Susan X. Li

Chapter 13 Satisficing DEA models under chance constraints

DEA (Data Envelopment Analysis) models and concepts are formulated here in terms of the “P-Models” of Chance Constrained Programming, which are then modified to contact the “satisficing concepts” of H.A. Simon. Satisficing is thereby added as a third category to the efficiency/inefficiency dichotomies that have heretofore prevailed in DEA. Formulations include cases in which inputs and outputs are stochastic, as well as cases in which only the outputs are stochastic. Attention is also devoted to situations in which variations in inputs and outputs are related through a common random variable. Extensions include new developments in goal programming with deterministic equivalents for the corresponding satisficing models under chance constraints.

An Analysis of Manufacturer‐Retailer Supply Chain Coordination in Cooperative Advertising*

In the literature of cooperative (co-op) advertising, the focus of the research is on a relationship in which a manufacturer is the leader and retailers are followers. This relationship implies the dominance of the manufacturer over retailers. Recent market trends have shown a shift in power from manufacturers to retailers. Retailers, as a result, may now possess equal or even greater power than a manufacturer in some instances when it comes to retailing. Based on this new market phenomenon, we intend to explore the role of co-op advertising in a manufacturer-retailer supply chain through brand name investments, local advertising expenditures, and sharing rules of advertising expenses. Two co-op advertising models are developed and compared. The first co-op advertising model is based on the traditional leader-follower relationship of a manufacturer and a retailer. The second model incorporates partnership into co-op advertising coordination. Business examples and managerial implications of the models have been discussed. A cooperative bargaining technique is utilized to implement the partnership co-op advertising model.

Chance Constrained Programming Formulations for Stochastic Characterizations of Efficiency and Dominance in DEA

Pareto-Koopmans efficiency in Data Envelopment Analysis (DEA) is extended to stochastic inputs and outputs via probabilistic input-output vector comparisons in a given empirical production (possibility) set. In contrast to other approaches which have used Chance Constrained Programming formulations in DEA, the emphasis here is on “joint chance constraints.” An assumption of arbitrary but known probability distributions leads to the P-Model of chance constrained programming. A necessary condition for a DMU to be stochastically efficient and a sufficient condition for a DMU to be non-stochastically efficient are provided. Deterministic equivalents using the zero order decision rules of chance constrained programming and multivariate normal distributions take the form of an extended version of the additive model of DEA. Contacts are also maintained with all of the other presently available deterministic DEA models in the form of easily identified extensions which can be used to formalize the treatment of efficiency when stochastic elements are present.

Chance constrained programming approaches to congestion in stochastic data envelopment analysis

Abstract The models described in this paper for treating congestion in DEA are extended by according them chance constrained programming formulations. The usual route used in chance constrained programming is followed here by replacing these stochastic models with their “deterministic equivalents.” This leads to a class of non-linear problems. However, it is shown to be possible to avoid some of the need for dealing with these non-linear problems by identifying conditions under which they can be replaced by ordinary (deterministic) DEA models. Examples which illustrate possible uses of these approaches are also supplied in an Appendix A .

Chance constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis

This paper replaces ordinary DEA formulations with stochastic counterparts in the form of a series of chance constrained programming models. Emphasis is on technical efficiencies and inefficiencies which do not require costs or prices, but which are nevertheless basic in that the achievement of technical efficiency is necessary for the attainment of ‘allocative’, ‘cost’ and other types of efficiencies.

Survey of mathematical programming models in air pollution management

Abstract This paper surveys the current state of the literature in management science/operations research approaches to air pollution management. After introducing suitable background we provide some of the institutional and legal framework needed to understand the continuing regulatory efforts in United States. Attention is then turned to mathematical programming models ranging from fairly simple deterministic linear programs to quite sophisticated stochastic models which have appeared in the literature dealing with these topics. This is followed by extensions reflecting some of the work we have undertaken in association with the Texas Natural Resource Conservation Commission, a regulatory agency in Texas. Application and potential use of models is the central theme of this survey. Issues for future research are presented at the end and an extensive list of publications is provided in the references at the end of the article. Principal air quality issues of local, national, and international concern are listed below in increasing order of difficulty based on the number of different types of pollutants and problems in quantification of the risks the pollutants pose: 1. 1. Stratospheric ozone depletion: one relatively easily controllable class of trace gases - ozone depleting chemicals, or ODCs, principally chloroflurocarbons (CFCs) — with relatively well quantified risks; 2. 2. Criteria pollutants: six common pollutants — ozone (O 3 ), carbon monoxide (CO), sulfur dioxide (SO 2 ), nitrogen dioxide (NO 2 ), lead (Pb), and particulate matter less than 10 microns in size (PM10) — regulated since 1970 in the U.S. and presenting relatively well quantified risks; 3. 3. Acid precipitation: two relatively easily controllable classes of trace gases — oxides of nitrogen (NO x ) and oxides of sulfur (SO x ) with relatively well quantified risks; 4. 4. Global warming/climate change: a few difficult to control trace gases — principally carbon dioxide (CO 2 ), methane (CH 4 ), nitrous oxide (N 2 O), and CFCs — with highly uncertain risks; 5. 5. Toxics or HAPS (hazardous air pollutants): hundreds of types of gaseous chemicals and particles with uncertain risks; 6. 6. Somewhat dated, but nevertheless useful, is the following reference: Glossary on Air Pollution (Copenhagen, World Health Organization, 1980).

Stochastic models and variable returns to scales in data envelopment analysis

Stochastic Data Envelopment Analysis (DEA) models were developed by taking random disturbances into account for the possibility of variations in input-output data structure. The stochastic efficiency measure of a Decision Making Unit (DMU) is defined via joint probabilistic comparisons of inputs and outputs with other DMUs, and can be characterized by solving a chance constrained programming problem. Deterministic equivalents are derived for both situations of multivariate symmetric random disturbances and a single random factor in the production relationships. An analysis of stochastic variable returns to scale is developed.

Managing buyer-seller system cooperation with quantity discount considerations

In this paper we explore cooperative relationships between two members in a simple buyer-seller system where the buyer is in a monopolistic position. We begin with a situation where the seller, as the leader, has the power to enforce his strategies on the buyer, but vice versa is not true. We then extend our analysis to a situation wherein the buyer can also influence the sellers decisions and address the issue of system cooperation. We illustrate the mutual incentives for cooperation and individual disincentives for cooperation. A quantity discount scheme is developed to implement a profit sharing mechanism for achieving equal division of additional cooperative system profits.

Dominance stochastic models in data envelopment analysis

Abstract In this paper stochastic models in data envelopment analysis (DEA) are developed by taking into account the possibility of random variations in input-output data, and dominance structures on the DEA envelopment side are used to incorporate the modelbuilders preferences and to discriminate efficiencies among decision making units (DMUs). The efficiency measure for a DMU is defined via joint dominantly probabilistic comparisons of inputs and outputs with other DMUs and can be characterized by solving a chance constrained programming problem. Deterministic equivalents are obtained for multivariate symmetric random errors and for a single random factor in the production relationships. The goal programming technique is utilized in deriving linear deterministic equivalents and their dual forms. The relationship between the general stochastic DEA models and the conventional DEA models is also discussed.

STOCHASTIC DEA MODELS WITH DIFFERENT TYPES OF INPUT-OUTPUT DISTURBANCES

This paper presents stochasticmodels in data envelopment analysis (DEA) for the possibilityof variations in inputs and outputs. Efficiency measure of adecision making unit (DMU) is defined via joint probabilisticcomparisons of inputs and outputs with other DMUs and can becharacterized by solving a chance constrained programming problem.By utilizing the theory of chance constrained programming, deterministicequivalents are obtained for both situations of multivariatesymmetric random disturbances and a single random factor in theproduction relationships. The linear deterministic equivalentand its dual form are obtained via the goal programming theoryunder the assumption of the single random factor. An analysisof stochastic variable returns to scale is developed using theidea of stochastic supporting hyperplanes. The relationshipsof our stochastic DEA models with some conventional DEA modelsare also discussed.