Optimal Layout of Transshipment Facility Locations

Optimal Layout of Transshipment Facility Locations on An Infinite Homogeneous Plane Weijun Xie1 and Yanfeng Ouyang∗2 1

School of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332 2 Department of Civil and Environmental Engineering, University of Illinois, Urbana-Champaign, Urbana, IL 61801

Abstract This paper studies optimal spatial layout of transshipment facilities and the corresponding service regions on an infinite homogeneous plane R2 that minimize the total cost for facility set-up, outbound delivery and inbound replenishment transportation. The problem has strong implications in the context of freight logistics and transit system design. This paper first focuses on a Euclidean plane and shows that a tight upper bound can be achieved by a type of elongated cyclic hexagons, while a cost lower bound based on relaxation and idealization is also obtained. The gap between the analytical upper and lower bounds is within 0.3%. This paper then shows that a similar elongated non-cyclic hexagon shape, with proper orientation, is actually optimal for service regions on a rectilinear metric plane. Numerical experiments are conducted to verify the analytical findings and to draw further insights.

1

Introduction

Problems related to facility location (e.g., fixed-charge location problems) and routing (e.g., travel salesman problem, or TSP) impose two fundamental yet distinct challenges to logistics system design. Due to their intrinsic complexity, these problems are typically handled separately in the literature (e.g., see Daskin, 1995 and Toth and Vigo, 2001 for complete reviews). Relatively fewer studies looked at the integrated “location-routing” problem with or without inventory considerations (e.g., Perl and Daskin, 1985; Shen and Qi, 2007). The location of distribution centers and the routing of outbound delivery vehicles are optimized simultaneously while inbound shipment (i.e., providing replenishment to these facilities) is assumed to be via direct visits (or more often, omitted from the model). Furthermore, most of these efforts focused on developing discrete mathematical programming models which can only numerically solve very limited-scale problem instances. In particular, little is known about the optimal facility lay-out, the suitable customer allocation, and the optimal vehicle tour in infinite homogeneous planes. Recently, Cachon (2014) studied a different location-routing model in a homogeneous Euclidean plane which optimizes the spatial layout of facilities that serve distributed customers (i.e., similar to a median problem) and the routing of ∗

Corresponding author, email: [email protected].

1

an inbound vehicle which visits these facilities (i.e., similar to a TSP problem). The model tried to minimize the total cost related to outbound customer access (i.e., direct shipment) and inbound replenishment transportation. This problem has strong implications on practical logistics systems design in real-world contexts. For example, transshipment is often used in a timber harvesting system, where the lumbers are collected by trucks to local processing mills, and then shipped out by train. Train capacity is generally orders of magnitudes larger than that of local trucks, and it is often sufficient to assume infinite capacity of a train and design a single train track route for a large area of forest. Another example is commuter transit system design in low-demand areas (e.g., building upon ideas from Nourbakhsh and Ouyang (2012)), where a bus (with sufficient capacity) collects passengers from a certain region at optimally located bus stops (where local passengers gather). Many similar examples can be found in the contexts of planning various mobile service systems; e.g., those related to vending trucks, concert band tours, and others. Mathematically, this problem can be described as follows (see Figure 1(a) for an illustration of the notation). We use a transshipment system to serve uniformly distributed customers (with demand density λ per area-time) on an infinite homogeneous Euclidean plane R2 . Transshipment facilities can be constructed anywhere with a prorated set-up and operational cost f per facilitytime. All facilities receive replenishment from a central depot, which is co-located at one of the facilities, through an inbound truck with infinite capacity. This truck will supply all facilities along one tour, incurring average transportation cost of C per distance1 . Without losing generality, we assume that the transshipment facilities are indexed along the tour of the inbound truck; i.e., the inbound truck starts from facility 1 (the depot), visits facilities sequentially in set N = {1, 2, · · · , N } before returning to facility 1, where N := |N | → ∞ is the total number of facilities. Facility i ∈ N is located at xi ∈ R2 to serve the customers in its service region Ai ∈ R2 through direct shipment, with a transportation cost of c per demand-distance. For each customer at x ∈ Ai , we use kx − xi k to denote the outbound travel distance. Moreover, for simplicity, the size of service region Ai is denoted by Ai = |Ai |, and we assume the inbound truck travels a distance of li within Ai . Since C all cost terms are relative, we further define κ = cλ f (area-demand-distance/facility) and r = cλ to denote the relative magnitudes of facility cost and inbound transportation cost as compared to the outbound cost. Some basic properties of this problem is readily available. For any given facility layout {xi : i ∈ N }, each customer should obviously choose the nearest facility for service and a tie may be broken arbitrarily. Thus, the set of service regions {Ai : i ∈ N } must form a Voronoi diagram (Okabe et al., 1992; Du et al., 1999), where  Ai = x ∈ R2 : kx − xi k ≤ kx − xj k, ∀j 6= i . (1) Moreover, for an optimal TSP tour along N facility locations in the Euclidean plane, it is easy to see that the optimal TSP tour has no crossover between any four facility locations; otherwise, a simple local perturbation can improve the solution. With this, the optimal solution of our problem must satisfy the following properties. Property 1. For any given facility locations {xi : i ∈ N }, 1

This assumption makes the replenishment frequency irrelevant to the optimization problem.

2

• the optimal service regions form a Voronoi diagram, i.e., (1) holds and [ Ai = R2 ;

(2)

i

• each service region Ai is a convex polygon (Okabe et al., 1992) with the number of sides ni ; T • for all x ∈ Ai Aj , kx − xi k = kx − xj k, and therefore, for all i ∈ N , li =

1 (kxi − xi−1 k + kxi − xi+1 k) ; 2

(3)

• the maximum distance maxx∈Ai kx − xi k is achieved by one of the corners of Ai if the service area Ai is bounded; • the line segment connecting xi and xi+1 is perpendicular to the interception line of Ai and Ai+1 . From Property 1, we can see that the boundary between any two adjacent facilities extends a triangle with either of the facility locations (e.g., ∆EBD and ∆GBD in Figure 1(b)). Moreover, these two triangles obviously must be identical (i.e., ∆EBD ∼ = ∆GBD). For simplicity, we define the following: Definition 1. Within each facility service region, • A “basic triangle” is the one extended by the facility location and one side of the service region border; e.g., ∆EBD or ∆GBD in Figure 1(b); • A “basic angle” is the internal angle of a basic triangle at the facility location; e.g., ∠BED or ∠BGD in Figure 1(b).

li

xi+1

B Ai

xi

outbound delivery

G

E xi -1

x

inbound truck tour

D

(a) Inbound and outbound transportation

(b) Basic triangle and basic angle

Figure 1: Notations for a transshipment system

3

In an infinite plane, a suitable objective is to find the optimal facility layout {xi : i ∈ N }, the service region partition {Ai : i ∈ N }, and the inbound truck tour that minimize the total system cost per unit area-time including facility set-up, outbound delivery and inbound replenishment cost per unit area-time. Obviously, the optimal number of facilities N → ∞; otherwise, the average outbound cost goes to infinity since at least one of the facilities would have an unbounded service region on this infinite plane. Hence, the optimization problem can be expressed as follows,  Z N  N κr X f X 1+κ kx − xi kdx + z= min lim N li N N ,{xi },{Ai } N →∞ P Ai i=1 i=1 Ai

(4)

i=1

s.t. (1) − (3). While extensive studies have been conducted in the literature to understand and solve the location and routing aspects of this problem, the analytical properties of the optimal location layout, customer partition, and vehicle routing strategies still remain unclear. In this paper, we fill these gaps by providing mathematical formulation of this problem and proving the optimal or near-optimal configuration of the transshipment facilities under different distance metrics. We first introduce a near-optimum shape for the Euclidean metric case – a cyclic hexagon with two equal “long” sides and four equal “short” sides. We also provide formulas to compute the size and shape of the service region (including the length of TSP tour within each shape), the best facility layout, and the best total system costs for this near-optimum configuration. An infeasible lower bound is introduced by relaxation and idealization, which shows that the proposed cyclic hexagonal shape yields a very small gap. After all these discussion, we shift our focus to a rectilinear metric plane and show that a cost lower bound can be achieved by a similar spatial configuration with elongated non-cyclic hexagons, which hence becomes exactly optimal, as long as they are properly oriented in the coordinate system. Numerical experiments are conducted to verify the correctness of our analytical results for both Euclidean and rectilinear metrics. Finally, we discuss the potential impacts of other factors such as inventory cost considerations. The remainder of the paper is organized as follows. Section 2 presents a brief summary of the related literature, and highlights the contributions of this paper. Section 3 derives cost upper bound and lower bound for the Euclidean metric case, and compares several different spatial configurations. Section 4 further discusses results for the rectilinear metric case. Section 5 presents the numerical experiments, analyses, and discussion. Section 6 concludes this paper.

2

Literature Review

Only limited literature has addressed some simpler versions of this problem, mainly in the form of large-scale travelling salesman problem, planar facility location problem or uniform quantization problem. On the routing side, it is well-known (Beardwood et al., 1959) that the optimal TSP tour length to visit N randomly √ distributed points in a bounded area of size A is almost always asymptotically proportional to N A by a constant k, when N is sufficiently large. Later, Johnson (1996) verified by studying the near-optimal Held-Karp bound that k ≈ 2.042. Daganzo (1984) proposed a swath strategy for building asymptotically near-optimum TSP tour, as illustrated in Figure 2. The decision essentially reduces to determining the swath width to balance the trade-off between longitudinal travel (which is somewhat inversely proportional to the swath width), and 4

local (or lateral) travel to reach customers (which increases monotonically with the swath width). These results hold for both Euclidean and rectilinear distance metrics (sometimes referred to as L2 and L1 metrics), and they shed light on the asymptotic behavior of our problem if we imagine the customers in each Ai , ∀i are spatially clustered at one point xi (instead of being spatially distributed). On the facility location side, Newell (1973) pointed out that warehouse service regions should ideally have “round” shapes in order to minimize outbound delivery costs, although he also acknowledged the fact that round regions do not form a spatial partition. Similarly, Gersho (1979) conjectured that hexagonal shape is generally optimal in a two dimensional space. Later, Newman (1982) and Haimovich and Magnanti (1988) respectively proved that under squared Euclidean metric and any positive power of Euclidean metric, regular hexagonal service regions are optimal for outbound customer service. These results, albeit very relevant, do not solve our problem because they hold only when the inbound routing cost is ignored. In fact, Geoffrion (1979) incorporated inbound freight cost into a warehousing location problem while approximating the service regions with identical squares, and also compared the effects of different cost components (fixed facility cost, outbound cost and inbound cost). Recently, Cachon (2014) compared three regular service region shapes (i.e., equilateral triangle, square and regular hexagon), showing that equilateral triangle tessellation is the best among these three when inbound transportation cost is relatively high. Carlsson and Jia (2014) further proposed several feasible tessellations, among which Archimedean spiral was proven to be asymptotically optimal under dominating inbound cost. This indicates that the consideration of inbound vehicle routing cost significantly affects the optimal spatial configuration of the transshipment system, making “round” service region shapes undesirable. Intuitively, inbound transportation cost tends to favor a design where facilities are clustered, which nevertheless twists the shape of the service region to become “irregular”. This is somewhat consistent with what we know about the “mobile facility problem” (Suzuki and Okabe, 1995), which deploys a given number of facilities to minimize customer access cost while restricting the distance of the inbound vehicle along its TSP tour (i.e., this problem does not try to optimize the number of facilities nor the total TSP distance). However, to the best of our knowledge, no general results have been revealed regarding the optimal facility layout and service region configuration that achieve the best trade-off among the inbound, outbound, and facility costs. Knowing the optimal shape of service regions would allow researchers to qualitatively approximate the outbound logistics cost in the facility service region, which significantly simplifies the modelling challenge. Hence, while the optimal shape was generally unknown, various studies had built their solution methods upon the “likely” optimal results. As early as the 1960s, Edwin and Michael (1964) compared four different market partition shapes (equilateral triangle, square, regular hexagon and circle) in a market equilibrium model to maximize each firm’s profit within its market region; however, they did not prove that regular hexagon should be the optimal choice of market layout. While solving transshipment problems, Daganzo (1992) assumed round service regions to approximate the outbound cost at each transshipment level, which was later proven to provide a tight cost lower bound for the actual optimal design (Ouyang and Daganzo, 2006). Qi and Shen (2010) developed several spatial partitioning approaches based on regular hexagonal heuristics to cover arbitrarily distributed demand points, and showed that the worst-case errors were bounded by a constant. Cui et al. (2010) and Li and Ouyang (2010) integrated regular hexagonal shapes into the facility location design under probabilistic facility disruptions.

5

(a) Swath construction

(b) Routing under Euclidean metric

(c) Routing under rectilinear metric

Figure 2: Illustration of swath strategy for building asymptotically near-optimum TSP tour (Source: Daganzo (1984))

3

The Euclidean Metric Case

In this section, we will try to first obtain a cost upper bound by constructing a reasonable feasible solution. Then we will derive a cost lower bound based on relaxation and idealization. After that, we show that the gap between these bounds is quite small, and hence the proposed feasible solution is near-optimum.

3.1

Upper Bound

To construct an upper bound to (4), we first consider a set of N ≥ 1 facilities N = {1, 2, · · · , N }. Facility i ∈ N serves a convex polygon service region Ai with ni ≥ 3 sides and area size Ai . Each polygon contains ni − 2 identical isosceles basic triangles, each with basic angle 2¯ αi , with outbound delivery only and 2 identical isosceles basic triangles, each with basic angle 2αi , which are passed by the inbound truck. The radial sides of all these triangles have an equal length of Ri , such that the ni -sided polygon is cyclic; i.e., it is circumscribed by a circle of radius Ri . See Figure 3(a) for an illustration. These two types of basic angles 2α ¯ i , 2αi satisfy (ni − 2)¯ αi + 2αi = π, α ¯i > 0 and αi > 0. Since the inbound truck must travel through the shortest distance within each service region, we must have αi ≥ α ¯ i which implies that π2 > αi ≥ nπi . We shall be careful, that with this construct, the set of such cyclic polygons (even when N → ∞) may or may not yet be feasible (i.e., forming a non-overlapping partition of R2 ). However, as we shall see later, the cost-minimizer among this type of cyclic polygons happens to be feasible. To see this, we note that the average inbound and outbound costs for such an ni -sided cyclic polygon region Ai , based on (4), can be obtained by considering each of its basic triangles; i.e.,   Z α¯ i Z αi 1 1 2 3 3 −3 3 3 −3 (ni − 2)κf Ri cos α ¯i cos tdt + κf Ri cos αi cos tdt + 2κrf Ai Ri cos αi , (5) Ai 3 3 −α ¯i −αi subject to ni2−2 Ri2 sin 2¯ αi + Ri2 sin 2αi = Ai and (ni − 2)α ¯ i + 2αi = π. Before deriving the optimal basic angles α ¯ i and αi that minimize (5), we introduce a function in the following lemma. 6

ɲi ɲi 2αi R

(a) Cyclic polygons

(b) A feasible tesselation

Figure 3: A feasible tessellation and cost upper bound Lemma 1. For any given ni ≥ 3 and r ∈ [0, ∞), h theimplicit equation H(ni , r, α) = 0 has one and only one root with respect to α in the domain nπi , π2 , which we denote by α∗ (ni , r), where   π α π − 2α π π − 2α π − 2α H(ni , r, α) = sin α cos log tan + − cos2 α sin log tan + ni − 2 4 2(ni − 2) ni − 2 4 2 π − 2α π − 2α π − 2α − r sin 2α sin − r(ni − 2) cos sin2 . (6) ni − 2 ni − 2 ni − 2 2

Now we are ready to show the optimal shape of the cyclic polygons defined above. Lemma 2. If an arbitrary service region takes the shape of an ni -sided cyclic polygon defined above and has a fixed area size Ai , then the cost function (5) is minimized when the basic angles and the radial side length take the following values: αi = α∗ (ni , r), α ¯i =

p 1 π − 2α∗ (ni , r) , Ri = Ai [(ni − 2) sin α ¯ i cos α ¯ i + sin 2αi ]− 2 . ni − 2

Moreover, the total cost for this service region becomes √ κf Ai Ai f+ , g(ni , r) where

(7)

1

3 sin α ¯ i (sin 2αi + (ni − 2) cos α ¯ i sin α ¯i) 2
g(ni , r) = . α ¯ π i sin α ¯ i + cos2 α ¯ i log tan 4 + 2 + 4r sin α ¯ i cos αi

(8)

Now we consider all N such ni -sided polygons, i ∈ N = P{1, 2, · · · , N }, each with the optimal shape (e.g., basic angle αi and radius Ri ). Again, let A = N i Ai denotes the total area occupied 7

by all N service regions. Since cost formula (7) holds for each service polygon i ∈ N , the average cost per unit area across these N service regions becomes: 3

N

N f X κf Ai2 + . A Ag(ni , r)

(9)

i=1

Proposition 1 below shows that, among all those shapes that form a spatial partition, (9) reaches a minimum value when ni = 6, ∀i. Proposition 1. For polygons that form a partition of the Euclidean plane, for all {ni }, N ≥ 1, r ≥ 3 1 1 PN P Ai2 A2 √ A2 √ A , we have 0, and A = N i=1 Ag(ni ,r) ≥ N g (PN ni /N,r) ≥ N g(6,r) . All equalities hold when i=1 i i=1 ni = 6 for all i. Hence, for all r ≥ 0, the following holds. s 1 1 Nf κf A 2 κf A 2 κ2 f 3 Nf ∗ P ≥ (9) ≥ +√ +√ ≥33 = zub . 2 (6, r) N A A 4g N g (6, r) Ng i=1 ni /N, r

(10)

The last inequality becomes an equality only when Ai = A/N, ∀i ∈ N . In summary, the last term of (10) can be achieved when and only when ni = 6, ∀i, and Ai =
2 2 A κ −3 [g(6, r)] 3 . This implies that identical 6-sided cyclic polygons (i.e., which we call “cyclic N = 2 hexagons”) is the cost minimizer among the class of cyclic polygons we have considered. Also, notice that 6-sided cyclic polygons can obviously form a spatial partition, and henceqthe last term ∗ =33 in (10) is achievable and feasible; see Figure 3(b) for an illustration. Hence, zub upper bound, and likely a tight upper bound, of (4).

3.2

κ2 f 3 4g 2 (6,r)

is an

Lower Bound

We now construct a cost lower bound which generalizes the asymptotic results in ?. Consider now the set of all solutions that incur a fixed inbound travel length l, a fixed number of facilities N that collectively cover the customers in an area of total size A and also satisfy the conditions in Property 1. The individual service regions in these solutions may take any shape, and some may not even be feasible if they do not form a spatial partition. The lowest possible total cost among these (relaxed) solutions will surely yield a cost lower bound. Note first that a circular shape minimizes the outbound cost (Ouyang and Daganzo, 2006) for any given size of service regions. Thus, if the inbound travel length l is larger than the total A diameters of N identical circles (each with area size N ), then the case degrades to a trivial one where the optimal cost is achieved when all N service regions take the shape of identical circles
1 A 2 of radius πN . This case is illustrated in Figure 4(a). We shall note that this case never yields a good cost lower bound since the circular shape of the service regions will be far from forming a spatial partition, and that we can always shift the facility locations and their service regions along the TSP tour to reduce the length of inbound truck (without changing the total service region size A nor the outbound costs).
1 Now we consider the more general case when l ≤ 2 NπA 2 (see Figure 4(b)). Recall from Property 1 that the service region of each facility should still be a Voronoi polygon. In case all 8

G

G B xi

xi+1

xi

I I’

I’’

xi

D xi+1

J

I’’

(c)

M M’

E

B

E

M’

G’ G M

B

xi

I

(b)

G

M xi+1 D

B

E

J

(a)

G’’

M xi+1 D J’

xi

D xi+1

E

G’ G

M’’

B

D’ xi

M D xi+1

E

J (d)

(e)

(f)

R

α xi

xi+1

(g)

Figure 4: Construction of a lower bound facilities are along a TSP line, two adjacent service regions should be separated by a boundary line that is perpendicular to the TSP tour (see Figure 4(b)), and each facility will only serve the customers within the two nearest boundaries. To minimize the outbound cost of each facility within its boundaries, the optimal service region should be the intersection of the area between the boundaries and a circle, such that the maximum delivery distance is minimized. This is easy to prove by contradiction, i.e., if this condition is not satisfied, we can always trade customers from a farther location (e.g., those between I 0 and I in Figure 4(b)) to nearer ones (e.g., those between J 0 and J in Figure 4(b)) so as to minimize delivery cost while keeping the total service region size unchanged. We now show that the lowest cost (for any fixed l, N, A) will be achieved when all N facilities are along a straight line. Otherwise, if there is an “elbow” facility along the TSP tour (e.g., facility i in Figure 4(c)), we can straighten this elbow without changing the total outbound cost within this facility’s service region (e.g., from Figure 4(c) to Figure 4(d)), since the optimal service region should be the intersection of the area between the boundaries and a circle. For example, for facility i, we can rotate line GI around xi until it is parallel to line M J. This rotation will force the customers in the pie-shaped area xi II 00 (stripe shaded in Figure 4(c)) to be relocated to xi GG00 (grid shaded in Figure 4(d)). Note these two areas must be exactly identical in shape and size, as they are part of the same circular region, and hence such customer relocation must be feasible. 9

Now, we are ready to prove that the optimal locations and service regions of the facilities should form a centroidal Voronoi tessellation, as shown in Figure 4(e); i.e., any facility should also be at the center of its service region. Otherwise, we can always perturb the facility location within the fixed service region to reduce the total outbound costs; e.g., if we move xi to the center of line BD, then the customers on the farther sides (e.g., those between G0 and G in Figure 4(e)) can be relocated to nearer ones (e.g., those between M 0 and M in Figure 4(e)) so as to minimize the delivery cost while keeping the total service region size unchanged. Next, we will show that the total outbound delivery cost for each facility i is convex over its service area size Ai and inbound tour length li . Or equivalently, we only need to show that any two adjacent service regions along the TSP tour must have the same service area size as well as the inbound tour length when the total outbound delivery cost is minimized. This can also be proven by perturbation. In Figure 4(f), for example, consider the nontrivial case where (Ai , li ) 6= (Ai+1 , li+1 ). We can always shift line M 0 D0 to pass the middle point of line segment BE while keeping the respective sizes of these two service regions unchanged. Since this perturbation essentially relocates some customers from farther locations (e.g., those between G0 and G) to nearer ones (e.g., the shadow region on the right-hand side of line M 0 D0 ), the total outbound delivery cost amongst these two service regions should decrease. This proves the convexity of the total outbound delivery cost for the facility i with its service area size Ai and inbound tour length li . Thus, by Jensen’s inequality, at optimality, service regions must be identical, and all facilities are evenly spaced along the straight TSP tour. This result is summarized in the following lemma.
1 Lemma 3. For a given l, N and A that satisfy l ≤ 2 NπA 2 , the lowest (per demand) system cost is achieved when (i) the TSP tour is a straight line along which all facilities lie evenly; (ii) the service regions of all facilities have the same size and shape; and (iii) each service region consists of two basic triangles and two pie shapes, as shown in Figure 4(g). Intuitively, the service region in Figure 4(g) is a special case of an ni -sided cyclic polygon with ni → ∞. It shall yield a cost lower bound because it obviously cannot form a spatial partition. To obtain such a lower bound, we express l as a function of N, A and α; i.e., l = 1 1 2 cos α (AN ) 2 (sin 2α + π − 2α)− 2 . Suppose the radius is R, then the lower bound can be achieved by solving the following minimization problem:   Z α Nf κf N 2 3 2 A 3 −3 3 min + R cos α cos tdt + (π − 2α) R + 2rR cos α , (11) A A 3 3 N −α subject to N R2 (sin 2α + π − 2α) = A. Note that for any given A, N , we can select the optimal α value (or the optimal shape of the service region). The first order condition of (11) with respect to α yields 2κf N sin α lim (n − 2) H(n, r, α) = 0, A (2 sin α cos α + π − 2α) n→∞ where H(n, r, α) is defined in (6). Since 0 < α < π2 , the above equation yields lim (n−2)H(n, r, α) = n→∞

0, which has one and only one solution by Lemma 1, α∗ (∞, r). Hence, when α = α∗ (∞, r), we have s 1 Nf κf A 2 κ2 f 3 (11) ≥ +√ ≥33 , 2 A 4g (∞, r) N g (∞, r) 10

where g(∞, r) := lim g(n, r) and g(n, r) is defined in (8). The last inequality becomes equality by n→∞
2 2 A κ −3 (g(∞, r)) 3 . choosing N = 2 The result is summarized in the following proposition. Proposition 2. Let α∗ (∞, r) be the root of lim (n − 2) H(n, r, α) = 0 and g(∞, r) := lim g(n, r). n→∞

n→∞

A cost lower bound to (4) under Euclidean metric is given by s κ2 f 3 33 . 4g 2 (∞, r)

3.3

(12)

Illustration: Impact of Service Region Shapes

The upper and lower bounds presented in the previous subsections are general. To illustrate this, we compare the minimal costs of three intuitive shapes of service regions that can form a spatial partition (Grunbaum and Shephard, 1977): triangles (n = 3), rectangles (n = 4), and hexagons (n = 6). For each shape, we fix ni = 3, 4, 6, ∀i, compute the optimal basic angles, and plug them into (10) (feasible cost upper bound) respectively. The comparisons of the optimal α and optimal costs of these three special cases are shown in Figure 5 against the cost lower bound (12) for ni = ∞. It can be seen that among these three intuitive shapes, the cyclic hexagon is the best. 1

5 Triangle Rectangle Hexagon

4

0.7

3.5

× 100

0.8

0.6 0.5 0.4 0.3

3

2.5 2 1.5

0.2

1

0.1

0.5

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Triangle Rectangle Hexagon

4.5

Z−Z LB Z

α−αLB α

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1

(a) Differences of optimal α between admissible shapes and the infinite polygon

0

0.2

0.4

r

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0.8

1

(b) Percentage cost differences between admissible shapes and the lower bound

Figure 5: Differences between admissible shapes and infinite polygon (an infeasible lower bound) It shall be noted that the differences of optimal angle α or optimal costs among these three shapes reduce as r increases, so when r is large enough, the spatial configuration of any one of these three shapes make no obvious difference. When r → 0, the relative gap between the n → ∞ lower bound and these feasible shapes grow. Fortunately, for the cyclic hexagon, the percentage cost gap remains quite small (0.3%), suggesting strongly that the cyclic hexagons are near-optimum shapes.

11

4

The Rectilinear Metric Case

4.1

Negligible Inbound Cost

Consider an arbitrary service region Ai on a rectilinear metric plane (see Figure 6). If we set the origin of the coordinate axes y1 , y2 at the facility location xi , the service region will be divided into four non-overlapping quadrant parts (i.e., Ai1 , Ai2 , Ai3 , Ai4 ), as shown in Figure 6(a). In each quadrant, there exists a line in the form of |y1 | + |y2 | =constant, such that the area of the resulting isosceles right-angled triangle equals that of the original quadrant part (see Figure 6(b)). Note that all points on such a line have an equal travel distance to the facility. We shall easily see that the outbound service cost for the four isosceles right triangles is lower than that for the original Ai , since our construct of these triangles can be done by simply re-locating some of the original customers to a nearer location (e.g., among the shaded areas in Figure 6(b)). Following similar arguments of Lemma 2, we can also see that the outbound cost is further minimized when all four isosceles right triangles have the same size; i.e., Ai1 = Ai2 = Ai3 = Ai4 = A4i . As such, the triangles form a square, as shown in Figure 6(c)), whose total outbound cost becomes √ 3 2 κf Ai2 . (13) 3

y2

xi

(a) Orignal service region

y2

y1

xi

(b) Reshape the service region

y2

y1

y1

xi

(c) Optimal shape

Figure 6: Illustration of relaxation of a service region with rectilinear metric NowP we consider a set of service regions N that form a partition, with a total area size equal to A = i∈N Ai . Since cost lower bound (13) holds for each region Ai , i ∈ N , the average facility and outbound cost per unit area-time satisfies the first equality below: r 3 √ √  1 N 2 3 2 X Ai2 Nf 2 A 2 Nf 3 κ f + κf ≥ + κf ≥3 . z(N, A) ≥ A 3 A A 3 N 18

(14)

i=1

The second inequality obviously holds due to function convexity, and it becomes equality when  √ − 2 3 2κ Ai = A/N, ∀i ∈ N ; the third inequality becomes equality by setting the value of A/N to . 6 The final lower bound in (14) is feasible and can be achieved when ni = 4, and Ai = 12

A N

=

 √ − 2 2κ 6

3

, ∀i ∈ N . This implies that identical square is the optimal shape for facility service regions under the rectilinear metric. Proposition 3. When inbound cost is negligible, the optimal shape of facility service region under the rectilinear metric is square with diagonals parallel to the coordinate axes.

4.2

Non-Negligible Inbound Cost

Similar to Section 3.2, we now construct a cost lower bound by considering all solutions that incur a fixed inbound travel length l, a fixed number of facilities N that collectively cover the customers in an area of total size A. See Figure 7 for an illustration. Note from Section 4.1 that the square shape minimizes the outbound cost for any given size of service regions. Thus, if the inbound travel A length l is larger than the total diagonal length of N identical squares (each with area size N ), then this case degrades to a trivial one where the optimal cost is achieved when all N service regions take the shape of identical squares, as shown in Figure 7(a). However, as discussed before, this case never yields a good cost lower bound since we can always shift the facility locations and their service regions along the TSP tour to reduce the length of inbound truck. 1 Now we consider the more general case when l ≤ (2N A) 2 . Overlaps among neighboring service regions are now inevitable, forcing facilities to serve farther customers (outside of the ideal squares of A ) and incur higher outbound cost. As such, minimizing the total overlapping area is equivalent size N to minimizing the total outbound cost for any given l, N, A. Note that for any two overlapping service areas, say i and i + 1 as shown in Figure 7(b), the overlap area will be minimized if we shift facility i+1 (and all facilities i+2, i+3, · · · ) along the 45 degree line so that segment xi xi+1 becomes parallel to one of the coordinate axes as shown in Figure 7(c). The inbound TSP distance will not change, but the outbound cost will decrease. We can repeat this for all neighboring facilities along the TSP tour, and a lower outbound cost for any given l, N, A will be achieved when all N facilities are along a straight line parallel to a coordinate axis. The rest of the argument is very similar to that in Section 3.2. When all facilities are along a straight TSP line, any two adjacent service regions should be separated by a boundary line that is perpendicular to the TSP tour (see Figure 7(d)); each facility will only serve the customers within the two nearest boundaries. To minimize the outbound cost of each facility within its boundaries, the optimal service region should be the intersection of the area between the two boundaries and a square shape; otherwise we can always perturb customer allocation to reduce outbound cost. Also note that the optimal locations and service regions of the facilities should form a centroidal Voronoi tessellation, as shown in Figure 7(d); i.e., any facility should also be at the center of its service region. Hence, all service regions must be identical, and all facilities are evenly spaced along the straight TSP tour. The above lower bound (mainly regarding outbound cost) holds for a fixed inbound cost (i.e., given values of l, N, A). The best lower bound can be obtained by choosing proper values of l, N, A to address the inbound and outbound cost trade-off. We consider the geometry in Figure 7(d) and express l as a function of N, A and α; i.e., 1

1

l = (2AN ) 2 (1 + 2 tan α)− 2 . Following the notation in Figure 7(d), the best lower bound can be achieved by finding the optimal

13

xi+1

x i+1

xi

xi

xi+1

xi

(a)

(b)

(c)

45°

R α xi

xi+1

(d)

Figure 7: Construction of a lower bound under rectilinear metric R and α that solves the following problem:  
Nf κf N 2 3 A 2 min + R cos α 2 + sin α + 6 sin α cos α + 2rR cos α , (15) A A 3 N
subject to 2N R2 cos2 α + sin 2α = A. It is easy to show from the p first order condition of (15) with respect to α, that there is a single optimizer α∗ = arctan(2r + (2r + 4r2 )) ∈ [0, π2 ). We write it as a function of r, and define a new function 1 3 3 (2 cos α∗ (r)) 2 (2 sin α∗ (r) + cos α∗ (r)) 2 . (16) g¯(r) = 3 sin (2α∗ (r)) − 2 cos (2α∗ (r)) + 4 Then we have

s 1 Nf κf A 2 κ2 f 3 (15) ≥ +√ ≥33 . A 4¯ g 2 (r) N g¯ (r)
− 2 2 A The last inequality becomes equality by choosing N = κ2 3 (¯ g (r)) 3 . We further note that the elongated hexagons in Figure 7(d), which achieve the cost lower bound, can also form a feasible spatial tessellation. Hence, it yields an optimal tessellation. This finding is summarized in the following proposition. Proposition 4. Under the rectilinearpmetric, the optimal shape of facility service region is elongated ∗ (r) = arctan(2r + hexagon. With αq (2r + 4r2 )) and g¯(r) as defined in (16), the optimal value of (4) is given by 3 3

κ2 f 3 . 4¯ g 2 (r)

14

5 5.1

Discussion Numerical Verification

To further verify our cost bounds, we solve a discrete mathematical program using a grid of M × M points denoted by set G (|G| = M 2 ). Each point i ∈ G denotes a customer as well as a candidate facility location. An arbitrary point o ∈ G represents the start point of the TSP tour. Since the discrete problem is very difficult, we use a simple simulated annealing algorithm (Golden and Skiscim, 1986; Wu et al., 2002) to solve it.2 Figure 8(a) shows the computation result for an instance under Euclidean metric, with M = 50, fi = 299.66, cij = 1, Cij = 12, ∀i, j ∈ G. We notice that most of facility service region shapes (especially those away from the boundaries) turn out to be cyclic hexagonal, exactly as what we would expect from Proposition 1. Figure 8(b) shows the result under the rectilinear distance metric, with the same parameters except for fi = 199.31, ∀i ∈ G and rectilinear distances. The facility service regions turn out to have noncyclic hexagonal shapes, again, as expected. For both cases, the finite boundaries of the 50 × 50 area do seem to influence some of the service region shapes, but such effect shall diminish when M → ∞. 49

44

44

39

39

34

34

29

29 y2

49

24

24

19

19

14

14

9

9

4

4

−1 −1

−1 −1

4

9

14

19

24

29

34

39

44

49

(a) Euclidean metric

4

9

14

19

24 y1

29

34

39

44

49

(b) Rectilinear metric

Figure 8: Results of numerical experiments on a discrete grid We also measure from Figure 8(a) the average basic angles for those cyclic hexagons in the center of the region, which turn out to be 18.8◦ and 52.3◦ . From (6) and (10), the theoretical number of facilities N ∗ ≈ 34, and the optimal basic angle values are 18.4◦ and 53.2◦ . We can see that the error between the theoretical result and the experimental measurement is only 0.9◦ . Similarly, the average basic angles for the non-cyclic hexagons in Figure 8(b) are around 45.0◦ , while the theoretical value is 44.5◦ , yielding a 0.5◦ difference. We anticipate that these minor errors will further diminish if we increase the size of the grid (so as to eliminate the influence of the boundaries), and discretize it into a finer resolution. 2 Simulated annealing is one of many approaches that can be used to solve the discrete problem above. Since the purpose of the numerical experiment is to illustrate the validity of the theoretical results, we choose to use simulated annealing for its simplicity

15

5.2

Impacts of Inventory Cost

Inventory cost is sometimes significant to a transshipment system. Attempts have been made to incorporate inventory considerations into discrete facility location models (Daskin et al., 2002; Chen et al., 2011) and location-routing models (Shen and Qi, 2007). In this section, we will consider cost for inbound inventory holding in transshipment facilities and discuss its impacts on the optimal system design (4). Since the demand rate for each facility is deterministic, constant over time, and proportional to the size of the service region, we can adopt a simple cycle inventory policy to determine the optimal inbound replenishment frequency, and assume that the fixed order cost (per order) and inventory holding cost (per item-time) are both constants. To simplify the formulas, 1 1 however, we express these cost coefficients in the form of bf κ 3 /λ and hf κ 3 , respectively, for some proper constants b and h. As such, b and h essentially represent the relative magnitudes of the fixed order cost and inventory holding cost as compared to the other costs (e.g., facility set-up cost, transportation cost). EOQ (economic order quantity) trade-off can be directly applied to determine 1 1 replenishment frequency, and the total inventory cost of the ith facility is f (2bhAi ) 2 κ 3 . Thus, (4) is now generalized as follows:  Z N N  1 1 f X κr X 2 z= min lim N kx − xi kdx + (2bhAi ) κ 3 + li . 1+κ N N ,{xi },{Ai } N →∞ P Ai i=1 i=1 Ai

(17)

i=1 1

1

We shall first note that the inventory cost term f (2bhAi ) 2 κ 3 is concave with respect to Ai . The optimal spatial tessellation patterns discussed in previous sections may no longer hold if the value of bh is large; e.g., in the extreme, if inventory cost dominates outbound and inbound transportation cost, it is beneficial to aggregate demand, and hence the optimal design should degenerate to a single facility that serves all customers on the plane. Hence, in general, inventory cost could have a significant impact on the optimal spatial facility layout. However, we shall also note from the proofs of all lemmas and theorems in Sections 3 and 4 that the optimality of the hexagonal spatial tessellations and facility layouts (under both metrics) will still hold as long as the objective function (17) remains convex with respect to Ai , ∀i. Such a condition could be achieved in many ways. First, it suffices if the parameter bh becomes facility specific, i.e., (bh)i , and it is dependent on Ai and satisfies the following second-order condition d2 (bh)i 2(bh)i A2i dA2i



d (bh)i − (bh)i − Ai dAi

2 ≥ 0.

For example, the fix order cost coefficient could be a convex function of the total served demand (Veinott, 1964), while the holding cost coefficient remain constant; e.g., for the ith facility, 1+β √ 1 (bh)i = Aβi , for some β ≥ 1. The inventory cost term in (17) becomes 2f κ 3 Ai 2 , which is now convex in Ai . Another obvious condition for (17) to remain convex is that the inventory cost term be dominated by other costs. In this case, the best tessellation for the Euclidean metric and the rectilinear metric remain elongated hexagons (as shown in previous sections), but the optimal value of A/N

16

may change. Simple algebra shows that closed-form formulas can still be found as follows:  √ i  h 1 1 −3 4 2 − 32  2 cos2 1 arccos (2bh/9) 4 g − 2 (n, r) g (n, r) (bh) , n = 6, ∞, Euclidean metric, κ A 3 3 h  i √ = 3 1 1 3 4 2 −2  N g¯ (r) (bh) 2 cos2 13 arccos (2bh/9)− 4 g¯− 2 (r) , L1 metric. 3 κ (18) Figure 9 illustrates the percentage differences in objective function (4) between the optimal solutions with and without considering inventory cost. It turns out that inventory cost has a slightly larger effect under the rectilinear metric than under Euclidean metric. We also note that for both metrics, when r grows, the inventory cost becomes less significant, so the cost difference becomes smaller. Conversely, when the value of bh grows larger, inventory cost plays a more important role and leads to a bigger cost difference.

% Difference

20

15

10

5

0 6 5

4

4 3

2

bh

2 0

1 0

r

(a) Euclidean metric

(b) Rectilinear metric

Figure 9: Percentage difference in system cost with and without inventory cost consideration.

6

Conclusion

This paper studies the optimal transshipment facility layout on a homogeneous plane R2 that minimizes total system cost for facility set-up, outbound customer delivery and inbound replenishment transportation. We first construct an upper bound by tessellating the plane with elongated cyclic hexagons under the Euclidean metric. Then we derive a cost lower bound which is achieved by an infeasible infinite cyclic polygon. We derive analytical formulas for both bounds, and show that the percentage gap between these two bounds is quite small (i.e., within 0.3%). We further illustrate the impact of service region shapes by comparing the performance of three special shapes (i.e., triangle, rectangle, hexagon), and show that elongated cyclic hexagon outperforms the others. To verify our analytical results, we observe the near-optimal spatial tessellation via a numerical experiment on a grid of points. The numerical results turn out to be very close to our analytical predictions. Finally, we extend our discussion to the rectilinear metric case and show that a similar non-cyclic hexagon shape becomes exactly optimal when they are properly oriented along the axes of the rectilinear coordinate system. 17

Future research can be conducted in several directions. For example, we may consider a finite capacity of the inbound truck and investigate its impacts on the optimal spatial facility layout, along the line of inquiry in Carlsson and Jia (2013). Moreover, we strongly suspect that the elongated cyclic hexagonal shape is actually optimal under the Euclidean metric, but we have only presented it as a feasible solution (which yields an upper bound). It will be ideal to further prove the optimal tessellation pattern for the Euclidean metric and other variations (e.g., other metrics). Finally, we have shown that when inventory cost becomes dominant, the spatial tessellation patterns presented in this paper may no longer be optimal. It will be interesting to find the optimal tessellation pattern for more general settings.

Acknowledgments This research was supported in part by the U.S. National Science Foundation through Awards EFRI - RESIN - 0835982, CMMI - 1234085 and CMMI - 0748067. The first author conducted this work while a graduate student at the University of Illinois. Professors Carlos Daganzo, Max Zuo-Jun Shen (UC Berkeley) and James Campbell (U of Missouri at St. Louis) provided helpful comments on an earlier version of the paper. The valuable comments from two anonymous reviewers are also gratefully acknowledged.

References Beardwood, J., Halton, J. H., Hammersley, J. M., 1959. The shortest path through many points. In: Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 55. Cambridge Univ Press, pp. 299–327. Cachon, G. P., 2014. Retail store density and the cost of greenhouse gas emissions. Management Science 60 (8), 1907–1925. Carlsson, J. G., Jia, F., 2013. Euclidean hub-and-spoke networks. Operations Research 61 (6), 1360–1382. Carlsson, J. G., Jia, F., 2014. Continuous facility location with backbone network costs. Transportation Science. In press. Chen, Q., Li, X., Ouyang, Y., 2011. Joint inventory-location problem under the risk of probabilistic facility disruptions. Transportation Research Part B 45 (7), 991–1003. Cui, T., Ouyang, Y., Shen, Z.-J. M., 2010. Reliable facility location under the risk of disruptions. Operations Research 58 (4), 998–1011. Daganzo, C., 1992. Logistics Systems Analysis. Springer, Heidelberg. Daganzo, C. F., 1984. The length of tours in zones of different shapes. Transportation Research Part B 18 (2), 135–145. Daskin, M., 1995. Network and Discrete Location: Models, Algorithms, and Applications. John Wiley, New York.

18

Daskin, M. S., Coullard, C. R., Shen, Z.-J. M., 2002. An inventory-location model: Formulation, solution algorithm and computational results. Annals of Operations Research 110 (1-4), 83–106. Du, Q., Faber, V., Gunzburger, M., 1999. Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review 41 (4), 637–676. Edwin, S., Michael, R., 1964. A model of market areas with free entry. The Journal of Political Economy 72 (3), 278–288. Geoffrion, A. M., 1979. Making better use of optimization capability in distribution system planning. AIIE Transactions 11 (2), 96–108. Gersho, A., 1979. Asymptotically optimal block quantization. IEEE Transactions on Information Theory 25 (4), 373–380. Golden, B. L., Skiscim, C. C., 1986. Using simulated annealing to solve routing and location problems. Naval Research Logistics Quarterly 33 (2), 261–279. Grunbaum, B., Shephard, G. C., 1977. Tilings by regular polygons. Mathematics Magazine, 227– 247. Haimovich, M., Magnanti, T. L., 1988. Extremum properties of hexagonal partitioning and the uniform distribution in Euclidean location. SIAM Journal on Discrete Mathematics 1 (1), 50–64. Johnson, D., 1996. Asymptotic experimental analysis for the Held-Karp traveling salesman bound DS Johnson* LA McGeochf EE Rothberg. In: Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete algorithms. Vol. 81. SIAM, p. 341. Li, X., Ouyang, Y., 2010. A continuum approximation approach to reliable facility location design under correlated probabilistic disruptions. Transportation Research Part B 44 (4), 535–548. Newell, G., 1973. Scheduling, location, transportation and continuum mechanics; some simple approximations to optimization problems. SIAM Journal of Applied Mathematics 25, 346–360. Newman, D., 1982. The hexagon theorem. IEEE Transactions on Information Theory 28 (2), 137– 139. Nourbakhsh, S. M., Ouyang, Y., 2012. A structured flexible transit system for low demand areas. Transportation Research Part B 46 (1), 204–216. Okabe, A., Boots, B., Sugihara, K., 1992. Spatial Tesselations: Concepts and Applications of Voronoi Diagrams. Wiley, Chichester, UK. Ouyang, Y., Daganzo, C., 2006. Discretization and validation of the continuum approximation scheme for terminal system design. Transportation Science 40 (1), 89–98. Perl, J., Daskin, M. S., 1985. A warehouse location-routing problem. Transportation Research Part B 19 (5), 381–396. Qi, L., Shen, Z.-J. M., 2010. Worst-case analysis of demand point aggregation for the Euclidean p-median problem. European Journal of Operational Research 202 (2), 434–443. 19

Shen, Z.-J. M., Qi, L., 2007. Incorporating inventory and routing costs in strategic location models. European Journal of Operational Research 179 (2), 372–389. Suzuki, A., Okabe, A., 1995. Using Voronoi diagrams. In Drezner Z.(edits) Facilty Location: A Survey of Applications and Methods. Springer, New York. Toth, P., Vigo, D., 2001. The vehicle routing problem. Society for Industrial and Applied Mathematics, Philadelphia. Veinott, A. F., 1964. Production planning with convex costs: A parametric study. Management Science 10 (3), 441–460. Wu, T.-H., Low, C., Bai, J.-W., 2002. Heuristic solutions to multi-depot location-routing problems. Computers & Operations Research 29 (10), 1393–1415.

20

Appendix A Appendix A.1

Proofs of the Lemmas and Propositions Proof of Lemma 1

H(ni ,r,α) sin α sin π−2α n −2 i

h



¯ i , r, α) = , we define a new function H(n h  ¯ i , r, α) has the same root as H(ni , r, α) with respect to α ∈ π , π . such that H(n ni 2

Proof. Since sin α > 0, sin π−2α ni −2 > 0, ∀α ∈

π π ni , 2

¯ i , r, α) as follows: From (6), we can express H(n   π π π − 2α ¯ − p (α) + q (α) , ≤ α < , r ≥ 0, ni ≥ 3, H(ni , r, α) = p ni − 2 ni 2

(19)

where π x cos2 x π log(tan + ), 0 < x < , sin x 4 2 2 π−2x π−2x r(ni − 2) cos ni −2 sin ni −2 π π , ≤ x < , r ≥ 0, ni ≥ 3. q(x) = −2r cos x − sin x ni 2

p(x) =

(20) (21)

π It is easy to show that dp(x) dx < 0 when 0 < x < 2 . Thus p(x) decreases strictly monotonically in the open interval (0, π2 ), and the first two terms in (19) increases monotonically with α. By the π π same token, we find that dq(x) dx > 0 when ni ≤ x < 2 . Hence, q(x) increases monotonically over x π π when ni ≤ x < 2 . h  ¯ i , r, α) increases monotonically over α ∈ π , π . Meanwhile, as H(ni , r, π ) = As such, H(n ni 2 ni
¯ ni , r, π = 1 , the implicit equation H(n ¯ i , r, α) = 0 has one and only one −mn cos nπi < 0 and H 2 h  2 π π root in the interval ni , 2 for all ni ≥ 3 and r ≥ 0.

Appendix A.2

Proof of Lemma 2

Proof. We take the first order derivative of (5) with respect to αi , and the first order condition yields αi ) d¯ αi dRi 2Ri3 dF (αi ) dRi dRi (ni − 2)Ri3 dF (¯ + + 2Ri2 F (αi ) + + 2κrf Ai cos αi dαi 3 d¯ αi dαi dαi 3 dαi dαi − 2κrf Ai Ri sin αi = 0, (22) Rx where F (x) = κf cos3 x −x cos13 t dt. Since (ni − 2)¯ αi + 2αi = π always holds, then we have (ni − 2)Ri2 F (¯ αi )

d¯ αi 2 =− , dαi ni − 2

(23)

and (22) can be simplified into the following F (αi ) −

F (¯ αi ) sin αi cos αi + κrf cos αi [(ni − 2) sin α ¯ i cos α ¯ i + 2 sin αi cos αi ] = 0. sin α ¯ i cos α ¯i

21

(24)



Rx x As F (x) = κf cos3 x −x cos13 t dt = κf cos3 x log tan x2 + π4 + tan cos x , and let α = αi , the left side of (24) can be rewritten as the following function   π α π − 2α π π − 2α 2 π − 2α H(ni , r, α) = sin α cos − cos2 α sin log tan + log tan + ni − 2 4 2(ni − 2) ni − 2 4 2 π − 2α π − 2α π − 2α − r sin 2α sin − r(ni − 2) cos sin2 . (25) ni − 2 ni − 2 ni − 2 Simple algebra will show that the lower bound of (5) is    3  ∗ π−2α∗ π π−2α∗ ∗ κf Ai2 1 + cos π−2α + 4r cos α cot log tan + ni −2 ni −2 4 2(ni −2) ,  1 π−2α∗ π−2α∗ 2 ∗ ∗ 3 2 cos α sin α + (ni − 2) cos ni −2 sin ni −2

(26)

where α∗ is the root of H(ni , r, α) = 0. In light of Lemma 1, this completes the proof.

Appendix A.3

Proof of Proposition 1

Proof. We minimize the right hand side of (9) as an unconstrained optimization problem by adding N P Ai = A into the objective with Lagrangian multiplier η; i.e., i=1

! √ N N X N f X κf Ai Ai + +η A− Ai . A 3Ag(ni , r) i=1

(27)

i=1

By the first order conditions, we have 3κf

η= s 2 A

N P

and Ai =

g 2 (ni , r)

i=1

i=1

Substitute Ai =

g 2 (ni ,r)A , ∀i N P g 2 (ni ,r)

g 2 (ni , r)A , ∀i. N P 2 g (ni , r)

into (9), we have

i=1

N

1

1

Nf κf A 2 N f X κf A 2 g 2 (ni , r) + = + zub (N, A) ≥ 3 N 2 N  12 , ∀{ni }, N, r. A A P 2 P 2 i=1 g (ni , r) g (ni , r) i=1

(28)

i=1

Numerical examination of the first order and second order derivatives of smooth function g 2 (n, r) 2 2 2 (n,r) < 0. Thus, g 2 (n, r) is concave and monotonically increasing over show that ∂g ∂n > 0, ∂ g∂n(n,r) 2 n ∈ [3, +∞). Thus, ! N N X X g 2 (ni , r) ≤ N g 2 ni /N, r , , ∀{ni }, N, r. (29) i=1

Note that Newman (1982) proved that

i=1

PN

i=1 ni

22

≤ 6N for any usual tessellation of the plane.

Thus, we further have N X i=1

3

1

1

Ai2 A2 A2 ≥√ P ≥√ , ∀{ni }, N, r. N Ag(ni , r) N g (6, r) n /N, r Ng i i=1

23

(30)