- Email: [email protected]
A practical heuristic for a large scale vehicle routing problem
32
European Journal of Operational Research 57 (1992) 32-38 North-Holland
Theory and Methodology
A practical heuristic for a large scale wehicle routing problem Anirudh Raghavendra, T.S. Krishnakumar, R. Muralidhar, D. Sarvanan B.M.S. College of Engineering, Bangalore - 560019, India
B.G. Raghavendra Department of Management Studies, Indian Institute of Science, Bangalore - 560012, India Received January 1989; revised February 1990
Abstract: We describe a variant of the traditional vehicle routing problem (VRP) associated with the pickup and dropping of employees of large organisations. The human factors involved result in a number of additional constraints to the VRP which are specific to this kind of situation. We have developed a heuristic algorithm which solves this type of VRP effectively. The adaptability of the algorithm is demonstrated through a real problem of a large public sector organisation in the city of Bangalore in India involving the daily transportation of about 12 000 employees from and to 410 pickup points in and around the city. The use of our heuristic method has indicated a savings of about 1400 km per day out of the daily coverage of 14 000 km by a fleet of 75 buses. Additional advantages include a reduction in the number of buses and drivers required per shift, improved seat utilisation and better fuel management. Keywords: Vehicle routing, transportation, heuristics, human factor constraints.
I. Introduction
Considerable research attention has been focussed on the vehicle routing and scheduling problem and its variants, especially in the last two decades. Several hundreds of papers have appeared in the literature describing the various versions of this type of problems and the solution seeking methods for them. An excellent survey paper highlighting the state-of-the-art, with references to nearly 700 papers, appeared in 1983 (Bodin et al. [2]). A number of journals have devoted special issues for the modelling and solution methods of these problems. A survey of time
window constrained routing and scheduling problems, to which the problem described in this paper partly belongs, has appeared relatively recently (Solomon and Desrosiers [5]). Yet, the diversity of the problems as reflected in the variety of applications, the non-availability of exact algorithms for solving many real world problems for optimal solutions and, most importantly, the significance of their relative potential in real world situations continue to make this category of problems one of special interest. A number of factors collectively describe a vehicle routing and scheduling problem. The size and type of the available vehicle fleet, the nature
0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A. Raghavendra et al. / Practical heuristic for a large scale VRP
and location of demands, vehicle capacity constraints, limitations on the direction of traveling and maximum routing-lengths/times, variation in operations, cost structure and the objective characterises each vehicle routing and scheduling problem (Bodin and Golden [1]). Since the resulting mathematical model invariably consists of a large number of integer 0-1 variables, obtaining a globally optimum solution is generally hard. Consequently, a wide variety of heuristic based procedures have been suggested. The existing solution strategies could be broadly categorised (see Bodin and Golden [1]) into the following five categories: (i) cluster first route second, (ii) route first cluster second, (iii) savings/insertion approach, (iv) interactive optimisation, and (v) mathematical programming based methods. Inspite of the discussion of several varieties of the vehicle routing and scheduling problem in literature, there are some real world problems that have not come to the attention of researchers but assume special significance due to their magnitude and complexity. The subject matter of this paper is one such problem concerning the pickup and dropping of employees of a large scale public sector organisation in India. Any optimality criteria suggested here should not only consider the objective of least cost transportation of the personnel but also human factors related to distances and duration of travel of the individual employees. This kind of vehicle routing problem (VRP) is analogous to the school bus problem (see Desrosiers et al. [3]), except that the size (with respect to the number of variables) and magnitude (with respect to the number of vehicles and nodes) is considerably high. It is also implicitly analogous to time window constrained routing problems (Solomon and Desrosiers [5]). We have developed a heuristic algorithm taking into explicit consideration the specific constraints related to these human factors. No claim is made on the efficiency of the heuristic in comparison with any other method. However, its effectiveness as compared to the existing practice in the real situation is highlighted. Details of the human factor related constraints, the algorithm and the results obtained are presented in this paper. In section 2, a profile of the case study organisation is given. Section 3 provides a description of the additional constraints associated with the human factors. In
33
Section 4 we briefly describe the algorithm, while in the following section the results obtained for the case study organisation are discussed. 2. Profile of the case study organisation
This study was conducted in a large public sector organisation of the Government of India having its headquarters at Bangalore, manufacturing telephones and a variety of telecommunication equipments. Its main manufacturing unit is situated on the outskirts of Bangalore while there are five other units elsewhere in India. The unit at Bangalore has six manufacturing divisions and two R & D wings attached to it. Its employee strength is around 18000 personnel. Approximately a third of the work force is provided housing facilities in a colony adjoining the factory premises. About 12000 of the remaining employees stay in different parts of the city. As a welfare measure, the organisation provides buses to pick up these employees for work and return them to their respective destinations after work. The manufacturing unit at Bangalore and the associated administrative offices work in five different shifts each day, six days a week. Of the five shifts, the one in the night is a lean shift with a small number of employees of the essential services category. In three of the remaining four shifts, the transport department of the organisation is also associated with decisions on sequencing of buses on various routes across the shifts, as the time gap between consecutive shifts is very small. If all the employees work at the same time, the fleet demand would be for at least 200 buses, as the average capacity of a bus is 60. Currently, the transport department has a fleet of 100 buses to satisfy the transportation needs of the 12000 employees. Of these, 75% of the buses are scheduled exclusively for employee pickup and dropping across the shifts. The rest are assigned for late pickups, school buses, emergency services and for maintenance. There are a total of 410 pickup points in and around the city of Bangalore from where the employees have to be picked up (and returned). The average direct distance of these points from the factory is about 16 km, the maximum being 27 km. The demand at the pickup points varies from a minimum of 5 to a maximum of 79. The round trip duration of a route depends upon the distance to be covered, traffic density
A. Raghavendra et al. / Practical heuristic for a large scale VRP
34
along the route and the number of pickup points assigned to it. The current practice of routing and scheduling of the buses is based purely on intuition and experience gained over several years of managing the system. The personnel involved in the allocation and scheduling of the routes had very little exposure to optimisation methods. Details of the various shift timings, the shiftwise demand in terms of the number of persons utilising the transport services, the number of pickup points, the number of routes, distance travelled by all the buses and the seat utilisation percentages are presented in Table 1. In few of the buses, there is a certain amount of overloading, in the sense that a higher number of employees is assigned than the seating capacity of the bus taking into account absenteeism. 3. The human factor constraints
As mentioned earlier, several versions of the formulation of the VRP as a 0-1 integer linear programming problem is available in literature. For ready reference, one version of the VRP (see Fisher and Jaikumar [4]) whose notations are used in this paper, is presented in the appendix. In case of routing of vehicles involving the transportation of employees, the human factors problem leads to the following additional constraints. The description is based on the requirements to be met while picking the employees for work. The
reverse sequence generally holds good for returning them after work. (i) In order to ensure an upper limit on the duration of travel time of the employees who board a bus at its first pickup point, the distance covered by a bus should not exceed a pre-specifled value D k. This is given by Y'~diiXijk<~O k
for all k.
(1)
Since the city is categorised into the main city and suburbs, the value of D k may be higher for routes involving pickup points in the suburbs. This type of indirect route duration restriction is also considered in other time window constrained VRPs (Solomon and Desrosiers [5]), generally to take care of transportation of perishable goods. However, we are including it here as a special constraint due to the human factors involved. (ii) At the time of pickup, an employee would be willing to travel 'inwards' only, i.e. towards the factory, rather than 'outwards', i.e. away from the factory. This is necessary since employees, while going to work, would be unwilling to travel in a direction away from the factory. For any two consecutive pickup points i and j, this requirement can be represented by the condition: djoX, jk >~0
dioXij k -
for all i, j, k.
(2)
(iii) When deviations from the shortest path to the factory are allowed to meet the organisation's objective of higher seat utilisation and lower costs,
Table 1 Details of routes, distances and seat utilisation Shift no.
Timings
I
0615-1415
3659
303
II
0730-1615
3999
313
III
0845-1730
3042
286
IV
1415-2215
975
242
V *
2215-0615
40
.
TOTAL
No. of commuters
11715
No. of pickup points
410 ~
No. of bus routes 64 (58) 66 (64) 53 (49) 30 (28) .
. 213 ÷ (199 ÷)
Distance per trip ÷ (km)
Seat utilisation (%)
1977.0 (1767.8) 2163.0 (1985.5) 1808.3 (1598.0) 1056.7 (949.1)
89.0 (95.6) 94.3 (95.2) 90.0 (96.0) 54.0 (59.0)
. 7005.0 ÷ (6300.4 + )
-
* Ignored in our study. # Distinct points. + Each bus route (trip) is repeated twice a day, once for pickup and once for returning of employees. The figures given are for one-way transport only. Note: Results obtained from the use of the heuristic are presented in parenthesis.
A. Raghavendra et aL / Practical heuristic for a large scale VRP
4. The heuristic
0 -Pick up points 0 -Central depot/factory
O
O
dii
Q (9
®
0
0
(9 O
0
° ~ ' /
®
Q 0
0
O
Figure 1
normally a limit is placed on such deviations. While D k ensures an upper limit on the total route distance covered, a limit on the maximum deviation from each pickup point is necessary. A pickup point j could be linked to a current pickup point i only if the difference between the resulting total distance and the shortest distance, from i to the factory, does not exceed a pre-specified value P. This results in a constraint of the form dijXijl, + djoXjo k - dioXio k <~P
35
for all i, j, k.
(3) This requirement is illustrated in Figure 1, from which it is seen that Xij k can take a value of 1 only if d~j + djo - d~o <~P.
The above mentioned additional constraints (1)-(3) uniquely distinguish this type of VRP from the classical problems addressed in literature. It is apparent that the resulting VRP model in real situations is a complex combinatorial problem, involving a large number of 0-1 variables and several constraints. We propose a heuristic method to tackle this problem, the details of which are described in the next section.
In view of the specific nature of the VRP described above, an intuitive algorithm was developed by us. The algorithm is essentially a tourbuilding method and takes care of the three additional constraints described in the earlier section. The basic input to this routing algorithm consists of two data files, one pertaining to the number of persons boarding the bus (demand) at each pickup point, shiftwise, and the other a distance matrix, giving the distances of direct travel from each pickup point to the other including the central point (factory). The algorithm is an aggregation of several route building procedures followed sequentially, till the entire demand at all the pickup points is met. At each step, the feasibility with respect to the special constraints described earlier as well as additional factors related to capacity utilization of the bus and such other factors are explicitly considered. An overview of this heuristic is presented in the form of a flowchart in Figure 2. Some of the salient features of this heuristic are as follows: (i) The heuristic works inwards, that is, points which are farthest are identified first and routes are developed from there. (ii) Points where the demand equalled or exceeded the capacity of a bus were considered first for exclusive assignment of routes. (iii) By keeping a check on the number of employees already assigned to the bus (NPCHK), the heuristic ensures that once the bus has N P C H K or more commuters, it returns to the factory without deviating, meeting additional demand at pickup points enroute, subject to feasibility. Certain other factors, not mentioned in Figure 2, were also considered. These are as follows: (a) Depending upon the geographical locations, there can be a set of points for which there is only one access or approach. The heuristic ensures that such 'land-locking' pickup points always go together in the sense that if one point is covered by a route the other(s) is also covered by the same route. (b) It is desirable that two vehicles need not go to the same point when one vehicle would satisfy the demand, keeping in view the practical
36
A. Raghavendra et al. / Practical heuristic for a large scale VRP
problems that may arise if some of the people at a particular pickup point are asked to use one bus while the rest use another. Therefore, the heuristic ensures that the demand at any pickup point is not split, unless the demand exceeded the capacity of a bus.
5.
Results
and
discussion
The heuristic described above was coded in FORTRAN on a DEC 1091 computer system. It was used on the data pertaining to each shift of the case study organisation and took approximately 45 s for execution per shift. The basic output consisted details of each route developed, the sequence and demand of pick up points covered, total distance covered by the route and related information. The results obtained for each
shift are summarised and presented in parenthesis in Table 1. Comparing the results obtained from the use of our heuristic with the existing situation (Table 1) the following significant contributions may be observed: There is a substantial decrease in the total distance covered; from 14 010 k m / d a y to 12 600 km/day which is about 10%. Reduction in the number of routes and consequently the number of buses required. Higher percentage of seat utilisation. - O t h e r incidental benefits include reduced maintenance cost for the buses, reduction in the demand for drivers and effective fuel utilisation. A further analysis and comparison of the expected journey time of employees at each pickup point with the existing situation indicated that the average one-way journey time per employee could be reduced by 8 minutes.
DATAFILE1
DATAFILE2
IDENTIFICATION OFPOINTS I WITHDEMAND> BUSCAPACITY ASSIGNROLITE.UPDATE I= tI IDENTIFY FARTHESTPOINT I ASSIGNROUTETOSTART.UPDATE ~
~
~
A]ERMINENEARESTFEASIBLEPOINT I
NO
SEQUENCEOFPICKUPPOINTS, DISTANCE,TIMEOFDEPARTURE ANDARRIVAL Figure 2
A. Raghavendra et al. / Practical heuristic for a large scale VRP
37
Table 2 Results of sensitivity analysis Scenario no.
Percentage absenteesim
NPAB
NPCHK
P (km)
Total distance (kin/day)
Reduction in distance (kin/day)
No. of routes
Reduction in routes (kin/day)
1 2 3 4 5 6 7 8 9
2 2 2 5 5 5 10 10 10
69 69 69 71 71 71 73 73 73
60 60 60 62 62 62 64 64 64
0.0 0.5 1.5 0.0 0.5 1.5 0.0 0.5 1.5
12728.6 12596.8 12574.0 12660.2 12556.0 12503.4 12421.6 12343.0 12224.4
1281.4 1413.2 1436.0 1349.8 1454.0 1506.6 1588.4 1667.0 1785.6
200 197 196 196 193 194 194 191 190
13 16 17 17 20 19 19 22 23
Once a computer program incorporating the heuristic was developed, the effects of varying some of the key decision parameters could be obtained under different scenarios. For example, the permissible deviation ( P ) as illustrated in constraint (3) could be varied. Further, in the basic run of the algorithm, no advantage was taken of the absenteeism factor. To study the effect of changes in some of the above parameters, our heuristic algorithm was used under different values of P, different absenteeism levels (2%, 5%, 10%) resulting in a higher number of persons assigned to a bus (NPAB) and different values of NPCHK. The results obtained under nine such scenarios are presented in Table 2. It may be observed from Table 2 that if the absenteeism factor is considered, the total distance covered and the number of routes would generally reduce further. Similarly allowing for higher deviations [value of P as illustrated in constraint (3)] would also result in less number of routes and lesser distance.
tate modifications, the computer program can be made PC based as a Decision Support System (DSS), so that it can be accessed and easily used by personnel in transport departments with no Operations Research background. Extending from the basic development of effective routing, other problems related to scheduling of vehicles across shifts and crew allocation can also be considered.
Acknowledgement The authors are thankful to the management and staff of Indian Telephone Industries Limited, Bangalore Complex, for all the facilities and cooperation extended during the course of this study. Suggestions and comments from referees on an earlier version of the paper are gratefully acknowledged.
Appendix 6. Conclusion A simple but effective heuristic has been developed for a specific type of VRP involving the transportation of employees. The use of this heuristic in the case study has shown scope for substantial monetary savings. Further extension and improvements in the algorithm to suit different environments are possible. A time constraint has been addressed implicitly in our case study by distance checks. It could be explicitly incorporated as an additional constraint. With appropri-
A model of the traditional VRP, cf. Fisher and Jaikumar [4]. The parameters and the variables are as follows: k = 1, 2 . . . . . K = number of vehicles/routes. i, j = 0, 1. . . . . n = number of pickup points with the convention that 0 (zero) denotes the factory which is the central dropping and pickup point. bk = capacity of vehicle k.
A. Raghavendra et al. / Practical heuristic for a large scale VRP
38
= number of persons alighting/ boarding (demand) at pickup point i (i 4: 0). = distance of direct travel from point i to point j. = m a x i m u m permissible distance for route k. = maximum permissible deviation in distance from any pickup point (measured as the difference between the distance of the direct route linking a pickup point to the factory and that via an added pickup point). 10 if point i is served by = route k, otherwise.
ai
dij
Dk P
Yik
10 if vehicle k is to travel directly from demand x,k point i to j, otherwise. The objective is to minimise (4)
~_~ dijXijk i,j,k S.t.
E ai Yik <~bk i
K,
Yik----- 1,
for all k
i=0, i = 1 . . . . . n,
(5) (6)
Yik = 0 o r 1
x/ik =
for all i, k,
(7)
for all j, k,
(8)
Y'~Xii~ =Yik for all i, k,
(9)
i
J
IS1-1,
Sc{1,...,n},
i,j
2~
Xi~l, = 0 or 1 for all i, j, k.
(10) (11)
References [1] Bodin, L., and Golden, B., "Classification in vehicle routing and scheduling", Networks 11 (1981) 97-108. [2] Bodin, L., Golden, B., Assad, A., and Ball, M., "The state of the art in routing and scheduling of vehicles and crew", Computers and Operations Research 10 (1983) 63-212. [3] Desrosiers, J., Ferland, J-A., Rousseau, J-M., Lapline, G., and Chapleau, "An overview of a school busing system", in: N.K. Jaiswal (ed.), Scientific Management of Transport Systems, North-Holland, Amsterdam (1981) 235-243. [4] Fisher, M.L., and Jaikumar, R., "A generalised assignment heuristic for vehicle routing", Networks 11 (1981) 109-124. [5] Solomon, M.M., and Desrosiers, J., "Time window constrained routing and scheduling problems", Transportation Science 22 (1988) 1-13.
European Journal of Operational Research 57 (1992) 32-38 North-Holland
Theory and Methodology
A practical heuristic for a large scale wehicle routing problem Anirudh Raghavendra, T.S. Krishnakumar, R. Muralidhar, D. Sarvanan B.M.S. College of Engineering, Bangalore - 560019, India
B.G. Raghavendra Department of Management Studies, Indian Institute of Science, Bangalore - 560012, India Received January 1989; revised February 1990
Abstract: We describe a variant of the traditional vehicle routing problem (VRP) associated with the pickup and dropping of employees of large organisations. The human factors involved result in a number of additional constraints to the VRP which are specific to this kind of situation. We have developed a heuristic algorithm which solves this type of VRP effectively. The adaptability of the algorithm is demonstrated through a real problem of a large public sector organisation in the city of Bangalore in India involving the daily transportation of about 12 000 employees from and to 410 pickup points in and around the city. The use of our heuristic method has indicated a savings of about 1400 km per day out of the daily coverage of 14 000 km by a fleet of 75 buses. Additional advantages include a reduction in the number of buses and drivers required per shift, improved seat utilisation and better fuel management. Keywords: Vehicle routing, transportation, heuristics, human factor constraints.
I. Introduction
Considerable research attention has been focussed on the vehicle routing and scheduling problem and its variants, especially in the last two decades. Several hundreds of papers have appeared in the literature describing the various versions of this type of problems and the solution seeking methods for them. An excellent survey paper highlighting the state-of-the-art, with references to nearly 700 papers, appeared in 1983 (Bodin et al. [2]). A number of journals have devoted special issues for the modelling and solution methods of these problems. A survey of time
window constrained routing and scheduling problems, to which the problem described in this paper partly belongs, has appeared relatively recently (Solomon and Desrosiers [5]). Yet, the diversity of the problems as reflected in the variety of applications, the non-availability of exact algorithms for solving many real world problems for optimal solutions and, most importantly, the significance of their relative potential in real world situations continue to make this category of problems one of special interest. A number of factors collectively describe a vehicle routing and scheduling problem. The size and type of the available vehicle fleet, the nature
0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A. Raghavendra et al. / Practical heuristic for a large scale VRP
and location of demands, vehicle capacity constraints, limitations on the direction of traveling and maximum routing-lengths/times, variation in operations, cost structure and the objective characterises each vehicle routing and scheduling problem (Bodin and Golden [1]). Since the resulting mathematical model invariably consists of a large number of integer 0-1 variables, obtaining a globally optimum solution is generally hard. Consequently, a wide variety of heuristic based procedures have been suggested. The existing solution strategies could be broadly categorised (see Bodin and Golden [1]) into the following five categories: (i) cluster first route second, (ii) route first cluster second, (iii) savings/insertion approach, (iv) interactive optimisation, and (v) mathematical programming based methods. Inspite of the discussion of several varieties of the vehicle routing and scheduling problem in literature, there are some real world problems that have not come to the attention of researchers but assume special significance due to their magnitude and complexity. The subject matter of this paper is one such problem concerning the pickup and dropping of employees of a large scale public sector organisation in India. Any optimality criteria suggested here should not only consider the objective of least cost transportation of the personnel but also human factors related to distances and duration of travel of the individual employees. This kind of vehicle routing problem (VRP) is analogous to the school bus problem (see Desrosiers et al. [3]), except that the size (with respect to the number of variables) and magnitude (with respect to the number of vehicles and nodes) is considerably high. It is also implicitly analogous to time window constrained routing problems (Solomon and Desrosiers [5]). We have developed a heuristic algorithm taking into explicit consideration the specific constraints related to these human factors. No claim is made on the efficiency of the heuristic in comparison with any other method. However, its effectiveness as compared to the existing practice in the real situation is highlighted. Details of the human factor related constraints, the algorithm and the results obtained are presented in this paper. In section 2, a profile of the case study organisation is given. Section 3 provides a description of the additional constraints associated with the human factors. In
33
Section 4 we briefly describe the algorithm, while in the following section the results obtained for the case study organisation are discussed. 2. Profile of the case study organisation
This study was conducted in a large public sector organisation of the Government of India having its headquarters at Bangalore, manufacturing telephones and a variety of telecommunication equipments. Its main manufacturing unit is situated on the outskirts of Bangalore while there are five other units elsewhere in India. The unit at Bangalore has six manufacturing divisions and two R & D wings attached to it. Its employee strength is around 18000 personnel. Approximately a third of the work force is provided housing facilities in a colony adjoining the factory premises. About 12000 of the remaining employees stay in different parts of the city. As a welfare measure, the organisation provides buses to pick up these employees for work and return them to their respective destinations after work. The manufacturing unit at Bangalore and the associated administrative offices work in five different shifts each day, six days a week. Of the five shifts, the one in the night is a lean shift with a small number of employees of the essential services category. In three of the remaining four shifts, the transport department of the organisation is also associated with decisions on sequencing of buses on various routes across the shifts, as the time gap between consecutive shifts is very small. If all the employees work at the same time, the fleet demand would be for at least 200 buses, as the average capacity of a bus is 60. Currently, the transport department has a fleet of 100 buses to satisfy the transportation needs of the 12000 employees. Of these, 75% of the buses are scheduled exclusively for employee pickup and dropping across the shifts. The rest are assigned for late pickups, school buses, emergency services and for maintenance. There are a total of 410 pickup points in and around the city of Bangalore from where the employees have to be picked up (and returned). The average direct distance of these points from the factory is about 16 km, the maximum being 27 km. The demand at the pickup points varies from a minimum of 5 to a maximum of 79. The round trip duration of a route depends upon the distance to be covered, traffic density
A. Raghavendra et al. / Practical heuristic for a large scale VRP
34
along the route and the number of pickup points assigned to it. The current practice of routing and scheduling of the buses is based purely on intuition and experience gained over several years of managing the system. The personnel involved in the allocation and scheduling of the routes had very little exposure to optimisation methods. Details of the various shift timings, the shiftwise demand in terms of the number of persons utilising the transport services, the number of pickup points, the number of routes, distance travelled by all the buses and the seat utilisation percentages are presented in Table 1. In few of the buses, there is a certain amount of overloading, in the sense that a higher number of employees is assigned than the seating capacity of the bus taking into account absenteeism. 3. The human factor constraints
As mentioned earlier, several versions of the formulation of the VRP as a 0-1 integer linear programming problem is available in literature. For ready reference, one version of the VRP (see Fisher and Jaikumar [4]) whose notations are used in this paper, is presented in the appendix. In case of routing of vehicles involving the transportation of employees, the human factors problem leads to the following additional constraints. The description is based on the requirements to be met while picking the employees for work. The
reverse sequence generally holds good for returning them after work. (i) In order to ensure an upper limit on the duration of travel time of the employees who board a bus at its first pickup point, the distance covered by a bus should not exceed a pre-specifled value D k. This is given by Y'~diiXijk<~O k
for all k.
(1)
Since the city is categorised into the main city and suburbs, the value of D k may be higher for routes involving pickup points in the suburbs. This type of indirect route duration restriction is also considered in other time window constrained VRPs (Solomon and Desrosiers [5]), generally to take care of transportation of perishable goods. However, we are including it here as a special constraint due to the human factors involved. (ii) At the time of pickup, an employee would be willing to travel 'inwards' only, i.e. towards the factory, rather than 'outwards', i.e. away from the factory. This is necessary since employees, while going to work, would be unwilling to travel in a direction away from the factory. For any two consecutive pickup points i and j, this requirement can be represented by the condition: djoX, jk >~0
dioXij k -
for all i, j, k.
(2)
(iii) When deviations from the shortest path to the factory are allowed to meet the organisation's objective of higher seat utilisation and lower costs,
Table 1 Details of routes, distances and seat utilisation Shift no.
Timings
I
0615-1415
3659
303
II
0730-1615
3999
313
III
0845-1730
3042
286
IV
1415-2215
975
242
V *
2215-0615
40
.
TOTAL
No. of commuters
11715
No. of pickup points
410 ~
No. of bus routes 64 (58) 66 (64) 53 (49) 30 (28) .
. 213 ÷ (199 ÷)
Distance per trip ÷ (km)
Seat utilisation (%)
1977.0 (1767.8) 2163.0 (1985.5) 1808.3 (1598.0) 1056.7 (949.1)
89.0 (95.6) 94.3 (95.2) 90.0 (96.0) 54.0 (59.0)
. 7005.0 ÷ (6300.4 + )
-
* Ignored in our study. # Distinct points. + Each bus route (trip) is repeated twice a day, once for pickup and once for returning of employees. The figures given are for one-way transport only. Note: Results obtained from the use of the heuristic are presented in parenthesis.
A. Raghavendra et aL / Practical heuristic for a large scale VRP
4. The heuristic
0 -Pick up points 0 -Central depot/factory
O
O
dii
Q (9
®
0
0
(9 O
0
° ~ ' /
®
Q 0
0
O
Figure 1
normally a limit is placed on such deviations. While D k ensures an upper limit on the total route distance covered, a limit on the maximum deviation from each pickup point is necessary. A pickup point j could be linked to a current pickup point i only if the difference between the resulting total distance and the shortest distance, from i to the factory, does not exceed a pre-specified value P. This results in a constraint of the form dijXijl, + djoXjo k - dioXio k <~P
35
for all i, j, k.
(3) This requirement is illustrated in Figure 1, from which it is seen that Xij k can take a value of 1 only if d~j + djo - d~o <~P.
The above mentioned additional constraints (1)-(3) uniquely distinguish this type of VRP from the classical problems addressed in literature. It is apparent that the resulting VRP model in real situations is a complex combinatorial problem, involving a large number of 0-1 variables and several constraints. We propose a heuristic method to tackle this problem, the details of which are described in the next section.
In view of the specific nature of the VRP described above, an intuitive algorithm was developed by us. The algorithm is essentially a tourbuilding method and takes care of the three additional constraints described in the earlier section. The basic input to this routing algorithm consists of two data files, one pertaining to the number of persons boarding the bus (demand) at each pickup point, shiftwise, and the other a distance matrix, giving the distances of direct travel from each pickup point to the other including the central point (factory). The algorithm is an aggregation of several route building procedures followed sequentially, till the entire demand at all the pickup points is met. At each step, the feasibility with respect to the special constraints described earlier as well as additional factors related to capacity utilization of the bus and such other factors are explicitly considered. An overview of this heuristic is presented in the form of a flowchart in Figure 2. Some of the salient features of this heuristic are as follows: (i) The heuristic works inwards, that is, points which are farthest are identified first and routes are developed from there. (ii) Points where the demand equalled or exceeded the capacity of a bus were considered first for exclusive assignment of routes. (iii) By keeping a check on the number of employees already assigned to the bus (NPCHK), the heuristic ensures that once the bus has N P C H K or more commuters, it returns to the factory without deviating, meeting additional demand at pickup points enroute, subject to feasibility. Certain other factors, not mentioned in Figure 2, were also considered. These are as follows: (a) Depending upon the geographical locations, there can be a set of points for which there is only one access or approach. The heuristic ensures that such 'land-locking' pickup points always go together in the sense that if one point is covered by a route the other(s) is also covered by the same route. (b) It is desirable that two vehicles need not go to the same point when one vehicle would satisfy the demand, keeping in view the practical
36
A. Raghavendra et al. / Practical heuristic for a large scale VRP
problems that may arise if some of the people at a particular pickup point are asked to use one bus while the rest use another. Therefore, the heuristic ensures that the demand at any pickup point is not split, unless the demand exceeded the capacity of a bus.
5.
Results
and
discussion
The heuristic described above was coded in FORTRAN on a DEC 1091 computer system. It was used on the data pertaining to each shift of the case study organisation and took approximately 45 s for execution per shift. The basic output consisted details of each route developed, the sequence and demand of pick up points covered, total distance covered by the route and related information. The results obtained for each
shift are summarised and presented in parenthesis in Table 1. Comparing the results obtained from the use of our heuristic with the existing situation (Table 1) the following significant contributions may be observed: There is a substantial decrease in the total distance covered; from 14 010 k m / d a y to 12 600 km/day which is about 10%. Reduction in the number of routes and consequently the number of buses required. Higher percentage of seat utilisation. - O t h e r incidental benefits include reduced maintenance cost for the buses, reduction in the demand for drivers and effective fuel utilisation. A further analysis and comparison of the expected journey time of employees at each pickup point with the existing situation indicated that the average one-way journey time per employee could be reduced by 8 minutes.
DATAFILE1
DATAFILE2
IDENTIFICATION OFPOINTS I WITHDEMAND> BUSCAPACITY ASSIGNROLITE.UPDATE I= tI IDENTIFY FARTHESTPOINT I ASSIGNROUTETOSTART.UPDATE ~
~
~
A]ERMINENEARESTFEASIBLEPOINT I
NO
SEQUENCEOFPICKUPPOINTS, DISTANCE,TIMEOFDEPARTURE ANDARRIVAL Figure 2
A. Raghavendra et al. / Practical heuristic for a large scale VRP
37
Table 2 Results of sensitivity analysis Scenario no.
Percentage absenteesim
NPAB
NPCHK
P (km)
Total distance (kin/day)
Reduction in distance (kin/day)
No. of routes
Reduction in routes (kin/day)
1 2 3 4 5 6 7 8 9
2 2 2 5 5 5 10 10 10
69 69 69 71 71 71 73 73 73
60 60 60 62 62 62 64 64 64
0.0 0.5 1.5 0.0 0.5 1.5 0.0 0.5 1.5
12728.6 12596.8 12574.0 12660.2 12556.0 12503.4 12421.6 12343.0 12224.4
1281.4 1413.2 1436.0 1349.8 1454.0 1506.6 1588.4 1667.0 1785.6
200 197 196 196 193 194 194 191 190
13 16 17 17 20 19 19 22 23
Once a computer program incorporating the heuristic was developed, the effects of varying some of the key decision parameters could be obtained under different scenarios. For example, the permissible deviation ( P ) as illustrated in constraint (3) could be varied. Further, in the basic run of the algorithm, no advantage was taken of the absenteeism factor. To study the effect of changes in some of the above parameters, our heuristic algorithm was used under different values of P, different absenteeism levels (2%, 5%, 10%) resulting in a higher number of persons assigned to a bus (NPAB) and different values of NPCHK. The results obtained under nine such scenarios are presented in Table 2. It may be observed from Table 2 that if the absenteeism factor is considered, the total distance covered and the number of routes would generally reduce further. Similarly allowing for higher deviations [value of P as illustrated in constraint (3)] would also result in less number of routes and lesser distance.
tate modifications, the computer program can be made PC based as a Decision Support System (DSS), so that it can be accessed and easily used by personnel in transport departments with no Operations Research background. Extending from the basic development of effective routing, other problems related to scheduling of vehicles across shifts and crew allocation can also be considered.
Acknowledgement The authors are thankful to the management and staff of Indian Telephone Industries Limited, Bangalore Complex, for all the facilities and cooperation extended during the course of this study. Suggestions and comments from referees on an earlier version of the paper are gratefully acknowledged.
Appendix 6. Conclusion A simple but effective heuristic has been developed for a specific type of VRP involving the transportation of employees. The use of this heuristic in the case study has shown scope for substantial monetary savings. Further extension and improvements in the algorithm to suit different environments are possible. A time constraint has been addressed implicitly in our case study by distance checks. It could be explicitly incorporated as an additional constraint. With appropri-
A model of the traditional VRP, cf. Fisher and Jaikumar [4]. The parameters and the variables are as follows: k = 1, 2 . . . . . K = number of vehicles/routes. i, j = 0, 1. . . . . n = number of pickup points with the convention that 0 (zero) denotes the factory which is the central dropping and pickup point. bk = capacity of vehicle k.
A. Raghavendra et al. / Practical heuristic for a large scale VRP
38
= number of persons alighting/ boarding (demand) at pickup point i (i 4: 0). = distance of direct travel from point i to point j. = m a x i m u m permissible distance for route k. = maximum permissible deviation in distance from any pickup point (measured as the difference between the distance of the direct route linking a pickup point to the factory and that via an added pickup point). 10 if point i is served by = route k, otherwise.
ai
dij
Dk P
Yik
10 if vehicle k is to travel directly from demand x,k point i to j, otherwise. The objective is to minimise (4)
~_~ dijXijk i,j,k S.t.
E ai Yik <~bk i
K,
Yik----- 1,
for all k
i=0, i = 1 . . . . . n,
(5) (6)
Yik = 0 o r 1
x/ik =
for all i, k,
(7)
for all j, k,
(8)
Y'~Xii~ =Yik for all i, k,
(9)
i
J
IS1-1,
Sc{1,...,n},
i,j
2~
Xi~l, = 0 or 1 for all i, j, k.
(10) (11)
References [1] Bodin, L., and Golden, B., "Classification in vehicle routing and scheduling", Networks 11 (1981) 97-108. [2] Bodin, L., Golden, B., Assad, A., and Ball, M., "The state of the art in routing and scheduling of vehicles and crew", Computers and Operations Research 10 (1983) 63-212. [3] Desrosiers, J., Ferland, J-A., Rousseau, J-M., Lapline, G., and Chapleau, "An overview of a school busing system", in: N.K. Jaiswal (ed.), Scientific Management of Transport Systems, North-Holland, Amsterdam (1981) 235-243. [4] Fisher, M.L., and Jaikumar, R., "A generalised assignment heuristic for vehicle routing", Networks 11 (1981) 109-124. [5] Solomon, M.M., and Desrosiers, J., "Time window constrained routing and scheduling problems", Transportation Science 22 (1988) 1-13.