Int. J. Bio-Inspired Computation, Vol. 4, No. 1, 2012

47

An application of genetic algorithm method for solving patrol manpower deployment problems through fuzzy goal programming in traffic management system: a case study Bijay Baran Pal* Department of Mathematics, University of Kalyani, Kalyani-741235, West Bengal, India E-mail: [email protected] *Corresponding author

Debjani Chakraborti Department of Mathematics, Narula Institute of Technology, Kolkata-700109, West Bengal, India E-mail: [email protected]

Papun Biswas Department of Electrical Engineering, JIS College of Engineering, Kalyani-741235,West Bengal, India E-mail: [email protected]

Anirban Mukhopadhyay Department of Computer Science and Engineering, University of Kalyani, Kalyani-741235, West Bengal, India E-mail: [email protected] Abstract: This article demonstrates a fuzzy goal programming (FGP) approach with the use of genetic algorithm (GA) for proper deployment of patrol manpower to various road-segment areas in urban environment in different shifts of a time period to deterring violation of traffic rules and thereby reducing the accident rates in a traffic control planning horizon. To expound the potential use of the approach, a case example of the city Kolkata, West Bengal, INDIA, is solved. Keywords: fuzzy goal programming; FGP; goal programming; GP; genetic algorithm; GA; membership function; traffic management. Reference to this paper should be made as follows: Pal, B.B., Chakraborti, D., Biswas, P. and Mukhopadhyay, A. (2012) ‘An application of genetic algorithm method for solving patrol manpower deployment problems through fuzzy goal programming in traffic management system: a case study’, Int. J. Bio-Inspired Computation, Vol. 4, No. 1, pp.47–60. Biographical notes: Bijay Baran Pal is a Professor at the Department of Mathematics, University of Kalyani, West Bengal, India. He received his MSc in Mathematics, University of Kalyani in 1979, and DIIT in Computational Mathematics and Computer Programming from Indian Institute of Technology, Kharagpur, in 1980. He was awarded a PhD by the University of Kalyani in 1988. He has published a number of research articles in different national and international journals including Elsevier Science. He is the Editorial Board Member of IJAMS and IJDAT journals, Inderscience. His research interests cover different areas of soft-computing for multiobjective decision analysis in uncertain environment.

Copyright © 2012 Inderscience Enterprises Ltd.

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B.B. Pal et al. Debjani Chakraborti is a Senior Lecturer in the Department of Mathematics, Narula Institute of Technology, Kokata, West Bengal, India. She received her MSc from the Department of Mathematics, University of Kalyani, India in 1999. She is working as a part-time research fellow under the supervision of Professor Bijay Baran Pal at Department of Mathematics, University of Kalyani, India. Her research interests pertain to different areas of multiobjective decision making. Papun Biswas is a Senior Lecturer in the Department of Electrical Engineering, JIS College of Engineering, Kalyani, West Bengal, India. He received his MTech in Electrical Engineering from the Department of Applied Physics, Calcutta University, India in 2007. He is working under the supervision of Professor Bijay Baran Pal at Department of Mathematics and Dr. Anirban Mukhyaypadhyay at Department of Computer Science and Engineering, University of Kalyani, India. His research interests pertain to different areas of soft and evolutionary computing in the area of fuzzy multiobjective decision making. Anirban Mukhopadhyay received his ME and PhD in Computer Science from Jadavpur University, Kolkata, India in 2004 and 2009, respectively. He is currently an Assistant Professor in the Department of Computer Science and Engineering, University of Kalyani, India. He has authored more than 50 research papers. He is the recipient of the University Gold Medal and Amitava Dey Memorial Gold Medal from Jadavpur University, India in 2004. He worked in German Cancer Research Centre, Heidelberg, Germany as a post-doctoral fellow during 2009–2010. His current research interests include soft computing and evolutionary computing, combinatorial optimisation, data mining, and bioinformatics.

1

Introduction

The recent trend of urbanisation of society with the increase of urban facilities, traffic load is increasing day-by-day owing to the increase in population of major cities, particularly in metropolitan cities, throughout the world. As a matter of fact, accident frequency with the increase of vehicle density, mainly in the major road-segments of urban areas, is increasing in an alarming rate in the recent years. It has now become a great challenge to traffic control sections of law enforcement departments for reduction of accidents by enforcing traffic laws. Here, it may be mentioned that although different effective measures like construction of new roads, expansion of existing roads, increase of the road segments, etc., have been taken into account by most of the metropolitan cities, traffic control departments are facing a lot of problems to enforce traffic laws for smooth functioning of traffic and thereby reducing accident frequency mainly in the major road segment areas, because of frequent overtaking tendency and hurried driving of vehicles of motorists even at the time of pick hours with heavy traffic load. From the mid-‘60s to early ‘70s of the last century, a considerable amount of efforts was devoted to develop analytical models (Crowther, 1964; Hanna and Gentel, 1971; Isaacs, 1967; Larson, 1969, 1972; McEwen, 1968; Shumate and Crowther, 1966) for allocation and deployment of police patrol manpower to control traffic by enforcing traffic laws. The different other methodological aspects studied then was surveyed by Chaiken and Larson (1972). But, most of developed models were inadequate there due to lack of efficient use of mathematical tools and development of executable computer programmes for analysing several performance measures and deployment strategy regarding enforcement of traffic laws.

During the late 1970s, patrol manpower planning problems were studied in Bodily (1978), Chaiken and Dormont (1978a), Chow (1976), Kolesar et al. (1975), Olson and Wright (1975) and Wilson and McLaren (1977) extensively and widely circulated in the literature. The sophisticated computer programmes were also developed then by the pioneer work of Chaiken and Dormont (1978b) in the field for solving the problems. But, most of models developed then were concerned with geographical patrol manpower allocation decisions. For which, their implementation is limited and not useful to wide range of traffic management problems. The first general mathematical model within the framework of goal programming (GP) (Ignizio, 1976; Lin, 1980; Romero, 1986, 1991; Zanakis and Smith, 1980) for state patrol deployment planning problems of a country was introduced by Lee et al. (1979). In their approach, the method of enumeration was used, which involves huge initial allocation to reach an optimal allocation decision. Again, in their model, hidden non-linearities (Awerbuch et al., 1976; Hannan, 1981; Soyster and Lev, 1978) in several performance measure functions inherently involved in developing patrol manpower deployment planning model were not taken into account. To overcome the above situation, Taylor et al. (1985) developed the gradient search technique within the framework of GP, where deployment policy goals in fractional form were taken into consideration. In their approach, searching of various possible solutions with the consideration of different initial points for making confidence of model solution was taken, which involves huge computational load to arrive at a satisfactory decision. Again, the approach presented there was illustrated by a hypothetical case example. The GP formulation for allocation and deployment of police patrol units was also studied by Basu (1997) in the past.

An application of genetic algorithm method for solving patrol manpower deployment problems Here, it may be mentioned that, in most of the previous studies, patrol manpower deployment activities were made by trial and error method and they were mostly concerned with problems of regional traffic control system. Now, in most of the real-life decision situations, it is to be observed that the decision parameters involved with the problem are often imprecisely known to the decision maker (DM) due to the expert’s ambiguous understanding of the nature of them. As a matter of fact, crisp definition of model parameters and use of conventional approaches create decision troubles to solving such multiobjective decision making (MODM) (Keeney and Raiffa, 1976) problems. To overcome the above difficulty, FGP approach (Pal and Moitra, 2003a; Tiwari et al., 1986) as an extension of fuzzy programming (FP) (Mohamed, 1997; Yang et al., 1991; Zimmerman, 1978) in the framework of conventional GP (Romero, 2004) have been studied in the past, and implemented to different real-world decision problems (Biswas and Pal, 2005; Pal and Moitra, 2003b; Slowinski, 1986). Now, in some practical decision situations, it is found that certain non-linearity in general form as well as in fractional form are frequently involved in defining various relationships among the parameters and decision variables of the problems. In such a case, the use of conventional approximation approaches (Pal et al., 2003; Pal et al., 2008) to FGP formulation of a problem involves computation load and often leads to local optimal solution rather than global one. To overcome the above situation, GAs (Goldberg, 1989; Holland, 1973) have appeared as the robust tools for searching satisfactory decisions of MODM problems. GAs to real-world MODM problems have been studied in Sakawa and Kobuta (2000) and Sakawa and Yauchi (2001) in the past. The use of GAs to decision problems in Stewart et al. (2004) and Zheng et al. (1996) in the framework of FGP have been studied in Pal et al. (2009) and Pal and Gupta (2009a, 2009b) in the recent past. However, the study on GA-based FGP approaches to real-life problem is at an early stage. Moreover, the uses of FGP as well as GAs to traffic control manpower allocation problems are yet to appear in the literature. Actually, the study on management science models for patrol manpower deployment problems is still very thin and not widely circulated in the literature. In this article, a FGP formulation of a patrol manpower deployment problem with several performance achievement functions which are inherently fractional in form is considered. A solution scheme based on GA is introduced to reach a satisfactory decision on the basis of priorities of achieving the objectives of the problem in the decision making environment. An illustrative case example of the Metropolitan city Kolkata in India is considered. The model solution is also compared with the solution obtained by using the conventional FP approach studied by Tiwari et al. (1987) previously to expound the potential use of the proposed approach.

2

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FGP problem formulations

In a fuzzy decision making environment, instead of crisp description of the objectives and structural resource constraints, fuzzy version of some or all of them are taken into account and that depend on the needs and desires of the DM in the decision making horizon. Actually, in the field of FP, instead of optimising objective functions directly, imprecisely defined aspiration level is assigned individually to each of them. Then, the objectives with aspiration levels are called fuzzy goals. In the FGP formulation of the problem, a full fuzzy version of the problem is considered to make the model a flexible one with regard to meeting the several criteria involved therein in the decision making context. Now, fuzzy goal description is presented in the following Section 2.1.

2.1 Definition of fuzzy goal Let bk be the imprecise aspiration level assigned to the kth objective Fk(X), k = 1, 2, ..., K. Then the fuzzy goal expressions generally take one of the following forms: Fk ( X) 2 bk and Fk ( X) 1 bk ,

where X is the vector of decision variables, and where 2 and 1 indicate the fuzziness of ≥ and ≤ restrictions, respectively, in the sense of Zimmermann (1987, 1991). In an FP approach, the fuzzy goals are characterised by their respective membership functions.

2.2 Characterisation of membership function Let t k and tuk be the lower- and upper-tolerance ranges, respectively, for achievement of the aspired level bk of the kth fuzzy goal. Then, the membership function, say μk(X), for the defined fuzzy goal expressions can be characterised as follows (Sakawa, 1993): For 2 type of restriction, μk(X) takes the form: 1 ⎧ ⎪ ⎪ F ( X) − ( bk − t μ k ( X) = ⎨ k tk ⎪ ⎪ 0 ⎩

k

)

, if Fk ( X) ≥ bk , if bk − t

k

≤ Fk ( X) < bk

, if Fk ( X) < bk − t

(1)

k

where (bk – t k ) represents the lower-tolerance limit for achievement of the stated fuzzy goal level bk. Again, for 1 type of restriction, μk(X) becomes: 1 , if Fk ( X) ≤ bk ⎧ ⎪ ⎪ ( b + t ) − Fk ( X) μ k ( X) = ⎨ k uk , if bk < Fk ( X) ≤ bk + tuk (2) tuk ⎪ ⎪ 0 , if Fk ( X) > bk + tuk ⎩

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where (bk + tuk) represents the upper-tolerance limit for achievement of the stated fuzzy goal level bk. From both the expressions in (1) and (2), it is to be followed that a value of μk(X) lies between 0 and 1, which actually represents the grade (called membership value) of achieving the aspire level of a defined fuzzy goal. Now, the FGP model formulation for the defined membership functions is presented in Section 2.3.

where d rk− is renamed for d k− to represent it at the rth − priority level, wrk (> 0) is the numerical weight associated

with d rk− and it designates the weight of importance of achieving the aspired level of the kth goal relative to other which are grouped together at the rth priority level, and − where wrk values are determined as (Pal et al., 2003): − wrk

2.3 FGP model formulation In the process of formulating the FGP model of the problem, the membership functions are transformed into membership goals by assigning the highest membership value (unity) as the aspiration level and introducing under- and over-deviational variables to each of them. Then, in the goal achievement function, the under-deviational variables of the membership goals are minimised on the basis of their importance of achieving the aspired goal levels in the decision making context. Now, in a MODM situation, since a multiplicity of goals are involved with the problem, and they often conflict each other for achievement of their aspired goal levels, the priority-based FGP (Pal and Moitra, 2003a) which is an extension of conventional GP (Ignizio, 1976) and widely used approach for goal achievement problems is taken into consideration. In priority-based FGP, the goals are rank ordered on the basis of the priorities of achieving the target levels of them. The goals which seem to be equally important from the view point of assigning priorities, they are included at the same priority level and numerical weights are given to them on the basis of weights of importance of achieving their aspired levels at the same priority level. The FGP model of the problem under a pre-emptive priority structure can be presented as (Pal and Moitra, 2003a): Find X (x1, x2, …, xn) so as to: Minimise: Z = [P1(d–), P2(d–), ..., Pr(d–), ..., PR(d–)] and satisfy Fk ( X) − ( bk − t t

k

) + d− − d+ =1 k

k

k

( bk + tuk ) − Fk ( X) + d − − d + = 1 k

tuk

d k− , d k+ ≥ 0,

k

(3)

k = 1, 2,..., K

where Z represents the vector of R priority achievement function, and d k− , d k+ are the under- and over-deviational variables of the kth goal. Pr(d–) is a linear function of the weighted under-deviational variables, where Pr(d–) is of the form (Pal and Moitra, 2003a): Pr (d – ) =

K

∑w k =1

− − rk d rk ,

k = 1, 2,..., K ; r = 1, 2, …, R ( R ≤ K )

⎧ 1 , for the defined μ k ( X) in (1) ⎪ ( t k )r =⎨ 1 ⎪ ( tuk ) , for the defined μ k ( X) in (2) ⎩ r

(4)

where (t k ) r and (tuk)r are used to present t k and tuk, respectively, at the rth priority level. It is worthy to mention here that the notion of pre-emptive priorities of the goals actually holds that the goals which are at the rth priority level Pr is preferred most for achievement of their aspired levels before consideration of achieving the goals included at the next priority Pr+1 regardless of any multiplier associated with Pr+1. Also, the relationship among the priorities is: P1 >>> P2 >>> . . . >>> Pr >>> . . . >>> PR ,

where >>> means ‘much greater than’ and implies that the goals at the first priority level (P1) are achieved to the extent possible before consideration of achieving the aspired levels of goals at the second priority level (P2), and so forth. Now, the decision variables and different types of parameters involved with the problem are defined in the following Section 2.4.

2.4 Definitions of variables and parameters 2.4.1 Decision variables 1

independent variables xij deployment of patrol units to the road-segment area i during the shift j.

2

dependent variables ARij reduction of accident rate contributed by the road-segment area i during the shift j PCij number of physical-contacts made by a patrolman in the road-segment area i during shift j SCij number of sight-contacts made by a patrolman in the road-segment area i during shift j.

2.4.2 Fuzzy resource parameters In most of the practical decision situations, it is to be realised to the fact that the DM is often faced with the problem of assigning the exact resource parameter values due to imprecise in nature of availability of resources as well as utilisation of them. To overcome the situation, the estimated values which may possibly take by the parameters in the sense of fuzziness of them are considered. That is, the

An application of genetic algorithm method for solving patrol manpower deployment problems possible situations of taking their individual values to the left or right to the estimated values are taken into account. The fuzzy resource parameters are defined as follows:

E

estimated total number of patrol units required for various road-segment areas in all the shifts of the time period

Ei

estimated total number of patrol units required in the road-segment area i in all the shift of the period

Fj

estimated total number of patrol units required during the shift j in all the road segment areas of the period

AR

total accident reduction level contributed by all the road-segment areas

ri

SC

goal level for the total sight contacts in all the road segment areas

si

goal level for sight-contacts at the road segment area i in all the shifts of the period

CE Estimated total cash expenditure for deployment of patrol units in all the road segment areas during all the shifts of the period. Crisp coefficients: estimated cost for deployment of a patrol unit to the road-segment area i during the shift j.

Now, for the defined variables and parameters, the algebraic structures of the fuzzy goals are described in the following Section 2.5.

2.5 Fuzzy goal description 2.5.1 Patrol manpower requirement goal A certain level of patrol manpower should be provided to deploy to the road-segment areas during different shifts of the period. The fuzzy goal expression can be presented as:

i =1

j =1

1

ij

1E

Segment-wise allocation Depending on the accident frequency and traffic density, a minimum number of patrol units need be deployed to each of the road-segment areas in all the shifts.

ij

2 Ei ,

i = 1, 2, ..., m

(6)

j =1

2

Shift-wise allocation For smooth functioning of traffic control in all the road segment areas, a certain number of patrol units need be provided in each shift to all the road-segment areas. The goal expression appears as: ij

goal level for physical contacts at the road-segment area i in all the shift of the period

∑ ∑X

∑x

2 Fj ,

j = 1, 2,..., n

(7)

i =1

pi

n

m

∑x

PC goal level for the total physical-contacts in all the road-segment areas in all the shifts

m

The goal expression takes the form:

m

estimated accident reduction level at the road-segment area i in all the shift of the period

Cij

51

(5)

2.5.2 Deployment performance goals A certain level of performance of each of the activities defined in the context of controlling traffic must have to be made by the patrol units to deter accidents by enforcing traffic laws and thereby smoothing the traffic flow during the period under consideration. Now, regarding the measure of performance against deployed patrol units, it may be mentioned that reduction of accident rates by making direct physical-contact for verification of genuine license of vehicles on road as well as enforcement of traffic rules through sight-contact are the main tasks in traffic management system. Here, it is to be observed that the reductions of accident rates as well as measures of other performances increase with the increase of patrol units to the road-segment areas. In the decision situation, it can easily be realised to the fact that increase of performance rate by reduction of accidents rate (i.e., increase of the rate of reduction of accidents) by increasing traffic personnel is an effective expression for smooth functioning of the traffic activities. The similar conceptual frame for measuring the performance of other criteria can also be defined. Actually, certain inverse relationships of different performance measuring criteria with deployment of patrol units can be effectively established here to formulating the model of the problem. In context to the above, the general mathematical expression for defining the relationships of xij with each of the defined ARij, PCij and SCij as performance measure function can be presented as: yij = aij −

bij xij

,

(8)

where yij ≥ 0 represent the performance measure function which is fitted against xij, xij ≥ 0 and integer, ∀i, j, and where aij and bij are the estimated parameters.

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Here, Two-point parameter estimation method studied in Christian and Baecher (2002), Englezos and Kalogerakis (2001) and Morales and Ruiz (2007) can be used to estimate the values of aij and bij to fit the parametric non-linear curve represented by yij in actual practice. In the expression (8), it is to be followed that when xij = 0, yij becomes infeasible. As such, yij = 0 is taken into consideration as and when xij = 0. Further, when xij increases, yij also increases. Again, after a certain value of xij, yij becomes asymptotic, i.e., it then takes the value aij approximately and that depends on the physical characteristics of the problem under consideration. The general graphical representation of the deployment efficiency measure function is presented in Figure 1. Figure1

Figure 2

An instance of patrolman on duty (see online version for colours)

Graphical representation of patrolmen deployment efficiency measure function

yij

aij

xij

0

The potential use of the expression in (8) at different phases of traffic operations is presented as follows.

1

2.5.2.1 Accident reduction goals The primary job of the deployed patrol units is to reduce the rate of accidents by enforcing traffic laws. Here, it may be mentioned that human beings are often found innocent and become victims of fatal accidents, because of horse racing in every day life and overtaking tendency at the time of crossing road-segment areas with regard to time bound pressure on attending activity places. As such, total elimination of accidents is a difficult task and which is imprecise in nature in actual practice. An instance of patrolman on duty is presented in Figure 2. Considering the above view point, it can easily be realised that the increase of accident reduction rate would be a meaningful expression to measure the effectiveness of deployment of patrol units for smoothing the traffic flow. Here, an increase of that rate takes place with the increase of patrol units. Now, following the expression in (8), reduction of accident rate can be presented as: ARij = Rij −

rij xij

,

It is to be followed from the expression in (9) that ARij increases non-linearly with the increase of xij,, and that would be hyperbolic in nature. Because, covering areas of road-segments are effectively not very large, and an increase of allocation of patrol manpower to a few units might become very effective with regard to controlling traffic efficiently in a road-segment area. Further, increase of patrol units beyond a certain limit would give no additional effective and eventually become asymptotic, because number of link roads of a road-segment area is normally limited and there are capacity limitations of link-roads to move vehicles. Now, the fuzzy goals are defined as follows.

(9)

where Rij and rij (Rij > rij) are the estimated parameters associated with ARij and xij, i = 1, 2 ,..., m; j= 1, 3, ..., n.

Segment-wise accident reduction goals The reduction of accident rate at the ith road-segment in all the shifts appears as: n

⎛

∑ ⎜⎜⎝ R

ij

rij ⎞ ⎟ 2 ri , xij ⎟⎠

−

j =1

2

i = 1, 2,..., m

(10)

Total accident reduction goal The goal expression for reduction of overall accident rate for all the road-segments areas during all the shifts of the period takes the form: m

n

⎛

∑∑ ⎜⎜⎝ R

ij

i =1 j =1

−

rij ⎞ ⎟ 2 AR xij ⎟⎠

(11)

2.5.2.2 Physical-content goals The most important task of a deployed patrol unit is to make physical-contact to deter traffic violations and accidents. Physical-contact indicates the direct contact to the vehicles for spot-checking of driving license, car registration, road-tax clearance, etc. Here, it may be mentioned that most of the unwanted situations are created there owing to bypassing attitude of unauthorised vehicles in traffic in the road-segment areas. An instance of physical-contact is presented in Figure 3.

An application of genetic algorithm method for solving patrol manpower deployment problems Figure 3

An instance of physical-contact (see online version for colours)

Figure 4

Similar to the expression in (9), the mathematical form of PCij can be obtained by following the expression in (8). The expression appears as: PCij = Pij −

pij xij

, i = 1, 2,..., m; j = 1, 3,..., n

1

n

⎛

pij ⎞ ⎟ 2 pi , ⎟ ij ⎠

∑ ⎜⎜⎝ P − x ij

j =1

2

SCij = Sij −

i = 1, 2,..., m

Sij X ij

, i = 1, 2,..., m ; j = 1, 2,..., n

(15)

where Sij and sij (Sij ≥ sij) are estimated parameters. The fuzzy goal expressions can be defined as follows. 1

Segment-wise sight-contact goals For smooth functioning of traffic operations, the goal expression for segment-wise sight-contact appears as: n

⎛

∑ ⎜⎜⎝ S

Segment-wise physical-contact goals Since the physical-contact operations are defined in terms of number contacts made there on the basis of traffic density, segment-wise physical-contact goal expression appears as:

An instance of sight-contact (see online version for colours)

Now, similar to the case of physical-contact, the expression of SCij takes the form:

(12)

where Pij and pij (Pij > pij) are the estimated parameters. Here, it is to be followed that the number of physical-contacts increases with the increase of xij, but at a decreasing rate. Then, the fuzzy goal expressions can be defined as follows:

53

ij

−

j =1

2

sij ⎞ ⎟ 2 si , xij ⎟⎠

i = 1, 2,..., m

(16)

Total sight-contact goal Similar to the total physical-contact goal, the goal expression here also appears as: m

(13)

⎛

n

∑∑ ⎜⎜⎝ S

ij

−

i =1 j =1

sij ⎞ ⎟ 2 SC xij ⎟⎠

(17)

Total physical-contact goal To measure the overall performance against deployment of patrol units, the goal expression for total physical-contact takes the form: m

n

⎛

∑∑ ⎜⎜⎝ P

ij

i =1 j =1

−

pij ⎞ ⎟ 2 PC xij ⎟⎠

(14)

2.5.2.3 Sight-contact goals Beyond physical-contact, sight-contact must be an integral part to controlling the vehicle speed, preventing against violation of general traffic rules, etc., at the time of crossing road-segment areas. An instance of sight-contact is presented in Figure 4.

2.5.3 Cash expenditure goal An estimated amount of money should have to be provided for deployment of patrol units. Here, a variation of cost is frequently involved owing to variation of allocation petrol units to the road-segment areas in different shifts of a period, and that depends on accident frequency and traffic load. The fuzzy goal expression appears as: m

n

∑∑ C x CE ij ij

(18)

i =1 j =1

Now, in the FGP model formulation, the membership goals of the defined fuzzy goals can easily be constructed by following the membership goal expressions in (3).

B.B. Pal et al.

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In the solution process, to overcome the computational difficulty (Awerbuch et al., 1976; Soyster and Lev, 1978) with non-linearity in model goals and to avoid the load with the use of traditional linear approximation approaches (Pal et al., 2003; Toksari, 2008), an GA approach as a goal satisfier (Goldberg, 1989) rather than objective optimiser for multiobjective decision analysis is introduced here in the solution search process. The GA method adopted in the process of solving the problem is presented in the following Section 3.

3

Step 3

The simple roulette-wheel scheme (Goldberg, 1989) is used for selecting two parents for mating purposes in the genetic search process. Step 4

Step 1

Step 2

Again, in the reproduction process, the arithmetic crossover for two selected parents M1, M2 can be defined as G1 = α1M1 + α2M2, G2 = α2M1 + α1M2, for producing two offspring G1 and G2 (∈S), where α1, α2 ≥ 0 with α1 + α2 = 1. Step 5

Step 6

The fitness function is defined as:

⎧⎪ K − − ⎫⎪ eval ( M v )r = ( Z ) r = ⎨ wrk d rk ⎬ , ⎩⎪ k =1 ⎭⎪v

∑

Termination The execution of the whole process terminates when the fittest chromosome is reported at a certain generation number in the solution search process.

Fitness function The fitness value of a chromosome is judged by the value of the goal achievement function considered for achieving aspired levels of goals under a given priority factor in the solution search process.

Mutation As in the conventional GA scheme, a parameter Pm of the genetic system is defined as the probability of mutation. The mutation operation is performed on a bit-by-bit basis, where for a random number r∈ [0, 1], a chromosome is selected for mutation provided that r < Pm.

Representation and initialisation. Let M denote the binary coded representation of a chromosome in a population as M = {g1, g2, ..., gn}. The population size is defined by pop-size, and pop-size chromosomes are randomly initialised in its search domain.

Crossover The parameter Pc is defined as the probability of crossover. The arithmetic crossover (single-point crossover) operation of a genetic system is applied here in the sense that the resulting offspring would carry very close characteristics of their parents and they always satisfy the constraints set S (≠ φ) defined in the decision making context. Here, a chromosome is selected as a parent, if for a defined random number r∈ [0, 1], r < Pc is satisfied.

GA-based solution method

The solution methods based on GAs for multiobjective decision analysis within the frameworks of conventional GP and FGP have been investigated by Zheng et al. (1996) and Gen et al. (1997), respectively, in the past. The GA methods to FGP problems have also been studied by Pal et al. (2009) and Pal and Gupta (2009b) in the recent past. The efficient uses of GA approaches to different real-life MODM problems have been studied in Pal and Gupta (2009a), Sakawa and Kobuta (2000), Taguchi et al. (1998) and Wang (2002) previously. The algorithmic steps of the GA scheme addressed for solving the proposed problem are presented as follows.

Selection

Now, the FGP formulation of the problem by defining the membership functions of the fuzzy goals and thereby solving the problem by employing the proposed GA scheme is demonstrated via the case example presented in Section 4.

(19)

where (Z)r represents the goal achievement function Z defined in (3) when achievement of goals under the rth priority level is considered, and where the subscript ‘v’ refers to the fitness value of the selected vth chromosome, v = 1, 2, ..., pop_size. The best chromosome with largest fitness value at each generation is determined as M r* = {min{eval(Mv)r | v = 1, 2, ..., pop_size}depending on searching out the best value of an objective.

4

A demonstrative case example

The manpower deployment problem of the Department of ‘Traffic Police’ of the Eastern Metropolitan City Kolkata in India is considered to illustrate the potential use of the proposed model. The required data were collected from the Annual Review Bulletin (2007) published by the Kolkata Traffic Police Department. Some other relevant information provided in the bulletins published in the previous few years were also taken into account. The four most accident-prone areas: BT Road, AJC Road, Strand Road and EM By-pass are selected to illustrate the proposed model.

An application of genetic algorithm method for solving patrol manpower deployment problems The geographical representations of the selected road-segment areas are shown in Figure5. Figure 5

Geographical representations of the selected road- segment areas of Kolkata (see online version for colours)

55

lower-tolerance (LT) limits, for the defined fuzzy goals and also the crisp parameter values are presented in Tables 1 to 4. Now, in the traffic control situation, it is to be followed that there is no additional effect for deployment of more than 5 patrolmen in any road-segment area during any shift. As such, the range of xij is considered as: 1 ≤ xi j ≤ 5, i = 1, 2,..., m; j = 1, 2,..., n

(20)

The values of different parameters involved with the traffic operations are estimated by using the conventional two-point estimation method (Englezos and Kalogerakis, 2001). Here, the execution is made in Excel-Window (Microsoft Excel 2007) and the obtained values are presented in Table 5. Now, using the data presented in Tables 1 to 5, the membership goals for the defined fuzzy goals can be constructed by following the expressions in (1) and (2). Here, the fuzzy goal for utilisation of patrol manpower takes the form: x11 + x12 + x13 + x14 + x21 + x22 + x23 + x24 + x31 + x32 + x33 + x34 + x41 + x42 + x43 + x44 … 1 49.

In the patrolmen deployment planning situation, since the overall yearly performance is measured on the basis of day-to-day performance, the period under consideration for traffic operations is considered a day. The record shows that the most of the accident cases occur during the time interval 6 to 30 hours to 22.00 hours. Generally, the four shifts: 6 am to 10 am, 10 am to 2 pm, 2 pm to 6 pm and 6 pm to 10 pm are arranged to deploy patrol manpower to different road-segment areas on the basis of traffic density and accident frequency. They are denoted by the shift numbers: 1, 2, 3 and 4, respectively. Also, the selected four road-segment areas are successively numbered as: 1, 2, 3, 4. Now, following the records in the Annual Review Bulletin (2007), the data for the aspiration levels (AL) with their associated tolerance limits, upper-tolerance (UT) and Table 1

μ1 = ( 56 − ( x11 + x12 + x13 + x14 + x21 + x22 + x23 + x24 + x31 + x32 + x33 + x34 + x41 + x42 + x43 + x44 ) ) 7

Then, the associated membership goal takes the form:

μ1 : 8 − 0.14 ( x11 + x12 + x13 + x14 + x21 + x22 + x23 + x24 + x31 + x32 + x33 + x34 + x41 + x42 + x43 +

)

(21)

x44 + d1−

− d1+

=1

In an analogous way, the membership goals of the other stated fuzzy goals can be determined. The membership goals appear as follows.

Data description for patrol manpower utilisation

Patrolmen allocation level Table 2

Then, for the given upper tolerance limit of the total patrol manpower allocation goal, the membership function of the stated fuzzy goal appears as:

E (AL, UT)

AR (AL, LT)

PC (AL, LT)

SC (AL, LT)

CE (AL, UT)

(49, 56)

(3.80, 2.00)

(750, 350)

(2,600, 700)

(3,580, 3,800)

Data description for segment-wise patrolmen allocation Ei (AL, LT)

Ri (AL, LT)

pi (AL, LT)

si (AL, LT)

1

(14,12)

(1.10, 0.55)

(210, 90)

(720, 180)

2

(14,12)

(0.96, 0.45)

(200, 85)

(700, 170)

3

(11,9)

(0.95, 0.40)

(180, 80)

(675, 160)

4

(10,8)

(0.91, 0.45)

(180, 80)

(650, 150)

Segment area (i)

56

B.B. Pal et al.

Table 3

Data description for shift-wise patrolmen allocation

Shift (j) Fj (AL, LT) Table 4

4.2 Accident reduction goals 4.2.1 Segment-wise accident reduction goals

1

2

3

4

(12, 10)

(14, 12)

(8, 6)

(15, 13)

Data description of cost cij ( in Rs ) for deployment of

patrol units

⎛ ⎜ ⎝

⎛ 0.24

μ10 : ⎜1.44 − 1.82 ⎜

⎝ x11

+

0.22 0.10 0.27 ⎞ ⎞ + + ⎟⎟ x12 x13 x14 ⎠ ⎟⎠

+

0.18 0.04 0.27 ⎞ ⎞ + + ⎟⎟ x22 x23 x24 ⎠ ⎟⎠

− + + d10 − d10 = 1,

Shift (j) Road segment (i)

1

2

3

4

1 2 3 4

75 80 65 70

75 80 65 70

75 80 65 70

75 80 65 70

⎛ ⎜ ⎝

⎛ 0.22

μ11 : ⎜1.37 − 1.96 ⎜

⎝ x21

− + + d11 − d11 = 1,

(24)

⎛ ⎛ 0.21 0.18 0.07 0.27 ⎞ ⎞ μ12 : ⎜1.29 − 1.82 ⎜ + + + ⎟⎟ ⎜ x32 x33 x34 ⎠ ⎟⎠ ⎝ x31 ⎝ − + + d12 − d12 = 1,

⎛ ⎜ ⎝

4.1 Patrolmen allocation goals 4.1.1 Segment-wise patrolmen allocation goals μ3 : ⎡⎣0.5 ( x21 + x22 + x23 + x24 ) − 6 ⎤⎦ + d3− − d3+ = 1, μ 4 : ⎡⎣0.5 ( x31 + x32 + x33 + x34 ) − 4.5⎤⎦ + d 4− − d 4+ = 1, μ5 : ⎡⎣0.5 ( x41 + x42 + x43 +

)

− d5+

⎝ x41

+

0.15 0.07 0.21 ⎞ ⎞ + + ⎟⎟ x42 x43 x44 ⎠ ⎟⎠

− + + d13 − d13 =1

μ 2 : ⎡⎣0.5 ( x11 + x12 + x13 + x14 ) − 6 ⎤⎦ + d 2− − d 2+ = 1,

x44 − 4 ⎤⎦ + d5−

⎛ 0.21

μ13 : ⎜1.35 − 2.17 ⎜

(22)

The goal expression takes the form:

=1

4.1.2 Shift-wise patrolmen allocation goals μ6 : ⎡⎣0.5 ( x11 + x21 + x31 + x41 ) − 5⎤⎦ + d 6− − d6+ = 1, μ7 : ⎡⎣0.5 ( x12 + x22 + x32 + x42 ) − 6 ⎤⎦ + d 7− − d7+ = 1,

(23)

μ8 : ⎡⎣0.5 ( x13 + x23 + x33 + x43 ) − 3⎤⎦ + d8− − d8+ = 1,

4.2.2 Total accident reduction goal

μ9 : ⎡⎣0.5 ( x14 + x24 + x34 + x44 ) − 6.5⎤⎦ + d9− − d9+ = 1

⎛ ⎛ 0.24 0.22 0.10 0.27 ⎞ ⎞ ⎜ ⎜ x + x + x + x ⎟⎟ 12 13 14 ⎜ ⎜ 11 ⎟⎟ ⎜ ⎜ 0.22 0.18 0.04 0.27 ⎟ ⎟ + + + ⎜ ⎜+ ⎟⎟ x21 x22 x23 x24 ⎟ ⎟ ⎜ ⎜ μ14 : ⎜1.48 − 0.56 ⎜ 0.21 0.18 0.07 0.27 ⎟ ⎟ ⎜ + + + ⎜+ ⎟⎟ x31 x32 x33 x34 ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ 0.21 0.15 0.07 0.21 ⎟ ⎟ ⎜ ⎜⎜ + ⎟⎟ + + + ⎜ x41 x42 x43 x44 ⎟⎠ ⎟⎠ ⎝ ⎝

(25)

− + − d14 =1 + d14

Table 5

Estimated parameter values associated with ARij, PCij, SCij

Road segment i

Shift j

Accident reduction (ARij)

Physical contacts (PCij)

Sight contacts (SCij)

a

b

a

b

a

b

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3

0.40 0.33 0.17 0.45 0.36 0.29 0.11 0.39 0.34 0.28 0.10 0.38 0.33 0.26 0.13

0.24 0.22 0.10 0.27 0.22 0.18 0.04 0.27 0.21 0.18 0.07 0.27 0.21 0.15 0.07

77 106 62 90 76 100 60 87 71 97 54 82 68 93 50

60 82 52 69 60 77 50 66 57 75 46 62 55 71 42

220 295 169 266 217 289 162 261 194 277 137 237 188 272 125

175 225 154 206 174 221 150 205 169 212 127 187 165 210 116

4

4

0.36

0.21

81

62

225

175

An application of genetic algorithm method for solving patrol manpower deployment problems

4.3 Physical-contact goals

4.4.2 Total sight-contact goal

4.3.1 Segment-wise physical-contact goals ⎛ ⎜ ⎝

⎛ ⎛ 175 225 154 206 ⎞ ⎞ ⎜ ⎜x + x + x + x ⎟⎟ 12 13 14 ⎜ ⎜ 11 ⎟⎟ ⎜ ⎜ 174 221 150 205 ⎟ ⎟ + + + ⎜ ⎜+ ⎟⎟ x21 x22 x23 x24 ⎟ ⎟ ⎜ ⎜ μ 24 : ⎜1.49 − 0.0005 ⎜ 169 277 127 187 ⎟ ⎟ ⎜ + + + ⎜+ ⎟⎟ ⎜ ⎜ x31 x32 x33 x34 ⎟ ⎟ ⎜ ⎜ 165 210 116 175 ⎟ ⎟ ⎜ ⎜⎜ + ⎟⎟ ⎟ + + + ⎜ ⎟ ⎝ x41 x42 x43 x44 ⎠ ⎠ ⎝

⎛ 60 82 52 69 ⎞ ⎞ + + + ⎟ ⎟⎟ ⎝ x11 x12 x13 x14 ⎠ ⎠

μ15 : ⎜1.96 − 0.008 ⎜

− + + d15 − d15 = 1,

⎛ ⎛ 60 77 50 66 ⎞ ⎞ μ16 : ⎜ 2.07 − 0.008 ⎜ + + + ⎟ ⎟⎟ ⎜ ⎝ x21 x22 x23 x24 ⎠ ⎠ ⎝ − + + d16 − d16 = 1,

(26)

⎛ ⎛ 57 75 46 62 ⎞ ⎞ μ17 : ⎜ 2.24 − 0.01⎜ + + + ⎟ ⎟⎟ ⎜ ⎝ x31 x32 x33 x34 ⎠ ⎠ ⎝ ⎛ ⎜ ⎝

⎛ 55 71 42 62 + + + ⎝ x41 x42 x43 x44

+

− d18

+ − d18

− + + d 24 − d 24 =1

μ 25 : (17.27 − 0.0045(75( x11 + x12 + x13 + x14 )

⎞⎞ ⎟ ⎟⎟ ⎠⎠

+ 80( x21 + x22 + x23 + x24 )

=1

− + + 70( x41 + x42 + x43 + x44 )) + d 25 − d 25 =1

Now, addressing four priority factors Pr (r = 1, 2, 3, 4) and following the procedure, the FGP model of the problem for achievement of the aspired levels of the stated fuzzy goals can be obtained by (3). The executable FGP model appears as: Find {xij | i = 1, 2, 3, 4; j = 1, 2, 3, 4} so as to: Minimise:

4.3.2 Total physical-contact goal ⎛ ⎛ 60 82 52 69 ⎜ ⎜x +x +x +x 12 13 14 ⎜ ⎜ 11 ⎜ ⎜ 60 77 50 66 + + + ⎜ ⎜+ x21 x22 x23 x24 ⎜ ⎜ μ19 : ⎜ 2.24 − 0.002 ⎜ 57 75 46 62 ⎜ + + + ⎜+ ⎜ ⎜ x31 x32 x33 x34 ⎜ ⎜ 55 71 42 62 ⎜ ⎜⎜ + + + + ⎜ ⎝ x41 x42 x43 x44 ⎝ +

+ − d19

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟ ⎟ ⎠⎠

(27)

(

Z = ⎡⎣ P1 0.0625d1− + 0.25d 2− + 0.25d3− + 0.25d 4−

(

− − − − + 1.96d11 + 1.81d12 + 2.1d13 P2 1.81d10

=1

− + 0.56d14 ) , P3 0.008d15− + 0.008d16− + 0.01d17−

(

+

(

⎛ ⎜ ⎝

= 1, ⎛ 174

μ 21 : ⎜1.43 − 0.001⎜

⎝ x21

+

221 150 205 ⎞ ⎞ + + ⎟⎟ x22 x23 x24 ⎠ ⎟⎠

− + + d 21 − d 21 = 1,

⎛ ⎛ 169 212 127 187 ⎞ ⎞ μ 22 : ⎜1.33 − 0.001⎜ + + + ⎟ ⎟⎟ ⎜ ⎝ x31 x32 x33 x34 ⎠ ⎠ ⎝ − + + d 22 − d 22 = 1,

⎛ ⎛ 165 210 116 175 ⎞ ⎞ + + + μ 23 : ⎜1.32 − 0.001⎜ ⎟ ⎟⎟ ⎜ ⎝ x41 x42 x43 x44 ⎠ ⎠ ⎝ − + + d 23 − d 23 =1

− + 0.0025d19

− + 0.0018d 20

(31)

− + 0.0018d 21

)

)

− ⎤ P4 0.0045d 25 ⎦,

⎛ ⎛ 175 225 154 206 ⎞ ⎞ + + + μ 20 : ⎜1.43 − 0.001⎜ ⎟ ⎟⎟ ⎜ ⎝ x11 x12 x13 x14 ⎠ ⎠ ⎝ +

− 0.01d18

− − − , + 0.0019d 22 + 0.002d 23 + 0.00052d 24

4.4.1 Segment-wise sight-contact goals

+ − d 20

)

+ 0.25d5− + 0.25d 6− + 0.25d 7− + 0.25d8− + 0.25d9− ,

4.4 Sight-contact goals

− d 20

(30)

+ 65( x31 + x32 + x33 + x34 )

26

− d19

(29)

4.5 Cash expenditure goal

− + + d17 − d17 = 1,

μ18 : ⎜ 2.65 − 0.01⎜

57

(28)

and satisfy the defined membership goal expressions in (21) to (30), subject to the system constraint in (20). Now, to employing the proposed GA scheme, the achievement function Z in (31) appears as the fitness function in the process of solving the problem. The number of generations = 300 is initially taken into account to conduct the experiment. In the genetic search process, the following parameter values are introduced. •

probability of crossover Pc = 0.8

•

probability of mutation Pm = 0.08

•

population size = 100

•

chromosome length = 220.

58

B.B. Pal et al.

The GA-based programme is designed in Programming Language C++. The execution is done in an Intel Pentium IV with 2.66 GHz clock-pulse and 1 GB RAM. The optimal solution is reached at 200 generations. The model solution is presented in Table 6. Table 6

Patrolmen allocation under the proposed approach Shift-wise allocation

6 am to 10 am

10 am to 2 pm

2 pm to 6 pm

6 pm to 10 pm

BT Road

3

5

2

4

AJC Road

3

4

2

5

Strand Road

3

3

2

3

EM By-pass

3

2

2

3

Road segment area

Priority

Table 8

Patrolmen allocation under the additive-FGP approach Shift-wise allocation

10 am to 2 pm

2 pm to 6 pm

6 pm to 10 pm

BT Road

1

4

5

2

AJC Road

3

5

1

1

Fuzzy goal achievement

Strand Road

5

1

1

1

EM By-pass

1

1

1

5

Resultant membership values and goal achievement Membership value

Note: If the additive- FGP approach studied by Tiwari et al. (1987) is used to solve the problem without linearising the defined fractional goals, where maximisation of sum of the defined membership functions μκ subject to μκ ≤ 1 and the system constraints in (20) is considered, then the solution obtained by employing the software LINGO (var. 6.0) is presented in Table 8.

6 am to 10 am

The resultant membership values and achievement of goals under the given priority structure are presented in Table7. Table7

The result shows that a most satisfactory decision is obtained here under the proposed FGP approach in term of achieving the aspired fuzzy goal levels of the objectives of the problem.

P1

µk = 1, k = 1, 2, ..., 9

All goals are fully achieved.

P2

µ10 = 0.99, µ11 = 0.99, µ12 = 0.83, µ13 = 0.80, µ14 = 0.94

Under achievement of the accident reduction goal of the four road-segment areas1, 2, 3 and 4 are 0.14%, 0.25%, 10% and 9.9%, respectively. Under achievement of the total accident reduction goal is 1.8%.

P3

µk = 1, k = 15, 16, ..., 24

All physical-contact and sight-contact goals are overly achieved.

P4

µ25 = 0.97

Increase of the cash expenditure is 0.0014%.

Road segment area

The graphical representation of the patrolmen allocation under the additive-FGP approach is shown in Figure 7. Figure 7

Graphical representation of the solution under the additive-FGP approach (see online version for colours)

The graphical representation of the model solutions is displayed in Figure 6. Figure 6

Graphical representation of the model solution (see online version for colours)

A comparison of the model solution with the result in Table 8 shows that a better patrol manpower deployment plan is achieved here under the proposed approach in the decision-making environment.

5

Conclusions

The main advantage of using the proposed GA-based solution approach is that the computational complexity arises with fractional objectives as well as computational load inherently involved there owing to the use of conventional linear approximation technique to solving the problem can be avoided here with the use of genetic search process. Again, since various objectives involved with the

An application of genetic algorithm method for solving patrol manpower deployment problems problem often conflict each other in achieving the aspired goal levels, the use of the GA as a global solution search method offers the most satisfactory decision in the decision making environment. Again, within the framework of the proposed FGP model, the other different patrolmen deployment measures, if needed there in a traffic control situation, can easily be incorporated without involving any computational difficulty. The problem of how the patrol manpower deployment planning can be made for controlling traffic in different emergency situations may be the problem in future studies. However, it is hoped that the solution approach presented here can contribute to future research works for developing more effective patrol manpower deployment strategies in different traffic control situations in the current complex traffic management systems.

Acknowledgements The authors are thankful to the anonymous reviewers as well as the Editor-in-Charge Professor K. Ohkura of IJBIC (special issue) for their valuable comments and suggestions which have led to improve the quality of presentation of the paper. The authors are also thankful to the Kolkata Traffic Police Department, Government of West Bengal, India, for providing data and various supports to carry out the study.

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