An Overview of a Break Assignment Problem Considering Area Coverage
Marin Lujak, ?lvaro Garcia Sánchez, Miguel Ortega Mier, Holger Billhardt
aa r X i v : . [ m a t h . O C ] J a n An Overview of a Break Assignment ProblemConsidering Area Coverage
Marin Lujak , Álvaro Garcia Sánchez , Miguel Ortega Mier , and HolgerBillhardt IMT Lille Douai, Univ. Lille, Unité de Recherche Informatique Automatique,F-59000 Lille, France, [email protected] , Department of Organization Engineering, Business Administration and Statistics,Universidad Politécnica de Madrid (UPM), 28006 Madrid, Spain, [email protected] , [email protected] University Rey Juan Carlos, calle Tulipán, s/n, Móstoles, Madrid, Spain, [email protected]
Abstract.
Prolonged focused work periods decrease efficiency with re-lated decline of attention and performance. Therefore, emergency fleetbreak scheduling should consider both area coverage by idle vehicles (re-lated to the fleet’s target arrival time to incidents) and vehicle crews’service requirements for breaks to avoid fatigue. In this paper, we pro-pose a break assignment problem considering area coverage (BAPCAC)addressing this issue. Moreover, we formulate a mathematical model forthe BAPCAC problem. Based on available historical spatio-temporal in-cident data and service requirements, the BAPCAC model can be usednot only to dimension the size of the fleet at the tactical level, but also todecide upon the strategies related with break scheduling. Moreover, themodel could be used to compute (suboptimal) locations for idle vehiclesin each time period and arrange vehicles’ crews’ work breaks consideringgiven break and coverage constraints.
Keywords: fleet scheduling, break assignment, emergency fleets
Usually, in emergency fleets (such as police and ambulance fleets), the relationbetween the dynamic spatio-temporal area coverage requirements and the num-ber of breaks and their duration is not taken into account for break assignmentand the breaks are regulated by (static) labor rules. Thus, it may happen thatthe fleet is undersized and recurrently falls behind its performance expectations.Even though a vehicle crew having a break can be assigned to assist anincident, its arrival time is delayed due to the preparation time necessary tostop the break and start the mission. This additional delay (often in terms ofminutes) can mean the difference between life and death both in the case ofurgent out-of-hospital patients and rapidly evolving disasters.To guarantee efficient and timely emergency assistance, a minimum numberof stand-by vehicles with in-vehicle emergency crews must always be available to
Marin Lujak et al. cover each area of interest, i.e., to assist a probable incident within a predefinedmaximum delay in the arrival. This can be done by dynamically transferringstand-by vehicles to (time- and area-dependent) locations that maximize thecoverage of a region of interest in each time period. These locations can be a setof strategically positioned depots, parking lots, terminals, garages, and similar.In the emergency fleet context with temporarily and geographically vary-ing workload, if the effective number and duration of previous work breaks hasnot been taken into account in the assignment of vehicles to incidents, it mayalso happen that a vehicle crew has worked during extended periods of timewithout any break or with interrupted breaks that were insufficiently long for ef-fective crew’s rest. Such a disparity between a too high workload and insufficientbreaks leads to increased fatigue that can have detrimental effects on emergencyassistance. The consequence is the feeling of being drained with related dete-rioration of attention and performance that may be hazardous for emergencyassistance. Hence, strategic fleet dimensioning performed before the fleet’s de-ployment should consider expected workload dynamics to guarantee frequentand sufficiently long time for rest for the vehicles’ crews.In this paper, we formulate this problem under the name of the break as-signment problem considering area coverage (BAPCAC). Based on the historicaldata, the BAPCAC problem consists in determining the vehicles with in-vehiclecrews necessary for optimal fleet’s deployment in each time period of a workshift,and their breaks’ type, number, and durations based on the legal constraints, andthe forecasted workload dynamics in terms of the area coverage by the fleet. Bythe area coverage, we consider that incidents appearing in the region of interestmay be potentially assisted within a predefined target delay in the arrival. More-over, the BAPCAC model relocates vehicles at each time period to the locationsthat best cover the forecasted incident demand. The objective of the BAPCACmodel is an efficient dimensioning and an effective choice of break coordinationstrategy in emergency vehicle fleets based on historical data while increasingboth the quality of service of emergency fleets as well as the well-being of thevehicles’ crew members while reducing their absenteeism due to fatigue.The paper is organized as follows. In Section 2, we describe the state-of-the-art practice in break scheduling and introduce the break assignment problemconsidering area coverage (BAPCAC) and its mathematical model. We concludethe paper with a discussion of the model in Section 3.
Breaks in emergency fleets are usually considered preemptive and in the caseof an incident, the closest available vehicle having the required equipment iscalled to assist the incident independently of the past crew’s break and workloaddynamics and if it is momentarily in a break or not. This approach may induce adelay in arrival of up to several minutes, which in critical cases can be hazardous.A review of the literature on personnel scheduling problems can be foundin [2]. Beer et al. [1] address a complex real-world break-scheduling problem for n Overview of a Break Assignment Problem Considering Area Coverage 3 supervisory personnel and present a scheduling system that can help professionalplanners create high-quality shift plans. Similarly, Di Gaspero et al. [4] alsoconsider the problem of scheduling breaks that fulfill different constraints abouttheir placement and lengths.Mesquita et al. [6] present different models and algorithms for the integratedvehicle and crew scheduling problem, where crews can be assigned to differentvehicles. In their algorithms, they consider several complicating constraints cor-responding to workload regulations for crews. Defraeye and Van Nieuwenhuyse[3] provide a state-of-the-art overview of research in the period 1991-2013 on per-sonnel staffing and scheduling approaches for systems with a stochastic demandwith a time-varying rate. Rekik et al. in [7] consider a break scheduling problemthat includes different forms of flexibility in terms of shift starting times, breaklengths and break placement. In the BAPCAC problem, we consider the F | V | W break model. Here, F stands for the fractionable breaks with an upper and lowerbound on the number of breaks that can be assigned to each vehicle crew; V stands for variable break lengths with an aggregated break length given, whichmay be split into sub-breaks for each vehicle crew. W stands for the workstretchduration that defines a lower and upper bound on the number of consecutive pe-riods of uninterrupted work before and after each sub-break can start, see, e.g.,[5]. We also assume that the plan of vehicle crew shifts that make a work-dayshift is given a priori.We assume that incidents appear, w.l.o.g., in a 2D square environment Env =[0 , l ] ⊂ R of side length l > . Environment Env is divided in mutuallyexclusive and nonempty rectangular unit areas e ⊂ Env of equal size such that e i ∩ e j = ∅ for i = j and S e ∈ Env e = Env . Vehicle agents are positioned andmay move only within the subset
J ⊆
Env of unit areas.Let T be a set of consecutive time periods { , . . . , |T |} , representing a vehiclecrew shift, where |T | is the cardinality of T . For each time period τ ∈ T , D eτ represents incident density or demand at area e ∈ Env . I ⊆
Env is a subset ofunit areas i ∈ I with positive incident density. Then, each vehicle a ∈ A canhave one of mutually exclusive vehicle states: { idle, on − break, occupied } : idle - a vehicle is waiting for a new incident assistance; on-break - currently on abreak. An idle vehicle is considered to be covering unit area i ⊂ I if it is locatedin any unit area from which it can arrive to unit area i within a predefined andgiven target arrival time ∆t , while for the vehicles on-break, the arrival time isincreased by an additional preparation delay.Let B = { b s , b l } be a set of short and long breaks respectively. The minimumnumber of short and long breaks is related to the duration of the shift, labor rules,and legal requirements. We track accumulated workload of each vehicle crewsince the last break at each time period τ ∈ T to comply with a given maximumallowed work time M AX wb before the start of a break b ∈ B . Each break hasa minimal duration ∆ MINb and a maximal duration ∆ MAXb , that specify therequired amount of break time based on legal requirements.Since the problem of area coverage is interconnected with the break assign-ment problem in the emergency fleet context, we approach it by a unique math-
Marin Lujak et al. ematical programming model in which the decisions on the number of vehiclesnecessary for the coverage, their locations, identities, and the break assignmentsare decided all at once for the whole workshift.
Sets and indices T time horizon; a set of time periods in a work shift; τ ∈ T A set of agents a ∈ A representing | A | capacitated vehicleswith vehicle crews B set of break types for example, { b s , b l } , where b s stands for short breaks, whereas b l stands for long breaks I set of unit areas i ∈ I with positive incident density J set of unit areas j ∈ J , apt for vehicle locations Parameters D iτ estimated demand and, thus, required minimum proportion of vehiclesfor unit area i ∈ I at time τ ∈ T ∆ MINb minimal duration of break b ∈ B∆ MAXb maximal duration of break b ∈ BIM jj ′ valued 1 if an agent can move from j ∈ J to j ′ ∈ J based onthe maximum transport time between 2 consecutive periods;0 otherwise N ij j ∈ J can assist an incident at i ∈ I within a given target arrival time, 0 otherwise ¯ N ij j ∈ J can assist an incidentat i ∈ I within a predefined target arrival time, 0 otherwise M AX wb maximum allowed work time before assigning break b ∈ B usual this time will be 3 hours for short and 4-6 hours for long breaks Decision variables x aτj valued 1 if agent a at time τ ∈ T is locatedat unit area j ∈ J ; 0 otherwise y aτ binary break assignment variable valued if agent a ∈ A is assigned a break at time τ ∈ T ; 0 otherwise z aτi real nonnegative coverage assignment variable representingthe part of density at incident unit area i ∈ I assigned to idle agent a ∈ A at time τ ∈ T , where ≤ z aτi ≤ z aτi real nonnegative coverage assignment variable representingthe part of density at incident unit area i ∈ I assigned to on-break agent a ∈ A at time τ ∈ T , where ≤ ¯ z aτi ≤ δ iτ real nonnegative density of area i ∈ I uncovered at time τ ∈ T α abτ binary variable valued 1 if agent a ∈ A starts a break of type b ∈ B at τ ∈ T ; 0 otherwise(BAPCAC): min w · X i ∈I ,τ ∈T δ iτ + (1 − w ) · X a ∈ A,τ ∈T (cid:0) − y aτ (cid:1) , (1) n Overview of a Break Assignment Problem Considering Area Coverage 5 such that: X a ∈ A (cid:0) z aτi + ¯ z aτi (cid:1) = D iτ − δ iτ , ∀ i ∈ I , τ ∈ T (2) z aτi ≤ X j ∈J | N ij =1 D iτ x aτj , ∀ a ∈ A, τ ∈ T , i ∈ I (3) ¯ z aτi ≤ X j ∈J | ¯ N ij =1 D iτ x aτj , ∀ a ∈ A, τ ∈ T , i ∈ I (4) z aτi ≤ D iτ (1 − y a,τ ) , ∀ a ∈ A, τ ∈ T , i ∈ I (5) ¯ z aτi ≤ D iτ y a,τ , ∀ a ∈ A, τ ∈ T , i ∈ I (6) X i ∈I z aτi ≤ , ∀ τ ∈ T , a ∈ A (7) X i ∈I ¯ z aτi ≤ , ∀ τ ∈ T , a ∈ A (8) X j ∈J x aτj = 1 , ∀ τ ∈ T , a ∈ A (9) x aτj + x a ( τ +1) j ′ ≤ IM jj ′ , ∀ a ∈ A, τ = { , |T | − } , j, j ′ ∈ J (10) τ + ∆ MINb X τ ′ = τ y aτ ′ ≥ ∆ MINb α abτ , ∀ a ∈ A, b ∈ B, τ = { , |T | − ∆ MINb } (11) y a ( τ + ∆ MAXb +1) + α abτ ≤ , ∀ a ∈ A, b ∈ B, τ = { , |T | − ∆ MAXb − } (12) τ + MAX wb X τ ′ = τ α abτ ′ = 1 , ∀ a ∈ A, b ∈ B, τ ∈ { , T −
M AX wb } (13) τ + MAX wbs X τ ′ = τ α ab s τ ′ + τ + MAX wbl X τ ′ = τ α ab s τ ′ ≤ , ∀ a ∈ A, τ ∈ { , T −
M AX wb l } (14) z aτi , ¯ z aτi , δ iτ ≥ , x aτj , y aτ , α abτ ∈ { , } , ∀ i ∈ I , j ∈ J , τ ∈ T , a ∈ A (15) Marin Lujak et al.
In (1), w is the weight assigned to the coverage mean such that ≤ w ≤ .Constraints (2) relate to the coverage of each unit area i by idle vehicles andvehicles on-break . Here, minimizing the arrival time to an incident has priorityover the performance of a break in its totality. If we exclude ¯ z aτi from the aboveand other constraints, our model turns into a non-preemptive break model.Constraints (3) and (4) limit the coverage of each idle agent and an agenton-break to at most density D iτ of unit area i if it is positioned within the traveltime defined by adjacency matrix N ij and ¯ N ij , respectively. These constraintsensure that part z aτi and ¯ z aτi of the coverage of area i by an idle agent and anagent on-break a respectively, is at most the sum of the densities of the areas j ∈ J within its reach at time τ ∈ T . Furthermore, constraints (5) and (6)limit idle coverage and the coverage on-break only to idle agents and the agentsat break, respectively. Moreover, constraints (7) guarantee that the incidentdensity covered by each idle agent sums up to at most 1. Similarly, constraints(8) capacitate the coverage for each agent on-break.Constraints (9) assign to each agent a at each time period τ a unique lo-cation j ∈ J , by means of x aτj , while constraints (10) relate agents’ positions x aτj in two consecutive time periods t , t + 1 based on the maximum distancetravelled established in IM jj ′ . Moreover, minimal and maximal break durationsare imposed by constraints (11) and (12). Constraints (13) assure that short andlong breaks start at latest after a maximum uninterrupted work time M AX wb ofconsecutive time periods without a break. Additionally, (14) specifies that longand short breaks are coordinated within the maximum allowed working times.Finally, constraints (15) are non-negativity constraints on the decision variables. Acknowledgements
This work has been partially supported by an STSM Grantfrom COST Action TD1409 “Mathematics for Industry Network (MI-NET)”.