A numerical method is proposed for a class of one-dimensional stochastic control problems with unbounded state space. This method solves an infinite-dimensional linear program, equivalent to the original formulation based on a stochastic differential equation, using a finite element approximation. The discretization scheme itself and the necessary assumptions are discussed, and a convergence argument for the method is presented. Its performance is illustrated by examples featuring long-term average and infinite horizon discounted costs, and additional optimization constraints.

The approximate maximum and minimum amounts of any phase in a complex mineral mixture can be determined by solving a linear programming problem involving chemical mass balance and X-ray powder diffraction (XRD) data. The chemical information necessary is the bulk composition of the mixture and an estimation of the compositional range of each of the minerals in the mixture. Stoichiometric constraints for the minerals may be used to reduce their compositional variation. If only a partial chemical analysis for the mixture is available, the maximum amounts of the phases may still be estimated; however, some or all of the stoichiometric constraints may not apply. XRD measurements (scaled using an internal standard) may be incorporated into the linear programming problem using concentration-intensity relations between pairs of minerals. Each XRD constraint added to the linear programming problem, in general, reduces the difference between the calculated maximum and minimum amounts of each phase. Because it is necessary to define weights in the objective function of the linear programming problem, the proposed method must be considered a model. For many mixtures, however, the solution is relatively insensitive to the objective function weights.

An example consisting of a mixture of montmorillonite, plagioclase feldspar, quartz, and opal-cristobalite illustrates the linear programming approach. Chemical information alone was used to estimate the mineral abundances. Because quartz and opal-cristobalite are not chemically distinct, it was only possible to determine the sum, quartz + opal-cristobalite, present in the mixture.

We give a new proof of rationality of stable commutator length (scl) of certain elements in surface groups: those represented by curves that do not fill the surface. Such elements always admit extremal surfaces for scl. These results also hold more generally for non-filling $1$–chains.

Dog vaccination is the key to controlling rabies in human populations. However, in countries like India, with large free-roaming dog populations, vaccination strategies that rely only on parenteral vaccines are unlikely to be either feasible or successful. Oral rabies vaccines could be used to reach these dogs. We use cost estimates for an Indian city and linear optimisation to find the most cost-effective vaccination strategies. We show that an oral bait handout method for dogs that are never confined can reduce the per dog costs of vaccination and increase vaccine coverage. This finding holds even when baits cost up to 10x the price of parenteral vaccines, if there is a large dog population or proportion of dogs that are never confined. We suggest that oral rabies vaccine baits will be part of the most cost-effective strategies to eliminate human deaths from dog-mediated rabies by 2030.

Extensive sheep farming systems make an important contribution to socio-economic well-being and the ‘ecosystem services’ that flow from large areas of the UK and elsewhere. They are therefore subject to much policy intervention. However, the animal welfare implications of such interventions and their economic drivers are rarely considered. Under Defra project AW1024 (a further study to assess the interaction between economics, husbandry and animal welfare in large, extensively managed sheep flocks) we therefore assessed the interaction between profit and animal welfare on extensive sheep farms. A detailed inventory of resources, resource deployment and technical performance was constructed for 20 commercial extensive sheep farms in Great Britain (equal numbers from the Scottish Highlands, Cumbria, Peak District and mid-Wales). Farms were drawn from focus groups in these regions where participative research with farmers added further information. These data were summarised and presented to a panel of 12 experts for welfare assessment. We used two welfare assessment methods one drawn from animal welfare science (‘needs’ based) the other from management science (Service Quality Modelling). The methods gave complementary results. The inventory data were also used to build a linear programme (LP) model of sheep, labour and feed-resource management month-by-month on each farm throughout the farming year. By setting the LP to adjust farm management to maximise gross margin under each farm's circumstances we had an objective way to explore resource allocations, their constraints and welfare implications under alternative policy response scenarios. Regression of indicators of extensification (labour per ewe, in-bye land per ewe, hill area per ewe and lambs weaned per ewe) on overall welfare score explained 0.66 of variation with labour and lambs weaned per ewe both positive coefficients. Neither gross margin nor flock size were correlated with welfare score. Gross margin was also uncorrelated with these indicators of extensification with the exception of labour/ewe, which was negatively correlated with flock size and hence with gross margin. These results suggest animal welfare is best served by reduced extensification while greater profits are found in flock expansion with reduced labour input per ewe and no increase in other inputs or in productivity. Such potential conflicts should be considered as policy adjusts to meet the requirements for sustainable land use in the hills and uplands.

In many countries, including the UK, the majority of domestic sows are housed in farrowing crates during the farrowing and lactation periods. Such systems raise welfare problems due to the close confinement of the sow. Despite the fact that many alternative housing systems have been developed, no commercially viable/feasible option has emerged for large scale units. Current scientific and practical knowledge of farrowing systems were reviewed in this study to identify alternative systems, their welfare and production potential. The aim was to establish acceptable trade-offs between profit and welfare within alternative farrowing systems. Linear programming (LP) was used to examine possible trade-offs and to support the design of welfare-friendly yet commercially viable alternatives. The objective of the LP was to optimise the economic performance of conventional crates, simple pens and designed pens subject to both managerial and animal welfare constraints. Quantitative values for constraints were derived from the literature. The potential effects of each welfare component on productivity were assessed by a group of animal welfare scientists and used in the model. The modelled welfare components (inputs) were extra space, substrate and temperature. Results showed that, when using piglet survival rate in the LP based on data drawn from the literature and incorporating costs of extra inputs in the model, the crates obtained the highest annual net margin and the designed pens and the pens were in second and third place, respectively. The designed pens and the pens were able to improve their annual net margin once alternative reference points, following expert-derived production functions, were used to adjust piglet survival rates in response to extra space, extra substrate and modified pen heating. The non-crate systems then provided higher welfare and higher net margin for sows and piglets than crates, implying the possibility of a win-win situation.

Zn deficiency arising from inadequate dietary intake of bioavailable Zn is common in children in developing countries. Because house crickets are a rich source of Zn, their consumption could be an effective public health measure to combat Zn deficiency. This study used Optifood, a tool based on linear programming analysis, to develop food-based dietary recommendations (FBR) and predict whether dietary house crickets can improve both Zn and overall nutrient adequacy of children’s diets. Two quantitative, multi-pass 24-h recalls from forty-seven children aged 2 and 3 years residing in rural Kenya were collected and used to derive model parameters, including a list of commonly consumed foods, median serving sizes and frequency of consumption. Two scenarios were modelled: (i) FBR based on local available foods and (ii) FBR based on local available foods with house crickets. Results revealed that Zn would cease to be a problem nutrient when including house crickets to children’s diets (population reference intake coverage for Zn increased from 89 % to 121 % in the best-case scenario). FBR based on both scenarios could ensure nutrient adequacy for all nutrients except for fat, but energy percentage (E%) for fat was higher when house crickets were included in the diet (23 E% v. 19 E%). This manoeuvre, combined with realistic changes in dietary practices, could therefore improve dietary Zn content and ensure adequacy for twelve nutrients for Kenyan children. Further research is needed to render these theoretical recommendations, practical.

The Pasture, Rangeland, Forage (PRF) insurance program is aimed to assist producers to manage the risk of forage loss due to the lack of precipitation. However, limited attention has been given to understanding the implications of policyholders’ coverage selection decisions. In this study, three alternative risk-efficient portfolio selection strategies are assessed in to the context of the PRF program. Proposed methods consider all the decision parameters and program restrictions, and they highlighted the underlying relationships between expected revenue, risk, and choice of the coverage parameters. Selection strategies are illustrated by examining the optimal coverage for a grid in South Texas.

The aim was to design culturally acceptable and healthy diets with reduced energetic share of ultra-processed foods (UPF%) at no cost increment and to evaluate the impact of the change in the UPF% on diet quality. Food consumption and price data were obtained from the Household Budget Survey (n 55 970 households) and National Dietary Survey (n 32 749 individuals). Linear programming models were performed to design diets in which the mean population UPF% was reduced up to 5 % with no cost increment relative to the observed costs. The models were isoenergetic or allowed the energy content to vary according to the UPF%, and they were not constrained to nutritional goals (nutrient-free models) or maximised the compliance with dietary recommendations (nutrient-constrained models). Constraints regarding food preference were introduced in the models to obtain culturally acceptable diets. The mean population UPF% was 23·8 %. The lowest UPF% attained was approximately 10 %. The optimised diet cost was up to 20 % cheaper than the observed cost, depending on the model and the income level. In the optimised diets, the reduction in the UPF% was followed by an increase in fruits, vegetables, beans, tubers, dairy products, nuts, fibre, K, Mg, vitamin A and vitamin C in the nutrient-constrained models, compared with the observed consumption in the population. There was little variation in most nutrients across the UPF% reduction. The UPF% reduction in the nutrient-free models impacted only trans-fat and added sugar content. UPF% reduction and increase in diet quality are possible at no cost increment.

Feeding cattle with on-pasture supplementation or feedlot diets can increase animal efficiency and system profitability while minimizing environmental impacts. However, cattle system profit margins are relatively small and nutrient supply accounts for most of the costs. This paper introduces a nonlinear profit-maximizing diet formulation problem for beef cattle based on well-established predictive equations. Nonlinearity in predictive equations for nutrient requirements poses methodological challenges in the application of optimization techniques. In contrast to other widely used diet formulation methods, we develop a mathematical model that guarantees an exact solution for maximum profit diet formulations. Our method can efficiently solve an often-impractical nonlinear problem by solving a finite number of linear problems, that is, linear time complexity is achieved through parametric linear programming. Results show the impacts of choosing different objective functions (minimizing cost, maximizing profit and maximizing profit per daily weight gain) and how this may lead to different optimal solutions. In targeting improved ration formulation on feedlot systems, this paper demonstrates how profitability and nutritional constraints can be met as an important part of a sustainable intensification production strategy.

We examined the feasibility of linear programming (LP) to develop diets that were economical, included traditional (cultural, non-market) foods and met the dietary reference intakes (DRI) in a Canadian Indigenous population. Diet optimisation using LP is a mathematical technique that can develop food-based dietary guidelines for healthy eating in Indigenous populations where food insecurity, availability and cost are important considerations. It is a means of developing nutritionally optimal food combinations that are based on economical and culture-specific foods. Observed food consumption data were derived using 24-h food recalls from the First Nations Food, Nutrition and Environment Study. The LP models were constructed to develop diets meeting DRI, cost and food constraints. Achieving the recommended food intake was not feasible in a model meeting all nutrient requirements. Models that met most nutrient requirements at reduced cost were designed for men and women, separately. In women, it was necessary to increase energy intake to meet most nutrient requirements. Nutrient requirements could not be met for fibre, linoleic and linolenic acids, vitamin D, Ca and K in both sexes, P in women, and Mg and vitamin A in men. Using LP to develop optimal diets for First Nations people, we found simultaneous achievement of all DRI was difficult, suggesting that supplementation might be necessary which goes against recommendations for individuals to meet their nutrient needs through healthy eating patterns. Additionally, to make diets feasible, programmes to reduce market food costs and to support First Nations people in traditional food harvesting are recommended.