Published on February 13, 2008
Dike Decisions Based on the Advanced Hydrological Prediction Service(AHPS): Dike Decisions Based on the Advanced Hydrological Prediction Service (AHPS) Contact: Lee Anderson, Meteorologist in Charge National Weather Service Grand Forks, North Dakota Phone: (701) 772-0720 (ext 642) E-mail: [email protected] Author: John Erjavec, Professor & Chair Department of Chemical Engineering University of North Dakota Phone: (701) 777- 4244 E-mail: [email protected] Introduction: Introduction The purpose of this material is to help city managers, city engineers, hydrologists and others to use the river crest predictions from the new Advanced Hydrological Prediction Service (AHPS). The methodology presented helps calculate the economic benefits from a dike that may prevent flooding of a city. Economics is only one factor in this decision making process, but it is clearly an important factor. ISSUE:: All river crest predictions, even those using the AHPS, are inherently uncertain. CONCLUSION: Dike decisions must take degree of certainty of predictions into account. ISSUE: Decision Making when dealing with Uncertainty: Decision Making when dealing with Uncertainty Situation: Exact outcome is NOT known Possible outcomes ARE known The probability that a particular outcome will occur is known or can be estimated for each outcome Economics is an important factor in the decision making process Example 1: Investment Opportunity: Example 1: Investment Opportunity Investment (i.e. COST) is $10,000 Return is uncertain: Return may be $100,000 There may be no return Summary of 2 Alternatives: Alternative 1: Do not invest Cost = $0, Benefits = $0 Alternative 2: Invest Cost = $10k, Benefits = uncertain Decision to be made: Should one invest? We Need a Decision Making Criterion Slide6: TYPICAL CRITERIA: One Alternative : Benefit / Cost Ratio > 1 Multiple Alternatives: Incremental Benefit Incremental Cost > 1 Slide7: TYPICAL CRITERIA (continued): PROBLEM: Since the outcomes can not be predicted with certainty, the actual Benefits are unknown. SOLUTION: Use EXPECTED benefits (i.e., use average benefits) Slide8: DEFINITION The Expected Benefit, E(B), for a course of action (alternative) is the weighted average of the benefits from the possible outcomes for that alternative. Each benefit value for a particular outcome is weighted by the probability of that outcome actually happening. Mathematically, if the outcomes are denoted by O1, O2, etc., then E(B) = BenefitO1x P(O1) + BenefitO2 x P(O2) + … where BenefitO1 is the benefit of Outcome1 , etc. and P(O1) is the probability of Outcome1 , etc. Slide9: CRITERION*: Examine ratio of Incremental Expected Benefit, DE(B), to Incremental Cost, DC Is DE(B) / DC > 1 ? If ratio is >1, choose higher cost alternative. If ratio is <1, choose lower cost alternative. *Remember, our situation is that we have two alternatives. Example 1 (Continued): Example 1 (Continued) Summary of 2 Alternatives: Alternative 1: Do not invest Cost = $0, Benefits = $0 Alternative 2: Invest Cost = $10k, Benefits = uncertain Outcomes for Alternative 2: Outcome A = Investment pays off BenefitA = $100,000 P(A) = 0.05 (that is, 5 times out of 100 it pays off) Outcome B = Investment does not pay off BenefitB = $0 P(B) = 0.95 (that is, 95 times out of 100 it does not pay off) Expected Value of Benefits = (0.05)x($100,0000) + (0.95)x($0) = $5,000 Example 1 (Continued): Example 1 (Continued) Apply Criterion: Incremental Expected Benefit = $5,000 (On average, we would get $5k per investment.) Incremental Cost = $10,000 Since: E (B) / C = $5k/$10k = 0.5 <1 Do NOT Invest Note: We would never actually get a $5,000 return; either we would get $100,000 or nothing. The expected benefit of $5,000 is the return that one would get “on average” from numerous $10,000 investments such as this. Example 2 (Fire Insurance for $250k House): Example 2 (Fire Insurance for $250k House) Outcomes (from fire rating bureau): O1 : No fire loss P(O1) = 0.985 O2 : $15k fire loss P(O2) = 0.010 O3 : $50k fire loss P(O3) = 0.004 O4 : $200k fire loss P(O4) = 0.001 Expected value of fire loss in any year: Expected Loss = ($0)(0.985) + ($15k)(0.010) + ($50k)(0.004) + ($200k)(0.001) = $550 Example 2 (Continued): Example 2 (Continued) Expected Value of fire loss = $550/yr Fire insurance (no deductible) = $750/yr Summary of Costs and Expected Benefits: Alt 1, No Insurance: Cost = $0/yr, Expected “Benefits” = -$550/yr (fire loss) Alt 2, Buy Insurance: Cost = $750/yr, Expected “Benefits” = $0/yr (No fire loss) Incremental Benefits = $0/yr – (-$550/yr) = $550/yr Incremental Costs = $750/yr - $0/yr = $750/yr Example 2 (Continued): Example 2 (Continued) Apply Criterion: E (B) / C = $550/$750 = 0.73 <1 Conclusion: The insurance is more costly (on average) than a fire. (This is the expected result, since we know that insurance companies make money). Decision: Should you buy insurance? Our criterion says that the less expensive alternative (no insurance) is the one that is the best. But we probably ought to buy the insurance anyway (assuming it is affordable), because being self-insured only makes sense if the worst outcome is not catastrophic. General Approach: General Approach List all reasonable alternative courses of action. “Do nothing” is usually included on the list. Determine the cost (investment) of each alternative on your list. List all outcomes for each alternative Determine the probability of each outcome Assign a value (benefit, which may be negative if it is a loss) to each outcome. Determine the Expected Value of the benefits for each alternative (course of action). Example 3 Why Use Incremental Approach? (Why Not Maximize Benefit/Cost Ratio?): Example 3 Why Use Incremental Approach? (Why Not Maximize Benefit/Cost Ratio?) Alt. Cost Benefit B/C 0 $0 $0 --- 1 $100 $200 2.00 2 $200 $350 1.75 3 $300 $475 1.58 4 $400 $575 1.42 5 $500 $650 1.30 All alternatives are good. They all have B/C ratios > 1 Alternative 1 has the highest B/C ratio, but it is not the best. Alternative 3 is the best. It keeps returning more in extra (incremental) benefits than the extra (incremental) costs. Increment D C D B DB/DC > Alt.1-Alt.0 $100 $200 2.00 > Alt.2-Alt.1 $100 $150 1.50 > Alt.3-Alt.2 $100 $125 1.25 > Alt.4-Alt.3 $100 $100 1.00 > Alt.5-Alt.4 $100 $75 0.75 General Approach (Continued): General Approach (Continued) Order the alternatives from least expensive to most expensive. Compare the alternatives pair-wise, starting with the least expensive, examining incremental benefits and incremental costs If B/C > 1, keep the more expensive alternative In this case, the increased benefits outweigh the increased costs of the more expensive alternative. If B/C < 1, keep the less expensive alternative The alternative that is kept is compared to the next alternative on the list, and the procedure is repeated until all alternatives have been examined, and only one remains. That alternative is the “winner” (final decision). Review the decision to make sure that the risk is not too great. (Being self insured only makes sense if the worst outcome(s) is not catastrophic.) General Approach Applied to Dike Decisions: General Approach Applied to Dike Decisions List all reasonable alternative dike levels. Start with the existing permanent dike level and end with the highest dike that could be built in the time allowed. Determine the cost of each dike level on your list. By difference, calculate the incremental cost of adding to the dike to get to the next level of protection. Determine the loss which would be incurred to the city if a river crest exceeded the level of the dike (by half of the interval between dike heights) for each dike height on your list. From AHPS, determine the probability of the river crest exceeding each dike height on your list. Determine the probability that the river crest will fall in each dike level range on your list by difference. General Approach Applied to Dike Decisions (Continued): General Approach Applied to Dike Decisions (Continued) Determine the Expected Loss for each increment of the dike, by multiplying the Loss to the city for flooding in a specific range times the probability that the river crest will fall in that range. Calculate the Incremental Expected Benefit / Incremental Cost ratio for each extra dike level under consideration. This is done by dividing the Expected Loss of Flooding which will be avoided by adding to the dike, by the incremental cost of building the dike higher (to the next level). If the ratio, E(B) / C, is greater than 1, the increment is worth adding. If the ratio, E(B) / C, is less than 1, the increment is not worth adding. Review the decision to make sure that the risk is not too great. (Being self insured only makes sense if the worst outcome(s) is not catastrophic.) General Approach Applied to Dike Decisions (Continued): General Approach Applied to Dike Decisions (Continued) Suggestions: Determine the cost of each dike level that is being evaluated. By difference, calculate the cost of adding to the dike to bring it to the next higher level, and put the information in a spreadsheet Determine the damage to the city that would be incurred if the city were not protected by a dike and the river crest hit a level halfway between the dike levels being considered. Include that information in the spreadsheet. Add the AHPS probability data to the spreadsheet (as per Examples 4-7). Calculate Expected benefits and Incremental E(B)/C ratios for each dike level. Use ratios to help guide dike level decision. Examples 4-7 (GF/EGF Flood Control): Examples 4-7 (GF/EGF Flood Control) Alternatives to Consider: 1: Do Nothing 2: 49’ Dike 3: 51’ Dike 4: 53’ Dike 5: 55’ Dike 6: 57’ Dike Examine incremental benefit / incremental cost Note: Incremental benefit is the EXPECTED flood loss which is avoided by building the dike higher. Benefits = Probability of flooding x Cost of Flood Damage Incremental cost is the cost of adding to the dike to make it higher (see figure on next slide). Dikes of Heights L1 and L2:: Dikes of Heights L1 and L2: Slide23: Examples 4-7 (GF/EGF Flood Control) *This data only needs to be collected once, and updated whenever it is deemed to be necessary. Slide24: Example 4: Very High Crest Predicted (Most likely crest of 43 feet) Slide25: Example 4 AHPS Output (Actual Probabilistic Outlook for Red River at East Grand Forks, MN – 2001) Example 4 (Continued): Example 4 (Continued) * These values were obtained by difference (see next slide) ** These values had to be obtained by extrapolation (see slide after next) Probability of Crest Between L1 and L2: Probability of Crest Between L1 and L2 Probability of Probability of Probability of River Cresting = River Cresting - River Cresting Between L1 & L2 Higher than L1 Higher than L2 AHPS Exceedance Probabilities Extrapolating AHPS Probabilities: Extrapolating AHPS Probabilities L 0.025 Assume predictions follow a Normal Distribution Mean, L 0.50 = Level for 50% Exceedance Probability Standard Deviation, s ~ (L 0.025 - L 0.50 )/2 where L 0.025 is the level which has a 2.5% chance of being exceeded. To find the probability of exceeding a level, L Calculate standard Normal deviate, z , which corresponds to how many standard deviations L is from the mean: z = (L – L 0.50)/s Look up probability for z using statistical tables of the Normal Dist. or use a spreadsheet (e.g. Excel NORMDIST function) L 0.50 Extrapolating AHPS Probabilities (For Example 4, Case: L = 55 ft): Extrapolating AHPS Probabilities (For Example 4, Case: L = 55 ft) L0.025 = 53 ft To find the probability of exceeding level, L = 55 ft Assume predictions follow a Normal Distribution Mean = L 0.50 (obtained from AHPS Output) = 43 ft for this example L 0.025 (also from AHPS Output) used to get s = 53 ft for this example Standard Deviation, s = (L 0.025 - L 0.50 )/2 = (53-43)/2 = 5.0 Calculate standard Normal deviate, z: z = (L – L 0.50)/s = (55-43)/5.0 = 2.4 Look up probability for z: Prob(L > 55 ft) = Prob(z > 2.4) = 0.0081= 0.81% Mean = 43 ft L = 55 ft Slide30: Example 4 (Continued) Slide31: Example 5: High Crest Predicted (Most likely crest of 40 feet) Slide32: Example 5 (Continued) * These probabilities were simulated assuming a Normal-distribution with a mean (L50%) = 40 ft, and a standard deviation, s = 5 ft. Slide33: Example 5 (Continued) Slide34: Example 6: Medium Crest Predicted (Most likely crest of 38 feet) Slide35: Example 6 (Continued) * These probabilities were simulated assuming a Normal-distribution with a mean (L50%) = 38 ft, and a standard deviation, s = 5 ft. Slide36: Example 6 (Continued) Slide37: Example 7: Low Crest Predicted (Most likely crest of 36 feet) Slide38: Example 7 (Continued) * These probabilities were simulated assuming a Normal-distribution with a mean (L50%) = 38 ft, and a standard deviation, s = 5 ft. Slide39: Example 7 (Continued) SUMMARY: SUMMARY The uncertainty of river level predictions is a key factor that must be taken into account in any dike decisions. The relative costs of building a dike versus the costs incurred by flood damage are also key factors in dike decisions. The appropriate economic criterion to use as part of the decision making process is the ratio of the incremental expected benefit (by avoiding flood damage) to the incremental cost of adding to the dike height. Even when the probability of a crest exceeding a high level is low (less than 1%, as in Example 3), it may still be worth building a dike to that level when the costs of damage are very high. This is true because the expected benefits are calculated as the (probability) x (potential damage).