# American Institute of Mathematical Sciences

• Previous Article
Performance analysis of transmission rate control algorithm from readers to a middleware in intelligent transportation systems
• NACO Home
• This Issue
• Next Article
A sufficient optimality condition for nonregular problems via a nonlinear Lagrangian
2012, 2(2): 333-355. doi: 10.3934/naco.2012.2.333

## Duration problem with multiple exchanges

 1 The University of Adelaide, Department of Applied Mathathematics, Adelaide, South Australia 5005, Australia 2 Institute of Mathematics and Computer Science, Wrocław University of Techechnology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland 3 Department of Business Administration, Aichi University, Nagoya Campus, Hiraike 4-60-6, Nakamura, Nagoya, Aichi 453-8777, Japan

Received  October 2011 Revised  May 2012 Published  May 2012

We treat a version of the multiple-choice secretary problem called the multiple-choice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the $m$--choice duration problem, there exists a sequence $(s_1,s_2,\ldots,s_m)$ of critical numbers such that, whenever there remain $k$ choices yet to be made, then the optimal strategy immediately selects a relatively best object if it appears at or after time $s_k$ ($1\leq k\leq m$). We also exhibit an equivalence between the duration problem and the classical best-choice secretary problem. A simple recursive formula is given for calculating the critical numbers when the number of objects tends to infinity. Extensions are made to models involving an acquisition or replacement cost.
Citation: Charles E. M. Pearce, Krzysztof Szajowski, Mitsushi Tamaki. Duration problem with multiple exchanges. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 333-355. doi: 10.3934/naco.2012.2.333
##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables," Dover, New York, 2000.  Google Scholar [2] K. Ano, Optimal selection problem with three stops, J. Oper. Res. Soc., 32 (1989), 491-504.  Google Scholar [3] D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Ratio prophet inequalities when the mortal has several choices, Ann. Appl. Probab., 12 (2002), 972-984. doi: 10.1214/aoap/1031863177.  Google Scholar [4] D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Two-choice optimal stopping, Adv. Appl. Probab., 36 (2004), 1116-1147. doi: 10.1239/aap/1103662960.  Google Scholar [5] D. Assaf and E. Samuel-Cahn, Simple ratio prophet inequalities for a mortal with multiple choices, J. Appl. Probab., 37 (4) (2000), 1084-1091. doi: 10.1239/jap/1014843085.  Google Scholar [6] Y. Chow, H. Robbins and D. Siegmund, "Great Expectations: The Theory of Optimal Stopping," Houghton Mifflin Co., Boston, 1971.  Google Scholar [7] E. Dynkin and A. Yushkevich, "Markov Process, Theorems and Problems," Plenum Press, New York, 1969.  Google Scholar [8] R. Eidukjavicjus, Optimalna ostanovka markovskoj cepi dvumia momentami ostanovki, Lit. Mat. Sb., 13 (1979), 181-183. Google Scholar [9] T. Ferguson, Who solved the secretary problem? Statist. Sci. 4 (1989), 282-296. doi: 10.1214/ss/1177012493.  Google Scholar [10] T. Ferguson, J. Hardwick and M. Tamaki, Maximizing the duration of owning a relatively best object, Strategies for sequential search and selection in real time (Amherst, MA, 1990). Contemp. Math., 125 (1992), 37-57. doi: 10.1090/conm/125/1160608.  Google Scholar [11] A. Frank and S. Samuels, On an optimal problem of Gusein-Zade, Stoch. Proc. Appl., 10 (1980), 299-317. doi: 10.1016/0304-4149(80)90013-7.  Google Scholar [12] J. P. Gilbert and F. Mosteller, Recognizing the maximum of a sequence, J. Am. Stat. Assoc., 61 (1966), 35-73.  Google Scholar [13] A. V. Gnedin, Best choice from the planar poisson process, Stoch. Proc. Appl., 111 (2004), 317-354. doi: 10.1016/j.spa.2003.12.005.  Google Scholar [14] A. V. Gnedin, Objectives in the best-choice problems, Sequential Analysis, 24 (2005), 1-11. doi: 10.1081/SQA-200056196.  Google Scholar [15] I. Gradshteyn and I. Ryzhik, "Tables of Integrals, Series, and Products," 6th Edition, Academic Press, San Diego, CA, 2000.  Google Scholar [16] S. M. Gusein-Zade, The problem of choice and the optimal stopping rule for a sequence of independent trials, Theory Probab. Appl., 11 (1966), 472-476. doi: 10.1137/1111050.  Google Scholar [17] G. W. Haggstrom, Optimal sequential procedures when more then one stop is required, Ann. Math. Stat., 38 (1967), 1618-1626. doi: 10.1214/aoms/1177698595.  Google Scholar [18] M. Hu, M. Tamaki and K. Ohno, A continuous time duration problem with an unknown number of options, Math. Japonica, 48 (1998), 233-237.  Google Scholar [19] A. Kurushima and K. Ano, On optimal stopping problems for possession-period maximization associated with Poisson arrival, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 2002), Sūrikaisekikenkyūsho Kōkyūroku No. 1306 (2003), 11-17. Available from: http://ci.nii.ac.jp/naid/110000167076/en  Google Scholar [20] A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations, J. Appl. Probab., 46 (2009), 402-414. doi: 10.1239/jap/1245676096.  Google Scholar [21] A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations, 13-th International Symposium on Dyn. Games and Appl., Wrocław 2008, 8 pages. Available from: http://home.imm.uran.ru/sector3/poland2008/isdg2008/prace/Kurushima121296568071203970.pdf  Google Scholar [22] R. Kühne and L. Rüschendorf, On optimal two-stopping problems, In"Limit theorems in probability and statistics. Fourth Hungarian colloquium on limit theorems in probability and statistics" ( Eds. I. Berkes et al.), Balatonlelle, Hungary, June 28-July 2, 1999, János Bolyai Mathematical Society, Budapest, II (2002), 261-271.  Google Scholar [23] A. Lehtinen, The best-choice problem with an unknown number of objects, Z. Oper.Res., 37 (1993), 97-106.  Google Scholar [24] V. V. Mazalov and M. Tamaki, Explicit solutions to the duration problem, Aichi Keiei Ronsyu, 147 (2003), 69-92. Google Scholar [25] V. V. Mazalov and M. Tamaki, An explicit formula for the limiting optimal gain in the full information duration problem, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 2002), Sūrikaisekikenkyūsho Kōkyūroku No. 1306 (2003), 217-222. Available from: http://ci.nii.ac.jp/naid/110000167076/en  Google Scholar [26] V. V. Mazalov and M. Tamaki, An explicit formula for the optimal gain in the full-information problem of owning a relatively best object, J. Appl. Probab., 43 (2006), 87-101. doi: 10.1239/jap/1143936245.  Google Scholar [27] V. V. Mazalov and M. Tamaki, Duration problem on trajectories, Stochastics, 79 (2007), 211-218. doi: 10.1080/17442500601072704.  Google Scholar [28] T. F. Móri, The random secretary problem with multiple choice, Ann. Univ. Sci. Budapest. de Rolando Eötvös Nominatae. Sect. Comput., 5 (1984), 91-102.  Google Scholar [29] A. G. Mucci, Differential equations and optimal choice problem, Ann. Stat., 1 (1973), 104-113. doi: 10.1214/aos/1193342386.  Google Scholar [30] A. G. Mucci, On a class of secretary problems, Ann. Probab., 1 (1973), 417-427. doi: 10.1214/aop/1176996936.  Google Scholar [31] M. L. Nikolaev, Zadacha vybora dvokh ob"ektov z minimal'nom sumarnym rangom, Izv. Vyss. Uchebn. Zaved. Mat., 166 (1976), 33-42. Google Scholar [32] M. L. Nikolaev, Ob odnom obobshchenii zadachi nailuchshego vybora, Teor. Veroyatn. Primen., 22, 191-194. on a generalization of the best choice problem, Theory Prob. Applications, 22 (1977), 187-190. doi: 10.1137/1122023.  Google Scholar [33] M. L. Nikolaev, Obobshchennyje posledovatelnyje procedury, Litov. Mat. Sb., 191 (1979), 35-44.  Google Scholar [34] M. Nikolaev, Optimal multi-stopping rules, Obozr. Prikl. Prom. Mat., 5 (1998), 309-348. Google Scholar [35] M. L. Nikolaev, On optimal multiple stopping of Markov sequences, Teor. Veroyatnost. i Primenen., 43 (1998), 374-382 in Russian; translation in Theory Probab. Appl., 43 (1999), 298-306.  Google Scholar [36] J. Petruccelli, On the best-choice problem when the number of observation is random, J. Appl. Probab., 20 (1983), 165-171. doi: 10.2307/3213731.  Google Scholar [37] Z. Porosiński, The full-information best choice problem with a random number of observations, Stoch. Proc. Appl., 24 (1987), 293-307. doi: 10.1016/0304-4149(87)90020-2.  Google Scholar [38] Z. Porosinski, On best choice problems having similar solutions, Stat. Probab. Lett., 56 (2002), 321-327. doi: 10.1016/S0167-7152(01)00201-2.  Google Scholar [39] J. Preater, On multiple choice secretary problem,, Math. Oper. Res., 19 (): 597.  doi: 10.1287/moor.19.3.597.  Google Scholar [40] E. Presman and I. Sonin, The best choice problem for a random number of objects, Theory Prob. Appl., 17 (1972), 657-668. doi: 10.1137/1117078.  Google Scholar [41] S. M. Ross, "Applied Probability Models with Optimization Applications," Holden-Day, San Francisco, Calif.-London-Amsterdam, 1970.  Google Scholar [42] M. P. Quine and J. S. Law, Exact results for a secretary problem, J. Appl. Probab., 33 (1996), 630-639. doi: 10.2307/3215345.  Google Scholar [43] M. Sakaguchi, Dowry problem and OLA policies, Rep. Stat. Appl. Res. Union Jap. Sci. Eng. JUSE, 25 (1978), 124-128. Google Scholar [44] M. Sakaguchi, Generalized secretary problems with three stops, Math. Japonica, 32 (1987), 105-122.  Google Scholar [45] S. M. Samuels, Secretary problems, In "Handbook of Sequential Analysis" (Eds. B. Ghosh and P. Sen), Marcel Decker, New York, 1991, 381-405.  Google Scholar [46] S. M. Samuels, Why do these quite different best-choice problems have the same solutions? Adv. Appl. Probab., 36 (2004), 398-416. doi: 10.1239/aap/1086957578.  Google Scholar [47] W. Stadje, On multiple stopping rules, Optimization, 16 (1985), 401-418. doi: 10.1080/02331938508843030.  Google Scholar [48] A. Suchwałko and K. Szajowski, Non standard, no information secretary problems, Sci. Math. Japonicae, 56 (2002), 443-456.  Google Scholar [49] K. Szajowski, Optimal choice problem of $a$-th object, Matem. Stos., in Polish, 19 (1982), 51-65.  Google Scholar [50] K. Szajowski, A two-disorder detection problem, Applicationes Mathematicae, 24 (1996), 231-241. Google Scholar [51] K. Szajowski and M. Tamaki, Shelf life of candidates in the generalized secretary problem, preprint, 2006, arXiv:0902.0232v2. Google Scholar [52] K. Szajowski, On a random number of disorders, Probab. Math. Statist., 31 (2011), 17-45.  Google Scholar [53] K. Szajowski, On stopping games when more than one stop is possible, In "Probability Methods in Discrete Mathematics" (Eds. V. F. Kolchin et al), Proceedings of the Fifth International Petrozavodsk Conference, May 2000. International Science Publishers, 2002, 57-72. Google Scholar [54] M. Tamaki, A secretary problem with double choice, J. Oper. Res. Soc. Jap., 22 (1979), 257-265.  Google Scholar [55] M. Tamaki, OLA policy and the best choice problem with random number of objects, Math. Japonica, 24 (1979), 451-457.  Google Scholar [56] M. Tamaki, C. E. Pearce and K. Szajowski, Multiple choice problems related to the duration of the secretary problem, Sūrikaisekikenkyūsho Kōkyūroku 1068 (1998), 75-86. Available from: http://ci.nii.ac.jp/naid/110001138655/en.  Google Scholar [57] J. Wilson, Optimal choice and assignment of the best $m$ of $n$ randomly arriving items, Stoch. Proc. Appl., 39 (1991), 325-343. doi: 10.1016/0304-4149(91)90086-R.  Google Scholar [58] M. Yasuda, On a stopping problem involving refusal and forced stopping, J. Appl. Probab., 20 (1983), 71-81. doi: 10.2307/3213721.  Google Scholar [59] M. Yasuda and K. Szajowski, Dynkin games and its extension to a multiple stopping model, Bulletin of the Japan Society for Industrial Mathematics 12 (2002), 17-28, in Japanese. Available from: http://www.math.s.chiba-u.ac.jp/ yasuda/accept/ouyou.pdf Google Scholar [60] A. J. Yeo and G. F. Yeo, Selecting satisfactory secretaries, Austral. J. Statist., 36 (1994), 185-198.  Google Scholar [61] G. F. Yeo, Duration of a secretary problem, J. Appl. Probab., 34 (1997), 556-558.  Google Scholar

show all references

##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables," Dover, New York, 2000.  Google Scholar [2] K. Ano, Optimal selection problem with three stops, J. Oper. Res. Soc., 32 (1989), 491-504.  Google Scholar [3] D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Ratio prophet inequalities when the mortal has several choices, Ann. Appl. Probab., 12 (2002), 972-984. doi: 10.1214/aoap/1031863177.  Google Scholar [4] D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Two-choice optimal stopping, Adv. Appl. Probab., 36 (2004), 1116-1147. doi: 10.1239/aap/1103662960.  Google Scholar [5] D. Assaf and E. Samuel-Cahn, Simple ratio prophet inequalities for a mortal with multiple choices, J. Appl. Probab., 37 (4) (2000), 1084-1091. doi: 10.1239/jap/1014843085.  Google Scholar [6] Y. Chow, H. Robbins and D. Siegmund, "Great Expectations: The Theory of Optimal Stopping," Houghton Mifflin Co., Boston, 1971.  Google Scholar [7] E. Dynkin and A. Yushkevich, "Markov Process, Theorems and Problems," Plenum Press, New York, 1969.  Google Scholar [8] R. Eidukjavicjus, Optimalna ostanovka markovskoj cepi dvumia momentami ostanovki, Lit. Mat. Sb., 13 (1979), 181-183. Google Scholar [9] T. Ferguson, Who solved the secretary problem? Statist. Sci. 4 (1989), 282-296. doi: 10.1214/ss/1177012493.  Google Scholar [10] T. Ferguson, J. Hardwick and M. Tamaki, Maximizing the duration of owning a relatively best object, Strategies for sequential search and selection in real time (Amherst, MA, 1990). Contemp. Math., 125 (1992), 37-57. doi: 10.1090/conm/125/1160608.  Google Scholar [11] A. Frank and S. Samuels, On an optimal problem of Gusein-Zade, Stoch. Proc. Appl., 10 (1980), 299-317. doi: 10.1016/0304-4149(80)90013-7.  Google Scholar [12] J. P. Gilbert and F. Mosteller, Recognizing the maximum of a sequence, J. Am. Stat. Assoc., 61 (1966), 35-73.  Google Scholar [13] A. V. Gnedin, Best choice from the planar poisson process, Stoch. Proc. Appl., 111 (2004), 317-354. doi: 10.1016/j.spa.2003.12.005.  Google Scholar [14] A. V. Gnedin, Objectives in the best-choice problems, Sequential Analysis, 24 (2005), 1-11. doi: 10.1081/SQA-200056196.  Google Scholar [15] I. Gradshteyn and I. Ryzhik, "Tables of Integrals, Series, and Products," 6th Edition, Academic Press, San Diego, CA, 2000.  Google Scholar [16] S. M. Gusein-Zade, The problem of choice and the optimal stopping rule for a sequence of independent trials, Theory Probab. Appl., 11 (1966), 472-476. doi: 10.1137/1111050.  Google Scholar [17] G. W. Haggstrom, Optimal sequential procedures when more then one stop is required, Ann. Math. Stat., 38 (1967), 1618-1626. doi: 10.1214/aoms/1177698595.  Google Scholar [18] M. Hu, M. Tamaki and K. Ohno, A continuous time duration problem with an unknown number of options, Math. Japonica, 48 (1998), 233-237.  Google Scholar [19] A. Kurushima and K. Ano, On optimal stopping problems for possession-period maximization associated with Poisson arrival, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 2002), Sūrikaisekikenkyūsho Kōkyūroku No. 1306 (2003), 11-17. Available from: http://ci.nii.ac.jp/naid/110000167076/en  Google Scholar [20] A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations, J. Appl. Probab., 46 (2009), 402-414. doi: 10.1239/jap/1245676096.  Google Scholar [21] A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations, 13-th International Symposium on Dyn. Games and Appl., Wrocław 2008, 8 pages. Available from: http://home.imm.uran.ru/sector3/poland2008/isdg2008/prace/Kurushima121296568071203970.pdf  Google Scholar [22] R. Kühne and L. Rüschendorf, On optimal two-stopping problems, In"Limit theorems in probability and statistics. Fourth Hungarian colloquium on limit theorems in probability and statistics" ( Eds. I. Berkes et al.), Balatonlelle, Hungary, June 28-July 2, 1999, János Bolyai Mathematical Society, Budapest, II (2002), 261-271.  Google Scholar [23] A. Lehtinen, The best-choice problem with an unknown number of objects, Z. Oper.Res., 37 (1993), 97-106.  Google Scholar [24] V. V. Mazalov and M. Tamaki, Explicit solutions to the duration problem, Aichi Keiei Ronsyu, 147 (2003), 69-92. Google Scholar [25] V. V. Mazalov and M. Tamaki, An explicit formula for the limiting optimal gain in the full information duration problem, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 2002), Sūrikaisekikenkyūsho Kōkyūroku No. 1306 (2003), 217-222. Available from: http://ci.nii.ac.jp/naid/110000167076/en  Google Scholar [26] V. V. Mazalov and M. Tamaki, An explicit formula for the optimal gain in the full-information problem of owning a relatively best object, J. Appl. Probab., 43 (2006), 87-101. doi: 10.1239/jap/1143936245.  Google Scholar [27] V. V. Mazalov and M. Tamaki, Duration problem on trajectories, Stochastics, 79 (2007), 211-218. doi: 10.1080/17442500601072704.  Google Scholar [28] T. F. Móri, The random secretary problem with multiple choice, Ann. Univ. Sci. Budapest. de Rolando Eötvös Nominatae. Sect. Comput., 5 (1984), 91-102.  Google Scholar [29] A. G. Mucci, Differential equations and optimal choice problem, Ann. Stat., 1 (1973), 104-113. doi: 10.1214/aos/1193342386.  Google Scholar [30] A. G. Mucci, On a class of secretary problems, Ann. Probab., 1 (1973), 417-427. doi: 10.1214/aop/1176996936.  Google Scholar [31] M. L. Nikolaev, Zadacha vybora dvokh ob"ektov z minimal'nom sumarnym rangom, Izv. Vyss. Uchebn. Zaved. Mat., 166 (1976), 33-42. Google Scholar [32] M. L. Nikolaev, Ob odnom obobshchenii zadachi nailuchshego vybora, Teor. Veroyatn. Primen., 22, 191-194. on a generalization of the best choice problem, Theory Prob. Applications, 22 (1977), 187-190. doi: 10.1137/1122023.  Google Scholar [33] M. L. Nikolaev, Obobshchennyje posledovatelnyje procedury, Litov. Mat. Sb., 191 (1979), 35-44.  Google Scholar [34] M. Nikolaev, Optimal multi-stopping rules, Obozr. Prikl. Prom. Mat., 5 (1998), 309-348. Google Scholar [35] M. L. Nikolaev, On optimal multiple stopping of Markov sequences, Teor. Veroyatnost. i Primenen., 43 (1998), 374-382 in Russian; translation in Theory Probab. Appl., 43 (1999), 298-306.  Google Scholar [36] J. Petruccelli, On the best-choice problem when the number of observation is random, J. Appl. Probab., 20 (1983), 165-171. doi: 10.2307/3213731.  Google Scholar [37] Z. Porosiński, The full-information best choice problem with a random number of observations, Stoch. Proc. Appl., 24 (1987), 293-307. doi: 10.1016/0304-4149(87)90020-2.  Google Scholar [38] Z. Porosinski, On best choice problems having similar solutions, Stat. Probab. Lett., 56 (2002), 321-327. doi: 10.1016/S0167-7152(01)00201-2.  Google Scholar [39] J. Preater, On multiple choice secretary problem,, Math. Oper. Res., 19 (): 597.  doi: 10.1287/moor.19.3.597.  Google Scholar [40] E. Presman and I. Sonin, The best choice problem for a random number of objects, Theory Prob. Appl., 17 (1972), 657-668. doi: 10.1137/1117078.  Google Scholar [41] S. M. Ross, "Applied Probability Models with Optimization Applications," Holden-Day, San Francisco, Calif.-London-Amsterdam, 1970.  Google Scholar [42] M. P. Quine and J. S. Law, Exact results for a secretary problem, J. Appl. Probab., 33 (1996), 630-639. doi: 10.2307/3215345.  Google Scholar [43] M. Sakaguchi, Dowry problem and OLA policies, Rep. Stat. Appl. Res. Union Jap. Sci. Eng. JUSE, 25 (1978), 124-128. Google Scholar [44] M. Sakaguchi, Generalized secretary problems with three stops, Math. Japonica, 32 (1987), 105-122.  Google Scholar [45] S. M. Samuels, Secretary problems, In "Handbook of Sequential Analysis" (Eds. B. Ghosh and P. Sen), Marcel Decker, New York, 1991, 381-405.  Google Scholar [46] S. M. Samuels, Why do these quite different best-choice problems have the same solutions? Adv. Appl. Probab., 36 (2004), 398-416. doi: 10.1239/aap/1086957578.  Google Scholar [47] W. Stadje, On multiple stopping rules, Optimization, 16 (1985), 401-418. doi: 10.1080/02331938508843030.  Google Scholar [48] A. Suchwałko and K. Szajowski, Non standard, no information secretary problems, Sci. Math. Japonicae, 56 (2002), 443-456.  Google Scholar [49] K. Szajowski, Optimal choice problem of $a$-th object, Matem. Stos., in Polish, 19 (1982), 51-65.  Google Scholar [50] K. Szajowski, A two-disorder detection problem, Applicationes Mathematicae, 24 (1996), 231-241. Google Scholar [51] K. Szajowski and M. Tamaki, Shelf life of candidates in the generalized secretary problem, preprint, 2006, arXiv:0902.0232v2. Google Scholar [52] K. Szajowski, On a random number of disorders, Probab. Math. Statist., 31 (2011), 17-45.  Google Scholar [53] K. Szajowski, On stopping games when more than one stop is possible, In "Probability Methods in Discrete Mathematics" (Eds. V. F. Kolchin et al), Proceedings of the Fifth International Petrozavodsk Conference, May 2000. International Science Publishers, 2002, 57-72. Google Scholar [54] M. Tamaki, A secretary problem with double choice, J. Oper. Res. Soc. Jap., 22 (1979), 257-265.  Google Scholar [55] M. Tamaki, OLA policy and the best choice problem with random number of objects, Math. Japonica, 24 (1979), 451-457.  Google Scholar [56] M. Tamaki, C. E. Pearce and K. Szajowski, Multiple choice problems related to the duration of the secretary problem, Sūrikaisekikenkyūsho Kōkyūroku 1068 (1998), 75-86. Available from: http://ci.nii.ac.jp/naid/110001138655/en.  Google Scholar [57] J. Wilson, Optimal choice and assignment of the best $m$ of $n$ randomly arriving items, Stoch. Proc. Appl., 39 (1991), 325-343. doi: 10.1016/0304-4149(91)90086-R.  Google Scholar [58] M. Yasuda, On a stopping problem involving refusal and forced stopping, J. Appl. Probab., 20 (1983), 71-81. doi: 10.2307/3213721.  Google Scholar [59] M. Yasuda and K. Szajowski, Dynkin games and its extension to a multiple stopping model, Bulletin of the Japan Society for Industrial Mathematics 12 (2002), 17-28, in Japanese. Available from: http://www.math.s.chiba-u.ac.jp/ yasuda/accept/ouyou.pdf Google Scholar [60] A. J. Yeo and G. F. Yeo, Selecting satisfactory secretaries, Austral. J. Statist., 36 (1994), 185-198.  Google Scholar [61] G. F. Yeo, Duration of a secretary problem, J. Appl. Probab., 34 (1997), 556-558.  Google Scholar
 [1] Andrew J. Steyer, Erik S. Van Vleck. Underlying one-step methods and nonautonomous stability of general linear methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2859-2877. doi: 10.3934/dcdsb.2018108 [2] Quan Zhou, Yabing Sun. High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021233 [3] Cong Qin, Xinfu Chen. A new weak solution to an optimal stopping problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4823-4837. doi: 10.3934/dcdsb.2020128 [4] Dong Li, Tong Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2011, 6 (4) : 681-694. doi: 10.3934/nhm.2011.6.681 [5] Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1 [6] Wenqing Bao, Xianyi Wu, Xian Zhou. Optimal stopping problems with restricted stopping times. Journal of Industrial & Management Optimization, 2017, 13 (1) : 399-411. doi: 10.3934/jimo.2016023 [7] Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431 [8] Yongki Lee, Hailiang Liu. Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 323-339. doi: 10.3934/dcds.2015.35.323 [9] Ryan Loxton, Qun Lin. Optimal fleet composition via dynamic programming and golden section search. Journal of Industrial & Management Optimization, 2011, 7 (4) : 875-890. doi: 10.3934/jimo.2011.7.875 [10] Mohammed Abdelghany, Amr B. Eltawil, Zakaria Yahia, Kazuhide Nakata. A hybrid variable neighbourhood search and dynamic programming approach for the nurse rostering problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2051-2072. doi: 10.3934/jimo.2020058 [11] Kevin Kuo, Phong Luu, Duy Nguyen, Eric Perkerson, Katherine Thompson, Qing Zhang. Pairs trading: An optimal selling rule. Mathematical Control & Related Fields, 2015, 5 (3) : 489-499. doi: 10.3934/mcrf.2015.5.489 [12] Haiying Liu, Wenjie Bi, Kok Lay Teo, Naxing Liu. Dynamic optimal decision making for manufacturers with limited attention based on sparse dynamic programming. Journal of Industrial & Management Optimization, 2019, 15 (2) : 445-464. doi: 10.3934/jimo.2018050 [13] Jakob Kotas. Optimal stopping for response-guided dosing. Networks & Heterogeneous Media, 2019, 14 (1) : 43-52. doi: 10.3934/nhm.2019003 [14] Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261 [15] Yan-An Hwang, Yu-Hsien Liao. Reduction and dynamic approach for the multi-choice Shapley value. Journal of Industrial & Management Optimization, 2013, 9 (4) : 885-892. doi: 10.3934/jimo.2013.9.885 [16] Yoshikazu Giga, Hirotoshi Kuroda. A counterexample to finite time stopping property for one-harmonic map flow. Communications on Pure & Applied Analysis, 2015, 14 (1) : 121-125. doi: 10.3934/cpaa.2015.14.121 [17] Hoi Tin Kong, Qing Zhang. An optimal trading rule of a mean-reverting asset. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1403-1417. doi: 10.3934/dcdsb.2010.14.1403 [18] Jingzhi Tie, Qing Zhang. Switching between a pair of stocks: An optimal trading rule. Mathematical Control & Related Fields, 2018, 8 (3&4) : 965-999. doi: 10.3934/mcrf.2018042 [19] Haibo Jin, Long Hai, Xiaoliang Tang. An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming. Journal of Industrial & Management Optimization, 2020, 16 (2) : 965-990. doi: 10.3934/jimo.2018188 [20] Zihui Liu, Xiangyong Zeng. The geometric structure of relative one-weight codes. Advances in Mathematics of Communications, 2016, 10 (2) : 367-377. doi: 10.3934/amc.2016011

Impact Factor: