Feasibility Mapping for Multi-attribute Decision Making

1. Informs/MC34.1 Feasibility Mapping for Multi-attribute Decision Making Liang Zhu, David Kazmer and Yanli Zhao Department of Mech. and Ind. Engineering University of Mass. Amherst November 2000

2. Introduction • Multi-attribute decision making • Coupled and competitive multiple objectives • Implicit preference over multiple attribute • Fuzzy constraint limits • Limited decision range in practical applications • Feasibility mapping • Visualization of the decision space and the performance space (attribute space) • Efficiency frontier for multi-attributes • Adjustable constraint limits

3. Standard form • ai1 x1 + ai2 x2 + ... + ainxnbi • aj1 x1 + aj2 x2 + ... + ajnxnbj • ak1 x1 + ak2 x2 + ... + aknxn = bk • x1, ..., xn 0, b1, ..., bm 0, • m = mi + mj + mk • Canonical form • x1+a11'xm+1+a12'xm+2+...+a1n'xm+n= b1' • x2+a21'xm+1+a22'xm+2+...+a2n'xm+n= b2' • ... • xm+am1'xm+1+am2'xm+2+...+amn'xm+n=bm' • x1, ..., xn+m  0, b1', ..., bm'  0 Linear Feasibility Mapping • Linear formulation • Objective in the constraint form • With the minimum acceptable performance level • Nonnegative decision variables • Convex feasible space • Solved by an extensive simplex method • Explore all adjacent extreme points • Store the connection graph

4. H3: x1 + 4x2 12 H2: x1 + x2 6 Objective E H1: x1 - 2x2 3 D G C F A B Case with 2 Decision Variables x2 H5: Primary constraint x2 0 Feasible Space x1 H4: Primary constraint x1 0

5. Independent variables Dependent variables Blocking distance E B C D Pivot Operation j = Min {bi/aij: bi/aij > 0} A

6. Q, Find the initial extreme point xs xind the new set of independent variables Push(xs, Q) xind existing in Q? Yes Yes Q= ? No No End xe Head[Q] xfxe pivots around ix, Push(xf, Q) ix  the next independent variable of xs No ix= ? Yes Pop(Q) Breadth-First Traversal

7. Algorithm Refinement • Initial extreme point • Two-phase method • Identification of an empty feasible space • Unbounded feasible space • aij 0when computing Min {bi/aij: bi/aij > 0} • Ignore the pivot at aij • Degeneracy • bi= 0 or tied blocking distance bi/aij • Alias for extreme point

8. Analysis • Correctness • All extreme points connected • Each extreme point reached in the shortest path • Time • Dominating pivot operation in O(mn) • Number of the pivots (extreme points) • Mean number  2n (Berenguer and Smith 1986) • NP Problem • More efficient than the exhaustive combinations • Sensitive to the number of decision variables

9. Option 2 3 1 0.25 Multi-attribute Space x2 B A • Duality in linear problems • Extreme points from x to y • Decision with multi-attribute space • Efficiency Frontier ABCD • Adjust the constraint limits • Options valuation • Inverse approach C Decision Space D x1 Cost A Attribute Space B C D Time (s)

10. Simulation Local linearization Linear patch Preferred solution Non-linear Systems • Simulation • Sampling within the decision range • Simple but expensive • Linear approximation • Local linearization • Depending on the initial point • Linear patch x2 Feasible space x1

11. Experimental Results y1 = 1-x12/2- x22/2 y2 = 1-x12/2- (x2-1)2/8 0.1  y1, y2 0.95 0  x1, x2 1

12. Conclusions • To establish the global feasibility mapping • An extensive simplex method for the linear model • Working on the approximation of non-linear systems • To support multi-attribute decision modeling and selection • Rational trade-off on multiple objectives • Enable the refinement on fuzzy constraints • Working on options valuation for successive decisions