Integrated redistricting, location-allocation and service sharing with intra-district service transfer to reduce demand overload and its disparity

Computers, Environment and Urban Systems 54 (2015) 132–143

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Computers, Environment and Urban Systems journal homepage: www.elsevier.com/locate/ceus

Integrated redistricting, location-allocation and service sharing with intra-district service transfer to reduce demand overload and its disparity Jeonghan Ko a,b,1, Ehsan Nazarian c, Yunwoo Nam d,⁎, Yin Guo e a

Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI 48109, USA Industrial Engineering, Ajou University, Republic of Korea c Global Business Processes, Walmart Home Office, Bentonville, AR 72716, USA d Community and Regional Planning, University of Nebraska-Lincoln, Lincoln, NE 68588, USA e Electrical Engineering & Computer Science, Case Western Reserve University, Cleveland, OH 44106, USA b

a r t i c l e

i n f o

Article history: Received 28 October 2014 Received in revised form 16 July 2015 Accepted 17 July 2015 Available online xxxx Keywords: Spatial modeling Redistricting Location-allocation Resource sharing Service transfer Overload disparity p-regions Judicial service GIS

a b s t r a c t Service demand overload has been one of the main concerns in district-based service planning, because it strongly affects service quality. Moreover, the overload problem usually involves overload disparity among districts. The disparity often results from outdated district boundaries not reflecting up-to-date spatial demand distributions. A lack of systematic methodologies, however, has hindered solving such overload and disparity problems despite the increasing availability of information on spatial service demand and supply. This paper presents a novel mathematical programming model to address the service demand overload problem by reorganizing services in multiple spatial scales. The mathematical program optimizes simultaneously (1) redistricting service areas, (2) allocating multiple service resources into service-providing units in each district, and (3) sharing services between service-providing units within a district. Information on geographically distributed units is used as the spatial data of the model. This new model integrates districting and location-allocation problems as a combined problem. A heuristic solution approach is also presented to solve large problem instances. As a case study, a judicial service overload problem is examined for a state court system in the United States. This new integrated approach enables efficient utilization of the geographically distributed service capacity. In addition, these new features of the model allow for better utilization of spatial information for practical service planning. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Spatial disparity in service demand overload (or capacity shortage) is a serious problem. Demand overload in service providers strongly affects the quality of service, a critical issue in service planning. However, many services, public or private, suffer from insufficient capacity because of problems such as downsizing and economic downturn. In some cases, overload is intentionally designed to avoid possible overcapacity and save costs. Furthermore, the service capacity often cannot be easily increased in the short-term. Although service providers may have sufficient overall capacity to meet the total demand, spatially uneven overload and (consequently) uneven service accessibility usually exist. Such examples include

⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Ko), [email protected] (E. Nazarian), [email protected] (Y. Nam), [email protected] (Y. Guo). 1 Mailing address: Ajou University, Industrial Engineering, 206, World cup-ro, Yeongtong-gu, Suwon-si, Gyeonggi-do 16499, Republic of Korea.

http://dx.doi.org/10.1016/j.compenvurbsys.2015.07.002 0198-9715/© 2015 Elsevier Ltd. All rights reserved.

more judicial filings than available judges in some regional courts (The National Center for State Courts) and physician shortages in rural and impoverished urban areas (Burkey, Bhadury, & Eiselt, 2012; US Department of Health and Human Services — Council on Graduate Medical Education, 2000; US General Accounting Office, 1995). This disparity often results from demographic or social changes. In some districtbased service systems such as public services in less populated areas or judicial services, this spatial inequality is not easy to address without redistricting or capacity adjustment. This is because in these services, little boundary permeability often exists or service users cannot easily find alternative service providers. One of the reasons for such spatial inequality is the lack of proper planning tools for decision makers to evaluate public service accessibility, adjust district boundaries, and set utilization levels in a cohesive manner. As pointed out in some research review papers and many geographical information systems (GIS)-related service research, the interrelationships among spatial service demand, utilization, and service outcomes are often neglected (Higgs, 2004; Mclafferty, 2003). The lack of integrated approaches to these inherently interwoven problems has hindered better service planning. Furthermore, despite the increasingly available diverse

J. Ko et al. / Computers, Environment and Urban Systems 54 (2015) 132–143

spatial information such as geographical service demand and supply data, the existing models in the literature lag in using such information. To overcome this problem, this study integrates three characteristics in organizing service providers: spatial redistricting, multiple service capacity allocation, and intra-district service transfer. Spatial redistricting is the reorganization of spatial units into geographical groups, called districts or regions. The spatial districting is of critical importance to the accessibility of potential service customers, because districting specifies the responsibility areas of each service-provider unit and the demand level for each provider. Location-allocation decisions of service capacity levels determine the level of supply at each service-providing unit or determine the location of such service-providing units. Intra-district service transfer is how the service capacity (supply) within each district is shared by transferring extra services in some service-providing units to overloaded ones. Thus, in addition to balancing workload between districts, the workload of facilities within each district is also balanced. This intra-district service transfer is a new concept proposed in this paper, and it is critical for minimizing utilization disparity among service-providing units and maximizing the efficient utilization of service capacities. To the best knowledge of the authors, the majority of the existing literature on redistricting has focused on workload balance only on the district level. In addition, this study pursues the minimization of work overload in each district, whereas the majority of previous redistricting studies have sought balances among districts. Only some previous studies have solved problems on the balanced demand or workforce distribution in each district. For the new approaches, this study develops a novel mathematical model that simultaneously considers redistricting, multiple locationallocation and capacity sharing for district-based service planning. The mathematical model is developed as a mixed-integer program (MIP). An effective heuristic solution method is also developed to calculate large-size problems. Information on geographically distributed units, such as service demand, location, supply, and cost are used as spatial data for the model. A judicial boundary problem in a US state is presented as a case study. The contributions of this study are summarized as follows. First, this paper presents a new integrated model of districting and multiple location-allocation problems. Many previous studies have focused on either districting or allocation. In this paper, the district generation (i.e., clustering of spatial units into districts of aggregated service locations and demand units) and multi-facility location-allocation (i.e., setting service supply levels of multiple service units in each district) are determined simultaneously. In other words, this paper extends the districting problem, also called p-regions or zonation problems (Duque, Church, & Middleton, 2011), to a problem of p-capacitated-regions with multiple location-allocation. Thus, this paper deals with a multi-scale spatial problem. Second, this paper presents a new mathematical model for capacity sharing combined with districting and location-allocation decisions by allowing the transfer of extra services in some service units to other service units with excessive demands. Most previous studies on service location problems have focused on opening or closing of service units depending on the service demand but have not utilized sharing service capacities among these facilities. This new integrated service transfer, however, is not simple to model. For instance, service transfer should not occur between extra-service facilities or between overloaded facilities, and a portion of transferred service may be used or lost during the transfer. Furthermore, the service transfer and districting of each unit are interwoven decisions. One of the innovative features of this study is to develop mathematical models that can consider the detailed conditions of such complex service transfer. Third, the new approaches and model in this study help optimize district-based service planning and will help advance scientific knowledge in related research fields. Moreover, the integrated approach enhances the use of spatial demand and supply information to a more sophisticated level.

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In addition, to the best knowledge of the authors, this study is the first attempt to address judicial redistricting, and actual judicial workload was analyzed and used for the case study. The rest of this paper is organized as follows. Section 2 reviews the related literature. Section 3 describes the mathematical programming models in this study. Section 4 describes the solution method. Section 5 presents numerical examples and their results and discussion. Section 6 concludes.

2. Literature review Location and redistricting problems can be categorized based on underlying assumptions, as summarized in some review papers. The health care location problem models were categorized into accessibility, adaptability, and availability models, and relevant formulations and analysis for each category were presented (Daskin & Dean, 2005). This study summarized the location set covering and maximal covering models as well as P-median models. A comprehensive literature review was performed on the methods for facility location and layout planning problems and their relevant mathematical models (Domschke & Krispin, 1997). A unified framework was proposed for location problems based on facilities, customers, and locations (Scaparra & Scutella, 2001). The healthcare related location and redistricting research has extended to combine other issues in healthcare, including dispatching emergency services (Toro-Díaz, Mayorga, Chanta, & McLay, 2013), ambulance allocation (Knight, Harper, & Smith, 2012) and emergency response locations (Li, Zhao, Zhu, & Wyatt, 2011). Facility location-allocation problems have been applied for a variety of services. The studies in these applications include electrical power redistricting and their physical special features for modeling (Bergey, Ragsdale, & Hoskote, 2003), sales territory planning with new models and solution methods such as a computational geometry based method (Kalcsics, Nickel, & Schröder, 2005; Salazar-Aguilar, Rios-Mercado, & Cabrera-Rios, 2009), liver allocation systems (Demirci, 2008; Kong, Schaefer, Hunsaker, & Roberts, 2008; Stahl, Kong, Shechter, Schaefer, & Roberts, 2005), organ transplantation (Bruni, Conforti, Sicilia, & Trotta, 2006), and environmental planning and supply chain management (Smith, Laporte, & Harper, 2009). Optimization models for emergency logistics were also analyzed and classified, including facility location, stock pre-positioning, and relief distribution (Caunhye, Nie, & Pokharel, 2012; Galindo & Batta, 2013). Recently, an electric car charging location problem has received attention due to the more penetration of electric cars to the market (Giménez-Gaydou, Ribeiro, Gutiérrez, & Antunes, accepted for publication; Jung, Chow, Jayakrishnan, & Park, 2014). A variety of covering problems were reviewed for models, solutions and their applications (Farahani, Asgari, Heidari, Hosseininia, & Goh, 2012). Recently, a new set of MIP models, called a family of p-regions models, have been published for spatial aggregation (districting). The notable studies include Duque et al. (2011), Duque, Anselin, and Rey (2012), Kim, Chun, and Kim (2015), and Li, Church, and Goodchild (2014a), each of which has a little different approaches. The p-regions problem was defined as a generic model for aggregating a number of small areas to p number of regions with minimizing region heterogeneity (Duque et al., 2011). This study also provided three new MIP models and compared their computational complexity. Duque et al. (2012) proposed MIP models and heuristics solution methods for clustering spatial areas to the maximum number of homogeneous regions with certain criteria values above predefined thresholds. A flow-based regionalization problem was also modeled to determine p functional centers and contiguous regions where areal units interact with a center (Kim et al., 2015). Li et al. (2014a) defined a p-compact-regions problem to additionally consider region compactness, and proposed a new efficient heuristics method of several steps including a solution seed selection, region growth, randomized greedy selection, and area reassignment.

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These studies and this paper have common features in terms of area aggregation into a certain number (p) of districts. The equity in the service level has been addressed in some redistricting and location-allocation problems. For instance, the maximum number of customers served among all facilities was minimized for p-locations and p-facilities to make weights of each facility as even as possible with dynamic programming and heuristics based solutions methods (Berman, Drezner, Tamir, & Wesolowsky, 2009). A mathematical model was proposed to optimize waiting time equity in the organ transplantation location problem (Bruni et al., 2006). A two-phase (generating feasible districts and then minimizing the deviation) algorithm was presented for a political redistricting problem to reduce the maximum deviation in the district population from the average (Garfinkel & Nemhauser, 1970). A network-based optimization program was proposed to achieve approximately equal populations in districts in a political districting problem (George, Lamar, & Wallace, 1997). A multi-objective p-median model was presented to analyze the tradeoffs between accessibility and equity in a location problem (Gerrard, 1994). Population equity in political redistricting was also studied (Forest, 2004; Mehrotra, Johnson, & Nemhauser, 1998). Equity in workspace sharing for mobile agents was also modeled (Pavone, Arsie, Frazzoli, & Bullo, 2009). Actual data of healthcare accessibility in the US were analyzed for efficiency and equity measures (Burkey et al., 2012). In addition, a hierarchical location model was studied for public service with the dual criteria of efficiency and equity (Smith, Harper, & Potts, 2012). Whereas the majority of the literature on location problems has deterministic assumptions, some studies considered diverse stochastic properties of demand or service time. Such stochastic properties include demand dynamics in multi-period urban hierarchy planning (Antunes, Berman, Bigotte, & Krass, 2009), stochastic demand and spatial congestion (Baron, Berman, & Krass, 2008), stochastic service time (Marianov & Serra, 2002), stochastic demand and immobile service problems (Wang, Batta, & Rump, 2002) and stochastic demand and congestion (Vidyarthi & Jayaswal, 2014). Because the location and redistricting problems are usually NP-hard, the use of MIP models and generic MIP solvers may not be the best option for large-scale problems. Therefore, some studies have improved MIP models or used heuristic solution methods. For instance, a network simplex algorithm was integrated with stochastic simulations and a genetic algorithm (Zhou & Liu, 2003). An efficient MIP model was proposed for multi-site land-use allocation by evaluating four integer programs in terms of speed and effectiveness for large data sets (Aerts, Eisinger, Heuvelink, & Stewart, 2003). An algorithm was presented to minimize the maximum deviation of district populations for political districting (Garfinkel & Nemhauser, 1970). Due to the inherent computational complexity, extensive heuristics methods were studied. A genetic algorithm and simulated annealing heuristic were compared for electrical power redistricting problems (Bergey et al., 2003). Other heuristics approaches include an ant colony algorithm for Euclidean location-allocation problems (Arnaout, 2013), variable neighborhood search combined with a genetic algorithm for a planar pmedian problem (Drezner, Brimberg, Mladenović, & Salhi, 2015), and particle swarm algorithm for an un-capacitated location-allocation problem (Ghaderi, Jabalameli, Barzinpour, & Rahmaniani, 2012). One of the important issues in such heuristics methods is how to move or swap areal units through the iterations of the heuristic solution process and how to reduce the computation time for checking the solution feasibility of a move or swap. An earlier study on political redistricting by Nagel (1965) included computer codes to improve an incumbent solution during calculation by moving a feasible unit from one district to another and described feasibility conditions for such move. Another earlier study by Ferligoj and Batagelj (1982) also included a method for interchanging units in hierarchical clustering having relational constraints by checking the induced subgraphs. To form better territories for an electoral districting problem, Horn (1995) proposed a procedure

based on local improvement by moving zone units on the borders of the territories. Duque, Ramos, and Suriñach (2007) reviewed a wide variety of supervised regionalization methods, and classified them to several groups according to spatial contiguity constraint types. Recently, Li, Church, and Goodchild (2014b) proposed a framework using an edgereassignment-based local search for a p-compact regions problem. Recently, a few studies have combined mathematical modeling of the location problem with the Geographical Information Systems (GIS). An interactive method created by coupling GIS software and a school redistricting mathematical model was presented, and a prototype and implementation issues are described as well as real-life examples (Caro, Shirabe, Guignard, & Weintraub, 2004). Integration of a mathematical model and GIS for sales territory planning was also proposed, and two solution approaches (classical approach with split resolution techniques and a computational geometry based approach) were compared (Kalcsics et al., 2005). The contribution of GIS to location research in areas of data, visualization, solution methods and theory was also pointed out in several studies (Murray, 2010). The advances of GIS integration with location problem algorithms were analyzed to help evaluate the performance of such mathematical models (Revelle & Eiselt, 2005). Whereas a great deal of studies have been devoted to the each area of location-allocation and redistricting-boundary problems, relatively less attention has been paid to linking these two research areas systematically. Furthermore, none of the aforementioned research has considered intra-district service transfer and capacity sharing in modeling overload problems. 3. Mathematical models This section presents the mathematical model of this study and necessary assumptions. This section also explains the mathematical equations and symbols in detail. 3.1. Assumptions To reflect the general conditions in district-based service planning and to focus on the essential mathematical relations, some assumptions are necessary. The following is the list of common assumptions not explicitly described with the equations in Section 3.2. • At most one service-providing facility, simply called a service facility, can exist in each spatial unit. • The number of districts is given. In district-based planning in practice, this number is often fixed or has a very limited range. • Candidate locations of service facilities are given. • A service facility with a positive net service balance over demand is not allowed to receive service from other facilities. This prevents a service facility with a net positive balance from borrowing service from another facility to transfer this service to other facilities. • A service facility with a negative net service balance is not allowed to send service. • For each facility, “its own service supply plus the sum of total service transferred to it” is allowed to exceed the original maximum capacity of the service facility. This reflects the situation in which transferred service temporarily serves the service facility. • Without loss of generality and for simpler equation display as in many other research papers, not all service cost coefficients are explicitly included in the equations. These cost terms can easily be included in the mathematical model if needed.

Some of these assumptions can be easily relaxed with added computational complexity for more general cases. For instance, the number of districts and service locations can be converted to variables without a significant change of the formulations in Section 3.2.

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3.2. Optimization formulation

3.3. Nomenclature

The combined redistricting-allocation problem with service transfer is formulated as a mixed integer program (MIP) as follows.

3.3.1. Indexes

Minimize z

ð1Þ

i, j m, n k

ð2Þ

3.3.2. Sets

Subject to z ≥ zkm X

∀ m ∈ M; k ∈ K

xki ¼ 1

∀ i∈I

k

X

xkm ≥ 1

ð3Þ

∀ k∈K

M K

m∈M

X

ukim ¼ xki

m∈M ukim ≤

xkm

∀ i ∈ I; k ∈ K

i X X

ð4Þ

∀ i ∈ I; m ∈ M; n ∈ M; m≠n; k ∈ K

α imn ¼ ðcim −cin Þ=ðcim þ cin Þ X X di  xki ≤ ð1 þ βÞ skm

∀ m≠n; ∀ k∈K

m∈M

skm ≤ L2

ð5Þ

max k k xkm  Smin m ≤ sm ≤ xm  Sm

vkmn þ vknm ¼ 0 vkmm ¼ 0

∀ m ∈ M; n ∈ M; m≠n; k ∈ K

∀ m ∈ M; k ∈ K ∀ m ∈ M; n ∈ M; m≠n; k ∈ K

k vk− mn ≤ vmn

∀ m ∈ M; n ∈ M; m≠n; k ∈ K

X k bm ¼ skm − di  ukim i k −bm −ð1−γ Þ

¼



∑ vknm n ∈ M; n≠m

kþ

k

∀ m ∈ M; k ∈ K

k− bm kþ bm k− bm kþ bm

≤

k bm

∀ m ∈ M; k ∈ K

≤

L2 ykm  k

k−

¼ bm −bm

kþ

vkþ mn ≤ bm

n ∈ M; n ≠ m kþ

bm ≥ vkþ mn ≤

Smax m

∀ m ∈ M; k ∈ K ð7Þ



qkij − qkji jjði; jÞ ∈ A jjð j;iÞ ∈ A r ki ¼ 1

∀ m ∈ M; k ∈ K bkm

∀ m ∈ M; k ∈ K ð8Þ

bkm+ bkm− ykm

∀m ∈ M; n ∈ M; k ∈ K

X

X

≥ xki −L1  r ki

∀ i ∈ I; k ∈ K

vkmn

∀ k∈K

ð9Þ

i

X

qkji ≤ ðL1 −1Þ  xki

vkmn+ vkmn− zkm

∀ i ∈ I; k ∈ K

jjð j;iÞ ∈ A

xki ;

ykm ;

ukim ;

ukim

skm

∀ m ∈ M; n ∈ M; m≠n; k ∈ K xkn

β γ

xki

∀ m ∈ M; n ∈ M; m≠n; k ∈ K

k−

bm ≤ vk− mn

r ki ∈f0; 1g

Set of all spatial units Set of spatial unit pairs that share at least one positive-length common border Set of spatial units with a possible service facility; M ⊆ I Set of districts

∀ i∈I; m∈M; k∈K

k kþ k− k− k k bm ; bm ; bm ; vkmn ; vkþ mn ; vmn ; z; zm ; qi j ∈R; ∀ m∈M; kþ kþ k ∀ m∈M; n∈M; k∈K; ði; bm ; vmn ; qi j ≥0 k− k− ∀ m∈M; n∈M bm ; vmn ≤0 ∀ m∈M; k∈K skm ∈fℤþ ; 0g

A large number ≥ |I| The total service supply limit Minimum service supply at the facility in spatial unit m Maximum service supply at the facility in spatial unit m Service demand of spatial unit i Generalized service cost per unit demand, defined between spatial unit i and the service facility in spatial unit m when i is served by the facility in m Relative cost ratio (cim − cin)/(cim + cin) among spatial unit i and the facilities in m and n, where n ≠ m; − 1 b αimn b 1; negative for n if i and m constitute the minimal cost pair, and positive for at least one n if i and m are not the minimal cost pair Allowable overload ratio for a district Rate of service loss in a service transfer; 0 ≤ γ b 1

3.3.4. Variables

∀ m ∈ M; k ∈ K  ≥ L2 ykm −1 ∀ m ∈ M; k ∈ K

X

X

ð6Þ

∀ m ∈ M; k ∈ K

bm ≥ bm

vþk mn

αimn

∀ m ∈ M; k ∈ K

k vkþ mn ≥ vmn

zkm

L1 L2 Smin m Smax m di cim

m∈M

k

Spatial unit representing a demand source Spatial unit with a service-providing facility District consisting of spatial units

3.3.3. Parameters

∀ i ∈ I; m ∈ M; k ∈ K

ukim ≤ 1−α imn  xkn

I A

135

n∈M; k∈K; ði; jÞ∈A

z qkij

jÞ∈A rik ð10Þ

Assignment of spatial units and districts; takes a value of 1 if and only if spatial unit i is assigned to district k and 0 otherwise Allocation of a spatial unit to a service facility within a district; takes a value of 1 if and only if the service facility in spatial unit m is the service-providing facility for spatial unit i within district k and 0 otherwise Service supply of the facility located at spatial unit m within district k; without loss of generality, it is defined as a nonnegative integer Net service balance of the service facility in spatial unit m within district k before service transfer; a real number defined as “a facility's own supply” minus “the directly allocated demand” Positive portion of net service balance bkm; max{bkm, 0} Negative portion of net service balance bkm; min{bkm, 0} Indicator variable of the sign of net service balance bkm; takes a value of 1 if and only if bkm is positive and 0 otherwise Transfer of service from the service facility in spatial unit m to that in spatial unit n within district k; a real number defined as positive for outgoing service and negative for incoming service Positive portion of service transfer vkmn; max{vkmn, 0} Negative portion of service transfer vkmn; min{vkmn, 0} Net overload in the service facility in spatial unit m within district k after services transfer; a real number Objective function term; z = max zkm Imaginary flow from spatial unit i to j in district k for contiguity evaluation; a non-negative real number defined for each pair of spatial units in A Imaginary flow sink in each district for contiguity evaluation; takes a value of 1 if and only if spatial unit i in district k is the sink and 0 otherwise

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3.4. Explanations of the equations Eqs. (1) and (2) define the objective function and together minimize the maximum overload among all service facilities. Here, variable zkm represents the net overload in a service facility in spatial unit m within district k after services transfer, and z = max zkm. Constraint (3) ensures that each spatial unit is assigned to only one district and that each district has at least one service facility. Constraint set (4) determines the service facility in a district for each spatial unit. Here, cim represents the generalized service cost per unit demand between spatial unit i and the service facility in spatial unit m, and αimn is the relative cost ratio among spatial unit i and the facilities in m and n. The first equation ensures that each spatial unit is served by only one service facility in the district to which the spatial unit belongs. The second equation ensures that the facility is assigned to the same district as the spatial unit. The third and fourth equations enforce that each spatial unit's demand is fulfilled by the minimal-cost service facility within a district. Note that the assigned facility is the minimal cost facility, and it depends on both cim values and district decisions. The related cost parameters can be calculated using GIS databases and other spatial data in advance. Constraint (5) sets the upper and lower bounds of service supply. Here, variable skm represents the service supply of a facility in spatial unit m inside district k. The first equation ensures that the total demand of each district is fulfilled by the sum of supplies from its service facilities at an allowable overload ratio β if overloaded. The second equation restricts the total service provided not to exceed the total supply limit. The third equation sets the service supply of each facility (variable skm) within its limits. Constraint set (6) defines the supply transfer between facilities in each district. Here, variable vkmn represents the service transfer from a facility in spatial unit m to that in spatial unit n within district k. It is defined positive for outgoing service and negative for incoming. The plus and minus signs in vkmn indicate the positive and negative portions, respectively. The first and second equations ensure that the service transfers between two facilities have correct signs and prevent service transfer from a facility to itself. The third and fourth equations with the sign restriction in Eq. (10) ensure that vkmn+ and vkmn− represent outgoing and incoming service transfers, respectively. These constraints are necessary to prevent non cost-efficient service sharing. The facilities with work overload are not allowed to receive more service than their overload and to transfer to other facilities. Using this method, the service for other overloaded facilities is provided only by service facilities with extra supply. Constraint set (7) defines the service balance and overload. Here, variable bkm expresses the net service balance of a facility before service transfer, calculated as its own supply minus the directly allocated demand. The first equation represents the net balance of service supply in each service facility before service transfer. The second equation expresses the overload in a facility using the net balance and total service transfer to this facility incorporating service loss or consumption during the transfer. The third and fourth equations with the sign restriction of Eq. (10) ensure that bkm+ and bkm− have the correct positive and negative signs according to the value of bkm. The fifth and sixth equations enforce that only one of bkm+ and bkm− becomes non-zero depending on the sign of bkm. The seventh equation specifies the value of bkm+ used in Constraint set (8) to impose an upper bound on the service transfer. Constraint set (8) restricts the bounds of the supply transfer. The first equation ensures that the sum of service transfers originating from each facility should not exceed its extra supply. The second equation ensures that outgoing service transfer occurs only from service facilities with extra supply. Although this equation is implied from the first equation in the set, this is included to clarify the variable relation. The third equation restricts incoming service transfer to service facilities with extra supply. The fourth equation prevents service transfer across districts; thus, facilities in each district serve only the spatial units within the district.

Constraint set (9), borrowed from Shirabe (2005) and extended to the multiple district case in this study, guarantees that spatial units constituting a district are contiguous. The left-hand side of the first equation is the difference between the inbound and outbound imaginary flows of spatial unit i. If this value is positive, the spatial unit is not considered an imaginary flow sink. The second equation ensures that each district includes only one sink. The third equation prohibits the imaginary flow between spatial units of different districts. Constraint set (10) restricts the variable ranges. In this MIP formulation, constraints for district compactness and convexity are not explicitly shown because they are application-specific. These constraints usually appear in non-linear forms in the literature and were included for a real-world case study shown in Section 5.2. The mathematical model in this study includes new developments that have not been reported in previous studies. First, detailed and practical conditions on service transfer are modeled. Constraint sets (6) to (8) prevent unnecessary service transfers that incur extra costs and result in a non-optimal solution. For instance, service transfer occurring between extra service facilities or between overload facilities would not improve capacity utilization but only incur unnecessary cost increases. In addition, without these constraints a service transfer from a facility to itself could generate a feasible solution. In order to prevent the aforementioned solutions, these constraints restrict the value and direction of service transfer between service facilities. For example, in Constraint set (8) the outgoing service transfer is allowed only with extra supply. Also, the incoming service transfer is prohibited in case of extra supply. Second, modeling of the closest facility to each spatial unit in Constraint (4) has an improved characteristic, which is being independent of decision on opening a new service facility. In most previous studies, the closest facility assignment included the decision on opening a new facility in the closest possible location to each spatial unit. This closest facility concept promotes district compactness if each district is generated by assigning the closest spatial units to a service facility in the district center. However, if there are more than one service facilities in a district, the compactness property may not be achievable. Hence, the focus of the proposed formulation is to assign the closest facility to each spatial unit and to utilize service transfer to smoothen overload inside districts with more than one service facility. 4. Solution methods This section describes a heuristics solution procedure to solve the problem. The models in Section 3 can be solved optimally for a midsized cases by a generic mixed integer programming (MIP) solvers, as shown in the next section (Section 5.1). For larger problem instances, it is generally not possible to solve such problems exactly using a generic MIP solvers due to the NP-hard nature of the problem. Thus, it is necessary to solve such problems using a heuristics method. Here, we provide a simulated annealing (SA) heuristics procedure. The details of the SA procedure are described as follows. Here, a common SA procedure is explained first. SA is a meta-heuristic that is widely used in combinatorial optimization problems (Ingber, Petraglia, Petraglia, & Machado, 2012; Kirkpatrick & Vecchi, 1983). The algorithm starts with an initial temperature (T0) as well as an initial solution (S0). S0 is considered at first the current and best solutions: SC and SB. The next solution as a neighbor, SN, is generated by a move from SC. The objective function as the cost of the system at neighbor solution, EN, is compared with that at current solution, EC. If EN is better than EC, the neighbor solution replaces the current solution and is further used to generate next neighbor solutions. If EN is worse than EC then the neighbor solution is accepted as the current solution with probability of exp(−ΔE/T) where T is equal to the current temperature. If EC is better than EB, then EB is replaced by EN. The procedure is repeated for MaxIter number of iterations, and at each iteration the current temperature is decreases smoothly by τ·T where τ is a real number between 0 and 1. This procedure guarantees that the solution is not locked at a local minimum by accepting worst

J. Ko et al. / Computers, Environment and Urban Systems 54 (2015) 132–143

solutions in each iteration with a probability. In other words, the common SA procedures found in the diverse literature to minimize this objective function is: 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12.

Set input parameters T0, τ, MaxIter, S0, Tmin; set T = T0. Calculate E0. Set SC = S0, SB = S0, EC = E0, EB = E0, iteration = 1. Generate an SN from SC and calculate EN. Let ΔE = EN − Ec. If ΔE ≤ 0 then accept the neighbor solution as the new current solution, set SC = SN; EC = EN and go to Step 9. If ΔE N 0 then generate a random value, ε, in [0, 1]. If exp(ΔE/T) N ε then accept the neighbor solution as the new current solution, set SC = SN and EC = EN. Otherwise, SC and EC remain unchanged and go to Step 10. If EC b EB then SB = SC and EB = EC. Otherwise, SB and EB remain unchanged. If Iteration = MaxIter then go to Step 11. Otherwise, set Iteration = Iteration + 1 and go to Step 4. Set T = τ·T and Iteration = 1. If T ≥ Tmin then go to Step 4. Otherwise, report SB and EB and Stop.

The proposed heuristic in this paper has the following special characteristics in order to properly reflect the structure of the proposed problem and to speed up the calculation by avoiding searching irrelevant solution space. Similar approaches to different problems can be found in the literature, for example (Duque et al., 2007; Ferligoj & Batagelj, 1982; Horn, 1995; Li et al., 2014b; Nagel, 1965). 4.1. Stage 1: initial solution At this stage, the existing assignment of spatial units to districts is considered as the initial solution to the problem. Also, for each service facility, the current service level is used as the initial service level. 4.2. Stage 2: neighbor solution generation A neighbor solution is generated by the replacement of one random spatial unit between two random districts. In this method, the two districts are chosen in random and in the first district one spatial unit is selected randomly, then the selected spatial unit is assigned to the second district. The new districts are verified for including at least one service facility. If there is no service facility in any of the new districts, the neighborhood solution generation is repeated until each district has at least one service facility. In addition, in each neighbor solution generation, a new solution for the service level in service facilities is generated. The neighbor solution for service level in each facility is generated by randomly generating an integer number that is between the minimum and maximum capacity of the service facility. Generating random numbers with fixed sum or limited sum necessitates specific algorithms and assumptions on the distribution of randomly generated numbers. Thus, the process of generating random numbers and distribution of random numbers varies by difference between maximum and minimum service capacities. In addition, the sum of these random numbers should be less than or equal to the maximum available service defined as a parameter in the problem. 4.3. Stage 3: objective function evaluation and procedure termination In this stage, the objective value of the neighbor solution should be compared with the objective function value of the current solution. The relevant calculations follow steps 4–9 of the aforementioned common SA procedure. The objective function consists of maximum overload in service facilities, penalty for violating the contiguity constraint of a district, and penalty for violating the supply and demand balance in a district. For each solution, the value of the maximum overload in each district is

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calculated by finding the minimum distance matching between spatial units and service facilities. Also, the value of the net transfer between service facilities is calculated but is not used in the objective function evaluation. For each service facility the sum of the demand of closest distance spatial units is calculated and the difference between the total demand and service level in that service facility will represent the overload in that service facility. The same calculation is carried out in all district and service facilities and the maximum value attained is the maximum overload part of the objective function. In order to include the contiguity constraint in the objective function of the SA algorithm, a large positive number M is used as the penalty for each district which violates this constraint. For each district in each solution all possible paths from each spatial unit to all the other spatial units in the same district is calculated. If for each two spatial units in the same district the value of all possible paths is zero, then contiguity constraint is violated in that district. The total number of districts violating the contiguity constraint is multiplied by M and added to the objective function. The same approach is used to guarantee the supply and demand balance in each district. As net transfers of service between service facilities in a district indicate that the total demand in this district is less than or equal to the total supply of service (with β considered) in that district, a large positive number M is used as penalty for districts violating this constraint. The total number of districts violating the demand fulfillment constraint is multiplied by M and added to the objective function. The service level of service facilities is constrained by the sum of the total service level defined. In order to speed up the algorithm in finding feasible service levels for facilities, a large sum of total supply was considered for the problem. However, in the final step of the heuristic, the optimal supply in each facility is revised to reduce the unused service in each service facility. For this purpose, the sum of supply of facilities was added to the objective function. Thus, the procedure results in the minima for service level of service facilities. Furthermore, the service level of each service facility is reduced to minimize the total service level while the supply and demand balance is not violated. By using a very large value for the initial temperature all possible changes are permissible in the neighborhood search. However, the value of T will decrease by multiplication with τ which is a random number between 0 and 1 in each iteration. The procedure is terminated when the maximum specified number of iterations is completed. In addition, it may be possible that a local minima does not show any improvement after several consecutive iterations. In this case, a maximum number of iterations without improvement is set to terminate the procedure. 5. Numerical and case studies This section presents two numerical studies. The first one (Section 5.1) illustrates the use of the methodologies and formulations in this paper through simple but intuitive numerical examples. The second one (Section 5.2) is a case study and shows a practical application of the methodology for judicial district boundary change in a state in the US. 5.1. Illustration of the methodology using a numerical example This first example solves a problem of 20 spatial units, 6 of which include service facilities. The map of the spatial units is shown in Fig. 1. Three districts were assumed for this example. The demand and supply capacity values are shown in Table 1. The total available service supply (L2) was set as 320 units, which was less than the total demand of 350 units; thus, overload should exist in some districts. The maximum allowable overload ratio β was set to 15%, and γ was assumed to be 10%. Table 2 shows the generalized service cost between spatial units and service facilities (cim). The MIP was solved using a generic MIP solver IBM CPLEX®, and the calculation took around 20 min on a PC with a 2.4 GHz CPU. The convexity

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16

17

18

16

20

19

17

A 11

12

11

14

13

7

1

8

3

2

14

15

4

9

10

C 2

3

4

5

Fig. 2. Districting result. Letters A, B and C represent districts.

5

Table 1 Demand and supply bounds at each spatial unit. Spatial unit (i)

Demand (di)

Min. supply (Smin m )

Max. supply (Smax m )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

12 10 20 10 14 24 17 15 30 12 22 10 14 10 25 23 28 15 16 23 350

– 0 – – 0 – – – 0 – 0 – 0 – – – – – 0 – 0

– 30 – – 90 – – – 30 – 90 – 30 – – – – – 90 – 360

and compactness of the districts were not explicitly considered for this illustrative example for simpler calculation and display. The SA heuristic was also verified for this numerical example. The heuristic converges to a near optimal solution for Temperature 200 and iterations = 50 within an hour. Increasing Temperature to 1000 and iterations to 250 would result in the optimal solution in 45 min. Please note that due to the randomness of the solution procedure, the result and time would vary. The results are shown in Fig. 2 and Table 3. Fig. 2 indicates that the constraints (including contiguity) are well-maintained. In district A, there are two service facilities, 2 and 11. Service facility 2 has one unit of extra supply before service transfer (bA11 = 1) that is transferred to Service facility 11, which has seven units overload. After this intradistrict service transfer and loss of transfer at a 10% rate, the resulting

Table 3 Allocation among spatial units and service facilities, and service balance, transfer and overload. District

A

Service facility Spatial units to which service is directly provided by each service facility

Sum of demands Service supply (skm) Service balance before service transfer (bkm) Sum of incoming service transfer Resulting overload (max{zkm, 0})

Table 2 Cost per unit demand between spatial units and service facilities (cim).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

20

B

13 8

1

Fig. 1. Map of the spatial units. Each rectangle represents a spatial unit. The shaded rectangles represent the spatial units that have service facilities.

Spatial unit

7

10

9

19

15

6 6

12

18

2 1 2

B 11 6 11 16 17

22 23 1 −1 0

97 90 −7 1 6.1

C

9 9

13 12 13

19 14 15 18 19 20

30 24 −6

24 18 −6

89 83 −6

5 3 4 5 7 8 10 88 82 −6

6

6

6

6

Service facility 2

5

9

11

13

19

100 0 75 175 275 180 112 125 180 269 236 202 206 266 361 316 300 309 347 407

375 275 200 100 0 437 340 224 160 103 447 361 301 224 202 480 407 361 316 300

269 180 125 103 160 300 200 75 0 100 293 202 141 103 180 320 250 214 202 236

202 236 283 361 447 103 125 224 293 388 0 100 175 300 425 103 160 224 316 412

250 206 202 236 301 224 141 103 141 224 175 75 0 125 250 180 112 103 160 246

407 347 316 300 316 382 301 224 202 214 316 224 160 100 160 275 175 100 0 100

16

A

11 6

17 12 7

1

2

18

19

13

14

8

9

3

20

B

15 10

4

5

Fig. 3. Districting result with the 2-stage solution approach.

J. Ko et al. / Computers, Environment and Urban Systems 54 (2015) 132–143 Table 4 Allocation among spatial units and service facilities, and service balance, transfer and overload with the 2-stage solution approach. District

A

B

Service Facility Spatial units to which service is directly provided

11 6 11 16

13 12 13 17

Sum of demands Service supply (skm) Service balance before service transfer (bkm) Sum of incoming service transfer Resulting overload (max{zkm, 0})

69 90 21

52 29 −23 21 4.1

19 14 15 18 19 20 89 68 −21

21

C 2 1 2 3 7

5 4 5

9 8 9 10

59 30 −29

24 75 51

57 28 −29

25.5 6.1

−51

25.5 6.1

overload in Facility 11 is 6.1 units. In District B, three service facilities (i.e., 9, 13, and 19) fulfill the district demand and have six units of overload each. In District C, the only Service facility 5 fulfills the district demand, and its overload is at six units. As shown in Table 3, in this numerical example, the maximum overload was 6.1 units after service transfer. This implies a 17% reduction of service shortage was achieved by using the service transfer. To demonstrate the effectiveness of the proposed integrated allocation-districting model, we compared the result with that of a two-stage solution approach (allocation and then districting). The demand-facility allocation is determined based on minimizing the maximum workload. Then the allocated units are grouped as districts by the criteria used in this paper. As expected, this comparison verified the impact of concurrent consideration of allocation and redistricting problems. Although the calculation time decreased by 40% for the two-stage model, the objective function increased (worsened) to more than 340%, as shown in Fig. 3 and Table 4. In general, a multi-staged approach restricts the solution search space by the values from the first stage; in this new example, the allocation in the first step restricted the overload values too high before districting. Another reason is that assignment of spatial units to their nearest service facility without districting constraints would result in exceeding the maximum capacity of the service facility. Thus, the facilities would be highly overloaded or underutilized because of improper distribution of the resources and demands. One benefit of imposing districting constraints on allocating service facilities to the spatial units is to balance the service-demand parity within district boundaries. For example, in our proposed models, spatial units 3 and 4 are assigned to service facility 5. However, if allocation stage is done without restriction on districting, spatial unit 3 is assigned

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to service facility 2 that is the closest service facility to it. This adds up 20 units of demand to service facility 2 whereas its total capacity is limited by 30 units of supply. Overall, the balanced distribution of demand and supply in district based service planning results in reduced district overload. 5.2. Redistricting judicial boundaries in a US state (Nebraska) We applied the developed methods to the judicial boundary redistricting problem for the county court system in the state of Nebraska in the US. Judicial workload and resource assessment were archived for some US states (The National Center for State Courts), and a countylevel court system in Nebraska (Jones, Kirven, & Tallarico, 2006) is chosen for this case study. As shown in Fig. 4, the state consists of 93 counties (spatial units), and a judicial district is a collection of counties under the county court system. In this county court system, each county has a county courthouse. However, only 35 counties currently have county courts with judges, called judge homes, and these courts are interpreted as the spatial units with service-providing facilities. The remaining 58 counties depend on judges who travel to the courthouses in each district. Currently, there are 12 judicial districts in the state, each of which consists of between 1 and 17 counties. The two counties with the two largest metro regions (cities of Lincoln and Omaha) have their own districts (i.e., Districts 3 and 4, respectively). This paper assumes that these two counties remain independent districts for political and administrative reasons. Travel distances between counties are considered the minimal travel distance for the judges. The goal of this case study is to verify the mathematical model in this paper through the case of spatial inequality of judicial work overload in the state of Nebraska. The status quo districting (Fig. 4) has experienced work overload and overcapacity disparity problems (University of Nebraska Public Policy Center, 2007), as shown in Fig. 5. The original workload data available for each county courthouse was converted to the workload level at each judge home. In this data conversion, without loss of generality, each county courthouse is assumed to be served by the closest judge home. Over the years, the characteristics of the judicial districts have changed, including the population, number of filings, and number of lawyers. A changed workload poses administrative and operational issues for county courts, and some courts have experienced more serious overload problems. Therefore, judges in the state have attempted to reorganize the current districts in order to equitably allocate the limited judicial resources as well as fairly and effectively provide court services to citizens. This case study compares three districting scenarios—Scenario 1: the status quo district structure, Scenario 2: redistricting without intradistrict service transfer, and Scenario 3: redistricting with service

Fig. 4. The status quo judicial districts in the case study. Different shades represent counties with/without judge homes.

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Fig. 5. The overload in judge homes in the status quo district structure. The workload and judicial resource data were extracted from 2006 Nebraska Judicial Workload Assessment study (Jones et al., 2006), and units are in yearly full-time equivalent (FTE) of judges. A positive number represents demand overload and a negative one indicates overcapacity.

transfer. It was also assumed that the current judge home locations were maintained, but their available judge full-time equivalent (FTE) can change. This assumption was applied to provide stability to the current judge home structure. The service transfer implies that some judges may travel beyond their initially assigned courthouses or that some work might be transferred between judge homes. For this judicial redistricting calculation, a heuristic solution method was used because of the NP-hardness and large problem size. The solution procedure for the formulations in Section 3.2 with additional convexity and compactness constraints was implemented as a customized simulated annealing heuristic using the mathematical programming language MATLAB©. This case study problem was also solved by a generic commercial MIP solver IBM OPL Cplex®. However, as expected, the MIP solver did not produce any feasible solution even after 24 h of computation due to the nature of the computational complexity of this problem. The customized simulated annealing heuristic in this study starts with the status quo district structure as the initial solution. In each iteration, two neighbor districts are randomly selected (e.g., D1 and D2). From each district, a county is selected randomly from the set of counties with common borders with the other district. For example, in district D1, county C1 (which is adjacent to at least one county in D2) is selected. Similarly, a county in D2 with these properties is randomly selected. These two counties are swapped between the two districts. Then, the solution with this new district structure is evaluated for acceptance based on the objective function of multi-criteria, including

contiguity, service overload, compactness, and convexity. All constraints of the MIP model are included in the objective function with penalty factors. A new solution with acceptable improvement in the objective function replaces the current one. The procedure continues for a specified number of iterations, starting temperature, and cooling rate. Fig. 6 shows the new districts and their overload with Scenario 3. Compared to Fig. 5, spatial units (counties) and service facilities (i.e., service-providing judge homes) are re-organized as new districts. Table 5 shows the summary statistics of the results from the three scenarios. In Scenario 3 (i.e., redistricting with service transfer), the total, average, and maximum overload all decrease significantly compared to the status quo and no-service transfer scenarios. The sum of the total overload is 5.3 units in the status quo case, but this decreases to 2.65 units in Scenario 3. In addition, whereas Scenario 2 does not decrease the number of overloaded courts from Scenario 1, Scenario 3 decreases this number from 14 to 10. The standard deviation and range of overload are also reduced in Scenario 3, demonstrating the effectiveness of the service transfer. Table 6 displays a district-level comparison of Scenario 1 (status quo) and Scenario 3 (redistricting with service transfer). The value of the maximum overload decreases in seven districts and increases in only one district, but the value of the increased overload is still smaller than the maximum overload of Scenario 1. In addition, four overloaded districts become non-overloaded in Scenario 3. The similar pattern of improvement is found for the overload range.

Fig. 6. Redistricting and new overload distribution with Scenario 3.

J. Ko et al. / Computers, Environment and Urban Systems 54 (2015) 132–143

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Table 5 Summary statistics for the three scenarios. Units: FTE. Scenario

Total overload

Average overload Max. overload

Number of courts with overload

Standard deviation of overload

Range of 'overload plus service capacity'

1 2 3

5.3 5.9 2.65

0.38 0.42 0.27

14 14 10

0.42 0.48 0.37

1.74 1.84 1.16

Status quo Redistricting without service transfer Redistricting with service transfer

0.98 0.92 0.5

Detailed results are shown in Table 7 for each judge home (i.e., service-providing facility). The table indicates that the overload/ overcapacity disparity is improved. In most judge homes, the absolute values of overload become smaller. The maximum overload is 0.98 in Facility 56 in the status quo district structure, which becomes 0.50 in Facility 66 in the new solution. From a comparison of Figs. 5 and 6 and the corresponding values in Tables 5 to 7, it is apparent that the new methodology significantly improves the overload amount and disparity. First, the new method reduces the total number of overloaded service facilities. In addition, the maximum overload is reduced. All these improved results are achieved with the same supply and demand as the status quo case. Therefore, these results demonstrate that the new methodology in this study can significantly reduce the overload throughout the service facilities. In summary, the analysis of the judicial district case clearly presents that the method developed in this paper is a useful tool for reducing and balancing the spatial inequality of judicial workloads as well as facilitating the efficient use of judicial resources in a state.

problem mainly resulted from local capacity insufficiency or overall district structure. Therefore, this study helps establish a spatial decision framework with rich spatial information and integrate optimization tools with the GIS. We also would like to point out the role of the closest facility constraint. This proximity constraint was added to reflect the situation in which service and demand matching is governed by certain rules. This is common in some services such as public healthcare or legal services because of regulations or administration purposes. The proximity is a generalization of such rules. Such rules may hinder a better natural supply and demand balance, and exacerbate the overload. This paper intends to address this more difficult problem. If we relax this closest facility assignment constraint, it is possible that optimal solution would improve due to a larger solution space. This allocation flexibility might also lead to the reduced effect of the intra-district service transfer for some problem instances.

6. Conclusions 5.3. Implications from the study and further discussion One implication of the case study is that the integrated model in the study enhances the usefulness of spatial service demand and supply information and analysis results. Currently, rich service information is often available from the increasing automated data collection in service providers. These extensive data will become more practically useful if they are used with a more-sophisticated planning model. In this case study, demand and supply information was used for redistricting, capacity allocation, and service transfer. Conversely, the model in this study can help identify additional data collection required for service improvement. For example, detailed cost data might be needed for service transfer in a geographically wide region. In addition, the model and results provide better spatial information for decision makers. Thus the decision makers can evaluate different spatial planning scenarios depending upon redistricting and service transfer conditions, and identify principal directions for overload improvement in a more comprehensive way. For example, the judges can examine whether the overload

This paper presented a new approach for an integrated redistricting, location-allocation and sharing problem to address work overload. This new approach integrates redistricting and discrete facility location problems, and incorporates a novel concept of intra-district service transfer to optimally use excessive service balance for other overloaded facilities. In this new method, extra supply in facilities of a district is transferred to overloaded facilities within the same district. The new approach was successfully applied to a judicial districting problem to significantly reduce possible work overload. This study also successfully developed new mathematical models and solutions procedures for the new integrated service-planning approach. The developed mathematical programming models can address the interrelated complicated districting, facility decision, and service transfer. To incorporate information on geographically distributed units, a GIS tool was adopted in data acquisition, analysis, and

Table 7 Overload comparison in the status quo and new districts. Units: FTE. Table 6 District-wide comparison of maximum overload and range between status quo districting and redistricting with service transfer. Unit: FTE. Maximum overload

Range of ‘overload plus overcapacity’

District

Status quo

Redistricting with service transfer

Status quo

Redistricting with service transfer

1 2 5 6 7 8 9 10 11 12

0.15 0.86 0.51 0.63 0 0 0.48 0.35 0.98 0.44

0.5 0.46 0.47 0 0 0 0 0 0.11 0

0.37 1.13 0.99 1.29 0.76 0.24 0.87 0.51 0.99 0.96

0 0.39 0.35 0.05 0 0.13 0 0 0.08 0.02

Note: The new district identification numbers are given in a way to provide high overlap with the current district numbering system. Districts 3 and 4 are not displayed because they are metro-area single-county districts.

Overload

Overload

Service facility

Status quo

Redistricting with service transfer

Service facility

Status quo

Redistricting with service transfer

1 2 10 14 17 19 21 22 23 24 27 34 40 41 45 51 56

0.35 −0.76 −0.39 −0.66 0.44 −0.13 −0.15 0.63 0.05 0.24 0.31 0.09 0.48 −0.34 −0.19 −0.01 0.98

−0.43 −0.49 0.21 −0.05 0 0 −0.36 0 0 0.11 0.46 0 0 0.12 −0.46 0 0

60 66 69 70 71 73 74 75 76 77 78 79 80 81 89 93

−0.37 −0.27 −0.16 −0.19 0.51 0.14 −0.22 −0.24 0.15 0.86 −0.45 −0.23 −0.48 −0.52 0.07 −0.27

−0.66 0.50 −0.03 −0.48 0.24 0.03 0 −0.40 0.47 0.44 0 −0.02 0 0 0.07 0

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management of the spatial data. A new approach with a heuristic solution method was successfully developed and applied. The method presented in this paper is expected to reduce work overload with less service supply, increasing service quality costeffectively. The accessibility of residents to public services is related to the quality of life in a region. To achieve an adequate level of service accessibility, local governments and regional policymakers have been concerned with the equitable and efficient use of limited resources, and this is particularly important in economic downturn. The method presented in this paper helps address such issues. The method of this study could be extended to various district-based service-planning problems. The case study focused on judicial workload imbalance, but further development of the case study may additionally consider political feasibility, implementation practicability, population change, filing trends, and county relationship history. The method in this paper can also be applied to other service-planning problems, such as adjusting school districts, relocating emergency-response units, designing police districts, and territory planning of public utilities.

Acknowledgment This research has been supported in part by the John C. and Nettie V. David Memorial Trust Fund through the Research Council at the University of Nebraska-Lincoln (2011-2012), National Science Foundation CMMI grant 1331633 (using part of a decomposition algorithm), and the research fund at Ajou University.

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