Capacity and Working Time Determination in a Reservation System

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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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• Endüstri Mühendisliği Seminerleri, 12.10.2012

### 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
• 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)

• 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)

• 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

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

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.
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
• 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 Definition
• 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.
Computational Complexity

USING

THEN:

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

where

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 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
• 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
• 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.
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
• 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