Multi-Criteria Decision Making under Uncertainty ...

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Multi-Criteria Decision Making under Uncertainty: Application to the California’s Sacramento-San Joaquin Delta Problem Soroush Mokhtari1, Kaveh Madani2, Ni-Bin Chang3 Department of Civil, Environmental and Construction Engineering, University of Central Florida, Orlando, Florida 32817; PH (407) 823 2317 1 [email protected]; [email protected]; [email protected] ABSTRACT Multi-criteria decision making problems are often associated with trade-offs between performances of the available alternative solutions under the considered decision making criteria. These problems become more complex when uncertainty is associated. This paper applies a Monte-Carlo multi-criteria decision making method for solving the California’s Sacramento-San Joaquin Delta problem in which the best alternative for exporting water from the Delta should be selected based on two main criteria, namely the economic cost and environmental performance. To deal with the uncertainty involved, the stochastic decision making problem is converted into numerous deterministic problems through a Monte-Carlo selection. Each deterministic problem is then solved using a range of multi-criteria decision making methods. The overall winning probabilities of the alternatives under each method are calculated and compared with the results obtained in previous studies which used conflict resolution methods, namely, non-cooperative game theory, fall-back bargaining, and social choice rules to suggest optimal solutions for the Delta problem.

INTRODUCTION Providing water for more than 22 million people and serving as a unique habitat of several endangered species and a major component of the states' civil infrastructure, the Sacramento-San Joaquin Delta is the heart of water supply system and a major support of California's trillion dollar economy and 27 billion dollar agriculture (CA DWR, 2008). The Delta is a web of 57 reclaimed islands and 700 miles of channels at the intersection of two of California's largest rivers. There are more than six counties in the Delta and five rivers flowing into it that altogether with their tributaries collect 45% of the state's runoff (Lund et al., 2007). Due to a special formation process, the Delta has a rich and productive soil. Thus, settlers began farming in this region shortly after the gold rush. Since the Delta is a low land, to protect their farmlands from floods levees have been built along the water channels. Today, most of the 1,150 square miles of Delta's area, laying 20 feet or more below

World Environmental and Water Resources Congress 2012

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surrounding water, is still reclaimed by such levees (Ingebritsen et al., 2000). Due to increasing agricultural and domestic water demand of southern California, the Bay Area, and the San Joaquin valley, several aqueducts were built at the southern end of the Delta from 1930 to 1960 (Lund et al., 2008). Nowadays, not only the neighboring area, but most of the California's is dependent to the water provided by the Delta (CA DWR, 2008). Passage of electricity and gas transmission lines and some major state highways through the Delta along with presence of several busy ports and natural gas extraction facilities, makes the Delta an important component of California's infrastructure (Ingebritsen et al., 2000; Madani and Lund, in press). Moreover, the Delta is a natural habitat for more than 500 wild species, 20 of which recognized endangered such as the delta smelt or Chinook salmon (CA DWR, 2008). Over decades, the increasing demands of competing sectors and decreasing water quality along with vulnerability of the delta to rising sea level and natural disasters such as possible earthquakes or hurricanes, have put the Delta's ability to meet the water demands in jeopardy, threatening the viability of the region. The economic cost of a very possible failure due to earthquake or other natural disasters can be up to 40 billion dollars in five years together with water export cut off for several months and disruption of power and road transmission lines (Lund et al., 2007). The Delta water quality is another major concern. Serving as a drainage area for agricultural, domestic and industrial runoffs over the past decades along with permeation of saltwater into the Delta, have led to alarming deteriorating water quality in the region (Lund et al., 2007). Moreover, drastic decline in wildlife and high extinction risk of endangered species have caused cry out of the environmentalists (CA DWR, 2008). A detailed research on Delta’s current situation and long-term solutions for the emerging crisis has been carried out by Lund et al. (2007). In their research nine feasible long-term solutions have been evaluated, considering the different environmental and economic criteria. Among the nine solutions, four were suggested for further investigation. These four alternative solutions are: (1) continuing the current water exports through the existing facilities (business as usual); (2) building a canal, tunnel, or pipeline to convey water around the delta (tunnel); (3) combination of the two previous strategies (dual conveyance); and (4) ending water exports (stop exports). These four scenarios were further investigated by Lund et al. (2008) considering two major criteria, i.e., the average cost and the likelihood of viable fish population (fish survival rate). Results of this assessment are presented in Table 1. Table 1. Estimated Performance Ranges of Different Water Export Alternatives under the Economic and Environmental Criteria (Lund et al., 2008) Economic Fish survival Criteria cost (%) Alternative (B$/year) Business as usual (BAU) Tunnel (T) Dual conveyance (DC) Stop exports (SE)

0.55-1.86 0.25-0.85 0.25-1.25 1.25-2.5

5-30 10-40 10-40 30-60


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Madani and Lund (2011) proposed a non-cooperative game theoretic approach for modeling the Delta’s multi-criteria multi-decision maker problem with four alternatives and two decision makers using a Monte-Carlo selection method for dealing with the uncertainty involved (performances are not unique numbers). Essentially, the problem is a stochastic decision making problem due to uncertain performances. Based on their suggested method the stochastic problem was converted to numerous deterministic decision making problems through a MonteCarlo selection. Assuming that each deterministic problem corresponds to a specific game structure, they used various non-cooperative game theory solutions to identify the best water export option. Their results suggested the current water export strategy (BAU) is not stable and once the parties decide to cooperate building a tunnel (T) becomes the most likely option. They identified dual conveyance (DC) as the second most likely water export strategy. Madani et al. (2011) studied the same problem using a bargaining approach. In their approach the decision making problem was modeled as a game in which parties bargain until a consensus is developed and one strategy is selected. Similarly, selection of a tunnel and dual conveyance were identified as the likely outcomes of the bargaining process. While the general findings of the two studies match, the selection probabilities of the alternatives were not equal, showing the sensitivity of the findings to the selection rules applied. While the difference may not be significant for small decision making problems, in larger problems inconsistencies become important. Shalikarian et al. (2011) used a different approach for modeling the Delta decision making problem. Instead of assuming that decisions are made in a noncooperative environment, in which parties may compete sometimes to increase their personal gains, they considered the problem as a social decision making problem in which parties can simply vote and rank the alternatives based on their different perspectives. The final decision is made based on the social choice or voting rules. While the results obtained based on voting rules may not be necessary optimal from the systems perspective, they are socially optimal. Application of the social choice rules to the Delta decision making problem also suggested that building a tunnel is the socially optimal solution, followed by the dual conveyance as the second socially optimal solution. The estimated selection probabilities of the alternatives based on social choice rules were different from the results of the two previous studies, highlighting the importance of using appropriate rules for solving decision making problems. To further investigate the effects of the choice of decision making rule on the final results of multi-criteria multi-decision maker problems this study uses the conventional multi-criteria decision making (MCDM) methods rooted in Operations Research (OR) to solve the Delta decision making problem as a benchmark example. In most of the OR-based MCDM methods, a single, all-powerful decision maker determines the fairest alternative considering all the objectives of different decision makers (Madani and Lund, 2011). In this paper five well-known MCDM method namely, Dominance, TOPSIS, SAW, Lexicographic, and Maximin are used to prescribe the optimal solution to the Delta decision making problem from the perspective of the central planner. Generally, prescriptive methods assume that all


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the stakeholders are compliant to the fair and unbiased decision maker while methods such as non-cooperative game theory and fallback bargaining are more descriptive, trying to describe the procedure of negotiations with emphasis on selfoptimizing behavior of the players (Madani and Lund, 2011).


METHOD To deal with uncertainty involved (the payoff of each alternative can be any number within the given intervals in Table 1), a Monte-Carlo random selection is used, following the method suggested by Madani and Lund (2011). Through the suggested method the stochastic decision making problem is converted to numerous deterministic problems that can be solved based on different MCDM methods. The suggested method in this study goes beyond selection of the winner (optimal) alternative in each round of selection. Instead, once a given deterministic problem is solved and the best solution is identified, the best solution is eliminated from the alternatives set and the problem is solved with the remaining (4-1=3) alternatives with the same performance values, specific to the same round of selection. The process is until all 4 alternatives are ranked for a given randomly selected performances. In the next section, each MCDM method is first described and then the results of the stochastic MCDM analysis are presented for that decision making method. The presented results are based on 100,000 rounds of selection, as results remain unchanged after 100,000 rounds of selection. MULTI-CRITERIA DECISION ANALYSIS Lexicographic. The method selects the alternative with the best performance under the most important criterion where ties are broken considering the performances of tying alternatives under next most important criterion. The procedure is continued until a unique alternative is found (Linkov et al., 2005). In order to determine the importance of each criterion, entropy weighting method is used (Chan et al., 1999; Yaghoubi et al., 2011). Based on this method, first the 'P' matrix (normalized matrix of payoffs) is formed: =

∑

where for j=1, 2, …, n: Pij is the normalized payoff of alternative 'i' under criteria 'j'; rij is the performance of alternative 'i' under criteria 'j'; 'm' and 'n' are the number of alternatives and criteria respectively. Then, using the P matrix, entropy of each criterion (Ej) can be calculated based on the following equation: =−

. ln


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where:

=

.


Weight of each criterion (

) is then calculated as: =

where:

∑

=1−

Table 2 indicates the results of the analysis based on the Lexicographic and entropy weighting methods. This table shows the probability of selection of each alternative at a given rank. After selection the alternative with the highest probability of winning (the best alternative), the process is continued to determine the winning probabilities of the other alternatives. Based on the Lexicographic method building a tunnel (T) is the best solution, followed by the dual conveyance (DC). The overall ranking of the four solutions based on this method is as follows: Tunnel> Dual conveyance > Stop export >Business as usual Table 2. The Winning Probabilities of the Alternatives Based on the Lexicographic Method BAU T DC NE

1st 3.39% 67.01% 29.23% 0.37%

2nd 17.18%

3ed 53.85%

4th

74.08% 8.74%

46.15%

100%

Simple Additive Weighting (SAW). According to this method the sum of weighted performances of alternatives under all criteria should be considered as the base for comparison. Thus, the SAW value of each alternative (SAWJ) can be calculated as follows: =

×

Table 3 indicates the results of the analysis based on the SAW method. The winning probabilities are similar to the estimated probabilities based on the Lexicographic method (Table 2). The overall ranking of the four solutions based on the SAW method is as follows: Tunnel > Dual conveyance > Stop export > Business as usual


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Table 3.The Winning Probabilities of the Alternatives Based on the SAW Method 1st 2.83% 67.27% 29.69% 0.21%

BAU T DC NE

2nd 15.43%

3ed 53.85%

4th

77.30% 7.28%

46.15%

100%

TOPSIS method. Based on this method, the distance of each alternative's performance from the best and worst performances under the same alternative should be calculated. The alternative with the least overall distance from the best performances under different criteria is the TOPSIS winner. Under this method, first the decision matrix is normalized. Normalized performance of alternative 'i' under criteria 'j' for TOPSIS method (Nij) can be calculated as follows: = ∑ The weighted normalized decision matrix (V) can be used to identify the best and worst performances under each criterion. The V-matrix can be formed as follows: =

.

Since, only two criteria are considered in this problem, the best and the worst performance in mathematical terms are: = min = max

| = 1,2, … , } | = 1,2, … , }

, max , min

Finally, the TOPSIS winner is the alternative with the largest =

+

Where: .

=

(

−

) .

=

(

−

)


:

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Application of the TOPSIS method to the Delta decision making problem yields the results showed in Table 4. The winning probabilities are similar to the estimated probabilities based on the two previous methods (Tables 2 and 3). The overall ranking of the four solutions based on this method is as follows:


Tunnel > Dual conveyance > Stop export > Business as usual Table 4.The Winning Probabilities of the Alternatives Based on the TOPSIS Method BAU T DC NE

1st 2.83% 67.11% 29.85% 0.21%

2nd 15.31%

3ed 53.85%

4th

77.32% 7.37%

46.15%

100%

Maximin. Based on this method the alternative that maximizes the minimum satisfaction of all criteria is the best option. In other words, minimum performance of all alternatives under each criterion should be considered. The alternative that yields the greatest minimum performance is the winner based on the Maximin method. Since the performances of alternatives for two considered criteria are not in the same units, normalization of data is required. The ranking of alternatives due to this method is as follows: Tunnel> Dual conveyance > Business as usual >Stop exports This ranking is different from the consistent rankings based on the previous three methods. While the other three methods suggest that the current water export strategy (BAU) is inferior to all other three solutions, the Maximin method suggests that BAU is still better than stopping the water exports (SE) due to the high costs of this solution. The estimated winning probabilities based on this method are different from the same based on the other three methods. Based on the Maximin method, building the tunnel is slightly preferred to the dual conveyance option. Table 5.The Winning Probabilities of the Alternatives Based on the Maximin Method BAU T DC NE

1st 10.12% 43.71% 42.72% 3.45%

2nd 23.68%

3ed 88.51%

69.87% 6.45%

100.00%

4th 100%

Dominance. Based on this method, the winner can be identified using a pair wise comparison. The least dominated alternative in pair wise comparisons with all other alternatives is the winner. To have such a comparison, performance of each pair of alternatives will be compared under different criteria. The alternative that has a better


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performance under more criteria is the dominant alternative. The overall ranking of the alternatives based on this method is as follows. This ranking is similar to the ranking based on the Maximin method.


Tunnel > Dual conveyance > Business as usual >Stop exports As shown in Table 6 application of this method has resulted in winning probabilities that are different from the same under the previous methods. It should be noted that unlike other methods, based on the Dominance method the summation of winning probabilities of the alternatives under each criterion is more than 100%, due to the possibility of ties. Table 6.The winning probabilities of the alternatives based on the Dominance method BAU T DC NE

1st 10.79% 72.45% 53.65% 13.20%

2nd 36.35%

3ed 88.51%

91.90% 42.10%

100.00%

4th 100%

SUMMARY AND CONCLUSIONS The results of the analysis based on the five different MCDM methods used in this study are summarized in Table 7. All methods suggest building a tunnel (T) as the optimal solution to the Delta problem and development of a dual conveyance (DC) system as the second best option. While three of the applied MCDM methods suggest that continuation of the water exports (BAU) is the worst strategy, two of the methods suggest that this strategy is still better than ending the water exports (SE) completely due to the high economic costs of this strategy despite its significant environmental benefits. Table 7. Summary of Results MCDM Methods Preference Order Lexicographic SAW TOPSIS MAXIMIN Dominance

T> DC > SE >BAU T> DC > SE >BAU T> DC > SE >BAU T> DC >BAU> SE T> DC >BAU> SE

Table 8 compares the overall ranking of the four considered alternatives using four different approaches, namely the stochastic game theoretic approach (Madani and Lund, 2011), the stochastic bargaining approach (Madani et al., 2011), the social stochastic fallback bargaining approach (Shalikaran et al., 2011), and the stochastic MCDM approach (this study). The differences between the results suggest that the


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final results of multi-criteria multi-decision making problems can be sensitive to the choice of the analysis method. Although, the differences between the results may not be significant for a small problem like the Delta problem, such differences become more important for more complex decision making problems. This highlights the importance of selecting the most appropriate analysis method that better reflects the reality of decision making problem. In case of participation of multiple stakeholders, MCDM and social choice decision making methods may not be appropriate for analyzing the decision making problem as these methods are not descriptive. Game theory and fallback bargaining methods seem more reliable for analyzing such situations. On the other hand, descriptive methods may fail to provide the best solutions when in practice the central planner has the authority to implement the solution. Table 8. Comparison of Results of Different Stochastic Decision Analysis Methods Category Method Results Game theory approach Non-cooperative solutions T>DC>BAU> SE (Madani and Lund, 2011) Unanimity FB T>DC>BAU>SE Fall-back Bargaining 1-Approval FB SE>T>DC>BAU (FB) methods 2-Approval FB T>DC>BAU=SE (Madani et al., 2011) FB with impasse T>DC>BAU=SE Borda score T>DC>SE=BAU Condorcet choice T>DC>BAU>SE Plurality rule SE>T>DC>BAU Social choice rules T>DC>BAU>SE (Shalikarian et al., 2011) Median voting rule Majoritarian compromise T>DC>BAU>SE Condorcet practical T>DC>BAU>SE Lexicographic T>DC>SE>BAU SAW T>DC>SE>BAU TOPSIS T>DC>SE>BAU MCDM methods MAXIMIN T>DC>BAU>SE Dominance T>DC>BAU>SE REFERENCES Al-Kloub, B., Al-Shemmeri, T., and Pearman, A. (1997). "The Role of Weighting in Multi-Criteria Decision Aid, And the Ranking of Water Projects in Jordan" European Journal of Operational Research, Vol. 99, p.p. 278-288. California Department of Water Resources (CA DWR). (2008). "Where Rivers Meet-The Sacramento-San Joaquin Delta" State Water Projects, Available at http://www.water.ca.gov/swp/delta.cfm Chan, L. K., Kao, H. P., Ng, A., and Wu, M. L. (1999). "Rating the Importance of Customer Needs in Quality Function Deployment by Fuzzy and Entropy


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Methods" International Journal of Production Research, Vol. 37 (11), pp. 2499- 2518. Ingebritsen, S. E., Ikehara, M. E., Galloway, D. L., and Jones, D. R. (2000). "Delta Subsidence in California the Sinking Heart of the State" USGS Online Publications, Fact Sheet 00500. Lund, J. R., Hanak, E., Fleenor, W., Howitt, R., Mount, J., and Moyle, P. (2007). “Envisioning Futures for the Sacramento-San Joaquin Delta” Public Policy Institute of California, CA. Lund, J. R., Hanak, E., Fleenor, W., Bennett, W., Howitt, R., Mount, J., and Moyle, P. (2008)."Comparing Futures for the Sacramento–San Joaquin Delta"Public Policy Institute of California, CA. Linkov, I., Varghese, A., Jamil, S., Seager, T., Kiker, G., and Bridges, T. (2005). "Multi-Criteria Decision Analysis: A Framework for Structuring Remedial Decisions at Contaminated Sites" Comparative Risk Assessment and Environmental Decision Making, NATO Science Series IV: Earth and Environmental Sciences, Vol. 38 (1), pp. 15-54. Madani, K., and Lund, J. R. (In Press). "California’s Sacramento-San Joaquin Delta Conflict: From Cooperation to Chicken" Journal of Water Resources Planning and Management. Madani, K., and Lund, J. R. (2011). "A Monte-Carlo Game Theoretic Approach for Multi-Criteria Decision Making Under Uncertainty" Advances in Water Resources, Vol. 35 (5), pp. 607-616. Madani, K., Shalikarian, L., and Naeeni, S. T. O. (2011). "Resolving HydroEnvironmental Conflicts Under Uncertainty Using Fallback Bargaining Procedure" Proceeding of the 2011 International Conference on Environment Science and Engineering (ICESE 2011), pp. 192-196, Bali Island, Indonesia. Shalikarian, L., Madani, K., and Naeeni S. T. O. (2011). “Finding the Socially Optimal Solution for California’s Sacramento-San Joaquin Delta Problem” Proceeding of the 2011 World Environmental and Water Resources Congress, pp. 3190-3197, ASCE, Palm Springs, California. Yaghoubi, N. M., Baradaran, V., and Shahraki, M. I. (2011). "Selecting Contractor With Cooperate VIKOR Model (Case Study Wheat Flour Mill)" Business and Management Review, Vol. 1 (7), pp. 20 – 27.


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