Routers and

On Optimal Location of Switches/Routers and Interconnection Z. Liu, Y. Gu, D. Medhi* Computer Science Telecommunications University of Missouri–Kansas City 5100 Rockhill Road Kansas City, MO 64110 USA Voice: +1 816 235-2006 E-mail: [email protected] Fax: +1 816 235-5159

November 1998 Abstract We address the computer communication network design problem of optimal location of switches/ routers as well as interconnection (OLSRI) of the core network. This problem is applicable, for example, in the network topological architecture design arising in ATM PNNI environment or IP/ATM networking environment. While the pure location design problem has received much attention in the literature, the combined location and interconnection problem, OLSRI, has received little attention in the literature. In this paper, we present two novel optimization formulations for two instances of the OLSRI design problem. We also present computational results for a number of examples, and present two extensions.

Keywords: computer communication network design, network topological architecture, combined location and interconnection problem, optimization formulation

16 pages (including figures)

* Author for Correspondence

1. Introduction We consider a wide-area networking environment where access networks or nodes are to be connected to a core network or where a two-level hierarchical architecture is applicable. We are particularly interested in the optimal location of the core network routers or switches which minimizes the access connectivity cost (the cost of the access network to the core network connectivity) and the switch/router placement cost as well as also addresses the cost of interconnecting the core network. We call this combined problem optimal location of switches/routers and interconnection (OLSRI) problem. An example of the two-level hierarchical architecture where this situation arizes is the two adjacent levels in ATM PNNI environment [2] which use the concept of peer groups. The nodes within a peer group is conceptually similar to an access network that needs to talk to other peer groups through the peer group leader in its group; thus, the peer group leader acts as the “core" node. In this case, the problem is how to organize a set of nodes into different peer groups as well as select the peer group leader (the “core" node) for each group while taking into account the interconnection cost. While ATM PNNI specification describes the functional architecture for the peer group concept, the OLSRI based design addresses the network topological architecture design problem faced by network providers. Here, we address the design problem for the adjacent two-level case. Another application of the OLSRI problem is in the organization of an integrated IP/ATM network. A group of ATM switches may be clustered to connect to another cluster of ATM switches through an IP-router based network where a set of possible sites for routers are available; in this case, the problem is which ATM switches should be homed to a specific router so that access/location as well interconnection cost between the routers are minimized. In this case, the ATM switches in a cluster can be thought of as in an access network environment. These scenarios appear to be similar to the classical computer communication network design problem called the concentrator location (CL) problem that has been around for over two decades [3, 5]. The basic idea behind the CL problem is that given a number of sources which can be connected to a set of possible concentrator sites, the goal is to find the optimal layout so that the sources to concentrators connectivity cost as well as the cost of locating concentrators is minimized given the requirement that a source needs to be connected to only a concentrator and a concentrator can have limits on how many sources it can handle. The basic idea is also applicable to the network design problem where access networks (similar to "sources") need to be connected to (yet to be located) core network nodes or gateways (similar to "concentrators"). This problem is essentially the same problem as the warehouse location problem often faced in production and operations management field [1]. On the other hand, OLSRI problem addresses the both location as well as interconnection problem together. Due to changing cost structure and domain of operations of networks (and network providers), OLSRI problem is more application than just the CL problem for networking scenarios of some network providers. Surprisingly, the combined location and interconnection network design problem has escaped the existing literature to our knowledge. In this paper, we consider 2

two instances of this problem and present novel optimization formulations. The first model is applicable to the ATM PNNI example we have described above while the second model is applicable to the IP/ATM scenario. The rest of the paper is organized as follows: in the next section, the optimization formulation of OLSRI applicable to ATM PNNI ("Model-A") is presented while in section 3 the formlation applicable to IP/ATM environment ("Model-B") is presented. In section 4, we present computional results for some example networks, and also discuss the advantage of OLSRI over the CL problem. We then consider two extensions in section 5. In the rest of the discussion, we will use the abstraction ‘site’ (or regular site) to denote the access network, or, nodes that needs to be connected to a core network. The core network can have routers and/or switches; to help visualize, in the first model we use the term ‘switch’ to relate to ATM PNNI environment while in the second model we will use the term ‘router’ to relate to the IP/ATM environment.

2. OLSRI: Model-A In the first case, we consider the scenario that every site can possibly be also a switching location. Thus, a site can be a regular site or a switching site. (In the figures, we will use a circle to denote a site and a diamond to denote a selected switching site.) Fig. 1 and Fig. 2 show two examples where sites are shown. If a switch is located at a site, it is assumed to have a finite ’source’ termination capacity (e.g. interfaces or ports) and can connect to other switches. We specifically consider the case where the selected switch sites are to be fully-interconnected (fully-mesh). We start with the notations for the given items.

N P Kj

Total number of sites Number of switch locations to be chosen (P

< N)

Maximum number of regular sites that can be connected to a switch located at j (if j is selected to be a switch site)

cij

Cost of connecting site i to site j (either regular or switch); the cost is assumed to be symmetrical, i.e.

bj

Cost of switch with maximum access link termination Kj if located at j

cij = cji .

We now consider the problem where a regular site is required to be connected to a site if the the site is chosen to be switch site. In this regard, we introduce the following 0/1 variables.

xij yj

1, if site i is connected to site j given that a switch is located at site j ; 0, otherwise (variable) 1 if site j is chosen for locating a switch; 0, otherwise Since cij

cji , thus, xij is indifferent from xji . Hence, the problem is restricted to the i and j ’s which are diagonal or upper diagonal, i.e., i  j with i; j = 1; :::; N: =

3

If site i is a regular site, then it is required that it be connected to one switch site; this can be related as

N X x

follows:

This is equivalent to

; for all regular site i:

ij

j =1

=1

N X x

ij

 1; for regular site i

(1)

Xx

ij

 1; for regular site i

(2)

j =1 N

j =1

On the other hand, if site j is chosen to be a switch site, then it needs to support at least one regular site (which could be itself), and, for the fully mesh of the backbone/core network, it is required to be connected to the other (P

, 1) switch sites; thus, we have the following relation N X x i=1

Due to symmetry (xij

=

ij

 P; for switch site j

(3)

xji ), we can combine this relation (3) with (1) by introducing the switch selection

variable yj to obtain the following:

N X x

ij

i=1

 Pyj + (1 , yj ); for all site j

Similarly, a switch site j can be connected to P

(4)

, 1 other switches (due to being fully-interconnected)

and can handle a maximum of Kj regular (access) sites; thus, we have

N X x i=1

ij

 Kj + P , 1 for all switch site j:

(5)

This time, combining this relation (5) with (2) and using symmetry, we get

N X x i=1

ij

 (Kj + P , 1)yj + (1 , yj ); for all site j:

(6)

We now move to determine the number of links that should exist. Each regular site should be connected to one switch for the access network part, and in the core network, all the

P

switches are to be fully

interconnected. Again, due to symmetry, this requirement can be reflected in the following by considering the diagonal and upper diagonal x’s

N X N X x i=1 j =i

ij

=

N + P (P , 1)=2: 4

(7)

Finally, we address the total number of that sites should be switching sites, and the fact that the regular site at i is connected to the switch site itself. Thus, we have

N X y

j

j =1

=

P

(8)

N X x

ii = P:

i=1

(9)

The goal is to minimize the total cost due to access as well as core network interconnection links and due to the selection of the switching sites. The objective function, then, is

f (x; y) =

N X N X c i=1 j =i

ij xij +

N X by j =1

j j

(10)

In this representation, cij can corresponds to either access links or interconnection links depending on

the selected xij along with the sites selected (yj ’s). The optimization problem is to minimize (10) under the requirements given in (4), (6), (7), (8), (9), and that the variables take only the values 0 or 1. Rearranging (4) and (6), we now summarize the formulation:

N X N X min c

x;y i=1 j =i ij xij + j =1 j j

N X x i=1

ij , (P , 1)yj

N X x i=1

N X by

ij , (Kj + P N X N X x i=1 j =i

ij

 1; j = 1; :::; N

, 2)yj  1; j = 1; :::; N =

N + P (P , 1)=2

N X y j =1 N

(4)

(6)

(7)

P

(8)

ii = P

(9)

j

=

Xx i=1

(10)

xij 2 f0; 1g; j = i; :::; N ; i = 1; :::; N

(11)

yj 2 f0; 1g; for j = 1; :::; N:

(12)

5

Thus, we have formulated the first case of the OLSRI problem as a linear integer programming problem. This formulation will be referred to as formulation-A. It may be noted that since the cost between two

sites is given using cij which need not be based on the Euclidean distance, the model is quite general. Typically, we assume that cii > 0 and is different for each i (i.e., cii 6= ckk if i 6= k). It may also be noted

P

P , the minimum number of switches to be chosen, should be in the order of N 2 = Nj=1 Kj for the optimization problem to be feasible. If Kj = K for all j , then for the feasibility of the problem, we must have P  dN=K e.

that

We briefly introduce two special instances of the OLSRI-A problem: Variation: A1 If in addition to the requirements described above, we are also given that Kj

=

K and bj = b for all

sites, then the following simplification takes place for the second part of the objective function (10) when

N X by

coupled with (8):

j =1

j b =

N X y j =1

j

=

bP:

Since this is a constant, the second part of the objective function need not be explicitly listed in the objective function, i.e.,

f (x; y) =

N X N X c i=1 j =i

ij xij :

Variation: A2 If a switch site is required to be connected to a regular site besides itself, as well as all the other core network, then (3) will change to

N X x

 P + 1; for switch site j

(13)

 (P + 1)yj + (1 , yj ); for all site j

(14)

i=1

ij

Consequently, (4) will change to

N X x i=1

ij

Note that with this condition, the number of switches P can not be more than N=2 to maintain feasibility of the problem.

6

3. OLSRI: Model-B This model is applicable to scenarios such as the IP/ATM case we have discussed in the introduction. In this case, the list for regular sites and the list for possible router sites are given to be separate. As before, we are operating under the requirement that the backbone network be fully interconnected. In the figures, we will use a circle to denote a regular site and a plain triangle to denote a possible router site, and a filled triangle to denote a selected router site. Fig. 3 shows an example of the initial regular sites and possible router location sites. We again start with the notations for the given information:

N M P Kj

Total number of sites Total number of possible router locations Total number of router locations to be chosen (P

< M)

Maximum number of regular sites that can be connected to a router located at

j = 1; :::; M (if j is

selected to be a router site)

gij ej

Cost of connecting regular site i to possible router site j Cost of router with maximum access link termination Kj if located at j with interfaces to other selected router sites

hjk

Cost of connecting possible router site j to possible router site k: this cost is assumed to be symmetrical,

i.e., hjk

=

hkj

In this case, we introduce three sets of 0/1 variables: one for the decision of connected regular site to a possible router site, the second set for the decision to select a router site, and the third set for the decision of interconnection in the core networks. Specifically, we have

wij

1, if regular site i is connected to possible router site

j if a router is located at site j ; 0, otherwise

(variable)

uj zjk

1 if possible router site j is chosen for locating a switch; 0, otherwise 1, if possible router site j is connected to possible router site k (if routers are located at both sites); 0, otherwise (variable) Here, the decision variables w are for i = 1; :::; N; j =

1

; :::; M , and the variables u for j = 1; :::; M:

For the interconnecting variables z , due to symmetry, we need to consider strictly upper diagonals ones, i.e,

j < k, ((k = j + 1; :::; M ); j = 1; :::; M , 1). For notational purpose, any zj1 ;j2 , where j1 appears to be greater than j2 , actually refers to zj2 ;j1 . We first start with the requirement that every site i needs to be connected to exactly a router site: M X w j =1

ij = 1; for all i 7

(15)

A router site j , if chosen, can accommodate only up to

Kj regular sites whereas a site not selected to be

a router site should not connect to any regular site. This can be given as follows incorporating the site selection variables uj ’s:

N X w

ij  Kj uj ; for all j

i=1

(16)

In the case of interconnecting part, a router site j , if chosen, requires to have exactly

P , 1) links to

(

other chosen router sites. This is only true for the selected router sites which can be reflected through the following equation for each j :

M X k=1;k6=j

zjk = (P , 1)uj ; for all j

Since the core network is fully-interconnected, thus only P (P

X X

active. Thus, we have

M ,1 M

j =1 k=j +1

Finally, only P router sites should be selected:

(17)

, 1)=2 interconnection variables should be

zjk = P (P , 1)=2:

M X u j =1

j

=

(18)

P:

(19)

There are three cost components for this problem: the cost of connecting sources to router sites, the cost of routers at each site (if selected), and the cost of interconnecting links in the core network. Thus, the objective function is written as follows:

N X M X g

ij wij +

i=1 j =1

M MX ,1 X M X eu j =1

j j+

j =1 k=j +1

hjk zjk :

(20)

Thus, the OLSRI model-B problem is to minimize the total cost given the constraints described above as well as the requirement that the variables only take the values 0 or 1. The complete formulation is summarized below:

min w;u;z

M N X X g i=1 j =1

ij wij +

M X w j =1

N X w

i=1 M

X

k=1;k6=j

M MX ,1 X M X eu j j+

j =1

ij = 1;

j =1 k=j +1

hjk zjk :

i = 1; :::; N

ij  Kj uj ;

j = 1; :::; M

zjk = (P , 1)uj ; j = 1; :::; M 8

(20)

(15)

(16)

(17)

X X

M ,1 M

zjk = P (P , 1)=2

j =1 k=j +1 M

Xu k=1

j

=

(18)

P

(19)

wij 2 f0; 1g; for i = 1; :::; N ; j = 1; :::; M

(21)

uj 2 f0; 1g; for j = 1; :::; N

(22)

zjk 2 f0; 1g; for k = j + 1; :::; M ; j = 1; :::; M , 1

(23)

This formulation is also a linear integer programming formulation and will be referred to as formulation-B. We like to point out that the constraints (15) and (16) along with the 0/1 requirements as in (21) and (22) and with the objective function consisting of the first two terms of (20) is the formulation for the classical concentrator-location problem discussed in [3, 5] which we will refer to also as the pure optimal location problem.

4. Design Results In this section, we present results on optimal OLSRI design for both problem formulations. First, we discuss for Model-A. To conduct this study, we randomly generated points on an x-y grid to generate sites. For example, Fig. 1 and Fig. 2 show the source locations for N For the cost cij we use the following relation cij

= 15

and N

= 25

networks, respectively.

 + d(i; j ) where d(i; j ) is the distance between site i and site j . We assume Kj = 5, for each j . The cost bj is randomly generated. Since Kj = 5, for the feasibility of the problem we need P  3 when N = 15 (or P  5 when N = 25). The optimization =

formulation is solved using the Mixed Integer Programming (MIP) solver of the tool CPLEX [4]. Note that

the cost cij = d(i; j ) is based only on distance. We will refer to such examples as distance-based examples. The first set of figures are for distance-based case when N = 15; when we set

 = 0, and

= 1,

specifically, refer to Fig. 4a, Fig. 4b and Fig. 4c for

P

; ;

= 3 4 5

respectively. Recall that in the figures,

selected switch sites are diamond shaped. From these figures, we see that the same set of three core nodes selected when P

=3

are also selected when P

= 4 and

P

= 5.

This may give the impression that the same

core nodes are always selected when P is incremented. This is only an artifact, and in general, the same set

of core nodes may not be included in the optimal design whenever P is incremented. To see this, consider

N = 25. For the distance based case, the optimal designs are shown in Fig. 6a, Fig. 6b and Fig. 6c for P = 5; 6; 7, respectively where we see that the same core nodes are not always selected when P

the case of

is incremented. For the above examples (for distance-based case), the optimal design layout appears to be somewhat intuitive (at least in some of them) as far as the selection of the core nodes are concerned. The optimal design layout discussed so far may also give the impression that by looking at the locations of the sites, it is 9

reasonably predictable where the likely sites of the core nodes are going to be located. This is not obvious

at all when the cij cost is not based purely on Euclidean distance. We consider the same 15-site and 25-site

problems but this time with 

= 10

and

The optimal designs are shown for N and for

N

= 25

with P

; ;

= 5 6 7

:

= 01

= 15

– these will be referred to as the skewed-distance cases.

with P

; ;

= 3 4 5

in Fig. 5a, Fig. 5b and Fig. 5c, respectively,

in Fig. 7a, Fig. 7b and Fig. 7c, respectively. For these skewed-distance

examples, we can see that the optimal OLSRI design is often non-intuitive. This also shows the advantage of the model formulations since the cost between two sites are generic and are not required to be based on the Euclidean distance. We next illustrate results for Model-B. In all cases, we now set Kj

= 6.

Recall that in this case, a set

of possible router sites are given — see the example in Fig. 3. We consider examples with both distance based and skewed-distance based for gij ’s and hjk ’s. We show optimal design for N = 30 nodes with M = 10 nodes when P is set to 5, 6 and 7 in Fig. 8a, Fig. 8b and Fig. 8c, respectively when the cost (gij and

hjk ) is based only on distance; the corresponding figures for the skewed-distance based cases are shown in

Fig. 9a, Fig. 9b and Fig. 9c. As with formulation-A, we observe that the same set of selected router sites may not always be selected when

P

is incremented. Also observe the difference in the optimal network

design between the distance-based and skewed-distance based cases. An obvious question is what advantage does OLSRI formulation provide over the classical concentratorlocation formulation discussed in [5, 3]. To compare, we show the the pure optimal location design for

N

= 30,

M

= 10

with P

= 5

corresponding to formulation-B in Fig. 10. As can be seen, the classical

formulation favors router sites that are closer to a cluster of sites which are in general not good for OLSRI network design since the interconnection cost of the core network is not taken into account (compare Fig. 10 to Fig. 8a). We now address the complexity of solving the models as faced by cplex MIP solver. We found that for 15-site examples for formulation-A, the solver obtains the optimal solution in less than a second. However, for 25-site examples, in some (not all) cases, the solver took several minutes to solve the problem. On inspecting the CPLEX output log for the specific cases where it took longer time to solve, we found that the actual optimal solution was obtained within a minute or so, but since MIP solver uses the branch and bound technique, it needed to prune all the other branches before reporting the optimality. Fortunately, cplex provides a parameter that can be set up before the run to limit the number of nodes to be visited in the branch and bound tree; in our case, we have used both 30,000 and 100,000, and have found that the optimal cost is almost always is found within about the first 10,000 nodes visited in the branch and bound implementation of the cplex solver (this takes about a minute). Another interesting phenomenon we observed is that it takes a lot longer to solve tighter problems for formulation-B; for example, with N

= 30,

M

= 10,

and since

Kj = 6, we need a minimum value of P to be 5 for the problem to be feasible – this is the ’tightness’ of the problem. This example (i.e., with P = 5) took several minutes to solve while the corresponding case with P = 7 took less than a minute to solve. 10

5. Extensions We discuss two general extension applicable to both Model-A and Model-B of the OLSRI problem.

5.1 Optimal Number of locations A natural extension would be to consider P , the number of locations to be selected, also as a variable

to determine optimal P instead of providing a specific value P as an input parameter (this should not be confused with optimal location of routers/switches). The primary disadvantage of developing a formulation for this extension is that the entire problem becomes a non-linear integer programming problem (NLIP). In general, NLIP problems are very hard to solve and, in fact, no general solution approach is known.

Fortunately, for the OLSRI problem, we conjecture that the optimal P is for the optimal OLSRI design with the smallest P for which the formulation is feasible. To illustrate, consider formulation-A with N

= 25, and

Kj = 5 for all j — in this case, the conjecture is that the optimal cost obtained by solving formulation-A with P = 5 is smaller than the optimal cost obtained with any other P > 5. To see this, consider the hypothetical case where we set cij = c and bj = b. Let us denote the optimal cost by o (P ) for formulation-A when P sites are to be selected. From formulation-A, using (10) along with (7), (8) for fixed values c and b, the optimal cost for fixed P becomes ohAi (P ) = cN + cP (P , 1)=2 + Pb: If now, we solve formulation-A for selection of P

+1

number of locations, the optimal cost changes to

ohAi (P + 1) = cN + c(P + 1)P=2 + (P + 1)b: Clearly, ohAi (P ) < ohAi (P

+ 1)

for positive cost components c and b showing that the optimal P is for the

smallest P for which the formulation-A is feasible. Another way to see this is that when we go from P to

P + 1, we need to add P additional links in the core network for the interconnection cost besides the cost

for the additional switch site. We also did several different runs with our randomly-generated test examples

and has empirically observed this to be true for general cij ’s and bj ’s. We have not found a pathological case so far that shows the conjecture to be untrue. Thus, for all practical purpose, the conjecture appears to hold true. For Model-B also, the same conjecture holds, i.e., the smallest value P for which the formulation-B

is feasible gives the optimal P . As before, we consider the hypothetical case where we assume gij

=

g,

ej = e, and hjk = h. If now, for a given P , we denote the optimal cost by ohB i(P ) for formulation-B, then using (20), (15), (18), and (19), we get

ohB i(P ) = gN + eP + hP (P , 1)=2: If we change the number of locations to be selected to P

+ 1,

we then have

ohBi (P + 1) = gN + e(P + 1) + h(P + 1)P=2: Again, we have ohB i (P ) < ohB i (P

+ 1).

11

5.2 Less than fully-interconnected core network In both formulation-A and formulation-B of the OLSRI problem, we have assumed the interconnected core network to be fully mesh. Although this assumption is valid in practice for several actual networks, it is desirable to have an approach that will work for the less-than fully interconnected case also. This is however not so easy to model since the connectivity (not full-connectivity) of the core network is somehow needed to be addressed to avoid networks being isolated into two or more sub-networks. On the other hand, it is easy to develop a heuristic based on the formulations we have already presented. Note that both formulations guarantee the connectivity (albeit, full-connectivity) of the core network. Thus, a simple heuristic can be developed if the goal is to connect the core network with, say, L links (where P

, 1  L  P (P , 1)=2).

This heuristic is referred to as the delete heuristic and it works as follows: Delete Heuristic:

1. Solve formulation-A (or formulation-B) 2. If the minimum number of core-network links, L, is specified to be P (P

, 1)=2, then stop; else go to

the step-3. 3. Sort all the core-network links in descending order of their cost (appropriate cij corresponding to optimal solution for formulation-A, or, appropriate hjk corresponding to the optimal solution for formulation-B) 4. While ( the number of remaining core-network links  L ) do Select the core-network link with the highest cost from the current list of links not yet tested Delete it if the overall network connectivity is not violated; else mark it and ignore if the connectivity is violated endwhile Thus, the delete heuristic provides a way to obtain the network topological architecture where the core network need not be fully-interconnected.

6. Conclusion The combined optimal location of routers/switches and interconnection problem is an important computer communication network design problem. This problem has applicability in ATM PNNI environment and IP/ATM networking environment for network topological architecture design; on the other hand, it has received little attention in the literature. We have presented two novel optimization formulations for two instances of the OLSRI problem. Through our computational results for several examples we have presented optimal network design configurations; further, we have also presented two extensions where we have shown how the OLSRI formulations can address the choice of optimal number of locations, and for extending to less than fully-interconnected core networks. 12

References [1] E. E. Adams, Jr. and R. J. Ebert, Production and Operations Management – 2nd Edition, Prentice-Hall, 1982. [2] ATM Forum, PNNI 1.0 Specification. March 1996. www.atmforum.com. [3] D. Bertsekas and R. Gallager, Data Networks – 2nd Edition, Prentice-Hall, 1992. [4] CPLEX Linear Optimizer 4.0.3 with Mixed Integer Solver. www.cplex.com. [5] M. Schwartz, Computer Communication Networks Design and Analysis, Prentice-Hall, 1977.

13

Fig. 1: 15-site example(A)

Fig. 4a: A-Design N

Fig. 2: 25-site example(A)

;P = 3

= 15

(distance-based)

Fig. 5a: A-Design N (skewed-distance-based)

Fig. 4b: A-Design N

= 15

;P = 4

Fig. 3: 30-site example(B)

Fig. 4c: A-Design N

(distance-based)

;P = 3

= 15

Fig. 5b: A-Design N

= 15

;P = 5

(distance-based)

;P = 4

(skewed-distance-based)

14

= 15

Fig. 5c: A-Design N

= 15

;P = 5

(skewed-distance-based)

Fig. 6a: A-Design N

;P = 5

= 25

(distance-based)

Fig. 7a: A-Design N (skewed-distance-based)

Fig. 6b: A-Design N

= 25

;P = 6

Fig. 6c: A-Design N

(distance-based)

;P = 5

= 25

Fig. 7b: A-Design N

= 25

;P = 7

(distance-based)

;P = 6

(skewed-distance-based)

15

= 25

Fig. 7c: A-Design N

= 25

;P = 7

(skewed-distance-based)

Fig. 8a: B-Design N

= 30

;P = 5

(distance-based)

Fig. 9a: B-Design N (skewed-distance-based)

Fig. 8b: B-Design N

= 30

;P = 6

Fig. 8c: B-Design N

(distance-based)

= 30

;P = 5

Fig. 9b: B-Design N

= 30

16

;P = 7

(distance-based)

;P = 6

Fig. 9c: B-Design N

(skewed-distance-based)

Fig. 10: Pure optimal location design N

= 30

= 30

= 30

;P = 7

(skewed-distance-based)

; P = 5 (distance-based)