A Multi-Objective Block Scheduling Model for theManagement of Surgical Operating Rooms: New

Solution Approaches via Genetic AlgorithmsDomenico Conforti, Francesca Guerriero and Rosita Guido

Laboratory of Decision Engineering for Health Care Delivery,DEIS, University of Calabria, Italy

Abstract—In this paper, a multi-objective optimization modeltackling the optimal planning and scheduling of surgical opera-tions is proposed. The model determines the assignment of timeslots to the surgical teams and schedules the elective inpatient sur-gical operations on the basis of clinical priorities. The proposedmulti-objective approach takes into account and suitable balancessome strategic and conflicting goals, related to the improvementof resources utilization and considering patient’s priority value.In order to find the Pareto frontier, a metaheuristic approachbased on the efficient implementation of genetic algorithmshas been proposed. This approach allows to obtain the set ofefficient solutions within reasonable computational time. Somepreliminary experiments based on real-life data are reported. Theresults demonstrate the effectiveness of the proposed approach,even though more extensive testing is necessary in order to finallyassess and validate its impact to health care systems.

I. INTRODUCTION

Within the broad set of hospital activities, surgical opera-tions are some of the most complex, in terms of proceduresinvolved, and expensive, in terms of resources allocated. More-over, they strongly impact on the overall quality of hospitalhealth care services delivery and, hence, on the quality of lifeof the relevant patients.

In this context, it is strategically important to study anddevelop methodologies aimed at improving the managementof the surgical operations, reducing delays and maximizingthe utilization of the available resources (mainly personnel andfacilities).

It is worthwhile to observe that the interest in the aforemen-tioned issues has been driven, among the others, by severalaspects. Recently, the demand of surgical treatments hasstrongly increased, without, in many cases, a correspondingincrease of the Operating Rooms (ORs) capacity. The ORsare, typically, shared resources; hence it is fairly importantto define impartial access policies. Surgery is an importantactivity in most hospitals, since it is estimated to generatearound two third of hospital revenues, and it accounts forapproximately 40% of hospital resource costs, including costsof personnel and facilities [1]. Inefficient and inaccurate plan-ning and scheduling of OR time implies delays of surgery orcancellations of procedures, which are costly both for patientsand hospital.

Moreover, several factors affect the optimal planning andmanagement of the operating theater (i.e., ORs and recovery

rooms). Indeed, the resources required to perform a surgicaloperation include personnel (surgeons, anesthetists, nurses,etc.) as well as facilities (specialized equipment, multiple ORs,presurgical and postsurgical capacity). In addition, differentsurgical specialties (SSs) and priorities for service need tobe taken into consideration [2]. Finally, multiple stakeholders(e.g. patients, OR managers, surgeons, anesthetists, nurses)act with conflicting interests, making the ORs planning andscheduling a very complex task, where achieving all objectivesat the same time is rather difficult or even impossible.

In this paper, a multi-objective optimization model has beenproposed in order to address the ORs planning and schedulingproblem. In particular, our proposal aims at improving theutilization of ORs, increasing the number of interventionsby considering patient’s priority value, reducing the underutilization of surgical teams by matching their preferences.

The rest of the paper is organized as follows. In Section II,we describe the problem under study, whereas the presentationof the proposed optimization model is given in Section III. InSection IV the solution approach used to determine the Paretofrontier is described, while experimental results are illustratedand analyzed in Section V. Finally, some concluding remarksand directions for future research are reported in Section VI.

II. PROBLEM DESCRIPTION

The effective and efficient performance of a surgical treat-ment is the result of several technical factors and the wellorganized collaboration of a team of health care operators.Beyond the therapeutic appropriateness of treatments, theoptimal organization and management of the hospital resourcesand planning of the activities are strategically important.

Under this respect, in this paper we draw our attention to themanagement of elective surgical cases by considering a blockscheduling approach [3]. In this case, each surgical team havealready assigned a set of time blocks (TBs), in which surgicalcases are arranged. In order to apply the approach, we needa so-called master surgical schedule (MSS), which cyclicallydefines when an OR will be available and the surgical teamavailable for each OR block. Then, surgical teams book thesurgical cases into the assigned time, with the constraint thatthe average duration of the operations fits in with the scheduledtime period.

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In terms of decision making structure, the planning andscheduling of surgery treatments can be classified accordingto the following hierarchical levels ([4], [5], [6]):• strategic: a “case mix planning problem” is addressed,

where the main objective is to determine the type ofavailable OR and its capacity, and distribute the OR timeamong the several surgical teams.

• tactical: on the basis of the OR time allocated to eachsurgical teams, the tactical level concerns the develop-ment of an MSS.

• operational: it concerns the scheduling of the individualelective case on daily basis, after a developed MSS [7].

We remark that the reported decisional levels are not indepen-dent one of each other. Indeed, the outcome of the planningactivities in one level is used as input for the subsequent level.A possible strategy to handle the problems belonging to theaforementioned decisional levels has been presented in [8] and[9], where a hierarchical three-phase approach, that relies onthe solution of three different optimization problems (one foreach phase), is developed to determine the weekly schedulingof ORs. The same problem has been considered in [10], wherea integer linear programming model incorporating all the threedecisional levels has been developed and tested.

In the proposed model, we face the tactical and operationallevels simultaneously. In fact, our contribution lies in thedevelopment of a new optimization model aiming at planningand scheduling elective surgeries in a block scheduling systemby determining how much time and when assign it to eachsurgical team, and how the surgical interventions are assignedto the ORs.

Fig. 1. Time blocks of an OR

Every OR has a number of TBs that may have differentdurations; moreover, aiming at simplifying the model formu-lation, these TBs are ordered for each OR, as represented inFig 1.

Different surgical specialties (SSs), priorities for service,multiple ORs, and type of operation are taken into account.In addition, diverse and conflicting objective functions areconsidered.

III. MODEL FORMULATION

In order to describe the proposed model formulation, aimingat optimizing the ORs planning and scheduling, it is useful tointroduce the following definitions and notations.

Data Sets• OR = {available operating rooms r, r = 1, . . . , R} ;• P = {patients i, i = 1, . . . , n} ;• T B = {time blocks j, j = 1, . . . ,m} ;

• SS = {surgical teams/specialties s, s = 1, . . . , S} .

Input Parameters• tjr, duration of block j of the OR r;• Tmin

s , minimum time required by each surgicalteam/specialty s,∀s ∈ SS;

• B|P|×|ORset|, matrix used to keep track of surgicalteam/specialty preferences about OR:

bir =

{1 if the OR r is preferred to operate on patient i0 otherwise

• Z|P|×|SS|, specifies the team/specialty in SS operating agiven patient:

zis =

{1 if the patient i is operated by s0 otherwise

• for each patient i ∈ P the following data are available:– d = expected surgery duration;– pr = assigned priority value;– w = waiting time in weeks;– TM = maximum waiting time, in weeks, before

intervention.Decision variablesWe define two decision variables: variables x are related to

patient’s surgical treatments, whereas variables y concern theassignment of TBs to SSs.

• xijr =

{1 if patient i is assigned to TB j of OR r0 otherwise.

• yjrs =

{1 if TB j of OR r is assigned to team/specialty s0 otherwise.

Objective FunctionsThe following four objective functions have been defined.Obj1 Maximize the utilization of all ORs

maxn∑

i=1

m∑j=1

R∑r=1

dixijr

Obj2 Maximize the number of scheduled patients withhigh priority values

minn∑

i=1

m∑j=1

R∑r=1

(1− xijr)pr( 1

Pmax−pri)

i

where Pmax = maxi∈P{pri}+ 1.

Obj3 Maximize the surgical teams/specialties’ preferences

maxn∑

i=1

m∑j=1

R∑r=1

birxijr

Obj4 Minimize the “under utilization” of each SS (whichis defined as the difference between the total timeassigned to each surgical team/specialty and the totaltime of the scheduled patients):

minS∑

s=1

m∑j=1

R∑r=1

tjryjrs−n∑

i=1

m∑j=1

R∑r=1

dixijr

Constraints

The result of the application of the optimization model isthe assignment of patients to ORs taking into account thefollowing constraints:

C1 Each patient can be assigned at most to one TBduring the planning horizon:

m∑j=1

R∑r=1

xijr ≤ 1 i = 1, . . . , n.

C2 The waiting time of patient i is limited by TMi , i.e.

patient i has to be operated if the waiting time wi isgreater than TM

i :

wi

1−m∑

j=1

R∑r=1

xijr

≤ TMi i = 1, . . . , n.

C3 The total time assigned to each surgicalteam/specialty s is at least equal to Tmin

s :

m∑j=1

R∑r=1

yjrstjr ≥ Tmins s = 1, . . . , S.

C4 Each TB j of a given OR r can be assigned only toa single surgical team/specialty s ∈ SS:

S∑s=1

yjrs ≤ 1 j = 1, . . . ,m, r = 1, . . . , R.

C5 The completion time of all patients operated bya given surgical team/specialty is bounded by theduration tjr of the assigned TB j of OR r:

n∑i=1

dizisxijr ≤ tjryjrs j = 1, . . . ,m

r = 1, . . . , R s = 1, . . . , S.

C6 Each patient i can be operated during a TB j of anOR r only if there is at least a surgical team/specialtyassigned to the same TB:

xijr ≤ yjrs i = 1, . . . , n, j = 1, . . . ,m

r = 1, . . . , R, s = 1, . . . , S.

C7 Binary constraints for the decision variables:

xijr ∈ {0, 1}, i = 1, . . . , n, j = 1, . . . ,m, r = 1, . . . , R

yjrs ∈ {0, 1}, j = 1, . . . ,m, r = 1, . . . , R, s = 1, . . . , S.

IV. THE SOLUTION APPROACH

To handle the proposed multi-objective optimization model,a solution approach based on genetic algorithms has beenconsidered. The aim is to find the set of Pareto/non-dominatedsolutions, i.e. solutions for which improvement in one ob-jective can only occur with the worsening of at least oneother objective. Evolutionary algorithms seem to be naturallywell-suited for solving multi-objective optimization problems,since their search mechanism exploits a population of potential

solutions, allowing to find several non-dominated solutionsduring the iterative search process.

The pioneering evolutionary approaches, based on thePareto’s non dominance concept, are the multi-objective opti-mization genetic algorithm (MOGA) developed by Fonsecaand Fleming [11] and the non-dominated sorting geneticalgorithm (NSGA) proposed by Srinivas and Deb ([12]).

In this paper, we have considered NSGA− II ([13])an enhanced version of the original NSGA, that allows todetermine an approximation of the Pareto solutions set (i.e.,Pareto frontier).NSGA− II has been defined with the aim toovercome some of the drawbacks of the previous version suchas the lack of elitism, the need to specify a sharing parameter,the high computational complexity of the non-dominated sort.

In what follows, we give a brief description of theNSGA− II; more details can be found in [13]. The al-gorithm uses the concept of non − dominated sorting inorder to assign fitness to individual solutions. In particular,given a population of solutions, the main aim of the non-dominated sorting procedure is to progressively identify thesets of non-dominated solutions, assigning them a rank andtemporarily ignoring them. Rank 1 is assigned to the first setof non-dominated individual, rank 2 to the second, etc. Thefitness of an individual solution is directly related to its rank.During the selection step, solutions with a lower rank value arepreferred with respect to solutions with a higher rank value.The crowding distance measure, that gives an indication of thenumber of solutions located close to a give individual, is usedin case of a tie.

The main operations executed by the NSGA− II arereported in Fig. 2.

Procedure NSGA− IISt. 0 Set t = 0 and randomly generate a population of

solutions Pt of size NSt. 1 By using the concept of non− dominated sorting,

sort Pt and assign fitness to individual solutionsSt. 2 Create new population of solution Qt by using

recombination operators and tournament selectionSt. 3 Repeat the following operations

St. 3.1 Create a combined population Rt = Pt+Qt

of size 2NSt. 3.2 By using the concept of non− dominated

sorting, sort Rt

St. 3.3 Create Pt+1, by considering the first Nsolutions belonging to Rt, on the basis oftheir non-dominations sorting rank

St. 3.4 Create a new population of individuals Qt+1

from Pt+1 by using recombination operatorsand tournament selection

Until a termination criterion is satisfied

Fig. 2. Representation of the main operations executed by the NSGA− II.

In particular, given an optimization problem, NSGA− II

starts with the creation of a random population of solutionsPt of size N (Step. 0 in Fig. 2).

In the Step. 1, the solutions are sorted by using the conceptof non-domination sorting, whereas in Step. 2, a new popu-lation of solutions Qt is created, by using a set of geneticoperators appropriate to the problem under consideration.

During the main loop (Step. 3), a new population Rt of size2N is determined, by merging Pt (i.e., the old population) andQt (i.e., the new population of solutions). The solutions be-longing to Rt are then sorted using the non-domination sortingprocedure. During the Step. 3.3, an intermediate populationPt+1 is determined by chosen the first N solutions (based ontheir ranking assignment) belonging to Rt. The main purposeof this mechanism is to prevent that non-dominated solutionsgenerated during the evolutionary process are easily lost insubsequent generations (the concept of elitism).

At the next step (i.e., Step. 3.4), the new population ofindividuals Qt+1 is determined by applying to Pt+1 recombi-nation operators and tournament selection. The same processis repeated for a pre-specified number of generations (i.e., thetermination criterion is met).

V. NUMERICAL RESULTS

In this section, the results obtained by the application ofthe NSGA− II are presented. The NSGA− II code hasbeen downloaded from the Kanpur Genetic Algorithm Lab atwww.iitk.ac.in/kangal/soft.htm.

The first step in implementing a genetic algorithm tacklingan optimization problem is to devise a suitable scheme forrepresenting a solution. For the problem at hand, a binaryrepresentation of a member of the population is an obvi-ous choice, since it represents the underlying 0 − 1 inte-ger variables. Hence, we use a `-bit binary string, where` represents the number of variables in the problem (i.e.,` = n×m×R+m×R×S). In all computational experiments,a crossover probability pc = 0.9 and a mutation probabilitypm = 1

` have been used.Regarding the quality and the diversity of the solutions

found by NSGA− II, the performance of the algorithmdepends mainly on two parameters: population size N andnumber of generation t. Consequently, a sensitivity analysis onN and t has been performed. In particular, we have consideredthree values for the population size, that is N = 100, 140,and 200, whereas the number of generations spans the valuest = 1000, 2000, and 4000. Since benchmark instances werenot available, the computational experiments have been carriedout by considering a set of randomly generated test problems.

For all combinations of the aforementioned parameters, thePareto frontier has been determined and evaluated. As anexample, in Fig. 3, we report the value of the first and thesecond objective functions obtained by varying the value ofN and t, in the considered ranges. A similar trend has beenobserved for the other objective functions.

In relation to the population size, Fig. 3 shows that thebest results are obtained by considering N = 200, whereasregarding the convergence rate of NSGA− II, it is evident

Fig. 3. Sensitivity analysis on N and t

that around the 4000−th iteration, the algorithm determines aset of non-dominated solutions, which seems to approximatethe Pareto frontier.

A. Case study

In this section, preliminary computational results obtainedusing the proposed model on a real data set are reported.

The General Hospital of Cosenza (Italy), having 10 ORs and12 SSs, provided real-world data and expertise in the domainof OR planning and scheduling.

The results, obviously, depend on several parameters such asnumber and duration of interventions, available ORs, durationof each TB, equipment availability, etc.

For the sake of simplicity, we consider a small scenarioon a week time horizon, where there are only 2 ORs, 5 SSsand 80 surgical operations to be performed. The TBs of thetwo ORs have different duration along the planning week. Itis well known that the surgery scheduling is characterized byuncertainty due to the nature of surgical procedures. Hence,we have considered for every SS an average surgery durationfor all similar operations.

The approximated Pareto frontier has been obtained byexecuting NSGA− II and letting N = 200 e t = 4000.The schedules, differing on how the TBs are assigned toeach surgical team/specialty and the surgical operations arescheduled, are widely distributed along the Pareto frontier.

In order to get an idea about the quality of the solutions,in Fig. 4, we report the one chosen by the decision maker.In particular, for each SS Fig. 4 gives the minimum timerequired, the duration of all interventions in list, and theduration of the scheduled surgical interventions (all figuresare in minutes). Making suitable comparisons, the solution ofFig. 4 outperforms the current schedule adopted by the hospital(which is manually performed), since the schedule chosen by

the decision maker allows to obtain an improvement in theOR utilization and in the number of scheduled operations.

Fig. 4. For each surgical specialty: minimum time required (red), durationof all interventions in list (blue), duration of scheduled interventions (green)

VI. CONCLUSION

In this paper, the rather complex problem of ORs manage-ment has been considered. A multiobjective optimation modelfor optimal scheduling and planning of surgical treatments,in the framework of a block scheduling system, has beenproposed.

The model allows to handle, in an integrated way, thetactical and operational decision making levels. Moreover,the multiobjective approach allows to take into account andbalance several important and strategic goals, which typicallyarise in this context.

Some preliminary computational experiments on a real-life case study seem quite encouraging. In fact, the optimalsolution chosen by the decision maker allows a more efficientuse of the resources (in terms of OR utilization), and thenumber of scheduled surgical interventions also increases.

In any case, a further and more extensive validation of theproposed model has to be carried out, in order to assess theperformance of the solution approach for a wider range ofcases.

ACKNOWLEDGMENT

This research work has been partially support by ItalianMinistry of University and Scientific Research by the specialfunding ”Bilancio UNICAL 2009”.

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