Linear matrix inequality-based robust model predictive control for time-delayed systems

This work addresses the solution to the problem of robust model predictive control (MPC) of systems with model uncertainty. The case of zone control of multi-variable stable systems with multiple time delays is considered. The usual approach of dealing with this kind of problem is through the inclusion of non-linear cost constraint in the control problem. The control action is then obtained at each sampling time as the solution to a non-linear programming (NLP) problem that for high-order systems can be computationally expensive. Here, the robust MPC problem is formulated as a linear matrix inequality problem that can be solved in real time with a fraction of the computer effort. The proposed approach is compared with the conventional robust MPC and tested through the simulation of a reactor system of the process industry.

Light energy management in peach: utilization, photoprotection , photodamage and recovery. Maximizing light absorption in orchard is not always the best solution

The relation between the intercepted light and orchard productivity was considered linear, although this dependence seems to be more subordinate to planting system rather than light intensity. At whole plant level not always the increase of irradiance determines productivity improvement. One of the reasons can be the plant intrinsic un-efficiency in using energy. Generally in full light only the 5 – 10% of the total incoming energy is allocated to net photosynthesis. Therefore preserving or improving this efficiency becomes pivotal for scientist and fruit growers. Even tough a conspicuous energy amount is reflected or transmitted, plants can not avoid to absorb photons in excess. The chlorophyll over-excitation promotes the reactive species production increasing the photoinhibition risks. The dangerous consequences of photoinhibition forced plants to evolve a complex and multilevel machine able to dissipate the energy excess quenching heat (Non Photochemical Quenching), moving electrons (water-water cycle , cyclic transport around PSI, glutathione-ascorbate cycle and photorespiration) and scavenging the generated reactive species. The price plants must pay for this equipment is the use of CO2 and reducing power with a consequent decrease of the photosynthetic efficiency, both because some photons are not used for carboxylation and an effective CO2 and reducing power loss occurs. Net photosynthesis increases with light until the saturation point, additional PPFD doesn’t improve carboxylation but it rises the efficiency of the alternative pathways in energy dissipation but also ROS production and photoinhibition risks. The wide photo-protective apparatus, although is not able to cope with the excessive incoming energy, therefore photodamage occurs. Each event increasing the photon pressure and/or decreasing the efficiency of the described photo-protective mechanisms (i.e. thermal stress, water and nutritional deficiency) can emphasize the photoinhibition. Likely in nature a small amount of not damaged photosystems is found because of the effective, efficient and energy consuming recovery system. Since the damaged PSII is quickly repaired with energy expense, it would be interesting to investigate how much PSII recovery costs to plant productivity. This PhD. dissertation purposes to improve the knowledge about the several strategies accomplished for managing the incoming energy and the light excess implication on photo-damage in peach. The thesis is organized in three scientific units. In the first section a new rapid, non-intrusive, whole tissue and universal technique for functional PSII determination was implemented and validated on different kinds of plants as C3 and C4 species, woody and herbaceous plants, wild type and Chlorophyll b-less mutant and monocot and dicot plants. In the second unit, using a “singular” experimental orchard named “Asymmetric orchard”, the relation between light environment and photosynthetic performance, water use and photoinhibition was investigated in peach at whole plant level, furthermore the effect of photon pressure variation on energy management was considered on single leaf. In the third section the quenching analysis method suggested by Kornyeyev and Hendrickson (2007) was validate on peach. Afterwards it was applied in the field where the influence of moderate light and water reduction on peach photosynthetic performances, water requirements, energy management and photoinhibition was studied. Using solar energy as fuel for life plant is intrinsically suicidal since the high constant photodamage risk. This dissertation would try to highlight the complex relation existing between plant, in particular peach, and light analysing the principal strategies plants developed to manage the incoming light for deriving the maximal benefits as possible minimizing the risks. In the first instance the new method proposed for functional PSII determination based on P700 redox kinetics seems to be a valid, non intrusive, universal and field-applicable technique, even because it is able to measure in deep the whole leaf tissue rather than the first leaf layers as fluorescence. Fluorescence Fv/Fm parameter gives a good estimate of functional PSII but only when data obtained by ad-axial and ab-axial leaf surface are averaged. In addition to this method the energy quenching analysis proposed by Kornyeyev and Hendrickson (2007), combined with the photosynthesis model proposed by von Caemmerer (2000) is a forceful tool to analyse and study, even in the field, the relation between plant and environmental factors such as water, temperature but first of all light. “Asymmetric” training system is a good way to study light energy, photosynthetic performance and water use relations in the field. At whole plant level net carboxylation increases with PPFD reaching a saturating point. Light excess rather than improve photosynthesis may emphasize water and thermal stress leading to stomatal limitation. Furthermore too much light does not promote net carboxylation improvement but PSII damage, in fact in the most light exposed plants about 50-60% of the total PSII is inactivated. At single leaf level, net carboxylation increases till saturation point (1000 – 1200 μmolm-2s-1) and light excess is dissipated by non photochemical quenching and non net carboxylative transports. The latter follows a quite similar pattern of Pn/PPFD curve reaching the saturation point at almost the same photon flux density. At middle-low irradiance NPQ seems to be lumen pH limited because the incoming photon pressure is not enough to generate the optimum lumen pH for violaxanthin de-epoxidase (VDE) full activation. Peach leaves try to cope with the light excess increasing the non net carboxylative transports. While PPFD rises the xanthophyll cycle is more and more activated and the rate of non net carboxylative transports is reduced. Some of these alternative transports, such as the water-water cycle, the cyclic transport around the PSI and the glutathione-ascorbate cycle are able to generate additional H+ in lumen in order to support the VDE activation when light can be limiting. Moreover the alternative transports seems to be involved as an important dissipative way when high temperature and sub-optimal conductance emphasize the photoinhibition risks. In peach, a moderate water and light reduction does not determine net carboxylation decrease but, diminishing the incoming light and the environmental evapo-transpiration request, stomatal conductance decreases, improving water use efficiency. Therefore lowering light intensity till not limiting levels, water could be saved not compromising net photosynthesis. The quenching analysis is able to partition absorbed energy in the several utilization, photoprotection and photo-oxidation pathways. When recovery is permitted only few PSII remained un-repaired, although more net PSII damage is recorded in plants placed in full light. Even in this experiment, in over saturating light the main dissipation pathway is the non photochemical quenching; at middle-low irradiance it seems to be pH limited and other transports, such as photorespiration and alternative transports, are used to support photoprotection and to contribute for creating the optimal trans-thylakoidal ΔpH for violaxanthin de-epoxidase. These alternative pathways become the main quenching mechanisms at very low light environment. Another aspect pointed out by this study is the role of NPQ as dissipative pathway when conductance becomes severely limiting. The evidence that in nature a small amount of damaged PSII is seen indicates the presence of an effective and efficient recovery mechanism that masks the real photodamage occurring during the day. At single leaf level, when repair is not allowed leaves in full light are two fold more photoinhibited than the shaded ones. Therefore light in excess of the photosynthetic optima does not promote net carboxylation but increases water loss and PSII damage. The more is photoinhibition the more must be the photosystems to be repaired and consequently the energy and dry matter to allocate in this essential activity. Since above the saturation point net photosynthesis is constant while photoinhibition increases it would be interesting to investigate how photodamage costs in terms of tree productivity. An other aspect of pivotal importance to be further widened is the combined influence of light and other environmental parameters, like water status, temperature and nutrition on peach light, water and phtosyntate management.

Enhanced Mixed Integer Programming Techniques and Routing Problems

Mixed integer programming is up today one of the most widely used techniques for dealing with hard optimization problems. On the one side, many practical optimization problems arising from real-world applications (such as, e.g., scheduling, project planning, transportation, telecommunications, economics and finance, timetabling, etc) can be easily and effectively formulated as Mixed Integer linear Programs (MIPs). On the other hand, 50 and more years of intensive research has dramatically improved on the capability of the current generation of MIP solvers to tackle hard problems in practice. However, many questions are still open and not fully understood, and the mixed integer programming community is still more than active in trying to answer some of these questions. As a consequence, a huge number of papers are continuously developed and new intriguing questions arise every year. When dealing with MIPs, we have to distinguish between two different scenarios. The first one happens when we are asked to handle a general MIP and we cannot assume any special structure for the given problem. In this case, a Linear Programming (LP) relaxation and some integrality requirements are all we have for tackling the problem, and we are ``forced" to use some general purpose techniques. The second one happens when mixed integer programming is used to address a somehow structured problem. In this context, polyhedral analysis and other theoretical and practical considerations are typically exploited to devise some special purpose techniques. This thesis tries to give some insights in both the above mentioned situations. The first part of the work is focused on general purpose cutting planes, which are probably the key ingredient behind the success of the current generation of MIP solvers. Chapter 1 presents a quick overview of the main ingredients of a branch-and-cut algorithm, while Chapter 2 recalls some results from the literature in the context of disjunctive cuts and their connections with Gomory mixed integer cuts. Chapter 3 presents a theoretical and computational investigation of disjunctive cuts. In particular, we analyze the connections between different normalization conditions (i.e., conditions to truncate the cone associated with disjunctive cutting planes) and other crucial aspects as cut rank, cut density and cut strength. We give a theoretical characterization of weak rays of the disjunctive cone that lead to dominated cuts, and propose a practical method to possibly strengthen those cuts arising from such weak extremal solution. Further, we point out how redundant constraints can affect the quality of the generated disjunctive cuts, and discuss possible ways to cope with them. Finally, Chapter 4 presents some preliminary ideas in the context of multiple-row cuts. Very recently, a series of papers have brought the attention to the possibility of generating cuts using more than one row of the simplex tableau at a time. Several interesting theoretical results have been presented in this direction, often revisiting and recalling other important results discovered more than 40 years ago. However, is not clear at all how these results can be exploited in practice. As stated, the chapter is a still work-in-progress and simply presents a possible way for generating two-row cuts from the simplex tableau arising from lattice-free triangles and some preliminary computational results. The second part of the thesis is instead focused on the heuristic and exact exploitation of integer programming techniques for hard combinatorial optimization problems in the context of routing applications. Chapters 5 and 6 present an integer linear programming local search algorithm for Vehicle Routing Problems (VRPs). The overall procedure follows a general destroy-and-repair paradigm (i.e., the current solution is first randomly destroyed and then repaired in the attempt of finding a new improved solution) where a class of exponential neighborhoods are iteratively explored by heuristically solving an integer programming formulation through a general purpose MIP solver. Chapters 7 and 8 deal with exact branch-and-cut methods. Chapter 7 presents an extended formulation for the Traveling Salesman Problem with Time Windows (TSPTW), a generalization of the well known TSP where each node must be visited within a given time window. The polyhedral approaches proposed for this problem in the literature typically follow the one which has been proven to be extremely effective in the classical TSP context. Here we present an overall (quite) general idea which is based on a relaxed discretization of time windows. Such an idea leads to a stronger formulation and to stronger valid inequalities which are then separated within the classical branch-and-cut framework. Finally, Chapter 8 addresses the branch-and-cut in the context of Generalized Minimum Spanning Tree Problems (GMSTPs) (i.e., a class of NP-hard generalizations of the classical minimum spanning tree problem). In this chapter, we show how some basic ideas (and, in particular, the usage of general purpose cutting planes) can be useful to improve on branch-and-cut methods proposed in the literature.

A farm-level programming model to compare the atmospheric impact of conventional and organic farming

A model is developed to represent the activity of a farm using the method of linear programming. Two are the main components of the model, the balance of soil fertility and the livestock nutrition. According to the first, the farm is supposed to have a total requirement of nitrogen, which is to be accomplished either through internal sources (manure) or through external sources (fertilisers). The second component describes the animal husbandry as having a nutritional requirement which must be satisfied through the internal production of arable crops or the acquisition of feed from the market. The farmer is supposed to maximise total net income from the agricultural and the zoo-technical activities by choosing one rotation among those available for climate and acclivity. The perspective of the analysis is one of a short period: the structure of the farm is supposed to be fixed without possibility to change the allocation of permanent crops and the amount of animal husbandry. The model is integrated with an environmental module that describes the role of the farm within the carbon-nitrogen cycle. On the one hand the farm allows storing carbon through the photosynthesis of the plants and the accumulation of carbon in the soil; on the other some activities of the farm emit greenhouse gases into the atmosphere. The model is tested for some representative farms of the Emilia-Romagna region, showing to be capable to give different results for conventional and organic farming and providing first results concerning the different atmospheric impact. Relevant data about the representative farms and the feasible rotations are extracted from the FADN database, with an integration of the coefficients from the literature.

Efficient certification of feasibility and objective value of linear programs and its applications

The use of linear programming in various areas has increased with the significant improvement of specialized solvers. Linear programs are used as such to model practical problems, or as subroutines in algorithms such as formal proofs or branch-and-cut frameworks. In many situations a certified answer is needed, for example the guarantee that the linear program is feasible or infeasible, or a provably safe bound on its objective value. Most of the available solvers work with floating-point arithmetic and are thus subject to its shortcomings such as rounding errors or underflow, therefore they can deliver incorrect answers. While adequate for some applications, this is unacceptable for critical applications like flight controlling or nuclear plant management due to the potential catastrophic consequences. We propose a method that gives a certified answer whether a linear program is feasible or infeasible, or returns unknown'. The advantage of our method is that it is reasonably fast and rarely answers unknown'. It works by computing a safe solution that is in some way the best possible in the relative interior of the feasible set. To certify the relative interior, we employ exact arithmetic, whose use is nevertheless limited in general to critical places, allowing us to rnremain computationally efficient. Moreover, when certain conditions are fulfilled, our method is able to deliver a provable bound on the objective value of the linear program. We test our algorithm on typical benchmark sets and obtain higher rates of success compared to previous approaches for this problem, while keeping the running times acceptably small. The computed objective value bounds are in most of the cases very close to the known exact objective values. We prove the usability of the method we developed by additionally employing a variant of it in a different scenario, namely to improve the results of a Satisfiability Modulo Theories solver. Our method is used as a black box in the nodes of a branch-and-bound tree to implement conflict learning based on the certificate of infeasibility for linear programs consisting of subsets of linear constraints. The generated conflict clauses are in general small and give good rnprospects for reducing the search space. Compared to other methods we obtain significant improvements in the running time, especially on the large instances.

Restructuring Justice: Moving Towards Gender-Responsive Programming and Alternatives to Incarceration for Non-Violent Female Offenders

The United States¿ Federal and State laws differentiate between acceptable (or, legal) and unacceptable (illegal) behavior by prescribing restrictive punishment to citizens and/or groups that violate these established rules. These regulations are written to treat every person equally and to fairly serve justice; furthermore, the sanctions placed on offenders seek to reform illegal behavior through limitations on freedoms and rehabilitative programs. Despite the effort to treat all offenders fairly regardless of social identity categories (e.g., sex, race, ethnicity, socioeconomic status, age, ability, and gender and sexual orientation) and to humanely eliminate illegal behavior, the American penal system perpetuates de facto discrimination against a multitude of peoples. Furthermore, soaring recidivism rates caused by unsuccessful re-entry of incarcerated offenders puts economic stress on Federal and State budgets. For these reasons, offenders, policy-makers, and law-abiding citizens should all have a vested interest in reforming the prison system. This thesis focuses on the failure of the United States corrections system to adequately address the gender-specific needs of non-violent female offenders. Several factors contribute to the gender-specific discrimination that women experience in the criminal justice system: 1) Trends in female criminality that skew women¿s crime towards drug-related crimes, prostitution, and property offenses; 2) Mandatory minimum sentences for drug crimes that are disproportionate to the crime committed; 3) So-called ¿gender-neutral¿ educational, vocational, substance abuse, and mental health programming that intends to equally rehabilitate men and women, but in fact favors men; and 4) The isolating nature of prison structures that inhibits smooth re-entry into society. I argue that a shift in the placement and treatment of non-violent female offenders is necessary for effective rehabilitation and for reducing recidivism rates. The first component of this shift is the design and implementation of gender- responsive treatment (GRT) rather than gender-neutral approaches in rehabilitative programming. The second shift is the utilization of alternatives to incarceration, which provide both more humane treatment of offenders and smoother reintegration to society. Drawing on recent scholarship, information from prison advocacy organizations, and research with men in an alternative program, I provide a critical analysis of current policies and alternative programs, and suggest several proposals for future gender- responsive programs in prisons and in place of incarceration. I argue that the expansion of gender-responsive programming and alternatives to incarceration respond to the marginalization of female offenders, address concerns about the financial sustainability of the United States criminal justice system, and tackle high recidivism rates.

Using Dynamic Programming Decomposition for Revenue Management with Opaque Products

Opaque products enable service providers to hide specific characteristics of their service fulfillment from the customer until after purchase. Prominent examples include internet-based service providers selling airline tickets without defining details, such as departure time or operating airline, until the booking has been made. Owing to the resulting flexibility in resource utilization, the traditional revenue management process needs to be modified. In this paper, we extend dynamic programming decomposition techniques widely used for traditional revenue management to develop an intuitive capacity control approach that allows for the incorporation of opaque products. In a simulation study, we show that the developed approach significantly outperforms other well-known capacity control approaches adapted to the opaque product setting. Based on the approach, we also provide computational examples of how the share of opaque products as well as the degree of opacity can influence the results.

Patient classification system: An integrated method for measuring nursing intensity and optimizing resource allocation

A patient classification system was developed integrating a patient acuity instrument with a computerized nursing distribution method based on a linear programming model. The system was designed for real-time measurement of patient acuity (workload) and allocation of nursing personnel to optimize the utilization of resources.^ The acuity instrument was a prototype tool with eight categories of patients defined by patient severity and nursing intensity parameters. From this tool, the demand for nursing care was defined in patient points with one point equal to one hour of RN time. Validity and reliability of the instrument was determined as follows: (1) Content validity by a panel of expert nurses; (2) predictive validity through a paired t-test analysis of preshift and postshift categorization of patients; (3) initial reliability by a one month pilot of the instrument in a practice setting; and (4) interrater reliability by the Kappa statistic.^ The nursing distribution system was a linear programming model using a branch and bound technique for obtaining integer solutions. The objective function was to minimize the total number of nursing personnel used by optimally assigning the staff to meet the acuity needs of the units. A penalty weight was used as a coefficient of the objective function variables to define priorities for allocation of staff.^ The demand constraints were requirements to meet the total acuity points needed for each unit and to have a minimum number of RNs on each unit. Supply constraints were: (1) total availability of each type of staff and the value of that staff member (value was determined relative to that type of staff's ability to perform the job function of an RN (i.e., value for eight hours RN = 8 points, LVN = 6 points); (2) number of personnel available for floating between units.^ The capability of the model to assign staff quantitatively and qualitatively equal to the manual method was established by a thirty day comparison. Sensitivity testing demonstrated appropriate adjustment of the optimal solution to changes in penalty coefficients in the objective function and to acuity totals in the demand constraints.^ Further investigation of the model documented: correct adjustment of assignments in response to staff value changes; and cost minimization by an addition of a dollar coefficient to the objective function. ^

Prenatal care content and utilization among pregnant women enrolled in managed care programs in Harris County, Texas

This research study was conducted as a descriptive study of prenatal care experiences of women enrolled in public and private managed care programs. The study's aim was to describe the demographic characteristics of the women in the study and to analyze and compare their prenatal care experiences. ^ The objective of this study was to examine the research question: Do pregnant women enrolled in Medicaid Managed Care receive the same level of care as women enrolled in other Managed Care Programs in Harris County, Texas? ^ The study population was a convenience sample of pregnant women enrolled in managed care programs who presented to one of the two hospital study sites for delivery of their infant. The study utilized a self administered survey to measure adequacy and content of prenatal care received by the women during this pregnancy. Adequacy of prenatal care utilization was determined based on the Kessner Index criteria of timing of initiation of care and number of visits. Content of care was measured by the number of different medical services the women reported they had received and the number of health information topics the women reported on which they had received information. Demographic characteristics were described with univariate and bivariate statistics of frequencies and cross tabulations. Associations were evaluated using measures of linear correlations. ^ Results from the study showed there is an association between enrollment in Medicaid Managed Care (public) and prenatal care received compared to women enrolled in other Managed Care Programs (private). The results were derived from statistical tests on data the postpartum women gave when they completed the self-administered survey. Provider type was a moderate predictor of quality and quantity of prenatal care. The results also indicate that in the study population, minority ethnicity, income and lower educational status were associated with intermediate and inadequate prenatal care. ^

Ethnographic and epidemiologic approaches for the programming of prenatal and birth services in Mesa los Hornos, Mexico

The investigator conducted an action-oriented investigation of pregnancy and birth among the women of Mesa los Hornos, an urban squatter slum in Mexico City. Three aims guided the project: (1) To obtain information for improving prenatal and maternity service utilization; (2) To examine the utility of rapid ethnographic and epidemiologic assessment methodologies; (3) To cultivate community involvement in health development.^ Viewing service utilization as a culturally-bound decision, the study included a qualitative phase to explore women's cognition of pregnancy and birth, their perceived needs during pregnancy, and their criteria of service acceptability. A probability-based community survey delineated parameters of service utilization and pregnancy health events, and probed reasons for decisions to use medical services, lay midwives, or other sources of prenatal and labor and delivery assistance. Qualitative survey of service providers at relevant clinics, hospitals, and practices contributed information on service availability and access, and on coordination among private, social security, and public assistance health service sectors. The ethnographic approach to exploring the rationale for use or non-use of services provided a necessary complement to conventional barrier-based assessment, to inform planning of culturally appropriate interventions.^ Information collection and interpretation was conducted under the aegis of an advisory committee of community residents and service agency representatives; the residents' committee formulated recommendations for action based on findings, and forwarded the mandate to governmental social and urban development offices. Recommendations were designed to inform and develop community participation in health care decision-making.^ Rapid research methods are powerful tools for achieving community-based empowerment toward investigation and resolution of local health problems. But while ethnography works well in synergy with quantitative assessment approaches to strengthen the validity and richness of short-term field work, the author strongly urges caution in application of Rapid Ethnographic Assessments. An ethnographic sensibility is essential to the research enterprise for the development of an active and cooperative community base, the design and use of quantitative instruments, the appropriate use of qualitative techniques, and the interpretation of culturally-oriented information. However, prescribed and standardized Rapid Ethnographic Assessment techniques are counter-productive if used as research short-cuts before locale- and subject-specific cultural understanding is achieved. ^

Part of the global DOC versus AOU (dissolved organic carbon/apparent oxygen utilization) data compilation, Pacific Ocean

Total organic carbon (TOC) was analyzed on four transects along 140°W in 1992 using a high temperature combustion/discrete injection (HTC/DI) analyzer. For two of the transects, the analyses were conducted on-board ship. Mixed-layer concentrations of organic carbon varied from about 80 µM C at either end of the transect (12°N and 12°S) to about 60 µM C at the equator. Total organic carbon concentrations decreased rapidly below the mixed-layer to about 38-40 µM C at 1000 m across the transect. Little variation was observed below this depth; deep water concentrations below 2000 m were virtually monotonic at about 36 µM C. Repeat measurements made on subsequent cruises consistently found the same concentrations at 1000 m or deeper, but substantial variations were observed in the mixed-layer and the upper water column above 400 m depth. Linear mixing models of total organic carbon versus sigmaT exhibited zones of organic carbon formation and consumption. TOC was found to be inversely correlated with apparent oxygen utilization (AOU) in the region between the mixed-layer and the oxygen minimum. In the mixed-layer, TOC concentrations varied seasonally. Part of the variations in TOC at the equator was driven by changes in the upwelling rate in response to variations in physical forcing related to an El Niño and to the passage of tropical instability waves. TOC export fluxes, calculated from simple box models, averaged 8±4 mmol C/m**2/day at the equator and also varied seasonally. These export fluxes account for 50-75% of the total carbon deficit and are consistent with other estimates and model predictions.

Magneto-optic current sensor with Faraday mirror for linear birefringence compensation

Fiber optic sensors have some advantages in subjects related with electrical current and magnetic field measurement. In spite of the optical fiber utilization advantages we have to take into account undesirable effects, which are present in real non-ideal optical fibers. In telecommunication and sensor application fields the presence of inherent and induced birefringence is crucial. The presence of birefringence may cause an undesirable change in the polarization state. In order to compensate the linear birefringence a promising method has been chosen. This method employs orthogonal polarization conjugation in the back propagation direction of the light wave in the fiber. A study and a simulation of an experimental setup are realized with the advantage of a significant sensitivity improvement.

Red Eléctrica de España uses integer programming for annual salary revisions

In this paper, we present a mixed-integer linear programming model for determining salary-revision matrices for an organization based on that organization?s general strategies. The solution obtained from this model consists of salary increases for each employee; these increases consider the employee?s professional performance, salary level relative to peers within the organization, and professional group. In addition to budget constraints, we modeled other elements typical of compensation systems, such as equity and justice. Red Eléctrica de España (REE), the transmission agent and operator of the Spanish electricity system, used the model to revise its 2010 and 2011 salary policies, and achieved results that were aligned with the company strategy. REE incorporated the model into the salary management module within its information system, and plans to continue to use the model in revisions of the module.

Analysis of linear trade models and relation to scale economies

We discuss linear Ricardo models with a range of parameters. We show that the exact boundary of the region of equilibria of these models is obtained by solving a simple integer programming problem. We show that there is also an exact correspondence between many of the equilibria resulting from families of linear models and the multiple equilibria of economies of scale models.