Journal of the Operational Research Society

, Volume 62, Issue 4, pp 627–637

A multi-period double coverage approach for locating the emergency medical service stations in Istanbul

Authors

  • A Başar
    • Sabanci University
  • B Çatay
    • Sabanci University
    • Sabanci University
Case-oriented Paper

DOI: 10.1057/jors.2010.5

Cite this article as:
Başar, A., Çatay, B. & Ünlüyurt, T. J Oper Res Soc (2011) 62: 627. doi:10.1057/jors.2010.5
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Abstract

We consider the multi-period location planning problem of emergency medical service (EMS) stations. Our objective 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 to develop a strategic plan that spans multiple periods. In order to solve this problem, we propose a Tabu Search approach. 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.

Keywords

maximal coveringmulti-period backup double coveringTabu Search

1. Introduction

It is very critical to optimally plan the locations of emergency medical service (EMS) stations, especially in heavily populated cities, since such decisions directly affect the extent of fatalities and injuries of the inhabitants. In the last 30 years, various models with different objective functions and constraints have been proposed in the academic literature, and various heuristic and metaheuristic methods have been developed to solve them efficiently. Typically, the objective function is to maximize a certain type of coverage with respect to constraints that restrict the total number of EMS stations and ensure a certain level of service quality. This problem is known as the maximal covering problem. An extensive review of the studies that address different variants of this problem may be found in Brotcorne et al (2003) and Goldberg (2004).

In this paper, we consider a new variant of the problem where we plan the locations of EMS stations over multiple periods with two types of service requirements. Essentially, our problem is to determine the optimal locations of EMS stations over a planning horizon of a number of periods with a given maximum number of stations that can be opened in each period. 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, the model 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. Therefore, 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 a station being busy when a call comes is relatively large.

This framework was motivated by a project conducted for the Directorate of Instant Relief and Rescue (DIRR) at the Istanbul Metropolitan Municipality (IMM). The project involved the investigation of the current service levels and the determination of the locations of EMS stations that IMM would open in the next few years under several scenarios. Within this scope, we present in this paper a mathematical model addressing the above-described situation and develop an efficient Tabu Search (TS) solution approach. The model parameters, the objective function and the constraints were determined according to the requirements specified by the authorities at DIRR. We will later describe the process that we went through for data collection and modelling along with the results obtained for Istanbul.

Single coverage problems have been extensively studied in the ambulance location literature, and various definitions of the term coverage have been introduced (eg Church and ReVelle, 1974; Daskin, 1983; Marianov and ReVelle, 1995). If a region is covered by a single EMS station, that region would not be able to receive any service if the team in that EMS station is busy responding to another call. Brotcorne et al (2003) provide a good review of various papers addressing this issue. Among them, Hogan and ReVelle (1986) present two backup coverage models (BACOP1 and BACOP2) and Gendreau et al (1997) propose the double standard model (DSM) planning the EMS stations in Montreal. BACOP1 maximizes the demand covered twice within the same coverage standard, whereas BACOP2 maximizes the weighted average of the demands covered once and covered twice, also within the same coverage standard. DSM, on the other hand, considers two coverage standards, t1 and t2 time units where t1<t2, and maximizes the demand covered twice within the coverage standard t1. DSM also limits the number of ambulances at each site and ensures that all demand is covered within t2 coverage standard and a certain proportion of the demand is covered within t1 coverage standard. Gendreau et al (1997) successfully applied a TS algorithm to solve this problem and obtained good results fast in comparison with the solutions obtained using a branch and bound algorithm with a limited number of iterations. Recently, Doerner et al (2005) have proposed Ant Colony Optimization to plan the EMS stations in Austria and compared its performance to that of TS of Gendreau et al (1997). Their experiments revealed that TS could find better results with less computational effort, especially in large problems. Harewood (2002) discusses the planning of EMS stations in Barbados using simulation techniques.

In this paper, we present a multi-period backup double covering model (namely, MPBDCM). The proposed model is a multi-period variant of BACOP1 and DSM; however, the mathematical formulation is quite different apart from being simply a multi-period extension. Multi-period location models have been widely studied and also referred to as dynamic location models in the literature, within a different framework though. For instance, many dynamic location models focus on several parameters that are likely to change during the operation of the located facility. The objective is to minimize the total cost over the planning horizon while making sure that the demand in each period is satisfied, no community is assigned to a closed facility, and the re-establishment of a facility that has been removed and the removal of a facility that has been established earlier during the planning horizon are avoided (Da Gama and Captivo, 1998). Two general problem types include the single-facility dynamic location problem (eg Ballou, 1968; Lodish, 1970; Wesolowsky, 1973; Drezner and Wesolowsky, 1991; Bastian and Volkmer, 1992; Andreatta and Mason, 1994) and multi-facility dynamic location problem (eg Scott, 1971; Warszawski, 1973; Wesolowky and Truscott, 1975; Rosenblatt, 1986; Frantzeskakis and Watson-Gandy, 1989; Chardaire et al, 1996). Drezner (1995) introduces the dynamic p-median problem. The objective is to minimize the total transportation costs while allowing changes in demand over time. Current et al (1997) address the dynamic location of multiple facilities when the total number of facilities is uncertain during the planning horizon. They propose two different objectives: minimization of expected opportunity loss and maximum regret.

More recently, relocation and redeployment models have been discussed in the literature as another type of dynamic location models (Repede and Bernardo, 1994; Gendreau et al, 2001; Rajagopalan et al, 2008). These models deal with operational level decisions depending on the changes in demand over time. Owing to demand fluctuations, relocation and redeployment decisions are generally revised daily or hourly in an attempt not to leave any demand point uncovered. Our study is different in the sense that we try to strategically determine the optimal locations of EMS stations over a planning horizon consisting of a number of periods, where each period is relatively long.

Although there are many dynamic models discussed in the literature, only Schilling (1980) attacks the multi-period location planning problem in a similar setting as ours. This paper proposes a model that considers the single coverage of EMS stations and the coverage in different periods are associated with weights representing the relative importance of the corresponding period. If a station is sited in a period, it cannot be closed in the following periods, as is the case in our study. Schilling presents a heuristic approach to find non-dominated solutions and investigates its performance using a two-period example.

In this paper, we formulate the MPBDCM as an integer program. A similar model for the single-period setting and some preliminary results are reported in Başar et al (2009b). Since MPBDCM is intractable for large instances, we propose a TS procedure to solve it efficiently. The performance of the proposed TS approach is investigated on randomly generated data with various characteristics. 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 multi-period backup double coverage model for locating EMS stations, the development of a TS approach to solve this problem and the application of this approach to a case study for the city of Istanbul.

The remainder of this paper is organized as follows. Section 2 is dedicated to the formulation of MPBDCM. Section 3 describes the proposed TS approach. The experimental study and the results are presented in Section 4. Section 5 depicts the case study carried out with IMM. Finally, the concluding remarks and future research directions are provided in Section 6.

2. Model formulation

The proposed MPBDCM incorporates 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. 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. We assume that if a demand region is double covered, then all the population in that region is double covered. Moreover, DIRR requires an EMS station opened in a certain period to remain open in the subsequent periods. This is a reasonable assumption when the costs associated with closing and reopening EMS stations are relatively large. Besides, it may not be socially easy to close an EMS station since it serves as a focus of community activity and provides a sense of security to residents (Badri et al, 1998). 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: illustration
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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 t1 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.

In the multiple coverage models in the literature typically 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 eg Brotcorne et al, 2003). It is possible to formulate our problem in a similar manner. However, this would bring in |T||M| more binary variables and |T||M| more constraints into the model, which are not necessary in our formulation.

We can prove that MPBDCM is NP-hard by considering a special case of the problem where t2 is sufficiently large such that each potential location site i can cover all demand regions in t2 time units and there is only one period. Then, the problem reduces to Maximal Covering Location Model (MCLM), which is known to be NP-hard (Berman and Krass, 2002).

One could argue that the objective function captures solely double coverage and the model may be biased to leave some regions totally uncovered since there are no ‘set-covering’ type constraints to guarantee the primary coverage of all regions in t1 or t2 time units. While Gendreau et al (1997) included such constraints in their DSM, nevertheless they pointed out that a feasible solution may not exist if the parameters are too restrictive. When the number of stations is limited and the single coverage of all regions is not possible, the coverage constraints may be relaxed, especially for remote lightly populated regions. In our case, DIRR has not specified any primary coverage requirements. However, our problem with Istanbul data would be infeasible if they did because their service time limits are quite tight. The numerical results on this issue are presented in Section 5.

Our model does not allow opening multiple stations in the same location since an additional EMS station will have no marginal contribution to the objective function value. Nonetheless, once the locations are determined by solving this model, the number of ambulances and equipment and personnel requirements may be determined at each location by taking into account various factors such as population sizes, age of the population, risk factors, touristic regions, industrial zones, etc. Furthermore, the objective function involves the simple sum of the coverages over the planning horizon. Although a weighted sum formulation may easily be incorporated, appropriate values for the weights need to be established. Note that the proposed TS approach may still be used in that case as well without any modifications. Finally, we observe that if we relax the requirement that once a station is opened it remains open throughout the whole planning period, constraints (5) disappear and the problem reduces to |T| single-period problems. As a matter of fact, these sub-problems can be efficiently solved with our approach by setting the number of periods to 1. The details of the single-period case may be found in Başar et al (2009a).

3. 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 neighbours 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 the 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. The interested reader is referred to Glover (1990) for a comprehensive tutorial.

3.1. Initialization

We investigated three initialization methods in our TS algorithm to observe their role on the overall solution quality: (i) a random method, (ii) a steepest-ascent method and (iii) a linear programming (LP) relaxation-based method. Our experiments showed that the steepest-ascent method is efficient in terms of both the solution quality and the computation time (detailed results can be found in Başar et al, 2009a). The steepest-ascent method is a greedy myopic approach where the stations are opened iteratively at locations that lead to largest improvement in the objective function value. When selecting the locations we examine the trade-off between opening one station and opening a pair of stations at once.

The details of the algorithm are as follows: First, a station is opened in the first period in the 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 corresponding location. We have also considered determining the best pair of stations that provides the largest coverage among all possible combinations; however, this requires significant computational effort in this and the following steps.

Next, we compare the improvement that can be achieved in the population double covered by opening two new stations consecutively using the above procedure and by adding one station only. In either case, we consider only the stations that provide the largest improvement and the comparison is performed on a per station basis. The alternative that gives the largest increase in the population double covered is accepted. 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. This procedure is repeated for all periods. The flowchart of the algorithm is given in Figure 1. In this figure, OFV refers to the objective function value and Covt1(Sτ) denotes the total population covered in t1 time units where (Sτ) is the set of open stations in the most recent period τ. OFVτ(S) is the OFV for solution S where S={S1,S2, …, Sτ}, That is, 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.
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Figure 1

Steepest-ascent procedure.

3.2. Description of the Tabu Search procedure

At each iteration, the TS procedure should maintain the feasibility of the number of stations with respect to constraint set (2) and the locations of the opened stations across time periods with respect to constraint set (5). Thus, if we can ensure that these two constraint sets are satisfied, constraint sets (3) and (4) will only be used to determine the double-covered regions and the objective function value can be computed accordingly. For this purpose, we propose a neighbourhood 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, that is, 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 only in periods τ, …, τ′−1.

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 that can be open in the corresponding period, and the numbers in the next three columns show the locations of the stations opened. In the neighbourhood 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 neighbourhood 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 to 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.
Table 1

Neighbourhood search structure

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Since we start with a feasible initial solution, we keep the feasibility with respect to constraints (2) by opening and closing one station. Alternatively, it is also possible to open multiple stations and close the same number of stations in a period without violating feasibility. However, this will increase the computational burden significantly. In addition, our preliminary experiments on some randomly generated data showed that such a move structure does not lead to better results.

The tabu list is the mechanism to avoid cycling in the neighbourhood 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 from occurring 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 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, 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 k2 consecutive iterations, we randomly close one station and open another one. We have experimented randomly closing and opening multiple stations; however, their performance was inferior. On the other hand, in some cases the current solution does not improve or deteriorate; that is, 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 that provides the least decrease in the 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 is provided in Figure 2. 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. Siterij 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 neighbourhood structure above. OFV* denotes the best-so-far OFV and TLiter1 and TLiter2 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 the same and counter2 keeps track of the consecutive number of iterations during which OFV* does not improve.
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Figure 2

Tabu search procedure.

4. Experimental study

To test the performance of the proposed TS approach we randomly generated 144 problem instances, considering four demand region configurations, four location site configurations, three configurations of maximum number of stations and three geographical distributions of demand regions. Three different types of geographical distributions of demand regions are illustrated in Figure 3. The whole geographical area to be covered is assumed to be a square. In the first type, a square area is divided into four 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 centre and the density of population decreases as we go away from the centre. 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.
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Figure 3

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, we use three different sets of values for the number of stations (Kt). The planning horizon is four 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 follows 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 min, respectively, in parallel with the requirements determined by the DIRR.

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. First, we investigate the performance of the initialization heuristic benchmarked against the solution obtained by OPL. We observe that the steepest-ascent heuristic gives fairly good initial solutions fast. The three problem types are solved in less than 11 s on the average and the average deviations are 4.65%, 4.88% and 4.82% for the problem types 1, 2 and 3, respectively. The deviation 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 15 min for problems with less than or equal to 300 potential locations and 30 min 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. In sum, 48 out of 144 instances could be solved optimally. These are small-sized problems with fewer potential locations, usually 25% of demand regions.

The TS 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 TS with respect to the number of iterations, we observed the solutions obtained for k3 values of 500, 1000 and 2500 iterations. In Table 2, 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 TS provides better average results than CPLEX.
Table 2

Average results for TS

Data type

CPLEX

Initial solution

TS 500 iterations

TS 1000 iterations

TS 2500 iterations

 

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

As can be seen in Table 2, the TS 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 TS performs as well as CPLEX in the worst case (0.00% for data type 2). Overall, TS results are robust with respect to different data types and TS is able to find better or comparable results for each data type in a relatively short amount of time. Even with a moderate computation time of 500 iterations, the average deviations are well below 0.5%.

The average time spent by the TS for 2500 iterations is significantly less than the average time CPLEX spends solving the instances. As a result, in we concentrate on the solutions obtained after 2500 iterations of TS and report the results with respect to the different properties of the instances (the detailed results for each instance are available in Başar et al, 2009a). 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 3, 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 TS 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 TS is only 0.17% and these solutions are found much faster than CPLEX.
Table 3

Average results with respect to the density of potential locations

Density (%)

CPLEX

TS 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

Finally in Table 4, we present the results with respect to the number of demand regions. As expected, TS 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 density 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. In addition, the average run time of CPLEX is almost the same for 400–500 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 TS outperforms CPLEX when the problem size is large, which is usually the case in real-world applications.
Table 4

Average results with respect to the number of demand regions

|M|

CPLEX

TS 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

We have also applied the TS algorithm for the single-period case. In this case, the TS approach can find solutions that are pretty close to the optimal solution or the best solution found by CPLEX. On the other hand, CPLEX performs relatively better when there is a single period compared with the multi-period case, in the sense that CPLEX can find the optimal solution in more instances in a reasonable time or it provides solutions with a lower optimality gap on the average. The results for the single-period case are reported in Başar et al (2009a).

5. Planning the locations of EMS stations in Istanbul

The proposed TS algorithm is applied to the 4-year location planning of the EMS stations in Istanbul based on the real data collected with the assistance of DIRR in Istanbul Metropolitan Municipality. Istanbul province has an area of 5196 km2 and a population of 11 914 848 (according to 2007 estimates of the Turkish Statistical Institute). It approximately spans 125 km on the east-west direction and 40 km on the north-south direction and consists of 28 districts. The Bosphorus Strait divides the city into 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) that constitute almost half of the total area but only 2% of the total population were excluded from the study, since they require an independent planning due to their geographical conditions. Furthermore, the district of ‘Prince's Islands’ that consists of five small islands was also excluded for the same reason. Hence, the total population we considered 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 neighbouring quarter must be covered by the station in that quarter for the neighbouring quarter to be considered covered. That is, if an EMS is declared to cover a quarter in 5 (8) min, the ambulance located at that EMS can reach all the points in the specified quarter within the 5 (8)-min time limit. The response across the two sides of the city is not allowed because of the unpredictable traffic conditions on the bridges. We assume a constant population over the 4-year planning horizon; however, the expected change in the population can easily be taken into account. Note that in the case of single coverage 111 stations are needed to serve all quarters in 5 min and 47 stations to serve them in 8 min.

The number of stations that can be operated in each year 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 TS implementation, rather than setting the number of iterations, we run both TS and CPLEX for 1200 s. 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 TS solution for the real data is illustrated in Figure 4. In this figure, •, ×, https://static-content.springer.com/image/art%3A10.1057%2Fjors.2010.5/MediaObjects/41274_2011_Article_BFjors20105_IEq1_HTML.gif and + symbolize the locations where EMS stations are opened in years 1, 2, 3 and 4, respectively. The striped areas are the regions excluded from the analysis. Dark and light coloured areas represent backup double-covered and uncovered regions in the specified period, respectively. The areas in white show the seas and the lakes.
Table 5

Computational results for Istanbul's data

Scenario

Year (t)

Kt

CPLEX

TS

   

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

https://static-content.springer.com/image/art%3A10.1057%2Fjors.2010.5/MediaObjects/41274_2011_Article_BFjors20105_Fig4_HTML.jpg
Figure 4

TS solution for the 4-year location planning of EMS stations in Istanbul: (a) Year 1; (b) Year 2; (c) Year 3 and (d) Year 4.

In Table 5, we observe that the results found by CPLEX are slightly better than the results obtained using TS. This may be due to the fact that Istanbul's data have certain special characteristics that are not commonly observed in the random data. First, the Asian and European sides are separated because the response of an EMS station by crossing the bridges is not allowed due to unpredictable traffic conditions. Thus, the problem can in fact be decomposed into two sub-problems that can be solved independently if the stations are allocated separately for each side. This property may facilitate the solution procedure of CPLEX. Second, 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.

6. Conclusion and future research

In this paper, we present a mathematical model for the multi-period planning of the locations of EMS stations, namely MPBDCM. Since this model is intractable for large-scale cases, we propose a TS solution approach to solve it efficiently. The initial solution to TS algorithm is obtained using a steepest-ascent heuristic. We test the performance of the TS algorithm on randomly generated data as well as the data we collected for Istanbul. The results show that the TS approach provides very good results in reasonable computation times compared to the results found by CPLEX.

Future research on this topic may focus on several areas. First of all, our model aims only at maximizing the total backup double-covered population. However, other requirements and restrictions may be imposed depending on the planning environment and the decisions involved. For instance, while maximizing the double-covered population a certain level of single coverage service may be guaranteed. In addition, we can relax the constraints that restrict closing a station once it has been opened in that location. In relation to this, the cost of opening and operating EMS stations of different sizes and in different locations will be different due to the construction, personnel and 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.

Acknowledgements

We thank the managers and personnel of the Directorate of Instant Relief and Rescue at the Istanbul Metropolitan Municipality for their valuable collaboration in collecting the data and sharing their ideas. We are grateful to the Directorate of Strategic Planning at the Istanbul Metropolitan Municipality for supporting this research. We also thank the two anonymous referees for their constructive comments.

Copyright information

© Operational Research Society 2011