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ANADOLU ÜNİVERSİTESİ Endüstri Mühendisliği Seminerleri, 12.10.2012. Capacity and Working Time Determination in a Reservation System. Deniz Türsel Eliiyi , Assoc. Prof. Dr. Izmir University of Economics, Department of Industrial Systems Engineering To appear in: Engineering Optimization.

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Capacity and working time determination in a reservation system

Capacity and Working Time Determinationin a ReservationSystem

DenizTürselEliiyi,

Assoc. Prof. Dr.

Izmir University of Economics,

Department of Industrial Systems Engineering

To appear in: Engineering Optimization


Outline
Outline

  • Preliminaries

  • Practical importance and motivation

  • Problem definition

  • Complexity results

  • An efficient heuristic algorithm

  • Computational results

  • Conclusion and Future work


Fixed job scheduling

Job 3(w3)

Job 1(w1)

Job 2(w2)

r1

r2

d1

r3

d2

d3

Time

Fixed Job Scheduling

  • Assumptions

  • All parameters known

  • No more arrivals

  • A m/c can process at most one job at a time

  • A job can be processed by at most one machine at a time

  • All machines are eligible to process all jobs

  • Machines are available at all times.

  • n jobs

    • Ready time: rj

    • Deadline: dj

    • Processing time: pj= dj- rj

    • Weight: wj(k)

  • Pa :Set of available jobs in interval a.

  • Identical parallel machines

    • Cost : ck


Fixed job scheduling the operational problem ofjs
Fixed Job SchedulingThe Operational Problem (OFJS)


Capacity and working time determination in a reservation system

Fixed 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)


Fixed job scheduling the tactical problem tfjs
Fixed Job SchedulingThe Tactical Problem (TFJS)


Capacity and working time determination in a reservation system

Fixed Job SchedulingThe Tactical Problem (TFJS)

  • Fleet 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 jobs

    • Gupta et al. (1979)

    • Eliiyi (2004): O(n log n) algorithm for arbitrary ck


Fjs np hard generalizations
FJS: NP-hard generalizations

  • Working 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)

M/c k

job 2(w2)

r1

d1

r2

d2

r3

d3

p1

p2

p3

S

T


Practical importance
Practical Importance

Areas of use include all kinds of reservation systems:

  • Tactical capacity planning of aircraft maintenance personnel

  • Hotel reservation systems / Renting bungalows

  • Car rental

  • Textile workshops

  • Operating room scheduling in hospitals

  • Bus Driver Scheduling Problem

  • Earth-observing satellites

  • Automated manufacturing systems


Previous work
Previous Work

Working Time Constraints:

  • Fischetti M., Martello S., Toth P., 1989 : Tactical

  • Eliiyi D.T., Azizoğlu M., 2009, 2011 : Operational

    Spread Time Constraints:

  • Fischetti M., Martello S., Toth P., 1987 : Tactical

  • Eliiyi D.T., Azizoğlu M., 2006, 2011 : Operational

    Eligibility Constraints:

  • Kroon L.G. et al. 1995 : Operational

  • Kroon L.G. et al. 1997 : Tactical

  • Kolen A.J.W., Kroon L.G., 1991 : Operational

  • Kolen A.J.W., Kroon L.G., 1992 : Tactical

  • Eliiyi D.T., Azizoğlu M., 2009 : Operational

  • Eliiyi D.T., Korkmaz A.G., Çiçek A.E., 2009 : Operational

    Nice 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.


Motivation
Motivation

  • Capacity planning of a reservation system directly affects total profit

  • Existing 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 deadlines

    • Requires rescheduling

  • Studies handle operational and tactical problems separately

  • Integrated decision ofcapacity planning and scheduling

    • Significantly important in systems showing seasonal demand changes

    • Eliiyi (2010): An iterative approach thatuses the operational model is proposed for determining the best capacity expansion level ina sewing workshop


Problem definition
Problem Definition

  • Three simultaneous decisions in a reservation environment:

    • the capacity level of the system

    • job-machine assignments

    • working time for eachmachine

  • Applications:

    • Multi-server data transfersystem where the servers have unit-time operating costs

    • Seasonal workforce paid on an hourly basis

    • Travel agency renting hotel rooms for its customers

  • Objective: Maximize the net profit while determiningthe number of servers and their respective working times as well as the processed job subset.

  • Working time: A decision variable


Problem definition1
Problem Definition

  • n jobs

    • Ready time: rj

    • Deadline: dj

    • Processing time: pj= dj- rj

    • Weight: wj

  • m: upper bound (external or internal) on the number of identical parallel machines

    • Operating cost per unit time (or rental costs): ck

  • Pa :Set of available jobs in interval a.



Computational complexity
Computational Complexity

USING

THEN:

Equivalent to FJS problem with generalweights, NP-hard in the strong sense (Eliiyi, Azizoglu, 2009)

where


Polynomially solvable special cases
Polynomially Solvable Special Cases

  • Limited number of machines, identical operating costs:

    • Problem reduces to the operational FJS

    • Can be solved in O(mn log n) time by a MCNF formulation.

  • Single machine:

    • Problem reduces to the operational FJS with single machine

    • Can be solved in O(n) time by a shortest path fomulation.


A simple effective heuristic approach o n log n nm
A simple & effective heuristic approachO(n log n + nm)

(S0) Index the potential m machines in nondecreasing order of their 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 kthmachine with |A| jobs, resulting in ZSP (k) = objective function value and XSP(k) = scheduled job set

If ZSP(k) ≥ 0 and XSP(k) then

ZLB = ZLB + ZSP(k), XLB = XLBXSP(k), update A

else

Go to (S2)

If A =  go to (S2)

(S2) Solution: ZLB , XLB


Computational experiment
Computational Experiment

  • n = 100, 200, 500, 1000, 2000

  • rj ~ U(0,200)

  • pj~ U(4,10)

  • Three levels for job weights:

    • wj = pj , j

    • wj~ 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}, k

  • 10 problem instances for each setting: 300 instances

  • PC with 4 GB Ram and 1.8 GHz, Windows 7

  • IBM ILOG CPLEX 12.1 solver for optimal solutions





Observations
Observations

  • The 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.


Observations1
Observations

  • The 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 work
Conclusion and Future Work

  • A new strongly NP-hard problem in a reservation system where the jobs have fixedready times and deadlines:

    • The objective is to maximize the net profit from the processedjob 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 times

  • Potential research for related problems:

    • Problem with side constraints (spread time, eligibility)

    • Both fixed and operating costs for machines