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# Capacity and Working Time Determination in a Reservation System - PowerPoint PPT Presentation

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 Determinationin a ReservationSystem

DenizTürselEliiyi,

Assoc. Prof. Dr.

Izmir University of Economics,

Department of Industrial Systems Engineering

To appear in: Engineering Optimization

• Preliminaries

• Practical importance and motivation

• Problem definition

• Complexity results

• An efficient heuristic algorithm

• Computational results

• Conclusion and Future work

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

• Processing time: pj= dj- rj

• Weight: wj(k)

• Pa :Set of available jobs in interval a.

• Identical parallel machines

• Cost : ck

Fixed Job SchedulingThe Operational Problem (OFJS)

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 SchedulingThe Tactical Problem (TFJS)

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

• Working 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

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

Working Time Constraints:

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

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

• 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.

• 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

• 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

• n jobs

• 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.

USING

THEN:

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

where

• 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 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

• 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

• 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.

• 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.

• 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