India’s Supplementary Nutrition Programme (SNP), under the Integrated Child Development Services, provides a Morning Snack and a Hot Cooked Meal to children aged 36–72 months through Anganwadis (day-care centres). This study assessed these meals against SNP standards (2012) and age-specific ICMR recommendations, when standards were unavailable, and explored the use of linear programming (LP) to improve nutrient quality of SNP meals. A cross-sectional survey documented the SNP-meals, ingredients, serving portions and other details, using questionnaires administered to Anganwadi workers and programme officials from purposively selected Anganwadis across 27 States and Union Territories. Nutrient composition was estimated using standard food composition tables, and State-specific food lists and retail prices were incorporated into the LP framework to identify foods that could improve nutrient content of SNP meals. Energy standards were met in 56% of States, while 22% fell more than 20% below recommendations. Protein standards were achieved in 74% of States; however, declined to 52% after adjusting for digestibility. Only 22% of States met the ICMR-based fat requirement. Zinc, iron and folate were largely met, whereas calcium and vitamins A, B6 and B12 were below recommendations in more than half the States. Meals were predominantly cereal-based with limited inclusion of nutrient-dense foods. LP identified foods that reduced nutrient gaps, although many improvements exceeded the per-child cost allocation. Overall, meals showed wide variability, with persistent gaps in fat and key micronutrients. LP provides a structured approach to enhance nutrient provision within the programme setting, although meeting all nutrient targets may require adjustments to cost and procurement strategies.

Pregnant women are at higher risk of vitamin D deficiency due to increased nutritional requirements and limited dietary sources. Fish is the major source of vitamin D, but its availability varies by region. This study aims to assess the availability of vitamin D-rich fish and develop food-based recommendations (FBR) for pregnant women in East Lombok, Indonesia. This comparative cross-sectional study was conducted in East Lombok, Indonesia, which is part of Action Against Stunting cohort of pregnant women. Twenty-five village markets in the area were scored based on availability of vitamin D-rich fish and number of fish sellers in each market and were categorised into high availability (HD) for the highest quartile and low availability (LD) for the lowest quartile. Quantum Geographic Information System software was used to identify each of respondents’ houses using 2·4 km buffer zones to either HD or LD markets. Dietary intake data were collected from 24-h dietary recalls and linear programming analysis using Optifood was used to identify problem nutrient and dietary inadequacy. No significant difference in vitamin D intake was found between HD and LD areas (P = 0·633). While both groups’ FBR ensure adequacy of Fe, Zn, vitamins A, B1, B2, B3, B6 and B12, Ca remain as dietary inadequacy in the LD group, suggesting that availability play a role in ensuring dietary adequacy. Future studies to develop nutrient-dense foods and improved fish availability mapping which consider non-static position of market (mobile vendor) are recommended.

This paper proposes a multi-objective programming method for determining samples of examinees needed for estimating the parameters of a group of items. In the numerical experiments, optimum samples are compared to uniformly and normally distributed samples. The results show that the samples usually recommended in the literature are well suited for estimating the difficulty parameters. Furthermore, they are also adequate for estimating the discrimination parameters in the three-parameter model, but not for the guessing parameters.

An algorithmic approach to test design, using information functions, is presented. The approach uses a special branch of linear programming, i.e. binary programming. In addition, results of some benchmark problems are presented. Within the same framework, it is also possible to formulate the problem of individualized testing.

A maximin model for IRT-based test design is proposed. In the model only the relative shape of the target test information function is specified. It serves as a constraint subject to which a linear programming algorithm maximizes the information in the test. In the practice of test construction, several demands as linear constraints in the model. A worked example of a text construction problem with practical constraints is presented. The paper concludes with a discussion of some alternative models of test construction.

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.