Review Article A Systematic Review of Linear Programming Techniques as Applied toDietOptimisation andOpportunities for Improvement Leticia Donkor ,1 Emmanuel Essien,2 and Nicole Sharon Afrifah 1 1Department of Food Process Engineering, School of Engineering Sciences, College of Basic and Applied Sciences, University of Ghana, P.O. Box LG 77, Legon, Accra, Ghana 2Department of Agricultural Engineering, School of Engineering Sciences, College of Basic and Applied Sciences, University of Ghana, P.O. Box LG 68, Legon, Accra, Ghana Correspondence should be addressed to Leticia Donkor; ldonkor005@st.ug.edu.gh Received 11 April 2023; Revised 20 July 2023; Accepted 24 August 2023; Published 7 September 2023 Academic Editor: Himadri Majumder Copyright © 2023 Leticia Donkor et al.Tis is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 pro- gramming ofer an alternative and rapid approach to developing products with nutritional/or sustainability considerations. Tis method has been extensively used in diet optimisation but does not sufciently address dietary problems with more than one objective function. Aim. Te review aimed to explore the extent of mathematical approaches to address dietary problems. Methodology. A systematic review approach was adopted for the research. Te 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. Te 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. Tese 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 malnu- trition 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 afects 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]. Tis 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 opti- misation techniques can better solve complex dietary problems than the tedious trial-and-error method. Diet optimisation has been identifed as one of the best approaches to address dietary problems and ensure that Hindawi Journal of Optimization Volume 2023, Article ID 1271115, 17 pages https://doi.org/10.1155/2023/1271115 https://orcid.org/0000-0003-0185-1512 https://orcid.org/0000-0003-0966-7592 mailto:ldonkor005@st.ug.edu.gh https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.1155/2023/1271115 sustainable diets are achieved for individuals or groups based on locally available and culturally specifc foods [7, 8]. Te Food and Agriculture Organisation (FAO) defnes a sustainable diet as a diet with a low environmental impact that is nutritionally adequate, accessible, economically fair, afordable, safe, and healthy [9].When planning food for any defned 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 pre- specifed 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 vari- able(s), and constraints [3, 11]. Jones and Tamiz [12] stated that the decision variables are factors that the decisionmaker 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, profts, etc. [14]. Linear programming is a mathematical approach that en- ables obtaining ideal solutions simultaneously while satis- fying several constraints [11]. Karlof [15] also defned linear programming as minimising or maximising a linear ob- jective function with a fnite number of linear equality and inequality constraints imposed on it. Diet optimisation models have been developed for diferent nutrient needs for diferent age groups that satisfy diferent constraints. In diet optimisation, nutrient-based references are translated into practical nutritionally optimum food combinations based on locally available foods within a defned geographic location [16]. However, diet optimisation goes beyond nutrition and cost. It includes acceptability and environmental friend- liness and incorporates all constraints defned for diet optimisation by the World Health Organisation [17]. Al- though 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. Tese are considered as an extension of LP problems [20], and they are continuous problems [21] that cut across areas of engi- neering, mathematics, economics, etc. and have more than one objective function to be addressed. When there are multiple objectives that confict with each other, multi- objective decision making is employed [22]. Some examples of multiobjective optimisation methods are the goal pro- gramming [23], weighted sum approach [24], and the ep- silon constraint (ε-constraint) methods. When goals are multiple and confict 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 optimisa- tion approaches like epsilon (ε) constraint [25] and weighted sum approach [24] methods can be used to address complex dietary problems. Tis paper therefore analyses diferent studies that used linear programming to address dietary problems. It high- lights the most considered objectives, cost, environment, deviations between observed and optimised patterns, and maximising nutrients. 2. Review Approach Te systematic research technique was explored for this review work. Te search selection method used for this review was the Preferred Reporting Items for the Sys- tematic Reviews and Meta-Analyses (PRISMA) frame- work [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]. Te PRISMA method followed a modifed version of the one used by van Dooren [11]. Te search method used for the literature search focused on using defned keywords in specifc search engines. Te language of selection was English, regardless of the origin or setting of the research work. Te method followed a structural approach briefy explained below. 2.1. Article Search by Keyword(s). Te search included the use of specifc keywords, “linear programming,” “sustainable diet,” and “diet optimisation” in Scopus, PubMed and Sci- ence Direct, within the search windows of 2000 – to date (2023). 2.2. Data Search and Evaluation. Te frst search using the defned terminologies in the search engines yielded 280 articles within the defned 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 sufcient details for further discussion, reducing the articles size to 137. Te 137 articles were accessed and further scrutinised for inclusion or exclusion. Duplicated publications were removed and accounted for as one. Tis 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. Te 73 articles retained were screened for analysis. Te criteria for the screening stage were the review of abstracts to ascertain if the article ft the description of having a defned objective, decision variable(s), and constraint(s). Tis resulted in the retention of 56 articles which were analysed for discussion. Te 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 2 Journal of Optimization aid in extracting data from the articles selected for the re- view. Figure 1 gives a schematic fow 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 se- lected articles showed the versatility of applying the tech- nique in diferent settings. Te 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]. Te reviewed articles were further classifed under the diferent objective functions defned 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 diferent 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 defned minimisation or max- imisation direction (Figure 2). Te frst component of a linear programming model is the objective function. According to Verly-Jr et al. [39], a linear programming model is defned by an objective function optimised and dependent on decision variables constrained by some defned constraints. In optimisation problems, objective functions are essential because they show how each variable contributes to the optimised value [57]. Te optimisation of the objective function could either be to minimise or maximise the function [14]. When re- searchers are concerned with profts and increasing revenue, setting an objective function will be to maximise profts. When a diet optimisation problem concerns cost, the ob- jective 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 func- tions, 6 maximised their objective functions, and 3 did not have a defnitive objective of minimisation or max- imisation. Objective functions are needed to show the direction of a linear programming optimisation model. Regardless of the direction, there could be diferent functions that can be minimised or maximised. For ex- ample, an objective could be to minimise cost, environ- mental 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], andMejos 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 defned study focus. According to Drewnowski and Specter [82], food prices remain one major factor afecting 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 defne 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 meanminimising deviations between an observed and modelled pattern. 3.2.2. Deviation Minimisation. Studies that sought to mini- mise the deviations between the observed and modelled pat- terns 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 defned 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. Tey then modelled a diet that met the constraints they defned. Te approach adopted by these studies was a good way to address dietary problems encountered in diferent settings for diferent 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, accept- ability, and nutrition). 3.2.3. Environmental Factor Minimisation. Furthermore, only 4 studies set to minimise environmental factors as their objective function (Table 3). Tere 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 con- tinue 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. Journal of Optimization 3 3.2.4. Maximisation. Six (6) studies maximised the nutri- tional requirements for defned populations, subject to defned constraints (Table 4). All these 6 studies set objective functions to maximise objective functions, thus not con- sidering other dimensions of a sustainable diet like envi- ronment, 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 defne 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 consid- eration for the environment nor a clearly defned cost di- mension.McMahon et al. [28] alsominimised 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 defned nutrients and food cost. Tis 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 ac- ceptability and the environment. 3.3. Decision Variables. Decision variables in mathemat- ical diet optimisation are important because they are the factors a decision maker can control while searching for an optimal solution for defned 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. Tough 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. Tis shows that the weights of diferent food items are essential in addressing any dietary problem. Records obtained through database search and other hand search (n=280) Records retrieved (n=137) Records not retrieved (n=143) Records obtained after removing duplicates (n=126) Records obtained for screening (n=102) Records accessed and assessed for inclusion (n=73) Articles included for extraction and data analysis (n=56) Duplicates excluded (n=11) Records excluded (n=24) Articles excluded (n=29) Articles excluded (n=17) Figure 1: A schematic representation of the procedure followed to retrieve publications for the review, showing the stepwise criteria used to eliminate and obtain the fnal articles. 4 Journal of Optimization Ta bl e 1: Su m m ar y of de ta ils re tr ie ve d fr om ar tic le s th at m in im ise d co st . Re fe re nc e O bj ec tiv e fu nc tio n (s ) D ec isi on va ri ab le (s ) C on st ra in t (s ) Fo cu s M at he m at ic al ap pr oa ch us ed M ai llo te ta l. [4 2] M in im um co st of di et th at m et in cr ea sin g le ve ls of nu tr iti on al co ns tr ai nt s A m ou nt of fo od an d en er gy co st N ut ri en t, so ci al ac ce pt ab ili ty D em on st ra te th at fo od s w ith go od nu tr iti on al qu al ity co m pa re d to th ei r co st ca n be ea sil y id en tif ed us in g th ei r nu tr ie nt pr of le s an d en er gy co st Li ne ar pr og ra m m in g Pi rič ki et al .[ 43 ] M in im um pr ic e A m ou nt of fo od in FB N ut ri en ts ,e ne rg y, an d pa la ta bi lit y (q ua nt ity of fo od co ns um ed ) D es ig n a di et th at co m bi ne s di fe re nt fo od gr ou ps an d ha s m in im um fa t( es pe ci al ly re du ce d sa tu ra te d fa tty ac id s) an d ch ol es te ro l Li ne ar pr og ra m m in g D ib ar ie ta l. [4 4] Lo w es tf or m ul at io n pr ic e W ei gh ts of th e ch os en co m m od iti es U N re co m m en da tio ns fo r th e m ac ro nu tr ie nt co nt en to f th er ap eu tic fo od in cl ud ed pa la ta bi lit y, te xt ur e, an d m ax im um fo od in gr ed ie nt w ei gh t cr ite ri a D es ig n a RU TF pr ot ot yp e fo r tr ea tin g w as tin g in Ea st A fr ic an ch ild re n an d ad ul ts Li ne ar pr og ra m m in g Br im bl ec om be et al .[ 45 ] T e m in im um co st of a di et A m ou nt of fo od co ns um ed N ut ri en t ad eq ua cy ,n ut ri en t de ns ity To ob se rv e th e di et ar y ch an ge re qu ir ed to ac hi ev e nu tr ie nt re qu ir em en ts at m in im um co st Li ne ar pr og ra m m in g Ry an et al .[ 46 ] M in im ise in gr ed ie nt co st of th e re ad y- to -u se th er ap eu tic fo od s In gr ed ie nt w ei gh t, th e us ag e ra te of fo od s av ai la bl e N ut ri tio na l, pr od uc t qu al ity (f oo d ta st e an d pr oc es sin g co ns id er at io ns ) N ov el re ad y- to -u se th er ap eu tic fo od s fo r Et hi op ia Li ne ar pr og ra m m in g D e C ar va lh o et al . [5 6] T e m in im um co st of po rr id ge m ix W ei gh t/a m ou nt of fo od C on st ra in ts on th ew ei gh to fs ta rc h, lim iti ng dr y m at te r to 25 % ,a lim it on th e to ta lm as s of re co ns tit ut ed po rr id ge ,a nd nu tr ie nt co ns tr ai nt s To ex te nd LP m et ho do lo gy to fo od fo rm ul at io n by se le ct in g in gr ed ie nt s to m ak e up fo od w ith ac ce pt ab le co ns ist en cy fo r th e in te nd ed co ns um er gr ou p Li ne ar pr og ra m m in g Pa rle sa k et al .[ 57 ] C os t- m in im ise d nu tr iti on al ly ad eq ua te fo od ba sk et A m ou nt of fo od C ul tu ra l, di et ar y gu id el in es , nu tr ie nt re co m m en da tio ns FB D G s Li ne ar pr og ra m m in g D ep tfo rd et al . [5 8] Le as t co st of di et En er gy an d nu tr ie nt sp ec if ca tio ns ,p re de fn ed gr ou ps w ith in ho us eh ol ds , po rt io n siz es ,a nd cu rr en cy co nv er sio n fa ct or s N ut ri en t re qu ir em en ts an d am ou nt s pe r m ea l A pp ly in g lin ea r pr og ra m m in g to un de rs ta nd be tte r ho w po ve rt y m ay af ec tp eo pl e’ s ab ili ty to m ee t th ei r nu tr iti on al sp ec if ca tio ns Li ne ar pr og ra m m in g Br ix i[ 59 ] Lo w co st N ut ri tio na lv al ue ,p ri ce ,a nd w at er ef ci en cy of su ita bl e in gr ed ie nt s N ut ri en t, fa vo ur ,c ro p w at er ef ci en cy Re ad y- to -u se th er ap eu tic fo od s op tim ise d at lo w co st us in g lo ca lly gr ow n cr op s Li ne ar pr og ra m m in g N yk än en et al .[ 1] T e su m of co st of ea tin g fo od (m in im um co st ) W ei gh t of ea ch fo od ,c os t of ea ch fo od En er gy an d nu tr ie nt re co m m en da tio ns ,m in im um de vi at io ns fr om th e fo od ba la nc e sh ee t fo r G ha na Fo od ba sk et fo r a fa m ily of 4 (m ot he r, fa th er ,a nd m al e an d fe m al e ch ild re n) th at ha s lo w co st Li ne ar pr og ra m m in g G ha za ry an [6 0] M in im ise co st Po rt io n (a m ou nt /w ei gh t) of th e fo od pr od uc t N ut ri en t (t ol er ab le le ve ls) ,a n up pe r lim it im po se d on th e qu an tit y of fo od pr od uc t A ch ie ve m in im um co st of di et w hi le sa tis fy in g so m e co ns tr ai nt s Li ne ar pr og ra m m in g Journal of Optimization 5 Ta bl e 1: C on tin ue d. Re fe re nc e O bj ec tiv e fu nc tio n (s ) D ec isi on va ri ab le (s ) C on st ra in t (s ) Fo cu s M at he m at ic al ap pr oa ch us ed Fa ks ov á et al .[ 38 ] T e m in im um co st of a di et T e am ou nt of fo od N ut ri en t de vi at io ns fr om ea tin g pa tte rn s O bt ai n a fo od ba sk et fo ra fa m ily of fo ur (m ot he r, hu sb an d, so n, an d da ug ht er ) Li ne ar pr og ra m m in g H am id et al .[ 29 ] M in im ise fo od co st T e po rt io n siz e of th e fo od N ut ri tio na l( am ou nt of nu tr ie nt ), ac ce pt ab ili ty (p or tio n siz e) To de te rm in e if an id ea ld ie tt ha t m ee ts nu tr ie nt in ta ke fo rp re gn an t w om en an d is af or da bl e ca n be cr ea te d fr om lo ca lly av ai la bl ef oo ds in M al ay sia Li ne ar pr og ra m m in g G ur m u et al .[ 61 ] T e m in im um co st of th e op tim ise d fo od ba sk et W ei gh t of fo od Es tim at ed en er gy re qu ir em en t, re co m m en de d m ac ro an d m ic ro nu tr ie nt re qu ir em en t D ev el op a ba sis fo r fo od -b as ed di et ar y gu id el in es fo r Et hi op ia Li ne ar pr og ra m m in g V er ly -J re ta l. [6 2] M in im ise co st w hi le m in im isi ng ne ga tiv e an d po sit iv e de vi at io ns A m ou nt of fo od C on st ra in ts on nu tr ie nt (R D Is ), co ns tr ai nt so n fo od gr ou p se le ct io n To es tim at e th e po ss ib ili ty of m ee tin g di et ar y re qu ir em en ts an d m ea su re th e co rr el at io n be tw ee n th e co st of m en us an d th ei r ad eq ua cy Li ne ar pr og ra m m in g A la in ie ta l. [6 3] Lo w es tc os t fo r ca nc er pr ev en tio n di et A m ou nt of fo od an d nu tr ie nt En er gy ,p or tio n siz e T is st ud y is ai m ed to bu ild a he al th y an d ba la nc ed m en u w ith m in im al co st ba se d on in di vi du al ne ed s an d fo cu se d on pr ev en tin g ca nc er (1 00 pe op le fr om a un iv er sit y) Li ne ar pr og ra m m in g Ib ra hi m et al .[ 64 ] T e m in im um co st of th e he al th ie st m en u fr om M cD on al d’ s A m ou nt of fo od Re co m m en de d nu tr ie nt s, th e lo w er an d up pe r bo un d fo r nu tr ie nt s To fn d th e m in im um co st of M cD on al d’ s he al th y co m bi na tio ns Li ne ar pr og ra m m in g La uk et al .[ 65 ] Lo w es tc os t A m ou nt of fo od in FB N ut ri en tr ec om m en da tio ns , ac ce pt ab ili ty (d ie ta ry pa tte rn s) O pt im ise fo od ba sk et fo r Es to ni an fa m ily of fo ur Li ne ar pr og ra m m in g M ej os et al .[ 30 ] T e m in im um co st of di et co m pl em en ta ry fe ed in g re co m m en da tio ns A m ou nt of fo od C on st ra in ts on nu tr ie nt s, th e lo w er an d up pe r lim its To re co gn ise pr ob le m nu tr ie nt s in co m pl em en ta ry di et s an d fo rm ul at e fe ed in g re co m m en da tio ns fo r ch ild re n ag ed 6 to 23 m on th s in th e ru ra l Ph ili pp in es Li ne ar pr og ra m m in g Ba ie ta l. [6 6] Le as t co st of di et s Q ua nt ity of fo od s C on st ra in ts on nu tr ie nt re qu ir em en ts (lo w er an d up pe r bo un ds ) Id en tif y po pu la tio ns w ho se nu tr ie nt ne ed sa re di f cu lt to m ee t w ith th e cu rr en t ch al le ng es of th e fo od sy st em s Li ne ar pr og ra m m in g FB — fo od ba sk et ,U N — U ni te d N at io ns ,F BD G s— fo od -b as ed di et ar y gu id el in es ,a nd RD Is — re co m m en de d di et ar y in ta ke s. 6 Journal of Optimization Ta bl e 2: Su m m ar y of de ta ils re tr ie ve d fr om ar tic le s th at m in im ise d de vi at io ns be tw ee n th e ob se rv at io n an d th e m od el le d fu nc tio n. Re fe re nc e O bj ec tiv e fu nc tio n (s ) D ec isi on va ri ab le (s ) C on st ra in t( s) Fo cu s M at he m at ic al ap pr oa ch us ed D ar m on et al .[ 47 ] M in im ise to ta ld ep ar tu re fr om th e m ea n fo od in ta ke A m ou nt /w ei gh t of fo od C on st ra in ts of en er gy co nt en t, fo od co ns tr ai nt s to en su re co m pa tib ili ty w ith ob se rv ed di et ar y pa tte rn s, co st co ns tr ai nt s se ta ta m ax im um le ve l Li ne ar pr og ra m m in g Fe rg us on et al .[ 38 ] M in im ise di fe re nc e in th e m ea n pe rc en ta ge of en er gy co nt ri bu te d by di fe re nt fo od gr ou ps be tw ee n m od el le d an d ob se rv ed di et s W ei gh ts of di fe re nt fo od s N ut ri en t (e ne rg y co ns tr ai nt ,R N Is fo r m ac ro an d m ic ro nu tr ie nt s) , ac ce pt ab ili ty (p or tio n siz e, pe rc en til es on fo od gr ou ps ) To de ve lo p a ri go ro us ,r ep ro du ci bl e, an d ob je ct iv e ap pr oa ch ba se d on lin ea r pr og ra m m in g an al ys is, w hi ch ca n be us ed to fo rm ul at e pr ac tic al FB D G s fo r hi gh -r isk po pu la tio ns Li ne ar pr og ra m m in g D ar m on et al .[ 32 ] M in im ise d di et ch an ge s ne ed ed to m ee t nu tr iti on al re qu ir em en ts Pa la ta bi lit y, nu tr iti on al ,c os t Is oe ne rg et ic di et s th at m ee t th e cu rr en t nu tr iti on re qu ir em en ts Li ne ar pr og ra m m in g M as se te t al .[ 10 ] M in im ise de pa rt ur e of op tim ise d di et fr om th e ob se rv ed qu an tit y of fo od ea te n by th e re fe re nc e po pu la tio n Q ua nt ity of fo od N ut ri tio na l( en er gy ,m ac ro ,a nd m ic ro nu tr ie nt co ns tr ai nt s) To us e m at he m at ic al op tim isa tio n to ol sf or di et ar y gu id el in es to pr ev en t ca nc er Li ne ar pr og ra m m in g M ai llo te ta l. [6 7] O pt im ise di et cl os e to ob se rv ed di et Q ua nt ity of fo od s D ie ta ry en er gy ,n ut ri tio na lt ar ge ts , m ax im al qu an tit ie s of fo od s, di et w ei gh t D es cr ib e di et ar y ch an ge s ne ed ed to ac hi ev e nu tr iti on al re co m m en da tio ns Li ne ar pr og ra m m in g C le rf eu ill e et al . [6 8] M in im ise de vi at io ns be tw ee n op tim ise d an d ob se rv ed di et W ei gh to ff oo ds N ut ri tio na la de qu ac y (s el ec te d nu tr ie nt s) ,t he co ns tr ai nt on fo od qu an tit ie s To es tim at e th e nu m be r of po rt io ns of th e di fe re nt m ilk -b as ed fo od ca te go ri es th at ft in to nu tr iti on al ly ad eq ua te di et s Li ne ar pr og ra m m in g— BA SA L m od el M et zg ar et al .[ 69 ] M in im ise th e su m of di fe re nc es in fo od in ta ke A m ou nt of fo od C os t (d oe s no te xc ee d a m ax im um le ve l), th e co ns tr ai nt on so m e fo od ca te go ri es (k ep ta ta m ax im um of 0) , nu tr ie nt (m in im um an d m ax im um le ve ls, D RI s) T is di et ar y op tim isa tio n pr og ra m us es co m m on fo od ch oi ce s to bu ild a su ita bl e di et (P al eo lit hi c di et ) Li ne ar pr og ra m m in g O ku bo et al .[ 16 ] M in im ise th e de vi at io n in fo od in ta ke be tw ee n th e ob se rv ed an d op tim ise d fo od in ta ke pa tte rn s Q ua nt ity of fo od N ut ri tio na l( m ee t D RI s) ,u pp er lim its of ea ch fo od To tr an sla te nu tr ie nt -b as ed re co m m en da tio ns in to re al ist ic nu tr iti on al ly op tim um co m bi na tio ns of fo od by in te gr at in g lo ca la nd cu ltu re -s pe ci fc fo od s Li ne ar pr og ra m m in g (in fu se d go al pr og ra m m in g) Pe ri gn on et al .[ 70 ] M in im ise de pa rt ur e fr om th e ob se rv ed di et Fo od pr ic e N ut ri tio n To ev al ua te th e co m pa tib ili ty am on g th e af or da bi lit y di m en sio ns of di et su st ai na bi lit y Li ne ar pr og ra m m in g H or ga n et al .[ 71 ] M in im isi ng di et ar y ch an ge s fr om th ei r cu rr en t re po rt ed in ta ke (t o m ee td ie ta ry re co m m en da tio ns an d G H G E ta rg et s) W ei gh t of fo od N ut ri en tc on st ra in ts ba se d on di et ar y re fe re nc e in ta ke ,c on st ra in t se to n m ea ta nd fs h, 25 % re du ct io n co ns tr ai nt on G H G E, th e lo w er an d up pe r lim it on in di vi du al fo od s To de te rm in e th e ra ng e of di et ar y ch an ge s th at ac hi ev e di et ar y re co m m en da tio ns an d re du ce G H G E (m ak in g lit tle ch an ge s to cu rr en t di et ar y in ta ke s) Li ne ar pr og ra m m in g Journal of Optimization 7 Ta bl e 2: C on tin ue d. Re fe re nc e O bj ec tiv e fu nc tio n (s ) D ec isi on va ri ab le (s ) C on st ra in t( s) Fo cu s M at he m at ic al ap pr oa ch us ed Sc ar bo ro ug h et al . [7 2] M in im ise de vi at io n be tw ee n th e co st of ob se rv ed an d m od el le d di et s A m ou nt of fo od D ie ta ry re co m m en da tio ns To m od el fo od gr ou p co ns um pt io n an d pr ic e of di et as so ci at ed w ith m ee tin g di et ar y re co m m en da tio ns w ith m in im um de vi at io n fr om cu rr en t di et to re de ve lo p FB D G s (s et tin g, U K ) N on lin ea r ge ne ra lis ed re du ce d gr ad ie nt al go ri th m M ai llo te ta l. [3 3] A nu tr iti on al ly ad eq ua te iso ca lo ri c di et th at st ay ed cl os e to th e ob se rv ed di et T e am ou nt of fo od av ai la bl e A cc ep ta bi lit y (f oo d m os tf re qu en tly ea te n) ,n ut ri en tc on st ra in t( ba se d on di et ar y re fe re nc e) N ut ri tio na lly ad eq ua te iso ca lo ri c di et (a m od el le d di et th at ca m e as cl os e as po ss ib le to th e co rr es po nd in g ob se rv ed di et ) Li ne ar pr og ra m m in g Ra ym on d et al . [7 3] A n af or da bl e di et th at ac hi ev es D RI s fo r se le ct ed nu tr ie nt s (o bj ec tiv e fu nc tio n w as to m in im ise de vi at io ns be tw ee n po pu la tio ns ’ fo od gr ou ps an d di et ar y st an da rd s) G ra m s of fo od A m ou nt of fo od us ed by th e po pu la tio n, so it do es no te xc ee d nu tr ie nt co ns tr ai nt ,a cc ep ta bi lit y (c on st ra in ts w er e se t on gr am s on ea ch fo od gr ou p) T e pr im ar y ob je ct iv e of th is st ud y w as to as ce rt ai n if a pr ac tic al an d af or da bl e di et th at m ee ts D RI s fo r so m e se le ct ed nu tr ie nt s ca n be de ve lo pe d fo r ru ra l6 –2 3- m on th -o ld ch ild re n in Ta nz an ia Li ne ar go al pr og ra m m in g K ra m er et al .[ 74 ] M in im isa tio n of ch an ge s to th e cu rr en t av er ag e di et W ei gh t of fo od N ut ri tio na lr eq ui re m en ts , en vi ro nm en ta lt ar ge ts T e m od el w as do ne to m im ic th e cu rr en t co ns um er be ha vi ou r Li ne ar pr og ra m m in g Ra ym on d et al . [7 5] M in im ise de vi at io n be tw ee n m od el le d an d ob se rv ed di et pa tte rn s w hi le m ee tin g di et ar y st an da rd s Q ua nt ity of fo od N ut ri tio na l( RN Is ) To es ta bl ish if a re al ist ic an d in ex pe ns iv e di et th at m ee ts se t nu tr iti on al go al s fo r ru ra lw om en (p re gn an t an d la ct at in g) ca n be fo rm ul at ed fr om lo ca lly av ai la bl e fo od s in Ta nz an ia Li ne ar pr og ra m m in g (u sin g go al pr og ra m m in g) Ba rr é et al .[ 76 ] M in im um de vi at io n fr om th e ob se rv ed di et Q ua nt iti es of fo od En vi ro nm en ta l, co ns tr ai nt on di et ar y m ac ro nu tr ie nt s an d RD A , ac ce pt ab ili ty (a co ns tr ai nt on qu an tit ie s of fo od su bg ro up s, bo vi ne m ea t, an d da ir y pr od uc ts co -c on st ra in ed ), co st (t o re m ai n lo w er or eq ua lt o ob se rv ed cu rr en t co st ) T e ob je ct iv e w as to ev al ua te th e im pa ct of nu tr ie nt bi oa va ila bi lit y an d co -p ro du ct io n lin ks co ns id er at io ns on th e di et ar y ch an ge s ne ed ed — es pe ci al ly re ga rd in g m ea t— to im pr ov e di et su st ai na bi lit y Li ne ar an d no nl in ea r pr og ra m m in g Br in k et al .[ 77 ] Su st ai na bl e di et st ha ta re cl os e to th e ob se rv ed pa tte rn s A m ou nt of fo od gr ou ps C on st ra in ts on fo od gr ou ps , en vi ro nm en ta lc on sid er at io ns ,t he m in im um an d m ax im um co ns tr ai nt so n nu tr ie nt sa nd en er gy , ac ce pt ab ili ty (c lo se ne ss of m od el le d di et s to cu rr en t pa tte rn ) To ob ta in he al th y an d su st ai na bl e fo od -b as ed di et ar y gu id el in es (F BD G s) fo r di fe re nt ta rg et gr ou ps in th e N et he rla nd s Li ne ar pr og ra m m in g K im an d K im [2 7] O pt im al nu tr ie nt le ve ls (m in im ise de vi at io ns ) A m ou nt of fo od T e up pe r an d lo w er lim its of ca lo ri es ,c on su m pt io n, an d am ou nt of nu tr ie nt A n ab so lu te op tim al de ci sio n th at pr ov id es th e be st po ss ib le nu tr ie nt co m bi na tio n Li ne ar pr og ra m m in g 8 Journal of Optimization Ta bl e 2: C on tin ue d. Re fe re nc e O bj ec tiv e fu nc tio n (s ) D ec isi on va ri ab le (s ) C on st ra in t( s) Fo cu s M at he m at ic al ap pr oa ch us ed Jo hn so n- D ow n et al .[ 78 ] T e de vi at io n be tw ee n m od el le d an d ob se rv ed di et G ra m of fo od C os t, nu tr ie nt re qu ir em en t (E A R) Sa tis fy m ed ic in e m ac ro an d m ic ro nu tr ie nt re qu ir em en ts in he al th y in di vi du al s ba se d on av ai la bl e fo od s co ns um ed by th e de fn ed po pu la tio n (a tm in im um co st ) Li ne ar pr og ra m m in g V er ly -J r et al .[ 39 ] O pt im ise d di et sw ith fo od qu an tit ie s at th e lo w es td ev ia tio n fr om th e ob se rv ed di et s T e am ou nt of fo od N ut ri en t( W H O gu id el in es fo r N C D s, re co m m en de d re qu ir em en ts ), ac ce pt ab ili ty (b ou nd ar ie s lim iti ng ch an ge s in fo od qu an tit ie s, ST RI C T an d FL EX m od el s on fo od s) ,G H G E (s te pw ise re du ct io n fr om 10 % ) Id en tif y th e di et ar y ch an ge s to im pr ov e nu tr iti on an d re du ce di et -r el at ed gr ee nh ou se ga s em iss io ns (G H G E) in Br az il, w ith co ns id er at io n gi ve n to fo od ha bi ts an d pr ic es Li ne ar pr og ra m m in g G az an et al .[ 79 ] M in im ise de vi at io n be tw ee n ob se rv ed an d op tim ise d di et s Q ua nt ity of fo od s C on st ra in t on re co m m en de d in ta ke s, 30 % re du ct io n of ca rb on im pa ct co ns tr ai nt Ex pl or e th e fe as ib ili ty of pl an t- ba se d “d ai ry -li ke ” pr od uc ts in ac hi ev in g su st ai na bl e di et s Li ne ar pr og ra m m in g Ro ca bo is et al .[ 80 ] M in im ise de vi at io n be tw ee n ob se rv ed an d m od el le d di et Q ua nt ity of fo od s C on st ra in ts on nu tr iti on al re qu ir em en ts an d en vi ro nm en ta l im pa ct ta rg et s D ev el op an ap pr oa ch (I N D IG O O )t o de sig n su st ai na bl e di et sw ith nu tr ie nt re qu ir em en ts an d ac hi ev e se t en vi ro nm en ta lt ar ge ts Li ne ar pr og ra m m in g V as ilo go u et al . [8 1] To m in im ise th e de vi at io n be tw ee n m od el le d an d ty pi ca ld ie tc on su m ed in A m er ic a Q ua nt ity of fo od N ut ri en ta nd fo od gr ou p co ns tr ai nt s A ss es s th e qu al ity of sim ul at ed fo od pa tte rn s th at ha ve re du ce d an im al pr ot ei n us in g N H A N ES da ta fr om 20 17 -2 01 8 M ix ed in te ge r lin ea r pr og ra m m in g G H G E— gr ee nh ou se ga se m iss io n, FB D G s— fo od -b as ed di et ar y gu id el in es ,D RI s— di et ar y re co m m en de d in ta ke s, U K — U ni te d K in gd om ,W H O — W or ld H ea lth O rg an isa tio n, N C D s— no nc om m un ic ab le di se as es , RN Is — re co m m en de d nu tr ie nt in ta ke s, EA R— es tim at ed av er ag e re co m m en da tio ns ,a nd IN D IG O O — in di vi du al di et in cl ud in g gl ob al ob je ct iv es op tim isa tio n. Journal of Optimization 9 Ta bl e 3: Su m m ar y of de ta ils re tr ie ve d fr om ar tic le s th at m in im ise d en vi ro nm en ta lf ac to rs . Re fe re nc e O bj ec tiv e fu nc tio n (s ) D ec isi on va ri ab le (s ) C on st ra in t( s) Fo cu s M at he m at ic al ap pr oa ch us ed C ol om bo et al .[ 48 ] Re du ce G H G E A m ou nt of fo od N ut ri tio n, af or da bi lit y, cu ltu ra l T e m od el th at ha d lo w er G H G E le ve ls an d w as nu tr iti on al ly ad eq ua te an d af or da bl e Li ne ar pr og ra m m in g La rr ea -G al le go s an d V áz qu ez -R ow e [4 9] M in im ise th e to ta la m ou nt of G H G E of op tim ise d di et s A m ou nt of fo od , ca rb on em iss io n fo r th e fo od s N ut ri tio na l, en vi ro nm en t, an d qu an tit y of fo od s Li ne ar pr og ra m m in g w ith th e in te gr at io n of M on te C ar lo sim ul at io n Fe rr ar ie ta l. [5 0] M in im ise G H G E w hi le sa tis fy in g nu tr iti on al (R D Is ), ac ce pt ab ili ty , an d he al th co ns tr ai nt A m ou nt of fo od N ut ri tio na l( nu tr ie nt an d en er gy ), ac ce pt ab ili ty (m ea n to ta la m ou nt of fo od an d be ve ra ge th at w as co ns tr ai ne d be tw ee n 80 an d 14 0% of ob se rv ed in ta ke an d pe rc en til e on ca lc ul at ed m ea n ob se rv ed di et ), an d he al th y co ns tr ai nt s (e st ab lis he d lo w er an d up pe r lim it) To cr ea te a su st ai na bl e an d he al th y It al ia n di et w ith lo w G H G E th at m ee ts di et ar y re qu ir em en ts an d re fe ct sc ur re nt fo od in ta ke pa tte rn s Li ne ar pr og ra m m in g To m pa et al .[ 40 ] Re du ce d di et ar y w at er fo ot pr in t an d m in im ise th e de vi at io n be tw ee n ob se rv ed an d m od el di et s W ei gh to fs ub gr ou ps C on st ra in ts on nu tr ie nt re qu ir em en ts ,c ul tu ra l ac ce pt ab ili ty ,a nd st ep w ise en vi ro nm en ta lr ed uc tio n D es ig n su st ai na bl e di et s th at m in im ise di et ar y w at er fo ot pr in t in H un ga ry Li ne ar pr og ra m m in g G H G E— gr ee nh ou se ga s em iss io n; RD Is — re co m m en de d di et ar y in ta ke s. 10 Journal of Optimization Ta bl e 4: Su m m ar y of de ta ils re tr ie ve d fr om ar tic le s th at m ax im ise d ob je ct iv e fu nc tio n. Re fe re nc e O bj ec tiv e fu nc tio n (s ) D ec isi on va ri ab le (s ) C on st ra in t( s) Fo cu s M at he m at ic al ap pr oa ch us ed Ra m be lo so n et al . [5 1] M ax im um nu tr iti on al re qu ir em en t Ed ib le gr am s of th e se le ct ed fo od s N ut ri en t In ve st ig at e th e nu tr iti on al qu al ity of FB fo od ai d de liv er ed in Fr an ce Li ne ar pr og ra m m in g W ils on et al .[ 35 ] M ax im um nu tr iti on al re qu ir em en t T e am ou nt of fo od N ut ri en t, co st A di et ar y pa tte rn th at m et th e es se nt ia l nu tr iti on al re qu ir em en ts fo r N ew Ze al an d m en Li ne ar pr og ra m m in g va n D oo re n et al . [5 2] M ax im ise m os t co ns um ed fo od s Pr ic e of fo od , am ou nt of fo od En er gy an d nu tr ie nt ,c os t, G H G E Fi nd lo w -p ri ce d di et sw ith lo w cl im at ei m pa ct ye t fu lf la ll nu tr iti on al ne ed s Li ne ar pr og ra m m in g do sS an to se ta l. [7 ] M ax im um nu tr iti on al re qu ir em en t N ut ri en td ev ia tio ns fr om ob se rv ed di et ar y pa tte rn s A di et th at do es no tv ar y m uc h fr om th e ob se rv ed di et ar y pa tte rn Li ne ar pr og ra m m in g M or ri so n et al . [4 1] M ax im ise th e ir on co nt en to ff oo d Q ua nt ity of fo od s N ut ri en t co ns tr ai nt ,i nt ak e co ns tr ai nt (m ed ia n po rt io n siz e, up pe r an d lo w er bo un ds on fr eq ue nc y) D ev el op st ra te gi es to ad dr es s an ae m ia in pr eg na nc y in ru ra lp la in s in N ep al Li ne ar pr og ra m m in g va n W on de re n et al .[ 53 ] M ax im ise th e ir on co nt en to fd ie ts Q ua nt ity of fo od s C on st ra in ts on D RI s Pl an a 2- w ee k m en u to es tim at e th e ir on bi oa va ila bi lit y of di fe re nt di et sf or w om en of re pr od uc tiv e ag e M ix ed in te ge r lin ea r pr og ra m m in g G H G E— gr ee nh ou se ga s em iss io n, FB — fo od ba sk et ,a nd D RI s— di et ar y re co m m en de d in ta ke s. Journal of Optimization 11 Ta bl e 5: Su m m ar y of de ta ils re tr ie ve d fr om ar tic le s th at di d no ts pe ci fy m in im isa tio n or m ax im isa tio n. Re fe re nc e O bj ec tiv e fu nc tio n (s ) D ec isi on va ri ab le (s ) C on st ra in t (s ) Fo cu s M at he m at ic al ap pr oa ch us ed Pa sic et al .[ 54 ] M in im ise va ri at io ns fr om th e de fn ed nu tr ie nt s( m ic ro an d m ac ro ), fo od co st Fo od co m m od iti es Re co m m en de d nu tr ie nt in ta ke ,u pp er lim its of nu tr ie nt s G oa lp ro gr am m in g Fe rg us on et al . [3 7] M ee tn ut ri en t ne ed s (w hi le co ns id er in g lo ca lly ap pr op ri at e FB R th at ar e lo w co st an d ca n en ha nc e nu tr ie nt ad eq ua cy ) A m ou nt of fo od ite m En er gy co nt en t, m in im um an d m ax im um nu m be ro fs er vi ng sf ro m fo od gr ou ps ,g ra m s of ea ch fo od ite m Fi nd re al ist ic fo od -b as ed re co m m en da tio ns (F BR s) an d as ce rt ai n ho w th ey co ul d en su re in ta ke ad eq ua cy fo r 12 nu tr ie nt s Li ne ar pr og ra m m in g (O pt ifo od so ftw ar e) M cM ah on et al .[ 28 ] M in im ise so di um in ta ke w hi le m ai nt ai ni ng io di ne in ta ke Re du ct io n in so di um co nt en t in di fe re nt fo od s N ut ri en ti nt ak e (a llo w ab le le ve ls, re co m m en de d di et ar y in ta ke ,u pp er lim its ) To ev al ua te th e in ta ke of so di um an d io di ne ag ai ns tr eq ui re m en ts an d th en m od el th e po ss ib le im pa ct s of sa lt- re du ct io n st ra te gi es on es tim at ed so di um an d io di ne in ta ke s in in di ge no us A us tr al ia n co m m un iti es Li ne ar pr og ra m m in g FB Rs — fo od -b as ed re co m m en da tio ns . 12 Journal of Optimization 3.4. Constraints. In linear programming, there is the need for constraints to be imposed, and these constraints must be respected to obtain the defned optimal solution for the objective function set [14]. Tese 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 re- quirements, environment, and the cost of diet [13, 39, 52]. All the studies had one thing in common: constraints that were defned 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 re- quirements 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 rec- ommended nutrient intakes (RNIs), as exhibited by Fer- guson 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 satisfed dietary reference intakes (DRIs), Ghazaryan [60] did the same for tolerable levels of selected nutrients, and McMahon et al. [28] imposed constraints to ensure allow- able 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 afordability, some studies [32, 48, 52] put a cost constraint to make it possible. Also, constraints on culture or acceptability can be actualised in diferent 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 con- straint 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 difcult to ensure in addressing dietary problems due to the varying needs of consumers and the diferent food items from one geographic area to an- other. 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. Tis 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 diferent settings. According to Beheshti et al. [85], these mathematical tools can simulate diferent scenarios in di- etary choices and other diet optimisation problems at low cost with minimal risk. Tough these studies either mini- mised or maximised an objective function, researchers had diferent 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 fnd optimal solutions for the ob- jectives they set, all studies (Table 1) used the linear pro- gramming approach. Linear programming has been used to address many diet-related problems [11, 14]. Tough many of the studies analysed in this write-up employed linear programming to address a problem they intended to solve, the articles did not defne mathematical equations that represent the physical problems. From fndings obtained from the review, LP has been successfully applied in solving diet optimisation problems. Tough 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. Tis 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 re- search 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 ofers room for 84% 11% 5% Number of articles Minimisation Maximisation Others Figure 2: Graph showing the distribution of direction of the objective functions from articles analysed. Journal of Optimization 13 more work. To achieve sustainable diets, researchers need to consider satisfying diferent conficting goals like nutrition, cost, environment, and acceptability, making this a multi- objective 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 re- searchers should identify when solving a multiobjective op- timisation problem. Te 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 pro- gramming is another approach to solving multiobjective optimisation problems by minimising deviations between the individual goals obtained and the set targets [88]. Tough 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 com- monly assume that multiobjective optimisation problems have conficting 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 felds, its application in addressing intricate dietary issues remains limited. It would beneft 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 defnitive defnition for sustainable diets, the diferent di- mensions, including nutrition, economics (cost), environ- ment, and sociocultural factors, make them complex. Te LP tool has been efciently used to minimise cost, maximise nutrient requirement, and minimise deviations between set targets and achieve objectives. Tis 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. Te tool has been efciently used to address dietary prob- lems by proposing diets that come at the least cost while meeting nutrient constraints or diets with low greenhouse gas emission values while meeting certain defned con- straints. 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