Report

Constructing the Decision Model Y. İlker TOPCU, Ph.D. www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info www.facebook.com/yitopcu twitter.com/yitopcu Decision Matrix • Alternative evaluations w.r.t. attributes are presented in a decision matrix • Entries are performance values • Rows represent alternatives • Columns represent attributes Example Car purchasing decision Selling price Fuel cons. Comfort K TL (cost) lt/100km (cost) 1-5 (benefit) a 40 10 5 b 30 9 4 c 35 8 4 Example Material Selection Problem • Consider a material selection problem for a complex 3D part which needs to be designed for operation in a high-temperature oxygen-rich environment, under extreme bending fatigue loads. The design is not intended for mass production. The decision maker (analyst) considers 6 alternative materials and 4 factors with the data shown below. The quantification of the corrosion resistance data has been previously done using an 11-point scale. Material selection factors Machinability Material Hardness rating(%) Cost($/lb) Corrosion resitance 1 420 25 5 Extremely high (0.865) 2 350 40 3 High (0.665) 3 390 30 3 High (0.665) 4 250 35 1.3 High (0.665) 5 600 30 2.2 Very high (0.745) 6 230 55 4 Average (0.5) Attributes • Benefit attributes Offer increasing monotonic utility. Greater the attribute value the more its preference • Cost attributes Offer decreasing monotonic utility. Greater the attribute value the less its preference • Nonmonotonic attributes Offer nonmonotonic utility. The maximum utility is located somewhere in the middle of an attribute range Global Performance Value • If solution method that will be utilized is performance aggregation oriented, performance values should be aggregated. • In this case • Performance values are normalized to eliminate computational problems caused by differing measurement units in a decision matrix • Relative importance of attributes are determined Normalization • Aims at obtaining comparable scales, which allow interattribute as well as intra-attribute comparisons • Normalized performance values have dimensionless units • The larger the normalized value becomes, the more preference it has Normalization Methods 1. Distance-Based Normalization Methods 2. Proportion Based Normalization Methods (Standardization) Distance-Based Normalization Methods If we define the normalized rating as the ratio between individual and combined distance from the origin (0,0,…,0) then the comparable rating of xij is given as (Yoon and Kim, 1989): rij(p) = (xij - 0) / xkj 0 k 1 m p 1/ p This equation is arranged for benefit attributes. Cost attributes become benefit attributes by taking the inverse rating (1/ xij) Distance-Based Normalization Methods • Normalization (p=1: Manhattan distance) • Vector Normalization (p=2: Euclidean distance) • Linear Normalization (p= : Tchebycheff dist.) m rij(1) = xij / x k 1 kj 2 m x rij(2) = xij / k 1 kj rij( ) = xij / max xkj , k 1,2,...,m rij() = min xkj , k 1,2,...,m / xij (BENEFIT ATTRIBUTE) (COST ATTRIBUTE) Proporiton-Based Normalization Methods The proportion of difference between performance value of the alternative and the worst performance value to difference between the best and the worst performance values (Bana E Costa, 1988; Kirkwood, 1997) rij = (xij – xj-) / (xj* – xj-) benefit attribute rij = (xj- – xij) / (xj- – xj*) cost attribute where * represents the best and – represents the worst (best: max. perf. value for benefit; min. perf. value for cost or ideal value determined by DM for that attribute) Example Transformation of Nonmonotonic Attributes to Monotonic • exp(–z2/2) exponential function is utilized for transformation where z = (xij – xj0) / sj xj0 is the most favorable performance value w.r.t. attribute j sj is the standard deviation of performance values w.r.t. attribute j Example Attribute Weighting • Most methods translate the relative importance of attributes into numbers which are often called as “weights” (Vincke, 1992) • Methods utilized for assignment of weights can be classified in two groups (Huylenbroeck, 1995; Munda 1993; AlKloub et al., 1997; Kleindorfer et al., 1993; Yoon and Hwang, 1995): • Direct Determination • Indirect Determination Weight Assignment Methods • Direct Determination • Rating, Point allocation, Categorization • Ranking • Swing • Trade-off • Ratio (Eigenvector prioritization) • Indirect Determination • Centrality • Regression – Conjoint analysis • Interactive Swing method • Given is a set of alternatives and a set of attributes. Let N be the number of attributes. 1. Determine the best and worst value of each attribute over the set of alternatives. 2. Create N+1 fictional alternatives. The first fictional alternative is the "worst-case" and has the worst value on every attribute. The next N fictional alternatives have the worst value on all but one of the attributes; on the remaining attribute, each alternative has the best value on one attribute. (Each of these alternatives has a different best than any of the others.) 3. Rank order the N+1 fictional alternatives. The ranks are determined by the decision-maker. The rank of the worst-case alternative will be N+1, and the rank of the best of the fictional alternatives will be 1. Swing method (Cont.) 4. Rate the N+1 fictional alternatives. The rating of the worstcase alternative will be 0, and the rating of the best of the fictional alternatives will be 100. The decision-maker must rate the others and these ratings should be coherent with the rankings. That is, if one fictional alternatives has a better rank than a second, the first should have a higher rating as well. An alternative's rating is the decision-maker's increase in satisfaction if he gives up the worst-case alternative and chooses this one instead. 5. Normalize the ratings by dividing each one by the sum of all the ratings. The normalized rating of the worst-case alternative will still be 0, and the sum of all the normalized ratings will equal 1. 6. The weight for each attribute is the normalized rating of the fictional alternative that has the best value on that attribute. Source: http://wiki.ece.cmu.edu/ddl/index.php/Swing_weighting