Tradeoff-based decomposition and decision-making in multiobjective programming
Introduction
Many decision problems in management or finance as well as various other areas require the simultaneous consideration of several criteria and, thus, are often modeled and solved using methods from multiobjective programming and multicriteria decision-making. A common characteristic of all these problems is that, in general, there does not exist a unique optimal solution, but a set of so-called Pareto solutions from among which the decision maker (DM) chooses based on personal preferences or additional criteria not included in the original optimization model. A great variety of approaches already exists to generate the Pareto set and to support the consecutive selection of solutions using multiple criteria decision analysis, many of which are reviewed in the surveys recently collected in [3], [14]. In practice, however, choosing a final preferred decision remains difficult especially if the numbers of objectives are too large to conduct a comprehensive but for the DM still manageable tradeoff analysis between the different decision criteria.
In remedy of this situation, we primarily see two possible ways out. First, to keep the number of objectives that are to be considered simultaneously at a manageable size, we may choose to restructure the original problem by an objective space or function decomposition and then coordinate between the resulting subproblems that are defined using only subsets of the original objective function components [1], [8], [12], [15], [16], [23], [42]. In this paper, we further advance this approach by a specific tradeoff-based decomposition that allows to more directly include a priori known preferences in the form of recent cone-based domination or preference models [17], [21], [27], [30], [32], [33], [40], which also provides a more flexible means to impose multiple tradeoff structures for individual subproblems to account for changing or variable preferences of the DM [7], [18].
Second, to facilitate the evaluation of any remaining tradeoffs that, initially, are unknown or only partially revealed, we may replace the final selection step by an interactive method [12], [20], [23], [29] which supports the decisions of the DM and enhances the ease of articulating preferences to reduce or more accurately identify a set of potential candidate solutions for the original problem. We deploy these benefits within the proposed decomposition framework by proposing a novel interactive decision-making procedure that coordinates additional tradeoff decisions within or between the different subproblems using a concept of approximate efficiency [22], [25], [39], which thereby enables the DM to work exclusively with the decomposed and smaller-sized subproblems to identify a preferred decision for the overall problem. While a series of analytical results establish a theoretical foundation for this procedure, its illustrative application to a quadri-objective problem from portfolio optimization suggests that the proposed method is an effective and promising new solution technique also for practical multiobjective programming and multicriteria decision-making.
Section snippets
Preliminaries
We briefly clarify the adopted notation and review the most important concepts used in this paper. The reader familiar with the area will typically skip most of this part and may only briefly glance over the discussion of tradeoffs beginning with Definition 2.3, before reading on in Section 3.
A multiobjective programming (MOP) model for a multicriteria decision-making (MCDM) problem is commonly described by three main elements, namely
- 1.
a (continuous or discrete, finite or infinite) set of
Tradeoff-based decomposition
To make full advantage of the benefits arising from the a priori specification of tradeoffs, that are discussed in the last paragraph of Section 2, we assume that the DM is willing and, maybe more critical, capable to provide the analyst or modeler with the tradeoff matrix T, or equivalently, with all maximal tradeoff ratios by answering the following question for each pair of objectives:
“What is the maximum decrease (or decay) you are willing to endure in objective to gain a
Tradeoff-based coordination and decision-making
Following the above decomposition that is based on a priori tradeoffs, we now present an interactive procedure that enables the DM to freely navigate between the resulting subproblems, explore their sets of efficient decisions and nondominated outcomes, and coordinate any remaining tradeoffs by making decisions on the associated coordination parameters . In particular, the smaller the values of in some , the more closely the corresponding outcomes will approximate a nondominated
Application to multiobjective portfolio optimization
In this section, we apply the tradeoff-based decomposition and coordination procedure developed so far to a small real-case decision problem from financial management. Following the seminal work by Markowitz [26] and Roy [34], much attention has been given to the bi-objective mean–variance (MV) formulation for portfolio optimization (for a comprehensive survey of MCDM approaches to this and other financial problems, see [36]). Most recently, a few researchers have also investigated several
Conclusion
We presented an interactive decision-making procedure for the solution of multicriteria optimization and decision problems that combines the highlighted concept of objective tradeoffs with recent advances in decomposition methods, cone-based preference modeling, and approximate efficiencies. Based on several theoretical results in Sections 3 Tradeoff-based decomposition, 4 Tradeoff-based coordination and decision-making and the illustrative application in Section 5, we conclude that this
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