Visualizing Production Surfaces in 3D Diagrams

During the last four decades Data Envelopment Analysis (DEA) has attracted considerable attention in the OR community. Using DEA, the efficiency frontier is constructed based on assumptions concerning the production possibility set rather than a priori defining a functional relationship between inputs and outputs. In this contribution, we propose an algorithm to visualize the efficiency surface in a 3D diagram and to extract isoquants from the efficient hull based on different RTS assumptions which might be particularly helpful for presentation purposes. In doing so, we extend the existing literature which has concentrated on the visualization of production frontiers in 2D diagrams to the visualization of efficient rather than fully efficient hulls in 3D diagrams. Displaying a fully efficient hull, however, does not reflect all properties of the production possibility set as weakly efficient frontier segments are missing

A multiobjective optimization approach to compute the efficient frontier in data envelopment analysis

Data envelopment analysis is a linear programming-based operations research technique for performance measurement of decision-making units. In this paper, we investigate data envelopment analysis from a multiobjective point of view to compute both the efficient extreme points and the efficient facets of the technology set simultaneously. We introduce a dual multiobjective linear programming formulation of data envelopment analysis in terms of input and output prices and propose a procedure based on objective space algorithms for multiobjective linear programmes to compute the efficient frontier. We show that using our algorithm, the efficient extreme points and facets of the technology set can be computed without solving any optimization problems. We conduct computational experiments to demonstrate that the algorithm can compute the efficient frontier within seconds to a few minutes of computation time for real-world data envelopment analysis instances. For large-scale artificial data sets, our algorithm is faster than computing the efficiency scores of all decision-making units via linear programming

The role of multiplier bounds in fuzzy data envelopment analysis

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.The non-Archimedean epsilon ε is commonly considered as a lower bound for the dual input weights and output weights in multiplier data envelopment analysis (DEA) models. The amount of ε can be effectively used to differentiate between strongly and weakly efficient decision making units (DMUs). The problem of weak dominance particularly occurs when the reference set is fully or partially defined in terms of fuzzy numbers. In this paper, we propose a new four-step fuzzy DEA method to re-shape weakly efficient frontiers along with revisiting the efficiency score of DMUs in terms of perturbing the weakly efficient frontier. This approach eliminates the non-zero slacks in fuzzy DEA while keeping the strongly efficient frontiers unaltered. In comparing our proposed algorithm to an existing method in the recent literature we show three important flaws in their approach that our method addresses. Finally, we present a numerical example in banking with a combination of crisp and fuzzy data to illustrate the efficacy and advantages of the proposed approach

Evaluating the quality of radiotherapy treatment plans with uncertainty using data envelopment analysis

External beam radiation therapy is a common treatment method for cancer. Radiotherapy is planned with the aim of achieving conflicting goals: while a sufficiently high dose of radiation is necessary for tumour control, a low dose of radiation is desirable to avoid complications in normal, healthy, tissue. This thesis aims to support the radiotherapy treatment planning process for prostate cancer by evaluating the quality of proposed treatment plans relative to previous plans. We develop a variable selection technique, autoPCA, to select the most relevant variables for use in our Data Envelopment Analysis (DEA) models. This allows us to evaluate how well plans perform in terms of achieving the conflicting goals of radiotherapy. We develop the uncertain DEA problem (uDEA) for the case of box uncertainty and show that for small problems this can be solved exactly. This study of uncertainty is motivated by the inherently uncertain nature of the treatment process. Robust DEA, uDEA and simulation are applied to prostate cancer treatment plans to investigate this uncertainty. We identify plans that have the potential to be improved, which clinicians then replan for us. Small improvements were seen and we discuss the potential difference this could make to planning cases that are more complex. To aid this, we develop a prototype software, EvaluatePlan, that assesses the efficiency of a plan compared to past treatment plans

Moving from Linear to Conic Markets for Electricity

We propose a new forward electricity market framework that admits heterogeneous market participants with second-order cone strategy sets, who accurately express the nonlinearities in their costs and constraints through conic bids, and a network operator facing conic operational constraints. In contrast to the prevalent linear-programming-based electricity markets, we highlight how the inclusion of second-order cone constraints enables uncertainty-, asset- and network-awareness of the market, which is key to the successful transition towards an electricity system based on weather-dependent renewable energy sources. We analyze our general market-clearing proposal using conic duality theory to derive efficient spatially-differentiated prices for the multiple commodities, comprising of energy and flexibility services. Under the assumption of perfect competition, we prove the equivalence of the centrally-solved market-clearing optimization problem to a competitive spatial price equilibrium involving a set of rational and self-interested participants and a price setter. Finally, under common assumptions, we prove that moving towards conic markets does not incur the loss of desirable economic properties of markets, namely market efficiency, cost recovery and revenue adequacy. Our numerical studies focus on the specific use case of uncertainty-aware market design and demonstrate that the proposed conic market brings advantages over existing alternatives within the linear programming market framework.Comment: Manuscript with electronic companion; submitted to Operations Researc

Convex Optimisation for Communication Systems

In this thesis new robust methods for the efficient sharing of the radio spectrum for underlay cognitive radio (CR) systems are developed. These methods provide robustness against uncertainties in the channel state information (CSI) that is available to the cognitive radios. A stochastic approach is taken and the robust spectrum sharing methods are formulated as convex optimisation problems. Three efficient spectrum sharing methods; power control, cooperative beamforming and conventional beamforming are studied in detail. The CR power control problem is formulated as a sum rate maximisation problem and transformed into a convex optimisation problem. A robust power control method under the assumption of partial CSI is developed and also transformed into a convex optimisation problem. A novel method of detecting and removing infeasible constraints from the power allocation problem is presented that results in considerably improved performance. The performance of the proposed methods in Rayleigh fading channels is analysed by simulations. The concept of cooperative beamforming for spectrum sharing is applied to an underlay CR relay network. Distributed single antenna relay nodes are utilised to form a virtual antenna array that provides increased gains in capacity through cooperative beamforming. It is shown that the cooperative beamforming problems can be transformed into convex optimisation problems. New robust cooperative beamformers under the assumption of partial and imperfect CSI are developed and also transformed into convex optimisation problems. The performance of the proposed methods in Rayleigh fading channels is analysed by simulations. Conventional beamforming to allow efficient spectrum sharing in an underlay CR system is studied. The beamforming problems are formulated and transformed into convex optimisation problems. New robust beamformers under the assumption of partial and imperfect CSI are developed and also transformed into convex optimisation problems. The performance of the proposed methods in Rayleigh fading channels is analysed by simulations