SABT ZELGELEMES

Deniz Trsel Eliiyi, Assoc. Prof. Dr.Izmir University of Economics, Department of Industrial Systems Engineering

To appear in: Engineering OptimizationCapacity and Working Time Determination in a Reservation System

ANADOLU NVERSTES Endstri Mhendislii Seminerleri, 12.10.2012 1Outline2PreliminariesPractical importance and motivationProblem definition Complexity resultsAn efficient heuristic algorithmComputational resultsConclusion and Future workFixed Job Scheduling3n jobsReady time: rjDeadline: djProcessing time: pj = dj - rjWeight: wj(k)Pa : Set of available jobs in interval a.Identical parallel machinesCost : ckJob 3 (w3)Job 1 (w1)r1d1Timer2Job 2 (w2) r3d2d3AssumptionsAll parameters knownNo more arrivalsA m/c can process at most one job at a timeA job can be processed by at most one machine at a timeAll machines are eligible to process all jobsMachines are available at all times.3Fixed Job SchedulingThe Operational Problem (OFJS)4

45Fixed Job SchedulingThe Operational Problem (OFJS)Algorithm for the number maximizing OFJS problem (Bouzina and Emmons, 1996)Algorithm for the weight maximizing OFJS problem (Bouzina and Emmons, 1996): Conversion to MCNF problem O(mn log n)

5Fixed Job SchedulingThe Tactical Problem (TFJS)6

6

7Fleet planning:Dantzig and Fulkerson (1954)Gertsbakh and Stern (1978) Computer wiring:Hashimoto and Stevens (1971): ck= c The minimum number of machines required to carry out all jobs = The maximum job overlap of the jobsGupta et al. (1979) Eliiyi (2004): O(n log n) algorithm for arbitrary ck

Fixed Job SchedulingThe Tactical Problem (TFJS)7FJS: NP-hard generalizations8Working Time:

Spread Time:

Sk : Start time of machine k Fk : Finish time of machine k

Eligibility: Each machine is eligible to process only a subset of jobs.

job 3 (w3)job 1 (w1)r1d1r2job 2 (w2) r3d2d3p1p2p3M/c kST8Practical Importance9 Areas of use include all kinds of reservation systems:Tactical capacity planning of aircraft maintenance personnel Hotel reservation systems / Renting bungalowsCar rentalTextile workshopsOperating room scheduling in hospitalsBus Driver Scheduling Problem Earth-observing satellitesAutomated manufacturing systems

Previous Work10Working Time Constraints:Fischetti M., Martello S., Toth P., 1989 : TacticalEliiyi D.T., Azizolu M., 2009, 2011 : OperationalSpread Time Constraints:Fischetti M., Martello S., Toth P., 1987 : TacticalEliiyi D.T., Azizolu M., 2006, 2011 : OperationalEligibility Constraints:Kroon L.G. et al. 1995 : OperationalKroon L.G. et al. 1997 : TacticalKolen A.J.W., Kroon L.G., 1991 : OperationalKolen A.J.W., Kroon L.G., 1992 : TacticalEliiyi D.T., Azizolu M., 2009 : OperationalEliiyi D.T., Korkmaz A.G., iek A.E., 2009 : OperationalNice Surveys:Kovalyov M.Y., Ng C.T., Cheng T.C.E., 2007, Fixed interval scheduling: Models, applications, computational complexity and algorithms, European Journal of Operational Research, 178, 331-342.Kolen A.J.W., Lenstra J.K., Papadimitriou C.H., Spieksma F.C.R., 2007, Interval scheduling: A survey, Naval Research Logistics, 54, 530 543.Motivation11Capacity planning of a reservation system directly affects total profitExisting studies in literature use the tactical FJS for capacity planning:Long term forecasts of job reservations necessary Ignores cancellations or possible changes in job ready times and deadlinesRequires reschedulingStudies handle operational and tactical problems separatelyIntegrated decision of capacity planning and schedulingSignificantly important in systems showing seasonal demand changesEliiyi (2010): An iterative approach that uses the operational model is proposed for determining the best capacity expansion level in a sewing workshop11Problem Definition12Three simultaneous decisions in a reservation environment:the capacity level of the system job-machine assignments working time for each machineApplications: Multi-server data transfer system where the servers have unit-time operating costsSeasonal workforce paid on an hourly basisTravel agency renting hotel rooms for its customersObjective: Maximize the net profit while determining the number of servers and their respective working times as well as the processed job subset.Working time: A decision variable12Problem Definition13n jobsReady time: rjDeadline: djProcessing time: pj= dj - rjWeight: wjm: upper bound (external or internal) on the number of identical parallel machinesOperating cost per unit time (or rental costs): ckPa : Set of available jobs in interval a.

13Mathematical Model 14

Computational Complexity15

USING

THEN:

where

Equivalent to FJS problem with general weights , NP-hard in the strong sense (Eliiyi, Azizoglu, 2009)16Limited number of machines, identical operating costs:

Problem reduces to the operational FJSCan be solved in O(mn log n) time by a MCNF formulation.

Single machine:

Problem reduces to the operational FJS with single machineCan be solved in O(n) time by a shortest path fomulation.Polynomially Solvable Special Cases

A simple & effective heuristic approach O(n log n + nm) 17(S0) Index the potential m machines in nondecreasing order oftheir ck. Index the jobs in nondecreasing order of their rj. Set ZLB = 0, XLB= , A = unassigned job set = {1,...,n}(S1)For k = 1,..., m:Formulate a shortest path problem for the kth machine with |A| jobs, resulting in ZSP (k) = objective function value and XSP(k) = scheduled job setIf ZSP(k) 0 and XSP(k) then ZLB = ZLB + ZSP(k), XLB = XLB XSP(k), update Aelse Go to (S2)If A = go to (S2)(S2) Solution: ZLB , XLBComputational Experiment18n = 100, 200, 500, 1000, 2000rj ~ U(0,200)pj ~ U(4,10)Three levels for job weights:wj = pj , jwj ~ U(4,10)wj ~ U(4,20)Two levels for operating costs:ck ~ U{1, 1.25, 1.5, 1.75, 2}, ck ~ U{0.5, 0.625, 0.75, 0.875, 1}, k10 problem instances for each setting: 300 instancesPC with 4 GB Ram and 1.8 GHz, Windows 7IBM ILOG CPLEX 12.1 solver for optimal solutionsResults 19

cResults 20

cResults 21

cObservations22The algorithm provides very high quality solutions in practically no time, especially for large instances: An average 1.8% optimality gap is attained over all instances. The optimality gap closes for larger instances, and the algorithm performs better than CPLEX for n = 2000.CPLEX could not solve 40 of the instances to optimality in the 1200-second time limit, for some it could not even obtain an initial lower bound for the problem.The optimal solution is obtained in 51 of the 300 instances, and for 34 instances the algorithm finds a better solution than CPLEX within the given time limit.Observations23The algorithm favors solutions with more number of used machines and more jobs processed.Applications may require high number of jobs with many machines, and the developed algorithm seems very promising in generating high quality solutions for very large problem instances. The algorithm performs robustly in terms of solution time for different levels of parameters including weight and cost. Conclusion and Future Work24A new strongly NP-hard problem in a reservation system where the jobs have fixed ready times and deadlines:The objective is to maximize the net profit from the processed job subset while determining the capacity level and the working times of the machines. A heuristic algorithm that performs excellently up to 2000 jobs in very small computation timesPotential research for related problems:Problem with side constraints (spread time, eligibility)Both fixed and operating costs for machinesThank you...