Management of Emergency Response Teams under Stochastic Demands

Iliya Markov

and Sacha Varone

HES, Haute Ecole de Gestion de Gen

eve, Campus de Battelle, B

atiment F, 7 route de Drize, 1227, Carouge, Switzerland

Decision Sciences, Haute Ecole de Gestion de Gen

eve,

Campus de Battelle, B

atiment F, 7 route de Drize, 1227, Carouge, Switzerland

Keywords:

Contractual Constraints, Decision Support Tool, Emergency Response Teams, Rare Skills, Stochastic

Optimization.

Abstract:

We propose a stochastic optimization model for the composition of emergency response teams. An emergency

intervention requires ﬁrst an evaluation of the situation which results in the need of different skills. People

involved in the response team must therefore comply with the required skills, be available, with a past and

future workload respecting contractual compliance. In addition, we must also anticipate the possibility of

future interventions that will require rare skills. It is the uncertain future demand for these skills that introduces

stochasticities to the system. Since shifting agents between emergencies may be impossible or impractical,

we would like to ensure that rare skills are not wasted but assigned to the emergency that most needs them.

We model this with a mixed integer linear program implemented in AMPL and capable of being solved in

real-time on common solvers.

1 INTRODUCTION

The composition of emergency response teams is a

task which is often done manually by a team lead who

evaluates the nature and seriousness of the accident at

hand. The team lead has a list of people on standby

that she can choose from. She knows their availabil-

ity, qualiﬁcations and working hours constraints, and

must compose the team in real time taking care that

scarce resources in terms of rare skills might be re-

quired in future interventions. The success of the in-

tervention thus rests on the knowledge of the team

lead. Her absence, in the case of illness for example,

may hinder the ﬂuent course of emergency response

operations. This highlights the importance of devel-

oping a decision support tool to aid the composition

of response teams and reduce the risk of disturbances

in the absence or replacement of the team lead. To our

knowledge, there is no existing tool for the real-time

composition of emergency response teams in optimal

or near-optimal way.

Emergencies are very often unpredictable. Their

critical nature, however, means that they must be eval-

uated and responded to very quickly. The key factor

in responding to emergency situations is not one of

monetary costs but of consistently meeting important

benchmarks of performance. Examples include inci-

dent response times, sufﬁcient personnel with appro-

priate skills, efﬁcient distribution and management of

the equipment, etc.

Road safety, particularly after a trafﬁc accident,

is a prime example of an emergency that requires an

immediate response. In addition to clearing the road

of vehicles and debris, which may require heavy ma-

chinery, a response team must close lanes, manage

trafﬁc and sometimes investigate accident causes, par-

ticularly when criminal behavior is suspected. Road

safety therefore requires a near immediate response

to prevent danger but also the presence of appropriate

skill levels in terms of forensics, ﬁrst aid, road main-

tenance, the operation of special machinery, etc. In

addition, the availability of agents, their contractual

working hours, overtime pay and resource utilization

must all be taken into account as they themselves have

an affect on what the optimal team composition is.

The stochastic nature of emergencies also affects

the optimal agent assignments, resource utilization,

etc. In the case of rare skills, for example, we would

have to consider future emergencies that occur while

the current emergency is being handled. Since shift-

ing agents between emergencies may be impossible

or impractical, we would like to ensure that rare skills

are not wasted but assigned to the emergency that

most needs them. This means that it could be opti-

mal for an agent to remain idle and wait for the next

emergency rather than being assigned now.

In this article we propose a mixed integer lin-

ear programming model whose objective function

and constraints accommodate the stochastic aspect of

emergencies. The team composition in terms of agent

Markov I. and Varone S..

Management of Emergency Response Teams under Stochastic Demands.

DOI: 10.5220/0004196001590167

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 159-167

ISBN: 978-989-8565-40-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

assignments must take into account the possibility of

future emergencies that may require, among other fac-

tors, rare agent skills. Our model is implemented

in AMPL and solved to optimality with a common

solver. The model is tested on randomly generated

data instances of realistic size and solves to optimal-

ity in real time. Future practical extensions and im-

plementations of the model may necessitate the de-

velopment of appropriate heuristics.

This article is structured as follows. After this in-

troduction, section 2 offers a review of related liter-

ature from several domains that our research touches

on. Section 3 develops a base stochastic model for

emergency team composition and then offers several

extensions to it. Section 4 describes the results of the

experimental testing. Finally, section 5 concludes and

stresses on further developments and applications.

2 LITERATURE REVIEW

Scheduling and assignment problems are some of the

most studied and applied problems in combinatorial

optimization. They are encountered in almost all

businesses with multiple employees especially where

large and complex task structures need to be efﬁ-

ciently executed. These problems are very often mod-

eled as integer or binary programs, which makes it rel-

atively easy to express the objectives and constraints.

On the other hand, they are very often NP-hard and

thus difﬁcult to solve. (Cattrysse and Wassenhove,

1992) present a survey of algorithms to solve rea-

sonable instances of the generalized assignment prob-

lem (GAP) to optimality. Large instances, however,

may necessitate the use of heuristics. (Pentico, 2007)

presents the most useful variations of the assignment

problem since the publication of the seminal work of

(Kuhn, 1955) on the Hungarian method.

A host of authors have contributed to the de-

velopment of variations of the assignment problem,

whether in terms of modeling or solution techniques.

(Caron et al., 1999) and (Volgenant, 2004) look at

the assignment problem with seniority and task prior-

ity constraints. (Volgenant, 2004) offers an improve-

ment to the coefﬁcient scaling approach of (Caron

et al., 1999) by successively re-optimizing assignment

problems of increasing size. The latter method has

the complexity of a standard linear assignment prob-

lem and can be successfully applied to a bottleneck

problem. (Felici and Mecoli, 2007) propose a version

of the GAP where agents exhibit join and split pref-

erences. The split preference problem can be formu-

lated as a minimum cost ﬂow on a suitable graph. It is

a polynomial time algorithm and can be solved with a

common solver in reasonable time. On the other hand,

the join preference problem is NP-hard and needs to

be solved by heuristics. (Gilberti and Righini, 2007)

design an integer programming model for the multi-

period allocation of duties in rapid intervention teams

with the objective of minimizing the maximum inter-

vention time. Moreover, the assignment has to be eq-

uitable so that the effort is evenly distributed among

the available teams.

Stochasticities in the assignment problem have

also been addressed in the literature. (Albareda-

Sambola and Fern

andez, 2000) study the GAP with

independent and identically distributed demands for

agents following a Bernoulli distribution. Their

model is based on probabilistic constraints and is

solved as a transportation problem. It proves to be

the best in terms of the value of the objective function

and in avoiding infeasibilities compared to the alter-

native approaches they consider. (Spoerl and Wood,

2004) model uncertainty in the elastic GAP

when

the amount of resources used by the agents follow a

normal distribution. They penalize resource overuse

and ﬁnd substantial value of the stochastic solution.

(Albareda-Sambola et al., 2006) study a two-stage

stochastic GAP where task availability is distributed

following a Bernoulli distribution. In the ﬁrst stage,

initial assignments are made before task availability

is known with certainty. After task availability is re-

vealed, reassignments may be made at a cost. The

authors construct a convex approximation of the non-

convex objective function and solve the problem us-

ing exact algorithms.

(Toktas et al., 2004) study stochastic resource-

constrained GAPs where the resource capacities fol-

low an unknown probability distribution which can

be sampled. They develop exact and approxi-

mate solution techniques based on Lagrangian re-

laxation to solve the collectively capacitated GAP

(CCGAP)

with three alternative recourse functions.

The stochastic approaches perform signiﬁcantly bet-

ter than those using expected values of the capacities

in a deterministic setting. In addition, approximate

stochastic programming techniques are as efﬁcient as

deterministic techniques. (Toktas et al., 2006) review

alternative methodologies for dealing with capacity

constraints under uncertainty of the resource capaci-

ties when stochastic programming is infeasible or un-

The elastic generalized assignment problem (EGAP),

addressed by (Brown and Graves, 1981), is an extension of

the GAP that treats capacity constraints as soft constraints,

with capacity overuse penalized in the objective function.

The collectively capacitated generalized assignment

problem is a version of the GAP with multiple resources

collectively capacitated for all agents, without the require-

ment for one-to-one matching between tasks and agents.

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

desirable, for example when it is very difﬁcult to de-

ﬁne a recourse function.

The emergency team composition problem be-

longs to the vast array of literature on disaster and

emergency planning. (Green and Kolesar, 2004), (Al-

tay and Green, 2006), and (Simpson and Hancock,

2009) offer detailed discussions on the development

of OR methodologies in emergency planning and dis-

aster response. They describe the City of New York

though its fruitful cooperation with the RAND Cor-

poration as the principal arena for much of the early

research on police, ﬁre and ambulance services allo-

cation and deployment. (Kolesar and Walker, 1974),

for example, develop a model and heuristic to deter-

mine the location of ﬁre trucks in New York. OR ap-

plications began in the context of improving perfor-

mance in organized environments. Emergencies, on

the other hand, represent the disruption of such de-

signs and highlight the practical needs of many orga-

nizations.

One of the ﬁrst inﬂuential papers outside the ur-

ban services group was (Brown and Vassiliou, 1993),

which presents a decision support system for the real-

time tactical allocation of response units in the con-

text of disaster relief following a major earthquake in

Greece. Similar research is carried out by other au-

thors as well. (Rolland et al., 2010) include work-

load and labor requirements, precedence constraints,

resource availability, and critical deadlines. Skills are

also considered by modeling them as team-task mis-

match costs in the objective function. (Barbaroso

glu

and Arda, 2004), (Beraldi et al., 2004), (Yi and Ku-

mar, 2007), and (Mete and Zabinsky, 2010), among

others, apply stochastic programming techniques to

the management of disasters of various natures. These

studies, however, do not necessarily belong to the as-

signment problem realm.

Emergency response planning, as brieﬂy de-

scribed above, generally has a space dimension and

involves routing problems. In terms of the manage-

ment of emergency teams, (Rolland et al., 2010), for

example, deal with the time-periodic scheduling of

assignments in the context of disaster response sit-

uations. Their contribution consists in proposing a

hybrid meta-heuristic which allows for the real-time

solution of the disaster recovery scheduling problem.

Unlike us, however, they work in a deterministic en-

vironment and view teams as already existing. Our

model on emergency team composition and manage-

ment therefore is a step forward in a new direction.

It allows us to both position ourselves in the area of

emergency response and develop our contribution of

a new approach in terms of team composition.

Even though optimal team composition is an area

of interest to OR, the focus has generally not been

on emergency response. (Boon and Sierksma, 2003),

for example, develop an assignment problem with

more than 20 additional constraints which is applied

to team composition in volleyball.

(Shipley and Johnson, 2009) treat the question

of choosing project team members who meet the

project’s goals for the preferred cognitive style. The

paper presents a fuzzy logic and an algorithm based

on belief in the fuzzy probability of a cognitive style

ﬁtting a deﬁned goal. Thus the paper addresses uncer-

tainty in decision making and facilitates the member

selection process. It illustrates how OR can adapt be-

havioral research techniques. (Woodruff, 2010) also

treats the problem of choosing optimal teams under

predeﬁned policies. It treats the players in terms

of their speciﬁc skills rather than interactions and

demonstrates that if satisfaction of the policy is dif-

ﬁcult then simply choosing a set of individually best

members will often not result in the best policy con-

formance. In other words, a team of the best will not

be the best team.

The model we propose in this article considers

workers with different skills. In this respect it touches

on a signiﬁcant body of literature that treats the prob-

lem of cross-trained workers from various perspec-

tives. (Campbell and Diaby, 2002) develop heuris-

tics for the model of (Campbell, 1999) which assigns

cross-trained workers across different departments at

the beginning of a shift. The departmental utility

function is non-linear and the resulting model can

be viewed as a variant of the GAP. It models cross-

trainings by specifying an agent’s capability as a per-

centage of a fully qualiﬁed agent in that task. (Camp-

bell, 2011) takes the problem of scheduling and as-

signment of cross-trained workers to a stochastic en-

vironment with variable demands. The second-stage

is a recourse on realized demands.

(Sayın and Karabatı, 2007) also build a framework

based on the non-linear nature of the departmental

objective function of (Campbell, 1999). They pro-

pose a two-stage model, where two objective func-

tions, departmental utility and skill improvement, are

considered in the assignment of cross-trained work-

ers. (Fowler et al., 2008) develop heuristics for the

model of (Wirojanagud et al., 2007) which determines

different stafﬁng decisions, such as hiring, ﬁring and

cross-training of workers, in order to minimize work-

force related costs over multiple periods. It is a multi-

period model where cross-training is done with the

purpose of satisfying labor requirements in each pe-

riod. In this article, we attempt to ﬁnd the best assign-

ment of agents with multiple skills by taking into ac-

count the need for rare skills in the future. The train-

ManagementofEmergencyResponseTeamsunderStochasticDemands

ing of agents is not an objective of our model.

3 PROBLEM FORMULATION

The methodology we propose here is based on a two-

stage stochastic linear programming framework. In

such a framework, the ﬁrst stage consists of a struc-

tural component which is ﬁxed while the second stage

varies with uncertainty. In general, the second stage

is represented by a collection of distinct future scenar-

ios which are governed by stochastic parameters. In

other words, the stochastic parameters have a discrete

distribution or are represented by a ﬁnite sample from

a continuous distribution. A number of studies con-

cern the selection of scenarios in stochastic programs.

(Jenkins, 2000), for example, proposes a mathemat-

ical programming model for the selection of a lim-

ited set of scenarios that have the maximum similarity

with all potential scenarios.

The ﬁrst-stage decision variables in a stochastic

program represent decisions that must be made now,

before the future state is known. These are proactive

decisions often concerned with planning. They are

independent of the values of the stochastic parame-

ters in the second stage. This property is known as

non-anticipativity, i.e. our actions now cannot take

advantage of knowledge of the future. The second-

stage decision variables, commonly referred to as re-

course or reactive variables, represent the actions we

take after the values of the stochastic parameters are

revealed. For every scenario, there is a set of recourse

variables linked to the ﬁrst-stage decision variables by

scenario-dependent constraints. Solving the stochas-

tic program thus yields an optimal ﬁrst-stage decision

under all scenarios, i.e. all considered realizations of

the stochastic parameters.

In a team composition framework, let’s consider

an emergency that occurs now. This emergency is

composed of a set of tasks, each of which requires

a set of skills to be simultaneously present in every

agent assigned to it. In the simplest case, an emer-

gency may only consist of one task. Moreover, a task

may only require one skill. An agent, in turn, may

possess a number of skills. Each tasks requires a num-

ber of skilled agents to be assigned to it.

We consider a set of tasks I indexed by i, a set

of agents J indexed by j, a set of skills S indexed by

s, and a set of resources R indexed by r. Moreover,

we consider a set of future emergencies F indexed by

f that develop while the current emergency is being

handled. Any emergency, be it the current one or the

future ones, is composed of a subset of I . The number

of tasks and the number of needed agents in the fu-

ture is not known a priori. It is stochastic and depends

on the particular emergency scenario that develops.

Therefore, we have to decide on the most appropriate

assignments so that the stafﬁng requirements of both

the current and the future emergencies are satisﬁed

with agents possessing the necessary skills and at the

lowest possible cost. The assignment of agents with

rare skills becomes critical here. Assigning such an

agent to a trivial task in the current emergency means

that his rare skills will not be available if a future

emergency needs them.

The index f designates the types of scenarios or,

in other words, the types of emergencies that can

develop while still handling the current emergency.

In a generalized framework, we assume that the

lapse between two consecutive emergencies of the

same type f follows an exponential distribution. The

probability of an occurrence during the handling of

the current emergency therefore does not depend

on the past due to the memoryless properties of the

exponential distribution. These probabilities can be

normalized to add up to one and used in the objective

function. Once a future emergency occurs, the system

is re-optimized taking care of any parameter changes

in terms of worked hours, availability, etc. Due to

the continuous nature of the exponential distribution

we can assume that two emergencies cannot occur at

the same time. The decision maker will thus need to

optimize with the occurrence of every emergency to

ﬁnd out the most appropriate assignments. We in-

troduce the following notation in terms of parameters:

i j

cost of assigning agent j to task i

skill qualiﬁcation of agent j (1 if skill s is

present, 0 otherwise)

skill requirement of task i (1 if skill s is re-

quired, 0 otherwise)

number of agents required for task i in the

current emergency

f i

number of agents required for task i in fu-

ture emergency f

use of resource r per agent assigned to task

total available amount of resource r

availability of agent j

d duration of the current emergency

duration of future emergency f

number of already worked hours by agent j

contractual number of working hours of

agent j

probability of occurrence of emergency f

All of these parameters are evaluated at the time

of every optimization (re-optimization). For exam-

ple, the available agents at any given optimization

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

step will not include those that are out handling an

emergency. In a realistic environment, therefore, this

optimization algorithm will be part of a larger system

that tracks the status of all agents and updates their

relevant parameters. We should also mention that the

costs in the above notation should not necessarily be

understood as monetary costs. These could be utili-

ties, hierarchy levels, execution efﬁciencies, response

times (depending on where the agents are located) etc.

A more detailed discussion of p

is in order. We

can imagine that there is an indeﬁnite number of dif-

ferent emergencies that can occur considering all the

different parameters that describe them. This is why

the model we describe below assumes that we have

already performed data mining on the historical data

we have and extracted several emergency types (in

terms of duration, tasks, etc.) that are closest to our

complete sample of emergencies. Looking at the fre-

quency of these emergencies, we can calculate the

probability that they develop while the current emer-

gency is being handled. Normalizing these probabili-

ties to add up to one, we obtain the parameters p

The decision variables in the current and future

emergencies are:

i j

assignment of agent j to task i in the current

emergency (binary)

f i j

assignment of agent j to task i in future

emergency f (binary)

The stochastic optimization model thus takes the fol-

lowing form:

min

∑

i∈I

∑

j∈J

i j

∑

f ∈F

∑

i∈I

∑

j∈J

i j

f i j

(1)

s.t.

∑

i∈I

i j

+ x

f i j

) 6 1, f ∈ F , j ∈ J (2)

i j

6 q

, i ∈ I , j ∈ J , s ∈ S (3)

f i j

6 q

, f ∈ F , i ∈ I , j ∈ J , s ∈ S (4)

∑

j∈J

i j

> n

, i ∈ I (5)

∑

j∈J

f i j

> n

f i

, f ∈ F , i ∈ I (6)

+ d)

∑

i∈I

i j

+ (h

+ d

)

∑

i∈I

f i j

6 H

f ∈ F , j ∈ J

(7)

∑

i∈I

∑

j∈J

i j

+ u

f i j

) 6 t

f ∈ F , r ∈ R

(8)

i j

6 a

, i ∈ I , j ∈ J (9)

f i j

6 a

, f ∈ F , i ∈ I , j ∈ J (10)

i j

, x

f i j

∈ {0, 1}, f ∈ F , i ∈ I , j ∈ J (11)

Expression (1) designates the minimization of the

cost of all assignments, i.e. the costs of assigning

agents to the current emergency and all future emer-

gencies according to their probability. Constraints (2)

impose that an agent is assigned to at most one task ei-

ther in the current emergency or (any or all of) the fu-

ture emergencies (as we do not know which one will

occur ﬁrst). As explained above, it may be optimal

for the agent to remain on standby and wait for the

next emergency. Constraints (3) ensure that an agent

is assigned to a task in the current emergency only

if he has all the necessary skills. Constraints (4) ex-

tend this to the future emergencies. Constraints (5)

and (6) require a minimum number of agents per task

in the current and future emergencies. Constraints (7)

establish that an agent can be assigned to the current

or future emergencies only if his contractual working

hours do not expire during the emergency to which

he is assigned. We have the usual assumption here

that working time only accrues during the handling

of an emergency. Standby is not considered as work-

ing time in our model. Moreover, we assume that the

agent should be present during the whole duration of

the emergency.

Constraints (8) impose a limit on resources. We

have multiple resources collectively capacitated for

all agents, an approach that appears as early as in

(Mazzola and Neebe, 1986). Given the fact that we

do not require a one-to-one matching between agents

and tasks, our model in its essence resembles the CC-

GAP proposed by (Toktas et al., 2004). In our model,

however, resource consumption is not agent speciﬁc.

It depends on the number of agents assigned but not

the speciﬁc agents that are assigned. Constraints (9)

and (10) specify agent availability and, ﬁnally, con-

straints (11) establish the variable domains.

The strict separation between the parameters re-

lating to the current emergency and those relating to

the future emergency types allows for a more ﬂexi-

ble model. Thinking about the emergencies that may

develop, we think of the emergency types that we de-

ﬁned above. Once an emergency develops, however,

it is no longer an emergency type but an actual emer-

gency. A quick assessment of it will allow us to at-

tribute the appropriate parameters in terms of what

tasks it is composed of, how many agents each task

needs and what the emergency’s duration is.

Tasks in the above model are in essence task

types. The set I describes all possible task types that

can occur

, such as road maintenance, forensics, etc.

Stochasticity in our model is manifested through the

types of emergency events in the scenarios. Based on

the event that occurs, some tasks will not exist (they

The deﬁnition of I can be based on historical analysis

of the emergency data.

ManagementofEmergencyResponseTeamsunderStochasticDemands

do not need any agents) or may need a certain number

of agents. More precisely, therefore, stochasticity in

the model is governed by the values of the parameters

f i

. In addition to the base model presented above,

some of the constraints can be substituted or supple-

mented by the alternatives below. [task types]

Overtime: We can modify the working hours

constraints (7) to allow overtime. Agent overtime will

then be penalized in the objective function. The idea

is similar to that of (Brown and Graves, 1981), cited

in (Spoerl and Wood, 2004), where they penalize

agent capacity under- and over-use. Working time, in

this sense, is agent capacity. We deﬁne the following

parameters:

cost of overtime of agent j

α, β weighting parameters

The decision variables associated with this ap-

proach are:

f j

overtime hours of agent j (linear)

−

f j

under-time hours of agent j (linear)

Introducing the above notation, the model be-

comes:

min α(Objective 1) + β

∑

f ∈F

∑

j∈J

f j

(12)

s.t. (2), (3), (4), (5), (6)

+ d

∑

i∈I

i j

+ d

∑

i∈I

f i j

− y

f j

+ y

−

f j

= H

f ∈ F , j ∈ J

(13)

f j

, y

−

f j

> 0, f ∈ F , j ∈ J (14)

(8), (9), (10), (11)

In addition to minimizing assignment costs, the

objective function (12) also minimizes the penalty as-

sociated with the overtime hours in constraints (13).

Capacitated overtime: We can also add maximum

allowable overtime on constraint (13). In the model

below, h

stands for the maximum allowable over-

time hours for agent j. The capacitated overtime

model thus becomes:

min Objective 12

s.t. (2), (3), (4), (5), (6), (13)

f j

6 h

, f ∈ F , j ∈ J (15)

(8), (9), (10), (11), (14)

Shared resources: Constraints (8) model task-

based resource consumption per agent, i.e. adding

an additional agent to any task increases that task’s

resource consumption. We may call these individual

resources because they are allocated to each agent

individually. An example of such resources are

face masks. However, we may want to model an

alternative resource consumption pattern that is not

linked to every incremental agent. The model below

is appropriate for resources such as vehicles that drive

the agents to the emergency. One vehicle should be

dispatched for every “n” agents regardless of which

task of the emergency they are assigned to. We call

these shared resources. To model them, we only need

to supply one additional parameter and two decision

variables:

number of agents per resource r

use (consumption) of resource r in the cur-

rent emergency (integer)

f r

use (consumption) of resource r in the fu-

ture emergency f (integer)

The alternative resource consumption model thus

becomes:

min Objective 1

s.t. (2), (3), (4), (5), (6), (7)

∑

i∈I

∑

j∈J

i j

6 k

, r ∈ R (16)

∑

i∈I

∑

j∈J

f i j

6 k

f r

, f ∈ F , r ∈ R (17)

+ v

f r

6 t

, f ∈ F , r ∈ R (18)

(9), (10), (11)

, v

f r

∈ integer, f ∈ F , r ∈ R (19)

Constraints (16) and (17) increment the resource

consumption variables v

and v

f r

based on how many

agents are assigned to each emergency

. Constraints

(18) then ensure that the resource availability is re-

spected. There could be different variations of this

constraint based on the speciﬁc circumstances or the

preferences of the decision maker.

Soft constraints: Depending on the policy of the

decision maker, some of the constraints above may be

softened. Whether two agents may be able to carry

out a task that requires three agents, whether these

agents will be able to execute the tasks if they are not

fully qualiﬁed or lack some of the necessary resources

are questions that should be analyzed by each of the

model’s users. A van intended to carry four people

may be able to carry ﬁve, but a ﬁre cannot be put out

without a ﬁre engine.

In the above model and its extensions, most con-

straints (apart from overtime) must be perfectly satis-

The reason we sum over the agent assignment variables

i j

and x

f i j

) as opposed to the parameters specifying the

number of required agents (n

and n

f i

), is to preserve the

ﬂexibility to soften constraints (5) and (6) as described be-

low.

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

ﬁed in order to ﬁnd a feasible solution for the emer-

gency team composition problem. However, in many

practical applications the constraints do not need to be

perfectly satisﬁed. In the case of soft constraints, the

constraint my be violated and the solution is still ac-

ceptable, although not as “desirable”. A common way

to specify soft constraints is by adding penalty vari-

ables to each soft constraint. These variables take the

value of zero when the constraint is satisﬁed and then

increase with the degree of constraint violation. There

is a penalty parameter associated with each penalty

variable. The resulting penalty function is minimized

in the objective.

In the model above, some of the constraints that

can be softened include the number of agents per task

constraints (5, 6) and the qualiﬁcation constraints (3,

4). In the ﬁrst case, the step may be necessary if an in-

sufﬁcient number of agents are available. The model

will include penalties on task under-stafﬁng which

may vary depending on task criticality. In the second

case, the step may be necessary if the workforce is not

sufﬁciently qualiﬁed yet. Given a penalty on not satis-

fying the qualiﬁcation constraints, the model will sat-

isfy the skill requirements of a subset of the tasks. For

the remaining tasks, under-qualiﬁed agents will be as-

signed. Softening both groups, it would make sense

to attach higher penalty to the number of agents con-

straint as assigning less than perfectly qualiﬁed agents

to a task is usually better than assigning no agents at

all.

Regarding the resource constraints (8, 18), in the

model above they are infeasibility constraints. They

are either satisﬁed or not and do not affect agent as-

signments. Our presumption is that a task consumes

the same amount of resource regardless of which par-

ticular agent is assigned to it. Therefore, being hard

our resource constraints serve two purposes. First,

they ensure that an assignment is feasible only if there

are enough available resources. Second, they allow

us to keep track of the used resources. Softening

the resource constraints will mean that agents may

be under-supplied with resources in some or all of

the scenarios. In particular, softening constraints (18)

may be sensible if we allow for “overcrowded” vehi-

cles, for example.

4 EXPERIMENTAL RESULTS

In order to test the performance of our model, we

carried numerical experiments on randomly gener-

ated instances of the stochastic model with capaci-

tated overtime constraints and two types of resources

– the individual resources modeled by constraints (8)

and the shared resources modeled by constraints (16)

through (19). That is, our model is composed of the

objective function (12) with α and β equal to 1, plus

constraints (2) - (6), (8) - (11), and (13) - (19). All of

our constraints except working hours remain as hard

constraints. Our idea was to generate problem in-

stances of realistic size as depicted in table 1 so as to

assess the model’s performance as if applied to a real-

istic setting. The tests were carried out on a 2.66 GHz

Intel Core 2 Quad PC with 8 Gb of memory running

a 64-bit Windows 7. The problem was modeled in

AMPL and we used an AMPL client associated with

the Gurobi 5.0 commercial solver.

Table 1: Random data. This table presents the key param-

eters used for the generation of random instances.

Parameters Size

Task types 15

Agents 300

Skills 10

Individual resources 10

Shared resources 4

Emergency scenarios 8

We generated a set of random instances involving

300 agents and 15 tasks. Each emergency consists of

a random subset of the tasks and each task requires a

random number of agents.

Most of the model’s parameters are based on uni-

form distributions, with rounding or ﬂooring where

necessary. As an exception. the number of required

agents per task is modeled using an exponential distri-

bution with a rate of 0.3 and then ﬂoored to the nearest

integer. The choice of the exponential distribution al-

lows tasks involving no agent with a high frequency.

Such tasks do not belong to the associated emergency.

This choice of parameter also allows occasional large

numbers. There are a total of 10 skills that we con-

sider, of which seven are frequent skills, present in

almost every agent, while the three remaining skills

are rare.

There are 10 types of individual resources, i.e.

resources modeled according to constraints (8), and

four types shared resources – those modeled by con-

straints (16) through (19). Finally, we have eight

future emergency scenarios. In every generated in-

stance, the probability of an emergency of each type

occurring during the handling of the current emer-

gency is sampled from a predeﬁned historical distri-

bution. These probabilities are then normalized to add

up to one. As explained above, in a realistic situation,

the model is run iteratively with every emergency oc-

currence taking into consideration the latest informa-

ManagementofEmergencyResponseTeamsunderStochasticDemands

tion on the model’s parameters.

As table 2 below shows, we generated 200 random

instances of the model. Since all of our constraints are

hard, seven of the instances (3.50%) turned out to be

infeasible.

Table 2: Instance statistics. This table presents various

characteristics of the randomly generated instances.

Before presolve Size

Number of instances 200

Feasible instances 193

Number of variables 45,336

Number of constraints 452,983

After presolve Mean St. Dev.

Number of variables 24,515.40 2,253.77

Number of constraints 4,803.12 38.24

Solution time (sec) 3.91 0.43

The table also shows that the original instances

have 45,336 variables and 452,983 constraints. After

AMPL’s presolve, however, the number of variables

is almost halved and the number of constraints is re-

duced by a factor of 90. The instances solve for an av-

erage of 3.91 seconds. Based on a t-distribution, the

95% conﬁdence interval for the mean is from 3.85 to

3.97 seconds. Using the free CBC 2.7.5 solver on the

same instances, the solution times generally remain

below 20 sec.

In a second experiment, we doubled the value of

each parameter in table 1. The resulting instances

have 325,336 variables and 6,455,890 constraints.

After presolve, the number of variables is reduced ap-

proximately three times and the number of constraints

is reduced to around 19,000, i.e. less than 1% of the

original value. Using Gurobi 5.0, the solution times

are between 20 and 30 sec, which is still a compara-

tively low value for the size of the problem we have.

The results therefore suggest that instances of re-

alistic size can be solved in reasonably short time

on a commercial or free solver without the use of

any heuristics. Further developments of the model

through the addition of client-speciﬁc constraints or

the softening of some of the constraints will inevitably

make the model more cumbersome and therefore im-

pact the solution time. Future testing and possible

commercial applications will show whether heuristics

or meta-heuristics need to be developed.

5 CONCLUSIONS

In this article, we proposed a stochastic optimization

model for the composition of emergency response

teams taking into account agent skills, availability,

contractual compliance and resource utilization. The

experimental results showed the viability of the model

as a ﬁrst step of a multi-stage project. The next steps

will see our model enriched with alternative objec-

tive functions and more constraints, including client

speciﬁc ones in order to increase the marketability of

our prospective tool. Finally, our model is intended to

evolve into a web-based decision support application

with on-demand access.

The potential market of our tool includes enter-

prises in the areas of emergency response services and

call centers. Most of the medium and large IT enter-

prises, especially the ones working with the banking

sector, ensure very short response times and should

thus be able to compose expert teams in real time.

There is one more message we want to convey

with our work. The generalized assignment problem

is NP-hard. Being a variation and a stochastic exten-

sion of the GAP, our model can be viewed as “dif-

ﬁcult” to solve. Researchers often tend to develop

heuristics when it comes to solving difﬁcult problems,

even of modest size. Nevertheless, we have shown

that before developing heuristics, it makes sense to

model the problem in a mathematical programming

language and try an available commercial solver, or a

free solver such as those from the COIN-OR project

(www.coin-or.org).

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ManagementofEmergencyResponseTeamsunderStochasticDemands