A Systematic Review of Linear Programming Techniques as Applied to Diet Optimisation and Opportunities for Improvement

Donkor, Leticia; Essien, Emmanuel; Affrifah, Nicole Sharon

doi:https://doi.org/10.1155/2023/1271115

Journal of Optimization

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Abstract Introduction Results and Discussion Conclusion Conflicts of Interest Acknowledgments References Copyright Related Articles

Review Article | Open Access

Volume 2023 | Article ID 1271115 | https://doi.org/10.1155/2023/1271115

A Systematic Review of Linear Programming Techniques as Applied to Diet Optimisation and Opportunities for Improvement

Leticia Donkor,¹Emmanuel Essien,²and Nicole Sharon Affrifah¹

Academic Editor: Himadri Majumder

Received11 Apr 2023

Revised20 Jul 2023

Accepted24 Aug 2023

Published07 Sept 2023

Abstract

Background. Food provides the required nutrients for adequate growth and development. However, meeting the recommended nutrients while considering environmental sustainability can be complicated and challenging. Previously, trial-and-error methods were used for product development, but these are tedious and time-consuming. Mathematical techniques such as linear programming offer an alternative and rapid approach to developing products with nutritional/or sustainability considerations. This method has been extensively used in diet optimisation but does not sufficiently address dietary problems with more than one objective function. Aim. The review aimed to explore the extent of mathematical approaches to address dietary problems. Methodology. A systematic review approach was adopted for the research. The major search engines used were Scopus, PubMed, and Science Direct, based on selected keywords. A stepwise structural method was used to obtain articles. Articles that contained the search keywords but applied in nonhuman cases were excluded. Duplicated articles were also excluded and accounted for as one. All articles were subjected to further review based on their abstract and complete titles before passing them for data analysis. Results. The total number of articles obtained from the search activity was 280. Fifty-six were retained after the criteria for inclusion were applied to them. Out of the 56 articles retained, only two studies used goal programming and nonlinear generalised mathematical approaches to address dietary problems. All other studies used the linear programming approach, focusing mainly on one or two constraints (nutrients and/or acceptability), highlighting the limitations of linear programming in addressing the multiple factors of a sustainable diet. Several researchers have proposed using multiobjective optimisation, an extension of linear programming, to address challenges with sustainable diets. These approaches can be further explored to address sustainable dietary problems.

1. Introduction

Good nutrition, a critical component of good health, is achieved when consumers access healthy diets. A population that consumes a healthy diet reduces the burden of malnutrition and its related diseases. However, access to a healthy diet is constrained by many factors, including cost, especially for low-income earners [1]. Socioeconomic status affects food choices and healthy diets due to the cost of food commodities [1–3], causing some consumers to purchase energy-dense over nutrient-dense foods. Healthy diets have been found to have a higher cost [4]. This and other factors like what consumers want and the environment pose dietary problems that need to be addressed to lessen the burden of malnutrition on the population while safeguarding the environment.

Trial-and-error and optimisation methods have been used to solve these dietary problems. Trial-and-error methods used for product development to resolve dietary problems mentioned above can be tedious [5] since they are typically completed manually [6]. Mathematical diet optimisation techniques can better solve complex dietary problems than the tedious trial-and-error method.

Diet optimisation has been identified as one of the best approaches to address dietary problems and ensure that sustainable diets are achieved for individuals or groups based on locally available and culturally specific foods [7, 8]. The Food and Agriculture Organisation (FAO) defines a sustainable diet as a diet with a low environmental impact that is nutritionally adequate, accessible, economically fair, affordable, safe, and healthy [9]. When planning food for any defined population, mathematical diet optimisation models can be used to design food plans that best look like the current eating patterns of the people while meeting prespecified nutrition and cost constraints [10].

From a literature search, the basic approach to diet optimisation is linear programming (LP), which has three main components: an objective function, decision variable(s), and constraints [3, 11]. Jones and Tamiz [12] stated that the decision variables are factors that the decision maker has control over and must be determined to solve a linear programming problem. Constraints are restrictions usually imposed on the decision variables [12, 13]. An objective function indicates the contribution of the decision variables to the value of the function to be optimised (minimised or maximised), with examples being cost, profits, etc. [14]. Linear programming is a mathematical approach that enables obtaining ideal solutions simultaneously while satisfying several constraints [11]. Karloff [15] also defined linear programming as minimising or maximising a linear objective function with a finite number of linear equality and inequality constraints imposed on it. Diet optimisation models have been developed for different nutrient needs for different age groups that satisfy different constraints. In diet optimisation, nutrient-based references are translated into practical nutritionally optimum food combinations based on locally available foods within a defined geographic location [16].

However, diet optimisation goes beyond nutrition and cost. It includes acceptability and environmental friendliness and incorporates all constraints defined for diet optimisation by the World Health Organisation [17]. Although linear programming methods can solve dietary problems, they can only optimise problems with a single objective function [11, 15]. As such, linear programming is not enough when there is more than one objective function to be optimised.

Moreover, what happens when an optimisation problem has multiple objective functions? Ferguson et al. [18], Jayaraman et al. [19], and Gazan et al. [3] proposed the multiobjective criteria approach to this question. These are considered as an extension of LP problems [20], and they are continuous problems [21] that cut across areas of engineering, mathematics, economics, etc. and have more than one objective function to be addressed. When there are multiple objectives that conflict with each other, multiobjective decision making is employed [22]. Some examples of multiobjective optimisation methods are the goal programming [23], weighted sum approach [24], and the epsilon constraint (ɛ-constraint) methods. When goals are multiple and conflict with each other, goal programming (GP) is used [23]. Goal programming (GP) is an extension of LP and a multiobjective optimisation tool used to minimise deviations between achievements and goals [19, 23]. Even beyond goal programming, other multiobjective optimisation approaches like epsilon (ɛ) constraint [25] and weighted sum approach [24] methods can be used to address complex dietary problems.

This paper therefore analyses different studies that used linear programming to address dietary problems. It highlights the most considered objectives, cost, environment, deviations between observed and optimised patterns, and maximising nutrients.

2. Review Approach

The systematic research technique was explored for this review work. The search selection method used for this review was the Preferred Reporting Items for the Systematic Reviews and Meta-Analyses (PRISMA) framework [26]. PRISMA can be used as a basis for reporting systematic reviews of other types of research and is suitable for evaluating published systematic reviews [26]. The PRISMA method followed a modified version of the one used by van Dooren [11]. The search method used for the literature search focused on using defined keywords in specific search engines. The language of selection was English, regardless of the origin or setting of the research work. The method followed a structural approach briefly explained below.

2.1. Article Search by Keyword(s)

The search included the use of specific keywords, “linear programming,” “sustainable diet,” and “diet optimisation” in Scopus, PubMed and Science Direct, within the search windows of 2000 – to date (2023).

2.2. Data Search and Evaluation

The first search using the defined terminologies in the search engines yielded 280 articles within the defined timeframe. Open-access articles were focused on, and articles that had linear programming but not diet optimisation were excluded. Also, annual meetings, poster presentations, and conference publications were also excluded because they did not provide sufficient details for further discussion, reducing the articles size to 137. The 137 articles were accessed and further scrutinised for inclusion or exclusion. Duplicated publications were removed and accounted for as one. This was followed by excluding articles that employed diet optimisation but for nonhuman settings, bringing the total number of articles to 73.

2.3. Data Screening and Analyses

The 73 articles retained were screened for analysis. The criteria for the screening stage were the review of abstracts to ascertain if the article fit the description of having a defined objective, decision variable(s), and constraint(s). This resulted in the retention of 56 articles which were analysed for discussion. The details extracted were the objective(s), decision variables, imposed constraints, study locations, and mathematical techniques used for diet optimisation. An Excel worksheet was used to aid in extracting data from the articles selected for the review. Figure 1 gives a schematic flow of the systematic method adopted for the review.

3. Results and Discussion

3.1. Summary of Findings

Two hundred and eighty (280) publications were obtained from PubMed, Scopus, and Science Direct. Fifty-six (56) empirical studies published from 2000 to date (2023) were obtained for data extraction and discussion (Figure 1). Results from reviewing the selected articles showed the versatility of applying the technique in different settings. The countries where mathematical diet optimisation has been applied include Korea [27], Australia [28], Malaysia [29], Philippines [16, 30], France [31, 32], New Zealand [33, 34], Ghana [1, 35], Kenya [36], Canada [10], Malawi [37], Czech Republic [38], Brazil [39], Hungary [40], and Nepal [41].

The reviewed articles were further classified under the different objective functions defined by researchers for discussion. Other details extracted from the reviewed articles include the decision variables, constraints, and modelling approach (Tables 1–5).

From the articles reviewed, researchers had different objectives they set out to achieve. Results showed 43 (84%) articles focused on minimising objective functions, 4 (11%) on maximising objective functions, and the remaining 3 (5%) did not have a clearly defined minimisation or maximisation direction (Figure 2).

The first component of a linear programming model is the objective function. According to Verly-Jr et al. [39], a linear programming model is defined by an objective function optimised and dependent on decision variables constrained by some defined constraints. In optimisation problems, objective functions are essential because they show how each variable contributes to the optimised value [57]. The optimisation of the objective function could either be to minimise or maximise the function [14]. When researchers are concerned with profits and increasing revenue, setting an objective function will be to maximise profits. When a diet optimisation problem concerns cost, the objective would be to minimise cost because researchers are interested in delivering healthy meals at minimum cost to consumers.

3.2. Objective Functions

Out of the 56 studies analysed for discussion, 47 articles minimised their objective functions, 6 maximised their objective functions, and 3 did not have a definitive objective of minimisation or maximisation. Objective functions are needed to show the direction of a linear programming optimisation model. Regardless of the direction, there could be different functions that can be minimised or maximised. For example, an objective could be to minimise cost, environmental function, or even deviations between an observation and a modelled function, as represented in Tables 1–5.

3.2.1. Cost Minimisation

Out of 48 studies that minimised their objective functions, 20 articles focused on minimising the cost of the modelled diet (Table 1). Highlighting a few, some of the studies minimised the cost of food baskets for a family [1, 38], others minimised the cost of RUTFs to treat malnutrition [44, 46, 59, 65], and Mejos et al. [30] minimised the cost of complementary feeding. All the studies that minimised the objective function (Table 1) obtained results that aligned with their defined study focus. According to Drewnowski and Specter [82], food prices remain one major factor affecting dietary quality, consumers’ choices, and corresponding dietary patterns. Hence, it is very valuable and important that these studies were directed at minimising the cost of diets. Although these studies highlighted in Table 1 did not clearly define any limitations, one of the major limitations of these studies in addressing complex sustainable diet issues is that only objective function was optimised. Furthermore, minimising an objective function does not only mean achieving a minimum diet cost, but it could also mean minimising deviations between an observed and modelled pattern.

3.2.2. Deviation Minimisation

Studies that sought to minimise the deviations between the observed and modelled patterns were 20 from the results obtained (Table 2). Some studies [10, 37, 43, 47, 54] [16, 27, 40, 39, 69, 73–76, 80, 81] assessed the dietary patterns of a defined population, modelled diets that meet nutritional requirements for the said population, and then set objective functions to minimise the deviation between the observed and patterns and modelled diets. From these studies, dietary intake data collected from the population served as observed data. They then modelled a diet that met the constraints they defined. The approach adopted by these studies was a good way to address dietary problems encountered in different settings for different population groups; they also considered only a single objective function, which made it impossible to meet the four main dimensions of a sustainable diet (cost, environment, acceptability, and nutrition).

3.2.3. Environmental Factor Minimisation

Furthermore, only 4 studies set to minimise environmental factors as their objective function (Table 3). There is an increasing concern for the environment due to growing consumption patterns, Patterson et al. [83]; Ferrari et al. [50]; Larrea-Gallegos and Vázquez-Rowe [49]; whereas Tompa et al. [40] minimised the water footprint of the optimised diet. Springmann et al. [84] highlighted that there is a tendency that the impact of consumers on the environment may worsen as the world population grows exponentially and dietary patterns continue to change. For this reason, it is necessary that research gears toward the minimisation of environmental factors to ensure consumers are considerate of them. However, like the studies that minimised cost, these studies also focused on minimising only environmental factors against certain constraints, which leaves the gap and question of the other dimensions of a sustainable diet.

3.2.4. Maximisation

Six (6) studies maximised the nutritional requirements for defined populations, subject to defined constraints (Table 4). All these 6 studies set objective functions to maximise objective functions, thus not considering other dimensions of a sustainable diet like environment, acceptability, and cost. Even though van Dooren et al. [52] considered nutrients, cost, and GHGE, they were considered constraints, further supporting the limitation of linear programming in addressing the four dimensions (cost, nutrient, acceptability, and environment) of a sustainable diet.

3.2.5. Others

Two (2) studies from the articles analysed did not clearly state their linear programming model equations to show the direction of the objective function (Table 5). Ferguson et al. [36] aimed at meeting nutrient needs while considering locally available food-based recommendations. Even though the study did not clearly define the objective function mathematically, the direction could be assumed as maximisation since it aimed at ensuring that recommended nutrient needs were met. However, there was no consideration for the environment nor a clearly defined cost dimension. McMahon et al. [28] also minimised sodium intake while maintaining iodine intake; it did not clearly indicate these mathematically. It could be assumed that the direction of the objective function was that of minimisation.

On the other hand, Pasic et al. [54] sought to minimise the deviations between defined nutrients and food cost. This was the single study in the 56 articles analysed that adopted a multiobjective approach to solve the dietary problem using the goal programming approach. Even though this study adopted, it only considered cost and nutrient, without acceptability and the environment.

3.3. Decision Variables

Decision variables in mathematical diet optimisation are important because they are the factors a decision maker can control while searching for an optimal solution for defined objectives [12, 14]. Data extracted from articles obtained and used for discussion showed that all studies had at least one decision variable. A common decision variable that ran through all these studies was the weight or amount of individual food available for optimisation (Table 1). Out of the 56 articles retained for data extraction and discussion, 54 had weight or amount of the individual food as a single decision variable, whereas 3 of the studies [1, 42, 52] had price as an additional decision variable. Though these 3 studies had cost as an extra decision variable, they set a limit to the total cost, which they did not want the modelled diet to exceed. This shows that the weights of different food items are essential in addressing any dietary problem.

3.4. Constraints

In linear programming, there is the need for constraints to be imposed, and these constraints must be respected to obtain the defined optimal solution for the objective function set [14]. These multiple constraints are linear equations and inequalities that must be respected for optimal solutions. Some of these constraints can be set on nutritional needs, food acceptability or cultural requirements, environment, and the cost of diet [13, 39, 52]. All the studies had one thing in common: constraints that were defined to ensure some nutrients.

Most nutrition-related studies have focused on achieving nutritional constraints, regardless of the objective function (Tables 1–5). Masset et al. [10] set constraints to ensure modelled diet meets the nutritional requirement for cancer treatment. Dibari et al. [44] and Ryan et al. [46] also set constraints to meet the nutritional requirements for ready-to-use-therapeutic foods, Brixi [59] imposed constraints that ensure that the diet modelled met the nutritional requirements necessary to treat acute malnutrition, and Morrison et al. [41] set constraints to ensure that maximum iron content of modelled diets is obtained in an attempt to address anaemia among women of pregnancy age in Nepal. Other forms of nutritional constraint can be on macro and micronutrient requirements in the form of meeting recommended nutrient intakes (RNIs), as exhibited by Ferguson et al. [38]; Raymond et al. [75]; Verly-Jr et al. [39]; and van Wonderen et al. [53]. Similarly, Okubo et al. [16] and Horgan et al. [71] imposed nutritional constraints that satisfied dietary reference intakes (DRIs), Ghazaryan [60] did the same for tolerable levels of selected nutrients, and McMahon et al. [28] imposed constraints to ensure allowable levels were achieved and not exceeded. In other studies [10, 39, 43, 47, 50, 61, 63, 77], constraints were imposed on energy requirements to ensure diets modelled were less energy dense. To ensure the optimised diets did not exceed a set cost budget to ensure affordability, some studies [32, 48, 52] put a cost constraint to make it possible.

Also, constraints on culture or acceptability can be actualised in different ways. To ensure that optimised diets still stayed within the consumption pattern of the population or target group, researchers imposed a constraint on portion size [37, 38, 43, 63]. Another means of imposing ensuring the acceptability of an optimised diet is to impose a constraint on each food group, as done by Metzgar et al. [69]; Okubo et al. [16]; Horgan et al. [71]; Raymond et al. [73]; Ghazaryan [60]; and Mejos et al. [30].

Out of the 56 articles analysed, only 10 had a component of acceptability factored into their optimisation problems. Acceptability constraints are difficult to ensure in addressing dietary problems due to the varying needs of consumers and the different food items from one geographic area to another. However, acceptability is vital in ensuring that the sociocultural aspects of sustainable diets are achieved. For example, van Doren et al. [52] stated one limitation: the nonconsideration of cultural or social factors for dietary choices. van Dooren et al. [52] considered three more constraints besides the nutrient: energy, cost, and GHGE. This implies that there is room for more than only nutrients to be considered as constraints.

Computational models have been extensively applied in diet optimisation, primarily linear programming, under different settings. According to Beheshti et al. [85], these mathematical tools can simulate different scenarios in dietary choices and other diet optimisation problems at low cost with minimal risk. Though these studies either minimised or maximised an objective function, researchers had different scenarios and settings for their works (Table 1). Except for Pasic et al. [54] and Scarborough et al. [72] who used goal programming and a nonlinear generalised reduced gradient algorithm to find optimal solutions for the objectives they set, all studies (Table 1) used the linear programming approach. Linear programming has been used to address many diet-related problems [11, 14]. Though many of the studies analysed in this write-up employed linear programming to address a problem they intended to solve, the articles did not define mathematical equations that represent the physical problems.

From findings obtained from the review, LP has been successfully applied in solving diet optimisation problems. Though the LP is a robust and widely applied algorithm [86] suitable for diet optimisation problems, it becomes limited when more than one objective function is to be optimised. This can be seen from Tables 1–3 as almost all the studies had only one objective. One of the major hurdles in using linear programming to address the complexity of sustainable diets lies in resolving the multiple objectives. Almost all the research discussed in the review has addressed one or two dimensions of a sustainable diet, but not all (Table 1). With current challenges being faced by global food systems to ensure the provision of sustainable diets, the question of what mathematical approach can be employed offers room for more work. To achieve sustainable diets, researchers need to consider satisfying different conflicting goals like nutrition, cost, environment, and acceptability, making this a multiobjective diet optimisation problem. While single-objective optimisation problems generate a unique optimal solution, multiobjective optimisation yields a set of solutions through the Pareto optimality theory [87], making it a key point researchers should identify when solving a multiobjective optimisation problem. The Pareto optimal solution set obtained from solving the multiobjective problem is nondominated [24]. Chiandussi et al. [87] further indicated that some methods that have been used to solve these types of problems include the linear combination of weights, the multiobjective genetic algorithm, and the ε-constraint methods. Goal programming is another approach to solving multiobjective optimisation problems by minimising deviations between the individual goals obtained and the set targets [88]. Though many scholars have solved multiobjective optimisation problems, one challenge that could be encountered with its applications is the burden of computation when the number of objectives increases [89]. According to Nakayama [90], the weighted sum approach is well known but reveals the decision maker’s subjective evaluation of the weights assigned. In addition, Zhen et al. [91] found that decision makers commonly assume that multiobjective optimisation problems have conflicting objectives when this is not always true. A review by Gazan et al. [3] showed the need for all the relevant aspects of a sustainable diet to be factored into mathematical diet optimisation. Although multiobjective optimisation has been widely used in various fields, its application in addressing intricate dietary issues remains limited. It would benefit future research to consider the various components of a sustainable diet and utilise a multiobjective optimisation approach.

4. Conclusion

Good nutrition is essential for obtaining nutrients to nourish the human body and ensure general well-being. However, current consumer dietary patterns raise concerns due to their environmental impact. As a result, there is increasing advocacy for sustainable diets which meet nutrient needs and consider the environment. Although there has been no definitive definition for sustainable diets, the different dimensions, including nutrition, economics (cost), environment, and sociocultural factors, make them complex.

The LP tool has been efficiently used to minimise cost, maximise nutrient requirement, and minimise deviations between set targets and achieve objectives. This handy tool can perfectly complement the tedious trial-and-error method used in addressing diet problems by providing the optimal solution for diet combinations in a shorter time. The tool has been efficiently used to address dietary problems by proposing diets that come at the least cost while meeting nutrient constraints or diets with low greenhouse gas emission values while meeting certain defined constraints. However, looking at how LP tools have extensively used, and some limitations highlighted, it is only able to solve problems with only one objective.

Therefore, challenges lie in its ability to solve more complex problems with multiple objective functions that may be conflicting, hence the advocacy for adopting multiobjective optimisation models in solving complex dietary problems with conflicting objectives.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This review paper was funded under a grant from Nestlé SA to the University of Ghana under the UG-Nestlé PhD Scholarship for Research Excellence.

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Copyright © 2023 Leticia Donkor et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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