Degree Of Suboptimality Research Articles

The Worst-Case Efficiency of Cost Sharing Methods in Resource Allocation Games

Resource allocation problems play a key role in many applications, including traffic networks, telecommunication networks, and economics. In most applications, the allocation of resources is determined by a finite number of independent players, each optimizing an individual objective function. An important question in all these applications is the degree of suboptimality caused by selfish resource allocation. We consider the worst-case efficiency of cost sharing methods in resource allocation games in terms of the ratio of the minimum guaranteed surplus of a Nash equilibrium and the maximal surplus. Our main technical result is an upper bound on the efficiency loss that depends on the class of allowable cost functions and the class of allowable cost sharing methods. We demonstrate the power of this bound by evaluating the worst-case efficiency loss for three well-known cost sharing methods: incremental cost sharing, marginal cost pricing, and average cost sharing.

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In This Issue

Stability of Constrained Adaptive Model Predictive Control Algorithms

AbstractRecently, suboptimality estimates for model predictive controllers (MPC) have been derived for the case without additional stabilizing endpoint constraints or a Lyapunov function type endpoint weight. The proposed methods yield a posteriori and a priori estimates of the degree of suboptimality with respect to the infinite horizon optimal control and can be evaluated at runtime of the MPC algorithm. Our aim is to design automatic adaptation strategies of the optimization horizon in order to guarantee stability and a predefined degree of suboptimality for the closed loop solution. Here, we present a stability proof for an arbitrary adaptation scheme and state a simple shortening and prolongation strategy which can be used for adapting the optimization horizon.

ROBUST MATHEMATICAL PROGRAMMING FOR NATURAL RESOURCE MODELING UNDER PARAMETRIC UNCERTAINTY

Inaccurate specification of model coefficients can lead to false or distorted findings in modeling investigations of natural resource management. Hence, this paper outlines a decision framework for optimization problems in which only the bounded set of outcomes for uncertain parameters is known. These models can be solved with standard mathematical programming software and are no larger than their deterministic equivalent. The robust approach is contrasted against deterministic analysis and is demonstrated for two applications regarding the management of natural resources. Deterministic plans are infeasible in at least 40% of cases when parameters vary from their point estimates. Inclusion of robust constraints immunizes against this infeasibility, thereby removing errors arising from false certainty. Additionally, incorporation of bounded parameters in the objective function yields interval-valued sets containing potential outcomes. However, this increase in the general relevance of model output introduces some degree of suboptimality as deterministic plans are buffered to proactively account for potential variability. The cost of robustness increases with the simulated spread of uncertain coefficients but may be reduced through accounting for the uncertainty aversion of decision makers.

Sales promotions and channel coordination

Consumer sales promotions are usually the result of the decisions of two marketing channel parties, the manufacturer and the retailer. In making these decisions, each party normally follows its own interest: i.e. maximizes its own profit. Unfortunately, this results in a suboptimal outcome for the channel as a whole. Independent profit maximization by channel parties leads to a lack of channel coordination with the implication of leaving money on the table. This may well contribute to the notoriously low profitability of sales promotions. This paper first shows analytically why the suboptimality occurs, and then presents an empirical demonstration, using a unique dataset from an Efficient Consumer Response (ECR) project; ECR is a movement in which parties work together to optimize the distribution channel). In this dataset, actual profit is only a small fraction of potential profit, implying that there is a large degree of suboptimality. It is important that (1) channel parties are aware of this suboptimality; and (2) that they have tools to deal with it. Solutions to the channel coordination problem should ensure that the goals of the individual channel parties are aligned with the goals of the channel as a whole. The paper proposes one particular agreement for this purpose, called proportional discount sharing. Application to the ECR data shows a win-win result for both the manufacturer and the retailer. Recognition of the channel coordination problem by the manufacturer and the retailer is the necessary starting point for agreeing on a way of solving it in a win-win fashion.

Dynamic evolution of economically preferred facilities

Abstract In a sustained development scenario, it is often the case that an investment is to be made over time in facilities that generate benefits. The benefits result from joint synergies between the facilities expressed as positive utilities specific to some subsets of facilities. As incremental budgets to finance fixed facility costs become available over time, additional facilities can be opened. The question is which facilities should be opened in order to guarantee that the overall benefit return over time is on the highest possible trajectory. This problem is common in situations such as ramping up a communication or transportation network where the facilities are hubs or service stations, or when introducing new technologies such as alternative fuels for cars and the facilities are fueling stations, or when expanding the production capacity with new machines, or when facilities are functions in a developing organization that is forced to make choices of where to invest limited funding. An intuitive strategy frequently used to evolve the set of facilities is a greedy approach that picks additional facilities which provide a highest rate of return on each budget increment. Such strategy is shown to have adverse impact by locking the system into future suboptimal solutions with total benefit that can be arbitrarily small compared to the optimal evolution sequence. It is shown here that the degree of suboptimality of such greedy strategy is extreme in that it can lead to the loss of the majority of future benefits. The main result here is the generation of the entire efficient frontier of optimal solution sets for different budget levels. This frontier contains information that guides the schedule of the optimal evolution of the set of facilities in future expansions. It is demonstrated that the efficient frontier forms a concave envelope, and the configurations on the frontier’s breakpoints are nested. The efficient frontier is generated by efficient and strongly polynomial algorithms. For n potential facility locations and for subsets of positive benefits of total size m we find the breakpoints of the efficient frontier using a parametric minimum cut procedure in time O ( nm log n 2 m ) , and all N configurations on the efficient frontier in time O( mn log n + N ).

Inf–sup control of discontinuous piecewise affine systems

AbstractThis paper considers the worst‐case optimal control of discontinuous piecewise affine (PWA) systems, which are subjected to constraints and disturbances. We seek to pre‐compute, via dynamic programming, an explicit control law for these systems when a PWA cost function is utilized. One difficulty with this problem class is that, even for initial states for which the value function of the optimal control problem is finite, there might not exist a control law that attains the infimum. Hence, we propose a method that is guaranteed to obtain a sub‐optimal solution, and where the degree of sub‐optimality can be specified a priori. This is achieved by approximating the underlying sub‐problems with a parametric piecewise linear program. Copyright © 2008 John Wiley & Sons, Ltd.

A new method to [formula omitted] robust filter design

In this paper a new method to H 2 robust filter design is proposed. Both continuous and discrete time settings are considered for systems subject to polytopic parameter uncertainty. Lower and upper bounds of the true cost are determined in order to evaluate the degree of sub-optimality of the proposed robust filter. The design method is based on the parametrization of all robust filters as a convex combination of Kalman filters associated to each vertex of the uncertainty domain. Among all feasible filters, the one minimizing a guaranteed H 2 cost of the estimation error is determined by a pure convex programming problem, expressed in terms of linear matrix inequalities (LMIs). The order of the robust filter is generally greater than the order of the plant, a fact that contributes to reduce conservatism. The proposed design technique is compared with other methods available in the literature. In several examples solved the proposed method outperforms all other designs.

Heuristic algorithms vs. linear programs for designing efficient conservation reserve networks: Evaluation of solution optimality and processing time

Systematic approaches to efficient reserve network design often make use of one of two types of site selection algorithm; linear programs or heuristic algorithms. Unlike with linear programs, heuristic algorithms have been demonstrated to yield suboptimal networks in that more sites are selected in order to meet conservation goals than may be necessary or fewer features are captured than is possible. Although the degree of suboptimality is not known when using heuristics, some researchers have suggested that it is not significant in most cases and that heuristics are preferred since they are more flexible and can yield a solution more quickly. Using eight binary datasets, we demonstrate that suboptimality of numbers of sites selected and biodiversity features protected can occur to various degrees depending on the dataset, the model design, and the type of heuristic applied, and that processing time is not dramatically different between optimal and heuristic algorithms. In choosing an algorithm, the degree of suboptimality may not always be as important to planners as the perception that optimal solvers have feasibility issues, and therefore heuristic algorithms might continue to be a popular tool for conservation planning. We conclude that for many datasets, feasibility of optimal algorithms should not be a concern and that the value of heuristic results can be greatly improved by using optimal algorithms to determine the degree of suboptimality of the results.

An applied stochastic model of the quality–quantity trade-off in the public health care sector

It is a striking feature of the many of the developing country public service sectors that the sectors in question often overproduce the quantity of services but underproduce the quality. This feature, which is exemplified in this paper, is rooted in a wide spectrum of economic and sociological factors ranging from the economic and sociological profile of the service receiving people to the asymmetric density of service-receiving population across their regions. This feature, we conjecture, is a source of a considerable degree of suboptimality in some of the developing countries. If our conjecture is correct, correcting such suboptimalities is likely to yield significant welfare improvements that could help speed up the process of development in the underdeveloped regions of the world. To analyze the supoopimalites in question, we will first develop a concept (and a model) of optimal quality in the public service sector, which indicates the level of quality that maximizes expected public satisfaction subject to available resources. Resources are used in an efficient manner to produce the service in question. The concept and the model in the paper make a needed contribution to the quality discourse by presenting a way of determining the quality improvements (or adjustments) necessary to achieve optimum in the public service sector. The paper presents an application (a case study) of this new concept in the public healthcare sector in Turkey, and explores the differences between the actual and optimal quality in the sector in question. It turns out that there is a considerable difference between the actual and optimal levels of quality (as well as those of quantity) in the Turkish public healthcare sector in an overpopulated city (Istanbul), indicating a significant overproduction of quantity and underproduction of quality. Thus, to achieve the optimal levels, the sector should increase quality and reduce quantity by a considerable margin. The quantified differences (gaps) between actual and optimal levels point out a considerable room for welfare improvement. Optimum-seeking adjustments closing these gaps could be shown to lead to considerable satisfaction and welfare gains, the measurement of which is worthy of future research.

Investment under uncertainty—Does competition matter?

Empirical applications of real options models in competitive environments implicitly exploit the optimality of myopic planning. In a seminal paper [Leahy, J.V., 1993. Investment in Competitive Equilibrium: the Optimality of Myopic Behavior. Quarterly Journal of Economics 108, 1105–1133] shows that the optimal investment strategy of a myopic planner who ignores market entries and exits of competitors, as well as the resulting price effects, constitutes a market equilibrium under rather general conditions. As a result, the calculation of optimal investment strategies is simplified considerably because competition does not have to be taken into account. In this paper, however, we demonstrate by using simulation experiments that myopic planning may lead to non-optimal investment strategies. This is due to the fact that it is difficult, or even impossible, to specify the correct or equivalent price process for the myopic investor using real world data. We quantify the degree of suboptimality and propose measures to reduce the error.

Average bit-error probability performance for optimum diversity combining of noncoherent FSK over rayleigh fading channels

We derive the performance of the optimum noncoherent frequency-shift keying diversity combining receiver and compare it with that which implements the more traditional noncoherent postdetection equal gain combining scheme to reveal the degree of suboptimality of the latter.

The impact of security layering on end-to-end latency and system performance in switched and distributed e-business environments

Contemporary e-business networks are increasingly implementing the multi-layer security scheme in order to provide a reasonable measure of security for their information systems. The implementation entails formation of a layered architecture (concentric security layers) using packet and application-level filters neither of which provides complimentary functions. The layered architecture provides convenient abstractions and increases the end-to-end latency that results into sub-optimal system performance. In this paper, we present the results of the experiment to quantify the latency introduced by security layering on end-to-end latency and investigate the resulting degree of sub-optimality of system performance in a distributed and switched e-business network.

Optimal Product Variety and Economic Growth: The Trade-off between Internal and External Economies of Scale

We develop a model to analyze the social optimality of growth and product variety. The model contains two sectors, one assembly sector producing a homogeneous consumption good, and one intermediate goods sector producing differentiated inputs. Both growth which results from R & D performed in the intermediate goods sector and the variety of differentiated intermediate goods are determined endogenously. We disentangle three effects of changes in variety: (i) a productivity effect, (ii) a business stealing effect, and (iii) a growth effect. The market provides too little variety and sub-optimally high growth if the productivity effect of variety is large relative to the market power of intermediate goods producers. If varieties are not very productive, the market provides a too low rate of growth, whereas variety may either be too low as well as too high. Decentralization of both the first-best and a second-best is considered. The model is calibrated to US data to assess the magnitude of required policy interventions and the degree of sub-optimality of growth and product variety. We show that in this framework growth in the market can indeed be sub-optimally high for plausible parameter values.

Normative design of organizations. II. Organizational structure

For pt.I. see ibid., p. 346-59. This paper presents a multiobjective structural optimization process of designing an organization to execute a specific mission. We provide mathematical formulations for optimization problems arising in Phases II and III of our organizational design process and polynomial algorithms to solve the corresponding problems. Our organizational design methodology applies specific optimization techniques at different phases of the design, efficiently matching the structure of a mission (in particular, the one defined by the courses of action obtained from mission planning) to that of an organization. It allows an analyst to obtain an acceptable tradeoff among multiple mission and design objectives, as well as between computational complexity and solution efficiency (desired degree of suboptimality).

Hierarchically accelerated dynamic programming for finite-state machines

A procedure called hierarchically accelerated dynamic programming (HADP) is presented which, at the cost of a degree of suboptimality, can significantly accelerate dynamic programming algorithms for discrete event systems modeled by finite-state machines (FSMs). The methodology is based. upon the (possibly iterated) dynamical abstraction of a given FSM by state aggregation in order to generate a so-called partition machine hierarchy. Necessary and sufficient conditions for the HADP procedure to generate globally optimal solutions are given as well as bounds on the degree of suboptimality of the method. A group of examples called the Broken Manhattan Grid problems is used to illustrate an implementation of HADP with two and three level hierarchies. A set of open problems is described concerning the construction and selection of the partition machine abstractions and the improvement of the estimation of HADP suboptimality.

Hydrologic Data Assimilation with the Ensemble Kalman Filter

Soil moisture controls the partitioning of moisture and energy fluxes at the land surface and is a key variable in weather and climate prediction. The performance of the ensemble Kalman filter (EnKF) for soil moisture estimation is assessed by assimilating L-band (1.4 GHz) microwave radiobrightness observations into a land surface model. An optimal smoother (a dynamic variational method) is used as a benchmark for evaluating the filter's performance. In a series of synthetic experiments the effect of ensemble size and non-Gaussian forecast errors on the estimation accuracy of the EnKF is investigated. With a state vector dimension of 4608 and a relatively small ensemble size of 30 (or 100; or 500), the actual errors in surface soil moisture at the final update time are reduced by 55% (or 70%; or 80%) from the value obtained without assimilation (as compared to 84% for the optimal smoother). For robust error variance estimates, an ensemble of at least 500 members is needed. The dynamic evolution of the estimation error variances is dominated by wetting and drying events with high variances during drydown and low variances when the soil is either very wet or very dry. Furthermore, the ensemble distribution of soil moisture is typically symmetric except under very dry or wet conditions when the effects of the nonlinearities in the model become significant. As a result, the actual errors are consistently larger than ensemble-derived forecast and analysis error variances. This suggests that the update is suboptimal. However, the degree of suboptimality is relatively small and results presented here indicate that the EnKF is a flexible and robust data assimilation option that gives satisfactory estimates even for moderate ensemble sizes.

A Practical Model for Generator Maintenance Scheduling with Transmission Constraints

In this paper an optimal maintenance scheduling of generating units in a power system has been developed with transmission network representation. Here a DC load flow has been embedded in the maintenance model to include network constraints resulting in a more practical maintenance schedule. The model developed here uses the minimization of system cost (production cost plus the unserved energy cost) as the objective criterion, whereas the reliability objective function used is the minimization of unserved energy. The optimization is achieved by integer linear programming. The incorporation of transmission network adds significant complexity to maintenance scheduling. The proposed model enables almost all practical maintenance scheduling constraints to be handled easily. The optimization has been carried out to minimize the cost function considering different cases (i.e., with and without incorporation of the transmission network). The effectiveness of the proposed method has been demonstrated by obtaining numerical results on sample and real scale test systems. A comparison of the cost objective function clearly indicates that the maintenance schedule obtained from the simple generation model alone is more expensive than the one with transmission, and that there is a considerable degree of suboptimality in the former case.

Removing the Need for QP in Constrained Predictive Control

Predictive control strategies allow for systematic constraint handling and performance optimization while at the same time giving guarantees for stability. However, the quadratic programming (QP) problems associated with constraint handling can be computationally expensive. A recent paper (Kouvaritakis et al. , 1999) introduced a novel alternative for uncertain systems which, for a low degree of suboptimality, converts the on-line optimization problem to one of determining the (unique) positive real root of a polynomial. The current paper examines the issue of computational efficiency versus degree of suboptimality for systems without uncertainty and develops an extension of this algorithm to reduce the degree of suboptimality for a small increase in the computational burden. It is claimed here that the proposed algorithm offers an efficient alternative to QP which should be preferred for applications with fast sampling.

Sequential testing algorithms for multiple fault diagnosis

We consider the problem of constructing optimal and near-optimal test sequences for multiple fault diagnosis. The computational complexity of solving the optimal multiple-fault isolation problem is super exponential, that is, it is much more difficult than the single-fault isolation problem, which, by itself, is NP-hard. By employing concepts from information theory and AND/OR graph search and by exploiting the single fault testing strategies of Pattipati et al. (1990), we present several test sequencing algorithms for the multiple fault isolation problem. These algorithms provide a trade-off between the degree of suboptimality and computational complexity. Furthermore, we present novel diagnostic strategies that generate a diagnostic directed graph, instead of a traditional diagnostic tree, for multiple fault diagnosis. Using this approach, the storage complexity of the overall diagnostic strategy reduces substantially. The algorithms developed herein have been successfully applied to several real-world systems.