Reduced sub-powers and the decision problem for finite algebras in arithmetical varieties

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the problem. We apply these results to the decision problem of interest and present illustrative examples.

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Logical Decision Problems and Complexity of Logic Programs

Fundamenta Informaticae ◽

10.3233/fi-1987-10102 ◽

1987 ◽

Vol 10 (1) ◽

pp. 1-33

Author(s):

Egon Börger ◽

Ulrich Löwen

Keyword(s):

Fixed Point ◽

Computational Complexity ◽

Decision Problem ◽

Decision Problems ◽

Complexity Classes ◽

Logic Programs ◽

Finite Structures ◽

Syntactic Restrictions

We survey and give new results on logical characterizations of complexity classes in terms of the computational complexity of decision problems of various classes of logical formulas. There are two main approaches to obtain such results: The first approach yields logical descriptions of complexity classes by semantic restrictions (to e.g. finite structures) together with syntactic enrichment of logic by new expressive means (like e.g. fixed point operators). The second approach characterizes complexity classes by (the decision problem of) classes of formulas determined by purely syntactic restrictions on the formation of formulas.

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A Framework for Modelling Economic Regional Location Processes Under Uncertainty

Journal of Quantitative Economics ◽

10.1007/s40953-021-00248-w ◽

2021 ◽

Author(s):

Roy Cerqueti ◽

Eleonora Cutrini

Keyword(s):

Theoretical Model ◽

Theoretical Analysis ◽

Site Selection ◽

Decision Problem ◽

Stochastic Theory ◽

Spatial Location ◽

Probabilistic Framework ◽

Location Patterns ◽

Regional Location ◽

Quantitative Tool

AbstractThis paper deals with the theoretical analysis of the spatial concentration and localization of firms and employees over a set of regions. In particular, it provides a simple site-selection theoretical model to describe the probabilistic framework of the location patterns. The adopted quantitative tool is the stochastic theory of urns. The model moves from the empirical evidence of the deviation of the spatial location of companies from the uniform distribution and of employees from the distribution of firms. Factors leading to such deviations are taken into consideration. Specifically, we formalize a decision problem grounded on the economic attributes of the regions and also on the distribution of the existing firms and employees in the territory. To our purpose, the site-selection model is presented as a stepwise process.

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Comparison of Judgment Scales of the Analytical Hierarchy Process — A New Approach

International Journal of Information Technology & Decision Making ◽

10.1142/s0219622019500044 ◽

2019 ◽

Vol 18 (02) ◽

pp. 445-463 ◽

Cited By ~ 15

Author(s):

Klaus D. Goepel

Keyword(s):

Analytical Hierarchy Process ◽

Analytic Functions ◽

Decision Problem ◽

Decision Criteria ◽

Maximum Weight ◽

Analytic Hierarchy ◽

New Approach ◽

Scale Functions ◽

Decision Method ◽

Hierarchy Process

The analytic hierarchy process (AHP) remains a popular multi-criteria decision method. One topic under discussion of AHP is the use of different scales to translate judgments into ratios. The author makes a new approach to compare different scale functions and to derive a recommendation for the application of scales. The approach is based on simple analytic functions and takes into consideration the number of criteria of the decision problem. A generalization of the so-called balanced scale is proposed, and a new adaptive-balanced scale is introduced. Scales are then categorized and compared based on weight boundaries and weight ratios, weight uncertainties, weight dispersion and number of decision criteria. Finally, a practical example of a decision hierarchy is presented applying the different scales. The results show that the generalized balanced scale improves weight dispersion and weight uncertainty in comparison to the fundamental AHP scale. The proposed adaptive-balanced scale overcomes the problem of a change of the maximum weight depending on the number of decision criteria.

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Two Remarks on Blackwell's Theorem

Journal of Applied Probability ◽

10.1017/s0021900200004459 ◽

2008 ◽

Vol 45 (02) ◽

pp. 580-586 ◽

Cited By ~ 1

Author(s):

Ehud Lehrer ◽

Eran Shmaya

Keyword(s):

Decision Maker ◽

Information Structure ◽

Decision Problem ◽

Partial Information ◽

The Other ◽

Actual State ◽

Information Structures ◽

Comparison Of Experiments ◽

Classical Characterization ◽

Blackwell's Theorem

In a decision problem with uncertainty a decision maker receives partial information about the actual state via an information structure. After receiving a signal, he is allowed to withdraw and gets zero profit. We say that one structure is better than another when a withdrawal option exists if it may never happen that one structure guarantees a positive profit while the other structure guarantees only zero profit. This order between information structures is characterized in terms that are different from those used by Blackwell's comparison of experiments. We also treat the case of a malevolent nature that chooses a state in an adverse manner. It turns out that Blackwell's classical characterization also holds in this case.

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Entry and Exit Decision Problem with Implementation Delay

Journal of Applied Probability ◽

10.1017/s0021900200004964 ◽

2008 ◽

Vol 45 (04) ◽

pp. 1039-1059 ◽

Cited By ~ 3

Author(s):

Marius Costeniuc ◽

Michaela Schnetzer ◽

Luca Taschini

Keyword(s):

Decision Problem ◽

Analytic Solution ◽

Time Lag ◽

Probabilistic Approach ◽

Stopping Time ◽

Stopping Times ◽

Maximization Problem ◽

Entry And Exit ◽

Brownian Motion With Drift ◽

Implementation Delay

We study investment and disinvestment decisions in situations where there is a time lagd> 0 from the timetwhen the decision is taken to the timet+dwhen the decision is implemented. In this paper we apply the probabilistic approach to the combined entry and exit decisions under the Parisian implementation delay. In particular, we prove the independence between Parisian stopping times and a general Brownian motion with drift stopped at the stopping time. Relying on this result, we solve the constrained maximization problem, obtaining an analytic solution to the optimal ‘starting’ and ‘stopping’ levels. We compare our results with the instantaneous entry and exit situation, and show that an increase in the uncertainty of the underlying process hastens the decision to invest or disinvest, extending a result of Bar-Ilan and Strange (1996).

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Posterior Implementability in a Two-Person Decision Problem

Econometrica ◽

10.2307/1911157 ◽

1987 ◽

Vol 55 (1) ◽

pp. 69 ◽

Cited By ~ 54

Author(s):

Jerry R. Green ◽

Jean-Jacques Laffont

Keyword(s):

Decision Problem

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