Freerk A. Lootsma

Multi-Criteria Decision Analysis via Ratio and Difference Judgement

1. Introduction. 2. SMART, direct rating. 3. The AHP, pairwise comparisons. 4. Scale sensitivity and rank preservation. 5. The alternatives in perspective. 6. Group decision making. 7. Resource allocation. 8. Scenario analysis. 9. Conflict analysis and negotiations. 10. Multi-objective linear programming. 11. MCDA in the hands of its masters. 12. Prospects of MCDA. Subject index. About the author.

Power Relations and Group Aggregation in the Multiplicative AHP and SMART

The relative power of the members in a group of decision makers can be incorporated in the multiplicative AHP via power coefficients in the logarithmic least squares whereby we analyse the pairwise comparison matrices. When each decision maker judges every pair of alternatives under each of the criteria, aggregation over the criteria and over the decision makers proceeds via a sequence of geometric-mean calculations which can be carried out in any order, at least with predetermined criterion weights and power coefficients. Hence, since we preserve the rank order of the alternatives, we avoid a deficiency of the original AHP. We also consider SMART, an additive method which is logarithmically related to the multiplicative AHP so that power relations can easily be incorporated in it. Finally, in order to illustrate the proposed model, we analyse a generalized version of the well-known example of Belton and Gear as well as the power relations between member countries of the European Community.

Group preference aggregation in the multiplicative AHP The model of the group decision process and Pareto optimality

Abstract A recent paper has focused awareness on group aggregation procedures in the AHP, showing that geometric mean aggregation violates the desirable social choice axiom of Pareto optimality. We show that this violation can be attributed to the representation used to model the group decision process, thereby questioning the legitimacy of the Pareto optimality axiom. We furthermore propose a geometric mean group aggregation procedure which satisfies all the social choice axioms suggested.

Design and implementation of optimization software

History of mathematical programming systems.- Scope of mathematical programming software.- Anatomy of a mathematical programming system.- Elements of numerical linear algebra.- A tutorial on matricial packing.- Pivot selection tactics.- An interactive query system for MPS solution information.- Modeling and solving network problems.- Integer programming codes.- Some considerations in using branch-and bound codes.- Quadratic programming.- Nonlinear programming using a general mathematical programming system.- The design and implementation of software for unconstrained optimization.- The GRG method for nonlinear programming.- Generalized reduced gradient software for linearly and nonlinearly constrained problems.- The ALGOL 60 procedure minifun for solving non-linear optimization problems.- An accelerated conjugate gradient algorithm.- Global optima without convexity.- Computational aspects of geometric programming.- A proposal for the classification and documentation of test problems in the field of nonlinear programming.- Guidelines for reporting computational experiments in mathematical programming.- COAL session summary.- List of participants.

Pairwise-comparison methods in multiple objective programming, with applications in a long-term energy-planning model

Abstract We propose to use pairwise comparisons within the framework of ideal-point or reference-point methods for multi-objective programming. The decision makers are requested to estimate the ratios which are acceptable for deviations from the ideal vector. Thereafter, we seek the nearest feasible solution using the Tchebycheff norm. In this paper we sketch the pairwise-comparison methods, some aspects of magnitude scaling, and the ideal-point methods. We show the results of our numerical experiments in long-term energy planning with nine objective functions. Finally, we present a preliminary evaluation of the combined method on the basis of its possible contribution to interactive decision analysis.

Stochastic and fuzzy PERT

Network-planning techniques are considered based on stochastic and fuzzy models of the activity durations. The stochastic versions of Pert are generally intractable, and they cannot be used to draw up tight plans for action. Fuzzy models are closer to reality, simpler to use, but theoretically not well established. The impact of a stochastic and a fuzzy version of Pert is illustrated via a numerical example. Finally, the representation of uncertainty in network planning is discussed, when the activity durations are estimated by human experts.

Multi-criteria analysis and budget reallocation in long-term research planning

Abstract We present the results of a feasibility study conducted to the order of the Commission of the European Communities (Directorate-General for Science, Research, and Development). We used a method of pairwise comparisons to rank and rate a number of non-nuclear energy research programs under criteria which were felt to be relevant for a European energy policy. Thereafter, we employed the final scores of the programs to calculate optimal reallocations of the research budget, assuming that there is only a small number of distinct support levels for each program. The main objective of the study was to experiment with multi-criteria analysis and to design a robust reallocation method, although there is no unique numerical scale to quantify verbal human judgement and no unique cost-benefit relationship. The reallocation method is based on an elaborate sensitivity analysis.

A model for the relative importance of the criteria in the Multiplicative AHP and SMART

Abstract We define the relative importance of any pair of criteria as the substitution rate between the relative gains and losses of the alternatives when we move along an indifference curve. Under the geometric-mean aggregation rule in the Multiplicative AHP and under the arithmetic-mean aggregation rule in SMART, the relative (not the marginal) substitution rate depends neither on the performance of the alternatives under the remaining criteria nor on the units of performance measurement. Hence, it provides a sound argument for distributed decision-making processes where those who judge the criteria are not the same actors as those who assess the performance of the alternatives. The definition has a plausible basis in the psycho-physical research on the relationship between physical stimuli and sensory responses, which shows that human beings are sensitive, not to marginal but to relative changes of the stimulus intensities.

State-of-the-art in parallel nonlinear optimization

Abstract This survey is concerned with variants of nonlinear optimization methods designed for implementation on parallel computers. First, we consider a variety of methods for unconstrained minimization. We consider a particular type of parallelism (simultaneous function and gradient evaluations), and we concentrate on the main sources of inspiration: conjugate directions, homogeneous functions, variable-metric updates, and multi-dimensional searches. The computational process for solving small and medium-size constrained optimization problems is usually based on unconstrained optimization. This provides a straightforward opportunity for the introduction of parallelism. In the present survey, however, we focus on promising approaches for solving large, well-structured constrained problems: dualization of problems with separable objective and constraint functions, and decomposition of hierarchical problems with linking variables (typical for Benders decomposition in the linear case). Finally, we outline the key issues in future computational studies of parallel nonlinear optimization algorithms.

Conflict resolution via pairwise comparison of concessions

Abstract We present a pairwise-comparison method to evaluate possible deals between two parties in mutual conflict. The basic step is the evaluation of a one-to-one deal, where each party offers exactly one concession. The trade-off of benefit and cost is judged in verbal terms which are subsequently converted into numerical values on a discrete geometric scale. The information to be used by a mediator between the two parties appears to be scale-independent. The trade-off analysis is illustrated via the example of an industrial labour dispute.