A contribution to the linear programming approach to joint cost allocation: Methodology and application

The linear programming (LP) approach has been commonly proposed for joint cost allocation purposes. Within a LP framework, the allocation rules are based on a marginal analysis. Unfortunately, the additivity property which is required to completely allocate joint costs fails in presence of capacity, institutional or environmental constraints.     

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.     

Allocation of dependent divisional resources

A resource allocation problem is considered with resources that are dependent in the sense that an allocation to an activity requires the application of several resources, except for certain activities which are divisional in the sense that an allocation to such an activity requires the use of only a single resource. Return and cost functions are assumed to be continuous and increasing, and the allocation variables are continuous. Conditions are given for the replacement of the continuous problem by an associated problem with discrete variables and a single constraint, and to a given degree of accuracy. The associated problem can be efficiently solved by dynamic programming. Certain divisional resource allocation problems with discrete variables and several linear constraints are shown to be equivalent to a discrete problem with a single constraint. A numerical example is given.     

基于“零和收益”的SBM效率分配方法及应用

在松弛变量度量(slacks-based measure,SBM)效率评价方法的基础上,首先明确投入(产出)要素固定的生产系统中,投入(产出)要素在各决策单元间的竞争性关系;然后采用比例分配策略对SBM无效决策单元的投入(产出)松弛进行效率分配,以构建一个基于零和收益的SBM(zero sum gains SBM,ZSG-SBM)效率分配方法;再通过分析ZSG-SBM模型与SBM模型效率评价结果的关系,给出了比例分配策略ZSG-SBM模型的求解方法;最后应用实例研究验证了本文模型在要素存在竞争性的复杂生产系统效率评价和资源分配中的优势。     

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.     

The freight allocation problem with lane cost balancing constraint

We consider a problem faced by a buying office for one of the largest retail distributors in the world. The buying office plans the distribution of goods from Asia to various destinations across Europe. The goods are transported along shipping lanes by shipping companies, many of which have collaborated to form strategic alliances; each lane must be serviced by a minimum number of companies belonging to a minimum number of alliances. The task involves purchasing freight capacity from shipping companies for each lane based on projected demand, and subject to minimum quantity requirements for each selected shipping company, such that the total transportation cost is minimized. In addition, the allocation must not assign an overly high proportion of freight to the more expensive shipping companies servicing any particular lane, which we call the lane cost balancing constraint.This study is the first to consider the lane cost balancing constraint in the context of freight allocation. We formulate the freight allocation problem with this lane cost balancing constraint as a mixed integer programming model, and show that even finding a feasible solution to this problem is computationally intractable. Hence, in order to produce high-quality solutions in practice, we devised a meta-heuristic approach based on tabu search. Experiments show that our approach significantly outperforms the branch-and-cut approach of CPLEX 11.0 when the problem increases to practical size and the lane cost balancing constraint is tight. Our approach was developed into an application that is currently employed by decision-makers at the buying office in question.     

Cost optimal allocation of rail passenger lines

We consider the problem of cost optimal railway line allocation for passenger trains for the Dutch railway system. At present, the allocation of passenger lines by Dutch Railways is based on maximizing the number of direct travelers. This paper develops an alternative approach that takes operating costs into account. A mathematical programming model is developed which minimizes the operating costs subject to service constraints and capacity requirements. The model optimizes on lines, line types, routes, frequencies and train lengths. First, the line allocation model is formulated as an integer nonlinear programming model. This model is transformed into an integer linear programming model with binary decision variables. An algorithm is presented which solves the problem to optimality. The algorithm is based upon constraint satisfaction and a Branch and Bound procedure. The algorithm is applied to a subnetwork of the Dutch railway system for which it shows a substantial cost reduction. Further application and extension seem promising.     

Measuring returns to scale in DEA models when the firm is regulated

In the standard framework of data envelopment analysis (DEA) models, the returns to scale are fully characterized using the multiplier on the convexity constraint of inefficient decision making units (DMU) using the projection of the input–output vector on the frontier. In this note, we investigate how the returns to scale measurements in DEA models are affected by the presence of regulatory constraints. These additional constraints change the role played by the convexity constraint. In order to avoid biased estimation of the returns to scale, we show that the interaction between the regulatory and the convexity constraints has to be taken into account.     

Radial DEA models without inputs or without outputs

In this paper we consider radial DEA models without inputs (or without outputs), and radial DEA models with a single constant input (or with a single constant output). We demonstrate that (i) a CCR model without inputs (or without outputs) is meaningless; (ii) a CCR model with a single constant input (or with a single constant output) coincides with the corresponding BCC model; (iii) a BCC model with a single constant input (or a single constant output) collapses to a BCC model without inputs (or without outputs); and (iv) all BCC models, including those without inputs (or without outputs), can be condensed to models having one less variable (the radial efficiency score) and one less constraint (the convexity constraint).     

Maximally productive input–output units

Technical or Pareto–Koopmans efficiency models can be based on ratios of weighted sums of outputs to weighted sums of inputs. Differing meanings have been considered for such factor weights. In this paper, we use value or cost rate meanings depending on model orientation. These meanings permit considering the simultaneous assignment of input and output factor weights along with optimal intensity values for a virtual composite unit constructed from the observed units. An optimization principle we call the winners-take-all criterion is proposed for determining the maximally productive unit(s). No assumptions are required on the internal transformation processes of the units. The model simultaneously determines the intensities and factor weights and results in indefinite quadratic programming problems that simplify to linear programming in certain cases. For the general case, genetic search is applied. Numerical illustrations are provided for faculty merit scoring and for the 15 hospital dataset of Sherman.     

给定模糊成本开支约束情况下的最优投入组合

传统的生产者行为分析,考虑了给定产量约束下的成本支出最小化和给定成本支出约束下的产量最大化两种情况.本文利用模糊规划方法,讨论了生产者的成本开支约束为模糊约束的情况下,如何确定最优投入组合以使其产量达到最大,即实现模糊意义下的生产者均衡问题.     

Asset-liability management for Czech pension funds using stochastic programming

It is possible to model a wide range of portfolio management problems using stochastic programming. This approach requires the generation of input scenarios and probabilities, which represent the evolution of the return on investment, the stream of liabilities and other random phenomena of the problem and respect the no-arbitrage properties. The quality of the recommended capital allocation depends on the quality of the input scenarios and a validation of results is necessary. Appropriate scenario generation techniques and output analysis methods are described in the context of defined contribution pension fund and applied to the specific model of a Czech pension fund. The numerical results indicate various components that influence the recommended investment decisions and the fund’s achievements. In particular, the initial balance sheet position of the pension fund is important for the optimal investment strategy because of the accounting rules embedded in the model and tracking of both the market and purchasing value of assets.     

Reconciling ratio analysis and DEA as performance assessment tools

The assessment of operational performance remains a fundamental challenge both in practice and in theory. Data envelopment analysis (DEA) is one method developed in production economic theory and applied by researchers to study groups of enterprises. In practice, individual enterprises almost universally rely on simple output–input ratios. Each approach has its strengths and weaknesses, but the theoretical connection between the two has not been fully articulated. This paper uses the framework of DEA to establish a mathematical relationship between DEA efficiency scores and corresponding ratio analysis. The relationship can be expressed as a product of seven components: technical efficiency, technical change, scale efficiency, input slack factor, input substitution factor, output slack factor and output substitution factor.     

基于非对称成本的设计竞赛博弈模型及奖金设置

论从项目业主的角度出发，利用博弈论和信息经济学理论和方法，在竞赛参与人成本不对称条件下，建立了一个项目业主和多个设计竞赛参与人组成的设计竞赛博弈模型；并在竞赛参与人成本函数分别为线性函数、凹函数和凸函数的条件下，研究对项目业主最优的竞赛奖金设置，研究表明：对于项目业主最优的奖金设置是单项奖金的方式。     

应急物资预置问题的可信性优化模型

基于可信性理论,研究了多受灾点、多出救点、多物资的应急设备选址和物资预置问题.考虑到运输费用、出救点的供应量、受灾点的需求量和道路容量的不确定性,用模糊变量来刻画,建立了模糊环境下应急物资预置的可信性优化模型以最小化期望总费用.当模型中的模糊变量相互独立且服从三角分布时,推导了总费用目标及服务质量和弧容量约束的解析表达式,从而将原模型转化为等价的确定模型.鉴于等价模型是一个混合整数规划,可采用Lingo软件编程求解.最后,数值算例演示所提建模思想.实验结果说明了所建模型的有效性.     

基于Black-Litterman框架的资产配置策略研究

本文基于Black-Litterman框架提出了中国股票市场投资中行业间资产配置的策略。因为宏观经济指标对于股票收益率有一定的解释能力,本文通过宏观经济变量对收益率序列建模并且用GJR-GARCH模型捕捉资产收益率变化的特征,得出的预测资产收益率及其方差作为Black-Litterman框架下的输入。最后通过实证结果表明,基于这种策略构建的投资组合收益率在一定条件下会优于基于市场均衡权重或者传统Markowitz框架下的投资策略。     

Geometric conditions for Kuhn–Tucker sufficiency of global optimality in mathematical programming

We present geometric criteria for a feasible point that satisfies the Kuhn–Tucker conditions to be a global minimizer of mathematical programming problems with or without bounds on the variables. The criteria apply to multi-extremal programming problems which may have several local minimizers that are not global. We establish such criteria in terms of underestimators of the Lagrangian of the problem. The underestimators are required to satisfy certain geometric property such as the convexity (or a generalized convexity) property. We show that the biconjugate of the Lagrangian can be chosen as a convex underestimator whenever the biconjugate coincides with the Lagrangian at a point. We also show how suitable underestimators can be constructed for the Lagrangian in the case where the problem has bounds on the variables. Examples are given to illustrate our results.     

Network DEA efficiency in input–output models: With an application to OECD countries

We undertake network efficiency analysis within an input–output model that allows us to assess potential technical efficiency gains by comparing technologies corresponding to different economies. Input–output tables represent a network where different sectoral nodes use primary inputs (endowments) to produce intermediate input and outputs (according to sectoral technologies), and satisfy final demand (preferences). Within the input–output framework it is possible to optimize primary inputs allocation, intermediate production and final demand production by way of non-parametric data envelopment analysis (DEA) techniques. DEA allows us to model the different subtechnologies corresponding to alternative production processes, to assess efficient resource allocation among them, and to determine potential output gains if inefficiencies were dealt with. The proposed model optimizes the underlying multi-stage technologies that the input–output system comprises identifying the best practice economies. The model is applied to a set of OECD countries.     

A cost allocation problem arising in hub–spoke network systems

This paper studies a cost allocation problem arising from hub–spoke network systems. When a large-scale network is to be constructed jointly by several agents, both the optimal network design and the fair allocation of its cost are essential issues. We formulate this problem as a cooperative game and analyze the core allocation, which is a widely used solution concept. The core of this game is not necessarily non-empty as shown by an example. A reasonable scheme is to allocate the cost proportional to the flow that an agent generates. We show that, if the demand across the system has a block structure and the fixed cost is high, this cost allocation scheme belongs to the core. Numerical experiments are given with real telecommunication traffic data in order to illustrate the usefulness of our analytical findings.