An improved formulation for the maximum coverage patrol routing problem

Computers & Operations Research 59 (2015) 1–10

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Computers & Operations Research journal homepage: www.elsevier.com/locate/caor

An improved formulation for the maximum coverage patrol routing problem İbrahim Çapar a, Burcu B. Keskin a,n, Paul A. Rubin b a b

Department of Information Systems, Statistics, and Management Science, The University of Alabama, Tuscaloosa, AL 35487-0226, United States Broad College of Business, Michigan State University, East Lansing, MI 48824, United States

art ic l e i nf o

a b s t r a c t

Available online 9 January 2015

We present an improved formulation for the maximum coverage patrol routing problem (MCPRP). The main goal of the patrol routing problem is to maximize the coverage of critical highway stretches while ensuring the feasibility of routes and considering the availability of resources. By investigating the structural properties of the optimal solution, we formulate a new, improved mixed integer program that can solve real life instances to optimality within seconds, where methods proposed in prior literature fail to find a provably optimal solution within an hour. The improved formulation provides enhanced highway coverage for both randomly generated and real life instances. We show an average increase in coverage of nearly 20% for the randomly generated instances provided in the literature, with a best case increase over 46%. Similarly, for the real life instances, we close the optimality gap within seconds and demonstrate an additional coverage of over 13% in the best case. The improved formulation also allows for testing a number of real life scenarios related to multi-start routes, delayed starts at the beginning of the shifts, and taking a planned break during the shift. Being able to solve these scenarios in short durations help decision and policy makers to better evaluate resource allocation options while serving public. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Mixed integer linear program Routing Resource allocation

1. Introduction Speeding, driving under the influence, and other aggressive driving behaviors are among the leading causes of highway crashes and fatalities. State officials continuously work to discourage such behaviors in several ways, including: (i) increased data-driven enforcement; (ii) technological advances, such as automated enforcement; and (iii) public information and education programs [7]. Data-driven enforcement involves developing strategic countermeasures and operational plans using locally collected data and hot spot information. Hot spots are defined as certain combinations of highway stretches and time of day with high frequencies of crashes. The visibility of law enforcement officers at hot spots is known to be one of the key deterrents to aggressive driving. Researchers help government entities in their data-driven enforcement efforts in two main ways. The first area is related to defining hot spots via clustering analysis of historical crash and citation data [2–4,6,11–14]. The second area in data-driven law enforcement, to which this paper contributes, is concerned with the use of predetermined hot

n

Corresponding author. Tel.: þ 1 205 348 8442. E-mail address: [email protected] (B.B. Keskin).

http://dx.doi.org/10.1016/j.cor.2014.12.002 0305-0548/& 2014 Elsevier Ltd. All rights reserved.

spot information in setting effective operational plans that enhance public safety [9,10]. We present here an improved formulation to the maximum covering and patrol routing problem (MCPRP) that was first defined by Keskin et al. [9]. In the MCPRP, certain locations on the highways are “hot” at certain times. The objective of the MCPRP is to maximize coverage of hot spots with a fixed set of state troopers, belonging to a single trooper post. Keskin et al. [9] formulate a mixed integer linear model to solve MCPRP. By demonstrating similarities to the team orienteering problem with time windows [5,16,17], Keskin et al. [9] show that MCPRP is NP-Hard and resort to local- and tabusearch based heuristics. Our improved formulation retains the original assumptions of the MCPRP formulation [9], including (i) that deterrence is not increased by the simultaneous presence of multiple state trooper vehicles, and (ii) that distinct hot spots at the same physical location (but different times) may be patrolled by different vehicles. In this paper, we first investigate the structural properties of the optimal solution of the MCPRP and prove that an optimal solution must exist in which no hot spot is patrolled by more than one state trooper. This result, together with other formulation strengthening techniques, leads to a significant reduction of the number of variables, and thus of the size of the solution space. Reformulating a problem and improving bounds are common techniques to solve

İ. Çapar et al. / Computers & Operations Research 59 (2015) 1–10

2

“hard” optimization problems in the literature [1,8,15]. With the improved formulation, we solve real life test instances and large randomly generated test instances to optimality within a short time, where Keskin et al. [9] fail to solve those instances to optimality. Moreover, we improve the coverage of the hot spots as much as 46.19% and 13.6% respectively for randomly generated and real life instances. This new model provides the state troopers with an optimal plan that is quickly computable. Additionally, the new formulation makes it possible to test a number of practical situations for state troopers. These scenarios include (i) letting state troopers start their patrol from their homes as opposed to the state trooper post (multi-depot vs single-depot); (ii) delaying the start time of the patrol due to other duties; and (iii) taking a planned break during the patrol. None of these scenarios would have been easily tested with the original MCPRP model. We computationally test these scenarios over extensive number of cases and propose insights to decision and policy makers regarding better resource allocation. The remainder of this paper is structured as follows. In Section 2, we present the new model, including necessary assumptions and notation. In Section 3, we discuss computational results based on randomly generated data and real data. In Section 4, we present several real life situations where the new model sheds insights on policy and strategy building. Finally, in Section 5, we provide our conclusions and recommendations for future work.

5. A state trooper can arrive at hot spot i prior to its start time ei, but the deterrence effect is only calculated starting at ei, i.e., only after the hot spot is “hot.” 6. Having multiple state troopers in the same hot spot at the same time would provide the same deterrence effect as a single trooper. 7. Travel speed is a constant and is set to 60 miles/hour in the numerical experiment. 8. Trooper cars are in continuous service during the shift. (Meals, refueling etc. are assumed to take place before or after the shift.) Before introducing the improved formulation for patrol routing problem (IPRP), we present the original patrol routing formulation Keskin et al. [9] for the sake of completeness: XX Maximize ðf ik  sik Þ ðMCPRPÞ iANkAK

Subject to: f ik þ t ij sjk r ð1  xijk ÞM ij ; X

ei

Our problem definition and assumptions are very similar to that of Keskin et al. [9]. For a given shift, we assume that there are N identified hot spots, where each hot spot iA N ¼ f1; …; Ng has earliest time ei and latest time li for its effective coverage duration, with ei o li . The dummy locations 0 and N þ 1 represent the state trooper post at the start and end of the shift, respectively. Without loss of generality, we assume that the shift begins at time 0. The objective of the patrol routing problem is to maximize the deterrence effect by finding the best patrol route and determining time spent at hot spots for each state trooper car k A K. We define our additional notation in Table 1. The problem setting includes the following assumptions and restrictions, as in Keskin et al. [9]: 1. Each state trooper division can be solved separately. 2. The model horizon is a single shift for one day. 3. State troopers start and end their shift at the state trooper post. Shift duration is enforced by the parameters e0 ¼ 0 and lN þ 1 ¼ T. 4. All of the state trooper cars (k A K) are identical.

and

8 ði; jÞ A E

ð1Þ

xijk r sik ;

8kAK

and

8 iA V

ð2Þ

xijk Z f ik ;

8kAK

and

8iAV

ð3Þ

þ

j A Δ ðiÞ

X

li

þ

j A Δ ðiÞ

sik r f ik ; X

2. Mathematical model

8kAK

8kAK

x0jk ¼ 1;

8iAV

and

ð4Þ

8kAK

ð5Þ

þ

j A Δ ð0Þ

X 

xi;N þ 1;k ¼ 1;

8kAK

ð6Þ

i A Δ ðN þ 1Þ

X

▵

iA



xijk ¼

ðjÞ

X

X

▵

iA

þ

xjik ;

8kAK

and

8 jA N

ð7Þ

ðjÞ

xijk ¼ yik ;

8kAK

8iAN

and

ð8Þ

þ

j A Δ ðiÞ

y0;k ¼ yN þ 1;k ¼ 1;

8kAK

ð9Þ

uikg þ uigk ryik ;

8iAV

and

k; g A K; g 4 k

ð10Þ

uikg þ uigk ryig ;

8iAV

and

k; g A K; g 4 k

ð11Þ

uikg þ uigk Zyik þ yig 1;

8iAV

and

k; g A K; g 4k

ð12Þ

f ik sig Mð1  uikg Þ r 0;

8iAV

and

k; g A K; g 4 k

ð13Þ

f ig  sik Mð1  uigk Þ r0;

8iAV

and

k; g A K; g 4 k

ð14Þ

Table 1 Notation. Problem parameters: V E Δ þ ðiÞ Δ  ðiÞ: t ij : T: Decision variables: xijk : sik Z 0 : f ik Z 0 : yik :

Set of hot spots and state trooper post, V ¼ N ⋃f0; N þ 1g. Set of arcs, E ¼ fði; jÞ : i; jA V; i a jg. Set of hot spots reachable from i A V within their time window, Δ þ ðiÞ ¼ fjA V : ði; jÞA E; ei þ t ij r lj g (see below for the definitions of e, t and l). Set of hot spots from which i A V is reachable, Δ  ðiÞ ¼ fj A V : ðj; iÞ A E; ej þ t ji r li g. Shortest travel time from iA V to jA V. End of shift. 1, if state trooper car k A K travels from hot spot i to j, ði; jÞ A E; 0, otherwise. Start of patrol at hot spot i A V by state trooper k A K. End of patrol at hot spot iA V by state trooper k A K. 1, if state trooper k A K patrols hot spot iA V; 0, otherwise.

İ. Çapar et al. / Computers & Operations Research 59 (2015) 1–10

sik ; f ik Z 0

and

xijk ; yik ; uikg A f0; 1g;

8 i; jA V

and k; g A K; g 4k

end of shift. The end of patrol time fik at hot spot i is (

ð15Þ f ik ¼

The objective of MCPRP is to maximize the total amount of patrol time that falls within the time window of a hot spot. Constraint (1) ensures the feasibility of a route. That is, if a state trooper car k visits hot spot j A V after a stop at hot spot i A V, then start time at hot spot j should be greater than or equal to the finish time at the prior hot spot i plus the travel time between i and j. Constraints (2)–(4) limit the patrol time within hot spot time window, i.e., ei r sik r f ik r li . Constraints (5) and (6) compel state trooper cars to start and end the shift at the state trooper post, respectively. Constraint (7) requires that each state trooper car k enters and exits any hot spot i the same number of times. Constraints (8) and (9) define yik. Constraints (10)–(14) ensure that if multiple cars patrol a single hot spot simultaneously, they contribute to the objective only once. Finally, constraint (15) defines the domains of the decision variables. Before stating our improved formulation, we have the following additional assumptions: Assumption 1. The travel times tij between distinct locations are strictly positive and satisfy the triangle inequality.

3

minfli ; T  t i;N þ 1 g; li ;

if i is the last hot spot visited on the route of k; otherwise:

Before stating our next result, we define a new term “transition” as follows: Definition 1. A transition is the change from one state trooper car to the next one at a particular hot spot, when two or more patrol cars patrol the hot spot during disjoint segments of its time window. Let the solution to MCPRP be represented by a set of patrol routes, S. We state the following lemma: Lemma 1. Given a solution S in which some hot spot iA N is served by two cars k; g A K; k a g, there is an alternative solution S 0 with total patrol time at least that of S and with one fewer transition. Proof. Without loss of generality, let us assume that in solution S car k precedes car g at hot spot i. In other words, ei r sik rf ik r sig r f ig r li :

Assumption 2. It is possible that a patrol car visits no hot spots. Both of these assumptions, in general, are satisfied by the real life practices of state troopers. If a patrol car is unused (i.e., visits no hot spots), the state trooper presumably is diverted to other tasks. We model this assumption by directly dispatching the patrol car on the dummy edge from hot spot 0 to hot spot N þ 1. Originally, (V, E) was a complete digraph. We redefine E to eliminate infeasibilities and improbable edges. That is, ði; jÞ A E if and only if one of the following holds:

 ði; jÞ ¼ ð0; N þ 1Þ: this is an edge used by an idle vehicle at the state trooper post;

 ði; jÞ ¼ fð0; jÞ : j A N and t 0j r lj g: hot spot j can be reached at start of shift before its window closes;

 ði; jÞ ¼ fði; N þ 1Þ : i A N and ei þ t i;N þ 1 r Tg: hot spot i can be patrolled in time to return to the post by end of shift; or

 ði; jÞ ¼ fi; j A N ; i a j; li þ t ij r lj g: after hot spot i's window closes, it is possible to reach hot spot j before its window closes; see Proposition 1 below. When all of the hot spots have equal priority, we have the following proposition from Keskin et al. [9]: Proposition 1 (Keskin et al. [9]). A car will leave a hot spot before its window closes only if it needs to return to the state trooper post by

Note that both k and g can be at the hot spot i for some duration. However, only the coverage from one of them contributes to the objective function. Let ði  1Þ be the hot spot visited by car g immediately prior to hot spot i, and let ðiþ 1Þ be the hot spot visited by car g immediately after hot spot i. Additionally, let j be the hot spot visited by car k immediately after hot spot i. Note that, any or all of j, ði 1Þ, ði þ 1Þ could be the state trooper post. Now, consider the following solution S 0 :

 All of the cars other than k and g follow the same routes in S 0 as

they do in S. Car k follows the same route in S 0 as it did in S until it reaches hot spot i. It stays at hot spot i from time sik to time fig, and then proceeds to hot spot ði þ 1Þ. From that point onward, car k follows the remainder of the route driven by car g in S. On the other hand, car g follows the same route in S 0 as it did in S up to and including hot spot ði 1Þ. From hot spot ði  1Þ, car g proceeds directly to hot spot j and follows the remainder of the route driven by car k in S.

The routes for k and g in solutions S and S 0 are given in Fig. 1. Since car numbering is irrelevant and route S was feasible, S 0 is clearly feasible. A subtle point is that this works only if ð0; N þ 1Þ A E, since it is possible that hot spot i is the first stop for car g ðði 1Þ ¼ 0Þ and the last stop for car k ðj ¼ N þ 1Þ. Patrol duration at hot spot i changes from f ik  sik þ f ig  sig to f ig  sik ; the net change is exactly Δ ¼ sig  f ik Z 0.

Fig. 1. The routes for k and g in solutions S and S 0 .

İ. Çapar et al. / Computers & Operations Research 59 (2015) 1–10

4

In solution S, car g leaves hot spot ði  1Þ no later than sig  t i  1;i and arrives at hot spot ðiþ 1Þ no earlier than f ig þ t i;i þ 1 . Car k leaves hot spot i at fik and arrives at hot spot j by f ik þ t ij . In S 0 , car k is free to leave hot spot i at time fig and arrives at hot spot ði þ 1Þ by f ig þ t i;i þ 1 , so patrol time for the remainder of its route in S 0 is identical to patrol time for the remainder of the route car g took in S. Meanwhile, in S 0 car g leaves hot spot ði  1Þ at or before sig  t i  1;i and arrives at hot spot j by sig  t i  1;i þ t i  1;j . Since, by the triangle inequality, t i  1;j r t i  1;i þ t i;j , it follows that sig  t i  1;i þt i  1;j r sig þ t ij : Therefore, the change to the patrol time at hot spot j is at worst ðf ik þt ij Þ ðsig þ t ij Þ ¼  Δ. Patrol time for the remainder of the route taken by g in S 0 is identical to that of car k in S. The overall change in patrol time from S to S 0 is therefore, at worst, equal to Δ  Δ ¼ 0. If t i  1;j o t i  1;i þ t ij (the fastest route from ði  1Þ to j does not pass through i) or if f ik þ t ij o ej (car k arrived early at hot spot j in solution S), the change to the patrol time is positive. The change from S to S 0 does not introduce any new transitions and eliminates the transition from car k to car g at hot spot i. □

In this formulation, constraints (1a-IPRP), (1b-IPRP), (1c-IPRP) guarantee schedule feasibility with respect to time considerations for each state trooper car k A K if car k visits hot spot jA V after a stop at hot spot i A V. Constraints (2-IPRP), (3-IPRP) and (4) refer to feasibility constraints with regard to time window restrictions. As in MCPRP, constraints (5) ensure all of the state trooper cars leave the state trooper post at the beginning of the shift (or remain idle at the post if j ¼ N þ1), and constraint (7) ensures their return to the post by the end of the shift. Finally, constraint (6) balances flow at each hot spot, i.e., each state trooper car k that visits hot spot i must leave. Constraints (8) and (9) define yik, 8 iA N and 8 k A K. Constraint (10-IPRP) establishes the structure of the visits to hot spots. Note that the constraints (1b-IPRP), (4) and (3-IPRP), combined with Assumption 1, preclude the occurrence of subtours. If the travel times among any group of hot spots were zero and their hotness ended at the same time, a subtour might have occurred. However, this is not possible in this problem context, and hence, subtours cannot arise.

3. Computational experiments

Proposition 2. An optimal solution exists containing no transitions. Proof. Given an optimal solution S containing q transitions, by Lemma 1, we can construct another feasible solution S 0 with objective value at least that of S (hence, optimal) containing q  1 transitions. Applying the lemma q times, we arrive at an optimal solution with zero transitions. □ Even though Proposition 1 is proposed by Keskin et al. [9], it is not reflected in the MCPRP formulation, but rather, utilized for heuristic development. We use Proposition 1 to further tighten constraints (1) of MCPRP [9] to t 0i x0ik r sik ;

8kAK

ðli þ t ij Þxijk rsjk ;

and

8 ð0; iÞ A E:

8 k A K; 8 i; jA N ;

f ik þ t i;N þ 1 xi;N þ 1;k rT;

8kAK

ð1a  IPRPÞ ði; jÞ A E:

ð1b  IPRPÞ

8 ði; N þ 1Þ A E:

ð1c  IPRPÞ

and

and

Using Proposition 2, we rewrite scheduling constraints (2–3) of MCPRP [9] as ei yik r sik ;

8kAK

and

8 i A V:

ð2  IPRPÞ

li yik Zf ik ;

8kAK

and

8 i A V:

ð3  IPRPÞ

Furthermore, we can replace the constraints related to visiting hot spots, (10)–(14) of MCPRP [9], with X yik r 1; 8 i A N : ð10  IPRPÞ kAK

Bounds on variables are as follows: sik ; f ik Z 0

and

xijk ; yik A f0; 1g;

8 i; jA V

and k A K: ð15  IPRPÞ

Constraints (4)–(9) of the original MCPRP remain in our formulation. We can now present the complete formulation of IPRP as follows: Maximize

XX

ðf ik  sik Þ

iANkAK

Subject to: Constraints ð1a  IPRPÞ; ð1b  IPRPÞ; ð1c  IPRPÞ; Constraints ð2  IPRPÞ; ð3  IPRPÞ; Constraints ð4Þ–ð9Þ; Constraint ð10  IPRPÞ; Constraint ð15  IPRPÞ:

ðIPRPÞ

3.1. Performance based experiments In order to evaluate the IPRP and compare its performance with the MCPRP, we repeat the experiments in [9]: (i) a set of randomly generated problems and (ii) a set of real life data. We implement both IPRP and MCPRP in the Java programming language using CPLEX 12.4 as the solver engine. We perform the computations on a computer with a 3.6 GHz Intel Xeon Processor and 64 GB RAM, using a time limit of one hour (3600 s). 3.1.1. Experiments with randomly generated data There are 480 randomly generated problem instances, with 20 instances for each combination of the number of hot spots (10, 20, or 40) and the number of state trooper cars (from one to eight). More precisely, for a given number of hot spots in an eight-hour shift, there are 20 randomly generated networks, each of which is tested with one through eight cars. Location and time window information of a hot spot is randomly picked from a list of real life hot spot locations and corresponding time windows. In Table 2, we present the number of optimal solutions found (# Opt.), average gap between upper and lower bound returned by CPLEX (ðUB LBÞ=LB), average solution time by IPRP and MCPRP (Sol. Time, in seconds), and average and maximum improvements in objective value over MCPRP by IPRP (Avg. and Max., respectively). The IPRP solves all of the test instances to optimality within a short time frame: less than 0.1 s with 10 hot spots; less than 1 s with 20 hot spots; and less than 4 min with 40 hot spots. Most other test instances with 40 hot spots are solved within ten to twenty seconds. The median solution times are 25, 34, and 20 s for 40 hot spots with six, seven, and eight cars, respectively. One test instance with 40 hot spots takes 3100 s with 6 cars and 2200 s with 7 cars that increase the average solution time for 6 and 7 cars. Additionally, increasing the number of state troopers beyond a certain level makes the problem easier to solve since the abundance of state troopers eliminates resource conflict. For instance, if there were 40 state troopers for 40 hot spots, the optimal solution would be trivial, dispatching one state trooper to each hot spot. Obviously, this would not be the case if there were a fixed charge for dispatching state troopers. Investigating the fixed charge patrol routing problem would be an interesting academic exercise. On the other hand, as can be seen in the second (gray portion) of Table 2, for MCPRP [9], CPLEX fails to close the gap between the lower and upper bound, and the solution duration hits the limit in every test instance, except with 10 hot spots and three cars. After

İ. Çapar et al. / Computers & Operations Research 59 (2015) 1–10

solving instances to optimality with IPRP, we notice that for small test instances with 10 and 20 hot spots, the lower bound returned by MCPRP is the optimal solution. For larger test instances with 40 hot spots and eight state trooper cars, however, the gap between the optimal solution returned by IPRP and the lower bound of Table 2 Performance of IPRP and MCPRP for random data. IPRP

MCPRP

Hot # # (UB-LB)/ spots Cars Opt. LB (%)

Sol. Time (s)

Improvement

# (UB-LB)/ Opt. LB (%)

Sol. Time (s)

Avg. (%)

Max. (%)

10

1 2 3 4 5 6 7 8

20 20 20 20 20 20 20 20

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

20 20 20 2 0 0 0 0

0.0 0.0 0.0 21.2 64.2 105.1 150.6 198.1

0.3 1.9 391.2 3262.2 3600.2 3600.3 3600.3 3600.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

20

1 2 3 4 5 6 7 8

20 20 20 20 20 20 20 20

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.3 0.3 0.3 0.4 0.6 0.6 0.8 0.5

20 20 1 0 0 0 0 0

0.0 0.0 6.2 38.1 74.7 116.6 158.2 202.8

2.2 20.8 3429.8 3600.4 3600.4 3600.6 3600.6 3600.9

0.0 0.0 0.0 0.0 0.0 0.1 0.5 1.0

0.0 0.0 0.0 0.4 0.2 1.0 5.1 8.7

40

1 2 3 4 5 6 7 8

20 20 20 20 20 20 20 20

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.4 1.3 6.4 16.2 25.1 188.4 224.5 49.4

20 6 0 0 0 0 0 0

0.0 5.2 24.5 46.1 73.9 110.3 145.1 184.9

20.5 2955.0 3601.3 3601.1 3602.0 3601.6 3602.1 3603.9

0.0 0.5 1.5 4.3 8.2 13.6 16.7 19.9

0.0 3.9 5.2 9.9 17.4 28.7 36.4 46.2

Mean Gap (%)

3.1.2. Experiments with real life data Real life data instances in [9] consist of hot spots from the rural areas of Jefferson (Jeff), Mobile (Mob), and Tuscaloosa (Tusc) counties in the state of Alabama. For each region, we run both MCPRP and IPRP for Monday, Friday, and Saturday as they are representative of the distribution of hot spots in a week. Furthermore, each day is split into three shifts: morning shift from 7:00 a.m. to 3:00 p.m. (“M”);

1.00

400

300

200

IPRP MCPRP

100

0

MCPRP grows as large as 46.2% with an average of 19.9%. As the number of state troopers increases, the size of the solution space for both models increases, resulting in a widening of the optimality gap (the difference between the upper and lower bound) for MCPRP but having little negative impact on IPRP performance. In fact, when the number of state troopers increased from seven to eight, we observed average solution times for IPRP decrease from 224.5 s to 49.4 s. In Fig. 2, we present the solution progress of IPRP and MCPRP for the random data instances. Fig. 2(a) shows the average percent gap for each model as a function of time. IPRP closes the optimality gap quickly while the average gap of MCPRP stays around 100%. Fig. 2 (b) depicts the fraction of problems on which each model proved optimality. IPRP finds optimal solutions for almost all test instances. On the other hand, MCPRP solves only 27% of test instances to optimality within one hour, and virtually none of the larger instances (20 or 40 hot spots, three or more cars). Fig. 2(c) shows the average ratio of the current incumbent value to the best known solution. Since the lower bound returned by MCPRP is the optimal solution for many of the instances, the overall ratio is close to 1 even for MCPRP. However, IPRP provides high quality solutions much faster than does MCPRP and proves their quality. Fig. 2(d) presents the fraction of problems on which each model attained the best known solution. While IPRP finds the best known solution very quickly in all cases, MCPRP fails to find the best known solution of approximately 30% of all test instances.

0

200

400

Fraction of Instances with Proven Optimality

Data set

600

0.75

0.50

IPRP MCPRP

0.25

0.00

0

Time (seconds)

200

400

600

Time (seconds)

1.00

0.75

0.50

IPRP MCPRP

0.25

0.00

0

200

400

Time (seconds)

600

Fraction of Instaces Achieving Best Solution

Mean Objective Value (Scaled)

5

1.00

0.75

0.50

IPRP MCPRP

0.25

0.00

0

200

400

600

Time (seconds)

Fig. 2. Comparison of IPRP and MCPRP for random test instances. (a) Comparison of mean gap, (b) Comparison of optimality, (c) Comparison of mean objective, and (d) Comparison of incumbent solution.

İ. Çapar et al. / Computers & Operations Research 59 (2015) 1–10

afternoon shift from 3:00 p.m. to 11:00 p.m. (“A”); and evening shift from 11:00 p.m. to 7:00 a.m. (“E”). The number of the hot spots varies from 6 to 27 in the real test instances. In Table 3, we provide the percentage improvement of the coverage provided by IPRP over the lower bound returned by MCPRP using the formula ðIPRP  MCPRPÞ=MCPRP  100: As in the randomly generated data experiments, we again observe that in many test cases, the lower bound returned by the MCPRP is the optimal solution. There are a limited number of test instances where the lower bound of MCPRP is not the optimal solution. However, there does not seem to be a pattern as to where the weaker or stronger performance of the MCPRP occurs. For instance, for the Friday afternoon shift at Jefferson county, the lower bounds of all test instances are optimal, except that there is 5.4% gap from the optimal solution with seven state troopers. Similar to Fig. 2, we also compare solution progress of IPRP and MCPRP for real test instances as a function of time. Since the comparison graphs are very similar, we do not show the graphs. As before, we determine that MCPRP struggles to close the optimality gap even though the best solution reported is the optimal one in most cases. Both in terms of solution quality and duration, IPRP is far superior. This performance gap results in up to 173 and 197 min of extra coverage in Jefferson and Mobile, respectively. In terms of run times with real data, the average solution time for IPRP is 3.42 s with maximum time climbing up to 39.7 s (o 1 min). On the other hand, the average solution time for MCPRP is 3203 s (4 53 min) and the maximum time is always at the one hour limit. The ability to solve IPRP to optimality in a reasonably short time conveys certain additional advantages beyond the obvious one, i.e., a guaranteed optimal solution. In planning routes, the state troopers might well wish to do sensitivity analysis, testing the incremental improvement in time window coverage per additional car. If the use of n cars covers a high fraction of the combined Table 3 Percentage improvement by IPRP over MCPRP. County

Shift

Patrol cars 1

2

3

4

5

6

7

8

Jeff

MM MA ME FM FA FE SM SA SE

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 1.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 5.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 13.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.1 13.6 0.0 0.0 5.4 0.0 0.0 0.0 0.0

0.3 13.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Mob

MM MA ME FM FA FE SM SA SE

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.2 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.0

0.0 0.0 0.0 0.0 0.0 0.0 1.1 0.0 0.0

MM MA ME FM FA FE SM SA SE

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 1.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Tusc

time windows, an additional car might be better utilized for other duties. The reduced solution time of IPRP compared to MCPRP facilitates running the routing model for the same county and shift repeatedly, with varying numbers of vehicles to determine the best plan with limited resources. The suboptimality of some of the results returned by MCPRP in [9] caused interesting anomalies. In three of our real world instances, adding a state trooper car resulted in a reduction in time window coverage. Fig. 3 illustrates this phenomenon for the Monday afternoon shift in Jefferson County. Obviously, this cannot happen in an optimal solution. Since IPRP always finds an optimal solution, adding another state trooper car can never hurt the solution. Given such anomalies, and not knowing whether an incumbent solution is actually optimal, MCPRP may not be trusted for a resource allocation sensitivity analysis.

4. Extensions, managerial insights, and discussion In this section, we demonstrate three real life uses of IPRP. In particular, we tested the scenarios related to starting patrols from homes; delayed shift starts; and mid-day diversions. In the remainder of this section, we first explain the additional data required to test these extensions and how these data are generated. Then, we compare each extension with the base case (i.e., starting from the state trooper post, patrolling the whole shift time). We present the results of the analysis and provide insights. 4.1. Scenario 1: Starting patrols from homes In some counties, it is possible for state troopers to start patrolling directly from their homes as opposed to starting at the state trooper post. From the modeling perspective, this is equivalent to the multi-start orienteering problem or the multi-depot vehicle routing problem. We first discuss how IPRP can be used to solve this extension, and next, we quantify the value of (or lack thereof) multi-start patrols. This comparison can be used by policy makers to decide strategically whether to let state troopers start patrolling from home or force them to start their patrol from the state trooper post. Regardless of where they start their patrol, the state troopers end their patrol at the state trooper post. 4.1.1. Data generation for home starts To demonstrate the capabilities of our new formulation, we use the data files from Jefferson County for Monday and Saturday. For starting the patrols from homes (i.e., multi-depot case), we need the home addresses of state troopers. The coverage time and collected benefits are highly dependent on the starting locations of the patrol routes as well as the location and timing of hot spots. Since we do not have actual home addresses of state troopers, the selection of starting points is very crucial. Percentage of Maximum Coverage

6

100%

90%

80%

IPRP MCPRP

70%

60%

1

2

3

4

5

6

7

8

Number of State Troopers Fig. 3. Jefferson County Monday Afternoon.

İ. Çapar et al. / Computers & Operations Research 59 (2015) 1–10

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On Saturday morning and Saturday afternoon, the first hot spots appear around 60 min mark. However, the farthest distance in the distance matrix requires only 47 min of travel time. So there is no difference between single depot and multiple home starts as cars depart from both cases has the same coverage benefit. Whereas on Monday Morning and Afternoon, there are locations that become hot in the beginning of the shift. Just by starting from the depot, you may not reach these hot spots. Hence, the coverage is highly sensitive to not only single versus multi-starts, but also the location and timing of hot spots. Based on this experimentation, better coverage with home-starts (multi-starts) is possible if the number of hot spots is larger or geographically distributed, or there are higher number of state troopers.

To adequately represent the starting locations, we pick home locations randomly from 92 zip codes in Jefferson County while using the population within each zip code as a probability weight. We assume that the starting point resides in the center of the zip code. If there are multiple starting points are selected in the same zip code (due to some zip codes having higher populations), we distribute these starting points within the zip code evenly. Recall that the maximum number of state troopers we have is 8. To reduce the variance in data generation, as we are adding one more home-start location, we keep the other locations fixed. For example, as we are generating the 5-state trooper instance, we keep the home locations of 4 state troopers from the previous data set and select one more from the pool of candidate locations. We then repeat this process ten times to create additional instances to test as to eliminate the data bias. In order to test this scenario with IPRP, we create a dummy hot spot location with zero (e.g., insignificantly small) minutes of hotness duration for each state trooper. Additionally, we append the original the distance matrix in the following way: (i) We set the distances from the state trooper post to dummy hot spots (state trooper homes) to zero. (ii) We set the distances from the state trooper post to real hot spot locations to a big value M. This ensures that direct travel from state trooper post to real hot spots are not possible. Finally, (iii) we set the distance from these dummy hot spots to real hot spots to the real distance between these new starting locations and hot spots. With this minimal data manipulation, and without any model changes, we solve this scenario with IPRP. A similar data manipulation would not have worked for MCPRP as an indirect consequence of its assumptions that multiple state troopers can be present at one hot spot and that a state trooper visiting a hot spot may leave prior to the ending duration of the hot spot.

4.2. Scenarios 2 and 3: diversion from patrols In both Scenarios 2 (delayed shift starts) and 3 (mid-day breaks), there are planned diversions from patrol, resulting in reduction in the patrol availability of state troopers. In Scenario 2, a subset of available state troopers may start their patrol later than the required shift start time. In Scenario 3, as opposed to shortening the shift duration at the beginning of the patrol (as in Scenario 2), we assume that a subset of the available state troopers takes a planned mid-day break. These diversions from patrols are essentially to mimic the other duties of state troopers such as attending court duties that state troopers are obliged to do. Obviously, the number of state troopers that need may be diverted will be less than the total number of available troopers. In Scenario 3, with limited number of state troopers and high number of mandatory duties, a state trooper may need to leave a serviced hot spot earlier than the end of the time window of the hot spot, violating Proposition 2. This is especially a problem with consecutive mandatory duties and long duration hot spots. In most real data (and in our test instances), there is at least two hours between two mandatory duties (from the end of one to the start of the other one) and the duration of hot spots are less than 90 min. Hence, we do not have any test instances violating Proposition 2.

4.1.2. Computational results for home starts We present the average coverage results starting from home over ten incidents for each shift in Table 4. As the number of cars increases for a particular day and shift, the total coverage of hot spots increases. The benefit of starting the routes from homes is more pronounced with a larger number of patrol cars. This is especially true when the number of hot spots is large and not all hot spots can be fully covered by routes starting from a single post. When the number of hot spots is low, having a single post may not be a deterrent to coverage, see for instance, Monday Evening or Saturday afternoon. Note that for the Monday morning shift, we have less coverage compared to the base case with three and four patrol cars. This is due to the random selection of the home locations of the state troopers over the ten data instances. In these data instances, the home locations of state troopers are farther away from the hot spots compared to the state trooper post, and hence, less coverage is realized. The sensitivity to starting locations diminishes as we add more state troopers to the experimentation.

4.2.1. Data generation for diversions When there is a need to divert one or more state troopers, we assume that these diverted state troopers attend court duties and start their remainder patrol directly from the court when their task at the court is completed. Therefore, we use the actual location of Jefferson Court House for court duties. Note that it is possible to add any other location to the model for such a diversion duty. Next, we add as many dummy hot spots as the diverted state troopers. The start time of the hot spots is set as the start time of the diversion activity and the duration of these hot spots are set as the delay time. For instance, for delayed shift starts, we set the dummy hot spot start time as zero. On the other hand, for scenario 3, the dummy hot spots (diversions) may take place anytime during the shift as long as the state troopers are guaranteed to get back to the state trooper post by the end of the shift. The distances from the

Table 4 Patrol coverage allowing home starts. Shifts

MonMor MonAft MonEve SatMor SatAft SatEve

HS

21 27 6 18 14 16

Patrol cars 3

4

5

6

7

8

527.3 (  0.5%) 718.8 (1.3%) 143.1 (0.0%) 478.5 (0.0%) 401.1 (0.0%) 526.9 (0.8%)

597.0 (  0.4%) 822.1 (1.4%) 143.1 (0.0%) 535.7 (0.0%) 415.3 (0.0%) 539.8 (0.9%)

647.5 (0.3%) 918.7 (2.0%) 143.1 (0.0%) 548.0 (0.0%) 415.3 (0.0%) 548.6 (1.1%)

688.8 (1.1%) 975.3 (2.6%) 143.1 (0.0%) 548.0 (0.0%) 415.3 (0.0%) 548.8 (1.2%)

732.4 (2.4%) 1022.6 (3.3%) 143.1 (0.0%) 548.0 (0.0%) 415.3 (0.0%) 549.5 (1.3%)

763.2 (2.4%) 1054.6 (3.4%) 143.1 (0.0%) 548.0 (0.0%) 415.3 (0.0%) 549.5 (1.3%)

İ. Çapar et al. / Computers & Operations Research 59 (2015) 1–10

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state trooper post to these dummy hot spots are set as the distance between the post and the court house. The distances between the dummy hot spots and the real ones are as the real distances. In order to ensure these dummy hot spots are visited, we set their corresponding assignment variables to 1 and set the start and end of the coverage to the start and end time of dummy hot spots. Note that the time spent at the dummy hot spots is not included in the total coverage. Additionally, we assume that if a state trooper has a diversion, the duration of the diversion is either 90, 120, or 150 min with an equal probability. Moreover, we use three different levels for how many state troopers can be diverted. In Table 5, we present the total number of state troopers in the first column, and the number of diversions in the next three columns. In low level, at most two state troopers are diverted from patrol, but with high level up to four state troopers may be diverted. In fixed level, regardless of the number of available state troopers, two state troopers are diverted from their shift. As in the previous extension, to eliminate any type bias in data generation, we randomly generated ten instances for diversions.

down in each case. It is more interesting to quantify this reduction and document changes based on low, high, and fixed levels of delays. When the number of hot spots is less than 20 and we have at least four state troopers, delay starting the shift at low and fixed levels (one or two troopers) does not influence the coverage. However, when there are higher number of hot spots as in Monday Morning and Monday Afternoon, the coverage hurts as much as 28% if two out of three or four troopers delay start their shift as in high and fixed levels. As the state trooper numbers increase to 6 or more, the effects of delayed shifts begin to diminish for low and fixed levels but for high levels (where we allow for four state troopers to delay start their shift), the instances with higher number of hot spots still experience reduced levels of coverage; for instance, 19.51% less in the best case on Monday Morning and 9.92% less on Monday afternoon.

Table 7 Comparison of coverage for mid-day breaks with base case. Low level

4.2.2. Computational results for scenario 2: Delayed shift In Table 6, we compare the results of the experimentation for delayed shift at low, high, and fixed levels with the base case (no delays). The percentage difference is calculated as ðBase Case Coverage  Delayed Start CoverageÞ  100: Base Case Coverage

ð16Þ

Clearly, since all of these cases are reducing the total available patrol time of the state troopers, the coverage compared to base case goes Table 5 Number of diversions for different settings. # of state troopers

Low level

High level

Fixed level

3 4 5 6 7 8

1 1 1 2 2 2

2 2 3 3 4 4

2 2 2 2 2 2

Patrol cars 3

4

5

6

7

8

 3.59%  7.23%  1.30%  6.06%  2.12%  2.52%

 0.66%  2.65% 0.00%  3.57%  0.34%  0.37%

 0.86%  2.98% 0.00%  1.57% 0.00%  0.29%

 2.81%  6.17% 0.00%  0.42% 0.00% 0.00%

 0.74%  3.54% 0.00% 0.00% 0.00% 0.00%

 2.77%  3.26% 0.00% 0.00% 0.00% 0.00%

High level MonMor MonAft MonEve Sat Mor SatAft SatEve

 11.32%  17.19%  5.36%  12.83%  11.49%  7.26%

 5.04%  8.51% 0.00%  6.41%  0.86%  0.90%

 4.56%  10.48% 0.00%  4.61%  0.06%  1.18%

 3.30%  8.88% 0.00%  1.27% 0.00% 0.00%

 2.62%  8.09% 0.00%  0.40% 0.00% 0.00%

 3.57%  5.93% 0.00%  0.42% 0.00% 0.00%

Fixed level MonMor MonAft MonEve Sat Mor SatAft SatEve

 11.32%  17.19%  5.36%  12.83%  11.49%  7.26%

 5.04%  8.51% 0.00%  6.41%  0.86%  0.90%

 3.45%  5.99% 0.00%  3.37% 0.00%  0.57%

 2.81%  6.17% 0.00%  0.42% 0.00% 0.00%

 0.74%  3.54% 0.00% 0.00% 0.00% 0.00%

 2.77%  3.26% 0.00% 0.00% 0.00% 0.00%

MonMor MonAft MonEve Sat Mor SatAft SatEve

Table 6 Comparison of coverage for delayed shift start with base case. Low level

Patrol cars 3

4

5

6

7

8

MonMor MonAft MonEve Sat Mor SatAft SatEve

 13.22%  11.93%  6.52%  0.64%  5.36%  2.59%

 10.88%  7.08% 0.00% 0.00% 0.00% 0.00%

 7.15%  4.63% 0.00% 0.00% 0.00% 0.00%

 12.03%  6.31% 0.00% 0.00% 0.00% 0.00%

 9.67%  3.85% 0.00% 0.00% 0.00% 0.00%

 8.50%  3.54% 0.00% 0.00% 0.00% 0.00%

High level MonMor MonAft MonEve Sat Mor SatAft SatEve

 28.08%  23.87%  27.48%  9.43%  13.63%  13.91%

 22.82%  14.85%  6.52%  0.57%  5.18%  2.66%

 28.34%  19.37%  6.52%  0.56%  5.18%  2.24%

 21.30%  12.68% 0.00% 0.00% 0.00% 0.00%

 24.97%  13.96% 0.00% 0.00% 0.00% 0.00%

 19.51%  9.92% 0.00% 0.00% 0.00% 0.00%

Fixed level MonMor MonAft MonEve Sat Mor SatAft SatEve

 28.08%  23.87%  27.48%  9.43%  13.63%  13.91%

 22.82%  14.85%  6.52%  0.57%  5.18%  2.66%

 17.41%  10.73% 0.00% 0.00% 0.00% 0.00%

 12.03%  6.31% 0.00% 0.00% 0.00% 0.00%

 9.67%  3.85% 0.00% 0.00% 0.00% 0.00%

 8.50%  3.54% 0.00% 0.00% 0.00% 0.00%

İ. Çapar et al. / Computers & Operations Research 59 (2015) 1–10 Table 8 Comparison of coverage for mid-day diversion with delayed start. Low level

Patrol cars 3

4

5

6

7

8

MonMor MonAft MonEve Sat Mor SatAft SatEve

11.10% 5.34% 5.58%  5.46% 3.43% 0.07%

11.46% 4.76% 0.00%  3.57%  0.34%  0.37%

6.78% 1.73% 0.00%  1.57% 0.00%  0.29%

10.48% 0.15% 0.00%  0.42% 0.00% 0.00%

9.89% 0.31% 0.00% 0.00% 0.00% 0.00%

6.27% 0.29% 0.00% 0.00% 0.00% 0.00%

High level MonMor MonAft MonEve Sat Mor SatAft SatEve

23.31% 8.78% 30.51%  3.75% 2.48% 7.73%

23.03% 7.45% 6.97%  5.87% 4.55% 1.81%

33.18% 11.03% 6.97%  4.07% 5.39% 1.09%

22.86% 4.34% 0.00%  1.27% 0.00% 0.00%

29.78% 6.82% 0.00%  0.40% 0.00% 0.00%

19.80% 4.43% 0.00%  0.42% 0.00% 0.00%

Fixed level MonMor MonAft MonEve Sat Mor SatAft SatEve

23.31% 8.78% 30.51%  3.75% 2.48% 7.73%

23.03% 7.45% 6.97%  5.87% 4.55% 1.81%

16.90% 5.31% 0.00%  3.37% 0.00%  0.57%

10.48% 0.15% 0.00%  0.42% 0.00% 0.00%

9.89% 0.31% 0.00% 0.00% 0.00% 0.00%

6.27% 0.29% 0.00% 0.00% 0.00% 0.00%

4.2.3. Computational results for scenario 3: Mid-day diversions In Table 7, we present the comparison of coverage with low, high, and fixed levels of mid-day diversions against the base case as in Eq. (16). As in delayed shift scenario, when state troopers are diverted mid-day, the available time for patrol gets reduced and there is less coverage compared to the base case. However, even with high and fixed levels of mid-day diversions, the coverage is only reduced up to 17.19% (as opposed to 28% in scenario 2). As the number of patrol cars increases, the reduction in coverage diminishes most of the time, except Monday Morning. For Monday Morning, even though it seems that there is no pattern in coverage, when we look into the raw coverage data (510.85; 595.65; 640.30; 662.45; 709.69; and 724.36 min, respectively, for 3–8 cars), the coverage increases with increasing number of patrol cars. However, while we are calculating the percentage difference, this increasing pattern is not reflected in Table 7. We also notice that the distribution of the timing of the hot spots is a factor in coverage. Even though there are less number of hot spots on Saturday, especially Saturday morning, we cannot match the base case coverage even with an increase in the number of state trooper cars. This is due to the fact that most hot spots occur during mid-shift on Saturday morning. Comparing the results of the delayed start with the mid-day diversion (scenario 2 versus scenario 3), we investigate the importance of timing of hot spots. In Table 8, we present the percentage difference between delayed start and mid-day diversions using the formula: ðCoverage with Mid  Day Diversions  Delayed Start CoverageÞ  100: Coverage with Mid  Day Diversions

The timing of the hot spots is early during the shifts on Monday Morning and Monday Afternoon. In fact, on Monday Morning, 10 HS out of 21 appear before the 90 min mark. Hence, when we allow state troopers to delay the start of their shift, we lose the ability to cover a number of hot spots. However, when we postpone the diversions later in the day, the coverage is not influenced as much, as can be seen from Table 8. For Monday Morning and Afternoon, mid-day diversions are more advantageous compared to delaying the start of shift: as much as 33% in the high levels. On the other

9

hand, during Saturday shifts, hot spots are more uniformly distributed in time and mostly later in the day. Hence, delaying the start of the shifts is more advantageous by as much as 5.87% (Saturday morning) compared to mid-day diversions. Clearly, the coverage is greatly influenced by the timing and geographical distribution of hot spots. With IPRP being flexible and running fast (under a few seconds), we can conduct these type of analyses and determine the best way to accommodate duties beyond patrolling with minimal reduction in the total coverage of hot spots.

5. Conclusions In this research, we present a new, stronger formulation for the maximum coverage patrol routing problem. The original formulation, MCPRP, and real life and randomly generated test instances are presented by Keskin et al. [9]. The new model, IPRP, can solve both real life and randomly generated test instances very quickly. Even for large instances, such as with 40 hot spots, IPRP finds optimal solutions within five minutes. We compare our model with that of Keskin et al. [9] and show that our model improves the existing solution by as much as 46.2% for randomly generated test instances. Additionally, for real life instances, the IPRP solution reaches more hot spots with better time window coverage. We also showcase three uses of IPRP that is very important to policy makers in real life: (i) allowing to start patrols from multiple locations; (ii) delay starting the shifts for some state troopers; and (iii) allowing for mid-day breaks for some state troopers. We quantify the impact of all of these cases in terms of coverage and highlight the importance of spatial and geographical distribution of hot spots. We note that all of these generalizations have been modeled separately. Combinations of these generalizations (e.g., diversion from patrols with multi-location starts) may require new mathematical models which is left as future work. Another possible future extension of this work relates to capacity planning. In this paper, we assume a fixed number of troopers (and cars) on any one shift. One possible extension would be to incorporate flexibility in the number of troopers available for patrol duties, with an objective function that captures trade-offs between hot spot coverage and other possible duties for troopers.

Acknowledgments The authors would like to thank the anonymous reviewers and the area editor for their valuable comments and suggestions to improve the paper. Dr. Burcu B. Keskin would like to acknowledge the support of TUBITAK 2221 program during the revision of this paper. References [1] Achterberg, Tobias, Thorsten Koch, Andreas Tuchscherer. 2008. On the effects of minor changes in model formulations. Konrad-Zuse-Zentrum für Informationstechnik Berlin. [2] Anderson T. Comparison of spatial methods for measuring road accident “hotspots”: a case study of London. J Maps 2006:55–63. [3] Chen HC, Quddus MA. Applying the random effect negative binomial model to examine traffic accident occurrence at signalized intersections. Accid Anal Prev 2003;35:253–9. [4] Cheng W, Washington SP. Experimental evaluation of hotspot identification methods. Accid Anal Prev 2005;37(5):870–81. [5] Feillet D, Dejax P, Gendreau M. Traveling salesman problem with profits. Transp Sci 2005;39:188–205. [6] Gatrell AC, Bailey TC, Diggle PJ, Rowlingson BS. Spatial point pattern analysis and its application in geographical epidemiology. Trans Inst Br Geogr 1996; 21(1):256–74. [7] GHSA. 〈http://www.ghsa.org/html/issues/speeding.html〉 Governer's Highway Safety Association; 2013.

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