Narrow Results

In equitable multiobjective optimization, all of the objectives are uniformly optimized, but in some cases, the decision maker believes that some of them should be uniformly optimized. In order to solve the proposed problem, we introduce the concept of equitable $A_P$-efficiency, where $P=\{P_1,P_2,\dots ,P_n\}$ is a partition of the index set of objective functions and the preference matrix $A_P$ is the direct sum of the matrices $A_1,A_2,\dots ,A_n$, in which $A_k$ is a preference matrix for the objective functions in the class $P_k$ for $k=1,2,\dots ,n$. We examine some theoretical and practical aspects of equitably $A_P$-efficient solutions and provide the some conditions that guarantee the relation of equitable $A_P$-dominance is a $P$-equitable rational preference.

Furthermore, we introduce the new problem with the preference matrix $A_P$ and we decompose it into a collection of smaller subproblems. In continuation, the subproblems are solved by the concept of equitable efficiency. Finally, two models are demonstrated to coordinate equitably efficient solutions of the proposed subproblems.

Multiplicative random processes in (not necessarily equilibrium or steady state) stochastic systems with many degrees of freedom lead to Boltzmann distributions when the dynamics is expressed in terms of the logarithm of the elementary variables. In terms of the original variables this gives a power-law distribution. This mechanism implies certain relations between the constraints of the system, the power of the distribution and the dispersion law of the fluctuations. These predictions are validated by Monte Carlo simulations and experimental data. We speculate that stochastic multiplicative dynamics might be the natural origin for the emergence of criticality and scale hierarchies without fine-tuning.

The paper studies the result of Zhu²¹ and establishes a relationship between the efficiency in data envelopment analysis (DEA) and the pareto optimality under multiple objective linear programming (MOLP). It is shown that the DEA/preference structure models in Zhu²¹ can be derived by traditional MOLP techniques. Incorporation of tradeoffs or value judgments is a direct result of using MOLP techniques. New uses of DEA are developed and described. The approach is applied to a set of Chinese cities.