Mathematical Programming

, Volume 1, Issue 1, pp 239–266 | Cite as

Problems and methods with multiple objective functions

Article

Abstract

LetA be a set of feasible alternatives or decisions, and supposen different indices, measures, or objectives are associated with each possible decision ofA. How can a “best” feasible decision be made? What methods can be used or experimented with to reach some decision?

The purpose of this paper is to attempt a synthesis of the main approaches to this problem which have been studied to date. Four different classes of approaches are distinguished: (1) aggregation of multiple objective functions into a unique function defining a complete preference order; (2) progressive definition of preference together with exploration of the feasible set; (3) definition of a partial order stronger than the product of then complete orders associated with then objective functions; and (4) maximum reduction of uncertainty and incomparability.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R.J. Aumann, “Utility theory without the completeness axiom,”Econometrica 30, No. 3 (1962).Google Scholar
  2. [2]
    R.J. Aumann, “Utility theory without the completeness axiom: a correction,”Econometrica 32, No. 1–2 (1964).Google Scholar
  3. [3]
    A. Auslender, “Méthodes numériques pour la résolution des problèmes d'optimisation avec contraintes,” Thèse: Faculté des Sciences de l'Université de Grenoble, 1969.Google Scholar
  4. [4]
    M. Barbut, “Note sur les ordres totaux à distance minimum d'une relation binaire donnée.”Revue Mathématiques et Sciences Humaines, No. 17.Google Scholar
  5. [5]
    R. Benayoun and J. Tergny, “Critères multiples en programmation mathématique: une solution dans le cas linéaire.”Revue Française d'Informatique et de Recherche Opérationnelle, 3ème année, no. V-2 (1969) 31–56.Google Scholar
  6. [6]
    R. Benayoun, B. Roy and B. Sussmann, “ELECTRE: une méthode pour guider le choix en présence de points de vue multiples,” SEMA (Metra International), Direction Scientifique, note de travail no. 49 (juin 1966).Google Scholar
  7. [7]
    R. Benayoun, J. de Montgolfier, J. Tergny and O. Larichev, “Linear programming with multiple objective functions: STEp Method (STEM),” 7th Mathematical Programming Symposium, The Hague, September 1970.Google Scholar
  8. [8]
    P. Bod, “Programmation linéaire dans le cas de plusieurs fonctions objectifs données simultanément (en hongrois).” Publications of the Mathematical Institute of the Hungarian Academy of Science (séries B) 8 (1963) 541–554.Google Scholar
  9. [9]
    G. Boldur, “Linear programming problems with complex decision conditions,” 7th Mathematical Programming Symposium, The Hague, September 1970.Google Scholar
  10. [10]
    G. Boldur, V. Ionescu and J. Stancu-Minasian, “Application de la théorie de l'utilité de la résolution des problèmes de programmation linéaire à plusieurs critères d'optimum,” Communication présentée à la session scientifique annuelle du Centre de Calcul Economique et Cybernétique Economique, Bucarest, Février 1969.Google Scholar
  11. [11]
    L.E. Briskin, “A method unifying multiple objective functions,”Management Science 12, no. 10 (1966).Google Scholar
  12. [12]
    P. Buffet, J.P. Gremy, M. Marc and B. Sussmann, “Peut-on choisir en tenant compte de critères multiples?: une méthode (ELECTRE) et trois applications.” METRA 6, no. 2 (1967).Google Scholar
  13. [13]
    A. Charnes and W. Cooper,Management models and industrial applications of linear programming, Vol. 1 (John Wiley and Sons, 1961).Google Scholar
  14. [41]
    A. Charnes, W. Cooper and Y. Ijiri, “Breakeven budgeting and programming to goals,”Journal of Accounting Research 1, no. 1 (1963).Google Scholar
  15. [15]
    A. Charnes, W. Cooper, R.J. Niehaus and A. Stedry, “Static and dynamic assignment models with multiple objectives, and some remarks on organization design,”Management Science 15, no. 8 (1969).Google Scholar
  16. [16]
    A. Charnes, W. Cooper, M.A. Keane, E.F. Snow and A.S. Walters, “A mixed goal programming model for PPBS in a consumer protection regulatory agency,” 7th Mathematical Programming Symposium, The Hague, September 1970.Google Scholar
  17. [17]
    B. Contini and S. Zionts, “Restricted bargaining for organizations with multiple objectives,”Econometrica 36, no. 2 (1968).Google Scholar
  18. [18]
    G. Debreu, “Topological methods in cardinal utility theory.” In:Mathematical methods in the social sciences, Eds. K.J. Arrow, S. Karlin and P. Suppes, 1959 (Stanford University Press, Stanford, 1960).Google Scholar
  19. [19]
    R.T. Eckenrode, “Weighting multiple criteria,”Management Science 12, no. 3 (1965).Google Scholar
  20. [20]
    H. Enzer, “On two nonprobabilistic utility measures for weapen systems,”Naval Research Logistics Quarterly 16, no. 1 (1969).Google Scholar
  21. [20bis]
    P. Fishburn,Decision and value theory (John Wiley and Sons, New York, 1970).Google Scholar
  22. [21]
    P. Fishburn, “Sensitivity of decisions to probability estimation errors: a reexamination,”Operations Research 16, no. 2 (1968).Google Scholar
  23. [22]
    P. Fishburn, “Preferences, summation, and social welfare functions,”Management Science 16, no. 3 (1969).Google Scholar
  24. [23]
    P. Fishburn,Utility theory for decision making (John Wiley and Sons, New York, 1970).Google Scholar
  25. [24]
    P. Fishburn, “Intransitive indifference in preference theory: a survey,”Operations Research 18, no. 2 (1970).Google Scholar
  26. [25]
    C. Fourgeaud, Lenclud and Ph. Sentis, “Critère de choix en avenir partiellement certain,”Revue Française d'Informatique et de Recherche Opérationnelle 14 (1968).Google Scholar
  27. [26]
    M. Freiner and L.S. Simon, “The evaluation of potential new product alternatives,”Management Science 13, no. 6 (1967).Google Scholar
  28. [27]
    S.I. Gass and T.L. Saaty, “The computational algorithm for the parametric objective function,”Naval Research Logistics Quarterly 2 (1955) 39–45.Google Scholar
  29. [28]
    A.M. Geoffrion, “Solving bicriterion mathematical programs,”Operations Research 15 (1967) 39–54.Google Scholar
  30. [29]
    A.M. Geoffrion, “Strictly concave parametric programming, parts I and II,”Management Science 13, 3 (1966) 244–253; and 5 (1967).Google Scholar
  31. [30]
    A.M. Geoffrion, “Proper efficiency and the theory of vector maximization,”Journal of Mathematical Analysis and Applications 22 (1968) 618–630.Google Scholar
  32. [31]
    A.M. Geoffrion, “Vector maximal decomposition programming,” 7th Mathematical Programming Symposium, The Hague, September 1970.Google Scholar
  33. [32]
    W. Gorman, “Separable utility and aggregation,”Econometrica 27 (1969).Google Scholar
  34. [33]
    M.D. Grigoriadis and K. Rittero, “A parametric method for semi-definite quadratic programs,”J. SIAM Control 7, 4 (1969) 559–577.Google Scholar
  35. [34]
    J.C. Holl, P. Leyrat and R. Benayoun, “Une modèle de gestion prévisionnelle des cadres d'entreprise.” Communication Congrès N.A.T.O., Porto, Juillet 1969.Google Scholar
  36. [35]
    Y. Ijiri,Management goals and accounting for control (North-Holland Publishing Company, Amsterdam, 1965).Google Scholar
  37. [36]
    E. Jacquet-Lagreze, “L'agrégation des opinions individuelles,”Informatique en Sciences Humaines 4 (1969).Google Scholar
  38. [37]
    E. Johnsen, “Studies in multiobjective decision models,” Suden Litteratur (1968).Google Scholar
  39. [38]
    R.F. Kirby, “A preferencing model for trip distribution,”Transportation Science 4, no. 1 (1970).Google Scholar
  40. [39]
    O. Larichev, “Queiques aspects des problèmes liés à la méthode des déclassements comparés,” SEMA (Metra International), Direction Scientifique, note de travail, no. 118 (mai 1970).Google Scholar
  41. [40]
    H. Le Boulanger and B. Roy, “L'entreprise face à la sélection et à l'orientation des projets de recherche: la méthodologic en usage dans le groupe Sema,”METRA 7, no. 4 (1968).Google Scholar
  42. [41]
    I.C. Lerman,Les bases de la classification automatique (Gauthier-Villars, Paris, 1970).Google Scholar
  43. [42]
    R.D. Luce and J.W. Tukey, “Simultaneous conjoint measurement: a new type of fundamental measurement,”Journal of Mathematical Psychology 1 (1964) 1–27.Google Scholar
  44. [43]
    R.D. Luce and H. Raiffa,Games and Decisions (John Wiley and Sons, New York, 1957).Google Scholar
  45. [44]
    C. Maier-Rothe and M.F. Stankard, Jr., “A linear programming approach to choosing between multi-objective alternatives,” 7th Mathematical Programming Symposium, The Hague, September 1970.Google Scholar
  46. [45]
    J.R. Miller, III, “The assessment of worth: a systematic procedure and its experimental validation,” Doctoral dissertation, Massachusetts Institute of Technology (June 1966).Google Scholar
  47. [46]
    J. de Montgolfier and J. Tergny, “Les décisions non totalement rationalisables. SEMA (Metra International),” Direction Scientifique, note de travail no. 109 bis, mars 1970).Google Scholar
  48. [47]
    J. Moore and N. Baker, “An analytical approach to scoring model design: application to research and development project selection,” Stanford University, The Institute of Management Sciences Meeting (1969).Google Scholar
  49. [48]
    F. Pechon, “Pour une intervention de l'ordinateur dans le processus de décision d'un jury,”METRA 9, no. 3 (1970).Google Scholar
  50. [49]
    J. Philip, “Algorithms for the vector maximization problem,” 7th Mathematical Programming Symposium, The Hague, September 1970.Google Scholar
  51. [50]
    H. Raiffa, “Preferences for multi-attributed alternatives,” The Rand Corp., Memorandum RM-5868-DOT Rc (April 1969).Google Scholar
  52. [51]
    Rousselot and Gastaut, “Techniques de préparation des décisions à caractère politique. Operational Research in the Public Section.” International Federation of Operational Research Societies, 3ème Session (1969).Google Scholar
  53. [52]
    B. Roy, “Pourquoi des approches multi-critères et comment? SEMA (Metra International),” Direction Scientifique, note de travail no. 108 (novembre 1969).Google Scholar
  54. [53]
    B. Roy, “Classement et choix en présence de points de vue multiples (la méthode ELECTRE).”Revue Française d'Informatique et de Recherche Opérationnelle, 2ème année, no. 8 (1968) 57–75.Google Scholar
  55. [54]
    B. Roy, “L'entreprise face à la sélection et l'orientation de projets de recherche et de développement - lère partie: Généralités.” Cahiers de l'Association Française pour l'Accroissement de la Productivité, no. 10 (février 1968).Google Scholar
  56. [55]
    B. Roy, “A propos de l'agrégation d'ordres complets: quelques considérations théoriques et pratiques.” Dans ”La Décision — Agrégation et dynamique des ordres de préférence,” Aix-en-Provence, 3–7 juillet 1967. Actes du Colloque du Centre National de la Recherche Scientifique, juillet 1967 (Editions du C.N.R.S.).Google Scholar
  57. [56]
    B. Roy,Algèbre moderne et théorie des graphes orientées vers les sciences économiques et sociale (ler tome: Dunod, Paris, 1969); 2ème tome: (1970).Google Scholar
  58. [56 bis]
    B. Roy et L. Bertier, “La méthode ELECTRE II (une méthode de classement en présence de critères multiples)”, Groupe METRA, Direction Scientifique, note de travail no. 142 (avril 1971).Google Scholar
  59. [57]
    S. Rudeanu, “Programmation bivalente à plusieurs fonctions économiques,”Revue Française d'Informatique et de Recherche Opérationnelle, 3ème année, no. V-2, (1969) 13–30.Google Scholar
  60. [58]
    D. Sapir, “Multi-objective linear programming,” Operations Research Center, College of Engineering, Berkeley, University of California (November 1966).Google Scholar
  61. [59]
    J. Saska, “Linear multiprogramming.”Economiko Matematiky Obzor 4 (1968) 359–373.Google Scholar
  62. [60]
    L. Savage,The foundations of statistics (John Wiley and Sons, New York, 1964).Google Scholar
  63. [61]
    Ph. Sentis, C. Fourgeaud and Lenclud, “Critère de choix en avenir partiellement incertain: note sur un algorithme de résolution,”Bulletin de Mathématiques Economiques, no. 1 (juin 1969).Google Scholar
  64. [62]
    M.F. Stankard, C. Maier-Rothe and S.K. Gupta, “Choosing between multiple objective alternatives: a linear programming approach,” Management Science Centre, University of Pennsylvania (December 1968).Google Scholar
  65. [63]
    M. Ternier, “L'étude pilote de préparation rationelle des décisions concernant les accidents de la route.” Bulletin du P.C.M. (Association Professionnelle des Ingénieurs des Ponts et Chaussées et des Mines) no. 5 (1969).Google Scholar
  66. [64]
    H. Terry, “Comparative evaluation of performance using multiple criteria,”Management Science 9, no. 3 (1963).Google Scholar
  67. [65]
    J. Vedder, “Planning problems with multidimensional consequences,”Journal of the American Institute of Planners (March 1970).Google Scholar
  68. [66]
    G. Zoutendijk,Methods of feasible directions (Elsevier, Amsterdam, 1960).Google Scholar

Copyright information

© North-Holland Publishing Company 1971

Authors and Affiliations

  • B. Roy
    • 1
  1. 1.Direction ScientifiqueGroupe METRAParisFrance

Personalised recommendations