In the Manpower Allocation Problem with Time Windows and Job-Teaming Constraints (MAPTWTC), we have a set of jobs located at various locations where each job requires a team of workers. Each job has a time window and a job duration, during which everyone on the team has to be present. The job requirement is satisfied if and only if the required composite team works for long enough duration within the job's time window. The objective of the problem is find a schedule to minimize a weighted sum of the total number of workers, the total travelling distances of all workers and their total waiting time. Two main approaches are proposed in the paper which are shown to be able to obtain very good performance

LI, Xi

LI, Yan Zhi

LIM, Andrew

RODRIGUES, Brian

Li, X.

Li, Y.Z.

Lim, A.

Rodrigues, B.

Hong Kong University of Science and Technology Institutional Repository

Manpower allocation with time windows and job teaming constraints

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Singapore Management UniversityInstitutional Knowledge at Singapore Management UniversityResearch Collection Lee Kong Chian School OfBusiness Lee Kong Chian School of Business11-2003Manpower Allocation with Time Windows andJob Teaming ConstraintsXi LIHong Kong University of Science and TechnologyYan Zhi LIHong Kong University of Science and TechnologyAndrew LIMHong University of Science and TechnologyBrian RODRIGUESSingapore Management University, brianr@smu.edu.sgFollow this and additional works at: https://ink.library.smu.edu.sg/lkcsb_researchPart of the Operations and Supply Chain Management CommonsThis Conference Paper is brought to you for free and open access by the Lee Kong Chian School of Business at Institutional Knowledge at SingaporeManagement University. It has been accepted for inclusion in Research Collection Lee Kong Chian School Of Business by an authorized administratorof Institutional Knowledge at Singapore Management University. For more information, please email libIR@smu.edu.sg.CitationLI, Xi; LI, Yan Zhi; LIM, Andrew; and RODRIGUES, Brian. Manpower Allocation with Time Windows and Job Teaming Constraints.(2003). Decision Sciences Institute Annual Meeting 2003, November 22-25. 1-6. Research Collection Lee Kong Chian School OfBusiness.Available at: https://ink.library.smu.edu.sg/lkcsb_research/2065MANPOWER ALLOCATION WITH TIME WINDOWS AND JOBTEAMING CONSTRAINTSX. Li1, Y.Z. Li2, A. Lim2 and B. Rodrigues31Dept of Computer Science,National University of Singapore,3 Science Drive 2,Singaporelixiaoc@comp.nus.edu.sg2Dept of IEEM,Hong Kong University of Science and Technology,Clear Water Bay, Hong Kong.{ieyanzhi,iealim}@ust.hk3School of Business,Singapore Management University,469 Bukit Timah Road,Singapore 259756br@smu.edu.sgABSTRACTIn the Manpower Allocation Problem with Time Windows and Job-Teaming Constraints(MAPTWTC), we have a set of jobs located at various locations where each job requires ateam of workers. Each job has a time window and a job duration, during which everyoneon the team has to be present. The job requirement is satisfied if and only if the requiredcomposite team works for long enough duration within the job’s time window. The ob-jective of the problem is find a schedule to minimize a weighted sum of the total numberof workers, the total travelling distances of all workers and their total waiting time. Twomain approaches are proposed in the paper which are shown to be able to obtain very goodperformance.Keywords: Workforce planning,Heuristics,Optimization,Work teamsINTRODUCTIONIn present-day business environments, groups of workers in the form of teams are common.We often see long lists of project team meetings in the busy schedules of investment bankers,mid-level business representatives or management. A team may consist of people withdifferent expertise. They can be easily deployed, and can be formed and disbanded flexibly.Moreover, team members may arrive at a job location separately, as is commonly the caseand since this allows for greater flexibility of deployment. In such situations, the efficientmanagement of such teams in the context of manpower allocation and scheduling becomesa priority[1].As an example, consider a media company which shoots programs at various sites. Eachsite may need a number of directors, coordinators, cameramen, lighting staff, recorders,technicians and stage workers. Programs can share staff members like recorders, cameramen,technicians and stage workers since their jobs are not quite unique. Each program shoothas its duration, and in particular, it may need a type of worker for just a fraction of theentire shooting duration (e.g. stage worker for stage setup). In such a setting, an efficientsharing scheme of staff members among the various programs may save the corporationstaffing costs. This staffing problem is more conspicuous in organizing large-scale activitiessuch as large conferences or sports events. Yet another potential application would be inmilitary planning for special forces teams comprising of members with different expertiseand who can arrive to form teams at different time epochs.1The rest of this paper is organized as follows. In next section we first formally state ourproblem. Two approaches to tackle this NP-hard problem are presented in the following twosections. And then experimental results of various approaches are compared and analyzed.In the final section we conclude our paper.PROBLEM STATEMENTIn the Manpower Allocation Problem with Time Windows and Job-Teaming Constraints(MAPTWTC), we have a set of projects or meetings or jobs (collectively termed jobs)located at various locations where each job requires a team of workers. We can have differentprofessions of workers in a team where the expertise make-up of the team can be given as arequirement for that job. All workers are sent from a single location called a depot. Eachjob has a time window that specifies the earliest time the job can begin and latest timeby which the job has to be completed. Each job has a duration, during which everyone onthe team has to be present (pre-emption is not allowed , for example, in a negotiation or aproject team meeting). The job requirement is satisfied if and only if the required compositeteam works for the required duration within the job’s time window. This implies that aftera worker comes to the site of a job, he/she has to wait for all his/her teammates to arriveand then, possibly wait until the start time of the job. They have to be at the job’s locationfor the entire duration of that job. We are given a map indicating the time of travel betweenevery pair of job locations and assume that every worker travels at the same speed. Theobjective of the problem is to minimize a weighted sum of the total number of workers, thetotal travelling distances of all workers and their total waiting time. The weights representthe relative importance of these numbers to the management.Let’s formally state our problems as follows. Givenm types of workers, T1, ..., Tm; n jobs,job i(i = 1, ..., n) is described as a three-tuple {[vi1, ..., vim], pi, [si, ei]} where [vi1, ..., vim]denotes requirements for each type of workers, pi is the duration of this job and [si, ei] is itstime window. All the workers start from a depot, denoted 0. Job i is located in location iand time to travel between locations i and j is dij, i, j = 0, 1, ...n, i 6= j. It’s possible thattwo locations have no direct link. We are asked to give a schedule to minimize the weightedsum of total number of different workers L,total travel time T and total waiting time W ofall workers, which can be denotes as αL+βT + γW,α+β+ γ = 1,where α, β, γ are relativeweights.This problem is NP-hard which can be proved by reducing the well-known NP-hardproblem TSP to it.SIMULATED ANNEALING ON CONSTRUCTION HEURISTICGiven a jobs sequence, we can form a schedule by some construction heuristics. In the firstapproach, we adopt a simulated annealing framework and propose two types of constructionheuristics–simple-append and block-insertion. In simulated annealing, the neighborhood ofa sequence is generated by block transposition and block reverse operators. We’ll sketchthese core components in this section.In simple-append, we read in jobs one by one from the the jobs sequence and construct aschedule incrementally. For job i, we find a set from all the current workers who can arriveat location i before (ei−pi),the latest starting time of job i. If more than enough workers areavailable for job i, we randomly choose what we need from the workers set. Otherwise wesimply hire new necessary workers. Job i is started when all the workers have arrived. If thearriving time is earlier than si, we set starting time to si. simple-append has the followingweakness. If in a worker’s optimal schedule, job i comes before job j while i comes after2j in the given permutation, for sure the optimal schedule can’t be found. block-insertionaddresses the defect to some extent.In block-insertion, for a job i just read in from the sequence, we compute an optimalsubset from the scheduled workers such that: 1) All the workers in the subset have a commontime interval that is within the job’s time window; 2) The subset results in a least numberof new hires for this job. The scheduled start time of job i is the start time of the commoninterval that makes the subset eligible. If more workers are needed, we hire new ones. Tocompute an optimal subset, we first search all the current workers’ schedules to see whetherthere exists a long enough interval for the insertion of job i. We collect all the intervals intoa set I and they apply the following O(|I|2) algorithm to find the optimal subset:• Sort starting and ending points of intervals in I together into an array in non-decreasing order. Create a stack.• Scan the array from beginning one bye one. If the scanned point is a starting pointof a interval, push it to the stack.• Otherwise, suppose it is end point e of interval v. Examine the stack from top tobottom to search for a starting point s s.t. (s, e) is a long enough interval. Evaluatethis interval and update the best found. Remove starting point s from the stack andproceed to scanning procedure.So now given a jobs permutation, we can construct a feasible schedule from it. Oursimulated annealing can be viewed as search on the permutation space. We find neighborsof the current permutation by block transposition and block reverse operators. Figure 1visualizes these two operators.We select the starting and end points of the selection regionrandomly. For n jobs, the whole search space has n! permutations. Block transposition andblock reverse can generate n4 and n2 neighbors respectively. So we will not jump betweendistant points in the search space with these two operators. In the simulated annealingFigure 1: Neighborhood operators Block Reverse and Block Transpositionscheme,we use annealing with at most five reheating times. See [2] for a detained discussionof simulated annealing strategies. In the local search, the size of neighborhood and thelargest distance between the nearest neighbors in the same neighborhood depends on theratio of current temperature and the starting temperature. Higher ratio results in moreneighbors in one neighborhood and more distant neighbors in the same neighborhood. The3simulate annealing on construction heuristics approach shows quite good performance inexperiments.NETWORK FLOW MODELSGiven a time scheduling of all jobs, we can show that the problem to minimize the totalnumber of workers can be transformed to a maximum flow problem(MFP). MFP can besolved optimally and efficiently. Thus our original problem will be changed to look for thebest one in the time scheduling space. In the following of this section we first illustrate thetransformation to MFP and then show how to find the best time scheduling.Suppose a fixed time scheduling for all jobs is given. We say job j is compatible to jobk if a worker can serve j in specified time interval and then arrive at location k no laterthan the specified start time of job k. We note that this relationship is anti-symmetric. OurManpower Flow Graph(MFG) consists of all jobs as nodes. There is a directed edge fromj to k if and only if pair (j, k) is compatible.Workers can travel along any directed path inMFG and every directed path denotes a feasible worker schedule. Notice that MFG is thesame for all types of workers. So the discussion in this section will focus on optimization ofone single worker type,say, Ti.We label the edges to represent worker flow from job to job. Edge e = (j, k) is labelledwith label(e) if there are label(e) workers of type Ti who perform job j and then travel tolocation k immediately. It is easy to see that the MFT is feasible if, for any job j, thefollowing two upper bounds hold:∑∀k, e=(k,j)∈MFGlabel(e) ≤ vij (1)∑∀k, e=(j,k)∈MFGlabel(e) ≤ vij (2)The first bound says that we require no more than vij workers type Ti to come to j and thesecond bound says that the number of Ti workers who depart from j can not be larger thanvij.If we are given a labelledMFG, it’s easy to recover schedules of type Ti workers. Since fora given fixed time schedule no cycle exists in the graph, we just need travel all the paths withnonzero labels to get all the schedules. If there exist unsatisfied job requirements, we simplyhire extra workers. We note that in such a sitting, the number of extra type Ti workersneeded for job j equals: vij −∑∀k, e=(k,j)∈MFG label(e). Summing up, for all jobs, the totalnumber of Ti workers needed for the given time schedule is:∑nj=1 vij −∑e∈MFG label(e).Because∑nj=1 vij is a fixed term, to get the minimum number of type Ti workers, we shalllabel MFG to maximize∑e∈MFG label(e).In order to apply maximum flow algorithms directly, we need to make a transformationfor our network flow model. We spit each node j to two nodes in(j) and out(j). For eachedge (j, k), add an edge (in(j), out(k)) with infinite capacity to the new graph,denotedas MFG. Source s and sink t are added to MFG and for each node j,add two edges(s, in(j)), (out(j), t) both with capacity vij.There is one-to-one correspondence between edges in MFT and edges in MFG whichare incident neither on s nor on t. Thus,∑e∈MFGlabel(e) =∑e∈MFG∀k,e6=(k,t)∧e6=(s,k)flow(e) = flow(MFG) (3)The first equality is established by correspondence between edges in the two graphs and thesecond equality can be easily seen as due to the mass balance constraint on the max-flowproblem.4We claim that any feasible flow in MFG corresponds to a feasible labelling in MFGbecause incoming edges to sink node enforce MFG upper bound (1) and outgoing edgesfrom source node enforce MFG upper bound (2). On the other hand, the mass balanceconstraint in MFG does not add any more restrictions to flow in MFG because for anynodes j, k and any feasible label of (∗, k) and (j, ∗) in MFG, upper bounds (1) and (2)dictate that the edges (s, in(k)) and (out(j), t) have sufficient capacity to balance the nodesin(k) and out(j). Thus, any feasible labelling inMFG has a corresponding balanced flow inMFG. In the light of that, a maximum flow in MFG corresponds to a maximum labellingin MFG.Meta-heuristic searches for optimal job time schedulesWe have shown that given a fixed scheduling, exact optimal total number of workers can beobtained. Since usually total number of workers is our most important aim, we can changeour problem to find the best time scheduling.Two meta-heuristic approach are implemented. One is Ant Colony Optimization(ACO)and the other is Tabu embedded Simulated Annealing(TSA). ACO is a meta-heuristic methodthat simulates the process of ants as they seek for food[3]. We have implemented a standardACO approach while it shows less powerful than TSA approach.The simulate annealing scheme of TSA is the same as presented in the solution usedon the construction heuristics. However, when we generate neighbors in TSA we firstcheck against tabu list and only investigate those neighbors that are not tabu(see [4] for anoverview of tabu search). A solution is represented by a set of time slot choices. The datais put in an array,choice[1..n],with choice[j] denoting the choice of time slot for job j. Togenerate the neighborhood, for each solution choice[1..n], the algorithm randomly selects atime slot within a range of the current solution. The higher the temperature is, the largerthe range size. For each possible choice of a specific job, the tabu list keeps a record on thetimes this choice has been chosen recently. If the times exceeds a limit, this choice is tabued.When making a neighborhood move on the current solution, we calculate how many of itschoices are tabued. If the number of tabued choices exceeds a certain preset ratio measurerelative to the total number of jobs, the neighbor is tabued. When too many choices aretabued, we will have a lot of neighbor moves rejected by the tabu data structure;if so,thealgorithm will forget part of the tabu list.EXPERIMENTAL RESULTSTest cases generationSince no benchmark data is available for the new MAPTWTC, we use newly generated testcases. In each test case, we specify a set of clusters. Each cluster is a circle with a givencentral point and a radius. The number of jobs in a unit area is constant in each cluster.Each cluster specifies the number of jobs in it, a time distribution of jobs and a workertype demand distribution. A important parameter is the ratio between job duration andjob time window, which influences the performance of our lower bound.Lower boundsThe lower bounds of number of hires are calculated using a relaxation of our networkflow model. There is one major relaxation in Relaxed Manpower F low Graph(RMFG)5compared with MFG. The definition of compatible relationship is relaxed. Without givena fixed time scheduling,if a worker is possible to perform job k immediately after job j, wesay job j is compatible to job k. We can see any feasible MFG must be a sub-graph ofthe RMFG; thus RMFG must have a maximal flow no less than any MFG does. So weobtain a lower bound by solving maximum flow on the RMFG. The bound is loose if theRMFG contains many cycles. A low ratio of job duration to time window leads to manycycles and thus loose bounds.Results analysisWe have implemented different approaches presented above and extensive experiments havebeen conducted. Due to space limitation, we may not give details of the experimentalresults. Here are conclusions drawn from the results.• In the scenarios of high ratio(≥ 0.8)–job duration to time window, all our algorithmsachieve solutions less than 10% over lower bounds. Since the lower bound is possiblynot tight, such performance is satisfactory. Among all the approaches, the simulatedannealing scheme on block-insertion heuristic(SABI) method achieves best perfor-mance.• In scenarios of lower ratio of job duration to time window, results of our algorithmshave considerable gaps to lower bounds. This is expected since many cycles exists inRMFG for lower ratios and thus we may often get infeasible solutions when seekingfor a lower bound. SABI approach also performs the best in this situation.CONCLUSIONSIn this paper, we studied the manpower allocation problem with time window and teamingconstraints. This is a new and relevant problem in manpower scheduling. Two approacheswere proposed to tackle this NP-hard problem. One is simulated annealing on constructionheuristics and the other utilizes network flow model and embeds it in the simulated annealingframework as well. The experiments showed our algorithms performed well.Future work may be done on tight lower bounds and more complicated MAPTWTCsuch as considering different costs for hiring different types of workers.References[1] Abboud, N., Inuiguchi, M., Sakawa, M., Uemura, Y., (1998), Manpower allocationusing genetic annealing, European Journal of Operational Research, 111:405-420[2] Dowsland K. (1993) Simulated annealing. In Reeves C. R. (1995) Modern HeuristicTechniques for Combinatorial Problems. Blackwell Scientific Publications, Oxford[3] Garbardella, L.M. (2000), An Ant Colony System Hybridized with A New Local Searchfor the Sequential Ordering Problem, Informs Journal on Computing 3, pp. 237 - 255.[4] Glover, F., Laguna, M., (1997), Tabu Search, Kluwer Academic Publishers6

Manpower Allocation with Time Windows and Job Teaming Constraints

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Manpower Allocation with Time Windows and Job Teaming Constraints

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