A solution of the bicriteria vehicle scheduling problems with time and area-dependent travel speeds

A solution of the bicriteria vehicle scheduling problemswith time and area-dependent travel speeds

Yang-Byung Park*

College of Mechanical and Industrial System Engineering, Kyung Hee University, Seochon 1, Kiheung, Yongin

Kyunggi-Do 449-701, South Korea

Abstract

This paper is concerned with the bicriteria vehicle scheduling problem with time and area-dependenttravel speeds, in which two con¯icting objectives are explicitly treated and the travel speed between twolocations depends on the passing area and time of day. The two objectives are the minimization of totalvehicle operation time and the minimization of total weighted tardiness. First, I construct a mixedinteger linear programming formulation, and present a heuristic named BC-saving algorithm (bicriteria-saving algorithm) that builds the vehicle schedules based on the savings computed. The results ofcomputational experiments show that the heuristic is very successful on a variety of test problems.Finally, I propose an interactive scheduling computer system based on the BC-saving algorithm to dealwith real complexity and subjectivity in the vehicle scheduling process. 7 2000 Published by ElsevierScience Ltd.

Keywords: Bicriteria vehicle scheduling problem; Area and time-dependent travel speeds; Minimization of weightedtardiness; Heuristic algorithm; Interactive scheduling computer system

1. Introduction

During the last 30 years, many researches have been devoted to the development ofoptimization and approximation algorithms for vehicle scheduling problems (VSP)(Desrochers, Lenstra & Savelsbergh, 1990). This interest is due to the practical importance ofe�ective and e�cient methods for managing physical distribution situation. However, the real-

Computers & Industrial Engineering 38 (2000) 173±187

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* Tel.: +82-331-201-2553; fax: +82-331-202-4004.E-mail address: [email protected] (Y.-B. Park).

life application of the algorithms developed has been very limited because of the followingreasons:

1. The travel times between locations are assumed to be constant in most VSP. However, innearly all metropolitan areas, vehicle speed is subject to rapid change during the day due tointricate road networks and heavy tra�c congestion (Hill & Benton, 1992).

2. The objective function in most VSP is to minimize the total travel distance (time). However,recently, the customer service to meet the required service time has been considered as animportant issue in VSP (Malmborg, 1996).

3. The algorithmic approaches to VSP do not solve the problems actually faced by vehicle ¯eetschedulers. It is apparent that real problems have complexities and elements of subjectivitywhich cannot be included in existing algorithms (Thangiah, Vinayagamoorthy & Gubbi,1993).

Hill and Benton (1992) proposed a parsimonious model for estimating time-dependent travelspeeds that alleviates both the data collection and data storage problems inherent in time-dependent travel speed vehicle scheduling models. They also discussed the issue of developingalgorithms to ®nd near-optimal vehicle schedules with time-dependent travel speeds.Malandraki and Daskin (1992) developed a nearest-neighbor search heuristic and a heuristicusing cutting planes for VSP, assuming that the travel time between two locations is discretelychanged according to two or three time intervals during a day. Considerable research has beendevoted to the time-dependent traveling salesman problem, a special model of VSP, takingaccount of the costs which vary depending on the time travel takes place. Some of the existingliterature are Wiel and Sahinidis (1995), Balakrishnan, Lucena and Wong (1992) and Lucena(1990), etc.Thangiah et al. (1993) developed two heuristics, a greedy heuristic and a genetic algorithm

based heuristic, to solve the vehicle routing problem with time deadlines whose objective is tominimize both the number of vehicles and the total vehicle travel distance. In recent years, theissue of vehicle scheduling with time window constraints has been gaining increasing attention.Some of the existing literature on the VSP with time window constraints are Balakrishnan(1993), Potvin and Rousseau (1993), Badeau, Guertin, Gendreau, Potvin and Taillard (1997)and Kohl and Madsen (1997), etc.The idea of allowing a scheduler to be part of the problem solving process has

provided interesting results in the vehicle scheduling domain. Potvin, Lapalme andRousseau (1994) developed an interactive-graphic computer system that facilitatesschedulers in their task of vehicle scheduling. Waters (1990) proposed a framework of anexpert system for vehicle scheduling and stated several important problems to overcomebefore the system can be designed. The development of expert systems is still at a veryearly stage. Some of the real di�culties in developing expert systems include the diversityof scheduling problems and lack of agreed facts and rules to form a knowledge base.This paper is concerned with the bicritera vehicle scheduling problems with time and area-

dependent travel speeds (BVSPTD) in which the ®rst two reasons stated earlier are re¯ected.The BVSPTD is de®ned as follow:

A set of geographically dispersed customers at known locations are to be serviced exactly once

Y.-B. Park / Computers & Industrial Engineering 38 (2000) 173±187174

by a set of vehicles with both capacity and travel time restrictions, starting from a centraldepot and eventually returning to the depot through customers such that all customers with aknown quantity of some commodity are fully serviced. The customers are speci®ed the duetimes and the latest allowable times (time deadlines) for servicing and given di�erent weightson tardiness. The travel times between two locations are changed depending upon the passingtime of day and area. Two objectives, the minimization of total vehicle operation time (traveltime plus service time at all customers plus waiting time at all customers) and the minimizationof total weighted tardiness which are frequently con¯icting, are considered. The latter one canbe understood as the maximization of customer service satisfaction. The importance of the twoobjectives is changed by the decision maker's preference.

Instances of the BVSPTD occur in retail distribution, raw material and parts supply, industrialrefuse collection, fuel oil delivery, etc.In this paper, for the BVSPTD, I present a mathematical formulation and a heuristic

algorithm, and propose an interactive scheduling computer system that deals with the thirdreason stated. Recently, the use of global positioning system (GPS) for vehicle managementhas been gaining increasing attention. However, it still remains on an important decisionmaking problem to schedule vehicles optimally before leaving a depot or on the way ofmoving, especially in delivery problems where the required quantity of commodity fromcustomers is known in advance.Section 2 presents a mixed integer linear programming of the BVSPTD and solve a small

example problem. Section 3 proposes a simple heuristic algorithm for the BVSPTD that buildsthe vehicle schedules based on the savings computed. Computational results of the proposedheuristic on test problems are discussed in Section 4. Section 5 proposes the development of aninteractive scheduling computer system for the BVSPTD that is based on the proposedheuristic. Finally, Section 6 concludes the paper.

2. Mathematical formulation of BVSPTD

2.1. Problem formulation

The BVSPTD is formulated as a mixed integer linear programming problem on the basis ofthe formulation given by Malandraki and Daskin (1992) for the case in which the travel timebetween nodes is a function of the starting time interval at the origin node. Once the timeinterval during which the vehicle starts from the origin node is known, the travel time to thedestination node is a known constant. It is assumed that the division of time intervals is samefor all physical links of nodes. The travel times between all pairs of nodes may be estimated byusing the Hill and Benton's (Hill & Benton, 1992) model for time-dependent vehicle travelspeeds.The formulation concerns the BVSPTD with the same type of V available vehicles. It

augments V virtual nodes, N� 1, . . . ,N� V, whose locations are same as the central depotnode, where N represents the number of nodes including a central depot. These virtual nodescorrespond to returning depot nodes for each of the V vehicles. Thus all vehicles start from a

Y.-B. Park / Computers & Industrial Engineering 38 (2000) 173±187 175

central depot and each vehicle has to return to its speci®ed return depot whose location is thesame as the central depot. The formulation considers deliveries to the customers, but it can beeasily modi®ed to model collections from the customers.The notation used in the formulationis summarized below:

ConstantsN number of nodes including a central depot (central depot = node 1)V available vehiclesM number of time intervalsdi delivery due time at node ili latest allowable delivery time at node ici quantity to deliver at node i; ci � 0 for i � 1, N� 1, . . . ,N� Vsi service time at node i; si � 0 for i � 1, N� 1, . . . ,N� VR maximum allowable vehicle return time to depotQ vehicle capacitytijm travel time from node i to j when starting at i during time interval m; t1, N�v, m � 0, for

v � 1, . . . , V and m � 1, . . . ,M; tijm � ti1m, for i � 2, . . . ,N and j � N� 1, . . . ,N� V andm � 1, . . . ,M

Um upper bound for time interval mwi tardiness weight of node iF weight of the minimization of total vehicle operation time �0RFR1)B a large number

Decision variables

xijm � f 1 if any vehicle travels directly from node i to j starting from i during time interval m0 otherwise

ai arrival time of any vehicle at node iqi quantity carried by a vehicle when arriving at node i

Min Z � FXVv�1

aN�v � �1ÿ F�XNi�2

wi max �ai ÿ di, 0� �1�

subject toXNi6�ji�1

XMm�1

xijm � 1, j � 2, . . .N� V �2�

XN�Vj6�ij�2

XMm�1

xijm � 1, i � 2, . . .N �3�


XN�Vj�2

XMm�1

x1jm � V �4�

aj ÿ ai ÿ Bxijmrtijm � si ÿ B, i � 1, . . . ,N; j � 2, . . . ,N� V; i6�j; m � 1, . . . ,M �5�

aj ÿ tijmRUm � Bÿ1ÿ xijm

�, i � 1, . . . ,N; j � 2, . . . ,N� V; i6�j; m � 1, . . . ,M �6�

aj ÿ tijmRUmÿ1xijm, i � 1, . . . ,N; j � 2, . . . ,N� V; i6�j; m � 1, . . . ,M �7�

qi ÿ qj ÿ BXMm�1

xijmrci ÿ B, i � 2, . . . ,N; j � 2, . . . ,N� V; i6�j �8�

qjXMm�1

x1jmRQ j � 2, . . . ,N �9�

aN�vRR v � 1, . . . ,V �10�

aiRli, i � 2, . . . ,N �11�

qi � 0, i � N� 1, . . . ,N� V �12�

a1 � 0 �13�The objective function (1) minimizes the weighted sum of total vehicle operation time (traveltime plus service time at all nodes plus waiting time at all nodes) and total weighed tardiness.The waiting time at nodes occurs when a vehicle awaits the next time interval for more rapidmovement. Constraint (2) and (3) ensure that each customer is visited exactly once. Theseconstraints and the integrality constraints of xijm impose that xijm is equal to either 0 or 1.Constraint (4) ensures that the maximum V vehicles are used, allowing some vehicles to movedirectly from the starting depot to its own return depot. In order to ensure that all vehiclesstart from the central depot and each vehicle returns to its speci®ed return depot, thefollowings should be set in the travel time matrix:

t1,N�v, m � 0, v � 1, . . . ,V; m � 1, . . . ,Mtijm � tilm, i � 2, . . . ,N; j � N� 1, . . . ,N� V; m � 1, . . . ,M

Constraints (5) compute the arrival time at node j. The objective function (1) ensures thatconstraints (5) apply with equality when xijm � 1 except in cases when waiting at i decreasesthe objective function value. Waiting at nodes occurs because the travel times between nodesare determined on the basis of the step functions of the starting time interval. Constraints (6)and (7) determine the time interval m corresponding to the departure time at node i.


Constraints (8) compute the quantity carried by a vehicle when arriving at node j (or whenleaving node i ). Constraints (9)±(11) impose the restrictions of vehicle capacity, vehicle returntime to depot, and delivery times at customers, respectively. Elimination of illegal sub-tours isguaranteed by constraint (5) and (8).The mathematical formulation of the BVSPTD de®nes an NP-hard problem involving many

variables and a large number of temporal and capacity constraints. Therefore, it will beextremely di�cult to get an optimal solution of the BVSPTD by solving the mathematicalformulation as the problem becomes large.

2.2. Example

A small example of the BVSPTD is solved to obtain an optimal solution by applying themathematical model. The example problem is consisted of seven nodes (a single depot and sixcustomers), where the service area is divided into nine rectangles and the daily service time isevenly divided into two time intervals. A vehicle capacity is set to 50 and the maximumallowable vehicle return time to depot is set to 50. Travel time from node i to j when startingat i during time interval m, tijm, is obtained by dividing an Euclidean distance between i and jby a known average travel speed in [i ] during m. [i ] represents a divided service area includingi. The complete set of input data for the example may be obtained from author upon request.The mathematical model for the example is formulated with a single objective function and

419 constraints, assuming three vehicles are available. The constructed mathematical model isexecuted with LINGO extended version (LINGO, 1994) on IBM compatible 586 PC running at133 MHz and 36M RAM. In order to facilitate the data input operation for executing LINGO, Iprogrammed a simple FORTRAN program that is capable of generating the equations ofmathematical model for the problem to solve and inputting their coe�cients and necessaryinformation to LINGO. Fig. 1 shows the optimal vehicle routes for the example when F � 0:3,whose total vehicle operation time is 11.05 and total weighted tardiness is 0.00.

3. A heuristic algorithm for BVSPTD

A heuristic named BC-saving algorithm (bicriteria-saving algorithm) is developed to solve theBVSPTD. It is based on the Clarke and Wright (1964) savings algorithm that was originallydesigned for the classical vehicle routing problems. The BC-saving algorithm starts withassigning a vehicle to each customer. The algorithm continues to construct the routes bycombining two distinct routes whose total saving is the largest. The combined route should notviolate the restrictions given.The total saving from a feasible combination of two routes is computed by summing the

savings in vehicle operation time and weighted tardiness with a consideration of theimportance of two objectives. Hence, the total saving is considered as a weighted sum of thesavings in two objectives.Let SVij be a saving in vehicle operation time and SWif be a saving in weighted tardiness.

Based on the assumptions that a service time at node is negligible and waiting at nodes is notallowed, and so the vehicle departure time from a node is equal to the arrival time to the node,


the total saving expected by connecting the last customer i to visit of one route to the ®rstcustomer j to visit of the other route, BCij, is computed by the following equation:

BCij � FSVij � �1ÿ F�SWij �14�

SVij � ti1�ai � � t1j�a1 � ÿ tij�ai � � b

b �

8>><>>:0 if a 0j � aj

VT�j� ÿ t1j�a1 � ÿX

�m, n�EP�j�tmn�a 0m � if a 0j 6�aj

SWij �

8>><>>:0 if a 0j � aj

WT�j� ÿX

�m, n�EP�j�wmmax

ÿa 0m ÿ dm, 0

�if a 0j 6�aj

where

a 0j arrival time at node j (also, departure time from node j ) which is newly determinedafter combining two routes,

Fig. 1. Optimal vehicle routes for example.


VT( j ) vehicle operation time of the route which includes node j before combining tworoutes,

WT( j ) total weighted tardiness of the route which includes node j before combining tworoutes,

P( j ) set of all paths except (1, j ) in the route which includes node j before combining tworoutes.

In the BC-saving algorithm, the savings for all pairs of nodes cannot be calculated in advanceas in the Clarke and Wright's algorithm, since the travel time between two nodes is variantdepending upon the passing area and time of day. Instead, at each iteration for combination,the savings are newly computed for all the feasible pairs of nodes that satisfy the restrictions.The procedural steps of the BC-saving algorithm are summarized as follows:

Step 1: Construct Nÿ 1 distinct routes in which each customer is served by a dedicatedvehicle. Compute the vehicle operation time and total weighted tardiness for each route.Step 2: Select two partially formed routes and perform the feasibility test for their combinedroute with respect to the restrictions of vehicle capacity, return time to depot, and deliverytimes at customers. The combination of two routes is to link the last customer i to visit ofone route to the ®rst customer j to visit of another route. Repeat the process of selectionand feasibility test for all the partially formed routes, and compute BCij of all the feasiblepairs of nodes for combination. If there exist no feasible pairs of nodes to link, go to Step 4.Step 3: Combine the two routes with the largest value of BCif into a single route, and go toStep 2.Step 4: Compute the improvement in the objective function (1) appeared in Section 2.1 byinterchanging adjacent pairs of nodes in a route, keeping the restrictions of vehicle returntime to depot and delivery times at customers. If there exist no feasible adjacent pairs ofnodes to interchange or no improvement is expected from the feasible interchanges, then goto Step 5. Otherwise, construct a new route by switching the positions of the adjacent pairthat shows the largest improvement. Repeat the interchange process to the newlyconstructed route until no more improvement is expected.Step 5: Apply Step 4 to the rest of the routes and stop. The vehicles are scheduled on thebasis of the ®nal vehicle routes obtained.

4. Computational experiments

A total of eight test problems with a single depot and 100 customers are constructed toevaluate the performance of the BC-saving algorithm on the BVSPTD. It is assumed that allthe vehicles available are homogeneous with a capacity of 100 and their return time to depot is100. In each test problem, the service area is equally divided into nine regular squares and thedaily service time is evenly divided into ®ve time intervals. All the experiments were performedon IBM compatible 586 running at 133 MHz and 36M RAM.The design of test problems highlights three factors that can a�ect the behavior of the BC-


saving algorithm. The factors include the geographical aspect of customer locations,distribution of delivery due times, and distribution of tardiness weights of customers. The dataused for the customer coordinates and demands are based on the data for the Solomon's(Solomon, 1987) test problems. Given the geographical and demand data, the test problems areconstructed by generating the due times and tardiness weights based on random uniformdistribution and clustered.The customer characteristics of eight test problems are summarized in Table 1. The clustered

locations mean that the customers are grouped into certain areas. For the clustered due times,the following distribution is used: 0±20 (10%), 20±50 (30%), 50±80 (50%), 80±100 (10%). Forthe clustered tardiness weights of customers, the following distribution is used: 1 (20%), 2(50%), 3 (30%).As a benchmark for evaluating the BC-saving algorithm, I selected the Balakrishnan's

(Balakrishnan, 1993) NNS algorithm that is a well-known heuristic for the vehicle routingproblems with soft time windows. This selection is because no explicit solutions for theBVSPTD have been found in the current literature. The due time at customer may beinterpreted as the time window whose lower and upper limits are equal. Thus, the formulaPO(j) in the NNS algorithm, used for choosing the customer to connect to the current partialroute, may be modi®ed as following:

Min PO�j� � Ftij�ai � � �1ÿ F�wj maxÿaj ÿ dj, 0

� �15�The notation used in the above formula is same as in Section 2.1.The ®rst experiment is aimed at analyzing the sensitivity of two objectives to changes in their

weights (importance). To this end, the BC-saving algorithm is applied to T000 problem withincreasing the weight of the minimization of total vehicle operation time, F, from 0.0 to 1.0 by0.1. Then the weight of the minimization of total weighted tardiness is decreased from 1.0 to0.0 by 0.1. The interchange process for improvement is repeated for each route ®ve times atmaximum. The result is presented in Table 2, and its graphical representation is shown inFig. 2. It is seen from Table 2 and Fig. 2 that two objectives are traded o� well according tothe changes in their weights. It is reasonable that the number of vehicles required is decreasedas F is increased.

Table 1Summary of the customer characteristics of eight test problems

Problem Location Due time Tardiness weight Service time

T000 Uniform Uniform Uniform 10T001 Uniform Uniform Clustered 10T010 Uniform Clustered Uniform 5

T011 Uniform Clustered Clustered 5T100 Clustered Uniform Uniform 10T101 Clustered Uniform Clustered 10T110 Clustered Clustered Uniform 5

T111 Clustered Clustered Clustered 5


The second experiment is aimed at evaluating the performance of the BC-saving algorithm.Table 3 shows the computational results obtained by applying both the BC-saving algorithmand the NNS algorithm to eight test problems with F � 0:0, 0.5 and 1.0. For comparing theBC-saving algorithm with the NNS algorithm, the relative comparison value is computed as(100 � the solution obtained by the BC-saving algorithm/the solution obtained by the NNSalgorithm). The value means the relative percentage of the solution of the BC-saving algorithmto the one of the NNS algorithm.The computational results are so impressive. The BC-saving algorithm is very successful in

most cases regardless of the weights of two objectives, showing the shorter total vehicleoperation time and less total weighted tardiness than the NNS algorithm. It is seen fromTable 3 that the total weighted tardiness obtained by the BC-saving algorithm is, on theaverage, only 4.2% of the one obtained by the NNS algorithm when F � 0:0, 21.9% whenF � 0:5, and 59.1% when F � 1:0: This outcome implies that the BC-saving algorithmperforms very well, especially with respect to the minimization of total weighted tardiness.It should be noted from Table 3 that the BC-saving algorithm often requires 1±2 vehicles

more than the NNS algorithm. This may be a drawback in applying the BC-saving algorithm.I did not perform the detailed experiments to examine the e�ect of customer characteristics onthe performance of the BC-saving algorithm. Nevertheless, from Table 3, it is not found that

Table 2Comparison of the solutions of BC-saving algorithm obtained by changing F on T000 problema

F 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

TOT 414.2 243.7 244.2 244.2 242.0 240.0 216.6 204.5 203.4 182.9 157.9TWT 112.6 232.9 232.2 232.2 319.4 184.9 316.4 353.3 430.8 693.8 1760.2NV 17 17 17 17 17 17 16 16 16 16 15

a TOT: total vehicle operation time, TWT: total weighted tardiness, NV: number of vehicles required.

Fig. 2. Sensitivity analysis of two objectives to changes in their weights on T000 problem.


the customer characteristics have any signi®cant impact on the performance of the BC-savingalgorithm.

5. Interactive scheduling computer system for BVSPTD

It is clear that the real problems of scheduling vehicles have complexities and subjectiveelements that cannot be included in the algorithmic approaches like the BC-saving algorithm.Thus, it is practically important to contrive a way to incorporate a combination of skills,intuition and experience of schedulers in the BC-saving algorithm based solution process. Oneway of doing this is to develop an interactive scheduling computer system whereby humans canplay some part in the scheduling process.In this section, I suggest a general framework for the interactive scheduling computer system

that is useful for the BVSPTD. The aim of this system is to enhance the e�ectiveness of theBC-saving algorithm by providing a user with the capability of incorporating the realcircumstances, subjective judgements, and common sense in the solution process, and thusobtain a practically improved solution.One of the major problems faced in the development of the interactive system for the

BVSPTD is a wide variety of local circumstances and requirements about vehicle scheduling.Furthermore, though there are many scheduling experts who can provide practical andtheoretical views, there is no agreed body of facts or rules that can be formed into a generally

Table 3Comparison of BC-saving algorithm and NNS algorithm on eight test problemsa

F � 0:0 F � 0:5 F � 1:0

TOT TWT NV TOT TWT NV TOT TWT NV

T000 BC-saving 414.2 112.6 17 240.0 184.9 17 157.9 1760.2 15

NNS 486.5 423.2 16 347.3 614.9 16 308.9 1661.1 15T001 BC-saving 383.8 120.3 16 212.7 340.1 17 170.2 1784.3 15

NNS 469.9 936.5 16 320.2 600.2 15 312.0 1268.0 15

T010 BC-saving 371.9 22.6 12 154.0 117.0 14 152.2 422.9 13NNS 392.9 332.1 12 362.9 293.9 13 317.4 768.9 12

T011 BC-saving 450.0 7.5 12 150.9 63.4 13 149.2 614.7 12NNS 426.3 812.3 11 396.9 550.7 11 290.4 853.8 11

T100 BC-saving 376.0 43.1 16 213.9 389.0 17 164.4 1234.3 16NNS 396.9 547.7 15 392.1 383.2 16 196.1 1913.3 14

T101 BC-saving 375.4 142.6 16 235.3 379.2 17 147.3 1621.7 14

NNS 489.1 447.3 16 404.9 424.2 16 240.5 1592.4 14T110 BC-saving 287.7 21.1 11 111.7 88.3 14 91.9 368.8 14

NNS 295.8 324.5 11 335.1 340.3 11 238.7 739.4 11

T111 BC-saving 339.2 24.1 12 110.0 20.5 15 96.6 172.3 15NNS 436.6 561.5 12 378.3 292.1 13 370.2 712.8 12

Average relative comparison value 88.1 4.2 102.5 44.6 21.9 112.1 46.8 59.1 109.5

a TOT: total vehicle operation time, TWT: total weighted tardiness, NV: number of vehicles required.


accepted interactive procedure. The alternative to a generalized interactive system for solving arange of the BVSPTD is to build a customized system for each application. Waters (1990)listed a number of di�culties to develop a computer-assisted interactive scheduling system.The suggested interactive scheduling computer system for the BVSPTD consists of two

phases: pre-scheduling and post-scheduling. Pre-scheduling is the phase in which the problemgiven is modi®ed taking into account the real circumstances before the BC-saving algorithm isapplied. Post-scheduling is the phase to improve the quality of the solution obtained byapplying the BC-saving algorithm.Some guidelines that may be formed into the interactive process between a scheduler and the

system are listed below, though they are open to argument:

1. Pre-scheduling guidelines(i) Cluster several customers who are close together into single visits.(ii) Set the prohibitive paths at speci®c time intervals depending upon the road conditions.(iii) Consider the alternative means of delivery (e.g., private shipping agent and parcelpost) for the outlying customers to whom long journeys are required.

Fig. 3. Procedural steps of the proposed interactive scheduling computer system.


(iv) Schedule the urgent customers with a very high tardiness weight as the ®rst visits byvehicles.(v) Schedule the dependent customers ®rst.

2. Post-scheduling guidelines(i) Schedule to serve more than one route consecutively by a single vehicle, if feasible withrespect to the restrictions given.(ii) Relax the input parameters such as the vehicle capacity, maximum allowable vehiclereturn time to depot, and due times and latest allowable delivery times at customers,within tolerable bounds.(iii) Combine the neighboring routes whose load quantities are very small into a singleroute, if feasible with respect to the restrictions given.(iv) Ensure that the routes do not cross each other when F � 1:(v) Perform the node interchange algorithms.

Future research is to embody such guidelines into the interactive process and eventuallydevelop a user-friendly interactive-graphic scheduling computer system for the BVSPTD,namely a decision support system for vehicle scheduling. The procedural steps of the proposedinteractive scheduling system are depicted in Fig. 3. Fig. 4 shows, as an example, an outputscreen of the interactive scheduling computer system, named IDSSTD (Interactive DecisionSupport System for Truck Dispatching), being developed by author.

Fig. 4. Graphic output of a ®nal solution by IDSSTD.


6. Conclusions

This paper presents a solution of the BVSPTD in which two con¯icting objectives areexplicitly treated and travel speed between two locations depends on the passing area and timeof the day. First, the BVSPTD is formulated as a mixed integer linear problem to obtain anoptimal solution. However, this NP-hard type problem becomes unsolvable as the number ofcustomers is increased because of the computation time and computer storage requirements.To overcome this problem, a simple and e�cient heuristic named BC-saving algorithm isdeveloped. The computational analysis of the BC-saving algorithm indicates that it proves tobe very successful on a variety of practical sized test problems of the BVSPTD, showing thecapability of trading o� two objectives. Finally, an interactive scheduling computer systembased on the BC-saving algorithm is proposed to deal with real complexity and subjectivity inthe vehicle scheduling process. The interactive system consists of two phases: pre-schedulingand post-scheduling. Some guidelines, that may be formed into the interactive process at eachphase, are listed.The BVSPTD represents an urban, congested environment more accurately than do its non-

temporal single criterion counterparts, but it is more di�cult to solve. More research needs tobe devoted to the development of solutions for the BVSPTD. The BC-saving algorithm needsalso to be tested on real data collected from congested road networks.

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Documents

A solution of the bicriteria vehicle scheduling problems with time and area-dependent travel speeds