A new compact linear programming formulation for choice network revenue management

The choice network revenue management model incorporates customer purchase behavioras a function of the offered products, and is the appropriate model for airline and hotel networkrevenue management, dynamic sales of bundles, and dynamic assortment optimization.The optimization problem is a stochastic dynamic program and is intractable. A certainty-equivalencerelaxation of the dynamic program, called the choice deterministic linear program(CDLP) is usually used to generate dyamic controls. Recently, a compact linear programmingformulation of this linear program was given for the multi-segment multinomial-logit (MNL)model of customer choice with non-overlapping consideration sets. Our objective is to obtaina tighter bound than this formulation while retaining the appealing properties of a compactlinear programming representation. To this end, it is natural to consider the affine relaxationof the dynamic program. We first show that the affine relaxation is NP-complete even for asingle-segment MNL model. Nevertheless, by analyzing the affine relaxation we derive a newcompact linear program that approximates the dynamic programming value function betterthan CDLP, provably between the CDLP value and the affine relaxation, and often comingclose to the latter in our numerical experiments. When the segment consideration sets overlap,we show that some strong equalities called product cuts developed for the CDLP remain validfor our new formulation. Finally we perform extensive numerical comparisons on the variousbounds to evaluate their performance.

Identifying strategies for mitigating the global warming impact of the EU-25 economy using a multi-objective input-output approach

Global warming mitigation has recently become a priority worldwide. A large body of literature dealing with energy related problems has focused on reducing greenhouse gases emissions at an engineering scale. In contrast, the minimization of climate change at a wider macroeconomic level has so far received much less attention. We investigate here the issue of how to mitigate global warming by performing changes in an economy. To this end, we make use of a systematic tool that combines three methods: linear programming, environmentally extended input output models, and life cycle assessment principles. The problem of identifying key economic sectors that contribute significantly to global warming is posed in mathematical terms as a bi criteria linear program that seeks to optimize simultaneously the total economic output and the total life cycle CO2 emissions. We have applied this approach to the European Union economy, finding that significant reductions in global warming potential can be attained by regulating specific economic sectors. Our tool is intended to aid policymakers in the design of more effective public policies for achieving the environmental and economic targets sought.

On the throughput-WIP trade-off in queueing systems, diminishing returns and the threshold property: A linear programming approach

We present a new unifying framework for investigating throughput-WIP(Work-in-Process) optimal control problems in queueing systems,based on reformulating them as linear programming (LP) problems withspecial structure: We show that if a throughput-WIP performance pairin a stochastic system satisfies the Threshold Property we introducein this paper, then we can reformulate the problem of optimizing alinear objective of throughput-WIP performance as a (semi-infinite)LP problem over a polygon with special structure (a thresholdpolygon). The strong structural properties of such polygones explainthe optimality of threshold policies for optimizing linearperformance objectives: their vertices correspond to the performancepairs of threshold policies. We analyze in this framework theversatile input-output queueing intensity control model introduced byChen and Yao (1990), obtaining a variety of new results, including (a)an exact reformulation of the control problem as an LP problem over athreshold polygon; (b) an analytical characterization of the Min WIPfunction (giving the minimum WIP level required to attain a targetthroughput level); (c) an LP Value Decomposition Theorem that relatesthe objective value under an arbitrary policy with that of a giventhreshold policy (thus revealing the LP interpretation of Chen andYao's optimality conditions); (d) diminishing returns and invarianceproperties of throughput-WIP performance, which underlie thresholdoptimality; (e) a unified treatment of the time-discounted andtime-average cases.

A multi-objective model for a multi-period distribution management problem

The problems arising in commercial distribution are complex and involve several players and decision levels. One important decision is relatedwith the design of the routes to distribute the products, in an efficient and inexpensive way.This article deals with a complex vehicle routing problem that can beseen as a new extension of the basic vehicle routing problem. The proposed model is a multi-objective combinatorial optimization problemthat considers three objectives and multiple periods, which models in a closer way the real distribution problems. The first objective is costminimization, the second is balancing work levels and the third is amarketing objective. An application of the model on a small example, with5 clients and 3 days, is presented. The results of the model show the complexity of solving multi-objective combinatorial optimization problems and the contradiction between the several distribution management objective.

Implementing TURF analysis through binary linear programming

This paper introduces the approach of using Total Unduplicated Reach and Frequency analysis (TURF) to design a product line through a binary linear programming model. This improves the efficiency of the search for the solution to the problem compared to the algorithms that have been used to date. The results obtained through our exact algorithm are presented, and this method shows to be extremely efficient both in obtaining optimal solutions and in computing time for very large instances of the problem at hand. Furthermore, the proposed technique enables the model to be improved in order to overcome the main drawbacks presented by TURF analysis in practice.

Restless bandits, linear programming relaxations and a primal-dual index heuristic

We develop a mathematical programming approach for the classicalPSPACE - hard restless bandit problem in stochastic optimization.We introduce a hierarchy of n (where n is the number of bandits)increasingly stronger linear programming relaxations, the lastof which is exact and corresponds to the (exponential size)formulation of the problem as a Markov decision chain, while theother relaxations provide bounds and are efficiently computed. Wealso propose a priority-index heuristic scheduling policy fromthe solution to the first-order relaxation, where the indices aredefined in terms of optimal dual variables. In this way wepropose a policy and a suboptimality guarantee. We report resultsof computational experiments that suggest that the proposedheuristic policy is nearly optimal. Moreover, the second-orderrelaxation is found to provide strong bounds on the optimalvalue.

Maximum likelihood Linear Programming Data Fusion for Speaker Recognition

Biometric system performance can be improved by means of data fusion. Several kinds of information can be fused in order to obtain a more accurate classification (identification or verification) of an input sample. In this paper we present a method for computing the weights in a weighted sum fusion for score combinations, by means of a likelihood model. The maximum likelihood estimation is set as a linear programming problem. The scores are derived from a GMM classifier working on a different feature extractor. Our experimental results assesed the robustness of the system in front a changes on time (different sessions) and robustness in front a change of microphone. The improvements obtained were significantly better (error bars of two standard deviations) than a uniform weighted sum or a uniform weighted product or the best single classifier. The proposed method scales computationaly with the number of scores to be fussioned as the simplex method for linear programming.

The Multiple-partners assignment game with heterogeneous sales and multi-unit demands: competitive equilibria

A multiple-partners assignment game with heterogeneous sales and multiunit demands consists of a set of sellers that own a given number of indivisible units of (potentially many different) goods and a set of buyers who value those units and want to buy at most an exogenously fixed number of units. We define a competitive equilibrium for this generalized assignment game and prove its existence by using only linear programming. In particular, we show how to compute equilibrium price vectors from the solutions of the dual linear program associated to the primal linear program defined to find optimal assignments. Using only linear programming tools, we also show (i) that the set of competitive equilibria (pairs of price vectors and assignments) has a Cartesian product structure: each equilibrium price vector is part of a competitive equilibrium with all optimal assignments, and vice versa; (ii) that the set of (restricted) equilibrium price vectors has a natural lattice structure; and (iii) how this structure is translated into the set of agents' utilities that are attainable at equilibrium.

The achievable region approach to the optimal control of stochastic systems

The achievable region approach seeks solutions to stochastic optimisation problems by: (i) characterising the space of all possible performances(the achievable region) of the system of interest, and (ii) optimisingthe overall system-wide performance objective over this space. This isradically different from conventional formulations based on dynamicprogramming. The approach is explained with reference to a simpletwo-class queueing system. Powerful new methodologies due to the authorsand co-workers are deployed to analyse a general multiclass queueingsystem with parallel servers and then to develop an approach to optimalload distribution across a network of interconnected stations. Finally,the approach is used for the first time to analyse a class of intensitycontrol problems.

Optimization of multiclass queueing networks with changeover times via the achievable region method: Part II, the multi-station case

We address the problem of scheduling a multi-station multiclassqueueing network (MQNET) with server changeover times to minimizesteady-state mean job holding costs. We present new lower boundson the best achievable cost that emerge as the values ofmathematical programming problems (linear, semidefinite, andconvex) over relaxed formulations of the system's achievableperformance region. The constraints on achievable performancedefining these formulations are obtained by formulatingsystem's equilibrium relations. Our contributions include: (1) aflow conservation interpretation and closed formulae for theconstraints previously derived by the potential function method;(2) new work decomposition laws for MQNETs; (3) new constraints(linear, convex, and semidefinite) on the performance region offirst and second moments of queue lengths for MQNETs; (4) a fastbound for a MQNET with N customer classes computed in N steps; (5)two heuristic scheduling policies: a priority-index policy, anda policy extracted from the solution of a linear programmingrelaxation.

Extracción de parámetros de un resonador FBAR por optimización

L’objectiu d’aquest projecte que consisteix a elaborar un algoritme d’optimització que permeti, mitjançant un ajust de dades per mínims quadrats, la extracció dels paràmetres del circuit equivalent que composen el model teòric d’un ressonador FBAR, a partir de les mesures dels paràmetres S. Per a dur a terme aquest treball, es desenvolupa en primer lloc tota la teoria necessària de ressonadors FBAR. Començant pel funcionament i l’estructura, i mostrant especial interès en el modelat d’aquests ressonadors mitjançant els models de Mason, Butterworth Van-Dyke i BVD Modificat. En segon terme, s’estudia la teoria sobre optimització i programació No-Lineal. Un cop s’ha exposat la teoria, es procedeix a la descripció de l’algoritme implementat. Aquest algoritme utilitza una estratègia de múltiples passos que agilitzen l'extracció dels paràmetres del ressonador.

Mapping sub-flows to p2mp LSPs

In previous work we proposed a multi-objective traffic engineering scheme (MHDB-S model) using different distribution trees to multicast several flows. In this paper, we propose a heuristic algorithm to create multiple point-to-multipoint (p2mp) LSPs based on the optimum sub-flow values obtained with our MHDB-S model. Moreover, a general problem for supporting multicasting in MPLS networks is the lack of labels. To reduce the number of labels used, a label space reduction algorithm solution is also considered

Equivalence of piecewise-linear approximation and Lagrangian relaxation for network revenue management

The network revenue management (RM) problem arises in airline, hotel, media,and other industries where the sale products use multiple resources. It can be formulatedas a stochastic dynamic program but the dynamic program is computationallyintractable because of an exponentially large state space, and a number of heuristicshave been proposed to approximate it. Notable amongst these -both for their revenueperformance, as well as their theoretically sound basis- are approximate dynamic programmingmethods that approximate the value function by basis functions (both affinefunctions as well as piecewise-linear functions have been proposed for network RM)and decomposition methods that relax the constraints of the dynamic program to solvesimpler dynamic programs (such as the Lagrangian relaxation methods). In this paperwe show that these two seemingly distinct approaches coincide for the network RMdynamic program, i.e., the piecewise-linear approximation method and the Lagrangianrelaxation method are one and the same.

Beyond Smith's rule: An optimal dynamic index, rule for single machine stochastic scheduling with convex holding costs

Most research on single machine scheduling has assumedthe linearity of job holding costs, which is arguablynot appropriate in some applications. This motivates ourstudy of a model for scheduling $n$ classes of stochasticjobs on a single machine, with the objective of minimizingthe total expected holding cost (discounted or undiscounted). We allow general holding cost rates that are separable,nondecreasing and convex on the number of jobs in eachclass. We formulate the problem as a linear program overa certain greedoid polytope, and establish that it issolved optimally by a dynamic (priority) index rule,whichextends the classical Smith's rule (1956) for the linearcase. Unlike Smith's indices, defined for each class, ournew indices are defined for each extended class, consistingof a class and a number of jobs in that class, and yieldan optimal dynamic index rule: work at each time on a jobwhose current extended class has larger index. We furthershow that the indices possess a decomposition property,as they are computed separately for each class, andinterpret them in economic terms as marginal expected cost rate reductions per unit of expected processing time.We establish the results by deploying a methodology recentlyintroduced by us [J. Niño-Mora (1999). "Restless bandits,partial conservation laws, and indexability. "Forthcomingin Advances in Applied Probability Vol. 33 No. 1, 2001],based on the satisfaction by performance measures of partialconservation laws (PCL) (which extend the generalizedconservation laws of Bertsimas and Niño-Mora (1996)):PCL provide a polyhedral framework for establishing theoptimality of index policies with special structure inscheduling problems under admissible objectives, which weapply to the model of concern.

Restless bandits, partial conservation laws and indexability

We show that if performance measures in a stochastic scheduling problem satisfy a set of so-called partial conservation laws (PCL), which extend previously studied generalized conservation laws (GCL), then the problem is solved optimally by a priority-index policy for an appropriate range of linear performance objectives, where the optimal indices are computed by a one-pass adaptive-greedy algorithm, based on Klimov's. We further apply this framework to investigate the indexability property of restless bandits introduced by Whittle, obtaining the following results: (1) we identify a class of restless bandits (PCL-indexable) which are indexable; membership in this class is tested through a single run of the adaptive-greedy algorithm, which also computes the Whittle indices when the test is positive; this provides a tractable sufficient condition for indexability; (2) we further indentify the class of GCL-indexable bandits, which includes classical bandits, having the property that they are indexable under any linear reward objective. The analysis is based on the so-called achievable region method, as the results follow fromnew linear programming formulations for the problems investigated.