The fair allocation of common fixed cost or revenue using DEA concept

The common fixed cost or revenue distribution amongst decision making units (briefly, DMUs) in an equitable way is one of the problems that can be solved by data envelopment analysis (DEA) concept. The motivation of this paper is common fixed cost or revenue allocation based on following three principles: First, allocation must be directly proportional to the elements (inputs and outputs) that are directly proportional to imposed common fixed cost or to obtained common fixed revenue. Second, allocation must be inversely proportional to the elements that are inversely proportional to common fixed cost or revenue. Finally, the elements that have no effect on common fixed cost or revenue must have no effect on allocation as well.     

Fixed cost and resource allocation based on DEA cross-efficiency

In many managerial applications, situations frequently occur when a fixed cost is used in constructing the common platform of an organization, and needs to be shared by all related entities, or decision making units (DMUs). It is of vital importance to allocate such a cost across DMUs where there is competition for resources. Data envelopment analysis (DEA) has been successfully used in cost and resource allocation problems. Whether it is a cost or resource allocation issue, one needs to consider both the competitive and cooperative situation existing among DMUs in addition to maintaining or improving efficiency. The current paper uses the cross-efficiency concept in DEA to approach cost and resource allocation problems. Because DEA cross-efficiency uses the concept of peer appraisal, it is a very reasonable and appropriate mechanism for allocating a shared resource/cost. It is shown that our proposed iterative approach is always feasible, and ensures that all DMUs become efficient after the fixed cost is allocated as an additional input measure. The cross-efficiency DEA-based iterative method is further extended into a resource-allocation setting to achieve maximization in the aggregated output change by distributing available resources. Such allocations for fixed costs and resources are more acceptable to the players involved, because the allocation results are jointly determined by all DMUs rather than a specific one. The proposed approaches are demonstrated using an existing data set that has been applied in similar studies.     

Allocating the fixed cost as a complement of other cost inputs: A DEA approach

In cost allocation problem, traditional DEA approaches allocate the fixed cost among a group of decision making units (DMUs), and treat the allocated cost as an extra input of each DMU. If costs except for the fixed cost are regarded as inputs in the cost allocation problem, then it is obvious that the fixed cost is a complement of other inputs rather than an extra independent input. Therefore it is necessary to combine the allocated cost with other cost measures in cost allocation problem. Based on this observation, this paper investigates the relationship between the allocated cost and the DEA efficiency score and develops a DEA-based approach to allocate the fixed cost among various DMUs. An example of allocating advertising expenditure between a car manufacturer and its dealers is presented to illustrate the method proposed in this paper.     

Resource allocation for branch network system with considering heterogeneity based on DEA method

In this paper, we proposed a new DEA approach to allocate the resource in branch network system which is not covered by the existing resource allocation works under a centralized decision-making environment. The branch network system is typically appears in multi-national or multi-regional corporations, which has many branches across multiple locations. Given the spatial distribution of the production, we imposed additional restrictions on resource allocation and divided the resource inputs into three groups: fixed inputs, regional inputs that allocated to the branches in the same area and common resource that an additional resource allocated to all the branches. Then, we generalize the model further to accommodate technological heterogeneity due to the difference in the geographical locations of the branches. And the objective of the proposed models is to maximize the gross profits of the entire organization, which is a natural assumption for a for-profit organization. Finally, an example was presented to illustrate the proposed approach with heterogeneous technology is more practically feasible and superior than the prior approach with homogeneous technology.     

Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings

Let be a Banach space, a weakly compact convex subset of and an asymptotically nonexpansive mapping. Under the usual assumptions on which assure the existence of fixed point for , we prove that the set of fixed points is a nonexpansive retract of . We use this result to prove that all known theorems about existence of fixed point for asymptotically nonexpansive mappings can be extended to obtain a common fixed point for a commuting family of mappings. We also derive some results about convergence of iterates.

     

A generalized cyclic iterative method for solving variational inequalities over the solution set of a split common fixed point problem

For solving the large-scale linear least-squares problem, we propose a block version of the randomized extended Kaczmarz method, called the two-subspace randomized extended Kaczmarz method, which does not require any row or column paving. Theoretical analysis and numerical results show that the two-subspace randomized extended Kaczmarz method is much more efficient than the randomized extended Kaczmarz method. When the coefficient matrix is of full column rank, the two-subspace randomized extended Kaczmarz method can also outperform the randomized coordinate descent method. If the linear system is consistent, we remove one of the iteration sequences in the two-subspace randomized extended Kaczmarz method, which approximates the projection of the right-hand side vector onto the orthogonal complement space of the range space of the coefficient matrix, and obtain the generalized two-subspace randomized Kaczmarz method, which is actually a generalization of the two-subspace randomized Kaczmarz method without the assumptions of unit row norms and full column rank on the coefficient matrix. We give the upper bound for the convergence rate of the generalized two-subspace randomized Kaczmarz method which also leads to a better upper bound for the convergence rate of the two-subspace randomized Kaczmarz method.

     

Resource allocation and target setting for parallel production system based on DEA

The paper proposes methodology for resource allocation and target setting based on DEA (data envelopment analysis). It deals with organization can be modeled as consisting of several production units, each of which has parallel production lines. The previous studies in the DEA literature only deal with reallocating/allocating organizational resources to production units and set targets for them. In their researches, the production unit is treated as a black box. In such circumstances, how to arrange the production at production unit level is not clear. This paper serves to generate resource allocation and target setting plan for each production unit by opening the black box. The proposed model exploits production information of production lines in generating production plans. The resulting plan has following characteristics: (1) the performance of each production lines are evaluated under common weights; (2) the weights chose for evaluation keep the efficiency of the entire unit not worse off; (3) the worst behaved production line in the production unit under evaluation are improved as much as possible. Finally, the real data of a production system extracted from extant literature are used to demonstrate the proposed method.     

A method for group decision-making based on determining weights of decision makers using TOPSIS

In general, weights of decision makers (DMs) play a very important role in multiple attribute group decision-making (MAGDM), how to measure the weights of DMs is an interesting research topic. This paper presents a new approach for determining weights of DMs in group decision environment based on an extended TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method. We define the positive ideal solution as the average of group decision. The negative ideal solution includes two parts: left and right negative ideal solution, which are the minimum and maximum matrixes of group decision, respectively. We give an example to illustrate the developed approach. Finally, the advantages and disadvantages of this study are also compared.     

Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping

Mathematical Programming - The existing algorithms for solving the convex minimization problem over the fixed point set of a nonexpansive mapping on a Hilbert space are based on algorithmic...     

A method for efficiently computing the number of codewords of fixed weights in linear codes

The problem of computing the number of codewords of weights not exceeding a given integer in linear codes over a finite field is considered. An efficient method for solving this problem is proposed and discussed in detail. It builds and uses a sequence of different generator matrices, as many as possible, so that the identity matrix takes disjoint places in them. The efficiency of the method is achieved by optimizations in three main directions: (1) the number of the generated codewords, (2) the check whether a given codeword is generated more than once, and (3) the operations for generating and computing these codewords. Since the considered problem generalizes the well-known problems “Weight Distribution” and “Minimum Distance”, their efficient solutions are considered as applications of the algorithms from the method.     

An approximation method for the efficiency set of multiobjective programming problems

In this paper so-called ε-approximations for the efficiency set of vector minimization problems are defined. A general generating algorithm for such E-approximations is given which will be modified for linear continuous problems by means of the Dual Simplex Method.     

Strong convergence of a hybrid steepest descent method for the split common fixed point problem

We study the convergence properties of an iterative method for a variational inequality defined on a solution set of the split common fixed point problem. The method involves Landweber-type operators related to the problem as well as their extrapolations in an almost cyclic way. The evaluation of these extrapolations does not require prior knowledge of the matrix norm. We prove the strong convergence under the assumption that the operators employed in the method are approximately shrinking.     

Regularization methods for accretive variational inequalities over the set of common fixed points of nonexpansive semigroups

In this paper, we introduce a regularization method based on the Browder–Tikhonov regularization method for solving a class of accretive variational inequalities over the set of common fixed points of a nonexpansive semigroup on a uniformly smooth Banach space. Three algorithms based on this regularization method are given and their strong convergence is studied. Finally, a finite-dimensional example is developed to illustrate the numerical behaviour of the algorithms.     

Improvement of efficiency intervals based on DEA by adjusting inputs and outputs

We improve the efficiency interval of a DMU by adjusting its given inputs and outputs. The Interval DEA model has been formulated to obtain an efficiency interval consisting of evaluations from both the optimistic and pessimistic viewpoints. DMUs which are not rated as efficient in the conventional sense are improved so that their lower bounds become as large as possible under the condition that their upper bounds attain the maximum value one. The adjusted inputs and outputs keep each other balanced by improving the lower bound of efficiency interval, since the lower bound becomes small if all the inputs and outputs are not proportioned. In order to improve the lower bound of efficiency interval, different target points are defined for different DMUs. The target point can be regarded as a kind of benchmark for the DMU. First, a new approach to improvement by adjusting only outputs or inputs is proposed. Then, the combined approach to improvement by adjusting both inputs and outputs simultaneously is proposed. Lastly, numerical examples are shown to illustrate our proposed approaches.     

Approach to group decision making based on determining the weights of experts by using projection method

The aim of this paper is to present a new approach for determining weights of experts in the group decision making problems. Group decision making has become a very active research field over the last decade. Especially, the investigation to determine weights of experts for group decision making has attracted great interests from researchers recently and some approaches have been developed. In this paper, the weights of experts are determined in the group decision environment via projection method. First of all, the average decision of all individual decisions is defined as the ideal decision. After that, the weight of expert is determined by the projection of individual decision on the ideal decision. By using the weights of experts, all individual decisions are aggregate into a collective decision. Then an ideal solution of alternatives of the collective decision, expressed by a vector, is determined. Further, the preference order of alternatives are ranked in accordance with the projections of alternatives on the ideal solution. Comparisons with an extended TOPSIS method are also made. Finally, an example is provided to illustrate the developed approach.     

Pareto set approximation by the method of adjustable weights and successive lexicographic goal programming

A nonlinear multiobjective optimization problem is considered. Two methods are proposed to generate solutions with an approximately uniform distribution in a Pareto set. The first method is supposed to find the solutions as minimizers of weighted sums of objective functions where the weights are properly selected using a branch and bound type algorithm. The second method is based on lexicographic goal programming. The proposed methods are compared with several metaheuristic methods using two and three-criteria tests and applied problems.     

Three-term conjugate gradient method for the convex optimization problem over the fixed point set of a nonexpansive mapping

Many constrained sets in problems such as signal processing and optimal control can be represented as a fixed point set of a certain nonexpansive mapping, and a number of iterative algorithms have been presented for solving a convex optimization problem over a fixed point set. This paper presents a novel gradient method with a three-term conjugate gradient direction that is used to accelerate conjugate gradient methods for solving unconstrained optimization problems. It is guaranteed that the algorithm strongly converges to the solution to the problem under the standard assumptions. Numerical comparisons with the existing gradient methods demonstrate the effectiveness and fast convergence of this algorithm.     

Common solutions to a system of variational inequalities over the set of common fixed points of demi-contractive operators

In this paper, we introduce an explicit parallel algorithm for finding common solutions to a system of variational inequalities over the set of common fixed points of a finite family of demi-contractive operators. Under suitable assumptions, we prove the strong convergence of this algorithm in the framework of a Hilbert space. The results obtained in this paper extend and improve the results of Tian and Jiang (2017), of Censor, Gibali and Reich (2012), and of many others.     

An iterative method for finding common solutions of equilibrium and fixed point problems

We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.