This theory provides a method for combining evidence Since objective uncertainty has already been extensively from different sources without prior knowledge of their distri- explored in works on classic probability, the decision-making butions, it is also possible to assign probability values to sets of under subjective uncertainty is the subject of this article, possibilities rather than to single events only, and it is unnecessary which extends one of the formal models that deals with it, to divide all the probability values among the events, once the the Mathematical Theory of Evidence or Dempster-Shafer remaining probability should be assigned to the environment and not to the remaining events, thus modeling more naturally certain Theory.
However, it has some pitfalls caused by the A key issue in dealing with knowledge representation is non-natural embodiment of the uncertainty in the results. This is accomplished combined have a high degree of conflict, or when they are by means of a new rule of combination of bodies of evidence that disjoint regarding the more believed hypothesis. This counter- embodies in the numeric results the unknown belief and conflict intuitive behavior limits the range of application of this theory, among the evidence, naturally modeling the epistemic reasoning.
Another differential is that there is Subjective Uncertainty is the uncertainty that comes from no need to divide all the probability among the events, once the scientific ignorance, uncertainty in measurement, impossibility remaining probability is assigned to the environment and not to of confirmation or observation, censorship, or other knowledge the remaining events.
These two differentials allow this theory deficiency. The pro- 10 7 3 cedures adopted by all rules of combination, are independent of evidence order exchangeability. Mass Function C. Belief Function The basic probability assignment, or Mass Function, assigns some quantity of belief to the elements of the Frame of The Belief Function, Bel, measures how much the infor- Discernment. Note that the belief in C is the sum of the mass of belief of B, Summarizing: 0.
Belief Interval his beliefs among the possibilities, the remaining 0. Weight of Conflict sources [7]. It also allows the use of contradicting the intuition, and making some authors as [12] evidence with high values of conflict, making useful evidence state as not advisable the combination of evidence with weight otherwise useless. We then ask to 10 people which The numeric value expressed by the Lateo represents a one of these hypotheses would be the right answer.
Each one of mobile mass of belief, that, in the absence of unknown belief these 10 people assigned most of their belief to an hypothesis and conflict among the evidence, could be associated with disjoint from the choice of the others, and little of their belief any element, or combination of elements, of the frame of to a common hypothesis. Considering all people with the same discernment. Combining evidence with most of their beliefs assigned to hypothesis, increases the uncertainty about them, at the same disjoints hypotheses time increasing also the belief upon the individually lesser believed one, once all the people agreed about it.
This quantity particular [14]. From an VI. Combining evidence with high degree of conflict A decision making process can be affected by these flaws leading to erroneous decisions.
If it chooses not to do the study the lower part of the tree , it can either build a large plant, sequential decisions a small plant, or no plant. If the decision is to build, the market will be either favorable. The payoffs for each of the possible consequences are listed along the right-hand side.
State-of-nature node by viewing and solving a number 1 has 2 branches coming out of it. We also note that the probability is. Take one The rest of the probabilities shown in parentheses in Figure A. For exam- decision at a time. Of course, you would expect to find a high probability of a favorable market given that the research indicated that the market was good. Any market research study is subject to error. The probability is much higher,. Finally, when we look to the payoff column in Figure A.
With all probabilities and payoffs specified, we can start calculating the expected monetary value of each branch.
We begin at the end or right-hand side of the decision tree and work back toward the origin. When we finish, the best decision will be known.
Thus, if the survey results are favorable, a large plant should be built. Continuing on the upper part of the tree and moving backward, we compute the expected value of conducting the market survey. If the market survey is not conducted. Thus, building a small plant is the best choice, given the marketing research is not performed. If the sur- vey results are favorable, Getz should build the large plant; if they are unfavorable, it should build the small plant.
Using Decision Trees in Ethical Decision Making Decision trees can also be a useful tool to aid ethical corporate decision making. The decision tree illustrated in Example A8, developed by Harvard Professor Constance Bagley, provides guidance as to how managers can both maximize shareholder value and behave ethically.
The tree can be applied to any action a company contemplates, whether it is expanding operations in a developing country or reducing a workforce at home. But Smithson also calculates that pollutants emitted from the plant, if unscrubbed, could damage the local fishing industry. This could cause a loss of millions of dollars in income as well as create health problems for local inhabitants. Action outcome Is it ethical? Don't do it Does action maximize s company Ye No returns?
Is action Is it ethical not to take s Don't do it action? Weigh the Ye legal? No harm to shareholders No versus benefits to Do it, other stakeholders. Figure A. Now, say Smithson proposes building a somewhat different plant, one with pollution controls, despite a negative impact on company returns.
Ethical decisions can be quite complex: What happens, for example, if a company builds a pol- luting plant overseas, but this allows the company to sell a life-saving drug at a lower cost around the world? Does a decision tree deal with all possible ethical dilemmas? No—but it does provide managers with a framework for examining those choices. These techniques are especially useful for making decisions under risk.
Many decisions in research and development, plant and equipment, and even new buildings and structures can be analyzed with these decision models. Problems in inventory control, aggregate planning, mainte- nance, scheduling, and production control are just a few other decision table and decision tree applications. When decision trees are involved, commercial packages such as DPL, Tree Plan, and Supertree provide flexibility, power, and ease. POM for Windows will also analyze trees but does not have graphic capabilities.
Using Excel OM Excel OM allows decision makers to evaluate decisions quickly and to perform sensitivity analysis on the results. Program A. To calculate the EVPI, find the best outcome for each scenario. For details on how to use this software, please refer to Appendix IV. The probabilities for these Medium-sized three possibilities are. The net profit or loss for the medium-sized No shop 0 0 0 or small shops for the various market conditions are given in the fol- lowing table. Building no shop at all yields no loss and no gain.
What Probabilities. Solved Problem A. Develop a decision tree that illustrates her decision alternatives as to whether to stock 5, 6, or 7 Demand is 7 cases s cases. On Our Companion Web site, www. Identify the six steps in the decision process.
What is the equally likely decision model? Give an example of a good decision you made that resulted in a 4. Discuss the differences between decision making under certainty, bad outcome. Also give an example of a bad decision you made under risk, and under uncertainty. Why was each decision good or bad? What is a decision tree? Explain how decision trees might be used in several of the 10 OM The expected value criterion is considered to be the rational crite- decisions. Is this true? Is it rational to con- 7.
What is the expected value of perfect information? What is the expected value under certainty? When are decision trees most useful? Identify the five steps in analyzing a problem using a decision tree. Why are the maximax and maximin strategies considered to be optimistic and pessimistic, respectively? The annual returns will depend on both the size of her station and a number of marketing factors related to the oil industry and demand for gasoline.
Assume each outcome is equally likely, then find the highest EMV. Weiss, Inc. His managers believe that there is a probability of 0. Determine the EMV of each decision.
Johnson do? Long-term demand for the product group is somewhat predictable, so the manufacturer must be concerned with the risk of choosing a process that is inappropriate. Chen Chung is VP of operations. He can choose among batch manufacturing or custom manufacturing, or he can invest in group technology.
Demand will be classified into four compartments: poor, fair, good, and excellent. The company has the option of not expanding. It normally relies on departmental forecasts and preregistration records to determine how many copies of a text are needed. Preregistration shows 90 operations management students enrolled, but bookstore manager Curtis Ketterman has second thoughts, based on his intuition and some historical evidence.
Curtis believes that the distribution of sales may range from 70 to 90 units, according to the following probability model: Demand 70 75 80 85 90 Probability. One product is a cheese spread sold to retail outlets. Susan Palmer must decide how many cases of cheese spread to manufacture each month. The probability that demand will be 6 cases is. Unfortunately, any cases not sold by the end of the month are of no value as a result of spoilage.
How many cases should Susan manufacture each month? The other option is to build a pilot plant and then decide whether to build a complete facility. Lau estimates a chance that the pilot plant will work. Lau faces a dilemma. Should he build the plant? Should he build the pilot project and then make a decision? Help Lau by analyzing this problem. Because Kellogg orders 10, tests per order, this would mean that there is a. Joseph Biggs owns his own sno-cone business and lives 30 miles from a California beach resort.
The sale of sno-cones is highly dependent on his location and on the weather. Boyer enjoys biking, but this is to be a business endeavor from which he expects to make a living. He can open a small shop, a large shop, or no shop at all. Because there will be a 5-year lease on the building that Boyer is thinking about using, he wants to make sure he makes the correct decision. Boyer is also thinking about hiring his old marketing professor to conduct a marketing research study to see if there is a market for his services.
The results of such a study could be either favorable or unfavorable. Develop a decision tree for Boyer. Risk Analysis, Vol. Multicriteria decision making MCDM necessitates to incorporate uncertainties in the decision-making process. The major thrust of this article is to extend the framework proposed by Yager 1 for multiple decisionmakers and fuzzy utilities payoffs. In addition, the concept of expert credibility factor is introduced. The proposed approach is demonstrated for an example of seismic risk management using a heuristic hierarchical struc- ture.
A step-by-step formulation of the proposed approach is illustrated using a hypothetical example and a three-story reinforced concrete building. In this article, the second strategy will be ex- plored for SRM.
Ambiguity is related to both ordered weighted averaging OWA. The belief functions require to be dis- Yager 1 defined decision making under uncer- counted before aggregation using the credibility of tainty DMUU as a class of decision problems, when DMs. In these problems, the major issue assessment. Cases 2—4 deal with uncertainties asso- is the representation of knowledge about the vari- ciated to the state pij. For case 5, the utility value able x. These sets , and Dempster-Shafer DS theory random cases are articulated in such a way that case 1 is a sub- sets , have been suggested for representing knowl- set of case 2, and case 2 is a subset of case 3, and so edge.
Any appropriate use of a given uncertainty on. Therefore, case 5 becomes the most complex case formulation in DMUU may lead to the selection of that encompasses all previous cases. The determination of V Ai depends on the The remainder of the article is organized as fol- payoffs of various criteria xij involved and knowl- lows.
The basic information about OWA operators edge pij of expert s about the state of the vari- is provided in Section 2. Section 3 provides a discus- able. This section also discusses a method have different levels of credibility about their knowl- to determine the credibility factor dk for an assess- edge of the state or value of the payoffs. In this arti- ment of a DM to discount belief function knowl- cle, we have chosen five cases in the context of SRM, edge about the state pij.
Section 5 provides a hypothetical Bayesian theory, in which the valuation function example for risk management to explain the above or a representative value V Ai is an expected value, five cases. The final section provides alternative Ai , the basic probability axiom holds, the summary and conclusions.
The DMUU problem 2. Transformation of Linguistic Degree of Optimism maximum logic. The OWA oper- Optimistic 0. Moderately conservative 0. For a given vector of payoffs for n cri- optimizes the Disp function entropy. These calcu- teria x1 , x2 ,. Therefore, it The or ness characterizes the degree to which the becomes a problem of decision making under igno- aggregation is like an or or and. Therefore, when rance. Using ME- is the generation of weights.
Decision Making Under Uncertainty 81 These weights represent the fraction of pij dis- where it can be shown that: tributed among x1 , x2 , x3 , x4. A narrow belief interval represents more precise probabilities.
DS Rule of Combination 3. The DS rule of combination has inter- Xl , i. The general relation between bpa esting characteristics, e. Extensions of DS Rule of Combination results than the DS rule of combination, but provides more specific results than the Yager rule of combina- Zadeh 24 has highlighted the shortcoming of tion that translates the specificity of a case, e.
The DP combination. The ma- jor differences among various rules of combination The DP rule is commutative and associative but are handling of the conflicting mass and the closed not idempotent. Discounted Evidence ferable belief model of Smets 25 uses open world as- sumption, i. A detailed discussion on this topic plicitly that all sources of information are equally can be found in Dezert and Smarandache.
The bodies of evidence obtained from dif- article, two modified rules, Yager and Dubois and ferent sources i. Yager Yg Rule of Combination.
Thus, dence. The formulation provided in of evidence, which does not change joint evidence Equation 10 is used for discounting of evidence. The Yager rule of combination is commu- require discounting. However, for multiple DMs, the tative, but not idempotent and associative. Kiremid- jian 30 proposed a technique to determine dk based 3. In this study, the DS rule of combination using disjunctive consensus. The normalization factor is derived is assigned to disjunction to avoid normalization.
Basics 2 8 0. Fuzzy sets qual- factor. A trian- DM 4 0. One important feature of fuzzy numbers sets is 0. Further, assume that the experi- 4. However, in many cases, the payoffs xij are either Note: The values of A and B are positive, if negative numbers are elicited linguistically or vaguely known. This type of used, the corresponding min and max values have to be selected.
Payoff xij for Two Alternatives and Corresponding clude center of area, first of maximums, last of Four Criteria maximums, and middle of maximums.
Different de- xij C1 C2 C3 C4 fuzzification techniques extract different levels of information, and consequently are prone to rank A1 0. For each alternative and corresponding criterion, payoff values xij are provided in Table IV. The denomi- 5. Case 1 nator in the above equations accounts for the area The disjoint probabilities for case 1 are sum- under the TFN.
Once mean and standard deviation marized in Table V. Case 2 0. As a result, Fig. Step 2: Define decision-making attitude, e. The native. Be- Step 3: The value for each alternative can be calcu- tween the orness intervals of [0. It implies that A2 is a desired alternative. The variations in the two alter- A1 A2 native valuations with respect to orness are provided in Figs. Finally, Figs. For all orness values, for both DST Fig. The 1], V A1 is higher than V A2 , and A1 is the pre- base of the TFN gives information on the impreci- ferred alternative that refers to as rank reversal sion vagueness.
The difference between maximum Fig. Case 4 on most likely value. Thus, an alternative ranking is The multiple DMs in the previous example were desired to handle this type of problem. Each of the bpa as- This result indicates that A2 is the preferred signed by the DM are discounted with the credibility alternative. Decision Making Under Uncertainty 87 1. Sensitivity analysis based on 0. Most of these buildings are currently operational and are required to be fur- 6.
Background ther assessed and upgraded to minimize seismic dam- Reported damages from recent global earth- age and improve life safety. The building vulnerability is due to effects of an earthquake or series of earthquakes. Sensitivity analysis of orness values without consideration of credibility a DS rule of combination for alternative A1 b DS rule of combination for alternative A2 c Yager rule of combination for alternative A1 d Yager rule of combination for alternative A2 e DP rule of combination for alternative A1 f DP rule of combination for alternative A2.
The framework shown in mitigating the anticipated losses? The SRM entails Fig. The risk assessment shown in Fig.
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