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An agent-based nurse rostering system under minimal staffing

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ARTICLE IN PRESS Int. J. Production Economics 114 (2008) 697– 713 Contents lists available at ScienceDirect Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe An agent-based nurse rostering system under minimal staffing conditions Michael V. Chiaramonte a,Ã, Laurel M. Chiaramonte b a b Department of Industrial Engineering, Ira A. Fulton School of Engineering, Arizona State University, Tempe, 22391 N Dietz Drive Maricopa, AZ 85287, USA Mike O’Callaghan Federal Hospital
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  ARTICLE IN PRESS An agent-based nurse rostering system under minimal staffingconditions Michael V. Chiaramonte a, Ã , Laurel M. Chiaramonte b a Department of Industrial Engineering, Ira A. Fulton School of Engineering, Arizona State University, Tempe, 22391 N Dietz Drive Maricopa, AZ 85287, USA b Mike O’Callaghan Federal Hospital, Nellis AFB, NV 89191, USA a r t i c l e i n f o Article history: Received 3 October 2007Accepted 20 March 2008Available online 29 March 2008 Keywords: Nurse schedulingstaff schedulingNurse rosteringAgent programming a b s t r a c t Nurse rostering is a complex problem. We propose a new heuristic using a competitiveagent-based negotiation that focuses on nurse preferences called competitive nurserostering (CNR). Unlike the existing literature, CNR models each nurse’s preferencefunctions separately and separates the cost minimization and preference maximizationproblems. CNR produces quality nurse rosters even though it cannot leverage extrastaffing. As an agent system, CNR can distribute computational requirements over severalcomputer systems, include other solution methods at various points in of the rosteringproblem, and act as a real-time scheduling system. These benefits are not naturallyinherent in centralized heuristic solutions.Published by Elsevier B.V. 1. Introduction Nurse scheduling can be broken into three phases: determining the number of nurses needed to work each shift of ascheduling period, developing schedules or shifts that cover that nursing demand, and finally rostering or assigning specificnurses to schedules or shifts to develop the final roster or work assignments. This paper focuses on the rostering phase of nurse scheduling by altering a simple initial temporary roster to create a final roster that better satisfies nurse preferences.Nurse rostering is an NP-Complete problem (Winstanley, 2004). This complexity makes it difficult to develop andemployautomated scheduling systems that can quickly publish quality rosters. The schedulerora scheduling system needsto consider organizational and legal scheduling rules and the preferences of the individuals being scheduled. This highlyconstrained problem is often solved at hospitals by hand.With the current nationwide nurse shortage, the development of quality nurse rosters is critical in providing adequaterecruitment and effective retention (Tierney, 2003). Surveying research indicates that the current poor quality of nurseschedules is one of the top factors leading to nurse job dissatisfaction (McIntosh et al., 2006). The care provided by thenursing corps consumes nearly 25–33% of a hospital’s budget (Welton et al., 2006). This critical segment of our healthcaresystem requires a stable nurse population. Therefore we need to develop scheduling systems that can effectively consider avariety of nurse preference inputs and quickly develop rosters that are favorable to nurses.Nurse scheduling has received a lot of attention in the literature. Traditional methods used to solve the nurse schedulingproblem include mathematical programs (Warner,1976;Bailey,1985;Miller et al.,1976). Due to the NP-Complete nature of  the problem these solutions are generally used to solve small problem instances. The high level of computational Contents lists available atScienceDirectjournal homepage:www.elsevier.com/locate/ijpe Int. J. Production Economics 0925-5273/$-see front matter Published by Elsevier B.V.doi:10.1016/j.ijpe.2008.03.004 Ã Corresponding author. Tel.: +14806262518. E-mail address: michael.chiaramonte@asu.edu (M.V. Chiaramonte).Int. J. Production Economics 114 (2008) 697–713  ARTICLE IN PRESS complexity of the nurse scheduling problem is widely recognized in the nurse scheduling literature and has led someauthors to develop heuristic solutions.Many of these heuristics have their roots in mathematical programs.Millar and Kiragu (1998)developed a networkprogramming model and recommended that it only be solved tothe first integer solution. This, theyassert, provided a goodsolution and avoided the unacceptably long solution time required to find an optimal solution. Another means to produce aheuristic using a mathematical model is the use of working sets. Here the solution space is reduced by selecting a smallerset of possible schedules that can be worked by each nurse. A mathematical model then selects the optimal solution forapplying those possible schedules (Bechtold and Brusco, 1995). Other hueristics include a variety of evolutionary modelssuch as genetic algorithms (Easton and Mansour, 1999), ant colony optimization (Gutjahr and Rauner, 2007), and tabu search (Dowsland, 1998).These traditional mathematical models and evolutionary heuristics use a single objective function to rate the quality of a nurse roster. This approach works well for problems of minimizing costs but not for handling nurse preferences. Using asingle objective function assumes that the stated preferences of nurses are equally scaled and rational. For example, if twonurses were asked to rate their preferences and nurse A rates a preference an eight and nurse B rates a preference a six, asingle objective function will treat nurse A’s preference as mores important as B’s even though this may not be true. It ispossible that nurse Atendsto overstate his preferences or that nurse B tends to understate them. Perhaps nurse B had otherpreferences that month that were more important so she artificially lowered her rating to a six to reflect this. Thesepossibilities make normalization of nurse preferences difficult.Winstanley published the only agent-based nurse scheduling paper we found (Winstanley, 2004). Winstanleyrecognized that both an agent framework and the nurse scheduling problem is naturally distributed. Using distributedintelligent agents in a cooperative framework accurately models the reality that each nurse and the organization have theirown set of goals. Winstanley used semi-autonomous nurse agents to create initial assignments. These nurse agentscommunicate with a constraint logic-programming agent to solve the overall problem. In a related problem,BenHassineand Ho (2007)cite the intuitively distributed characteristics of personnel scheduling as a motivating factor for their agent-based meeting scheduling system.Like the aforementioned agent-based solutions, we propose an agent-based nurse rostering system to model theintuitive distribution of the problem. We use agents to focus on each nurse’s preference goals and one agent to maintainoverall roster feasibility. We call our system a competitive nurse rostering (CNR) algorithm. CNR contributes to the body of nurse scheduling literature in three ways: first, it is the first competitive agent methodology used on the problem: second,it isolates and uses each nurse’s preference utility model within a system design to focus solely on nurse preferences: andthird, it separates the organizational cost minimization problem and the nurse preference rostering problem.CNR is designed to model nurse shift trading — a natural occurrence in shift work. Unlike existing methods, CNR implements a competitive agent-based simulation where agents representing each nurse compete for days off by tradingshifts in an auction. The CNR implementation leverages the intuitively distributed nature of the rostering problem andaffords two key benefits not inherent in more traditional methods. First, CNR correctly handles each nurse’s statedpreference ratings by avoiding direct comparisons between nurse preferences and by separating the preference problemfrom the cost problem. Second, it is not encumbered by the highly constrained complexity of mathematical programmingapproaches thus allowing for the inclusion of many more preference considerations.Our implementation of CNR considers three factors for nurse preferences: informal request-offs (ROs), preferredsequence length of days off, and preference for specific days-of-the-week (DOW) off. ROs are days the nurses request off butdo not want to take vacation days for. The granting of ROs are not guaranteed. Preferred sequence lengths allow nurses toindicate whether they like schedules that tend to have 2, 3, or 4 days off in a row.The research described in this paper is designed to demonstrate that the distribution of rostering objectives (e.g.organizational costs, overall feasibility, and nurse preferences) to software agents is a feasible and a potentially powerfulidea. The CNR agent paradigm presents the foundation for a system that affords many of the benefits of agent-basedscheduling methods. These benefits include the ability to naturally distribute computation, the ability to realize real-timerescheduling, and the ability to integrate other solution methods to help with problem solving (Shen, 2002).This paper is divided into three more sections. Section 2 outlines the CNR auction paradigm including detaileddescriptions of the software agents. Section 3 discusses our motivating case study including the experimental design andstaffing rules used in our experiments. Section 4 includes the results of our experiments and a discussion of theimplications and future research. 2. Competitive nurse rostering (CNR) paradigm CNR is designed to model a phenomenon often seen in shift work–shift trading. When a nurse wants a dayoff that is notin their schedule she will often solicit other nurses to trade shifts. CNR automates this process in a computer systemdesigned to improve simple rosters with respect to nurse preferences. CNR starts with an initial roster that is feasible and istreated as if it is minimally staffed. The implementation of CNR described in this paper is designed for the Mike O’CallaghanFederal Hospital’s Air Force Medical Surgical Unit (MOFH-AFMSU). MOFH-AFMSU rosters its nurses to 12-h day and nightshifts. All the nurses in the ward are Registered Nurses (RNs) trained in-house to one of three levels: level one nurses M.V. Chiaramonte, L.M. Chiaramonte / Int. J. Production Economics 114 (2008) 697–713 698  ARTICLE IN PRESS include new nurses who still require supervision, level two includes nurses who are no longer new to the unit but have notyet been cleared for charge nurse duty, level three nurses are cleared for charge nurse duty. Charge nurses are responsiblefor all nursing activity in the ward for the shift they are assigned and must be present at all times.The algorithm relies on two types of software agents: an Auction Control Agent (ACA) and Nurse Broker Agents (BAs).The ACA is responsible for soliciting sales, soliciting bids, ensuring the feasibility of staffing numbers, and determiningalgorithm termination. The BAs are responsible for modeling the preferences of the nurses they represent, deciding whichshifts to sell, deciding which sales to bid on, and ensuring their individual schedules are feasible.All work shifts are traded as stints, a concept introduced byMillar and Kiragu (1998). They define a stint as a series of consecutive work or off day shifts. The stints available in this implementation of CNR are detailed inTable 1. 2.1. The auction control agent (ACA) The ACA is responsible for controlling the flow of the CNR algorithm. During the initialization of the CNR algorithm theACA initializes control variables and receives information on the required demand levels for each shift. Once the ACA isinitialized it reads the nurse roster from a data file. Using this initial roster, the ACA creates the data elements that store thecurrent staffing levels in the initial roster and creates the BAs that are required to represent all the nurses in the roster.A sequence list in the ACA maintains the BAs in a specific order determined by the order of the nurses in the initial roster.Finally, the ACA passes initializing information to the BAs.The ACA is composed of two major datasets: the demand picture and the current auction offer. The demand picture isstored in four arrays, two for each the dayand night shifts. The dayshift arrays are depicted inFig.1. The four arrays consistof four sets of variables: the minimal day shift staffing ( D rj ), the day shift’s current staffing (Ad rj ), the minimal night shiftstaffing ( N  rj ), and the night shift’s current staffing (An rj ). In these variables r  is the training level from the set of the three in-house nurse-training levels and j is the day of the scheduling period from the set of days in the scheduling period ( J  ). Thus, D rj is the minimum number of nurses trained to level r that are required to work day j . Nurses of lower training levels canbe substituted by nurses of higher training levels.The auction offer is a collection of stints; one stint is the set of shifts being auctioned off, called the sale item, the otherstints are stored in a list called the currency. All the stints in the auction offer are on-duty stints. By using this auction offerformat nurse BAs are trading workdays. The auction offer is depicted inFig. 2.The auction offer’s currency list is used to determine the highest bid in the auction. When developing the auction offerthe selling BA will place every stint that they are willing to accept in trade for the sale item into the currency list. The Table 1 Millar and Kiragu’s (1998)stints as they are adapted to the manning rules at Mike O’callaghan Federal HospitalOff-duty stints On-duty stintsO D N DNOO DD NN DDNOOO DDD NNNOOOO D indicates a day shift assignment, N  a night shift assignment and O indicates a day off. Thus, DDN is a stint of 3 consecutive days where a nurse works theday shift on the first 2 days and the night shift on the last day. Fig.1. The dayshift demand picture arraysfor a 1-weekperiod. Each shift traded in the CNR algorithmmust satisfy the minimal demand requirements forevery day of the scheduling period ( j ) and every nurse level ( r  ). In this figure, D rj is the minimal number of nurses trained to at least level r  that arerequired to work the day shift on day j . Ad rj is the current total number of nurses at level r  assigned to the day shift on day j of the scheduling period. M.V. Chiaramonte, L.M. Chiaramonte / Int. J. Production Economics 114 (2008) 697–713 699  ARTICLE IN PRESS currency list is then sorted so that the trade that will result in the greatest improvement to the selling nurse’s schedule islisted first. The bid highest up on the list is awarded with a sale.Aside from initialization, the ACA has four major functions: soliciting bids, soliciting sales, closing sales, and closingthe auction. The ACA runs a sequential auction. In this auction, sales are solicited from one BA at a time. At itscompletion a new sale is solicited from the next BA. To control the sequence of the auction the ACA tracks the sequence of all the BAs in a list. Sequencing instills the CNR algorithm with a hierarchy present in the workplace. This hierarchy allowsthe model to give nurses with more seniority the ability to attempt sales first when most nurses will still have shifts totrade.When soliciting a sale the ACA passes an emptyauction offer tothe solicited BA. After the BA returns the auction offer tothe ACA there are two tasks that the ACA must perform: first, it must verify there is a valid sale item; second, if there is asale item, it must check the staffing demand associated with that sale item. To verify that a valid auction offer was returnedthe ACA will receive a control flag from the BA indicating if the BA has a stint to sell. If the flag returns false then the ACAwill solicit a sale from the next nurse in the sequence.When the ACA receives a valid auction offer it prescreens the bidders to help ensure that nursing demand levels aremaintained. To do this the ACA checks the training level of the selling nurse and the current staffing. During this check theACA looks for the following conditions in the demand picture:(1) Ad rj 4 D rj or An rj 4 N  rj , where r  is the selling nurse’s level and for all j in the set of days in the sale item.(2) Ad rj ¼ D rj or An rj ¼ N  rj , where r  is the selling nurse’s level and for all j in the set of days in the sale item.The first condition indicates that there are more nurses of the selling nurses training level than are required. This doesnot mean that there are more nurses working than are needed. Forexample, a shift that requires five nurses and onlyone of those nurses is required to be a level three nurse may have five nurses working and two are level three nurses. In this casethere is a surplus of one level three nurse. When there is a surplus of nurses at a specific training level and one of thosenurses is attempting to sell shifts at auction, the ACA will open the auction up to nurses of all levels.The second condition indicates that the shifts being sold have the minimum number of nurses assigned that are trainedto at least the same level as the selling nurse. Under this condition the ACA opens the auction up to only those biddingnurses that are trained to at least the same level as the selling nurse. This ensures that the nurse that is no longer workingthose shifts is replaced by a nurse that is trained to at least the same level.After obtaining an auction offer the ACA opens the auction up to bids from the nurses that are feasible after checkingdemand conditions. These bidding nurses are approached by the ACA in the order they appear in the sequence list. In thislist, the selling nurse is always first. The first potential bidding nurse is second. When soliciting a bid, the ACA will supplythe bidding nurse with the auction offer and the index of the current highest bid in the currency list. The BA will indicate if they are bidding, if they are bidding they will the include index of their bid in the currency list.Once a bid is received the ACA must perform another check of the demand picture. This check is exactly the same as thecheck performed after soliciting a sale except that it is performed on the bid, not on the sale item. If the bid should fail thedemand check the ACA will solicit the same nurse again for a new bid.After the first pass through the sequence of bidding nurses, the ACAwill check to see if there is a highest bid. If there is ahighest bid, the ACA will solicit new bids from all the bidding nurses except the nurse that placed the current highest bid.These new bids must be higher than the current highest bid. This process will be repeated until a pass through thesequence of bidding nurses results in no change to the highest bid.Once the bidding has ended the ACA will inform the bidding and selling BAs and the BAs will then update theirschedules. The ACA will also adjust the demand picture as necessary. After the sale is closed the ACA will move the selling Fig. 2. The auction offer is used to pass the stint that is for sale (the sale item) to the bidding BAs. The auction offer contains a sorted list of stints(currency list) that the selling BA is willing to take in trade for the sale item. The currency list is sorted from best to worst so that trading the stint at thetop of the list for the sale item will improve the seller’s schedule the most. M.V. Chiaramonte, L.M. Chiaramonte / Int. J. Production Economics 114 (2008) 697–713 700
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