An optimal-control based integrated model of supply chain

Problems of supply chain scheduling are challenged by high complexity, combination of continuous and discrete processes, integrated production and transportation operations as well as dynamics and resulting requirements for adaptability and stability analysis. A possibility to address the above-named issues opens modern control theory and optimal program control in particular. Based on a combination of fundamental results of modern optimal program control theory and operations research, an original approach to supply chain scheduling is developed in order to answer the challenges of complexity, dynamics, uncertainty, and adaptivity. Supply chain schedule generation is represented as an optimal program control problem in combination with mathematical programming and interpreted as a dynamic process of operations control within an adaptive framework. The calculation procedure is based on applying Pontryagin’s maximum principle and the resulting essential reduction of problem dimensionality that is under solution at each instant of time. With the developed model, important categories of supply chain analysis such as stability and adaptability can be taken into consideration. Besides, the dimensionality of operations research-based problems can be relieved with the help of distributing model elements between an operations research (static aspects) and a control (dynamic aspects) model. In addition, operations control and flow control models are integrated and applicable for both discrete and continuous processes.supply chain, model of supply chain scheduling, optimal program control theory, Pontryagin’s maximum principle, operations research model,

On the Optimal Taxation of Common-Pool Resources

Recent research developments in common-pool resource models emphasize the importance of links with ecological systems and the presence of non-linearities, thresholds and multiple steady states. In a recent paper Kossioris et al. (2008) develop a methodology for deriving feedback Nash equilibria for non-linear differential games and apply this methodology to a common-pool resource model of a lake where pollution corresponds to benefits and at the same time affects the ecosystem services. This paper studies the structure of optimal state- dependent taxes that steer the combined economic-ecological system towards the trajectory of optimal management, and provides an algorithm for calculating such taxes.Differential Games, non-linear Feedback Nash Equilibria, Ecosystems, Optimal State-dependent Tax

A MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEMS WITH STATE AND MIXED CONSTRAINTS

Here we derive a variant of the nonsmooth maximum principle for optimal control problems with both pure state and mixed state and control constraints. Our necessary conditions include a Weierstrass condition together with an Euler adjoint inclusion involving the joint subdifferentials with respect to both state and control, generalizing previous results in [M.d.R. de Pinho, M.M.A. Ferreira, F.A.C.C. Fontes, Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems. ESAIM: COCV 11 (2005) 614-632]. A notable feature is that our main results are derived combining old techniques with recent results. We use a well known penalization technique for state constrained problem together with an appeal to a recent nonsmooth maximum principle for problems with mixed constraints

Reduced order modelling of bone resorption and formation.

The bone remodelling process, performed by the Bone Multicellular Unit (BMU) is a key multi-hierarchically regulated process, which provides and supports various functionality of bone tissue. It is also plays a critical role in bone disorders, as well as bone tissue healing following damage. Improved modelling of bone turnover processes could play a significant role in helping to understand the underlying cause of bone disorders and thus develop more effective treatment methods. Moreover, despite extensive research in the field of bone tissue engineering, bonescaffold development is still very empirical. The development of improved methods of modelling the bone remodelling process should help to develop new implant designs which encourage rapid osteointegration. There are a number of limitations with respect to previous research in the field of mathematical modelling of the bone remodelling process, including the absence of an osteocyte loop of regulation. It is within this context that this research presented in this thesis utilises a range of modelling methods to develop a framework for bone remodelling which can be used to improve treatment methods for bone disorders. The study concentrated on dynamic and steady state variables that in perspective can be used as constraints for optimisation problem considering bone remodelling or tissue remodelling with the help of the grafts/scaffolds.The cellular and combined allosteric-regulation approaches to modelling of bone turnover, based on the osteocyte loop of regulation, have been studied. Both approaches have been studied different within wide range of rate parameters. The approach to the model validation has been considered, including a statistical approach and parameter reduction approach. From a validation perspective the cellular class of modes is preferable since it has fewer parameters to validate. The optimal control framework for regulation of remodelling has been studied. Future work in to improve the models and their application to bone scaffold design applications have been considered. The study illustrates the complexity of formalisation of the metabolic processes and the relations between hierarchical subsystems in hard tissue where a relatively small number of cells are active. Different types/modes of behaviour have been found in the study: relaxational, periodical and chaotic modes. All of these types of behaviour can be found, in bone tissue. However, a chaotic or periodic modes are ones of the hardest to verify although a number of periodical phenomena have been observed empirically in bone and skeletal development. Implementation of the allosteric loop into cellular model damps other types of behaviour/modes. In this sense it improves the robustness, predictability and control of the system. The developed models represent a first step in a hierarchical model of bone tissue (system versus local effects). The limited autonomy of any organ or tissue implies differentiation on a regulatory level as well as physiological functions and metabolic differences. Implementation into the cellular phenomenological model of allosteric-like loop of regulation has been performed. The results show that the robustness of regulation can be inherited from the phenomenological model. An attempt to correlate the main bone disorders with different modes of behaviour has been undertaken using Paget’s disorder in bone, osteoporosis and some more general skeleton disorders which lead to periodical changes in bone mass, reported by some authors. However, additional studies are needed to make this hypothesis significant. The study has revealed a few interesting techniques. When studying a multidimensional phenomenon, as a bone tissue is, the visualisation and data reduction is important for analysis and interpretation of results. In the study two novel technical methods have been proposed. The first is the graphical matrix method to visualise/project the multidimensional phase space of variables into diagonal matrix of regular combination of two-dimensional graphs. This significantly simplifies the analysis and, in principle, makes it possible to visualise the phase space higher than three-dimensional. The second important technical development is the application of the Monte-Carlo method in combination with the regression method to study the character and stability of the equilibrium points of a dynamic system. The advantage of this method is that it enables the most influential parameters that affect the character and stability of the equilibrium point to be identified from a large number of the rate parameters/constants of the dynamic system. This makes the interpretation of parameters and conceptual verification of the model much easier

Cooperation or competition : dilemma for resource managers in sustainable wildlife utilisation

Keywords: analytical modelling; Associated Private Nature Reserves; consumptive use; elephants; Kruger National Park; land productivity; non-consumptive use; waterpoints; Savanna ecosystem model; South Africa. Wildlife as part of biodiversity is a global natural resource. However, landowners have some control over the future of wildlife on their land. Wildlife could be managed by the state or private landowners. The survival of the wildlife and their habitats is determined by how these landowners decide to use the land and the renewable resources on it. Some complication come into place given that wildlife usually roam on land held over by more than one owner providing more challenge to its management. In addition, wildlife as a natural resource has multiple uses that generate revenues for the betterment of the landowners. The uses could be consumptive or non-consumptive. Each landowner has multiple objectives which might be conflicting which poses even a greater challenge to the sustainable wildlife management. To meet their objectives wildlife managers use management tools. Some of the tools used include constructing or closing of artificial waterpoints, fire management, fencing, and population manipulation through culling/hunting or translocation of animals. However, use of these management tools can lead to unintended or opposite effects if they are not well understood. There are direct and indirect effects of the tools on biodiversity. Landowners could be tempted to excessively use some of the tools in order to achieve their objectives. In addition, most studies have concentrated on either the ecological or economic effects of the wildlife management tools. For the landowner, it is essential that he comprehends both the ecological and economic effects of the wildlife management tools for the sustainable management of wildlife, a contribution of this study. The main objective of this study is to assess the ecological and economic implications of some wildlife management tools on the landowners’ welfare. I use simple ecological economic analytical models based on the Pontryagin’s maximum principle to perform the analyses. The Savanna ecosystem model which is a spatially explicit, process-oriented model is also used to further explore the effects of one of the wildlife management tools on landowner’s multiple objectives. One of the tools that is analyzed in this thesis is the improvement of land productivity through increasing of vegetation quality. Given that, it is usually not easy to increase the land size in response to increased incentives, some landowners might consider increasing the land productivity. The results show that utilization of wildlife can contribute to wildlife conservation and enhancement of welfare as a result of investment by landowners into habitat quality improvement. However, the use of a wildlife management tool has direct and indirect effects as demonstrated by another framework presented in this thesis on waterpoints. Waterpoints are used by wildlife managers to supplement natural water supplies which in turn support herbivore populations, like elephants. A private oriented landowner may be interested only in maximization of profits or personal benefits either from elephant offtake and/or tourism revenue, thus might ignore the negative effects that could be brought about by elephants to biodiversity. In such case, the game reserve management as the authority entrusted with sustainable management of the game reserve should use economic instruments such as subsidies or payments for the compliant landowners and/or taxes or charges for the non-compliant landowners to encourage compliance with sustainable wildlife management practices. The Savanna ecosystem model is used to explore the effects of waterpoints on elephant density (representing an economic objective) and biodiversity (representing an ecological objective). The model is used to analyze the differential impact of waterpoints on the Kruger National Park’s regions under 26 waterpoints manipulation scenarios. The model is also used to analyze elephant impact on vegetation biomass diversity in four regions of Kruger National Park. The results showed that constructing (or closing) extra waterpoints in one region does not necessarily translate into higher (or lower) elephant densities in that region, but the effect depends on the vegetation and other conditions of the region in comparison to neighbouring regions. In one of the regions, the model showed that there is a trade-off between elephant density and vegetation biomass diversity. In another region, elephants’ effect on vegetation biomass diversity follows the intermediate disturbance hypothesis, whilst in other regions the relationship is positive. The intermediate disturbance hypothesis postulates that there would be a higher diversity of vegetation structure at intermediate elephant densities whilst at extreme levels of both low and high disturbance the diversity would be reduced. The model thus suggests that different strategies should be adopted for different regions, e.g., an adaptive management strategy could be used for one of the regions where waterpoints are switched on and off depending on the elephant density. Another wildlife management tool that is analyzed is the use of physical barriers like fences. Physical barriers could be utilised by landowners to separate different wildlife uses which might be conflicting. Landowners or game reserve management are often faced with the decision whether to undertake consumptive (hunting) and/or non-consumptive (tourism) use on their properties. A theoretical model is constructed to examine these cases. The results show that that the two uses can be undertaken in the same contiguous area if the consumptive use is not dominating. In conclusion, what emerges from this work is that given that the landowner’s welfare is not only affected by his own actions but also his neighbours’ modi operandi, then the landowner should consider all levels of cooperation with his neighbours in order to fully maximize his welfare. This includes cooperation in terms of which management tool(s) he and/or his neighbour should use. The frameworks presented in this thesis could be used by landowners (both state and private) to analyze the effects of their management actions on their welfare. </p

Consumer-Resource Dynamics: Quantity, Quality, and Allocation

CITATION: Getz, W. M. & Owen-Smith, N. 2011. Consumer-resource dynamics : quantity, quality, and allocation. PLoS ONE, 6(1): e14539, doi:10.1371/journal.pone.0014539.The original publication is available at http://journals.plos.org/plosoneBackground: The dominant paradigm for modeling the complexities of interacting populations and food webs is a system of coupled ordinary differential equations in which the state of each species, population, or functional trophic group is represented by an aggregated numbers-density or biomass-density variable. Here, using the metaphysiological approach to model consumer-resource interactions, we formulate a two-state paradigm that represents each population or group in a food web in terms of both its quantity and quality. Methodology and Principal Findings: The formulation includes an allocation function controlling the relative proportion of extracted resources to increasing quantity versus elevating quality. Since lower quality individuals senesce more rapidly than higher quality individuals, an optimal allocation proportion exists and we derive an expression for how this proportion depends on population parameters that determine the senescence rate, the per-capita mortality rate, and the effects of these rates on the dynamics of the quality variable. We demonstrate that oscillations do not arise in our model from quantity-quality interactions alone, but require consumer-resource interactions across trophic levels that can be stabilized through judicious resource allocation strategies. Analysis and simulations provide compelling arguments for the necessity of populations to evolve quality-related dynamics in the form of maternal effects, storage or other appropriate structures. They also indicate that resource allocation switching between investments in abundance versus quality provide a powerful mechanism for promoting the stability of consumer-resource interactions in seasonally forcing environments. Conclusions/Significance: Our simulations show that physiological inefficiencies associated with this switching can be favored by selection due to the diminished exposure of inefficient consumers to strong oscillations associated with the wellknown paradox of enrichment. Also our results demonstrate how allocation switching can explain observed growth patterns in experimental microbial cultures and discuss how our formulation can address questions that cannot be answered using the quantity-only paradigms that currently predominate. © 2011 Getz, Owen-Smith.http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0014539Publisher's versio

OPTIMAL CONTROL OF ION EXCHANGE PROCESS FOR CHROMATE REMOVAL AND PROTEIN A CHROMATOGRAPHY FOR ANTIBODY EXTRACTION

Ion exchange resins are widely used in the extraction of hazardous chemicals as well as the recovery of precious molecules. Therefore, an early breakthrough from the resin system can lead to toxic compounds affecting the drinking water quality or inefficient use of costly resins. Hence, accurate modeling of the ion exchange process and control strategy can enable decisions that assist in avoiding leakage when facing fluctuations in the inlet contaminant concentrations. In this work, the ion exchange process is modeled via the method of moments where the system uncertainties are captured via stochastic modeling using Ito processes. The flow rate is controlled to optimize the resin performance by maximizing its dynamic removal efficiency. The process runs more efficiently with a well-controlled varying flow rate rather than a constant flow, a standard industrial practice. The optimal control reveals that introducing the feed at a high flow rate followed by a decreasing flow can achieve significant removal of the target molecules and increase the efficiency of the purification process. This work has wide applicability ranging from chromate removal from water to extracting antibodies with a costly affinity chromatography resin

A mathematical framework for evo-devo dynamics

Funding: This work was funded by an ERC Consolidator Grant to A. Gardner (grant no. 771387), by the School of Biology of the University of St Andrews, and by a John Templeton Foundation grant to K.N. Laland and T. Uller (grant ID 60501).Natural selection acts on phenotypes constructed over development, which raises the question of how development affects evolution. Classic evolutionary theory indicates that development affects evolution by modulating the genetic covariation upon which selection acts, thus affecting genetic constraints. However, whether genetic constraints are relative, thus diverting adaptation from the direction of steepest fitness ascent, or absolute, thus blocking adaptation in certain directions, remains uncertain. This limits understanding of long-term evolution of developmentally constructed phenotypes. Here we formulate a general, tractable mathematical framework that integrates age progression, explicit development (i.e., the construction of the phenotype across life subject to developmental constraints), and evolutionary dynamics, thus describing the evolutionary and developmental (evo-devo) dynamics. The framework yields simple equations that can be arranged in a layered structure that we call the evo-devo process, whereby five core elementary components generate all equations including those mechanistically describing genetic covariation and the evo-devo dynamics. The framework recovers evolutionary dynamic equations in gradient form and describes the evolution of genetic covariation from the evolution of genotype, phenotype, environment, and mutational covariation. This shows that genotypic and phenotypic evolution must be followed simultaneously to yield a dynamically sufficient description of long-term phenotypic evolution in gradient form, such that evolution described as the climbing of a fitness landscape occurs in “geno-phenotype” space. Genetic constraints in geno-phenotype space are necessarily absolute because the phenotype is related to the genotype by development. Thus, the long-term evolutionary dynamics of developed phenotypes is strongly non-standard: (1) evolutionary equilibria are either absent or infinite in number and depend on genetic covariation and hence on development; (2) developmental constraints determine the admissible evolutionary path and hence which evolutionary equilibria are admissible; and (3) evolutionary outcomes occur at admissible evolutionary equilibria, which do not generally occur at fitness landscape peaks in geno-phenotype space, but at peaks in the admissible evolutionary path where “total genotypic selection” vanishes if exogenous plastic response vanishes and mutational variation exists in all directions of genotype space. Hence, selection and development jointly define the evolutionary outcomes if absolute mutational constraints and exogenous plastic response are absent, rather than the outcomes being defined only by selection. Moreover, our framework provides formulas for the sensitivities of a recurrence and an alternative method to dynamic optimization (i.e., dynamic programming or optimal control) to identify evolutionary outcomes in models with developmentally dynamic traits. These results show that development has major evolutionary effects.Publisher PDFPeer reviewe