Bi-objective goal programming for balancing costs vs. nutritional adequacy.

Introduction: Linear programming (LP) is often used within diet optimization to find, from a set of available food commodities, the most affordable diet that meets the nutritional requirements of an individual or (sub)population. It is, however, not always possible to create a feasible diet, as certain nutritional requirements are difficult to meet. In that case, goal programming (GP) can be used to minimize deviations from the nutritional requirements in order to obtain a near feasible diet. With GP the cost of the diet is often overlooked or taken into account using the ε-constraint method. This method does not guarantee to find all possible trade-offs between costs and nutritional deficiency without solving many uninformative LPs. Methods: We present a method to find all trade-offs between any two linear objectives in a dietary LP context that is simple, does not solve uninformative LPs and does not need prior input from the decision maker (DM). This method is a bi-objective algorithm based on the NonInferior Set Estimation (NISE) method that finds all efficient trade-offs between two linear objectives. Results: In order to show what type of insights can be gained from this approach, two analyses are presented that investigate the relation between cost and nutritional adequacy. In the first analysis a diet with a restriction on the exact energy intake is considered where all nutrient intakes except energy are allowed to deviate from their prescription. This analysis is especially helpful in case of a restrictive budget or when a nutritionally adequate diet is either unaffordable or unattainable. The second analysis only relaxes the exact energy intake, where the other nutrients are kept within their requirements, to investigate how the energy intake affects the cost of a diet. Here, we describe in what situations the so-called more-for-less paradox takes place, which can be induced by requiring an exact energy intake. Conclusion: To the best of our knowledge, we are the first to address how to obtain all efficient trade-offs of two linear objectives in a dietary LP context and how this can be used for analyses.