Zakopane, 12.09.2016

1. Maciej Nowak, Tadeusz TrzaskalikProcedura interaktywna oparta na współczynnikach wymiany dla dwukryterialnego dyskregnego zadania programowania dynamicznego Zakopane, 12.09.2016

2. Agenda Introduction Trade-off analysis in discrete problems Multiobjective discrete dynamic programming Single criterion dynamic programming – algorithm 1 k-th optimal realization of the process – algorithm 2 Interactive procedure – algorithm 3 Numerical example Interactive procedure – algorithm 4 Final remarks

3. 2.Trade-off analysis in discrete problems • When solving a bi-criteria problem interactively, we can use one of two basic paradigms. • According to the first one, called the absolute paradigm, the decision maker discloses his/her preferences using criteria values. • In the second one, called the relative paradigm, the preferences are expressed using trade-offs.

4. 2.Trade-off analysis in discrete problems (cont.) • A trade-off is calculated for a pair of alternatives and a fixed pair of criteria • We shall assume that the alternatives are evaluated with respect to two criteria, whose values are maximized • We consider the alternative ai and discuss the consequences of the replacement of this alternative by the alternative aj

5. 2.Trade-off analysis in discrete problems (cont.) • An analysis of the trade-off for criteria (f1,f2) makes sense if the alternative ai is evaluated as better than the alternative aj with respect to one criterion, but as worse with respect to the other criterion • We assume that the evaluation of the alternative ai with respect to the criterion f1 is better than the respective evaluation of aj, while the situation is reversed with respect to the criterion f2: f1(ai) >f1(aj) and f2(ai) <f2(aj) • In this case, the trade-off determines the value of the increase of f2 per unit of decrease of f1 when ai is replaced by aj: (1)

6. 3.Multiobjective discretedynamic programming Notation • T– number of stages of the decision process under consideration, • yt– state of the process at the beginning of stage t (t=1,…,T), • Yt– finite set of process states at stage t, • xt– feasible decision at stage t, • Xt(yt) – finite set of decisions feasible at stage t, when the process was in state ytYt at the beginning of this stage, • yt+1= Ωt(yt, xt) – transition function in stage t • dt(yt, xt) – realization of the process at stage t • d=(d1,…,dt) realization of the process • D – (finite) set of all realizations of the process

7. 3.Multiobjective discretedynamic programming (cont.) • M – number of considered criteria • Ftm(dt) – m-th stage criterion function at stage t (m=1,…,M, t=1,…,T) • Fm(d) = ∑ Ftm(dt) - m-th multistage criterion function • F(d) = [F1,…,FK]’ – vector criteria function • ≥ - relation of domination • F(ď)≥ F(d) – realization ď dominates realization d • d* - non-dominated realization: • D* - set of efficient realizations. The problem of finding D* is called vector optimization problem.

8. 4.Single criterion dynamic programming • ft(dt) – stage criterion function • f(d)=∑ft(dt) – criterion function Algorithm 1 Optimality equations: • for t=T gT(yT)=max{fT(yT,xT): xTXT(yT)} (2) • for t=T-1,…,1 gt(yt)=max{ft(yt,xt)+gt+1(Ωt(yt,xt): xtXt(yt)}(3)

9. 5.k-th optimal realization of the process • D={d(1),d(2),…,d(K)} • f(d(i))>f(d(i+1)) Algorithm 2 k-th optimality equations: • for t=T, i=1,…,k gT(i)(yT)=maxi{fT(yT,xT): xTXT(yT)} (4) • for t=T-1,…,1 gt(i)(yt)=maxi{ft(yt,xt)+g(j)t+1(Ωt(yt,xt)), (5) xtXt(yt),j=1,…,i}

10. 6.Interactive procedure – algorithm 3 • Applying Algorithm 1 identify process realization which is optimal with respect to criterion f1, calculate the value of criterion f2 and assume this realization to be a proposal solution d0. • Present the proposal process realization d0 to the DM and ask him/her, whether he/she accepts it. If the answer is „yes” – go to (10). • Ask the DM to specify the extent to which the value of f1 can be decreased in order to improve the value of f2; let us assume that that f10 is the minimal value of f1, which is accepted by the DM. • Use Algorithm 2 to identify the set of process realizations D0 satisfying the constraint specified by the DM in step 3; calculate values of f2 and eliminate dominated solutions from D0. If D0=, inform the DM that it is not possible to improve f2 by decreasing f1 to the level f1* and go back to step (2). • For each process realization dkD0 calculate trade-offs using the following formula: (6)

11. 6.Interactive procedure – algorithm 3 (cont.) • Identify the process realization d’D0 maximizing the value of trade-off. Present the process realization d’ to the DM. • Ask the DM whether the value of f2 obtained for process realization d’ is acceptable. If the answer is „yes”, assume d0=d’ and go to (10). • Eliminate from the set D0 all process realizations for which the value of f2 is not greater that the value obtained for process realization d’. If the set D0, go to (6). • Inform the DM that it is not possible to obtain the solution for which the value of f2 is greater than f2(d’) and the value of f1 is greater than f10, assume d0 = d’ and go to (2). • Assume d0 to be the final solution.

12. 7.Numerical example [2; 37] [1; 9] 1 4 7 [2; 8] [5; 29] [7; 22] [4; 7] [10; 16] [7; 6] [14; 11] [10; 5] 2 5 8 [17; 7] [14; 4] [20; 4] [22; 2] [25; 2] [19; 3] [31; 1] 3 [25; 1] 6 9

13. 7.Numerical example (cont.) • The optimal solution with respect to criterion f1:d0 = [3; 6; 9]; f1(d*) = 56; f2(d*) = 2 • The DM does not accept the solution: the value of f2 is too low • The DM specifies the acceptable value of the first criterion f10=40 • Process realizations to be analyzed:

14. 7.Numerical example (cont.) • The new proposal d’ is presented to the DM:d’ = [2; 6; 9]; f1(d’) = 48; f2(d’) = 8 • The DM does not accept the solution: the value of f2 is too low

15. 7.Numerical example (cont.) • The new proposal d’ is presented to the DM:d’ = [2; 6; 8]; f1(d’) = 42; f2(d’) = 9 • The DM does not accept the solution: the value of f2 is too low • The DM is informed that it is not possible to improve f2 without decreasing f1 below f10=40 • The DM confirms decides to accept solutions for which f1(dk)30

16. 7.Numerical example (cont.) • The current proposal: d0 = [2; 6; 8]; f1(d0) = 42; f2(d0) = 9 • New solutions identified: • The DM accepts the new proposal.

17. 8. Interactive procedure – algorithm 4 • Applying Algorithm 1 identify process realization which is optimal with respect to criterion f1, calculate the value of criterion f2 and assume this realization to be a proposal realizationd0. • Present the proposal process realization d0 to the DM and ask him/her, whether he/she accepts it. If the answer is „yes” – go to (11). • Use Algorithm 2 to identify the set D0 grouping a specified number ofprocess realizations that are next after realization d0in the ranking with respect to criterion f1, eliminating dominated solutions. If D0=, inform the DM that it is not possible to improve f2 by decreasing f1 and go back to (2). • For each process realization dkD0 calculate trade-offs using the following formula: (7) 5. Identify the process realization d’D0 maximizing the value of trade-off. Present the process realization d’ to the DM.

18. 8.Interactive procedure – algorithm 4 (cont.) • Ask the DM whether the value of f1 obtained for process realization d’ is acceptable. If the answer is yes – go to (9). • Eliminate from the set D0 all process realizations for which the value of f1 is not greater that the value obtained for process realization d’. If D0, go to (6). • Inform the DM that it is not possible to obtain the solution for which the value of f1 is greater than f1(d’) and the value of f2 is greater than f2(d0) and ask the DM once more whether the value f1(d’) is acceptable. If the answer is no – go to 2. • Ask the DM whether the value of f2 obtained for process realization d’ is acceptable. If the answer is yes, assume d0 = d’ and go to (11). • Assume d0 = d’ and go to 3. • Assume d0 to be the final solution.

19. 9.Final remarks • The crucial point of the analysis is generating near optimal solution in dynamic programming problem (k-th optimal solutions. • The next step in our research will be considering problems involving more than 2 criteria and perform trade-off analysis • The method could be performed for stochastic dynamic problems. Anyway it could be much more complicated to find trade-offs.

21. About the journalMultiple Criteria Decision Making • Access: Print and online edition. • Multiple Criteria Decision Making (MCDM) journal was founded at the University of Economics in Katowice in 2011. • It is a continuation of a books edited by Tadeusz Trzaskalik and Tomasz Wachowicz since 2006. • MCDM is a double-blinded-reviewed annual. • In addition to theoretical and empirical research, the journal presents real-world applications and case studies and the software developments that support MCDM problems.

22. Databases • EBSCOhost: http://www.ebscohost.com (database: Business Source Complete), • Proquest: http://www.proquest.com/, • CEJSH: http://cejsh.icm.edu.pl/, • BazEkon: http://kangur.uek.krakow.pl/.

23. Scientific Board • Gregory Kersten, Canada • Carlos Romero, Spain • Roman Słowiński, Poland • Ralph Steuer,USA, • Tomasz Szapiro, Poland • Tadeusz Trzaskalik, Poland, • Gwo Hsiung Tzeng, Taiwan, • Edmundas Zavadskas, Lithuania.

24. Among revieweras • Rafael Caballero, Spain, • Petr Fiala, Czech Republic, • Luiz F. Autran M. Gomez, Brazil, • Josef Jablonsky Czech Republic, • Ignacy Kaliszewski,Poland • Włodzimierz Ogryczak,Poland, , • Jaroslav Ramik, Czech Republic, • Roman Słowiński, Poland, • Tetsuzo Tanino,Japan, • Leonas Ustinovichius Lithuania, • Lidija Zadnik-Stirn,Slovenia, • Kazimierz Zaras, Canada

25. Austria Bernhard Boehm Mikulaš Luptáčik Brazil Eppie E. Clark Luis Gomez Lanndon A. Ocampo Canada Belaid Aouni Christian Fonteix Hamdjatou Kane Laszlo Nandor Kiss Silvre Massebeuf Jean Marc Martel Kazimierz Zaraś Among authors

26. Czech Republik Helena Broz Jaroslav Charouz, Petr Fiala Jana Hanclova Josef Jablonsky Jana Kalcevova Jaroslav Ramík Tomas Subrt, Finland Kaisa Miettinen France Joseph Hanna Hongkong Sydney CK Chu James K. Ho S.S. Lam Christina SY Yuen Among authors

27. Italy Salvatore Greco Benedetto Matarazzo Lithuania Sigitas Mitkus Galina Ševčenko Eva Trink?nien? Vaidotas Šarka Edita Šarkiene, , Leonas Ustinovichius Malysia Azilah Anis Rafikul Islam Among authors

28. Poland Marcin Anholcer Tadeusz Banek Tomasz Błaszczyk Jakub Brzostowski Marek Chmielewski Cezary Dominiak Alicja Ganczarek-Gamrot Barbara Glensk Dorota Górecka Agnieszka Gładysz Barbara Gładysz Wit Jakuczun Miłosz Kadziński Ignacy Kaliszewski Bogumił Kamiński Leszek Klukowski Grzegorz Koloch Edward Kozłowski Lech Kruś Bogumiła Krzeszowska Dorota Kuchta Dominik Kudyba Tomasz Kuszewski Among authors

29. Jerzy Michnik Marek Miszczyński Dorota Miszczyńska Bogusław Nowak Maciej Nowak Włodzimierz Ogryczak Dmitry Podkopaev Ewa Roszkowska Jaideep Roy Paweł Rotter Agata Sielska Sebastian Sitarz Jan Skorupiński Andrzej Skulimowski Roman Słowiński Olena Sobotka Honorata Sosnowska Jarosław Stańczyk Małgorzata Szałucka Tomasz Szapiro Krzysztof Targiel Eugeniusz Toczyłowsk Tadeusz Trzaskalik Małgorzata Trzaskalik Wyrwa Among authors

30. Grażyna Trzpiot Justyna Urbańska Tomasz Wachowicz Paweł Wieszała Maciej Wolny Ireneusz Wyrwa Sylwia Zawadzka Mateusz Zawisza Piotr Zielniewicz Russia E.M. Askerov ,M.A. Emelin Dmitry Kochin Igor Rudynsky Slovenia Petra Grošelj Lidija Zadnik Stirn Among authors

31. Spain Rafael Caballero José Enrique Padilla García Trinidad Gómez Mónica Hernández María Amparo León Francisca Garcí Lopera Julína Molina Beatriz Rodríguez, Fátima Pérez Taiwan Yuh-Wen Chen Moussa Larbani Gwo Hsiung Tseng Among authors

32. Tunezja Habibi Chabchoub Abdelbasset Essabri Mariem Gzara Sahnoun Imen Mouna Mezghani Frikha Hela Moalla Loukil Taicir Turkey Írem Uçal Sari USA Vincent Y. Blouin, Markus Hirschberger Brian J. Hunt Denise Mills Yue Qi, Mark Ridgley Ralph Steuer Margaret M. Wiecek Among authors