Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are sparse.
We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. SNOPT is a particular implementation that makes use of a semidefinite QP solver. It is based on a limited-memory quasi-Newton approximation to the Hessian of the Lagrangian and uses a reduced-Hessian algorithm (SQOPT) for solving the QP subproblems. It is designed for problems with many thousands of constraints and variables but a moderate number of degrees of freedom (say, up to 2000). An important application is to trajectory optimization in the aerospace industry. Numerical results are given for most problems in the CUTE and COPS test collections (about 900 examples).

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SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

Analytic hierarchy process : An overview of applications

In this paper, we will propose a Slacks-Based measure (SBM) of efficiency in DEA. This scalar measure deals directly with the input surplus and the output shortage of the decision making unit (DMU) concerned. It is unit invariant and monotone decreasing with respect to input surplus and output shortage. Furthermore, this measure is decided only by consulting with the reference set of the DMU and is not affected by statistics over the whole data set. The new measure has a close connection with other measures proposed so far, e.g., CCR and BCC. The dual side of this model can be interpreted as profit maximization, in contrast to the ratio maximization of the CCR model. Numerical experiments show its validity as an efficiency measurement tool and its compatibility with other measures of efficiency.

A Slacks Based Measure of Efficiency in Data Envelopment Analysis

A slacks-based measure of super-efficiency in data envelopment analysis

Economists have shown that large and persistent differences in productivity levels across businesses are ubiquitous. This finding has shaped research agendas in a number of fields, including (but not limited to) macroeconomics, industrial organization, labor, and trade. This paper surveys and evaluates recent empirical work addressing the question of why businesses differ in their measured productivity levels. The causes are manifold, and differ depending on the particular setting. They include elements sourced in production practices -- and therefore over which producers have some direct control, at least in theory -- as well as from producers' external operating environments. After evaluating the current state of knowledge, I lay out what I see are the major questions that research in the area should address going forward. (JEL D24, G31, L11, M10, O30, O47)

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What Determines Productivity

This paper covers some of the past accomplishments of DEA and its future prospects. It starts with the engineering-science definitions of efficiency and uses the duality theory of linear programming to show how, in DEA, they can be related to the Pareto-Koopmens definitions used in welfare economics. Some of the models that have now been developed for implementing these concepts are then described and properties of these models are examined for weaknesses and strengths along with different measures of distance that may be used to determine their optimal values. Relations between the models are also demonstrated en route to delineating paths for future developments. These include extensions to different objectives such as satisfactory vs. full (or strong) efficiency. They also include extensions from efficiency to effectiveness evaluations of performances as well as extensions to evaluate social-economic performances of countries and other entities where inputs and outputs give way to other categories in which increases and decreases are located in the numerator or denominator of the ratio (=engineering-science) definition of efficiency in a manner analogous to the way output (in the numerator) and input (in the denominator) are usually positioned in the fractional programming form of DEA. Beginnings in each of these extensions are cited and the role of applications in bringing further possibilities to the fore is highlighted.

DEA: Past Accomplishments and Future Prospects

In this contribution, first the concept of returns to growth (RTG) of a high-tech firm facing hyper-competition in the new economy is introduced by describing a proportional relationship between growth in inputs and growth in outputs using the growth efficiency (GE) model of Sengupta (2002). Second, both technology- and value-based methods are suggested for estimating the RTG behavior of high-tech firms. Third, though the GE concept seems closely related to the notion of total factor productivity change, this link remains unexplored: we suggest a link between both concepts. Finally, our empirical application to the Indian hardware computer industry reveals that first, companies operating under increasing returns to scale (RTS) may exhibit constant or decreasing RTG; second, companies showing constant RTS may exhibit increasing or decreasing RTG; and third, companies showing decreasing RTS may exhibit constant or increasing RTG. These findings imply that RTS estimates need not provide proper information regarding the growth strategy behavior of high-tech companies.

Returns to Growth in a Nonparametric DEA Approach

The existing methods in the data envelopment analysis (DEA) approach to determination of returns to scale (RTS) of firms apply only to production technologies exhibiting constant returns to scale and variable returns to scale (VRS). In this chapter, we present a brief review of the characterization of the RTS of firms using VRS DEA production technologies.

Determining returns to scale in the vrs dea model

POS data are inputted at a step S1. At a step S2, the inputted POS data are put in order. For example, a daily sale quantity data set for each commodity is prepared by removing the day(s) in which no stock exists at the time of opening and the days(s) in which commodity(ies) is out of stock at opening. At a step S3, unsteady commodities are removed. At a step S4, basic statistical quantities of the daily sales quantity of each commodity, such as a mean value, standard deviation, maximum value, minimum value, distortion value, peak value, Geary value, etc., are calculated. At a step S5, the data are classified into a plurality of types determined in advance such as a Poisson type, a normal type, a Causal type, and other types. An optimum supplementary ordering quantity is determined on the basis of the classification described above.

Sale quantity characteristics classification system and supplementary ordering system

This paper proposes a new search model to deal with public sector decisions in which intangible factors need to be considered along with tangible ones. Such problems are often found in a wide range of administrative investigations and criminal inspections in Japan. In an effort to manage such problems, this article first constructs an Analytic Hierarchy Process (AHP) model to deal with the intangible factors. Then, we reinforce the AHP model by incorporating tangible factors which are not included in tlte model. Tlte probit and/or logit models are applied to test the statistical significance of this combination, based on a past data set. This model can be considered as a two-stage procedure in the sense that the AHP results (the first stage) are utilized to construct a statistical model in the second stage, which aims at obtaining a better probability of detection. Although this study can foresee the probability of detection for individual objects based on this probability model, there is another important practical issue, i.e. the scheduling of associated officials. Therefore, this study proposes a scheduling model using mathematical programming methodology. Finally, the proposed model was applied to the investigation work of the Post-Clearance Audit Department of Japanese Customs. It was found that the probability of finding incorrect declarations would be improved from the current 60% to 75-80%. While our problem is not related to criminal activities directly, this study predicts that our approach might be applicable for other governmental investigations and inspections in the scope of this empirical study.

Kaoru Tone

Papers

DEA: Past Accomplishments and Future Prospects

Returns to Growth in a Nonparametric DEA Approach

Determining returns to scale in the vrs dea model

Sale quantity characteristics classification system and supplementary ordering system

A Search Model with Subjective Judgements : Auditing of Incorrect Tax Declarations