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Decision making under Uncertainty example problems
A decision problem, where a decision-maker is aware of various possible states of nature but has insufficient informa�on to assign any probabili�es of occurrence to them, is termed as decision- making under uncertainty. A decision under uncertainty is when there are many unknowns and no possibility of knowing what could occur in the future to alter the outcome of a decision.
We feel uncertainty about a situa�on when we can't predict with complete confidence what the outcomes of our ac�ons will be. We experience uncertainty about a specific ques�on when we can't give a single answer with complete confidence.
Launching a new product, a major change in marke�ng strategy or opening your first branch could be influenced by such factors as the reac�on of compe�tors, new compe�tors, technological changes, changes in customer demand, economic shi�s, government legisla�on and a host of condi�ons beyond your control. These are the type of decisions facing the senior execu�ves of large corpora�ons who must commit huge resources.
The small business manager faces, rela�vely, the same type of condi�ons which could cause decisions that result in a disaster from which he or she may not be able to recover.
A situa�on of uncertainty arises when there can be more than one possible consequences of selec�ng any course of ac�on. In terms of the payoff matrix, if the decision-maker selects A1, his payoff can be X11, X12, X13, etc., depending upon which state of nature S1, S2, S3, etc., is going to occur.
Methods of Decision Making under Uncertainty
The methods of decission making under certainity are.There are a variety of criteria that have been proposed for the selec�on of an op�mal course of ac�on under the environment of uncertainty. Each of these criteria make an assump�on about the a�tude of the decision-maker.
Maximin Criterion: This criterion, also known as the criterion of pessimism, is used when the decision-maker is pessimis�c about future. Maximin implies the maximisa�on of minimum payoff. The pessimis�c decision-maker locates the minimum payoff for each possible course of ac�on. The maximum of these minimum payoffs is iden�fied and the corresponding course of ac�on is selected. This is explained in the following example :
Example : Let there be a situa�on in which a decision-maker has three possible alterna�ves A1, A2 and A3, where the outcome of each of them can be affected by the occurrence of any one of the four possible events S1, S2, S3 and S4. The monetary payoffs of each combina�on of Ai and Sj are given in the following table:
monetary payoffs of each combina�on of Ai and Sj
Solu�on: Since 17 is maximum out of the minimum payoffs, the op�mal ac�on is A2.
Maximax Criterion: This criterion, also known as the criterion of op�mism, is used when the decision-maker is op�mis�c about future. Maximax implies the maximisa�on of maximum payoff. The op�mis�c decision-maker locates the maximum payoff for each possible course of ac�on. The maximum of these payoffs is iden�fied and the corresponding course of ac�on is selected. The op�mal course of ac�on in the above example, based on this criterion, is A3.
Regret Criterion: This criterion focuses upon the regret that the decision-maker might have from selec�ng a par�cular course of ac�on. Regret is defined as the difference between the best payoff we could have realised, had we known which state of nature was going to occur and the realised payoff. This difference, which measures the magnitude of the loss incurred by not selec�ng the best alterna�ve, is also known as opportunity loss or the opportunity cost.
From the payoff matrix (given in § 12.6), the payoffs corresponding to the ac�ons A1, A2, ...... An under the state of nature Sj are X1i, X2j, ...... Xnj respec�vely. Of these assume that X2j is maximum. Then the regret in selec�ng Ai, to be denoted by Rij is given by X2j - Xij, i = 1 to m. We note that the regret in selec�ng A2 is zero. The regrets for various ac�ons under different states of nature can also be computed in a similar way.
The regret criterion is based upon the minimax principle, i.e., the decision-maker tries to minimise the maximum regret. Thus, the decision-maker selects the maximum regret for each of the ac�ons and out of these the ac�on which corresponds to the minimum regret is regarded as op�mal. The regret matrix of example can be wri�en as given below:
regret matrix
From the maximum regret column, we find that the regret corresponding to the course of ac�on is A3 is minimum. Hence, A3 is op�mal.
Hurwicz Criterion: The maximax and the maximin criteria, discussed above, assumes that the decision-maker is either op�mis�c or pessimis�c. A more realis�c approach would, however, be to take into account the degree or index of op�mism or pessimism of the decision-maker in the process of decision-making. If a, a constant lying between 0 and 1, denotes the degree of op�mism, then the degree of pessimism will be 1 - a. Then a weighted average of the maximum and minimum payoffs of an ac�on, with a and 1 - a as respec�ve weights, is computed. The ac�on with highest average is regarded as op�mal.
