Diet models with linear goal programming: impact of achievement functions

Area of Science:

• Nutritional Science
• Operations Research
• Mathematical Modeling

Background:

• Diet models using goal programming (GP) are crucial for balancing nutritional, palatability, and cost constraints.
• The choice of achievement function significantly impacts the outcomes of GP diet models.
• Existing achievement functions present limitations in flexibility and solution diversity.

Purpose of the Study:

• To provide methodological insights into various achievement functions for GP diet models.
• To introduce and describe the extended GP (EGP) achievement function.
• To demonstrate EGP's capability to generate multiple solutions from a single dataset.

Main Methods:

• Illustrating the mechanics of achievement functions using small numerical examples.
• Applying the EGP achievement function to a comprehensive diet problem involving 144 foods and 19 nutrients.
• Modeling nutritional constraints using fuzzy sets within the EGP framework.

Main Results:

• The selection of an achievement function demonstrably influences the results of diet optimization models.
• MinSum functions can lead to solutions sensitive to weight adjustments and concentrated deviations.
• MinMax functions distribute deviations more evenly but may result in numerous small deviations.

Conclusions:

• The extended GP (EGP) achievement function integrates MinSum and MinMax approaches, offering compromises.
• EGP allows decision-makers to obtain a spectrum of solutions with diverse properties from a single dataset.
• This flexibility enhances the utility of GP models in diet design by accommodating varied user preferences and constraints.