page_title: Multiple-criteria decision analysis
section_title: Generating nondominated solutions
hierarchical_section_title: Multiple-criteria decision analysis / Foundations, concepts, definitions / Generating nondominated solutions
caption_attribution_description: English: Figure 3. Projecting points onto the nondominated set with an Achievement Scalarizing Function
context_page_description: Multiple-criteria decision-making or multiple-criteria decision analysis is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making. Conflicting criteria are typical in evaluating options: cost or price is usually one of the main criteria, and some measure of quality is typically another criterion, easily in conflict with the cost. In purchasing a car, cost, comfort, safety, and fuel economy may be some of the main criteria we consider – it is unusual that the cheapest car is the most comfortable and the safest one. In portfolio management, we are interested in getting high returns while simultaneously reducing risks; however, the stocks that have the potential of bringing high returns typically carry high risk of losing money. In a service industry, customer satisfaction and the cost of providing service are fundamental conflicting criteria.
  In our daily lives, we usually weigh multiple criteria implicitly and we may be comfortable with the consequences of such decisions that are made based on only intuition.
context_section_description: There are several ways to generate nondominated solutions.  We will discuss two of these.  The first approach can generate a special class of nondominated solutions whereas the second approach can generate any nondominated solution.
  Weighted sums (Gass & Saaty, 1955)
  If we combine the multiple criteria into a single criterion by multiplying each criterion with a positive weight and summing up the weighted criteria, then the solution to the resulting single criterion problem is a special efficient solution.  These special efficient solutions appear at corner points of the set of available solutions.  Efficient solutions that are not at corner points have special characteristics and this method is not capable of finding such points.  Mathematically, we can represent this situation as
  max  wᵀ.q =  wᵀ.f(x), w> 0
  subject to
  x ∈ X
  By varying the weights, weighted sums can be used for generating efficient extreme point solutions for design problems, and supported (convex nondominated) points for evaluation problems.
  Achievement scalarizing function (Wierzbicki, 1980)
  Achievement scalarizing functions also combine multiple criteria into a single criterion by weighting them in a very special way.  They create rectangular contours going away from a reference point towards the available efficient solutions.  This special structure empower achievement scalarizing functions to reach any efficient solution.  This is a powerful property that makes these functions very useful for MCDM problems.
  Mathematically, we can represent the corresponding problem as
  Min s(g, q, w, ρ) =  Min  {maxᵢ [(gᵢ − qᵢ)/wᵢ ] + ρ ∑ᵢ (gᵢ − qᵢ)},
  subject to
  q ∈ Q
  The achievement scalarizing function can be used to project any point (feasible or infeasible) on the efficient frontier. Any point (supported or not) can be reached. The second term in the objective function is required to avoid generating inefficient solutions. Figure 3 demonstrates how a feasible point, g₁, and an infeasible point, g₂, are projected onto the nondominated points, q₁ and q₂, respectively, along the direction w using an achievement scalarizing function.  The dashed and solid contours correspond to the objective function contours with and without the second term of the objective function, respectively.
caption: Figure 3. Projecting points onto the nondominated set with an Achievement Scalarizing Function
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