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1325:{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}\hline &f_{1}(\cdot )&f_{2}(\cdot )&\cdots &f_{j}(\cdot )&\cdots &f_{q}(\cdot )\\\hline a_{1}&f_{1}(a_{1})&f_{2}(a_{1})&\cdots &f_{j}(a_{1})&\cdots &f_{q}(a_{1})\\\hline a_{2}&f_{1}(a_{2})&f_{2}(a_{2})&\cdots &f_{j}(a_{2})&\cdots &f_{q}(a_{2})\\\hline \cdots &\cdots &\cdots &\cdots &\cdots &\cdots &.\cdots \\\hline a_{i}&f_{1}(a_{i})&f_{2}(a_{i})&\cdots &f_{j}(a_{i})&\cdots &f_{q}(a_{i})\\\hline \cdots &\cdots &\cdots &\cdots &\cdots &\cdots &\cdots \\\hline a_{n}&f_{1}(a_{n})&f_{2}(a_{n})&\cdots &f_{j}(a_{n})&\cdots &f_{q}(a_{n})\\\hline \end{array}}}
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zero. If the difference exceeds the preference threshold it is considered to be significant. Therefore, the unicriterion preference degree is equal to one (the maximum value). When the difference is between the two thresholds, an intermediate value is computed for the preference degree using a linear interpolation.
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actions according to the decreasing values of their positive flow scores. The second one is obtained by ranking the actions according to the increasing values of their negative flow scores. The
Promethee I partial ranking is defined as the intersection of these two rankings. As a consequence, an action
368:
While it can be used by individuals working on straightforward decisions, the
Promethee & Gaia is most useful where groups of people are working on complex problems, especially those with several criteria, involving a lot of human perceptions and judgments, whose decisions have long-term impact.
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is being globally preferred by all the other actions. An ideal action would have a positive preference flow equal to 1 and a negative preference flow equal to 0. The two preference flows induce two generally different complete rankings on the set of actions. The first one is obtained by ranking the
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are respectively the indifference and preference thresholds. The meaning of these parameters is the following: when the difference is smaller than the indifference threshold it is considered as negligible by the decision maker. Therefore, the corresponding unicriterion preference degree is equal to
339:
Rather than pointing out a "right" decision, the
Promethee and Gaia method helps decision makers find the alternative that best suits their goal and their understanding of the problem. It provides a comprehensive and rational framework for structuring a decision problem, identifying and quantifying
405:
The applications of
Promethee and Gaia to complex multi-criteria decision scenarios have numbered in the thousands, and have produced extensive results in problems involving planning, resource allocation, priority setting, and selection among alternatives. Other areas have included forecasting,
348:
The basic elements of the
Promethee method have been first introduced by Professor Jean-Pierre Brans (CSOO, VUB Vrije Universiteit Brussel) in 1982. It was later developed and implemented by Professor Jean-Pierre Brans and Professor Bertrand Mareschal (Solvay Brussels School of Economics and
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4848:{\displaystyle {\begin{array}{cc}P_{j}(d_{j})=\left\{{\begin{array}{lll}0&{\text{if}}&|d_{j}|\leq q_{j}\\\\{\frac {|d_{j}|-q_{j}}{p_{j}-q_{j}}}&{\text{if}}&q_{j}<|d_{j}|\leq p_{j}\\\\1&{\text{if}}&|d_{j}|>p_{j}\\\end{array}}\right.\end{array}}}
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The descriptive approach, named Gaia, allows the decision maker to visualize the main features of a decision problem: he/she is able to easily identify conflicts or synergies between criteria, to identify clusters of actions and to highlight remarkable performances.
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When a preference function has been associated to each criterion by the decision maker, all comparisons between all pairs of actions can be done for all the criteria. A multicriteria preference degree is then computed to globally compare every couple of actions:
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4544:{\displaystyle {\begin{array}{cc}P_{j}(d_{j})=\left\{{\begin{array}{lll}0&{\text{if}}&|d_{j}|\leq q_{j}\\\\{\frac {1}{2}}&{\text{if}}&q_{j}<|d_{j}|\leq p_{j}\\\\1&{\text{if}}&|d_{j}|>p_{j}\\\end{array}}\right.\end{array}}}
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Promethee has successfully been used in many decision making contexts worldwide. A non-exhaustive list of scientific publications about extensions, applications and discussions related to the
Promethee methods was published in 2010.
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It has unique advantages when important elements of the decision are difficult to quantify or compare, or where collaboration among departments or team members are constrained by their different specializations or perspectives.
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It has particular application in decision making, and is used around the world in a wide variety of decision scenarios, in fields such as business, governmental institutions, transportation, healthcare and education.
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4294:{\displaystyle {\begin{array}{cc}P_{j}(d_{j})=\left\{{\begin{array}{lll}{\frac {|d_{j}|}{p_{j}}}&{\text{if}}&|d_{j}|\leq p_{j}\\\\1&{\text{if}}&|d_{j}|>p_{j}\\\end{array}}\right.\end{array}}}
1769:. Six different types of preference function are proposed in the original Promethee definition. Among them, the linear unicriterion preference function is often used in practice for quantitative criteria:
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4083:{\displaystyle {\begin{array}{cc}P_{j}(d_{j})=\left\{{\begin{array}{lll}0&{\text{if}}&|d_{j}|\leq q_{j}\\\\1&{\text{if}}&|d_{j}|>q_{j}\\\end{array}}\right.\end{array}}}
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M. Behzadian; R.B. Kazemzadeh; A. Albadvi; M. Aghdasi (2010). "PROMETHEE: A comprehensive literature review on methodologies and applications". European
Journal of Operational Research.
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1968:{\displaystyle P_{k}(x){\begin{cases}0,&{\text{if }}x\leq q_{k}\\{\frac {x-q_{k}}{p_{k}-q_{k}}},&{\text{if }}q_{k}<x\leq p_{k}\\1,&{\text{if }}x>p_{k}\end{cases}}}
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Promethee I is a partial ranking of the actions. It is based on the positive and negative flows. It includes preferences, indifferences and incomparabilities (partial preorder).
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Based on mathematics and sociology, the
Promethee and Gaia method was developed at the beginning of the 1980s and has been extensively studied and refined since then.
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J.P. Brans (1982). "L'ingénierie de la décision: élaboration d'instruments d'aide à la décision. La méthode PROMETHEE". Presses de l’Université Laval.
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Promethee II is a complete ranking of the actions. It is based on the multicriteria net flow. It includes preferences and indifferences (preorder).
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dimensional space. The GAIA plane is the principal plane obtained by applying a principal components analysis to the set of actions in this space.
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Prioritization – Determining the relative merit of members of a set of alternatives, as opposed to selecting a single one or merely ranking them.
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As a consequence the notion of preference function is introduced to translate the difference into a unicriterion preference degree as follows:
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of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be
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B. Mareschal; J.P. Brans (1988). "Geometrical representations for MCDA. the GAIA module". European
Journal of Operational Research.
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J.P. Brans & P. Vincke (1985). "A preference ranking organisation method: The PROMETHEE method for MCDM". Management
Science.
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The Promethee II complete ranking is obtained by ordering the actions according to the decreasing values of the net flow scores.
1546:. Of course, these differences depend on the measurement scales used and are not always easy to compare for the decision maker.
379:– The selection of one alternative from a given set of alternatives, usually where there are multiple decision criteria involved.
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According to the definition of the multicriteria preference degree, the multicriteria net flow can be disaggregated as follows:
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The prescriptive approach, named Promethee, provides the decision maker with both complete and partial rankings of the actions.
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be a consistent family of q criteria. Without loss of generality, we will assume that these criteria have to be maximized.
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its conflicts and synergies, clusters of actions, and highlight the main alternatives and the structured reasoning behind.
3478:{\displaystyle \phi _{k}(a_{i})={\frac {1}{n-1}}\displaystyle \sum _{a_{j}\in A}\{P_{k}(a_{i},a_{j})-P_{k}(a_{j},a_{i})\}}
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3905:{\displaystyle P_{j}(d_{j})={\begin{cases}0&{\text{if }}d_{j}\leq 0\\1&{\text{if }}d_{j}>0\end{cases}}}
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Deciding which resources are the best with the available budget to meet SPS quality standards (STDF –
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D-Sight for Academics: Collaborative Decision-Making (CDM) Software For Academics based on PROMETHEE
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In order to position every action with respect to all the other actions, two scores are computed:
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evaluations. Each line corresponds to an action and each column corresponds to a criterion.
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The positive and negative preference flows are aggregated into the net preference flow:
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Some uses of Promethee and Gaia have become case-studies. Recently these have included:
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It may require cleanup to comply with Knowledge (XXG)'s content policies, particularly
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Management, ULB Université Libre de Bruxelles), including extensions such as GAIA.
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is globally preferred to all the other actions while the negative preference flow
2630:{\displaystyle \phi ^{-}(a)={\frac {1}{n-1}}\displaystyle \sum _{x\in A}\pi (x,a)}
2536:{\displaystyle \phi ^{+}(a)={\frac {1}{n-1}}\displaystyle \sum _{x\in A}\pi (a,x)}
3296:{\displaystyle \phi (a_{i})=\displaystyle \sum _{k=1}^{q}\phi _{k}(a_{i}).w_{k}}
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The basic data related to such a problem can be written in a table containing
2137:{\displaystyle \pi (a,b)=\displaystyle \sum _{k=1}^{q}P_{k}(a,b)\cdot w_{k}}
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Decision situations to which the Promethee and Gaia can be applied include:
400:– Settling disputes between parties with apparently incompatible objectives
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is the difference between the evaluations of two actions for criterion
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394:– Putting a set of alternatives in order from most to least preferred
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Preference Ranking Organization METHod for Enrichment of Evaluations
4961:{\displaystyle P_{j}(d_{j})=1-e^{-{\frac {d_{j}^{2}}{2s_{j}^{2}}}}}
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User manual for Visual PROMETHEE, a guide to all PROMETHEE methods
181:"Preference ranking organization method for enrichment evaluation"
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Please help to demonstrate the notability of the topic by citing
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Multiple Criteria Decision Analysis: State of the Art Surveys
3553:, has the same interpretation as the multicriteria net flow
1454:{\displaystyle d_{k}(a_{i},a_{j})=f_{k}(a_{i})-f_{k}(a_{j})}
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is a positive non-decreasing preference function such that
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AMIA Systems: Visualize, Quantify and Optimize your flows
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2432:{\displaystyle \pi (a_{i},a_{j})+\pi (a_{j},a_{i})\leq 1}
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will be made between all the actions for each criterion:
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A major contributor to this article appears to have a
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2972:{\displaystyle \phi ^{-}(a_{i})\leq \phi ^{-}(a_{j})}
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2900:{\displaystyle \phi ^{+}(a_{i})\geq \phi ^{+}(a_{j})}
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Smart-Picker Pro implementing PROMETHEE and FLOWSORT
3589:but is limited to one single criterion. Any action
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may be too technical for most readers to understand
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3050:{\displaystyle \phi (a)=\phi ^{+}(a)-\phi ^{-}(a)}
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388:– Allocating resources among a set of alternatives
3060:Direct consequences of the previous formula are:
3181:{\displaystyle \sum _{a_{i}\in A}\phi (a_{i})=0}
5050:J. Figueira; S. Greco & M. Ehrgott (2005).
1720:{\displaystyle P_{k}:\mathbb {R} \rightarrow }
8:
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1664:{\displaystyle \pi _{k}(a_{i},a_{j})=P_{k}}
421:Selecting new route for train performance (
66:Learn how and when to remove these messages
27:Promethee & Gaia, tools for management
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305:Learn how and when to remove this message
287:Learn how and when to remove this message
271:, without removing the technical details.
232:Learn how and when to remove this message
118:Learn how and when to remove this message
323:geometrical analysis for interactive aid
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2339:{\displaystyle \pi (a_{i},a_{j})\geq 0}
406:talent selection, and tender analysis.
3753:{\displaystyle {\vec {\phi }}(a_{i})=}
2281:{\displaystyle \sum _{k=1}^{q}w_{k}=1}
5133:CoDE: PROMETHEE & GAIA Literature
269:make it understandable to non-experts
7:
3546:{\displaystyle \phi _{k}(a_{i})\in }
2288:. As a direct consequence, we have:
554:{\displaystyle F=\{f_{1},..,f_{q}\}}
493:{\displaystyle A=\{a_{1},..,a_{n}\}}
3489:The unicriterion net flow, denoted
2174:represents the weight of criterion
2808:will be as good as another action
1512:{\displaystyle d_{k}(a_{i},a_{j})}
25:
5123:D-Sight: PROMETHEE based software
3616:can be characterized by a vector
47:This article has multiple issues.
5016:Multi-criteria decision analysis
3113:{\displaystyle \phi (a_{i})\in }
2746:{\displaystyle \phi ^{-}(a_{i})}
2676:{\displaystyle \phi ^{+}(a_{i})}
248:
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98:. Please discuss further on the
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2037:Multicriteria preference degree
321:and its descriptive complement
145:may not meet Knowledge (XXG)'s
55:or discuss these issues on the
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500:be a set of n actions and let
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5138:PROMETHEE & GAIA web site
2640:The positive preference flow
3582:{\displaystyle \phi (a_{i})}
147:general notability guideline
2227:{\displaystyle w_{k}\geq 0}
5179:
1762:{\displaystyle P_{k}(0)=0}
154:reliable secondary sources
143:The topic of this article
5021:Ordinal Priority Approach
583:{\displaystyle n\times q}
5006:Decision-making software
325:are better known as the
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430:The mathematical model
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2201:. It is assumed that
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2194:{\displaystyle f_{k}}
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2167:{\displaystyle w_{k}}
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2025:{\displaystyle p_{j}}
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1998:{\displaystyle q_{j}}
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1539:{\displaystyle f_{k}}
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364:Uses and applications
96:neutral point of view
4871:
4567:
4317:
4106:
3928:
3799:
3764:
3620:
3593:
3557:
3493:
3313:
3206:
3127:
3067:
2988:
2911:
2839:
2812:
2785:
2757:
2714:
2687:
2644:
2548:
2454:
2353:
2295:
2238:
2205:
2178:
2151:
2049:
2009:
1982:
1776:
1731:
1678:
1561:
1523:
1467:
1351:
1342:pairwise comparisons
1336:Pairwise comparisons
597:
568:
504:
443:
5113:Italferr Case Study
5026:Pairwise comparison
4952:
4933:
398:Conflict resolution
386:Resource allocation
5054:. Springer Verlag.
4974:Promethee rankings
4958:
4938:
4919:
4845:
4843:
4836:
4541:
4539:
4532:
4291:
4289:
4282:
4080:
4078:
4071:
3902:
3897:
3770:
3750:
3606:
3579:
3543:
3475:
3474:
3386:
3293:
3292:
3178:
3152:
3110:
3047:
2969:
2897:
2825:
2798:
2770:
2743:
2700:
2673:
2627:
2626:
2607:
2533:
2532:
2513:
2429:
2336:
2278:
2224:
2191:
2164:
2134:
2133:
2022:
1995:
1965:
1960:
1759:
1717:
1661:
1536:
1509:
1451:
1322:
1320:
580:
551:
490:
327:Promethee and Gaia
149:
5163:Decision analysis
4954:
4797:
4732:
4725:
4619:
4493:
4428:
4421:
4369:
4243:
4191:
4184:
4032:
3980:
3877:
3847:
3773:{\displaystyle q}
3632:
3364:
3361:
3130:
2592:
2589:
2498:
2495:
1940:
1894:
1884:
1817:
1550:Preference degree
315:
314:
307:
297:
296:
289:
242:
241:
234:
216:
144:
128:
127:
120:
91:with its subject.
70:
16:(Redirected from
5170:
5100:
5099:
5095:
5089:
5088:
5084:
5078:
5077:
5073:
5067:
5066:
5062:
5056:
5055:
5047:
4967:
4965:
4964:
4959:
4957:
4956:
4955:
4953:
4951:
4946:
4932:
4927:
4918:
4896:
4895:
4883:
4882:
4854:
4852:
4851:
4846:
4844:
4840:
4837:
4833:
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4820:
4815:
4814:
4805:
4798:
4795:
4785:
4781:
4780:
4768:
4763:
4762:
4753:
4745:
4744:
4733:
4730:
4726:
4724:
4723:
4722:
4710:
4709:
4699:
4698:
4697:
4685:
4680:
4679:
4670:
4664:
4659:
4655:
4654:
4642:
4637:
4636:
4627:
4620:
4617:
4596:
4595:
4583:
4582:
4550:
4548:
4547:
4542:
4540:
4536:
4533:
4529:
4528:
4516:
4511:
4510:
4501:
4494:
4491:
4481:
4477:
4476:
4464:
4459:
4458:
4449:
4441:
4440:
4429:
4426:
4422:
4414:
4409:
4405:
4404:
4392:
4387:
4386:
4377:
4370:
4367:
4346:
4345:
4333:
4332:
4300:
4298:
4297:
4292:
4290:
4286:
4283:
4279:
4278:
4266:
4261:
4260:
4251:
4244:
4241:
4231:
4227:
4226:
4214:
4209:
4208:
4199:
4192:
4189:
4185:
4183:
4182:
4173:
4172:
4167:
4166:
4157:
4151:
4135:
4134:
4122:
4121:
4089:
4087:
4086:
4081:
4079:
4075:
4072:
4068:
4067:
4055:
4050:
4049:
4040:
4033:
4030:
4020:
4016:
4015:
4003:
3998:
3997:
3988:
3981:
3978:
3957:
3956:
3944:
3943:
3911:
3909:
3908:
3903:
3901:
3900:
3888:
3887:
3878:
3875:
3858:
3857:
3848:
3845:
3824:
3823:
3811:
3810:
3779:
3777:
3776:
3771:
3759:
3757:
3756:
3751:
3743:
3742:
3730:
3729:
3714:
3713:
3701:
3700:
3679:
3678:
3666:
3665:
3647:
3646:
3634:
3633:
3625:
3615:
3613:
3612:
3607:
3605:
3604:
3588:
3586:
3585:
3580:
3575:
3574:
3552:
3550:
3549:
3544:
3518:
3517:
3505:
3504:
3484:
3482:
3481:
3476:
3467:
3466:
3454:
3453:
3441:
3440:
3425:
3424:
3412:
3411:
3399:
3398:
3385:
3378:
3377:
3362:
3360:
3346:
3338:
3337:
3325:
3324:
3302:
3300:
3299:
3294:
3291:
3290:
3275:
3274:
3262:
3261:
3251:
3246:
3224:
3223:
3187:
3185:
3184:
3179:
3168:
3167:
3151:
3144:
3143:
3119:
3117:
3116:
3111:
3085:
3084:
3056:
3054:
3053:
3048:
3037:
3036:
3015:
3014:
2978:
2976:
2975:
2970:
2965:
2964:
2952:
2951:
2936:
2935:
2923:
2922:
2906:
2904:
2903:
2898:
2893:
2892:
2880:
2879:
2864:
2863:
2851:
2850:
2834:
2832:
2831:
2826:
2824:
2823:
2807:
2805:
2804:
2799:
2797:
2796:
2779:
2777:
2776:
2771:
2769:
2768:
2752:
2750:
2749:
2744:
2739:
2738:
2726:
2725:
2709:
2707:
2706:
2701:
2699:
2698:
2682:
2680:
2679:
2674:
2669:
2668:
2656:
2655:
2636:
2634:
2633:
2628:
2606:
2590:
2588:
2574:
2560:
2559:
2542:
2540:
2539:
2534:
2512:
2496:
2494:
2480:
2466:
2465:
2438:
2436:
2435:
2430:
2419:
2418:
2406:
2405:
2384:
2383:
2371:
2370:
2345:
2343:
2342:
2337:
2326:
2325:
2313:
2312:
2287:
2285:
2284:
2279:
2271:
2270:
2260:
2255:
2233:
2231:
2230:
2225:
2217:
2216:
2200:
2198:
2197:
2192:
2190:
2189:
2173:
2171:
2170:
2165:
2163:
2162:
2143:
2141:
2140:
2135:
2132:
2131:
2104:
2103:
2093:
2088:
2031:
2029:
2028:
2023:
2021:
2020:
2004:
2002:
2001:
1996:
1994:
1993:
1974:
1972:
1971:
1966:
1964:
1963:
1957:
1956:
1941:
1938:
1924:
1923:
1905:
1904:
1895:
1892:
1885:
1883:
1882:
1881:
1869:
1868:
1858:
1857:
1856:
1840:
1834:
1833:
1818:
1815:
1788:
1787:
1768:
1766:
1765:
1760:
1743:
1742:
1726:
1724:
1723:
1718:
1698:
1690:
1689:
1670:
1668:
1667:
1662:
1654:
1653:
1641:
1640:
1628:
1627:
1615:
1614:
1599:
1598:
1586:
1585:
1573:
1572:
1545:
1543:
1542:
1537:
1535:
1534:
1518:
1516:
1515:
1510:
1505:
1504:
1492:
1491:
1479:
1478:
1460:
1458:
1457:
1452:
1447:
1446:
1434:
1433:
1418:
1417:
1405:
1404:
1389:
1388:
1376:
1375:
1363:
1362:
1331:
1329:
1328:
1323:
1321:
1314:
1313:
1301:
1300:
1281:
1280:
1268:
1267:
1248:
1247:
1235:
1234:
1220:
1219:
1207:
1206:
1195:
1194:
1141:
1140:
1128:
1127:
1108:
1107:
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1075:
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1047:
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1022:
1021:
965:
964:
952:
951:
932:
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919:
918:
899:
898:
886:
885:
871:
870:
858:
857:
846:
845:
829:
828:
816:
815:
796:
795:
783:
782:
763:
762:
750:
749:
735:
734:
722:
721:
710:
709:
687:
686:
661:
660:
635:
634:
614:
613:
603:
589:
587:
586:
581:
560:
558:
557:
552:
547:
546:
525:
524:
499:
497:
496:
491:
486:
485:
464:
463:
310:
303:
292:
285:
281:
278:
272:
252:
251:
244:
237:
230:
226:
223:
217:
215:
174:
138:
137:
130:
123:
116:
112:
109:
103:
89:close connection
81:
80:
73:
62:
40:
39:
32:
21:
5178:
5177:
5173:
5172:
5171:
5169:
5168:
5167:
5153:
5152:
5109:
5104:
5103:
5097:
5096:
5092:
5086:
5085:
5081:
5075:
5074:
5070:
5064:
5063:
5059:
5049:
5048:
5044:
5039:
5001:Decision making
4997:
4989:
4981:
4976:
4934:
4909:
4887:
4874:
4869:
4868:
4842:
4841:
4835:
4834:
4824:
4806:
4799:
4792:
4786:
4783:
4782:
4772:
4754:
4736:
4734:
4727:
4714:
4701:
4700:
4689:
4671:
4665:
4660:
4657:
4656:
4646:
4628:
4621:
4614:
4603:
4587:
4574:
4565:
4564:
4538:
4537:
4531:
4530:
4520:
4502:
4495:
4488:
4482:
4479:
4478:
4468:
4450:
4432:
4430:
4423:
4410:
4407:
4406:
4396:
4378:
4371:
4364:
4353:
4337:
4324:
4315:
4314:
4288:
4287:
4281:
4280:
4270:
4252:
4245:
4238:
4232:
4229:
4228:
4218:
4200:
4193:
4186:
4174:
4158:
4152:
4142:
4126:
4113:
4104:
4103:
4077:
4076:
4070:
4069:
4059:
4041:
4034:
4027:
4021:
4018:
4017:
4007:
3989:
3982:
3975:
3964:
3948:
3935:
3926:
3925:
3896:
3895:
3879:
3872:
3866:
3865:
3849:
3842:
3832:
3815:
3802:
3797:
3796:
3786:
3762:
3761:
3734:
3721:
3705:
3692:
3670:
3657:
3638:
3618:
3617:
3596:
3591:
3590:
3566:
3555:
3554:
3509:
3496:
3491:
3490:
3458:
3445:
3432:
3416:
3403:
3390:
3369:
3350:
3329:
3316:
3311:
3310:
3282:
3266:
3253:
3215:
3204:
3203:
3197:
3159:
3135:
3125:
3124:
3076:
3065:
3064:
3028:
3006:
2986:
2985:
2956:
2943:
2927:
2914:
2909:
2908:
2884:
2871:
2855:
2842:
2837:
2836:
2815:
2810:
2809:
2788:
2783:
2782:
2760:
2755:
2754:
2730:
2717:
2712:
2711:
2690:
2685:
2684:
2660:
2647:
2642:
2641:
2578:
2551:
2546:
2545:
2484:
2457:
2452:
2451:
2445:
2410:
2397:
2375:
2362:
2351:
2350:
2317:
2304:
2293:
2292:
2262:
2236:
2235:
2208:
2203:
2202:
2181:
2176:
2175:
2154:
2149:
2148:
2123:
2095:
2047:
2046:
2039:
2012:
2007:
2006:
1985:
1980:
1979:
1959:
1958:
1948:
1935:
1926:
1925:
1915:
1896:
1889:
1873:
1860:
1859:
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1841:
1836:
1835:
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1681:
1676:
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1632:
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1606:
1590:
1577:
1564:
1559:
1558:
1552:
1526:
1521:
1520:
1496:
1483:
1470:
1465:
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1438:
1425:
1409:
1396:
1380:
1367:
1354:
1349:
1348:
1338:
1319:
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1305:
1292:
1290:
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1272:
1259:
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1186:
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1177:
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1132:
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1099:
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1038:
1025:
1023:
1013:
1010:
1009:
1001:
996:
991:
986:
981:
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970:
969:
956:
943:
941:
936:
923:
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903:
890:
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862:
849:
847:
837:
834:
833:
820:
807:
805:
800:
787:
774:
772:
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754:
741:
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726:
713:
711:
701:
698:
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678:
676:
671:
652:
650:
645:
626:
624:
605:
595:
594:
566:
565:
538:
516:
502:
501:
477:
455:
441:
440:
437:
432:
409:
404:
366:
346:
311:
300:
299:
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293:
282:
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265:help improve it
262:
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151:
139:
135:
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113:
107:
104:
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15:
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11:
5:
5176:
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5135:
5130:
5125:
5120:
5115:
5108:
5107:External links
5105:
5102:
5101:
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5068:
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3079:
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3018:
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2019:
2015:
1992:
1988:
1976:
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1962:
1955:
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313:
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85:
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45:
44:
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35:
26:
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14:
13:
10:
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4:
3:
2:
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2159:
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2100:
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2079:
2075:
2070:
2064:
2061:
2058:
2052:
2045:
2044:
2043:
2036:
2034:
2017:
2013:
1990:
1986:
1953:
1949:
1945:
1942:
1932:
1929:
1920:
1916:
1912:
1909:
1906:
1901:
1897:
1886:
1878:
1874:
1870:
1865:
1861:
1853:
1849:
1845:
1842:
1830:
1826:
1822:
1819:
1809:
1806:
1800:
1792:
1784:
1780:
1772:
1771:
1770:
1756:
1753:
1747:
1739:
1735:
1711:
1708:
1705:
1691:
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1471:
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1377:
1372:
1368:
1359:
1355:
1347:
1346:
1345:
1343:
1335:
1310:
1306:
1297:
1293:
1287:
1277:
1273:
1264:
1260:
1254:
1244:
1240:
1231:
1227:
1216:
1212:
1203:
1199:
1191:
1187:
1179:
1174:
1169:
1164:
1159:
1154:
1149:
1137:
1133:
1124:
1120:
1114:
1104:
1100:
1091:
1087:
1081:
1071:
1067:
1058:
1054:
1043:
1039:
1030:
1026:
1018:
1014:
1006:
1003:
998:
993:
988:
983:
978:
973:
961:
957:
948:
944:
938:
928:
924:
915:
911:
905:
895:
891:
882:
878:
867:
863:
854:
850:
842:
838:
825:
821:
812:
808:
802:
792:
788:
779:
775:
769:
759:
755:
746:
742:
731:
727:
718:
714:
706:
702:
691:
683:
679:
673:
665:
657:
653:
647:
639:
631:
627:
618:
610:
606:
593:
592:
591:
577:
574:
571:
562:
543:
539:
535:
532:
529:
526:
521:
517:
510:
507:
482:
478:
474:
471:
468:
465:
460:
456:
449:
446:
434:
429:
424:
420:
417:
413:
412:
411:
407:
399:
396:
393:
390:
387:
384:
381:
378:
375:
374:
373:
370:
363:
361:
357:
354:
350:
343:
341:
337:
333:
330:
328:
324:
320:
309:
306:
291:
288:
280:
270:
266:
260:
257:This article
255:
246:
245:
236:
233:
225:
214:
211:
207:
204:
200:
197:
193:
190:
186:
183: –
182:
178:
177:Find sources:
171:
167:
163:
159:
155:
148:
141:
132:
131:
122:
119:
111:
101:
97:
92:
90:
84:
75:
74:
69:
67:
60:
59:
54:
53:
48:
43:
34:
33:
30:
19:
5093:
5082:
5071:
5060:
5051:
5045:
4990:
4987:Promethee II
4982:
3488:
3305:
3198:
3190:
3059:
2980:
2639:
2446:
2146:
2040:
1977:
1673:
1553:
1463:
1339:
563:
438:
408:
403:
371:
367:
358:
355:
351:
347:
338:
334:
331:
326:
322:
318:
316:
301:
283:
274:
258:
228:
219:
209:
202:
195:
188:
176:
114:
105:
86:
63:
56:
50:
49:Please help
46:
29:
4979:Promethee I
435:Assumptions
158:independent
5037:References
5031:Preference
1340:At first,
192:newspapers
166:redirected
52:improve it
4915:−
4907:−
4770:≤
4712:−
4687:−
4644:≤
4466:≤
4394:≤
4216:≤
4005:≤
3860:≤
3723:ϕ
3694:ϕ
3687:…
3659:ϕ
3630:→
3627:ϕ
3561:ϕ
3529:−
3523:∈
3498:ϕ
3430:−
3380:∈
3366:∑
3355:−
3318:ϕ
3255:ϕ
3234:∑
3210:ϕ
3154:ϕ
3146:∈
3132:∑
3096:−
3090:∈
3071:ϕ
3034:−
3030:ϕ
3026:−
3008:ϕ
2992:ϕ
2949:−
2945:ϕ
2941:≤
2920:−
2916:ϕ
2873:ϕ
2869:≥
2844:ϕ
2723:−
2719:ϕ
2649:ϕ
2609:π
2601:∈
2594:∑
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2243:∑
2219:≥
2121:⋅
2076:∑
2053:π
1913:≤
1871:−
1846:−
1823:≤
1700:→
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1423:−
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692:⋅
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666:⋅
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640:⋅
619:⋅
575:×
329:methods.
277:June 2014
222:June 2014
156:that are
108:June 2014
100:talk page
58:talk page
18:PROMETHEE
5157:Category
4995:See also
4861:Gaussian
3876:if
3846:if
1939:if
1893:if
1816:if
423:Italferr
5011:D-Sight
4096:V-shape
3918:U-shape
3306:Where:
392:Ranking
344:History
263:Please
206:scholar
170:deleted
4557:Linear
2147:Where
1978:where
1674:where
377:Choice
208:
201:
194:
187:
179:
162:merged
4307:Level
3789:Usual
3760:in a
213:JSTOR
199:books
168:, or
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2907:and
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2005:and
1946:>
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439:Let
317:The
185:news
2835:if
416:WTO
267:to
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