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#2002 Operational Research Society Ltd. All rights reserved. 0160-5682/02 $15.00 www.palgrave-journals.com/jors

Corrigendum

Hirotsu N and Wright M (2002). Using a Markov process model of an association football match to determine the optimal timing of substitution and tactical decisions Journal of the Operational Research Society (2002) 53, 1174. doi:10.1057/palgrave.jors.2601414

This paper contains the following errors, for which we apologise. 1. Equations (8) and (9) should be replaced by

Om ðT þ dtÞ 8 Am Om ðT Þdt þ Om ðT Þ > > > > > > < ¼ max > Am Os ðT Þdt þ Os ðT Þ > > > > > :

Os ðT þ dtÞ 8 As Os ðT Þdt þ Os ðT Þ > > < ¼ max > A O ðT Þdt þ Om ðT Þ > : s m

not changing tactics ðtactics S ! SÞ changing tactics ðtactics S ! MÞ ð11Þ

3. Equations (12) and (13) should be replaced by

no substitution should yet be made with time T remaining now making the substitution ðM ! SÞ at time T remaining ð8Þ

Om ðT þ dtÞ 8 Am Om ðT Þdt þ Om ðT Þ > > > > >

Ð TDT > > > þ T DT 1DT As Os ðtÞdt > 1 2 : þ Os ðT  DT1  DT2 Þ

not changing tactics ðM ! MÞ changing tactics ðM ! SÞ ð12Þ

Os ðT þ dtÞ ¼ As Os ðT Þdt þ Os ðT Þ

a substitution has already been made and there is time T remaining

ð9Þ

2. Equations (10) and (11) should be replaced by

Os ðT þ dtÞ 8 As Os ðT Þdt þ Os ðT Þ not changing tactics > > > > ðS ! SÞ > > > Ð > < T A O ðtÞdt changing tactics TDT1 s s ¼ max Ð T DT1 > þ T DT DT Am Om ðtÞdt ðS ! MÞ > > 1 2 > > > þOm ðT  DT1  DT2 Þ > > : ð13Þ

Om ðT þ dtÞ 8 Am Om ðT Þdt þ Om ðT Þ not changing tactics > > < ðtactics M ! MÞ ¼ max > A O ðT Þdt þ Os ðT Þ changing tactics > : m s ðtactics M ! SÞ ð10Þ

Reference 1 Hirotsu N and Wright M (2002). Using a Markov process model of an association football match to determine the optimal timing of substitution and tactical decisions. J Opl Res Soc 53: 88–96. University of Lancaster

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