Lexicographic Goal Programming Model for Police Patrol ... - CiteSeerX

Information and Management Sciences Volume 18, Number 2, pp. 173-188, 2007

Lexicographic Goal Programming Model for Police Patrol Cars Deployment in Metropolitan Cities Dinesh K. Sharma

Debasis Ghosh

University of Maryland Eastern Shore

National Informatics Centre

U.S.A.

Ministry of Communications & Information Technology India Avinash Gaur UP Technical University India

Abstract In this paper, we propose an optimal deployment of police patrol cars for the department of traffic police. The lexicographic goal programming model was used to formulate the police patrol car deployment problem. In this study, different road segments and shifts were considered. To demonstrate the model, we have focused on the metropolitan city, Delhi (Central), as a case study. Sensitivity analysis on the budget and number of patrol cars has been performed to identify the best possible solution.

Keywords: Lexicographic Goal Programming, Manpower Allocation, Police Patrol Car.

1. Introduction The police patrol car (PPC) deployment model is designed to help traffic police departments determine the number of patrol cars to have on duty per shift and road segment. In the department, a patrol is defined as a period of time, generally eight hours, when a PPC is on duty. To deploy the patrol cars initially, information is required; such as the number of cars assigned, the average fraction of time cars are busy on service calls, the average number of cars available to respond to calls, and the average total response time. The PPC allocation model deploys car-hours to shifts, where a shift is defined as a combination of a specific patrol on a designated day for a road segment. To avoid traffic Received May 2006; Revised and Accepted October 2006. Supported by ours.

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violations, disorders and accidents, the patrol cars are deployed according to accident frequency and traffic density for each road segment and shift. The initial deployment may be redistributed if a particular road segment is found to have a higher accident occurrence than originals estimated. As a result, the goal is to deploy the patrol cars optimally from the asset. Many models have been developed to address this problem [1-4]. Chaiken and Dormont [5, 6] developed a model for the deployment of the urban police car. In their study, the deployment is treated as a queuing system and the patrolmen may not be available until called. Thus, the queuing system model is not applicable because the initial deployment of patrolmen in metropolitan cities is the primary issue and the reply to the request for patrolmen is the secondary issue. Hanna and Gentel [7] indicated that the number of traffic violations and accidents decreased as the number of patrolmen assigned to an area increased. Their model is difficult to implement with a small budget and limited number of patrolmen. In these complex decision making situations, multiobjective optimization techniques such as lexicographic goal programming (LGP) are more appropriate for problem solving. The LGP is an extension of linear programming (LP), was originally introduced by Charnes and Cooper [8] and further presented by Ijiri [9], Lee [10], Ignizio [11], and others. This technique was developed to handle multi-criteria situations within the general framework of LP. The essence of this technique is the achievement of a “satisfactory” solution, which comes closest to meeting the stated goals according to the given constraints of the problem. In LGP, instead of trying to optimize the objective function directly, the deviation between the goals and what can be achieved within the given set of constraints are minimized. Using LGP, a number of studies have been made on police patrol deployment problems. Lee et al. [12] have developed a model for the deployment of patrolmen in Nebraska State. Taylor et al. [13] and Basu and Ghosh [14] used a non-linear relationship for the accident rate reduction and estimated the result using two-point estimation. However, in two-point estimation, the standard error is relatively large. Hence, the best allocation may not be possible. In this paper, we present a lexicographic goal programming (LGP) model for the optimal deployment of police patrol cars for the department of traffic police in a metropolitan city, Delhi (Central), India. In the study, different road segments and shifts were considered. To demonstrate the model, we have used estimated data from the traffic police

Lexicographic Goal Programming Model for Police Patrol Cars Deployment in Metropolitan Cities

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patrol department. Sensitivity analysis on the budget and number of patrol cars has been performed to identify the best possible solution. The remainder of the paper is organized as follows. Section 2 describes the LGP model formulation for the problem. Section 3 demonstrates the proposed model via a case example. Section 4 presents the results obtained from the case example and identifies the most acceptable solution in the decision-making process. Section 5 concludes.

2. Model Formulation The general LGP as defined by Ignizio [11] can be presented as: Find X(x1 , x2 , x3 , . . . . . . , xn ) so as to M inimize

:

− − P1 (w1+ d+ 1 + w 1 d1 )

M inimize

:

− − P2 (w2+ d+ 2 + w 2 d2 )

....................................... ....................................... M inimize

:

− − Pi (wi+ d+ i + w i di )

....................................... ....................................... M inimize

:

+ + − − Pm (wm dm + w m dm )

Subject to, + fi (X) + d− i − di = bi + and xj ≥ 0, d− i , di = 0

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

Here, m represents number of goal constraints, b i represents the target level of the i-th goal, X represents the vector of n-decision variables, w i+ and wi− (≥ 0) represents the numerically weights associated with the under-and over-deviational variables d − i and d+ i (≥ 0) respectively. In the police patrol cars (PPCs) problem, the city road segments have been divided into three types: City-link Road Segments, Intra-city Road Segments, and By-passes. The total working hours of each PPC is divided into three equal parts called shifts. In the LGP model of the problem, the following notations are defined.

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2.1 Indices i

index for the road segment {i = 1, 2, . . . , I}.

j

index for the shift {j = 1, 2, . . . , J).

2.2 Variables and parameters Xij

The number of police patrol cars assigned to road segment i in shift j.

R

Total number of road segments in a city.

S

Total number of shifts in a day.

P

Total number of patrol cars available.

B

Total available budget per day for all patrol cars.

Ci

Average daily budget for each patrol car in road segment i.

aij

Estimated value for the minimum requirement of patrol cars for road segment i in shift j.

bij

Estimated value for the maximum requirement of patrol cars for road segment i in shift j.

Lt

Total number of patrol-cars at the t-th link road segments (t = 1, 2, . . . , T ).

Du

Number of road segments having same traffic density.

Mi

The minimum number of police patrol cars assigned to road segment i.

2.3 The goals The goals set in order of importance can be defined as follows: (i) minimize the allocation of police patrol cars (ii) minimize the total budget (iii) ensure sufficient number of cars for busy road segments during a specific period (iv) satisfy lesser busy road segments and minimum shift requirements 2.4 Goal constraints In order to formulate the model of the problem, the following goal constraints have been considered. 1. Patrol Cars The total available patrol cars are assigned to the road segments during the shifts


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every day. The total patrol cars attainment goal can be expressed as: I X J X

+ Xij + d− 1 − d1 = P.

(1)

i=1 j=1

2. Budget The management provides daily allowance to each PPC for fuel charges. The allowance depends upon the road length and surface condition of the road segments. The goal equation for the total budget can be expressed as: I X J X

+ Ci Xij + d− 2 − d2 = B.

(2)

i=1 j=1

3. Lower and Upper Bounds To ensure enough patrol cars in all road segments, there should be lower and upper bounds on the patrol cars on those road segments. Bounds on the decision variables are given according to the accident frequency and traffic density in each road segment and shift. So, the goal equations can be written as: + Xij + d− p − dp = aij ,

(3)

+ Xij + d− q − dq = bij

(4)

where i ∈ {1, 2, . . . , I}, j ∈ {1, 2, . . . , J}, p = 3, 4, . . . , IJ and q = IJ + 1, . . . , 2IJ. 4. Minimum Shift Requirement The traffic in the metropolitan cities cannot be estimated uniformly because road segments have different traffic density and accident frequency. The average number of cars available to respond to calls for service depends on the preventive patrol frequency, i.e. travel time from the dispatch of a patrol car until its arrival at the incident scene, the probability that a call will enter in a queue, average time in queue calls, and the average total response time. The minimum number of patrol cars is assigned according to the traffic density and accident frequency in each road segment. For minimum shift requirements, the following road segments are considered. (i) Road segments which are linked together J X

+ (Xtj + Xtj + · · · + Xtj ) + d− p − dp = Lt ,

j=1

where t ∈ {1, 2, . . . , I} and p = 2IJ + 1, . . . , I(2J + 1).

∀t

(5)

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(ii) Road segments having same traffic density m X

+ (Xuj + Xuj + · · · + Xuj ) + d− p − dp = Du ,

∀u

(6)

j=1

where m ∈ {1, 2, . . . , J}, u ∈ {1, 2, . . . , I} and p = 2IJ + I + 1, . . . , 2I(J + 1). (iii) Each road segment J X

+ Xij + d− p − dp = Mi ,

∀i

(7)

j=1

where p = 2I(J + 1) + 1, . . . , I(2J + 3). To demonstrate the usefulness of the proposed LGP model, the following case study has been considered.

3. Case Study Delhi is a metropolis in northern India. It borders the Indian state of Uttar Pradesh on the south and Haryana on the west. In Delhi, traffic is more complicated when compared to other cities in India. With incremental growth in the population and improvements in styles of living, the number of vehicles is increasing drastically. Heavy traffic on the roads has increased traffic violations and accidents, making travel inconvenient for citizens in the city. It is the responsibility of traffic police departments to make proper rules and regulations for vehicle drivers. To encourage drivers to follow rules and avoid traffic violations, disorders, and accidents, the traffic police departments allocate their police patrol cars (PPCs) for patrolling busy roads at busy times to other roads. In these cases, departments will try to follow the optimal allocation of their cars to patrol roads so that PPCs can attend calls from any road segment in each shift. In this study, we have estimated the data after discussions with the Delhi (Central) traffic police patrol department. Most of the road segments have high traffic density because of the limited road area. The city road segments were divided into three types as follows: (i) City-link Road Segments: The city-link road segments are the only way to enter the city. These road segments always have high traffic density.


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(ii) Intra-city Road Segments: The intra-city road segments usually have high traffic density between 8 A.M. to 9 P.M. These roads are the only way to move within the city. (iii) By-passes The bypasses connect various parts of the city to other nearby states or cities to avoid traffic violations and high traffic density. The total working hours of a PPC is divided into three equal parts called shifts. The shifts are divided as: First Shift

: Xi1 = Between 6 am to 2 pm

Second Shift

: Xi2 = Between 2 pm to 10 pm

Third Shift

: Xi3 = Between 10 pm to 6 am

In the LGP, all constraints are considered as flexible goal constraints by introducing under-deviational and over-deviational variables to each of the goals, and priorities are assigned to the goals according to their importance in the decision making situation. The decision variables and the number of roads considered in the problem formulation are defined below: Table 1. Decision Variable Description. Xij The number of patrol cars assigned to the road segment i in the shift j City-link Road Segment 1 X1j G.T. Road 2 X2j Gurgaon Road 3 X3j Mathura Road 4 X4j Rohtak Road Intra-city 1 X5j Aurobindo Marg 2 X6j Baba Kharag Singh Marg 3 X7j Gurugovind Singh Marg 4 X8j Lodhi Road 5 X9j J.L. Nehru Marg 6 X10j Janpath 7 X11j L.B. Shastri Marg 8 X12j M.G. Marg 9 X13j Sansad Marg 10 X14j S. Bhagat Singh Marg 11 X15j Sardar Patel Marg 12 X16j Zakir Hussain Marg By-pass 1 X17j Outer Ring Road

To formulate the model, the following goal constraints have been considered.

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3.1. Constraints Patrol Cars A total of 60 patrol cars are assigned to the 17 road segments (i = 1, 2, . . . , 17) during first, second, and third shift (j = 1, 2, 3) every day. The total patrol cars goal can be expressed as: 17 X 3 X

+ Xij + d− 1 − d1 = 60.

(1)

i=1 j=1

Budget

The management provides a daily allowance to each PPC, which includes fuel charges, and maintenance of the PPC. The allowance depends upon the road length and surface condition of the road segments. The maximum available daily total budget is Rs. 5500. The goal equation for the total budget can be expressed as: 3 17 X X

+ Ci Xij + d− 2 − d2 = 5500.

(2)

i=1 j=1

Table 2. Variable Costs (Road Segment wise). Road Segment No. (i) 1 2 3 4 5 6 7 8 9

Variable Cost (Ci ) 115 115 115 112 112 100 100 100 112

Road Segment No. (i) 10 11 12 13 14 15 16 17

Variable Cost (Ci ) 115 115 112 112 112 112 115 115

Lower and Upper Bounds In Delhi (Central), the total number of patrol cars is 60. To ensure patrol cars are on important busy road segments, there should be minimum number of patrol cars on those road segments. Thus, the bounds on the decision variables, according to the accident frequency and traffic density in each road segment in each shift, can be written as a ≤ Xij ≤ b, where a and b are the estimated values for the mini/max requirement of the patrol cars at the road segment i at the shift j. In the present problem, it can be written as: 2 ≤ Xi1 ≤ 4,

i = 1, 2, 3, 4, 10, 14


1 ≤ Xi1 ≤ 3,

i = 5, 6, 7, 11, 12, 13, 15, 16

1 ≤ Xi1 ≤ 2,

i = 8, 9, 17

2 ≤ Xi2 ≤ 4,

i = 1, 2, 3, 10, 14

1 ≤ Xi2 ≤ 3,

i = 4, 5, 6, 8, 11, 12, 13, 16

1 ≤ Xi2 ≤ 2,

i = 7, 9, 15, 17

0 ≤ Xi3 ≤ 4,

i = 10

0 ≤ Xi3 ≤ 3,

i = 1, 15

0 ≤ Xi3 ≤ 2,

i = 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17

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The above defined bounds can be presented as the following goal equations: + X11 + d− 3 − d3 = 4

X21 + X31 + X41 +

d− 5 d− 7 d− 9

− − −

d+ 5 d+ 7 d+ 9

(3)

+ X11 + d− 4 − d4 = 2

(4)

=4

(5)

(6)

=4

(7)

=4

(9)

+ X21 + d− 6 − d6 = 2 + X31 + d− 8 − d8 = 2 + X41 + d− 10 − d10 = 2 + X10,1 + d− 12 − d12 = 2 + X14,1 + d− 14 − d14 = 2 + X51 + d− 16 − d16 = 1 + X61 + d− 18 − d18 = 1 + X71 + d− 20 − d20 = 1 + X11,1 + d− 22 − d22 = 1 + X12,1 + d− 24 − d24 = 1 + X13,1 + d− 26 − d26 = 1 + X15,1 + d− 28 − d28 = 1 + X16,1 + d− 30 − d30 = 1 + X81 + d− 32 − d32 = 1 + X91 + d− 34 − d34 = 1 + X17,1 + d− 36 − d36 = 1 + X12 + d− 38 − d38 = 2 + X22 + d− 40 − d40 = 2 + X32 + d− 42 − d42 = 2 + X10,2 + d− 44 − d44 = 2 + X14,2 + d− 46 − d46 = 2

+ X10,1 + d− 11 − d11 = 4

(11)

+ X14,1 + d− 13 − d13 = 4 + X51 + d− 15 − d15 =3 + X61 + d− 17 − d17 = 3 + X71 + d− 19 − d19 = 3 + X11,1 + d− 21 − d21 = 3 + X12,1 + d− 23 − d23 = 3 + X13,1 + d− 25 − d25 = 3 + X15,1 + d− 27 − d27 = 3 + X16,1 + d− 29 − d29 = 3 + X81 + d− 31 − d31 = 2 + X91 + d− 33 − d33 = 2 + X17,1 + d− 35 − d35 = 2 + X12 + d− 37 − d37 = 4 + X22 + d− 39 − d39 = 4 + X32 + d− 41 − d41 = 4 + X10,2 + d− 43 − d43 = 4 + X14,2 + d− 45 − d45 = 4

(13) (15) (17) (19) (21) (23) (25) (27) (29) (31) (33) (35) (37) (39) (41) (43) (45)

(8) (10) (12) (14) (16) (18) (20) (22) (24) (26) (28) (30) (32) (34) (36) (38) (40) (42) (44) (46)

182


+ X42 + d− 47 − d47 = 3

(47)

+ X42 + d− 48 − d48 = 1

(48)

+ X52 + d− 49 − d49 = 3

(49)

+ X52 + d− 50 − d50 = 1

(50)

(51)

+ X62 + d− 52 − d52 = 1 + X82 + d− 54 − d54 = 1 + X11,2 + d− 56 − d56 = 1 + X12,2 + d− 58 − d58 = 1 + X13,2 + d− 60 − d60 = 1 + X16,2 + d− 62 − d62 = 1 + X72 + d− 64 − d64 = 1 + X92 + d− 66 − d66 = 1 + X15,2 + d− 68 − d68 = 1 + X17,2 + d− 108 − d108 = + X13 + d− 72 − d72 = 3 + X23 + d− 74 − d74 = 2 + X53 + d− 76 − d76 = 2 + X73 + d− 78 − d78 = 2 + X93 + d− 80 − d80 = 2 + X12,3 + d− 82 − d82 = 2 + X14,3 + d− 84 − d84 = 2 + X17,3 + d− 86 − d86 = 2

(52)

+ X62 + d− 51 − d51 = 3 + X82 + d− 53 − d53 = 3 + X11,2 + d− 55 − d55 = 3 + X12,2 + d− 57 − d57 = 3 + X13,2 + d− 59 − d59 = 3 + X16,2 + d− 61 − d61 = 3 + X72 + d− 63 − d63 = 2 + X92 + d− 65 − d65 = 2 + X15,2 + d− 67 − d67 = 2 + X17,2 + d− 69 − d69 = 2 + X10,3 + d− 71 − d71 = 4 + X15,3 + d− 73 − d73 = 3 + X43 + d− 75 − d75 = 4 + X63 + d− 77 − d77 = 2 + X83 + d− 79 − d79 = 2 + X11,3 + d− 81 − d81 = 2 + X13,3 + d− 83 − d83 = 2 + X16,3 + d− 85 − d85 = 2

(53) (55) (57) (59) (61) (63) (65) (67) (69) (71) (73) (75) (77) (79) (81) (83) (85)

(54) (56) (58) (60) (62) (64) (66) (68) 1

(70) (72) (74) (76) (78) (80) (82) (84) (86)

Minimum Shift Requirement The traffic in the metropolitan cities cannot be estimated uniformly. All the road segments have different traffic densities and accident frequencies. Hence, a minimum number of patrol cars are assigned to each road segment according to the traffic density and accident frequency of each segment. (i) Road segments which are linked together: 2 X

+ (X7j + X11j + X16j ) + d− 87 − d87 = 22

(87)

+ (X5j + X15j ) + d− 88 − d88 = 10

(88)

+ (X8j + X17j ) + d− 89 − d89 = 36

(89)

j=1 3 X

j=1 3 X

j=1


3 X

183

+ (X4j + X8j ) + d− 90 − d90 = 28

(90)

+ (X17j + X18j + X24j ) + d− 91 − d91 = 22

(91)

+ (X3j + X11j + X12j ) + d− 92 − d92 = 8

(92)

+ (X3j + X13j ) + d− 93 − d93 = 8

(93)

+ (X5j + X13j ) + d− 94 − d94 = 26

(94)

+ (X1j + X9j + X14j ) + d− 95 − d95 = 16

(95)

+ (X6j + X10j ) + d− 96 − d96 = 16

(96)

j=1 3 X

j=1 2 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1

(ii) Road segments having the same (approximately) traffic density: 2 X

+ (X2j + X5j + X11j ) + d− 97 − d97 = 18

(97)

+ (X1j + X4j ) + d− 98 − d98 = 16

(98)

+ (X6j + X10j ) + d− 99 − d99 = 22

(99)

j=1 3 X

j=1 3 X

j=1 3 X

+ (X15j + X16j ) + d− 100 − d100 = 20

(100)

j=1

(iii) Each road segment: 3 X

+ (X1j ) + d− 101 − d101 = 18

(101)

+ (X2j ) + d− 102 − d102 = 16

(102)

+ (X3j ) + d− 103 − d103 = 18

(103)

+ (X4j ) + d− 104 − d104 = 15

(104)

j=1 3 X

j=1 3 X

j=1 3 X

j=1

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3 X

+ (X5j ) + d− 105 − d105 = 14

(105)

+ (X6j ) + d− 106 − d106 = 19

(106)

+ (X7j ) + d− 107 − d107 = 19

(107)

+ (X8j ) + d− 108 − d108 = 9

(108)

+ (X9j ) + d− 109 − d109 = 9

(109)

+ (X10j ) + d− 110 − d110 = 18

(110)

+ (X11j ) + d− 111 − d111 = 8

(111)

+ (X12j ) + d− 112 − d112 = 16

(112)

+ (X13j ) + d− 113 − d113 = 6

(113)

+ (X14j ) + d− 114 − d114 = 12

(114)

+ (X15j ) + d− 115 − d115 = 8

(115)

+ (X16j ) + d− 116 − d116 = 10

(116)

+ (X17j ) + d− 117 − d117 = 6

(117)

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 3 X

j=1 + and Xij , d− k , dk ≥ 0 where 1 ≤ i ≤ 28, 1 ≤ j ≤ 3, 1 ≤ k ≤ 117.

Priority Definition The priority structure of the model is defined as follows: P1: minimize the allocation of police patrol cars P2: minimize the total budget


185

P3: ensure sufficient number of cars for busy road segments during a specific period P4: satisfy lesser busy road segments and minimum shift requirements Priority Structure M inimize P1 (d+ 1) M inimize P2 (d+ 2) M inimize P3 2

13 X

d+ i +

i=3

M inimize P4 2

35 X

d+ i

i=31

62 61 30 29 46 45 X X X X X X + − + − + − d− di + dj + di + dj + di + dj + j j=48 i=47 j=16 i=15 j=38 i=37 j=4 14 X

+

!

36 X

d− j

+

j=32

73 X

d+ i

i=63

+

72 X

d− j

j=64

+

86 X

d+ l

l=74

+

117 X

d+ l

l=87

!

.

Where i represent all odd suffixed numbers in the deviational variables, j represents all even suffixed numbers and l represents all continuous numbers.

4. Computational Results The problem was executed using the GP package developed in Visual C++. The algorithm developed by Ignizio [11] was followed to develop the package. A total 39 iterative steps were executed to obtain the final result. Table 3. Allocation of Patrol Cars in each Road Segment in each Shift. Segment i → Shift j ↓ 1 2 3

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

2 2 1

2 2 0

2 2 0

2 1 0

1 1 0

1 1 0

1 1 0

1 1 0

1 1 0

2 2 0

1 1 0

1 1 0

1 1 0

2 2 0

1 1 0

1 1 0

1 1 0

5. Sensitivity Analysis The result in Table 3 shows the possible allocation of patrol cars to the road segments stated above. The sensitivity analysis on budget and patrol cars has been done. Table 4 shows the variation on budget amount, the number of patrol cars and accordingly deviations in the result.

186


Table 4. Sensitivity Analysis. Variation of the number of Patrol Cars 40

Variation of budget amount 5250

50 60

5250 5000

60

5250

Deviation of Results There is no such allocation made for road segments X72 , X81 , X91 , X15,2 and X17,1 . All allocation has been made as in Table 3. There is no such allocation made for road segments X72 and X81 . There is not much difference in allocation of cars as in Table 3

Table 4 shows that the best solution has a requirement of a budget of Rs.5250 and patrol 50 cars. The priority achievements are given in the Table 5. Table 5. Priority Achievement. Priority P1 P2 P3 P4

Achievements Achieved Achieved Achieved Not Achieved

Conclusions Reduces number of patrol cars from 60 to 50. Reduces total budget from Rs. 5500 to Rs. 5250. Satisfy high density road requirements during peak periods Requirement at the third shift and related goals are not achieved

6. Conclusion This paper presents the development and testing of a lexicographic goal programming (LGP) approach for the deployment of police patrol cars (PPCs) to road segments in varying shifts. Although it is not possible to obtain a guaranteed optimal solution, we demonstrate that a satisfactory solution can be achieved. Other factors have not been included in the present model due to the non-availability of data. The present study was done to demonstrate the role of LGP in a PPC deployment problem, and showing the reallocation of the PPCs at different road segments in shifts to minimize traffic violations and accidents. The model may be further utilized to help management of traffic police departments in order to achieve optimal allocation of existing PPCs depending upon given goal constraints and priorities. The solution shows that all the goals have been achieved given the importance of various road segments. A similar model can also be applied in other cities. This model can assist the administration in their planning of manpower allocations. The model may be improved by adding factors like accident reduction goal and other safety factors to the original model. The priority structure may


187

also be modified in that case.

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Authors’ Information Dr. Dinesh K. Sharma received his Ph.D. in Operations Research from the Chaudhary Charan Singh University at Meerut, India. He is currently a Professor of Quantitative methods & Computer Applications in the Department of Business, Management & Accounting at the University of Maryland Eastern Shore. His research interests include system design and analysis, multi-objective programming, nonlinear programming, and application of operations research to business & industry. He has over 60 journal publications. Some of Dr. Sharma’s journal publications are in Yugoslav Journal of Operations Research, The Spanish Journal of Operations Research, Journal of Business and Information Technology, Journal of Applied Mathematics and Computing, International Journal of Production Economics, International Journal of Logistics, International Modelling and Simulation to name a few. He is a member of Decision Sciences Institute and Operational Research Society of India. Department of Business, Management & Accounting University of Maryland Eastern Shore, Princess Anne, MD 21853, U.S.A. Dr. Debasis Ghosh is a scientist with the National Informatics Center, Ministry of Communication and Information Technology, Government of India. He received his Ph.D. degree form the University

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of Kalyani, India. His published research work include papers on multi-objective programming, inventory theory, transportation problem and spatial database system which have appeared in OPSEARCH, Optimization, Journal of Applied Mathematics and Computing, International Journal of Production Economics, International Journal of Logistics, International Journal of Modelling and Simulation, The Spanish Journal of Operations Research, Journal of Business and Information Technology, International Journal of Information Technology and Decision Making. National Informatics Centre, Ministry of Communications & Information Technology, Government of India, Calcutta, West Bengal, India. Dr. Avinash Gaur is an Assistant Professor in the Department of Applied Sciences and Humanities at RKG Institute of Technology (U.P. Technical University), Ghaziabad, U.P. (India). He received his Ph.D. in Mathematics from C.C.S. University at Meerut, U.P. (India). His research has focused on Mathematical programming and applied business studies. Dr. Gaur is a member of Operations research Society of India (ORSI) and he is also reviewer of some reputed International Journal. Department of Applied Sciences & Humanities (Mathematics), RKG Institute of Technology (UP Technical University), Ghaziabad, UP, India.