MULTI-CRITERIA DECISION-MAKING IN MANAGEMENT UNDER CONDITIONS OF RISK AND Karel CHOBOT VSB Technical University of Ostrava, Ostrava, Czech Republic, EU, k.chobot@seznam.cz Abstract Decision-making processes cross all sequential managerial functions. The quality of outcomes from these processes is decisive for the prosperity and viability of the managed (entrepreneurial) entity (unit). It depends on the quality of the information available and its use for making a decision. For the sake of decision-making process quality, a manager should not particularly if the management of entrepreneurial entities is concerned rely only on their experience and intuition when making a decision. They should not omit the possibility of obtaining information for decision-making by means of existing theoretical methodological concepts, generalizing on an exact basis the formal-logical part of decision-making processes under various conditions. These conditions include situations when it is necessary to consider more criteria for decisionmaking with different weights, hierarchy and mutual relations, at various levels of certainty or risk and conditions when the probability of risk factor effects is only known up to the uncertainty, where the influence of randomly acting factors cannot be statistically estimated. In other words: The quality of managerial decision-making processes particularly those with strategic relations, long-term influence on the efficiency of the managed entity, connected with terms such as multi-criteria, risk and uncertainty can be significantly supported by using the theory of decision-making following, among other things, from the game theory situation analogy and strategic behaviour. The following text tries to specify in detail the above mentioned knowledge and document it in a simple example of investment decision-making. Keywords: Multi-criteria decision making, risk, uncertainity 1. INTRODUCTION Decision-making situations can be classified from numerous points of view [1]. The number of players and the presence of a random mechanism can be considered the most important risks and uncertainties in the decision-making theory. Decision-making situations with the presence of random players can be divided into conflicts where at least two intelligent players play (a player that strives to maximize its win unlike the random mechanisms or a non-intelligent player), and decision-making under uncertainty and risk. Decisionmaking under risk is considered a situation when an intelligent and a non-intelligent player play the game and the intelligent player knows the probability distribution according to which the random mechanism chooses its strategies. If the intelligent player does not know the probability distribution of random mechanism strategies, then it is decision-making under uncertainty. The essence of decision-making theory models is models of games against nature [2]. Strategies applied by a decision-maker are called alternatives. The opponent is the real states of the circumstances (random mechanisms) which influence the selected decision. The result of every alternative under the given states of the circumstances is a profit or loss (so called "payoff"), most frequently expressed in money. As many payoffs correspond to every state of the circumstances, as many alternatives of the solution are considered. In fact, it is difficult to reliably determine the amount of payoffs for individual alternatives in various states of the circumstances. The decision-maker usually estimates these themselves or by means of experts [1,3]. The success of the decision-making process depends on the level of accuracy of such estimates. The making of a certain decision (alternative) with a corresponding profit (payoff) is governed by the decision-maker s intention and approach to the issue or priorities of a company. It is not always suitable to select an alternative with the
maximum or minimum payoff (depending on whether costs or revenues are concerned). Aside from payoff amounts, the decision-maker is also influenced by the states of the circumstances at the time of decision implementation. 2. MULTI-CRITERIA MODEL OF DECISION-MAKING THEORY The majority of decision-making situations are characterized by many different criteria; therefore, it can be useful to define a decision-making model in the multi-criteria way. With multi-criteria models of decisionmaking theory it is necessary to distinguish situations with certainty, risk and uncertainty. Methods of multicriteria decision-making first require setting the weights of individual evaluation criterion. The more important the criterion, the higher the weight [1-2]. The importance of criteria is associated, e.g. with the goals in view of the company they reflect. The multi-criteria model of decision-making theory [3] shows a situation in which the decision-maker can select from several alternatives which are influenced by the implementation of one of many states of circumstances. The payoff of every alternative and the states of the circumstances are then determined by a multicomponent vector. It would be necessary to find such a function for the multicriteria model of decision-making theory which determines a comprehensive alternative evaluation according to all the criteria. To express such an analytical function is difficult, therefore it is possible to use criteria aggregation or comprehensive alternative evaluation as a help in multi-criteria decision-making. In the case of criteria aggregations the multi-criteria model of decision-making theory is converted to a single-criterion model. It means that every payoff vector is converted to a simple evaluation of alternatives corresponding to the payoffs of the single-criterion model. Then, a suitable method for solving the single-criterion model is used. We can select the best alternative by means of a suitable rule for its solution. The rule must best respect the characteristics of the decision-making situation. It is necessary to realize in all the cases that the aggregated payoff need not be practically interpretable. For the comprehensive alternative evaluation, such an alternative is selected as the best which includes the best results of individual alternatives for all the possible states of the circumstances. Every rule creates a vector of criterion values and the best alternative is the one evaluated with the best criterion value. 2.1. Solving multi-criteria decision-making models under risk There are two procedures for solving decision-making models under risk following from the Bayesian principle which lead to the same results. They are: the rule of maximum expected payoff the rule of minimum expected loss. These are decision-making situations where two players play; in a simplified case, one of them is intelligent and the other representing a random mechanism. [1] He states that an optimum strategy under risk is the behaviour which results in maximizing the mean value of win. A decision-making situation can be described by the following matrix: A a a,..., a... a 11... 1n m1 mn. The behaviour of the non-intelligent player is determined by the probability function defining columns of the matrix. It is represented by vector p p1, p2,..., p n. Its components show the probability that the non-intelligent player will choose the j th column of the matrix j th strategy. The intelligent player selects (according to the sentence on the mean value of the win) such strategy or line as optimum, for which the expression i a ij p j n j 1 takes the maximum value. People do not always decide in a way to maximize mean values of strategies, but to maximize utility from these strategies. Bernoulli s utility function u(x) follows from the fact that the utility increment is directly proportional to the increment of the x amount, but indirectly proportional to the x amount. The utility function is highly individual and expresses the decision-maker s attitude to the risk. They can have an aversion to risk, tend to risk or take a neutral position to risk. The decision-maker with aversion
to risk selects few risky alternatives. The decision-maker with a tendency to risk selects alternatives which have a chance for particularly good results, but which also bear a high risk of losses. The aversion and tendency to risk of a decision-maker with a neutral attitude are in balance. The dependence of utility vs. criterion will be concave with a decision-maker with an aversion to risk and vice versa, it will be convex with a decision-maker with the tendency to risk. The linear dependence of utility vs. criterion can be expected with the decision-maker having a neutral attitude to risk. It is necessary to realize that the utility function does not express a decision-maker s overall attitude to risk but only to a certain evaluation criterion. It follows that in multi-criteria decision-making the utility vs. risk function can have several convex, concave and linear parts depending on the set of individual criteria and decision-maker s attitude to these individual criteria. The rule of expected mean value and variance uses two basic characteristics of probability distribution of the risky alternative evaluation criterion, where the risky alternative variance comes out as a level of risk. The bigger the risky alternative variance, the higher the risk of this alternative. It is followed from the presumption that the decision-maker evaluates more the risky alternatives: With a higher expected value of the selected evaluation criterion and prefers them to alternatives with a lower expected value. With a lower risk and prefers them to alternatives with a higher risk. This rule does not enable complete arrangement of risky alternatives with regard to the selected evaluation criterion but only enables dividing the set of non-dominated alternatives (The alternative is called nondominated if there is no better alternative to it with the intent that some of the criteria values could be improved without worsening the values of other criteria. In contrast, an alternative is called dominated if there is an alternative which has all the values of criteria at least as good as the original one and at least one value is better.) which are better than the other alternatives. It is a suitable tool for the determination of risky alternatives in case their probability distributions of selected criterion are approximately symmetric. One of the tools for showing consequences or risky alternatives with regard to the selected evaluation criterion is decision-making matrices. They can only be used if risk factors influencing consequences of risky alternatives are of a discrete nature [1]. Lines of the decision-making matrix show individual alternatives for decision-making and columns show combinations of risk factor values with regard to the selected evaluation criterion. If risk factors are of continuous nature, or if the number of risk factors is higher than two, the decision-making matrix is no longer a suitable tool for showing consequences of risky alternatives. In such cases, it is recommended to use the simulation by the Monte Carlo method to determine the probability distribution of the selected evaluation criterion of risky alternatives. The Monte Carlo method is best performed by PC. A risky situation is created in every simulation step. After creating a sufficient number of risky situations and determining corresponding values of risky alternatives the computer software provides a result in the form of probability distribution of the given criterion (project). 2.2. Solving multi-criteria decision-making models under (complete) uncertainty These are decision-making situations in which a random mechanism acts as a non-intelligent player. However, in this case the intelligent player does not know the probability distribution of strategies of the random mechanism. There are several approaches to decision-making under uncertainty according to which it is possible to determine an optimum strategy for the intelligent player. The principle of indifference, so called Bernoulli Laplace s criterion. The intelligent player presumes that probabilities by means of which the non-intelligent player selects their strategies are equal. In this way, the decision-making situation is converted to decision-making under risk. Such strategy is selected as optimum which leads to maximizing the mean value. The minimax approach, so called Wald s criterion. According to this criterion the intelligent player selects such a strategy which will bring them the maximum minimum of profit regardless of the strategy selected by the non-intelligent player. This criterion is highly pessimistic. A rational decision-maker uses it if they want to ensure minimum risk for their decision. The loss minimax approach, so called Savage s criterion. In its essence, it is Wald s criterion when the intelligent player selects such a strategy as
optimum which will bring the maximum minimum of losses. The loss is defined as a profit when the strategy of the other player is known, decreased by the maximum from this profit. Minimax approach. When using this optimistic criterion, the player presumes that the non-intelligent opponent will select such a strategy which will maximize the intelligent player s performance. Medium optimism approach, so called Hurwicz s criterion. Such strategy is considered to be optimum which maximizes the sum of its value multiplied by an "optimism coefficient" and its addition to one. This coefficient takes values from (0.1) interval. The optimism criterion 0 is Wald s criterion. If the coefficient equals 1, it is the minimax approach. The optimism coefficient is highly subjective. Probability trees represent consequences of risky alternatives at a certain time sequence. Risk factors are expressed by means of nodes. Edges of these nodes represent potential results of risky alternatives including their probabilities. Consequences of risky alternatives are expressed by the ends of branches of the probability tree. Probability trees serve particularly for displaying discrete risk factors. Continuous random quantities must be approximated by means of several discrete values. The application of probability trees requires some simplification of issues solved such as the approximation of uncertain quantities by means of determination estimates. Decision-making trees, as in the previous case, can be implemented as a sequence of nodes and edges of an oriented chart. Decision-making trees serve as a tool for displaying a multi-stage decision-making process or they can be used for the determination of an optimum decision-making strategy. In multi-stage decision-making processes the optimum strategy represents a sequence of optimum decisions in individual stages of the decision-making process. When selecting an optimum strategy, we follow from the end of the decision-making tree. It is necessary to determine expected utilities (values) of an evaluation criterion for situation nodes and select an alternative with the highest utility of the respective criterion. We determine an optimum decision-making strategy by gradual repetition of these two steps for all parts of the decision-making tree from the end of the tree to its beginning. The weakness of decision-making trees is that they are mono-criterion. When assessing a decision-making situation in multi-criteria decision-making, it is necessary to create as many decision-making trees as the given criteria. 3. EXAMPLE OF INVESTMENT DECISION - MAKING A manager of a company should decide which of five projects should be implemented. He solves project implementation both for the conditions of risk, and the conditions of uncertainty. In this case, each of the projects is characterized by price, the 1 st milestone deadline, current work in progress, the end of project deadline and penalty for the failure to adhere to individual deadlines. Characteristics of individual projects are summarized in Table 1. The probability of success in project fulfilment was estimated on the basis of work in progress by an expert. The following probabilities were determined for individual projects: A - 0.02, B - 0.00, C - 0.16, D - 0.19 and E - 0.48. This means that project A has a 20% chance for timely fulfilment of the 1 st milestone with the given work in progress (%) and stated deadline for project termination (KT) without any penalty. A higher risk expressed by the rate of variance was accepted for the methods of decision-making under risk or for the method of expected mean value and variance. As seen from the results summarized in Table 2, project E has been evaluated as the most advantageous project for implementation according to all the decision-making methods. The situation is more complicated with the work in progress criterion, see Table 3. The project selection analysis for decision-making under risk is significantly influenced by the estimate of project success probability on the basis of the current work in progress. Project E seemed to be the most advantageous for implementation under the condition of risk. Project B was determined as the most suitable in decision-making under uncertainty. Project E ranked second and third in the selection. When project E was subjected to a more detailed examination under conditions of uncertainty in the case of work in progress criterion, second place was determined according to Wald s, Savage s and Hurwicz s criteria. In the case of Wald s criterion, we would count on the second biggest minimum of profit of all the projects. Savage s criterion would tell us that it will be the second least loss-making project based on work in progress. On the basis of Hurwicz s criterion with an optimism coefficient equal to 0.2, i.e. rather pessimistic
estimate of future work in progress, project E would be again the second best choice. Selecting project E as the second most successful project based on work in progress can also be substantiated by the fact that Bernoulli Laplace criterion was influenced by the same probability of success of individual projects (in this case the probability for Bernoulli Laplace criterion was calculated at the rate of 0.2 for all the projects). The maximax analysis was too optimistic. Table 4 was compiled on the basis of results stated for the term of handover criterion. Project E was selected as the most suitable one for the case of decision-making under risk. In the cases of decision-making under uncertainty dominated strategies were better. According to the term of handover it was not possible to say in the case of decision-making under uncertainty which of the stated projects should be implemented. Table 1. Characteristics of individual projects Project Price (EUR) 1 st milestone (KT) End (KT) A 93860 50 6 0 B 63160 48 6 0 C 114280 1 7 1 D 117700 2 7 1 E 226000 52 6 35 Work in progress (%) Penalty for failure to meet the deadline 0.05% per day of delay, max. 5% of the total price Table 2. Order of success in projects based on meeting or not the 1 st milestone 1 st milestone Order of project success Maximizing the mean value of the win 1271 V. 0 16337 II. 20965 I. 96489 RISK Expected mean value and variance 45 V. 0 45 II. 45 I. 53 Bernoulli - Laplace 18772 V. 12632 22856 II. 23540 I. 45200 Wald s 51623 V. 28422 79996 II. 88275 I. 146900 Savage s 6423 V. -16778 34796 II. 43075 I. 101700 II. Maximax 93860 V. 63160 114280 117700 I. 226000 Hurwicz s (alpha 0.2) 60070 V. 35370 86853 II. 94160 I. 162720 Table 3. Order of success in projects based on production work in progress Work in progress Order of project success Maximizing the mean value of the win 13 V. 0 84 II. 91 I. 327 RISK Expected mean value and variance 0.94 V. 0 2.96 II. 3.17 I. 4.43 Bernoulli - Laplace II. 38 I. 42 32 V. 30 34 Wald s V. 0 I. 100 1 1 II. 35 Savage s V. -42 I. 58-41 -41 II. -7 Maximax II. 190 I. 210 160 V. 150 170 Hurwicz s (alpha 0.2) 76 I. 144 65 V. 61 II. 89
Table 4. Order of success of projects based on the term of handover Handover Order of project success Maximizing the mean value of the win 0.13 V. 0 1.25 II. 1.52 I. 3.87 RISK Expected mean value and variance 0.16 V. 0 0.5 II. 0.55 I. 0.87 Bernoulli - Laplace Wald s Savage s Maximax Hurwicz s (alpha 0.2) Table 5. Order of success of projects based on criterion aggregation Criterion aggregation Order of project success II. Maximizing the mean value of the win 6148 V. 0 92690 124367 I. 521646 RISK Expected mean value and variance 22 V. 0 74 II. 80 I. 187 Bernoulli - Laplace 18772 V. 12632 22856 II. 23540 I. 45200 Wald s 13351 V. 6491 26469 II. 32072 I. 44494 - V. - Savage s 31849 38709-18731 II. -13128 I. -706 Maximax 93860 V. 63160 114280 II. 117700 I. 226000 Hurwicz s (alpha 0.2) 61656 V. 40492 79156 II. 83449 I. 153397 All the decision-making criteria were summarized individually on the basis of revenues and costs following from their fulfilment or non-fulfilment. Criteria aggregated in this way were processed. Project E was determined as the most suitable project for implementation based on the results in Table 5. Although in the case of the work in progress and term of handover criteria the results of project evaluations led to the nomination of a different project or to the impossibility to decide, project E was determined as the most suitable project based on results. 4. CONCLUSION In this thesis, the approach to multi-criterion decision-making under conditions of risk and uncertainty was shown. A procedure resulting in the preference of one project which should be implemented at the expense of the others was presented in a specific example. The whole situation would change, provided that the result should be the establishment of a project portfolio, in order that they would surpass revenues from one project. The situation with the work in progress criterion when projects in the second and third place were also evaluated, clearly showed the necessity for analyzing individual criteria (procedures) in overall relations and on the basis of the evaluator s preferences. LITERATURE [1] Fort, J., Dědina, J., Hrůzová, H.: Manažerské rozhodování, Ekopress,s.r.o., 2003. [2] Doubravová, H: Diplomová práce Využití teorie her při řešení konfliktních situací, Jihočeská Universita v Českých Budějovicích, 2007. [3] Brožová, H.: Vícekriteriální model teorie rozhodování, Odborné konference 2009.