a tabu search approach for scheduling hazmat shipments

Computers & Operations Research 34 (2007) 1328–1350www.elsevier.com/locate/cor

A tabu search approach for scheduling hazmat shipments�

Pasquale Carotenutoa, Stefano Giordanib,∗, Salvatore Ricciardellib, Silvia Rismondoa,b

aIstituto di Technologie Industriali e Automazione—Sezione di Roma, Consiglio Nazionale delle Ricerche,Via del fosso del cavaliere 100, I-00133 Rome, Italy

bDipartimento di Ingegneria dell’Impresa, Università di Roma “Tor Vergata”, Via del Politechnico 1, I-00133 Rome, Italy

Available online 22 July 2005

Abstract

Vehicle routing and scheduling are two main issues in the hazardous material (hazmat) transportation problem. Inthis paper, we study the problem of managing a set of hazmat transportation requests in terms of hazmat shipmentroute selection and actual departure time definition. For each hazmat shipment, a set of minimum and equitable riskalternative routes from origin to destination points and a preferred departure time are given. The aim is to assign aroute to each hazmat shipment and schedule these shipments on the assigned routes in order to minimize the totalshipment delay, while equitably spreading the risk spatially and preventing the risk induced by vehicles travelingtoo close to each other. We model this hazmat shipment scheduling problem as a job-shop scheduling problem withalternative routes. No-wait constraints arise in the scheduling model as well, since, supposing that no safe area isavailable, when a hazmat vehicle starts traveling from the given origin it cannot stop until it arrives at the givendestination. A tabu search algorithm is proposed for the problem, which is experimentally evaluated on a set ofrealistic test problems over a regional area, evaluating the provided solutions also with respect to the total route riskand length.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Hazmat transportation problem; Job-shop scheduling; Tabu search algorithm

� This work has been partially supported by grant CNR 02.00171.37 from the Italian National Research Council(CNR—GNDRCIE), and commissioned by Protezione Civile the Italian Civil Protection Body.∗ Corresponding author. Tel.: +39 06 7259 7358; fax: +39 06 7259 7305.

E-mail addresses: [email protected] (P. Carotenuto), [email protected] (S. Giordani),[email protected] (S. Ricciardelli), [email protected] (S. Rismondo).

0305-0548/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2005.06.004

http://www.elsevier.com/locate/cor

mailto:[email protected]




P. Carotenuto et al. / Computers & Operations Research 34 (2007) 1328–1350 1329

1. Introduction

The transportation of hazardous materials (hazmats) is a constantly increasing activity that plays aremarkable role in the economy of industrialized countries. It is estimated that, at present, about 4 billiontons of hazmats are transported annually at the worldwide level [1]; in Italy, about 74 million tonsare transported annually on trucks [2]. Given the extent, the risks associated with accidents involvinghazardous material shipments have gained considerable attention, encouraging research on hazardousmaterial transportation. Research in this area focuses on two major issues. The first one is related torisk assessment associated with shipping hazmats along a specified link of a transportation network; thesecond concerns hazmat routing, that is, finding minimum and equitable risk routes for hazmat shipments.

Ample literature has focused on risk assessment, and much work in this direction has already beendone by modeling risk probability distribution over given areas, taking into account the risk related to thecarried substance and the transport mode [3], the environmental conditions [4] and the utilized route. Theestimation and analysis of risk associated with the transportation activities of hazardous materials (andwastes) have been carried out using various techniques. Some of these techniques are direct measuresof the consequences (e.g. expected annual fatalities), while others are based on constructed indices forcomparative measures. A survey on risk modeling can be found in [5].

The second main line of research deals with finding low-risk routes, and at the same time with theequitable distribution of the risk. This aspect has to be considered to ensure that the risk connected to thehazardous material shipment is spread equitably throughout the whole crossed area.

Several models have been proposed in the literature that allow us to determine routes of minimum riskunder the tie of equity; see, for example, the model proposed by Gopalan et al. [6] and Current and Ratick[7]. Gopalan et al. [8] deal with the equitable distribution of the risk by considering a strip around everypath (area of impact) and by considering the population included in this area as equally subject to theconsequences of the accident.

Taking into account an equitable distribution of the risk, different methods have been proposed in theliterature to find, given a pair of origin–destination points, a set of alternative routes, also in contextsdifferent from that of hazmat transportation (e.g., see [9–13]). However, these methods only allow us toequitably spread the risk spatially over a region, but do not consider the effect of simultaneous travelingof hazmat vehicles over the network.

In fact, when finding routes of minimum total risk it is supposed that the probability of an incident permile for a vehicle, on a given link, is independent from the other vehicles traveling on the network. Thisis a reasonable assumption only if any two vehicles are, at any given time, sufficiently far apart. On thecontrary, if at a given time, two vehicles are close to each other, and one of them has an accident, thereis a high probability that the other one will also be involved and have an accident as well. Therefore,when two vehicles are not far apart, the total risk induced by them on the population living in the vicinityis not simply given by the sum of the risks induced on that population by each of them, but is muchhigher.

For example, let us consider two hazmat vehicles traveling along two routes intersecting on a certainpoint, and suppose that both vehicles cross the intersection point at about the same time. The populationin the area centered in the intersection point is intuitively exposed to a risk whose magnitude is greaterthan that suffered in the case in which the two vehicles cross that point at rather different times.

Hence, to maintain the independent incident probability assumption, and thus the hypothesis of riskadditivity, we have to guarantee that at any given time any two vehicles must be sufficiently far apart.

1330 P. Carotenuto et al. / Computers & Operations Research 34 (2007) 1328–1350

This can be addressed by suitably scheduling hazmat vehicles on the assigned routes, and the integrationof routing and scheduling decisions seems to be potentially significant in this regard.

In this paper, we study the routing and scheduling of hazmat shipments, where one has to select a routeand find a departure time for each hazmat shipment, such that, at any given time, no hazmat vehicle istoo (spatially) close to another one.

We consider the shipment of hazardous materials from the production/distribution plants (the origins)to the customer plants (the destinations), where the materials are employed. A centralized decision maker,such as a public authority, needs to plan daily routes, and daily departure times for a given set of hazardousmaterial shipment requests, with the aim of optimizing a given objective function; once the hazmattransportation requests have been collected, with their preferred departure times, the authority needs toassign to each shipment a route from its origin point to its destination point, and an actual departure timeno earlier than the preferred one.

In Italy, for instance, there is the Protezione Civile, a public body that coordinates people and infrastruc-tures in order to handle emergencies caused by natural or catastrophic events, with the aim of protectinglives, goods, and the environment. Even though the Protezione Civile is not currently a central authoritythat determines and enforces hazmat shipments and, therefore, does not fix routes and schedules, it isnonetheless interested in evaluating the impact and the benefit of hazmat shipment planning activities atthe research level. On the other hand, the restrictions of fixing routes and schedules could be relaxed ifembedded in a DSS for planning hazmat shipments. Indeed, by showing that imposing shipment routesand schedules strictly might considerably reduce risks, an authority may consider possible amendmentsto its hazmat transportation management policy. Currently, the Italian hazmat transportation policy is inline with the European agreement concerning the international transportation of dangerous goods by road(Accord Dangereuses par Route—ADR) (see the Italian infrastructure and transport ministerial decreeof June 20, 2003; G.U. n. 156, July 8, 2003).

We consider a two-stage approach to finding a solution to the routing and scheduling hazmat shipmentproblem. In the first stage, a set of minimum and equitable risk alternative routes is generated; in particular,these routes are assumed to equitably spread the risk spatially among the zones of the geographical regionwhere the transportation network is embedded. In the second stage, a route, among the generated ones inthe first stage, and a departure time, no less than the preferred one, are assigned to each hazmat shipmentrequest, so as to assure that at any given time any two hazmat vehicles must be sufficiently far apart, andminimize the sum of the hazmat shipment delays evaluated with respect to the minimum arrival timesat destinations; in particular, each of these latter values is equal to the preferred (earliest) departure timeplus the minimum travel time required to go from the origin to the destination along one of the availableroutes.

In this work, we focus our attention on this hazmat shipment scheduling problem addressed in thesecond stage. The proposed two-stage approach is devised to act at a planning level within a regionalnetwork. In this regard we consider a short planning horizon where we may assume a negligible datavariability over time.

To the best of our knowledge, the problem of scheduling hazmat shipments has received very littleattention until recently. For example, Cox and Turnquist [14] propose a model to find the departure timefor any given route which minimizes the delay due to time-of-day restrictions. The model assumes thatthe link travel time is a random variable and a recursive algorithm is given.

For the hazmat shipment scheduling problem we focus on, we propose a job-shop scheduling model.Indeed, the proposed model is an extension of the job-shop scheduling problem (JSP) where each job (i.e.,


a hazmat shipment) has alternative routes; that is, each job has different alternative chains of operations,and the problem consists in selecting, for each job, one of the alternative routes (i.e., a chain (sequence) ofoperations), and in scheduling each job with the assigned route. No-wait constraints arise in the schedulingmodel as well, since, supposing that no safe area is available, when a hazmat vehicle starts traveling fromthe given origin it cannot stop until it arrives at the given destination. The objective in the schedulingmodel is to find a solution, that is, a job route assignment and a (no-wait) job schedule, such that the totaljob completion time is minimized, implying the minimization of the total hazmat shipment delay.

Job-shop scheduling is one of the most basic models of scheduling theory. In the classic model (e.g., see[15]) known to be NP-hard, a job-shop consists of a set of dedicated machines that perform job operations.There is a set of jobs, where each job has a pre-specified processing order through the machines, that is,each job is composed of an ordered list (i.e., a chain or sequence) of operations with each one requiring apre-specified machine of the job-shop. The other assumptions are that no machine can process more thanone job operation at a time, the processing of operations cannot be interrupted, and no two operations ofthe same job can be processed at the same time.

Over time, the classic model has been extended to consider machine alternatives for individual opera-tions. For example, some of the JSP extensions are:

• the multi-purpose machine job-shop scheduling problem (MPM-JSP), where each operation may beexecuted on exactly one of several different machines (e.g., see [16]);

• the flexible job-shop scheduling problem (FLEX-JSP), where operations may have different durationsdepending on the different machines (e.g., see [17]);

• the multi-mode job-shop scheduling problem (MMJSP), where each operation may be executed in oneout of several different execution modes, where each mode identifies a set of resources (i.e., machines),and specifies the time period required for the execution of the operation with these resources (e.g., see[18]).

A second direction in which the classic job-shop model has been extended deals with the job routings,ranging from simple chains of operations to general directed acyclic graphs (e.g., see [19]). In [19], Kisstudies the job-shop scheduling problem with processing alternatives (AJSP), where job routings aredirected acyclic graphs with or-subgraphs. As for solution methods, most of the job-shop models areheuristically solved by means of some kind of meta-heuristic approach (e.g., see [17,20,21]).

The model we study in this paper is a special case of the AJSP, where each job has alternative routes(i.e., operation chains), and exactly one of them must be chosen for job execution. Nevertheless, ourmodel contains a further extension, i.e., the no-wait constraints which require the operations of a job tobe executed without a waiting time between two consecutive operations (e.g., see the survey in [22]).

We propose a tabu search algorithm to heuristically solve the proposed scheduling model, and finda solution to the hazmat shipment scheduling problem. The tabu search algorithm is experimentallyevaluated on a set of realistic test problems in a regional area, evaluating the provided solutions also withrespect to the total route risk and length. Moreover, we propose a simple lower bound for the job-shopscheduling model to evaluate the effectiveness of the proposed tabu search algorithm.

The paper is organized as follows. In Section 2, we describe the two-stage approach we considerfor routing and scheduling hazmat shipments, and provide a formal definition of the hazmat shipmentscheduling problem. In Section 3, we propose a job-shop scheduling model for the hazmat shipmentscheduling problem. In Sections 4 and 5, we provide the solution approach and a lower bound for the


job-shop scheduling model, respectively. Finally, Section 6 is devoted to the experimental analysis of theproposed model and algorithm.

2. Problem definition

Hazmats are often employed in customer plants that may be located far from the production/distributionplants. Therefore, transportation authorities should deal with the problem of managing hazmat shipments.Once the daily hazmat transportation requests (HTRs for the sake of simplicity) are collected, the au-thority has to define how the shipments must go from the origins (production/distribution plants) to thedestinations (customer plants), and at what time they have to leave the origins, with the main aim ofexecuting these HTRs in the safest way.

We consider a two-stage approach for the problem of finding daily routes and daily departure times. Inthe first stage, a set of minimum and equitable risk routes are defined for each HTR in order to (spatially)spread the risk equitably over the population; in the second stage, one of the defined routes and a departuretime no less than the preferred (earliest) one are assigned to each HTR in order assure that, at any giventime, any two hazmat vehicles must be sufficiently far apart and minimize the sum of the hazmat shipmentdelays. Fig. 1 describes the general schema of the two-stage approach.

More formally, let G = (N, L) be a (road) transportation network, where N is the set of nodes repre-senting geographically spread locations and L represents the set of direct links between pairs of nodes.Let J = {1, . . . , n} be a given set of n HTRs. Each HTR j ∈ J is identified by an origin–destinationpair of nodes, with Oj, Dj ∈ N being the origin and the destination, respectively, and has a preferred(earliest) departure time rj for leaving the origin Oj .

In the first stage, for each HTR j, the problem of finding kj routes of minimum total risk that alsoguarantee the equitable spread of risk over the population has to be solved. A solution for this problem(which is out of the scope of this paper) can be found, for example, by using the approach proposed in[13]. In that paper, the problem is formulated as a constrained k-shortest path problem by limiting therisk sustained by the population located in the proximity of each populated link of the network; this latterproblem is heuristically solved by a greedy randomized algorithm that, at each run, constructs iterativelya feasible solution (i.e., fulfilling the risk threshold constraints on populated links) by adding one path(route) at a time, randomly selected from a candidate path set, to a partial (feasible) solution formed bya subset of k′ < k feasible paths.

Fig. 1. The two-stage approach.


In this paper, we study the problem addressed in the second stage, that is, the hazmat shipment schedulingproblem which is defined as follows.

A set J = {1, . . . , n} of n HTRs defined on the road network G = (N, L) is given, with Oj , Dj ∈ N

being the origin and the destination of HTR j. For each HTR j, a preferred (earliest) departure time rjfor leaving the origin Oj , and a predefined set �j = {�1j , . . . , �kj j } of kj �1 alternative routes on thenetwork G, each one going from origin Oj ∈ N to destination Dj ∈ N , are given.

Executing HTR j corresponds to shipping the hazmat with a vehicle traveling on a selected route in�j , with the actual departing time sj being no less than the preferred departure time rj . We suppose thatonce the hazmat vehicles have departed, they have to travel on the assigned route without stopping atintermediate nodes. The aim is to execute all the HTRs minimizing the total HTR shipment delay withrespect to the minimum arrival times at destinations, while guaranteeing that at any given time no hazmatvehicle is too spatially close to another one.

While the given routes are assumed to be chosen so as to assure a certain equity of risk distributionover the population of the region crossed by the hazmat shipments, in order to guarantee the independentincident probability assumption among vehicles and reduce the risk magnitude due to the hazmat vehiclesimultaneous traveling, it is required that, at any given time, the distance between any two hazmat vehiclesis not less than a certain physical (substance-dependent) distance �.

In order to represent this concept, we consider a �-circle around each (route) point x of the given routes(i.e., a circle of radius � centered at x), and require that, at any given time, at most one hazmat vehiclemay travel (strictly) inside the �-circle of each point x. In general, we may have distinct values �j of � foreach HTR j. For simplicity, we will refer to the case with �j = �, but the scheduling model we are goingto describe can handle also the more general case.

3. A job-shop scheduling model

In this section, we give a job-shop scheduling model for the hazmat shipment scheduling problem.We consider each alternative route �hj of HTR j being formed by a sequence of nhj �2 (consecutive)

route segments �1hj , . . . , �i

hj , . . . , �nhj

hj , each one of at most � length with ��, and centered in route points

x1hj =Oj, . . . , x

ihj , . . . , x

nhj

hj =Dj , respectively, and consider a �-circle �ihj around each point xi

hj . Hence,we have that the physical distance between two consecutive points in the route �hj is at most �.

The value of � controls the segmentation of route traveling, and we require that, at any given time,at most one hazmat vehicle may travel along each route segment. The smaller the �, the more detailedthe segmentation and, hence, the traveling representation, but, at the same time, the larger the size of thecorresponding scheduling model instance. However, � should be no greater than �, otherwise two hazmatvehicles traveling along two consecutive route segments will be (unnecessary) maintained at distancegreater than �. Fig. 2 depicts the �-circles centered in the route segment center points of route �hj of HTRj supposing that the physical distance � between two consecutive points is equal to �.

The execution of HTR j along route �hj corresponds to the execution of job j composed of a sequence

(or chain)(o1hj , . . . , o

ihj , . . . , o

nhj

hj

)of nhj job operations, where operation oi

hj represents the traveling on

route segment �ihj centered in point xi

hj with operation processing time pihj equal to the traveling time.

Let � be the union set of the route segment center points of all the alternative routes for all the HTRs.We underline that � is in general distinct from the node set N of the network G.


xihj

x1hj = Oj

xnhjhj = Dj

Fig. 2. �-circles around points or route �hj of HTR j, with � = �.

�

�

xihj

x1hj = Oj

xnhjhj = Dj

�lkq

�ikq

x1kq = Oqxnkqkq = Dq

xlkq

Fig. 3. Operations oihj

and olkq

are incompatible.

Recall that in the hazmat shipment scheduling problem, it is required that, at any given time, at mostone hazmat vehicle may travel (strictly) inside the �-circle centered in any route point; within the routesegmentation, this means that traveling at the same time on two route segments centered in route pointsxihj , x

lkq , respectively, with the physical distance between these points being less than � (i.e., at least one

of the �-circles �ihj , �l

kq contains both of those two points), is forbidden; hence, operations oihj , o

lkq are

incompatible and cannot be done simultaneously (e.g., see Fig. 3).The operation incompatibility relationships are modeled by considering a set M = � of machines (i.e.,

associating one machine to each point x ∈ �) in the job-shop scheduling model, and by supposing thateach operation oi

hj is a multi-processor task that requires the subset Mihj ⊆ M of machines associated

to point xihj and to the points that are at distance less than � (�j for the general case of distinct � values)

from xihj . Hence, two operations oi

hj , olkq are incompatible if they share at least one machine, that is,

Mihj ∩ Ml

kq �= ∅.Since the hazmat vehicle is supposed to travel without stops, the sequence (or chain) of operations of the

same job must be done with no-wait, that is, without waiting times between two consecutive operations.Moreover, each HTR j has a predefined set �j = {�1j , . . . , �kj j } of kj �1 alternative routes; hence, eachjob j has kj alternative routes Rj = {�1j , . . . , �kj j }, (each one representing an operation chain), andexactly one of them must be selected for the execution of job j.


Therefore, the problem of finding a departure time, and assigning a route to each HTR can be modeledas a generalization of the multi-processor no-wait job-shop scheduling problem, where a job has differentalternative routes on the shop.We call this problem a multi-processor no-wait job-shop scheduling problemwith alternative job routes.

Recall that for each HTR j the hazmat shipment cannot start before the preferred departure time rj ;hence, in the scheduling model, each job has a given release date. Finally, the objective function of thehazmat shipment scheduling problem we wish to minimize is the total HTR shipment delay with respectto minimum shipment arrival times at destinations. Since the shipment delay of each HTR is definedas the difference between its actual shipment arrival time and its minimum shipment arrival time, withthe latter being equal to the preferred (earliest) shipment departure time plus the minimum travel timeto go from the origin to the destination along one of the available routes, the objective of the job-shopscheduling model is to minimize the total job completion time.

Formally, the JSP we consider can be defined as follows:

• a set M = � of machines is given,• a set J of n jobs is given, with rj being the release date of job j ∈ J ,• a set Rj = {�1j , . . . , �kj j } of kj �1 routes (i.e., operation chains) are available for each job j ∈ J ,

where �hj is a chain (o1hj , . . . , o

nhj

hj ) of nhj job operations.

• each operation oihj , for its execution, requires a dedicated subset Mi

hj ⊆ M of machines for a given

processing time pihj .

The remaining assumptions are:

• no machine can process more than one operation at a time;• the processing of an operation cannot be interrupted;• no two operations of the same job can be done simultaneously.

The problem is to select a route �hj ∈ Rj for each job j, i.e., a chain of operations to be executed withno waiting time between two consecutive operations, and a starting time for each job, no less than itsrelease date, such that the total job completion time is minimized.

Let X : {1, . . . , n} → N be a route assignment function, representing the selection of one route �X(j)j

in Rj for each job j.The pair (X, (X)) defines a solution to our problem, where (X) is a feasible schedule of the job

operations of the selected routes. In particular, (X) is feasible if the operations of a job are executedwith no-wait according to the order (operation precedence relationship) given by the job operation chain,the operations sharing a machine are sequenced on that machine, and job starting times are not less thanjob release dates. Let Cj(X, (X)) be the completion time of job j. Since the scheduling objective is tominimize the function z(X, (X)) = ∑

j∈J Cj (X, (X)), we have that, given X, the scheduling problemconsists in finding a feasible schedule ∗(X), such that z(X, (X)) is minimized; let z∗(X) be the optimalsolution value.

Therefore, the overall problem is

z∗ = minX

{z∗(X)} = min(X,(X))

{z(X, (X))}.


4. Solution algorithm

We propose a hierarchical approach to finding a solution to our JSP, based on the idea that when,for each job, one out of its available routes is selected, then the problem becomes a no-wait job-shopscheduling problem. In particular, we propose a two-level algorithm, where the first-level procedure isa tabu search algorithm dealing with the definition of a route assignment X, and the second-level one isa constructive heuristic algorithm for determining feasible job starting times given the assigned routes,that is a feasible schedule (X).

Let us first describe the latter one.

4.1. The constructive heuristic

Given a route assignment X, the objective of the constructive heuristic is to find a feasible schedule(X) for the no-wait job-shop scheduling problem with job release dates while minimizing the totaljob completion time. In this section, we describe a greedy randomized algorithm for finding such aschedule.

The algorithm, called GR, uses the partial schedule concept, that is, it starts with an empty partialschedule, and at each iteration enlarges it by identifying the set of the unscheduled jobs, and then insertingone out of these jobs into the partial schedule. Therefore, algorithm GR finds a solution in n iterations.

Let (X, J\NS) be the partial schedule obtained by scheduling all the operations of jobs in J\NS,where NS is the set of unscheduled jobs. At each iteration, algorithm GR selects a job j from the set NS,and schedules it at the earliest possible starting time sj (X)�rj , such that operation oi

X(j)j of job j is

executed after operation olX(q)q of a scheduled job q, if oi

X(j)j and olX(q)q are mutually incompatible, that

is, they share at least one machine (i.e., MiX(j)j ∩ Ml

X(q)q �= ∅).

At the beginning, NS is equal to the set of jobs J and the solution value z is equal to zero. In a genericiteration, in order to select a candidate job to be inserted in the partial schedule, for each job j ∈ NS, theincremental cost cj (X) due to the addition of job j to the partial schedule (X, J\NS) is computed. Then,algorithm GR randomly selects a job of a restricted candidate set RCS ⊆ NS of jobs whose incrementalcosts are not greater than cmin + (cmax − cmin), where cmin and cmax are the minimum and maximumincremental cost values, respectively, among jobs in NS, and is a real value between 0 and 1 that controlsthe cardinality of the restricted candidate set RCS. Finally, algorithm GR is run several times and the bestsolution found is taken.

The value of parameter is not fixed a priori, but is guided by the solution values found during theprevious runs. In particular, this is accomplished by using the rule proposed in [23], where parameter ischosen randomly according to a probability distribution that is updated periodically, by assigning higherprobabilities to values of , which allow us to find solutions whose values are, on average, closer to thebest solution found. Fig. 4 shows a sketch of algorithm GR.

For each job j ∈ NS, the incremental cost cj (X) is equal to the completion time of job j givenby the sum of the duration dj (X) of job j and its earliest starting time sj (X) in the new partialschedule:

cj (X) = sj (X) + dj (X).


Fig. 4. Algorithm GR.

Due to the no-wait constraints, the duration dj (X) can be computed as follows:

dj (X) =nX(j)j∑i=1

piX(j)j .

We now explain how we compute the earliest starting time sj (X) of job j. For the sake of readability,we omit the index X(j) identifying the route (operation chain) assigned to j. Let sk

j be the starting time ofthe kth operation of job j; the no-wait constraints imply that

skj = s1

j +k−1∑i=1

pij ,

with s1j = sj (X).

If operation okj is incompatible with a scheduled operation ol

q , we choose to schedule operation okj after

the completion time of operation olq . Therefore, sk

j has to be greater than, or equal to, the time in which

operation olq is completed, i.e.,

s1j +

k−1∑i=1

pij �s1

q +l∑

h=1

phq ,

and, hence,

s1j �s1

q +l∑

h=1

phq −

k−1∑i=1

pij .

This latter condition should be valid for all operations olq that are incompatible with ok

j , and, hence,

minimizing s1j implies

sj (X) = s1j = max

⎧⎨⎩rj , max{

(okj ,ol

q ):Mkj ∩Ml

q �=∅}{

s1q +

l∑h=1

phq −

k−1∑i=1

pij

}⎫⎬⎭ .


4.2. The tabu search algorithm

In this section, we describe the first-level procedure of our two-level algorithm. The procedure is atabu search algorithm called TS. Tabu search [24,25], like other meta-heuristic approaches, is based onthe local search paradigm. Starting from a given initial solution S0 (e.g., found by a constructive heuristicalgorithm), at each iteration t, the method moves from solution St−1 to the best solution in its neighborhoodN(St−1) (i.e., the set of solutions obtained from St−1 by applying all the possible moves), even if thiscauses a deterioration in the objective function value (in order to escape from the local optimum). Toavoid cycling, some solutions of the neighborhood are declared forbidden, or tabu, for a given numberof iterations. The search stops whenever a given stopping rule is satisfied. The method can be enhancedby incorporating several features to allow the intensification and diversification of the search process.

Typically, tabu solutions are declared by forbidding some of the possible moves for a certain numberof iterations, and, in doing so, the concept of memory, which records the search process history, is used.Commonly, the most used type of memory is the recency-based one that sets as tabu a move for a givennumber of iterations (tabu tenure).

In our algorithm TS, we also consider the frequency-based memory, by which also moves that frequentlyoccur in the search process are considered as tabu in order to diversify the search and to help the searchprocess explore a larger portion of the solution space. Furthermore, algorithm TS considers also theaspiration criteria for removing the status tabu for a move if a specific criterion is satisfied. Next, wedescribe the main features of algorithm TS sketched in Fig. 5.

The initial feasible solution is generated by algorithm GR, with the route assignment function Xselecting the minimum length route for each job.

The neighborhood N((X, (X))) is defined as the set of solutions obtained from solution (X, (X))

with a move (j ′, �), where j ′ is the job with the greater value (1 − vj )fj (X) among all the jobs in J, withfj (X) being the flow time fj (X) = (sj (X) + dj (X) − rj ) of job j, and vj the frequency of choosing amove involving job j, with respect to the total number of tabu search iterations t. The move (j ′, �) consistsin substituting the route X(j ′) assigned to job j ′ with the new route �, and finding a new schedule withthis new job route assignment by applying algorithm GR; the value of move (j ′, �) is the total completiontime of this schedule.

Fig. 5. Algorithm TS.


Fig. 6. Example: the available routes for the HTRs.

According to the classic recency-based implementation of tabu search algorithms, at iteration t a move(j ′, �) is tabu only if � was the selected route for job j ′ in a solution visited during the last tabu-tenureiterations. Let T ((X, (X)); t) ⊆ N((X, (X))) be the set of tabu moves at iteration t.

In the search process, diversification strategies are taken into account by implementing frequency-basedmemory (with the frequency values vj ); this allows us to penalize the moves that occur frequently in thesearch process, and, hence, to encourage the analysis of new unexplored solutions.

We consider a further diversification strategy that is useful in avoiding a deterministic tabu searchalgorithm behavior. In fact, according to the N((X, (X))) definition, the job selection occurs by meansof two distinct modalities. With a certain probability p, we select the job corresponding to the maximumpenalized flow time, while with probability (1 − p) one out of the jobs in J is randomly selected.

Moreover, we consider the following aspiration criteria. We remove the tabu status of a move if thevalue of the move is not greater than � times the current best solution value, with � ∈ [1, 1.3]. Wheneverall the moves are tabu, and do not satisfy the aspiration criteria, the tabu status of the oldest move isoverridden. Let A((X, (X)); t) ⊆ T ((X, (X)); t) be the set of tabu moves at iteration t whose tabustatus is removed.

Finally, algorithm TS is stopped whenever the number of iterations t reaches the iteration maximumnumber MAX_ITER.

4.3. Algorithm description

We describe a simple example explaining how algorithm TS works. We consider n = 2 HTRs andkj = k = 2 routes for each HTR j. All the routes have the same length equal to 3�, and segmented into4 route segments, each one of length at most � = �; the first and the last segments of each route are oflength �/2, and the others are of length �. In particular (see Fig. 6), let (y1, y3, y5, y8) be the sequence ofthe route segment center points of route �11 and let (y1, y4, y6, y8) be the sequence of the route segmentcenter points of route �21 of HTR 1; let (y2, y3, y4, y7) be the sequence of the route segment center pointsof route �12 and let (y2, y5, y6, y7) be the sequence of the route segment center points of route �22 ofHTR 2. Therefore, the set of route segment center points is � = {y1, y2, y3, y4, y5, y6, y7, y8}.


We suppose that the hazmat vehicles have the same speed, and, hence, let � be the time required by avehicle to travel along a route segment of length �. Moreover, we suppose all the HTRs have the samedeparture time r (e.g., r = 0).

Therefore, the given problem instance translates into the following JSP instance with a set M = � of8 machines and a set J of 2 jobs, where:

• job 1 is composed of 2 alternative routes, that is, the operation chains �11 = (o1

11, o211, o

311, o

411

)and

�21 = (o1

21, o221, o

321, o

421

), with the following processing times and machine requirements:

oihj pi

hj Mihj

o111 �/2 y1

o211 � y3

o311 � y5

o411 �/2 y8

oihj pi

hj Mihj

o121 �/2 y1

o222 � y4

o321 � y6

o421 �/2 y8

• job 2 is composed of 2 alternative routes, that is, operation chains �12 = (o1

12, o212, o

312, o

412

)and

�22 = (o1

22, o222, o

322, o

422

), with the following processing times and machine requirements:

oihj pi

hj Mihj

o112 �/2 y2

o212 � y3

o312 � y4

o412 �/2 y7

oihj pi

hj Mihj

o122 �/2 y2

o222 � y5

o322 � y6

o422 �/2 y7

We set the parameters of algorithm TS as follows: MAX_ITER = 2; tabu-tenure = 1. The main stepsof the algorithm are listed as follows:

• Generate initial solution (X, (X)): The initial feasible solution is generated by algorithm GR, with theroute assignment X selecting for each job j the route �

j=X(j) with minimum job duration (i.e., the job

routes �1=�11 = (

o111, o

211, o

311, o

411

)and �

2=�12 = (

o112, o

212, o

312, o

412

)for jobs 1 and 2, respectively).

Algorithm GR sets the job starting times s1(X) = 0 and s2(X) = �, in order to sequence job operationso2

11 and o212 on the shared machine y3. The solution (X, (X)) is considered as the current best known

solution S∗, and its value z(X, (X)) = ∑j∈J Cj (X, (X)) is equal to 7�.

• Iter. t = 1: Given the solution (X, (X)) = ({�11, �12}, ({�11, �12})), algorithm TS selects the job j ′that has the maximum value for (1 − vj )fj (X), where vj and fj (X) are the frequency and the flowtime of job j, respectively. Since v1 = v2 = 0 and f1(X) = 3� and f2(X) = 4�, algorithm TS selectsjob 2 and considers the neighborhood N(X, (X)) obtained with the set {(2, �)} of moves with � ∈{�12, �22}\{�12}. Therefore, N(X, (X)) is composed of the single solution ({�11, �22}, ({�11, �22}))obtained from solution (X, (X)) with the move (2, �22).

• Both the tabu set T ((X, (X)); t) and the aspirant set A((X, (X)); t) are empty at iteration t=1. There-fore the best solution in N(X, (X))/T ((X, (X)); t) ∪ A((X, (X)); t) is ({�11, �22}, ({�11, �22}))with job starting times s1({�11, �22})=s2({�11, �22})=0 and solution value z({�11, �22}, ({�11, �22}))=6�. Note that this solution assures that job operations o3

11 and o222 are sequenced on the shared

machine y5.


O1 y1

y3

y5

y4

y6

D2 y7

Route assigned to HTR 1

Route assigned to HTR 2

O2 y2

D1 y8

Fig. 7. Example: the routes assigned to the HTRs.

Since the selected solution S′ = ({�11, �22}, ({�11, �22})) is better than the best known solution S∗,we update S∗ with the new solution S′.

• Iter. t =2: Starting from solution (X, (X))=S′, algorithm TS selects job 1 since v2 > v1 and f1(X)=f2(X) = 3�. Therefore, the algorithm considers the neighborhood N(X, (X)) composed of the singlesolution ({�21, �22}, ({�21, �22})) obtained from solution (X, (X)) with the move (1, �21).

• Both the tabu set T ((X, (X)); t) and the aspirant set A((X, (X)); t) are empty at iteration t=2. There-fore the best solution in N(X, (X))/T ((X, (X)); t) ∪ A((X, (X)); t) is ({�21, �22}, ({�21, �22})),with job starting times s1({�21, �22}) = 0 and s2({�21, �22}) = � in order to sequence job operationso3

21 and o322 on the shared machine y6; the solution value z({�21, �22}, ({�21, �22})) is equal to 7�.

At round t = 2, the selected solution S′ = ({�21, �22}, ({�21, �22})) is not better than the best knownsolution S∗; hence, S∗ is not updated with the new solution S′.

• Iter t = 3: Algorithm TS stops since the number of iterations t is greater than MAX_ITER = 2.

In this simple example, the best solution found S∗ corresponds to that computed at iteration 1 ofalgorithm TS. According to S∗, route �11 is assigned to job 1 and route �22 to job 2, and the job startingtimes are s1({�11, �22}) = s2({�11, �22}) = 0; the solution value z({�11, �22}, ({�11, �22})) is 6�.

Therefore, according to the solution of the job-shop scheduling model, routes �11 and �22 are assignedto HTR 1 and 2, respectively (see Fig. 7), and the HTRs actual departing times s1 and s2 are set to 0.Since both routes �11 and �22 are of minimum length (duration) among the available routes for HTRs 1and 2, respectively, the total hazmat shipment delay is zero.

5. Lower bound computation

To evaluate the solution algorithm for the JSP, we compute a lower bound on the optimal total jobcompletion time. The lower bound is obtained by relaxing the precedence relations between operations ofa job. Therefore, the relaxed problem is similar to a no-wait open-shop with multi-processor operations,with the extension that there are alternative operation sets for each job.


By removing the latter extension, we obtain the problem Pno-alt with no alternative operation sets,where each job j ∈ J is formed by a set {o1

j , . . . , onj

j } of nj operations, with operation oij simultaneously

requiring all the machines in the set Mij ⊆ M for pi

j time. Now, by relaxing the machine simultaneous

requirement constraint for each multi-processor operation oij and the constraint that no two operations

of the same job can be simultaneously executed, and by allowing operation preemptions, problem pno-altdecomposes into |M| independent single-machine scheduling problems, where, for each mth problem, we

have a set J of preemptable jobs with release date rj and processing time pj (m)= j (m)=∑nj

i=1{pij |m ∈

Mij } for each job j ∈ J , and the goal is to minimize the total job completion time. According to the well-

known machine scheduling notation, let OPTm[1|rj , pmtn, pj (m) = j (m)| ∑ Cj(m)] be the optimalsolution value of the mth single-machine scheduling problem, where Cj(m) is the completion time ofjob j. Therefore, a lower bound LBno-alt on the optimal total job completion time for problem Pno-altmay be obtained by taking the maximum among the optimal solution values of these |M| (independent)single-machine scheduling problems and the minimum total job completion time in the case in which, inthe original problem, we have no resource (machine) constraints. Hence, we have that

LBno-alt = min lb (1)

s.t.

OPTm

[1|rj , pmtn, pj (m) = j (m)|

∑Cj(m)

]� lb, m = 1, . . . , |M|; (2)

|J |∑j=1

(rj + pj)� lb, (3)

where pj= ∑nj

i=1 pij is the (total) job processing time of job j, and the left-hand side of constraint (3) is

the minimum total job completion if we relax resource constraints.We now generalize that lower bound computation, reconsidering, for each job j, the necessary choice

of one out of kj alternative operation sets O1j , . . . , Okj j for that job, with Ohj = {o1hj , . . . , o

nhj

hj } beinga set of nhj job operations, for h = 1, . . . , kj . More formally, a lower bound LB of our scheduling modeldefined in Section 4 is the optimum value of this problem:

LB = min lb (4)

s.t.

OPT m

⎡⎣1|rj , pmtn, pj (m) =

kj∑h=1

�hj hj (m)|∑

Cj(m)

⎤⎦ � lb, m = 1, . . . , |M|; (5)

|J |∑j=1

⎛⎝rj +

kj∑h=1

�hjphj

⎞⎠ � lb; (6)

kj∑h=1

�hj = 1, j = 1, . . . , |J |; (7)


�hj ∈ {0, 1}, j = 1, . . . , |J |, h = 1, . . . , kj , (8)

where

• �hj is a decision variable which is equal to 1 if operation set Ohj is assigned to job j, and 0 otherwise;• hj (m) is the total processing time of job j on machine m ∈ M , if job j is executed with the hth

alternative operation set Ohj ; that is, hj (m) = ∑nhj

i=1{pihj |m ∈ Mi

hj };• phj

is the (total) job processing time of job j, if job j is executed with the hth alternative operation set

Ohj ={o1hj , . . . , o

nhj

hj

}; that is, p

hj= ∑nhj

i=1 pihj .

Constraints (7) force, for each job j, the selection of exactly one operation set among its kj alternatives.In particular, each mth constraint of (5) can be substituted with the following set of constraints in order

to obtain an integer linear program (ILP) formulation for the problem of determining the lower boundLB:

|J |∑j=1

Cj(m)� lb; (9)

T∑t=1

yjt (m) =kj∑

h=1

�hj hj (m), j = 1, . . . , |J |; (10)

|J |∑j=1

yjt (m)�1, t = 1, . . . , T ; (11)

(t − 1)yjt (m)�rj , j = 1, . . . , |J |, t = 1, . . . , T ; (12)

tyjt (m)�Cj(m), j = 1, . . . , |J |, t = 1, . . . , T ; (13)

rj �Cj(m), j = 1, . . . , |J |; (14)

yjt (m) ∈ {0, 1}, j = 1, . . . , |J |, t = 1, . . . , T . (15)

where:

• yjt (m) is a decision variable which is equal to 1 if job j is processed on machine m during the unit timeinterval [t − 1, t], and 0 otherwise;

• T is an upper bound on the time required to execute all the jobs.

Constraint (9) states that the total job completion time of any solution to the mth single-machineproblem cannot be greater than lb. Constraints (10) state that the number of unit time executions for job jmust be equal to the total processing time of job j on machine m ∈ M . Constraints (11) state that machinem cannot execute more than one job at a time. Constraints (12) force unit time executions for job j tobe done after job release date rj . Constraints (13) and (14), respectively, state that the completion timeCj(m) of job j on machine m cannot be less than the finish time of the last unit time execution for job j,nor (obviously) less than rj .


6. Experimental results

In this section, we report and analyze the experimental results by evaluating the performance of thetabu search algorithm TS. The algorithm is coded in C language and run on a 2.53 GHz Pentium 4 PC.

We experimented the proposed algorithm on an Italian regional road network using realistic data. Inparticular, we considered the region of Lazio containing the city of Rome, which has an extension of17,203 km2 and a population of about 5 million inhabitants. The resulting network G = (N, L) has 311nodes and 441 links, and contains both highways and local regional roads. The experimental analysis hasbeen done considering two different �-radius values for the “impact zone”. These are � = 1 and 3 km,which are close (one smaller and the other greater than) to the value of 2.4 km (1.5 miles) that in [8] hasbeen used as the �-radius for the benzyl chloride hazmat.

We consider instances with n = 5, 10 HTRs, and kj = k = 3, 5, 10 routes for each HTR j. For eachHTR, these given routes are a set of k minimum and equitable risk. In particular, since the generationof these routes is out of the scope of this paper, these minimum and equitable risk routes are generatedaccording to the technique proposed in [13], which was briefly discussed in Section 2. The origin anddestination points of the HTRs belong to two distinct circular areas with a 10 km radius, in order to avoidgenerating trivial instances where routes do not intersect. The average length of the generated routes isequal to 149.6 km, while the average risk increment between two routes for the same origin–destinationpair is equal to 6.6%.

For each (n, k) combination, we consider 5 instances with the preferred departure times rj , for eachHTR j, uniformly randomly generated in the 8:00–9:00 a.m. time window.

We evaluate the TS algorithm varying the maximum number of iterations MAX_ITER. In particular,we consider three versions of algorithm TS, called V1, V2 and V3, with MAX_ITER = n, nk, and 2nk,respectively. For all these algorithm versions, the tabu tenure is equal to 0.2 MAX_ITER.

Table 1 shows all the results obtained by considering � = 1 km, and � = � (i.e., “test cases # 1”). In thefirst column, the number n of HTRs is listed, and in the second one, the number k of available routes foreach HTR is reported. Column three represents the algorithm version. The results in columns four, five,six and seven represent the total hazmat shipment delay in minutes. In particular, the fourth column liststhe average value “Ave. GR” of the solutions found by algorithm GR. From column five to column seven,the minimum, the average and the maximum values (i.e., “Min. TS”, “Ave. TS”, and “Max. TS”) of thesolutions found by algorithm TS are given, respectively. Column eight represents the average relative gap“Ave. LB gap %”, in percentage, between the total completion time of the job-shop schedule determinedby algorithm TS and the lower bound LB on the optimal solution value which is computed by solvingthe linear relaxation of the IPL formulation given in Section 5. In column nine, we give, in percentage,the average relative improvement “Ave. TS imp. %” of the solution values given by algorithm TS withrespect to the initial solution values found by algorithm GR (i.e., 100 (Ave. GR − Ave. TS)/Ave. GR).Column 10 shows the average computational time of algorithm TS. Finally, column 11 lists the averagegap in percentage, “Ave. L gap %”, between the total length L of the assigned routes and the minimumtotal route length Lmin of the shortest assignable routes (i.e., 100 (L−Lmin)/Lmin); analogously, column12 lists the gap “Ave. R gap %”, in percentage, between the total risk R of the assigned routes with respectto the minimum total route risk Rmin of the least risk assignable routes (i.e., 100 (R − Rmin)/Rmin).Therefore, in the last two columns, we evaluate solutions also with respect to the total length and the totalrisk of the assigned routes, whose minimization is sought by both hazmat carriers and the public agencydevoted to planning routes for hazmat shipments.


Table 1Heuristic comparison: test cases # 1 (� = 1 km and � = �)

n k Version Ave. GR Min. TS Ave. TS Max. TS Ave. LB Ave. TS Ave. time (s) Ave. L Ave. Rgap % imp. % gap % gap %

5 3 V1 32.2 16 29.0 47 0.9 9.9 2.8 20.6 3.7V2 28.6 11 25.0 39 0.8 12.6 8.1 23.0 2.0V3 36.4 18 26.4 38 0.8 27.5 15.8 21.6 1.4

5 5 V1 36.6 14 23.0 36 0.7 37.2 4.8 18.2 1.0V2 36.4 13 20.2 33 0.6 44.5 23.9 18.0 2.4V3 36.4 5 15.8 29 0.5 56.6 46.0 16.1 1.6

5 10 V1 45.6 17 25.4 32 0.8 44.3 10.4 15.7 2.1V2 40.2 4 18.0 28 0.6 55.2 83.7 14.8 2.6V3 46.2 4 15.6 28 0.5 66.2 149.3 15.8 2.9

10 3 V1 206.4 118 159.4 204 2.5 22.8 46.8 22.4 0.8V2 200.4 146 168.0 210 2.5 16.2 144.4 23.2 2.1V3 231.0 106 143.6 206 2.2 37.8 287.6 25.7 2.1

10 5 V1 236.6 104 151.8 197 2.4 35.8 85.4 19.2 2.3V2 195.8 101 116.6 126 1.8 40.4 222.8 17.9 2.3V3 221.4 91 111.8 147 1.7 49.5 429.5 20.4 3.9

10 10 V1 221.2 103 134.2 158 2.1 39.3 177.5 22.2 8.9V2 235.0 68 102.4 128 1.6 56.4 535.1 22.0 9.3V3 231.0 78 106.4 131 1.6 53.9 885.5 21.7 10.0

The average gap “Ave. LB gap %” between the values of the solutions found by algorithm TS and thelower bound LB values (see column eight of Table 1) is 1.4% on average, with a maximum value of 2.5%,while the average improvement “Ave. TS imp. %” of these solution values with respect to those obtainedwith algorithm GR (see column nine) is 39.2% on average, with a maximum value of 66.2%.

Clearly, “Ave. LB gap %” values decrease, and “Ave. TS imp. %” values almost always increase ifwe increase the number MAX_ITER of iterations executed by algorithm TS, with algorithm version V3being, in practice, the best one in terms of algorithm effectiveness. On the contrary, algorithm efficiencydecreases with MAX_ITER, with algorithm version V3, in general, requiring much more CPU times thanthe other versions.

The average values “Ave. TS” of the solution values found by algorithm TS decrease with increasingvalues of k, while, obviously, they increase with n.

As for the quality of the solutions with respect to total route length and risk attributes (see table columns11 and 12), we may notice that, on average, the “Ave. L gap %” is 19.9% and the “Ave. R gap %” is 3.4%.These values reflect the fact that the given routes are generated with the aim of minimizing the risk andnot the length of the routes.

We also analyze the impact on the performance of algorithm TS due to the increase in the numberof segments into which the routes are decomposed, i.e., by decreasing the maximum length � of a routesegment. Recall that decreasing � implies both an increase in the number of machines, and an increasein job operations in the scheduling model. In particular, we list in Table 2 (organized as Table 1) theresults of algorithm version V1 for the cases with � = 1 km and � = �/2. We refer to these cases as“test cases # 2”.


Table 2Heuristic comparison: test cases # 2 (� = 1 km and � = �/2)

n k Ave. GR Min. TS Ave. TS Max. TS Ave. LB Ave. TS Ave. time (s) Ave. L Ave. Rgap % imp. % gap % gap %

5 3 35.4 12 26.4 39 0.8 25.42 9.6 18.9 1.95 35.4 13 23.6 35 0.7 33.33 16.8 18.8 1.2

10 48.4 11 23.0 40 0.7 52.48 34.9 15.6 2.8

10 3 189.2 110 142.6 178 2.2 24.63 168.4 22.3 0.85 202.4 107 134.2 167 2.1 33.70 297.1 19.3 1.6

10 224.4 115 131.0 143 2.1 41.62 641.6 20.5 9.2

Comparing results in Table 2 (i.e., test cases # 2) with those in Table 1 (i.e., test cases # 1 with algorithmversion V1), we have that, for � = 1 km, the average values of the “Ave. LB gap %” are 1.4% and 1.6%for � = �/2 and � = �, respectively, while the “Ave. TS imp. %” is 35.2% when � = �/2, and it is 31.5%when � = �. Concerning the “Ave. TS” values, we notice that the total hazmat shipment delays for n = 5are on average 24.3 min (i.e., 4.9 min per HTR) and 25.8 min (i.e., 5.2 min per HTR) for � = �/2 and� = �, respectively; for n = 10, they are 135.9 and 148.5 (i.e., 13.6 and 14.8 min per HTR), respectively,for � = �/2 and � = �. Also the “Ave. L gap %” and the “Ave. R gap %” values in Table 2 are, on average,similar to the corresponding values listed in Table 1. On the other hand, as one may expect, the CPUtimes significantly increase with decreasing values of �: for n = 5, the “Ave. time (s)” values are, onaverage, equal to 20.4 s for test cases # 2 with � = �/2 (see Table 2), compared to 6.0 s for � = � (seeTable 1, algorithm version V1); for n = 10, they are, on average, equal to 369.0 s (for � = �/2), comparedto 103.2 s (for � = �).

Table 3 (organized as Table 1) refers to all the results obtained considering � = 3 km and � = � (i.e.,“test cases # 3”). Comparing these results with those for test cases # 1 shown in Table 1, the average“Ave. LB gap %” is 5.1% with respect to 1.4%; the “Ave. TS imp. %” values are, on average, equal to25.9% compared to 39.2%. Clearly, a greater value of � (compared to the one for test cases # 1) implies alower number of operations in the scheduling model and, consequently, lower algorithm computationaltimes. In fact, for n = 5, the average “Ave. time (s)” is equal to 9.9 s (for � = 3 km and � = �; seeTable 3), compared to 30.3 s (for � = 1 km and � = �; see Table 1); for n = 10, the “Ave. time (s)” is,on average, equal to 91.3 s (for � = 3 km and � = �; see Table 3), compared to 312.7 s (for � = 1 kmand � = �; see Table 1). Also for test cases # 3, the values of the solutions (see column six “Ave. TS” ofTable 3), on average, decrease with increasing values of k, while, obviously, they increase with n. Com-paring these solution values (obtained for � = 3 km and � = �) with the corresponding values listed inTable 1 (obtained for � = 1 km and � = �), we may observe that the former ones are greater than the latterones, reflecting the fact that the degree of incompatibility among hazmat traveling operations increaseswith �. For the “Ave. L gap %” and “Ave. R gap %” values, we do not notice significant variations withrespect to those listed in Table 1.

Note that, in the job-shop scheduling model, the number |M| of machines (i.e., the number of routesegments) and the total number of job operations increase by decreasing �, since the maximum distance �between the center points of two consecutive segments of a route decreases with �. At the same time, the


Table 3Heuristic comparison: test cases # 3 (� = 3 km and � = �)

n k Version Ave. GR Min. TS Ave. TS Max. TS Ave. LB Ave. TS Ave. time (s) Ave. L Ave. Rgap % imp. % gap % gap %

5 3 V1 110.2 60 86.4 118 2.6 21.6 0.6 23.9 1.9V2 110.2 68 83.4 109 2.5 24.3 1.7 24.3 2.4V3 117.4 54 81.4 110 2.5 30.7 3.5 22.3 3.2

5 5 V1 109.0 69 90.0 116 2.7 17.4 1.2 17.1 1.7V2 120.6 71 83.4 113 2.5 30.8 5.8 19.2 2.4V3 126.2 60 73.0 88 2.2 42.2 11.2 18.9 3.4

5 10 V1 124.6 71 92.0 120 2.8 26.2 2.5 20.8 1.6V2 122.6 68 74.6 83 2.3 39.2 21.3 18.4 2.3V3 126.6 68 74.2 79 2.3 41.4 41.5 16.8 3.0

10 3 V1 690.4 529 607.4 732 8.6 12.0 10.3 32.1 2.0V2 758.6 492 577.2 646 8.2 23.9 28.7 30.3 2.2V3 710.4 519 578.2 625 8.2 18.6 57.8 28.3 1.9

10 5 V1 747.6 559 617.6 686 8.8 17.4 18.3 30.5 0.6V2 688.8 511 555.4 605 8.0 19.4 50.1 30.4 2.7V3 669.2 440 488.4 534 7.1 27.0 90.4 24.0 3.0

10 10 V1 627.6 495 508.6 541 7.4 19.0 35.8 22.2 7.7V2 639.2 422 479.8 503 7.0 24.9 164.1 22.6 10.2V3 663.2 440 462.4 487 6.8 30.3 366.2 24.1 9.0

fraction |Mihj |/|M| of machines simultaneously required by a job operation oi

hj , on average, decreasesby decreasing �, and, hence, the number of operation conflicts also decreases. This implies a reduction ofthe total hazmat shipment delay when decreasing � (e.g., compare the results for �= 3 and 1 km). Indeed,for � → 0 the conflicts tend to vanish, and in an optimal solution each hazmat shipment tends to startat the earliest (i.e., preferred) time with the assigned shortest available route and then the total hazmatshipment delay tends to zero.

Finally, we also analyze results on another set of test problems (i.e., “test cases # 4”), where for eachorigin–destination pair, two HTRs are given. In this latter scenario, we consider � = 1 km and � = �, (asfor test cases #1), but we double each HTR; that is, beside each HTR j, we consider another HTR j ′,which has the same preferred departure time, the same origin–destination pair, and, obviously, the sameavailable routes of HTR j. Therefore, for test cases # 4, besides the problem of sequencing hazmat vehiclescrossing a certain �-circle, we also have to sequence HTRs defined for the same origin–destination pair.

The results of the experimentation for test cases # 4 are summarized in Table 4 (organized as Table 1),reporting solution values obtained by applying only the version V1 of algorithm TS. The “Ave. LB %”is, on average, 5.4% with respect to 1.6% for test cases # 1 (obtained with algorithm TS version V1). Fortest cases # 4 the improvement (Ave. TS imp. %) of the solutions found by algorithm TS with respectto the ones found by algorithm GR is on average equal to 20.1%, compared to 31.5% for test cases # 1.On the other hand, comparing the “Ave. TS” values shown in Table 4 for test cases # 4 with the “Ave.TS” values for test cases # 1 listed in Table 1 (algorithm version V1), we notice that by doubling theHTRs, the hazmat shipment delay of a HTR, on average, moves from 5.2 min (for n = 5; see Table 1) to


Table 4Heuristic comparison: test cases # 4 (� = 1 km and � = �, double HTRs)

n k Ave. GR Min. TS Ave. TS Max. TS Ave. LB Ave. TS Ave. time (s) Ave. L Ave. Rgap % imp. % gap % gap %

10 3 296.8 194 239.2 296 3.6 19.4 21.5 21.2 3.25 302.8 160 225.4 293 3.4 25.6 38.2 21.2 1.5

10 304.2 144 216.4 256 3.3 28.9 83.0 17.1 2.1

20 3 1227.6 908 1051.8 1156 7.5 14.3 162.6 23.5 2.05 1237.2 924 1029.2 1132 7.4 16.8 329.0 19.2 1.4

10 1181.4 895 994.4 1132 7.2 15.8 967.1 23.5 8.1

22.7 min (for n=10; see Table 4), and from 14.8 min (for n=10; see Table 1) to 51.2 min (for n=20; seeTable 4). Computational times for test cases # 4 are much greater than those for test cases # 1. Finally,for test cases # 4 the assigned routes show an average gap for the total length with respect to Lmin

(i.e., “Ave. L gap %”) equal to 20.9%, and an average gap for the total risk with respect to Rmin (i.e.,“Ave. R gap %”) equal to 3.0%, compared to 19.7% and 3.1%, respectively, for test cases # 1 (algorithmversion V1).

7. Conclusions

In this paper, we consider the planning problem of routing and scheduling hazmat shipments. Wepropose a two-stage approach. The first-stage defines a set of minimum and equitable routes from theorigins to the destinations, where hazmat transportation requests are employed. This set of routes is theinput for the second stage, devoted to selecting one of the available routes for each hazmat shipment,and scheduling them in order to prevent the risk induced by vehicles traveling close to each other. Thepaper focuses on the hazmat shipment scheduling problem addressed in the second stage, with the aimof minimizing the total hazmat shipment delay. We propose a no-wait job-shop scheduling model withalternative job routes and a tabu search algorithm, which are experimentally evaluated on realistic testproblems defined on an Italian regional road network, evaluating the provided solutions also with respectto minimum route length and minimum route risk.

The two-stage model has been devised in order to act at a planning level within a fixed scenario anddoes not take into account variations of traffic density of both hazmat and non-hazmat vehicles and/orexposed population that may happen over time, and then does not take into account possible variations ofthe risk magnitude over time. This can be a reasonable assumption when hazmat shipments are performedin a regional network and the planning horizon is limited. Under the assumption to know in advance thetraffic density variations of both hazmat and non-hazmat vehicles, a future research option could be amulti-period version of the model proposed in this paper.

In the model we assume that trucks cannot stop, for example, for truck drivers’ rest requirements.This assumption is typical for a regional network like the one we considered in our experimentation. Forlarger networks, truck drivers’ rest requirements should be taken into account. With few complications,these requirements can be taken into account also by the proposed scheduling model: for example, some


route points may indeed represent also rest areas, where rest operations may take place; rest operationson these points represent rest periods with (operation) durations equal to the minimum rest time period,while no-wait constraints between these (rest) operations and their immediate successors are removed inthe scheduling model. Moreover, the �-circle centered at a rest point may be conveniently resized if a restoperation takes place at that point, without affecting the scheduling model.

Acknowledgements

The authors are grateful to the Guest Editor and the Referees for their valuable comments and sugges-tions that contributed to improving the quality of the paper.

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a tabu search approach for scheduling hazmat shipments

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