Approximation and Visualization of Interactive Decision Maps Short course of lectures Alexander V. Lotov Dorodnicyn Computing Center of Russian Academy.
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Approximation and Visualization of Interactive Decision Maps Short course of lectures Alexander V. Lotov Dorodnicyn Computing Center of Russian Academy of Sciences and Lomonosov Moscow State University
Lecture 3. Interactive Decision Maps technique Plan of the lecture 1.Few words concerning the psychology of decision making 2.Why VISUALIZATION is needed? 3.Requirements to visualization 4.Why decision maps satisfy the requirements? 5.Interactive Decision Maps -- main concepts 6.Feasible Goals Method 7.The first real-life application of the Pareto frontier visualization for a large number of criteria
Features of human thinking It is well known that a human being cannot simultaneously handle very many objects (it has been experimentally proven that the number of objects should not exceed the magical number seven plus or minus two). This statement is true in the case of letters, words, sentences, paragraphs and even alternatives. Thus, a human being cannot think simultaneously about hundreds or thousands objective points of the Pareto frontier approximation.
Famous experiment There is a sequence of alternatives A1, A2, A3,…,Ak, which are described by two criteria: an important and a less important. A2 is much better than A1 in the sense of the second criterion, but a bit worse in the sense of the second criterion. The same for A3 and A2, etc. A human being usually prefers A2 to A1, A3 to A2, etc., i.e.
Simple methods used by people 1.Instead of several criteria, people often use only two of them. By this, they simplify the problem. 2.Another simple method. Even having a long list of possible alternatives in front, a human being may be unable to find the best one. Instead, he/she often somehow selects a small number of alternatives from the list and compares them. Though the most preferred one may be selected from this short list, such an approach results in missing most of the Pareto optimal solutions; one of them may be better than the selected one.
Mental models Studies in the field of human psychology have resulted in a fairly complicated picture of a human decision making process. In particular, the concept of a mental model of reality that provides the basis of decision making has been proposed and experimentally proven. The mental models have at least three levels that describe the reality in different ways: Level of logical thinking, Level of images, and Level of subconscious processes. Preferences are connected to processes of all three levels. A conflict between the mental levels may be one of the reasons of the well-known non-transitive behavior of people (both in experiments and in real-life situations).
Coordinating the levels A large part of human mental activities is related to the coordination of the levels. To settle the conflict between the levels, time is required. Psychologists assure that sleeping is used by the brain to coordinate the mental levels. (Compare with the proverb: ``The morning is wiser than the evening''). In his famous letter on making a tough decision, Benjamin Franklin advised to spend several days to make a choice. It is known that group decision and brainstorming sessions are more effective if they last at least two days.
Coordinating the levels in MOO problems Thus, to settle the conflict between the levels of one’s mental model in finding a balance between different objectives in a multi-objective optimization problem, he/she needs to keep information on the problem in his/her brains for a sufficiently long time. Such opportunity is provided by a posteriori methods. The absence of method-related time pressure is an important advantage of them. In contrast, other approaches require fast answers to the questions on preferences.
VISUALIZATION — why it is needed? Visualization is a transformation of symbolic data into geometric information that must aid in the formation of mental picture of the symbolic data.
Effectiveness of visualization As a proverb says: “A picture is worth a thousand words.” Another estimate of the role of visualization is given by Wierzbicki and Nakamori in their book “Creative Space”, Springer, Berlin, 2005. To their opinion, “a picture is worth a ten thousands words.”
Important feature: Visualization can influence all levels of human thinking!
Visualization in a posteriori methods Since visualization can influence all levels of thinking, visualization of the Pareto frontier can support the mental search for the most preferred Pareto optimal solution. Such a search may be logically imperfect, but acceptable for all levels of human mentality. Visualization of the Pareto frontier can be repeated as many times as the DM wants to and can last as long as needed. The question that we consider is how visualization can be effectively used in the field of multi-objective optimization, namely, in a posteriori methods.
Requirements that must be satisfied by a visualization technique To be effective, a visualization technique must satisfy some requirements, which include (i) simplicity, that is, visualization must be immediately understandable, (ii) persistence, that is, the graphs must linger in the mind of the beholder, and (iii) completeness, that is, all relevant information must be depicted by the graphs.
Visualization of the Pareto frontier in the bi-objective case satisfies the requirements
Tradeoff information given by the graph of the Pareto frontier in a clear form can be accessed immediately and, if the frontier is not too complicated, can linger in the mind of the DM for a relatively long time. In any case, the one can explore the curve as long as needed. Finally, the graph provides full information on the objective values and their mutual dependence along the tradeoff curve. Thus, visualization of the Pareto frontier in the bi-objective case satisfies the requirements.
Visualization of the Pareto frontier in the case of three objectives The question is: what kind of visualization can be used in MOO problems in the case of more than two criteria?
Decision maps A decision map is a collection of bi- criterion slices of the Pareto frontier. It is a tool for visualization of the Pareto frontier in the case of three criteria.
A question The question arises: Why not to display the three dimensional graphs instead of the decision maps? Let us consider a dynamic model of long-time development of a national economy with such objectives as economic growth C*, maximal (in time) pollution level Z* and maximal (in time) unemployment U*.
An example of a three dimensional graph Objective points 1-10 are depicted in the graph
An example of the related decision map Tradeoff curves (bi-objective slices of the EPH) are given in the graph; values of U* are given near the associated tradeoff curves.
Comparison of the three-dimensional graphs with the decision maps One can find much more information on the tradeoff curves than in the three-dimensional graph (except the tradeoff curve corresponding to U* = 0 that is actually given in the three-dimensional graph since it belongs to the slice provided by the plane U* = 0). In the decision map, other slices are seen in a clear form, too. The tradeoff rates are visible at any point of the tradeoff curves. One can easily estimate the zones with qualitatively different tradeoff rates. One can see the total tradeoff between any two points that belong to the same tradeoff curve as well as the total tradeoff between any two points that have the same value of C* or Z*.
Properties of the decision maps First of all, let us note that tradeoff curves do not intersect in a decision map (though they may sometimes coincide). Due to this, they look like contour lines of topographic maps. Indeed, a value of a third objective (related to a particular tradeoff curve) plays the role of the height level related to a contour line of a topographic map.
For example, a tradeoff curve describes such combinations of values of the first and the second objectives that are feasible for a given constraint imposed on the value of the third objective (like ‘places lower, than...'' or ‘places higher, than...''). Moreover, one can easily estimate which values of the third objective are feasible for a given combination of the first and of the second objectives (like ‘height of this particular place is between...''). If the distance between tradeoff curves is small, this could mean that there is a steep ascent or descent in values, that is, a small move in the plane of two objectives is related to a substantial change in the value of the third objective.
Why the requirements are met? Thus, decision maps are fairly similar to topographic maps. Thus, one can use topographic maps for the evaluation of the effectiveness of the visualization given by decision maps. Topographic maps have been used for a long time and educated people usually understand information displayed without difficulties. Experience of application of topographic maps shows that they are simple enough to be immediately understood, persistent enough not to be forgotten by people after their exploration is over, and complete enough to provide information on the levels of particular points in the map. The analogy between decision maps and topographic maps asserts that decision maps satisfy the above requirements.
Old methods for constructing the decision maps The simplest approach to constructing and displaying a decision map is based on a direct conversion of the multi-objective problem to a series of bi-objective problems: one has to select any two objectives f i and f j to be minimized. Then, the following bi-objective problem is considered Here the values of must be given. Various methods for constructing bi-objective Pareto frontier can be used for solving this problem. As a result, one obtains the Pareto frontier for two selected objectives for given constraints on other objectives. One has to note, however, that this is true only in the case if the plane does indeed cut the Pareto frontier.
To get a full picture of the Pareto frontier in this way, one needs to specify a grid in the space of m-2 objectives and solve a bi-objective problem for any point of the grid. For example, in case of five objective functions with 10 possible constraints for any of m-2=3 objectives, we have to construct the Pareto frontier for 1000 bi-objective problems, which naturally is a tremendous task. In addition, one has to visualize these 1000 bi-objective frontiers somehow. For this reason, researchers usually apply this approach only in the case of m=3 or, sometimes, m=4 and restrict to a dozen bi-objective problems. As to visualization, one usually can find a convenient way for the displaying about a dozen tradeoff curves. One can prove that tradeoff curves obtained in this way do not intersect for m=3 (though they can touch each other).
Interactive Decision Maps (IDM) technique The IDM technique provides interactive display of the decision maps for three to seven criteria. It is based on visualization of decision maps by overlapping bi-criterion slices of the EPH approximated in advance. The IDM technique is the development of ideas of the NISE method for the case of more than two criteria.
The principle differences between the IDM technique and the NISE method are: the EPH is approximated instead of the Pareto frontier; slices of the Pareto frontier are visualized as frontiers of bi-criterion slices of the EPH.
The requirements to visualization are met by the IDM since it displays the decision maps, which satisfy such requirements. In particular, the IDM-produced decision maps are complete since they can display information on Pareto frontiers with any desired precision. On-line calculation of decision maps in the IDM technique provides additional options. One can: change objectives located on axes, change the number of tradeoff curves on decision maps, zoom the picture, and change graphic features of the display such as the color of the background, colors of the slices, etc.
Since the IDM technique satisfies the above requirements, the decision maker can consider the decision maps mentally as long as needed and select the most preferred objective point from the whole set of Pareto optimal solutions. If some features of the graphs are forgotten, he/she can look at the frontier again and again.
Feasible Goals Method (FGM) A preferred point of the Pareto frontier can be specified by the user directly on computer display. It is considered as a goal. Since the goal is feasible, a Pareto-optimal decision can be found that results in the goal. It can be found by solving a special single- criterion optimization problem.
Single-criterion optimization used in the FGM Let us consider the multi-objective minimization problem y = f (x)→min, x X. Let y’ be the goal point specified by the user. Since the EPH was approximated, the graph of the Pareto frontier is an approximation, too. It means that the goal point specified by the user is only approximately feasible or Pareto-optimal.
For this reason, we consider the goal y’ as a “reference point”, which is used in a special goal- related optimization problem for computing the related Pareto-optimal decision: where y = f (x), x X, the values are small positive parameters. Since the goal y’ is close to the Pareto frontier, the Pareto-optimal feasible goal y* = f (x*) is close to the goal specified by the user. Such a procedure was proposed by A.Wierzbicki in 1981.
Feasible Goals Method/Interactive Decision Maps technique Combination of such idea with the IDM results in the FGM/IDM technique.
The first real-life application of Pareto frontier visualization: specification of national goals in the USSR
In the 1980s, the State Planning Agency of the Soviet Union has started the design of a computer- based decision support system for a long-term national economy planning. The DSS was based on application of the hierarchical system of dynamic input-output models that described the development of the USSR economy with different levels of aggregation.
The model The most aggregated (upper level) model described the possible long-time development of the USSR production system. It was a dynamic input-output model, in the framework of which 17 production industries were considered. Yearly outputs of production industries were defined to be equal to the sum of investments, exports minus imports, final consumption, and raw materials consumption of all other industries (balance economic model of the type proposed by the Nobel prize winner W.Leontiev).
Balance of the product of the i-th industry, where x i is the product, parameters a ij are coefficients of direct production consumption, is y i the final consumption. Decision variables x i and inv i are non-negative. Coefficients γ i describe labor requirements per unit of production, values x i max are constraints imposed by capital limitations. The parameters of the model depend on time.
Decision variables, which are time-dependent, are related to production of industries (it resulted in distribution of the labor force among industries), allocation of investments between industries, etc. Feasible labor statistics were projected, and the capacities of production industries depended upon investment. The delay between investment and the resulting capacity growth was given, its value depended on the industry.
The criteria The upper level model was used for identification of long- term national social-economic goals for a time period of 15 years. For the particular goals, values of several performance indicators of the national economic system were used. They included consumption of several population groups, development of health care and educational systems, etc. Seven criteria were considered in total
In the early variant of the DSS, officials of the State Planning Agency had to identify the particular goals on the basis of their experience, without any computer support. As a rule, the goals identified by them were not feasible. Then, some optimization software was used to compute a feasible criterion vector as close as possible to the identified goal. Usually, such feasible criterion vectors were distant from the goals identified by the officials, and so it turned out that the identified goals had nothing to do with the reality. Experience – 1
The officials were disappointed with such results. After several attempts, they refused to use the DSS. It seems that the officials regarded such results as undermining their prestige since it might be attributed to their incompetence. It was clear for the DSS developers that an additional decision support tool was needed to help officials to identify goals that are close to feasible performance vectors. They decided to use the FGM and the related software.
At that time (at the very beginning of the 1980s) the State Planning Agency was unable to get personal computers, and so the authors had to approximate the feasible goals set by the mainframe computer and to print out a large number of graphs that contained collections of bi-criterion tradeoff curves. The officials of the State Planning Agency studied the album of feasible social-economic goals by themselves with his help. Indeed, it turned out that the album of graphs worked sufficiently well without any support from the authors. In total, the officials used the album for more than three years. Experience – 2
Perestrojka in the USSR destroyed the planning system, and application of the DSS was halted.