CRP 834: Decision Analysis

Week Eight Notes 1

Plan Evaluation Methods

Monetary-based technique • • • Financial Investment Appraisal Cost-effective analysis Cost-benefit analysis Multicriteria technique • Check-list of criteria • • • Goals-achievement matrix Planning balance sheet analysis Concordance-Discordance Statistical Technique • Correspondence analysis (principal component analysis) • The information theory (entropy) Optimization techniques • Multi-Objective Programming 2

Review of Cost-Benefit Analysis

– Monetary-based technique – When evaluating over time, need to consider discount rate – The concept of consumer surplus 3

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Cost-Benefit Analysis—Examples

Case 1: The Simplest Case The planner is asked to design a project to provide 100 mgd of usable water, and there is but one feasible source.

• There are only two sensible designs – one with higher first costs, but lower OMR costs, and the other with lower first costs but higher OMR costs. 4

Case 1 (cont’d) Example: Economic life of structure – 50 years (with discounting) First Cost Annual OMR Cost PV of OMR Cost 3% 6% Total PV, first Cost & OMR 3% 6% Design A $5,000,000 $100,000 $2,573,000 $1,576,000 $7,573,000 $6,576,000 Design B $3,000,000 $200,000 $5,146,000 $3,152,000 $8,146,000 $6,152,000 5

Case 1 (cont’d) • (Warning) Need to consider benefits and costs realized at different time.

– Inflation – Risk and uncertainty on the rate of return – Internal rates of return: • NB pv =(B 0 -C 0 )+ (B 1 -C 1 ) /(1+r)+ (B 2 -C 2 )/(1+r) 2 +… • 0=(B 0 -C 0 )+ (B 1 -C 1 ) /(1+i)+ (B 2 -C 2 )/(1+i) 2 +… • Where r is the interest rate, and i is the internal rate of return.

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Case 2: Two Potential Reservoirs (Q) How to design the project for the optimal costs and benefits?

Reservoir Storage – Yield Data Storage 10 3 acre feet 25 50 75 100 125 150 175 200 Res. I 35 67 83 90 97 103 107 110 Y d mgd Res. II 60 77 32 100 110 115 120 125 140 120 100 80 60 40 II I 20 0 0 20 40 60 80 100 120 Reservoir Storage 140 160 180 200 Goal: To produce output of 100 mgd of usable water at least cost!

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Case 2 (Cont’d) Reservoir-Cost Data Cost = Capital Cost + PV of OMR cost ( 50 year period, discount rate 5%) 25 Storage 10 3 acre feet 25 50 75 100 125 150 175 200 Cost ($10 5 ) Res. I Res. II 2.7

4.5

5.3

6.3

7.5

10.0

15.0

25.0

3.5

5.5

7.5

10.0

15.0

25.0

20 15 10 5 II I 0 0 20 40 60 80 100 120 Reservoir Storage 140 160 180 200 8

Case 2 (Cont’d) • Step 1: Estimate cost function – Ci=f(xi), where Ci = cost, xi=reservoir capacity • Step 2: Estimate Production (yield) – Yi=f(xi), where yi = yield, xi=reservoir capacity • Step 3: Establish Iso-Output Function of combinations of Reservoirs I and II for 100 mgd usable water • Step 4: Establish Iso-Output Function of combinations of Reservoirs I and II • Step 5: Find the least cost combination of reservoirs using the point of tangency 9

Case 2 (Cont’d) Iso-Output Function of combinations of Reservoirs I and II Combinations of Reservoirs I and II that will provide 100 mgd of usable water Storage 10 3 acre feet 0 25 50 75 100 Res. I Output mgd 0 35 67 83 90 Storage 10 3 acre feet 100 30 12 6 4 Res. II Output mgd 100 65 33 17 10 Total Output mgd 100 100 100 100 100 Iso-Cost Function of combinations of Reservoirs I and II Combinations of Reservoirs I and II with total cost of $ 5.5 million.

Res. I Res. II Total Capacity 10 3 acre feet 75 62 43 20 17 Cost $ M 5.5

5.0

4.0

3.0

2.0

Capacity 10 3 acre feet 0 3 10 17 25 Cost $ M 0 0.5

1.5

2.5

3.5

Cost $ M 5.5.

5.5

5.5

5.5

Case 2 (Cont’d) 25 50 75 100 Iso-output curve =100 mgd Iso-cost curve = $ 5.5 M Least cost combination is at tangency of two curves: Res. I =50 *10 3 Res. I =12 *10 3 acre feet –$ 4.5 M acre feet --$ 1.5 M 75 25 100 50 11

Case 3: Two Reservoirs without specifying fixed water supply MC I MC II  MP MP II I Points of tangency 1.

Scale line Scale line 150 mgd 50 mgd 100 mgd 75 mgd 25 125 mgd 50 75 100 Res. II - Storage Capacity 10 3 acre 12

Case 4: The optimal yield at a minimum cost with budget constraint MC=MB PV Costs and Benefits ($ M) Scale of output at which slopes of benefit and total cost functions are equal.

Gross Benefit Total Cost Capital Cost 25 50 75 100 125 Maximize NB = B(y) - C(y) Subject to: K(y) < B (Budget Constraint) 13

Multicriteria - Basic Problem

Definition: a multicrtieria decision problem is a situation in which, having defined a set of actions (A) and a consistent family (F) of criteria on A, one wishes – to determine a subset of actions considered to be the best with respect to F – to divide A into subsets according to some norms (sorting problems) – to rank the actions of A from best to worst (ranking problems) 14

Balance Sheet of Project Evaluation

•Can be viewed as a particular application of the social cost-benefit approach to evaluation • Developed by Lichfield and widely used in England • Considers all benefits and all costs with respect to all community goals in one enumeration • Presents a complete set of social accounts, with respect to different goals, and for consumers and producers • The costs and benefits are recorded as capital (once for all) items or annual (continuing) items. Types of evaluation considered: monetary, quantitative but non-monetary, intangible, and time.

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Check List Criteria

•Ranks appropriate alternative proposals on an ordinal basis in relation to a number of specific criteria.

• Widely used by professional land-use planners 16

Goals-Achievement Matrix

•Developed by Hill (1965) – M Hill. 1965. A Goals-Achievement Matrix for Evaluating Alternative Plans. Application to Transportation Plans. Journal of the American Institute of Planners , Vol. 34, No. 1, 19-29.

•This method attempts to determine the extent to which alternative plans will achieve a predetermined set of goals or objectives •Costs and benefits are always defined in terms of achievement.

– Benefits represent progress toward the defined objectives, while costs represent retrogression from defined objectives.

– The basic difference between PBSA and GAM is that GAM only considers costs and benefits with reference to well stated objectives, and to well defined incidence groups.

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Concordance-Discordance analysis

•It consists of a pair-wise comparisons based on calculated indicators of concordance or discordance.

•The concordance index reflects the relative dominance of a certain competing plan, and the discordance index shows the degree to which the outcomes of a certain plan are worse than the outcomes of a competing plan.

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Concordance-Discordance analysis

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Concordance-Discordance analysis—example

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Multi Objective Programming (MOP)

• Basic Concepts and Definition of MOP • Non-Dominated Solutions • Generating/Incorporating Methods – Weight Method – Constraint method • MOP under uncertainty 21

Why MOP?

• Multiple decision makers • Multiple evaluating criteria • Wider range of alternatives  More realistic analysis of problems * In fact, any optimization is (or should be) multiobjective!!!!

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MOP Formulation (Vector Optimization Problem)  [

Z

1

x



R p g x j i

 0,  0, i=1..m

j

 1..

n Z

2

Z p

Where

Z(x)

= p-dimensional objective function

x

= feasible region in decision space 23