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4. Decision & Risk Analysis (Fach) / n (Lektion)
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- Reduced Cost The reduced cost for any decision variable with value 0 in the optimal solutionindicates how much better that coefficient must be before that variable enters at apositive level. The reduced cost for any decision variable at its upper bound in theoptimal solution indicates how much worse its coefficient must be before it willdecrease from its upper bound. The reduced cost for any variable between 0 and itsupper bound in the optimal solution is irrelevant.
- Shadow Price The term shadow price is an economic term. It indicates the change in the optimalvalue of the objective when the right side of some constraint changes by one unit. The idea is that a constraint “costs” the company by keeping the objective from beingbetter than it would be. A shadow price indicates how much the company would be willingto pay (in units of the objective) to “relax” a constraint
- Resource Availability and Shadow Prices Resource Availability and Shadow PricesIf a resource constraint is binding in the optimalsolution, the company is willing to pay up to someamount, the shadow price, to obtain more of theresource. This is because the objective improvesby having more of the resource. However, there istypically a decreasing marginal effect: As the companyowns more and more of the resource, the shadowprice tends to decrease. This is usually because otherconstraints become binding, which causes extraunits of this resource to be less useful (or not usefulat all).
- The Effect of Constraints on the Objective The Effect of Constraints on theObjectiveIf a constraint is added or an existing constraintbecomes more constraining (for example, less ofsome resource is available), the objective can onlyget worse; it can never improve. The easiest way tounderstand this is to think of the feasible region.When a constraint is added or an existing constraintbecomes more constraining, the feasible regionshrinks, so some solutions that were feasible before,maybe even the optimal solution, are no longer feasible.The opposite is true if a constraint is deletedor an existing constraint becomes less constraining.In this case, the objective can only improve; it cannever get worse. Again, the idea is that when a constraintis deleted or an existing constraint becomesless constraining, the feasible region expands. In thiscase, all solutions that were feasible before are stillfeasible, and there are some additional feasible solutionsavailable.
- The Value of Information The Value of InformationThe amount you should be willing to spend forinformation is the expected increase in EMV youcan obtain from having the information. If the actualprice of the information is less than or equal tothis amount, you should purchase it; otherwise, theinformation is not worth its price. In addition, informationthat never affects your decision is worthless,and it should not be purchased at any price. Finally,the value of any information can never be greaterthan the value of perfect information that would eliminateall uncertainty.
- Formula for EVPI EVPI = EMV with 1free2 perfect information − EMV without information Information that has noeffect on the ultimatedecision is worthless.
- risk tolerance The risk tolerance for an exponential utility function is a single number that specifiesan individual’s aversion to risk. The higher the risk tolerance, the less risk averse theindividual is.
- assess a person’s (or company’s) exponential utility function To assess a person’s (or company’s) exponential utility function, only one number, thevalue of R, needs to be assessed. There are a couple of tips for doing this. First, it has beenshown that the risk tolerance is approximately equal to that dollar amount R such that thedecision maker is indifferent between the following two options:■■ Option 1: Obtain no payoff at all.■■ Option 2: Obtain a payoff of R dollars or a loss of R/2 dollars, depending on the flipof a fair coin.For example, if you are indifferent between a bet where you win $1000 or lose $500, withprobability 0.5 each, and not betting at all, your R is approximately $1000. From this criterionit certainly makes intuitive sense that a wealthier person (or company) ought to have alarger value of R. This has been found in practice.A second tip for finding R is based on empirical evidence found by Ronald Howard,a prominent decision analyst. Through his consulting experience with large companies, hediscovered tentative relationships between risk tolerance and several financial variables:net sales, net income, and equity. [See Howard (1988).] Specifically, he found that R wasapproximately 6.4% of net sales, 124% of net income, and 15.7% of equity for the companieshe studied. For example, according to this prescription, a company with net sales of$30 million should have a risk tolerance of approximately $1.92 million.
- Rational decision makers ? Rational decision makers are sometimes willing to violate the EMV maximization criterionwhen large amounts of money are at stake. These decision makers are willing to sacrificesome EMV to reduce risk. Are you ever willing to do so personally? Consider the followingscenarios. most researchers believe that if certain basicbehavioral assumptions hold, people are expected utility maximizers—that is, they choosethe alternative with the largest expected utility.
- Risk Aversion When large amounts of money are at stake, most of usare risk averse, at least to some extent. We are willingto sacrifice some EMV to avoid risk. The exact way thisis done, using utility functions and expected utility, canbe difficult to implement in real situations, but the idea issimple. If you are an EMV maximizer, you are indifferentbetween a gamble with a given EMV and a sure dollaramount equal to the EMV of the gamble. However, ifyou are risk averse, you prefer the sure dollar amountto the gamble. That is, you are willing to accept a suredollar amount that is somewhat less than the EMV ofthe gamble, just to avoid risk. The more EMV you arewilling to give up, the more risk averse you are
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