A backup double covering model and tabu search solution approach for

locating emergency medical stations†

Ayfer Başar, Bülent Çatay*, Tonguç Ünlüyurt

Sabanci University, Faculty of Engineering and Natural Sciences

Tuzla, 34956, Đstanbul

Abstract: The location planning of emergency medical service (EMS) stations is crucial,

especially in populated cities with heavy traffic conditions like Istanbul. In this paper, we

propose both a single-period and a multi-period problem, two variants of the well-known

Maximal Covering Location Problem, which require two types of services in the location

planning of the EMS stations. The objective of the single-period problem is to maximize the

total population serviced within t1 and t2 minutes (t1 < t2) using two distinct emergency service

stations where the total number of stations is limited. Similar to single-period problem, our

objective in the multi-period problem is to maximize the total population serviced by two

distinct stations within two different response time limits over a multi-period planning

horizon. Our aim is to provide a backup station in case no ambulance is available in the closer

station and develop a strategic plan that spans multiple periods. In order to solve these two

problems, we propose a Tabu Search approach using different initialization approaches. We

demonstrate the effectiveness of the proposed approach on randomly generated data. We also

implement our approach to the case of Istanbul to determine the locations of EMS stations in

the metropolitan area. Our computational study on both randomly generated data and real data

for Istanbul shows that the proposed approach provides optimal/near-optimal solutions for the

two problems.

Keywords: Maximal covering, backup double covering, tabu search, emergency medical

service.

† This work has been partially supported by Istanbul Metropolitan Municipality “My Project Istanbul” program * Corresponding author. Tel.: +90 216.483.9531; fax: +90 216.483.9550. E-mail addresses: catay@sabanciuniv.edu (B. Çatay), ayferbasar@su.sabanciuniv.edu (A. Başar), tonguc@sabanciuniv.edu (T. Ünlüyurt)

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1. INTRODUCTION

The location planning of emergency medical service (EMS) stations is crucial, especially in

populated cities with heavy traffic conditions since an effective planning of these stations

directly affects human life protection. The fatalities and disabilities caused by illnesses,

accidents, etc. may be reduced by making the right planning decisions. In recent years, EMS

has become more important for cities having a fast growing population such as Istanbul.

In the last 30 years, a lot of research effort has been spent in the literature to plan the

locations of both the fire brigade and EMS stations. Brotcorne et al. (2003) and Goldberg

(2004) provide a good review of these studies. Generally, mathematical programming has

been employed to address these problems. Optimal solutions of several problems described in

the literature can be found by means of advancements in computer technology, Operations

Research (OR) methods, and the available software. However, more complicated problems of

larger size require efficient solution techniques to obtain good solutions.

In this paper, we formulate a new Backup Double Covering Model (BDCM) where the

objective is to maximize the total population serviced within t1 and t2 minutes (t1 < t2) using

two distinct EMS stations. The proposed BDCM is conceptually similar to Maximal Covering

Location Model (MCML) in Church and ReVelle (1974), Double Coverage Model (DCM) in

Gendreau et al. (1997), and Backup Coverage Model (BCM) in Hogan and ReVelle (1986);

its formulation, however, is different.

Also, we present a multi-period backup double covering model (namely, MPBDCM)

for the locations of EMS stations over multiple periods. Essentially, we have a problem where

we determine the optimal locations of EMS stations for a number of periods with a given

maximum number of EMS stations that can be opened in each period. In MPBDCM, we

require that an EMS station remains open until the end of the planning horizon in the same

location once it has been opened. In this respect, MPDCM can be considered as a strategic

decision problem for public decision makers. The objective of this problem is to maximize

the total population serviced from two distinct EMS stations within two different response

time limits. So, our goal is to maximize the total “backup double covered” population where

backup double coverage means that a location can be reached from two EMS stations within

two distinct time limits. This objective is especially appropriate if the probability of being

busy when a call comes is relatively large. The proposed MPBDCM is a multi-period variant

of BACOP1 introduced by Hogan and ReVelle (1986) and Double Standart Model (DSM)

3

presented by Gendreau et al. (1997); however, the mathematical formulation is quite different

apart from being simply a multi-period extension.

We propose a Tabu Search (TS) approach to efficiently solve the proposed two

problems . TS is a well-known metaheuristic capable of finding good solutions fast. We

implement three initialization methods in our TS algorithm to observe their role on the overall

solution quality for BDCM: (i) a random method, (ii) a steepest-ascent method, and (iii) a

linear programming (LP) relaxation-based method. Since the steepest-ascent approach

provides good solutions fast, we implement it for solving MPBDCM as well. The

performance of TS is tested on randomly generated data. Furthermore, the approach is applied

for the location planning of the EMS stations in the Istanbul metropolitan area.

The organization of the paper is as follows. The relevant literature is reviewed in

Section 2. Section 3 is dedicated to the formulation of BDCM and MPBDCM. Section 4

describes the initialization methods and Section 5 presents the proposed TS approach. The

experimental study and the results are discussed in Section 6. Section 7 depicts the case study

carried out with Istanbul Metropolitan Municipality. Finally, the concluding remarks and

future research directions are provided in Section 8.

2. LITERATURE REVIEW

The problem of effectively planning EMS systems has attracted the attention of many

researchers and a vast amount of publications exists in the literature. Various solution and

modeling approaches have been proposed for solving different variants of this problem.

Especially, single coverage problems have been extensively studied in the ambulance location

literature and various definitions of the term coverage have been introduced. We can classify

the EMS station planning models in the literature in various ways. One basic classification is

the deterministic and probabilistic models. Since the probabilistic models are beyond the

scope of this study, we will review the deterministic literature.

The very basic deterministic model is the Set Covering Model (SCM) proposed by

Toregas et al. (1971). The objective of the model is to find the minimum number of EMS

stations covering all demand points at least once. This model does not take into account the

populations of these points. Although SCM is a theoretically important model, it does not

capture some practical constraints, e.g. when an ambulance departs for responding to a service

aid, other demand points covered by this ambulance are no longer covered until the

ambulance is available (Brotcorne et al., 2003).

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Church and ReVelle (1974) proposed the MCLM to maximize the population or the

number of demand points covered by with a limited number of stations. By using this model,

the efficiency of the available services (rate of total coverage) can easily be measured.

Moreover, the additional cost of locating more stations and the additional coverage achieved

by these stations may be compared in strategic decision-making.

Schilling et al. (1979) developed the Tandem Equipment Allocation Model (TEAM)

as an extension of MCLM. In TEAM, the objective is to maximize the population covered

with two different service types where the number of stations for each type of service is

limited. This model also requires that if a service type is assigned to a demand point, the other

service must also be assigned.

SCM, MCLM, and TEAM all try to maximize single coverage of demand points by

the ambulances. In other words, if an ambulance is busy serving a demand point, other

demand points covered by this ambulance will no longer be covered. To overcome this

drawback, multiple coverage models have been proposed in the literature, which consider

both location planning and ambulance related constraints. The Modified Maximal Covering

Location Model (MMCLM) in Daskin and Stern (1981) maximizes the covered population.

The model includes a second objective which maximizes the demand points covered multiple

times. Two different variants of MMCLM were presented by Hogan and Revelle (1986) as the

DCM. In the first type of DCM (namely, BACOP1), the population covered at least twice is

maximized within the same coverage standard given a limited number of stations. In the

second type (namely, BACOP2), the objective function is to maximize the weighted average

of, the demands covered once and multiple times. All these three problems are based on the

multiple coverage of demand points in a single critical travel time restriction.

Gendreau et al. (1997) introduced the Double Standard Model (DSM) which

maximizes the demand covered multiple times using two different travel time restrictions.

Their objective is to maximize the demand covered at least twice in the shorter travel time

limit. The constraints include a set covering requirement of all demand points within the

longer travel time limit and a given proportion of the population has to be covered within the

shorter travel time limit. An important difference of DSM from the other deterministic

models is the assignment of multiple ambulances to the same station. However, there is an

upper bound on the number ambulances to be assigned to each station.

By the help of the models proposed in the literature, several applied studies have been

conducted for the location planning of EMS stations and various heuristic and metaheuristic

methods have been developed to solve them efficiently. For instance, Gendreau et al. (1997)

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proposed a TS algorithm for the location planning of the EMS stations in Montreal. Their

approach provided good results fast in comparison with the solutions obtained using a branch-

and-bound algorithm with a limited number of iterations. Harewood (2002) discussed the

planning of EMS stations in Barbados using simulation techniques. Recently, Doerner et al.

(2005) have proposed an Ant Colony Optimization (ACO) to plan the EMS stations in Austria

and compared its performance against the TS approach of Gendreau et al. (1997). Their

results revealed that TS could find better results especially for large problems within a shorter

computational time. In another recent study, Jia et al. (2007) have proposed a Genetic

Algorithm, a Lagrangian Relaxation (LR) approach, and greedy heuristics for EMS planning

in Los Angeles and compared their performances.

All the models discussed above are proposed for single-period location planning of

EMS stations. In fact, there are only a few studies in the literature that attack the multi-period

planning. Schilling (1980) discusses multi-period planning for dynamic location modeling of

EMS stations. The proposed model considers single coverage and the coverage in different

periods are associated with weights that represent the relative importance of the

corresponding period. Schilling presents a heuristic to find nondominated solutions and

investigates its performance using a two-period example. Very recently, Rajagopalan et al.

(2008) have introduced a multi-period set covering location model for dynamic redeployment

of ambulances and proposed a TS approach to solve it. The objective in this model is to

minimize the number of EMS stations while guaranteeing certain coverage. The locations of

the EMS stations that have been opened are allowed to change from period to period

depending on the demand in each period.

In this paper, we formulate the DCM and MPBDCM as an integer program. Since the

problem is intractable for large instances, we propose a TS procedure to solve the models

efficiently. The performance of the proposed TS approach is investigated on randomly

generated data with various characteristics and initial solutions. Furthermore, we also apply

this approach for solving the location planning of the EMS stations in the Istanbul

metropolitan area. As such, the contributions of this study may be summarized as the

formulation of a single-period and a multi-period backup double coverage model for locating

EMS stations, the development of a TS approach to solve these problems, and the application

of this approach to a case study for the city of Istanbul.

In what follows is the description and the mathematical formulation of our problems.

Then, we will discuss the TS approach developed for solving this problem.

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3. MODEL FORMULATION

In this section, we formulate the proposed BCDM and MPBDCM. Both the BDCM and

MPBDCM incorporate two types of service requests to be fulfilled. The motivation in using a

double covering model is to provide a backup station in case no ambulance is available in the

closer station.

3.1. Proposed BDCM

The objective of the BDCM is to maximize the total population serviced within t1 and t2 time

units (t1 < t2) using two distinct emergency service stations where the total number of stations

is limited. If a demand region is covered by an emergency service station, we assume that the

whole population of this demand point is covered. In the multiple coverage models in the

literature two sets of coverage related decision variables are utilized: one to define if a region

is covered at least once and the other if a region is covered at least twice. In addition, these

models include a set of constraints to ensure that a region cannot be covered at least twice if it

is not covered at least once (see e.g. Brotcorne et al, 2003). Following the same approach, the

mathematical model is as follows:

Notation:

M set of demand regions (j ∈ M)

N set of potential location sites (i ∈ N)

K the maximum number of EMS stations to be opened

Pj population of region j

otherwise 0

units time within region thereach can location in station theif 1 1

=,

tji,aij

otherwise 0

units time within region thereach can location in station theif 1 2

=,

tji,bij

Decision Variables:

otherwise 0

location in located isstation a if 1

=,

i,xi

7

otherwise 0

covered double is region if 1

=,

j,y j

otherwise 0

units timein covered is region if 1 11

=,

tj,y j

(BDCM1) Maximize ∑∈

=Mj

jj yPZ (1)

subject to ,KxNi

i ≤∑∈

(2)

,01 ≥−∑

∈ji

Ni

ij yxa Mj∈∀ (3)

,01 ≥−−∑

∈jji

Ni

ij yyxb Mj∈∀ (4)

,1

jj yy ≥ Mj∈∀ (5)

∈ ix {0,1} Ni∈∀ (6)

∈ , 1

jj yy {0,1} Mj∈∀ (7)

The objective (1) of BDCM1 maximizes the population double covered with a backup

station. Constraint (2) imposes the total number of stations that can be opened. Constraints (3)

ensure that any demand point must be covered in t1 time units in order to be covered multiple

times. If a demand point is covered in t1 time units, it is also covered in t2 time units by the

same station due to t1<t2. Therefore, constraints (4) express the requirement of coverage by at

least two different stations in t2 time units. Constraints (5) ensure that a demand point can not

be double covered if it is not covered in t1 time units. Constraints (6) and (7) make sure that all

the decision variables are binary.

(BDCM2) Maximize ∑∈

=Mj

jj yPZ (1)

subject to ,KxNi

i ≤∑∈

(2)

,0≥−∑∈

ji

Ni

ij yxa Mj∈∀ (3)

,02 ≥−∑∈

ji

Ni

ij yxb Mj∈∀ (4)

∈ ix {0,1} Ni∈∀ (5)

∈ jy {0,1} Mj∈∀ (6)

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We formulate BDCM2 to solve the problem about location planning of EMS stations

more efficiently. Although BDCM1 and BDCM2 solve the same covering problem, BDCM2

has |M| less binary decision variables and |M| less constraints. We can prove that BDCM2, the

model which we address as BDCM henceforward, is NP-hard as follows: Consider a special

case of BDCM where t2 is sufficiently large such that each potential site i can cover all

demand regions in t2. Then, the problem reduces to a MCLM, which is known to be NP-hard

(Berman and Krass, 2002).

3.2. Proposed MPBDCM

Similar to BDCM, the proposed MPBDCM incorporates two types of service requests to be

fulfilled to provide a backup station in case no ambulance is available in the closer station.

Our objective is to maximize the total population serviced within t1 and t2 time units (t1 < t2)

using two distinct EMS stations where the total number of stations is limited in each period.

Moreover, if an EMS station is opened in a certain period, it must remain open in the

subsequent periods. This is a reasonable assumption when the costs associated with closing

and reopening EMS stations are relatively large. For this strategic planning problem, the

appropriate planning horizon may vary for different environments depending on various

factors such as the total population, the planning horizon of the budgets, growth strategies,

etc. but usually spans a few years. The mathematical formulation of the problem is as follows:

Additional Notation:

Kt the maximum number of EMS stations that can be open in period t (Kt ≤ Kt+1)

Pjt population of region j in period t

Decision Variables:

otherwise 0

periodin location in opened isstation a if 1

=,

ti,xit

otherwise 0

periodin covered double is region if 1

=,

tj,y jt

(MPBDCM) Maximize ∑∑∈ ∈Tt Mj

jtjt yP (1)

subject to ,tNi

it Kx ≤∑∈

Tt∈∀ (2)

9

,0≥−∑∈

jtit

Ni

ij yxa Mj∈∀ , Tt∈∀ (3)

,02 ≥−∑∈

jtit

Ni

ij yxb Mj∈∀ , Tt∈∀ (4)

,01, ≥− −tiit xx Ni∈∀ , Tt∈∀ (5)

∈ itx {0,1} Ni∈∀ , Tt∈∀ (6)

∈ jty {0,1} Mj∈∀ , Tt∈∀ (7)

The objective (1) is to maximize the total population double covered in all the periods.

Constraints (2) restrict the total number of stations that can be open in each period.

Constraints (3) ensure that, in any period t, a demand region j can be double covered (yjt=1) if

it is covered (at least once) in t1 time units. Furthermore, if a demand region is covered in t1

time units, it is also covered in t2 time units by the same station due to the relationship t1<t2.

Hence, constraints (4) make sure that, in any period t, a demand region j is double covered

(yjt=1) only if it is covered by at least two distinct stations in t2 time units. Constraints (5)

ensure that if a station is opened in any period, it remains open in the subsequent periods.

Thus, the total number of open stations at each period is larger than or equal to the number of

open stations in any previous period. Constraints (6) and (7) define the binary decision

variables.

When there is only one planning period MPBDCM reduces to BDCM; hence it is NP-

hard as well.

4. INITIALIZATION ALGORITHMS

We investigate three initialization approaches for BDCM to evaluate their contribution to the

final solution obtained by the TS procedure. Among these three approaches, one that provides

efficient solutions fast is implemented for MPBDCM.

4.1. Initialization for BDCM

Random approach: In order to find an initial solution very fast, we propose the random

method. In this method, we randomly select K stations among N potential location sites using

a uniform distribution U[1, N].

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Figure 1. Steepest-ascent approach for BDCM

Steepest-ascent approach: In this method, first a station is opened in the potential location that

covers the largest population in t1 time units. Then, the station maximizing the double covered

population in t2 time units is opened. In the next step, we compare the solution obtained by

opening two new stations consecutively using the same procedure to the solution obtained by

adding another station to the existing solution. The comparison is based on the increase in the

objective function value per station opened and the alternative that gives the largest increase

is accepted. We repeat this procedure until the total number of stations opened reaches K or

the whole population is double covered. The flowchart of the algorithm is given in Figure 1.

In this chart Z_t1(S) denotes the total population covered in t1 time units where S is the set of

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opened stations and Z(S) is the objective function value (OFV) associated with S. Imp1 and

Imp2 refer to the improvement in the OFV achieved by opening one and two additional EMS

stations, respectively.

Figure 2. LP relaxation-based approach

LP relaxation-based approach: In this method, we first solve the LP relaxation of the model

and fix xi = 1 if its optimal value is 1. Then, the largest non-binary xi is fixed to 1 and the LP

relaxation is solved again. This procedure is repeated until the total number of stations opened

reaches K or the whole population is double covered. The flowchart of the algorithm is given

in Figure 2.

4.2. Initialization for MPBDCM

Since our experimental study has shown that the steepest-ascent approach performs well on

BDCM, the same approach is adopted in MPBDCM by repeating the procedure for all periods

successively. We continue in the same way until the total number of stations opened reaches

the maximum allowable number (Kt) or the whole population has been double covered.

Note that if only one more station can be opened, it is opened at the location that improves the

double covered population most without evaluating opening two stations alternative. The

flowchart of the algorithm is given in Figure 3. In this figure, OFV refers to the objective

function value and Cov t 1 Sτ denotes the total population covered in t1 time units where Sτ is

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the set of open stations in the most recent period τ. OFVτ(S) is the OFV for solution S, where

S = {S1, S2, …, Sτ}, i.e. S consists of τ sets of open stations, each set corresponding to the

associated period. Imp1 and Imp2 refer to the improvement in the OFV achieved by opening

one station and two stations, respectively.

Figure 3. Steepest-ascent procedure for MPBDCM

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5. PROPOSED TABU SEARCH ALGORITHM

TS is a local search technique that was originally developed by Glover (1977). Using an

initial feasible solution, TS investigates the neighbors of the existing solution at each iteration

in an attempt to improve the incumbent best solution. It avoids the repetition of the same

solutions by maintaining a tabu list of moves that are forbidden in the short run because they

lead either to sub-optimal solutions or to solutions that have already been explored. TS

accepts a tabu move only if it satisfies a pre-specified aspiration criterion. TS has been

successfully applied to solve various combinatorial optimization problems, e.g. traveling

salesman problem (Malek et al., 1989), quadratic assignment problem (Skorin-Kapov, 1990),

scheduling (Widmer, 1991), vehicle routing problem (Gendreau et al., 1994), p-median

problem (Rolland, 1996), etc. The interested reader is referred to Glover (1990) for a

comprehensive tutorial.

The most crucial mechanism in the TS is the neighborhood structure since it directly

affects the search procedure and the quality of the final solution as well as the computational

complexity of the algorithm. In our TS algorithm both for BDCM and MPBDCM, the

neighborhood structure is determined as simultaneously closing and opening one station at

each iteration. This neighborhood structure maintains the feasibility of the solution generated

at each move. Our preliminary analysis for BDCM on closing and opening one station vs.

multiple stations has revealed that one-station neighborhood alternative performed

significantly better. At each iteration, the TS procedure of MPBDCM should maintain the

feasibility of the number of stations with respect to constraint set (2) and of the stations that

are open in any period with respect to constraint set (5). For this purpose, we propose a

neighborhood search structure in which we close one station in a location and open a new one

in another location simultaneously in period τ. Then, two situations may occur as a result of

this move: in the current multi-period solution, a station may already be open in this new

location in one of the subsequent periods, i.e. in period τ’ where τ’ is between τ+1 and T, or no

station is opened in that location in any of the periods. In the latter case, the set of open

stations are updated for periods τ+1,…,T and in the former case, the set of open stations

changes onlyin periods τ,…,τ’-1.

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Table 1. Neighbourhood search structure for MPBDCM

An example for the two possible moves is illustrated in Table 1. The first column is

the period, the second column gives the maximum number of stations than can be open in the

corresponding period, and the numbers in the next three columns show the locations of the

stations opened. In the neighborhood case 1, we consider switching the location of one station

from location 4 to location 8 in period 2. Then, the same change must also be applied to

period 3. In period 4, however, no exchange is necessary since a station is already open in

location 8. In the neighborhood case 2, the station in location 4 is moved to location 10. Since

there exists no station in location 10 in any of the periods, this change needs also be applied to

the subsequent periods. Note that in either case, we do not consider the closure of stations 1,

2, or 3 in period 2 since this move has already been investigated in period 1. At each iteration,

the TS algorithm examines all such possible moves in all periods and determines the

candidate solution as the one which provides the largest increase in the OFV.

The tabu list is the mechanism to avoid cycling in the neighborhood during the search

procedure. This is achieved by keeping certain moves in a tabu list for a number of iterations

so as to prevent the same moves to occur repeatedly. The number of iterations during which a

move is kept in the tabu list is referred to as the tabu tenure. A move which is tabu at one

iteration may no longer be tabu in a later iteration and could become tabu again as the search

progresses. If the tabu tenure is too small, cycling may still occur since the moves remain

forbidden for a short period of time. On the other hand, the quality of the solutions may

deteriorate when the tabu tenure is too large since the moves leading to better solutions may

be forbidden for a long time. Once the algorithm determines the best move, it first looks up

the tabu list. If the move is tabu then it checks whether the aspiration criterion is satisfied: if

the move leads to a solution which is better than the best we currently have (best-so-far) then

it is accepted.

Another important mechanism is the tabu move structure. We have considered two

tabu move types: keeping the list of stations closed and opened separately and keeping closed

t Kt Current Solution Neighborhood Case 1 Neighborhood Case 2

1 3 1-2-3- 1-2-3- 1-2-3-

2 5 1-2-3- 4 -5 1-2-3- 4 -5-8 1-2-3- 4 -5-10

3 7 1-2-3- 4 -5-6-7 1-2-3- 4 -5-6-7-8 1-2-3- 4 -5-6-7-10

4 9 1-2-3- 4 -5-6-7-8-9 1-2-3- 4 -5-6-7-8-9 1-2-3- 4 -5-6-7-8-9-10

15

and opened station pairs in a single list. In the former case, opening a recently closed station

or closing a recently opened station is forbidden whereas in the latter case, the exact station

pair to be opened and closed simultaneously is forbidden. After an initial experimental study,

we have observed that using two separate tabu lists, one of which for the station closed and

the other for the station opened, provided better solutions. Hence, we have adopted that

strategy.

In many cases, the TS may get trapped at a local optimum when it can no longer find a

solution better than the best-so-far. To overcome this undesirable situation we have adopted a

diversification strategy: if the best-so-far solution cannot improve after a number of k2

consecutive iterations, we randomly close one station and open another. We have

experimented randomly closing and opening multiple station; however, their performance was

inferior. On the other hand, in some cases the current solution does not improve or deteriorate;

i.e. cycling occurs and the same current objective function repeats. This happens when

multiple potential locations have the same coverage properties. To avoid this cycling, we use

a mechanism to jump to another solution resulting in with the least decrease in the current

objective function value instead of accepting the solution which gives the same current

objective function value when the current objective function value repeats for k1 consecutive

iterations.

Finally, an outer iteration limit is imposed as the stopping criterion. The algorithm

stops when k3 iterations have been completed or the whole population has been double

covered. The flowchart of the TS procedure described for BDCM and MPBDCM is provided

in Figure 4. In this figure, Siter denotes the solution in iteration iter, that is it corresponds to the

list of open stations in each period for the current solution. Siteri-j denotes the solution that is

obtained from Siter by closing the station in location j in a certain period and opening a

station in location i instead as explained in the description of neighborhood structure

above. OFV* denotes the best-so-far OFV and 1iterTL and 2

iterTL are the tabu lists for opened

and closed stations, respectively for the current iteration. iter counts the number of TS

iterations while counter1 keeps track of the consecutive number of iterations during which

OFV remains same and counter2 keeps track of the consecutive number of iterations during

which OFV* does not improve.

16

Let TABU = ØDetermine (i*, j*) = argmax OFV(Siter-1

i-j)

Siter = Siter-1i*-j*

OFViter = OFV(Siter-1i*-j*)

Update tabu list

{i*} TL1iter

or {j*} TL

2iter?

Select (i,j) randomly.Siter=Siter

i-j , OFViter= OFV(Siter) and update tabu list.Set counter2 = 0.

OFViter > OFV* ?

OFV* = OFViter

counter1= 0counter2= 0

No

OFViter > OFV* ?

counter1 = k1 ?

counter2 = k2 ?

iter ≤ k3 ?

Let S0 be an initial feasible solution.OFV

* = OFV(S0) and set iter = 1.

Terminate

counter2 = counter2 +1

OFViter = OFViter-1 ?

counter1 = counter1+1

YesNo

iter=iter+1counter1 = 0

Determine (i*, j*) = argmax {[(OFV(Siteri-j) - OFViter ] < 0}

Let Siter = Siteri*-j* , OFViter = OFV(Siter) Update tabu list and set counter1 = 0

Yes

No

No

No

Yes

No

No

Yes

Siter = Siter-1i*-j*

OFViter = OFV(Siter-1i*-j*)

Update tabu list

Yes

Yes

Yes

(i*, j*) = argmax OFV(Siter-1i-j)

(i, j) TABU

TABU = TABU (i*, j*)

∉

Figure 4. Tabu search procedure for BDCM and MPBDCM

17

6. EXPERIMENTAL STUDY

To test the performance of the proposed TS approach we randomly generate 192 problem

instances for BDCM and 144 problem intances for MPBDCM. We consider three different

types of geographical distributions of demand regions as illustrated in Figure 5. The whole

geographical area to be covered is assumed to be a square. In the first type, a square area is

divided into 4 square zones with equal areas where the demand regions and potential locations

are distributed uniformly in each zone. The second type has four square telescopic zones with

1:3:5:7 space proportions in which each zone contains 25% of demand regions and potential

locations. This corresponds to a single center and the density of population decreases as we go

away from the center. The third type of layout involves two central zones each with 25% of

demand regions and potential locations, and a larger zone with 50% of demand regions and

potential locations where the space proportions are 1:1:7.

Figure 5.Geographical distributions of demand regions (a) type 1, (b) type 2, and (c) type 3

Our data set includes problems with 200, 300, 400, and 500 demand regions. The

number of potential locations is set to 100%, 75%, 50% and 25% of the number of demand

regions. The potential locations are selected randomly among demand regions. When the

number of potential locations is equal to the number of demand regions, the complete double

coverage of the whole population is guaranteed if the number of stations is sufficiently large.

For each demand region-potential location configuration in BDCM problem instances, we

consider four different values of the number of stations, depending on the problem size. These

values are 15, 20, 25, and 30 for the case of 200 demand regions; 20, 25, 30, and 35 for the

case of 300 demand regions; 20, 30, 40, and 50 for the case of 400 demand regions; and 30,

40, 50, and 60 for the case of 500 demand regions. For each demand region-potential location

18

configuration in MPBDCM intances, we use three different sets of values for the number of

stations (Kt). The planning horizon is 4 periods. The maximum number of stations in period 1

(K1) are drawn from uniform distributions U[5-15], U[10-20], U[15-25], and U[20-30] for

the data with 200, 300, 400, and 500 demand regions, respectively. K2, K3, and K4 are

determined by adding a random number of stations to the previous period’s number. The

additional number of stations in each period follow uniform distributions U[2-6], U[3-9], U[4-

12], and U[5-15] for the data with 200, 300, 400, and 500 demand regions, respectively.

The edge length of the square region is determined according to the number of demand

regions so that the demand regions and potential locations are not concentrated in a small area

or too dispersed to cause a large number of regions that do not have any means of being

double covered at all. We assume a constant speed of 40 km/h for the ambulances and aij and

bij values are obtained by using these data.The populations of the demand regions are

generated randomly according to an exponential distribution with mean 1000. The values of t1

and t2 are set equal to 5 and 8 minutes, respectively, in parallel with the requirements

determined by the Directorate of Instant Relief and Rescue (DIRR) at Istanbul Metropolitan

Municipality.

The proposed TS procedure is coded in Microsoft Visual C++ 6.0 and executed on 2.8

GHz Intel Pentium with 3.25 GB of RAM. All the problem instances are solved using OPL

Studio 5.5 with ILOG CPLEX v.11 on the same processor.

6.1. Evaluation of the Results for BDCM

First, we investigate the performance of the three different initialization heuristics desribed for

BDCM benchmarked against the solution obtained by OPL. We observe that, the steepest-

ascent heuristic gives fairly good initial solutions fast. An average deviation of 6.04%,

4.94%, and 5.52% from the optimal are obtained for problem types 1, 2, and 3, respectively,

in less than 4 seconds. The LP relaxation-based heuristic gives a deviation of 2.98%, 1.47%,

and 1.89% on the average for types 1, 2, and 3, respectively. However, the computional time

is significantly larger as expected. The deviaton is calculated as (OPL solution/Heuristic

solution)-1. Note that the OPL solutions are obtained by running the software using its default

setting and the maximum run time is set to 5 minutes for problems with less than 300

potential locations and 10 minutes for the larger problems. Note also that OPL could not find

the optimal in 9 instances of type 1 and 5 instances of type 2 and 3 within the required time

limits. The instances that cannot be solved to optimality have relatively larger sizes with 400

and 500 demand regions. In sum, 173 out of 192 problems were solved to optimality.

19

Table 2. Average results for TS1, TS2, and TS3

Data

Type

Initial

Solution 500 iterations 1000 iterations 2500 iterations 5000 iterations

Avg

%Dev

Avg

Time

Avg

%Dev

Avg

Time

Avg

%Dev

Avg

Time

Avg

%Dev

Avg

Tim

Avg

%Dev

Avg

Time

TS1

1 44.36 <1 0.69 42 0.42 82 0.19 202 0.10 401

2 35.68 <1 0.74 42 0.54 79 0.34 192 0.17 377

3 39.97 <1 0.66 42 0.47 81 0.26 198 0.18 390

TS2

1 6.04 3.5 0.63 44 0.41 84 0.18 205 0.12 403

2 4.94 3.5 0.55 41 0.33 79 0.24 192 0.17 378

3 5.52 3.3 0.56 42 0.41 81 0.27 198 0.20 390

TS3

1 2.98 81 0.37 121 0.28 162 0.11 282 0.08 483

2 1.47 70 0.27 108 0.24 146 0.18 259 0.14 448

3 1.89 61 0.29 100 0.26 138 0.19 255 0.15 450

The parameters of TS applied explained earlier are set according to some initial

experimental study as follows for BDCM problem intances: tabu tenure=7, k1=5, k2=15. To

make an overall assessment of the performance of TS with respect to the number of iterations,

we observed the solutions obtained for k3 values of 500, 1000, 2500, and 5000 iterations. In

Table 2, we report the average results of all problem instances. Each entry in the table

corresponds to the average of 64 instances ( 4 demand regions * 4 potential locations * 4

number of stations). TS1, TS2, and TS3, respectively, refer to the initialization approaches as

random, steepest-ascent, and LP-relaxation, respectively. “Avg %Dev” shows the average of

the deviations from the OPL solutions computed using the formula above and “Avg Time” is

the average computation time.

As seen in Table 2, all three TS approaches provide very good results in comparison

with the solutions found by OPL, independent from the data type of the geographical

distributions of the demand regions. Since TS3 starts from a good initial solution, its

performance is slightly better than TS1 and TS2. However, the difference between different

algorithms reduces as the search progresses with the increasing number of iterations. At the

end of 5000 iterations, no significant difference exists neither on the deviations nor on the

computation times and the average deviations do not exceed 0.2% of the mostly optimal

solutions given by OPL. Even with a moderate computation time of 500 iterations, the

average deviations are well below 1%.

The detailed results for each problem instance are provided in the Appendices:

Appendix A, B, and C, respectively, reports the results of 5000 iterations for the three types of

geographical distributions of the demand regions, type 1, 2, and 3, respectively. We observe

20

that the TS algorithms solve almost half of the problems to optimality. In additon, for data

type 1, TS algorithms were able to achieve better solutions than those obtained by OPL within

the specified time limit for some instances.

We have noted that OPL fails to find the optimal solutions in large problems with

large number of potential station locations. Then, the contribution of the TS approach may be

questioned since the optimal solutions can be found by using a reduced set of potential

locations. However, by decreasing the number of potential locations we reduce the feasible

region which, in turn, gives optimal solutions that may be significantly worse than the

optimal/near optimal solutions that could have been achieved in the full-sized potential

locations case. This may be easily noticed in the results provided in Appendix A. Consider the

case with |M|=300 demand regions and K=20 stations: when each demand region is also a

potential station location (100% case) the optimal solution is 223,396 whereas when the

number of potential locations is reduced to 150 (50% of demand regions) the optimal solution

turns to 200,086. Thus, scaling down the problem size results in approximately 10% loss in

the total coverage. However, TS1 is able to find a solution with only 0.29% deviation from

the optimal. Therefore, since the TS approach is able to find optimal/near optimal solutions,

the total coverage it provides can be significantly better when each demand region is

considered as a potential location as compared to finding an optimal solution by reducing the

number of locations.

6.2. Evaluation of the Results for MPBDCM

Since the steepest-ascent initialization heuristic is observed to give fairly good solutions very

fast for BDCM, we decide to apply only this method for MPBDCM. First, we investigate the

performance of the steepest-ascent solution against the solution obtained by OPL. The three

problem types are solved in less than 11 seconds on the average and the average deviations

are 4.65%, 4.88%, and 4.82% for the problem types 1, 2, and 3, respectively. Note that the

OPL solutions are obtained by running the software using its default setting and the maximum

run time is set to 15 minutes for problems with less than or equal to 300 potential locations

and 30 minutes for the larger problems. Note also that OPL could find the optimal solution in

only 13 instances of type 1 (27%), 20 instances of type 2 (42%), and 15 instances of type 3

(31%) within the required time limits out of 48 instances for each data type. The instances that

can be solved optimally have relatively smaller sizes with 200 and 300 demand regions. In

sum, 96 out of 144 problems could not be solved optimally.

21

Table 3. Average results for TS2

CPLEX Initial Solution 500 iterations 1000 iterations 2500 iterations

Data

Type

Avg

Time

Avg

%Dev

Avg

Time

Avg

%Dev

Avg

Time

Avg

%Dev

Avg

Time

Avg

%Dev

Avg

Time

1 954 4.65 9.9 0.15 117.6 -0.15 225.4 -0.48 548.6

2 852 4.88 10.5 0.31 116.9 0.20 223.3 0.00 542.6

3 929 4.82 10.9 0.44 123.0 0.29 235.2 -0.08 571.6

The TS2 parameters are set according to an extensive preliminary experimental study

as follows: tabu tenure=8, k1=5, k2=20. To make an overall assessment of the performance of

TS2 with respect to the number of iterations, we observed the solutions obtained for k3 values

of 500, 1000 and 2500 iterations. In Table 3, we report the average results of all problem

instances with respect to the data type and the number of iterations. Each entry in the table

corresponds to the average of 48 instances for each geographical distribution type. “Avg

%Dev” shows the average of the deviations from the CPLEX solutions computed using the

formula above and “Avg Time” is the average computation time. The negative average gaps

indicate that the TS2 provides better average results than CPLEX.

As seen in Table 3, TS2 approach provides very good results in comparison with the

solutions found by CPLEX, independent from the data type of the geographical distribution of

the demand regions. At the end of 2500 iterations, no significant difference exists for three

different types of data neither on the deviations nor on the computation times. The average

percentage deviations show that TS2 performs as good as CPLEX in the worst case (0.00%

for data type 2). Overall, TS2 results are robust with respect to different data types and TS2 is

able to find better or comparable results for each data type in a relatively short amount of time

time. Even with a moderate computation time of 500 iterations, the average deviations are

well below 0.5%.

Table 4. Average results with respect to the density of potential locations

Density

CPLEX 2500 iterations

% Optimal found Avg Time Avg %Dev Avg Time

25% 100.0 43 0.17 200

50% 19.4 760 0.19 441 75% 11.1 1286 -0.18 685

100% 2.8 1558 -0.93 891

22

Table 5. Average results with respect to the number of demand regions

|ΜΜΜΜ|

CPLEX 2500 iterations

% Optimal found Avg Time Avg %Dev Avg Time

200 52.8 491 0.13 83

300 30.6 872 0.04 266

400 25.0 1145 -0.36 643

500 25.0 1140 -0.55 1225

The average time spent by the TS2 for 2500 iterations is significantly less than the

average time CPLEX spends for solving the instances. So, in Tables 4 and 5 we concentrate

on the solutions obtained after 2500 iterations of TS2 and report the results with respect to the

different properties of the instances (the detailed results are available in Appendix D, E and

F). In these tables, “CPLEX” column reports the percentage of instances that CPLEX was

able to find the optimal solution and the average time spent by CPLEX. In Table 4, we present

the results with respect to the density of the location sites where density is the ratio of the

number of potential locations to the number of demand regions. We see CPLEX finds the

optimal solution in all instances when the density is 25%. As the density increases, the

percentage of optimal solutions found by CPLEX decreases and the results obtained by the

TS2 become better than those of CPLEX. Note that when the density is large, the feasible

region enlarges and it is possible to find solutions with better coverage. Note also that even

when the density is 25%, the gap of TS2 is only 0.17% and these solutions are found much

faster than CPLEX.

Finally in Table 5, we present the results with respect to the number of demand

regions. As expected, TS2 outperforms CPLEX on the average for the instances with larger

number of demand regions. This table should be read with caution since CPLEX finds optimal

solutions when the intensity is 25% regardless of the number of demand regions. That is why

25% of the instances with 400 and 500 regions are solved to optimality by CPLEX. Also, the

average run time of CPLEX is almost the same for 400 to 500 demand region cases since in

many instances CPLEX stops at the time limit without finding the optimal solution. In

summary, the results in these two tables show that TS2 outperforms CPLEX when the

problem size is large, which is usually the case in real world applications.

23

7. PLANNING THE LOCATIONS OF EMS STATIONS IN ISTANBUL

Istanbul province has an area of 5,196 km2 and a population of 11,914,848 (according to 2007

estimates of the Turkish Statistical Institute). It approximately spans 125 kms on the east-west

direction and 40 kms on the north-south direction and consists of 28 districts. The Bosphorus

Strait divides the city in two parts, namely the European side and the Asian side which are

connected by two suspension bridges. Three outmost districts (two in the European and one in

the Asian side) which constitute almost half of the total area but only 2% of the total

population are excluded from the study since they require an independent planning due to

their geographical conditions. Hence, the total population we consider is 11,674,632.

Istanbul is a large city with a dense population and heavy traffic conditions. As

requested by DIRR, we used the administrative quarters as the demand regions for the

accuracy of results because the population data of each quarter is available through the census

of Turkish Statistical Institute. This corresponds to a total of 710 quarters, 243 in the Asian

and 467 in the European side. The reachability data aij and bij are directly collected through

extensive interviews conducted with the experienced ambulance drivers of DIRR rather than a

deduction from the distance information and average ambulance speed. t1 and t2 are taken as 5

and 8, respectively. Each quarter is a potential EMS location where we assume that the station

will be centrally located in a quarter and the farthest point in a neighboring quarter must be

covered by the station in that quarter for the neighboring quarter to be considered covered.

That is, if an EMS is declared to cover a quarter in 5 (8) minutes, the ambulance located at

that EMS can reach all the points in the specified quarter within 5 (8) minute time limit. The

response across the two sides of the city is not allowed because of the unpredictable traffic

conditions on the bridges.

Figure 6. Optimal solution of BDCM for Istanbul with 35 EMS stations

24

7.1. Results for BDCM

The number of EMS stations to be located in Istanbul was provided as 35 by DIRR for 2008.

CPLEX solved this problem to optimality in 38 seconds. We believe that this rather short

solution time is due to the fact that Istanbul data have certain characteristics that cannot be

seen in the random data. Firstly, the Asian and European sides are separated and in fact, the

problem can be decomposed into two sub-problems that can be solved independently if the

stations are allocated separately for each side. Secondly, the reachability data is very dense at

some central areas having many small quarters with high population whereas the opposite is

true in some areas where the quarters are large and sparsely populated. This characteristic

resembles to data type 3 except that there are more than two dense areas in Istanbul. Hence,

CPLEX is able to arrive at the optimal solution faster in this multiple clustered population

environment, in parallel with our observation in the experimental study.

Table 6. Comparison of the results for Istanbul data

K

Optimal TS2

% Coverage Time % Coverage %Dev Time

35 74.73 38 74.73 0.00 482 50 84.28 90 84.05 0.27 695 60 88.59 102 88.33 0.30 843 70 92.06 71 91.54 0.57 984

To solve the BDCM using Istanbul’s data we utilized TS2 approach and set the

iteration limit k3=1000 as earlier. TS2 finds the optimal solution for this problem. The results

are given in Table 6 and the optimal solution is illustrated in Figure 6. In this figure, yellow

colored zones show the Istanbul metropolitan area, striped zones are the three districts that are

excluded in the study, the red “♦” symbolizes the EMS station locations, and green colored

zones represent the double covered regions. In order to observe the sensitivity of the results to

different station numbers, we also solved the problem for K = 50, 60, and 70. These values, in

fact, reflect the expansion plans of DIRR in the next three years. The corresponding results

are included in Table 6 as well. We observe that the TS2 procedure provides near optimal

solutions. Nevertheless, the computation times of the fixed 1000 iterations are larger

compared to those of CPLEX. Note, however, that it is not our aim to compete with a state-of-

the-art software. We only use CPLEX to benchmark the performance of our approach.

25

Table 7. Computational results for Istanbul’s data

Scenario

Year

(t) Kt

CPLEX TS2

Population

covered

%

Coverage

Population

covered

%

Coverage

Real

1 35 8,663,839 74.21 8,637,091 73.98 2 50 9,832,316 84.22 9,761,408 83.61 3 60 10,321,413 88.41 10,238,983 87.70 4 70 10,701,764 91.67 10,677,445 91.46

1

1 30 8,162,343 69.92 8,145,641 69.77 2 40 9,128,446 78.19 9,085,618 77.82 3 50 9,801,607 83.96 9,772,468 83.71 4 60 10,326,157 88.45 10,309,793 88.31

2

1 40 9,092,654 77.88 9,082,764 77.80 2 50 9,811,566 84.04 9,798,521 83.93 3 60 10,328,737 88.47 10,309,439 88.31 4 70 10,707,822 91.72 10,688,654 91.55

3

1 45 9,448,031 80.93 9,417,091 80.66 2 55 10,092,226 86.45 10,048,945 86.08 3 65 10,550,610 90.37 10,518,665 90.10 4 75 10,894,939 93.32 10,839,259 92.84

7.2. Results of the MPBDCM for Istanbul

The number of stations that can be operated in each year (2008, 2009, 2010 and 2011) is

provided by DIRR as Kt=(35,50,60,70). However, we solved MPBDCM with three additional

Kt scenarios to observe the sensitivity of the results to these values. In our TS2

implementation rather than setting the number of iterations, we run both TS2 and CPLEX for

1200 seconds. Note that in none of the problems, CPLEX was able to reach to optimality

within this time limit. The results are given in Table 5 and the TS2 solution for the real data is

illustrated in Figure 7. In this figure, , , , and symbolize the locations where

EMS stations are opened in years 1, 2, 3, and 4, respectively. Brown colored areas show the

districts excluded from the analysis. Green and yellow colored areas represent backup double

covered and uncovered regions in the specified period, respectively.

In Table 7, we observe that the results found by CPLEX are slightly better than the

results obtained using TS2 due to the fact that Istanbul’s certain special characteristics

desribed above.

26

(a) Year 1

(b) Year 2

(c) Year 3

(d) Year 4

Figure 7. TS solution for the 4-year location planning of EMS stations in Istanbul

27

8. CONCLUSION AND FUTURE RESEARCH

In this paper, we present the Single-Period Backup Double Covering Model and Multi-Period

Backup Double Covering Model to plan the locations of EMS stations. Since the models are

intractable for large-scale cases, we propose a TS solution approach. To obtain an initial

solution to the TS procedure, we experiment three different initialization methods with

different computational complexity on BDCM: random, steepest-ascent, and LP relaxation-

based methods. Then we utilize the steepest-ascent method in solving MPBDCM since it

provides good intial results fast. We test the performance of the TS algorithms on randomly

generated data as well as the data we collected for Istanbul. The results show that our TS

approach provides optimal/near optimal solutions for both models based on the comparisons

performed against the OPL Studio 5.5 with CPLEX 11.0. The advantage of using the TS

approach is two-fold: first, it can efficiently solve problems which consider larger numbers of

potential locations and achieve better coverage of population than that can be obtained by

reducing the size of potential locations and obtaining the optimal solution through OPL (or

any other IP solver). Second, even if OPL solves the problem optimally TS may still be

preferable since it is an easy to use generic code and accessible to everyone while OPL is an

expensive licensed program.

Our models aim only at maximizing the double covered population. However, other

requirements and restrictions may be imposed depending on the planning environment and the

TS approach may be modified accordingly. In this study, we assumed that all stations are

equivalent. However, the cost of opening and operating EMS stations of different sizes and in

different locations will be different due to the construction, personnel, ambulance costs as

well as the cost of land. Thus, a multi-objective model may be developed to consider these

costs in addition to the coverage of population. A more robust planning may include the

planning of the ambulances and their crews.

28

APPENDIX A. Solutions of BDCM for geographical distribution type 1

OPL Solution TS1 TS2 TS3

||||M|||| ||||N|||| K OFV

CPU

Time*

%

Dev

CPU

Time

%

Dev

CPU

Time % Dev

CPU

Time

200 200

15 143799 27.78 0.01 81.83 0.01 82.64 0.01 104.78

20 181996 6.51 0.38 108.78 0.00 108.55 0.00 133.95

25 199117 4.56 0.00 126.20 0.00 126.25 0.00 144.09

30 203783 4.84 0.00 140.88 0.00 141.17 0.00 161.53

200 150

15 140098 6.01 0.00 60.31 0.00 60.47 0.00 84.38

20 172243 5.28 0.00 80.23 0.00 80.95 0.00 89.08

25 191763 3.56 0.00 93.20 0.00 93.25 0.00 106.38

30 201323 3.31 0.12 100.36 0.05 100.66 0.00 102.42

200 100

15 136441 4.84 0.00 38.44 0.00 38.88 0.00 48.58

20 168349 4.28 0.00 50.69 0.00 50.97 0.00 61.84

25 190357 2.78 0.00 58.16 0.00 57.98 0.00 63.92

30 197638 2.84 0.00 63.02 0.00 63.25 0.00 67.00

200 50

15 123343 2.75 0.00 16.95 0.00 17.06 0.00 21.72

20 152191 2.31 0.00 21.75 0.00 21.77 0.00 27.67

25 168491 2.54 0.00 24.02 0.00 24.05 0.00 24.97

30 174802 2.32 0.00 25.45 0.00 25.56 0.00 26.44

300 300

20 223396 244.84 0.29 247.69 0.57 248.30 0.42 323.63

25 260263 124.92 0.29 309.09 0.57 314.03 0.95 387.94

30 287068 124.14 0.54 360.03 0.62 356.63 0.02 433.19

35 304948 75.34 0.63 393.75 0.45 394.92 0.38 501.64

300 225

20 211110 28.26 0.00 183.05 0.00 183.36 0.00 248.44

25 246565 16.03 0.00 232.80 0.00 233.59 0.00 281.56

30 274764 7.09 0.15 270.80 0.15 269.97 0.15 313.34

35 291795 8.10 0.13 297.02 0.07 303.39 0.10 342.72

300 150

20 200086 11.07 0.00 115.91 0.00 117.22 0.00 130.70

25 239045 5.06 0.00 147.98 0.00 147.23 0.00 168.17

30 266266 5.32 0.00 171.36 0.03 172.77 0.00 185.20

35 283656 4.07 0.00 191.09 0.13 193.92 0.00 206.53

300 75

20 184908 4.09 0.00 53.56 0.00 53.02 0.00 59.58

25 219793 3.03 0.00 66.23 0.00 64.89 0.00 68.25

30 244597 3.29 0.00 74.73 0.00 72.14 0.00 78.08

35 263743 3.56 0.00 78.75 0.00 76.33 0.00 199.69

29

400 400

20 224756 600.00 -0.28 406.89 -0.28 409.41 -0.28 563.30

30 304207 600.00 -0.24 682.28 -0.60 687.20 -0.27 888.03

40 371986 600.00 0.32 868.95 0.27 867.53 0.47 1061.44

50 414350 402.29 0.45 993.80 0.57 1010.30 0.58 1089.84

400 300

20 215202 44.59 0.22 288.25 0.00 292.78 0.00 438.17

30 293598 76.31 0.46 494.11 0.61 506.70 0.30 656.50

40 353921 45.76 0.24 637.70 0.21 647.27 0.62 741.92

50 397890 13.60 0.03 728.11 0.37 740.50 0.22 777.17

400 200

20 209740 11.79 0.17 194.75 0.24 187.09 0.39 274.64

30 284549 9.35 0.37 314.09 0.37 316.50 0.00 401.36

40 342616 9.60 0.00 402.86 0.00 400.81 0.00 448.11

50 386531 6.57 0.29 452.86 0.29 459.97 0.38 467.97

400 100

20 197095 5.35 0.00 84.62 0.00 84.34 0.00 93.83

30 265797 3.78 0.00 135.73 0.00 136.47 0.00 141.98

40 319101 4.50 0.00 167.91 0.00 168.00 0.00 171.33

50 357477 4.53 0.00 179.70 0.00 180.33 0.00 449.80

500 500

30 311188 600.00 -0.03 1059.40 0.57 1047.44 -0.58 1414.27

40 388461 600.00 -0.48 1417.66 0.21 1430.14 -0.37 1826.59

50 455336 600.00 0.45 1678.95 0.50 1674.42 0.34 2052.98

60 497061 600.00 0.42 1867.59 0.33 1864.28 0.24 2113.00

500 375

30 299447 600.00 -0.09 772.38 -0.09 765.61 0.25 1020.47

40 376199 600.00 0.37 1040.28 0.13 1039.39 -0.09 1253.61

50 442540 96.75 0.17 1231.67 0.19 1232.73 0.17 1355.70

60 486812 23.60 0.45 1369.39 0.48 1369.88 0.32 1548.98

500 250

30 292636 101.31 0.34 487.69 0.04 489.20 0.20 605.83

40 368549 162.01 0.00 629.30 0.34 654.30 0.03 771.65

50 429430 66.92 0.00 780.19 0.01 782.30 0.00 859.10

60 472568 10.78 0.10 859.61 0.45 856.45 0.25 950.14

500 125

30 282310 6.07 0.00 218.33 0.00 217.36 0.05 252.68

40 354614 5.01 0.00 283.48 0.00 283.66 0.00 309.00

50 407915 6.31 0.37 325.03 0.00 325.05 0.00 341.42

60 445166 5.34 0.00 341.39 0.00 341.06 0.00 356.52

Average 281200 113.70 0.10 401.23 0.12 402.56 0.08 482.73 * A value of 600 seconds indicates that OPL could not find the optimal solution. The OFV reported is the best integer solution.

30

APPENDIX B. Solutions of BDCM for geographical distribution type 2

OPL Solution TS1 TS2 TS3

||||M|||| ||||N|||| K OFV

CPU

Time* % Dev

CPU

Time

%

Dev

CPU

Time % Dev

CPU

Time

200 200

15 171366 6.09 0.47 80.39 0.00 80.50 0.06 100.89

20 191066 6.56 0.27 99.59 0.00 99.20 0.00 119.75

25 202226 4.60 0.10 115.42 0.45 116.48 0.10 133.17

30 203824 3.79 0.00 128.73 0.00 129.34 0.00 134.33

200 150

15 165586 5.07 0.00 60.03 0.00 60.44 0.00 76.00

20 188164 3.26 0.00 75.44 0.00 76.09 0.00 82.67

25 198713 3.06 0.03 87.08 0.00 87.64 0.03 88.75

30 200878 3.10 0.00 94.47 0.00 94.59 0.00 96.64

200 100

15 163023 4.01 0.00 38.30 0.00 38.30 0.00 47.20

20 186515 2.54 0.00 48.25 0.00 48.42 0.00 52.14

25 197734 2.76 0.00 54.42 0.00 54.55 0.00 56.53

30 200782 2.51 0.00 59.72 0.00 59.83 0.00 60.59

200 50

15 144415 2.51 0.00 17.25 0.00 17.23 0.00 23.16

20 162616 2.28 0.00 21.05 0.00 21.13 0.00 25.94

25 172223 2.29 0.00 22.72 0.00 22.70 0.00 24.73

30 172238 2.56 0.00 23.44 0.00 23.64 0.00 24.59

300 300

20 249902 40.07 0.76 231.75 0.00 233.86 0.74 295.45

25 277422 43.60 0.17 287.83 0.52 287.91 0.78 357.70

30 296912 16.78 0.31 335.47 0.40 335.58 0.00 418.42

35 308272 17.51 0.26 369.28 0.10 367.86 0.03 460.11

300 225

20 234277 19.06 0.26 166.91 0.08 166.81 0.32 235.38

25 260969 10.07 0.26 209.75 0.31 210.44 0.00 252.78

30 281546 7.07 0.00 248.31 0.00 248.48 0.00 260.25

35 294690 6.59 0.05 269.59 0.27 270.52 0.01 301.34

300 150

20 230181 9.34 0.06 107.92 0.06 107.13 0.13 139.75

25 252965 7.07 0.19 134.66 0.19 134.69 0.21 160.06

30 271144 4.85 0.05 155.64 0.01 155.50 0.05 173.36

35 284595 4.57 0.00 171.19 0.17 170.73 0.00 186.22

300 75

20 219047 3.84 0.00 51.41 0.00 51.13 0.00 60.78

25 237619 3.32 0.00 62.33 0.00 62.05 0.00 66.78

30 253245 3.26 0.00 69.64 0.00 69.80 0.00 71.42

35 262452 3.57 0.00 72.75 0.00 72.97 0.00 175.06

31

400 400

20 288228 600.00 0.98 427.67 0.00 424.22 0.00 569.38

30 357225 600.00 0.21 649.69 0.35 644.25 0.01 806.98

40 399661 109.31 0.60 798.61 0.63 802.89 0.63 925.64

50 420422 21.12 0.37 921.53 0.34 928.44 0.21 1021.05

400 300

20 280778 45.23 0.00 310.20 0.00 313.53 0.13 437.38

30 347483 93.31 0.17 469.11 0.17 471.03 0.34 568.42

40 392716 18.64 0.23 589.47 0.45 583.61 0.33 651.88

50 415562 11.14 0.14 672.61 0.09 674.97 0.03 750.45

400 200

20 277845 32.79 0.15 199.14 0.43 197.28 0.15 274.55

30 338887 56.59 0.00 305.41 0.00 308.30 0.01 398.53

40 383054 19.45 0.25 373.08 0.18 372.58 0.23 405.53

50 409321 9.12 0.00 420.73 0.03 418.33 0.00 444.70

400 100

20 267473 8.28 0.06 91.86 0.00 92.08 0.00 112.88

30 329276 6.39 0.22 128.64 0.16 128.25 0.23 133.14

40 367542 4.32 0.00 154.38 0.00 153.50 0.00 156.73

50 382899 4.57 0.00 166.52 0.00 166.38 0.00 406.17

500 500

30 378984 600.00 0.34 996.44 0.25 987.75 0.96 1341.03

40 433805 600.00 0.54 1319.91 1.00 1308.39 0.30 1680.27

50 478138 600.00 0.72 1591.47 0.81 1630.59 0.75 1928.25

60 497942 84.85 0.54 1786.19 0.61 1809.28 0.57 2019.05

500 375

30 368249 275.78 0.14 737.08 0.67 734.86 0.00 915.06

40 423618 63.60 0.18 957.41 0.19 960.34 0.59 1135.09

50 460266 20.79 0.66 1159.09 0.44 1151.41 0.27 1297.98

60 486184 14.56 0.32 1294.16 0.33 1290.06 0.00 1465.16

500 250

30 363210 29.75 0.00 456.80 0.75 457.63 0.13 604.42

40 417640 18.28 0.43 601.27 0.00 601.66 0.39 716.50

50 454390 12.87 0.33 706.92 0.00 712.22 0.16 727.84

60 480012 9.64 0.00 803.14 0.17 801.03 0.16 858.84

500 125

30 341980 6.90 0.00 203.45 0.34 202.08 0.08 268.91

40 390890 5.51 0.00 256.91 0.00 255.42 0.00 275.42

50 424959 6.04 0.00 297.20 0.00 295.05 0.00 300.11

60 447382 5.81 0.00 317.27 0.00 315.97 0.00 320.67

Average 306916 66.44 0.17 377.09 0.17 377.64 0.14 448.12

32

APPENDIX C. Solutions of BDCM for geographical distribution type 3

OPL Solution TS1 TS2 TS3

||||M|||| ||||N|||| K OFV

CPU

Time* % Dev

CPU

Time

%

Dev

CPU

Time % Dev

CPU

Time

200 200

15 163804 5.85 0.00 82.03 0.00 82.02 0.00 101.75

20 186315 7.60 0.14 106.41 0.38 106.55 0.54 138.48

25 199601 6.82 0.66 124.02 0.35 123.28 0.35 156.02

30 203824 4.03 0.00 136.33 0.00 136.36 0.00 151.61

200 150

15 155858 6.09 0.00 61.16 0.00 60.45 0.33 82.81

20 179463 4.50 0.00 78.50 0.00 78.44 0.00 88.34

25 195710 4.81 0.00 89.39 0.00 89.58 0.00 94.83

30 203448 3.75 0.01 97.61 0.00 97.58 0.00 101.31

200 100

15 152150 4.84 0.21 38.30 0.34 38.30 0.31 46.44

20 176268 4.03 0.27 48.70 0.16 48.72 0.29 57.66

25 192319 3.81 0.04 55.86 0.00 56.20 0.00 63.00

30 202000 2.81 0.00 61.30 0.00 61.17 0.00 62.86

200 50

15 140012 2.51 0.00 17.25 0.00 17.13 0.33 19.06

20 162294 2.71 0.00 21.17 0.13 21.13 0.00 24.91

25 179666 2.81 0.00 22.89 0.00 22.97 0.00 24.81

30 188327 2.53 0.00 23.53 0.00 23.47 0.00 25.50

300 300

20 236271 172.39 0.14 241.58 0.00 240.25 0.00 308.64

25 267345 19.92 0.33 303.52 0.00 302.98 0.00 373.70

30 289241 11.90 0.88 346.11 0.63 346.28 0.35 428.94

35 305228 9.09 0.39 375.92 1.00 379.56 0.00 421.53

300 225

20 222954 16.37 0.07 172.22 0.51 173.64 0.07 220.52

25 253660 10.70 0.00 218.38 0.00 219.75 0.00 257.91

30 273811 12.45 0.16 255.39 0.03 254.64 0.09 312.19

35 292600 7.34 0.05 279.31 0.07 279.16 0.26 302.94

300 150

20 216185 8.90 0.00 112.38 0.00 112.52 0.01 138.30

25 245814 6.59 0.00 140.70 0.00 140.39 0.00 164.36

30 266637 6.32 0.00 161.42 0.00 160.91 0.00 191.98

35 284290 5.70 0.15 175.67 0.00 175.92 0.27 182.31

300 75

20 203715 3.56 0.00 50.98 0.00 51.05 0.00 58.83

25 227240 3.32 0.00 63.75 0.00 64.08 0.00 67.72

30 245060 3.79 0.00 72.45 0.00 72.00 0.00 73.63

35 260582 3.31 0.00 75.97 0.00 75.53 0.00 171.70

33

400 400

20 283477 136.03 0.03 432.06 0.14 432.48 0.10 567.05

30 351208 600.00 0.45 660.48 0.45 663.80 0.93 827.55

40 399178 600.00 0.28 817.72 0.03 818.56 0.24 974.88

50 419467 426.85 0.38 940.61 0.50 945.56 0.38 1044.83

400 300

20 271276 23.26 0.05 310.51 0.00 314.34 0.00 430.77

30 337229 106.01 0.49 479.52 0.82 477.72 0.63 556.16

40 392772 14.35 0.56 590.75 1.15 576.48 0.02 633.00

50 415001 9.06 0.29 666.61 0.14 658.38 0.08 737.33

400 200

20 266215 9.85 0.00 196.95 0.00 197.30 0.00 275.23

30 330533 10.65 0.00 306.55 0.16 306.58 0.42 316.88

40 380561 7.32 0.51 375.70 0.56 378.55 0.00 397.36

50 409594 8.73 0.06 428.61 0.06 430.38 0.10 465.86

400 100

20 249080 4.01 0.00 92.03 0.00 91.91 0.00 121.73

30 308866 5.82 0.00 136.03 0.00 135.08 0.08 140.05

40 355454 5.03 0.00 159.59 0.00 159.47 0.00 166.59

50 378518 4.07 0.00 176.25 0.00 176.74 0.00 346.30

500 500

30 371928 600.00 1.23 1021.75 1.06 1026.34 0.58 1328.42

40 430904 600.00 0.26 1377.86 0.92 1380.53 0.12 1602.44

50 477936 600.00 0.46 1648.88 0.36 1639.36 0.36 1915.09

60 498766 93.56 0.35 1825.63 0.56 1818.84 0.23 1979.41

500 375

30 357526 26.20 0.00 735.88 0.00 735.59 0.08 905.05

40 410106 19.53 0.99 998.30 0.31 1004.81 0.26 1183.97

50 451464 17.67 0.48 1228.44 0.33 1236.81 0.77 1303.45

60 478222 16.04 0.29 1358.16 0.40 1360.27 0.07 1464.14

500 250

30 346555 20.43 0.27 471.42 0.44 469.13 0.44 558.58

40 397479 17.37 0.01 632.31 0.09 630.64 0.30 703.09

50 440291 13.15 0.40 764.13 0.23 767.80 0.13 819.39

60 468022 12.70 0.08 856.61 0.00 851.09 0.00 901.95

500 125

30 328346 7.10 0.00 209.59 0.35 208.64 0.35 224.98

40 379386 6.04 0.00 275.03 0.00 275.17 0.00 289.66

50 418272 5.09 0.00 319.45 0.00 319.41 0.00 325.06

60 442174 7.06 0.00 335.84 0.00 335.08 0.00 339.66

Average 300273 68.81 0.18 389.68 0.20 389.61 0.15 449.38

34

APPENDIX D. Solutions of MPBDCM for geographical distribution type 1

OPL Solution TS

M N Kt OFV

CPU

Time

%

Dev

CPU

Time

200 200

12,18,21,24 664082 900.00 0.04 154.06

10,16,20,26 624125 900.00 0.17 153.97

6,10,16,22 497920 900.00 1.68 117.33

200 150

5,10,16,20 454896 900.00 0.12 80.86

11,14,20,25 592985 900.00 0.09 111.87

13,19,22,26 655664 900.00 -0.13 123.42

200 100

8,13,16,21 508325 900.00 0.35 57.69

14,19,25,28 669630 242.78 0.30 83.27

5,9,15,20 426690 900.00 -0.20 49.61

200 50

5,11,14,19 400416 18.07 1.06 22.38

7,12,18,23 464264 19.87 0.00 27.56

11,15,20,24 531020 12.03 0.13 31.05

300 300

17,24,27,33 993514 1800.00 -0.57 483.15

15,23,31,36 993678 1800.00 -0.54 513.05

16,25,34,37 1028901 1800.00 -0.48 533.80

300 225

15,23,29,37 940757 900.00 -0.92 373.84

11,16,23,31 793761 900.00 -0.58 294.25

17,24,27,30 942031 900.00 -0.24 335.92

300 150

15,21,28,35 893299 900.00 0.29 230.30

14,19,24,27 812196 900.00 0.20 183.22

10,13,17,24 650215 900.00 0.55 141.27

300 75

16,24,31,35 876019 25.39 0.00 108.64

12,15,23,28 709353 23.06 0.14 81.92

16,24,27,33 850202 40.76 0.08 102.53

400 400

21,26,32,39 1149377 1800.00 -2.25 996.35

23,33,39,47 1275855 1800.00 -3.97 1210.03

18,29,33,41 1151319 1800.00 -2.62 1022.03

400 300

22,33,44,53 1311283 1800.00 0.70 973.86

21,28,35,45 1191729 1800.00 -0.41 824.84

19,29,33,39 1138394 1800.00 -0.17 770.47

400 200

15,26,35,46 1098953 900.00 0.55 518.50

18,27,31,38 1072306 900.00 0.36 451.01

23,32,43,49 1261673 900.00 -0.01 598.20

400 100

15,20,32,44 965867 41.60 0.33 209.44

24,31,43,49 1181145 22.09 0.23 263.17

20,28,39,49 1114730 53.95 0.00 250.77

35

500 500

26,31,42,56 1393943 1800.00 -3.64 1908.25

20,25,30,40 1119239 1800.00 -3.92 1531.86

20,31,36,43 1233703 1800.00 -4.43 1742.59

500 375

22,31,40,49 1303229 1800.00 -1.49 1450.94

21,29,38,46 1258808 1800.00 -0.43 1359.52

25,34,41,48 1335593 1800.00 -3.44 1486.85

500 250

30,37,45,56 1479398 900.00 -0.62 1074.22

26,37,47,60 1473407 900.00 -0.15 1109.53

22,33,40,46 1296753 900.00 -0.02 879.87

500 125

27,32,38,53 1312838 132.50 0.27 437.47

23,31,36,41 1198884 126.96 0.52 368.23

30,40,45,59 1456724 50.90 0.15 499.92

Average 973939 954.37 -0.48 548.60

* A value of 900 and 1800 seconds indicates that OPL could not find the optimal solution. The OFV reported is the best integer solution.

36

APPENDIX E. Solutions of MPBDCM for geographical distribution type 2

OPL Solution TS

M N K (t) OFV

CPU

Time % Dev

CPU

Time

200 200

15,19,25,28 762180 278.07 0.62 166.69

14,17,20,24 732181 900.00 0.31 146.88

14,19,23,28 751125 900.00 0.09 163.66

200 150

5,8,11,16 509305 900.00 -0.10 65.72

9,13,18,22 652703 161.14 0.00 97.36

9,15,20,25 675786 294.31 0.00 107.36

200 100

13,17,20,26 709014 16.34 0.00 73.31

10,13,17,20 645853 19.31 0.00 59.31

12,18,24,29 719603 17.12 0.00 80.12

200 50

8,13,16,21 554559 21.93 0.00 26.91

11,16,22,28 614123 10.12 0.00 33.86

11,15,19,22 596190 11.09 0.01 29.83

300 300

16,20,28,34 1062967 1800.00 1.09 444.50

10,13,17,20 838436 1800.00 0.17 270.34

16,20,26,30 1039159 1800.00 -0.41 408.20

300 225

16,21,29,33 1011372 900.00 0.18 325.69

12,18,24,32 938055 900.00 0.01 301.53

17,25,31,38 1055414 900.00 0.26 368.80

300 150

17,21,24,27 956009 498.60 0.57 181.66

12,21,30,38 966117 900.00 0.35 219.97

20,27,30,38 1049674 102.98 0.05 233.47

300 75

12,18,21,27 846448 28.60 0.04 80.20

14,21,29,34 919504 12.34 0.03 99.22

14,18,21,26 860011 25.73 0.07 80.47

400 400

18,27,32,40 1352608 1800.00 0.23 969.25

20,27,38,45 1408138 1800.00 -0.14 1080.02

16,27,35,42 1344542 1800.00 -0.38 997.02

400 300

19,23,27,31 1233217 1800.00 -1.03 590.37

19,24,28,38 1290155 1800.00 -0.32 675.59

20,29,36,42 1380294 1800.00 0.32 768.62

400 200

24,33,41,51 1447463 900.00 0.52 583.28

25,30,37,46 1412716 900.00 -0.12 534.87

16,24,30,41 1260691 900.00 -0.09 455.52

400 100

21,25,35,46 1301799 87.84 0.06 228.75

22,31,39,50 1358311 83.29 0.01 251.25

17,25,36,46 1272775 353.65 0.77 224.67

37

500 500

29,37,43,57 1673219 1800.00 -2.43 2191.97

22,34,39,51 1574777 1800.00 -2.27 1939.70

23,32,39,46 1579086 1800.00 0.03 1792.49

500 375

29,34,41,48 1621454 1800.00 -0.29 1420.06

28,37,46,61 1683903 1800.00 0.32 1672.98

22,34,46,58 1601596 1800.00 -0.53 1583.27

500 250

24,39,41,48 1595029 900.00 0.10 872.23

23,33,38,49 1535222 900.00 0.12 846.95

22,30,45,60 1572688 900.00 0.41 983.30

500 125

30,35,49,59 1577070 30.53 0.13 460.17

21,28,33,44 1370856 107.54 0.78 347.78

30,44,56,65 1638666 26.09 0.24 508.12

Average 1136501 851.80 0.00 542.57

38

APPENDIX F. Solutions of MPBDCM for geographical distribution type 3

OPL Solution TS

M N K (t) OFV

CPU

Time

%

Dev

CPU

Time

200 200

6,10,14,17 535918 900.00 -0.12 101.23

5,11,15,21 547281 900.00 -0.34 114.83

6,9,13,16 512518 900.00 0.13 101.59

200 150

13,19,25,29 714592 134.95 0.00 127.77

12,17,21,24 676820 692.79 0.05 111.03

10,13,18,24 622755 900.00 -0.04 104.75

200 100

5,9,13,16 475693 377.48 0.00 43.81

8,14,20,24 606743 900.00 0.07 65.53

11,15,18,23 628394 900.00 0.04 66.28

200 50

12,18,21,27 624668 11.31 0.00 34.28

5,9,12,16 433656 11.42 0.39 20.22

12,15,20,23 595144 13.00 0.00 31.00

300 300

13,18,27,31 946080 1800.00 -1.47 418.17

18,23,28,32 1045133 1800.00 0.70 457.58

13,20,25,34 975830 1800.00 0.47 440.28

300 225

18,27,32,40 1051962 900.00 0.15 391.20

13,17,26,33 910011 900.00 0.15 308.16

13,20,29,33 945210 900.00 0.37 324.55

300 150

15,24,27,32 945279 900.00 0.18 210.91

11,16,19,22 774737 900.00 0.28 146.83

14,19,22,30 874129 900.00 0.05 186.69

300 75

16,25,32,36 922229 9.40 0.00 105.41

14,17,26,35 847054 9.56 0.00 92.62

12,17,22,30 798919 9.65 0.05 83.94

400 400

19,29,33,38 1289322 1800.00 -4.82 957.69

21,30,35,40 1379788 1800.00 -1.27 1021.39

16,24,33,42 1289322 1800.00 -1.30 996.47

400 300

18,26,37,44 1327737 1800.00 0.47 768.78

19,26,37,44 1322902 1800.00 -0.31 770.28

24,34,40,50 1448035 1800.00 0.10 898.55

400 200

24,31,41,47 1407394 900.00 0.73 546.55

23,33,41,50 1416856 900.00 0.69 565.91

18,28,38,42 1313738 900.00 0.11 485.09

400 100

23,27,39,48 1282556 30.01 0.00 244.08

19,23,34,42 1193778 26.75 0.13 210.86

25,34,41,48 1341362 10.42 0.01 250.47

39

500 500

22,29,35,50 1527222 1800.00 0.06 1871.81

21,30,43,57 1562472 1800.00 -1.63 2139.94

30,45,52,61 1779809 1800.00 -0.27 2504.25

500 375

22,37,42,51 1561084 1800.00 0.04 1493.78

29,35,42,53 1611786 1800.00 0.23 1542.05

28,38,50,55 1658133 1800.00 0.55 1647.69

500 250

30,43,57,64 1685074 900.00 0.61 1152.37

23,36,46,60 1561193 900.00 0.44 1046.64

23,32,37,50 1472266 900.00 0.21 871.69

500 125

27,34,47,54 1495893 27.67 0.09 448.66

20,34,41,48 1410702 32.10 0.31 400.66

30,45,50,62 1589951 7.90 0.06 511.69

Average 1102898 929.26 -0.08 571.58

40

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