GDH Claassen

I Dumitrescu

L-M Rousseau

P. Eveborn

R Lougee-Heimer

SV Begur

V Hemmelmayr

W Jang

M. Gamst

T. Sejr Jensen

Crossref

A branch-and-price algorithm for the long-term home care scheduling problem

In several countries, home care is provided for certain citizens living at home. The long-term home care scheduling problem is to generate work plans such that a high quality of service is maintained, the work hours of the employees are respected, and the overall cost is kept as low as possible. We propose a branchand-price algorithm for the long-term home care scheduling problem. The pricingproblem generates a one-day plan for an employee, and the master problem merges the plans with respect to regularity constraints. The method is capable of generating plans with up to 44 visits during one week

Gamst, Mette

Jensen, Thomas Sejr

Online Research Database In Technology

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 20, 2017A branch-and-price algorithm for the long-term home care scheduling problemGamst, Mette; Jensen, Thomas SejrPublished in:Operations Research Proceedings 2011Link to article, DOI:10.1007/978-3-642-29210-1_77Publication date:2012Link back to DTU OrbitCitation (APA):Gamst, M., & Jensen, T. S. (2012). A branch-and-price algorithm for the long-term home care schedulingproblem. In D. Klatte, H-J. Lüthi, & K. Schmedders (Eds.), Operations Research Proceedings 2011: SelectedPapers of the International Conference on Operations Research (pp. 483-488). Springer.  (Operations ResearchProceedings). DOI: 10.1007/978-3-642-29210-1_77A branch-and-price algorithm for the long-termhome care scheduling problemM. Gamst and T. Sejr JensenAbstract In several countries, home care is provided for certain citizens living athome. The long-term home care scheduling problem is to generate work plans suchthat a high quality of service is maintained, the work hours of the employees arerespected, and the overall cost is kept as low as possible. We propose a branch-and-price algorithm for the long-term home care scheduling problem. The pricingproblem generates a one-day plan for an employee, and the master problem mergesthe plans with respect to regularity constraints. The method is capable of generatingplans with up to 44 visits during one week.1 IntroductionIn several countries, home care is provided for certain citizens living at home. Homecare offers cleaning, grocery shopping, help with personal hygiene and medicine,etc. The long-term home care scheduling problem is to generate work plans span-ning a longer period of time, such that a high quality of service is maintained, thework hours of the employees are respected, and the overall cost is kept as low aspossible.Quality of service consists of the following. Regularity: all visits at a citizenshould be conducted at the same time of the day and by the same (small group of)employee(s) in order for the citizen to feel safe. Skill set requirements: certain taskscan only be performed by a subset of the employees due to skill requirements.All time windows are soft, i.e., preferred visit times and employee work hourscan be violated at a cost. Such violations are denoted “busyness”. The overall costof a solution consists of a linear combination of travel time between visits, qualityof service, and busyness.The long-term home care scheduling problem is NP-hard and is typically ap-proached in one of two ways in the literature: (1) plans for employees for a singleday are generated. This corresponds to a modified VRPTW and is denoted the dailyUniversity of Southern Denmark, DK-5230 Odense M, e-mail: gamst@man.dtu.dk andthomassejr@gmail.com12 M. Gamst and T. Sejr Jensenplanning problem [1, 4, 5]. (2) Otherwise a Periodic VRPTW is solved, i.e., plansare made for several days, but regularity constraints are ignored [2, 6, 7].We propose a branch-and-price (BP) algorithm for the full long-term home carescheduling problem. The pricing problem generates a one-day plan for an employee,and the master problem merges the plans with respect to regularity constraints. Themethod is capable of generating plans with up to 44 visits during one week. Thistruly illustrates the complexity of the problem.2 Exact solution algorithmThe problem considers a given period of time consisting of L ∈ N days. Time isdiscretized into time steps, which together span L. The set of employees is denotedE . For each employee j ∈E is given a set of days H j and time windows [a jh,b jh],h∈H j, that specify the work hours of j.The set of visits is denoted V , the set of citizens is C, and the set of visits atcitizen c is Vc ⊆ V . Each visit j ∈ V is repeated after p j amount of time. The visitrepetitions are scheduled independently of each other and are denoted activities.Let A be the set of all activities and A j ⊆ A the set of activities for visit j ∈ V .Two consecutive activities from visit j are denoted (i,k) ∈ A j. An activity i ∈ Ahas attached a prioritized list of employees to conduct the activity, denoted pri. Theprioritized list represents the skill set requirement. The duration of i ∈ A is di andthe time window is [ai,bi]. The travel time between two activities i, j ∈ A is ci j ≥ 0.Finally, busyness, i.e., the amount of time employee j is late for conducting visit i,is denoted bi j ≥ 0.Dantzig-Wolfe decomposing the problem results in a pricing problem, whichgenerates a daily schedule for a given employee on a given day, and a master prob-lem, which merges the daily schedules into an overall feasible solution.The daily schedule for an employee is denoted a path and contains an orderedlist of activities with attached starting times. The overall solution consists of pathsfor appropriate employees on appropriate days, covering all activities. Let si ∈ Ndenote the starting time of activity i ∈ A. Recall that the objective function consistsof travel time, quality of service, and busyness. We aggregate these into a single,weighted objective function. Let w be the non-negative weight vector. The objectivefunction which is to be minimized consists of:1. Travel time (TT) between activities.2. Busyness (B), i.e., how late an employee j is for conducting an activity i withrespect to travel times and time windows.3. Employee priority (EP).4. Employee regularity (ER), i.e., the number of different employees at a citizen.5. Visit regularity (VR), i.e., if the time between two consecutive activities (i,k) ∈A j differs from p j.A branch-and-price algorithm for the long-term home care scheduling problem 3Let p be a path and P the set of all generated paths. Let xp ∈{0,1} denote whetheror not path p is part of the solution, and let uik ≥ 0 denote the difference in the start-ing times between two consecutive activities (i,k) ∈ A j. Each path p has a numberof constants attached: δ i jBp ≥ 0 denotes the amount of busyness for employee j andactivity i, δ ip ∈ {0,1} denotes if activity i is visited, δ isp ≥ 0 denotes the start timeat activity i, δ i jp ∈ {0,1} denotes if employee j visits activity i in the path, andδ jhp ∈ {0,1} denotes if the path is generated for employee j on day h ∈ H j. Themaster problem is formulated as:min ∑p∈P∑i∈A(∑k∈AwT T cikδ ipδ kpxp + ∑j∈E(wB δ i jBp xp+ wEP pri( j) δ i jp xp))+∑c∈C∑j∈EwER y jc + ∑j∈V∑(i,k)∈A jwV R uik (1)s. t. ∑p∈Pδ ipxp = 1 ∀i ∈ A (2)∑p∈Pδ ispxp + p j − ∑p∈Pδ kspxp ≤ uik ∀ j ∈V, ∀(i,k) ∈ A j (3)∑p∈Pδ kspxp− (∑p∈Pδ ispxp + p j)≤ uik ∀ j ∈V, ∀(i,k) ∈ A j (4)∑p∈Pδ i jp xp ≤ y jc ∀c ∈C, ∀v ∈Vc,∀i ∈ Av, ∀ j ∈ E (5)∑p∈Pδ jhp xp ≤ 1 ∀ j ∈ E, ∀h ∈ H j (6)xp ∈ {0,1} ∀p ∈ P (7)uik ≥ 0 ∀ j ∈V, ∀(i,k) ∈ A j (8)y jc ∈ {0,1} ∀c ∈C,∀ j ∈ E (9)The objective function (1) minimizes a weighted sum of travel times, busyness,employee priorities, employee regularity and visit regularity. Constraints (2) ensurethat every activity is visited. Constraints (3) and (4) measure visit regularity. Con-straints (5) measure employee regularity. Constraints (6) ensure that at most onepath per employee per day is part of a solution. The number of columns in the mas-ter problem is reduced by fixing daily visits to days: if a visit must be repeated everyday, then the corresponding activities are fixed to day 1, 2, 3, etc., respectively. Thepricing problem only allows such activities to be part of paths on appropriate days.Let pi (2)i ∈ R be the dual of constraints (2), pi (3)ik ≤ 0 the dual of (3), pi (4)ik ≤ 0 thedual of (4), pi (5)ic j ≤ 0 the dual of (5), and pi (6)jh ≤ 0 the dual of (6). The pricing problemis solved for each employee j ∈ E on each day h ∈ H j. The reduced cost of visitingactivity i ∈ Av,v ∈Vc,c ∈C is:c¯ijh = wEP pri( j)−pi (2)i −pi (5)ic j −{si(pi(3)ik −pi(4)ik ) ∃k ∈ A : (i,k) ∈ Av0 otherwise4 M. Gamst and T. Sejr JensenActivity i ∈ A is visited exactly once, hence the reduced cost for employee j ∈ E onday h ∈ H j is defined as:c¯ jh = ∑i∈A(c¯ijh +wBbi j + ∑k∈AwT T cik)≤ pi (6)jh (10)Recall that bi j ≥ 0 is the amount of busyness for employee j at activity i and thatcik ≥ 0 is the travel time between activities i,k ∈ A. Now, pi (6)jh is a constant, so ifthe pricing problem generates a path where c¯ jh ≤ pi (6)jh , then the path has negativereduced cost and the corresponding column is added to the master problem. Thepricing problem is recognized as a shortest path problem with time constraints andpotentially negative edge weights. This is also denoted the Elementary Shortest PathProblem with Resource Constrained (ESPPRC), which is NP-hard. We solve theproblem to optimality using the labeling algorithm in [3]. Initially, we try to solvethe pricing problem heuristically using the labeling algorithm, where only a singlelabel is stored at each activity.Branching is necessary when the optimal solution in a branch node is fractional.The following strategy is finite and eventually ensures a feasible solution. Fractionalsolutions occur when:An employee j visits a citizen c fractionally. Two branching children are gener-ated with added cut:(y jc = 0)resp.(y jc = 1).An activity i is visited by several employees j, j′ or on several days h,h′. Twobranching children are generated with rules:(∑p∈P δ ipδ jhp xp = 0)resp.(∑p∈P δ ipδ j′h′pxp = 0). The pricing problem ensures that employee j (resp. j′) never visits activityi on day h (resp. h′).An employee j travels on edges (ik),(ik′) on a given day h a fractional numberof times. Let i be the first activity, from which employee j travels on different edgesor at different times. Let constant δ sikp denote whether or not path p travels fromi to k at time sik. Two branching children are generated with the following rules:(∑p∈P δ jhp δ sikp xp = 0)resp.(∑p∈P δ jhp δ sik′p xp = 0). The pricing problem ensures thatemployee j never travels from i to k (resp. k′) at time sik (resp. sik′ ) on day h.The branching rules do not complicate the pricing problem, because they eitherconsider different columns (y jc) or consist of rules, which can trivially be handled byforbidding appropriate extensions in the labeling algorithm.3 Computational ResultsThe BP algorithm is implemented using the framework COIN Bcp [8] and testedon an Intel 2.13GHz Xeon CPU with 4 cores and 8 GB RAM. Note that test resultsstem from using a single core. CPLEX 12.1 is used as standard MIP solver.A branch-and-price algorithm for the long-term home care scheduling problem 5The BP algorithm is tested on a number of real-life benchmark instances pro-vided by Papirgrden, a home care center in Funen, Denmark. Time is discretizedinto either 5 or 10 minute time steps. The objective weights are as follows: W T T =500, wB = 750, wEP = 0, wER = 750 ·5/τ and wV R = 50, where τ denotes the size ofa time step. We have tested two algorithms: (1) The exact BP algorithm as describedand (2) a heuristic BP algorithm where columns are only generated heuristically.Instance Master Problem Tree Gap Sol. Time Master Problem Tree Sol. TimCols Rows Size Value Sec. Cols Rows Size Value Sec.1-20-5 6327 199 1 0.00 27000.0 3.97 1794 199 1 38750.0 0.751-25-5 8671 250 4687 0.79 29750.0 1801.33* 608 250 1 44500.0 0.181-30-5 12883 295 2079 3.17 38000.0 1801.97* 2852 295 1 86000.0 1.451-40-5 28365 397 171 0.00 43000.0 812.94 903 397 1 67500.0 0.551-50-5 31480 493 27 110.91 87000.0 1803.35* 1235 493 1 97500.0 1.451-80-5 8737 787 1 333.89 259250.0 1906.50* 2954 787 1 259250.0 10.962-20-5 8917 346 5 0.00 27000.0 8.58 1917 346 1 31750.0 0.662-25-5 10089 432 2405 0.94 29750.0 1803.39* 1416 432 3 34750.0 0.272-30-5 16383 512 695 1.41 38400.0 1803.29* 5637 512 481 73500.0 117.372-80-5 8261 1354 1 297.49 237500.0 1908.73* 12677 1354 185 237500.0 388.091-20-10 4877 199 59 0.00 17250.0 6.37 1286 199 1 24750.0 0.291-25-10 5507 250 277 0.00 18125.0 27.36 324 250 1 29625.0 0.111-30-10 10934 295 2449 0.32 25300.0 1801.07* 1845 295 1 54250.0 0.711-40-10 14037 397 63 0.00 27750.0 90.16 733 397 1 48750.0 0.321-50-10 19907 493 3 0.00 36500.0 337.04 1002 493 1 67500.0 0.711-55-10 19912 544 3 0.00 60375.0 123.64 1409 544 1 86875.0 0.691-58-10 40434 571 111 73.60 64450.0 1803.82* 2270 571 1 106650.0 1.891-80-10 19711 787 1 224.63 161100.0 1847.78* 2255 787 1 161100.0 4.442-20-10 5913 346 39 0.00 17250.0 9.19 1770 346 1 20750.0 0.432-25-10 6594 432 81 0.00 18125.0 16.30 714 432 3 23125.0 0.152-30-10 10109 512 1245 0.20 25200.0 1802.52* 4585 512 751 48000.0 115.492-40-10 15937 684 77 0.00 27250.0 162.93 3267 684 7 37500.0 1.002-50-10 25457 850 21 0.00 35750.0 654.53 6914 850 191 61250.0 58.382-55-10 29331 936 49 133.70 80625.0 1841.08* 6090 936 67 80625.0 21.942-58-10 21593 984 31 160.61 96750.0 1826.54* 7242 984 143 96750.0 76.812-80-10 22136 1354 3 201.76 149750.0 2475.74* 8552 1354 221 149500.0 265.50Table 1 Results for the exact BP (left hand side) and heuristic BP (right hand side) for instancesnamed “|E|−|A|−τ”, τ =time step size. Tests marked with * exceeded the 30 minute time bound.Test results are summarized in Table 1. As can be seen from the left hand size ofthe Table, only few instances with 5 minute time steps can be solved to optimalitywithin half an hour. A coarser discretization helps, but the BP algorithm still suffersfrom a large time usage. The number of columns is large, which is caused partlyby large time windows and partly by busyness, i. e., that time windows may beviolated. The tree size grows very large for some instances not solved to optimality,hence branching also constitutes a bottleneck. As can be seen from the right handside of the Table, the heuristic BP algorithm is generally faster. Few instances sufferfrom large tree sizes and many columns, but the far majority of instances are solved6 M. Gamst and T. Sejr Jensenin seconds. Unfortunately, the objective values generally suffer from the heuristicapproach.Improving the exact BP approach would require methods for reducing the num-ber of columns and for improving the bounds to prune larger parts of the tree.The authors attempted stabilizing the value of dual variables using an interior pointmethod [9], but with no avail. Other stabilization methods could be investigated, asbetter values for the dual variables could reduce the number of generated columns.The authors also tried different primal and incumbent heuristics for improving thebounds with little luck. Future work could continue on such heuristics or on chang-ing the branching strategy.4 ConclusionIn this paper, we presented a BP algorithm for the long-term home care schedul-ing problem. The NP-hard pricing problem consisted of calculating a work plan ona given day for a given employee and the master problem merged the plans intoan overall optimal solution. The BP algorithm was tested on a number of real-lifeinstances and was capable of only solving smaller instances due to the large num-ber of combinations of visits, visit times and employees. This truly illustrates thecomplexity of the problem.Acknowledgements We would like to thank Papirgden for sharing knowledge and real-life datafor home care scheduling, and Villum-Kann-Rasmussen foundation for supporting this work.References1. Begur S. V., Miller D. M., Weaver J. R.: An Integrated Spatial DSS for Scheduling and Rout-ing Home-Health-Care Nurses. Interfaces 27, 35–48 (1997)2. Claassen G.D.H., Hendriks Th.H.B.: An application of Special Ordered Sets to a periodicmilk collection problem. European Journal of Operational Research 180, 754 – 769, (2007)3. Dumitrescu I., Boland N.: Improved preprocessing, labelling and scaling algorithms for theweight-constrained shortest path problem. Networks 42, 135–153 (2004)4. Eveborn P., Flisberg P., Ro¨nnqvist M.: Laps Care - an operational system for staff planningof home care. European Journal of Operational Research 171, 962–976 (2006)5. Hansen A. D., Rasmussen M. S., Justensen T., Larsen J.: The Home Care Crew Schedul-ing Problem. In 1st International Conference on Applied Operational Research volume 1 ofLecture Notes in Management Science, 1–9 (2008)6. Hemmelmayr V., Doerner K., Hartl R., Savelsbergh M.: Delivery strategies for blood productssupplies. OR Spectrum 31, 707–725 (2008)7. Jang W., Lim H. H., Crowe T. J., Raskin G., Perkins T. E.: The Missouri Lottery Optimizes ItsScheduling and Routing to Improve Efficiency and Balance. Interfaces 36, 302–313 (2006)8. Lougee-Heimer R.: The Common Optimization INterface for Operations Research. IBMJournal of Research and Development 47, 57–66 (2003)9. Rousseau L.-M., Gendreau M., Feillet D.: Interior point stabilization for column generation.Operations Research Letters 35, 660–668 (2007)

http://orbit.dtu.dk/ws/files/12377283/A_branch_and_price_algorithm.pdf

A branch-and-price algorithm for the long-term home care scheduling problem

Abstract

Similar works

Full text

Available Versions

Crossref

Online Research Database In Technology