raw material allocation Weber problem, (6)(9), if each input type is considered independently. That is, for input type , the associated points are known in advance,
. Further, one knows that the closest input to the production
facility site would be utilized as this will minimize total cost, stipulated in (6). Thus, identification of the associated Voronoi diagram for each input type provides a delineation of space that reflects which point would be served if the production facility was located in an associated
15
Voronoi polygon. Consider then the Voronoi diagram associated with input type , where :
{(
)
|
Whereas all points
(
)(
)
(
)(
(
)
)}
(14)
are considered in (11), the Voronoi polygon defined in (14) is only
associated with the subset having to do with type . This means that for the Voronoi diagram associated with type , we have the following:
(15)
⋃ (
⋂
In theory then, the
)
(16)
different Voronoi diagram layers each tell us something about which points
would be served by the production facility, if it is sited in a particular Voronoi polygon,
.
However, the optimal location of the production facility is dependent on serving each of the input types and all demand, so individual Voronoi diagram layers are of limited use. The potential benefit for solution is integration of the independent input layers.
Given the set of types , the result is | | Voronoi diagram layers, one for each set of generator points
. Integration of these layers is possible through vector-based overlay. de Berg et al.
(2008), Wei and Murray (2014) and Murray (2018) characterize the overlay operator as resulting in a maximally connected subset of faces that do not overlap. Specifically, a set of faces, , are identified having the properties that ⋃
and
. Deriving the set of faces
essentially requires evaluation of overlapping Voronoi polygons in order to create new polygons that represent areas of intersection. In set theoretic terms, the power set function,
( ),
enumerates combinations of objects, Voronoi polygons in this case. The number of potential 16
intersecting combinations is obtained using the power set, and is {
of members in set . Here,
| | }.
| |
, where | | is the number
Using the power set,
sub-set combinations, but many do not overlap, e.g.
⋂
( ), this identifies
| |
. Accordingly, the interest is in
intersections, so an operator is necessary that enumerates all possible face combinations. The operator ( ) is defined here to return all non-empty face combinations for
as
follows:
( )
{
|
⋂
⋃
( )
( )
}
(17)
The overlay operator, ( ), therefore produces the following set of faces:
( )
{|
( )
( )
⋃
}
(18)
Wei and Murray (2014) review polygon overlay in terms of computational complexity. The intent here is to merely indicate that unique faces can be identified through overlay,
Lemma 1. A face type ,
( ), corresponds to the closest allocation
, where
of each input
.
Proof: For any location in face ( implies that
( ).
(
)(
)
(
) )(
, suppose there exists ).
,
that is closer. This
However, this contradicts the definition of a Voronoi
polygon given in (14). Thus, face indicates the closest allocation for any input type k.
17
Given the set of faces,
( ), then we know that each face represents an area for which the
closest resource (input) type is known. Accordingly, identifying the optimal allocation along with the optimal production facility location involves evaluation of each face, within which the Weber point can be identified because the closest
resource/demand points are known. If each face
is evaluated, it can then be determined which face contains the optimal location for the production facility.
Figure 3. Pseudo code for exact algorithm to solve multiple-source, raw material allocation Weber problem.
In each face, the best facility site is that obtained using Weiszfeld algorithm for the associated closest allocations (e.g.,
) for each type . Evaluation of objective (6) for each face
indicates which one represents the optimal allocation scheme, and subsequently the lowest total transportation costs. The pseudo code for the algorithm associated with this process is given in Figure 3. After initialization, the Voronoi diagram for each resource type is identified, as specified in (14). This gives the set of Voronoi polygons, . With this set, the overlay operator is then applied, ( ), giving the set of faces, . For each face the optimal allocation variables,
,
can be identified through inspection. Using this for each face, the best facility site corresponding to the closest inputs for each type along with all demand is found using the Weiszfeld algorithm, (10), giving the location that best serves the face, (
), along with its objective value,
specified in (6). Upon termination, the optimal facility location, ( 18
), is identified.
,
Theorem 1. The solution algorithm given in Figure 3 identifies the optimal solution for the multiple-source, raw material allocation Weber problem, (6)-(9).
Proof: By lemma 1, face indicates the best allocation for any point ( for each input type , the best location in face , (
)
, implying
. These allocations along with the demand are used in (10) to find )
, with convergence and optimality proven in Weiszfeld
(1937) (see also Ostrech 1978, Weiszfeld and Plastria 2009, Gorner and Kanzow 2016) within the tolerance . Suppose there exists a face , but this contradicts ⋃
with a lower objective (6). This would imply that
. Thus, since every face is evaluated in Figure 3, the global
optimal must be identified.
Application Results
Three different empirical applications are utilized to demonstrate the nuances of the multiplesource, raw material allocation form of the Weber problem structured in this paper. Performance of the proposed solution algorithm is also reported and compared to results obtained using a commercial nonlinear solver, LINGO (ver. 18, Extended).
The first two applications are associated with the region shown in Figure 4, approximately 675 square kilometers in size (roughly 27 x 25 kilometers), with two resource types: raw material 1 (3 total) and raw material 2 (39 total). In the first application, there is one market to be served. Each resource type is depicted as an independent layer in Figure 4 along with the market layer. The total number of resource points is 42, so | | | |
. The number of markets is one, so
. For illustrative purposes, the amount of material shipped is the same for all locations, for each . The types in Figure 4 indicate
{raw material 1,raw material 2}, so | |
It is assumed that only one resource per type is needed, so 19
. The challenge is to
.
simultaneously find the best resource allocations along with the best facility location. The algorithm outlined in Figure 3 was implemented in Python and run in Windows 10 Pro on an Intel Xeon CPU E5-2650 v3 (2.30 GHz) with 96GB RAM desktop computer. ArcGIS (ver. 10.5) was utilized for data creation, management, manipulation, analysis and display along with ArcPy functionality. Total solution time was 9.24 seconds, with convergence of the Weiszfeld algorithm set at
.
Figure 4. Potential resource types and market layers.
One of the first steps in the solution algorithm (Figure 3) involves derivation of Voronoi polygons for each resource type, specification of which is defined in (12) along with conditions (13) and (14). Accordingly, Figure 5 details the Voronoi diagram for each resource type. For example, in Figure 5a the Voronoi polygons correspond to the least cost raw material 1 provider for each generator point. Similarly, Figure 5b indicates the least cost resource for raw material 2 for any location in the region. Voronoi diagrams for the two raw material inputs are shown for the region in Figure 6. Integration of the two layers, Figures 5a and 5b through overlay, (16), is necessary to obtain the set of faces. These faces are indicated in Figure 7, with 55 faces in this case (e.g., | |
). Each face represents a unique resource allocation if the industrial facility is
located within the face.
20
Figure 5. Closest assignment cost delineation (Voronoi polygon) by type.
Figure 6. Voronoi diagram layers by resource type.
21
Figure 7. Faces along with resource points by type and market.
In order to demonstrate an iteration of the algorithm, Figure 8 shows one face along with the closest (least cost) allocations to the two resource types and the market. The Weiszfeld initial and intermediate locations are depicted along with the best final location. Convergence was achieved in 179 iterations, with a total cost, objective (4), of 22,519.41 meters. The optimal location is detailed in Figure 9, with an objective (4) of 5,250.02 meters (obtained after 2,225 iterations after 11.84 seconds). One can observe the allocation as well as location decisions made in this case, representing the solution of lowest total transportation costs. For comparison purposes, the multiple-source, raw material allocation Weber problem, (6)-(9), was structured in LINGO and solved. The obtained solution had an objective of 11,863.63 meters, requiring 2.06 seconds (1,255 iterations). This is approximately 126% above the true optimum.
22
Figure 8. Intermediate and final Weiszfeld locations for a face (along with allocation).
Figure 9. Optimal facility location along with resource allocations.
A second application considers the same raw material input options as the previous case but now involves four markets, and is shown in Figure 10. Thus, the total number of resource points is | |
42 and the number of markets is | |
4. The local Weber solutions (initial and
intermediate) for each face are indicated in Figure 11. The optimal solution in the four market case is given in Figure 12, along with associated allocations. Total transportation distances, objective (6), are 36,890.55 meters, requiring 18.40 seconds of processing time (789 iterations). 23
Across the other faces (Figure 11), the objective averaged 46,719.63 meters with a high of 63,762.11 meters. Solution using LINGO identified a site and allocation with an objective value of 39,072.72 (requiring 0.63 seconds to perform 678 iterations). This solution is approximately 6% higher than the optimum found using the Voronoi based algorithm.
Figure 10. Application involving four markets.
Figure 11. Individual face evaluations.
24
Figure 12. Optimal solution involving four markets.
The third siting application involves fruit juice production in San Joaquin County, California. An entrepreneur has identified a market niche for juice(s) that is high quality, chemical free, 100% fruit made from locally grown sources. Based on this, they are seeking to lease or purchase a factory property for juice production. The juice will be distributed by Riley’s, a small local grocery chain known for high quality products. San Joaquin County is located east of San Francisco with primary cities being Stockton and Lodi. The 2018 total population of San Joaquin County is approximately 752,660 spread across more than 1,400 square miles. Crop producer information was obtained from the San Joaquin County Agricultural Commissioner (2017). Eight different fruit farms types were identified as raw material inputs, so | |
8. The types are as
follows, with the number of indicated farms noted: cherry (870), apple (91), peach (91), watermelon (41), berries (38), apricot (18), nectarine (14) and plum (13), and are shown in Figure 13. The total number of potential resource points is therefore | |
1,176. There are four
Riley’s, representing the markets for the finished product in this region (| | that material shipped is the same for all locations,
4). It is assumed
for each . The juice factory requires
one farm to be selected for each raw material input type. Only one resource per type is needed, so
.
25
Figure 13. Juice processing facility siting.
26
Figure 14. Resulting faces for juice processing facility application.
27
Figure 15. Optimal location and associated allocations for juice processing facility.
The Voronoi based algorithm identifies 4,274 unique faces (Figure 14) for evaluation. The total solution time in this case was 21.73 seconds. The objective was 96,226.07 meters, meaning that the total transportation costs would be a function of this total distance. The solution and 28
associated allocations are depicted in Figure 15. For comparison purposes, solution using LINGO required 65.37 seconds (4,987 iterations), with the identified site resulting in an objective of 101,636.67 meters. This is nearly 6% above the optimum found using the developed solution algorithm.
As a final basis of comparison, randomly generated application instances were created. In total, 14 unique problems were considered. The total number of resource locations range from | |
4,991 to | |
99,997, the number of types span from | |
of markets vary between | |
1 and | |
5 to | |
15, and the number
9. Consistent with previous problems, the amount
of material shipped is the same for all locations. Solution summaries are reported in Table 1, indicating objective function value (feet) and solution time (seconds) along with other information for both the Voronoi based algorithm and LINGO. Problem instance particulars are noted in the Details column. Most of the 14 applications in Table 1 could not be solved for various reasons using LINGO. However, the Voronoi algorithm developed in the paper solved all problems optimally, requiring less than 15 minutes in the most computationally challenging case (R14). For some instances, the LINGO solver did find a local optima, but only came within 107% of optimality once (R4) and was over 225% in the worst case (R7). Computationally, the Voronoi based algorithm is fast and efficient, enabling all problems to be solved optimally.
29
Table 1. Comparison of developed exact approach to commercial solver. Voronoi based algorithm (exact/optimal) ID
Details
LINGO (nonlinear approximate)
Objective
Iterations
Faces, |L|
Time (sec)
Objective
Iterations
Time (sec)
16,546.10
352,185
23,171
89.76
27,079.71
13,244
1,127.08
181,925.91
135,154
18,340
50.93
269,866.08
11,451
1,067.72
486,407.54
74,615
23,167
44.10
-
15,679
4,816.30
755,205.22
58,372
21,494
36.24
807,902.37
12,834
1,245.29
346,382.10
290,919
49,298
119.90
652,771.48
12,800
1,325.67
230,786.39
281,068
75,449
131.84
358,727.59
16,233
3,465.42
268,596.14
466,675
46,563
129.60
604,945.71
33,826
11,163.26
391,927.53
588,417
94,082
181.94
-
17,378
11,697.64
318,782.12
648,629
151,601
375.77
-
48,201
40442.02
292,281.48
1,191,639
107,990
300.01
|K|=5
R1
|I|=4999 |M|=1 |K|=5
R2
|I|=4997 |M|=3 |K|=5
R3
|I|=4996 |M|=6 |K|=5
R4
|I|=4991 |M|=9 |K|=10
R5
|I|=4997 |M|=3 |K|=15
R6
|I|=4997 |M|=3 |K|=5
R7
|I|=9997 |M|=3 |K|=10
R8
|I|=9997 |M|=3 |K|=15
R9
|I|=9997 |M|=3
R10
|K|=5
30
*
|I|=29997 |M|=3 |K|=5
R11
|I|=39994
436,870.10
538,248
177,676
415.80
**
213,059.92
1,801,972
232,292
812.62
**
341,157.06
637,402
174,557
537.11
**
175,377.26
3,071,005
412,692
891.25
**
|M|=6 |K|=5
R12
|I|=49997 |M|=3 |K|=5
R13
|I|=59994 |M|=6 |K|=5
R14
|I|=99997 |M|=3
- Problem terminated with local optima, but infeasible solution * Problem had no feasible solution after 20 hours of processing ** Problem could not be solved due to insufficient computer memory
Discussion
The motivation for the paper was to demonstrate that legacy interpretations of the Weber problem mischaracterize important problem nuances. Although researchers have added problem elements not addressed by Weber, e.g. negative weights and barriers, virtually all have neglected a rich set of problem elements that have been overlooked (Church, 2019). One of these elements is the presence of multiple localized sources of a given raw material, the primary subject of this paper. Weber recognized that such sources are likely to exist for any type of raw material, whether it be sheet metal, iron ore, etc. The application results demonstrate the richness associated with this broader interpretation. However, perhaps a more significant takehome message is that the use of the classically conceived Weber problem, (4)-(5), will result in
31
substantially increased total transportation cost if there are indeed multiple options from which to choose from for a given resource type. For example, in the single market application, the location derived using (4)-(5) would have an objective value of 29,572.48 meters when evaluated as a multiple-source, raw material allocation Weber problem, (6)-(9). This is 563% higher than the optimum of 5,250.02 shown in Figure 9. Similarly, for the four market application, the location derived using (4)-(5) has a total transportation cost of 49,871.30 meters, which is 135% higher than the more efficient facility location identified using multiplesource, raw material allocation Weber problem, (6)-(9), of 36,890.55 meters (Figure 12). The point then is that one could not simply use all resource inputs to find a good solution if only a subset of inputs is actually necessary. Correct model specification and solution matters.
A second point that needs further discussion is that the formulation of the multiple-source, raw material allocation Weber problem introduced in (6)-(9) is indeed new and unique. While it accounts for allocation to a pre-specified number of resources inputs, it is dramatically different from the location-allocation (multi-facility Weber) problem discussed in Cooper (1963). One issue is that the location-allocation problem focuses on multiple facilities, but here we only seek one facility. Further, the location-allocation problem requires that all resource/demand points are allocated to a facility. This is not the case with the multiple-source, raw material allocation Weber problem, (6)-(9).
A third point is associated with computational processing. The Voronoi based algorithm is an exact approach, capable of identifying an optimal solution. As problem size grows, it appears to scale up rather well. There is clearly an overhead associated with derivation of the Voronoi diagram as well as overlay to obtain unique faces. Such effort proved to be well worth it when examining the larger problem instances in Table 1. Of course, the implementation reported in this paper relied on Arcpy functionality for this, which is part of the ArcGIS commercial software package. In addition to being costly, it is likely to be computationally inefficient in many ways. Part of this is associated with use through Arcpy, but also implementation. This becomes an important issue when you consider that 36.40% of solution time on average for the application instances in Table 1 was devoted to Voronoi and intersection processing (nearly 70% for R11 32
and R13). The two example problems (| | (| |
42) and the juice production facility application
1180) spent over 95% and 80%, respectively, on this GIS processing (Voronoi diagram and
overlay). Native programing or open source options may prove particularly valuable in reducing solution time. Of course this was not an issue for the problem applications considered in this paper.
A final note is that there are a number of assumptions applied to the classic Weber problem, and for consistency and comparison they were also assumed in the detailed approach. Of course, extension of the multiple-source, raw material allocation Weber problem, (6)-(9), is possible to account for a range of additional considerations, such as price variation, quality issues, supply disparities, etc. This is left for future research.
Conclusions
This paper discussed and reviewed the problem addressed by Weber (1909) along the lines originally intended. Most interpretations of the classical Weber problem are overly restrictive and narrow. Specifically, not addressed in previous research are realities of selection from among a set of options for inputs in serving one or more markets. This no doubt complicates the planning problem because the siting decision is dependent on the locations to be utilized and quantities being transported. The complexities of multiple potential input options increases the difficulty of the problem substantially, necessitating new solution approaches. Introduced in the paper is the multiple-source, raw material allocation Weber problem, (6)-(9), to simultaneously account for input selection (allocation) and facility siting (location). Further, an algorithm capable of exact solution was developed for identifying an optimal production facility location. This was accomplished through the use of GIS and computational geometry techniques. These techniques effectively exploit spatial knowledge, and as a result enable the problem to be solved optimally. Application results were presented to highlight the distinctions of what Weber
33
(1909) actually described, but also to demonstrate the utility and computational efficiency of the developed solution algorithm.
34
References Berman, O., Drezner, Z., & Wesolowsky, G. O. (2007). The transfer point location problem. European Journal of Operational Research, 179(3), 978-989. Bischoff, M., & Klamroth, K. (2007). An efficient solution method for Weber problems with barriers based on genetic algorithms. European Journal of Operational Research, 177(1), 22-41. Bjelland, B. D., Montello, D. R., Fellmann, J. D., Getis, A., & Getis, J. (2014). Human Geography: Landscape of Human Activities, 12th edition. New York: McGraw Hill. Bozarth, C. B., & Handfield, R. B. (2019). Introduction to Operations and Supply Chain Management, 5th edition. Pearson. Brimberg, J., Drezner, Z., Mladenović, N., & Salhi, S. (2014). A new local search for continuous location problems. European Journal of Operational Research, 232(2), 256-265. Canbolat, M. S., & Wesolowsky, G. O. (2010). The rectilinear distance Weber problem in the presence of a probabilistic line barrier. European Journal of Operational Research, 202(1), 114-121. Canbolat, M. S., & Wesolowsky, G. O. (2012). On the use of the Varignon frame for single facility Weber problems in the presence of convex barriers. European Journal of Operational Research, 217(2), 241-247. Capello, R. (2014). Classical contributions: Von Thunen, Weber, Christaller, Losch. In, Handbook of Regional Science, edited by M.M. Fischer and P. Nijkamp, 507-526. Berlin: SpringerVerlag. Carrizosa, E., Muñoz-Márquez, M., & Puerto, J. (1998). The Weber problem with regional demand. European Journal of Operational Research, 104(2), 358-365. Christaller, W. (1933). Die zentralen Orte in Süddeutschland: eine ökonomisch-geographische Untersuchung über die Gesetzmässigkeit der Verbreitung und Entwicklung der Siedlungen mit städtischen Funktionen.
35
Church, R. L. (2019). Understanding the Weber location paradigm. In, Festschrift for Zvi Drezner, edited by H.A. Eiselt and V. Marianov, 69-88. New York: Springer. Church, R. L., & Murray, A. T. (2009). Business site selection, location analysis, and GIS. Hoboken, NJ: John Wiley & Sons. Cooper, L. (1963). Location-allocation problems. Operations Research, 11(3), 331-343. Drezner, T., Drezner, Z., & Schobel, A. (2018). The Weber obnoxious facility location model: a big arc small arc approach. Computers & Operations Research, 98, 240-250. Drezner, Z., & Guyse, J. (1999). Application of decision analysis techniques to the Weber facility location problem. European Journal of Operational Research, 116(1), 69-79. de Berg, M., Cheong, O., Van Kreveld, M. & Overmars, M. (2008). Computational Geometry: Algorithms and Applications, 3rd edition. Berlin: Springer. Feng, X., & Murray, A. T. (2018). Allocation using a heterogeneous space Voronoi diagram. Journal of Geographical Systems, 20(3), 207-226. Francis, R. L., McGinnis, L. F. & White, J. A. (1992). Facility layout and location: an analytical approach, second edition. Englewood Cliffs, NJ: Prentice-Hall. Gorner, S., & Kanzow, C. (2016). On Newton’s Method for the Fermat–Weber Location Problem. Journal of Optimization Theory and Applications, 170(1), 107-118. Haines, V. G. (1970). Business relocation: a guide to moving a business. Random House Business. Hoover, E. M. (1937). Location theory and the shoe and leather industries. Cambridge: Harvard University Press. Hotelling, H. (1990). Stability in competition. In, The Collected Economics Articles of Harold Hotelling, pp. 50-63. New York: Springer. Irawan, C. A., Salhi, S., Luis, M., & Azizi, N. (2017). The continuous single source location problem with capacity and zone-dependent fixed cost: Models and solution approaches. European Journal of Operational Research, 263(1), 94-107. Isard, W. (1956). Location and space-economy. New York: Wiley. 36
Isard, W. (1960). Methods of Regional Analysis: an Introduction to Regional Science. Cambridge: MIT. Ivanov, D., Tsipoulanidis, A., & Schönberger, J. (2019). Global supply chain and operations management. A Decision-Oriented Introduction to the Creation of Value, 2nd edition. Springer. Kelly, L. M. (1969). Optimal distance configurations. In Recent Progress in Combinatorics. Academic Press New York. Klose, A., & Drexl, A. (2005). Facility location models for distribution system design. European Journal of Operational Research, 162(1), 4-29. Kuhn, H. W., & Kuenne, R. E. (1962). An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics. Journal of Regional Science, 4(2), 21-33. Launhardt, W. (1872). Kommercielle Tracirung der Verkehrswege. Architekten-und Ingenieurverein. Love, R. F., Morris, J. G., & Wesolowsky, G. O. (1988). Facilities location: models and methods. North-Holland, NY, NY. Lösch, A. (1940). The Economics of Location, 1954. New Haven: Yale University Press. Miehle, W. (1958). Link-length minimization in networks. Operations Research, 6(2), 232-243. Murray, A. T. (2010). Advances in location modeling: GIS linkages and contributions. Journal of Geographical Systems, 12(3), 335-354. Murray, A. T. (2018). Complexities in spatial center derivation. Transactions in GIS, 22(6), 13351350. Murray, A. T., Matisziw, T. C., Wei, H., & Tong, D. (2008). A geocomputational heuristic for coverage maximization in service facility siting. Transactions in GIS, 12(6), 757-773. Okabe, A., Boots, B., Sugihara, K. & Chiu, S.N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, second edition. New York: Wiley.
37
Ostresh Jr, L. M. (1978). On the convergence of a class of iterative methods for solving the Weber location problem. Operations Research, 26(4), 597-609. Perreur, J. (1998). Industrial location theory in German thought–Launhardt and Weber. Recherches Économiques de Louvain/Louvain Economic Review, 64(1), 75-96. Pinto, J. V. (1977). Launhardt and location theory: Rediscovery of a neglected book. Journal of Regional Science, 17(1), 17-29. Plastria, F. (2016a). How bad can the centroid be?. European Journal of Operational Research, 252(1), 98-102. Plastria, F. (2016b). Up-and downgrading the euclidean 1-median problem and knapsack Voronoi diagrams. Annals of Operations Research, 246(1-2), 227-251. Reid, R. D., & Sanders, N. R. (2016). Operations Management: an integrated approach, 6th edition. New York: Wiley. San Joaquin County Agricultural Commissioner (2017). https://www.sjgov.org/department/agcomm/gis_crop_data (accessed 3/4/19). Schöbel, A., & Scholz, D. (2014). A solution algorithm for non-convex mixed integer optimization problems with only few continuous variables. European Journal of Operational Research, 232(2), 266-275. Schultz, G. P. (1969). Facility planning for a public service system: Domestic solid waste collection. Journal of Regional Science, 9(2), 291-307 Simpson, T. (1750). The doctrine and application of Fluxions, Printed by J. Nourse, London. Stevenson, W. J. (2018). Operations Management, 13th edition. New York: McGraw Hill. Vardi, Y., & Zhang, C. H. (2001). A modified Weiszfeld algorithm for the Fermat-Weber location problem. Mathematical Programming, 90(3), 559-566. Varignon, P. (1687). Projet d’une nouvelle mechanique. Chez C. Jombert, Paris.
38
Venkateshan, P., Ballou, R. H., Mathur, K., & Maruthasalam, A. P. (2017). A Two-echelon joint continuous-discrete location model. European Journal of Operational Research, 262(3), 1028-1039. Vergin, R. C., & Rogers, J. D. (1967). An algorithm and computational procedure for locating economic facilities. Management Science, 13(6), B-240-B-254. von Thunen, J. V. (1826). Der isolierte Staat. Beziehung auf Landwirtschaft und Nationalökonomie. Weber, A. (1909). Uber den Standort der Industrien (Theory of the location of industries) (translated with an introduction and notes by C.J. Friedrich, 1929). University of Chicago Press, Chicago. Wei, H., Murray, A. T., & Xiao, N. (2006). Solving the continuous space p-centre problem: planning application issues. IMA Journal of Management Mathematics, 17(4), 413-425. Wei, R., & Murray, A. T. (2014). Evaluating polygon overlay to support spatial optimization coverage modeling. Geographical Analysis, 46(3), 209-229. Weiszfeld, E. (1937). Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Mathematical Journal, First Series, 43, 355-386. Weiszfeld, E., & Plastria, F. (2009). On the point for which the sum of the distances to n given points is minimum. Annals of Operations Research, 167(1), 7-41. Wersan, S. J., Quon, J. E., & Charnes, A. (1962). Systems analysis of refuse collection and disposable practices. American Public Works Association. Year-Book, 195-211. Wesolowsky, G. O. (1993). The Weber problem: history and perspectives. Location Science, 1(1): 5-24. Wesolowsky, G. O., & Love, R. F. (1972). A nonlinear approximation method for solving a generalized rectangular distance Weber problem. Management Science, 18(11), 656-663. Yao, J., & Murray, A. T. (2013). Continuous surface representation and approximation: spatial analytical implications. International Journal of Geographical Information Science, 27(5), 883-897. 39
Yao, J., & Murray, A. T. (2014). Serving regional demand in facility location. Papers in Regional Science, 93(3), 643-662.
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Single Facility Siting Involving Allocation Decisions Alan T. Murray , Richard L. Church , Xin Feng PII: DOI: Reference:
S0377-2217(20)30084-9 https://doi.org/10.1016/j.ejor.2020.01.047 EOR 16301
To appear in:
European Journal of Operational Research
Received date: Accepted date:
5 December 2018 21 January 2020
Please cite this article as: Alan T. Murray , Richard L. Church , Xin Feng , Single Facility Siting Involving Allocation Decisions, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.01.047
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Highlights
Reviews classic Weber problem for industrial facility location Formulates extended interpretation of Weber problem Develops an exact solution algorithm based on the use of GIS Empirical applications demonstrate utility of developed approach
Single Facility Siting Involving Allocation Decisions Alan T. Murray Richard L. Church Xin Feng
Department of Geography University of California at Santa Barbara Santa Barbara, CA 93106, USA
(Email: [email protected])
December 27, 2019
1
Abstract
Single facility siting is often viewed as the most basic of location planning problems. It has been approached by many researchers, across a range of disciplines, and has a rich and distinguished history. Much of this interest reflects the general utility of single facility siting, but also the mathematical and computational advancements that have been made over past decades to support better decision making. This paper discusses a recently rediscovered form of Weber’s classic single facility location problem that is both important and relevant in contemporary planning and decision making. This form of the Weber problem involves locating a production plant where there are multiple sources of each needed raw material (input) distributed throughout a region. This means that the selection of a given raw material source may vary depending on the plant location. In essence, this makes the problem non-convex, even when locating only one production plant. We review elements of the Weber problem that have been addressed in the literature along with proposed solution techniques. In doing so, we highlight elements of the problem originally noted by Weber, but to date have not been operationalized in practice – allocation selection among multiple sources of given raw material inputs. A problem formulation involving allocation decisions for this generalization is derived and an optimal solution approach is developed. Application results demonstrate the significance of addressing important planning characteristics and the associated nuances that result.
Keywords: Location; Weber; spatial optimization; Voronoi diagram; GIS
2
Introduction
There are two basic forms of location models that have been proposed in the literature: those defined on a network and those involving continuous space (e.g., Cartesian plane, sphere, etc.). Although both problem domains are important, location science (or analysis or modeling) can trace its roots to formalisms first defined on a Cartesian plane specifically involving the Euclidean distance measure. Approaches used to solve continuous space problems and network based problems rarely intersect due to significant differences in representation, though there are exceptions such as the noted mixed use of continuous and discrete solution approaches by Brimberg et al. (2014). One of the true classic location problems recognized in operations research, economics and production and operations management is referred to as the Weber problem, although others have also been credited with its introduction, such as Fermat, Torricelli, Cavalieri, and Simpson (see Wesolowsky 1993). As proposed by Weber (1909), the problem is defined on a continuous plane and involves finding the best location for a factory or production plant so as to minimize associated transportation costs in shipping raw materials or components to the factory and in shipping the final product to retail centers, where distances are measured using the Euclidean metric. Accordingly, the transportation costs are viewed as critically important under conditions where land acquisition and labor costs are constant across a region. What has made this basic problem of such great interest is that economic efficiency is formalized, and that the importance of spatial location is inherently central to notions of efficiency. Further, the simplicity of this problem along with mathematical specificity make clear that it reflects many siting and planning contexts, and can be extended in different ways, as demonstrated in Francis et al. (1992), Carrizosa et al. (1998), Berman et al. (2007), Bischoff and Klamroth (2007), Canbolat and Wesolowsky (2010, 2012) and Venkateshan et al. (2017), among others.
Of great significance is that the work of Weber (1909) remains relevant today, arguably more so than it ever has been. The problem has been portrayed as one where a factory is to be sited 3
given two localized raw material inputs and one market to which the final products are to be shipped. This is commonly referred to as the Weber locational triangle (see for example Isard 1960, Wesolowsky 1993, Capello 2014, among others). Weber (1909) actually referred to this as a locational figure as he purposely made his drawings as simple as possible to convey the issue at hand, in this case highlighting the need for locating one of more facilities dependent on localized raw materials in order to serve retail centers. Accordingly, the simplest form could involve just three points, two raw material sources and one market, to illustrate the basic concept. His descriptions of this problem clearly indicate that each raw material may be sourced from a number of discrete locations and that the selection of a given needed raw material would be from that location which would involve the least transport cost in bringing the necessary amounts to the facility. Hence, raw materials would be allocated in order to minimize transport costs, and those allocations would be a function of the positioning of the plant. Whereas, the literature has treated the single plant location problem of Weber (1909) as focusing only on location (no material allocations), the real intent was to describe a true location-allocation problem. This is the subject of this paper. Surprisingly given the widespread interest in Weber (1909), this interpretation to address resource allocation was only recently uncovered in a review by Church (2019).
In the following sections we formulate what many describe as the classic Weber problem. We follow this with a discussion of how this problem has been solved, review its relationship to the center of gravity problem, and highlight the use of associated models in the fields of business and productions and operations management. This is followed by an overview of how the Weber problem has been extended to add relevant features for specific applications. We then develop a model formulation that includes the allocation of possible raw material or component sources in selecting a location for a production facility in order to serve a set of demands and markets. A solution approach is then derived that exploits spatial properties using Voronoi diagrams. Application results are presented to highlight the capabilities of both the new model formulation as well as the solution approach. The paper ends with discussion and concluding comments.
4
Background
Consider the illustrative example of the problem attributed to Weber (1909): find the production plant location, (
), that minimizes total costs, where costs are a function of the
amount of materials shipped and the distance it must be transported. This simple problem can be expressed mathematically as:
Minimize )
√( √(
)
(
(
)
√(
)
(
)
)
(1)
where there are two required raw materials (or components) and one market. For example, point 1 is the source location of raw material 1 at ( second needed raw material at (
), point 2 is the source location for the
), and point 3 is the location of the market (or point of
demand) at (
). Associated with each point is attribute information, such as the amount to
be shipped ( ,
and
market, (
), (
) from/to the factory and the location of the raw materials and ) and (
), respectively. Thus, what remains is finding the optimal
location for siting the production plant, (
). One option is the physical analog frame
attributed to Varignon (1687). A second approach is the use of isotims (lines of equal cost from a given source or to a given market) and isodapanes (lines of equal total transport cost from the associated sources and to the markets), as proposed by Weber (1909) and employed by others (e.g., Isard 1956). A third is an iterative approach outlined in Weiszfeld (1937) obtained through calculus using derivatives. This has been widely used, with one slight modification to handle the case when a trial solution or a final solution coincides with one of the input/output points (Wesolowsky and Love 1972, Ostresh 1978). Worthy of note is that the Weiszfeld algorithm has been rediscovered since its original appearance by Miehle (1958), Kuhn and Kuenne (1962), Cooper (1963), and Vergin and Rodgers (1967), among others.
5
Related to the simple form of the Weber problem, (1), is the following approach which minimizes weighted squared distances:
Minimize
((
)
(
) )
((
)
(
) )
((
)
(
) )
The difference in (2) is the squaring of Euclidean distance, which effectively removes the radicals. This is the center of gravity problem. Through calculus using derivatives, a closed form solution is possible. The result is an optimal solution for (2) as follows:
(3a) (3b)
Although most transport cost functions are reasonably approximated by the weighted Euclidean distance in (1), this is not true for weighted squared distances in (2). Plastria (2016a) and others have noted the significant problems associated with cost approximation using squared distances (see also Church and Murray 2009). Nevertheless, the closed form solution in (3) has been appealing to use for many due to its simplicity and ease of application. Unfortunately (and incorrectly!), some call (1) a center of gravity problem (e.g., Ivanov et al. 2019), which means that there appears to be some confusion between these two models (see also Plastria 2016a). Of course, mathematically they are not the same nor will they identify the same location as an optimal solution (Wesolowsky and Love 1972, Church and Murray 2009, Plastria 2016a). Interestingly, AIMMS recently released a “Center of Gravity Navigator” as a “quantitative method for locating distribution facilities based upon distance and the weight of demand/or supply.” 1 It is unclear as to whether they actually support solving the center gravity model or
1
https://aimms.com/english/news-archive/aimms-launches-center-gravity-app/
6
(2)
rather the Weber problem. Whichever is the case, both the center of gravity and the Weber approaches are central models in production and operations management, with most textbooks describing how to solve such models along with examples of how they can be applied in an industrial setting (see Stevenson 2018, Bozarth and Handfield 2019, Ivanov et al. 2019, among others). Given that the Weber problem is virtually always noted in distribution system design (e.g., Klose and Drexl 2005), described as a principal approach in warehouse, production, and distribution design in textbooks (e.g., Stevenson 2018, Bozarth and Handfield 2019, Ivanov et al. 2019), and that support tools have been designed to solve it with interesting map interfaces (e.g., Murray 2018), it is clear that it is a de facto analytical approach to site a production facility and/or adjust a location within a supply chain.
Worth highlighting as well is that even though most of the applications of single facility siting have focused on the use of Euclidean distance, a number have proposed the use of other metrics, including rectilinear or Manhattan distance (Wersan et al. 1962, Vergin and Rodgers 1967, Love et al. 1988. Francis et al. 1992). In fact, many textbooks in production and operations management cover the use of both Euclidean and rectilinear distance measure in the application of the Weber problem (e.g., Reid and Sanders 2016, Ivanov et al. 2019). Research on solution and extension of the Weber problem has been extensive. Work continues to explore solution properties, and ways to make methods more computationally efficient (e.g., Vardi and Zhang 2001, Gorner and Kanzow 2016). However, problem nuances, such as regret (Drezner and Guyse 1999), regional representation (Yao and Murray 2013, 2014), demand uncertainty (Plastria 2016b), and multi-facility extension (Irawan et al. 2017, Drezner et al. 2018), among others, continue to be addressed in order to better reflect the realities of encountered planning contexts. The Weber problem is indeed a fundamental approach utilized across many different disciplines and applied in a range of planning contexts.
One of the key elements raised by Weber (1909) was that there could be several sources of a given raw material or product component, and that it would be most efficient to use that source
7
which offered the closest transport. That is, Weber (1909) noted that his location construct would require allocation decisions when multiple sources of a given needed input material were present. This is an important aspect of the original work that appears to have been completely overlooked until recently (see Church 2019). Most researchers have viewed Weber as focusing on location only, creating a cost function that was convex and solvable using a descent algorithm like that of Weiszfeld (1937) and Ostresh (1978). However, these approaches in and of themselves cannot completely address the situation when multiple sources of a given raw material are present as this creates a cost service that is non-convex. The reason is that allocation decisions must be simultaneously addressed as well.
Mathematical Model
The locational triangle characterization of the Weber problem was mathematically structured in (1). This problem can be generalized for any number of raw material inputs or market outputs. Consider the following notation:
index of resource (raw material)/demand (market) points total amount to be shipped to/from point (
(
)
)
geographic coordinates of point
location to site production facility
Again, of particular importance is (
), the problem decision variables. These indicate the
location, as a coordinate pair, where the production facility is to be sited. With this notation, the traditionally conceived version of the Weber problem is:
8
Minimize
∑
√(
)
(
)
(4) (5)
The objective, (4), seeks the minimum total weighted distance from the sited facility to all resource and demand points. This function reflects costs associated with the transportation of materials and goods. Constraints (5) formalize that the decision variables could be positive or negative, depending on the coordinate system for the region being evaluated.
The model formulation, (4)-(5), is a necessary step in structuring this problem, but in order for it to be of practical use and significance, a solution method is needed. In this particular case, methods for solving (4)-(5) do exist, and will be discussed in the next section.
Explicitly structured in (4)-(5) is that all resources and markets interact with the sited production facility, impacting where it should be located. That is, points representing input and output destinations all contribute to the objective function, (4), adding to the total transportation costs structured as weighted distance. A situation arises, however, when a particular resource has multiple options. If we think about an individual resource type, then perhaps there are alternative possibilities. For example, say that resource type 1 corresponds to raw material 1. There may be many vendors capable of supplying raw material 1. The production facility has need for only one vendor, the one that is least costly, assuming no difference in quality, pricing, etc. However, the one that is the most cost efficient depends on the location of the production facility because transportation costs are the primary concern. The implication is that in addition to the location decision, (
), there is also an allocation decision for each resource input. The
traditionally conceived formulation, (4)-(5), is therefore incomplete without taking into account alternative input options.
9
Consider this additional notation:
set of resource (input) points set of demand (market) points index of resource types (entire set ) set of points providing resource of type number of type
assignments to be made
{
The index refers to resource (raw material) and/or demand (market) points. Some of the points are resource inputs, , while others are demand (markets),
. Further, the set of
resources can be partitioned by types, where for any resource type ⋃
we have
. Without loss of generality, it can be further stipulated that where
resource
. The introduced variable,
and for all
⋂
, represents the allocation decision for each
. This notation allows for a more general formulation that includes multiple input
options for each type of resource.
The multiple-source, raw material allocation form of the Weber problem is introduced and formulated as follows:
Minimize
∑
Subject to
∑
√(
)
(
)
∑
√(
)
(
)
(6) (7)
10
(8) {
}
(9)
The objective, (6), remains consistent with the specification in (4)-(5), and (1). However, it now includes allocation decision variables so that only utilized points of actual service are included in the measurement of total cost, and subsequently locational siting. Constraints (7) indicate a prespecified number of members in each resource (input) type. Decision variable conditions are stipulated in constraints (8) and (9).
To illustrate the nuances of this new formulation, (6)-(9), let us return to something akin to the Weber location triangle example. In this case, assume that the number of types, | |, is two, where
1 (raw material 1, e.g., timber) and
one point from each type is needed, e.g.,
2 (raw material 2, e.g., coal), and that only for each . Figure 1 depicts a representative
situation, where there are two options for raw material 1 ( material 2 (
) and three options for raw
). There are also two markets to be served. The location and associated
amounts to be transported are known in advance. The additional complexity means possible choices for each raw material. Not all potential resource destinations will actually be utilized once the facility has been sited, so it is important to include in the decision making process the fact that there are alternatives to select from. This selection, or allocation, will necessarily impact what will ultimately be the best location for siting the production facility.
11
Figure 1. Extension of the Weber locational triangle.
Formalizing the example, it is possible to visualize the decision variables, unknowns, in this particular case. Figure 2 includes all decision variables for the model that would result, where the lines represent the allocations, many of which are potential allocation variables. The optimal production facility location, (
), would depend on which input points are included in the
system. The efficiency criteria in objective (6) provides a specification for identifying the best location for the facility, with the associated allocations reflecting only those points that are interacting with the facility. This represents the total transportation costs. If only one supplier of raw material 1 is necessary, then the associated since
1 type constraint (7) would be
. Similarly, if one supplier of raw material 2 is needed, then the associated
type constraint (7) would be
, 2
. Collectively, this means that two resource
(input) points are only to be considered in facility siting as ∑
, but which two is part of
the overall decision making process. Further, it is not just any two, but rather one from the set of type 1 (raw material 1) and one from the set of type 2 (raw material 2). In addition, the two markets are to be served.
12
Figure 2. Visualization of model decision variables.
Consistent with the classic Weber problem, if we assume that resource supplies are unlimited along with indistinguishable pricing among alternatives, then the least cost resource of each type would be selected based upon the location of the production facility.
Solution
The formal mathematical specification of the problem is clearly important, and highlights the nuances of production facility siting that are overlooked in classical interpretations of the Weber problem. Of course, the issue of being able to solve this problem to identify the best production facility location is also critical. It is likely that the classical interpretation of the Weber problem has been appealing precisely because it is amenable to exact (optimal) solution. The Weber locational triangle, or that reflected in (4)-(5), can be solved using a physical analog (or apparatus), the so called Varignon frame (see Varignon 1687, Wesolowsky 1993, Bjelland et al. 2014). Another possibility is isotims and isodapanes detailed in Weber (1909). Commercial and open source software for nonlinear optimization too may be an option. The primary approach, 13
however, for solving the Weber problem exactly has been that of Weiszfeld (1937) (see Wesolowsky and Love 1972, Ostresh 1978, Wesolowsky 1993, Church and Murray 2009, Weiszfeld and Plastria 2009). This method is based on the derivatives of (4) with respect to and with respect to . This information can be employed to structure an iterative scheme using an initial estimate/guess of the production facility location, ( be revised, or updated, to (
[ ]
[ ]
[ ]
[ ]
). This estimate can then
) and continues to be updated until it converges to the
optimal location.
The iterative scheme of Weiszfeld (1937) for solving (4)-(5) is based on the derivatives of (4) as follows:
∑
(
[
]
[
]
)
(
∑ √( [ ]
)
( [ ]
∑ )
( [ ]
Equation (10) relies on the estimate ( [
]
[
]
)
( [ ]
)
)
( [ ]
)
∑ √( [ ]
(
√( [ ]
)
). In theory
√( [ ]
)
[ ]
[ ]
)
(10)
) to then revise/update the next iteration,
, but in practice this process terminates when two consecutive
estimates are essentially the same. That is, √(
[
]
[ ])
(
[
]
[
])
, where
is a tolerance value representing zero.
The Weiszfeld solution approach reflected in equation (10), or its variants and physical analogs, assume that all inputs and outputs are to be serviced. However, the multiple-source, raw material allocation form of the Weber problem structured in (6)-(9) recognizes that some inputs will not be used, and therefore should not influence the production facility location selected. This means that the Weiszfeld approach cannot be directly applied in this case, unless of course the input and output points to be served are known in advance. Unfortunately this is not the case. However, the solution approach developed below exploits the partitioning of space in such a manner so as to reflect different service configurations. 14
Before presenting the developed solution approach, some preliminary methods used to delineate and define continuous space are introduced, namely Voronoi diagram and vector overlay. The Voronoi diagram has proven to be an efficient approach for allocation and partitioning of space, representing access, service assignments and trade areas (Feng and Murray 2018). A formal mathematical presentation of the Voronoi diagram may be found in Okabe et al. (2000). The Voronoi diagram is a widely relied upon method in computational geometry, location modeling and GIScience for partitioning space. Given a set of generator points , an individual Voronoi polygon associated with each generator point is defined in set theoretic terms as:
̂
{(
)
|
where any point, (
(
)(
)
(
), in the region
)(
)
(
)}
(11)
is contained within the Voronoi polygon of the closest
generator point. Further conditions are that the combination of all Voronoi polygons is equal to the region and that no two Voronoi polygons overlap. Formally, the conditions are:
⋃ ̂ ̂ ⋂̂
(12) (
)
(13)
This can be relevant for solving the multiple-source, raw material allocation Weber problem, (6)(9), if each input type is considered independently. That is, for input type , the associated points are known in advance,
. Further, one knows that the closest input to the production
facility site would be utilized as this will minimize total cost, stipulated in (6). Thus, identification of the associated Voronoi diagram for each input type provides a delineation of space that reflects which point would be served if the production facility was located in an associated
15
Voronoi polygon. Consider then the Voronoi diagram associated with input type , where :
{(
)
|
Whereas all points
(
)(
)
(
)(
(
)
)}
(14)
are considered in (11), the Voronoi polygon defined in (14) is only
associated with the subset having to do with type . This means that for the Voronoi diagram associated with type , we have the following:
(15)
⋃ (
⋂
In theory then, the
)
(16)
different Voronoi diagram layers each tell us something about which points
would be served by the production facility, if it is sited in a particular Voronoi polygon,
.
However, the optimal location of the production facility is dependent on serving each of the input types and all demand, so individual Voronoi diagram layers are of limited use. The potential benefit for solution is integration of the independent input layers.
Given the set of types , the result is | | Voronoi diagram layers, one for each set of generator points
. Integration of these layers is possible through vector-based overlay. de Berg et al.
(2008), Wei and Murray (2014) and Murray (2018) characterize the overlay operator as resulting in a maximally connected subset of faces that do not overlap. Specifically, a set of faces, , are identified having the properties that ⋃
and
. Deriving the set of faces
essentially requires evaluation of overlapping Voronoi polygons in order to create new polygons that represent areas of intersection. In set theoretic terms, the power set function,
( ),
enumerates combinations of objects, Voronoi polygons in this case. The number of potential 16
intersecting combinations is obtained using the power set, and is {
of members in set . Here,
| | }.
| |
, where | | is the number
Using the power set,
sub-set combinations, but many do not overlap, e.g.
⋂
( ), this identifies
| |
. Accordingly, the interest is in
intersections, so an operator is necessary that enumerates all possible face combinations. The operator ( ) is defined here to return all non-empty face combinations for
as
follows:
( )
{
|
⋂
⋃
( )
( )
}
(17)
The overlay operator, ( ), therefore produces the following set of faces:
( )
{|
( )
( )
⋃
}
(18)
Wei and Murray (2014) review polygon overlay in terms of computational complexity. The intent here is to merely indicate that unique faces can be identified through overlay,
Lemma 1. A face type ,
( ), corresponds to the closest allocation
, where
of each input
.
Proof: For any location in face ( implies that
( ).
(
)(
)
(
) )(
, suppose there exists ).
,
that is closer. This
However, this contradicts the definition of a Voronoi
polygon given in (14). Thus, face indicates the closest allocation for any input type k.
17
Given the set of faces,
( ), then we know that each face represents an area for which the
closest resource (input) type is known. Accordingly, identifying the optimal allocation along with the optimal production facility location involves evaluation of each face, within which the Weber point can be identified because the closest
resource/demand points are known. If each face
is evaluated, it can then be determined which face contains the optimal location for the production facility.
Figure 3. Pseudo code for exact algorithm to solve multiple-source, raw material allocation Weber problem.
In each face, the best facility site is that obtained using Weiszfeld algorithm for the associated closest allocations (e.g.,
) for each type . Evaluation of objective (6) for each face
indicates which one represents the optimal allocation scheme, and subsequently the lowest total transportation costs. The pseudo code for the algorithm associated with this process is given in Figure 3. After initialization, the Voronoi diagram for each resource type is identified, as specified in (14). This gives the set of Voronoi polygons, . With this set, the overlay operator is then applied, ( ), giving the set of faces, . For each face the optimal allocation variables,
,
can be identified through inspection. Using this for each face, the best facility site corresponding to the closest inputs for each type along with all demand is found using the Weiszfeld algorithm, (10), giving the location that best serves the face, (
), along with its objective value,
specified in (6). Upon termination, the optimal facility location, ( 18
), is identified.
,
Theorem 1. The solution algorithm given in Figure 3 identifies the optimal solution for the multiple-source, raw material allocation Weber problem, (6)-(9).
Proof: By lemma 1, face indicates the best allocation for any point ( for each input type , the best location in face , (
)
, implying
. These allocations along with the demand are used in (10) to find )
, with convergence and optimality proven in Weiszfeld
(1937) (see also Ostrech 1978, Weiszfeld and Plastria 2009, Gorner and Kanzow 2016) within the tolerance . Suppose there exists a face , but this contradicts ⋃
with a lower objective (6). This would imply that
. Thus, since every face is evaluated in Figure 3, the global
optimal must be identified.
Application Results
Three different empirical applications are utilized to demonstrate the nuances of the multiplesource, raw material allocation form of the Weber problem structured in this paper. Performance of the proposed solution algorithm is also reported and compared to results obtained using a commercial nonlinear solver, LINGO (ver. 18, Extended).
The first two applications are associated with the region shown in Figure 4, approximately 675 square kilometers in size (roughly 27 x 25 kilometers), with two resource types: raw material 1 (3 total) and raw material 2 (39 total). In the first application, there is one market to be served. Each resource type is depicted as an independent layer in Figure 4 along with the market layer. The total number of resource points is 42, so | | | |
. The number of markets is one, so
. For illustrative purposes, the amount of material shipped is the same for all locations, for each . The types in Figure 4 indicate
{raw material 1,raw material 2}, so | |
It is assumed that only one resource per type is needed, so 19
. The challenge is to
.
simultaneously find the best resource allocations along with the best facility location. The algorithm outlined in Figure 3 was implemented in Python and run in Windows 10 Pro on an Intel Xeon CPU E5-2650 v3 (2.30 GHz) with 96GB RAM desktop computer. ArcGIS (ver. 10.5) was utilized for data creation, management, manipulation, analysis and display along with ArcPy functionality. Total solution time was 9.24 seconds, with convergence of the Weiszfeld algorithm set at
.
Figure 4. Potential resource types and market layers.
One of the first steps in the solution algorithm (Figure 3) involves derivation of Voronoi polygons for each resource type, specification of which is defined in (12) along with conditions (13) and (14). Accordingly, Figure 5 details the Voronoi diagram for each resource type. For example, in Figure 5a the Voronoi polygons correspond to the least cost raw material 1 provider for each generator point. Similarly, Figure 5b indicates the least cost resource for raw material 2 for any location in the region. Voronoi diagrams for the two raw material inputs are shown for the region in Figure 6. Integration of the two layers, Figures 5a and 5b through overlay, (16), is necessary to obtain the set of faces. These faces are indicated in Figure 7, with 55 faces in this case (e.g., | |
). Each face represents a unique resource allocation if the industrial facility is
located within the face.
20
Figure 5. Closest assignment cost delineation (Voronoi polygon) by type.
Figure 6. Voronoi diagram layers by resource type.
21
Figure 7. Faces along with resource points by type and market.
In order to demonstrate an iteration of the algorithm, Figure 8 shows one face along with the closest (least cost) allocations to the two resource types and the market. The Weiszfeld initial and intermediate locations are depicted along with the best final location. Convergence was achieved in 179 iterations, with a total cost, objective (4), of 22,519.41 meters. The optimal location is detailed in Figure 9, with an objective (4) of 5,250.02 meters (obtained after 2,225 iterations after 11.84 seconds). One can observe the allocation as well as location decisions made in this case, representing the solution of lowest total transportation costs. For comparison purposes, the multiple-source, raw material allocation Weber problem, (6)-(9), was structured in LINGO and solved. The obtained solution had an objective of 11,863.63 meters, requiring 2.06 seconds (1,255 iterations). This is approximately 126% above the true optimum.
22
Figure 8. Intermediate and final Weiszfeld locations for a face (along with allocation).
Figure 9. Optimal facility location along with resource allocations.
A second application considers the same raw material input options as the previous case but now involves four markets, and is shown in Figure 10. Thus, the total number of resource points is | |
42 and the number of markets is | |
4. The local Weber solutions (initial and
intermediate) for each face are indicated in Figure 11. The optimal solution in the four market case is given in Figure 12, along with associated allocations. Total transportation distances, objective (6), are 36,890.55 meters, requiring 18.40 seconds of processing time (789 iterations). 23
Across the other faces (Figure 11), the objective averaged 46,719.63 meters with a high of 63,762.11 meters. Solution using LINGO identified a site and allocation with an objective value of 39,072.72 (requiring 0.63 seconds to perform 678 iterations). This solution is approximately 6% higher than the optimum found using the Voronoi based algorithm.
Figure 10. Application involving four markets.
Figure 11. Individual face evaluations.
24
Figure 12. Optimal solution involving four markets.
The third siting application involves fruit juice production in San Joaquin County, California. An entrepreneur has identified a market niche for juice(s) that is high quality, chemical free, 100% fruit made from locally grown sources. Based on this, they are seeking to lease or purchase a factory property for juice production. The juice will be distributed by Riley’s, a small local grocery chain known for high quality products. San Joaquin County is located east of San Francisco with primary cities being Stockton and Lodi. The 2018 total population of San Joaquin County is approximately 752,660 spread across more than 1,400 square miles. Crop producer information was obtained from the San Joaquin County Agricultural Commissioner (2017). Eight different fruit farms types were identified as raw material inputs, so | |
8. The types are as
follows, with the number of indicated farms noted: cherry (870), apple (91), peach (91), watermelon (41), berries (38), apricot (18), nectarine (14) and plum (13), and are shown in Figure 13. The total number of potential resource points is therefore | |
1,176. There are four
Riley’s, representing the markets for the finished product in this region (| | that material shipped is the same for all locations,
4). It is assumed
for each . The juice factory requires
one farm to be selected for each raw material input type. Only one resource per type is needed, so
.
25
Figure 13. Juice processing facility siting.
26
Figure 14. Resulting faces for juice processing facility application.
27
Figure 15. Optimal location and associated allocations for juice processing facility.
The Voronoi based algorithm identifies 4,274 unique faces (Figure 14) for evaluation. The total solution time in this case was 21.73 seconds. The objective was 96,226.07 meters, meaning that the total transportation costs would be a function of this total distance. The solution and 28
associated allocations are depicted in Figure 15. For comparison purposes, solution using LINGO required 65.37 seconds (4,987 iterations), with the identified site resulting in an objective of 101,636.67 meters. This is nearly 6% above the optimum found using the developed solution algorithm.
As a final basis of comparison, randomly generated application instances were created. In total, 14 unique problems were considered. The total number of resource locations range from | |
4,991 to | |
99,997, the number of types span from | |
of markets vary between | |
1 and | |
5 to | |
15, and the number
9. Consistent with previous problems, the amount
of material shipped is the same for all locations. Solution summaries are reported in Table 1, indicating objective function value (feet) and solution time (seconds) along with other information for both the Voronoi based algorithm and LINGO. Problem instance particulars are noted in the Details column. Most of the 14 applications in Table 1 could not be solved for various reasons using LINGO. However, the Voronoi algorithm developed in the paper solved all problems optimally, requiring less than 15 minutes in the most computationally challenging case (R14). For some instances, the LINGO solver did find a local optima, but only came within 107% of optimality once (R4) and was over 225% in the worst case (R7). Computationally, the Voronoi based algorithm is fast and efficient, enabling all problems to be solved optimally.
29
Table 1. Comparison of developed exact approach to commercial solver. Voronoi based algorithm (exact/optimal) ID
Details
LINGO (nonlinear approximate)
Objective
Iterations
Faces, |L|
Time (sec)
Objective
Iterations
Time (sec)
16,546.10
352,185
23,171
89.76
27,079.71
13,244
1,127.08
181,925.91
135,154
18,340
50.93
269,866.08
11,451
1,067.72
486,407.54
74,615
23,167
44.10
-
15,679
4,816.30
755,205.22
58,372
21,494
36.24
807,902.37
12,834
1,245.29
346,382.10
290,919
49,298
119.90
652,771.48
12,800
1,325.67
230,786.39
281,068
75,449
131.84
358,727.59
16,233
3,465.42
268,596.14
466,675
46,563
129.60
604,945.71
33,826
11,163.26
391,927.53
588,417
94,082
181.94
-
17,378
11,697.64
318,782.12
648,629
151,601
375.77
-
48,201
40442.02
292,281.48
1,191,639
107,990
300.01
|K|=5
R1
|I|=4999 |M|=1 |K|=5
R2
|I|=4997 |M|=3 |K|=5
R3
|I|=4996 |M|=6 |K|=5
R4
|I|=4991 |M|=9 |K|=10
R5
|I|=4997 |M|=3 |K|=15
R6
|I|=4997 |M|=3 |K|=5
R7
|I|=9997 |M|=3 |K|=10
R8
|I|=9997 |M|=3 |K|=15
R9
|I|=9997 |M|=3
R10
|K|=5
30
*
|I|=29997 |M|=3 |K|=5
R11
|I|=39994
436,870.10
538,248
177,676
415.80
**
213,059.92
1,801,972
232,292
812.62
**
341,157.06
637,402
174,557
537.11
**
175,377.26
3,071,005
412,692
891.25
**
|M|=6 |K|=5
R12
|I|=49997 |M|=3 |K|=5
R13
|I|=59994 |M|=6 |K|=5
R14
|I|=99997 |M|=3
- Problem terminated with local optima, but infeasible solution * Problem had no feasible solution after 20 hours of processing ** Problem could not be solved due to insufficient computer memory
Discussion
The motivation for the paper was to demonstrate that legacy interpretations of the Weber problem mischaracterize important problem nuances. Although researchers have added problem elements not addressed by Weber, e.g. negative weights and barriers, virtually all have neglected a rich set of problem elements that have been overlooked (Church, 2019). One of these elements is the presence of multiple localized sources of a given raw material, the primary subject of this paper. Weber recognized that such sources are likely to exist for any type of raw material, whether it be sheet metal, iron ore, etc. The application results demonstrate the richness associated with this broader interpretation. However, perhaps a more significant takehome message is that the use of the classically conceived Weber problem, (4)-(5), will result in
31
substantially increased total transportation cost if there are indeed multiple options from which to choose from for a given resource type. For example, in the single market application, the location derived using (4)-(5) would have an objective value of 29,572.48 meters when evaluated as a multiple-source, raw material allocation Weber problem, (6)-(9). This is 563% higher than the optimum of 5,250.02 shown in Figure 9. Similarly, for the four market application, the location derived using (4)-(5) has a total transportation cost of 49,871.30 meters, which is 135% higher than the more efficient facility location identified using multiplesource, raw material allocation Weber problem, (6)-(9), of 36,890.55 meters (Figure 12). The point then is that one could not simply use all resource inputs to find a good solution if only a subset of inputs is actually necessary. Correct model specification and solution matters.
A second point that needs further discussion is that the formulation of the multiple-source, raw material allocation Weber problem introduced in (6)-(9) is indeed new and unique. While it accounts for allocation to a pre-specified number of resources inputs, it is dramatically different from the location-allocation (multi-facility Weber) problem discussed in Cooper (1963). One issue is that the location-allocation problem focuses on multiple facilities, but here we only seek one facility. Further, the location-allocation problem requires that all resource/demand points are allocated to a facility. This is not the case with the multiple-source, raw material allocation Weber problem, (6)-(9).
A third point is associated with computational processing. The Voronoi based algorithm is an exact approach, capable of identifying an optimal solution. As problem size grows, it appears to scale up rather well. There is clearly an overhead associated with derivation of the Voronoi diagram as well as overlay to obtain unique faces. Such effort proved to be well worth it when examining the larger problem instances in Table 1. Of course, the implementation reported in this paper relied on Arcpy functionality for this, which is part of the ArcGIS commercial software package. In addition to being costly, it is likely to be computationally inefficient in many ways. Part of this is associated with use through Arcpy, but also implementation. This becomes an important issue when you consider that 36.40% of solution time on average for the application instances in Table 1 was devoted to Voronoi and intersection processing (nearly 70% for R11 32
and R13). The two example problems (| | (| |
42) and the juice production facility application
1180) spent over 95% and 80%, respectively, on this GIS processing (Voronoi diagram and
overlay). Native programing or open source options may prove particularly valuable in reducing solution time. Of course this was not an issue for the problem applications considered in this paper.
A final note is that there are a number of assumptions applied to the classic Weber problem, and for consistency and comparison they were also assumed in the detailed approach. Of course, extension of the multiple-source, raw material allocation Weber problem, (6)-(9), is possible to account for a range of additional considerations, such as price variation, quality issues, supply disparities, etc. This is left for future research.
Conclusions
This paper discussed and reviewed the problem addressed by Weber (1909) along the lines originally intended. Most interpretations of the classical Weber problem are overly restrictive and narrow. Specifically, not addressed in previous research are realities of selection from among a set of options for inputs in serving one or more markets. This no doubt complicates the planning problem because the siting decision is dependent on the locations to be utilized and quantities being transported. The complexities of multiple potential input options increases the difficulty of the problem substantially, necessitating new solution approaches. Introduced in the paper is the multiple-source, raw material allocation Weber problem, (6)-(9), to simultaneously account for input selection (allocation) and facility siting (location). Further, an algorithm capable of exact solution was developed for identifying an optimal production facility location. This was accomplished through the use of GIS and computational geometry techniques. These techniques effectively exploit spatial knowledge, and as a result enable the problem to be solved optimally. Application results were presented to highlight the distinctions of what Weber
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(1909) actually described, but also to demonstrate the utility and computational efficiency of the developed solution algorithm.
34
References Berman, O., Drezner, Z., & Wesolowsky, G. O. (2007). The transfer point location problem. European Journal of Operational Research, 179(3), 978-989. Bischoff, M., & Klamroth, K. (2007). An efficient solution method for Weber problems with barriers based on genetic algorithms. European Journal of Operational Research, 177(1), 22-41. Bjelland, B. D., Montello, D. R., Fellmann, J. D., Getis, A., & Getis, J. (2014). Human Geography: Landscape of Human Activities, 12th edition. New York: McGraw Hill. Bozarth, C. B., & Handfield, R. B. (2019). Introduction to Operations and Supply Chain Management, 5th edition. Pearson. Brimberg, J., Drezner, Z., Mladenović, N., & Salhi, S. (2014). A new local search for continuous location problems. European Journal of Operational Research, 232(2), 256-265. Canbolat, M. S., & Wesolowsky, G. O. (2010). The rectilinear distance Weber problem in the presence of a probabilistic line barrier. European Journal of Operational Research, 202(1), 114-121. Canbolat, M. S., & Wesolowsky, G. O. (2012). On the use of the Varignon frame for single facility Weber problems in the presence of convex barriers. European Journal of Operational Research, 217(2), 241-247. Capello, R. (2014). Classical contributions: Von Thunen, Weber, Christaller, Losch. In, Handbook of Regional Science, edited by M.M. Fischer and P. Nijkamp, 507-526. Berlin: SpringerVerlag. Carrizosa, E., Muñoz-Márquez, M., & Puerto, J. (1998). The Weber problem with regional demand. European Journal of Operational Research, 104(2), 358-365. Christaller, W. (1933). Die zentralen Orte in Süddeutschland: eine ökonomisch-geographische Untersuchung über die Gesetzmässigkeit der Verbreitung und Entwicklung der Siedlungen mit städtischen Funktionen.
35
Church, R. L. (2019). Understanding the Weber location paradigm. In, Festschrift for Zvi Drezner, edited by H.A. Eiselt and V. Marianov, 69-88. New York: Springer. Church, R. L., & Murray, A. T. (2009). Business site selection, location analysis, and GIS. Hoboken, NJ: John Wiley & Sons. Cooper, L. (1963). Location-allocation problems. Operations Research, 11(3), 331-343. Drezner, T., Drezner, Z., & Schobel, A. (2018). The Weber obnoxious facility location model: a big arc small arc approach. Computers & Operations Research, 98, 240-250. Drezner, Z., & Guyse, J. (1999). Application of decision analysis techniques to the Weber facility location problem. European Journal of Operational Research, 116(1), 69-79. de Berg, M., Cheong, O., Van Kreveld, M. & Overmars, M. (2008). Computational Geometry: Algorithms and Applications, 3rd edition. Berlin: Springer. Feng, X., & Murray, A. T. (2018). Allocation using a heterogeneous space Voronoi diagram. Journal of Geographical Systems, 20(3), 207-226. Francis, R. L., McGinnis, L. F. & White, J. A. (1992). Facility layout and location: an analytical approach, second edition. Englewood Cliffs, NJ: Prentice-Hall. Gorner, S., & Kanzow, C. (2016). On Newton’s Method for the Fermat–Weber Location Problem. Journal of Optimization Theory and Applications, 170(1), 107-118. Haines, V. G. (1970). Business relocation: a guide to moving a business. Random House Business. Hoover, E. M. (1937). Location theory and the shoe and leather industries. Cambridge: Harvard University Press. Hotelling, H. (1990). Stability in competition. In, The Collected Economics Articles of Harold Hotelling, pp. 50-63. New York: Springer. Irawan, C. A., Salhi, S., Luis, M., & Azizi, N. (2017). The continuous single source location problem with capacity and zone-dependent fixed cost: Models and solution approaches. European Journal of Operational Research, 263(1), 94-107. Isard, W. (1956). Location and space-economy. New York: Wiley. 36
Isard, W. (1960). Methods of Regional Analysis: an Introduction to Regional Science. Cambridge: MIT. Ivanov, D., Tsipoulanidis, A., & Schönberger, J. (2019). Global supply chain and operations management. A Decision-Oriented Introduction to the Creation of Value, 2nd edition. Springer. Kelly, L. M. (1969). Optimal distance configurations. In Recent Progress in Combinatorics. Academic Press New York. Klose, A., & Drexl, A. (2005). Facility location models for distribution system design. European Journal of Operational Research, 162(1), 4-29. Kuhn, H. W., & Kuenne, R. E. (1962). An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics. Journal of Regional Science, 4(2), 21-33. Launhardt, W. (1872). Kommercielle Tracirung der Verkehrswege. Architekten-und Ingenieurverein. Love, R. F., Morris, J. G., & Wesolowsky, G. O. (1988). Facilities location: models and methods. North-Holland, NY, NY. Lösch, A. (1940). The Economics of Location, 1954. New Haven: Yale University Press. Miehle, W. (1958). Link-length minimization in networks. Operations Research, 6(2), 232-243. Murray, A. T. (2010). Advances in location modeling: GIS linkages and contributions. Journal of Geographical Systems, 12(3), 335-354. Murray, A. T. (2018). Complexities in spatial center derivation. Transactions in GIS, 22(6), 13351350. Murray, A. T., Matisziw, T. C., Wei, H., & Tong, D. (2008). A geocomputational heuristic for coverage maximization in service facility siting. Transactions in GIS, 12(6), 757-773. Okabe, A., Boots, B., Sugihara, K. & Chiu, S.N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, second edition. New York: Wiley.
37
Ostresh Jr, L. M. (1978). On the convergence of a class of iterative methods for solving the Weber location problem. Operations Research, 26(4), 597-609. Perreur, J. (1998). Industrial location theory in German thought–Launhardt and Weber. Recherches Économiques de Louvain/Louvain Economic Review, 64(1), 75-96. Pinto, J. V. (1977). Launhardt and location theory: Rediscovery of a neglected book. Journal of Regional Science, 17(1), 17-29. Plastria, F. (2016a). How bad can the centroid be?. European Journal of Operational Research, 252(1), 98-102. Plastria, F. (2016b). Up-and downgrading the euclidean 1-median problem and knapsack Voronoi diagrams. Annals of Operations Research, 246(1-2), 227-251. Reid, R. D., & Sanders, N. R. (2016). Operations Management: an integrated approach, 6th edition. New York: Wiley. San Joaquin County Agricultural Commissioner (2017). https://www.sjgov.org/department/agcomm/gis_crop_data (accessed 3/4/19). Schöbel, A., & Scholz, D. (2014). A solution algorithm for non-convex mixed integer optimization problems with only few continuous variables. European Journal of Operational Research, 232(2), 266-275. Schultz, G. P. (1969). Facility planning for a public service system: Domestic solid waste collection. Journal of Regional Science, 9(2), 291-307 Simpson, T. (1750). The doctrine and application of Fluxions, Printed by J. Nourse, London. Stevenson, W. J. (2018). Operations Management, 13th edition. New York: McGraw Hill. Vardi, Y., & Zhang, C. H. (2001). A modified Weiszfeld algorithm for the Fermat-Weber location problem. Mathematical Programming, 90(3), 559-566. Varignon, P. (1687). Projet d’une nouvelle mechanique. Chez C. Jombert, Paris.
38
Venkateshan, P., Ballou, R. H., Mathur, K., & Maruthasalam, A. P. (2017). A Two-echelon joint continuous-discrete location model. European Journal of Operational Research, 262(3), 1028-1039. Vergin, R. C., & Rogers, J. D. (1967). An algorithm and computational procedure for locating economic facilities. Management Science, 13(6), B-240-B-254. von Thunen, J. V. (1826). Der isolierte Staat. Beziehung auf Landwirtschaft und Nationalökonomie. Weber, A. (1909). Uber den Standort der Industrien (Theory of the location of industries) (translated with an introduction and notes by C.J. Friedrich, 1929). University of Chicago Press, Chicago. Wei, H., Murray, A. T., & Xiao, N. (2006). Solving the continuous space p-centre problem: planning application issues. IMA Journal of Management Mathematics, 17(4), 413-425. Wei, R., & Murray, A. T. (2014). Evaluating polygon overlay to support spatial optimization coverage modeling. Geographical Analysis, 46(3), 209-229. Weiszfeld, E. (1937). Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Mathematical Journal, First Series, 43, 355-386. Weiszfeld, E., & Plastria, F. (2009). On the point for which the sum of the distances to n given points is minimum. Annals of Operations Research, 167(1), 7-41. Wersan, S. J., Quon, J. E., & Charnes, A. (1962). Systems analysis of refuse collection and disposable practices. American Public Works Association. Year-Book, 195-211. Wesolowsky, G. O. (1993). The Weber problem: history and perspectives. Location Science, 1(1): 5-24. Wesolowsky, G. O., & Love, R. F. (1972). A nonlinear approximation method for solving a generalized rectangular distance Weber problem. Management Science, 18(11), 656-663. Yao, J., & Murray, A. T. (2013). Continuous surface representation and approximation: spatial analytical implications. International Journal of Geographical Information Science, 27(5), 883-897. 39
Yao, J., & Murray, A. T. (2014). Serving regional demand in facility location. Papers in Regional Science, 93(3), 643-662.
40