page_title: Expected value of including uncertainty
section_title: Example
hierarchical_section_title: Expected value of including uncertainty / Example
caption_attribution_description: English: "Influence Diagram depiction of an EVIU model"
context_page_description: In decision theory and quantitative policy analysis, the expected value of including uncertainty is the expected difference in the value of a decision based on a probabilistic analysis versus a decision based on an analysis that ignores uncertainty.
context_section_description: The diagram at right is an influence diagram for deciding how early the decision maker should leave home in order to catch a flight at the airport.  The single decision, in the green rectangle, is the number of minutes that one will decide to leave prior to the plane's departure time.  Four uncertain variables appear on the diagram in cyan ovals: The time required to drive from home to the airport's parking garage (in minutes), time to get from the parking garage to the gate (in minutes), the time before departure that one must be at the gate, and the loss (in minutes) incurred if the flight is missed.  Each of these nodes contains a probability distribution, viz:
  Time_to_drive_to_airport   := LogNormal(median:60,gsdev:1.3)
  Time_from_parking_to_gate  := LogNormal(median:10,gsdev:1.3)
  Gate_time_before_departure := Triangular(min:20,mode:30,max:40)
  Loss_if_miss_the_plane     := LogNormal(median:400,stddev:100)
  Each of these distributions is taken to be statistically independent.  The probability distribution for the first uncertain variable, Time_to_drive_to_airport, with median 60 and a geometric standard deviation of 1.3, is depicted in this graph:
  The model calculates the cost (the red hexagonal variable) as the number of minutes (or minute equivalents) consumed to successfully board the plane.  If one arrive too late, one will miss one's plane and incur the large loss (negative utility) of having to wait for the next flight.  If one arrives too early, one incurs the cost of a needlessly long wait for the flight.
  Models that utilize EVIU may use a utility function, or equivalently they may utilize a loss function, in which case the utility function is just the negative of the loss function.  In either case, the EVIU will be positive.  The main difference is just that with a loss function, the decision is made by minimizing loss rather than by maximizing utility.  The example here uses a loss function, Cost.
  The definitions for each of the computed variables is thus:
  Time_from_home_to_gate := Time_to_drive_to_airport + Time_from_parking_to_gate + Loss_if_miss_the_plane
  Value_per_minute_at_home := 1
  Cost := Value_per_minute_at_home * Time_I_leave_home +
             (If Time_I_leave_home < Time_from_home_to_gate Then Loss_if_miss_the_plane Else 0)
  The following graph displays the expected value taking uncertainty into account (the smooth blue curve) to the expected utility ignoring uncertainty, graphed as a function of the decision variable.
  When uncertainty is ignored, one acts as though the flight will be made with certainty as long as one leaves at least 100 minutes before the flight, and will miss the flight with certainty if one leaves any later than that.  Because one acts as if everything is certain, the optimal action is to leave exactly 100 minutes (or 100 minutes, 1 second) before the flight.
  When uncertainty is taken into account, the expected value smooths out (the blue curve), and the optimal action is to leave 140 minutes before the flight.  The expected value curve, with a decision at 100 minutes before the flight, shows the expected cost when ignoring uncertainty to be 313.7 minutes, while the expected cost when one leaves 140 minute before the flight is 151 minutes.  The difference between these two is the EVIU:
  In other words, if uncertainty is explicitly taken into account when the decision is made, an average savings of 162.7 minutes will be realized.
caption: [influence diagram of EVIU model
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