Decision Theory in Operations Research

Decision making can be defined as a process of selecting the best strategy from various alternatives. Decision making is a rational analysis of the problem which helps the management to reach some decision. In other words, Decision making is a process which results in the selection from a set of alternatives (course of action) that course of action which is considered to meet the objectives of the decision making more satisfactorily than others. Decision theory refers to the process which is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision. It is concerned with identifying the best decision to take, assuming an ideal decision maker who is fully informed, able to compute with perfect accuracy, and fully rational. The practical application of this prescriptive approach (how people actually make decisions) is called decision analysis, and aimed at finding tools and methodologies to help people make better decisions. In this course , the students will learn :

• Decision Making Under Risk

• EMV, EOL, EVPI

• Decision making Under Uncertainty

• Maximin, Maximax, Baye's/Laplace, Hurwitz Coefficient, Minimax Regret

• Decision Tree

  Decision making can be classified according to the following:

  1. Decision making under risk -Decision under probabilistic uncertainty issues are those in which one of several outcomes can result from a given action depending on the state of nature, and these states occur with known probabilities. There are outcome uncertainties, and the probabilities associated with these are known precisely. Multiple possible outcomes of each alternative can be identified, and the probability of occurrence can be attached to each. . In DECISION UNDER risk each alternative will have one of several possible consequences, and the probability of occurrence for each consequence is known. Therefore, each alternative is associated with a probability distribution, and a choice among probability distributions.In this situation, the decision make faces several states of nature. Probabilities could be assigned to future events by reference to similar previous experience and information.Knowing the probability distribution of the states of nature, the best decision is to select that course of action which has the largest expected pay off value.For problems involving risk situation, the most widely used decision criterion for evaluating the alternative courses of action is the expected monetary value (EMV) (or expected payoff). The objective of decision making here is to optimize the expected payoff which may mean either maximization of expected profit or minimization of expected regret.

  a)EMV: - EMV stands for Expected Monetary Value. The EMV  for given course of action is the sum  of possible  payoffs of the alternative ,each weighted by the probability of that payoff occurring. The expected monetary value prepares the analyst to calculate the expected value. For each decision alternative and select the alternative giving the best expected value.In other words, it is the sum of weighted payoffs for the alternative. The weight for a payoff is the probability of associated ( related) states of nature.

  b)EOL:-  EOL stands for Expected Opportunity Loss.It is the relative loss following a given action conditional upon a given event occurring. The EOL for given courses of action is the sum of possible opportunity losses of the alternative each weighted by the probability of that opportunity loss occurring.

  c)EVPI:- EVPI stands for Expected value with Perfect Information. The EVPI is the expected or average return if the decision maker has perfect information. Expected value of perfect information is the maximum amount decision maker would be willing to pay to acquire  perfect information as to which event would occur.

  EVPI   = EPPI –EMV(max.)

  Where EPPI =Expected payoff under perfect information

       Or,      EVPI= Sum of (prob. × Maximum payoff of each row)

  2. Decision making under uncertainty- Decision under information imperfection issues are those in which one of several outcomes can result from a given action depending on the state of nature, and these states occur with imperfectly specified probabilities. There are outcome uncertainties, and the probabilities associated with these are not all known precisely. Imperfections in knowledge of the utility of the various event outcomes may exist as well. Multiple outcomes for each alternative can be identified, there is no knowledge of the probability to be attached to each. When the probability distributions are unknown, one speaks about DECISION UNDER uncertainty  Decision theory recognizes that the ranking produced by using a criterion has to be consistent with the decision maker's objectives and preferences. The theory offers a rich collection of techniques and procedures to reveal preferences and to introduce them into models of decision. It is not concerned with defining objectives, designing the alternatives or assessing the consequences; it usually considers them as given from outside, or previously determined. Given a set of alternatives, a set of consequences, and a correspondence between those sets, decision theory offers conceptually simple procedures for  selecting the best strategy.The various criterions used in decision making under uncertaintyare :-

  1. Maximin Criteria- This is also called Waldian criterion because it was suggested by Abraham Wald. This criterion suggests the decision to take that course of action which maximizes the minimum possible payoff. The decision selects the minimum outcome within every course of action and then selects that strategy which has the maximum value. Since this decision criterion locates the alternative strategy that has the least possible payoff it is also known as a pessimistic decision criterion. 

  2. Maximax Criterion – This criteria  is based on the optimistic approach of decision maker. The decision maker using this criterion locates the maximum payoff each act, and then selects that act as optimal act which corresponds to maximum payoff of the above located maximum payoffs.

  3. Baye’s/Laplace Criterion –This criterion was developed by Thomas Baye and Simor de Laplace. It is based on the principal of equal likelihood. In this criterion  equal probabilities are assigned  to each environment state. Then   the expected value for each alternative is calculated  by multiplying each outcome by its probability and then adding them up. Ultimately  the alternative with the highest expected value is selected.

  4. Hurwicz  Criterion –This was suggested by Hurwicz.He introduced the coefficient of optimism   (α ) .This coefficient lies between 0 and 1.When α is close to 1 the decision maker is optimistic about the future and when it is close to 0 the decision maker is pessimistic about the future.In this criterion the decision maker take into account both the maximum and minimum payoff of each act and assigns them weights depending upon his degree of optimism. The alternative for which the weighted sum of both the payoff is maximum is selected as optimal act.

  H=α (Maximum payoff  )+ (1-α) (minimum payoff )

  α = degree of optimism

  5. Minimax Regret  Criterion –This criterion was suggested by Leonard Savage. The decision maker using this criterion locates the maximum regret for each act and then selects that act which has minimum regret in the above located maximum regrets.