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Treatments of Risks and Uncertainty in Projects The availability of partial or imperfect information about a problem leads to two new category of decision-making techniques Decisions under risk (In terms of a probability function) Decisions under Uncertainty (No probability function is secure)

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Decisions under risk Decisions under risk are usually based on one of the following criteria Expected Value Combined Expected value and variance Known Aspiration level Most likely occurrence of a future state

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Expected Value Criterion Expressed in terms of either actual money or its utility Decision Maker’s attitude towards the worth or utility of money is important The final decision should ultimately be made by considering all pertinent factors that affect the decision maker’s attitude towards the utility of money The drawback of this is that use of expected value criterion may be misleading fro the decisions that are applied only a few number of times i.e small sample sizes

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Example 1 A preventive maintenance policy requires making decisions about when a machine (or a piece of equipment) should be serviced on a regular basis in order to minimize the cost of sudden breakdown The decision situation is summarized as follows. A machine in a group of n machines is serviced when it breaks down. At the end of T periods, preventive maintenance is performed by servicing all n machines. The decision problem is to determine the optimum T that minimizes the total cost per period of servicing broken machines and applying preventive maintenance

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Let p t be the probability that machine would break down in period t Let n t be the random variable representing the number of broken machines in the same period. C 1 is the cost of repairing a broken machine C 2 the preventive maintenance of the machine The expected cost per period can be written as Where E{n t } is the expected number of broken machines in period t. n t is a binominal random variable with parameter (n,p t ), E{n t }=np t The necessary condition for T * to minimize EC(T) are EC(T * -1)>= EC(T*) and EC(T * +1)>= EC(T*)

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To illustrate the above formulation, suppose c 1 =Rs.100, c 2 =Rs.10 and n=50 The values of p t and EC(T) are tabulated below T * Tptpt Cumulative p t EC(T) 10.050500 20.070.05375 30.100.12366.7 40.130.22400 50.180.35450

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Expected Value-Variance Criterion We indicated that the expected value criterion is suitable for making “long-run” decisions To make it work for the short-run decision problems Expected Value-Variance criterion is used A possible criterion reflecting this objective is Max E[Z]-k*var[z] Where z is a random variable for profit and k is a constant referred to as risk aversion factor Risk aversion factor k is an indicator of the decision maker’s attitude towards excessive deviation from the expected values.

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Applying this criteria to example 1 we get C t is the variance of EC(T) This criteria has resulted in a more conservative decision that applies preventive maintenance every period compared with every third period previously TpTpT pT2pT2 Cum. p T Cum. p T 2 EC(T)+varc T 10.050.002500500 20.070.00490.050.00256312 30.100.01000.120.00746622 40.130.01690.220.01746731 50.180.03240.350.03436764

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Aspiration Level Criterion This method does not yield an optimal decision in the sense of maximizing profit or minimizing cost It is a means of determining acceptable courses of action Most Likely Future Criterion Converting the probabilistic situation into deterministic situation by replacing the random variable with the single value that has the highest probability of occurrence

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Decisions under uncertainty They assume that there is no probability distributions available to the random variable. The methods under this are The Laplace Criterion The Minimax criterion The Savage criterion The Hurwicz criterion

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Laplace Criterion This Criterion is based on what is known as the principle of insufficiency a i is the selection yielding the largest expected gain Selection of the action a i * corresponding where 1/n is the probability that

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Example 2 A recreational facility must decide on the level of supply it must stock to meet the needs of its customers during one of the holiday. The exact number of customers is not known, but it is expected to be of four categories:200,250,300 or 350 customers. Four levels of supplies are thus suggested with level i being ideal (from the view point of the costs) if the number of customer falls in category i. Deviation from these levels results in additional costs either because extra supplies are stocked needlessly or because demand cannot be satisfied. The table below provides the costs in thousands of dollars a1, a2, a3 and a4 are the supplies level Customer Category a15101825 a287823 a321181221 a430221915

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Solution by Laplace Criterion E{a1} = (1/4)(5+10+18+25) = 14.5 E{a2} = (1/4)(8+7+8+23) = 11.5 E{a3} = (1/4)(21+18+12+21) = 18.0 E{a4} = (1/4)(30+22+19+15) = 21.5 Thus the best level of inventory according to Laplace criterion is specified by a2.

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Minimax (Maxmini) Criterion This is the most conservative criterion since it is based on making the best out of the worst possible conditions If the outcome v(a i, θ j ) represents loss for the decision maker, then, for, a i the worst loss regardless of what θ j may be is max θ j [v(a i, θ j )] The minimax criterion then selects the action a i associated with min a i max θ j [v(a i, θ j )] Similarly if v(a i, θ j )] represents gain, the criterion selects the action a i associated with max a i min θ j [v(a i, θ j )] This is called the maxmini criterion

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Applying this criterion to the Example 2 Thus the best level of inventory according to this criterion is specified by a3 Customer Category Max Supply a15101825 a287823 a321181221 a43022191530 Minimax value

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Savage Minimax Regret criterion This is an extremely conservative method The Savage Criterion introduces what is called as regret matrix which is defined as r(a i, θ j ) = { if v is profit if v is loss

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Applying this criteria to Example 2 The regret matrix is shown below Thus the best level of inventory according to this criterion is specified by a2 Customer Category Max Supply a10310 a230088 a316114616 a4251511025 minimax

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Hurwicz Criterion This Criterion represents a range of attitudes from the most optimistic to the most pessimistic The Hurwicz criterion strikes a balance between extreme pessimism and extreme optimism by weighing the above two conditions by the respective weights α and 1- α, where 0<= α<=1 If v(a i, θ j ) represents profit, select the action that yields If v(a i, θ j ) represents cost, select the action that yields

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Applying this criterion to Example 2 Set α =0.5 Resolving with α =0.75 for selecting between a1 and a2 minmaxαmin+(1-α)max 52515 72315 122116.5 153022.5 minimum

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