Risk-Based ROI, Capital Budgeting, and Portfolio Optimization in the Department of Defense https://www.dau.edu
If only one descriptor is used to rank the objects, then it is possible to defne a total order in P . In general, given x, y ϵ P , if q i ( x ) ≤ q i ( y ) Ɐ i , then x and y are said to be comparable. However, if two descriptors are used simultaneously, the following could happen: q 1 ( x ) ≤ q 1 ( y ) and q 2 ( x ) > q 2 ( y ). In such a case, x and y are said to be incomparable (denoted by x || y ). If several objects aremutually incomparable, set P is called a partially ordered set or poset . Note that since comparisons are made for each criterion, no normalization is required. Anonparametric ranking technique canbe used toperformranking decisions from the available information without using any aggregation criterion. However, while it cannot always provide a total order of objects, it does provide an interesting overall picture of the relationships among objects. A useful approach to produce a ranking is based on the concept of the average rank of each object in the set of linear extensions of a poset (De Loof, De Baets, &DeMeyer, 2011). Since the algorithms suggested for calculating such average ranks are exponential in nature (De Loof et al., 2011), special approximations have been developed, such as the Local Partial OrderModel (LPOM; Bruggemann, Sorensen, Lerche, & Carlsen, 2004), the extended LPOM (LPOMext; Bruggemann & Carlsen, 2011), or the approximation suggested by De Loof et al. (2011). From the Hasse diagram, several sets can be derived (Bruggemann & Carlsen, 2011). If x ϵ P , 1. U ( x ), the set of objects incomparable with x : U ( x ):= { y ϵ P : x || y } 2. O ( x ), the down set: O ( x ): = { y ϵ P : y ≤ x } 3. S ( x ), the successor set: S ( x ): = O ( x )−{ x } 4. F ( x ), the up set: F ( x ):= { y ϵ P : x ≤ y } Then, the following average rank indices are defned: a. LPOM ( x ) = (| S ( x )| + 1)×( n + 1)÷( n + 1 − | U ( x )|) < p y b. LPOMext ( x )= | O ( x )|+ y ϵ Σ U ( x ) p < y + p y > where n is the number of objects,
| V | defnes the cardinality of the set V , p < y = | O ( x ) ∩ U ( y )|, p > y = | F ( x ) ∩ U ( y )|, and y
ϵ U ( x )
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Defense ARJ, January 2020, Vol. 27No. 1 : 60-107
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